Equivariant ordinary homology and cohomology
Steven R. Costenoble
Homology-chap1.tex
\chapter{$RO(G)$-graded Ordinary Homology and Cohomology}
\label{chap:rog}
%
\section*{Introduction}

A construction of $RO(G)$-graded ordinary homology and cohomology was announced
in \cite{LMM:roghomology} and given in \cite{GM:eqhomotopy}.
In this chapter we give another construction, which first appeared
in \cite{Wan:gcwv}, for finite groups,
in \cite{Le:emspaces} for compact Lie groups,
and is outlined in \cite[Chapter~X]{May:alaska}.
We also give more details about
the behavior of these theories and their dual theories. 
Some of this material is well-known, but some
is new, particularly the material on the dual theories, on subgroups, and on products. 

At the end of Chapter~\ref{chap:ordone} we shall generalize this construction to give a theory
graded on a larger group. In a very precise sense, 
the theory defined in this chapter is the local
case of the more general one. We shall discuss various constructions in detail in this chapter,
where the setting is somewhat simpler,
to allow us to concentrate in the next several chapters
on those things that need to be changed when we generalize.

Throughout we understand all spaces to be $k$-spaces.
It is more common to restrict to compactly generated spaces, that is,
weak Hausdorff $k$-spaces, but the extra generality is needed
in subsequent chapters when we discuss parametrized spaces
\cite{MaySig:parametrized}.
Our ambient group $G$ is a compact Lie group and subgroups are
understood to be closed.
We write $\K$ for the category of $k$-spaces, $G\K$ for the category
of $G$-spaces and $G$-maps, $\K_*$ for the category of based
$k$-spaces, and $G\K_*$ for the category of based $G$-spaces
(with $G$-fixed basepoints) and basepoint-preserving $G$-maps.

For Bredon's integer-graded equivariant cellular homology the
appropriate notion of a ``$G$-complex'' is a $G$-CW complex in the
sense of \cite{Br:cohomology} and \cite{Ill:homology}.
This is a $G$-space with cells
of the form $G/H\times D\sp n$ where $H\subset G$ and $G$ acts
trivially on the unit $n$-disc $D\sp n$. The cell complexes we use
to construct the $RO(G)$-graded extension of Bredon homology 
have cells of the form $G\times_H D(W)$ where $D(W)$ is the unit
disc in a (possibly nontrivial) representation $W$ of $H$. 


\section{$G$-CW($V$) complexes}\label{sec:vcomplexes}

Fix a finite dimensional orthogonal
representation $V$ of $G$. If $i \geq 0$, denote by $V\!+i$ the
orthogonal representation $V\!\oplus  \Real\sp i$. If  
$V$ contains a copy of $\Real\sp i$,
let $V\!-i$ be the orthogonal complement of any copy of $\Real^i$
in $V$. Let $D(V)$ denote the unit disc in $V$, let $S(V)$
denote $\bndry D(V)$, let $S^V$ denote the one-point compactification of
$V$ with the compactification point as basepoint,
and let $|V|$ denote the (real) dimension of $V$.


\begin{definition}\label{def:vcomplex}
 A {\em $G$-CW($\,V$\!) complex} is a $G$-space $X$ together with a
decomposition $X = \colim_n X\sp n$ such that
 \begin{enumerate}
 \item\label{item:vzerocells}
$X\sp 0$ is a disjoint union of $G$-orbits $G/K$ such that $K$ acts
trivially on $V$, and
 \item\label{item:vattachment}
$X\sp n$ is obtained from $X\sp {n-1}$ by
attaching ``cells'' of the form $G \times_K D(V\!-|V|+n)$ where
$|V|-n \leq |V^K|$. The
attaching maps have the form  $G\times_H S(V\!-|V|+n) \to X\sp
{n-1}$.
 \end{enumerate} 
Similarly, a relative $G$-CW($V$) complex is a
pair $(X,A)$ where $X = \colim_nX\sp n$, $X\sp 0$ is the disjoint
union of $A$ with orbits as in (\ref{item:vzerocells}), and cells are attached as in
(\ref{item:vattachment}). 
For notational convenience, and to remind ourselves of the role of $V$,
we shall also write $X^{V+n}$ for $X^{|V|+n}$.
If $X = X^{V+n}$, then we say that $X$ is {\em $(V+n)$-dimensional}.

If we allow cells of any dimension to be attached at each stage,
we get the weaker notion of a {\em $V$-cell complex.}
If $X$ is a $V$-cell complex with cells only of dimension
less than or equal to $|V|+n$, we say again that
$X$ is $(V+n)$-dimensional.
 \end{definition}

 Note that, if $V = 0$, then a $G$-CW($V$) complex is just a $G$-CW
complex in the sense of \cite{Br:cohomology} and \cite{Ill:homology}.

\begin{definition}\label{def:basedvcomplex}
A {\em based $G$-CW($\,V$\!) complex} is a based $G$ space
$X$ with $G$-fixed basepoint $*$ such that $(X,*)$ is a
relative $G$-CW($V$) complex.
 \end{definition}

Notice that, if $(X,A)$ is a relative $G$-CW($V$) complex, then
$X/A$ is a based $G$-CW($V$) complex.

\begin{examples}\label{ex:vcomplexes}
Definitions~\ref{def:vcomplex} and \ref{def:basedvcomplex} are
motivated by the following examples.

 \begin{enumerate}
 \item\label{item:vbundle}
As in \cite{Pu:rogbordism}, \cite{Ko:rogbordism}, and \cite{Wan:ROGbordism}, 
we say that a
$G$-vector bundle $p\colon E \to B$ is a {\em $V$-bundle} if for each 
$b \in B$, $p\sp {-1}(b)$ is isomorphic to $V$ as a representation of
$G_b$.
Let $Tp$ denote the Thom space of $p$ and $*$ its basepoint.
If $B$ is an ordinary $G$-CW complex, then $Tp$ has an
evident structure as a based $G$-CW($V$) complex. Corresponding
to each $n$-cell in $B$ is a $(V\!+n)$-cell in $Tp$.

 \item\label{item:vmanifold}
Referring again to the references above,
we say that a smooth
$G$-manifold $M$ is a {\em $V$-manifold} if its tangent bundle is
a $V$-bundle. 
Associated to any $G$-tri\-angu\-lation $T$ of $M$  is
the {\em dual cell structure}. The top-dimensional cells
in the dual cell structure are the orbits of
the closed stars of the original vertices in the first barycentric
subdivision, while the lower dimensional  cells are intersections of
these. If $G$ is finite, these cells form a $G$-CW($V$) structure on $M$.
Corresponding to each $n$-cell in $T$ is a $(V-n)$-cell in the dual
complex. 
In Section~\ref{sec:dualvcomplexes} we shall discuss what
happens when $G$ is infinite.

 \item If $G$ is finite and if $X$ and $Y$ are, respectively, a $G$-CW($V$)
and a $G$-CW($W$) complex, then $X \times Y$ is a
$G$-CW($V\oplus W$) complex. In particular, if $X$ is a $G$-CW complex, then 
$X \times D(V)$ has the structure of a $G$-CW($V$) complex. This permits one to
replace a $G$-CW complex by a homotopy equivalent $G$-CW($V$) complex for any $V$.

 \item In general, if $X$ is a $G$-CW($V$) complex and $Y$ is a
$K$-CW($W$) complex, then $X\times Y$ is a
$(G\times K)$-CW($V\oplus W$) complex, where $G$ acts trivially on $Y$ and $W$, and $K$
acts trivially on $X$ and $V$.

 \end{enumerate}
The first two of these examples are the geometric motivations
for the Thom Isomorphism and Poincar\'e Duality theorems that we shall discuss
in the last section of this chapter. 
 \end{examples}

We now want to show that Definition~\ref{def:vcomplex} gives a good cellular theory.
To do this, we first need appropriate equivariant homotopy groups.

\begin{definition}\label{def:VHomotopySet}
 Let $f\colon X\to Y$ be a $G$-map and let $K$ be a subgroup of $G$. 
We define the
{\em $K$-equivariant $(V+i)$th homotopy set of $f$},
$\pi_{V+i}^K (f)$ ($i$ may be negative where it makes sense), to be the
set of $G$-homotopy classes of commutative diagrams
 \[
 \xymatrix{
  G\times_K S(V+i) \ar[r] \ar[d] & X \ar[d]^{f} \\
  G\times_K D(V+i) \ar[r] & Y. \\
 }
 \]
 This is functorial in $f$ in the evident way. We say that
$\pi_{V+i}^K(f)$ is {\em trivial} if it coincides with 
$\Image (\pi_{V+i}^K(1_X) \to \pi_{V+i}^K(f))$,
i.e., if every diagram as above is homotopic to one in which we can find
a lift $G\times_K D(V+i)\to X$.
 \end{definition}

\begin{definition}
 A $G$-map $f\colon X\to Y$ is a {\em $(V+n)$-equivalence} if
$\pi_{V+i}^K(f)$ is trivial for all $K$ and all $i\leq n$ for which the set is
defined. 
The map $f$ is a  {\em weak $V$-equivalence} if it is a $(V+n)$-equivalence for
all $n$.
\end{definition}

This is a natural generalization of the definition of $n$-connected map given in
\cite{Wh:homotopy} (see his II.3.1 and \S II.7).
In the case $V=0$ we should address the difference between our essentially
unbased homotopy groups and the usual based groups, but it is not hard
to see that our group $\pi_i(f)$ vanishes if and only if all the based homotopy groups $\pi_i(Ff)$ of all the homotopy fibers of $f$ vanish. 
At the end of this section we shall
discuss the relationship between $(V+n)$-equivalence and other notions of
equivalence. In particular, we shall show that $f$ is a weak $V$-equivalence 
if and only if it is a weak $G$-equivalence.

We have the following variant of the ``homotopy extension and
lifting property'' of \cite{May:foundations}.

\begin{lemma}[H.E.L.P.]\label{lem:HELP}
 Let $r\colon Y \to Z$ be a $(V+n)$-equivalence.
Let $(X,A)$ be a relative 
%$G$-CW($\,V$\!) complex 
$V$-cell complex
with cells of dimension
$\leq |V|+ n$. If the following diagram commutes without the
dashed arrows, then there exist maps $\tilde g$ and $\tilde h$
making the diagram commute.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\times I \ar'[d][dd] \ar[dl]_(.6)h & & 
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g \ar[dd]  \\
   & Z & & Y \ar[ll]_(.3)r \\
  X \ar[ur]^f \ar[rr]_-{i_0} & & X\times I \ar@{-->}[ul]^(.6){\tilde h} & &
   X \ar[ll]^-{i_1} \ar@{-->}[ul]^(.6){\tilde g} \\
 }
 \]
The result remains true when $n = \infty$.
 \end{lemma}

\begin{proof}
 The proof is by induction over the cells of $X$
not in $A$, so it suffices to consider the case
$A = G\times_K S(V+i)$ and $X = G\times_K D(V+i)$.
As in the picture on the left of Figure~\ref{fig:HELP},
identify $D(V+i)\union [S(V+i)\times I]$ with $D(V+i)$ and use
$f$ and $h$ to define a map $G\times_K D(V+i)\to Z$ that lifts to $g$ on its boundary:
 \[
 \xymatrix{
  G\times_K S(V+i) \ar[r]^-g \ar[d] & Y \ar[d]^{r} \\
  G\times_K D(V+i) \ar[r]_-{f\union h} & Z. \\
 }
 \]
By assumption, this diagram is homotopic to one in which we can
find a lift $k\colon G\times_K D(V+i)\to Y$. Let $H$ be the homotopy of $g$ and $J$ be the homotopy of $f\union h$, as in Figure~\ref{fig:HELP}.
Now distort the cylinder in that figure to become the picture on the right.
On the far right end of this new cylinder, $k$ and $H$ combine to give a map $\tilde g\colon G\times_K D(V+i)\to Y$, while the whole cylinder describes a map
$\tilde h\colon G\times_K D(V+i)\times I\to Z$. It is easy to see that these maps make the diagram in the statement of the lemma commute.
 \end{proof}
\begin{figure}
 \centerline{\includegraphics[scale=0.8]{HELP}}
 \caption{}\label{fig:HELP}
\end{figure}

\begin{theorem}[Whitehead]\label{thm:VWhitehead}
 \begin{enumerate} \item[] % force first real item to be on a new line

\item If $f\colon Y\to Z$ is a $(V+n)$-equivalence and $X$ is a
$(V+n-1)$-dimensional 
$V$-cell complex,
then  $f_*\colon \pi G\K(X,Y) \to \pi G\K(X,Z)$
is an isomorphism, where $\pi G\K(-,-)$ denotes
the set of homotopy classes of $G$-maps.  
It is an epimorphism if $X$ is
$(V+n)$-dimensional.

\item If $f\colon Y\to Z$ is a weak $V$-equivalence and $X$ is a 
$V$-cell complex,
then $f_*\colon \pi G\K(X,Y) \to \pi G\K(X,Z)$ is an
isomorphism.  Therefore, any weak $V$-equivalence of 
$V$-cell complexes
is a $G$-homotopy equivalence.

 \end{enumerate}
 \end{theorem}

\begin{proof}
 For the first part, apply the  H.E.L.P. lemma to the pair $(X,\emptyset)$ to get
surjectivity and to the pair $(X\times I, X\times\bndry I)$ to get
injectivity. The second part follows in the same way from the last statement of
the H.E.L.P.\ lemma.
 \end{proof}

\begin{definition}
 A map $f\colon X\to Y$ of $G$-CW($V$) complexes is {\em cellular} if 
$f(X\sp {V+n})\subset Y\sp {V+n}$ for each $n$.
 \end{definition}

\begin{theorem}[Cellular Approximation]
 If $f\colon X\to Y$ is a map of $G$-CW($\,V$\!) complexes that is
cellular on a subcomplex $A\subset X$, then $f$
is $G$-homotopic rel $A$ to a cellular map.
 \end{theorem}

\begin{proof}
 This follows easily from the H.E.L.P. lemma once we know that the inclusion
$Y\sp {V+n}\to Y$ is a $(V+n)$-equivalence.  By the usual induction this reduces to
showing that there are no essential $G$-maps 
$G/H\times(D(V),S(V)) \to G/K\times(D(V+k), S(V+k))$
for $k>0$.  But this follows from obstruction theory (see \cite{Br:cohomology}), the
obstructions vanishing for dimensional reasons.
 \end{proof}

\begin{theorem}[Approximation by $G$-CW($V$) complexes]
 Let X be a $G$-space. Then there exists a $G$-CW($\,V$\!)
complex $\Gamma X$ and a weak $V$-equivalence $\Gamma X\to X$.
 \end{theorem}

\begin{proof}
 We use the usual technique of killing homotopy
groups.  We start with a union of 0-cells $\Gamma_0 X$ and a $G$-map
$f_0\colon \Gamma_0 X\to X$ such that $\pi_0^K(f_0)$ is trivial for
all $K$ acting trivially on $V$.  Inductively, we assume that a
$(V-|V|+n-1)$-connected map $f_{n-1}\colon \Gamma_{n-1} X\to X$ has been
constructed, and then attach cells to $\Gamma_{n-1} X$ in order
to trivialize $\pi_{V-|V|+n}^K(f_{n-1})$ for all $K$. We then take 
$\Gamma X = \colim_n \Gamma_n X$.
 \end{proof}

By Whitehead's Theorem, the map $\Gamma X\to X$ is unique
up to canonical homotopy equivalence.  In addition, if $f\colon X\to
Y$ is a $G$-map then there is a $G$-map 
$\Gamma f\colon \Gamma X\to \Gamma Y$, unique up to $G$-homotopy, such that the
following diagram commutes up to $G$-homotopy:
 \[
 \xymatrix{
  {\Gamma X} \ar[r]^{\Gamma f} \ar[d] & {\Gamma Y} \ar[d] \\
  X \ar[r]_{f} \ar[r] & Y
  }
 \]

In the remainder of this section we discuss
the relationship between $(V+n)$-equivalence and other kinds of
equivalence.
The following definition first appeared in
\cite{Wan:gcwv} and was also used in \cite{Le:emspaces}.

\begin{definition}
 \begin{enumerate} \item[]
 \item
A {\em dimension function} $n^*$ is a function from the set of conjugacy
classes of subgroups of $G$ to the integers. We write $n^K$ for the value of
$n^*$ at the conjugacy class of $K$. If $V$ is a representation of $G$, we
write $|V^*|+n$ for the dimension function whose value at $K$ is $|V^K|+n$.

 \item
Let $n^*$ be a dimension function. A $G$-map $f\colon X\to Y$ is said to be an
{\em $n^*$-equivalence} if $f^K\colon X^K\to Y^K$ is a nonequivariant
$n^K$-equivalence for all $K\subset G$.

 \end{enumerate}
 \end{definition}

\begin{theorem}\label{thm:vequivalence}
A $G$-map $f\colon X\to Y$ is a $(V+n)$-equivalence if and only if it is a
$(|V^*|+n)$-equivalence. The map $f$ is a weak $V$-equivalence if and only if it is a
weak $G$-equivalence.
 \end{theorem}

The proof is based on the following two lemmas.

\begin{lemma}\label{lem:vdisk}
The pair $(D(V+n),S(V+n))$ can be given the structure of a pair of $G$-CW complexes 
in which the cells of the form $G/K\times D^k$ satisfy $k \leq |V^K|+n$.
 \end{lemma}

\begin{proof}
That $(D(V+n),S(V+n))$ can be given the structure of a pair of $G$-CW complexes 
follows from the fact that
smooth $G$-manifolds can be triangulated \cite{Ill:triangulation}.
If $G/K\times D^k$ is a cell, then $eK\times D^k \subset D(V+n)^K$, which
clearly implies that $k\leq |V^K| + n$.
 \end{proof}

\begin{lemma}\label{lem:trivdisk}
The pair $(D(n),S(n))$ is $G$-homotopy equivalent to a pair of $G$-CW($\,V$\!) complexes
of dimension $V - |V^G| + n$.
 \end{lemma}

\begin{proof}
If $V_G = V - V^G$, we shall first show that $D(V_G)$ is $G$-homotopy equivalent
to a $G$-CW($V$) complex of dimension $V - |V^G|$. 
Assuming this, we can then 
triangulate $(D(n),S(n))$ to get a pair of $G$-CW complexes 
of dimension $n$, and the product
$(D(n)\times D(V_G), S(n)\times D(V_G))$ will be a pair of $G$-CW($V$) complexes of
dimension $V-|V^G|+n$ that is $G$-homotopy equivalent to $(D(n),S(n))$.

If $G$ is finite, it is easy to put the correct structure on $D(V_G)$
(see Example~\ref{ex:vcomplexes}(\ref{item:vmanifold})). 
The following argument is necessary
when $G$ is infinite.

We'll prove a slightly more general result (necessary for an inductive
argument): If $H$ is a subgroup of $G$,
$W$ is a sub-$H$-representation of $V$, and $V-W$ contains at least $i$ $H$-trivial
summands, then the pair $G\times_H(D(V-i),S(V-W-i)\times D(W))$ is $G$-homotopy
equivalent to a pair of $G$-CW($V$) complexes of dimension $V-i$
(we consider $S(V-W-i)\times D(W) \subset D(V-W-i)\times D(W) \iso D(V-i)$
as usual). 
We shall prove this by induction on the subgroup $H$.

The beginning of the induction is the case where $H = e$. In this
case, $(D(V-i),S(V-W-i)\times D(W))$ can be given a nonequivariant triangulation of
dimension $|V|-i$, and
crossing with $G$ gives the necessary structure.

Now assume the result for all proper subgroups of $H$. We may as well assume
that $H=G$, since an $H$-CW($V$) structure on $Y$ gives a
$G$-CW($V$) structure on $G\times_H Y$. First assume that $W$ contains a
trivial summand $1$ and consider the pairs
$(D(V-i-1), S(V-W-i)\times D(W-1))$ and $(D(1),S(1))$. If we can show that
the first pair is $G$-homotopy equivalent to a $G$-CW($V$) complex of dimension
$V-i-1$, then, using
an ordinary triangulation of $D(1)$, we get a $G$-CW($V$) complex of dimension
$V-i$ equivalent to
$D(V-i) = D(V-i-1)\times D(1)$, and it is not hard to see that 
$S(V-W-i)\times D(W)$ is equivalent to a subcomplex. A similar argument holds
if $V-W-i$ contains a trivial summand. 

This leaves the case where $V-i$
contains no trivial summands.
It suffices to show that $(S(V-i),S(V-W-i)\times D(W))$ 
is equivalent to a $G$-CW($V$)
complex of dimension $V-i-1$,
since we then get $D(V-i)$ by attaching one $(V-i)$-cell, the
interior.  Now give $(S(V-i),S(V-W-i)\times D(W))$ 
an ordinary $G$-triangulation, which will
have dimension $|V|-i-1$. Taking
the dual cells, we get a ``cell structure'' in which the cells
have the form $G\times_K D(V-V(G/K)-j)$ where $V(G/K)$ is the tangent representation
at the identity coset in $G/K$ and $j \geq i + 1$. 
Notice that $K$ is a proper subgroup of $G$, since $V-i$ has no trivial
summands. Now, by the inductive hypothesis, we can approximate the pair
 \begin{multline*}
 G\times_K [(D(V-V(G/K)-j),S(V-V(G/K)-j))\times D(V(G/K))] \\
 = G\times_K [D(V-j),S(V-V(G/K)-j)\times D(V(G/K))]
 \end{multline*}
by a pair of $G$-CW($V$) complexes of dimension $V-j$,
and attach that approximating pair
(using cellular approximation of the attaching map) instead of
the original dual cell. This constructs a $G$-CW($V$) pair of dimension $V-i$,
$G$-homotopy
equivalent to $(S(V-i),S(V-W-i))$ as required.
 \end{proof}

\begin{proof}[Proof of \ref{thm:vequivalence}]
Suppose first that $f\colon X\to Y$ is a $(V+n)$-equivalence, and that
$K\subset G$. Let 
$i\leq |V^K|+n$ and consider the diagram
 \[
 \xymatrix{
  G\times_K S(i) \ar[r]^-{\alpha} \ar[d] & X \ar[d]^{f} \\
  G\times_K D(i) \ar[r]^-{\beta} & Y
  }
 \]
By Lemma~\ref{lem:trivdisk}, the pair $(D(i),S(i))$ is $K$-homotopy equivalent to a pair of
$K$-CW($V$) complexes of dimension $V-|V^K|+i \leq V + n$. The H.E.L.P. lemma then
implies that there is a $\tilde\beta\colon G\times_K D(i)\to X$ extending
$\alpha$ such that $f\circ\tilde\beta \hmtpc \beta$. Thus,
$f^K\colon X^K\to Y^K$ is a $(|V^K|+n)$-equivalence
for every $K$, hence $f$ is a $(|V^*|+n)$-equivalence.

Conversely, suppose that $f\colon X\to Y$ is a $(|V^*|+n)$-equivalence. Let
$i\leq n$ and consider the diagram
 \[
 \xymatrix{
  G\times_K S(V+i) \ar[r]^-{\alpha} \ar[d] & X \ar[d]^{f} \\
  G\times_K D(V+i) \ar[r]^-{\beta} & Y
  }
 \]
By Lemma~\ref{lem:vdisk}, 
the pair $G\times_K(D(V+i),S(V+i))$ can be given the structure of
a  pair of $G$-CW complexes in which the cells of the form $G/L\times D^k$
satisfy $k \leq |V^L|+i \leq |V^L|+n$. By induction on the cells
we can again find a $G$-map
$\tilde\beta\colon G\times_K D(V+i)\to X$ extending
$\alpha$ such that $f\circ\tilde\beta \hmtpc \beta$. Thus,
$f\colon X\to Y$ is a $(V+n)$-equivalence.

The last statement of the theorem is an immediate consequence of the first.
 \end{proof}

Here is another consequence of Lemmas~\ref{lem:vdisk} and \ref{lem:trivdisk}:

\begin{theorem}\label{thm:vhomotopytype}
A $G$-space $X$ has the $G$-homotopy type of a $G$-CW complex if and only if it
has the $G$-homotopy type of a $G$-CW($\,V$\!) complex.
 \end{theorem}


\section{Ordinary homology and cohomology}\label{sec:vhomology}

We now turn to the construction of ordinary equivariant
homology and cohomology graded on $RO(G)$. 
The {\em orbit category} $\orb{G}$ is the topological category
whose objects are the orbit spaces $G/H$ and whose morphisms
are the $G$-maps between them.
We topologize the mapping
set $\orb G(G/H,G/K)$ as the space $(G/K)^H$.
Write $\sorb G$ for
the {\em stable orbit category}, the category of $G$-orbits and 
$G$-homotopy clases of stable
$G$-maps between them (see \cite{LMM:roghomology} and 
\cite[\S V.9]{LMS:eqhomotopy}). As in \cite{LMM:roghomology}
we take a {\em Mackey functor} to be an additive functor 
$\sorb G \to \Ab$ where $\Ab$ is the category of abelian groups.
(see also \cite{Dr:reps}, \cite{Le:Greenfunctors},
and \cite{GM:eqhomotopy}). 
If we want to specify the group $G$, we shall refer to a {\em $G$-Mackey functor}.
Mackey functors can be either covariant or
contravariant. When $G$ is finite the variance is irrelevant because
$\sorb G$ is self-dual (\cite{Le:Greenfunctors}, \cite{LMS:eqhomotopy}). 
However, when $G$ is infinite the
variance is important, since $\sorb G$ is not self-dual. We shall
write a bar above or below to indicate variance; for example, $\Mackey T$ will
denote a contravariant functor, and $\MackeyOp S$ will denote a covariant one.

This definition is a special case of the following more general definition.

\begin{definition}\label{def:indres}
Let $\A$ be a small preadditive category (that is, its hom sets
are Abelian groups and composition is bilinear). 
Define an {\em $\A$-Mackey functor}
to be an additive functor from $\A$ to $\Ab$, the category of abelian
groups. We shall write $\Mackey T$ for a contravariant Mackey functor,
and $\MackeyOp S$ for a covariant Mackey functor. We define
 \[
 \Hom_{\A}(\Mackey T, \Mackey U)
 \]
to be the group of natural transformations from $\Mackey T$ to $\Mackey U$, and
similarly for covariant Mackey functors. We define
 \[
 \Mackey T \tensor_{\A} \MackeyOp S = \MackeyOp S \tensor_{\A} \Mackey T
 = \int\nolimits^{a\in\A} \Mackey T(a)\tensor\MackeyOp S(a)
 = \left[ \sum_{a} \Mackey T(a)\tensor \MackeyOp S(a) \right] \Big/ \sim,
 \]
where the sum extends over all objects $a$ in $\A$ and we impose on the sum
the equivalence relation generated by
 \[
 (\alpha^* x)\tensor y \sim x\tensor (\alpha_* y)
 \]
when $\alpha\colon a\to a'$ is a map in $\A$, $x\in \Mackey T(a')$ and
$y\in \MackeyOp S(a)$.

Let $\B$ be another small preadditive category and let
$F\colon\A\to\B$ be an additive functor. If $\Mackey U$ is a contravariant
$\B$-Mackey functor and $\MackeyOp S$ is a covariant $\B$-Mackey functor,
define
 \[
 F^*\Mackey U = \Mackey U\circ F
 \]
and
 \[
 F^*\MackeyOp S = \MackeyOp S\circ F.
 \]
If $\Mackey T$ is a contravariant $\A$-Mackey functor, define the 
contravariant $\B$-Mackey functor $F_*\Mackey T$ by
 \[
 (F_*\Mackey T)(b) = \left[ \sum_{b\to F(a)} \Mackey T(a) \right] \Big/ \sim
 \]
where the sum extends over all maps in $\B$ of the form $b\to F(a)$, and we impose on
the sum the equivalence relation generated by
 \[
 t_{b\to F(a) \xrightarrow{F(\alpha)} F(a')} \sim (\alpha^* t)_{b\to F(a)}
 \]
for $\alpha\colon a\to a'$ in $\A$ and $t\in \Mackey T(a')$.
Similarly, if $\MackeyOp R$ is a covariant $\A$-Mackey functor, define the
covariant $\B$-Mackey functor $F_*\MackeyOp R$ by
 \[
  (F_*\MackeyOp R)(b) = \left[ \sum_{F(a)\to b} \MackeyOp R(a) \right] \Big/ \sim
 \]
with the equivalence relation generated by
 \[
 r_{F(a')\xrightarrow{F(\alpha)}F(a)\to b} \sim (\alpha_* r)_{F(a)\to b}.
 \]
 \end{definition}

These definitions could be made more generally, not requiring the categories
$\A$ and $\B$ to be preadditive. In all of our applications the categories
will be preadditive and the functors additive, so it is simpler to just assume
these conditions from the outset.

We have the following very useful examples of Mackey functors.
If $a$ is an object of $\A$, let $\Mackey A_a$ be the $\A$-Mackey functor defined by
$\Mackey A_a = \A(-,a)$, and let $\MackeyOp A^a$ be defined by
$\MackeyOp A^a = \A(a,-)$.

The following facts are standard and straightforward to prove.

\begin{proposition}\label{prop:indresadjunction}
\begin{enumerate} \item[]
\item
For any object $a$ of $\A$ we have the following isomorphisms, 
natural in $a$ as well as in $\Mackey T$ or $\MackeyOp S$.
 \[
 \Hom_{\A}(\Mackey A_a, \Mackey T) \iso \Mackey T(a)
 \]
 \[
 \Mackey A_a \tensor_{\A} \MackeyOp S \iso \MackeyOp S(a)
 \]
 \[
 \Hom_{\A}(\MackeyOp A^a, \MackeyOp S) \iso \MackeyOp S(a)
 \]
 \[
 \MackeyOp A^a \tensor_{\A} \Mackey T \iso \Mackey T(a)
 \]
\item
The functor $F_*$ is left adjoint to $F^*$. That is,
 \[
 \Hom_{\B}(F_*\Mackey T, \Mackey U) \iso \Hom_{\A}(\Mackey T, F^*\Mackey U)
 \]
and similarly for covariant Mackey functors.
\item
We have isomorphisms
 \[
 F_*\Mackey T \tensor_{\B} \MackeyOp S \iso \Mackey T\tensor_{\A} F^*\MackeyOp S
 \]
and
 \[
 \Mackey U \tensor_{\B} F_*\MackeyOp R \iso F^*\Mackey U \tensor_{\A} \MackeyOp R.
 \]
\item
We have the following isomorphisms, natural in $a$.
 \[
 F_*\Mackey A_a \iso \Mackey A_{F(a)}
 \]
and
 \[
 F_*\MackeyOp A^a \iso \MackeyOp A^{F(a)}
 \]
\item
$F^*$ is exact and $F_*$ is right exact. \qed
 \end{enumerate}
 \end{proposition}

Returning to the context of $G$-Mackey functors, we shall be using quite
often the Mackey functors $\Mackey A_{G/K}$ and $\MackeyOp A^{G/K}$.
We think of these as {\em free} Mackey functors because of the isomorphisms
 \[
  \Hom_{\sorb G}(\Mackey A_{G/K}, \Mackey T) \iso \Mackey T(G/K)
 \]
and
 \[
  \Hom_{\sorb G}(\MackeyOp A^{G/K}, \MackeyOp S) \iso \MackeyOp S(G/K),
 \]
given in the preceding proposition.
The isomorphism is given in each case by sending a homomorphism $f$ to
$f(1_{G/K})$.
Thus, we think of $\{1_{G/K}\}$ as a basis for either $\Mackey A_{G/K}$
or $\MackeyOp A^{G/K}$.
In general, we shall use the term {\em free Mackey functor} for a
direct sum of Mackey functors of the form $\Mackey A_{G/K}$ or
$\MackeyOp A^{G/K}$.
If, for example, $\Mackey T = \sum_i \Mackey A_{G/K_i}$, we think of the elements
$1_{G/K_i} \in \Mackey T(G/K_i)$ as forming a basis of $\Mackey T$.
The Mackey functors $\Mackey A_{G/K}$ and $\MackeyOp A^{G/K}$ are
also known as the canonical projectives, as the isomorphisms above make
clear that they are projectives in the categories of contravariant and
covariant Mackey functors, respectively.
As we shall see, these Mackey functors play much the same central role in the theory
of equivariant homology and cohomology as does the group $\Z$ nonequivariantly.


Now let $X$ be a $G$-CW($V$) complex with skeleta $X^{V+n}$. Then clearly
 \[
 X^{V+n}/X^{V+n-1} = \Wedge G_+\smsh_{H_i} S^{V+n},
 \]
where the wedge runs over the $(V+n)$-cells of $X$.

\begin{definition}\label{def:vchains}
Let $X$ be a $G$-CW($V$) complex. The {\em cellular chain complex of $X$}, 
$\Mackey {C}_{V+*} (X)$, is the differential graded contravariant Mackey functor
specified on objects by
 \[
 \Mackey {C}_{V+n} (X)(G/H) 
  = \{ G/H_+ \smsh  S\sp {V+n}, X\sp {V+n}/X\sp {V+n-1} \}_G,
 \]
where $\{-,-\}_G$ denotes the group of stable $G$-maps.
The differential is given in
the usual way by the $G$-map 
$X\sp {V+n}/X\sp {V+n-1}\to \susp X\sp{V+n-1}/X\sp {V+n-2}$.
 \end{definition}

$\Mackey C_{V+n}(X)$ is a free Mackey functor. Precisely, if
 \[
 X^{V+n}/X^{V+n-1} = \Wedge G_+\smsh_{H_i} S^{V+n}
 \]
as above, then
 \[
 \Mackey C_{V+n}(X) \iso \sum \Mackey A_{G/H_i}.
 \]
A basis is given by the set of inclusions
 \[
  G_+\smsh_{H_j} S^{V+n} \includesin \Wedge G_+\smsh_{H_i} S^{V+n}.
 \]
It follows that
 \[
 \Hom_{\sorb G}(\Mackey{C}_{V+n}(X), \Mackey T)
  = \sum \Mackey T(G/H_i)
 \]
and
 \[
 \Mackey{C}_{V+n}(X) \tensor_{\sorb G} \MackeyOp S 
  = \sum \MackeyOp S(G/H_i).
 \]
In both
cases the induced differential can be understood as coming from the attaching
maps of the cells.

\begin{definition}\label{def:vhomology}
 Let $\Mackey T$ be a contravariant Mackey functor and let  
$\MackeyOp S$ be a covariant Mackey functor.
Let $X$ be a $G$-CW($V$) complex.
We define the {\em $(V+n)$th ordinary homology} of
$X$, with coefficients in $\MackeyOp S$, to be
 \[
 H\sp G_{V+n}(X;\MackeyOp S) =
  H_{V+n}( \Mackey{C}_*(X) \tensor_{\sorb G} \MackeyOp S).
 \]
 and we define the {\em $(V+n)$th ordinary cohomology} of $X$, with
coefficients in $\Mackey T$, to be
 \[
 H_G\sp {V+n} (X;\Mackey T) = 
  H_{V+n}(\Hom_{\sorb G}( \Mackey{C}_*(X), \Mackey T)).
 \]
Homology is covariant in $X$ and we consider it contravariant in
$G$-linear isometries $V\to V'$.
Cohomology is contravariant in $X$ and we consider it covariant
in $G$-linear isometries $V\to V'$.
\end{definition}

Definitions~\ref{def:vchains} and \ref{def:vhomology} 
extend to relative $G$-CW($V$) complexes in the obvious way.
If $X$ is a $G$-CW($V$) complex and $A$ is a subcomplex, then
$\Mackey C_*(X,A) \iso \Mackey C_*(X)/\Mackey C_*(A)$ and the usual
homological algebra gives long exact sequences (with any
coefficients)
 \[
 \cdots\to H\sp G_{V+n}(A)\to H\sp G_{V+n}(X)\to H\sp G_{V+n}(X,A)
 \to H\sp G_{V+n-1}(A) \to\cdots
 \]
and
 \[
 \cdots\to H_G\sp {V+n}(X,A) \to H_G\sp {V+n}(X) \to H_G\sp {V+n}(A)
  \to H_G\sp {V+n+1}(X,A) \to\cdots.
 \]
 As usual, if $X$ is a
based $G$-CW($V$) complex, we define the reduced homology
and cohomology of $X$ by
 \[
 \tilde H\sp G_{V+n}(X) = H\sp G_{V+n}(X,*)
 \]
and
 \[
 \tilde H_G\sp {V+n}(X) = H_G\sp {V+n}(X,*).
 \]
If $(X,A)$ is a relative $G$-CW($V$) complex, then
 \[
 H^G_{V+n}(X,A) \iso \tilde H^G_{V+n}(X/A)
 \]
and
 \[
 H_G^{V+n}(X,A) \iso \tilde H_G^{V+n}(X/A)
 \]

We extend to  arbitrary $G$-spaces by approximating, up to weak
$G$-equivalence, by  $G$-CW($V$) complexes. We make homology and cohomology
functorial on homotopy classes of $G$-maps by using cellular approximation of
$G$-maps. 
Following \cite[XIII.1]{May:alaska},
given $V$ and $W$, two representations of $G$, we define
 \[
 \tilde H\sp G_{V\ominus W}(X) = \tilde H\sp G_V ( \susp\sp W X )
 \]
and
 \[
 \tilde H_G\sp {V\ominus W}(X) = \tilde H_G\sp V ( \susp\sp W X )
 \]
where $V\ominus W$ is the ``formal difference'' of $V$ and $W$.
Loosely speaking, this gives a theory graded on $RO(G)$,
since the groups $\tilde H\sp G_{V\ominus W}(X)$ and
$\tilde H_G\sp {V\ominus W}(X)$
are determined up to isomorphism by the equivalence class of $V\ominus W$ in $RO(G)$.
However, if we want to avoid sign ambiguities coming from the phrase
``up to isomorphism,'' we should view these groups as
functors on triples $(X,V,W)$.
We will use the well-established term ``$RO(G)$-graded'' to describe
these theories from now on.

The main properties of $RO(G)$-graded ordinary
homology and cohomology are summarized in the following theorem.
These properties are generalizations of those given by
Wirthm\"uller in \cite{Wi:duality}. The algebra of Mackey functors and the associated
notation used in the last five parts of the theorem will be explained in
Sections~\ref{sec:vsubgroups} and \ref{sec:vproducts},
where those parts will be proved.

\begin{theorem}[Ordinary $RO(G)$-Graded Homology and Cohomology]\label{thm:vhomology}
Let $X$ be a based $G$-space, let $\alpha$ denote $V\ominus W$, let 
$\MackeyOp S$ and $\Mackey T$ be respectively a covariant and a contravariant
Mackey functor. Then the abelian groups $\tilde H^G_{\alpha}(X;\MackeyOp S)$
and $\tilde H_G^{\alpha}(X;\Mackey T)$ are homotopy
functors on the category of triples $(X,V,W)$ and satisfy the
following properties.

\begin{enumerate}
\item
The functor $\tilde H^G_\alpha(-;\MackeyOp S)$ is exact on cofiber sequences
and send wedges to direct sums.
The functor $\tilde H_G^\alpha(-;\Mackey T)$ is exact on cofiber sequences
and send wedges to products.

\item
There are suspension isomorphisms
 \[
  \sigma_Z\colon \tilde H^G_V(X;\MackeyOp S) 
     \xrightarrow{\iso} \tilde H\sp G_{V\oplus Z}(\susp^Z X;\MackeyOp S)
 \]
and
 \[
 \sigma_Z\colon \tilde H_G^V(X;\Mackey T) 
     \xrightarrow{\iso} \tilde H_G^{V\oplus Z}(\susp^Z X;\Mackey T).
 \]
These isomorphisms are natural in the sense of \cite[XIII.1.1]{May:alaska}
and satisfy $\sigma_{Z'}\circ \sigma_Z = \sigma_{Z\oplus Z'}$.


\item\label{item:vdimension}
(Dimension Axiom)
If we restrict $\alpha$ to be an integer, then these groups coincide with
Bredon's ordinary homology and cohomology 
\upn{(}\cite{Br:cohomology} and \cite{Ill:homology}\upn{)}. In
particular, they satisfy the dimension axiom in the following form:
As functors on the stable orbit category, we have isomorphisms
 \[
 \tilde H\sp G_n(-_+;\MackeyOp S) \iso
  \begin{cases} \MackeyOp S & \text{if $n=0$} \\
                 0 & \text{if $n\neq 0$}
  \end{cases}
 \]
and
 \[
 \tilde H_G\sp n(-_+;\Mackey T) \iso
   \begin{cases} \Mackey T & \text{if $n=0$} \\
                  0 & \text{if $n\neq 0$}
   \end{cases}
 \]
for integers $n$.

\item\label{item:vWirthmuller}
(Wirthm\"uller Isomorphism)
If $K \subset G$ there is an isomorphism
 \[
 \tilde H_{\alpha}^G(G_+\smsh_K X;\MackeyOp S)
 \iso  \tilde H_{\alpha|K}^K(X;\MackeyOp S|K),
 \]
and similarly in cohomology.
In cohomology, the composite
 \[
 \tilde H_G^\alpha(X;\Mackey T) 
 \to \tilde H_G^\alpha(G_+\smsh_K X;\Mackey T)
 \iso \tilde H_K^{\alpha|K}(X;\Mackey T|K),
 \]
where the first map is induced by the projection $G/K\to *$, is
called restriction to $K$ and is denoted by $a \mapsto a|K$.
In homology, if $G$ is finite, there is a similar restriction map given by the
composite
 \[
 \tilde H_{\alpha}^G(X;\MackeyOp S)
 \to \tilde H_{\alpha}^G(G_+\smsh_K X;\MackeyOp S)
 \iso \tilde H_{\alpha |K}^K(X;\MackeyOp S|K),
 \]
where the first map is induced by the stable map $*\to G/K$ dual to the projection.
When $G$ is infinite, there is again a restriction in homology, but with a
shift in dimension. We shall discuss this further in Section~\ref{sec:vsubgroups}.

\item\label{item:vfixedsets}
If $K$ is a normal subgroup of $G$ then there is a natural restriction
to fixed sets 
 \[
  \tilde H_{\alpha}^G(X;\MackeyOp S) \to
  \tilde H_{\alpha^K}^{G/K}(X^K;\MackeyOp S^K),  
 \]
and similarly in cohomology; we shall
denote this $a \mapsto a \sp K$.
This restriction respects suspension in the sense that the following
diagram commutes:
 \[
 \xymatrix{
  \tilde H_{V}^G(X;\MackeyOp S) \ar[r]^-{\sigma_Z} \ar[d]_{(-)^K}
    & \tilde H_{V\oplus Z}^G(\susp^Z X;\MackeyOp S) \ar[d]^{(-)^K} \\
  \tilde H_{\alpha^K}^{G/K}(X^K;\MackeyOp S^K) \ar[r]_-{\sigma_{Z^K}}
    & \tilde H_{V^K \oplus Z^K}^{G/K}(\susp^{Z^K}X^K;\MackeyOp S^K)
  }
 \]
Restriction to fixed sets 
respects restriction to subgroups in the sense that, if $K\subset L$,
then $(a|L)^K = a^K |(L/K)$.
It also has the property that the map
 \[
 \xymatrix{
 \MackeyOp S(G/G) \iso \tilde H_0^G(S^0;\MackeyOp S) 
   \ar[r]^-{(-)^K} &  \tilde H_0^{G/K}(S^0;\MackeyOp S^K) \iso \MackeyOp S^K(G/G)
 }
 \]
coincides with the natural projection.
We shall also write $a^K$ for $(a|NK)^K$ when $K$ is not normal in $G$.

\item\label{item:vcupproduct} 
There is a natural associative cup product
 \[
 -\cup - \colon \tilde H^{\alpha}_G(X;\Mackey T) \tensor
 \tilde H^{\beta}_G(Y;\Mackey U) 
  \to \tilde H^{\alpha+\beta}_G(X\smsh Y;\Mackey T\boxprod \Mackey U). 
 \]
If $\Mackey T$ is a ring (i.e., a Green functor) and
$Z\to X\smsh Y$ is a map (e.g., a diagonal $X/(A\cup B)\to X/A\smsh X/B$), 
we also write $x\cup y$ for the induced map
 \[
 -\cup - \colon \tilde H^{\alpha}_G(X;\Mackey T) \tensor
 \tilde H^{\beta}_G(Y;\Mackey T) \to \tilde H^{\alpha+\beta}_G(Z;\Mackey T).
 \]
This cup product satisfies $(x \cup y)|K = (x|K) \cup (y|K)$
and $(x \cup y)\sp K = (x\sp K) \cup (y\sp K)$.

\item\label{item:veval}
There is an evaluation map
 \[
 \langle -,- \rangle \colon \tilde H^{\alpha}_G(X;\Mackey T) \tensor
  \tilde H_{\alpha+\beta}^G(X;\MackeyOp S) 
  \to \tilde H_\beta^G(S^0; \Mackey T\mixprod\MackeyOp S).
 \]
This evaluation is natural: if $f\colon X\to Y$, then
$\langle y, f_*(x)\rangle = \langle f^*(y), x\rangle$. It satisfies
$\langle x,a\rangle|K = \langle x|K, a|K\rangle$ and 
$\langle x,a\rangle^K = \langle x^K, a^K\rangle$.

\item\label{item:vcapproduct}
More generally, there is a cap product
 \[
 -\cap - \colon \tilde H^{\alpha}_G(Y;\Mackey T) \tensor
 \tilde H_{\alpha + \beta}^G(X\smsh Y;\MackeyOp S) 
 \to \tilde H_{\beta}^G(X;\Mackey T \mixprod \MackeyOp S).
 \]
This cap product is natural in the usual sense: if $f\colon X\to X'$ and
$g\colon Y\to Y'$
then $y'\cap (f\smsh g)_*(z) = f_*(g^*(y')\cap z)$.
It satisfies $(w\cup y)\cap z = w \cap (y \cap z)$,
$(y \cap z)|K = (y|K)\cap (z|K)$, and
$(y\cap z)^K = y^K \cap z^K$.
Further, $\langle (x\cup y), z \rangle = \langle x, y\cap z\rangle$.
If $d\colon Z\to X\smsh Y$ is a map
(e.g., a diagonal $X/(A\cup B)\to X/A\smsh X/B$),
we also write $y\cap z$ for $y\cap d_*(z)$.
 \end{enumerate}
 \end{theorem}

The first three parts of the theorem are proved in the usual way. The
remaining parts will be proved in Sections~\ref{sec:vsubgroups} and \ref{sec:vproducts},
where they will also be restated more generally in terms of spectra.

\begin{remark}
The Wirthm\"uller isomorphism in homology given in
part (\ref{item:vWirthmuller}) above is
the map of the same name discussed in \cite[\S{II.6}]{LMS:eqhomotopy},
although at first they may look different. 
The isomorphism given in \cite{LMS:eqhomotopy} is
 \[
 E^G_*(G\halfsmsh_K Y) \iso E^K_*(\Susp^{V(G/K)} Y)
 \]
for any $G$-spectrum $E$ and $K$-spectrum $Y$, where $V(G/K)$ is the tangent
representation at the identity coset of $G/K$.
The two isomorphisms agree because the restriction 
to $K$ of the $G$-spectrum
representing ordinary $G$-homology does
not represent ordinary $K$-homology, but ordinary $K$-homology with a shift in grading
of $V(G/K)$. We shall have more to say about this in the next section.
 \end{remark}

\begin{definition}
If $\tilde h^G_*(-)$ is a reduced $RO(G)$-graded homology theory in the sense of 
\cite{Wi:duality} or \cite[XIII.1]{May:alaska},
write $\MackeyOp h^G_\alpha$ for the covariant Mackey functor whose value at
$G/K$ is $\tilde h^G_\alpha(G/K_+)$. Similarly, if $\tilde h_G^*(-)$ is a reduced $RO(G)$-graded
cohomology theory, let $\Mackey h_G^\alpha$ denote the contravariant Mackey
functor whose value at $G/K$ is $\tilde h_G^\alpha(G/K_+)$.
 \end{definition}

The Atiyah-Hirzebruch spectral sequence generalizes as follows
(cf. \cite[\S 12]{Bo:spectralsequences}).

\begin{theorem}[Atiyah-Hirzebruch Spectral Sequence]
Suppose that $\tilde h^G_*(-)$ is a reduced $RO(G)$-graded homology theory.  Let
$\alpha$ and $\beta$ be elements of $RO(G)$. Then there is a 
strongly convergent spectral sequence
 \[
 E^2_{p,q} = \tilde H^G_{\alpha+p}(X;\MackeyOp h^G_{\beta+q})
  \convto \tilde h^G_{\alpha+\beta+p+q}(X).
 \]
Similarly, if $\tilde h_G^*(-)$ is a reduced $RO(G)$-graded cohomology theory,
there is a conditionally convergent spectral sequence
 \[
 E_2^{p,q} = \tilde H_G^{\alpha+p}(X;\Mackey h_G^{\beta+q})
  \convto \tilde h_G^{\alpha+\beta+p+q}(X).
 \]
 \end{theorem}

\begin{proof}
This spectral sequence arises in the usual way:
If $\alpha = V\ominus W$, let $X^{\alpha+n}$ denote the
$(V+n)$-skeleton of a $G$-CW($V$) approximation of
$\susp^W X$.
This skeletal filtration leads to an exact couple, upon applying
either $\tilde h_G^*(-)$ or $\tilde h_*^G(-)$, which gives the spectral sequence.
 \end{proof}
 
Behind this proof are the useful facts that
 \[
 \Mackey C_{V+n}(X,*)\tensor_{\sorb G} \MackeyOp S
   \iso \tilde H^G_{V+n}(X^{V+n}/X^{V+n-1}; \MackeyOp S)
 \]
and
 \[
 \Hom_{\sorb G}(\Mackey C_{V+n}(X,*), \Mackey T)
  \iso \tilde H_G^{V+n}(X^{V+n}/X^{V+n-1}; \Mackey T).
 \]

\begin{corollary}[Uniqueness of Ordinary $RO(G)$-Graded Homology]\label{cor:vCohomUniqueness}
Let $\MackeyOp S$ be a covariant Mackey
functor. Let $\tilde h^G_*(-)$ be a reduced $RO(G)$-graded homology theory obeying the
dimension axiom as in 
part~(\ref{item:vdimension}) of Theorem~\ref{thm:vhomology}, with 
$\MackeyOp h^G_0 \iso \MackeyOp S$.
Then there is a natural isomorphism $\tilde h^G_*(-) \iso \tilde H^G_*(-;\MackeyOp S)$.
Similarly, if $\Mackey T$ is a contravariant Mackey functor and 
$\tilde h_G^*(-)$ is a reduced
$RO(G)$-graded cohomology theory obeying the dimension axiom with 
$\Mackey h_G^0 \iso \Mackey T$, then $\tilde h_G^*(-) \iso \tilde H_G^*(-;\Mackey T)$.
 \end{corollary}

\begin{proof}
There is an Atiyah-Hirzebruch spectral sequence
 \[
 E^2_{p,q} = \tilde H^G_{\alpha+p}(X;\MackeyOp h^G_q)
  \convto \tilde h^G_{\alpha+p+q}(X)
 \]
(and similarly for cohomology).
The spectral sequence collapses to the line $q=0$ when $\tilde h^G_*(-)$
satisfies the dimension axiom.
 \end{proof}

Also useful is a universal coefficients spectral sequence.
The canonical projectives, used as coefficients here, 
are the universal coefficients, in the same sense
that $\Z$ is in nonequivariant ordinary homology.
Write $\Mackey H^G_\alpha(X)$ for the contravariant Mackey functor given by
 \[
 \Mackey H^G_\alpha(X)(G/K) = \tilde H^G_\alpha(X;\MackeyOp A^{G/K}).
 \]
Note that, with this notation, we have
 \[
 \Mackey C_{V+n}(X,*) \iso \Mackey H^G_{V+n}(X^{V+n}/X^{V+n-1}).
 \]
$\Tor^{\sorb G}$ and $\Ext_{\sorb G}$ 
below are the derived functors of $\tensor_{\sorb G}$ and
$\Hom_{\sorb G}$ respectively.

\begin{theorem}[Universal Coefficients Spectral Sequence]
If $\MackeyOp S$ is a covariant Mackey functor and $\Mackey T$ is a
contravariant Mackey functor, there are spectral sequences
 \[
 E^2_{p,q} = \Tor^{\sorb G}_p(\Mackey H^G_{\alpha+q}(X), \MackeyOp S)
 \convto
 \tilde H^G_{\alpha+p+q}(X;\MackeyOp S)
 \]
and
 \[
 E_2^{p,q} = \Ext_{\sorb G}^p(\Mackey H^G_{\alpha+q}(X), \Mackey T)
 \convto
 \tilde H_G^{\alpha+p+q}(X;\Mackey T).
 \]
\end{theorem}

\begin{proof}
These are constructed in the usual way by taking 
resolutions of $\MackeyOp S$
and $\Mackey T$ and then taking the spectral sequences of the bigraded complexes
obtained from the chain complex of $X$ and the resolutions.
 \end{proof}

\begin{remark}
Bredon's original integer-graded theories can be defined using unstable
versions of all of the above. In particular, we use the homotopy orbit category
$h\orb G$ rather than the stable orbit category. A useful consequence
for our theories is that there is
an ``unstable'' universal coefficients spectral sequence
in integer grading:
 \[
  E^2_{p,q} = \Tor^{h\orb G}_p(\Mackey H^G_{q}(X), \MackeyOp S)
 \convto
 \tilde H^G_{p+q}(X;\MackeyOp S).
 \]
Here,
 \[
 \Mackey H^G_q(X)(G/K) = \tilde H^G_q(X;\MackeyOp B^{G/K})
 \]
where $\MackeyOp B^{G/K}(G/J) = \Z[G/K,G/J]_G$, the free abelian group on
the set of $G$-homotopy classes of unstable $G$-maps.
Looking at orbits, one can see that, in fact,
 \[
 \Mackey H^G_q(X)(G/K) = \tilde H^G_q(X;\MackeyOp B^{G/K}) 
   \iso \tilde H_q(X^K/W_0K; \Z)
 \]
where $W_0K$ is the identity coset of $WK = NK/K$.
 \end{remark}


\section{The representing spectra}

In this section we discuss the equivariant spectra representing ordinary
homology and cohomology. 
There are now a number of models of equivariant spectra available.
For our purposes in this section it does not matter much which we use, because they
all give equivalent stable categories. The orthogonal spectra
of \cite{MM:orthogonal} have good formal properties, but
the prespectra of \cite{LMS:eqhomotopy} and \cite{MM:orthogonal}
are perhaps best suited for discussions of representing cohomology theories.
Moreover, we shall need to use presepectra later
in the parametrized case.
We shall use the term ``spectrum'' to mean any convenient model,
but where we have to be specific we shall use $G$-prespectra.
We borrow freely from the notations used in \cite{LMS:eqhomotopy}.

We begin with the spectrum $H\Mackey T$ representing
ordinary cohomology with coefficients in $\Mackey T$. Constructions have
previously been given or announced in several places, including
\cite{LMM:roghomology},
\cite{CW:loopspaces}, \cite{Le:emspaces},
and \cite{GM:eqhomotopy}.
By Corollary~\ref{cor:vCohomUniqueness}, $H\Mackey T$ is unique
up to weak $G$-equivalence, and is characterized by its homotopy:
 \[
 \Mackey\pi^G_n(H\Mackey T) \iso
 \begin{cases}  \Mackey T & \text{if $n = 0$} \\
                0 & \text{if $n\neq 0$,} \end{cases}
 \]
for integers $n$. Here,
 \[
 \Mackey\pi^G_n(H\Mackey T)(G/K) =
   \pi^K_n(H\Mackey T) = [\susp^n\susp^\infty G/K_+, H\Mackey T]_G
 \]
where $\susp^\infty X$ is the suspension spectrum associated to the based
$G$-space $X$, and $[-,-]_G$ denotes the homotopy group of stable $G$-maps of
spectra (i.e., the maps we obtain by inverting weak equivalences). It suffices therefore to construct a $G$-spectrum with that homotopy;
we shall of course call such a spectrum an {\em Eilenberg-Mac\,Lane
spectrum} of type $\Mackey T$.

\begin{theorem}
If $\Mackey T$ is a contravariant Mackey functor, there exists a $G$-CW
Eilenberg-Mac\,Lane spectrum of type $\Mackey T$. This 
$G$-CW spectrum is unique up to
$G$-homotopy equivalence.
 \end{theorem}

\begin{proof}
As mentioned above, constructions of this spectrum have been given elsewhere
several times. We give here a simple construction that we can modify later
to construct other interesting spectra.

First, it is clear that
there exists an orbit-wise epimorphism $\Mackey F\to \Mackey T$ where
$\Mackey F$ is a free contravariant Mackey functor,
 \[
 \Mackey F = \sum_i \Mackey A_{G/H_i}.
 \]
Let
 \[
 X^0 = \Wedge \susp^\infty (G/H_i)_+ = \Wedge (G/H_i)_+\smsh S^0.
 \]
Then $X^0$ is connective and
 \[
 \Mackey\pi^G_0 (X^0) \iso \Mackey F.
 \]
We can now kill the kernel of $\Mackey F\to \Mackey T$ by attaching 1-cells,
by which we mean taking the cofiber of a map 
$\Wedge (G/H_j)_+\smsh S^0 \to X^0$. This gives a spectrum $X^1$ with
$\Mackey\pi^G_0(X^1) \iso \Mackey T$. We then kill the higher
homotopy groups inductively by attaching higher dimensional cells.
This gives a $G$-CW Eilenberg-Mac\,Lane spectrum of type $\Mackey T$. Since
Eilenberg-Mac\,Lane spectra are unique up to weak $G$-equivalence, $G$-CW 
Eilenberg-Mac\,Lane spectra are unique up to $G$-homotopy equivalence.
 \end{proof}

Eilenberg-Mac\,Lane spectra behave well on passage to subgroups and fixed-sets.
Recall from \cite[I.3.7]{LMS:eqhomotopy} that, 
if $E$ is a $G$-spectrum indexed on a complete $G$-universe $\U$, then there is a
$WK$-spectrum $E^K$ indexed on $\U^K$, called the $K$-fixed point spectrum,
given on the prespectrum level by $E^K(V) = E(V)^K$ for each $WK$-invariant
subspace $V\subset \U^K$.
There is an adjunction
 \[
  [D, E^K]_{WK} \iso [\epsilon^\sharp D, E]_{NK}.
 \]
Here, $D$ is a $WK$-spectrum indexed on $\U^K$
and $\epsilon^\sharp D$ is the $NK$-spectrum obtained by first letting $NK$ act on
$D$  via the projection $\epsilon\colon NK\to WK$, then expanding the universe
from $\U^K$ to $\U$.

\begin{proposition}\label{prop:emsubgroups}
Let $H\Mackey T$ be a $G$-Eilenberg-Mac\,Lane spectrum of type $\Mackey T$, and
let $K$ be a subgroup of $G$. Then $H\Mackey T$, regarded as a $K$-spectrum, is
again an Eilenberg-Mac\,Lane spectrum, with
 \[
 [\susp^\infty K/L_+, H\Mackey T]_K \iso \Mackey T(G/L)
 \]
for $L\subset K$. Also,
$(H\Mackey T)^K$ is a $WK$-Eilenberg-Mac\,Lane spectrum, with
 \[
 [\susp^\infty WK/(L/K)_+, (H\Mackey T)^K]_{WK} \iso \Mackey T(G/L)
 \]
for $K \subset L \subset NK$.
 \end{proposition}

\begin{proof}
For the first claim, we have
 \begin{align*}
 [\susp^n\susp^\infty K/L_+, H\Mackey T]_K
   &\iso [\susp^n\susp^\infty G/L_+, H\Mackey T]_G \\
   &\iso \pi_n^L(H\Mackey T) \\
   &\iso \begin{cases} \Mackey T(G/L) & \text{if $n = 0$} \\
                       0 & \text{if $n \neq 0$.}
         \end{cases}
 \end{align*}
For the second, we have
 \begin{align*}
 [\susp^n\susp^\infty WK/(L/K)_+, (H\Mackey T)^K]_{WK}
  &\iso [\susp^n\susp^\infty NK/L_+, H\Mackey T]_{NK} \\
  &\iso [\susp^n\susp^\infty G/L_+, H\Mackey T]_G \\
  &\iso \pi_n^L(H\Mackey T)
 \end{align*}
again.
 \end{proof}

Now for homology.  Let $\MackeyOp S$ be a covariant
Mackey functor;
we shall construct a spectrum $H\MackeyOp S$ representing ordinary
homology with coefficients in $\MackeyOp S$.
(The existence of such spectra was noted in
\cite[XIII.4]{May:alaska}, where they were denoted by $J\MackeyOp S$.)
By Corollary~\ref{cor:vCohomUniqueness}, 
$H\MackeyOp S$ is unique up to weak $G$-equivalence, and is characterized by its
homotopy in this sense:
 \[
 [S^n, G/K_+\smsh H\MackeyOp S]_G \iso
 \begin{cases}  \MackeyOp S(G/K) & \text{if $n = 0$} \\
                 0 & \text{if $n\neq 0$,}
  \end{cases}
 \]
for integers $n$, naturally in $G/K$.
It suffices therefore to construct a $G$-spectrum with that homotopy;
we shall call such a spectrum a {\em Mac\,Lane-Eilenberg
spectrum} of type $\MackeyOp S$.

In order to relate this characterization of Mac\,Lane-Eilenberg spectra to that
of Eilenberg-Mac\,Lane spectra, recall from 
\cite[II.6.3 \& \S III.2]{LMS:eqhomotopy} that
the dual of the orbit $G/K$ is $D(G/K_+) = G\halfsmsh_K S^{-V(G/K)}$ where $V(G/K)$
is the tangent $K$-representation at $eK\in G/K$. Thus,
 \[
 [S^n, G/K_+\smsh H\MackeyOp S]_G \iso [\susp^n D(G/K_+), H\MackeyOp S]_G.
 \]

\begin{definition}\label{def:dualhomotopy}
If $X$ is a $G$-spectrum define the
{\em $n$th dual homotopy groups} of $X$ by
 \[
 \dualhmtpy_n^K(X) = [\susp^n D(G/K_+), X]_G 
   \iso [S^n, G/K_+\smsh X]_G.
 \]
Also, let $\MackeyOp \dualhmtpy^G_n(X)$ be the covariant Mackey functor defined by
 \[
 \MackeyOp\dualhmtpy^G_n(X)(G/K) = \dualhmtpy_n^K(X).
 \]
 \end{definition}

With this notation we can rewrite the
characterization of $H\MackeyOp S$ as
 \[
 \MackeyOp\dualhmtpy^G_n(H\MackeyOp S) \iso
 \begin{cases}  \MackeyOp S & \text{if $n = 0$} \\
                 0 & \text{if $n\neq 0$.}
 \end{cases}
 \]
If $G$ is finite, $V(G/K) = 0$ for all $K$ and $D(G/K_+) = \susp^\infty G/K_+$.
As a consequence, Mac\,Lane-Eilenberg spectra are
Eilenberg-Mac\,Lane spectra, and the two notions are coextensive.
However, when $G$ is infinite
they are quite different.

\begin{theorem}
If $\MackeyOp S$ is a covariant Mackey functor, there exists a $G$-CW
Mac\-\,Lane-Eilenberg spectrum of type $\MackeyOp S$. This 
$G$-CW spectrum is unique up
to $G$-homotopy equivalence.
 \end{theorem}

\begin{proof}
We ``dualize'' the construction of Eilenberg-Mac\,Lane spectra we gave above.
First, it is clear that
there exists an orbit-wise epimorphism $\MackeyOp F\to \MackeyOp S$ where
$\MackeyOp F$ is a free covariant Mackey functor,
 \[
 \MackeyOp F = \sum_i \MackeyOp A^{G/H_i}.
 \]
Let
 \[
 X^0 = \Wedge D((G/H_i)_+),
 \]
then, by duality,
 \[
 \MackeyOp\dualhmtpy^G_n(X^0) \iso 
  \begin{cases} 0 & \text{if $n<0$,} \\
                \MackeyOp F & \text{if $n=0$}.
  \end{cases}
 \]
We can now kill the kernel of $\MackeyOp F\to \MackeyOp S$ by attaching ``dual''
1-cells, by which we mean taking the cofiber of a map 
$\Wedge D((G/H_j)_+)\smsh S^0 \to X^0$. This gives a spectrum $X^1$ with
$\MackeyOp\dualhmtpy^G_0(X^1) \iso \MackeyOp S$ naturally in $G/K$. We then kill the
higher dual homotopy groups $\MackeyOp\dualhmtpy^G_n(X^n)$ inductively by
attaching higher dimensional dual cells. This gives a Mac\,Lane-Eilenberg
spectrum of type $\MackeyOp S$ of the $G$-homotopy type of a $G$-CW spectrum;
since Mac\,Lane-Eilenberg spectra are unique up to weak $G$-equivalence, $G$-CW
Mac\,Lane-Eilenberg spectra are unique up to $G$-homotopy equivalence.
 \end{proof}

We shall return to the idea of ``dual'' cells in the next section.

Mac\,Lane-Eilenberg spectra do not behave well on passage to fixed-sets
(however, see Corollary~\ref{cor:mefixedsets}), but
they behave reasonably well on passage to subgroups.

\begin{proposition}\label{prop:mesubgroups}
Let $H\MackeyOp S$ be a $G$-Mac\,Lane-Eilenberg spectrum of type $\MackeyOp S$,
and let $K$ be a subgroup of $G$. Then the $K$-spectrum 
$\susp^{V(G/K)}H\MackeyOp S$
is again a Mac\,Lane-Eilenberg spectrum, with
 \[
 \MackeyOp\dualhmtpy^K_0(\susp^{V(G/K)}H\MackeyOp S)(K/L) =
 [D(K/L_+), \susp^{V(G/K)}H\MackeyOp S]_K \iso \MackeyOp S(G/L)
 \]
for $L\subset K$.
 \end{proposition}

\begin{proof}
 \begin{align*}
 [\susp^nD(K/L_+),\susp^{V(G/K)}H\MackeyOp S]_K 
    &\iso [\susp^n K\halfsmsh_L S^{-V(K/L)},\susp^{V(G/K)}H\MackeyOp S]_K \\
    &\iso [\susp^n \susp^{-V(G/K)}K\halfsmsh_L S^{-V(K/L)},H\MackeyOp S]_K \\
    &\iso [\susp^n K\halfsmsh_L S^{-V(G/L)},H\MackeyOp S]_K \\
    &\qquad \text{since $V(G/K)\dirsum V(K/L) \iso V(G/L)$} \\
    &\iso [\susp^n G\halfsmsh_L S^{-V(G/L)},H\MackeyOp S]_G \\
    &\iso [\susp^n D(G/L_+), H\MackeyOp S]_G.
 \end{align*}
 \end{proof}



\section{Dual complexes and the dual theories}\label{sec:dualvcomplexes}

If $G$ is finite, then the spectrum $H\Mackey T$ representing ordinary
cohomology also represents ordinary homology, so these theories are dual in the
usual sense. However, if $G$ is infinite, then ordinary homology and cohomology
are represented by different spectra, and so are not dual theories.
In this case, there are two more theories to consider, the duals of the ordinary
theories.
Following convention, we write theories defined on spectra without tildes.

\begin{definition}
If $\Mackey T$ is a contravariant Mackey functor, let 
{\em dual (ordinary) homology theory},
$\H^G_*(-;\Mackey T)$,
be the homology theory on spectra represented by $H\Mackey T$.
If $\MackeyOp S$ is a covariant Mackey
functor, let 
{\em dual (ordinary) cohomology theory},
$\H_G^*(-;\MackeyOp S)$, be the cohomology theory on spectra
represented by $H\MackeyOp S$.
If $X$ is a based $G$-space we write
$\widetilde \H^G_*(X;\Mackey T) = \H^G_*(\susp^\infty X;\Mackey T)$
and similarly for cohomology.
If $X$ is an unbased $G$-space we write
$\H^G_*(X;\Mackey T) = \widetilde\H^G_*(X_+;\Mackey T)$ and
similarly for cohomology and for pairs.
 \end{definition}

Once again, if $G$ is finite, $\H^G_*(-;\Mackey T) = H^G_*(-;\MackeyOp T)$
where $\MackeyOp T$ denotes $\Mackey T$ considered as a covariant functor by
the self-duality of $\sorb G$. Similarly,
$\H_G^*(-;\MackeyOp S) = H_G^*(-;\Mackey S)$. Therefore, the
discussion in this section is relevant only for infinite groups.

The dual theories satisfy a list of properties related to those of ordinary homology
and cohomology; we state most of them for spectra for convenience.
We use repeatedly the fact mentioned in the preceding section,
that $D(G/K_+) = G\halfsmsh_K S^{-V(G/K)}$.

\begin{theorem}\label{thm:dualvhomology}
Let $X$ be a $G$-spectrum.
Let $\alpha$ denote $V\ominus W$ and let 
$\MackeyOp S$ and $\Mackey T$ be respectively a covariant and a contravariant
Mackey functor. Then the theories $\H_G^{\alpha}(X;\MackeyOp S)$
and $\H^G_{\alpha}(X;\Mackey T)$ satisfy the
following properties.

\begin{enumerate}
\item (Dimension Axiom)
They satisfy the following dimension axioms.
 \[
 \H^G_n(D(G/K_+);\Mackey T) \iso
  \begin{cases} \Mackey T(G/K) & \text{if $n = 0$,} \\
                0 & \text{if $n\neq 0$,}
  \end{cases}
 \]
 \[
 \H_G^n(D(G/K_+);\MackeyOp S) \iso
  \begin{cases} \MackeyOp S(G/K) & \text{if $n = 0$,} \\
                0 & \text{if $n\neq 0$,}
  \end{cases}
 \]
for integers $n$, naturally on the stable orbit category.

\item\label{item:dualvWirthmuller} (Wirthm\"uller Isomorphism)
If $K \subset G$ and $X$ is a $K$-spectrum, there is an isomorphism
 \[
 \H_{\alpha}\sp G(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\Mackey T)
 \iso  \H_{\alpha|K}\sp K(X;\Mackey T|K),
 \]
and similarly in dual cohomology.
In dual homology, the composite
 \begin{align*}
 \H^G_\alpha(X;\Mackey T) 
 &\to \H^G_\alpha(D(G/K_+)\smsh X; \Mackey T) \\
 &\iso \H^G_\alpha(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\Mackey T) \\
 &\iso \H^K_{\alpha|K}(X;\Mackey T|K),
 \end{align*}
where the first map is induced by the dual of the projection $G/K\to *$, is
called restriction to $K$ and is denoted by $a \mapsto a|K$.
There is a similar restriction in cohomology, but with a
shift in dimension. We shall discuss this further in Section~\ref{sec:vsubgroups}.

\item\label{item:dualvfixedsets}
If $K$ is a normal subgroup of $G$ then there is a natural restriction
to fixed sets 
\[
 \H_{\alpha}\sp G(X;\Mackey T) \to
 \H_{\alpha \sp K}\sp {G/K}(\Phi^K(X);\Mackey T\sp K) , 
\]
and similarly in dual cohomology; we shall
denote this $a \mapsto a \sp K$.
(We shall discuss the construction $\Phi^K$ in the next section;
if $X$ is a $G$-space then $\Phi^K(\susp^\infty X) \hmtpc \susp^\infty X^K$.)
This restriction respects suspension and restriction to subgroups in
the same way that restriction to fixed sets does for ordinary homology and cohomology,
and its evaluation at a point again gives the projection.
We shall also write $a^K$ for $(a|NK)^K$ when $K$ is not normal in $G$.

\item\label{item:dualvcupproduct} 
There is a natural cup product
 \[
 -\cup - \colon H\sp{\alpha}_G(X;\Mackey T) \tensor
 \H\sp{\beta}_G(Y;\MackeyOp S) 
 \to \H\sp{\alpha+\beta}_G(X\smsh Y; \Mackey T\mixprod \MackeyOp S). 
 \]
This cup product is associative in the sense that
$(x\cup y)\cup z = x\cup(y\cup z)$ for
$x\in H_G^*(X)$, $y\in H_G^*(Y)$ and $z\in\H_G^*(Z)$.
This cup product also satisfies $(x \cup y)|K = (x|K) \cup (y|K)$
and $(x \cup y)\sp K = (x\sp K) \cup (y\sp K)$.
If $d\colon Z\to X\smsh Y$ is a map we also write
$x\cup y$ for $d^*(x\cup y)$.

\item\label{item:dualveval}
There are evaluation maps
 \[
 \langle -,- \rangle \colon \H^{\alpha}_G(X;\MackeyOp S) \tensor
  \H_{\alpha+\beta}^G(X;\Mackey T) \to H_\beta^G(S; \MackeyOp S\mixprod\Mackey T)
 \]
and
 \[
 \langle -,- \rangle \colon H^{\alpha}_G(X;\Mackey T) \tensor
  \H_{\alpha+\beta}^G(X;\Mackey U) \to \H_\beta^G(S; \Mackey T\boxprod\Mackey U).
 \]
These evaluations are natural in the usual sense: if $f\colon X\to Y$, then
$\langle y, f_*(x)\rangle = \langle f^*(y), x\rangle$. They also satisfy
$\langle x,a\rangle|K = \langle x|K, a|K\rangle$ and 
$\langle x,a\rangle^K = \langle x^K, a^K\rangle$.

\item\label{item:dualvcapproduct}
If $X$ and $Y$ are $G$-spectra,
there are cap products
 \[
 -\cap - \colon \H\sp{\alpha}_G(Y;\MackeyOp S) \tensor
 \H_{\alpha + \beta}\sp G(X\smsh Y;\Mackey T) \to 
 H_{\beta}^G(X;\MackeyOp S \mixprod \Mackey T)
 \]
and
 \[
 -\cap - \colon H\sp{\alpha}_G(Y;\Mackey T) \tensor
 \H_{\alpha + \beta}\sp G(X\smsh Y;\Mackey U) \to 
 \H_{\beta}^G(X;\Mackey T \boxprod \Mackey U). 
 \]
These cap products are natural in the usual sense: if $f\colon X\to X'$
and $g\colon Y\to Y'$,
then $y'\cap (f\smsh g)_*(z) = f_*(g^*(y')\cap z)$.
They also satisfy $(x\cup y)\cap a = x \cap (y \cap a)$,
$(x \cap a)|K = (x|K)\cap (a|K)$, and
$(x\cap a)\sp K = x \sp K \cap a \sp K$.
Further, $\langle x\cup y, a \rangle = \langle x, y\cap a\rangle$.
If $d\colon Z\to X\smsh Y$ is a map we also write
$y\cap z$ for $y\cap d_*(z)$.
 \end{enumerate}
 \end{theorem}

The dimension axiom follows from the definitions of the dual theories.
The remaining properties will be proved in the next two sections.

In the remainder of this section we shall show that the dual theories 
can be computed from chain complexes
based on ``dual cell complexes,'' which we now define.
Although we shall see important examples of such cell complexes on the space level,
the theory of these complexes is intrinsically stable.

\begin{definition}
Let $V$ be a virtual representation of $G$.
A {\em dual $V$-cell} is a spectrum of the form
$D(G/K_+)\smsh CS^{V+n}$ for some (possibly negative) integer $n$,
where $C$ denotes the cone.
The {\em boundary} of such a cell is $D(G/K_+)\smsh S^{V+n}$,
and its {\em dimension} is $|V|+n+1$.
\end{definition}

\begin{definition}
Let $V$ be a virtual representation of $G$.
A {\em dual $V$-cell spectrum} is a filtered $G$-spectrum
\[
 X = \colim_{n\geq 0} X_n,
\]
where $X_0 = *$ and, for $n>0$, each $X_n$ is obtained from
$X_{n-1}$ by attaching a wedge of dual $V$-cells along their boundaries.
We call $\{X_n\}$ the {\em sequential filtration} of $X$.
A dual $V$-cell spectrum $X$ is a
{\em dual $G$-CW($V$) spectrum}
if each cell of dimension $n$ is attached only to cells of dimension
less than $n$; for such a spectrum we let the {\em skeleton}
$X^n$ be the subspectrum consisting of all the cells of $X$ of
dimension no more than $n$, and call $\{X^n\}$ the
{\em skeletal filtration} of $X$.
As with $G$-CW($V$) complexes, we shall usually write $X^{V+n}$ for $X^{|V|+n}$.
If $(X,A)$ is a pair of $G$-spectra, we define a {\em relative}
dual $V$-cell or $G$-CW($V$) structure on $(X,A)$ in the obvious way,
by taking $X_0 = A$.
A $G$-map $X\to Y$ of dual $G$-CW($V$) spectra is {\em cellular} if it
respects the skeletal filtrations.
 \end{definition}


\begin{examples}\label{ex:dualvcomplexes}
\begin{enumerate} \item[]
\item
As an example of a dual $G$-CW spectrum, we have the 
Mac\,Lane-Eilenberg spectrum
$H\MackeyOp S$ constructed in the preceding section.

\item
The dual of a finite $G$-CW($V$) spectrum has a dual $G$-CW($-V$)
structure.
Explicitly, if $X$ is a finite $G$-CW($V$) spectrum, then the duals
$D(X/X^{V+n})$ give a skeletal filtration of $DX$
as a dual $G$-CW($-V$) spectrum.

\item\label{item:dualvmanifold}
If $V$ is a representation of $G$ and
$M$ is a smooth  $V$-dimensional $G$-manifold with a $G$-triangulation,
then the dual cell structure on $M$
(as described in Example~\ref{ex:vcomplexes}(\ref{item:vmanifold})) gives
a dual $G$-CW($V$) structure to $\susp^\infty M_+$. This is the
motivation for the following space level definition.
 \end{enumerate}
 \end{examples}

\begin{definition}
Let $V$ be a representation of $G$.
 A {\em dual $G$-CW($\,V$\!) complex} is a $G$-space $X$ together with a
decomposition $X = \colim_n X\sp n$ such that:
 \begin{enumerate}
 \item 
$X\sp 0$ is a disjoint union of 0 or more $G$-orbits $G/K$ such that $G/K$ is
discrete and $K$ acts trivially on $V$.
 \item 
$X\sp n$ is obtained from $X\sp {n-1}$ by
attaching ``dual cells'' of the form $G \times_K D(V - V(G/K) - |V| + n)$ where 
$V - V(G/K) - |V| + n$ is an actual $K$-representation. The
attaching maps have the form  
$G\times_K S(V - V(G/K) - |V| + n) \to X\sp {n-1}$.
Notice that this sphere may be empty even if $n>0$, in which case attaching the
cell means taking the disjoint union with $G/K$.
 \end{enumerate} 
Similarly, a relative $G$-CW($V$) complex is a
pair $(X,A)$ where $X = \colim_nX\sp n$, $X\sp 0$ is the disjoint
union of $A$ with orbits as in (1), and cells are attached as in (2).
We shall often write $X^{V+n}$ for $X^{|V|+n}$.
If $X = X^{V+n}$, we shall say that $X$ is
{\em dually $(V+n)$-dimensional}.
 \end{definition}

\begin{remark}
If $X$ is a dual $G$-CW($V$) complex, then $\susp^\infty X_+$ is a dual
$G$-CW($V$) spectrum, since
 \[
 \susp^\infty G\halfsmsh_K S^{V - V(G/K) - |V| + n} = D(G/K_+)\smsh S^{V - |V| + n}.
 \]
An example of this was given in \ref{ex:dualvcomplexes}(\ref{item:dualvmanifold}). 
The dual cell structure
on a smooth closed $V$-dimensional $G$-manifold $M$ is a dual $G$-CW($V$)
structure.
 \end{remark}

An interesting way of thinking of the distinction between $G$-CW($V$) complexes
and dual $G$-CW($V$) complexes is that, in the latter case, an orbit
$G/K$ contributes its geometric dimension $V(G/K$) to the dimension of a
cell. That is, $G\times_K D(W)$ is considered to be $(W+V(G/K))$-dimensional,
which is its natural geometric dimension. On the other hand, in an ordinary
$G$-CW($V$) complex a cell $G\times_K D(W)$ is just $W$-dimensional, ignoring
the dimension of $G/K$.

Now for the appropriate homotopy groups.
in the stable context, we make the following definitions.

\begin{definition}\label{def:gendualhomotopy}
If $V$ is a virtual representation of $G$ and $X$ is a $G$-spectrum, define
the {\em $V$th dual homotopy groups} of $X$ by
 \[
 \dualhmpty^K_V(X) = [S^V, G/K_+\smsh X]_G \iso [\susp^VD(G/K_+), X]_G.
 \]
Also, let $\MackeyOp \dualhmtpy^G_V(X)$ be the covariant Mackey functor defined by
 \[
 \MackeyOp\dualhmtpy^G_V(X)(G/K) = \dualhmtpy_V^K(X).
 \]
 \end{definition}
 
This generalizes Definition~\ref{def:dualhomotopy}.

\begin{definition}\label{def:dualVequivalence}
Let $V$ be a virtual representation of $G$.
A map of $G$-spectra $f\colon X\to Y$ is a {\em dual $(V+n)$-equivalence} if 
 \[
 f_*\colon \MackeyOp\dualhmpty^G_{V+i}(X) \to \MackeyOp\dualhmpty^G_{V+i}(Y)
 \]
is an isomorphism for all $i < n$ and an epimorphisms for $i=n$. 
It is a {\em dual weak $V$-equivalence}
if it is a dual $(V+n)$-equivalence for all $n$.
 \end{definition}

For spaces things are a little more delicate, stemming from the fact
that duality is essentially a stable phenomenon.
Because we are ultimately interested in how dual cells map into spaces,
we make the following definition, using the homotopy sets
defined in Definition~\ref{def:VHomotopySet}.

\begin{definition}
Let $X$ and $Y$ be $G$-spaces. A $G$-map
$f\colon X\to Y$ is a {\em dual $(V+n)$-equivalence}
if $\pi^K_{V-V(G/K)+i}(f)$ is trivial
for all $K$ and all $i\leq n$ for which $V-V(G/K)+i$ is an actual
representation of $K$.
The map $f$ is called a {\em dual weak $V$-equivalence} if it
is a dual $(V+n)$-equivalence for all $n$.
\end{definition}

The restriction that $V-V(G/K)+i$ has to be an actual representation of $K$
has the result that dual weak $V$-equivalence for spaces is weaker than weak 
$G$-equivalence. Below we shall see that these concepts coincide for
$G$-spectra, however.

As in Section~\ref{sec:vcomplexes} (see also \cite[\S I.5]{LMS:eqhomotopy}),
we can now prove the following.
(Except where noted, the following results have analogues for
spaces.)

\begin{lemma}[H.E.L.P.]\label{lem:dualHELP}
 Let $r\colon Y \to Z$ be a dual $(V+n)$-equivalence.
Let $(X,A)$ be a relative 
dual $V$-cell spectrum
with cells of dimension
$\leq |V|+ n$. If the following diagram commutes without the
dashed arrows, then there exist maps $\tilde g$ and $\tilde h$
making the diagram commute.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\times I \ar'[d][dd] \ar[dl]_(.6)h & & 
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g \ar[dd]  \\
   & Z & & Y \ar[ll]_(.3)r \\
  X \ar[ur]^f \ar[rr]_-{i_0} & & X\times I \ar@{-->}[ul]^(.6){\tilde h} & &
   X \ar[ll]^-{i_1} \ar@{-->}[ul]^(.6){\tilde g} \\
 }
 \]
The result remains true when $n = \infty$.
\qed
 \end{lemma}
 
\begin{theorem}[Whitehead]
\begin{enumerate} \item[]
\item
 If $f\colon Y\to Z$ is a dual $(V+n)$-equivalence of $G$-spectra, and $X$ is a
dually $(V+n-1)$-dimensional dual $G$-CW(\,$V$\!) spectrum, then  
$f_*\colon [X,Y]_G \to [X,Z]_G$ is an isomorphism. 
It is an epimorphism if $X$ is
dually $(V+n)$-dimensional.

\item
 If $f\colon Y\to Z$ is a dual weak $V$-equivalence and $X$ is
a dual $G$-CW(\,$V$\!) complex, then $f_*\colon [X,Y]_G \to [X,Z]_G$ is an
isomorphism.  In particular any dual weak $V$-equivalence of 
dual $G$-CW(\,$V\!$) spectra
is a $G$-homotopy equivalence. \qed

\end{enumerate}
\end{theorem}


\begin{theorem}[Cellular Approximation]
If $f\colon X\to Y$ is a map of dual $G$-CW(\,$V\!$) spectra, then $f$ is $G$-homotopic
to a cellular map.
 \end{theorem}

\begin{proof}
This follows from the H.E.L.P. lemma once we know that
the inclusion $Y^{V+n}\to Y$ is a dual $(V+n)$-equivalence.
This reduces to showing that there are no essential $G$-maps
$G\times_H (D(V-V(G/H)),S(V-V(G/H))) \to
G\times_K (D(V-V(G/K)+k),S(V-V(G/K)+k))$
for $k>0$. This follows from obstruction theory, the obstructions
vanishing for dimensional reasons. The point is that
$G\times_H (V-V(G/H))$ and $G\times_K(V-V(G/K))$ have
the same fixed-point dimensions.
\end{proof}


\begin{theorem}[Approximation by Dual $G$-CW($V$) Spectra]
 Let X be a $G$-spectrum. Then there exists a dual $G$-CW(\,$V\!$)
spectrum $\Gamma' X$ and a dual weak $V$-equivalence $\Gamma' X\to X$.
 \end{theorem}

\begin{proof}
The proof is the same as for ordinary $G$-CW($V$) spectra,
except that we kill the appropriate dual homotopy groups.
\end{proof}

The following two results do not seem to have analogues for spaces.

\begin{proposition}
A $G$-spectrum $X$ has the $G$-homotopy type of a $G$-CW spectrum if and only
if it has the $G$-homotopy type of a dual $G$-CW(\,$V$\!) spectrum.
 \end{proposition}

\begin{proof}
If $X$ is a dual $G$-CW($V$) spectrum, then $X$ has the $G$-homotopy type of a
$G$-CW spectrum, since $D(G/K_+)$ has the homotopy type of a $G$-CW spectrum
\cite[III.2.6]{LMS:eqhomotopy}.

Conversely, to see that a $G$-CW spectrum has the $G$-homotopy type of a dual
$G$-CW($V$) spectrum, it suffices to show that $G/K_+\smsh S^n$ has the $G$-homotopy
type of a dual $G$-CW($V$) spectrum. For this, consider the dual spectrum
$D(G/K_+\smsh S^n)$; this has the $G$-homotopy type of a $G$-CW($-V$) spectrum
by Theorem~\ref{thm:vhomotopytype}. 
Hence $G/K_+\smsh S^n = D(D(G/K_+\smsh S^n))$ has the $G$-homotopy
type of a dual $G$-CW($V$) spectrum.
 \end{proof}

\begin{corollary}\label{cor:dualweakequivalence}
A $G$-map of spectra $X\to Y$ is a weak $G$-equivalence if and only if it is a
dual weak $V$-equivalence.
 \qed
 \end{corollary}


Now let $X$ be a dual $G$-CW($V$) spectrum. Let 
$\MackeyOp C^G_{V+*}(X) = \MackeyOp C_{V+*}(X)$, the 
{\em covariant chains} of $X$, be given by
 \[
 \MackeyOp C_{V+n}(X) = \MackeyOp\dualhmpty^G_{V+n}(X^{V+n}/X^{V+n-1}).
 \]

\begin{theorem}
There are natural isomorphisms
 \[
 \H^G_{V+n}(X;\Mackey T) 
   \iso H_{V+n}(\MackeyOp C_*(X)\tensor_{\sorb G} \Mackey T)
 \]
and
 \[
 \H_G^{V+n}(X;\MackeyOp S)
  \iso H_{V+n}(\Hom_{\sorb G}(\MackeyOp C_*(X), \MackeyOp S)).
 \]
 \end{theorem}

\begin{proof}
Construct the spectral sequence obtained by applying $\H^G_*$ or
$\H_G^*$ to the dual skeletal filtration of $X$. The
$E^2 = E^\infty$ (or $E_2=E_\infty$) term consists, by the dimension axiom,
of the groups on the right above.
We could also use the direct argument of \cite[15.2 \& 19.5]{May:concise}.
 \end{proof}

Once again, behind this proof are the useful identifications
 \[
 \MackeyOp C_{V+n}(X)\tensor_{\sorb G} \Mackey T
  \iso \H^G_{V+n}(X^{V+n}/X^{V+n-1}; \Mackey T)
 \]
and
 \[
 \Hom_{\sorb G}(\MackeyOp C_{V+n}(X), \MackeyOp S)
  \iso \H_G^{V+n}(X^{V+n}/X^{V+n-1}; \MackeyOp S).
 \]

The spectral sequence used in the preceding proof is, of course, a dual
Atiyah-Hirzebruch spectral sequence.

\begin{theorem}[Dual Atiyah-Hirzebruch Spectral Sequence]
Let $h_G^*(-)$ be an $RO(G)$-graded cohomology theory and 
$h^G_*(-)$ its dual theory, and let $X$ be a dually bounded below
$G$-spectrum. Then there is a strongly convergent spectral sequence
 \[
 E^2_{p,q} = \H_{\alpha+p}^G(X;\Mackey h_G^{-\beta-q})
  \convto h^G_{\alpha+\beta+p+q}(X)
 \]
and a conditionally convergent spectra sequence
 \[
 E_2^{p,q} = \H^{\alpha+p}_G(X;\MackeyOp h^G_{-\beta-q})
 \convto h_G^{\alpha+\beta+p+q}(X).
 \]
Here, as usual, $\Mackey h_G^*$ denotes the
contravariant Mackey functor with 
 \[
 \Mackey h_G^*(G/K) = h_G^*(G/K) \iso h^G_{-*}(D(G/K_+)),
 \] 
and
$\MackeyOp h^G_*$ is the covariant Mackey functor with
 \[
 \MackeyOp h^G_*(G/K) = h^G_*(G/K) \iso h_G^{-*}(D(G/K_+)).
 \]
\qed
 \end{theorem}

For the universal coefficients spectral sequence, we write
 \[
 \MackeyOp\H_*^G(X)(G/K) = \H_*^G(X;\Mackey A_{G/K}).
 \]
With this notation, we have
 \[
 \MackeyOp C_{V+n}(X) \iso \MackeyOp\H^G_{V+n}(X^{V+n}/X^{V+n-1}).
 \]

\begin{theorem}[Dual Universal Coefficients Spectral Sequence]
\label{thm:dualucss}
If $\MackeyOp S$ is a covariant Mackey functor and $\Mackey T$ is a
contravariant Mackey functor, there are spectral sequences
 \[
 E^2_{p,q} = \Tor^{\sorb G}_p(\MackeyOp\H^G_{\alpha+q}(X), \Mackey T)
 \convto
 \H^G_{\alpha+p+q}(X;\Mackey T)
 \]
and
 \[
 E_2^{p,q} = \Ext_{\sorb G}^p(\MackeyOp\H^G_{\alpha+q}(X), \MackeyOp S)
 \convto
 \H_G^{\alpha+p+q}(X;\MackeyOp S).
 \]
\qed
\end{theorem}


\section{Subgroups}\label{sec:vsubgroups}

In this section we prove parts (\ref{item:vWirthmuller}) and (\ref{item:vfixedsets}) of
Theorem~\ref{thm:vhomology} and the corresponding parts 
of Theorem~\ref{thm:dualvhomology}. 

\subsection*{Restriction to Subgroups}

\begin{definition}
Let $K$ be a subgroup of $G$ and let $\sorb K$ and $\sorb G$ be the
respective stable orbit categories.
Let $i_K^G = G\halfsmsh_K- \colon \sorb K\to \sorb G$ be the induction functor
defined in \cite[II.4.1]{LMS:eqhomotopy}. Its effect on objects is
$G\halfsmsh_K \susp^\infty K/L_+ = \susp^\infty (G\times_K(K/L))_+
= \susp^\infty G/L_+$.
If $\Mackey T$ is a contravariant $G$-Mackey functor, let
$\Mackey T|K$ denote $(i_K^G)^*\Mackey T = \Mackey T\circ i_K^G$,
as in Definition~\ref{def:indres}. 
Similarly, if $\MackeyOp S$ is a
covariant Mackey functor, let $\MackeyOp S|K = (i_K^G)^*\MackeyOp S$.
If $\Mackey C$ is a contravariant $K$-Mackey functor, let 
$G\times_K \Mackey C$ denote $(i_K^G)_*\Mackey C$,
as defined in Definition~\ref{def:indres}.
If $\MackeyOp D$ is a covariant $K$-Mackey functor, let 
$G\times_K\MackeyOp D = (i_K^G)_*\MackeyOp D$.
 \end{definition}
 
It follows from Proposition~\ref{prop:indresadjunction} that
 \[
 \Hom_{\sorb G}(G\times_K \Mackey C, \Mackey T)
 \iso \Hom_{\sorb K}(\Mackey C, \Mackey T|K)
 \]
for any contravariant $K$-Mackey functor $\Mackey C$ and contravariant $G$-Mackey
functor $\Mackey T$. The same adjunction holds for covariant Mackey functors.
We also have the isomorphisms
 \[
 (G\times_K \Mackey C)\tensor_{\sorb G} \MackeyOp S
 \iso \Mackey C\tensor_{\sorb K} (\MackeyOp S|K)
 \]
if $\MackeyOp S$ is a covariant $G$ Mackey functor, and
 \[
 (G\times_K \MackeyOp D)\tensor_{\sorb G} \Mackey T
  \iso \MackeyOp D \tensor_{\sorb K} (\Mackey T|K).
 \]
Finally, we have the calculations
 \[
 G\times_K \Mackey A_{K/J} \iso \Mackey A_{G/J}
 \]
and
 \[
 G\times_K \MackeyOp A^{K/J} \iso \MackeyOp A^{G/J}.
 \]

The following is 
part (\ref{item:vWirthmuller}) of Theorem~\ref{thm:vhomology},
restated with spectra for convenience.

\begin{theorem}[Wirthm\"uller Isomorphisms]
Let $X$ be a $G$-spectrum.
If $K \subset G$ there are natural isomorphisms
 \[
 H_{\alpha}\sp G(G\halfsmsh_K X;\MackeyOp S)
 \iso  H_{\alpha|K}\sp K(X;\MackeyOp S|K)
 \]
and
 \[
 H^{\alpha}_G(G\halfsmsh_K X;\Mackey T)
 \iso  H^{\alpha|K}_K(X;\Mackey T|K).
 \]
 \end{theorem}

\begin{proof}
If $V$ is a representation of $G$ and $X$ is a $K$-CW($V$) spectrum,
then $G\halfsmsh_K X$ is a $G$-CW($V$) spectrum with corresponding cells. 
Define a map 
 \[
  G\times_K\Mackey C^K_{V+n}(X) \to \Mackey C^G_{V+n}(G\halfsmsh_K X)
 \]
as follows. Let $c_{G/L\xrightarrow{d}G\times_K(K/M)}$ be an element
of $G\times_K\Mackey C^K_{V+n}(X)$, where
 \[
 c\colon K/M_+\smsh S^{V+n}\to X^{V+n}/X^{V+n-1}.
 \] 
Send
$c_{G/L\xrightarrow{d}G\times_K(K/M)}$ to
 \[
 (G\halfsmsh_K c)\circ(d\smsh 1) \colon
  G/L_+\smsh S^{V+n} \to
  (G\halfsmsh_K X^{V+n})/(G\halfsmsh_K X^{V+n-1}).
 \]
The isomorphism $G\times_K \Mackey A_{K/J} \iso \Mackey A_{G/J}$
then implies that
 \[
 \Mackey C^G_*(G\halfsmsh_K X) \iso G\times_K \Mackey C^K_*(X).
 \]
Using Proposition~\ref{prop:indresadjunction}, 
we see that there are chain isomorphisms
 \[
 \Mackey C^G_*(G\halfsmsh_K X)\tensor_{\sorb G} \MackeyOp S
 \iso \Mackey C^K_*(X)\tensor_{\sorb K} (\MackeyOp S|K)
 \]
and
 \[
 \Hom_{\sorb G}(\Mackey C^G_*(G\halfsmsh_K X), \Mackey T)
 \iso \Hom_{\sorb K}(\Mackey C^K_*(X), \Mackey T|K).
 \]
These isomorphisms induce the Wirthm\"uller isomorphisms.
 \end{proof}

The following restatement of part (\ref{item:dualvWirthmuller}) of
Theorem~\ref{thm:dualvhomology} gives
the analogues of the Wirthm\"uller isomorphisms for the dual theories.

\begin{theorem}[Dual Wirthm\"uller Isomorphisms]
If $K \subset G$ and $X$ is a $K$-spectrum, there are natural isomorphisms
 \[
 \H_{\alpha}\sp G(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\Mackey T)
 \iso  \H_{\alpha|K}\sp K(X;\Mackey T|K)
 \]
and
 \[
 \H^{\alpha}_G(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\MackeyOp S)
 \iso  \H^{\alpha|K}_K(X;\MackeyOp S|K).
 \]
 \end{theorem}

\begin{proof}
If $V$ is a representation of $G$ and $X$ is a dual $K$-CW($V$) spectrum, then
$G\halfsmsh_K (S^{-V(G/K)}\smsh X)$ is a dual $G$-CW($V$) spectrum with corresponding
cells. We use here the fact that
 \begin{align*}
 G\halfsmsh_K (S^{-V(G/K)}\smsh D(K/L_+))
  &= G\halfsmsh_K (S^{-V(G/K)}\smsh (K\halfsmsh_L S^{-V(K/L)})) \\
  &= G\halfsmsh_L( S^{-V(G/K)}\smsh S^{-V(K/L)}) \\
  &= G\halfsmsh_L S^{-V(G/L)} \\
  &= D(G/L_+).
 \end{align*}
As in the proof of the preceding theorem, 
this leads to the desired isomorphisms on the chain level.
 \end{proof}


While we gave chain-level proofs of the Wirthm\"uller isomorphisms,
it is illuminating to see them on the represented level as well. 
In the following discussion, $X$ will always represent a spectrum. The
ordinary cohomology Wirthm\"uller isomorphism is just the adjunction
 \[
 [G\halfsmsh_K X,H\Mackey T]_G \iso [X,(H\Mackey T)|K]_K
 \]
together with the equivalence $(H\Mackey T)|K \iso H(\Mackey T|K)$ shown in
Proposition~\ref{prop:emsubgroups}
(we are suppressing the grading for clarity). 
The ordinary homology isomorphism is given by
 \[
 [S,G\halfsmsh_K X \smsh H\MackeyOp S]_G
 \iso [S, X\smsh \susp^{V(G/K)}(H\MackeyOp S)|K]_K
 \iso [S, X\smsh H(\MackeyOp S|K)]_K.
 \]
Here, the first isomorphism is shown in \cite[II.6.5]{LMS:eqhomotopy} and the second is
given by the equivalence 
$\susp^{V(G/K)}(H\MackeyOp S)|K \iso H(\MackeyOp S|K)$ shown in
Proposition~\ref{prop:mesubgroups}. 
That these isomorphisms agree with the ones constructed on the chain level above
follows by comparing their behavior on 
the filtration quotients $X^{V+n}/X^{V+n-1}$, which are just wedges of 
(suspensions of) orbits.

The dual cohomology isomorphism is
 \begin{align*}
 [G\halfsmsh_K (S^{-V(G/K)}\smsh X), H\MackeyOp S]_G
  &\iso [S^{-V(G/K)}\smsh X, (H\MackeyOp S)|K]_K \\
  &\iso [S^{-V(G/K)}\smsh X, S^{-V(G/K)}\smsh H(\MackeyOp S|K)]_K \\
  &\iso [X, H(\MackeyOp S|K)]_K.
 \end{align*}
The dual homology isomorphism is
 \begin{align*}
 [S, G\halfsmsh_K(S^{-V(G/K)}\smsh X)\smsh H\Mackey T]_G
  &\iso [S, \susp^{V(G/K)}(S^{-V(G/K)}\smsh X)\smsh (H\Mackey T)|K]_K \\
  &\iso [S, X\smsh H(\Mackey T|K)]_K.
 \end{align*}


In Theorem~\ref{thm:vhomology} we defined the 
ordinary cohomology restriction map $a \mapsto a|K$ to be
the composite
 \[
 H_G^\alpha(X;\Mackey T) 
 \to H_G^\alpha(G\halfsmsh_K X;\Mackey T)
 \iso H_K^{\alpha|K}(X;\Mackey T|K)
 \]
where the first map is induced by the projection $G/K\to *$. 
On the represented level,
it is easy to see that this is simply the forgetful map
 \[
 [X,H\Mackey T]_G \to [X,(H\Mackey T)|K]_K \iso [X,H(\Mackey T|K)]_K.
 \]
It is also easy to see that, when evaluated at the sphere spectrum $S$, the map
 \[
 \Mackey T(G/G) \iso H_G^0(S;\Mackey T) \to H_K^0(S;\Mackey T|K) 
  \iso \Mackey T(G/K)
 \]
is the same as the map induced by the projection $G/K\to G/G$.
In ordinary homology, the obvious forgetful map is this:
 \[
 [S,X\smsh H\MackeyOp S]_G \to [S,X\smsh (H\MackeyOp S)|K]_K
 \iso [S,X\smsh \susp^{-V(G/K)}H(\MackeyOp S|K)]_K.
 \]
This gives a map
 \[
 H^G_\alpha(X;\MackeyOp S) 
 \to H^G_{\alpha|K + V(G/K)}(X;\MackeyOp S|K)
 \]
which we shall call restriction to $K$. When $G$ is finite, $V(G/K) = 0$, and
restriction behaves as expected. In terms of the Wirthm\"uller
isomorphism, this restriction can be written as the composite
 \begin{align*}
 H^G_\alpha(X;\MackeyOp S)
 &\to H^G_\alpha(D(G/K_+)\smsh X;\MackeyOp S) \\
 &\iso H^G_\alpha(G\halfsmsh_K(S^{-V(G/K)}\smsh X); \MackeyOp S) \\
 &\iso H^K_{\alpha|K}(S^{-V(G/K)}\smsh X; \MackeyOp S|K) \\
 &\iso H^K_{\alpha|K+V(G/K)}(X; \MackeyOp S|K).
 \end{align*}
Here, the first map is induced by the map $S\to D(G/K_+)$ dual to the projection
$G/K\to G/G$. When $G$ is finite, the map
 \[
 \MackeyOp S(G/G) \iso H^G_0(S;\MackeyOp S) \to H^K_0(S;\MackeyOp S|K)
 \iso \MackeyOp S(G/K)
 \]
agrees with the map induced by the stable map $G/G\to G/K$ dual to the
projection.


In Theorem~\ref{thm:dualvhomology}
we defined the dual homology restriction map $a\mapsto a|K$ to be the
composite
 \[
 \H^G_\alpha(X;\Mackey T) 
 \to \H^G_\alpha(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\Mackey T)
 \iso \H^K_{\alpha|K}(X;\Mackey T|K),
 \]
where the first map is induced by the dual of the projection $G/K\to G/G$.
On the represented level, this is the forgetful map
 \[
 [S,X\smsh H\Mackey T]_G
  \to [S,X\smsh (H\Mackey T)|K]_K \iso [S,X\smsh H(\Mackey T|K)]_K.
 \]
In dual cohomology, the forgetful map is
 \[
 [X,H\MackeyOp S]_G \to [X,(H\MackeyOp S)|K]_K
  \iso [X,\susp^{-V(G/K)}H(\MackeyOp S|K)]_K,
 \]
giving a forgetful map of the form
 \[
 \H_G^\alpha(X;\MackeyOp S) \to \H_K^{\alpha|K-V(G/K)}(X;\MackeyOp S|K).
 \]
This restriction can be written as the composite
 \begin{align*}
 \H_G^\alpha(X;\MackeyOp S)
 &\to \H_G^\alpha(G/K_+\smsh X;\MackeyOp S) \\
 &\iso \H_G^\alpha(G\halfsmsh_K(S^{-V(G/K)}\smsh S^{V(G/K)}\smsh X);\MackeyOp S)\\
 &\iso \H_K^{\alpha|K}(S^{V(G/K)}\smsh X; \MackeyOp S|K) \\
 &\iso \H_K^{\alpha|K-V(G/K)}(X;\MackeyOp S|K).
 \end{align*}

\subsection*{Restriction to Fixed Sets}

We shall now restate and prove part (\ref{item:vfixedsets}) of 
Theorem~\ref{thm:vhomology}, and the
corresponding part of Theorem~\ref{thm:dualvhomology}. 
Until further notice, $K$ will be a {\em normal} subgroup of $G$.
We first give some definitions and constructions.

\begin{definition}
If $\MackeyOp S$ is a covariant $G$-Mackey functor and $K$ is a normal
subgroup of $G$, let
$\MackeyOp S_K$ be the sub-Mackey functor of $\MackeyOp S$ generated by the
elements in the groups $\MackeyOp S(G/L)$ for those $L$ not containing $K$. Let
$\MackeyOp S^K$ be the $G/K$-Mackey functor defined by
 \[
 \MackeyOp S^K(G/L) = \MackeyOp S(G/L)/\MackeyOp S_K(G/L)
 \]
for $K\subset L\subset G$.
If $\Mackey T$ is a contravariant Mackey functor, define $\Mackey T_K$ and
$\Mackey T^K$ in exactly the same way.
 \end{definition}

\begin{remarks}\label{rems:mackeyfixedsets}
 \begin{enumerate} \item[]
 \item
The construction $\MackeyOp S^K$ has already been considered by algebraists. 
In \cite[5.1]{TW:mackey} Th\'evenaz and Webb
write $\MackeyOp S_+$ for $\MackeyOp S^K$, and show that this construction
is left adjoint to the {\em inflation} functor $\Inf_{G/K}^G$ defined by
 \[
 (\Inf_{G/K}^G \MackeyOp S)(G/L) =
 \begin{cases} \MackeyOp S((G/K)/(L/K)) & \text{if $K\subset L$} \\
               0 & \text{if $K\not\subset L$.}
 \end{cases}
 \]
Notice that there is a short exact sequence
 \[
  0 \to \MackeyOp S_K \to \MackeyOp S \to \Inf_{G/K}^G (\MackeyOp S^K) \to 0
 \]
where the rightmost map $\MackeyOp S \to \Inf_{G/K}^G \MackeyOp S^K$ is adjoint
to the identity map on $\MackeyOp S^K$.
 \item
If $G$ is finite and $\Mackey T$ is a contravariant Mackey functor, then we can
regard $\MackeyOp T$ as a covariant Mackey functor via the self-duality of
$\sorb G$. The two functors $\Mackey T^K$ and $\MackeyOp T^K$ so obtained
are again dual, so the definition is consistent.
 \item
If $K$ is not normal in $G$, we let
$\MackeyOp S^K$ denote $(\MackeyOp S|NK)^K$.
There are places below where we must pay careful attention to
the restriction from $G$ to $NK$.
 \end{enumerate}
 \end{remarks}

We shall also need the following construction, discussed in detail in
\cite[II.9]{LMS:eqhomotopy} 
(see also \cite{Ad:prereq}, \cite{Co:thomspectra}, and
\cite{GM:eqhomotopy}).

\begin{definition}\label{def:geometricfixed}
Let $\F[K]$ denote the family of subgroups
 \[
 \F[K] = \{ L\subset G \mid K\subset L \}.
 \]
Let $E\F[K]$ be the classifying space of $\F[K]$, and let $\tilde E\F[K]$ be
the cofiber in 
 \[
 E\F[K]_+ \to S^0 \to \tilde E\F[K].
 \]
The $G$-CW complex $\tilde E\F[K]$ is characterized up to $G$-homotopy
by its fixed-sets:
 \[
 \tilde E\F[K]^L \hmtpc
 \begin{cases} S^0 & \text{if $K \subset L$} \\
               * & \text{if $K \not\subset L$.}
 \end{cases}
 \]
If $E$ is a $G$-spectrum, define a $G/K$-spectrum $\Phi^K(E)$ by
 \[
 \Phi^K(E) = (\tilde E\F[K]\smsh E)^K.
 \]
 \end{definition}

The construction $\Phi^K$ (called the {\em geometric fixed-point construction}
in \cite{GM:eqhomotopy})
behaves better in many ways than the ordinary 
$K$-fixed point construction. In particular, if $X$ is a based $G$-space, then
 \[
 \Phi^K(\susp^\infty X) \hmtpc \susp^\infty X^K,
 \]
and if $E$ and $E'$ are $G$-spectra, then
 \[
 \Phi^K(E\smsh E') \hmtpc \Phi^K(E)\smsh \Phi^K(E')
 \]
(these are \cite[II.9.9 and II.9.12]{LMS:eqhomotopy}). Notice that $\Phi^K$ induces a
restriction
 \[
 \Phi^K\colon [\susp^\infty X, \susp^\infty Y]_G
  \to [\susp^\infty X^K, \susp^\infty Y^K]_{G/K},
 \]
which, when $X$ is compact, is just the obvious restriction to fixed-sets
 \[
 \colim_V [\susp^V X, \susp^V Y]_G
  \to \colim_{V} [\susp^{V^K} X^K, \susp^{V^K} Y^K]_{G/K}.
 \]

The following calculation is the motivation for the definition of $\MackeyOp S^K$ and 
underlies the definition of restriction to fixed-sets.

\begin{proposition}\label{prop:burnsidefixedsets}
If $K$ is normal in $G$, then
 \[
 (\MackeyOp A^{G/L})^K \iso \{(G/L)^K, -\}_{G/K}
  = \begin{cases} \{G/L, -\}_{G/K} & \text{if $K\subset L$} \\
                  0 & \text{if $K\not\subset L$}
    \end{cases}
 \]
and
 \[
 (\Mackey A_{G/L})^K \iso \{-,(G/L)^K\}_{G/K}
  = \begin{cases} \{-, G/L\}_{G/K} & \text{if $K\subset L$} \\
                  0 & \text{if $K\not\subset L$}
    \end{cases}
 \]
for any subgroup $L$ of $G$.
 \end{proposition}

\begin{proof}
Notice first a simple fact that we shall use many times:
since $K$ is a normal subgroup of $G$,
$(G/L)^K$ is either $G/L$ or empty, depending on whether
$K\subset L$ or not.

Now consider the following map. Let $J\subset G$, and consider the
composite
 \begin{align*}
 \MackeyOp A^{G/L}(G/J)
 &= [\susp^\infty G/L_+, \susp^\infty G/J_+]_G \\
 &\to [\Phi^K(\susp^\infty G/L_+), \Phi^K(\susp^\infty G/J_+)]_{G/K} \\
 &\iso [\susp^\infty (G/L_+)^K, \susp^\infty (G/J_+)^K]_{G/K}.
 \end{align*}
Since $(G/J)^K$ is $G/J$ if $K\subset J$ or empty if $K\not\subset J$,
this defines a map $\MackeyOp A^{G/L} \to \Inf_{G/K}^G\{(G/L)^K,-\}_{G/K}$,
and hence a map 
 \begin{align*}
 (\MackeyOp A^{G/L})^K \to \{(G/L)^K,-\}_{G/K}
 \end{align*}
by Remarks~\ref{rems:mackeyfixedsets}.
By \cite[V.9.4]{LMS:eqhomotopy}, 
$\MackeyOp A^{G/L}(G/J) = \sorb G(G/L,G/J)$ is a free abelian group on
equivalence classes of diagrams
 \[
 G/L \xleftarrow{\phi} G/H \xrightarrow{\chi} G/J
 \]
where $H$ runs through the subgroups of $L$ with $W_L H$ finite,
$\phi$ is the projection, and $\chi$ is a space-level $G$-map.
(The corresponding stable map is $\chi\circ\phi^*$, where $\phi^*$ is
the transfer associated to $\phi$.)
Similarly, if $K\subset J$, the group
$[\susp^\infty (G/L_+)^K, \susp^\infty G/J_+ ]_{G/K}$ is a
free abelian group on equivalence classes of diagrams
 \[
 (G/L)^K \xleftarrow{\phi} G/H \xrightarrow{\chi} G/J
 \]
where $H$ runs through the groups such that $K\subset H\subset L$ and $W_L H$ is finite,
$\phi$ is the projection, and $\chi$ is a space-level $G$-map.
From this description it is easy to see that the map from
$\MackeyOp A^{G/L}(G/J)$ to $\{(G/L)^K,G/J\}_{G/K}$
defined above is an epimorphism. Further, if $K\not\subset H$, then the
generator $G/L\from G/H\to G/J$ of $(\MackeyOp A^{G/L})(G/J)$
must map to 0, since $(G/H)^K = \emptyset$. It
follows that the kernel of the map is precisely $(\MackeyOp A^{G/L})_K(G/J)$,
showing that $(\MackeyOp A^{G/L})^K \iso \{(G/L)^K,-\}_{G/K}$ as
claimed.

The proof for contravariant systems is almost identical.
 \end{proof}

We can now define the restriction to fixed-sets.
The following is part~(\ref{item:vfixedsets}) of
Theorem~\ref{thm:vhomology}, restated for spectra for convenience.

\begin{theorem}[Restriction to Fixed-Sets]\label{thm:vfixedsets}
If $X$ is a $G$-spectrum and
$K$ is a normal subgroup of $G$, there is a natural restriction
to fixed sets 
 \[
 (-)^K\colon H_{\alpha}\sp G(X;\MackeyOp S) \to
 H_{\alpha \sp K}\sp {G/K}(\Phi^K(X);\MackeyOp S\sp K).
 \]
This restriction respects suspension in the sense that the following
diagram commutes:
 \[
 \xymatrix{
  {\tilde H_{\alpha}\sp G(X;\MackeyOp S)} \ar[r]^-{\sigma_Z} \ar[d]_{(-)^K}
   & {\tilde H_{\alpha + Z}\sp G(\susp^Z X;\MackeyOp S)} \ar[d]^{(-)^K} \\
  {\tilde H_{\alpha^K}\sp {G/K}(\Phi^K(X);\MackeyOp S^K)} \ar[r]^-{\sigma_{Z^K}}
   & {\tilde H_{\alpha^K + Z^K}\sp {G/K}(\susp^{Z^K} \Phi^K(X);\MackeyOp S^K)}
  }
 \]
(and similarly for cohomology).
It respects restriction to subgroups in the sense that, if $K\subset L$,
then $(a|L)^K = (a^K)|(L/K)$.
It also has the property that the map
 \[
 \MackeyOp S(G/G) \iso H_0^G(S;\MackeyOp S) 
   \xrightarrow{(-)^K} H_0^{G/K}(S;\MackeyOp S^K) \iso \MackeyOp S^K(G/G)
 \]
is the quotient map from the definition of
$\MackeyOp S^K$, as is the similar map in cohomology.
 \end{theorem}

\begin{proof}
We give a chain level construction of this restriction; later
we shall discuss the represented equivalent.

The construction is really the observation that, if $X$ is a $G$-CW($V$)
spectrum, then $\Phi^K(X)$
is a $(G/K)$-CW($V^K$) complex with corresponding cells.
More precisely, a cell $\susp^{\infty}(G_+\smsh_H S^{V+n})$
in $X^{V+n}/X^{V+n-1}$ either contributes
nothing to $\Phi^K(X)$ (if $K\not\subset H$) or gives a cell 
$\susp^\infty(G_+\smsh_H S^{V^K+n})$
(if $K\subset H$). From Proposition~\ref{prop:burnsidefixedsets} it follows that
 \[
 \Mackey C^G_{V+i}(X)^K \iso \Mackey C^{G/K}_{V^K+i}(\Phi^K(X)),
 \]
the map given by restriction of stable maps to fixed-sets (using $\Phi^K$).

In cohomology, we can now define restriction to fixed-sets
to be the map induced by the restriction
 \begin{multline*}
 \Hom_{\sorb G}(\Mackey C^G_{V+*}(X), \Mackey T)
 \to \Hom_{\sorb{G/K}} (\Mackey C^G_{V+*}(X)^K, \Mackey T^K)\\
 \iso \Hom_{\sorb{G/K}} (\Mackey C^{G/K}_{V^K+*}(\Phi^K(X)), \Mackey T^K).
 \end{multline*}
In homology, we use the map
 \[
 \Mackey C^G_{V+*}(X)\tensor_{\sorb G}\MackeyOp S
 \to \Mackey C^G_{V+*}(X)^K\tensor_{\sorb {G/K}}\MackeyOp S^K
 \iso \Mackey C^{G/K}_{V^K+i}(\Phi^K(X))\tensor_{\sorb {G/K}}\MackeyOp S^K.
 \]
Here, the map
$\Mackey C\tensor_{\sorb G}\MackeyOp S
 \to \Mackey C^K\tensor_{\sorb {G/K}}\MackeyOp S^K$
is induced by the projections $\Mackey C(G/H)\to \Mackey C^K(G/H)$ for
$K\subset H$ (and $\Mackey C(G/H)\to 0$ for $K\not\subset H$), 
and the similar projections for
$\MackeyOp S$.

The stated properties of restriction to fixed-sets are now clear, except
possibly for the fact that it respects restriction to subgroups. This follows
from its naturality and the fact that it respects the Wirthm\"uller 
isomorphisms, in the sense that, if $K\subset L$, then 
the following diagram commutes:
 \[
 \xymatrix{
  H^G_{\alpha}(G\halfsmsh_L X;\MackeyOp S) \ar[r]^-{\iso} \ar[d]_{(-)^K}
   & H^L_{\alpha|L}(X;\MackeyOp S|L) \ar[d]^{(-)^K} \\
  H^{G/K}_{\alpha^K}(G\halfsmsh_L \Phi^K(X); \MackeyOp S^K) \ar[r]^-{\iso}
   & H^{L/K}_{\alpha^K|L}(\Phi^K(X);\MackeyOp S^K|L).
  }
 \]
Here we are using the identifications $(\alpha|L)^K = \alpha^K|L$
and  $(\MackeyOp S|L)^K \iso \MackeyOp S^K|L$. 
The latter isomorphism follows when you
notice that any basic stable map $G/J\to G/L$ with $K\not\subset J$ may be
factored (in the sense used in the proof of Proposition~\ref{prop:burnsidefixedsets})
as $G/J\from G/M\to G/L$ with $M\subset L$ and $K\not\subset M$.
It is straightforward to check that the diagram above commutes
on the chain level.
 \end{proof}

The dual results are similar.
The following is a restatement of part~(\ref{item:dualvfixedsets})
of Theorem~\ref{thm:dualvhomology}.

\begin{theorem}[Restriction to Fixed-Sets in the Dual Theories]
\label{thm:dualvfixedsets}
If $X$ is a $G$-spectrum and
$K$ is a normal subgroup of $G$, there is a natural restriction
to fixed sets 
 \[
 \H_{\alpha}\sp G(X;\Mackey T) \to
 \H_{\alpha \sp K}\sp {G/K}(\Phi^K(X);\Mackey T\sp K).
 \]
Restriction respects suspension and restriction to subgroups 
as in Theorem~\ref{thm:vfixedsets}, and its
evaluation at a point again gives the projection.
 \end{theorem}

\begin{proof}
The dual restriction is constructed and its properties proved in the same way as for the
ordinary theories, using the equivalence
 \[
 \Phi^K(D(G/L_+)) \hmtpc D((G/L_+)^K) =
 \begin{cases} D(G/L_+) & \text{if $K\subset L$} \\
               * & \text{if $K\not\subset L$,}
 \end{cases}
 \]
which follows from the fact that $\Phi^K$ preserves orbits and smash products
(and, hence, duality).
This leads to the natural isomorphism
 \[
 \MackeyOp C^G_{V+i}(X)^K \iso \MackeyOp C^{G/K}_{V^K+i}(\Phi^K(X)),
 \]
and the rest follows as before.
 \end{proof}

Now let us see how restriction to fixed sets is represented on the spectrum level.
Let $X$ be a $G$-spectrum for this discussion.
Let $H$ denote either $H\MackeyOp S$ or $H\Mackey T$. In cohomology we have
the map
 \[
 [X, H]_G \to [\Phi^K(X), \Phi^K(H)]_{G/K},
 \]
and in homology we have the map
 \[
 [S, X\smsh H]_G \to [\Phi^K(S), \Phi^K(X\smsh H)]_{G/K}
 \iso [S, \Phi^K(X)\smsh \Phi^K(H)]_{G/K}.
 \]
(Notice that $\Phi^K(\susp^V X) \hmtpc \susp^{V^K}\Phi^K(X)$, so the grading
behaves correctly.)
We therefore need to analyze the spectra $\Phi^K(H\MackeyOp S)$ and
$\Phi^K(H\Mackey T)$. The following results give us what we need.

\begin{proposition}\label{prop:fixedsethomotopy}
Let $X$ be a connective $G$-spectrum and let $K$ be a normal
subgroup of $G$. Then $\Phi^K(X)$ is a connective $(G/K)$-spectrum and
 \[
 (\Mackey\pi^G_0 X)^K \iso \Mackey\pi^{G/K}_0\Phi^K(X).
 \]
%Further, the map $\Mackey\pi^G_0 X \to \Inf_{G/K}^G\Mackey\pi^{G/K}_0\Phi^K(X)$
%adjoint to the isomorphism is given by application of $\Phi^K$. 
 \end{proposition}

\begin{proof}
Consider first the case $X = \susp^\infty G/J_+$. We have
$\Mackey\pi^G_0(\susp^\infty G/J_+) = \Mackey A_{G/J}$ by definition. On the other hand,
$\Phi^K(\susp^\infty G/J_+) \hmtpc \susp^\infty (G/J_+)^K$, so
Proposition~\ref{prop:burnsidefixedsets} implies that
 \[
 \Mackey\pi^{G/K}_0\Phi^K(\susp^\infty G/J_+) \iso
  (\Mackey A_{G/J})^K \iso
  (\Mackey\pi^G_0\susp^\infty G/J_+)^K.
 \]
%Further, the map
%$\Mackey\pi^G_0\susp^\infty G/J_+ \to 
% \Inf_{G/K}^G \Mackey\pi^{G/K}_0\Phi^K(\susp^\infty G/J_+)$
%adjoint to this isomorphism is clearly given by application of $\Phi^K$.

Now let $X$ be any connective $G$-spectrum. Choose an epimorphism
$\Mackey F\to \Mackey\pi^G_0 X$, where $\Mackey F$ is a free Mackey functor.
Let $E_0 = \Wedge \susp^\infty (G/J_i)_+$ be a wedge of orbits such that
$\Mackey\pi^G_0 E_0 \iso \Mackey F$ and let $E_0\to X$ be a map realizing the
epimorphism $\Mackey F\to \Mackey\pi^G_0 X$. Let $X_1$ be the cofiber of the
map $E_0\to X$. Then $X_1$ is 1-connected. Let $\Mackey R\to \Mackey\pi^G_1 X_1$
be an epimorphism with $\Mackey R$ free. 
Let $E_1 = \Wedge \susp^\infty\susp (G/J_j)_+$ be such that
$\Mackey\pi^G_1 E_1 \iso \Mackey R$, and let $E_1\to X_1$ realize the epimorphism
$\Mackey R\to \Mackey\pi^G_1 X_1$. We then have an exact sequence
 \[
 \Mackey R \to \Mackey F \to \Mackey\pi^G_0 X \to 0
 \]
realized as
 \[
 \Mackey\pi^G_1 E_1 \to \Mackey\pi^G_0 E_0 \to \Mackey\pi^G_0 X \to 0.
 \]
Now apply $\Phi^K$ to all of these spectra. $\Phi^K$ preserves (co)fibrations, so we
have once again an exact sequence
 \[
 \Mackey\pi^{G/K}_1 \Phi^K(E_1) \to \Mackey\pi^{G/K}_0 \Phi^K(E_0)
  \to \Mackey\pi^{G/K}_0 \Phi^K(X) \to 0.
 \]
On the other hand, the first part of this proof implies that
$\Mackey\pi^{G/K}_1 \Phi^K(E_1) \iso \Mackey R^K$ and
$\Mackey\pi^{G/K}_0 \Phi^K(E_0) \iso \Mackey F^K$, so we have the exact sequence
 \[
 \Mackey R^K \to \Mackey F^K \to \Mackey\pi^{G/K}_0 \Phi^K(X) \to 0.
 \]
Finally, the functor $\Mackey T\mapsto \Mackey T^K$ is right exact (being a left
adjoint), which implies that
 \[
 \Mackey\pi^{G/K}_0 \Phi^K(X) \iso (\Mackey\pi^G_0 X)^K
 \]
as desired.
%The statement about the map
%$\Mackey\pi^G_0 X \to \Inf_{G/K}^G\Mackey\pi^{G/K}_0\Phi^K(X)$
%now follows from the first part of the proof.
 \end{proof}


\begin{corollary}\label{cor:emfixedsets}
Let $\Mackey T$ be a contravariant $G$-Mackey functor and let $K$ be a normal
subgroup of $G$.
Then $\Phi^K(H\Mackey T)$ is connective and
 \[
 \Mackey\pi^{G/K}_0\Phi^K(H\Mackey T) \iso \Mackey T^K.
 \]
Hence, there is a $G/K$-map
$\Phi^K(H\Mackey T)\to H(\Mackey T^K)$ that is an isomorphism
in $\Mackey\pi_0^{G/K}$.
 \qed
 \end{corollary}

\begin{proposition}\label{prop:dualfixedsethomotopy}
Let $X$ be a dually connective $G$-spectrum (i.e.,
$\MackeyOp \dualhmtpy^G_n X = 0$ for $n<0$) and let
$K$ be a normal subgroup of $G$. Then
$\Phi^K(X)$ is dually connective and
 \[
 (\MackeyOp\dualhmpty^G_0 X)^K \iso \MackeyOp\dualhmpty^{G/K}_0\Phi^K(X).
 \]
%Further, the map 
%$\MackeyOp\dualhmpty^G_0 X \to \Inf_{G/K}^G\MackeyOp\dualhmpty^{G/K}_0\Phi^K(X)$
%adjoint to the isomorphism is given by application of $\Phi^K$. 
 \end{proposition}

\begin{proof}
The proof is almost identical to that of Proposition~\ref{prop:fixedsethomotopy};
we need only replace all references to orbits $\susp^\infty G/J_+$ with their
duals $D(G/J_+)$.
We use here the fact that $\Phi^K(D(G/J_+)) \hmtpc D((G/J_+)^K)$, which follows from
the fact that $\Phi^K$ preserves spheres and smash products.
\end{proof}

\begin{corollary}\label{cor:mefixedsets}
Let $\MackeyOp S$ be a covariant $G$-Mackey functor and let $K$
be a normal subgroup of $G$. Then $\Phi^K(H\MackeyOp S)$ is dually connective and
 \[
 \MackeyOp\dualhmtpy^{G/K}_0\Phi^K(H\MackeyOp S) \iso \MackeyOp S^K.
 \]
Hence, there is a $G/K$-map
$\Phi^K(H\MackeyOp S) \to H(\MackeyOp S^K)$ that is an isomorphism in
$\MackeyOp\dualhmtpy^{G/K}_0$.
 \qed
 \end{corollary}

The maps in Corollaries~\ref{cor:emfixedsets} and \ref{cor:mefixedsets}
complete the spectrum-level representations of the
fixed-set maps. For example, in ordinary homology the fixed-set map is the
composite
 \begin{align*}
 [S, X\smsh H\MackeyOp S]_G 
 &\to [\Phi^K(S), \Phi^K(X\smsh H\MackeyOp S)]_{G/K} \\
 &\iso [S, \Phi^K(X)\smsh \Phi^K(H\MackeyOp S)]_{G/K} \\
 &\to [S, \Phi^K(X)\smsh H(\MackeyOp S^K)]_{G/K}.
 \end{align*}
The other three restriction maps are represented in similar ways.
That these maps defined on the spectrum level agree with the maps defined on
the chain level follows by comparing their effects on the filtration quotients
$X^{V+n}/X^{V+n-1}$ (for a $G$-CW($V$) structure or a dual structure, as
appropriate).

Finally, we should say a word about what happens if $K$ is not normal in $G$.
In ordinary cohomology and dual homology, there is no problem. We simply
let $a^K = (a|NK)^K$ and everything works as expected.
In ordinary homology and dual cohomology we make the same definition,
but now there will be a shift in dimension because of the restriction to
$NK$. Precisely, in ordinary homology we have
 \[
 (-)^K\colon H^G_{\alpha}(X;\MackeyOp S)
  \to H^{WK}_{\alpha^K + V(G/NK)^K}(\Phi^K(X);\MackeyOp S^K)
 \]
and in dual cohomology we have
 \[
 (-)^K\colon \H_G^{\alpha}(X;\MackeyOp S)
  \to \H_{WK}^{\alpha^K - V(G/NK)^K}(\Phi^K(X);\MackeyOp S^K).
 \]


\section{Products}\label{sec:vproducts}

In this section we prove parts~(\ref{item:vcupproduct}),
(\ref{item:veval}) and (\ref{item:vcapproduct}) of
Theorem~\ref{thm:vhomology}, and the corresponding parts of
Theorem~\ref{thm:dualvhomology}. 
Again, we work with spectra.

\subsection*{Cup Products}

The product of two orbits is not an orbit; if $G$ is infinite it
is not even a disjoint union of orbits. Thus, if $X$ and $Y$ are
$G$-CW spectra, their product does not have a canonical $G$-CW
structure. However, it does have a canonical $(G\times G)$-CW
structure. Because of this observation, it is convenient to define and
work with an {\em external} cup product, pairing a $G$-spectrum $X$
and a $K$-spectrum $Y$ to get a $(G\times K)$-spectrum $X\smsh Y$. In
the case of two $G$-spectra we can internalize by restricting to the
diagonal $G\subset G\times G$, using the restriction to subgroups
discussed in detail in the preceding section.

We begin by defining the appropriate tensor product of Mackey
functors (see also \cite{Dr:reps} and \cite{Le:Greenfunctors}).

\begin{definition}
Let $p\colon \sorb G\times{\sorb K} \to \sorb{G\times K}$
be the functor taking the pair $(\susp^\infty G/J_+, \susp^\infty K/L_+)$ 
to $\susp^\infty G/J_+ \smsh \susp^\infty K/L_+ = \susp^\infty (G\times K)/(J\times L)_+$.
Let $\Mackey T$ be a contravariant $G$-Mackey functor and let $\Mackey U$ be a
contravariant $K$-Mackey functor. Let $\Mackey T\tensor \Mackey U$ denote the 
$(\sorb G\times{\sorb K})$-functor defined by
 \[
  (\Mackey T\tensor \Mackey U)(G/J, K/L) = \Mackey T(G/J)\tensor \Mackey U(K/L).
 \]
Define the contravariant $(G\times K)$-Mackey functor
$\Mackey T \exboxprod \Mackey U$ (the {\em external box product}) by
 \[
 (\Mackey T \exboxprod \Mackey U) = p_*(\Mackey T\tensor \Mackey U),
 \]
where $p_*$ is defined in Definition~\ref{def:indres}.
If $\Mackey T$ and $\Mackey U$ are both $G$-Mackey functors, we define the
{\em (internal) box product} to be
 \[
  \Mackey T \boxprod \Mackey U = (\Mackey T\exboxprod \Mackey U) | G
 \]
where the restriction is to the diagonal subgroup $G \subset G\times G$.
 \end{definition}

The following facts are easy to check.

\begin{proposition}\label{prop:burnsideboxproduct}
\begin{enumerate} \item[]
 \item $\Mackey A_{G/J} \exboxprod \Mackey A_{K/L} \iso \Mackey A_{G/J\times K/L}$.
More generally, if $E$ is a $G$-space and $F$ is a $K$-space then
$\{-,E\}_G\exboxprod \{-,F\}_K \iso \{-,E\times F\}_{G\times K}$.
 \item $\Mackey A_{G/G} \boxprod \Mackey T \iso \Mackey T$
for any $\Mackey T$.
 \item If $X$ is a $G$-CW(\,$V\!$) spectrum and $Y$ is a $K$-CW(\,$W\!$) spectrum, then
 \[
 \Mackey C^G_{V+*}(X) \exboxprod \Mackey C^K_{W+*}(Y)
  \iso \Mackey C^{G\times K}_{V\oplus W + *} (X\smsh Y).
 \]
 \end{enumerate}
 \qed
 \end{proposition}

Now let $X$ be a $G$-spectrum and let $Y$ be a $K$-spectrum.
The external box product induces a natural chain map
 \[
 \Hom_{\sorb G}(\Mackey C^G_*(X), \Mackey T) \tensor
 \Hom_{\sorb K}(\Mackey C^K_*(Y), \Mackey U)
 \to
 \Hom_{\sorb {G\times K}}(
    \Mackey C^G_*(X)\exboxprod \Mackey C^K_*(Y), 
    \Mackey T \exboxprod \Mackey U).
 \]
This induces the (external) cup product
 \[
 -\cup - \colon 
 H\sp{\alpha}_ G(X;\Mackey T) \tensor
 H\sp{\beta}_ K(Y;\Mackey U) \to H\sp{\alpha+\beta}_
 {G\times K}(X\smsh Y);\Mackey T\exboxprod \Mackey U).
 \]

When $G = K$, we can follow the external cup product with the restriction to the
diagonal subgroup $G\subset G\times G$ to get the cup product
 \[
 -\cup - \colon 
 H\sp{\alpha}_ G(X;\Mackey T) \tensor
 H\sp{\beta}_ G(Y;\Mackey U) \to H\sp{\alpha+\beta}_
 {G}(X\smsh Y;\Mackey T\boxprod \Mackey U).
 \]
Further, if $d\colon Z\to X\smsh Y$ is a $G$-map,
applying $d^*$ gives us the cup product
 \[
 -\cup - \colon 
 H\sp{\alpha}_ G(X;\Mackey T) \tensor
 H\sp{\beta}_ G(Y;\Mackey U) \to H\sp{\alpha+\beta}_
 {G}(Z;\Mackey T\boxprod \Mackey U).
 \]

A contravariant $G$-Mackey functor $\Mackey T$ is a {\em ring} 
(also called a {\em Green functor}) if there is an associative multiplication 
$\mu\colon \Mackey T\boxprod \Mackey T\to \Mackey T$. When $\Mackey T$ is a ring, 
there is a cup product
\[
 -\cup - \colon 
 H\sp{\alpha}_ G(X;\Mackey T) \tensor
 H\sp{\beta}_ G(Y;\Mackey U) \to H\sp{\alpha+\beta}_
 {G}(Z;\Mackey T).
\]
In particular, when $Z = Y = \susp^\infty X$ and $d$ is the diagonal,
we have the expected associative internal cup product
 \[
 -\cup - \colon 
 \tilde H\sp{\alpha}_ G(X;\Mackey T) \tensor
 \tilde H\sp{\beta}_ G(X;\Mackey T) \to 
  \tilde H\sp{\alpha+\beta}_{G}(X;\Mackey T).
 \]
 
For example, $\Mackey A_{G/G}$ is a ring.
Notice also that
$\Mackey A_{G/G} \boxprod \Mackey T \iso \Mackey T$ for any contravariant
$G$-Mackey functor $\Mackey T$. We say that $\Mackey T$ is a {\em unital} ring if
there is a unit map $\eta\colon \Mackey A_{G/G}\to \Mackey T$ such that the 
following diagram commutes.
 \[
  \xymatrix@C-1em{
   {\Mackey A_{G/G}\boxprod \Mackey T} \ar[rr]^-{\eta\boxprod 1} \ar[dr]_{\iso}
    && {\Mackey T\boxprod \Mackey T} \ar[dl]^{\mu} \\
    & {\Mackey T}
   }
 \]
In particular, $\Mackey A_{G/G}$ is a unital ring. Every
contravariant $G$-Mackey functor $\Mackey T$ is a {\em module} over
$\Mackey A_{G/G}$, in the sense that there is a unital and associative pairing
$\Mackey A_{G/G}\boxprod \Mackey T\to \Mackey T$ (namely, the isomorphism).

These cup products have the following properties, 
most of which were also listed in
Theorem~\ref{thm:vhomology}.

\begin{theorem}\label{thm:vcupproduct}
The cup product defined above generalizes the nonequivariant cup product and
satisfies the following.
 \begin{enumerate}
 \item It is natural: $f^*(x)\cup g^*(y) = (f\smsh g)^*(x\cup y)$ for the external
product.
 \item It is associative: $(x\cup y)\cup z = x\cup (y\cup z)$ when we identify
gradings using the obvious identification $(U\dirsum V)\dirsum W \iso
U\dirsum(V\dirsum W)$.
 \item It is commutative: If $x\in H^\alpha_G(X;\Mackey T)$ and
$y\in H^\beta_K(Y;\Mackey S)$ then $x\cup y = \iota(y\cup x)$ where $\iota$ is the
obvious isomorphism between
$H^{\alpha+\beta}_{G\times K}(X\smsh Y;\Mackey T\exboxprod \Mackey S)$
and
$H^{\beta+\alpha}_{K\times G}(Y\smsh X;\Mackey S\exboxprod \Mackey T)$; in
particular, $\iota$ uses the isomorphism of $\alpha+\beta$ and $\beta+\alpha$ that
switches the direct summands.
 \item It is unital: there exists an element $1\in H^0_G(S;\Mackey A_{G/G})$ such
that $1\cup x = x$ for any $x\in H^\alpha_G(X;\Mackey T)$ (using 
the identification $\Mackey A_{G/G}\boxprod \Mackey T \iso \Mackey T$).
 \item\label{item:vcupsubgroup}
 It respects restriction to subgroups: 
$(x|I)\cup (y|J) = (x\cup y)|(I\times J)$ in the external product.
 \item\label{item:vcupfixedsets}
 It respects restriction to fixed sets:
$x^I \cup y^J = (x\cup y)^{I\times J}$ in the external product.
 \end{enumerate}
 \end{theorem}

The proofs are all standard except for the last two points.

\begin{proof}[Proof of Part (\ref{item:vcupsubgroup}) of Theorem \ref{thm:vcupproduct}]
(Refer to the beginning of Section~\ref{sec:vsubgroups}
for the algebra involved in the construction of the
restriction to subgroups.) Let $I\subset G$ and $J\subset K$. Since restriction from
$G$ to $I$ is defined as the composite
 \[
 H_G^\alpha(X;\Mackey T) 
 \to H_G^\alpha(G\halfsmsh_I X;\Mackey T)
 \iso H_I^{\alpha|I}(X;\Mackey T|I)
 \]
and the cup product is natural, it suffices to show that the cup product respects the
Wirthm\"uller isomorphism. 
From the commutativity of the diagram
 \[
 \xymatrix{
  {\sorb I \times \sorb J} \ar[r]^{p} \ar[d]_{i_I^G\times i_J^K}
   & {\sorb{I\times J}} \ar[d]^{i_{I\times J}^{G\times K}} \\
  {\sorb G \times \sorb K} \ar[r]_{p}
   & {\sorb{G\times K}} }
 \]
it follows that there is a natural isomorphism
 \[
 (G\times_I \Mackey C)\exboxprod (K\times_J \Mackey D)
 \iso (G\times K)\times_{I\times J} (\Mackey C\exboxprod \Mackey D).
 \]
This diagram and the adjunctions that follow from 
Proposition~\ref{prop:indresadjunction} also lead to a natural quotient map
 \[
 (\Mackey T|I)\exboxprod (\Mackey U|J) \to (\Mackey T\exboxprod \Mackey U)|(I\times J)
 \]
(not necessarily an isomorphism). Together these give the following
diagram.
 \[
 \def\objectstyle{\scriptstyle}
 \def\labelstyle{\scriptstyle}
 \xymatrix{
  {\Hom_{\sorb G}(G\times_I\Mackey C, \Mackey T)\tensor
    \Hom_{\sorb K}(K\times_J\Mackey D, \Mackey U)}
    \ar[r]^-{\iso} \ar[d]
    & {\Hom_{\sorb I}(\Mackey C, \Mackey T|I)\tensor
        \Hom_{\sorb J}(\Mackey D, \Mackey U|J)} \ar[d] \\
  {\Hom_{\sorb{G\times K}}((G\times_I\Mackey C) \exboxprod (K\times_J\Mackey D),
      \Mackey T \exboxprod \Mackey U)} \ar[d]_{\iso}
    & {\Hom_{\sorb {I\times J}}(\Mackey C\exboxprod \Mackey D,
      (\Mackey T|I) \exboxprod (\Mackey U|J))} \ar[d] \\
  {\Hom_{\sorb {G\times K}}((G\times K)\times_{I\times J}
       (\Mackey C \exboxprod \Mackey D), \Mackey T \exboxprod \Mackey U)}
    \ar[r]^-{\iso}
    & {\Hom_{\sorb {I\times J}}(\Mackey C\exboxprod \Mackey D,
      (\Mackey T \exboxprod \Mackey U)|(I\times J))}
  }
 \]
It is an elementary exercise to show that this diagram
commutes. Letting $\Mackey C = \Mackey C_*^I(X)$ and
$\Mackey D = \Mackey C_*^J(Y)$ shows that the cup product respects the
Wirthm\"uller isomorphism on the chain level, so does so also in cohomology. 
 \end{proof}

\begin{proof}[Proof of Part (\ref{item:vcupfixedsets}) of Theorem \ref{thm:vcupproduct}]
(Refer to the middle of Section~\ref{sec:vsubgroups}
for the algebra involved in the construction of
restriction to fixed points.) 
By part~(\ref{item:vcupsubgroup}) we know that the cup product respects the
restriction to the normalizers $NI$ and $NJ$, so we may as well assume that
$I$ is normal in $G$ and $J$ is normal in $K$.
There is a natural quotient map (not necessarily an isomorphism)
 \[
 \Mackey T^I \exboxprod \Mackey U^J \to (\Mackey T\exboxprod \Mackey U)^{I\times J}.
 \]
However, Propositions~\ref{prop:burnsidefixedsets} and 
\ref{prop:burnsideboxproduct} imply that
 \[
 \Mackey A_{G/H}^I \exboxprod \Mackey A_{K/L}^J \iso 
  (\Mackey A_{G/H}\exboxprod \Mackey A_{K/L})^{I\times J},
 \]
with the consequence that
 \[
 \Mackey C_*^G(X)^I\exboxprod \Mackey C_*^K(Y)^J \iso
  \Mackey C_*^{G/I\times K/J}(\Phi^{I\times J}(X\smsh Y)).
 \]
The fact that cup product respects restriction to fixed sets now follows from the
following commutative diagram, in which $\Mackey C = \Mackey C_*^G(X)$ and
$\Mackey D = \Mackey C_*^K(Y)$.
 \[
 \def\objectstyle{\scriptstyle}
 \def\labelstyle{\scriptstyle}
  \xymatrix{
   {\Hom_{\sorb G}(\Mackey C, \Mackey T)\tensor
     \Hom_{\sorb K}(\Mackey D, \Mackey U)} \ar[r] \ar[d]
     & {\Hom_{\sorb {G/I}}(\Mackey C^I, \Mackey T^I)\tensor
          \Hom_{\sorb {K/J}}(\Mackey D^J, \Mackey U^J)} \ar[d] \\
   {\Hom_{\sorb {G\times K}}(\Mackey C \exboxprod \Mackey D,
      \Mackey T \exboxprod \Mackey U)} \ar[d]
     & {\Hom_{\sorb {{G/I}\times {K/J}}}(\Mackey C^I\exboxprod \Mackey D^J,
         \Mackey T^I \exboxprod \Mackey U^J)} \ar[d] \\
   {\Hom_{\sorb {{G/I}\times {K/J}}}((\Mackey C \exboxprod \Mackey D)^{I\times J},
      (\Mackey T \exboxprod \Mackey U)^{I\times J})} \ar[r]^{\iso}
     & {\Hom_{\sorb {{G/I}\times {K/J}}}(\Mackey C^I\exboxprod \Mackey D^J,
       (\Mackey T \exboxprod \Mackey U)^{I\times J})}
  }
 \]
 \end{proof}

\begin{remark} As the proofs make clear, parts
(\ref{item:vcupsubgroup}) and (\ref{item:vcupfixedsets})
of Theorem~\ref{thm:vcupproduct} are stated
a bit loosely. In fact,
 \[
 (x|I)\cup(y|J) \in H_{G\times K}^*(X\smsh Y; 
			(\Mackey T|I)\exboxprod (\Mackey U|J))
 \]
while
 \[
 (x\cup y)|(I\times J) \in H_{G\times K}^*(X\smsh Y;
  (\Mackey T\exboxprod \Mackey U)|(I\times J)).
 \]
The theorem should say that, when we apply the map induced by the quotient map
$(\Mackey T|I)\exboxprod (\Mackey U|J) \to 
(\Mackey T\exboxprod \Mackey U)|(I\times J)$,
the element $(x|I)\cup(y|J)$ maps to $(x\cup y)|(I\times J)$. A similar comment
applies to part (\ref{item:vcupfixedsets}).
 \end{remark}


The external cup product is represented by a $(G\times K)$-map
$H\Mackey T \smsh H\Mackey U \to H(\Mackey T\exboxprod \Mackey U)$
that is an isomorphism in $\Mackey\pi_0^{G\times K}$. 
To construct this map purely on
the spectrum level takes a bit
of work. We begin by constructing the map for the internal product.

\begin{proposition}\label{prop:smshhomotopy}
Let $X$ and $Y$ be connective $G$-spectra. Then
$X\smsh Y$ is a connective $G$-spectrum with
$\Mackey\pi^G_0(X\smsh Y) \iso \Mackey\pi^G_0(X) \boxprod \Mackey\pi^G_0(Y)$.
\end{proposition}

\begin{proof}
That the smash product of two connective
$G$-spectra is connective is standard, so we concentrate on the statement
about homotopy. The method of proof is essentially the same as for
Proposition~\ref{prop:fixedsethomotopy}.

First consider the case
$X = \susp^\infty G/I_+$ and $Y = \susp^\infty G/J_+$.
$X\smsh Y = \susp^\infty(G/I\times G/J)_+$, so
 \[
 \Mackey\pi^G_0(X\smsh Y) =
 \Mackey\pi^G_0(\susp^\infty(G/I\times G/J)_+) =
 \{-,G/I\times G/J\}_G \iso
 \Mackey A_{G/I} \boxprod \Mackey A_{G/J}
 \]
by Proposition~\ref{prop:burnsideboxproduct}.

Now let $X$ and $Y$ be any connective $G$-spectra.
As in the proof of Proposition~\ref{prop:fixedsethomotopy},
let $E_0 = \Wedge\susp^\infty (G/K_i)_+\to X$ be a map that is an
epimorphism in $\Mackey\pi^G_0$, and let $X_1$ be its cofiber.
Let $E_1 = \Wedge\susp^\infty\susp (G/K_j)_+\to X_1$ be an
epimorphism in $\Mackey\pi^G_1$. Then we have an exact sequence
 \[
 \Mackey\pi^G_1 (E_1) \to \Mackey\pi^G_0 (E_0) \to \Mackey\pi^G_0 (X) \to 0.
 \]
Let $F_0$ and $F_1$ be chosen similarly, using $Y$ in place of $X$.
Using the fact that smash product preserves (co)fibrations it follows
that we have an exact sequence
 \[
 (\Mackey\pi^G_0(E_0) \boxprod \Mackey\pi^G_1(F_1)) \dirsum
  (\Mackey\pi^G_1(E_1) \boxprod \Mackey\pi^G_0(F_0))
  \to \Mackey\pi^G_0(E_0) \boxprod \Mackey\pi^G_0(F_0)
  \to \Mackey\pi^G_0(X\smsh Y) \to 0.
 \]
Here we have used the first part of the proof to identify, for example,
$\Mackey\pi^G_0(E_0\smsh F_0)$ with 
$\Mackey\pi^G_0(E_0)\boxprod \Mackey\pi^G_0(F_0)$.
It now follows from the right exactness of the box product that
 \[
 \Mackey\pi^G_0(X\smsh Y) \iso \Mackey\pi^G_0(X) \boxprod \Mackey\pi^G_0(Y)
 \]
as claimed.
 \end{proof}

\begin{corollary}\label{cor:emproduct}
Let $\Mackey T$ and $\Mackey U$ be $G$-Mackey functors.
Then $H\Mackey T\smsh H\Mackey U$ is a connective $G$-spectrum, and
 \[
 \Mackey\pi^G_0(H\Mackey T\smsh H\Mackey U) \iso
  (\Mackey T \boxprod \Mackey U).
 \]
 \qed
 \end{corollary}

To deal with the external product we first need a little more algebra.

\begin{definition}
Let $N$ be a normal subgroup of $G$ and let $p\colon G\to G/N$ be the quotient
map. Let $p^\sharp\colon \sorb{G/N}\to \sorb G$ be the functor
mentioned just before Proposition~\ref{prop:emsubgroups}; its effect on orbits
is to consider a $(G/N)$-orbit to be a $G$-orbit via $p$. With the notation
of Definition~\ref{def:indres}, let
$p_*\Mackey T = (p^\sharp)^*\Mackey T$ when
$\Mackey T$ is a $G$-Mackey functor.
If $\Mackey S$ is a $(G/N)$-Mackey functor, let $p^*\Mackey S = (p^\sharp)_*\Mackey S$.
 \end{definition}

The following results are consequences of Proposition~\ref{prop:indresadjunction}.

\begin{lemma}
\begin{enumerate} \item[]
\item
 The functor $p^*$ is left adjoint to $p_*$.
\item
 $p^*\Mackey A_{(G/N)/(L/N)} \iso \Mackey A_{G/L}$.
\item
 $p^*$ is right exact.
\qed
\end{enumerate}
\end{lemma}

\begin{proposition}
Let $X$ be a connective $(G/N)$-spectrum. Then
$p^\sharp X$ is a connective $G$-spectrum with
 \[
 \Mackey\pi^G_0(p^\sharp X) \iso p^*\Mackey\pi^{G/N}_0 X.
 \]
\end{proposition}

\begin{proof}
The line of argument is again much like that of
Proposition~\ref{prop:fixedsethomotopy}.
Let $E_0 = \Wedge\susp^\infty (G/K_i)_+\to X$ be a map that is an
epimorphism in $\Mackey\pi^G_0$, and let $X_1$ be its cofiber.
Let $E_1 = \Wedge\susp^\infty\susp (G/K_j)_+\to X_1$ be an
epimorphism in $\Mackey\pi^G_1$. The facts that $p^\sharp$ preserves
(co)fibrations, that $p^\sharp\susp^\infty G/L_+ = \susp^\infty G/L_+$ for
$N\subset L\subset G$, and the preceding lemma imply that there is an exact
sequence
 \[
 p^*\Mackey\pi^{G/N}_1 E_1 \to p^*\Mackey\pi^{G/N}_0 E_0 
 \to \Mackey\pi^G_0(p^\sharp X) \to 0.
 \]
The right exactness of $p^*$ now implies that
$\Mackey\pi^G_0(p^\sharp X) \iso p^*\Mackey\pi^{G/N}_0 X$
as claimed.
 \end{proof}

\begin{corollary}
Let $\Mackey T$ be a $(G/N)$-Mackey functor. 
Then $p^\sharp H\Mackey T$ is a connective $G$-spectrum with
$\Mackey\pi^G_0(p^\sharp H\Mackey T) \iso p^*\Mackey T$.
Therefore, there is a map
$p^\sharp H\Mackey T \to H(p^*\Mackey T)$
that is an isomorphism in $\Mackey\pi^G_0$.
 \qed
 \end{corollary}


In fact, the map $p^\sharp H\Mackey T \to H(p^*\Mackey T)$
can be constructed more neatly as follows.
Begin with the map $\Mackey T \to p_*p^*\Mackey T$ that is the unit of the adjunction.
This induces a $(G/N)$-map
$H\Mackey T \to H(p_*p^*\Mackey T) \hmtpc H(p^*\Mackey T)^N$.
This last isomorphism was shown in
Proposition~\ref{prop:emsubgroups}.
The desired $G$-map is the one adjoint to this last map.

Now consider the two quotient maps $p_1\colon G\times K\to G$ and
$p_2\colon G\times K\to K$. If $E$ is a $G$-spectrum and $F$ is a $K$-spectrum,
$E\smsh F$ is the $(G\times K)$-spectrum
$p_1^\sharp E \smsh p_2^\sharp F$.
If $\Mackey T$ is a $G$-Mackey functor
and $\Mackey U$ is a $K$-Mackey functor, there is then a map
 \[
 H\Mackey T \smsh H\Mackey U = p_1^\sharp H\Mackey T \smsh p_2^\sharp H\Mackey U
  \to H(p_1^*\Mackey T) \smsh H(p_2^*\Mackey U)
 \]
Corollary~\ref{cor:emproduct} then gives a map
 \[
 H(p_1^*\Mackey T) \smsh H(p_2^*\Mackey U)
 \to H(p_1^*\Mackey T \boxprod p_2^*\Mackey U)
 \iso H(\Mackey T \exboxprod \Mackey U),
 \]
where the isomorphism 
$p_1^*\Mackey T \boxprod p_2^*\Mackey U \iso \Mackey T \exboxprod \Mackey U$
is easily checked from the definitions. Putting these together, we have a map
 \[
 H\Mackey T \smsh H\Mackey U \to H(\Mackey T \exboxprod \Mackey U)
 \]
that is the identity on $\Mackey\pi^{G\times K}_0$. 
That this represents the product constructed on the
chain level follows by considering its effect on the
quotients $X^{V+m}/X^{V+m-1}$ and $Y^{W+n}/Y^{W+n-1}$.


Now we turn to part~(\ref{item:dualvcupproduct}) of
Theorem~\ref{thm:dualvhomology}. Recalling the discussion at the
beginning of this section, the product of a $G$-CW space
with a dual $G$-CW space has a complicated ``mixed'' structure,
even viewed as a $(G\times G)$-space. 
The disadvantages of working with this mixed structure seem to outweigh the
advantages of working with an external product, so we shall not
take this route.

We first need the following algebra.

\begin{definition}
If $\MackeyOp S$ is a covariant $G$-Mackey functor and
$\Mackey T$ is a contravariant $G$-Mackey functor, define
a covariant $G$-Mackey functor 
$\MackeyOp S \mixprod \Mackey T = \Mackey T\mixprod \MackeyOp S$ by
 \[
 (\MackeyOp S \mixprod \Mackey T)(G/L)
 = \left[\sum_{G/K \to (G/J)\times (G/L)}
    \MackeyOp S(G/K)\tensor \Mackey T(G/J) \right]\Big/\sim
 \]
where the direct sum runs over all stable $G$-maps
$G/K \to (G/J)\times (G/L)$, and we impose 
the equivalence relation generated by
 \[
 (s\tensor t)_{G/K' \xrightarrow{f} G/K \to G/J\times G/L 
     \xrightarrow{g\times 1} G/J'\times G/L}
 \sim (f_*s \tensor g^*t)_{G/K \to G/J\times G/L}.
 \]
 \end{definition}

\begin{remark}
It is possible, if a bit awkward, to define $\mixprod$ along the lines of
$\boxprod$, using the general Definition~\ref{def:indres}. To do this, we first
notice that $\sorb G^{\text{op}}$ is isomorphic to the category consisting
of the duals $D(G/K_+)$ of the orbits of $G$, and all stable $G$-maps between them.
We introduce a category ${}_2\sorb G$ whose objects are all the
smash products $\susp^\infty(G/K_+)\smsh D(G/J_+)$, and whose maps are
all the stable $G$-maps between them. There is then a functor
$s\colon \sorb G\times \sorb G^{\text{op}} \to {}_2\sorb G$
given by the smash product. Also, there is an inclusion
$d\colon \sorb G\to {}_2\sorb G$ that takes $\susp^\infty(G/K_+)$ to
$\susp^\infty(G/K_+)\smsh D(G/G_+) \hmtpc \susp^\infty(G/K_+)$. With these
functors defined, we have
 \[
 \MackeyOp S \mixprod \Mackey T = d^*s_*(\MackeyOp S \tensor \Mackey T)
 \]
where we consider $\Mackey T$ to be a covariant functor
on $\sorb G^{\text{op}}$. From this description it is easy to see, for example,
that $\mixprod$ is right exact.
\end{remark}

When $G$ is finite,
$\MackeyOp S \mixprod \Mackey T \iso \Mackey S \boxprod \Mackey T$,
using the self-duality of $\orb G$ to change the variance of $\MackeyOp S$.

The following facts follow from the definitions.

\begin{lemma}
If $\MackeyOp S$, $\Mackey T$, and $\Mackey U$ are $G$-Mackey functors, then
 \[
 (\MackeyOp S \mixprod \Mackey T) \tensor_{\sorb G} \Mackey U
 \iso \MackeyOp S \tensor_{\sorb G} (\Mackey T \boxprod \Mackey U)
 \]
and
 \[
 (\MackeyOp S \mixprod \Mackey T) \mixprod \Mackey U
 \iso \MackeyOp S \mixprod (\Mackey T \boxprod \Mackey U).
 \]
Further,
 \[
 \MackeyOp A^{G/K} \mixprod \Mackey A_{G/J} = \{G/K, -\times G/J\}_G,
 \]
and
 \[
 \MackeyOp S \mixprod \Mackey A_{G/G} \iso \MackeyOp S.
 \]
\qed
 \end{lemma}

In order to define the
cup product pairing ordinary and dual cohomology, we shall need the following
result.

\begin{proposition}\label{prop:mixedsmshhomotopy}
Let $X$ be a connective $G$-spectrum and let $Y$ be a dually connective $G$-spectrum.
Then $X\smsh Y$ is dually connective, and
 \[
 \MackeyOp\dualhmtpy^G_0(X\smsh Y) \iso 
 \Mackey\pi^G_0(X)\mixprod \MackeyOp\dualhmtpy^G_0(Y).
 \]
\end{proposition}

\begin{proof}
The proof proceeds along the lines of Proposition~\ref{prop:smshhomotopy}. It requires first
the following computation.
 \begin{align*}
 \MackeyOp\dualhmtpy^G_0( G/K_+\smsh D(G/L_+) )
 &= [D(-_+), G/K_+\smsh D(G/L_+)]_G \\
 &\iso [\susp^\infty G/L_+, \susp^\infty({-}\times G/K)_+]_G \\
 &= \Mackey A_{G/K} \mixprod \MackeyOp A^{G/L}.
 \end{align*}
From this and the right exactness of $\mixprod$, the calculation follows.
\end{proof}

Now consider two $G$-spectra, $X$ and $Y$. Let $\Gamma_V(X)$ be a
$G$-CW($V$) approximation of $X$, let $\Gamma'_W(Y)$ be a
dual $G$-CW($W$) approximation
of $Y$, and let $\Gamma'_{V+W}(X\smsh Y)$ be a dual $G$-CW($V+W$) approximation
of $X\smsh Y$.
The skeletal filtrations of $\Gamma_V(X)$ and $\Gamma'_W(Y)$ give a filtration of
$\Gamma_V(X)\smsh \Gamma'_W(Y)$ in which the filtration quotients are wedges of
spectra of the form $\susp^{V+W+n}G/K_+\smsh D(G/L_+)$. Now, there are no
nontrivial stable maps of the form $D(G/J_+)\to \susp^k G/K_+\smsh D(G/L_+)$
if $k>0$; we can see this by dualizing to a stable map of ordinary orbits.
Therefore, there is a filtration-preserving map
 \[
  \iota\colon \Gamma'_{V+W}(X\smsh Y) \to \Gamma_V(X)\smsh \Gamma'_W(Y),
 \]
approximating the identity,
unique up to homotopies raising filtration level by at most one.
Applying $\MackeyOp\dualhmtpy^G_{V+W+*}$ to the filtration quotients then
gives a chain map
 \[
  \iota_*\colon \MackeyOp C_{V+W+*}(X\smsh Y)
   \to \Mackey C_{V+*}(X) \mixprod \MackeyOp C_{W+*}(Y),
 \]
well-defined up to chain homotopy 
(we use Proposition~\ref{prop:mixedsmshhomotopy} to identify the dual homotopy
of the target).

Given the chain map $\iota_*$ it is now easy to define the cup product
 \[
 -\cup-\colon
  H_G^\alpha(X;\Mackey T) \tensor
  \H_G^\beta(Y;\MackeyOp S) \to
  \H_G^{\alpha+\beta}(X\smsh Y; \Mackey T\mixprod \MackeyOp S).
 \]
It is the map induced by the following composite of chain maps.
 \begin{align*}
 \Hom_{\sorb G}(\Mackey C_{V+*}(X), \Mackey T) \tensor {}
 &\Hom_{\sorb G}(\MackeyOp C_{W+*}(Y), \MackeyOp S) \\
 &\to \Hom_{\sorb G}(\Mackey C_{V+*}(X) \mixprod \MackeyOp C_{W+*}(Y),
                           \Mackey T \mixprod \MackeyOp S ) \\
 &\xrightarrow{(\iota_*)^*}
   \Hom_{\sorb G}(\MackeyOp C_{V+W+*}(X\smsh Y), \Mackey T \mixprod \MackeyOp S)
 \end{align*}

\begin{theorem}\label{thm:dualvcupproduct}
The cup product defined above generalizes the nonequivariant cup product and
satisfies the following.
 \begin{enumerate}
 \item It is natural: $f^*(x)\cup g^*(y) = (f\smsh g)^*(x\cup y)$.
 \item It is associative: $(x\cup y)\cup z = x\cup (y\cup z)$ when we identify
gradings using the obvious identification $(U\dirsum V)\dirsum W \iso
U\dirsum(V\dirsum W)$.
 \item It is unital: the element $1\in H^0_G(*;\Mackey A_{G/G})$ 
of Theorem~\ref{thm:vcupproduct} satisfies
$1\cup x = x$ for any $x\in \H^\alpha_G(X;\MackeyOp S)$ (using 
the identification $\Mackey A_{G/G}\mixprod \MackeyOp S \iso \MackeyOp S$).
 \item\label{item:dualvcupsubgroup}
 It respects restriction to subgroups: 
$(x|K)\cup (y|K) = (x\cup y)|K$.
 \item\label{item:dualvcupfixedsets}
 It respects restriction to fixed sets:
$x^K \cup y^K = (x\cup y)^K$.
 \end{enumerate}
 \end{theorem}

Again, the proofs are standard except for the last two points.

\begin{proof}[Proof of part (\ref{item:dualvcupsubgroup}) of Theorem \ref{thm:dualvcupproduct}]
By naturality, it suffices to show that the cup product respects the
Wirthm\"uller isomorphisms. Precisely, assume that $X$ and $Y$ are $K$-spectra.
It suffices to show that the following composites agree.
 \begin{align*}
 H_G^\alpha(G\halfsmsh_K X;\Mackey T) &{}\tensor 
     \H_G^\beta(G\halfsmsh(S^{-V(G/K)}\smsh Y);\MackeyOp S) \\
  &\to \H_G^{\alpha+\beta}((G\halfsmsh_K X)\smsh(G\halfsmsh(S^{-V(G/K)}\smsh Y));
       \Mackey T\mixprod \MackeyOp S) \\
  &\to H_G^{\alpha+\beta}(G\halfsmsh_K(X\smsh S^{-V(G/K)}\smsh Y);
      \Mackey T \mixprod \MackeyOp S) \\
  &\iso \H_K^{(\alpha+\beta)|K}(X\smsh Y; (\Mackey T\mixprod \MackeyOp S)|K)
 \end{align*}
and
 \begin{align*}
 H_G^\alpha(G\halfsmsh_K X;\Mackey T) &{}\tensor 
     \H_G^\beta(G\halfsmsh(S^{-V(G/K)}\smsh Y);\MackeyOp S) \\
  &\iso H_K^{\alpha|K}(X;\Mackey T|K) \tensor
    \H_K^{\beta|K}(Y;\MackeyOp S|K) \\
  &\to \H_K^{(\alpha+\beta)|K}(X\smsh Y; (\Mackey T|K)\mixprod(\MackeyOp S|K)) \\
  &\to \H_K^{(\alpha+\beta)|K}(X\smsh Y; (\Mackey T\mixprod \MackeyOp S)|K)
 \end{align*}
Here, as usual, we use a projection
 \[
 (\Mackey T|K)\mixprod(\MackeyOp S|K) \to (\Mackey T\mixprod \MackeyOp S)|K
 \]
that is not necessarily an isomorphism. It is straightforward to check that the 
composites agree on the chain level.
\end{proof}

\begin{proof}[Proof of part (\ref{item:dualvcupfixedsets}) of Theorem \ref{thm:dualvcupproduct}]
We may assume that $K$ is normal in $G$.
Once again, we have a projection
 \[
 \Mackey T^K \mixprod \MackeyOp S^K \to (\Mackey T\mixprod \MackeyOp S)^K
 \]
that is not necessarily an isomorphism. However, we can calculate that
 \[
 (\Mackey A_{G/J})^K \mixprod (\MackeyOp A^{G/L})^K
  \iso (\Mackey A_{G/J} \mixprod \MackeyOp A^{G/L})^K
 \]
and we have the following commutative diagram.
 \[
 \xymatrix{
  {\MackeyOp C^{G/K}_*(\Phi^K(X\smsh Y))} \ar[r] \ar[dd]_{\iso}
  & {\Mackey C^{G/K}_*(\Phi^K(X)) \mixprod \MackeyOp C^{G/K}_*(\Phi^K(Y))} \ar[d]^\iso \\
  & {\Mackey C^G_*(X)^K \mixprod \MackeyOp C^G_*(Y)^K} \\
  {\MackeyOp C^G_*(X\smsh Y)^K} \ar[r]
  & {(\Mackey C^G_*(X)\mixprod\MackeyOp C^G_*(Y))^K} \ar[u]_\iso
  }
 \]
From this it is easy to see that the cup product does respect restriction to
fixed points.
\end{proof}

This cup product is represented on the spectrum level by a map
 \[
 H\Mackey T\smsh H\MackeyOp S \to H(\Mackey T\mixprod \MackeyOp S)
 \]
that is an isomorphism in $\MackeyOp\dualhmtpy^G_0$.
That this map exists follows from Proposition~\ref{prop:mixedsmshhomotopy}, which
implies that $H\Mackey T\smsh H\MackeyOp S$ is dually connective and that
 \[
 \MackeyOp\dualhmtpy^G_0(H\Mackey T\smsh H\MackeyOp S) \iso
   \Mackey T \mixprod\MackeyOp S.
 \]
In fact, this pairing is associative in the sense that the following diagram
commutes up to homotopy.
 \[
 \xymatrix{
  H\Mackey T \smsh H\Mackey U \smsh H\MackeyOp S \ar[d] \ar[r]
  & H\Mackey T \smsh H(\Mackey U \mixprod \MackeyOp S) \ar[d] \\
  H(\Mackey T\boxprod \Mackey U)\smsh H\MackeyOp S \ar[r]
  & H((\Mackey T\boxprod\Mackey U)\mixprod\MackeyOp S)
  }
 \]
Here, we use the identification 
$(\Mackey T\boxprod\Mackey U)\mixprod\MackeyOp S \iso
\Mackey T\mixprod(\Mackey U\mixprod\MackeyOp S)$.

\begin{remark}
There are several other products that we have not written down here, but
may be interesting. For example, there is a homology pairing
 \[
 \H^G_*(X;\Mackey T)\tensor \H^K_*(Y;\Mackey U) \to
  \H^{G\times K}_*(X\smsh Y; \Mackey T \exboxprod \Mackey U).
 \]
This can be constructed on the chain level, by noticing that the
smash product of a dual $G$-CW spectrum with a dual $K$-CW spectrum
has an obvious dual $(G\times K)$-CW structure. It is represented on
the spectrum level by the map
$H\Mackey T\smsh H\Mackey U \to H(\Mackey T\exboxprod \Mackey U)$
already considered.

More intriguing, perhaps, is another consequence of the observation above about
smash products of dual spectra. There is an external pairing
 \[
 \H_G^*(X;\MackeyOp R) \tensor \H_K^*(Y;\MackeyOp S)
  \to \H_{G\times K}^*(X\smsh Y; \MackeyOp R \exboxprod \MackeyOp S),
 \]
where $\MackeyOp R \exboxprod \MackeyOp S$ is defined in essentially
the same way as for contravariant Mackey functors. This pairing is represented
by a map
$H\MackeyOp R \smsh H\MackeyOp S \to H(\MackeyOp R \exboxprod\MackeyOp S)$.
When $G=K$ and we try to internalize this pairing, 
we get a shift of dimension. Precisely, the restriction from
$G\times G$ to the diagonal subgroup $G$ involves a shift of dimension
of $V(G\times G/G) \iso L(G)$, the Lie algebra of $G$. So, we get an
internal pairing
 \[
 \H_G^\alpha(X;\MackeyOp R) \tensor \H_G^\beta(Y;\MackeyOp S)
  \to \H_G^{\alpha+\beta - L(G)}(X\smsh Y; \MackeyOp R \boxprod \MackeyOp S).
 \]
This internal pairing is represented by a $G$-map
 \[
 H\MackeyOp R \smsh H\MackeyOp S \to \susp^{-L(G)}H(\MackeyOp R \boxprod\MackeyOp S).
 \]
\end{remark}

\subsection*{Evaluations and Cap Products}

We now construct the evaluations and cap products given in 
Theorem~\ref{thm:vhomology} and
Theorem~\ref{thm:dualvhomology}.
We deal with evaluation maps and cap products simultaneously by considering
a pairing that generalizes both.
Suppose that $X$, $Y$ and $Z$ are $G$-spectra, and we are given a $G$-map
$d\colon Z\to X\smsh Y$. We define a chain map
 \[
 d_*\colon \Mackey C_{V+W+*}(Z) \to \Mackey C_{V+*}(X)\boxprod \Mackey C_{W+*}(Y)
 \]
as follows. Let $\Gamma_{V+W}(Z)$ be a $G$-CW($V+W$) approximation of $Z$,
$\Gamma_V(X)$ be a $G$-CW($V$) approximation of $X$, and
$\Gamma_W(Y)$ be a $G$-CW($W$) approximation of $Y$. The skeletal filtrations of
$\Gamma_V(X)$ and $\Gamma_W(Y)$ give a filtration of $\Gamma_V(X)\smsh \Gamma_W(Y)$,
and we can find a filtration-preserving approximation to $d$,
 \[
 \tilde d\colon \Gamma_{V+W}(Z)\to \Gamma_V(X)\smsh\Gamma_W(Y).
 \]
That such an approximation exists follows from the fact that there are no nontrivial
$G$-maps $\susp^\infty G/K_+ \to \susp^k \susp^\infty(G/L\times G/J)_+$
if $k>0$. Similarly, any two approximations of $d$ will be homotopic through a map
that raises filtration level by at most one. If we now apply $\Mackey\pi^G_{V+W+*}$
to the filtration quotients we get the desired chain map $d_*$, which is well-defined
up to chain homotopy.

Given the chain map $d_*$ we can define a pairing
 \[
 \langle -,- \rangle_d \colon
  H_G^\alpha(Y;\Mackey T) \tensor H^G_{\alpha+\beta}(Z;\MackeyOp S)
  \to H^G_\beta(X; \Mackey T\mixprod \MackeyOp S).
 \]
Taking suspensions as necessary, we may assume that $\alpha = W+*$ and
$\beta = V+*$. The pairing is induced by the following composite of chain maps:
 \begin{align*}
 \Hom_{\sorb G}(\Mackey C_{W+*}(Y),&\Mackey T)
  \tensor (\MackeyOp C_{V+W+*}(Z)
             \tensor_{\sorb G}\MackeyOp S) \\
&\xrightarrow{d_*}
 \Hom_{\sorb G}(\Mackey C_{W+*}(Y),\Mackey T)
  \tensor ((\Mackey C_{V+*}(X)\boxprod \Mackey C_{W+*}(Y))
   \tensor_{\sorb{G}} \MackeyOp S) \\
&\iso
 \Hom_{\sorb G}(\Mackey C_{W+*}(Y),\Mackey T)
  \tensor (\Mackey C_{V+*}(X)\tensor_{\sorb G}
   (\Mackey C_{W+*}(Y)\mixprod \MackeyOp S)) \\
&\to
 \Mackey C_{V+*}(X)\tensor_{\sorb G}
  (\Mackey T \mixprod \MackeyOp S).
 \end{align*}
The last map above is the evaluation map.

\begin{theorem}\label{thm:vpairing}
The pairing
$\langle -,- \rangle_d \colon
  H_G^\alpha(Y;\Mackey T) \tensor H^G_{\alpha+\beta}(Z;\MackeyOp S)
  \to H^G_\beta(X; \Mackey T\mixprod \MackeyOp S)$
has the following properties.
\begin{enumerate}
\item
 It is natural in the following sense. Assume given $G$-maps
$f\colon Z\to Z'$, $g\colon X\to X'$ and $h\colon Y\to Y'$, and
a $G$-map $d'\colon Z'\to X'\smsh Y'$ making the following diagram
$G$-homotopy commutative.
 \[
 \xymatrix{
  Z \ar[r]^-d \ar[d]_f & X\smsh Y \ar[d]^{g\smsh h} \\
  Z' \ar[r]^-{d'} & X'\smsh Y'
  }
 \]
Then 
 \[
 \langle y',f_*(z)\rangle_{d'} = g_*\langle h^*(y'), z\rangle_d.
 \]
\item
It is associative in the following sense. Assume given $d\colon Z\to X\smsh Y$
and $d'\colon X\to E\smsh F$, so we also have
$(d'\smsh 1)\circ d \colon Z\to E\smsh F\smsh Y$. Then
 \[
 \langle f\cup y, z \rangle_{(d'\smsh 1)\circ d} = 
  \langle f, \langle y, z \rangle_d \rangle_{d'},
 \]
where $\cup$ is the external cup product.
\item
It respects restriction to subgroups in the sense that
\[
 \langle y, z \rangle_d | K = \langle y|K, z|K \rangle_{d|K}
\]
in $H^K_{\beta|K}(X; (\Mackey T\mixprod \MackeyOp S)|K)$.
\item
It respects restriction to fixed sets, in the sense that
\[
 (\langle y, z \rangle_d)^K = \langle y^K, z^K \rangle_{\Phi^K(d)}
\]
in $H^{NK/K}_{\beta^K}(\Phi^K(X); (\Mackey T\mixprod \MackeyOp S)^K)$.
 \end{enumerate}
 \end{theorem}

\begin{proof}
The naturality and associativity follow easily from the definitions.

To show that the pairing respects restriction to subgroups, consider first
the following diagram.
 \[
 \xymatrix{
  Z \ar[r]^-d \ar[d] & X\smsh Y \ar[d] \\
  D(G/K_+)\smsh Z \ar[r]^-{d\smsh 1} \ar@{=}[d] & (D(G/K_+)\smsh X)\smsh Y \\
  D(G/K_+)\smsh Z \ar[r] & (D(G/K_+)\smsh X)\smsh (G/K_+\smsh Y) \ar[u]
  }
 \]
The vertical arrows in the top square come from the map
$S\to D(G/K_+)$ dual to the projection $G/K\to *$.
The bottom horizontal arrow uses the diagonal map 
$D(G/K_+)\to D(G/K_+)\smsh G/K_+$, and the upwards arrow is given
by the projection $G/K\to *$. By the commutativity of this diagram and
the naturality of the pairing, it suffices to show that the pairing respects
the Wirthm\"uller isomorphisms. So, we change our point of view and consider
$K$-spectra $d\colon Z\to X\smsh Y$ and the induced map of $G$-spectra
$G\halfsmsh_K Z\to G\halfsmsh_K(X\smsh Y)\to (G\halfsmsh_K X)\smsh (G\halfsmsh_K Y)$.
We need to show that the following diagram commutes.
 \[
 \xymatrix{
  H_G^\alpha(G\halfsmsh_K Y;\Mackey T) \tensor H^G_{\alpha+\beta}(G\halfsmsh_K Z;\MackeyOp S)
  \ar[r] \ar[dd]_{\iso}
  & H^G_\beta(G\halfsmsh_K X; \Mackey T\mixprod \MackeyOp S) \ar[d]^{\iso} \\
  & H^K_{\beta|K}(X; (\Mackey T\mixprod \MackeyOp S)|K) \\
  H_K^{\alpha|K}(Y;\Mackey T|K) \tensor H^K_{(\alpha+\beta)|K}(Z;\MackeyOp S|K) \ar[r]
  & H^K_{\beta|K}(X; (\Mackey T|K)\mixprod (\MackeyOp S|K)) \ar[u]
  }
 \]
But this is easy to check using the chain level description of the
Wirthm\"uller isomorphisms.

Finally, to see that the pairing respects restriction to fixed sets, it now
suffices to assume that $K$ is normal in $G$. In that case it is easy to write out
the necessary diagram on the chain level and check that it commutes.
\end{proof}

\begin{proof}[Proof of Parts (\ref{item:veval}) and (\ref{item:vcapproduct})
of Theorem \ref{thm:vhomology}]
The evaluation map referred to in part~(\ref{item:veval}) 
can actually be defined for any $G$-spectrum $X$, and is the pairing associated
with the identity map $X\to S\smsh X$. Its properties follow from
Theorem~\ref{thm:vpairing}. The cap product referred to in part~(\ref{item:vcapproduct})
is the pairing associated with the 
identity map $X\smsh Y\to X\smsh Y$.
Again, its properties follow from Theorem~\ref{thm:vpairing}.
\end{proof}

The evaluations and cap products of Theorem~\ref{thm:dualvhomology} are constructed
in a very similar way. The only real difference is this. Given a map
$d\colon Z\to X\smsh Y$, we let $\Gamma'_{V+W}(Z)$ be a {\em dual} $G$-CW($V$)
spectrum approximating $Z$, we let $\Gamma_{V}(X)$ be a $G$-CW($V$) spectrum
(of the ordinary kind)
approximating $X$, and we let $\Gamma'_W (Y)$ be a dual $G$-CW($V$) spectrum
approximating $Y$. We can then find a filtration-preserving
approximation to $d$, using the fact that there are no nontrivial maps
$D(G/K_+)\to \susp^k G/L_+\smsh D(G/J_+)$ when $k>0$, which follows upon dualizing
to ordinary orbits. When looking at the chains, we also use the fact that
 \[
 \MackeyOp \dualhmtpy^G_0(G/L_+\smsh D(G/J_+)) \iso
 \MackeyOp A^{G/L} \mixprod \Mackey A_{G/J},
 \]
as we saw when discussing cup products.
We use this $d$ to get one of the evaluations and one of the cap products;
we get the other by using a dual structure on $X$ and an ordinary
structure on $Y$. 
With these comments the constructions and proofs go through exactly as in
the ordinary case.

Let us now look at these pairings on the spectrum level.
We begin with the pairing 
 \[
 \langle -,- \rangle_d \colon
  H_G^\alpha(Y;\Mackey T) \tensor \H^G_{\alpha+\beta}(Z;\Mackey U)
  \to \H^G_\beta(X; \Mackey T\boxprod\Mackey U)
 \]
underlying one of the evaluations and cap products
of Theorem~\ref{thm:dualvhomology}. Again, we assume given a $G$-map
$d\colon Z\to X\smsh Y$. Let $y\colon Y\to \susp^\alpha H\Mackey T$ represent
an element of $H_G^\alpha(Y;\Mackey T)$ and let
$z\colon S^{\alpha+\beta}\to Z\smsh H\Mackey U$ represent an element of
$\H^G_{\alpha+\beta}(Z;\Mackey U)$. Then 
$\langle y, z \rangle_d \in \H^G_\beta(X; \Mackey T\boxprod\Mackey U)$ is
represented by the composite
 \begin{align*}
 S^{\alpha+\beta} \xrightarrow{z}
 Z\smsh H\Mackey U \xrightarrow{d\smsh 1}{}&
 X\smsh Y\smsh H\Mackey U \\ 
 &\xrightarrow{1\smsh y\smsh 1}
 X\smsh \susp^\alpha H\Mackey T \smsh H\Mackey U \to
 X\smsh \susp^\alpha H(\Mackey T\boxprod \Mackey U).
 \end{align*}
To show that this defines the same pairing as we defined using chains,
we take approximations $\Gamma'_{V+W}(Z)$, $\Gamma'_{V}(X)$ and
$\Gamma_W (Y)$, and consider the effect of the spectrum level pairing
on the filtration quotients. The only tricky part of this verification is the
identification of 
\[
 \H^G_{V+W+n}((\Gamma'_V(X)\smsh\Gamma_W(Y))^{V+W+n}/(\Gamma'_V(X)\smsh\Gamma_W(Y))^{V+W+n-1};
 \Mackey U).
\]
This comes down to a calculation of
$\H^G_0(D(G/K_+)\smsh G/L_+; \Mackey U)$:
 \begin{align*}
 \H^G_0(D(G/K_+)\smsh G/L_+; \Mackey U)
 &= [S, D(G/K_+)\smsh G/L_+\smsh H\Mackey U]_G \\
 &\iso [\susp^\infty G/K_+, G/L_+\smsh H\Mackey U]_G \\
 &= \pi^K_0(G/L_+\smsh H\Mackey U) \\
 &\iso (\Mackey A_{G/L} \boxprod \Mackey U)(G/K) \\
 &\iso \MackeyOp A^{G/K} \tensor_{\sorb G}  (\Mackey A_{G/L} \boxprod \Mackey U),
 \end{align*}
the next-to-last isomorphism coming from Proposition~\ref{prop:smshhomotopy}. Thus,
 \begin{multline*}
 \H^G_{V+W+n}((\Gamma'_V(X)\smsh\Gamma_W(Y))^{V+W+n}/(\Gamma'_V(X)\smsh\Gamma_W(Y))^{V+W+n-1};
 \Mackey U) \\
 \iso \sum_{p+q=n}\MackeyOp C_{V+p}(X) \tensor_{\sorb G}
  (\Mackey C_{W+q}(Y) \boxprod \Mackey U),
 \end{multline*}
which is exactly what is needed in the chain-level description of the pairing.

The pairing underlying the evaluation and cap product of
Theorem~\ref{thm:vhomology} and the other evaluation and
cap product of Theorem~\ref{thm:dualvhomology} use the map
$H\Mackey T \smsh H\MackeyOp S \to H(\Mackey T\mixprod \MackeyOp S)$
used earlier in discussing the cup product.
We can represent the pairing
 \[
 \langle -,- \rangle_d \colon
  H_G^\alpha(Y;\Mackey T) \tensor H^G_{\alpha+\beta}(Z;\MackeyOp S)
  \to H^G_\beta(X; \Mackey T\mixprod \MackeyOp S)
 \]
as follows.
Let $y\colon Y\to \susp^\alpha H\Mackey T$ represent an element of
$H_G^\alpha(Y;\Mackey T)$ and let
$z\colon S^{\alpha+\beta}\to Z\smsh H\MackeyOp S$ represent an element of
$H^G_{\alpha+\beta}(Z;\MackeyOp S)$. Then
$\langle y, z \rangle_d \in H^G_\beta(X; \Mackey T\mixprod \MackeyOp S)$
is represented by the composite
 \begin{align*}
 S^{\alpha+\beta} \xrightarrow{z}
 Z\smsh H\MackeyOp S \xrightarrow{d\smsh 1}{}&
 X\smsh Y\smsh H\MackeyOp S \\ 
 &\xrightarrow{1\smsh y\smsh 1}
 X\smsh \susp^\alpha H\Mackey T \smsh H\MackeyOp S \to
 X\smsh \susp^\alpha H(\Mackey T\mixprod \MackeyOp S).
 \end{align*}
We can verify that this agrees with the chain-level definition by considering
its effects on filtration quotients, as usual. The pairing
 \[
 \langle -,- \rangle_d \colon
  \H_G^\alpha(Y;\MackeyOp S) \tensor \H^G_{\alpha+\beta}(Z;\Mackey T)
  \to H^G_\beta(X; \MackeyOp S\mixprod \Mackey T)
 \]
is defined similarly.


\section{The Thom isomorphism and Poincar\'e duality}\label{sec:vThomIso}

\subsection*{The Thom isomorphism}

\begin{definition}
Let $p\colon E\to B$ be a $V$-bundle as in 
Example~\ref{ex:vcomplexes}(\ref{item:vbundle}).
Let $D(p)$ and $S(p)$ denote the disk and sphere bundles of $p$, respectively,
and let $T(p) = D(p)/S(p)$ be the quotient space.
A {\em Thom class} for $p$ is an element
$t_p\in \tilde H_G^V(T(p);\Mackey A_{G/G})$ such that, for each
$G$-map $b\colon G/K\to B$,
 \begin{align*}
 b^*(t_p) \in &\tilde H_G^V(T(b^*p);\Mackey A_{G/G}) \\
 &\iso \tilde H_G^V(\susp^V G/K_+;\Mackey A_{G/G}) \\
 &\iso A(K)
 \end{align*}
is a generator.
 \end{definition}

\begin{remark}
A Thom class must live in ordinary cohomology, not dual cohomology.
The isomorphism 
$ \tilde H_G^V(\susp^V G/K_+;\Mackey A_{G/G})
\iso A(K)$
would not hold if $G$ were infinite and we were to replace $H_G^V$ with the dual
cohomology $\H_G^V$.
 \end{remark}

A Thom class for $p$ is related to Thom classes for the fixed-point bundles
$p^K\colon E^K\to B^K$ as follows.

\begin{proposition}\label{prop:vThomclass}
The following are equivalent for a cohomology
class $t\in \tilde H_G^V(T(p);\Mackey A_{G/G})$.
 \begin{enumerate}
 \item
$t$ is a Thom class for $p$.
 \item
For every subgroup $K\subset G$,
$t|K\in \tilde H_K^V(T(p);\Mackey A_{K/K})$
is a Thom class for $p$ as a $K$-bundle.
 \item
For every subgroup $K\subset G$, 
$t^K\in \tilde H_{WK}^{V^K}(T(p^K); \Mackey A_{WK/WK})$
is a Thom class for $p^K$ as a $WK$-bundle.
 \item
For every subgroup $K\subset G$,
$t^K|e \in \tilde H^{|V^K|}(T(p^K);\Z)$
is a Thom class for $p^K$ as a nonequivariant bundle.
 \end{enumerate}
 \end{proposition}

\begin{proof}
That $(2) \Rightarrow (1)$ is trivial.
To show that $(1) \Rightarrow (2)$,
let $b\colon K/J\to B$ be a $K$-map,
let $\tilde b\colon G/J\to B$ be the extension of $b$ to a $G$-map, and
consider the following diagram. 
\[
 \xymatrix{
  \tilde H_G^V(T(p);\Mackey A_{G/G}) \ar[r] \ar[d]_{\tilde b^*}
   & \tilde H_K^V(T(p);\Mackey A_{K/K}) \ar[d]^{b^*} \\
  \tilde H_G^V(T(\tilde b^*p);\Mackey A_{G/G}) \ar[r]_{\iso}
   & \tilde H_K^V(T(b^*p);\Mackey A_{K/K})
 }
\]
We need to show that the image of $t$, under the composite across the top
and right, is a generator of the copy of $A(J)$ on the bottom right. However,
by definition the image of $t$ is a generator in the bottom left, and the
bottom map is an example of the Wirthm\"uller isomorphism.

To show that $(2) \Rightarrow (3)$,
it suffices to assume that $K$ is normal
in $G$ (otherwise, begin by restricting to $NK$).
Let $b\colon G/L\to B^K$ where $K\subset L\subset G$, and consider the following
diagram:
\[
 \xymatrix{
  \tilde H_G^V(T(p);\Mackey A_{G/G}) \ar[r] \ar[d]_{b^*}
   & \tilde H_{G/K}^{V^K}(T(p^K);\Mackey A_{(G/K)/(G/K)}) \ar[d]^{b^*} \\
  \tilde H_G^V(T(b^*p);\Mackey A_{G/G}) \ar[r] \ar[d]_{\iso}
   & \tilde H_{G/K}^{V^K}(T(b^*p^K);\Mackey A_{(G/K)/(G/K)}) \ar[d]^{\iso} \\
  \tilde H_G^V(\susp^V G/L_+;\Mackey A_{G/G}) \ar[r]_-{\iso}
   & \tilde H_{G/K}^{V^K}(\susp^{V^K} G/L_+;\Mackey A_{(G/K)/(G/K)})
  }
\]
Again, we need to show that the image of $t$ across the top and down the
right is a generator. By definition, the image down the left is a generator,
and the bottom map is an isomorphism because restriction to fixed sets
respects suspension (Theorem~\ref{thm:vfixedsets}).

That $(3) \Rightarrow (4)$ follows from the equivalence
of (1) and (2).
That $(4) \Rightarrow (1)$
follows from the fact that, for every $K$,
an element of $A(K) \iso \colim_W[S^W,S^W]_K$ is a generator if and only
if its restriction to $\colim_W[S^{W^J},S^{W^J}]$ is a unit (that is, $\pm 1\in\Z$)
for each subgroup $J$ of $K$.
 \end{proof}

%In the following we use the definition of an orientable bundle
%given in \ref{def:orientable}.

In the statement of the Thom isomorphism theorem we use the following
notion of orientability (as in, for example, \cite{Wan:class}).
(The ideas of orientability and orientation are generalized and discussed
fully in \cite{CMW:orientation}.)

\begin{definition}
A $V$-bundle $p\colon E\to B$ is {\em orientable} if,
for every subgroup $H$ of $G$ and every $b\in B^H$,
every loop in $B^H$ at $b$ lifts to an orthogonal $H$-map of the fiber over $b$
that is homotopic to the identity through orthogonal $H$-maps.
\end{definition}

\begin{theorem}[Thom Isomorphism]\label{thm:vthomiso}
If $p\colon E\to B$ is an orientable $V$-bundle, then there exists a Thom class
$t_p\in \tilde H_G^V(T(p);\Mackey A_{G/G})$. For any Thom class $t_p$,
the maps
 \[
 t_p\cup - \colon H_G^\alpha(B;\Mackey T)
 \to \tilde H_G^{\alpha+V}(T(p);\Mackey T)
 \]
and
 \[
 t_p \cup - \colon \H_G^\alpha(B;\MackeyOp S)
 \to \tilde \H_G^{\alpha+V}(T(p);\MackeyOp S)
 \]
are isomorphisms.
 \end{theorem}

\begin{proof}
The existence of $t_p$ and the fact that multiplication by $t_p$ gives an
isomorphism is clear in the special case that $p$ is a bundle over an orbit.
The general case follows, as it does nonequivariantly, by a Mayer-Vietoris
patching argument (see \cite{MS:characteristicclasses} or 
\cite{CW:duality}).
 \end{proof}

This Thom isomorphism can be generalized easily to orientable spherical
$V$-fibrations.
In fact, in Section~\ref{sec:genThom} we prove a much more general
Thom isomorphism for arbitrary spherical $G$-fibrations.

\subsection*{Poincar\'e duality}

We now outline a proof of Poincar\'e duality for compact oriented $V$-manifolds.
The noncompact case can be handled in the usual way using cohomology with
compact supports, as in \cite{MS:characteristicclasses} or 
\cite{CW:duality}.

\begin{definition}
Let $M$ be a closed $V$-manifold. A {\em fundamental class} of $M$ is
a class $[M]\in \H_V^G(M;\Mackey A_{G/G})$ such that, for each 
point $m\in M$ with
$G$-orbit $Gm \subset M$ and tangent plane $\tau_m\iso V$,
the image of $[M]$ in
 \begin{align*}
 \H_V^G(M,M-Gm;{}&\Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^G(G\times_{G_m} S^{V-V(G/G_m)};
    \Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^{G_m}(S^V; \Mackey A_{G_m/G_m}) \\
 &\iso A(G_m)
 \end{align*}
is a generator.
 \end{definition}

The next-to-last isomorphism above is the Wirthm\"uller isomorphism, which would not
be true in this form if $G$ were infinite and we
were to use $H_V^G$ in place of $\H_V^G$.
Thus, a fundamental class must live in dual homology, not ordinary homology.

Note that the map 
$\H_V^G(M;\Mackey A_{G/G}) \to \widetilde\H_V^{G_m}(S^{\tau_m}; \Mackey A_{G_m/G_m})$
used above is the same as the following composite:
 \[
 \H_V^G(M;\Mackey A_{G/G}) \to \H_V^{G_m}(M;\Mackey A_{G_m/G_m})
   \to \H_V^{G_m}(M,M-m;\Mackey A_{G_m/G_m})
 \]

The fundamental class $[M]$ is related to fundamental classes of the fixed
submanifolds $M^K$ as follows. The proof is similar to that of
Proposition~\ref{prop:vThomclass}.

\begin{proposition}\label{prop:vFundclass}
The following are equivalent for a dual homology
class $\mu\in \H_V^G(M;\Mackey A_{G/G})$.
 \begin{enumerate}
 \item
$\mu$ is a fundamental class for $M$ as a $G$-manifold.
 \item
For every subgroup $K\subset G$,
$\mu|K \in \H_V^K(M;\Mackey A_{K/K})$ is a fundamental class for
$M$ as a $K$-manifold.
 \item
For every subgroup $K\subset G$,
$\mu^K \in \H_{V^K}^{WK} (M^K; \Mackey A_{WK/WK})$ is a fundamental class for
$M^K$ as a $WK$-manifold.
 \item
For every subgroup $K\subset G$,
$\mu^K|e \in H_{|V^K|}(M^K;\Z)$ is a fundamental class for $M^K$
as a nonequivariant manifold. \qed
 \end{enumerate}
\end{proposition}

\begin{theorem}[Poincar\'e Duality]
A closed $V$-dimensional $G$-manifold $M$ is orientable if and only if
it has a fundamental class $[M]\in\H_V^G(M;\Mackey A_{G/G})$.
If $M$ does have a fundamental class, then
 \[
 -\cap [M] \colon H_G^\alpha(M;\Mackey T) \to \H_{V-\alpha}^G(M;\Mackey T)
 \]
and
 \[
 -\cap [M] \colon \H_G^\alpha(M;\MackeyOp S) \to H_{V-\alpha}^G(M;\MackeyOp S)
 \]
are isomorphisms.
 \end{theorem}

\begin{proof}
The standard geometric proof, as given in 
\cite{MS:characteristicclasses} or \cite{CW:duality}, works
in this case.
 \end{proof}

As with the Thom isomorphism theorem, we will lift both the orientability
assumption and the assumption that $M$ is $V$-dimensional in
Section~\ref{sec:genThom}.

We could also prove Poincar\'e duality from the Thom isomorphism by using
the representing spectra. For this, we use the fact that, if $M$ is embedded
in a $G$-representation $W$ with normal bundle $\nu$, then the dual of $M_+$
is $\susp^{-W}T\nu$ \cite{LMS:eqhomotopy}. 
Thus, we have the following chain of isomorphisms, where the first
is the Thom isomorphism.
 \begin{align*}
 H_G^\alpha(M;\Mackey T)
 &\iso \tilde H_G^{\alpha+W-V}(T\nu;\Mackey T) \\
 &\iso [T\nu, \susp^{\alpha+W-V} H\Mackey T]_G \\
 &\iso [\susp^W D(M_+), \susp^{\alpha+W-V} H\Mackey T]_G \\
 &\iso [S^W, \susp^{\alpha+W-V}M_+\smsh H\Mackey T]_G \\
 &\iso \H^G_{V-\alpha}(M;\Mackey T)
 \end{align*}
Similarly, we have the following chain of isomorphisms.
 \begin{align*}
 \H_G^\alpha(M;\MackeyOp S)
 &\iso \tilde \H_G^{\alpha+W-V}(T\nu;\MackeyOp S) \\
 &\iso [T\nu, \susp^{\alpha+W-V} H\MackeyOp S]_G \\
 &\iso [\susp^W D(M_+), \susp^{\alpha+W-V} H\MackeyOp S]_G \\
 &\iso [S^W, \susp^{\alpha+W-V}M_+\smsh H\MackeyOp S]_G \\
 &\iso H^G_{V-\alpha}(M;\MackeyOp S)
 \end{align*}
As usual, we can take as a fundamental class 
$[M] \in \H^G_{V}(M;\Mackey A_{G/G})$ the image of a Thom class
in $\tilde H_G^{W-V}(T\nu;\Mackey A_{G/G})$. The maps displayed above
can then be shown to be given by capping with this fundamental class.

If $M$ is a compact orientable $V$-manifold with boundary, then we can prove relative,
or Lefschetz, duality. We state the results.

\begin{definition}
Let $M$ be a compact $V$-manifold with boundary. 
A {\em fundamental class} of $M$ is
a class $[M,\bndry M]\in \H_V^G(M,\bndry M;\Mackey A_{G/G})$ such that, for each 
point $m\in M-\bndry M$ with
$G$-orbit $Gm \subset M$ and tangent plane $\tau_m\iso V$,
the image of $[M,\bndry M]$ in
 \begin{align*}
 \H_V^G(M,&M-Gm;\Mackey A_{G/G}) \\
 &\iso \H_V^G(G\times_K D(\tau_m-V(G/K)), G\times_K S(\tau_m-V(G/K));
    \Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^G(\susp^{\tau_m}D(G/G_m)_+); \Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^{G_m}(S^{\tau_m}; \Mackey A_{G_m/G_m}) \\
 &\iso A(G_m)
 \end{align*}
is a generator.
 \end{definition}

There is an obvious relative version of Proposition~\ref{prop:vFundclass}.
Finally, we have the following relative duality.

\begin{theorem}[Lefschetz Duality]
A compact $V$-dimensional $G$-manifold $M$ is orientable if and only if
it has a fundamental class $[M,\bndry M]$.
If $M$ does have a fundamental class, then the following are all
isomorphisms.
 \[
 -\cap [M,\bndry M] \colon H_G^\alpha(M;\Mackey T) \to
   \H_{V-\alpha}^G(M,\bndry M;\Mackey T)
 \]
 \[
 -\cap [M,\bndry M] \colon H_G^\alpha(M,\bndry M;\Mackey T) \to
   \H_{V-\alpha}^G(M;\Mackey T)
 \]
 \[
 -\cap [M,\bndry M] \colon \H_G^\alpha(M;\MackeyOp S) \to
   H_{V-\alpha}^G(M,\bndry M;\MackeyOp S)
 \]
 \[
 -\cap [M,\bndry M] \colon \H_G^\alpha(M,\bndry M;\MackeyOp S) \to
   H_{V-\alpha}^G(M;\MackeyOp S)
 \]
% \qed
 \end{theorem}

Homology-chap2.tex
\chapter{Parametrized Homotopy Theory and Fundamental Groupoids}
\label{chap:homotopytheory}

\section*{Introduction}

In this chapter we discuss some needed background material on
parametrized spaces and spectra. Much of this is summarized from
\cite{MaySig:parametrized}. We add some material on {\em lax maps,}
which requires additional discussion, beyond that provided by May and Sigurdsson, of the Moore path
fibration functor.

We also summarize material on the fundamental groupoid.
The equivariant fundamental groupoid was considered by
tom~Dieck in \cite{tD:transfgroups}.
It was used in \cite{CMW:orientation}
to give a theory of equivariant orientations,
where it provides the necessary machinery to keep track of
the local representations appearing in a $G$-vector bundle
or a smooth $G$-manifold.
In particular, a collection of local representations assembles into
what we call a representation of the fundamental groupoid.
Such representations can be thought of as the natural dimensions
of $G$-vector bundles or $G$-manifolds, and provide
the grading for our extension of ordinary homology and cohomology.

The stable orbit category played a prominent role
in Chapter~\ref{chap:rog}. In our extended theory a similar role
is played by the {\em stable fundamental groupoid}, which we introduce and discuss in
the last two sections of this chapter.


%%%%%%%%%%%%%%%%%%%%%

\section{Parametrized spaces and lax maps}

In \cite{MaySig:parametrized} May and Sigurdsson give a careful exposition of the homotopy theory
of parametrized spaces and spectra. We recall the parts we need and refer the reader to \cite{MaySig:parametrized} for further details.

\begin{definition}
Let $B$ be a compactly generated $G$-space.
\begin{enumerate}
\item
Let $G\ParaU B$ be the category of {\em $G$-spaces over $B$}:
Its objects are pairs $(X,p)$ where $X$ is a $G$-$k$-space
and $p\colon X\to B$ is a $G$-map.
A map $(X,p)\to (Y,q)$ is a $G$-map
$f\colon X\to Y$ such that $q\circ f = p$, i.e., a $G$-map over $B$.
\item
Let $G\Para B$ be the category of {\em ex-$G$-spaces over $B$}:
Its objects are triples $(X,p,\sigma)$ where $(X,p)$ is a $G$-space
over $B$ and $\sigma$ is a section of $p$,
i.e., $p\circ\sigma$ is the identity.
A map $(X,p,\sigma)\to (Y,q,\tau)$ is a section-preserving $G$-map
$f\colon X\to Y$ over $B$, i.e., a $G$-map over and under $B$.
\end{enumerate}
When the meaning is clear, we shall write just $X$ for $(X,p)$ or $(X,p,\sigma)$. If $(X,p)$ is a space over $B$, we write
$(X,p)_+$ for the ex-$G$-space obtained by adjoining a disjoint section.
We shall also write $X_+$ for $(X,p)_+$, a notation that
May and Sigurdsson denigrate but should not cause confusion in context.
\end{definition}

As May and Sigurdsson point out, in order to make these categories closed symmetric monoidal we need $B$ to be weak Hausdorff but cannot require it of the spaces over $B$. Hence we assume that $B$ is compactly generated (i.e., it is a $k$-space and weakly Hausdorff) but assume of the spaces over it only that they are $k$-spaces.

On either category we can define a model category structure in which a map
$f$ is a weak equivalence, fibration, or cofibration if it is one as
a map of $G$-spaces, ignoring $B$. Here, as usual, a weak equivalence is
a weak equivalence on all components of all fixed sets, a fibration is
an equivariant Serre fibration, and a cofibration is a retract of
a relative $G$-CW complex. This is what May and Sigurdsson call the
$q$-model structure and they call these classes of maps $q$-equivalences, $q$-fibrations, and $q$-cofibrations.
However, they point out serious technical difficulties with this model structure, stemming from the fact that these cofibrations do not interact well with fiberwise homotopy. To rectify this, they define another model structure, the $qf$-model structure, with the same weak equivalences, but with a more restrictive class of cofibrations, the $qf$-cofibrations, and a corresponding notion of $qf$-fibration.
In any case, we are entitled to invert these weak equivalences, getting the homotopy categories
$\Ho G\ParaU B$ and $\Ho G\Para B$.
If $X$ and $Y$ are ex-$G$-spaces over $B$ we write
$[X,Y]_{G,B}$ for the set of maps in $\Ho G\Para B$.

In this context, inverting the weak equivalences introduces many more maps than are evident if one pays attention only to the fiber-preserving maps over $B$. For example, let $(X\times I,h)$ and $(Y,q)$ be spaces over $B$,
where $h\colon X\times I\to B$ does not necessarily factor through $X$.
Let $i_0, i_1\colon X\to X\times I$ be the inclusions of the two endpoints and let $p_0 = hi_0$ and $p_1 = hi_1$. Then the maps
$i_0\colon (X,p_0)\to (X\times I,h)$ and $i_1\colon (X,p_1)\to (X\times I,h)$ are both $q$-equivalences.
(That we really want these to be weak equivalences is implicit
already in Dold's axiom {\bf CYL} in \cite{Dold:parahomology}.)
Hence, if $f\colon (X,p_0)\to (Y,q)$ is a map over $B$, we have in the homotopy category the map
\[
 [f]\circ [i_0]^{-1}\circ [i_1]\in [(X,p_1), (Y,q)]_{G,B},
\]
which might not be represented by any fiberwise map $(X,p_1)\to (Y,q)$ over $B$.

There are a couple of useful ways of viewing this example.
Conceptually, the pair $(f,h)$ specifies a map $f\colon X\to Y$
and a homotopy $h\colon qf\to p_1$. This is an example of what we
shall call a {\em lax map} from $(X,p_1)$ to $(Y,q)$.
Technically, it helps to view $(f,h)$ as specifying a map over $B$
from $(X,p_1)$ to the Hurewicz fibration associate with $q$.
It will be most convenient to use the following variant of the associated fibration, given in \cite[\S8.3]{MaySig:parametrized}. Recall that the space of
{\em Moore paths} in $B$ is
\[
 \Lambda B = \{ (\lambda, l) \in B^{[0,\infty]}\times [0,\infty)
                \mid \lambda(r) = \lambda(l) \text{ for $r\geq l$} \}.
\]
We usually write $\lambda$ for $(\lambda,l)$ and $l_\lambda$ for $l$, the
{\em length} of $\lambda$.
Moore paths are useful largely because they have an associative composition:
Suppose $\lambda$ and $\mu$ are Moore paths with $\lambda(\infty) = \mu(0)$. Then we write $\lambda\mu$ for the Moore path with length
$l_{\lambda\mu} = l_\lambda + l_\mu$ given by
\[
 (\lambda\mu)(t) =
  \begin{cases}
    \lambda(t) & \text{if $t\leq l_\lambda$} \\
    \mu(t - l_\lambda) & \text{if $t\geq l_\lambda$.}
  \end{cases}
\]
(We reverse the order of composition here compared to
\cite[\S8.3]{MaySig:parametrized}, as it will be more useful to us in this order.)
If $p\colon X\to B$, the {\em Moore path fibration} $LX = L(X,p)$ is given by
\[
 LX = X\times_B \Lambda B
  = \{ (x,\lambda)\in X\times \Lambda B \mid p(x) = \lambda(0) \}.
\]
We define the projection $Lp\colon LX \to B$ using the endpoint projection: $Lp(x,\lambda) = \lambda(\infty)$. If $X$ is an ex-space with section $\sigma$, we define a section $L\sigma$ of $LX$ by $L\sigma(b) = (\sigma(b),b)$, where we write $b$ for the path of length 0 at the point $b$. There is an inclusion $\iota\colon X\to LX$ given by $\iota(x) = (x,p(x))$
and composition of paths defines a map
$\chi\colon L^2 X \to LX$, $\chi(x,\lambda,\mu) = (x,\lambda\mu)$.
With these maps, $(L,\iota,\chi)$ is a monad.
The unit $\iota$ is a $G$-homotopy equivalence under (but not over) $B$, hence a $q$-equivalence, and the projection $LX\to B$ is a Hurewicz fibration, hence a $q$-fibration and a $qf$-fibration.

For simplicity, restrict now to the ex-space case. (For the unbased case, just ignore sections in the following discussions.)
Because $\iota$ is a $q$-equivalence, it induces an isomorphism
\[
 [X,Y]_{G,B} \iso [X,LY]_{G,B}
\]
for any $(X,p,\sigma)$ and $(Y,q,\tau)$. So, we can represent more maps in $[X,Y]_{G,B}$ by considering maps from $X$ to $LY$ over $B$. A map $X\to LY$ over $B$ is given by a pair $(f,\lambda)$ where $f\colon X\to Y$ and $\lambda\colon X\to \Lambda B$ are $G$-maps under $B$, with $\lambda(x)(0) = qf(x)$ and $\lambda(x)(\infty) = p(x)$ for all $x$. 
We can view $\lambda$ as specifying a {\em Moore homotopy} rel $\sigma(B)$ from $q\circ f$ to $p$, by which we mean a $G$-map
\[
 \lambda\colon X\times [0,\infty] \to B
\]
and a $G$-invariant map
\[
 l_\lambda\colon X\to [0,\infty)
\]
for which each $\lambda(x,-)$ is a Moore path of length $l_\lambda(x)$, and such that $\lambda(-,0) = qf$, $\lambda(-,\infty) = p$, and $\lambda(\sigma(b),-)$ is the 0-length path at $b$ for every $b\in B$.

\begin{definition}\label{def:laxmap}
A {\em lax map} $(X,p,\sigma)\laxto (Y,q,\tau)$ is a map $X\to LY$ under and over $B$. It can also be thought of as a pair $(f,\lambda)$ where $f\colon X\to Y$ is a $G$-map under $B$ and $\lambda\colon qf\to p$ is a Moore homotopy rel $\sigma(B)$.
\end{definition}

In this context we shall call maps under and over $B$ {\em strict} maps. We can also think of strict maps as those lax maps in which the homotopies have zero length.
To avoid confusion we will use ``$\laxto$" in the text to indicate lax maps,
reserving ``$\to$" for strict maps. (In diagrams we will use ordinary arrows
and rely on context.)

We have an associative composition of lax maps: If
$(f,\lambda)\colon (X,p,\sigma)\laxto (Y,q,\tau)$ and
$(g,\mu)\colon (Y,q,\tau) \laxto (Z,r,\upsilon)$, then
\[
 (g,\mu)\circ (f,\lambda) = (gf, (\mu f)\lambda),
\]
where $\mu f\colon rgf \to qf$ is the evident Moore homotopy.
Thus, we can form the category $G \Lax B$ of ex-$G$-spaces and lax maps
(similarly, we have the category $G\LaxU B$ of $G$-spaces over $B$ and lax maps).
Thinking of lax maps as strict maps $X\to LY$, we have
\[
 G\Lax{B}(X,Y) = G\Para{B}(X,LY) \quad\text{and}\quad
 G\LaxU{B}(X,Y) = G\ParaU{B}(X,LY)
\]
and we topologize $G\Lax{B}$ and $G\LaxU{B}$ using these identifications.
The composite of $f\colon X\to LY$ and $g\colon Y\to LZ$ is then the composite
\[
 X \xrightarrow{f} LY \xrightarrow{Lg} L^2 Z \xrightarrow{\chi} LZ.
\]
Thus, the category of lax maps is the Kleisli category of the monad $L$
(see \cite{Kle:KleisliCat}, \cite[\S VI.5]{Mac:categories}, and \cite[\S5]{Str:MonadsI}).

One of the points of the Kleisli category is that the
inclusion $G\Para B\to G\Lax B$ has a right adjoint
$\bar L\colon G\Lax B\to G\Para B$. On objects it is given by $L$,
and, if $f\colon X\to LY$ represents a lax map, $\bar Lf$ is the composite
\[
 LX \xrightarrow{Lf} L^2Y \xrightarrow{\chi} LY.
\]
Both the inclusion $G\Para B\to G\Lax B$ and
$\bar L\colon G\Lax B\to G\Para B$ are continuous.

There is a notion of homotopy in $G\Lax B$, defined using lax maps out of cylinders $X\smsh_B I_+$ or paths in $G\Lax{B}(X,Y)$.
We've already noted that $\iota\colon X\to LX$ is a weak equivalence;
it's not hard to see that, considered as a lax map, it is a lax homotopy equivalence,
i.e., a homotopy equivalence in $G\Lax{B}$. Its inverse is the map
$\kappa\colon LX\laxto X$ represented by the identity $1\colon LX\to LX$.
We record the following useful facts.

\begin{lemma}\label{lem:laximpliesweak}
\begin{enumerate}\item[]
\item
If $f\colon X\to Y$ is a strict map that is also a lax homotopy
equivalence, then it is a weak equivalence (i.e., a $q$-equivalence).
\item
If $f\colon X\laxto Y$ is a lax homotopy equivalence,
then its representing map $\hat f\colon X\to LY$ is also a 
lax homotopy equivalence, hence a weak equivalence.
\end{enumerate}
\end{lemma}

\begin{proof}
To show the first claim, note that $\bar L f = Lf\colon LX\to LY$
is a fiberwise homotopy equivalence, because $\bar L$, being continuous,
takes lax homotopy equivalences to fiberwise homotopy equivalences.
It then follows, from the following diagram and the fact that $\iota$ is
a weak equivalence, that $f$ is a weak equivalence:
\[
 \xymatrix{
  X \ar[d]_\iota \ar[r]^f & Y \ar[d]^\iota \\
  LX \ar[r]_{Lf} & LY
 }
\]

For the second claim, an easy calculation shows that 
$\hat f = \bar L f\circ \iota$. Again, $\bar L$ takes the lax homotopy
equivalence $f$ to a fiberwise homotopy equivalence, hence a
lax homotopy equivalence. Combined with the fact that $\iota$ is
a lax homotopy equivalence, we get that $\hat f$ is a lax homotopy equivalence.
\end{proof}

If $Y$ is any ex-$G$-space, then $LY$ is $q$-fibrant.
If $X$ is $q$-cofibrant, we then have
\begin{align*}
 [X,Y]_{G,B}
  &\iso [X,LY]_{G,B} \\
  &\iso \pi G\Para B(X,LY) \\
  &\iso \pi G\Lax B(X,Y),
\end{align*}
where $\pi$ denotes homotopy classes of maps.
In other words, if $X$ is $q$-cofibrant and $Y$ is any space over $B$, then
$[X,Y]_{G,B}$ is precisely the set of lax homotopy classes of lax
maps from $X$ to $Y$.

We should point out that the category $G\Lax B$ is wanting in several respects. It does not, in general, have limits. In particular, it has no terminal object.
It also lacks colimits in general, although any diagram of strict maps has a colimit in $G\Lax B$, namely the image of the colimit in $G\Para B$ (because $G\Para B\to G\Lax B$ is a left adjoint).
It is, therefore, not a candidate to be a model category. We shall use it primarily as a place to represent maps in $\Ho G\Para B$ that cannot easily be represented in $G\Para B$.

We end this section with some results about lax cofibrations.

\begin{definition}
A lax map $i\colon A\laxto X$ is a {\em lax cofibration} if it satisfies the lax homotopy extension property, meaning that we can always fill in the dashed arrow in the following commutative diagram of lax maps:
\[
 \xymatrix{
  A \ar[r]^-{i_0} \ar[d]_i & A\smsh_B I_+ \ar[d]\ar@/^/[ddr] \\
  X \ar[r] \ar@/_/[drr] & X\smsh_B I_+ \ar@{-->}[dr] \\
   && Y
 }
\]
\end{definition}

We shall show that the lax cofibrations are precisely the strict maps that are cofibrations under $B$.

\begin{lemma}
If $(i,\lambda)\colon A\laxto X$ is a lax cofibration, then
$\lambda$ is a constant homotopy, so $i$ is a map over $B$.
\end{lemma}

\begin{proof}
Suppose that $(i,\lambda)\colon (A,p,\sigma)\laxto (X,q,\tau)$ is a lax cofibration. Construct $\tilde Mi$ as the following variant of the mapping cylinder: As a $G$-space, let
$\tilde Mi = X\union_i (A\smsh_B I_+)$, the pushout along the inclusion $i_0\colon A \to A\smsh_B I_+$. Give $\tilde Mi$ the evident section, but define $r\colon \tilde Mi\to B$ as follows:
\begin{align*}
 r(x) &= q(x)
 			\qquad\text{if $x\in X$} \\
 r(a,s) &= \lambda(a, l_\lambda(a)\cdot s)
 			\qquad\text{if $(a,s)\in A\smsh_B I_+$}.
\end{align*}
Define a lax map $(\tilde\imath,\tilde\lambda)\colon A\smsh_B I_+\laxto \tilde Mi$ by letting $\tilde\imath$ be the usual map and defining $\tilde\lambda$ as follows:
\begin{align*}
 \tilde\lambda(a,s,t) &= \lambda(a, l_\lambda(a)\cdot s + t) \\
 l_{\tilde\lambda}(a,s) &= l_\lambda(a)\cdot(1 - s).
\end{align*}
Note that the restriction of $(\tilde\imath,\tilde\lambda)$ to $A\smsh 1$ is a strict map over $B$.
Now, from the assumption that $A\to X$ is a lax cofibration, we can fill in a lax map $f$ in the following diagram:
\[
 \xymatrix{
  A \ar[r] \ar[d]_{(i,\lambda)}
    & A\smsh_B I_+ \ar[d]\ar@/^/[ddr]^{(\tilde\imath,\tilde\lambda)} \\
  X \ar[r] \ar@/_/[drr] & X\smsh_B I_+ \ar@{-->}[dr]^f \\
   && \tilde Mi
 }
\]
Consider the rightmost triangle restricted to $A\smsh 1$. From the definition of composition and the fact that the restriction of $\tilde\imath$ is strict, it follows that the map $A\smsh 1\laxto X\smsh 1$ must be strict, from which it follows that the original map $A\laxto X$ must be strict.
\end{proof}

\begin{theorem}\label{thm:laxcofib}
A lax map $i\colon A\laxto X$ is a lax cofibration if and only $i$ is a strict map and a cofibration under $B$.
\end{theorem}

\begin{proof}
Suppose that $i\colon A\laxto X$ is a lax cofibration.
The lemma above shows that $i$ is a strict map. We now form $Mi=X\union_A (A\smsh_B I_+)$, the usual mapping cylinder of $i$ in $G\Para B$, which is a subspace of $X\smsh_B I_+$. From the definition of lax cofibration we get a lax retraction $r\colon X\smsh_B I_+\laxto Mi$. Forgetting the maps to $B$, this retraction under $B$ shows that $i$ is a cofibration under $B$.

Conversely, let $i\colon A\to X$ be a strict map and a cofibration under $B$.
Let $p\colon X\to B$ be the projection.
Let $q\colon Mi\to B$ be the mapping cylinder of $i$ and let
$r\colon X\smsh_B I_+ \to Mi$ be a retraction under $B$.
We want to make $r$ into a lax retraction.
Define $h\colon (X\smsh_B I_+)\times [0,\infty]\to B$ by
\[
 h(x,s,t) =
  \begin{cases}
   qr(x,s-t) & \text{if $t \leq s$} \\
   q(x) & \text{if $t\geq s$}
  \end{cases}
\]
The map $h$ is clearly continuous, but is not quite what we need.
Let $u\colon X\to I$ be a $G$-invariant map such that $u^{-1}(0) = A$,
e.g., $u(x) = \sup\{t - p_2 r(x,t) \mid t\in I\}$ where $p_2$ is projection to $I$.
Define $\lambda\colon (X\smsh_B I_+)\times [0,\infty]\to B$ by
\[
 \lambda(x,s,t) =
  \begin{cases}
   h(x,s,t/u(x)) & \text{if $u(x) > 0$} \\
   q(x) & \text{if $u(x) = 0$}
  \end{cases}
\]
and let
\[
 l_\lambda(x,s) = su(x).
\]
Assuming that $\lambda$ is continuous, the pair $(r,\lambda)$ defines a lax retraction, from which it follows that $i$ is a lax cofibration. So, we need to check that $\lambda$ is continuous.

Continuity at points $(x,s,t)$ such that $u(x) \neq 0$ is clear, so we need to check continuity at points $(a,s,t)$ with $a\in A$. For such a point we have $\lambda(a,s,t) = p(a)$. Let $V$ be any neighborhood of $p(a)$ in $B$. Because $I\times [0,\infty]$ is compact, we can find a neighborhood $U$ of $a$ such that $U\times I\times [0,\infty] \subset h^{-1}(V)$. It follows that
$U\times I\times [0,\infty] \subset \lambda^{-1}(V)$, showing that $\lambda$ is continuous.
\end{proof}

We are now entitled to the following result, in which $C_B$ denotes the mapping cone over $B$ and $/_B$ indicates the quotient over $B$.

\begin{proposition}
Suppose that $i\colon A\to X$ is a strict map that is a cofibration under $B$. Then the natural map $c\colon C_B i\to X/_B A$ is a lax homotopy equivalence.
\end{proposition}

\begin{proof}
Because $i$ is a lax cofibration,
the obvious map
$k\colon X\union_A (A\smsh_B I_+) \to C_B i$ over $B$ extends to a
lax map $(h,\lambda)\colon X\smsh_B I_+\laxto C_B i$.
Note that $\lambda(a,t)$ is a 0-length path for every $a\in A$ and $t\in I$.
If $(h_1,\lambda_1)$ is the restriction to
$X\smsh 1$, note that $h_1$ takes $A$ to the section, hence induces a lax map
$\bar h_1\colon X/_B A \laxto C_B i$, which we claim is a lax homotopy inverse to $c$.

The composite $c\circ h\colon X\smsh_B I_+\laxto X/_B A$ factors through
$k\colon X/_B A\smsh_B I_+\laxto X/_B A$, which clearly gives a lax homotopy from the identity to $c\circ \bar h_1$. (This uses the fact that $\lambda$ is constant on $A$.)

On the other hand, it is easy to extend $h$ to a lax homotopy
$h'\colon C_B i \smsh_B I_+\to C_B i$ from the identity to $\bar h_1\circ c$.
\end{proof}

We also have the following results on the homotopy invariance
of pushouts and colimits.

\begin{proposition}
Suppose given the following diagram in which the horizontal maps
are strict, the vertical maps are lax homotopy equivalences, and
$i$ and $i'$ are lax cofibrations.
\[
 \xymatrix{
  X \ar[d] & A \ar[l]_i \ar[r] \ar[d] & Y \ar[d] \\
  X' & A' \ar[l]^{i'} \ar[r] & Y'
 }
\]
Then the induced map of pushouts
$Y\union_A X \laxto Y'\union_{A'} X'$ is a lax homotopy equivalence.
\end{proposition}

\begin{proof}
The proof of the classical result, as in
\cite[7.5.7]{Br:topologyBook}, depends only on formal properties of
the homotopy extension property and homotopies, and can
be adapted to the context of $G\Lax B$.
The precursor results to this result also appear in the more recent
\cite[\S6.5]{May:concise}; Peter tells us that the result itself
will appear in the second edition.
\end{proof}

\begin{proposition}
Suppose given the following diagram in which the horizontal maps
are lax cofibrations and the vertical maps are lax homotopy equivalences.
\[
 \xymatrix{
  X_0 \ar[r] \ar[d] & X_1 \ar[r] \ar[d] & X_2 \ar[r] \ar[d] & \cdots \\
  Y_0 \ar[r] & Y_1 \ar[r] & Y_2 \ar[r] & \cdots
 }
\]
Then the induced map $\colim_n X_n \laxto \colim_n Y_n$ is a
lax homotopy equivalence.
\end{proposition}

\begin{proof}
This follows from
\cite[7.4.1]{Br:topologyBook}, specifically the special case
considered in Step~1 of its proof.
(It would also follow from the last proposition
of \cite[\S6.5]{May:concise} using details
about the forms of the homotopies used in the proof.)
Again, the proof generalizes to our context.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%

\section{The fundamental groupoid}\label{sec:fundgrpd}

The following definition was given in \cite{CMW:orientation}.

\begin{definition}
Let $B$ be a $G$-space. The {\em equivariant fundamental groupoid}
$\Pi_G B$ of $B$ is the category whose objects are the $G$-maps
$x\colon G/H\to B$ and whose morphisms $x\to y$, $y\colon G/K\to B$,
are the pairs $(\omega,\alpha)$, where
$\alpha\colon G/H\to G/K$ is a $G$-map and $\omega$ is an equivalence
class of paths $x\to y\circ \alpha$ in $B^H$.
As usual, two paths are equivalent if they are homotopic rel endpoints.
Composition is induced by composition of maps of orbits and the usual
composition of path classes.
\end{definition}

Recall that $\orb G$ is the category of $G$-orbits and $G$-maps.
We have a functor
$\pi\colon \Pi_G B\to \orb G$ defined by
$\pi(x\colon G/H\to B) = G/H$ and $\pi(\omega,\alpha) = \alpha$.
We topologize the mapping sets in $\Pi_G B$ as in
\cite[3.1]{CMW:orientation} so, in particular, $\pi$ is continuous.
(The details of the topology are not important here.)
For each subgroup $H$, the subcategory
$\pi^{-1}(G/H)$ of objects mapping to $G/H$ and morphisms mapping to
the identity is a copy of $\Pi(B^H)$, the nonequivariant fundamental
groupoid of $B^H$ (with discrete topology).
$\Pi_G B$ itself is not a groupoid in the usual
sense, but a ``cat\'egories fibr\'ees en groupoides'' \cite{Gr:groupefond}
or a ``bundle of groupoids'' \cite[5.1]{CMW:orientation} over $\orb G$
(which we usually shorten to {\em groupoid over $\orb G$}).

The {\em homotopy fundamental groupoid} of $B$ is the category
$h\Pi_G B$ obtained by replacing each morphism space with its
set of path components. Equivalently, we identify any two morphisms
that are homotopic in the following sense:
Maps $(\omega,\alpha)$, $(\xi, \beta)\colon x\to y$, where
$x\colon G/H\to B$ and $y\colon G/K\to B$, are homotopic
if there is a homotopy
$j\colon G/H\times I\to G/K$ from $\alpha$ to $\beta$
and a homotopy $k\colon G/H\times I\times I\to B$ from
a representative of $\omega$ to a representative of $\xi$
such that
$k(a,0,t) = x(a)$ and $k(a,1,t) = yj(a,t)$ for all
$a\in G/H$ and $t\in I$.
Thus, $h\Pi_G B$ coincides with tom~Dieck's
``discrete fundamental groupoid'' \cite[10.9]{tD:transfgroups}.
When $G$ is finite, $h\Pi_G B = \Pi_G B$.

The main reason we mention the homotopy fundamental groupoid now is to make the following connection: $h\Pi_G B$ is isomorphic to the full subcategory of
$\Ho G\Para B$ determined by the objects $x\colon G/H\to B$.
This follows from the description of $\Ho G\Para B$ in terms of lax maps given in the preceding section.

\begin{definition}\label{def:ortho}
Let $\ortho G{}$ be the category whose objects are the
orthogonal $G$-vector bundles over orbits of
$G$ and whose morphisms are the equivalence classes of orthogonal
$G$-vector bundle maps between them.
Here, two maps are equivalent if they are $G$-bundle homotopic,
with the homotopy inducing the constant homotopy on base spaces.
Let $\pi\colon \ortho G{} \to \orb G$ be the functor taking the
bundle $p\colon E\to G/H$ to
its base space $G/H$, and taking a bundle
map to the underlying map of base spaces. Let $\ortho Gn$ be the full
subcategory of $\ortho G{}$ consisting of the $n$-dimensional bundles.
 \end{definition}

We topologize the mapping sets in $\ortho G{}$ as in
\cite[4.1]{CMW:orientation}. The map
\[
 \pi\colon \ortho G{}(p,q)\to \orb G(\pi(p),\pi(q))
\]
is then a bundle with discrete fibers.

The categories $\ortho G{}$ and $\ortho Gn$ are not small, but have small skeleta
which we call
$\orthosmall G{}$ and $\orthosmall Gn$,
obtained by choosing one representative in each equivalence class.
An explicit choice is given in \cite[2.2]{CMW:orientation}.
The following is \cite[2.7]{CMW:orientation}.

\begin{proposition} \label{prop:bundlereps}
A $G$-vector bundle $p\colon E\to B$ determines by pullbacks a map
\[
 p^*\colon \Pi_G B \to \ortho G{}
\]
 over $\orb G$. A $G$-vector bundle map
$(\tilde f,f)\colon p\to q$, with $\tilde f\colon E\to E'$ the map of total
spaces and $f\colon B\to B'$ the map of base spaces, determines a natural
isomorphism $\tilde f_*\colon p^*\to q^*f_*$ over the identity.
If $(\tilde h, h)\colon (\tilde f_0,f_0)\to (\tilde f_1,f_1)$ is a homotopy of
$G$-vector bundle maps, then
$(\tilde f_1)_* = (p')^*h_* \circ (\tilde f_0)_*$.
\qed
\end{proposition}

The map $p^*$ is an example of a {\em representation} of
the fundamental groupoid $\Pi_G B$:

\begin{definition}
An {\em $n$-dimensional orthogonal representation} of $\Pi_G B$ is a
continuous functor $\gamma\colon \Pi_G B\to \orthosmall Gn$
over $\orb G$.
If $\gamma$ is an $n$-dimensional orthogonal representation of
$\Pi_G B$ and
$\gamma'$ is an $n$-dimensional orthogonal representation of
$\Pi_G B'$, then a {\em map} from $\gamma$ to $\gamma'$ is a pair
$(f,\eta)$ where $f\colon B\to B'$ is a $G$-map and
$\eta\colon \gamma\to \gamma'\circ f_*$ is a natural isomorphism
over the identity of $\orb G$.
\end{definition}

For example, if $V$ is an orthogonal $H$-representation, then
we write $G\times_H \Vrep$ for the orthogonal representation of
$\Pi_G(G/H)$ associated to the bundle
$G\times_H V\to G/H$.
If $B$ is a $G$-space and $V$ is $G$-representation, we also write
$\Vrep$ for the pullback to $\Pi_G B$ of the representation
$\Vrep$ of $\Pi_G(*) = \orb G$.

In addition to $\orthosmall G{}$, there are several other small categories
that are useful to use as the targets of representations of $\Pi_G B$.
They fit into the following commutative diagram
(see \cite[\S\S 18 \& 19]{CMW:orientation} for details).
\[
 \xymatrix{
  {\orthosmall Gn} \ar[r] \ar[d] & {\PLsmall Gn} \ar[r] \ar[d] &
      {\Topsmall Gn} \ar[r] \ar[d] & {\spheresmall Gn} \ar[d] \\
  {s\orthosmall Gn} \ar[r] \ar[d] & {s\PLsmall Gn} \ar[r] \ar[d] &
      {s\Topsmall Gn} \ar[r] \ar[d] & {s\spheresmall Gn} \ar[d] \\
  {v\orthosmall Gn} \ar[r] & {v\PLsmall Gn} \ar[r]  &
      {v\Topsmall Gn} \ar[r]  & {v\spheresmall Gn} 
 }
\]
To form the top row,
we take the categories of $G$-vector bundles over orbits, with morphisms
being, respectively, the fiber-homotopy classes of $G$-vector bundle maps, PL $G$-bundle
maps, topological $G$-bundle maps, and spherical $G$-fibration maps.
In the case of $\spheresmall Gn$, what we mean is that the objects are bundles, but a map
from $G\times_H V$ to $G\times_K W$ is a based
fiber-homotopy class of homotopy equivalences
$G\times_H S^V\to G\times_K S^W$.
The middle row is the stabilization of the top row and
the bottom row gives virtual bundles.
Maps into these categories give us, respectively orthogonal, PL, topological, and
spherical representations, and their stable and virtual analogues.

In particular, spherical representations arise naturally from spherical fibrations.
Here, by a spherical fibration we mean the following
(see also \cite[\S 23]{CMW:orientation}).

\begin{definition}
A {\em spherical $G$-fibration} is a sectioned $G$-fibration $p\colon E\to B$ such that
each fiber $p^{-1}(b)$ is based $G_b$-homotopy equivalent to $S^V$ for some
orthogonal $G_b$-representation $V$.
\end{definition}

The various categories over $\orb G$ that we've discussed, including
$\Pi_G B$ and the twelve categories in the diagram above, are all
groupoids over $\orb G$.
Associated to a groupoid $\R$ over $\orb G$, there is a classifying
$G$-space $B\R$, defined in \cite[\S 20]{CMW:orientation}.
The following classification result is proved for finite groups as
\cite[24.1]{CMW:orientation} and extended to compact Lie groups
in \cite{Co:monoids}.

\begin{theorem}\label{thm:classification}
The $G$-space $B\R$ classifies representations of $\Pi_G B$ in $\R$.
That is, for $G$-CW complexes $B$, $[B, B\R]_G$ is in natural bijective
correspondence with the set of isomorphism classes of representations
$R\colon \Pi_G B\to \R$.
\end{theorem}


%%%%%%%%%%%%%%%%%%%%%

\section{Parametrized spectra}\label{sec:paraspec}

May and Sigurdsson give a very careful treatment of equivariant parametrized spectra
in \cite{MaySig:parametrized}.
They concentrate on orthogonal $G$-spectra over $B$, which give a model category
with good formal properties.
They also discuss $G$-prespectra over $B$, which
arise more naturally as representing objects for cohomology theories.
As in the first chapter,
it is therefore more convenient for us to use the latter, keeping in mind that the two kinds
of spectra give equivalent stable categories.
As explained in \cite{MaySig:parametrized}, we need to restrict our parametrizing
spaces to be compactly generated and have the homotopy types of $G$-CW complexes so that all of the
functors and adjunctions we need pass to homotopy categories.
We make this restriction from this point on.

Let $\RO(G)$ be the collection of all finite-dimensional representations of $G$. (May and Sigurdsson consider collections closed under direct sum but not necessarily containing all representations. For our purposes we do not need the greater generality.) The following definition is
\cite[11.2.16 \& 12.3.6]{MaySig:parametrized}.

\begin{definition}
\begin{enumerate}
\item[]
\item
A {\em $G$-prespectrum $E$ over $B$} consists of a collection of
ex-$G$-spaces $E(V)\to B$, one
for each $V\in\RO(G)$, together with structure
maps given by $G$-maps
\[
 \sigma\colon \susp_B^W E(V) \to E(V\oplus W)
\]
over $B$.
The $\sigma$ are required to be unital and transitive in the usual way.
\item
An {\em $\Omega$-$G$-prespectrum} is a $G$-prespectrum in which each
ex-$G$-space $E(V)$ is $qf$-fibrant and the adjoint structure maps
\[
 \tilde\sigma \colon E(V) \to \Loop_B^W E(V\oplus W)
\]
are $G$-weak equivalences.
\end{enumerate}
We let $G\PreSpec{} B$ denote the category of $G$-prespectra over $B$ and $G$-maps between them, i.e., levelwise $G$-maps over $B$ that respect the structure maps.
\end{definition}

May and Sigurdsson
define a stable model structure on $G\PreSpec{} B$
\cite[\S 12.3]{MaySig:parametrized}.
The stable equivalences
({\em $s$-equivalences} for short) are the fiberwise stable equivalences,
where the fibers are those of levelwise fibrant approximations.
The fibrant objects are exactly the $\Omega$-$G$-prespectra.
%They distinguish a full subcategory of {\em excellent} $G$-prespectra,
%the $\Omega$-$G$-prespectra $E$ such that each
%$E(V)$ has the $G$-homotopy type of a $G$-CW complex, each
%$E(V)\to B$ is a well-sectioned Hurewicz fibration, and
%each $\sigma\colon \susp^W E(V)\to E(V\oplus W)$ is
%a cofibration over $B$. The classical homotopy category
%of excellent $G$-prespectra, formed by taking homotopy classes
%of maps, is equivalent to $\Ho G\PreSpec{} B$.

We write $[E,F]_{G,B}$ for the group of stable $G$-maps
between two $G$-prespectra over $B$,
that is, $\Ho G\PreSpec{} B(E,F)$.
By \cite[12.4.5]{MaySig:parametrized}, $[E,F]_{G,B}$ is stable in the sense that, if
$V$ is any representation of $G$ then there is an isomorphism
\[
 [E,F]_{G,B} \iso [\susp_B^V E, \susp_B^V F]_{G,B}.
\]
It is useful to note that these groups are stable in a stronger sense: they are
stable under suspension by any spherical fibration over $B$.
The following is a direct consequence of \cite[15.1.5]{MaySig:parametrized}.

\begin{proposition}\label{prop:stability}
Let $\xi$ be a spherical $G$-fibration over
a $G$-CW complex $B$ and let
$\susp^\xi$ denote the fiberwise smash product with the total space of $\xi$.
Then, if $E$ and $F$ are any two $G$-prespectra over $B$ we have a natural isomorphism
\[
 [E,F]_{G,B} \iso [\susp^\xi E, \susp^\xi F]_{G,B}.
\]
\qed
\end{proposition}

The following functors and adjunctions are discussed in detail in
\cite{MaySig:parametrized}.
Much of that book is aimed at showing that these and other relationships descend to homotopy categories.
We have the functor $\susp_B^\infty\colon G\Para B \to G\PreSpec{} B$ taking an ex-$G$-space $X$ over $B$ to the prespectrum with
$(\susp_B^\infty X)(V) = \Susp_B^V X$.
If $X$ and $Y$ are ex-$G$-spaces over $B$, we write
\[
 \{X,Y\}_{G,B} = [\Susp_B^\infty X, \Susp_B^\infty Y]_{G,B}
\]
for the group of stable maps from $X$ to $Y$.
More generally, if $W$ is a representation of $G$, we have the
{\em shift desuspension} functor
$\susp^\infty_W\colon G\Para B \to G\PreSpec{} B$ defined by
\[
 \susp^\infty_W X(V) =
  \begin{cases}
   \susp_B^{V-W}X & \text{when $W\subset V$} \\
   B & \text{when $W\not\subset V$.}
  \end{cases}
\]
This functor is left adjoint to the evaluation functor $\Omega^\infty_W$ given by evaluation at $W$.
The adjunction descends to the homotopy categories, by
\cite[12.6.2]{MaySig:parametrized}. We insert here the standard warning:  Functors used on homotopy categories are the derived functors, obtained by first taking a cofibrant or fibrant approximation as appropriate. In particular, on the homotopy category, $\Omega^\infty_W$ does not return the $W$th space of a prespectrum but the $W$th space of a stably equivalent $\Omega$-$G$-prespectrum.

Suppose that $\alpha\colon A\to B$ is a $G$-map. Then
there are functors
\[
 \alpha^*\colon G\Para {B} \to G\Para A
\]
and
\[
 \alpha^*\colon G\PreSpec{}{B} \to G\PreSpec{} A
\]
given by taking pullbacks.
There are also functors
\[
 \alpha_!\colon G\Para A \to G\Para {B}
\]
and
\[
 \alpha_!\colon G\PreSpec{} A\to G\PreSpec{} {B}
\]
given by composition with $\alpha$ and identification of base points;
$(\alpha_!,\alpha^*)$ is a Quillen adjoint pair.
Moreover, $\alpha^*$ and $\alpha_!$ both commute with suspension, in the sense that
\[
 \susp_A^\infty \alpha^* X \iso \alpha^*\susp_{B}^\infty X
\]
and
\[
 \susp_{B}^\infty \alpha_! X \iso \alpha_!\susp_A^\infty X.
\]

Another useful property is the natural homeomorphism
 \[
  X\smsh_{B} \alpha_!Y \homeo \alpha_!(\alpha^*X\smsh_A Y)
 \]
for ex-$G$-spaces $X$ over $B$ and $Y$ over $A$.
(The smash product used is the usual
fiberwise smash product.)
This homeomorphism descends to an isomorphism in the homotopy category.
Similarly, if $E$ is a prespectrum over $B$, then
 \[
  E\smsh_{B} \alpha_!Y \iso \alpha_!(\alpha^*E\smsh_A Y).
 \]
We shall apply this most often to the case of
$\rho\colon B\to *$, the projection to a point.
(We shall use $\rho$ generically for any projection to a point.)
$\rho$ induces functors
$\rho^*\colon G\PreSpec{}{} \to G\PreSpec{} B$ and
$\rho_!\colon G\PreSpec{} B\to G\PreSpec{}{}$. Notice that
 \[
 \rho^*\susp^\infty Y = \susp^\infty_B \rho^* Y = \susp^\infty_B (B\times Y)
 \]
for a based $G$-space $Y$, and
 \[
 \rho_!\susp^\infty_B X = \susp^\infty \rho_! X = \susp^\infty (X/\sigma(B))
 \]
for an ex-$G$-space $(X,p,\sigma)$ over $B$.


%%%%%%%%%%%%%%%%%%%%%%%

\section{Lax maps of prespectra}

We would like to define a category of lax maps of prespectra,
similar to the category $G\Lax B$. May and Sigurdsson
extended $L$ to a functor on prespectra in
\cite[\S13.3]{MaySig:parametrized}, but there is a mistake in their
definition. Moreover, they were not trying to make 
their $L$ a monad, so, for example, took no care to allow for an
associative composition.
We begin by giving a definition that does give us a monad,
so that we can again define the category of lax maps to be the
Kleisli category of $L$.

The first difficulty with extending $L$ to a functor on prespectra is that, if $K$ is an ex-space, there is no canonical map $LK\smsh_B S^V\to L(K\smsh_B S^V)$. So, we need to define an ad hoc map, which we call $\beta^V$.
(May and Sigurdsson write $\beta_V$ for their similar map.) In defining $\beta^V$, we shall think of $S^V = D(V)/S(V)$, so that a $v\in S^V$ will have $\|v\|\leq 1$. Here we differ from May and Sigurdsson; the difference is unimportant, but we find this choice easier to work with here.
We first choose any homeomorphism $\phi\colon [0,\infty]\to [1/2,1]$ with
$\phi(0) = 1$ and $\phi(\infty) = 1/2$.
Now we define $\beta^V$ by the following formula, in which we write
$l$ for $l_\lambda$ to simplify the notation:
\[
 \beta^V((x,\lambda)\smsh v) =
  \begin{cases}
    ( x\smsh v, \lambda )
    & \text{if $\|v\| \leq 1/2$}
  \\[\bigskipamount]
   \left( x\smsh \phi(\phi^{-1}(\|v\|) - l)\frac{v}{\|v\|}, \lambda \right)
    & \text{if $1/2 < \|v\| \leq \phi(l)$}
  \\[\bigskipamount]
   \left(
     \sigma\lambda(l-\phi^{-1}(\|v\|)),
      \lambda|^{l}_{l-\phi^{-1}(\|v\|)} \right)
    & \text{if $\|v\| \geq \phi(l)$}
  \end{cases}
\]
Here, as in \cite{MaySig:parametrized}, we write $\lambda|_a^b$ for the Moore path of length $b-a$ given by $t\mapsto \lambda(a+t)$.

Now, to extend $L$ to prespectra we need to deal with the fact that 
$\beta^V\circ\beta^W \neq \beta^{V\dirsum W}$, so we do as in 
\cite[\S13.3]{MaySig:parametrized} and 
consider prespectra indexed on a fixed countable cofinal sequence
$\W = \{0 = V_0 \subset V_1\subset\cdots\}$.
Write $W_i = V_i-V_{i-1}$
(note that our indexing of the $W_i$ is slightly different from that
of May and Sigurdsson).
As explained in \cite[13.3.5]{MaySig:parametrized},
the category of prespectra indexed on $\W$ gives us the same
stable homotopy category as usual.
If $X$ is a $G$-prespectrum, we define an indexed prespectrum $LX$
to have spaces $(LX)(V_i) = L(X(V_i))$ and
structure maps
$\sigma_i\colon LX(V_i)\smsh_B S^{W_{i+1}}\to LX(V_{i+1})$ given by the composites
\[
 \xymatrix{
 LX(V_i)\smsh_B S^{W_{i+1}}
  \ar[r]^{\beta^{W_{i+1}}} &
 L(X(V_i)\smsh_B S^{W_{i+1}})
  \ar[r] &
 LX(V_{i+1}).
 }
\]
In general, define the structure maps
$\Susp^{V_j-V_i}LX(V_i) \to LX(V_j)$ to be the composite
$\sigma_{j-1}\circ\sigma_{j-2}\circ\cdots\circ\sigma_i$.

\begin{proposition}\label{prop:LSpecMonad}
The levelwise inclusion $\iota\colon X\to LX$ and levelwise
composition $\chi\colon L^2 X\to LX$ make $(L,\iota,\chi)$
a monad on the category of $\W$-indexed $G$-prespectra over $B$.
\end{proposition}

\begin{proof}
The main difficulty is showing that $\iota$ and $\chi$ are actually
maps of prespectra, and the definition of $\beta^V$ was crafted to
make them so. The fact that $\iota$ is a map of prespectra follows from
the fact that the following diagram commutes for any ex-$G$-space $K$:
\[
 \xymatrix@C-3em{
  & K\smsh_B S^V \ar[dl]_{\iota\smsh 1} \ar[dr]^{\iota} \\
  LK\smsh_B S^V \ar[rr]_-{\beta^V} && L(K\smsh_B S^V)
 }
\]
The fact that $\chi$ is a map of prespectra comes down to the commutativity
of the following diagram:
\[
 \xymatrix{
  L^2K \smsh_B S^V \ar[r]^{(\beta^V)^2} \ar[d]_{\chi\smsh 1}
    & L^2(K\smsh_B S^V) \ar[d]^{\chi} \\
  LK \smsh_B S^V \ar[r]_{\beta^V} & L(K\smsh_B S^V)
 }
\]
The verification that these diagrams commute is by
straightforward, if tedious, calculation using the definition of $\beta^V$.

That $(L,\iota,\chi)$ is a monad now follows directly
from the analogous fact for spaces.
\end{proof}

By analogy with spaces, we now make the following definition.

\begin{definition}
The category of {\em lax maps of $G$-prespectra over $B$}, denoted
$G\LaxPreSpec{}B$, is the Kleisli category of the monad $L$
on the category of $\W$-indexed $G$-prespectra over $B$.
\end{definition}

Explicitly, then, a lax map $X\laxto Y$ is a strict map
$X\to LY$. The composite of the maps $f\colon X\laxto Y$ and 
$g\colon Y\laxto Z$ is the following composite of strict maps:
\[
 X \xrightarrow{f} LY \xrightarrow{Lg} L^2 Z \xrightarrow{\chi} LZ.
\]
A lax map of prespectra can be thought of as a collection of lax maps
of ex-spaces at each level, compatible under suspension, using $\beta^{W_i}$ to interpret what compatibility means. Composition of lax maps is given levelwise
by composition of lax maps of ex-spaces.

Since $LY$ is a levelwise fibrant approximation but not, in general, a $qf$-fibrant approximation of $Y$, we don't expect to represent all stable maps by lax maps. However, lax maps give us just enough more freedom than strict maps to help us with the cellular theory we develop in the next chapter.

There are a number of cases where we will want to know that a functor
on strict maps extends to one on lax maps.
The general context is this: Suppose that $\C$ and $\D$
are categories,
$(S,\eta,\mu)$ is a monad on $\C$, and $(T,\theta,\nu)$ is a monad
on $\D$. Write $\C_S$ for the Kleisli category of $S$ and $\D_T$ for
that of $T$, so $\C_S(X,Y) = \C(X,SY)$ and $\D_T(X,Y) = \D(X,TY)$.
We ask what we need to extend a functor $F\colon\C\to\D$ to a functor $\C_S\to \D_T$.
The answer is the following.

\begin{definition}
A {\em map of monads} $(F,\psi)\colon (\C,S)\to (\D,T)$
consists of a functor $F\colon \C\to \D$ and a natural transformation
$\psi\colon FS\to TF$ such that the following diagrams commute:
\[
 \xymatrix@C-1.5em{
  & F \ar[dl]_{F\eta} \ar[dr]^{\theta F} \\
  FS \ar[rr]_-{\psi} && TF
 }
\]
\[
 \xymatrix{
  FS^2 \ar[r]^{\psi^2} \ar[d]_{F\mu}
   & T^2 F \ar[d]^{\nu F} \\
  FS \ar[r]_\psi & TF
 }
\]
(This is what Street calls a {\em monad opfunctor} in
\cite{Str:MonadsI}. His monad functors have the natural transformation
going in the other direction, and induce maps of Eilenberg-Moore categories.)
\end{definition}

The general theory of \cite{Str:MonadsI} shows that a
map of monads $(F,\psi)\colon (\C,S)\to (\D,T)$ gives
an extension of $F$ to $\bar F\colon \C_S\to \D_T$.
Explicitly, $\bar F(X) = F(X)$ on objects and, if
$f\colon X\to SY$ represents a map in
$\C_S(X,Y)$, then $\bar F(f)$ is the composite
\[
 FX \xrightarrow{Ff} FSY \xrightarrow{\psi} TFY.
\]
It is straightforward to check that this defines a functor
extending $F$.
In fact, though we shall not need it, extensions of $F$
are in one-to-one correspondence with natural transformations
$\psi\colon FS\to TF$ such that $(F,\psi)$ is a map of monads.

We have already seen one example of a map of monads, namely
\[
 (\susp^V_B, \beta^V)\colon (G\Para B,L) \to (G\Para B,L).
\]
The diagrams shown in the proof of Proposition~\ref{prop:LSpecMonad}
can be rewritten as the diagrams showing that $(\susp^V_B,\beta^V)$
is a map of monads:
\[
 \xymatrix@C-1.5em{
  & \susp^V_B \ar[dl]_{\susp^V_B\iota} \ar[dr]^{\iota\susp^V_B} \\
  \susp^V_B L \ar[rr]_-{\beta^V} && L\susp^V_B
 }
\]
\[
 \xymatrix{
  \susp^V_B L^2 \ar[r]^{(\beta^V)^2} \ar[d]_{\susp^V_B \chi}
   & L^2 \susp^V_B \ar[d]^{\chi \susp^V_B} \\
  \susp^V_B L \ar[r]_{\beta^V} & L\susp^V_B
 }
\]
Thus, we get the following.

\begin{proposition}
The suspension functor $\susp^V_B$ extends to a continuous functor we shall
again call $\susp^V_B\colon G\Lax B\to G\Lax B$.
\qed
\end{proposition}

Explicitly,
the suspension of a map $f\colon X\laxto Y$ is given by the composite
\[
 \susp^V_B X \xrightarrow{\susp^V_B f} \susp^V_B LY
   \xrightarrow{\beta^V} L\susp^V_B Y.
\]
Note, however, that $\susp^W_B\susp^V_B \neq \susp^{V\dirsum W}_B$
in general, because $\beta^W\circ\beta^V \neq \beta^{V\dirsum W}$.

We insert here a useful fact about lax cofibrations.

\begin{corollary}\label{cor:suspcofibr}
Suspension preserves lax cofibrations. I.e., if $A\to X$ is
a lax cofibration of ex-$G$-spaces, then so is $\susp^V_B A\to \susp^V_B X$.
\end{corollary}

\begin{proof}
Let $i\colon A\to X$ be a lax cofibration.
Because $\susp^V_B$ is a left adjoint (on $G\Para B$),
it preserves colimits, hence $\susp^V_B Mi \iso M(\susp^V_B i)$,
where $Mi$ denotes the usual (fiberwise) mapping cylinder.
Because $i$ is a lax cofibration, $Mi$ is a retract of $X\smsh_B I_+$
in $G\Lax B$,
hence $M(\susp^V_B i) \iso \susp^V_B Mi$ is a retract of $\susp^V_B X\smsh I_+$
in $G\Lax B$. Therefore $\susp^V_B i$ is a lax cofibration.
\end{proof}

We would also like to extend the functors $\susp^\infty_Z$
to lax maps.

\begin{remark}\label{rem:indexedSuspension}
Note that, because we are using indexed prespectra, the functor
$\susp^\infty_Z$ is now defined by
\[
 \susp^\infty_Z X(V_i) =
  \begin{cases}
   \susp_B^{V_i-Z}X & \text{when $Z\subset V_i$} \\
   B & \text{when $Z\not\subset V_i$.}
  \end{cases}
\]
Let $\bar Z$ represent the smallest $V_i$ such that $Z\subset V_i$.
Then $\susp^\infty_Z K = \susp^\infty_{\bar Z}\susp_B^{\bar Z-Z}K$.
From this it is clear that, in the context of indexed prespectra,
$\susp^\infty_Z$ still has a right adjoint, but that right adjoint
is $\Loop_B^{\bar Z-Z} \Loop^\infty_{\bar Z}$.
\end{remark}

\begin{definition}
The natural transformation
$\beta_Z^\infty \colon \susp^\infty_Z L \to L \susp^\infty_Z$
is defined as follows. Let $k$ be the least integer such that
$Z\subset V_k$. For $i\geq k$ we let
\[
 \beta_Z^\infty
 = \beta^{W_i}\circ \beta^{W_{i-1}}\circ\dots\circ\beta^{W_{k+1}}
   \circ \beta^{V_k-Z}
 \colon \susp_B^{V_i-Z}L \to L\susp_B^{V_i-Z},
\]
using the decomposition
\[
 V_i - Z = (V_k - Z)\dirsum W_{k+1}\dirsum \cdots \dirsum W_i.
\]
For $i<k$, $\beta_Z^\infty$ is the trivial map between trivial ex-spaces.
\end{definition}

\begin{proposition}
The natural transformation $\beta_Z^\infty$ defines a map of monads
$(\susp^\infty_Z, \beta_Z^\infty)\colon
 (G\Para B, L) \to (G\PreSpec{}B, L)$.
Hence, $\susp^\infty_Z$ extends to a continuous functor
$\susp^\infty_Z\colon G\Lax B \to G\LaxPreSpec{}B$.
\end{proposition}

\begin{proof}
We first need to verify that $\beta_Z^\infty$ is a map of prespectra.
For this we need the following to commute:
\[
 \xymatrix{
  \susp_B^{W_{i+1}}\susp_B^{V_i-Z} LK
    \ar[r]^{\susp\beta_Z^\infty}
    \ar@{=}[d]
   & \susp_B^{W_{i+1}} L\susp_B^{V_i-Z} K \ar[d]^{\beta^{W_{i+1}}} \\
  \susp_B^{V_{i+1}-Z} LK
    \ar[r]_{\beta_Z^\infty}
   & L\susp_B^{V_{i+1}-Z} K
 }
\]
We defined $\beta_Z^\infty$ precisely to make this diagram commute.

Now we need to know that $(\susp^\infty_Z,\beta_Z^\infty)$ is a map
of monads. That $\beta_Z^\infty$ preserves units and multiplication
follows by iterating the corresponding facts about the $\beta^V$s.
\end{proof}

\begin{corollary}\label{cor:suspPreservesHomotopy}
If $f\colon X\laxto Y$ is a lax homotopy equivalence of ex-$G$-spaces,
then $\susp^\infty_Z f\colon \susp^\infty_Z X\laxto \susp^\infty_Z Y$
is a a lax homotopy equivalence of $G$-prespectra over $B$.
\qed
\end{corollary}

The proof of the following is almost identical to that
of Corollary~\ref{cor:suspcofibr}.

\begin{corollary}\label{cor:suspcofibrspec}
If $A\to X$ is
a lax cofibration of ex-$G$-spaces, then 
$\susp^\infty_Z A\to \susp^\infty_Z X$ is a lax cofibration
of $G$-prespectra over $B$.
\qed
\end{corollary}

We will use this result, together with the fact that pushouts
of lax cofibrations are lax cofibrations, to construct
lax cofibrations of $G$-prespectra over $B$.
The following homotopy invariance results are proved the same way
as for the corresponding results for ex-$G$-spaces.

\begin{proposition}\label{prop:hominvpushouts}
Suppose given the following diagram
of $G$-prespectra, in which the horizontal maps
are strict, the vertical maps are lax homotopy equivalences, and
$i$ and $i'$ are lax cofibrations.
\[
 \xymatrix{
  X \ar[d] & A \ar[l]_i \ar[r] \ar[d] & Y \ar[d] \\
  X' & A' \ar[l]^{i'} \ar[r] & Y'
 }
\]
Then the induced map of pushouts
$Y\union_A X \laxto Y'\union_{A'} X'$ is a lax homotopy equivalence.
\qed
\end{proposition}

\begin{proposition}\label{prop:hominvcolimits}
Suppose given the following diagram of $G$-prespectra over $B$,
in which the horizontal maps are strict and
are lax cofibrations, and the vertical maps are lax homotopy equivalences.
\[
 \xymatrix{
  X_0 \ar[r] \ar[d] & X_1 \ar[r] \ar[d] & X_2 \ar[r] \ar[d] & \cdots \\
  Y_0 \ar[r] & Y_1 \ar[r] & Y_2 \ar[r] & \cdots
 }
\]
Then the induced map $\colim_n X_n \laxto \colim_n Y_n$ is a
lax homotopy equivalence.
\qed
\end{proposition}

We mentioned earlier, and it's easy to show, that
$\iota\colon X\to LX$ is a lax homotopy equivalence if $X$ is
an ex-$G$-space. The same is true if $X$ is a $G$-prespectrum over $B$,
but the proof is not as straightforward.
(Note that it's easy to show that $\iota$ is a stable equivalence.)

\begin{proposition}
If $X$ is a $G$-prespectrum over $B$, then $\iota\colon X\to LX$ is
a lax homotopy equivalence.
Its inverse is the lax map $\kappa\colon LX\laxto X$ represented
by the identity map $1\colon LX\to LX$.
\end{proposition}

\begin{proof}
The composite $\kappa\iota\colon X\laxto X$ is the map
represented by $\iota\colon X\to LX$, i.e., the identity map
in $G\LaxPreSpec{}B$.

The composite $\iota\kappa\colon LX\laxto LX$ is represented
by the map $L\iota\colon LX\to L^2X$. This is not the identity map,
but we will show that it is (fiberwise) homotopic to $\iota L$,
which does represent the identity.
To this end, for any ex-$G$-space $K$ we define a homotopy
$H\colon \iota L\to L\iota$,
\[
 H\colon LK\smsh_B [0,\infty]_+ \to L^2K,
\]
natural in $K$.
Here, we use $[0,\infty]$ rather than the homeomorphic $[0,1]$
to make the formulas defining $H$ simpler:
\[
 H(x,\lambda,t) =
  \begin{cases}
    (x, \lambda|_0^{l-t}, \lambda|_{l-t}^l )
       & \text{if $t\leq l$}
    \\[\bigskipamount]
    (x, \lambda|_0^0, \lambda|_0^l)
       & \text{if $t\geq l$.}
  \end{cases}
\]
In order for $H$ to define a map of prespectra $LX\to L^2X$,
we need the following diagram to commute
(where we have in mind $K = X(V_i)$ and $V = W_{i+1}$):
\[
 \xymatrix{
  {\susp^V_B LK\smsh_B I_+} \ar[r]^-{\susp^V H} \ar[d]_{\beta^V}
   & {\susp^V_B L^2 K} \ar[d]^{(\beta^V)^2} \\
  L\susp^V_B K\smsh_B I_+ \ar[r]_-H
   & L^2 \susp^V_B K
 }
\]
To check that this diagram commutes is now a computation based on
the formulas for $\beta^V$ and $H$, which we will leave to the reader.
\end{proof}

We can now state the prespectrum version of Lemma~\ref{lem:laximpliesweak};
the proof is exactly the same.

\begin{lemma}\label{lem:speclaximpliesweak}
\begin{enumerate}\item[]
\item
If $f\colon X\to Y$ is a strict map that is also a lax homotopy
equivalence, then it is a stable equivalence.
\item
If $f\colon X\laxto Y$ is a lax homotopy equivalence,
then its representing map $\hat f\colon X\to LY$ is a 
lax homotopy equivalence, hence a stable equivalence.
\qed
\end{enumerate}
\end{lemma}


\begin{corollary}
$\beta^\infty_Z\colon \susp^\infty_Z LK\to L\susp^\infty_Z K$
is a lax homotopy equivalence.
\end{corollary}

\begin{proof}
Write $\kappa\colon LK\laxto K$ again for the lax map of
ex-$G$-spaces represented by the identity $1\colon LK\to LK$.
We know that $\kappa$ is a lax homotopy equivalence (inverse
to $\iota\colon K\laxto LK$), hence
$\susp^\infty_Z\kappa\colon \susp^\infty_Z LK\laxto \susp^\infty_Z K$ is
a lax homotopy equivalence. By the preceding lemma,
$\widehat{\susp^\infty_Z\kappa}$ is also a lax homotopy equivalence,
but it is easy to see from the definitions that
$\widehat{\susp^\infty_Z\kappa} = \beta^\infty_Z$.
\end{proof}


Finally, some results about computing the stable homotopy groups used to define stable equivalence of $G$-prespectra.
Recall from \cite[12.3.4]{MaySig:parametrized} that the homotopy groups of a $G$-prespectrum $X$ are the stable homotopy groups of the fibers of a levelwise fibrant approximation of $X$, for which we can use $LX$. So, for each $b\in B$ and $H\subset G_b$, we have
\begin{align*}
 \pi_q^H(X_b)
   &\iso \colim_i \pi_q^H(\Omega^{V_i} LX(V_i)_b) \\
   &\iso \colim_i \pi H\Para *(S^{q+V_i}, LX(V_i)_b) \\
   &\iso \colim_i \pi H\Para B (b_! S^{q+V_i}, LX(V_i)) \\
   &\iso \colim_i \pi H\Lax B(b_! S^{q+V_i}, X(V_i)).
\end{align*}
Thus, we can represent an element of $\pi_q^H(X_b)$ as a lax $H$-map $(f,\lambda)\colon b_! S^{q+V_i}\to X(V_i)$ for some $i$. The equivalence relation is then the one generated by lax homotopy and suspension of lax maps.

We also have the following result that simplifies computing the stable homotopy groups of an ex-$G$-space:
If $K$ is an ex-$G$-space, $b\in B$, and $H\subset G_b$, then
\begin{align*}
 \pi_q^H((\susp^\infty_B K)_b)
  &\iso \pi_q^H((L\susp^\infty_B K)_b) \\
  &\iso \pi_q^H((\susp^\infty_B LK)_b) \\
  &\iso \pi_q^H(\susp^\infty_H LK_b) \\
  &\iso \colim_V \pi H\Para * (S^{q+V}, \susp^V LK_b)
\end{align*}
is just the stable homotopy group of the (non-parametrized) $H$-space $LK_b$.
This also follows from \cite[13.7.4]{MaySig:parametrized}.

%%%%%%%%%%%%%%%%%%%%%

\section{The stable fundamental groupoid}\label{sec:stabgrpd}

As mentioned earlier, we need a stable version of the fundamental
groupoid. We use the following definition.

\begin{definition}
The {\em stable fundamental groupoid} of a $G$-space $B$ is the category
$\stab\Pi_G B$ whose objects are the $G$-maps $x\colon G/H\to B$ (the same
objects as the unstable fundamental groupoid) and whose maps
from $(G/H,x)$ to $(G/K,y)$ are the stable $G$-maps
$\{(G/H,x)_+,(G/K,y)_+\}_{B,G}$.
\end{definition}

Note that, by forgetting $B$ (i.e., applying $\rho_!$, where $\rho\colon B\to *$), there is a functor
$\phi\colon\stab\Pi_G B\to \sorb G$ from the stable fundamental
groupoid to the stable orbit category.

To supplement this conceptual definition of the stable fundamental groupoid,
we give a computational description generalizing the one given in the case
of $G$ finite in \cite{CW:duality}. If $(G/H,x)$ and $(G/K,y)$ are two objects in
$\stab\Pi_G X$, from the comments at the end of the preceding section we have
\[
 \stab\Pi_G X(x,y)
  \iso \pi_0^H (\susp^\infty_H (L(G/K,y)_+)_x).
\]
Write $L(x,y)$ for the unbased $H$-space consisting of pairs
$(gK, \lambda)$ where $gK\in G/K$ and $\lambda$ is a Moore path from $y(gK)$
to $x(eH)$.
$(L(G/K,y)_+)_x$ is the disjoint union of $L(x,y)$ and a component contractible to the basepoint (coming from
the disjoint section added to $(G/K,y)$), so we may safely replace $(L(G/K,y)_+)_x$ with
$L(x,y)_+$.

By the general calculation of \cite[V.9.3]{LMS:eqhomotopy},
$\pi_0^H(\susp^\infty_H L(x,y)_+)$ is the free abelian group generated by
the equivalence classes of $H$-maps
$a\colon H/J\to L(x,y)$ for those subgroups $J$ of $H$ such that
the Weyl group $W_H J = N_H J/J$ is finite.
A map $a$ is equivalent to $a'\colon H/J'\to L(x,y)$ if there is an
$H$-homeomorphism $\xi\colon H/J\to H/J'$ such that $a'\circ\xi \hmtpc a$.
In other words, there is a generator for each element of
$\pi_0(L(x,y)^J)/W_H J$, with $J$ running through a set of representatives
of the conjugacy classes of subgroups of $H$ with finite Weyl group.

Now, from the definition of $L(x,y)$, an $H$-map map $a\colon H/J\to L(x,y)$
is equivalent to a pair $(\beta,\lambda)$ such that $\beta$ is an $H$-map
and the following diagram commutes
up to the Moore $H$-homotopy $\lambda\colon y\beta\to x\alpha $:
\[
 \xymatrix{
  H/H \ar[dr]_x & H/J \ar[l]_\alpha \ar[r]^\beta & G/K \ar[dl]^y \\
  & B
  }
\]
Using the $G$-homeomorphism $G\times_H(H/J) \homeo G/J$, $(\beta,\lambda)$
in turn gives a triple $(\alpha,\beta,\lambda)$ where $\alpha$ and
$\beta$ are now $G$-maps and the following diagram commutes up to the
$G$-homotopy $\lambda$:
\[
 \xymatrix{
  G/H \ar[dr]_x & G/J \ar[l]_\alpha \ar[r]^\beta & G/K \ar[dl]^y \\
  & B
  }
\]
Analyzing when two such diagrams are to be considered equivalent, we get the following
description of $\stab\Pi_G B$.

\begin{proposition}\label{prop:geostable}
If $x\colon G/H\to B$ and $y\colon G/K\to B$, then
$\stab\Pi_G B(x,y)$ is
the free abelian
group on the set of equivalence classes $[\alpha,\beta,\lambda]$
as in the following diagram:
\[
 \xymatrix{
  G/H \ar[dr]_x & G/J \ar[l]_\alpha \ar[r]^\beta & G/K \ar[dl]^y \\
  & B
  }
\]
Here, $J\subset H$ with $W_H J$ finite,
$\alpha$ and $\beta$ are $G$-maps, and $\lambda$ is a Moore $G$-homotopy
from $y\circ\beta$ to $x\circ\alpha$.
We say that $(\alpha,\beta,\lambda)$ is equivalent to
$(\alpha',\beta', \lambda')$ when we have a $G$-homeomorphism
$\xi\colon G/J'\to G/J$ such that the diagram below coherently
homotopy commutes, meaning that there exist $G$-homotopies
$j\colon G/J'\times I\to G/H$ from $\alpha'$ to $\alpha\xi$,
$k\colon G/J'\times I\to G/K$ from $\beta'$ to $\beta\xi$, and
$m\colon G/J'\times [0,\infty]\times I\to B$ such that
$m(gJ',-,t)$ is a Moore path for each $g$ and $t$, with
\begin{enumerate}
\item
$m(-,-,0) = \lambda'$ and $m(-,-,1) = \lambda\xi$, and
\item
$m(-,0,-) = yk$ and $m(-,\infty,-) = xj$.
\end{enumerate}
\[
 \xymatrix@R=2ex{
  & G/J' \ar[dl]_{\alpha'} \ar[dr]^{\beta'} \ar[dd]^\xi \\
  G/H \ar[dddr]_x && G/K \ar[dddl]^y \\
  & G/J \ar[ul]_\alpha \ar[ur]^\beta \\
  \\
  & B
  }
\]
\qed
\end{proposition}

Given a triple $[\alpha,\beta,\lambda]$ as in the preceding proposition, let
$z = x\circ\alpha$. It follows that a free generator of $\stab\Pi_G B(x,y)$
can be written as a diagram of the form
\[
 x \from z \to y.
\]
The corresponding map in $\stab\Pi_G B(x,y)$ is the composite of
the transfer $\tau\colon \susp^\infty_G G/H_+\to \susp^\infty_G G/K_+$ associated with the projection $\alpha$
and the unstable map $z\to y$ over $B$ specified by the pair $(\beta,\lambda)$.

To describe composition,  suppose that
$[\alpha,\beta, \lambda]\colon x\to y$ and
$[\gamma,\delta,\mu]\colon y\to z$. We form the following
diagram, in which $P$ makes the top diamond a pullback.
\[
 \xymatrix@C-2em@R-2ex{
 && P \ar[dl] \ar[dr] \\
 & G/J \ar[dl]_\alpha \ar[dr]^\beta && G/L \ar[dl]_\gamma \ar[dr]^\delta \\
 G/H \ar[ddrr]_x && G/K \ar[dd]^y && G/M \ar[ddll]^z \\
 \\
 && B
 }
\]
If $G$ is finite then $P$ is a disjoint union of orbits of $G$,
say $P = \coprod_i G/N_i$. By composition, each orbit gives us a diagram
\[
 \xymatrix{
  G/H & G/N_i \ar[l]_{\alpha_i} \ar[r]^{\beta_i} & G/M
  }
\]
and composition of the paths $\mu$ and $\lambda$ gives a homotopy
$\lambda_i\colon z\beta_i\to x\alpha_i$. The composite in $\stab\Pi_G B$ is then
\[
 [\gamma,\delta,\mu]\circ [\alpha,\beta,\lambda]
  = \sum [\alpha_i,\beta_i, \lambda_i].
\]

In general, $P$ is a manifold over $G/H$ and we can write
$P = G\times_H P'$ where $P'$ is an $H$-manifold.
If $t(P')\in \pi^H_0(\susp^\infty_G P'_+)$ is the transfer associated with
the projection $P'\to *$, then, by
\cite[V.9.3]{LMS:eqhomotopy}, we can write $t(P')$ as a finite sum
\[
 t(P') = \sum n_i [w'_i\colon H/N_i\to P'].
\]
Thus, the transfer map $t(P)\colon\susp^\infty_G G/H_+\to \susp^\infty_G P_+$ can be written as the finite sum
\[
  t(P) = \sum n_i [w_i\colon G/N_i\to P].
\]
As above, each $w_i$ gives a triple $(\alpha_i,\beta_i, \lambda_i)$ and
the composite in $\stab\Pi_G B$ is
\[
 [\gamma,\delta,\eta]\circ [\alpha,\beta,\omega]
  = \sum n_i [\alpha_i,\beta_i, \lambda_i].
\]
This describes the composite of two generators, so we can now extend
bilinearly to describe composition in general.


%%%%%%%%%%%%%%%%%%%%%

\section{A technical result}

In preparation for defining cellular chains in the next chapter,
we need to prove a technical result.
We continue working with a fixed parametrizing space $B$.

\begin{definition}\label{def:parasphere}
Let $x\colon G/K\to B$ be a $G$-map and let
$V$ be a representation of $K$.
Consider $G\times_K S^V$ to be an ex-$G$-space over $G/K$, with section
$\tau\colon G/K\to G\times_K S^V$ the inclusion of the compactification points,
and define
$S^{V,x} = x_! (G\times_K S^V)$.
Explicitly, it is defined by the following diagram,
in which the top square is a pushout and the composite down the right side is the identity on $B$.
 \[
 \xymatrix{
 G/K \ar[r]^{x} \ar[d]_{\tau} & B \ar[d] \\
 G\times_K S^V \ar[r] \ar[d] & S^{V,x} \ar[d] \\
 G/K \ar[r]_{x} & B
 }
 \]
\end{definition}

If $\gamma$ is a spherical representation of $\Pi_G B$,
$x\colon G/K\to B$, and $\gamma(x) = G\times_K S^V$,
we write $S^{\gamma,x}$ for $S^{V,x}$.

\begin{definition}
Let $\gamma$ be a stable spherical representation of $\Pi_G B$. Define a category
$\stab\Pi_\gamma B$ to have as objects the maps $x\colon G/K\to B$ (i.e., the
same objects as $\Pi_G B$), and as morphisms
 \[
 \stab\Pi_\gamma B(x,y) = \{ S^{\gamma,x}, S^{\gamma,y} \}_{G,B},
 \]
the group of stable maps.
\end{definition}

Note that $\stab\Pi_0 B = \stab\Pi_G B$. The purpose of this section is to
prove the following result.

\begin{theorem}\label{thm:piXinvariance}
For any $\gamma$,
the category $\stab\Pi_\gamma B$ is naturally isomorphic to $\stab\Pi_G B$.
\end{theorem}

A proof of this result when $G$ is finite appeared in
\cite[Lemma 4.2]{CW:duality}.
The idea of the proof given there
is that a stable map from $S^{\gamma,x}$ to $S^{\gamma,y}$,
viewed as a lax map,
involves paths from $x$ to $y$ through fixed sets $X^J$, and applying $\gamma$
to such a path determines a $J$-isomorphism between $\gamma(x)$ and
$\gamma(y)$. In other words, in the eyes of the subgroups that matter, the two
representations are identical, and we get the same maps as if they really
were the same representation.

We now give a different proof that works for
arbitrary compact Lie groups. We shall embed both categories in a larger
category and show that the images are identical. This will be a much modified
version of tom Dieck's approach to maps between spheres in
\cite[II.4]{tD:transfgroups}.
We need a preliminary definition.

\begin{definition}
Let $X$ and $Y$ be nonequivariant spaces and
let $f\colon X\to Y$ be a nonequivariant map.
Let $\Z\pi_1(f)$ denote the contravariant functor on $\stab\Pi Y$ given by
 \[
 \Z\pi_1(f)(y) = \{(*,y)_+, (X,f)_+\}_Y
 \]
for every map $y\colon *\to Y$.
\end{definition}

We use this notation because
 \[
 \Z\pi_1(f)(y) \iso\pi_0(\susp^\infty_Y L(y,f)_+) \iso \Z\pi_0(L(y,f)).
 \]
($\Z S$ denotes the free abelian group generated by the set $S$.)
Similar to the notation in Section~\ref{sec:stabgrpd},
$L(y,f)$ here denotes the fiber over $y$ of the Moore path fibration associated to $f$,
so that
$\pi_0(L(y,f)) \iso \pi_1(f,y)$ is the usual $\pi_1$ of a map.
Notice that
if $Y=*$ then $\Z\pi_1(f) = \Z\pi_0(X)$.

Here is the larger category in which we shall embed $\stab\Pi_G B$
and $\stab\Pi_\gamma B$.

\begin{definition}
Define $\Degs B$ to be the category whose objects
are those of $\Pi_G B$, but whose morphisms from $x\colon G/H\to B$ to
$y\colon G/K\to B$ are given by
 \[
 \Degs B(x,y) = \prod_{(J)\in\Gamma G}
     \Nat(\Z\pi_1(x^J), \Z\pi_1(y^J))^{WJ}
 \]
where $\Gamma G$ denotes the set of conjugacy classes of closed subgroups of $G$.
Here, $x^J$ denotes the map $x^J\colon (G/H)^J\to B^J$ and $\Nat$
denotes the group of natural transformations. $WJ$ acts on this group of
natural transformations in the obvious way, and we take only the $WJ$-invariant
transformations.
\end{definition}

Note that, if $J$ and $J'$ are conjugate, then there is a canonical
identification
 \[
 \Nat(\Z\pi_1(x^J), \Z\pi_1(y^J))^{WJ} \iso
   \Nat(\Z\pi_1(x^{J'}), \Z\pi_1(y^{J'}))^{WJ'}.
 \]
Notice also that, if $B = *$ is just a point, then
 \[
 \Degs(*)(G/H,G/K) = \prod_{(J)\in\Gamma G}
  \Hom_{WJ}(\Z\pi_0(G/H)^J, \Z\pi_0(G/K)^J).
 \]

Now, there is a functor
 \[
 \delta\colon \stab\Pi_G B\to \Degs B
 \]
that is the identity on objects, and on a morphism $f\colon x\to y$ is given
by
 \[
 \delta(f) = \prod_{(J)} f^J_*.
 \]
We think of $\delta$ as recording the degrees of the fixed-set maps $f^J$, but
it also records the path data associated with $f$, thinking in terms of lax maps.

Our next goal is to define a similar functor
$\delta_\gamma\colon\stab\Pi_\gamma B \to \Degs B$.
To this end, consider a map in $\stab\Pi_\gamma B(x,y)$ and, using
the discussion at the end of the preceding section, represent it by
a lax map
 \[
 f\colon S^{\gamma,x}\smsh S^V\to S^{\gamma,y}\smsh S^V.
 \]
Taking $J$-fixed points we get a lax map
 \[
 f^J\colon S^{\gamma^J,x^J}\smsh S^{V^J} \to
    S^{\gamma^J,y^J}\smsh S^{V^J}.
 \]
Consider now a point $z\in B^J$ and an unstable lax map $\eta\colon z\to x^J$
representing a generator of $\Z\pi_1(x^J)(z)$. Write $W = \gamma(x)_{\eta(z)}$,
the fiber of $\gamma(x)$ over $\eta(z)$.
Then the composite of $f^J$ and $\eta$ is
a stable map  $f^J\circ\eta\colon z\to y^J$ represented by the lax map $(F,\Lambda)$ where
$F\colon S^{W^J,z}\smsh S^{V^J}\to S^{\gamma^J,y^J}\smsh S^{V^J}$ and $\Lambda$ is a Moore homotopy from $y^J\circ F$ to $z$. $F$ is homotopic to a
map transverse to the zero section, so we may assume that $F$ itself is transverse.
$F^{-1}(y^J)$ is then a finite set of points in $W^J\oplus V^J$; let $w$ be one
such point. The restriction of $F$ to $w$ determines a linear isomorphism
 \[
 \bar F\colon W^J\dirsum V^J\to \gamma(y)^J_v\dirsum V^J
 \]
where $v = F(w)\in y^J$.
On the other hand, $\Lambda$ determines a path from $y\circ v\colon G/J\to B$ to
$x\circ\eta\colon G/J\to B$
in $B^J$, and applying $\gamma$ to this path determines
a $J$-isomorphism $W\dirsum V\iso \gamma(y)_v\dirsum V$ which then determines
another isomorphism
 \[
 r\colon W^J\dirsum V^J\iso \gamma(y)^J_v\dirsum V^J.
 \]
We now let $\epsilon$ be the degree of the composite $r^{-1}\circ F$
(i.e., $\epsilon = \pm 1$). Let $\omega$ be the path in $B^J$ from $z$ to
$y(v)$ determined by $\eta$ and $\Lambda$; we now can let
 \[
 \delta_\gamma (f)_J (\eta) = \sum_w \epsilon\omega.
 \]
The usual transversality arguments show that this is independent of the choice
of representative $F$ transverse to the zero section.
It is now not difficult to check that $\delta_\gamma(f)_J$ is a well-defined
natural transformation for each $J$. Moreover, using the functoriality of
$\gamma$, it is not hard to show that $\delta_\gamma$ is a functor.

What $\delta_\gamma$ really does is record the degrees of the various fixed-set
maps $f^J$, using $\gamma$ and the path data associated with $f$ to identify the
source and target representations.

\begin{lemma}\label{lem:deltainclusion}
The image of $\delta$ is contained within the image of $\delta_\gamma$.
\end{lemma}

\begin{proof}
Let $x\colon G/H\to B$.
As in the comment after Proposition~\ref{prop:geostable},
a generator of $\stab\Pi_G B(x,y)$ can be written in the form
 \[
 x \xrightarrow{\tau} z \to y
 \]
where $\tau$ is the transfer map associated with a projection $G/J\to G/H$.
If we replace the map $z\to y$ with the map $S^{\gamma,z}\to S^{\gamma,y}$
induced by $\gamma$, we get a map in $\stab\Pi_\gamma B(x,y)$ that has the same
image under $\delta_\gamma$ as the original map  $x\to y$ did under $\delta$.
 \end{proof}

The construction used in this proof will turn out to define an isomorphism of
$\stab\Pi_G B$ with $\stab\Pi_\gamma B$, but we shall not prove this directly.

Our next task is to show that both $\delta$ and $\delta_\gamma$ are injective.
For this it suffices to restrict our attention to the set of maps between two
objects $x$ and $y$. We shall actually show that a composite into another set
is injective.

\begin{definition}
Suppose that $x\colon G/H\to B$. Let $\Phi H$ denote the set of $H$-conjugacy
classes of subgroups $J\subset H$ with $|W_H J|<\infty$. Let
 \[
 e\colon \Degs B(x,y) = \prod_{(J)\in\Gamma G}
     \Nat(\Z\pi_1(x^J), \Z\pi_1(y^J))^{WJ}
 \to
 \prod_{(J)\in\Phi H} \Z\pi_1(y^J,x(eH))^{W_H J}
 \]
be the homomorphism given by
 \[
 e(\eta)_{(J)} = \eta_J(eH)
 \]
where $eH$ denotes the element in $\pi_1(x^J,x(eH))$ given by the element
$eH\in G/H$ and the constant path at $x(eH)$. Note that $\eta_J(eH)$
is well-defined, independent of the choice of $J$ in its $H$-conjugacy class,
but that it does vary in the $G$-conjugacy class. That is, $e$ is well-defined,
but it need not be the case that $e(\eta)_{(J)} = e(\eta)_{(J')}$ if $J$ and
$J'$ are conjugate in $G$ but not in $H$. Now define
 \[
 d = e\circ\delta\colon \stab\Pi_G B(x,y)\to
  \prod_{(J)\in\Phi H} \Z\pi_1(y^J,x(eH))^{W_H J}
 \]
and
 \[
 d_\gamma = e\circ\delta_\gamma\colon \stab\Pi_\gamma B(x,y) \to
  \prod_{(J)\in\Phi H} \Z\pi_1(y^J,x(eH))^{W_H J}.
 \]
\end{definition}

We shall show that both $d$ and $d_\gamma$ are injective, from which it follows
that $\delta$ and $\delta_\gamma$ are injective. The remaining arguments are
obstruction theory arguments, and we first need a result generalizing
\cite[II.4.9]{tD:transfgroups}.

\begin{lemma}
Suppose that $W$ is a finite group, $S$ is a finite $W$-set, and
$\epsilon\colon W\to \{\pm 1\}$ is a homomorphism. Write $\Z_\epsilon$ for $\Z$
with $W$ acting through $\epsilon$, $\Z[S]$ for the $W$-module generated by
$S$, and $\Z_\epsilon[S]$ for the $W$-module given by the action of $W$ on $S$
and the action of $W$ on $\Z$ through $\epsilon$. Suppose that $X$ is a
$W$-complex and that $A\subset X$ is a subcomplex, such that
 \begin{enumerate}
 \item $\dim X = n > 0$ and $\dim A \leq n - 2$;
 \item $W$ acts freely on $X-A$; and
 \item $H_n(X;\Z) = \Z_\epsilon$.
 \end{enumerate}
Then $H^n(X;\Z_\epsilon[S]) \iso \Z[S]$, and the composite
 \[
 H^n_W(X,A;\Z_\epsilon[S]) \to H^n(X,A;\Z_\epsilon[S])
  \to H^n(X;\Z_\epsilon[S])
 \]
is injective, with image the set of multiples of
$T = \sum_{w\in W} w \in \Z[W]$.
 \end{lemma}

Here, $H^n_W(-;\Z_\epsilon[S])$ denotes Bredon cohomology with coefficients
$\Z_\epsilon[S]$ concentrated at $W/e$.

\begin{proof}
It suffices to show the case in which $W$ acts transitively on $S$.
That
 \[
 H^n(X;\Z_\epsilon[S]) \iso \Hom(H_n(X),\Z_\epsilon[S])
 \iso \Z[S]
 \]
is clear. Since
$\dim A \leq n-2$, it is also clear that we have the isomorphisms
 \[
 H^n(X,A;\Z_\epsilon[S]) \iso H^n(X;\Z_\epsilon[S])
 \]
and
 \[
 H^n_W(X,A;\Z_\epsilon[S]) \iso H^n_W(X;\Z_\epsilon[S]).
 \]
Now let $X^k$
denote the $k$-skeleton of $X$. We have the following exact sequences of
$W$-modules:
 \begin{multline*}
 0\from H^n(X;\Z_\epsilon[S]) \from \Hom(H_n(X^n,X^{n-1}),\Z_\epsilon[S])
\\
  \from \Hom(H_{n-1}(X^{n-1},X^{n-2}),\Z_\epsilon[S])
 \end{multline*}
and
 \begin{multline*}
 0\from H^n_W(X;\Z_\epsilon[S]) \from \Hom_W(H_n(X^n,X^{n-1}),\Z_\epsilon[S])
\\
  \from \Hom_W(H_{n-1}(X^{n-1},X^{n-2}),\Z_\epsilon[S]).
 \end{multline*}
Now if $F$ is any free $W$-module, there is a surjection
 \[
 \nu\colon \Hom(F,\Z_\epsilon[S])\to \Hom_W(F,\Z_\epsilon[S])
 \]
given by
$f\mapsto \sum_{w\in W} wfw^{-1}$. This induces a surjection
 \[
 \nu\colon H^n(X;\Z_\epsilon[S])\to H^n_W(X;\Z_\epsilon[S]).
 \]
On the other
hand, the inclusion of $\Hom_W$ in $\Hom$ induces the map
\[
 t\colon H^n_W(X;\Z_\epsilon[S])\to H^n(X;\Z_\epsilon[S])
\]
in which we are
interested, and $t\circ\nu$ is multiplication by $T$. Now observe that, since
$\nu$ is surjective, the trivial $W$-module $H^n_W(X;\Z_\epsilon[S])$ is
cyclic, so must be isomorphic to $\Z$ or $\Z/k$ for some $k$. On the other hand,
the image of $t$ is the set of multiples of $T$, so is isomorphic to $\Z$.
Therefore, $H^n_W(X;\Z_\epsilon[S])$ must also be isomorphic to $\Z$, and $t$
is injective with image the multiples of $T$.
 \end{proof}

Now we do the obstruction theory. If $V$ is a representation of $H$, let
$\Phi(V)$ be the set of conjugacy classes of isotropy groups of $V$ that are
also in $\Phi H$.

\begin{lemma}\label{lem:obstrtheory}
Let $V$ be a representation of $H$ such that
$|V^H| \geq 2$, and such that
$|V^L| \leq |V^K|-2$ whenever $K\subset L$ and $V^L\neq V^K$. Let $z\in B^H$ and
let $y\colon G/K\to B$.
 \begin{enumerate}
 \item Let $f\colon \Susp^V z_+\to \Susp^V y_+$ be a map in $H\Lax B$.
Let $T = \sum_{w\in W_H J} w \in \Z[W_H J]$ as above.
If $d'\equiv d(f)_J \mod (T)$ in the group $\Z\pi_1(y^J,z)^{W_H J}$, then there
exists another map $f'\colon \Susp^V z_+\to \Susp^V y_+$ such that
 \[
 f'|(S^V)^{>J} = f|(S^V)^{>J}
 \]
and $d(f')_J = d'$.
 \item Let $f$ and $f'\colon \Susp^V z_+\to \Susp^V y_+$ be maps in $H\Lax B$
such that their restrictions to $(S^V)^{>J}$ are homotopic in
$(W_H J)\Lax {B^{>J}}$. Then $d(f)_J \equiv d(f')_J \mod (T)$.
 \item Two maps $f, f'\colon \Susp^V z_+\to \Susp^V y_+$ in $H\Lax B$ are
homotopic if and only if $d(f)_J = d(f')_J$ for all $(J)\in\Phi(V)$.
 \end{enumerate}
 \end{lemma}

\begin{proof}
This is fairly standard obstruction theory, once we observe that we can replace
the target $y$ with the associated fibration $Ly$ and then look at $H$-maps from
$S^V$ into $\susp^V L(z,y)_+$. According to
obstruction theory, if $f$ and $f'$ are two $H$-maps from $(S^V)^{(J)}$ to
$\susp^V L(z,y)_+$ whose restrictions to $(S^V)^{>(J)}$ are homotopic, then the obstruction
to extending the homotopy to all of $(S^V)^{(J)}$ is an element of
 \[
 H^n_{WJ}((S^V)^J, (S^V)^{>J}; \pi_n((\Susp^V L(z,y)_+)^J))
 \]
where $n = |V^J|$. It
is not hard to see that the preceding lemma applies to allow us to consider this
obstruction as an element of
$H^n((S^V)^J; \pi_n((\Susp^V L(z,y)_+)^J))^{WJ} \iso \Z\pi_1(y^J,z)$,
where it is a multiple of $T$.
But this must be the nonequivariant obstruction to extending the homotopy, which
is $d(f)_J - d(f')_J$. The rest of the argument is standard.
 \end{proof}

\begin{lemma}\label{lem:gammaobstrtheory}
Let $V$ be a representation of $H$ such that
$|V^H| \geq 2$, and such that
$|V^L| \leq |V^K|-2$ whenever $K\subset L$ and $V^L\neq V^K$. Let $z\in B^H$ and
let $y\colon G/K\to B$. If $x\colon G/H\to B$ has $x(eH) = z$, write
$V(z) = \gamma(x)_z$.
 \begin{enumerate}
 \item Let
$f\colon \Susp^V\Susp^{V(z)} z_+\to \Susp^V\Susp^{\gamma(y)} y_+$
be a map in $H\Lax B$.
Let $T = \sum_{w\in W_H J} w \in \Z[W_H J]$ as above.
If $d'\equiv d_\gamma(f)_J \mod (T)$ in the group
$\Z\pi_1(y^J,z)^{W_H J}$, then there exists another map
$f'\colon \Susp^V\Susp^{V(z)} z_+\to \Susp^V\Susp^{\gamma(y)} y_+$ such that
 \[
 f'|(S^{V(z)}\smsh S^V)^{>J} = f|(S^{V(z)}\smsh S^V)^{>J}
 \]
and $d_\gamma(f')_J = d'$.
 \item Let $f$ and
$f'\colon \Susp^V\Susp^{V(z)} z_+\to \Susp^V\Susp^{\gamma(y)} y_+$
be maps in $H\Lax B$
such that their restrictions to $(S^{V(z)}\smsh S^V)^{>J}$ are homotopic in
$(W_H J)\Lax{B^{>J}}$. Then $d_\gamma(f)_J \equiv d_\gamma(f')_J \mod (T)$.
 \item Two maps
$f, f'\colon \Susp^V\Susp^{V(z)} z_+\to \Susp^V\Susp^{\gamma(y)} y_+$
in $H\Lax B$ are
homotopic if and only if $d_\gamma(f)_J = d_\gamma(f')_J$ for all
$(J)\in\Phi(V\oplus V(z))$.
 \end{enumerate}
 \end{lemma}

\begin{proof}
This is identical to the preceding lemma once we notice that $\gamma$ allows us to
identify the two $WJ$-modules
$\pi_n((\Susp^V\Susp^{V(z)} L(z,y)_+)^J)$ and
$\pi_n((\Susp^V L(z,\Susp^{\gamma(y)} y_+))^J)$,
where $n = |V(z)^J\oplus V^J|$. (This uses the same sort of argument as in the
construction of $\delta_\gamma$.)
 \end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:piXinvariance}]
Lemmas~\ref{lem:obstrtheory} and \ref{lem:gammaobstrtheory}
imply that $d$ and $d_\gamma$ are injective, hence so are
$\delta$ and $\delta_\gamma$. Lemma~\ref{lem:deltainclusion}
showed that the image of $\delta$ is
contained in the image of $\delta_\gamma$, and so we have an injective functor
$\alpha\colon\stab\Pi_G B\to \stab\Pi_\gamma B$. It remains to show that this
functor is surjective on morphisms. Fix objects $x\colon G/H\to B$
and $y\colon G/K\to B$, and suppose that
$f\colon \Susp^V S^{\gamma,x} \to \Susp^V S^{\gamma,y}$ represents a map
in $\stab\Pi_\gamma B(x,y)$. It is easy to find a map
$f_0\in\stab\Pi_G B(x,y)$ such that $d(f_0)_H = d_\gamma(f)_H$. By induction on the
orbit types in $\Phi(V\dirsum \gamma(x)_{eH})$ we can now modify $f_0$, using
Lemmas~\ref{lem:obstrtheory} and \ref{lem:gammaobstrtheory}
and the fact that $d = d_\gamma\circ\alpha$, to construct
a map $f'\in\stab\Pi_G B(x,y)$ such that $d(f')_J = d_\gamma(f)_J$ for all
$(J)\in \Phi(V\oplus \gamma(x)_{eH})$. It follows from
Lemma~\ref{lem:gammaobstrtheory} that
$\alpha(f') = f$. Therefore $\alpha$ is surjective on morphisms, and hence is
an isomorphism of categories. That the construction of $\alpha$ is natural is
clear from the definitions of $\delta$ and $\delta_\gamma$.
 \end{proof}

Homology-chap3.tex
\chapter{$G$-CW Parametrized Spaces and Spectra}
\label{chap:ordone}
%
\section*{Introduction}

As we've already mentioned, $RO(G)$-graded ordinary cohomology
is not adequate to give Poincar\'e duality for $G$-manifolds
except in very restrictive settings.
To get Poincar\'e duality we need to extend to a theory
indexed on representations of $\Pi X$.
(For simplicity of notation we shall now write
$\Pi X$ for $\Pi_G X$.)
That is, the homology and cohomology of $X$ should be graded
on representations of $\Pi X$.
A construction of the $RO(\Pi X)$-graded theory for finite $G$
was given in \cite{CW:duality}, and the theory was used in
\cite{CW:spivaknormal} and \cite{CW:simpleduality}
to obtain $\pi$-$\pi$ theorems for
equivariant Poincar\'e duality spaces and equivariant simple Poincar\'e duality spaces.
In this chapter we give a construction for all compact Lie groups.

The construction uses CW spaces or spectra in which the cells are
modeled on disks of varying representations, as specified by a stable
spherical representation of $\Pi X$.
It will be convenient for spaces---and essential for spectra---to have the
representation carried by a parametrizing space. Thus, we work in the categories
of spaces and spectra parametrized by a fixed basespace $B$ and
grade on $RO(\Pi B)$.

At the end of \cite{MaySig:parametrized}, May and Sigurdsson discuss the
difficulties inherent in trying to develop a good theory of CW objects in
the parametrized context. For parametrized spaces, the problem comes down to
the fact that CW spaces are not generally fibrant. The usual technique for producing
cellular fibrant approximations, based on the small object argument, does not
give CW spaces.
On the other hand, the Moore path fibration approximation does not preserve
cellular objects. It is not at all clear that there is a construction
that will give CW fibrant approximations for all parametrized spaces.
The result is that, for the CW spaces that arise from our constructions, we
cannot expect that maps between them can be approximated up to fiberwise
homotopy by CW maps. However, if we are willing to relax our requirements
a bit, we can approximate maps by {\em lax} maps
that preserve skeleta, which is quite
sufficient for our purposes here.

For CW spectra, there is an additional problem, which we can call the problem
of {\em stability.} This is essentially the fact that the fibrant parametrized
prespectra are the $\Omega$-prespectra, which are levelwise fibrant but also
have the property that $E(V)\to \Omega^W_B E(V\dirsum W)$ is a weak equivalence.
The problem arises in the following form: The stable homotopy groups of a fibrant
prespectrum $E$ can be computed using maps of prespectra
$G/H_+\smsh S^n \to E_b$.
For a general prespectrum, however, even one that is levelwise fibrant,
we need to use a colimit of the form
$\colim_W \pi H\Para *(\Sigma^\infty_W S^{n+W}, E(W)_b)$.
Thus, maps out of a single cell $G/H_+\smsh S^n$ do not detect stable homotopy.
May and Sigurdsson point out that, in the nonparametrized case, this
problem is addressed by using Lewis-May spectra \cite{LMS:eqhomotopy}.
They sketch a theory of Lewis-May type parametrized spectra, which would address
the problem of stability, but it's not at all clear in that context how to address
the fibrancy problem that still exists.
Again, it will suffice for our purposes to get around these problems by
working with lax maps, but also working with
certain pro-systems of CW spectra, rather than single spectra, when approximating
maps.


\section{$G$-CW($\gamma$) complexes}

Fix a compactly generated $G$-space $B$ of the homotopy type of a $G$-CW complex,
and a stable spherical representation $\gamma$ of $\Pi B$.
In this section we describe a theory of CW complexes in $G\ParaU B$,
the cells being locally modeled on $\gamma$.
At the end of the section we describe a similar theory for $G\Para B$.
We begin by describing the cells we use.

\begin{definition}
A {\em $\gamma$-cell} is an object $e = (G\times_H D(V),p)$ in $G\ParaU B$ such that
$\gamma(G\times_H 0,p)$ is stably spherically isomorphic to $G\times_H (V+n)$
for some possibly negative integer $n$.
The {\em dimension} of $e$ is $|V|$.
The {\em boundary} of $e$ is $\bndry e = (G\times_H S(V),p)$.
\end{definition}

Because $G\ParaU B$ has pushouts, we can now speak of attaching a cell $e$ to a space $X$ over $B$, given an attaching map $\bndry e\to X$ over $B$.

\begin{definition}
A {\em $G$-CW($\gamma$) complex} is a filtered object
\[
 (X,p) = \colim (X^n,p^n)
\]
in $G\ParaU B$, where $X^0$ is a union of 0-dimensional $\gamma$-cells, and each
$(X^n,p^n)$, $n>0$, is obtained from $(X^{n-1},p^{n-1})$ by attaching
$n$-dimensional $\gamma$-cells.
If $(X,A,p)$ is a pair of spaces over $B$, then we define a
{\em relative $G$-CW($\gamma$) structure} on $(X,A)$ in the obvious way.
We shall also write $X^{\gamma+n}$ for $X^{|\gamma|+n}$.
If we allow cells of any dimension to be attached at each stage,
we get the weaker notion of a {\em $\gamma$-cell complex.}
\end{definition}

\begin{examples}\label{ex:gammacomplexes}
 \begin{enumerate}\item[]

\item If $V$ is a representation of $G$ and
$X$ is a $G$-CW($V$) complex, then, considering $X$ as a space over itself,
$X$ has the structure of a $G$-CW($\Vrep$) complex.

\item If $M$ is any smooth $G$-manifold, let $\tau$ be the tangent
representation of $\Pi M$.
If $G$ is finite then the dual of a smooth $G$-triangulation
(as in Example~\ref{ex:vcomplexes}) gives an explicit $G$-CW($\tau$) structure on
$M$, again considering $M$ as a space over itself.
If $G$ is infinite then the dual gives a {\em dual} $G$-CW($\tau$) structure
of the sort we shall discuss later.

\item If $p\colon E \to B$ is a $G$-vector bundle, let $\rho$ be
the associated representation of $\Pi B$ as in \ref{prop:bundlereps}.
Let $S^p$ be the fiberwise one-point compactification of the bundle,
with basepoint section $\sigma\colon B\to S^p$, so $(S^p,p,\sigma)$ is an
ex-$G$-space.
If $B$ is a $G$-CW complex,
then $(S^p,\sigma(B))$ has an evident structure as a relative
$G$-CW($\rho$) complex. We use $(S^p,\sigma(B))$ as a convenient
model for the Thom space of $p$ because, unlike the Thom space, it
comes with a natural parametrization.
 \end{enumerate}
 \end{examples}


In order to accomplish our next goal---that of proving cellular
approximation and Whitehead theorems---we need
appropriate homotopy groups.

\begin{definition}\label{def:ehomotopy}
 Let $f\colon X\laxto Y$ be a lax map and let
$e = (G \times _H D(V),p)$ be a cell. We define $\pi_e (f)$ to
be the set of homotopy classes of commutative diagrams of lax maps
of the following form:
 \[
 \xymatrix{
  {\bndry e} \ar[r] \ar[d] & X \ar[d]^f \\
  e \ar[r] & Y.
 }
 \]
 This is functorial in $f$ in the evident way. We say that
$\pi_e(f)$ is {\em trivial} if it coincides with
$\Image (\pi_e(1_X) \to \pi_e(f))$,
i.e., if every diagram is homotopic to one in which we can find
a lift $e\laxto X$.
 \end{definition}

Note that, because $\bndry e\to e$ is a lax cofibration
by Theorem~\ref{thm:laxcofib}, we can say that
$\pi_e(f)$ is trivial if every diagram as above is
homotopic {\em rel $\bndry e$} to one in which we can find a lift.

\begin{definition}
 A lax map $f\colon X\laxto Y$ is a
{\em $(\gamma + n)$-equivalence} if $\pi_e(f)$ is trivial for all
$\gamma$-cells $e$ of dimension
$\leq |\gamma| + n$; $f$ is a {\em weak $\gamma$-equivalence} if it is a
$(\gamma + n)$-equivalence for all $n$.
 \end{definition}

We insert the following reassuring result.
In its statement, we write {\em $\gamma(b)$-equivalence} for $V$-equivalence,
where $\gamma(b)$ is stably equivalent to $G\times_{G_b} S^V$.

\begin{theorem}\label{thm:gammaequivalence}
Let $f\colon X\laxto Y$ be a lax map.
The following are equivalent:
\begin{enumerate}
\item
$f$ is a weak $\gamma$-equivalence.
\item
For each $b\in B$, $f_b\colon LX_b\to LY_b$ is a
weak $\gamma(b)$-equivalence.
\item
The underlying map of spaces $f$ is a weak $G$-equivalence.
\end{enumerate}
\end{theorem}

\begin{proof}
A $\gamma$-cell $e$ with $p(0) = b$ is lax homotopy equivalent
to the cell $e'$ of the same dimension with $p(e') = b$.
It follows that $f$ is a weak $\gamma$-equivalence if and only if
each map of fibers $f_b\colon LX_b\to LY_b$ is a
weak $\gamma(b)$-equivalence.

Theorem~\ref{thm:vequivalence} tells us that $f_b$ is a weak
$\gamma(b)$-equivalence if and only if $f_b$ is a weak $G_b$-equivalence.
Comparison of the long exact sequences of the fibrations
$LX\to B$ and $LY\to B$ shows that each $f_b$ is
a weak equivalence if and only if $f\colon X\to Y$ is a weak $G$-equivalence
of $G$-spaces.
\end{proof}

We have the following variant of the ``homotopy extension and
lifting property''.

\begin{lemma}[H.E.L.P.]\label{lem:HELPParaSpace}
 Let $r\colon Y \laxto Z$ be a $(\gamma+n)$-equivalence.
Let $(X,A)$ be a relative
%$G$-CW($\gamma$) complex
$\gamma$-cell complex
with cells of dimension
$\leq |\gamma|+n$. If the following diagram commutes in $G\LaxU B$ without the
dashed arrows, then there exist lax maps $\tilde g$ and $\tilde h$
making the diagram commute.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\times I \ar'[d][dd] \ar[dl]_(.6)h & &
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g \ar[dd]  \\
   & Z & & Y \ar[ll]_(.3)r \\
  X \ar[ur]^f \ar[rr]_-{i_0} & & X\times I \ar@{-->}[ul]^(.6){\tilde h} & &
   X \ar[ll]^-{i_1} \ar@{-->}[ul]^(.6){\tilde g} \\
 }
 \]
The result remains true when $n = \infty$.
 \end{lemma}

\begin{proof}
We can use the same proof as that given for
Lemma~\ref{lem:HELP}, understanding all maps to be in the lax category.
\end{proof}

To state the Whitehead theorem, we need the following notations.
Write $\pi(G\LaxU B)(X,Y)$ for the set of lax homotopy classes
of maps $X\laxto Y$. If $A\subset X$ and $f\colon A\laxto Y$,
write $\pi(G\LaxU B)(X,Y)/f$ for the set of lax extensions of $f$
modulo lax homotopy rel $A$.

\begin{theorem}[Whitehead]\label{thm:ParaWhitehead}
 \begin{enumerate}\item[]

\item If $r\colon Y\laxto Z$ is a $(\gamma+n)$-equivalence and $X$ is a
$(|\gamma|+n-1)$-dimensional
$\gamma$-cell complex,
then
\[
 r_*\colon \pi(G\LaxU B)(X,Y) \to \pi(G\LaxU B)(X,Z)
\]
is an isomorphism.
It is an epimorphism if $X$ is
$(|\gamma|+n)$-dimensional.

\item If $r\colon Y\laxto Z$ is a weak $\gamma$-equivalence and $X$ is a
$\gamma$-cell complex,
then
\[
 r_*\colon \pi(G\LaxU B)(X,Y) \to \pi(G\LaxU B)(X,Z)
\]
is an
isomorphism.  In particular any weak $\gamma$-equivalence of $G$-CW($\gamma$) complexes
is a lax homotopy equivalence.

\item If $r\colon Y\laxto Z$ is a $(\gamma+n)$-equivalence, $(X,A)$ is a
relative $(|\gamma|+n-1)$-dimensional
$\gamma$-cell complex,
and $f\colon A\laxto Y$ is a lax map,
then
\[
 r_*\colon \pi(G\LaxU B)(X,Y)/f \to \pi(G\LaxU B)(X,Z)/rf
\]
is an isomorphism.
It is an epimorphism if $(X,A)$ is
$(|\gamma|+n)$-dimensional.

\item If $r\colon Y\laxto Z$ is a weak $\gamma$-equivalence, $(X,A)$ is a
relative $\gamma$-cell complex,
and $f\colon A\laxto Y$ is a lax map,
then 
\[
 r_*\colon \pi(G\LaxU B)(X,Y)/f \to \pi(G\LaxU B)(X,Z)/rf
\]
is an
isomorphism.

\end{enumerate}
\end{theorem}

\begin{proof}
This follows from the H.E.L.P. lemma in exactly the same way as
in Theorem~\ref{thm:VWhitehead}.
 \end{proof}

\begin{definition}
A lax map $f\colon X\laxto Y$ of $G$-CW($\gamma$) complexes is {\em cellular} if
$f(X^n)\subset Y^n$ for each $n$.
Cellular lax maps of relative $G$-CW($\gamma$)
complexes are defined similarly.
 \end{definition}

\begin{theorem}[Cellular Approximation]\label{thm:cellapprox}
 If $f\colon X\laxto Y$ is a lax map of $G$-CW($\gamma$) complexes that is
cellular on a subcomplex $A\subset X$, then $f$
is lax homotopic rel $A$ to a cellular map.
A similar statement holds for relative $G$-CW($\gamma$) complexes.
 \end{theorem}

\begin{proof}
 This follows easily from the  H.E.L.P. lemma once we show that the inclusion
$Y^{\gamma+n}\to Y$ is a $(\gamma+n)$-equivalence.
This reduces to showing that, if 
$K\subset H\subset G$, $V$ is a representation of $K$,
$(M,\bndry M)$ is a $K$-manifold contained in $D(V)$,
and $V'$ is a representation of $H$ stably spherically equivalent to $V$
as a representation of $K$, then there are no essential $K$-maps
$(M,\bndry M)\to G\times_H(D(V'+k),S(V'+k))$ for $k>0$.
This follows from ordinary obstruction theory, using a $K$-CW decomposition of $M$,
the point being that $V$ and $V'$ have the same fixed-point dimensions.
\end{proof}

\begin{theorem}[Approximation by $G$-CW($\gamma$) complexes]\label{thm:ghomotopytype}
 Let $(X,A,p)$ be any pair of objects in $G\ParaU B$. Then there exists a 
relative $G$-CW($\gamma$)
complex $(\Gamma X,A)$ and a weak equivalence $\Gamma X\to X$ under $A$ in $G\ParaU B$.
 \end{theorem}

\begin{proof}
As for $G$-CW($V$) complexes, we kill homotopy groups.
Start with $\Gamma_0 X$, a union of $A$ and 0-cells, and a map
$f_0\colon \Gamma_0 X\to X$ in $G\ParaU B$ with $\pi_e(f_0)$ trivial for
all 0-dimensional cells $e$.  Inductively, we assume that an
$(n-1)$-equivalence $f_{n-1}\colon (\Gamma_{n-1} X,q_{n-1})\to (X,p)$ has been
constructed (with $f_{n-1}$ a strict map).
Consider an $n$-dimensional $\gamma$-cell $e$
and a diagram of lax maps representing an element of
$\pi_e(f_{n-1})$:
 \[
 \xymatrix{
  {\bndry e} \ar[r]^-{(\alpha,\lambda)} \ar[d] & \Gamma_{n-1}X \ar[d]^{f_{n-1}} \\
  e \ar[r]^-{(\beta,\mu)} & X.
 }
 \]
Let $e'$ be the cell with the same underlying space as $e$ but mapping
to $B$ by the map $p\beta$, so $\bndry e$ maps to $B$ by $q_{n-1}\alpha$.
Replacing $e$ with $e'$ gives us the following diagram of strict maps:
 \[
 \xymatrix{
  {\bndry e'} \ar[r]^-{\alpha} \ar[d] & \Gamma_{n-1}X \ar[d]^{f_{n-1}} \\
  e' \ar[r]^-{\beta} & X.
 }
 \]
Use $\alpha$ to attach $e'$ to $\Gamma_{n-1}X$ and use $\beta$ to map the
result to $X$. It is easy to see that this kills the original element of
$\pi_e(f_{n-1})$.
Attach cells in this way to trivialize $\pi_e(f_{n-1})$ for arbitrary $n$-dimensional cells
$e$. We then take $\Gamma X = \colim_n \Gamma_n X$.
 \end{proof}

Note that, by Whitehead's theorem, the map $\Gamma X\to X$ is unique
up to canonical lax homotopy equivalence.  In addition, if
$f\colon X\laxto Y$ is a lax map, then there is a lax map
$\Gamma f\colon \Gamma X\laxto \Gamma Y$, unique up to lax homotopy,
such that the following
diagram commutes up to lax homotopy:
 \[
 \xymatrix{
  {\Gamma X} \ar[r]^{\Gamma f} \ar[d] & {\Gamma Y} \ar[d] \\
  X \ar[r]_{f} \ar[r] & Y
  }
 \]

Turning to $G\Para B$, we make the following definition.

\begin{definition}
An {\em ex-$G$-CW($\gamma$) complex},
or a $G$-CW($\gamma$) complex in $G\Para B$, is an
ex-$G$-space $(X,p,\sigma)$ together with a relative
$G$-CW($\gamma$) structure on $(X,\sigma(B),p)$
as a pair in $G\ParaU B$.
{\em Ex-$\gamma$-cell complexes} are defined similarly.
\end{definition}

Equivalently, we can think of an ex-$G$-CW($\gamma$) complex
as built out of $\gamma$-cells of the form
$e_+ = (G\times_H D(V), p)_+$.
The following are special cases of relative versions of results
of this section.

\begin{theorem}[Whitehead]
 \begin{enumerate}\item[]

\item If $r\colon Y\laxto Z$ is a $(\gamma+n)$-equivalence
of ex-$G$-spaces and $X$ is a
$(|\gamma|+n-1)$-dimensional
ex-$\gamma$-cell complex,
then
$r_*\colon \pi G\Lax B(X,Y) \to \pi G\Lax B(X,Z)$
is an isomorphism.
It is an epimorphism if $X$ is
$(|\gamma|+n)$-dimensional.

\item If $r\colon Y\laxto Z$ is a weak $\gamma$-equivalence
of ex-$G$-spaces and $X$ is an
ex-$\gamma$-cell complex,
then $r_*\colon \pi G\Lax B(X,Y) \to \pi G\Lax B(X,Z)$ is an
isomorphism.  In particular any weak $\gamma$-equivalence of ex-$G$-CW($\gamma$) complexes
is a lax homotopy equivalence.
\qed

\end{enumerate}
\end{theorem}

\begin{theorem}[Cellular Approximation]
 If $f\colon X\laxto Y$ is a lax map of ex-$G$-CW($\gamma$) complexes that is
cellular on a subcomplex $A\subset X$, then $f$
is lax homotopic rel $A$ to a cellular map.
\qed
\end{theorem}

\begin{theorem}[Approximation by $G$-CW($\gamma$) complexes]
 Let $X$ be any object in $G\Para B$. Then there exists an 
ex-$G$-CW($\gamma$)
complex $\Gamma X$ and a weak equivalence $\Gamma X\to X$ in $G\Para B$.
\qed
\end{theorem}



%%%%%%%%%%%%%%%
\section{$G$-CW($\gamma$) prespectra}


We continue to work with a fixed space $B$
but we now allow $\gamma$ to be a virtual spherical representation of $\Pi B$.
We begin by describing the cells we use to build $G$-CW prespectra,
which will be shift desuspensions of certain based cells.
As in the preceding section, we need to allow our cells to map to $B$ by arbitrary maps.
We will also need our collection of cells to be closed under fiberwise suspension.
This leads us to the following definition.

\begin{definition}\label{def:prespeccell}
Let $V$ be a representation of a subgroup $H$ of $G$. Write $D^{V+1}$ for the
unit disc of $V\dirsum\Real$, write $S^V_+$ for the upper hemisphere of its
unit sphere, and write $*$ for its north pole.
Suppose given a $G$-map $f\colon G\times_H D^{V+1}\to B$.
Write $D^{V+1}_f$ for the pushout in the following diagram.
\[
 \xymatrix{
  G\times_H S^V_+ \ar[r]^-f \ar[d] & B \ar[d] \\
  G\times_H D^{V+1} \ar[r] & D^{V+1}_f
 }
\]
The map $f$ makes $D^{V+1}_f$ into an ex-$G$-space in the obvious way.
Write $S^V_f\subset D^{V+1}_f$ for the ex-$G$-space given by
the unit sphere.
A {\em $\gamma$-cell} is then a $G$-prespectrum of the form
$\susp^\infty_W D^{V+1}_f$ where $\gamma(f(G\times_H *))$
is virtually spherically isomorphic to $G\times_H(V-W+n)$
for some possibly negative integer $n$.
The {\em dimension} of such a cell is $|V|-|W|+1$, and its
{\em boundary} is $\susp^\infty_W S^{V}_f$.
We will also use the term {\em $(\gamma+n)$-cell} to refer to a
$\gamma$-cell of dimension $|\gamma|+n$.
\end{definition}

\begin{definition}
A {\em $\gamma$-cell prespectrum} is a filtered $G$-prespectrum
\[
 X = \colim_{n\geq 0} X_n
\]
over $B$, where $X_0=*$ and, for $n>0$, each $X_n$ is obtained from
$X_{n-1}$ by attaching a wedge of $\gamma$-cells along their boundaries.
We call $\{X_n\}$ the {\em sequential filtration} of $X$.
A $\gamma$-cell prespectrum $X$ is a {\em $G$-CW($\gamma$) prespectrum}
if each cell of dimension $n$ is attached only to cells of dimension smaller than $n$;
for such a prespectrum we let the {\em skeleton} $X^n$ be the subprespectrum consisting of all cells of $X$ of dimension no more than $n$, and call $\{X^n\}$ the {\em skeletal filtration.}
We also write $X^{\gamma+n}$ for $X^{|\gamma|+n}$.
If $(X,A)$ is a pair of $G$-prespectra, we define a {\em relative}
$\gamma$-cell or $G$-CW($\gamma$) structure
on $(X,A)$ in the obvious way, by taking $X_0 = A$.
\end{definition}

When looking at the filtration quotients $X^{\gamma+n}\ParaQuot B X^{\gamma+n-1}$,
we will need the following result.

\begin{proposition}\label{prop:cellquotient}
If $e = \susp^\infty_W D^{V+1}_f$ is a $\gamma$-cell, then
$e\ParaQuot B \bndry e$ is the boundary sphere
of a $\gamma$-cell of one dimension higher.
\end{proposition}

\begin{proof}
Take $D^{V+1}\times D^1$ as a model of $D^{V+2}$, with
$S^{V+1}_+$ corresponding to
$S^V\times D^1 \union D^{V+1}\times 1$.
Let $g\colon D^{V+1}\times D^1\to B$ be the
projection to $D^{V+1}$ composed with $f$.
Then it is straightforward to check that
$D^{V+1}_f\ParaQuot B S^V_f \homeo S^{V+1}_g$.
\end{proof}

We mentioned that our cells are closed under fiberwise suspension.
Specifically, we have the following.

\begin{proposition}
If $e = \susp^\infty_W D^{V+1}_f$ is a $\gamma$-cell, then so
is $\susp^\infty_{Z+W}\Omega^{Z+W}_B e$ for any $Z$.
\end{proposition}

\begin{proof}
First notice that
$\susp^\infty_{Z+W}\Omega^{Z+W}_B e =
 \susp^\infty_{Z+W}\susp^Z_B D^{V+1}_f$,
so it suffices to show that
$\susp^Z_B D^{V+1}_f$ is homeomorphic to $D^{Z+V+1}_g$ for some $g$.
For this purpose, we take as our model of suspension the homeomorphism
\[
 \susp^Z K \homeo D^Z\times K/[\bndry D^Z\times K \union D^Z\times *].
\]
We also use the homeomorphism $D^{V+1} \homeo D^V\times D^1$, with
$S^V_+ \homeo \bndry D^V\times D^1\union D^V\times 1$.
We can then see
that $\susp^Z_B D^{V+1}_f \homeo B\union D^Z\times D^V\times D^1/\sim$,
where $(z,v,t) \sim f(v,t)$ if either $z\in\bndry D^Z$, $v\in\bndry D^V$,
or $t = 1$. In other words, $\susp^Z_B D^{V+1}_f \homeo D^{Z+V+1}_g$
where $g\colon D^Z\times D^V\times D^1\to B$ is the composite of $f$ with the
projection to $D^V\times D^1$.
\end{proof}

The cells $e$ and $\susp^\infty_{Z+W}\Omega^{Z+W}_B e$ are related by
the following result.

\begin{proposition}\label{prop:costabequiv}
If $K$ is any ex-$G$-space, then the natural map
\[
 \epsilon\colon \susp^\infty_{Z+W}\Omega^{Z+W}_B\susp^\infty_W K
 \to \susp^\infty_W K
\]
 is a weak equivalence.
Further, if $X$ is any $G$-prespectrum over $B$, $\epsilon^*$
induces the suspension map
$G\Para B(K, X(W))\to G\Para B(\susp^Z_B K, X(Z+W))$
up to natural isomorphism.
\end{proposition}

\begin{proof}
That $\epsilon$ is a weak equivalence follows from the fact that
it is a levelwise identity map when evaluated at large enough representations.
The second statement follows by chasing adjunction isomorphisms, as in the
following diagram:
\[
 \xymatrix{
  G\PreSpec{} B(\susp^\infty_W K, X) \ar[r]^-\iso \ar[d]_{\epsilon^*}
   & G\Para B(K, X(W)) \ar[d] \\
  G\PreSpec{} B(\susp^\infty_{Z+W}\Omega^{Z+W}_B\susp^\infty_W K,X)
   \ar[r]^-\iso & G\Para B(\susp^Z_B K, X(Z+W))
  }
\]
\end{proof}

Because of the preceding result, we call
$\susp^\infty_{Z+W}\Omega^{Z+W}_B e\to e$
a {\em costabilization of $e$ of degree $Z$}
when $e = \susp^\infty_W D^{V+1}_f$.
In general, we form a costabilization of a $\gamma$-cell
prespectrum by replacing each cell by one of its costabilizations.
Precisely, we make the following definition.

\begin{definition}
Let $f\colon X\to Y$ be a map of $\gamma$-cell prespectra.
We say that $f$ is a {\em costabilization of $Y$}
if $f$ respects the sequential filtrations, and
$X$ and $f$ are given inductively as follows:
For $n>0$, if $Y_n$ is obtained from $Y_{n-1}$ by attaching the wedge
of cells $\Wedge_\alpha e_\alpha$, then $X_n$ is obtained from $X_{n-1}$
by attaching a corresponding wedge of cells $\Wedge_\alpha e'_\alpha$,
and there are costabilization maps $e'_\alpha\to e_\alpha$ (of possibly
varying degrees) such that the following diagram commutes,
in which the front and back faces are the pushout diagrams defining
$X_n$ and $Y_n$, respectively:
\[
 \xymatrix@=1em{
  & {\Wedge \bndry e_\alpha} \ar[rr] \ar'[d][dd]
    && Y_{n-1} \ar[dd] \\
  {\Wedge\bndry e'_\alpha} \ar[ur] \ar[dd] \ar[rr]
    && X_{n-1} \ar[ur]_(.4){f_{n-1}} \ar[dd] \\
  & {\Wedge e_\alpha} \ar'[r][rr]
    && Y_n \\
  {\Wedge e'_\alpha} \ar[ur] \ar[rr]
    && X_n \ar[ur]_{f_n}
 }
\]
We define a {\em relative} costabilization $(X,A)\to (Y,A)$ similarly.
By abuse of language, we will also refer to $X$ as a costabilization of $Y$.
\end{definition}

A costabilization of a prespectrum is very closedly related to it.
Here is one weak aspect of that relationship.


\begin{proposition}
If $f\colon X\to Y$ is a costabilization of the $G$-CW($\gamma$)
prespectrum $Y$, then $f$ is a weak equivalence.
\end{proposition}

\begin{proof}
If $Y$ is a finite $G$-CW($\gamma$) prespectrum,
then $f$ is a levelwise identity map when evaluated at large enough representations,
hence is a weak equivalence.
This implies the general
case, because the homotopy groups of $Y$ are the colimits
of the homotopy groups of its finite subspectra.
\end{proof}

We now define the appropriate homotopy groups for our purpose
(cf. \cite[\S12.3]{MaySig:parametrized}).

\begin{definition}
Let $X$ be a $G$-prespectrum over $B$, let $H\subset G$,
and let $b\in B^H$.
Suppose that $\gamma(b)$ is virtually spherically equivalent
to $G\times_H (V-W)$.
We define
\[
 \pi^H_{\gamma(b)+i}(X_b)
  = \colim_Z \pi^H_{Z+V+i}(X_b(Z+W))
  = \colim_Z \pi H\Para *(S^{Z+V+i},X_b(Z+W)),
\]
the stable homotopy group of the homotopy fiber $X_b$ of $X$ over $b$.
\end{definition}

Note that, for the homotopy fiber, we can use the fiber $LX_b$, so we
can also write
\begin{align*}
 \pi^H_{\gamma(b)+i}(X_b)
 & = \colim_Z \pi G\Lax B(S^{Z+V+i,b},X(Z+W)) \\
 & \iso \colim_Z \pi G\LaxPreSpec{}B (\susp^\infty_{Z+W} S^{Z+V+i,b}, X),
\end{align*}
where $S^{Z+V+i,b} = b_! S^{Z+V+i}$, thinking of $b\colon G/H\to B$.
(More precisely, we should take the colimit over a sequence $Z_k$
such that $\{Z_k+W\}$ is an indexing sequence for the prespectrum $LX$, as in \S\ref{sec:paraspec}.)

\begin{definition}
Let $f\colon X\laxto Y$ be a lax map. We say that $f$ is a {\em $(\gamma+n)$-equivalence}
if, for all $b$ and $H$, $f_*\colon \pi^H_{\gamma(b)+i}(X_b) \to \pi^H_{\gamma(b)+i}(Y_b)$ is an isomorphism for $i < n$ and an epimorphism for $i=n$.
We say that $f$ is a {\em weak $\gamma$-equivalence} if it is a $(\gamma+n)$-equivalence
for all $n$.
\end{definition}

As expected, weak $\gamma$-equivalence is the same as the usual weak $G$-equivalence:

\begin{theorem}\label{thm:weakgammaequivalence}
A lax map $f\colon X\laxto Y$ of $G$-prespectra over $B$ is a weak $\gamma$-equivalence
if and only if it is a weak $G$-equivalence.
\end{theorem}

\begin{proof}
Suppose $\gamma(b) = G\times_H S^{V-W}$. Then, to say $f$ is a weak equivalence
is to say that $f_b\colon \susp^W LX_b\to \susp^W LY_b$ is a weak
$V$-equivalence for all $b$, which is equivalent to saying that
$\loops^V\susp^W LX_b\to \loops^V\susp^W LY_b$ is a weak equivalence for all $b$.
On the other hand, $\loops^V$ and $\susp^W$ induce equivalences of the stable
category, so $\loops^V\susp^W LX_b\to \loops^V\susp^W LY_b$ is a weak
equivalence if and only if $LX_b\to LY_b$ is.
As defined in \cite[\S12.3]{MaySig:parametrized}, a weak $G$-equivalence is a
(stable) weak equivalence on all (homotopy) fibers over $B$. 
The result follows.
\end{proof}

Here is the appropriate variant of the homotopy extension and lifting property.

\begin{lemma}[H.E.L.P.]\label{lem:HELPParaSpec}
 Let $r\colon Y \laxto Z$ be a $(\gamma+n)$-equivalence of $G$-prespectra over $B$.
Let $(X,A)$ be a relative $\gamma$-cell prespectrum with cells of dimension
$\leq |\gamma|+n$.
Suppose that the following diagram commutes, with the maps to $Y$ and $Z$ being lax maps.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\smsh_B I_+ \ar[dl]_(.6)h & &
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g   \\
   & Z & & Y \ar[ll]_r \\
  X \ar[ur]^f  \\
 }
 \]
Then there exists a relative costabilization $(X',A)\to (X,A)$ and
lax maps $\tilde g$ and $\tilde h$
making the following diagram commute, where $f'$ is the composite
$X'\to X\laxto Z$.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\smsh_B I_+ \ar'[d][dd] \ar[dl]_(.6)h & &
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g \ar[dd]  \\
   & Z & & Y \ar[ll]_(.3)r \\
  X' \ar[ur]^{f'} \ar[rr]_-{i_0} & & X'\smsh_B I_+ \ar@{-->}[ul]^(.6){\tilde h} & &
   X' \ar[ll]^-{i_1} \ar@{-->}[ul]^(.6){\tilde g} \\
 }
 \]
The result remains true when $n = \infty$.
\end{lemma}

\begin{proof}
We begin with the special case of a cell
\[
 (X,A) = (\susp^\infty_W D^{V+i}_\phi, \susp^\infty_W S^{V+i-1}_\phi),
\]
where $\phi\colon G\times_H D^{V+i}\to B$
and $\gamma(\phi(G\times_H *))$ is virtually spherically isomorphic to
$G\times_H(V-W)$. For simplicity, restrict from $G$ to $H$,
write $\phi$ again for the $H$-map
obtained by restricting $\phi$ to $D^{V+i}$, and write $b = \phi(*)$.
Restrict $f$ similarly.
Using $\phi$ to define necessary homotopies, it is not hard to see that
the pair
$(D^{V+i}_\phi,S^{V+i-1}_\phi)$ is lax homotopy equivalent to
$(D^{V+i,b},S^{V+i-1,b})$ and, further, that we have
\begin{align*}
 \colim_U{}& \pi H\Lax B((\susp^U_B D^{V+i}_\phi,\susp^U_B S^{V+i-1}_\phi),
         (Z(U+W),Y(U+W)))
\\
  &\iso \colim_U \pi H\Lax B((D^{U+V+i,b},S^{U+V+i-1,b}),
         (Z(U+W),Y(U+W)))
\\
  &= \pi^H_{\gamma(b)+i}(Z_b,Y_b).
\end{align*}
Now, by our assumptions, $\pi^H_{\gamma(b)+i}(Z_b,Y_b) = 0$.
Hence, for sufficiently large $U$,
we can apply the argument in the proofs of Lemma~\ref{lem:HELP}
and Lemma~\ref{lem:HELPParaSpace},
with appropriate modifications,
to $(X',A') = (\susp^U_B D^{V+i}_\phi,\susp^U_B S^{V+i-1}_\phi)$
to find the needed $\tilde g$ and $\tilde h$.

For a general relative $G$-CW($\gamma$) prespectrum $(X,A)$,
we proceed by induction on the sequential filtration, constructing
$X'$, $\tilde g$, and $\tilde h$ all at the same time.
Assume that we have constructed $X'_k$ and defined the maps
$\tilde g$ and $\tilde h$ on this costabilization of $X_k$.
Consider one of the cells $e$ attached to $X_k$ to obtain $X_{k+1}$.
The attachment map $\bndry e\to X_k$ may not factor through
$X'_k$, but it does factor through some finite subcomplex of $X_k$.
Each cell in that finite subcomplex has been costabilized to some degree
in $X'_k$.
By taking a costabilization of $e$ of degree at least as large as each
of these degrees, we obtain a cell $e'$ for which the map
$\bndry e'\to X_k$ does factor through $X'_k$.
By the first part of this proof, there is a further costabilization
$e''$ over which we can extend $\tilde g$ and $\tilde h$.
Attaching this new cell to $X'_k$, and doing similarly for all the
cells at this stage, we obtain $X'_{k+1}$, a costabilization of $X_{k+1}$
over which we can extend $\tilde g$ and $\tilde h$.
By induction, we can construct $X'$ as desired.
\end{proof}

In order to state our version of the Whitehead theorem, we need the
following definition.
First, notice that, by a construction similar to that in the proof of H.E.L.P.,
if $X'\to X$ and $X''\to X$ are two costabilizations of $X$, then there
exists a common costabilization $X'''\to X'$ and $X'''\to X''$.

\begin{definition}
If $X$ is a $G$-CW($\gamma$) prespectrum, the
collection of costabilizations of $X$ forms a pro-system
we write as ${}_*X = \{{}_\alpha X\}$.
We call this the {\em costabilization system} of $X$.
\end{definition}

In order for a costabilization system to constitute a set,
we must restrict costabilizations of cells to degrees in some universe,
and we do so tacitly from now on.

\begin{theorem}[Whitehead I]\label{thm:SpecWhiteheadI}
Let $X$ be a $\gamma$-cell prespectrum with costabilization system
${}_*X$,
let $Y$ and $Z$ be $G$-prespectra over $B$, and let
$f\colon Y\to Z$ be a lax map.

\begin{enumerate}

\item If $f$ is a $(\gamma+n)$-equivalence and $X$ is
$(\gamma+n-1)$-dimensional, then
\[
 f_*\colon \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Y) \to
  \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Z)
\]
is an isomorphism.
It is an epimorphism if $X$ is
$(\gamma+n)$-dimensional.

\item If $f$ is a weak $\gamma$-equivalence
and $X$ is an arbitrary $\gamma$-cell prespectrum, then
\[
 f_*\colon \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Y) \to
  \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Z)
\]
is an
isomorphism.

 \end{enumerate}
 \end{theorem}

\begin{proof}
This follows from the H.E.L.P.\ lemma in essentially the same way as for
Theorems~\ref{thm:VWhitehead} and~\ref{thm:ParaWhitehead}.
\end{proof}

Although weak equivalences of $\gamma$-cell complexes need not be
(lax) homotopy equivalences, we do have the following.

\begin{definition}
Let $\pro\pi G\LaxPreSpec{}B$ denote the pro-category of
$\pi G\LaxPreSpec{}B$.
Thus, if $X$ and $Y$ are $\gamma$-cell complexes with costabilization systems
${}_*X$ and ${}_*Y$, then
\[
 \pro\pi G\LaxPreSpec{}B( {}_*X, {}_*Y )
  = \lim_\beta \colim_\alpha \pi G\LaxPreSpec{}B ( {}_\alpha X, {}_\beta Y ).
\]
We say that $X$ and $Y$ are {\em pro-homotopy equivalent}
if ${}_*X$ and ${}_*Y$ are isomorphic in $\pro\pi G\LaxPreSpec{}B$.
We say that $f\colon X\to Y$ is a pro-homotopy equivalence if
it induces an isomorphism of ${}_*X$ and ${}_*Y$ in $\pro\pi G\LaxPreSpec{}B$.
\end{definition}

Note the following important simplification: Since
$Y$ is terminal in ${}_*Y$ and every ${}_\alpha Y$ maps
to $Y$ by a weak equivalence, by Theorem~\ref{thm:SpecWhiteheadI} we have
\[
 \lim_\beta \colim_\alpha \pi G\LaxPreSpec{}B ( {}_\alpha X, {}_\beta Y )
  \iso \colim_\alpha \pi G\LaxPreSpec{}B ( {}_\alpha X, Y ).
\]
It is in this way that a lax map $f\colon X\laxto Y$ induces a pro-map
$\hat f\colon {}_*X \to {}_*Y$.

\begin{theorem}[Whitehead II]
If $f\colon X\to Y$ is a weak $\gamma$-equivalence of $\gamma$-cell complexes,
then $f$ is a pro-homotopy equivalence.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:SpecWhiteheadI}, $f$ induces an isomorphism
\[
 f_*\colon \colim_\beta \pi G\LaxPreSpec{}B ( {}_\beta Y, X )
  \xrightarrow{\iso}
  \colim_\beta \pi G\LaxPreSpec{}B ( {}_\beta Y, Y )
\]
Let $\hat g\in \colim_\beta \pi G\LaxPreSpec{}B ( {}_\beta Y, X )$
be the element such that $f_*(\hat g) = 1$.
Then it is easy to check that $\hat f\circ \hat g$ is the identity
in $\pro\pi G\LaxPreSpec{}B ( {}_*Y, {}_*Y )$.

Let $g\colon {}_\beta Y\to X$ be a representative of $\hat g$.
Because $f$ and all of the maps in ${}_* Y$ are weak equivalences,
so is $g$. Just as above, we can therefore find a right inverse
$\hat h$ to $\hat g$. By the usual argument, $\hat h = \hat f$
(in $\pro\pi G\LaxPreSpec{}B ( {}_*X, {}_*Y ))$ and $\hat g$ is
a two-sided inverse to $\hat f$.
\end{proof}

\begin{theorem}[Cellular Approximation of Maps]\label{thm:mapapprox}
If $f\colon X\laxto Y$ is a lax map of $G$-CW($\gamma$) prespectra
that is cellular on a subcomplex $A\subset X$, then
there exists an $\alpha$ such that the composite
${}_\alpha X\to X\laxto Y$ is homotopic rel $A$ to a cellular map.
\end{theorem}

\begin{proof}
This follows from the H.E.L.P. lemma
(applied to each cell in an induction on the sequential filtration
of $(X,A)$) once we know that the inclusion
$Y^{\gamma+n}\to Y$ is a $(\gamma+n)$-equivalence.
But, in fact, $Y^{\gamma+n}(W)\to Y(W)$ is a $(W+\gamma+n)$-equivalence,
by an argument similar to that given for
Theorem~\ref{thm:cellapprox}.
\end{proof}

\begin{theorem}[Approximation by $G$-CW($\gamma$) Prespectra]\label{thm:specapprox}
Let $X$ be any $G$-prespectrum over $B$. Then there exists
a $G$-CW($\gamma$) prespectrum $\Gamma X$ and a weak equivalence
$\Gamma X\to X$ in $G\PreSpec{}B$.
\end{theorem}

\begin{proof}
Let $\Gamma_0 X$ be the trivial prespectrum over $B$ and
$f_0\colon \Gamma_0 X\to X$ the unique (strict) map.
We repeatedly kill relative homotopy groups: Suppose $n>0$ and
that $f_{n-1}\colon\Gamma_{n-1}\to X$ has been constructed
with $\Gamma_{n-1}$ being a $G$-CW($\gamma$) prespectrum.
For each subgroup $H$ and each $b\in B^H$, choose representatives
of the elements of the relative stable homotopy groups
$\pi^H_{\gamma(b)+i}(\Gamma_{n-1}X\to X)$. We can represent an element
by a diagram of lax maps of the following form:
\[
 \xymatrix{
  S^{Z+V+i-1,b} \ar[d]\ar[r]^-\phi & \Gamma_{n-1}X(Z+W) \ar[d] \\
  D^{Z+V+i,b} \ar[r]_-{\bar\phi} & X(Z+W)
 }
\]
By the preceding result, for sufficiently large $Z$ we may assume
that $\phi$ lands in the $(\gamma+i-1)$-skeleton of $\Gamma_{n-1}X$.
Now, $\bar\phi$ may be only a lax map. However, if we let
$\psi$ be the composite of $\bar\phi$ with the projection
$X(Z+W)\to B$, we can replace the diagram above with the following
diagram of strict maps:
\[
 \xymatrix{
  S^{Z+V+i-1}_\psi \ar[d]\ar[r] & \Gamma_{n-1}X(Z+W) \ar[d] \\
  D^{Z+V+i}_\psi \ar[r] & X(Z+W)
 }
\]
(Here we think of the whole upper hemisphere $S_+^{Z+V+i-1}$
as mapping to the single point $b$.)
Taking adjoints, we get the following strict diagram of prespectra:
\[
 \xymatrix{
  {\susp^\infty_{Z+W}S^{Z+V+i-1}_\psi} \ar[d]\ar[r] & \Gamma_{n-1}X \ar[d] \\
  {\susp^\infty_{Z+W}D^{Z+V+i}_\psi} \ar[r] & X \\
 }
\]
Now we use these diagrams to attach cells to $\Gamma_{n-1}X$ to
get the $G$-CW($\gamma$) prespectrum $\Gamma_n X$ and the map
$f_n\colon \Gamma_n X \to X$.

We take $\Gamma X = \colim_n \Gamma_n X$ with the induced map
$f\colon \Gamma X\to X$.
Any element of the relative stable homotopy group
$\pi^H_{\gamma(b)+i}(\Gamma X\to X)$ comes from some
$\pi^H_{\gamma(b)+i}(\Gamma_n X\to X)$, but, by construction, the
map
\[
 \pi^H_{\gamma(b)+i}(\Gamma_n X\to X) \to
\pi^H_{\gamma(b)+i}(\Gamma_{n+1} X\to X)
\]
is 0. It follows that
$\pi^H_{\gamma(b)+i}(\Gamma X\to X) = 0$, hence that
$f$ is a weak $\gamma$-equivalence.
\end{proof}

We end this section with a summary. Let $\CWLaxPS G \gamma B$ denote
the full subcategory of $\pro\pi G\LaxPreSpec{}B$ with objects the
costabilization systems associated with $G$-CW($\gamma$)
prespectra. Recall that $[X,Y]_{G,B} = \Ho G\PreSpec{}{B}(X,Y)$.
The following results have either already been proven or follow
in standard ways from what we have already shown.

\begin{theorem}\label{thm:cwspecsummary}
\begin{enumerate}\item[]
\item
If $X$ and $Y$ are $G$-CW($\gamma$) prespectra over $B$, then
\begin{align*}
 [X,Y]_{G,B} &\iso \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Y) \\
  &\iso \pro\pi G\LaxPreSpec{}B({}_*X, {}_*Y) \\
  &= \CWLaxPS G \gamma B({}_*X, {}_*Y).
\end{align*}

\item
If $f\colon X\laxto Y$ is a lax map of $G$-CW($\gamma$) prespectra over $B$, then
there exist an $\alpha$ such that the composite ${}_\alpha X\to X\laxto Y$
is (lax) homotopic to a cellular map. Moreover, any two such cellular
approximations are (after further costabilization) cellularly homotopic.

\item
If $X$ is an arbitrary $G$-prespectrum over $B$, there exists a
$G$-CW($\gamma$) prespectrum $\Gamma X$ and a (strict) weak equivalence
$\Gamma X\to X$. A choice of such a cellular approximation for each
prespectrum gives a well-defined equivalence of categories
\[
 \Ho G\PreSpec{}B \xrightarrow{\iso} \CWLaxPS G \gamma B.
\]
An inverse equivalence is given by taking a representative map
${}_\alpha Y\to Z$ in $\colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha Y,Z)$
to the composite $Y\to {}_\alpha Y\to Z$, where the first map is the inverse
of the weak equivalence ${}_\alpha Y\to Y$.
\qed
\end{enumerate}
\end{theorem}

%%%%%%%%%%%%%%%

\section{CW objects and the $qf$-model structure}

We would like to compute certain
total left derived functors on homotopy categories
using $G$-CW($\gamma$) objects where we would usually use cofibrant
approximations. The problem we run into is that $G$-CW($\gamma$)
objects are not usually cofibrant in the $qf$-model structures
defined in \cite{MaySig:parametrized}.
We shall show that $G$-CW($\gamma$) objects are lax homotopy equivalent
to cofibrant objects, and that this is usually good enough.

We consider $G$-CW($\gamma$) complexes in $G\ParaU{}B$ first.
Recall that May and Sigurdsson define two model structures on $G\ParaU{}B$,
the $q$-model structure and the better-behaved $qf$-model structure, with
the same weak equivalences.
The cofibrant objects in the $q$-model structure
are retracts of $q$-cell complexes, where
$q$-cells have the form
$(G/H\times D^n,G/H\times S^{n-1},p)$ for arbitrary $p$. In other
words, $q$-cell complexes are our 0-cell complexes.
On the other hand, the cofibrant objects of the $qf$-model structure,
the $qf$-cofibrant objects, are retracts of $qf$-cell complexes,
where the $qf$-cells include those cells
$(G/H\times D^n,G/H\times S^{n-1},p)$ for which
the inclusion
$(G/H\times S^{n-1},p)\to (G/H\times D^n,p)$ is a cofibration over $B$,
meaning that it has the homotopy extension property
in the category of strict maps over $B$
\cite[7.2.6]{MaySig:parametrized}.

\begin{proposition}
Let $(X,A,p)$ be a relative $\gamma$-cell complex. Then
$p\hmtpc \bar p$ rel $A$ where $(X, A, \bar p)$ is a relative
$qf$-cell complex.
Hence $(X,A,p)$ is lax homotopy equivalent rel $A$ to
$(X,A,\bar p)$.
\end{proposition}

\begin{proof}
We first point out that a $\gamma$-cell of the form
$(G\times_H D(V),G\times_H S(V))$ can be triangulated
\cite{Ill:triangulation}, allowing us to give $(X,A,p)$
the structure of a relative $q$-cell complex.

Now let $(X,p) = \colim_n (X_n,p_n)$, where $X_0$ is the disjoint union
of $A$ and some orbits, and $X_{n+1}$ is obtained from $X_n$ by attaching
$q$-cells. We construct $\bar p = \colim_n \bar p_n$
and a homotopy $p\hmtpc \bar p$ inductively,
beginning with $\bar p_0 = p_0$. Suppose then that
$\bar p_n$ has been defined on $X_n$ with
$p_n \hmtpc \bar p_n$ rel $A$
and $(X_n,A,\bar p_n)$ a relative $qf$-cell complex.
Consider a typical cell $(e,\bndry e,q)$ attached to
$X_n$ to form $X_{n+1}$, with attaching map
$\chi\colon\bndry e\to X_n$. We have a homotopy
$p_n\chi\hmtpc \bar p_n\chi$, which we can extend to a homotopy
$q\hmtpc q'$ with $q'|\bndry e = \bar p_n\chi$.
Now define $\bar q\colon e\to B$ by the formula
\[
 \bar q(v) =
 \begin{cases}
  q'(2v) & \text{if $\|v\| \leq 1/2$} \\
  q'(v/\|v\|) & \text{if $\|v\| \geq 1/2$}
 \end{cases}
\]
There is an obvious homotopy $q'\hmtpc \bar q$ rel $\bndry e$, and
$(e,\bndry e,\bar q)$ is a $qf$-cell.
With a little more work we can combine homotopies to get a homotopy
$q\hmtpc \bar q$ which agrees with the homotopy $p_n\chi\hmtpc \bar p_n\chi$
on $\bndry e$.
Define $\bar p_{n+1} = \bar q$ on this cell, and do similarly
with all cells. Then $p_{n+1} \hmtpc \bar p_{n+1}$, by a homotopy
extending the already constructed homotopy $p_n\hmtpc\bar p_n$, and
$(X_{n+1},A,\bar p_{n+1})$ is a relative $qf$-cell complex.

The final claim of the proposition is the observation that
$p\hmtpc \bar p$ rel $A$ implies that $(X,A,p)$ is lax homotopy equivalent
rel $A$ to $(X,A,\bar p)$.
\end{proof}

The argument for $G$-CW($\gamma$) prespectra is similar in concept,
but a little more delicate in the details.
In the $qf$-model structure for $G$-prespectra, cofibrant spectra are again retracts
of $qf$-cell prespectra. The $qf$-cells include those of the form
$\susp^\infty_W(G/H\times D^{n},G/H\times S^{n-1},p)_+$
where $(G/H\times D^{n},G/H\times S^{n-1},p)$ is a $qf$-cell as above
\cite[12.4.3 \& 12.1.7]{MaySig:parametrized}.

\begin{proposition}
Let $X$ be a $\gamma$-cell prespectrum. Then $X$ is lax homotopy equivalent
to a $qf$-cell prespectrum.
\end{proposition}

\begin{proof}
Recall from the preceding section
that the cells of $X$ have the form $\susp^\infty_W D^{V+1}_f$.
Using \cite{Ill:triangulation} once again, we may triangulate $D^{V+1}$ and so
give $X$ the structure of a $q$-cell prespectrum, one in which the cells have the form
$\susp^\infty_W (G/H\times D^n,p)_+$ where
$(G/H\times D^n,G/H\times S^{n-1},p)$ is not necessarily
a $qf$-cell.

Now let $\{X_n\}$ be the sequential filtration of $X$, so $X_{n+1}$ is obtained from
$X_n$ by attaching a wedge of $q$-cells $\Wedge e_\alpha$.
We will inductively construct a $qf$-cell prespectrum $\bar X = \colim_n\bar X_n$
and a compatible sequence of maps $X_n\laxto \bar X_n$, each of which is a lax homotopy equivalence.
To start, we take $\bar X_0 = X_0$. Suppose then that
a lax homotopy equivalence $X_n\laxto \bar X_n$
has been constructed and consider a typical cell
$e_\alpha = \susp^\infty_W (G/H\times D^{k},q)_+$ that will be attached
to form $X_{n+1}$.
The attaching map of $e$ gives a map $\chi\colon G/H\times S^{k-1} \to X_n(W)$.
Let $(f,\lambda)\colon X_n(W)\to \bar X_n(W)$ be the already constructed lax map at level $W$.
Similar to the proof of the preceding proposition, we can find a map
$\bar q\colon G/H\times D^{k}\to B$ such that $\bar q|G/H\times S^{k-1} = pf\chi$,
$q\hmtpc \bar q$ by a homotopy agreeing with $\lambda$,
and $(G/H\times D^{k},\bar q)$ is a $qf$-cell.
Let $\bar e_\alpha = \susp^\infty_W (G/H\times D^{k},\bar q)_+$,
define an attaching map $G/H\times S^{k-1}\to \bar X_n(W)$ by $f\chi$,
and do similarly for each cell at this stage.
We then have the following diagram in which the horizontal maps
are strict and the vertical maps lax:
\[
 \xymatrix{
  {\Wedge e_\alpha} \ar[d] & {\Wedge\bndry e_\alpha} \ar[l] \ar[r] \ar[d] & X_n \ar[d] \\
  {\Wedge \bar e_\alpha} & {\Wedge\bndry \bar e_\alpha} \ar[l] \ar[r] & \bar X_n
 }
\]
%\[
% \xymatrix@=1em{
%  & {\Wedge \bndry\bar e_\alpha} \ar[rr] \ar'[d][dd]
%    && {\bar X_{n}} \ar[dd] \\
%  {\Wedge\bndry e_\alpha} \ar@{=}[ur] \ar[dd] \ar[rr]
%    && X_{n} \ar[ur] \ar[dd] \\
%  & {\Wedge\bar e_\alpha} \ar'[r][rr]
%    && {\bar X_{n+1}} \\
%  {\Wedge e_\alpha} \ar[ur] \ar[rr]
%    && X_{n+1} \ar[ur]
% }
%\]
Let $\bar X_{n+1}$ be the pushout of the bottom row.
By Proposition~\ref{prop:hominvpushouts},
the induced map $X_{n+1}\to \bar X_{n+1}$ is a lax homotopy equivalence.
By Proposition~\ref{prop:hominvcolimits}, the resulting map $X\to \bar X$ is also a lax homotopy equivalence.
\end{proof}

Here is how we shall make use of these results. Let $\C$ be one
of $G\ParaU B$, $G\Para B$, or $G\PreSpec{}B$, with its $qf$-model structure,
and let $\D$ be any category.
Let $F\colon\C\to\D$ be a functor that takes weak equivalences
between $qf$-cofibrant objects to isomorphisms. Then we form
$\Lder F\colon \Ho\C\to \D$, the left derived functor of $F$,
in the usual way: For each object $X$ of $\C$ we choose a
$qf$-cofibrant $QX$ with an acyclic
fibration $QX\to X$, making $Q$ functorial up to homotopy
and $QX\to X$ natural.
On objects, then, $(\Lder F)(X) = FQX$.

\begin{proposition}
Let $\C$ and $F$ be as above. Suppose, in addition, that
$F$ takes lax homotopy equivalences to isomorphisms.
If $X$ is a $G$-CW($\gamma$) object in $\C$,
then the map $QX\to X$ induces a natural isomorphism
$(\Lder F)X \iso FX$.
\end{proposition}

\begin{proof}
By the appropriate one of the two preceding propositions,
there is a lax homotopy equivalence $X'\laxto X$,
with $X'$ $qf$-cofibrant.
This gives us a strict map $X'\to LX$
that is a lax homotopy equivalence
(Lemma~\ref{lem:speclaximpliesweak}).
Consider the following diagram of weak equivalences:
\[
 \xymatrix{
  QX \ar[d] \ar[r] & QLX \ar[d] \\
  X \ar[r] & LX & X' \ar[l] \ar[ul]
 }
\]
The map $X'\to QLX$ is a lift up the acyclic fibration $QLX\to LX$.
The maps $QX\to QLX$ and $X'\to QLX$ are weak equivalences between
cofibrant objects, hence are taken to isomorphisms by $F$.
The maps $X\to LX$ and $X'\to LX$ are lax homotopy equivalences,
so are also taken to isomorphisms by $F$.
It follows that $F$ takes the map $QX\to X$ to an isomorphism.
\end{proof}

What follows are the most important examples we need, beginning
with a relatively trivial one that shows the pattern of the applications.
In these examples we will prefix left derived functors with $\Lder$ for clarity,
but subsequently we will follow common usage and drop the prefix.

\begin{example}
Consider $(-)_+\colon G\ParaU B\to G\Para B$, the functor that
adds a disjoint section. It is Quillen left adjoint to the forgetful functor.
Further, there is a map of monads
\[
 ((-)_+, \psi)\colon (G\ParaU B, L) \to (G\Para B, L),
\]
where $\psi\colon (LK)_+\to L(K_+)$ is the inclusion.
(Note that the only difference between $(LK)_+$ and $L(K_+)$ is
that the latter includes points of the form $(b,\lambda)$ with
$b$ in the adjoined disjoint section and $l_\lambda > 0$.)
Thus, $(-)_+$ extends to a continuous functor $G\LaxU B\to G\Lax B$.
We conclude that the composite $G\ParaU B\to G\Para B\to \Ho G\Para B$,
because it factors through $G\LaxU B\to G\Lax B$,
takes both weak equivalences between cofibrant spaces and lax homotopy
equivalences to isomorphisms. Hence,
$\Lder (-)_+\colon \Ho G\ParaU B\to \Ho G\Para B$ takes any $G$-CW($\gamma$) complex $X$
to an object $\Lder (X)_+ \iso X_+$.
(Note: This example really is trivial, because $(-)_+$ preserves
all weak equivalences.)
\end{example}

\begin{example}
Consider $\susp^V_B\colon G\Para B\to G\Para B$.
It is left Quillen adjoint to $\loops^V_B$ by
\cite[7.3.1]{MaySig:parametrized}.
We have the map of monads $(\susp^V_B, \beta^V)$, which allows
us to extend $\susp^V_B$ to a continuous functor
$\susp^V_B\colon G\Lax B\to G\Lax B$.
It follows that the derived functor
$\Lder\susp^V_B\colon\Ho G\Para B\to \Ho G\Para B$
satisfies $\Lder\susp^V_B X \iso \susp^V_B X$ for any
ex-$G$-CW($\gamma$) complex $X$.
Note that $\susp^V_B X$ has the structure of an
ex-$G$-CW($\gamma+V$) complex, with cells in one-to-one correspondence
with those of $X$. The point is that this correspondence survives
after passage to homotopy categories.
\end{example}

\begin{example}\label{ex:suspaprespec}
Consider $\susp^V_B\colon G\PreSpec{}B\to G\PreSpec{}B$.
It is left Quillen adjoint to $\loops^V_B$ by
\cite[12.6.4]{MaySig:parametrized};
in fact, the adjunction is a Quillen equivalence.
Computation shows that $\beta^V$ defines
a map of $G$-prespectra $\susp^V_B LX \to L\susp^V_B X$
and that $(\susp^V_B,\beta^V)$ is a map of monads,
so $\susp^V_B$ preserves lax homotopy equivalence.
If follows that the derived functor
$\Lder\susp^V_B\colon\Ho G\PreSpec{}B\to \Ho G\PreSpec{}B$
satisfies $\Lder\susp^V_B X \iso \susp^V_B X$ for any
$G$-CW($\gamma$) prespectrum $X$.
Note again that $\susp^V_B X$ has the structure of
a $G$-CW($\gamma+V$) prespectrum.
\end{example}

\begin{example}
Consider $\susp^\infty_B \colon G\Para B\to G\PreSpec{}B$.
This is Quillen left adjoint to $\loops^\infty_B$
by \cite[12.6.2]{MaySig:parametrized}.
We have the map of monads $(\susp^\infty_B, \beta^\infty_0)$,
so $\susp^\infty_B$ preserves lax homotopy equivalence.
It follows that the derived functor
$\Lder\susp^\infty_B\colon \Ho G\Para B\to \Ho G\PreSpec{}B$
satisfies $\Lder\susp^\infty_B X \iso \susp^\infty_B X$
for any ex-$G$-CW($\gamma$) complex $X$.
This suspension prespectrum is not quite
a $G$-CW($\gamma$) prespectrum; we shall return to this issue
after the next example.
\end{example}

\begin{example}
If $Z$ is one of the indexing representations used in
forming $G\LaxPreSpec{}B$, the discussion in the preceding
example generalizes directly to show that
$\Lder\susp^\infty_Z X \iso \susp^\infty_Z X$
for any ex-$G$-CW($\gamma$) complex $X$.
In general, as in Remark~\ref{rem:indexedSuspension},
let $\bar Z$ be the smallest indexing representation such that
$Z\subset \bar Z$ and write
$\susp^\infty_Z = \susp^\infty_{\bar Z}\susp^{\bar Z-Z}_B$.
We conclude that
$\Lder\susp^\infty_Z X \iso 
\Lder\susp^\infty_{\bar Z}\Lder\susp^{\bar Z-Z}_B X
\iso \susp^\infty_{\bar Z}\susp^{\bar Z-Z}_B X
\iso \susp^\infty_Z X$,
because $\susp^{\bar Z-Z}_B$ takes
ex-$G$-CW($\gamma$) complexes to ex-$G$-CW($\gamma+\bar Z-Z$) complexes.
\end{example}

Now let's deal with the problem that, if $X$ is an ex-$G$-CW($\gamma$)
complex, then $\susp^\infty_B X$ doesn't quite have the structure of
a $G$-CW($\gamma$) prespectrum. The problem is the difference between
the unbased cells used for spaces and the based cells used for prespectra.
However, we have the following result:

\begin{proposition}\label{prop:spacesToSpec}
If $X$ is an ex-$G$-CW($\gamma$) complex, then
$\susp^\infty_1 \susp_B X$ is a $G$-CW($\gamma$) prespectrum
whose cells are in one-to-one correspondence with those of $X$.
\end{proposition}

\begin{proof}
Consider the attachment of a typical $\gamma$-cell
$(G\times_H D(V),p)_+$ to an ex-$G$-space $Y$ over $B$, as in the following
pushout diagram:
\[
 \xymatrix{
  G\times_H S(V)_+ \ar[r]^-\xi \ar[d] & Y \ar[d] \\
  G\times_H D(V)_+ \ar[r] & Z
 }
\]
Suspending, we get the following diagram,
also a pushout:
\[
 \xymatrix{
  {\susp_B (G\times_H S(V))_+} \ar[r]^-{\susp \xi} \ar[d]
     & {\susp_B Y} \ar[d] \\
  {\susp_B (G\times_H D(V))_+} \ar[r] & {\susp_B Z}
 }
\]
Now consider $D^{V+1} \homeo D(V)\times D^1$ with
$S^V_+$ corresponding to $D(V)\times 1$. Let
$f\colon D^{V+1}\to B$ be given by the composite
of the projection $D(V)\times D^1\to D(V)$ and
$p\colon D(V)\to B$.
Writing $S^V = S(V)\times D^1 \union D(V)\times S^0$,
let $\hat\xi\colon S^V_f\to \susp_B Y$ be defined by
\[
 \hat\xi(v,t) =
  \begin{cases}
   (\xi(v),t) & \text{if $\|v\| = 1$,} \\
   \sigma p(v) & \text{if $t = \pm 1$.}
  \end{cases}
\]
(Here, we think of $\susp_B Y$ as a quotient of
$Y\times D^1$.)
Then we also have the following pushout diagram:
\[
 \xymatrix{
  S^V_f \ar[r]^-{\hat\xi} \ar[d] & \susp_B Y \ar[d] \\
  D^{V+1}_f \ar[r] & \susp_B Z
 }
\]
Replacing each cell of $\susp_B X$ in this way makes
$\susp^\infty_1 \susp_B X$ a $G$-CW($\gamma$) prespectrum.
Moreover, it is clear from the construction that the natural
map $\susp^\infty_1 \susp_B X\to \susp^\infty_B X$ preserves
filtrations and induces a one-to-one correspondence between
the cells of $\susp^\infty_1\susp_B X$ and those of $X$.
\end{proof}

Note that, by Proposition~\ref{prop:costabequiv}, the natural map
$\susp^\infty_1 \susp_B X\to \susp^\infty_B X$ is a weak equivalence.
It follows that $\Lder \susp^\infty_B X \iso \susp^\infty_1\susp_B X$
in $\Ho G\PreSpec{} B$.
In this way we can give $\Lder\susp^\infty_B X$ a cell structure
corresponding to that of $X$.

\begin{example}
Let $\alpha\colon A\to B$ be a map of base spaces.
Recall that we have a functor
$\alpha_!\colon G\Para A \to G\Para B$
given by composition with $\alpha$ and identification of base points.
It is Quillen left adjoint to the functor $\alpha^*$ given by
taking pullbacks.
If $(X,p,\sigma)$ is an ex-$G$-space, write
$\alpha_!(X,p,\sigma) = (\alpha_!X,q,\tau)$,
as in the following diagram, in which the top square is a pushout:
\[
 \xymatrix{
  A \ar[d]_\sigma \ar[r]^\alpha & B \ar[d]^\tau \\
  X \ar[d]_p \ar[r] & \alpha_! X \ar[d]^q \\
  A \ar[r]_\alpha & B
 }
\]
Let us write $[x]$ for the image in $\alpha_! X$ of $x\in X$,
and $[b]$ for $\tau(b)$. Then $[\sigma(a)] = [\alpha(a)]$ for
$a\in A$.
Define $\psi\colon \alpha_!(LX) \to L(\alpha_!X)$ by the formulas
\begin{align*}
 \psi[x,\lambda] &= ([x],\alpha\lambda) \quad\text{and} \\
 \psi[b] &= ([b],b).
\end{align*}
That $\psi$ is well-defined means that the formulas agree on $A$:
\[
 \psi[\sigma(a),a] = ([\sigma(a)],\alpha(a)) = ([\alpha(a)],\alpha(a))
 = \psi[\alpha(a)].
\]
It's now straightforward to check that $(\alpha_!,\psi)$ is a map of
monads, so $\alpha_!$ preserves lax homotopy equivalence.
It follows that the derived functor
$\Lder\alpha_!\colon \Ho G\Para A\to \Ho G\Para B$ satisfies
$\Lder\alpha_!X \iso \alpha_!X$ for any ex-$G$-CW($\gamma$) complex $X$.
It is also easy to see that, if $\gamma = \alpha^*(\delta)$
for some representation $\delta$ of $\Pi B$, then
$\alpha_!X$ is an ex-$G$-CW($\delta$) complex.

The same statements are true (and simpler) for 
$\Lder\alpha_!\colon \Ho G\ParaU A \to \Ho G\ParaU B$,
where $\alpha_!(X,p) = (X,\alpha p)$.
\end{example}

\begin{example}\label{ex:basechangeprespec}
Let $\alpha\colon A\to B$ again be a map of base spaces,
and consider $\alpha_!\colon G\PreSpec{}A\to G\PreSpec{}B$.
Since $\alpha_!$ is defined levelwise, it is easy to check
that the transformation $\psi$ from the preceding example
again gives a map of monads $(\alpha_!,\psi)$,
hence that $\alpha_!$ preserves lax homotopy equivalencs.
It follows that the derived functor
$\Lder\alpha_!\colon \Ho G\PreSpec{}A\to \Ho G\PreSpec{}B$
satisfies $\Lder\alpha_! X\iso \alpha_! X$ for any
$G$-CW($\gamma$) prespectrum $X$.
If $\gamma = \alpha^*(\delta)$, then $\alpha_! X$ is
a $G$-CW($\delta$) prespectrum.
\end{example}

\begin{example}
Let $\iota\colon H\to G$ be the inclusion of a subgroup,
let $A$ be an $H$-space,
and consider the equivalence $\iota_!\colon H\Para A\to G\Para{G\times_H A}$
given by $\iota_!X = G\times_H X$ (see \cite[7.4.7]{MaySig:parametrized}).
This equivalence preserves the monad $L$, meaning that there
is a natural isomorphism $\iota_! LX \iso L\iota_!X$
that is a map of monads.
Therefore, $\iota_!$ preserves lax homotopy equivalences
and the derived functor $\Lder\iota_!$ satisfies
$\Lder\iota_!X\iso \iota_!X$ for any $H$-CW($\gamma$) complex $X$.
\end{example}

\begin{example}
Let $\iota\colon H\to G$ be the inclusion of a subgroup,
let $A$ be an $H$-space,
and consider the equivalence $\iota_!\colon H\PreSpec{} A\to G\PreSpec{}{G\times_H A}$
of \cite[12.6.10]{MaySig:parametrized}), given by applying the functor $\iota_!$
of the preceding example levelwise.
Again, $\iota_!$ preserves $L$, so preserves lax homotopy equivalences,
hence $\Lder\iota_!X\iso \iota_!X$ for any $H$-CW($\gamma$) prespectrum $X$.
\end{example}

\begin{example}
Let $H$ be a normal subgroup of $G$ and consider the fixed-set
functor $(-)^H\colon G\Para B\to (G/H)\Para{B^H}$.
As pointed out in the proof of \cite[7.4.3]{MaySig:parametrized},
$(-)^H$ preserves weak equivalences, so passes immediately to
homotopy categories.
However, it is useful to note also that
there is a natural isomorphism $(LX)^H \iso L(X^H)$
which is a map of monads, hence $(-)^H$ also extends to a functor
on lax maps.
\end{example}

\begin{example}
Let $H$ be a normal subgroup of $G$ and consider the
functor $\Phi^H\colon G\PreSpec{}B\to (G/H)\PreSpec{}{B^H}$
given by $\Phi^H(X) = (\tilde E\F[H]\smsh_B X)^H$
(see Definition~\ref{def:geometricfixed}; we will discuss
this functor in the parametrized case further in the next chapter).
As in \cite[14.4.2]{MaySig:parametrized}, $(-)^H$ preserves
weak equivalences; as in the preceding example, it
also preserves lax homotopy equivalences. 
To deal with smashing with $\tilde E\F[H]$,
we need to generalize the natural transformation $\beta^V$.
For any unbased $G$-space $K$, consider the based space $\tilde K$,
the cofiber of the map $K_+\to S^0$, or the unbased suspension.
We write $\tilde K = K\times I/\sim$ where $(a,0)\sim(b,0)$ and $(a,1)\sim(b,1)$
for every $a$ and $b\in K$. (We have in mind $K = E\F[H]$,
but this also applies to $K = S(V)$, where $\tilde K = S^V$.)
For $X$ an ex-$G$-space, define a map
$\beta^K\colon (LX)\smsh_B \tilde K \to L(X\smsh_B\tilde K)$
by
\[
 \beta^K((x,\lambda)\smsh(a,t)) =
  \begin{cases}
   (x\smsh(a,t), \lambda) & \text{if $t\leq 1/2$} \\
   (x\smsh(a,\phi(\phi^{-1}(t)-l)), \lambda)
     & \text{if $1/2 < t \leq \phi(l)$} \\
   (\sigma\lambda(l-\phi^{-1}(t)), \lambda|_{l-\phi^{-1}(t)}^l)
     & \text{if $t\leq \phi(l)$}.
  \end{cases}
\]
Here, $\phi$ is as in the definition of $\beta^V$ and $l = l_\lambda$.
Calculation identical to that for $\beta^V$ shows that
$(-\smsh_B\tilde K, \beta^K)$
is a map of monoids on $G\Para B$.
As in Example~\ref{ex:suspaprespec}, we can now check that
$(-\smsh_B\tilde K,\beta^K)$ defines a map of monoids
on $G\PreSpec{}B$ as well, so $-\smsh_B\tilde K$ preserves
lax homotopy equivalence of $G$-prespectra.
Hence, the total left derived functor of $-\smsh_B\tilde E\F[H]$
agrees with $-\smsh_B\tilde E\F[H]$ on $G$-CW($\gamma$) prespectra.
Putting this together with the fact that $(-)^H$ also preserves
lax homotopy equivalences, we get that
$\Phi^H(X)$ can be computed on $G$-CW($\gamma$) prespectra
as expected.
\end{example}


%%%%%%%%%%%%%%%

\section{Ordinary homology and cohomology}\label{sec:ordinarytheories}

We shall now construct the cellular chains of $G$-CW($\gamma$) spaces and prespectra,
and we begin with spaces.
As usual, we work with a fixed basespace $B$ and representation $\gamma$
of $\Pi B$, which we assume to be a stable spherical representation when dealing with
spaces, and a virtual spherical representation when dealing with prespectra.
The following
is a specialization of Definition~\ref{def:indres}.

\begin{definition}
A {\em (contravariant) $\stab\Pi B$-Mackey functor}
is a contravariant additive functor $\Mackey T\colon \stab\Pi B\to \Ab$.
A {\em covariant $\stab\Pi B$-Mackey functor} is a covariant additive functor
$\MackeyOp S\colon \stab\Pi B\to \Ab$.
\end{definition}

If $(X,A)$ is a pair of $G$-spaces over $B$,
we write $X\ParaQuot B A$ for the ex-$G$-space
$X_+\ParaQuot B A_+$
over $B$ obtained by taking the fiberwise quotient.

\begin{lemma}\label{lem:skeletalquotient}
Let $(X,A,p)$ be a relative $G$-CW($\gamma$) complex. Then
 \[
 X^{\gamma+n}\ParaQuot B X^{\gamma+n-1} \hmtpc \Wedge_x S^{\gamma+n,px},
 \]
where the wedge (over $B$) runs through the centers $x$ of
the $(\gamma+n)$-cells of $X$
and the equivalence is lax homotopy equivalence.
(Here we think of $x\colon G/H\to X$ and $px$ denotes the composite
$px\colon G/H\to B$.)
 \end{lemma}

\begin{proof}
$X^{\gamma+n}\ParaQuot B X^{\gamma+n-1}$ consists of a copy of $B$,
the image of the disjoint
section, with a copy of each $(\gamma+n)$-cell of $X$ adjoined via
the composite of its attaching map and the projection to $B$.
The attaching map for a cell, as a map into $B$, may be deformed to be the
constant map at the image of the center of the cell, and then the
assertion is clear.
 \end{proof}


\begin{definition}\label{def:gammachains}
Let $(X,A)$ be a relative $G$-CW($\gamma$) complex.
Let $\Mackey C_{\gamma+*}(X,A)$ (which we also
write as $\Mackey C_{\gamma+*}^G(X,A)$ when we need to emphasize the group involved)
be the chain complex of contravariant $\stab\Pi B$-Mackey functors defined by
 \begin{align*}
 \Mackey C_{\gamma+n}(X,A)(b)
   &= \{ S^{\gamma+n,b}, X^{\gamma+n}\ParaQuot B X^{\gamma+n-1} \}_{G,B} \\
   &= [ \susp^\infty_B S^{\gamma+n,b},
           \susp^\infty_B (X^{\gamma+n}\ParaQuot B X^{\gamma+n-1}) ]_{G,B} \\
   &\iso \pi^H_{\gamma(b)+n}(\susp^\infty_B (X^{\gamma+n}\ParaQuot B X^{\gamma+n-1})_b)
 \end{align*}
if $b\colon G/H\to B$.
This is obviously a contravariant functor on $\stab\Pi_\gamma B$, and we use
the isomorphism of Theorem~\ref{thm:piXinvariance} to consider it as a functor on
$\stab\Pi B$. Let $\bndry\colon \Mackey C_{\gamma+n}(X,A)\to \Mackey C_{\gamma+n-1}(X,A)$ be
the natural transformation induced by the composite
 \[
 X^{\gamma+n}\ParaQuot B X^{\gamma+n-1} 
  \to \Susp_B (X^{\gamma+n-1}\ParaQuot B A)
  \to \Susp_B (X^{\gamma+n-1}\ParaQuot B X^{\gamma+n-2}).
 \]
 As usual, we write
 $\Mackey C_{\gamma+*}(X)$ for $\Mackey C_{\gamma+*}(X,\emptyset)$.
 \end{definition}

If $A\subset X$ is a subcomplex, then clearly
$\Mackey C_{\gamma+*}(A)$ is a direct summand of
$\Mackey C_{\gamma+*}(X)$, and
$\Mackey C_{\gamma+*}(X,A) \iso 
\Mackey C_{\gamma+*}(X)/\Mackey C_{\gamma+*}(A)$.

Turning to prespectra, we first record the following result.

\begin{lemma}\label{lem:skeletalspecquotient}
Let $X$ be a $G$-CW($\gamma$) prespectrum. Then
 \[
 X^{\gamma+n}\ParaQuot B X^{\gamma+n-1} \hmtpc \Wedge_x \susp^\infty_B S^{\gamma+n,px},
 \]
where the wedge (over $B$) runs through the centers $x$ of
the $(\gamma+n)$-cells of $X$
and the equivalence is lax homotopy equivalence.
 \end{lemma}

\begin{proof}
This follows from Proposition~\ref{prop:cellquotient} and the
argument of Lemma~\ref{lem:skeletalquotient}.
\end{proof}

 \begin{definition}
 Let $X$ be a $G$-CW($\gamma$) prespectrum.
Let $\Mackey C_{\gamma+*}(X)$
be the chain complex of contravariant $\stab\Pi B$-Mackey functors defined by
\begin{align*}
 \Mackey C_{\gamma+n}(X)(b)
   &= [ \susp^\infty_B S^{\gamma+n,b}, X^{\gamma+n}\ParaQuot B X^{\gamma+n-1} ]_{G,B} \\
   &\iso \pi^H_{\gamma(b)+n}((X^{\gamma+n}\ParaQuot B X^{\gamma+n-1})_b)
 \end{align*}
if $b\colon G/H\to B$.
Again, we use
the isomorphism of Theorem~\ref{thm:piXinvariance} to consider this as a functor on
$\stab\Pi B$.
Let $\bndry\colon \Mackey C_{\gamma+n}(X)\to \Mackey C_{\gamma+n-1}(X)$ be
the natural transformation induced by the composite
 \[
 X^{\gamma+n}\ParaQuot B X^{\gamma+n-1} \to \Susp_B X^{\gamma+n-1}
  \to \Susp_B (X^{\gamma+n-1}\ParaQuot B X^{\gamma+n-2}).
 \]
 \end{definition}

For both spaces and prespectra, it is clear that $(\Mackey C_{\gamma+*}(-),\bndry)$
is functorial on cellular maps. Shortly we shall discuss what happens
with more general maps, but we first make some other observations.
The following says that every prespectrum in a costabilization
system give the same chain complex.

\begin{proposition}\label{prop:costabiso}
Let $f\colon X\to Y$ be a costabilization of the $G$-CW($\gamma$)
prespectrum $Y$. Then $f$ induces an isomorphism
$\Mackey C_{\gamma+*}(X) \iso \Mackey C_{\gamma+*}(Y)$.
\end{proposition}

\begin{proof}
This follows from the one-to-one correspondence of the cells
and Proposition~\ref{prop:costabequiv}.
\end{proof}

We now relate $G$-CW($\gamma$) complexes and prespectra.
If $X$ is a $G$-CW($\gamma$) complex, note that the chain complex
of $X$ is defined in terms of the filtration quotients of the induced
filtration on $\susp^\infty_B X_+$. However, this filtration is
not quite the skeletal filtration of
a $G$-CW($\gamma$) structure, stemming from the fact that
the cells used in $X$ are unbased. 
However, we have already seen how to get around this problem in
Proposition~\ref{prop:spacesToSpec}:
$\susp^\infty_1\susp_B X_+$ is a $G$-CW($\gamma$) prespectrum
naturally weakly equivalent to $\susp^\infty_B X_+$ and
with cells in one-to-one correspondence to those of $X$.
That correspondence give us the following.

\begin{proposition}\label{prop:spacesToSpecChains}
If $X$ is a $G$-CW($\gamma$) complex, then
$\susp^\infty_1\susp_B X_+$ is a $G$-CW($\gamma$) prespectrum
whose chain complex is naturally isomorphic to that of $X$.
\qed
\end{proposition}

In view of this result, we now concentrate on prespectra.

If $\MackeyOp S$ is a covariant $\stab\Pi B$-Mackey functor, and
$\Mackey T$ and $\Mackey U$ are contravariant $\stab\Pi B$-Mackey functors,
then Definition~\ref{def:indres} defines for us the groups
 \[
 \Hom_{\stab\Pi B}(\Mackey T,\Mackey U)
 \]
and
 \[
 \Mackey T \tensor_{\stab\Pi B} \MackeyOp S.
 \]

The coefficient systems for our ordinary homology and cohomology theories
will be covariant and contravariant $\stab\Pi B$-Mackey functors, respectively.
Since a classical Mackey functor is a contravariant functor on
$\stab{\orb G}$, it gives rise to a coefficient
system on $B$ by composition with $\stab\phi\colon\stab\Pi B\to \stab{\orb G}$.
Systems that do not factor through $\stab{\orb G}$
we should call {\em local coefficient systems on $B$}.

Recall that a $\stab\Pi B$-Mackey functor is said to be
{\em free} if it is a sum of functors of the form $\stab\Pi B(-,b)$. The
following follows from Lemma~\ref{lem:skeletalspecquotient}.

\begin{proposition}
$(\Mackey C_{\gamma+*}(X),\bndry)$ is a chain complex of free
$\stab\Pi B$-Mackey functors.
If $\Mackey T$ is a $\stab\Pi B$-Mackey functor then
 \[
 \Hom_{\stab\Pi B}(\Mackey C_{\gamma+n}(X),\Mackey T) \iso
     \Dirsum_x \Mackey T(x)
 \]
where the direct sum runs over the centers $x$ of the $(\gamma+n)$-cells of $X$.
(For simplicity of notation, we write $\Mackey T(x)$ for $\Mackey T(p(x))$, where
$p(x)$ is the image of $x$ in $B$.)
Similarly, if $\MackeyOp S$ is a covariant $\stab\Pi B$-Mackey functor then
 \[
 \Mackey C_{\gamma+n}(X) \tensor_{\stab\Pi B} \MackeyOp S \iso
     \Dirsum_x \MackeyOp S(x)
 \]
where again $x$ runs over the centers of the $(\gamma+n)$-cells of $X$.
 \qed \end{proposition}

\begin{definition}
 Let $X$ be a $G$-CW($\gamma$) prespectrum, let $\MackeyOp S$ be a
covariant $\stab\Pi B$-Mackey functor, and let $\Mackey T$ be a
contravariant $\stab\Pi B$-Mackey functor. Let
 \[
 H_{\gamma+n}^G(X;\MackeyOp S) =
  H_{n}(\Mackey C_{\gamma+*}(X)\tensor_{\stab\Pi B} \MackeyOp S)
 \]
and
 \[
 H^{\gamma+n}_G(X;\Mackey T) =
  H_{n}(\Hom_{\stab\Pi B}(\Mackey C_{\gamma+*}(X), \Mackey T)).
 \]
 \end{definition}

Given an arbitrary $G$-prespectrum $X$ over $B$, we now define
 \[
 H_{\gamma}^G(X;\MackeyOp S) = H_{\gamma}^G(\Gamma X;\MackeyOp S)
 \]
and
 \[
 H^{\gamma}_G(X;\Mackey T) = H^{\gamma}_G(\Gamma X;\Mackey T),
 \]
 where $\Gamma X$ is a $G$-CW($\gamma$) approximation of
$X$.
Much as in the classical context, it is a consequence of
Theorem~\ref{thm:cwspecsummary},
using Proposition~\ref{prop:costabiso}, that these groups
are well-defined and functorial on the homotopy category.

In view of Proposition~\ref{prop:spacesToSpecChains},
if $(X,A)$ is a pair of $G$-spaces over $B$, we write
$H^G_{\gamma+n}(X,A;\MackeyOp S)$ for
$H^G_{\gamma+n}(\susp^\infty_B \Gamma X\ParaQuot B\Gamma A; \MackeyOp S)$, where we approximate the pair $(X,A)$ by a pair
$(\Gamma X,\Gamma A)$ of $G$-CW($\gamma$) complexes,
and similarly for cohomology.



The main properties of the theories just defined can
be summarized in the following theorem,
generalizing Theorem~\ref{thm:vhomology}.

\begin{theorem}\label{thm:ghomology}
Let $X$ be a $G$-prespectrum over $B$,
let $\gamma$ be a virtual spherical representation of $\Pi B$,
and let $\MackeyOp S$ and
$\Mackey T$ be respectively a covariant and a contravariant
$\stab\Pi B$-Mackey functor.
Then the abelian groups $ H^G_{\gamma}(X;\MackeyOp S)$
and $ H_G^{\gamma}(X;\Mackey T)$ are
functors on $\Ho G\PreSpec{}B$
and are also natural in $\gamma$ and the coefficient system.
Further, they satisfy the
following properties.

\begin{enumerate}
\item $ H^G_{\gamma}(X;\MackeyOp S)$ is exact on cofiber sequences and
sends wedges to direct sums. $ H_G^{\gamma}(X;\Mackey T)$ is exact on cofiber sequences
and sends wedges to products.

\item For each $W$ there are natural isomorphisms
\[
 \sigma^W\colon  H^G_{\gamma}(X;\MackeyOp S)\to  H^G_{\gamma+ W}(\susp_B^W X;\MackeyOp S)
\]
and
\[
 \sigma^W\colon  H_G^{\gamma}(X;\Mackey T)\to  H_G^{\gamma+ W}(\susp_B^W X;\Mackey T).
\]
$\sigma^0 = \id$ and
$\sigma^Z\circ \sigma^W = \sigma^{W+ Z}$
for every pair of representations $W$ and $Z$.

\item The $\sigma$ are natural in $W$ in the following sense
(cf \cite[XIII.1.1]{May:alaska}).
If
$\eta\colon W\to W'$ is a $G$-linear isomorphism, then the following diagrams commute.
\[
 \xymatrix{
 { H^G_{\gamma}(X;\MackeyOp S)} \ar[r]^{\sigma^{W'}} \ar[d]_{\sigma^{W}} &
   { H^G_{\gamma+ W'}(\susp_B^{W'}X;\MackeyOp S)} \ar[d]^{ H_{\id+\eta}^G(\id)} \\
 { H^G_{\gamma+ W}(\susp_B^W X;\MackeyOp S)} \ar[r]_{(\susp_B^\eta\id)_*} &
   { H^G_{\gamma+ W}(\susp_B^{W'} X;\MackeyOp S)} \\
 }
\]
\[
 \xymatrix{
 { H_G^{\gamma}(X;\Mackey T)} \ar[r]^{\sigma^{W}} \ar[d]_{\sigma^{W'}} &
   { H_G^{\gamma+ W}(\susp_B^{W}X;\Mackey T)} \ar[d]^{ H^{\id+\eta}_G(\id)} \\
 { H_G^{\gamma+ W'}(\susp_B^{W'} X;\Mackey T)} \ar[r]_{(\susp_B^\eta\id)^*} &
   { H_G^{\gamma+ W'}(\susp_B^{W} X;\Mackey T)} \\
 }
\]

\item 
If $\alpha\colon A\to B$ and $X$ is a $G$-prespectrum over $A$,
then
\[
 H^G_\gamma(\alpha_!X; \MackeyOp S)
  \iso H^G_{\alpha^*\gamma}(X; \alpha^*\MackeyOp S)
\]
and
\[
 H_G^\gamma(\alpha_!X; \Mackey T)
  \iso H_G^{\alpha ^*\gamma}(X; \alpha ^*\Mackey T).
\]

\item (Dimension Axiom)
We have the following natural isomorphisms of functors of $b$ in $\stab\Pi B$:
\[
 H^G_{\gamma+n}(\susp^\infty_B S^{\gamma,b}; \MackeyOp S) \iso
  \begin{cases} \MackeyOp S(b) & \text{if $n = 0$,} \\
                0 & \text{if $n\neq 0$,}
  \end{cases}
\]
and
\[
 H_G^{\gamma+n}(\susp^\infty_B S^{\gamma,b}; \Mackey T) \iso
  \begin{cases} \Mackey T(b) & \text{if $n = 0$,} \\
                0 & \text{if $n\neq 0$,}
  \end{cases}
\]
for integers $n$.
(On the left, we use the identification of
$\stab\Pi_\gamma B$ with $\stab\Pi B$.)


\item\label{item:gWirthmuller} (Wirthm\"uller Isomorphism)
If $K \subset G$ there is an isomorphism
 \[
  H_{\gamma}\sp G(G\halfsmsh_K X;\MackeyOp S)
 \iso   H_{\gamma|K}\sp K(X;\MackeyOp S|K).
 \]
and similarly in cohomology.
In cohomology, the composite
 \[
  H_G^\gamma(X;\Mackey T)
 \to  H_G^\gamma(G\halfsmsh_K X;\Mackey T)
 \iso  H_K^{\gamma|K}(X;\Mackey T|K),
 \]
where the first map is induced by the projection $G/K\to *$, is
called restriction to $K$ and is denoted by $a \mapsto a|K$.
In homology, if $G$ is finite, there is a similar restriction map given by the
composite
 \[
  H_{\gamma}\sp G(X;\MackeyOp S)
 \to  H_{\gamma}\sp G(G\halfsmsh_K X;\MackeyOp S)
 \iso  H_{\gamma |K}\sp K(X;\MackeyOp S|K),
 \]
where the first map is induced by the stable map $*\to G/K$ dual to the projection.
When $G$ is infinite, there is again a restriction in homology, but with a
shift in dimension, just as in the $RO(G)$-graded case.

\item\label{item:gfixedsets}
If $K$ is a normal subgroup of $G$, then there is a natural restriction
to fixed sets
\[
  H_{\gamma}\sp G(X;\MackeyOp S) \to
  H_{\gamma \sp K}\sp {G/K}(\Phi^K X;\MackeyOp S\sp K)
\]
and similarly in cohomology; we shall
denote this $a \mapsto a \sp K$.
This restriction respects suspension in the sense that the following
diagram commutes:
 \[
 \xymatrix{
   H_{\gamma}\sp G(X;\MackeyOp S) \ar[r]^-{\sigma_V} \ar[d]_{(-)^K}
    &  H_{\gamma + V}\sp G(\susp_B^V X;\MackeyOp S) \ar[d]^{(-)^K} \\
   H_{\gamma^K}\sp {G/K}(\Phi^K X;\MackeyOp S^K) \ar[r]_-{\sigma_{V^K}}
    &  H_{\gamma^K + V^K}\sp {G/K}(\susp_{B^K}^{V^K}\Phi^K X;\MackeyOp S^K)
  }
 \]
Restriction to fixed sets
respects restriction to subgroups in the sense that, if $K\subset L$,
then $(a|L)^K = a^K |(L/K)$.
We shall also write $a^K$ for $(a|NK)^K$ when $K$ is not normal in $G$.

\item\label{item:gcupproduct}
There is a natural associative cup product
 \[
 -\cup - \colon  H\sp{\gamma}_ G(X;\Mackey T) \tensor
  H\sp{\delta}_ G(Y;\Mackey U) \to  H\sp{\gamma+\delta}_
 G(X\smsh_B Y;\Mackey T\boxprod \Mackey U).
 \]
If $\Mackey T$ is a $\stab \Pi B$-ring, then there is a cup product
 \[
 -\cup - \colon  H\sp{\gamma}_G(X;\Mackey T) \tensor
  H\sp{\delta}_G(X;\Mackey T) \to  H\sp{\gamma+\delta}_G(X;\Mackey T).
 \]
This cup product satisfies $(x \cup y)|K = (x|K) \cup (y|K)$
and $(x \cup y)\sp K = x\sp K \cup y\sp K$.

\item\label{item:geval}
There is an evaluation map
 \[
 \langle -,- \rangle \colon  H^{\gamma}_G(X;\Mackey T) \tensor
   H_{\gamma+\delta}^G(X;\MackeyOp S) \to
       H_\delta^G(\susp^\infty_B B_+; \Mackey T\mixprod\MackeyOp S).
 \]
This evaluation is natural in the usual sense: if $f\colon X\to Y$, then
$\langle y, f_*(x)\rangle = \langle f^*(y), x\rangle$. It satisfies
$\langle x,a\rangle|K = \langle x|K, a|K\rangle$ and
$\langle x,a\rangle^K = \langle x^K, a^K\rangle$.

\item\label{item:gcapproduct}
There is a cap product
 \[
 -\cap - \colon  H^{\gamma}_G(Y;\Mackey T) \tensor
  H_{\gamma + \delta}^G(X\smsh_B Y;\MackeyOp S)
 \to  H_{\delta}^G(X;\Mackey T \mixprod \MackeyOp S).
 \]
This cap product is natural in the usual sense: if $f\colon X\to X'$ and
$g\colon Y\to Y'$
then $y'\cap (f\smsh g)_*(a) = f_*(g^*(y')\cap a)$.
It satisfies $(x\cup y)\cap a = x \cap (y \cap a)$,
$(y \cap a)|K = (y|K)\cap (a|K)$, and
$(y\cap a)^K = y^K \cap a^K$.
Further, $\langle x\cup y, a \rangle = \langle x, y\cap a\rangle$.
If $d\colon W\to X\smsh_B Y$ is a map
(e.g., the suspension of a diagonal $Z/_B(C\cup D)\to Z/_B C\smsh_B Z/_B D$),
we also write $y\cap a$ for $y\cap d_*(a)$.
 \end{enumerate}
 \end{theorem}

The first five parts of this theorem follow easily from the definitions,
with the help of Examples~\ref{ex:suspaprespec}
and~\ref{ex:basechangeprespec}.
The remaining parts are proved in much the same way as was Theorem~\ref{thm:vhomology};
the proofs will be discussed in the following chapter.

\begin{definition}
Let $b\colon G/H\to B$ be a $G$-map and let
$V$ be a representation of $H$. In \ref{def:parasphere} we defined the space
$S^{V,b}$ over $B$.
By \cite[\S 15.1]{MaySig:parametrized}, we have an inverse
spectrum $S^{-V,b}$ over $B$ .
Generalizing, if $\alpha$ is a virtual
representation of $\Pi B$, we have a $G$-spectrum
$S^{\alpha,b}$ over $X$.
\end{definition}

\begin{definition}\label{def:homologymackey}
\begin{enumerate}\item[]
\item
If $h^G_*(-)$ is a generalized $RO(G)$-graded homology theory on ex-$G$-spaces over $B$
(we shall make precise what we mean by this in the following chapter)
and $\alpha = \beta\ominus\gamma$ is a formal difference of spherical
representations of $\Pi B$,
write $\MackeyOp h^G_\alpha$ for the covariant Mackey functor defined by
\[
 \MackeyOp h^G_\alpha (b\colon G/H\to B) = h^G_\beta(b)(S^{\gamma,b}).
\]
This is a functor on $\stab\Pi_{-\alpha} B$, and we make it a functor
on $\stab\Pi B$ using the identification in
Theorem~\ref{thm:piXinvariance}.
For a cohomology theory $h_G^*$,
we define $\Mackey h_G^\alpha$ similarly.
\item
If $h^G_*(-)$ is a generalized $RO(G)$-graded homology theory on $G$-spectra over $B$
and $\alpha$ is a virtual
representation of $\Pi B$,
write $\MackeyOp h^G_\alpha$ for the covariant Mackey functor defined by
\[
 \MackeyOp h^G_\alpha (b\colon G/H\to B) = h^G_0(S^{-\alpha,b}).
\]
For a cohomology theory $h_G^*$,
we define $\Mackey h_G^\alpha$ similarly.
\end{enumerate}
\end{definition}

The Atiyah-Hirzebruch spectral sequence generalizes as follows
(cf. \cite[\S 12]{Bo:spectralsequences}).

\begin{theorem}[Atiyah-Hirzebruch Spectral Sequence]
Suppose that $h^G_*(-)$ is an $RO(G)$-graded homology theory on
ex-$G$-spaces over $B$ or $G$-prespectra over $B$.
Let $\alpha$ be an element of $RO(G)$ and let
$\beta$ be a formal difference of spherical representations of $\Pi B$.
Let $X$ be an ex-$G$-space or $G$-prespectrum over $B$ that is bounded below.
Then there is a strongly convergent spectral sequence
 \[
 E^2_{p,q} = H^G_{\alpha-\beta+p}(X;\MackeyOp h^G_{\beta+q})
  \convto h^G_{\alpha+p+q}(X).
 \]
Similarly, if $h_G^*(-)$ is an $RO(G)$-graded cohomology theory
on ex-$G$-spaces over $B$ or $G$-prespectra over $B$,
there is a conditionally convergent spectral sequence
 \[
 E_2^{p,q} = H_G^{\alpha-\beta+p}(X;\Mackey h_G^{\beta+q})
  \convto h_G^{\alpha+p+q}(X).
 \]
\qed
\end{theorem}

\begin{corollary}[Uniqueness of Ordinary $RO(\Pi B)$-Graded Homology]\label{cor:tCohomUniqueness}
Let $h^G_*(-)$ be an $RO(G)$-graded homology theory on ex-$G$-spaces over $B$
or $G$-prespectra over $B$
and let $\beta$ be a formal difference of spherical representations of $\Pi B$.
Suppose that, for integers $n$,
\[
 \MackeyOp h^G_{\beta+n} = 0\qquad\text{for $n\neq 0$}
\]
Then, for $\alpha\in RO(G)$, there is a natural isomorphism
\[
 h^G_\alpha(-) \iso H^G_{\alpha-\beta}(-;\MackeyOp h^G_\beta).
\]
There is a similar statement for cohomology theories.
 \end{corollary}

\begin{proof}
There is an Atiyah-Hirzebruch spectral sequence
 \[
 E^2_{p,q} = H^G_{\alpha-\beta+p}(X;\MackeyOp h^G_{\beta+q})
  \convto h^G_{\alpha+p+q}(X)
 \]
(and similarly for cohomology).
The spectral sequence collapses to the line $q=0$.
(We could also use the direct argument of \cite[15.2 \& 19.5]{May:concise}.)
 \end{proof}

Also useful is a universal coefficients spectral sequence.
Write $\Mackey H^G_\alpha(X)$ for the contravariant $\stab\Pi B$-Mackey functor given by
 \[
 \Mackey H^G_\alpha(X)(b\colon G/K\to B) =
   H^G_\alpha(X;\stab\Pi B(b,-)).
 \]
$\Tor_*^{\stab\Pi B}$ and $\Ext^*_{\stab\Pi B}$
below are the derived functors of $\tensor_{\stab\Pi B}$ and
$\Hom_{\stab\Pi B}$ respectively.

\begin{theorem}[Universal Coefficients Spectral Sequence]
If $\MackeyOp S$ is a covariant $\stab\Pi B$-Mackey functor and $\Mackey T$ is a
contravariant $\stab\Pi B$-Mackey functor, and if $\alpha$
is a virtual representation of $\Pi B$,
there are spectral sequences
 \[
 E^2_{p,q} = \Tor^{\stab\Pi B}_p(\Mackey H^G_{\alpha+q}(X), \MackeyOp S)
 \convto
 H^G_{\alpha+p+q}(X;\MackeyOp S)
 \]
and
 \[
 E_2^{p,q} = \Ext_{\stab\Pi B}^p(\Mackey H^G_{\alpha+q}(X), \Mackey T)
 \convto
 H_G^{\alpha+p+q}(X;\Mackey T).
 \]
\end{theorem}

\begin{proof}
These are constructed in the usual way by taking
resolutions of $\MackeyOp S$
and $\Mackey T$ and then taking the spectral sequences of the bigraded complexes
obtained from the chain complex of $X$ and the resolutions.
 \end{proof}

Finally, a word or two about obstruction theory.
For ex-$G$-spaces over $B$, obstruction theory works pretty much as expected,
if we look for obstructions to obtaining lax maps rather than strict
maps. That is, we have results like the following:
Let $X$ be a $G$-CW($\gamma$) complex and let $Y$ be a $G$-space over $B$.
Let $f\colon X^{\gamma+n}\laxto Y$ be a lax map. Then there is an
obstruction class
\[
 [c^{\gamma+n+1}(f)]\in H^{\gamma+n+1}(X;\Mackey\pi_{\gamma+n}(Y))
\]
such that $f|X^{\gamma+n-1}$ extends to a lax map on $X^{\gamma+n+1}$
if and only if $[c^{\gamma+n+1}(f)] = 0$.
We can, for example, adapt the presentation of
\cite[\S V.5]{Wh:homotopy}.

For spectra over $B$, we find obstructions to finding lax maps out of
costabilizations. That is, we have results like the following:
Let $X$ be a $G$-CW($\gamma$) prespectrum and let $Y$
be a $G$-prespectrum over $B$.
Let $f\colon X^{\gamma+n}\laxto Y$ be a lax map. Then there is
an obstruction class
\[
 [c^{\gamma+n+1}(f)]\in H^{\gamma+n+1}(X;\Mackey\pi_{\gamma+n}(Y))
\]
such that $f|X^{\gamma+n-1}$ extends to a lax map on 
${}_\alpha X^{\gamma+n+1}$ for some costabilization
${}_\alpha X$ of $X$
if and only if $[c^{\gamma+n+1}(f)] = 0$.


Homology-chap4.tex
\chapter{Parametrized Homology and Cohomology}
%
\section*{Introduction}

At the end of the preceding chapter we defined parametrized ordinary
homology and cohomology theories. Our first goal in this chapter
is to discuss how these theories are represented, and
we begin by discussing general parametrized theories.
To deal with homology it is useful to have a duality
connecting homology and cohomology theories, but fiberwise duality,
the most obvious, does not do the job. We introduce
a new duality, which we call {\em homological duality}, that suits
our needs better.

With this machinery in place, we discuss the parametrized prespectra
representing ordinary homology and cohomology. As in the nonparametrized
case, when $G$ is infinite these spectra give rise to two more theories,
the dual ordinary homology and cohomology theories.
We show how these dual theories can be calculated from chain complexes
arising from dual $G$-CW structures.

Finally, for both the ordinary theories and their duals,
we discuss passage to subgroups, restriction to fixed sets, and products.
We complete our program by giving general
Thom isomorphism and Poincar\'e duality theorems.


%%%%%%%%%%%%%%%%%%%
\section{Representing homology and cohomology theories}

We begin by defining carefully what we mean by ``$RO(G)$-graded''
parametrized homology and cohomology theories.
We concentrate on reduced theories on ex-$G$-spaces and theories
on spectra.

In \cite{Dold:parahomology} Dold gave axioms
for nonequivariant homology and cohomology theories defined on
spaces parametrized by $B$.
Several authors, including Wirthm\"uller \cite{Wi:duality}, have given
axioms for $RO(G)$-graded theories defined on $G$-spaces;
we follow those given more recently by May in \cite[XIII.1]{May:alaska}.
We combine the two approaches to give axioms for
equivariant theories defined on parametrized spaces.

Let $\RO(G)$ be the category of finite-dimensional
linear representations of $G$ and $G$-linear isometric
isomorphisms between them.
Say that two maps $V\to W$ are homotopic if the associated $G$-maps
$S^V\to S^W$ are stably homotopic. Let $h\RO(G)$ denote the resulting
homotopy category.
(Comparing with \cite{May:alaska}, we are assuming that we are using a complete
$G$-universe $\U$. Variants for smaller universes are possible, but we are interested
in duality, which at present requires a complete universe.)

As usual, fix a suitable basespace $B$.
The following definition also appears in \cite[\S21.1]{MaySig:parametrized}.

\begin{definition}\label{def:parametrizedtheory}
An {\em $RO(G)$-graded parametrized (reduced) homology theory} consists of a 
functor
\[
 \tilde h^G_*\colon \Op{h\RO(G)}\times \Ho G\Para B \to \Ab,
\]
written $(V,X) \mapsto \tilde h^G_V(X)$ on objects. 
Similarly, an {\em $RO(G)$-graded parametrized (reduced)
cohomology theory} consists of a functor
\[
 \tilde h_G^*\colon h\RO(G)\times \Op{\Ho G\Para B} \to \Ab.
\]
These functors are required to satisfy the following
axioms.
 \begin{enumerate}
\item For each $V$, the functor $\tilde h^G_V$ is exact on cofiber sequences
and sends wedges to direct sums. The functor $\tilde h_G^V$ is exact on cofiber sequences
and sends wedges to products.
\item For each $W$ there are natural isomorphisms
\[
 \sigma^W\colon \tilde h^G_V(X)\to \tilde h^G_{V\dirsum W}(\susp_B^W X)
\]
and
\[
 \sigma^W\colon \tilde h_G^V(X)\to \tilde h_G^{V\dirsum W}(\susp_B^W X).
\]
$\sigma^0 = \id$ and
$\sigma^Z\circ \sigma^W = \sigma^{W\dirsum Z}$
for every pair of representations $W$ and $Z$.
\item The $\sigma$ are natural in $W$ in the following sense. If
$\alpha\colon W\to W'$ is a map in $h\RO(G)$, then the following diagrams commute.
\[
 \xymatrix{
 {\tilde h_V^G(X)} \ar[r]^-{\sigma^{W'}} \ar[d]_{\sigma^{W}} &
   {\tilde h_{V\dirsum W'}^G(\susp_B^{W'} X)} \ar[d]^{\tilde h_{\id\dirsum\alpha}^G(\id)} \\
 {\tilde h_{V\dirsum W}^G(\susp_B^W X)} \ar[r]_{(\susp_B^\alpha\id)_*} &
   {\tilde h_{V\dirsum W}^G(\susp_B^{W'} X)} \\
 }
\]
\[
 \xymatrix{
 {\tilde h^V_G(X)} \ar[r]^-{\sigma^{W}} \ar[d]_{\sigma^{W'}} &
   {\tilde h^{V\dirsum W}_G(\susp_B^{W} X)} \ar[d]^{\tilde h^{\id\dirsum\alpha}_G(\id)} \\
 {\tilde h^{V\dirsum W'}_G(\susp_B^{W'} X)} \ar[r]_{(\susp_B^\alpha\id)^*} &
   {\tilde h^{V\dirsum W'}_G(\susp_B^{W} X)} \\
 }
\]
 \end{enumerate}
\end{definition}
\bigskip

As in \cite{May:alaska} we extend the grading to ``formal differences'' $V\ominus W$ by setting
\[
 \tilde h^G_{V\ominus W}(X) = \tilde h^G_V(\susp_B^W X)
\]
and similarly in cohomology. Rigorously, we extend the grading from
$\Op{h\RO(G)}$ to $\Op{h\RO(G)}\times h\RO(G)$. Now, $RO(G)$ can be obtained from
$\RO(G)$ by considering $V\ominus W$ to be equivalent to $V'\ominus W'$ when there
is a $G$-linear isometric isomorphism
\[
 \alpha\colon V\dirsum W'\to V'\dirsum W.
\]
Given such an isomorphism we get the following diagram.
\[
 \xymatrix{
  {\tilde h^G_{V'}(\susp_B^{W'} X)} \ar[r]^-{\sigma^W} \ar[d] &
   {\tilde h^G_{V'\dirsum W}(\susp_B^{W'\dirsum W} X)} \ar[d]^{\tilde h^G_\alpha(\susp^\tau\id)} \\
  {\tilde h^G_V(\susp_B^W X)} \ar[r]_-{\sigma^{W'}} &
   {\tilde h^G_{V\dirsum W'}(\susp_B^{W\dirsum W'} X)}
 }
\]
Here, $\tau\colon W\dirsum W' \to W'\dirsum W$ is the transposition map.
Thus,
\[
 \tilde h^G_{V\ominus W}(X) \iso \tilde h^G_{V'\ominus W'}(X).
\]
(There is a similar isomorphism in cohomology.)
It is in this sense
that we can think of grading on $RO(G)$, but note that the isomorphism depends on the
choice of $\alpha$, so is not canonical.

Dold pointed out that
$\tilde h^G_*(-)$ and
$\tilde h_G^*(-)$ restrict to functors on $\Pi B$. 
Actually, the suspension isomorphisms imply that
homology and cohomology theories extend to stable maps, and so we 
get restrictions to functors on the stable fundamental groupoid $\stab\Pi B$.
The restrictions to $\stab\Pi B$ give $RO(G)$-graded collections of
$\stab\Pi B$-Mackey functors
that we shall call the {\em coefficient systems} of the theories and denote by
$\MackeyOp h^G_*$ and $\Mackey h_G^*$.

As usual, if $X$ is an unbased space over $B$, or $(X,A)$ is a pair of
based or unbased $G$-spaces over $B$, we define
 \[
  h^G_*(X) = \tilde h^G_*(X_+)
 \]
and
 \[
  h^G_*(X,A) = \tilde h^G_*(C_B i)
 \]
where $i\colon A\to X$ is the inclusion and $C_B i$ is its cofiber over $B$. 
We use a similar convention for cohomology.

If we replace $\Ho G\Para B$ with $\Ho G\PreSpec{}B$ in
Definition~\ref{def:parametrizedtheory}, we
get the definitions of parametrized homology and cohomology theories
on prespectra over $B$.

We are interested in how these theories are represented. 
It is well-known how to represent parametrized cohomology
theories using spectra over $B$ 
(see, for example, \cite{Cl:paraspectra} and \cite{CP:paraspectra}).
The following is Theorem~21.2.3 in \cite{MaySig:parametrized},
and follows from Brown's representability theorem in the usual way.

\begin{theorem}\label{thm:cohomrep}
If $\tilde h_G^*(-)$ is an $RO(G)$-graded cohomology theory on $\Ho G\Para B$, then it is
represented by a $G$-prespectrum $E$ over $B$, in the sense
that, for each $V$, there is a natural isomorphism
 \[
 \tilde h_G^V(X) \iso [\Susp_B^\infty X,\Susp_B^V E]_{G,B}.
 \]
The prespectrum $E$ is determined by $h$ up to non-unique equivalence.
Similarly, if
$h_G^*(-)$ is an $RO(G)$-graded cohomology theory on $\Ho G\PreSpec{} B$, then it is
represented by a $G$-prespectrum $E$ over $B$, in the sense
that, for each $V$, there is a natural isomorphism
 \[
 h_G^V(X) \iso [X,\Susp_B^V E]_{G,B}
 \]
The prespectrum $E$ is determined by $h$ up to unique equivalence.
\qed
\end{theorem}

We now consider homology theories.
Curiously, previously to \cite{MaySig:parametrized} and this paper,
there appears to have been no discussion in the literature
of representing homology theories on parametrized
spaces. The next proposition gives a way of defining a homology theory using
a parametrized prespectrum. Before that we should say a few words about
smash products. (We summarize the more detailed discussion \cite{MaySig:parametrized}.)

The fiberwise smash product behaves
poorly from the point of view of homotopy theory; it does not pass directly
to homotopy categories. However, the {\em external} fiberwise smash product does
behave well. If $D$ is a $G$-prespectrum over $A$ and $E$ is a $G$-prespectrum
over $B$ we can form the external smash product $D\exsmsh E$ over $A\times B$.
(Since we are using prespectra and not orthogonal spectra, the smash product
we use is either ``external'' with respect to indexing spaces as well, or
is a ``handicrafted'' smash product \cite[13.7.1]{MaySig:parametrized}.)
The internal smash product of spectra $D$ and $E$ in $\Ho G\PreSpec{} B$
is then defined by
$D\smsh_B E = \Delta^*(D\exsmsh E)$,
where $\Delta\colon B\to B\times B$ is the diagonal. 
As usual, this notation is convenient but misleading.
In the homotopy category, $\Delta^*$ and $\exsmsh$ must be understood as
derived functors of their point-set level precursors. In particular,
$\Delta^*$ is defined by first applying a fibrant approximation functor and
then applying the point-set level $\Delta^*$.
Fortunately, May and Sigurdsson show that the familiar compatibility
relations among the smash product and the change of basespace functors
remain true in the homotopy category.

\begin{proposition}
If $E$ is a $G$-prespectrum over $B$ then the groups
 \[
 \tilde E^G_V(X) = [S^V, \rho_!(E\smsh_B X)]_G
 \]
define an $RO(G)$-graded homology theory on $\Ho G\Para B$
(taking $X$ to be an ex-$G$-space) or on $\Ho G\PreSpec{} B$
(taking $X$ to be a prespectrum over $B$).
Here, $\rho\colon B\to *$ denotes projection to a point.
\qed
\end{proposition}

\begin{proof}
The main problem is to see that $\tilde E^G_V(X)$ is exact on cofiber
sequences and takes wedges to direct sums. Write
$\rho_!(E\smsh_B X) = \rho_!\Delta^*(E\exsmsh X)$.
As in \cite{MaySig:parametrized},
each of $\rho_!$, $\Delta^*$, and $E\exsmsh -$ 
is exact on $\Ho G\PreSpec{} B$ and also takes wedges to wedges.
The remainder of the proof is as in \cite[XIII.2.6]{May:alaska}.
\end{proof}

At first glance, this is not an obvious way to define a homology
theory using a spectrum. In particular, the use of
$\rho_!$ on the right might not be expected,
given that $\rho_!$ is a left adjoint. One
justification is that it works: We do get a homology theory and,
moreover, we will be able to show that every homology theory
is representable in this way.
Here is, perhaps, a better justification.
The homology and cohomology theories determined by a parametrized
spectrum $E$ should, locally, reflect the nonparametrized theories
given by the fibers of $E$. To make this precise, we first record
a change of basespace result.

\begin{proposition}\label{prop:hombasechange}
Let $\alpha\colon A\to B$ be a $G$-map, let $X$ be an ex-$G$-space
over $A$, and let $E$ be a $G$-prespectrum over $B$. Then
\[
 \tilde E^G_V(\alpha_! X) \iso (\widetilde{\alpha^*E})^G_V(X)
\]
and
\[
 \tilde E_G^V(\alpha_! X) \iso (\widetilde{\alpha^*E})_G^V(X).
\]
\end{proposition}

\begin{proof}
We begin with cohomology, which is a little more obvious:
\begin{align*}
 \tilde E_G^V(\alpha_! X)
 &= [\alpha_! X, \susp^V_B E]_{G,B} \\
 &\iso [X, \alpha^*(\susp^V_B E)]_{G,A} \\
 &\iso [X, \susp^V_A \alpha^*E]_{G,A} \\
 &= (\widetilde{\alpha^*E})_G^V(X).
\end{align*}
For homology we have the following, using one of the
stable equivalences listed in Section~\ref{sec:paraspec}:
\begin{align*}
 \tilde E^G_V(\alpha_! X)
 &= [S^V, \rho_!(E\smsh_B \alpha_! X)]_G \\
 &\iso [S^V, \rho_!\alpha_!(\alpha^*E\smsh_A X)]_G \\
 &= [S^V, \rho_!(\alpha^*E\smsh_A X)]_G \\
 &= (\widetilde{\alpha^*E})^G_V(X).
\end{align*}
\end{proof}

We can now say that these theories behave correctly
over individual points of $B$:

\begin{corollary}
Let $E$ be a $G$-prespectrum over $B$, let $b\colon G/H\to B$,
and let $X$ be a based $H$-space. Then
\[
 \tilde E^G_V(b_!(G\halfsmsh_H X)) \iso (\tilde E_b)^H_V(X)
\]
and
\[
 \tilde E_G^V(b_!(G\halfsmsh_H X)) \iso (\tilde E_b)_H^V(X).
\]
\qed
\end{corollary}

Thus, the homology and cohomology theories defined by $E$
restrict over each point $b$ of $B$ to the nonparametrized
theories defined by the fiber $E_b$.

Specializing further, we can examine the coefficient systems
of the homology and cohomology theories determined by $E$.

\begin{corollary}
If $E$ is a $G$-prespectrum over $B$, then the coefficient systems
of its associated homology and cohomology theories are given by
\[
 \MackeyOp E^G_V(b\colon G/H\to B)
  \iso \pi^H_{V-V(G/H)}(E_b)
\]
and
\[
 \Mackey E_G^V(b) \iso \pi^H_{-V}(E_b),
\]
where $V(G/H)$ is the tangent $H$-representation at $eH\in G/H$.
\end{corollary}

\begin{proof}
The calculation for cohomology follows directly from the definitions
and the preceding corollary.
The calculation for homology uses the
isomorphism of \cite[II.6.5]{LMS:eqhomotopy}.
\end{proof}

Finally, we give one more useful application
of Proposition~\ref{prop:hombasechange}:
The groups $\tilde E^G_*(X,p,\sigma)$ and $\tilde E_G^*(X,p,\sigma)$
depend only on the spectrum $p^*E$ parametrized by $X$. To see
this, note first that 
$(X,p,\sigma) = p_!(X',q,\tau)$ where $X' = X\union_{\sigma(B)} X$, 
$q\colon X'\to X$ is the evident map, and $\tau\colon X\to X'$ takes $X$ onto
the second copy of $X$ in $X'$. From this it follows that
\[
 \tilde E^G_V(X) = \tilde E^G_V(p_! X')
  \iso (\widetilde{p^*E})^G_V(X')
\]
and similarly for cohomology.
In particular, if $p^*E$ is trivial, in the sense that it is
equivalent to $\rho^* D = X\times D$ for a nonparametrized
$G$-prespectrum $D$, then (assuming that $X$
is well-sectioned)
 \[
 \tilde E^G_V(X) \iso \tilde D^G_V(\rho_! X') \iso \tilde D^G_V(X/\sigma(B))
 \]
is just the usual $D$-homology of $X/\sigma(B)$, and
 \[
 \tilde E_G^V(X) \iso \tilde D_G^V(X/\sigma(B))
 \]
is just the usual $D$-cohomology of $X/\sigma(B)$.


%%%%%%%%%%%%%%%
\section{Homological duality}

Let $E$ be a $G$-prespectrum parametrized by $B$. 
We would like to relate the homology and cohomology theories determined by
$E$ by finding, for each suitably ``finite'' $G$-prespectrum $X$ over $B$,
a dual prespectrum $\bar D_B X$ over $B$ such that
$E_G^*(X) \iso E^G_{-*}(\bar D_B X)$.
It will be convenient to generalize as follows.
Let $A$ and $B$ be basespaces, let
$W$ be a $G$-prespectrum over $A$, and let $E$ be a $G$-prespectrum over $A\times B$.
Writing in terms of mapping sets, we would like a natural isomorphism
of the form
\[
 [W\exsmsh X, E]^G_{A\times B} \iso [W, (1\times\rho)_!(1\times\Delta)^*(E\exsmsh \bar D_B X)]_{G,A}.
\]
Here, $\exsmsh$ denotes the external smash product while,
as is our custom, $\rho$ denotes projection to a point.
We are most interested in the case $A = *$, but the extra generality comes at no
cost and gives a considerably more satisfying result.
We call this kind of duality {\em homological duality}
to distinguish it from the better-known fiberwise duality
(see, for example, \cite{Cl:paraspectra} or \cite{CP:paraspectra} as well as
\cite{MaySig:parametrized}).
What we call homological duality, May and Sigurdsson have seen fit
to call Costenoble-Waner duality, for which we thank them.

Homological duality does not fit the usual formalism of duality in closed symmetric
monoidal categories---fiberwise duality does. However, we can set up a very similar
formalism for our purposes. 
May and Sigurdsson \cite{MaySig:parametrized}
have developed a more general notion of duality, in what they call closed symmetric bicategories,
that includes both fiberwise duality and homological duality as special cases.

Recall from
\cite{MaySig:parametrized} that there is an external function spectrum functor
$\bar F$: If $Y$ is a $G$-prespectrum over $B$ and $Z$ is a $G$-prespectrum over
$A\times B$, then $\bar F(Y,Z)$ is a $G$-prespectrum over $A$.
$\bar F(Y,-)$ is right adjoint to $-\exsmsh Y$.

\begin{definition}
If $X$ is a $G$-prespectrum over $B$, let $\bar D_B X = \bar F(X,\Delta_! S_B)$
where $S_B$ is the sphere spectrum over $B$ (i.e., $\susp^\infty_B B_+$) and
$\Delta\colon B\to B\times B$ is the diagonal.
\end{definition}

We shall justify this definition over the course of this section.
It's interesting to note that the fiberwise dual of $X$ can be written
as $\bar F(X,\Delta_* S_B)$ where $\Delta_*$ is the right adjoint of $\Delta^*$.
We should reiterate our standard warning at this point: 
All of our constructions are
done in the stable category. The point-set definition of
$\bar F(X,\Delta_! S_B)$ gives a misleading picture of $\bar D_B X$,
so we shall rely heavily on the formal properties of the functors we use,
as given in \cite{MaySig:parametrized}.

Evaluation, the counit of the $\exsmsh$-$\bar F$ adjunction, gives a map
\[
 \epsilon\colon \bar D_B X\exsmsh X \to \Delta_! S_B.
\]

The following innocuous-looking lemma is a key technical point in this circle of ideas.

\begin{lemma}
$(1\times\rho\times 1)_!(1\times\Delta\times 1)^*(E\exsmsh\Delta_! S_B) \iso E$
in the stable category, for every $G$-prespectrum $E$ over $A\times B$.
\end{lemma}

\begin{proof}
Here and elsewhere we use the commutation relation $j^*f_! \iso g_!i^*$ that holds
when the following is a pullback diagram and at least one of $f$ or $j$ is a fibration:
\[
 \xymatrix{
  C \ar[r]^g \ar[d]_i & D \ar[d]^j \\
  A \ar[r]_f & B
 }
\]
(This is \cite[13.7.7]{MaySig:parametrized}.) The proof of the present lemma would be
considerably easier if the fibration condition were not required, but
the result if false in general without it.
The following chain of isomorphisms proves the lemma.
\begin{align*}
 (1\times\rho\times 1)_!&(1\times\Delta\times 1)^*(E\exsmsh\Delta_! S_B) \\
  &\iso (1\times\rho\times 1)_!\Delta_{A\times B\times B}^*
     [(1\times 1\times\rho)^*E\exsmsh (\rho\times 1\times 1)^* \Delta_! S_B] \\
  &= (1\times\rho\times 1)_![(1\times 1\times\rho)^*E\smsh_{A\times B\times B} 
     (\rho\times 1\times 1)^*\Delta_! S_B] \\
  &\iso (1\times\rho\times 1)_![(1\times 1\times\rho)^* E\smsh_{A\times B\times B}
     (1\times\Delta_!)(\rho\times 1)^* S_B] \\
  &\iso (1\times\rho\times 1)_!(1\times\Delta)_!
     [(1\times\Delta)^*(1\times 1\times\rho)^*E\smsh_{A\times B} (\rho\times 1)^*S_B] \\
  &\iso E\smsh_{A\times B} S_{A\times B} \\
  &\iso E
\end{align*}
Here, $\Delta$ generally denotes the diagonal map $B\to B\times B$, but
$\Delta_{A\times B\times B}$ denotes the diagonal map on $A\times B\times B$.
\end{proof}

\begin{definition}
Let $X$ and $Y$ be $G$-prespectra over $B$, let $W$ be a $G$-prespectrum over $A$,
and let $E$ be a $G$-prespectrum over $A\times B$.
\begin{enumerate}
\item
If $\epsilon\colon Y\exsmsh X\to \Delta_! S_B$ is a $G$-map over $B\times B$, let
\[
 \epsilon_\sharp\colon [W,(1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y)]_{G,A}
  \to [W\exsmsh X, E]_{G,A\times B}
\]
be defined by letting $\epsilon_\sharp(f)$ be the composite
\begin{align*}
 W\exsmsh X
  &\xrightarrow{f\smsh 1} (1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y)\exsmsh X \\
  &\iso (1\times\rho\times 1)_!(1\times\Delta\times 1)^*(E\exsmsh Y\exsmsh X) \\
  &\xrightarrow{1\smsh\epsilon} 
    (1\times\rho\times 1)_!(1\times\Delta\times 1)^*(E\exsmsh \Delta_! S_B) \\
  &\iso E.
\end{align*}
\item
If $\eta\colon S\to \rho_!\Delta^*(X\exsmsh Y) = \rho_!(X\smsh_B Y)$ is a $G$-map, let
\[
 \eta_\sharp\colon [W\exsmsh X, E]_{G,A\times B} \to 
   [W, (1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y)]_{G,A}
\]
be defined by letting $\eta_\sharp(f)$ be the composite
\begin{align*}
 W \iso W\exsmsh S
  &\xrightarrow{1\smsh\eta} W\exsmsh\rho_!\Delta^*(X\exsmsh Y) \\
  &\iso (1\times\rho)_!(1\times\Delta)^*(W\exsmsh X\exsmsh Y) \\
  &\xrightarrow{f\smsh 1} (1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y).
\end{align*}
\end{enumerate}
\end{definition}

\begin{definition}
A $G$-prespectrum $X$ over $B$ is {\em homologically finite}
if
\[
 \epsilon_\sharp\colon [W, (1\times\rho)_!(1\times\Delta)^*(E\exsmsh \bar D_B X)]_{G,A}
  \to [W\exsmsh X, E]_{G,A\times B}
\]
is an isomorphism for all $W$ and $E$,
where $\epsilon\colon \bar D_B X\exsmsh X\to \Delta_! S_B$ is the evaluation map.
\end{definition}

We have the following characterizations of the dual,
similar to \cite[III.1.6]{LMS:eqhomotopy}.

\begin{theorem}\label{thm:dualcharacterization}
Let $X$ and $Y$ be $G$-prespectra over $B$. The following are equivalent.
\begin{enumerate}
\item\label{item:XIsFinite}
$X$ is homologically finite and $Y\iso \bar D_B X$
in the stable category.

\item\label{item:epsilon}
There exists a map $\epsilon\colon Y\exsmsh X\to \Delta_! S_B$
such that
\[
 \epsilon_\sharp\colon [W, (1\times\rho)_!(1\times\Delta)^*(E\exsmsh \bar Y)]_{G,A}
  \to [W\exsmsh X, E]_{G,A\times B}
\]
is an isomorphism for all $W$ and $E$.

\item\label{item:eta}
There exists a map $\eta\colon S\to \rho_!\Delta^*(X\exsmsh Y)$ such that
\[
 \eta_\sharp\colon [W\exsmsh X, E]_{G,A\times B} \to 
   [W, (1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y)]_{G,A}
\]
is an isomorphism for all $W$ and $E$.

\item\label{item:both}
There exist maps $\epsilon\colon Y\exsmsh X\to \Delta_! S_B$ and
$\eta\colon S\to \rho_!\Delta^*(X\exsmsh Y)$ such that
the following two composites are each the identity in the stable category:
\begin{align*}
 X
 &\xrightarrow{\eta\smsh 1} \rho_!\Delta^*(X\exsmsh Y)\exsmsh X \\
 &\iso (\rho\times 1)_!(\Delta\times 1)^*(X\exsmsh Y\exsmsh X) \\
 &\xrightarrow{1\smsh\epsilon} (\rho\times 1)_!(\Delta\times 1)^*(X\exsmsh\Delta_! S_B) \\
 &\iso X
\end{align*}
and
\begin{align*}
 Y
 &\xrightarrow{1\smsh\eta} Y \exsmsh \rho_!\Delta^*(X\exsmsh Y) \\
 &\iso (1\times\rho)_!(1\times\Delta)^*(Y\exsmsh X\exsmsh Y) \\
 &\xrightarrow{\epsilon\smsh 1} (1\times\rho)_!(1\times\Delta)^*(\Delta_! S_B\smsh Y) \\
 &\iso Y
\end{align*}

\end{enumerate}

Moreover, when these conditions are satisfied, the following are also true:
The maps $\epsilon_\sharp$ and $\eta_\sharp$ are inverse isomorphisms;
the adjoint of $\epsilon\colon Y\exsmsh X\to \Delta_! S_B$ is an
equivalence $\tilde\epsilon\colon Y\to \bar D_B X$;
and the maps $\epsilon\gamma$ and $\gamma\eta$ satisfy the same conditions
with the roles of $X$ and $Y$ reversed, where $\gamma$ is the twist map.
\end{theorem}

\begin{proof}
(\ref{item:XIsFinite}) implies (\ref{item:epsilon}) trivially.

Suppose that (\ref{item:both}) is true.
The following diagram shows that $\epsilon_\sharp\eta_\sharp(f) = f$
for any $f\colon W\exsmsh X\to E$:
\[
 \def\objectstyle{\scriptstyle}
 \def\labelstyle{\scriptstyle}
 \xymatrix{
  W\exsmsh X \ar[d]_{1\smsh\eta\smsh 1} \\
  W\exsmsh \rho_!\Delta^*(X\exsmsh Y)\exsmsh X \ar[d]_\iso \\
  (1\times\rho\times 1)_!(1\times\Delta\times 1)^*(W\exsmsh X\exsmsh Y\exsmsh X)
    \ar[r]^{f\smsh 1} \ar[d]_{1\smsh 1\smsh\epsilon}
  & (1\times\rho\times 1)_!(1\times\Delta\times 1)^*(E\exsmsh Y\exsmsh X)
    \ar[d]^{1\smsh\epsilon} \\
  (1\times\rho\times 1)_!(1\times\Delta\times 1)^*(W\exsmsh X\exsmsh\Delta_! S_B)
    \ar[r]^{f\smsh 1} \ar[d]_\iso
  & (1\times\rho\times 1)_!(1\times\Delta\times 1)^*(E\exsmsh\Delta_! S_B) \ar[d]^\iso \\
  W\exsmsh X \ar[r]^{f} & E
 }
\]
On the other hand, the following diagram shows that
$\eta_\sharp\epsilon_\sharp(g) = g$ for any
$g\colon W \to (1\times\rho)_!(1\times\Delta^*)(E\exsmsh Y)$:
\[
% \def\objectstyle{\scriptstyle}
% \def\labelstyle{\scriptstyle}
 \xymatrix{
  W \ar[r]^-{g} \ar[d]_{1\smsh\eta}
   & (1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y) \ar[d]^{1\smsh\eta} \\
  W\exsmsh\rho_!\Delta^*(X\exsmsh Y) \ar[r]^-{g\smsh 1} \ar[d]_\iso
   & (1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y)\exsmsh\rho_!\Delta^*(X\exsmsh Y) \ar[d]^\iso \\
  (1\times\rho)_!(1\times\Delta)^*(W\exsmsh X\exsmsh Y) \ar[r]^-{g\smsh 1}
   & (1\times\rho\times\rho)_!(1\times\Delta\times\Delta)^*(E\exsmsh Y\exsmsh X\exsmsh Y)
     \ar[d]^{1\smsh\epsilon \smsh 1} \\
   & (1\times\rho\times\rho)_!(1\times\Delta\times\Delta)^*(E\exsmsh\Delta_!S_B\exsmsh Y)
     \ar[d]^\iso \\
   & (1\times\rho)_!(1\times\Delta)^*(E\exsmsh Y)
  }
\]
Here we use the natural isomorphism 
$(1\times\rho)_!(1\times\Delta)^*(1\times\rho\times 1\times 1)_!(1\times\Delta\times 1\times 1)^*
\iso (1\times\rho\times\rho)_!(1\times\Delta\times\Delta)^*$, which follows from the
isomorphism
$(1\times\Delta)^*(1\times\rho\times 1\times 1)_! 
\iso (1\times\rho\times 1)_!(1\times 1\times\Delta)^*$.
We also use the isomorphism
$(1\times\rho)_!(1\times\Delta)^*(1\times 1\times 1\times\rho)_!(1\times 1\times 1\times\Delta)^*
\iso (1\times\rho\times\rho)_!(1\times\Delta\times\Delta)^*$, which is true for a similar reason.
Therefore, $\epsilon_\sharp$ and $\eta_\sharp$ are inverse isomorphisms, so
(\ref{item:both}) implies both (\ref{item:epsilon}) and (\ref{item:eta}).

Assume now that (\ref{item:epsilon}) is true.
Taking $W = S$ and $E = X$, let $\eta\colon S\to\rho_!\Delta^*(X\exsmsh Y)$ be the
map such that $\epsilon_\sharp(\eta) = 1$, the identity map of $X$.
The first composite in (\ref{item:both}) is just $\epsilon_\sharp(\eta)$,
so is the identity by construction.
The first of the diagrams above shows that $\epsilon_\sharp\circ\eta_\sharp = 1$,
so $\eta_\sharp$ is the inverse isomorphism to $\epsilon_\sharp$.
Now take $W = Y$ and $E = \Delta_! S_B$.
Using the identification $(1\times\rho)_!(1\times\Delta)^*(\Delta_!S_B\exsmsh Y) \iso Y$,
we have $\epsilon_\sharp(1) = \epsilon$, so $\eta_\sharp(\epsilon) = 1$.
Written out, this says that the second composite in (\ref{item:both}) is also
the identity. Thus, (\ref{item:epsilon}) implies (\ref{item:both}).

Now suppose that (\ref{item:eta}) is true.
Take $W = Y$ and $E = \Delta_! S_B$ and
let $\epsilon\colon Y\exsmsh X\to \Delta_!S_B$ be the map such that
$\eta_\sharp(\epsilon) = 1$.
The remainder of the proof that (\ref{item:eta}) implies (\ref{item:both})
is now similar to the preceding argument.

Assuming (\ref{item:both}), hence (\ref{item:epsilon}), let
$\tilde\epsilon\colon Y\to \bar D_B X$ be the adjoint of
$\epsilon\colon Y\exsmsh X\to \Delta_! S_B$.
A reasonably straightforward diagram chase shows that $\tilde\epsilon$
is an equivalence with inverse the composite
\begin{align*}
 \bar D_B X
  &\xrightarrow{1\smsh\eta} \bar D_B X\exsmsh \rho_!\Delta^*(X\exsmsh Y) \\
  &\iso (1\times\rho)_!(1\times\Delta)^*(\bar D_B X\exsmsh X\exsmsh Y) \\
  &\xrightarrow{\epsilon\smsh 1} (1\times\rho)_!(1\times\Delta)^*(\Delta_! S_B\exsmsh Y) \\
  &\iso Y.
\end{align*}
(\ref{item:XIsFinite}) now follows. 

The claim in the theorem about $\epsilon\gamma$ and $\gamma\eta$ follows from
(\ref{item:both}) by symmetry. The other claims in the last part of the theorem
have been shown along the way.
\end{proof}

\begin{corollary}
If $X$ is homologically finite then so is $\bar D_B X$, and
$\bar D_B\bar D_B X \iso X$.
\qed
\end{corollary}

We have the following useful compatibility result.

\begin{proposition}\label{prop:pushdual}
If $X$ is a homologically finite $G$-prespectrum over $B$ and
$\beta\colon B\to B'$ is a $G$-map, then $\beta_! X$ is homologically
finite and $\beta_!\bar D_B X \iso \bar D_{B'}\beta_! X$.
\end{proposition}

\begin{proof}
Let $E$ be a $G$-prespectrum over $A\times B'$.
We first show that, for any $G$-prespectrum $Y$ over $B$, we have an isomorphism
\[
 (1\times\rho)_!(1\times\Delta)^*(E\exsmsh\beta_! Y)
  \iso (1\times\rho)_!(1\times\Delta)^*[(1\times\beta)^*E\exsmsh Y].
\]
Once more, we need to detour carefully around commutation relations we wish
were true but need not be in general. The following chain of isomorphisms works:
\begin{align*}
 (1\times\rho)_!(1\times\Delta)^*(E\exsmsh \beta_! Y)
  &\iso (1\times\rho)_![E\smsh_{A\times B'}(\rho\times 1)^*\beta_! Y] \\
  &\iso (1\times\rho)_![E\smsh_{A\times B'}(1\times\beta)_!(\rho\times 1)^*Y] \\
  &\iso (1\times\rho)_!(1\times\beta)_![(1\times\beta)^*E\smsh_{A\times B}(\rho\times 1)^*Y] \\
  &\iso (1\times\rho)_!(1\times\Delta)^*[(1\times\beta)^*E\exsmsh Y]
\end{align*}
Let $W$ be a $G$-prespectrum over $A$. We now have the following chain of isomorphisms:
\begin{align*}
 [W,(1\times\rho)_!(1\times\Delta)^*(E\exsmsh\beta_! \bar D_B X)]_{G,A}
 &\iso [W,(1\times\rho)_!(1\times\Delta)^*[(1\times\beta)^*E\exsmsh \bar D_B X]]_{G,A} \\
 &\iso [W\exsmsh X, (1\times\beta)^*E]_{G,A\times B} \\
 &\iso [W\exsmsh \beta_! X, E]_{G,A\times B'}
\end{align*}
Tracing through the adjunctions, we can see that this is $\epsilon'_\sharp$ where
$\epsilon'$ is the composite
\begin{align*}
 \beta_!Y\exsmsh\beta_!X &\iso (\beta\times\beta)_!(Y\exsmsh X) \\
  &\xrightarrow{\epsilon} (\beta\times\beta)_!\Delta_! S_B \\
  &\iso \Delta_!\beta_!S_B \\
  &\to \Delta_!S_{B'},
\end{align*}
where the last map is induced by the adjoint of the isomorphism
$S_B \xrightarrow{\iso} \beta^*S_{B'}$.
The proposition now follows from
Theorem~\ref{thm:dualcharacterization}.
\end{proof}

In particular, if $X$ is homologically finite over $B$,
then $\rho_!X$ is a finite nonparametrized spectrum and
$D\rho_! X \iso \rho_!\bar D_B X$.
Thus, the homological dual of $X$ is essentially the ordinary dual of the
underlying nonparametrized spectrum $\rho_! X$, except that it is constructed
so as to be parametrized by $B$.
For example, we have the following sequence of results, giving examples
of duals of many spaces.
(Our original proofs were modeled on
\cite[\S III.4]{LMS:eqhomotopy} and were rather involved.
Peter informs us that he no longer trusts the argument of that section
and we're not confident that our arguments were correct, either.
Given that May and Sigurdsson have since provided better proofs,
we shall defer to them for the proofs of the following results.)

We first introduce the following notion for spaces.

\begin{definition}
Let $X$ and $Y$ be ex-$G$-spaces over $B$.
We say that $X$ and $Y$ are {\em $V$-dual} if there exist
weak maps
\[
 \epsilon\colon Y\exsmsh X\to \Delta_! S_B \smsh S^V
\]
and
\[
 \eta\colon S^V\to \rho_!(X\smsh_B Y)
\]
such that the following diagrams stably commute:
\[
 \xymatrix{
  S^V\smsh X \ar[r]^-{\eta\smsh 1} \ar[d]_\gamma
   & (\rho\times 1)_![(1\times\rho)^*X\smsh_{B\times B}(Y\exsmsh X)]
      \ar[d]^{1\smsh\epsilon} \\
  X\smsh S^V 
   & (\rho\times 1)_![(1\times\rho)^*X\smsh_{B\times B}\Delta_! S_B \smsh S^V]
     \ar[l]^-\iso
  }
\]
and
\[
 \xymatrix{
  Y\smsh S^V \ar[r]^-{1\smsh\eta} \ar[dd]_\gamma
   & (1\times\rho)_![(Y\exsmsh X)\smsh_{B\times B}(\rho\times 1)^*Y]
     \ar[d]^{\epsilon\smsh 1} \\
   & (1\times\rho)_![(\Delta_!S_B\smsh S^V)\smsh_{B\times B}(\rho\times 1)^*Y]
     \ar[d]^\iso \\
  S^V\smsh Y \ar[r]_{\sigma\smsh 1} & S^V\smsh Y 
 }
\]
Here, $\gamma$ is transposition and $\sigma(v) = -v$.
\end{definition}

\begin{proposition}\cite[18.3.2]{MaySig:parametrized}
If $X$ and $Y$ are $V$-dual, then $\susp^\infty_B X$ is homologically finite
and $\bar D_B X \iso \susp^{-V}_B\susp^\infty_B Y$.
\qed
\end{proposition}

If $(X,A)\to B$ is any pair of (unbased) spaces over $B$, let $C_B(X,A)$ denote the
fiberwise mapping cone of $A\to X$ over $B$, with section given by the conepoints.
In particular, $C_B(X,\emptyset) = X_+$.

Let $X$ be a compact $G$-ENR and let $p\colon X\to B$. Take an embedding of $X$ as
an equivariant neighborhood retract in a $G$-representation $V$.
Let $r\colon N\to X$ be a $G$-retraction of an open neighborhood of $X$ in $V$.
Via $r$, we think of $N$ as a space over $X$, hence over $B$.

\begin{theorem}\label{thm:GENRduality}\cite[18.5.2]{MaySig:parametrized}
Let $(X,A)$ be a compact $G$-ENR pair over $B$ and let $V$ and $N$ be as above.
Then $C_B(X,A)$ is $V$-dual to $C_B(N-A,N-X)$, hence
$\susp^\infty_B C_B(X,A)$ is homologically finite with 
$\bar D_B\susp^\infty_B C_B(X,A) \iso \susp^{-V}\susp^\infty_B C_B(N-A, N-X)$.
\qed
\end{theorem}

As usual, we can specialize to smooth compact manifolds.
Let $M$ be a smooth compact $G$-manifold and let $p\colon M\to B$ be a $G$-map.
Take a proper embedding of $M$ in
$V'\times[0,\infty)$ for some $G$-representation $V'$,
with $\bndry M$ embedded in $V'\times 0$ and $M$ meeting
$V'\times 0$ transversely.
Write $V = V'\oplus\Real$.
Write $\nu$ for the normal bundle of the embedding of $M$ 
in $V'\times[0,\infty)$ and write
$\bndry\nu$ for the normal bundle of the embedding of $\bndry M$ in $V'$,
so $\bndry\nu = \nu|\bndry M$.
Write $S^\nu$ for the sectioned bundle over $M$ formed by taking the one-point
compactification of each fiber of $\nu$, the section being given by the
compactification points. Let $S^\nu_B = p_!S^\nu$.
In particular, $S^\nu_* = T\nu$, the Thom space of $\nu$.
We have the following version of Atiyah duality.

\begin{theorem}\label{thm:manifoldduality}\cite[18.7.2]{MaySig:parametrized}
Let $M$ be a compact $G$-manifold and let $p\colon M\to B$ be a $G$-map.
Embed $M$ in a $G$-representation $V$ as above.
Then $\susp_B^\infty M_+$ is homologically finite, with dual
$\susp^{-V}S^\nu_B\PQ{B}S^{\bndry\nu}_B$,
and $\susp_B^\infty M\PQ{B}\bndry M$ is homologically finite with dual
$\susp^{-V}S^\nu_B$.
The cofibration sequence
\[
 \bndry M_+ \to M_+ \to M\PQ{B}\bndry M \to \susp_B \bndry M_+
\]
is $V$-dual to the cofibration sequence
\[
 \susp_B S^{\bndry\nu}_B \from S^\nu_B\PQ{B}S^{\bndry\nu}_B
  \from S^\nu_B \from S^{\bndry\nu}_B.
\]
\qed
\end{theorem}

Note: In \cite{MaySig:parametrized}, Atiyah duality is actually shown first
and used to prove the more general duality theorem for $G$-ENRs.

Using homological duality we can now show that parametrized homology
theories are representable.

\begin{theorem}\label{thm:homrep}
If $h^G_*(-)$ is an $RO(G)$-graded parametrized homology theory 
on $\Ho G\PreSpec{} B$, then it is
represented by a $G$-prespectrum $E$ over $B$, in the sense
that, for each $V$, there is a natural isomorphism
 \[
  h^G_V(X) \iso [S^V,\rho_!(E\smsh_B X)]_G.
 \]
\end{theorem}

\begin{proof}
Define a cohomology theory $h_G^*(-)$ on homologically finite objects in
$\Ho G\PreSpec{} B$ by letting
\[
 h_G^\alpha(X) = h^G_{-\alpha}(\bar D_B X).
\]
By \cite[20.8.6]{MaySig:parametrized}, which can be generalized to the equivariant
context as discussed in \cite[\S 21.5]{MaySig:parametrized},
$h_G^*(-)$ is represented by a $G$-prespectrum $E$ over $B$.
This spectrum therefore represents
$h^G_*(-)$ on homologically finite objects, 
hence on finite cell spectra by \cite[18.2.1]{MaySig:parametrized}. 
The spectrum $E$ therefore represents $h^G_*(-)$ on all cell spectra by taking colimits,
hence it represents $h^G_*(-)$ on all of $\Ho G\PreSpec{}B$.
\end{proof}


%%%%%%%%%%%%%%%
\section{The spectra representing ordinary theories}


Let $B$ be a $G$-space and let $\MackeyOp S$ and $\Mackey T$ be, respectively,
a covariant and a contravariant $\stab\Pi B$-Mackey functor.
Fix a virtual spherical representation $\gamma$ of $\Pi B$.
We get reduced $RO(G)$-graded parametrized
homology and cohomology theories on spaces
or spectra parametrized by
$B$ (as in Definition~\ref{def:parametrizedtheory}) by sending
the pair $(V,X)$ to $\tilde H^G_{\gamma+V}(X;\MackeyOp S)$ or
to $\tilde H_G^{\gamma+V}(X;\Mackey T)$.
By Theorems~\ref{thm:homrep} and~\ref{thm:cohomrep},
there exist $G$-prespectra $H\MackeyOp S_\gamma$ and $H\Mackey T^\gamma$,
parametrized by $B$, representing these two theories.

%If $\gamma+V$ and $\delta+W$ are equivalent for some linear representations 
%$V$ and $W$ of $G$, then each choice of equivalence gives a natural isomorphism
%of homology or cohomology theories, hence an equivalence of parametrized $G$-prespectra
%$\susp^{-V}H\MackeyOp S_\gamma \hmtpc \susp^{-W}H\MackeyOp S_\delta$ and
%$\susp^V H\Mackey T^\gamma \hmtpc \susp^W H\Mackey T^\delta$.
%Thus, these spectra are interrelated, but we will not here explore these
%relationships further. 

If $X$ is a $G$-space over $A$, the canonical map $A\to B\Pi_G A$
induces a natural isomorphism
\[
 \tilde H^G_{\gamma}(X\to A;\MackeyOp S) \iso \tilde H^G_{\gamma}(X\to B\Pi_G A;\MackeyOp S)
\]
and similarly in cohomology. On the right, $\gamma$ and $\MackeyOp S$ denotes the corresponding objects
under the equivalence $\Pi_G(B\Pi_G A) \iso \Pi_G A$.
As a consequence, the spectra $H\MackeyOp S_\gamma$ and $H\Mackey T^\gamma$
are pullbacks of spectra over $B\Pi_G A$,
and it is sometimes convenient to think of them
as parametrized by $B\Pi_G A$ rather than $A$.

It follows from the dimension axiom in Theorem~\ref{thm:ghomology}
that each fiber of $H\MackeyOp S_\gamma$ or $H\Mackey T^\gamma$ is
a Mac\,Lane-Eilenberg spectrum or an Eilenberg-Mac\,Lane spectrum,
respectively, as in Chapter~\ref{chap:rog}.

If $E$ is any $G$-spectrum over $B$, define a 
contravariant functor $\Mackey\pi_\gamma E$
on $\stab\Pi B$ by
\[
 \Mackey\pi_\gamma E(b) = E_G^0(S^{\gamma,b}) = [S^{\gamma,b}, E]_{G,B}.
\]
This obviously defines a functor on
$\stab\Pi_\gamma B$, and we make it a functor on
$\stab\Pi B$ using the isomorphism of Theorem~\ref{thm:piXinvariance}.
It follows from obstruction theory
(see the end of \S\ref{sec:ordinarytheories})
that $H\Mackey T^\gamma$ is characterized, up to isomorphism in the stable category over $B$,
by the calculation
\[
 \Mackey\pi_{\gamma+n} H\Mackey T^\gamma \iso
 \begin{cases} \Mackey T & n = 0 \\ 0 & n \neq 0. \end{cases}
\]
We call any spectrum satisfying such an isomorphism a
{\em parametrized Eilenberg-Mac\,Lane spectrum.}

Similarly, we define a covariant functor $\MackeyOp\tau_\gamma E$
on $\stab\Pi B$ by
\[
 \MackeyOp\tau_{\gamma} E(b) = E^G_0(S^{\gamma,b})
  = [S, \rho_!(E\smsh_B S^{\gamma,b})]_G
  \iso [\bar D_B S^{\gamma,b}, E]_{G,B}.
\]
We then have the following calculation:
\[
 \MackeyOp\tau_{\gamma+n} H\MackeyOp S_\gamma \iso
  \begin{cases} \MackeyOp S & n = 0 \\ 0 & n \neq 0. \end{cases}
\]
We call $H\MackeyOp S_\gamma$ a 
{\em parametrized Mac\,Lane-Eilenberg spectrum},
and it follows from the dual obstruction theory to be outlined
in the following section that $H\MackeyOp S_\gamma$ is
characterized up to stable isomorphism by the calculation above.

We can now define theories dual to the $RO(\Pi B$)-graded ordinary theories as we did for the $RO(G)$-graded
ordinary theories in Section~\ref{sec:dualvcomplexes}.
As there, the distinctions we make exist only when $G$ is not finite.

\begin{definition}
Let $B$ be a $G$-space.
If $\Mackey T$ is a contravariant $\stab\Pi B$-Mackey functor, let 
{\em dual ordinary homology theory},
$\H^G_{\gamma+*}(-;\Mackey T)$,
be the homology theory on spaces or spectra over $B$ represented by $H\Mackey T^{-\gamma}$.
If $\MackeyOp S$ is a covariant $\stab\Pi B$-Mackey
functor, let 
{\em dual ordinary cohomology theory},
$\H_G^{\gamma+*}(-;\MackeyOp S)$, be the cohomology theory on spaces or spectra over $B$
represented by $H\MackeyOp S_{-\gamma}$.
 \end{definition}

These theories satisfy a list of properties.

\begin{theorem}\label{thm:dualghomology}
Let $X$ be a $G$-prespectrum parametrized by $B$.
Let $\gamma$ be a virtual spherical representation of $\Pi B$ and let 
$\MackeyOp S$ and $\Mackey T$ be respectively a covariant and a contravariant
$\stab\Pi B$-Mackey functor. Then the theories $\H_G^{\gamma+*}(X;\MackeyOp S)$
and $\H^G_{\gamma+*}(X;\Mackey T)$ satisfy the
following properties.

\begin{enumerate}
\item 
If $\alpha\colon A\to B$ and $X$ is a $G$-prespectrum over $A$, then
\[
 \H^G_\gamma(\alpha_!X;\Mackey T) 
  \iso \H^G_{\alpha^*\gamma}(X;\alpha^*\Mackey T)
\]
and
\[
 \H_G^\gamma(\alpha_!X;\MackeyOp S)
  \iso \H_G^{\alpha^*\gamma}(X;\alpha^*\MackeyOp S).
\]

\item 
(Dimension Axiom)
We have the following natural isomorphisms of functors of $b$
in $\stab\Pi B$:
 \[
 \H^G_{\gamma+n}(\bar D_B\susp^\infty_B S^{-\gamma,b};\Mackey T) \iso
  \begin{cases} \Mackey T(b) & \text{if $n = 0$,} \\
                0 & \text{if $n\neq 0$,}
  \end{cases}
 \]
 \[
 \H_G^{\gamma+n}(\bar D_B\susp^\infty_B S^{-\gamma,b};\MackeyOp S) \iso
  \begin{cases} \MackeyOp S(b) & \text{if $n = 0$,} \\
                0 & \text{if $n\neq 0$,}
  \end{cases}
 \]
for integers $n$.

\item\label{item:dualgWirthmuller} (Wirthm\"uller Isomorphism)
If $K \subset G$ and $X$ is a $K$-spectrum over $B$, there is an isomorphism
 \[
 \H_{\gamma}\sp G(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\Mackey T)
 \iso  \H_{\gamma|K}\sp K(X;\Mackey T|K),
 \]
and similarly in dual cohomology.
In dual homology, if $X$ is a $G$-spectrum over $B$, the composite
 \begin{align*}
 \H^G_\gamma(X;\Mackey T) 
 &\to \H^G_\gamma(D(G/K_+)\smsh X; \Mackey T) \\
 &\iso \H^G_\gamma(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\Mackey T) \\
 &\iso \H^K_{\gamma|K}(X;\Mackey T|K),
 \end{align*}
where the first map is induced by the dual of the projection $G/K\to *$, is
called restriction to $K$ and is denoted by $a \mapsto a|K$.
There is a similar restriction in cohomology, but with a
shift in dimension. We shall discuss this further in Section~\ref{sec:gsubgroups}.

\item\label{item:dualgfixedsets}
If $K$ is a normal subgroup of $G$, then there is a natural restriction
to fixed sets 
\[
 \H_{\gamma}\sp G(X;\Mackey T) \to
 \H_{\gamma \sp K}\sp {G/K}(\Phi^K(X);\Mackey T\sp K) , 
\]
and similarly in dual cohomology; we shall
denote this $a \mapsto a \sp K$.
This restriction respects suspension and restriction to subgroups in
the same way that restriction to fixed sets does for ordinary homology and cohomology,
and its evaluation at a point again gives the projection.
We shall also write $a^K$ for $(a|NK)^K$ when $K$ is not normal in $G$.

\item\label{item:dualgcupproduct} 
There is a natural associative cup product
 \[
 -\cup - \colon H\sp{\gamma}_G(X;\Mackey T) \tensor
 \H\sp{\delta}_G(Y;\MackeyOp S) 
 \to \H\sp{\gamma+\delta}_G(X\smsh_B Y; \Mackey T\mixprod \MackeyOp S). 
 \]
This cup product satisfies $(x \cup y)|K = (x|K) \cup (y|K)$
and $(x \cup y)\sp K = (x\sp K) \cup (y\sp K)$.
If $d\colon W\to X\smsh_B Y$ is a map we also write
$x\cup y$ for $d^*(x\cup y)$.


\item\label{item:dualgeval}
There are evaluation maps
 \[
 \langle -,- \rangle \colon \H^{\gamma}_G(X;\MackeyOp S) \tensor
  \H_{\gamma+\delta}^G(X;\Mackey T) \to H_\delta^G(\susp^\infty_B B_+; \MackeyOp S\mixprod\Mackey T)
 \]
and
 \[
 \langle -,- \rangle \colon H^{\gamma}_G(X;\Mackey T) \tensor
  \H_{\gamma+\delta}^G(X;\Mackey U) \to \H_\delta^G(\susp^\infty_B B_+; \Mackey T\boxprod\Mackey U).
 \]
These evaluations are natural in the usual sense: if $f\colon X\to Y$, then
$\langle y, f_*(x)\rangle = \langle f^*(y), x\rangle$. They also satisfy
$\langle x,a\rangle|K = \langle x|K, a|K\rangle$ and 
$\langle x,a\rangle^K = \langle x^K, a^K\rangle$.

\item\label{item:dualgcapproduct}
There are cap products
 \[
 -\cap - \colon \H\sp{\gamma}_G(Y;\MackeyOp S) \tensor
 \H_{\gamma + \delta}\sp G(X\smsh_B Y;\Mackey T) \to 
 H_{\delta}^G(X;\MackeyOp S \mixprod \Mackey T)
 \]
and
 \[
 -\cap - \colon H\sp{\gamma}_G(Y;\Mackey T) \tensor
 \H_{\gamma + \delta}\sp G(X\smsh_B Y;\Mackey U) \to 
 \H_{\delta}^G(X;\Mackey T \boxprod \Mackey U). 
 \]
These cap products are natural in the usual sense: if $f\colon X\to X'$
and $g\colon Y\to Y'$,
then $y'\cap (f\smsh g)_*(a) = f_*(g^*(y')\cap a)$.
They also satisfy $(x\cup y)\cap a = x \cap (y \cap a)$,
$(x \cap a)|K = (x|K)\cap (a|K)$, and
$(x\cap a)\sp K = x \sp K \cap a \sp K$.
Further, $\langle x\cup y, a \rangle = \langle x, y\cap a\rangle$.
If $d\colon W\to X\smsh_B Y$ is a map we also write
$y\cap a$ for $y\cap d_*(a)$.
 \end{enumerate}
 \end{theorem}

The dimension axiom follows from the definitions of the dual theories.
The remaining properties will be proved in the remainder of this chapter.


%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dual $G$-CW objects}

Before we discuss the properties of the various ordinary theories
we now have, we show how the dual ordinary theories can be computed
from chain complexes. These chain complexes come from
``dual $G$-CW complexes'' over $B$. Here are two motivating
examples.

\begin{examples}
\begin{enumerate}\item[]

\item\label{item:dualgmanifold}
If $M$ is a smooth $G$-manifold,
consider the cell structure on $M$ dual to a $G$-triangulation,
as in Example~\ref{ex:gammacomplexes}.
The cell dual to a $k$-simplex of the form
$G/H\times \Delta^k$, with barycenter $m$,
has the form $G\times_H D(\tau(m)-k-V(G/H))$,
where $\tau$ denotes the tangent representation of $\Pi_G M$
and $V(G/H)$ denotes the tangent plane to $eH$ in $G/H$.
We will want to think of this cell as a
dual $(\tau-k)$-dimensional cell.
Notice that the geometric dimension of the $G$-manifold
$G\times_H(\tau(m)-k-V(G/H))$ is $|\tau(m)|-k$.
Thus, when considering a dual cell of the form
$G\times_H D(V)$, we will be interested in its geometric
dimension $|V|+|V(G/H)|$ rather than just $|V|$.

\item
If $X$ is a homologically finite parametrized prespectrum, then
$\H_G^{\gamma}(X;\MackeyOp S) \iso H^G_{-\gamma}(\bar D_B X;\MackeyOp S)$.
Suppose further that $\bar D_B X \hmtpc Y$ where $Y$ is a finite
$G$-CW($-\gamma$) prespectrum.
Then $Y^{-\gamma+n}/_B Y^{-\gamma+n-1}$ is equivalent to a wedge
of spheres of the form $G\halfsmsh_H S^{-\gamma(b)+n,b}$.
The duals of the prespectra $Y/_B Y^{-\gamma+n}$ give a filtration
of $\bar D_B Y \hmtpc X$ whose filtration quotients are
equivalent to wedges of prespectra of the form
$\bar D_B(G\halfsmsh_H S^{-\gamma(b)+n,b})$, which we want
to think of as having dimension $\gamma-n$. However,
\[
 \bar D_B(G\halfsmsh_H S^{-\gamma(b)+n,b})
  \hmtpc G\halfsmsh_H S^{\gamma(b)-n-V(G/H),b}.
\]
(This calculation follows from Atiyah duality,
Theorem~\ref{thm:manifoldduality}.)
Thus, we again need to take into account the dimension of $G/H$
and say that the ``dual dimension'' of
$G\halfsmsh_H S^{V,b}$ is $V+V(G/H)$.

\end{enumerate}
\end{examples}


\subsection*{Dual $G$-CW($\gamma$) complexes}

We begin with parametrized spaces.
As usual when dealing with spaces, we assume that $\gamma$
is a stable spherical representation of $\Pi B$.

\begin{definition}
\begin{enumerate} \item[]
\item
A {\em dual $\gamma$-cell} is an object
$e = (G\times_H D(V),p)$ in
$G\ParaU B$ such that
$\gamma(G\times_H 0, p)$ is stably spherically isomorphic
to $G\times_H(V+V(G/H)+n)$ for some possibly negative integer $n$.
The {\em dimension} of $e$ (as a dual cell) is $|V|+|V(G/H)|$.
The {\em boundary} of $e$ is 
$\bndry e = (G\times_H S(V),p)$.

\item
A {\em dual $G$-CW($\gamma$) complex} is a filtered object
\[
 (X,p) = \colim(X^n,p^n)
\]
in $G\ParaU B$, where $X^0$ is a union of 0-dimensional dual $\gamma$-cells,
and each $(X^n,p^n)$, $n>0$, is obtained from $(X^{n-1},p^{n-1})$
by attaching $n$-dimensional dual $\gamma$-cells.
If $(X,A,p)$ is a pair of spaces over $B$, then we define
a {\em relative dual $G$-CW($\gamma$) structure} on $(X,A)$ in the obvious way.
We shall also write $X^{\gamma+n}$ for $X^{|\gamma|+n}$.
If we allow cells of any dimension to be attached at each stage, we get
the weaker notion of a {\em dual $\gamma$-cell complex.}

\end{enumerate}
\end{definition}

We shall use again the homotopy sets $\pi_e(f)$ defined
in Definition~\ref{def:ehomotopy}.

\begin{definition}
A lax map $f\colon X\laxto Y$ is a
{\em dual $(\gamma+n)$-equivalence}
if $\pi_e(f)$ is trivial for all dual $\gamma$-cells $e$
of dimension $\leq |\gamma|+n$;
$f$ is a {\em dual weak $\gamma$-equivalence} if it is
a dual $(\gamma+n)$-equivalence for all $n$.
\end{definition}

Dual weak $\gamma$-equivalence is implied by, but
in general is not equivalent to,
weak $G$-equivalence. The problem is that, for a given
$b\colon G/H\to B$, it may be impossible to find a
dual $\gamma$-cell with $b$ as its center.
This problem disappears on stabilization, as we shall see
when we discuss prespectra below.

However, dual $\gamma$-cell complexes behave well.
The following version of the homotopy extension and lifting property
is proved in the same way as previous versions.

\begin{lemma}[H.E.L.P.]
 Let $r\colon Y \laxto Z$ be a dual $(\gamma+n)$-equivalence.
Let $(X,A)$ be a relative
dual $\gamma$-cell complex
with cells of dimension
$\leq |\gamma|+n$. If the following diagram commutes in $G\LaxU B$ without the
dashed arrows, then there exist lax maps $\tilde g$ and $\tilde h$
making the diagram commute.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\times I \ar'[d][dd] \ar[dl]_(.6)h & &
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g \ar[dd]  \\
   & Z & & Y \ar[ll]_(.3)r \\
  X \ar[ur]^f \ar[rr]_-{i_0} & & X\times I \ar@{-->}[ul]^(.6){\tilde h} & &
   X \ar[ll]^-{i_1} \ar@{-->}[ul]^(.6){\tilde g} \\
 }
 \]
The result remains true when $n = \infty$.
\qed
\end{lemma}

Using the notation of Theorem~\ref{thm:ParaWhitehead}, the following
follows from the H.E.L.P. lemma in the usual way.

\begin{theorem}[Whitehead]\label{thm:DualParaWhitehead}
 \begin{enumerate}\item[]

\item If $r\colon Y\laxto Z$ is a dual $(\gamma+n)$-equivalence and $X$ is a
$(|\gamma|+n-1)$-dimensional
dual $\gamma$-cell complex,
then
\[
 r_*\colon \pi(G\LaxU B)(X,Y) \to \pi(G\LaxU B)(X,Z)
\]
is an isomorphism.
It is an epimorphism if $X$ is
$(|\gamma|+n)$-dimensional.

\item If $r\colon Y\laxto Z$ is a dual weak $\gamma$-equivalence and $X$ is a
dual $\gamma$-cell complex,
then 
\[
 r_*\colon \pi(G\LaxU B)(X,Y) \to \pi(G\LaxU B)(X,Z)
\]
is an
isomorphism.  In particular any dual weak $\gamma$-equivalence of dual $G$-CW($\gamma$) complexes
is a lax homotopy equivalence.

\item If $r\colon Y\laxto Z$ is a dual $(\gamma+n)$-equivalence, $(X,A)$ is a
relative $(|\gamma|+n-1)$-dimensional
dual $\gamma$-cell complex,
and $f\colon A\laxto Y$ is a lax map,
then
\[
 r_*\colon \pi(G\LaxU B)(X,Y)/f \to \pi(G\LaxU B)(X,Z)/rf
\]
is an isomorphism.
It is an epimorphism if $(X,A)$ is
$(|\gamma|+n)$-dimensional.

\item If $r\colon Y\laxto Z$ is a dual weak $\gamma$-equivalence, $(X,A)$ is a
relative dual $\gamma$-cell complex,
and $f\colon A\laxto Y$ is a lax map,
then 
\[
 r_*\colon \pi(G\LaxU B)(X,Y)/f \to \pi(G\LaxU B)(X,Z)/rf
\]
is an
isomorphism.
\qed

\end{enumerate}
\end{theorem}

Thus, although dual weak $\gamma$-equivalence is, in general, weaker than
weak $G$-equivalence, for dual $G$-CW($\gamma$) complexes these notions
concide and are equivalent to lax homotopy equivalence.
The point is that $G$-CW($\gamma$) complexes are limited in what
cells they can use, and this limitation
allows dual weak $\gamma$-equivalence to detect weak $G$-equivalence.

\begin{definition}
A lax map $f\colon X\laxto Y$ of dual $G$-CW($\gamma$) complexes is {\em cellular} if
$f(X^n)\subset Y^n$ for each $n$.
Cellular lax maps of relative dual $G$-CW($\gamma$)
complexes are defined similarly.
 \end{definition}

\begin{theorem}[Cellular Approximation]\label{thm:dualcellapprox}
 If $f\colon X\laxto Y$ is a lax map of dual $G$-CW($\gamma$) complexes that is
cellular on a subcomplex $A\subset X$, then $f$
is homotopic rel $A$, in $G\LaxU B$, to a cellular map.
A similar statement holds for relative dual $G$-CW($\gamma$) complexes.
 \end{theorem}

\begin{proof}
 This follows from the  H.E.L.P. lemma once we show that the inclusion
$Y^{\gamma+n}\to Y$ is a dual $(\gamma+n)$-equivalence.
This reduces to showing that, if 
$K\subset H\subset G$, $V$ is a representation of $K$ containing $V(G/K)$,
$(M,\bndry M)$ is a $K$-manifold contained in $D(V-V(G/K))$,
and $V'$ is a representation of $H$
containing $V(G/H)$ and stably spherically equivalent to $V$
as a representation of $K$, then there are no essential $G$-maps
$G\times_K(M,\bndry M)\to G\times_H(D(V'-V(G/H)+k),S(V'+V(G/H)+k))$ for $k>0$.
This follows from ordinary obstruction theory, using a $G$-CW decomposition of $G\times_K M$,
the point being that $G\times_K(V-V(G/K))$ 
and $G\times_H(V'-V(G/H))$ have the same (geometric) fixed-point dimensions.
\end{proof}

The following is proved exactly as was Theorem~\ref{thm:ghomotopytype},
except that we kill dual homotopy groups.

\begin{theorem}[Approximation by $G$-CW($\gamma$) complexes]
 Let $(X,A,p)$ be any pair of objects in $G\ParaU B$. Then there exists a 
relative dual $G$-CW($\gamma$)
complex $(\Gamma' X,A)$ and a dual weak $\gamma$-equivalence $\Gamma' X\to X$ under $A$ in $G\ParaU B$.
\qed
\end{theorem}

Note what this theorem does {\em not} say: Not every space over $B$ is
weakly equivalent to a dual $G$-CW($\gamma$) complex.
Even so, Whitehead's theorem still implies that the map
$\Gamma' X\to X$ is unique up to canonical lax homotopy equivalence
and that $\Gamma'$ is functorial up to lax homotopy.

As we did for $G$-CW($\gamma$) complexes, we can apply these results
to ex-spaces.

\begin{definition}
A {\em dual ex-$G$-CW($\gamma$) complex} is an ex-$G$-space $(X,p,\sigma)$
together with a relative dual $G$-CW($\gamma$) structure
on $(X,\sigma(B),p)$.
\end{definition}

The Whitehead theorem and the theorems on cellular approximation of maps
and ex-spaces now follow as they did for ex-$G$-CW($\gamma$) complexes.


\subsection*{Dual $G$-CW($\gamma$) prespectra}

Now let $\gamma$ be a virtual spherical representation of $\Pi B$.
Recall the definition of the ex-$G$-space $D^{V+1}_f$ given in
Definition~\ref{def:prespeccell}.

\begin{definition}
A {\em dual $\gamma$-cell} is a $G$-prespectrum of the form
$\susp^\infty_W D^{V+1}_f$, where $\gamma(f(G\times_H *))$
is virtually spherically isomorphic to $G\times_H(V-W+V(G/H)+ n)$
for some possibly negative integer $n$.
The {\em dimension} of such a cell is $|V|-|W|+|V(G/H)|+1$.
We will also use the term {\em dual $(\gamma+n)$-cell} to refer
to a dual $\gamma$-cell of dimension $|\gamma|+n$.
\end{definition}

\begin{definition}
A {\em dual $\gamma$-cell prespectrum} is a filtered $G$-prespectrum
\[
 X = \colim_{n\geq 0} X_n
\]
over $B$, where $X_0=*$ and, for $n>0$, each $X_n$ is obtained from
$X_{n-1}$ by attaching a wedge of dual $\gamma$-cells along their boundaries.
We call $\{X_n\}$ the {\em sequential filtration} of $X$.
A dual $\gamma$-cell prespectrum $X$ is a {\em dual $G$-CW($\gamma$) prespectrum}
if each cell of dimension $n$ is attached only to cells of dimension smaller than $n$;
for such a prespectrum we let the {\em skeleton} $X^n$ be the subprespectrum consisting of all cells of $X$ of dimension no more than $n$, and call $\{X^n\}$ the {\em skeletal filtration.}
We also write $X^{\gamma+n}$ for $X^{|\gamma|+n}$.
If $(X,A)$ is a pair of $G$-prespectra, we define a {\em relative}
dual $\gamma$-cell or $G$-CW($\gamma$) structure
on $(X,A)$ in the obvious way, by taking $X_0 = A$.
\end{definition}

The notion of a {\em costabilization} of a dual $\gamma$-cell prespectrum
is exactly as for an ordinary $\gamma$-cell prespectrum. Likewise, we can
talk about the costabilization system of a given dual $\gamma$-cell
prespectrum. Every map in the costabilization system is a weak equivalence.

The relevant homotopy groups are the covariant Mackey functors
$\MackeyOp \dualhmpty_{\gamma+n} X$. Note that, if $b\colon G/H\to B$
and $\gamma(b) = G\times_H(V-W)$,
\[
 \MackeyOp \dualhmpty_{\gamma+n}X(b)
  = [\bar D_B S^{\gamma+n,b},X]_{G,B}
  \iso [b_!\bar D_B S^{\gamma(b)+n}, X]_{G,B}
  \iso [\susp^{V-W}D(\susp^\infty G/H_+), X_b]_H.
\]
Thus,
\[
 \MackeyOp \dualhmpty_{\gamma+n}X(b)
  \iso \dualhmpty^H_{V-W}(X_b)
  \iso \dualhmpty^H_V(\susp^W X_b)
\]
in the notation of Definition~\ref{def:gendualhomotopy}.

\begin{definition}
Let $f\colon X\laxto Y$ be a lax map. We say that
$f$ is a {\em dual $(\gamma+n)$-equivalence} if
$\MackeyOp\dualhmtpy_{\gamma+i}(X)\to \MackeyOp\dualhmtpy_{\gamma+i}(Y)$
is an isomorphism for $i<n$ and an epimorphism for $i=n$.
We say that $f$ is a {\em dual weak $\gamma$-equivalence}
if it is a $(\gamma+n)$-equivalence for all $n$.
\end{definition}

From the discussion above, $f$ is a dual weak $\gamma$-equivalence if and only if
each map of fibers $f_b\colon X_b\to Y_b$ is a dual weak $(V-W)$-equivalence
(where $\gamma(b) = G\times_H(V-W)$). By this we mean that
$\susp^W X_b\to \susp^W Y_b$ is a dual weak $V$-equivalence in the sense of
Definition~\ref{def:dualVequivalence}.
Using Corollary~\ref{cor:dualweakequivalence}
we now get the following.

\begin{theorem}
A lax map $f\colon X\laxto Y$ of $G$-prespectra over $B$ is
a dual weak $\gamma$-equivalence if an only if it is a weak equivalence.
\qed
\end{theorem}

The following form of the H.E.L.P. lemma is shown in the same way
as Lemma~\ref{lem:HELPParaSpec}.

\begin{lemma}[H.E.L.P.]
 Let $r\colon Y \laxto Z$ be a dual $(\gamma+n)$-equivalence of $G$-prespectra over $B$.
Let $(X,A)$ be a relative dual $\gamma$-cell prespectrum with cells of dimension
$\leq |\gamma|+n$.
Suppose that the following diagram commutes, with the maps to $Y$ and $Z$ being lax maps.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\smsh_B I_+ \ar[dl]_(.6)h & &
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g   \\
   & Z & & Y \ar[ll]_r \\
  X \ar[ur]^f  \\
 }
 \]
Then there exists a relative costabilization $(X',A)\to (X,A)$ and
lax maps $\tilde g$ and $\tilde h$
making the following diagram commute, where $f'$ is the composite
$X'\to X\laxto Z$.
 \[
 \xymatrix{
  A \ar[rr]^-{i_0} \ar[dd] & & A\smsh_B I_+ \ar'[d][dd] \ar[dl]_(.6)h & &
  A \ar[ll]_-{i_1} \ar[dl]_(.6)g \ar[dd]  \\
   & Z & & Y \ar[ll]_(.3)r \\
  X' \ar[ur]^{f'} \ar[rr]_-{i_0} & & X'\smsh_B I_B \ar@{-->}[ul]^(.6){\tilde h} & &
   X' \ar[ll]^-{i_1} \ar@{-->}[ul]^(.6){\tilde g} \\
 }
 \]
The result remains true when $n = \infty$.
\qed
\end{lemma}

We have the following version of the Whitehead theorem.

\begin{theorem}[Whitehead]
Let $X$ be a dual $\gamma$-cell prespectrum with costabilization system
${}_*X$,
let $Y$ and $Z$ be $G$-prespectra over $B$, and let
$f\colon Y\laxto Z$ be a lax map.

\begin{enumerate}

\item If $f$ is a dual $(\gamma+n)$-equivalence and $X$ is
$(\gamma+n-1)$-dimensional, then
\[
 f_*\colon \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Y) \to
  \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Z)
\]
is an isomorphism.
It is an epimorphism if $X$ is
$(\gamma+n)$-dimensional.

\item If $f$ is a dual weak $\gamma$-equivalence
and $X$ is an arbitrary $\gamma$-cell prespectrum, then
\[
 f_*\colon \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Y) \to
  \colim_\alpha \pi G\LaxPreSpec{}B({}_\alpha X, Z)
\]
is an
isomorphism.

\item
If $f\colon X\to Y$ is a dual weak $\gamma$-equivalence of dual $\gamma$-cell complexes,
then $f$ is a pro-homotopy equivalence.

\end{enumerate}
\end{theorem}

\begin{theorem}[Cellular Approximation of Maps]
If $f\colon X\laxto Y$ is a lax map of dual $G$-CW($\gamma$) prespectra
that is cellular on a subcomplex $A\subset X$, then
there exists an $\alpha$ such that the composite
${}_\alpha X\to X\laxto Y$ is homotopic rel $A$ to a cellular map.
\end{theorem}

\begin{proof}
This follows from the H.E.L.P. lemma
(applied to each cell in an induction on the sequential filtration) once we know that the inclusion
$Y^{\gamma+n}\to Y$ is a dual $(\gamma+n)$-equivalence.
But, in fact, $Y^{\gamma+n}(W)\to Y(W)$ is a dual $(W+\gamma+n)$-equivalence,
by an argument similar to that given for
Theorem~\ref{thm:dualcellapprox}.
\end{proof}

The following is proved in the same way as Theorem~\ref{thm:specapprox}.

\begin{theorem}[Approximation by Dual $G$-CW($\gamma$) Prespectra]
Let $X$ be any $G$-prespectrum over $B$. Then there exists
a dual $G$-CW($\gamma$) prespectrum $\Gamma' X$ and a weak equivalence
$\Gamma' X\to X$ in $G\PreSpec{}B$.
\end{theorem}


\subsection*{Cellular dual homology and cohomology}

\begin{definition}
Let $X$ be a dual $G$-CW($\gamma$) prespectrum over $B$.
The {\em dual chain complex} of $X$ is the chain complex of
covariant $\Pi B$-Mackey functors given by
\[
 \MackeyOp C^G_{\gamma+n}(X) =
 \MackeyOp C_{\gamma+n}(X) =
    \MackeyOp\dualhmtpy_{\gamma+n}(X^{\gamma+n}/_B X^{\gamma+n-1}).
\]
\end{definition}

Using the direct arguments of
\cite[15.2 \& 19.5]{May:concise}, we can now show the following.

\begin{theorem}
If $X$ is a dual $G$-CW($\gamma$) prespectrum over $B$ then
we have the following isomorphisms, natural in cellular maps:
\[
 \H^G_{\gamma+n}(X;\Mackey T) \iso
   H_{n}(\MackeyOp C_{\gamma+*}(X)\tensor_{\stab\Pi B}\Mackey T)
\]
and
\[
 \H_G^{\gamma+n}(X;\MackeyOp S) \iso
   H_{n}(\Hom_{\stab\Pi B}(\MackeyOp C_{\gamma+*}(X),\MackeyOp S)).
\]
\qed
\end{theorem}

Using the results above, showing that we can approximate any $G$-prespectrum
over $B$ with a dual $G$-CW($\gamma$) prespectrum and that we can approximate
any map with a cellular map, we conclude that we can calculate the dual
homology and cohomology functors via dual chain complexes.
We are then entitled to the following spectral sequences.

\begin{theorem}[Dual Atiyah-Hirzebruch Spectral Sequence]
Suppose that $h^G_*(-)$ is an $RO(G)$-graded homology theory on
ex-$G$-spaces over $B$ or $G$-prespectra over $B$
and that $h_G^*(-)$ is its dual theory.
Let $\alpha$ be an element of $RO(G)$ and let
$\beta$ be a formal difference of spherical representations of $\Pi B$.
Let $X$ be an ex-$G$-space or $G$-prespectrum over $B$ that is bounded below.
Then there is a strongly convergent spectral sequence
 \[
 E^2_{p,q} = \H_{\alpha-\beta+p}^G(X;\Mackey h_G^{-\beta-q})
 \convto h^G_{\alpha+p+q}(X)
 \]
and a conditionally convergent spectral sequence
 \[
 E_2^{p,q} = \H^{\alpha-\beta+p}_G(X;\MackeyOp h^G_{-\beta-q})
 \convto h_G^{\alpha+p+q}(X).
 \]
Here, $\Mackey h_G^*$ and 
$\MackeyOp h^G_*$ are as in Definition~\ref{def:homologymackey}.
\qed
 \end{theorem}

For the universal coefficients spectral sequence, we write
 \[
 \MackeyOp\H_*^G(X)(b\colon G/K\to B) = \H_*^G(X;\stab\Pi B(-,b)).
 \]

\begin{theorem}[Dual Universal Coefficients Spectral Sequence]
\label{thm:dualgucss}
If $\MackeyOp S$ is a covariant $\stab\Pi B$-Mackey functor and $\Mackey T$ is a
contravariant $\stab\Pi B$-Mackey functor, there are spectral sequences
 \[
 E^2_{p,q} = \Tor^{\stab\Pi B}_p(\MackeyOp\H^G_{\alpha+q}(X), \Mackey T)
 \convto
 \H^G_{\alpha+p+q}(X;\Mackey T)
 \]
and
 \[
 E_2^{p,q} = \Ext_{\stab\Pi B}^p(\MackeyOp\H^G_{\alpha+q}(X), \MackeyOp S)
 \convto
 \H_G^{\alpha+p+q}(X;\MackeyOp S).
 \]
\qed
\end{theorem}


%%%%%%%%%%%%%%%%
\section{Subgroups}\label{sec:gsubgroups}

In this section we prove parts \ref{item:gWirthmuller} and \ref{item:gfixedsets} of
Theorem~\ref{thm:ghomology} and the corresponding parts 
of Theorem~\ref{thm:dualghomology}. 
Since we shall be using several different groups, we shall write
$\Mackey C^G_*(X)$ for the $G$-chain complex of $X$.

\subsection*{Restriction to Subgroups}

Let $\iota\colon K\to G$ be the inclusion of a (closed) subgroup.
in \cite[\S 14.3]{MaySig:parametrized}, May and Sigurdsson define
the forgetful functor 
\[
 \iota^*\colon G\PreSpec{} B \to K\PreSpec{} B
\]
and, for $A$ a $K$-space, an induction functor 
\[
 \iota_!\colon K\PreSpec{} A\to G\PreSpec{}{G\times_K A},
\]
an analogue of the space level functor that takes a $K$-space
$Z$ to the $G$-space $G\times_K Z$.
In fact, $\iota_!$ is an equivalence of categories with inverse
$\nu^*\circ\iota^*$, where $\nu\colon A\to G\times_K A$ is the
$K$-map $\nu(a) = [e,a]$. If $\epsilon\colon G\times_K B\to B$ is the
map $\epsilon[g,b] = gb$, we let
\[
 i^G_K = \epsilon_! \iota_! \colon K\PreSpec{}B\to G\PreSpec{}B.
\]
(Note that this makes sense only when $B$ is a $G$-space.)
Then $i^G_K$ is left adjoint to $\iota^*$: If $X$ is a $K$-spectrum over $B$
and $E$ is a $G$-spectrum over $B$, then
\[
 [ i^G_K X, E ]_{G,B} \iso [X, (\iota_!)^{-1}\epsilon^* E]_{K,B} 
   \iso [X, \iota^* E]_{K,B}.
\]
The second isomorphism follows from the natural isomorphism
$(\iota_!)^{-1}\epsilon^* \iso \nu^*\iota^*\epsilon^* \iso \iota^*$,
the last isomorphism being a special case of
\cite[14.3.3]{MaySig:parametrized}.
We shall also write $G\halfsmsh_K X$ for $i^G_K X$
and $E|K$ for $\iota^* E$. With this notation we have
\[
 [G\halfsmsh_K X, E]_{G,B} \iso [X, E|K]_{K,B}.
\]

\begin{definition}
Let $B$ be a $G$-space.
Let $K$ be a subgroup of $G$ and let $\stab\Pi_K B$ and $\stab\Pi_G B$ be the
stable fundamental groupoids of $B$ thought of as a $K$-space and as a $G$-space,
respectively.
Let $i_K^G = G\halfsmsh_K- \colon \stab\Pi_K B\to \stab\Pi_G B$
be the restriction to $\stab\Pi_K B$ of $i_K^G$ defined as above.
In particular, its effect on an object $b\colon K/L\to B$ is
\[
 i_K^G(b) = G\times_K b\colon G/L = G\times_K (K/L) \to G\times_K B\to B.
\]
If $\Mackey T$ is a contravariant $\stab\Pi_G B$-Mackey functor, let
$\Mackey T|K$ denote $(i_K^G)^*\Mackey T = \Mackey T\circ i_K^G$
as in Definition~\ref{def:indres}. 
Similarly, if $\MackeyOp S$ is a
covariant Mackey functor, let $\MackeyOp S|K = (i_K^G)^*\MackeyOp S$.
If $\Mackey C$ is a contravariant $\stab\Pi_K B$-Mackey functor, let 
$G\times_K \Mackey C$ denote $(i_K^G)_*\Mackey C$.
If $\MackeyOp D$ is a covariant $\stab\Pi_K B$-Mackey functor, let 
$G\times_K\MackeyOp D = (i_K^G)_*\MackeyOp D$.
 \end{definition}
 
It follows from Proposition~\ref{prop:indresadjunction} that
 \[
 \Hom_{\stab\Pi_G B}(G\times_K \Mackey C, \Mackey T)
 \iso \Hom_{\stab\Pi_K B}(\Mackey C, \Mackey T|K)
 \]
for any contravariant $\stab\Pi_K B$-Mackey functor $\Mackey C$ 
and contravariant $\stab\Pi_G B$-Mackey
functor $\Mackey T$. The same adjunction holds for covariant Mackey functors.
We also have the isomorphisms
 \[
 (G\times_K \Mackey C)\tensor_{\stab\Pi_G B} \MackeyOp S
 \iso \Mackey C\tensor_{\stab\Pi_K B} (\MackeyOp S|K)
 \]
if $\MackeyOp S$ is a covariant $\stab\Pi_G B$-Mackey functor, and
 \[
 (G\times_K \MackeyOp D)\tensor_{\stab\Pi_G B} \Mackey T
  \iso \MackeyOp D \tensor_{\stab\Pi_K B} (\Mackey T|K).
 \]
Finally, we have the calculations
 \[
 G\times_K \Mackey A_b \iso \Mackey A_{G\times_K b}
 \]
and
 \[
 G\times_K \MackeyOp A^{b} \iso \MackeyOp A^{G\times_K b}.
 \]
Here, $b\colon K/L\to B$, $\Mackey A_b = \stab\Pi_K B(-,b)$,
and $\MackeyOp A^b = \stab\Pi_K B(b,-)$.

\begin{definition}
Let $\gamma$ be a representation of $\Pi_G B$ and let
$K$ be a subgroup of $G$. We define $\gamma|K$ to be the
representation of $\Pi_K B$ whose value on
$b\colon K/L\to B$ is the restriction to $K/L$
of $\gamma(G\times_K b)$ over $G/L$.
Its values on morphisms is defined by restriction similarly.
\end{definition}

The following is 
part \ref{item:gWirthmuller} of Theorem~\ref{thm:ghomology}.

\begin{theorem}[Wirthm\"uller Isomorphisms]
If $K\subset G$ and 
$X$ is a $K$-spectrum over the $G$-space $B$,
there are natural isomorphisms
 \[
 H_{\gamma}\sp G(G\halfsmsh_K X;\MackeyOp S)
 \iso  H_{\gamma|K}\sp K(X;\MackeyOp S|K)
 \]
and
 \[
 H^{\gamma}_G(G\halfsmsh_K X;\Mackey T)
 \iso  H^{\gamma|K}_K(X;\Mackey T|K).
 \]
\qed
 \end{theorem}

The proof is essentially the same as the similar statement for
the $RO(G)$-graded theories,
as is the proof of the following version for the dual theories,
which is part \ref{item:dualgWirthmuller} of
Theorem~\ref{thm:dualghomology}.

\begin{theorem}[Dual Wirthm\"uller Isomorphisms]
If $K \subset G$ and $X$ is a $K$-spectrum over the $G$-space $B$, 
there are natural isomorphisms
 \[
 \H_{\gamma}\sp G(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\Mackey T)
 \iso  \H_{\gamma|K}\sp K(X;\Mackey T|K)
 \]
and
 \[
 \H^{\gamma}_G(G\halfsmsh_K (S^{-V(G/K)}\smsh X);\MackeyOp S)
 \iso  \H^{\gamma|K}_K(X;\MackeyOp S|K).
 \]
\qed
 \end{theorem}


On the represented level, the
ordinary cohomology Wirthm\"uller isomorphism is just the adjunction
 \[
 [G\halfsmsh_K X,H\Mackey T^\gamma]_{G,B} \iso [X,(H\Mackey T^\gamma)|K]_{K,B}
 \]
together with the equivalence $(H\Mackey T^\gamma)|K \iso H(\Mackey T|K)^{\gamma|K}$.
This equivalence follows from the calculation
\[
 \Mackey\pi^K_{\gamma|K+n} (H\Mackey T^\gamma |K)
 \iso (i_K^G)^*(\Mackey\pi^G_{\gamma+n} H\Mackey T^\gamma)
 \iso \begin{cases} \Mackey T|K & n = 0 \\ 0 & n \neq 0 \end{cases}
\]
and the characterization of $H(\Mackey T|K)$ by its homotopy groups.
The ordinary homology isomorphism is given by
\begin{align*}
 [S,\rho_!(H\MackeyOp S_\gamma \smsh_B G\halfsmsh_K X) ]_G
 &\iso [S,G\halfsmsh_K \rho_!((H\MackeyOp S_\gamma)|K \smsh_B X)]_G \\
 &\iso [S, \susp^{V(G/K)}\rho_!((H\MackeyOp S_\gamma)|K \smsh_B X)]_K \\
 &\iso [S, \rho_!(\susp^{V(G/K)}(H\MackeyOp S_\gamma)|K \smsh_B X)]_K \\
 &\iso [S, \rho_!(H(\MackeyOp S|K)_{\gamma|K}\smsh_B X)]_K.
\end{align*}
Here, the second isomorphism is shown in \cite[II.6.5]{LMS:eqhomotopy} and the final is
given by the equivalence 
$\susp^{V(G/K)}(H\MackeyOp S_\gamma)|K \iso H(\MackeyOp S|K)_{\gamma|K}$,
which follows from the characterization of Mac\,Lane-Eilenberg spectra and
the argument of Proposition~\ref{prop:mesubgroups}. 
That these isomorphisms agree with the ones constructed on the chain level above
follows by comparing their behavior on 
the filtration quotients $X^{\gamma+n}/_B X^{\gamma+n-1}$, which are just wedges of 
suspensions of orbits.

The dual homology and cohomology isomorphisms now follow as in the
$RO(G)$-graded case.


\subsection*{Restriction to Fixed Sets}

We shall now restate and prove part \ref{item:gfixedsets} of 
Theorem~\ref{thm:ghomology}, and the
corresponding part of Theorem~\ref{thm:dualghomology}. 
Until further notice, $K$ will be a {\em normal} subgroup of $G$.
We first give some definitions and constructions.

\begin{definition}
If $\MackeyOp S$ is a covariant $\stab\Pi_G B$-Mackey functor and $K$ is a normal
subgroup of $G$, let
$\MackeyOp S_K$ be the sub-Mackey functor of $\MackeyOp S$ generated by the
elements in the groups $\MackeyOp S(b\colon G/L\to B)$ for those $L$ not containing $K$. Let
$\MackeyOp S^K$ be the $\stab\Pi_{G/K} B^K$-Mackey functor defined by
 \[
 \MackeyOp S^K(b) = \MackeyOp S(b)/\MackeyOp S_K(b)
 \]
for $b\colon G/L\to B$, where $K\subset L\subset G$.
If $\Mackey T$ is a contravariant $\stab\Pi_G B$-Mackey functor, define $\Mackey T_K$ and
$\Mackey T^K$ in exactly the same way.
 \end{definition}

If $E$ is a $G$-spectrum over $B$ then $E^K$ is a $G/K$-spectrum over $B^K$
\cite[\S 14.4]{MaySig:parametrized}.
We define the geometric fixed-point construction
\[
 \Phi^K(E) = (\tilde E\F[K]\smsh E)^K
\]
as in Definition~\ref{def:geometricfixed}.
In this context, $\Phi^K$ continues to enjoy the properties listed
after that definition.

We also have the following analogue of Proposition~\ref{prop:burnsidefixedsets}.

\begin{proposition}\label{prop:Xburnsidefixedsets}
If $K$ is normal in $G$ and $b\colon G/L\to B$, then
 \[
 (\MackeyOp A^b)^K \iso \{(b^K)_+, -\}_{G/K,B^K}
  = \begin{cases} \{b_+, -\}_{G/K,B^K} & \text{if $K\subset L$} \\
                  0 & \text{if $K\not\subset L$}
    \end{cases}
 \]
and
 \[
 (\Mackey A_b)^K \iso \{-,(b^K)_+\}_{G/K,B^K}
  = \begin{cases} \{-, b_+\}_{G/K,B^K} & \text{if $K\subset L$} \\
                  0 & \text{if $K\not\subset L$.}
    \end{cases}
 \]
 \end{proposition}

\begin{proof}
The proof is essentially the same as the proof of
Proposition~\ref{prop:burnsidefixedsets}, using
Proposition~\ref{prop:geostable}.
 \end{proof}

The following is part~\ref{item:gfixedsets} of
Theorem~\ref{thm:ghomology}.

\begin{theorem}[Restriction to Fixed-Sets]\label{thm:gfixedsets}
If $X$ is a $G$-spectrum over $B$ and
$K$ is a normal subgroup of $G$, there is a natural restriction
to fixed sets 
 \[
 (-)^K\colon H_{\gamma}\sp G(X;\MackeyOp S) \to
 H_{\gamma\sp K}\sp {G/K}(\Phi^K(X);\MackeyOp S\sp K).
 \]
This restriction respects suspension in the sense that the following
diagram commutes:
 \[
 \xymatrix{
  {H_{\gamma}\sp G(X;\MackeyOp S)} \ar[r]^-{\sigma_V} \ar[d]_{(-)^K}
   & { H_{\gamma + V}\sp G(\susp^V X;\MackeyOp S)} \ar[d]^{(-)^K} \\
  { H_{\gamma^K}\sp {G/K}(\Phi^K(X);\MackeyOp S^K)} \ar[r]^-{\sigma_{V^K}}
   & { H_{\gamma^K + V^K}\sp {G/K}(\susp^{V^K} \Phi^K(X);\MackeyOp S^K)}
  }
 \]
(and similarly for cohomology).
It respects restriction to subgroups in the sense that, if $K\subset L$,
then $(a|L)^K = (a^K)|(L/K)$.
\qed
 \end{theorem}

The proof is again essentially the same as that of
Theorem~\ref{thm:vfixedsets}.

The dual results are similar.
The following is a restatement of part~\ref{item:dualgfixedsets}
of Theorem~\ref{thm:dualghomology}
and is proved just as in the $RO(G)$-graded case.

\begin{theorem}[Restriction to Fixed-Sets in the Dual Theories]
\label{thm:dualgfixedsets}
If $X$ is a $G$-spectrum over $B$ and
$K$ is a normal subgroup of $G$, there is a natural restriction
to fixed sets 
 \[
 \H_{\gamma}\sp G(X;\Mackey T) \to
 \H_{\gamma \sp K}\sp {G/K}(\Phi^K(X);\Mackey T\sp K)
 \]
and similarly in dual cohomology.
This restriction respects suspension and restriction to subgroups 
as in Theorem~\ref{thm:gfixedsets}.
\qed
 \end{theorem}


As in the $RO(G)$-graded case, the key to representing restriction to fixed sets
on the spectrum level is the following result.

\begin{proposition}\label{prop:gemfixedsets}
Let $\Mackey T$ be a contravariant $\stab\Pi_G B$-Mackey functor and let $K$ be a normal
subgroup of $G$.
Then
 \[
 \Mackey\pi^{G/K}_{\gamma^K+n}\Phi^K(H\Mackey T^\gamma) \iso 
  \begin{cases} \Mackey T^K & n = 0 \\ 0 & n < 0. \end{cases}
 \]
Hence, there is a $G/K$-map
$\Phi^K(H\Mackey T^\gamma)\to H(\Mackey T^K)^{\gamma^K}$ over $B$
that is an isomorphism
in $\Mackey\pi_{\gamma^K}^{G/K}$.
Similarly, let 
$\MackeyOp S$ be a covariant $\stab\Pi_G B$-Mackey functor and let $K$
be a normal subgroup of $G$. Then 
 \[
 \MackeyOp\dualhmtpy^{G/K}_{\gamma^K+n}\Phi^K(H\MackeyOp S_\gamma) \iso 
 \begin{cases} \MackeyOp S^K & n = 0 \\ 0 & n < 0 \end{cases}.
 \]
Hence, there is a $G/K$-map
$\Phi^K(H\MackeyOp S_\gamma) \to H(\MackeyOp S^K)_{\gamma^K}$ that is an isomorphism in
$\MackeyOp\dualhmtpy^{G/K}_{\gamma^K}$.
\end{proposition}

\begin{proof}
The calculations of the homotopy groups follow from
Corollaries~\ref{cor:emfixedsets} and \ref{cor:mefixedsets}
applied to each fiber.
\end{proof}


Finally, as in the $RO(G)$-graded case, there can be a dimension shift
when restricting to fixed sets by a subgroup that is not normal in $G$.
As before, we let $a^K = (a|NK)^K$ if $K$ is not normal.
In ordinary cohomology and dual homology there is no dimension shift,
but in ordinary homology the map is
 \[
 (-)^K\colon H^G_{\gamma}(X;\MackeyOp S)
  \to H^{WK}_{\gamma^K + V(G/NK)^K}(\Phi^K(X);\MackeyOp S^K)
 \]
and in dual cohomology the map is
 \[
 (-)^K\colon \H_G^{\gamma}(X;\MackeyOp S)
  \to \H_{WK}^{\gamma^K - V(G/NK)^K}(\Phi^K(X);\MackeyOp S^K).
 \]


%%%%%%%%%%%%%%%%%
\section{Products}\label{sec:gproducts}

In this section we prove parts~\ref{item:gcupproduct},
\ref{item:geval} and \ref{item:gcapproduct} of
Theorem~\ref{thm:ghomology}, and the corresponding parts of
Theorem~\ref{thm:dualghomology}. 

\subsection*{Cup Products}

As in the $RO(G)$-graded case,
it is convenient to define and
work with an {\em external} cup product.
We use the external smash product of parametrized spectra
that takes a $G$-spectrum $X$ over $B$ and
a $G'$-spectrum $X'$ over $B'$ and gives the
$G\times G'$-spectrum $X\exsmsh X'$ over $B\times B'$.
When $G = G'$ and $B = B'$, we can internalize by restricting
along the diagonals of both $G\times G$ and $B\times B$.

Mimicking the $RO(G)$-graded case,
we defining the tensor product of Mackey
functors as follows.

\begin{definition}
Let $B$ be a $G$-space and let $B'$ be a $G'$-space.
Let $p\colon \stab\Pi_G B\times \stab\Pi_{G'}B' \to 
\stab\Pi_{G\times G'}(B\times B')$ be the functor given by
the external smash product.
In particular, if $b\colon G/H\to B$ and
$b'\colon G'/H' \to B'$ then
\[
 p(b\times b') = b\times b'\colon G/H \times G'/H'
  = (G\times G')/(H\times H') \to B\times B'.
\]
Let $\Mackey T$ be a contravariant $\stab\Pi_G B$-Mackey functor and let 
$\Mackey U$ be a
contravariant $\stab\Pi_{G'}B'$-Mackey functor. Let $\Mackey T\tensor \Mackey U$ denote the 
$(\stab\Pi_G B\times \stab\Pi_{G'}B')$-functor defined by
 \[
  (\Mackey T\tensor \Mackey U)(b,b') = \Mackey T(b)\tensor \Mackey U(b').
 \]
Define the contravariant $\stab\Pi_{G\times G'}(B\times B')$-Mackey functor
$\Mackey T \exboxprod \Mackey U$ (the {\em external box product}) by
 \[
 (\Mackey T \exboxprod \Mackey U) = p_*(\Mackey T\tensor \Mackey U),
 \]
where $p_*$ is defined as in Definition~\ref{def:indres}.
We define the
{\em (internal) box product} in two stages.
If $\Mackey T$ is a $\stab\Pi_G B$-Mackey functor and $\Mackey U$ 
is a $\stab\Pi_G B'$-Mackey functor (same group $G$), we first define
\[
 \Mackey T\exboxprod_G \Mackey U = (\Mackey T\exboxprod\Mackey U)|G
\]
where we restrict along the diagonal $G\subset G\times G$.
If $\Mackey T$ and $\Mackey U$ are both $\stab\Pi_G B$-Mackey functors, we define
\[
 \Mackey T\boxprod \Mackey U = (\Mackey T\exboxprod_G \Mackey U)|B
\]
where we restrict along the diagonal of $B\times B$.
 \end{definition}

The following facts are easy to check.

\begin{proposition}\label{prop:gburnsideboxproduct}
\begin{enumerate} \item[]
 \item $\Mackey A_{b} \exboxprod \Mackey A_{b'} \iso \Mackey A_{b\times b'}$
if $b\colon G/H\to B$ and $b'\colon G'/H'\to B'$.
%More generally, if $E$ is a $G$-space and $F$ is a $K$-space then
%$\{-,E\}_G\exboxprod \{-,F\}_K \iso \{-,E\times F\}_{G\times K}$.
 \item $\Mackey A_{G/G} \exboxprod_G \Mackey T \iso \Mackey T$
for any $\Pi_G B$-Mackey functor $\Mackey T$, where we think of
$\Mackey A_{G/G}$ as a $\stab\Pi_G(*)$-functor.
 \item If $X$ is a $G$-CW($\gamma$) spectrum over $B$
  and $X'$ is a $G'$-CW($\gamma'$) spectrum, then
 \[
 \Mackey C^G_{\gamma+*}(X) \exboxprod \Mackey C^{G'}_{\gamma'+*}(X')
  \iso \Mackey C^{G\times G'}_{\gamma\oplus \gamma' + *} (X\exsmsh X').
 \]
 \end{enumerate}
 \qed
 \end{proposition}

Now let $X$ be a $G$-spectrum over $B$ and let $X'$ be a $G'$-spectrum over $B'$.
The external box product induces a natural chain map
\begin{multline*}
 \Hom_{\stab\Pi_G B}(\Mackey C^G_*(X), \Mackey T) \tensor
 \Hom_{\stab\Pi_{G'}B'}(\Mackey C^{G'}_*(X'), \Mackey U) \\
 \to
 \Hom_{\stab\Pi_{G\times G'}(B\times B')}(
    \Mackey C^G_*(X)\exboxprod \Mackey C^{G'}_*(X'), 
    \Mackey T \exboxprod \Mackey U).
\end{multline*}
This induces the (external) cup product
 \[
 -\cup - \colon 
 H\sp{\gamma}_ G(X;\Mackey T) \tensor
 H\sp{\gamma'}_ {G'}(X';\Mackey U) \to H\sp{\gamma+\gamma'}_
 {G\times G'}(X\exsmsh X');\Mackey T\exboxprod \Mackey U).
 \]

When $G = G'$, we can follow the external cup product with the restriction to the
diagonal subgroup $G\subset G\times G$ to get the cup product
 \[
 -\cup - \colon 
 H\sp{\gamma}_ G(X;\Mackey T) \tensor
 H\sp{\gamma'}_ G(X';\Mackey U) \to H\sp{\gamma+\gamma'}_G
   (X\exsmsh X';\Mackey T\exboxprod_G \Mackey U).
 \]
Further, if $B = B'$ and $d\colon Y\to \Delta^*(X\exsmsh X')$ is a $G$-map,
where $\Delta$ is the diagonal $B\to B\times B$,
applying $d^*$ gives us the cup product
 \[
 -\cup - \colon 
 H\sp{\gamma}_ G(X;\Mackey T) \tensor
 H\sp{\gamma'}_ G(X';\Mackey U) \to H\sp{\gamma+\gamma'}_G
 (Y;\Mackey T\boxprod \Mackey U).
 \]
There are clearly other variations, but the ones above appear to
be among the most useful.

A contravariant $\stab\Pi_G B$-Mackey functor $\Mackey T$ is a {\em ring} 
if there is an associative multiplication 
$\mu\colon \Mackey T\boxprod \Mackey T\to \Mackey T$. When $\Mackey T$ is a ring, 
there is a cup product
\[
 -\cup - \colon 
 H\sp{\gamma}_ G(X;\Mackey T) \tensor
 H\sp{\gamma'}_ G(X';\Mackey T) \to H\sp{\gamma+\gamma'}_
 {G}(Y;\Mackey T)
\]
when $d\colon Y\to X\smsh_B X'$ is a $G$-map over $B$.
In particular, when $Y = X = X' = \susp^\infty_B K$ and $d$ is the diagonal,
we have the expected associative internal cup product
 \[
 -\cup - \colon 
 \tilde H\sp{\gamma}_ G(K;\Mackey T) \tensor
 \tilde H\sp{\gamma'}_ G(K;\Mackey T) \to 
   \tilde H\sp{\gamma+\gamma'}_G (K;\Mackey T).
 \]

These cup products have the following properties, 
most of which were also listed in
Theorem~\ref{thm:ghomology}.

\begin{theorem}\label{thm:gcupproduct}
The cup products defined above satisfy the following.
 \begin{enumerate}
 \item Naturality: $f^*(x)\cup g^*(y) = (f\smsh g)^*(x\cup y)$ for the external
product.
 \item Associativity: $(x\cup y)\cup z = x\cup (y\cup z)$.
 \item Commutativity: If $x\in H^\gamma(X;\Mackey T)$ and
$y\in H^{\gamma'}_{G'}(X';\Mackey S)$ then $x\cup y = \iota(y\cup x)$ where $\iota$ is the
obvious isomorphism between
$H^{\gamma+\gamma'}_{G\times G'}(X\exsmsh X';\Mackey T\exboxprod \Mackey S)$
and
$H^{\gamma'+\gamma}_{G'\times G}(X'\exsmsh X;\Mackey S\exboxprod \Mackey T)$.
 \item Unit: there exists an element $1\in H^0_G(S;\Mackey A_{G/G})$
(where $S$ is the sphere spectrum over a point) such
that $1\cup x = x$ for any $x\in H^\alpha_G(X;\Mackey T)$ (using 
the identification $\Mackey A_{G/G}\exboxprod_G \Mackey T \iso \Mackey T$).
 \item\label{item:gcupsubgroup}
 Restriction to subgroups: 
$(x|I)\cup (y|J) = (x\cup y)|(I\times J)$ in the external product.
 \item\label{item:gcupfixedsets}
 Restriction to fixed sets:
$x^I \cup y^J = (x\cup y)^{I\times J}$ in the external product.
 \end{enumerate}
 \qed
 \end{theorem}

The proofs are all standard except for the last two points,
whose proofs are formally identical to those in Theorem~\ref{thm:vcupproduct}.

The external cup product is represented by a $(G\times G')$-map
$H\Mackey T^\gamma \exsmsh H\Mackey U^{\gamma'} 
\to H(\Mackey T\exboxprod \Mackey U)^{\gamma+\gamma'}$
that is an isomorphism in $\Mackey\pi_{\gamma+\gamma'}^{G\times G'}$;
the internal cup product is represented by a $G$-map
$H\Mackey T^\gamma \smsh_B H\Mackey U^{\gamma'} 
\to H(\Mackey T\boxprod \Mackey U)^{\gamma+\gamma'}$
that is an isomorphism in $\Mackey\pi_{\gamma+\gamma'}^G$.
These maps are constructed as in the $RO(G)$-graded case, the
calculations done in that case giving the necessary fiberwise
calculations here.


Now we turn to part~\ref{item:dualgcupproduct} of
Theorem~\ref{thm:dualghomology}.

\begin{definition}
If $\MackeyOp S$ is a covariant $\stab\Pi_G B$-Mackey functor and
$\Mackey T$ is a contravariant $\stab\Pi_G B$-Mackey functor, define
a covariant $\stab\Pi_G B$-Mackey functor 
$\MackeyOp S \mixprod \Mackey T = \Mackey T\mixprod \MackeyOp S$ by
 \[
 (\MackeyOp S \mixprod \Mackey T)(y)
 = \left[\sum_{c \to a\times b}
    \MackeyOp S(a)\tensor \Mackey T(b) \right]\Big/\sim
 \]
where the direct sum runs over all stable $G$-maps
$c \to a\times b$ over the diagonal $B\to B\times B$, and we impose 
the equivalence relation generated by
 \[
 (s\tensor t)_{c' \xrightarrow{f} c \xrightarrow{} a\times b 
     \xrightarrow{g\times 1} a'\times b}
 \sim (f_*s \tensor g^*t)_{c \to a\times b}.
 \]
 \end{definition}

The following facts follow from the definitions.

\begin{lemma}
If $\MackeyOp S$, $\Mackey T$, and $\Mackey U$ are $\stab\Pi_G B$-Mackey functors, then
 \[
 (\MackeyOp S \mixprod \Mackey T) \tensor_{\stab\Pi_G B} \Mackey U
 \iso \MackeyOp S \tensor_{\stab\Pi_G B} (\Mackey T \boxprod \Mackey U)
 \]
and
 \[
 (\MackeyOp S \mixprod \Mackey T) \mixprod \Mackey U
 \iso \MackeyOp S \mixprod (\Mackey T \boxprod \Mackey U).
 \]
Further,
 \[
 \MackeyOp A^{b} \mixprod \Mackey A_{a} = \{b_+, \Delta^*(-\exsmsh a_+)\}_{G,B}
 \]
where the latter can be thought of as the group of stable maps over the diagonal
$B\to B\times B$.
\qed
 \end{lemma}
 

Now consider two $G$-spectra, $X$ and $Y$ over $B$. Let $\Gamma_\alpha(X)$ be a
$G$-CW($\alpha$) approximation of $X$, let $\Gamma'_\beta(Y)$ be a dual
$G$-CW($\beta$) approximation
of $Y$, and let $\Gamma'_{\alpha+\beta}(X\smsh_B Y)$ be a 
dual $G$-CW($\alpha+\beta$) approximation
of $X\smsh_B Y$.
As in the $RO(G)$-graded case, there is a filtration-preserving map
 \[
  \iota\colon \Gamma'_{\alpha+\beta}(X\smsh_B Y) \to \Gamma_\alpha(X)\smsh_B \Gamma'_\beta(Y),
 \]
approximating the identity,
unique up to homotopies raising filtration level by at most one.
This gives a chain map
 \[
  \iota_*\colon \MackeyOp C_{\alpha+\beta+*}(X\smsh_B Y)
   \to \Mackey C_{\alpha+*}(X) \mixprod \MackeyOp C_{\beta+*}(Y),
 \]
well-defined up to chain homotopy.
Given the chain map $\iota_*$ it is now easy to define the cup product
 \[
 -\cup-\colon
  H_G^\alpha(X;\Mackey T) \tensor
  \H_G^\beta(Y;\MackeyOp S) \to
  \H_G^{\alpha+\beta}(X\smsh_B Y; \Mackey T\mixprod \MackeyOp S).
 \]
The properties of this product listed in
Theorem~\ref{thm:dualghomology} can now be shown as in the $RO(G)$-graded case.


This cup product is once again represented on the spectrum level by a map
 \[
 H\Mackey T^\alpha\smsh_B H\MackeyOp S_\beta 
 \to H(\Mackey T\mixprod \MackeyOp S)_{\alpha+\beta}.
 \]

\subsection*{Evaluations and Cap Products}

The evaluations and cap products of 
Theorem~\ref{thm:ghomology} and
Theorem~\ref{thm:dualghomology}
are based on properties of cell complexes over $B$ and
their products.
The details are formally the same as in Section~\ref{sec:vsubgroups}
and are omitted here.


%%%%%%%%%%%%%%%%%%%
\section{The Thom Isomorphism and Poincar\'e Duality}\label{sec:genThom}

\subsection*{The Thom Isomorphism}

Recall that
a spherical $G$-fibration is a sectioned $G$-fibration $p\colon E\to B$
such that, for each $b\in B$, $p^{-1}(b)$ is based $G_b$-homotopy
equivalent to $S^V$ for some linear representation $V$ of $G_b$.

\begin{definition}
Let $p\colon E\to B$ be a spherical $G$-fibration and let
$\gamma$ be the associated spherical representation of $\Pi_G B$.
A {\em Thom class} for $p$ is an element
$t_p\in \tilde H_G^\gamma(E;\Mackey A_{G/G})$ such that, for each
$G$-map $b\colon G/K\to B$,
 \begin{align*}
 b^*(t_p) \in{} &\tilde H_G^{\gamma(b)}(b^*E;\Mackey A_{G/G}) \\
 &\iso \tilde H_G^{\gamma(b)}(G\times_K S^V;\Mackey A_{G/G}) \\
 &\iso \tilde H_K^{V}(S^V;\Mackey A_{K/K}) \\
 &\iso A(K)
 \end{align*}
is a generator.
Here, $V$ is a representation of $K$ such that
$\gamma(b) \hmtpc G\times_K S^V$.
 \end{definition}

Since a global Thom class is characterized by its local behavior, and
locally a spherical fibration is a $V$-fibration, the next proposition
follows from the $RO(G)$-graded analogue, Proposition~\ref{prop:vThomclass}.

\begin{proposition}\label{prop:gThomclass}
The following are equivalent for a class $t\in \tilde H_G^\gamma(E;\Mackey A_{G/G})$.
 \begin{enumerate}
 \item
$t$ is a Thom class for $p$.
 \item
For every subgroup $K\subset G$,
$t|K\in \tilde H_K^{\gamma|K}(E;\Mackey A_{K/K})$
is a Thom class for $p$ as a $K$-fibration.
 \item
For every subgroup $K\subset G$, 
$t^K\in \tilde H_{WK}^{\gamma^K}(E^K; \Mackey A_{WK/WK})$
is a Thom class for $p^K\colon E^K\to B^K$ as a $WK$-fibration.
 \item
For every subgroup $K\subset G$,
$t^K|e \in \tilde H^{|\gamma^K|}(E^K;\Z)$
is a Thom class for $p^K$ as a nonequivariant fibration.
 \end{enumerate}
 \qed
 \end{proposition}


\begin{theorem}[Thom Isomorphism]
If $p\colon E\to B$ is any $G$-spherical fibration
and $\gamma$ is the associated spherical representation of $\Pi_G B$, 
then there exists a Thom class
$t_p\in \tilde H_G^\gamma(E;\Mackey A_{G/G})$. For any Thom class $t_p$,
the maps
 \[
 t_p\cup - \colon H_G^\alpha(B;\Mackey T)
 \to \tilde H_G^{\alpha+\gamma}(E;\Mackey T)
 \]
and
 \[
 t_p \cup - \colon \H_G^\alpha(B;\MackeyOp S)
 \to \tilde \H_G^{\alpha+\gamma}(E;\MackeyOp S)
 \]
are isomorphisms.
 \end{theorem}

\begin{proof}
As in the proof of Theorem~\ref{thm:vthomiso},
the theorem is clear in the special case that $p$ is a bundle over an orbit.
The general case follows, as it does in the nonequivariant case for
twisted coefficients, by a Mayer-Vietoris patching argument.
The key point is that we can choose a compatible collection of local
classes because the action of $\Pi_G B$ on $\gamma$ is the same
as the action on the fibers of $p$.
 \end{proof}

\subsection*{Poincar\'e Duality}

We are now in a position to describe Poincar\'e duality for 
arbitrary compact smooth $G$-manifolds.
(Again, the noncompact case can be handled using cohomology with
compact supports.)

\begin{definition}
Let $M$ be a closed smooth $G$-manifold and let $\tau$ be the tangent representation
of $\Pi_G M$, i.e., the representation associated with the tangent bundle. 
Think of $M$ as a $G$-space over itself in the following.
A {\em fundamental class} of $M$ is
a class $[M]\in \H_\tau^G(M;\Mackey A_{G/G})$ such that, for each 
point $m\in M$, thought of as the map
$m\colon G/G_m\to M$
 with image $Gm$, and tangent plane $\tau(m) = G\times_{G_m}V$,
the image of $[M]$ in
 \begin{align*}
 \H_\tau^G(M,M-Gm;{}&\Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^G(G\times_{G_m}S^{V-V(G/G_m)};
    \Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^{G_m}(S^V; \Mackey A_{G/G}) \\
 &\iso A(G_m)
 \end{align*}
is a generator.
 \end{definition}

The fundamental class $[M]$ is related to fundamental classes of the fixed
submanifolds $M^K$ as in
Proposition~\ref{prop:vFundclass}.

\begin{proposition}\label{prop:gFundclass}
The following are equivalent for a dual homology class 
$\mu\in \H_\tau^G(M;\Mackey A_{G/G})$.
 \begin{enumerate}
 \item
$\mu$ is a fundamental class for $M$ as a $G$-manifold.
 \item
For every subgroup $K\subset G$,
$\mu|K \in \H_{\tau|K}^K(M;\Mackey A_{K/K})$ is a fundamental class for
$M$ as a $K$-manifold.
 \item
For every subgroup $K\subset G$,
$\mu^K \in \H_{\tau^K}^{WK} (M^K; \Mackey A_{WK/WK})$ is a fundamental class for
$M^K$ as a $WK$-manifold.
 \item
For every subgroup $K\subset G$,
$\mu^K|e \in H_{|\tau^K|}(M^K;\Z)$ is a fundamental class for $M^K$
as a nonequivariant manifold. \qed
 \end{enumerate}
\end{proposition}

We now have sufficient machinery in place to prove Poincar\'e duality for
arbitrary $G$-manifolds either geometrically, along the lines of
\cite{MS:characteristicclasses} or \cite{CW:duality}, or
homotopically as in Section~\ref{sec:vThomIso}.
The homotopical approach uses, of course, the duality
of Theorem~\ref{thm:manifoldduality}.

\begin{theorem}[Poincar\'e Duality]
Every closed smooth $G$-manifold $M$ 
has a fundamental class $[M]\in\H_\tau^G(M;\Mackey A_{G/G})$, and
 \[
 -\cap [M] \colon H_G^\alpha(M;\Mackey T) \to \H_{\tau-\alpha}^G(M;\Mackey T)
 \]
and
 \[
 -\cap [M] \colon \H_G^\alpha(M;\MackeyOp S) \to H_{\tau-\alpha}^G(M;\MackeyOp S)
 \]
are isomorphisms.
\qed
 \end{theorem}


If $M$ is a compact $G$-manifold with boundary, then we get relative,
or Lefschetz, duality.

\begin{definition}
Let $M$ be a compact $G$-manifold with boundary, with tangent representation $\tau$. 
A {\em fundamental class} of $M$ is a dual homology
class $[M,\bndry M]\in \H_\tau^G(M,\bndry M;\Mackey A_{G/G})$ such that, for each 
point $m\in M-\bndry M$,
thought of as the map
$m\colon G/G_m\to M$
 with image $Gm$, and tangent plane $\tau(m) = G\times_{G_m}V$,
the image of $[M,\bndry M]$ in
 \begin{align*}
 \H_\tau^G(M,M-Gm;{}&\Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^G(G\times_{G_m}S^{V-V(G/G_m)};
    \Mackey A_{G/G}) \\
 &\iso \widetilde\H_V^{G_m}(S^V; \Mackey A_{G/G}) \\
 &\iso A(G_m)
 \end{align*}
is a generator.
 \end{definition}

There is an obvious relative version of Proposition~\ref{prop:gFundclass}.

\begin{theorem}[Lefschetz Duality]
Every compact smooth $G$-manifold $M$ has a fundamental class 
$[M,\bndry M]\in\H_V^G(M,\bndry M;\Mackey A_{G/G})$,
and the following are all isomorphisms.
 \[
 -\cap [M,\bndry M] \colon H_G^\alpha(M;\Mackey T) \to
   \H_{\tau-\alpha}^G(M,\bndry M;\Mackey T)
 \]
 \[
 -\cap [M,\bndry M] \colon H_G^\alpha(M,\bndry M;\Mackey T) \to
   \H_{\tau-\alpha}^G(M;\Mackey T)
 \]
 \[
 -\cap [M,\bndry M] \colon \H_G^\alpha(M;\MackeyOp S) \to
   H_{\tau-\alpha}^G(M,\bndry M;\MackeyOp S)
 \]
 \[
 -\cap [M,\bndry M] \colon \H_G^\alpha(M,\bndry M;\MackeyOp S) \to
   H_{\tau-\alpha}^G(M;\MackeyOp S)
 \]
 \qed
 \end{theorem}


Homology-defs.tex
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Homology-intro.tex
\chapter*{Introduction}
\label{chap:intro}
%

Poincar\'e duality lies at the heart of the homological study of manifolds.
In the presence of the action of a group, it is well-known that Poincar\'e
duality fails to hold in ordinary, integer-graded (Bredon-Illman) equivariant homology.
(See \cite{Br:cohomology} and \cite{Ill:homology} for the construction
of these homology theories;
see \cite{DR:surgery} and \cite{Lu:transfgroups} for the failure
of Poincar\'e duality.)
It is our purpose here to extend the Bredon theory to one in which
a form of
Poincar\'e duality holds for all smooth compact $G$-manifolds, where
$G$ is any compact Lie group.
We did this for $G$ finite in \cite{CW:duality}, where we showed
that, for finite group actions and Mackey functor
coefficient systems, Bredon-Illman theory extends
to a theory graded on $RO(\Pi_G B)$, the group of
virtual representations of
the equivariant fundamental groupoid $\Pi_G B$ of a $G$-space $B$,
in which Poincar\'e duality holds for all smooth compact $G$-manifolds. 
That theory is an
extension of a well-known intermediate theory: $RO(G)$-graded equivariant ordinary
(co)homology 
(see \cite{LMM:roghomology}, \cite{Le:Hurewicz},
\cite{Le:emspaces} and \cite[Chapters~X, XI and~XIII]{May:alaska}).
In \cite{CW:spivaknormal} and \cite{CW:simpleduality}
we applied the theory to obtain $\pi$-$\pi$ theorems for
equivariant Poincar\'e duality spaces and equivariant simple Poincar\'e duality spaces.
In other words,
our equivariant ordinary theory is sensitive enough to 
yield a surgery theory that
parallels the non-equivariant theory \cite{Wal:surgery} modulo the usual transversality
issues (\cite{Was:difftopology}, \cite{Pe:pseudo}).

The $RO(G)$-graded extension of Bredon-Illman homology dates back to
1981 and is interesting in its own right: 
It is necessary to extend to $RO(G)$ grading to get
theories represented by equivariant spectra 
(\cite{LMS:eqhomotopy}, \cite{MM:orthogonal}),
and so they provide a minimal substrate in which we can take advantage
of equivariant
Spanier-Whitehead duality. 
However, the documentation of $RO(G)$-graded ordinary homology is
sparse and incomplete.
In particular, its behaviour with regard to products, restriction to fixed sets,
and restriction to subgroups is nowhere discussed in detail.
(An outline of the product structure is given \cite[XIII.5]{May:alaska}.)

When $G$ is finite, the $RO(G)$-graded theory does yield Poincar\'e duality for smooth
$G$-manifolds that are modeled on a single representation of $G$ 
(see \cite{Pu:rogbordism},
\cite{Ko:rogbordism}, and \cite{Wan:ROGbordism}).
However, as noted in \cite{May:alaska},
ordinary homology and cohomology need not be dual when $G$ is an infinite
compact Lie group---if $DX$ is the $G$-spectrum dual to $X$, $H^G_V(X)$ and
$H_G^{-V}(DX)$ need not be isomorphic. Thus, equivariant
Poincar\'e duality between ordinary homology and ordinary cohomology cannot be
expected to hold
when $G$ is not finite, even for $G$-manifolds modeled on a single representation.
Although ordinary homology and cohomology are not dual, each
possesses a dual theory. These theories, which we refer to as
dual ordinary cohomology and homology, are mentioned in
\cite{May:alaska} but not described in any detail.

In short, the state of the art to date is this:
There are $RO(G)$-graded ordinary homology and cohomology theories
defined for compact Lie groups $G$.
Poincar\'e duality holds with these theories only when $G$ is finite and
the manifold is modeled on a single representation of $G$.
When $G$ is finite, there is an extension of the $RO(G)$-graded theory
to an $RO(\Pi_G B)$-graded theory, in which Poincar\'e duality holds
for all smooth compact $G$-manifolds.
This raises the following questions: How does Poincar\'e duality work
in the $RO(G)$-graded theories when $G$ is infinite?
How do the $RO(\Pi_G B)$-graded theories extend to the case when $G$ is infinite,
and how does Poincar\'e duality work in this context?
The goal of the present work is to answer these questions while also
giving a more complete and coherent account of what is already known.

In Chapter~1 we give
a reasonably complete account of $RO(G)$-graded
ordinary homology and cohomology as well as their associated dual theories,
both geometrically (from the cellular viewpoint) and homotopically (from the
represented viewpoint). In particular, we find that the dual theories are
also cellular theories, arising from ``dual'' cell
complexes. 
We then discuss products, restriction to subgroups,
and restriction to fixed sets---accounts that have not appeared
heretofore in the literature.
Finally, we show that
Poincar\'e duality for manifolds modeled on a single representation
holds for general compact Lie groups in that, 
if $M$ is compact, equivariantly orientable (in a sense to be made precise),
and modeled locally on the $G$-representation $V$, one has
duality isomorphisms like
$H^W_G(M,\bndry M) \iso \H_{V-W}^G(M)$,
where $\H^G_*$ is the dual theory.

%In Chapter~2 we review the basic facts about the equivariant fundamental groupoid
%and its representations as given in detail in \cite{CMW:orientation}.
%In the case of $RO(G)$-graded theories the coefficient systems used are
%based on the stable orbit category; likewise, in the $RO(\Pi_G X)$-graded
%theories the coefficient systems used are based on a stable version of
%the equivariant fundamental groupoid of $X$.
%The bulk of Chapter~2 is devoted to this stable equivariant fundamental groupoid
%and its properties.

%With the necessary machinery in hand, we proceed in Chapter~3 to
%construct the $RO(\Pi_G X)$-graded ordinary homology and cohomology theories.
%In Chapter~4 we discuss the representing objects,
%which are equivariant parametrized spectra.
%We rely heavily on the recent carefully worked out theory
%of \cite{MaySig:parametrized}, about which more below.
%In Chapter~5 we describe the dual theories, products, restriction to subgroups,
%and restriction to fixed sets.
%The chapter culminates in proofs of the Thom isomorphism theorem for
%equivariant spherical fibrations and Poincar\'e duality
%for arbitrary smooth compact $G$-manifolds.
%A word about orientability:
%The reader will in due course notice that the $RO(\Pi_G X)$-graded theories are by
%their nature twisted, so that the Thom
%isomorphism and Poincar\'e duality theorems
%require no orientability assumptions. Indeed,
%restricting attention to orientable $G$-manifolds and spherical fibrations in
%this context gives us no advantage: The fundamental groupoid
%is woven as deeply into the theory of orientations as it is in the general
%theory, so we get no simplification by considering the orientable
%case.
%    
%We would like to thank L.~Gaunce Lewis for conversations on
%numerous topics, including the dual
%theories we discuss here. And, of course, we would like to
%thank J.~Peter May, with whom we have collaborated for many years on the
%theory of equivariant orientations, the ideas of which underlie this whole project.
%We are also greatly indebted to Peter and to Johann Sigurdsson for \cite{MaySig:parametrized}. It became clear some time ago that the representing objects of $RO(\Pi_G X)$-graded theories ought to be parametrized spectra. We naively thought that it would be straightforward to set up a well-behaved model category of such. At about that time Peter, with other applications in mind, had begun working out the details of this model structure, and was soon joined by Johann. They discovered that the details were much more delicate than any of us had thought. The present work could only be completed once \cite{MaySig:parametrized} was; our use of parametrized spectra now relies on the foundation they set. We are pleased that, in turn, they saw fit to use some of the ideas developed here.

Our development of the $RO(\Pi_G B)$-graded theories is done in the context
of parametrized spaces and spectra. In Chapter~2 we review facts we need about
parametrized homotopy theory, relying heavily on work by
May and Sigurdsson
\cite{MaySig:parametrized}. However, we emphasize a point of view
not taken by May and Sigurdsson: that, rather than concentrating on maps
that strictly commute with projection to the basespace, we should take
seriously maps that commute only up to specified homotopies in the
basespace. These maps, which we call {\em lax maps}, we discuss in detail
in Chapter~2.

In Chapter~2 we also review basic facts about the equivariant fundamental
groupoid $\Pi_G B$ and its representations, as developed in detail in
\cite{CMW:orientation}.
In the case of $RO(G)$-graded ordinary theories it is well-known that
the coefficient systems we
use must be Mackey functors, i.e., functors on the stable orbit category.
In the $RO(\Pi_G B)$-graded theory the fundamental groupoid replaces
the orbit category, so we must also have a stable version of $\Pi_G B$.
The last two sections of Chapter~2 discuss this stable fundamental groupoid
and some of its properties.

In order to define ordinary homology and cohomology as cellular theories,
we must have well-behaved CW objects to work with in the categories
of parametrized spaces and spectra.
May and Sigurdsson point out the difficulties of developing
a theory of CW parametrized objects at the end of \cite{MaySig:parametrized}.
In Chapter~3 we lay out such a theory in detail, answering some of the
questions raised by May and Sigurdsson. The theory requires
some compromises, particularly in our results about approximating 
arbitrary maps by cellular maps. The compromise in the case of maps
of spaces is to allow lax maps; in the case of maps of spectra we must
also allow ``costabilization'' of the source spectrum.
These compromises make the theory less satisfying than in the
nonparametrized case, but these CW objects are still adequate to use in defining cellular
homology and cohomology, which we do at the end of Chapter~3.

In the first part of Chapter~4 we discuss the way in which parametrized
homology and cohomology theories are represented by parametrized spectra.
This is known for cohomology
(see, for example, \cite{Cl:paraspectra} and \cite{CP:paraspectra})
but seems to be new for homology.
Homology and cohomology theories should be connected by a duality,
but the obvious fiberwise duality does not work for this.
We introduce instead a kind of duality we call
{\em homological duality}. May and Sigurdsson have done us the honor
of calling this Costenoble-Waner duality in \cite{MaySig:parametrized}
and give there a very nice development of it in a very general context.

The last half of Chapter~4 develops the dual ordinary theories as cellular
theories. We then discuss, for all of the ordinary theories, products,
restriction to subgroups, and restriction to fixed sets.
The chapter culminates in the Thom isomorphism theorem for equivariant
spherical fibrations and the Poincar\'e duality theorem for smooth compact
$G$-manifolds.
A word about orientability: The $RO(\Pi_G B)$-graded theories are, 
by their nature, twisted, so that the Thom isomorphism and
Poincar\'e duality theorems require no orientability assumptions.
Indeed, restricting to orientable fibrations or manifolds in this
context gives us no advantage---the fundamental groupoid is
woven so deeply into the theory of equivariant orientations
\cite{CMW:orientation} that there is no significant simplification
in considering the orientable case.

\subsection*{History}
This project has been in development for long enough that it is worth
saying a few words about its history and relation to other works.
We have long believed that equivariant Thom isomorphism and
Poincar\'e duality theorems require homology and cohomology theories
that take into account the varying local representations given
by the fibers of a general equivariant vector bundle.
This belief led to the work with Peter May on equivariant orientations
that became \cite{CMW:orientation},
and early versions of the orientation theory, as well as versions
of $RO(\Pi_G B)$-graded ordinary homology and cohomology,
appeared in \cite{CW:duality} and \cite{CW:thomiso}.
(See \cite{May:thomiso} for a different approach to Thom isomorphisms
suggested by Peter.)

While we were working on \cite{CMW:orientation} we had the present
work in mind as well.
Early on, we did not give parametrized homotopy theory a prominent role,
but as we worked out the details it became clearer that we should.
In an earlier version of this manuscript
\cite{CW:homologypreprint} we used parametrized spectra while
discussing the objects representing the $RO(\Pi_G B)$-graded
theories, but our naive belief was that the theory of parametrized spectra
was an easy generalization of the nonparametrized theory.
At about the same time we posted that preprint, Peter had started looking
at parametrized homotopy theory for other reasons, and he warned us
that he had already found serious pitfalls and that the parametrized
theory was definitely {\em not} an easy generalization of the
nonparametrized case. Peter was joined in his efforts by
Johann Sigurdsson and their work became \cite{MaySig:parametrized}.
We are pleased that they incorporated some of the ideas from our
preprint, particularly our notion of homological duality,
and we are indebted to them for providing a firm foundation on which
we could build the present work.
Once \cite{MaySig:parametrized} appeared it became clear that
we needed to substantially rewrite our earlier manuscript. The
present version uses parametrized homotopy theory much more extensively,
and we have attempted to do so with the care that Peter has enjoined
us to take.
In several places \cite{MaySig:parametrized} gives better proofs
of results from our earlier preprint, so we have replaced our proofs
with references to theirs.

\subsection*{In Memoriam}
Our good friend L. Gaunce Lewis died in 2006 of brain cancer.
We had many enlightening conversations with Gaunce about compact
Lie groups, Mackey functors, and the dual theories, among other topics.
This work would have benefited from the close reading he would
have given it. We miss him.

\today


Homology.tex
% Equivariant ordinary homology and cohomology
% by Costenoble and Waner
%
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
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%
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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
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%
\input Homology-defs.tex
%
\begin{document}
%
\frontmatter
\title{Equivariant ordinary homology and cohomology}

\author{Steven R. Costenoble}
\address{Department of Mathematics\\103 Hofstra University\\
   Hempstead, NY 11549}
\email{Steven.R.Costenoble@hofstra.edu}

\author{Stefan Waner}
\email{Stefan.Waner@hofstra.edu}

\subjclass[2000]{Primary 55N91;
Secondary 55M05, 55N25, 55P20, 55P42, 55P91, 55R70, 55R91, 57R91}

\date{\today}
%
\maketitle
\tableofcontents

\mainmatter
\include{Homology-intro}
\include{Homology-chap1}
\include{Homology-chap2}
\include{Homology-chap3}
\include{Homology-chap4}

\backmatter
%\bibliographystyle{amsplain}
%\bibliography{Topology}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{10}

\bibitem{Ad:prereq}
J.~F. Adams, \emph{Prerequisites (on equivariant stable homotopy) for
  {C}arlsson's lecture}, Algebraic topology, Aarhus 1982 (Aarhus, 1982),
  Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp.~483--532.
  \MR{MR764596 (86f:57037)}

\bibitem{Bo:spectralsequences}
J.~M. Boardman, \emph{Conditionally convergent spectral sequences}, Homotopy
  invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., vol.
  239, Amer. Math. Soc., Providence, RI, 1999, pp.~49--84. \MR{MR1718076
  (2000m:55024)}

\bibitem{Br:cohomology}
G.~E. Bredon, \emph{Equivariant cohomology theories}, Lecture Notes in
  Mathematics, No. 34, Springer-Verlag, Berlin, 1967. \MR{MR0214062 (35
  \#4914)}

\bibitem{Br:topologyBook}
R.~Brown, \emph{Elements of modern topology}, McGraw-Hill Book Co., New York,
  1968. \MR{MR0227979 (37 \#3563)}

\bibitem{Cl:paraspectra}
M.~Clapp, \emph{Duality and transfer for parametrized spectra}, Arch. Math.
  (Basel) \textbf{37} (1981), no.~5, 462--472. \MR{MR643290 (83i:55010)}

\bibitem{CP:paraspectra}
M.~Clapp and D.~Puppe, \emph{The homotopy category of parametrized spectra},
  Manuscripta Math. \textbf{45} (1984), no.~3, 219--247. \MR{MR734840
  (85c:55007)}

\bibitem{Co:thomspectra}
S.~R. Costenoble, \emph{The structure of some equivariant {T}hom spectra},
  Trans. Amer. Math. Soc. \textbf{315} (1989), no.~1, 231--254. \MR{MR958887
  (89m:57038)}

\bibitem{Co:monoids}
\bysame, \emph{Classifying spaces of bundles of monoids}, preprint, 2002.

\bibitem{CMW:orientation}
S.~R. Costenoble, J.~P. May, and S.~Waner, \emph{Equivariant orientation
  theory}, Homology Homotopy Appl. \textbf{3} (2001), no.~2, 265--339
  (electronic), Equivariant stable homotopy theory and related areas (Stanford,
  CA, 2000). \MR{MR1856029 (2002j:55016)}

\bibitem{CW:loopspaces}
S.~R. Costenoble and S.~Waner, \emph{Fixed set systems of equivariant infinite
  loop spaces}, Trans. Amer. Math. Soc. \textbf{326} (1991), no.~2, 485--505.
  \MR{MR1012523 (91k:55015)}

\bibitem{CW:duality}
\bysame, \emph{Equivariant {P}oincar\'e duality}, Michigan Math. J. \textbf{39}
  (1992), no.~2, 325--351. \MR{MR1162040 (93d:55007)}

\bibitem{CW:spivaknormal}
\bysame, \emph{The equivariant {S}pivak normal bundle and equivariant surgery},
  Michigan Math. J. \textbf{39} (1992), no.~3, 415--424. \MR{MR1182497
  (94d:57058)}

\bibitem{CW:thomiso}
\bysame, \emph{The equivariant {T}hom isomorphism theorem}, Pacific J. Math.
  \textbf{152} (1992), no.~1, 21--39. \MR{MR1139971 (93c:55005)}

\bibitem{CW:simpleduality}
\bysame, \emph{Equivariant simple {P}oincar\'e duality}, Michigan Math. J.
  \textbf{40} (1993), no.~3, 577--604. \MR{MR1236180 (94m:55006)}

\bibitem{CW:homologypreprint}
\bysame, \emph{Equivariant ordinary homology and cohomology}, preprint,
  arXiv:math/0310237v1 [math.AT], 2003.

\bibitem{Dold:parahomology}
A.~Dold, \emph{Chern classes in general cohomology}, Symposia Mathematica, Vol.
  V (INDAM, Rome, 1969/70), Academic Press, London, 1971, pp.~385--410.
  \MR{MR0276968 (43 \#2707)}

\bibitem{DR:surgery}
K.~H. Dovermann and M.~Rothenberg, \emph{Equivariant surgery and
  classifications of finite group actions on manifolds}, Mem. Amer. Math. Soc.
  \textbf{71} (1988), no.~379, viii+117. \MR{89h:57032}

\bibitem{Dr:reps}
A.~W.~M. Dress, \emph{Contributions to the theory of induced representations},
  Algebraic $K$-theory, II: ``Classical'' algebraic $K$-theory and connections
  with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972),
  Springer, Berlin, 1973, pp.~183--240. Lecture Notes in Math., Vol. 342.
  \MR{MR0384917 (52 \#5787)}

\bibitem{GM:eqhomotopy}
J.~P.~C. Greenlees and J.~P. May, \emph{Equivariant stable homotopy theory},
  Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp.~277--323.
  \MR{MR1361893 (96j:55013)}

\bibitem{Gr:groupefond}
A.~Grothendieck, \emph{R{\^e}vetements {{\'e}}tale et groupe fondemental
  {(SGA1)}}, Lecture Notes in Math., vol. 224, Springer-Verlag, 1971.

\bibitem{Ill:homology}
S.~Illman, \emph{Equivariant singular homology and cohomology. {I}}, Mem. Amer.
  Math. Soc. \textbf{1} (1975), no.~issue 2, 156, ii+74. \MR{MR0375286 (51
  \#11482)}

\bibitem{Ill:triangulation}
\bysame, \emph{The equivariant triangulation theorem for actions of compact
  {L}ie groups}, Math. Ann. \textbf{262} (1983), no.~4, 487--501. \MR{MR696520
  (85c:57042)}

\bibitem{Kle:KleisliCat}
H.~Kleisli, \emph{Every standard construction is induced by a pair of adjoint
  functors}, Proc. Amer. Math. Soc. \textbf{16} (1965), 544--546. \MR{MR0177024
  (31 \#1289)}

\bibitem{Ko:rogbordism}
C.~Kosniowski, \emph{A note on {$RO(G)$} graded {$G$} bordism theory}, Quart.
  J. Math. Oxford Ser. (2) \textbf{26} (1975), no.~104, 411--419. \MR{MR0388418
  (52 \#9254)}

\bibitem{Le:Greenfunctors}
L.~G. Lewis, Jr., \emph{The theory of {G}reen functors}, unpublished
  manuscript, 1981.

\bibitem{Le:emspaces}
\bysame, \emph{Equivariant {E}ilenberg-{M}ac {L}ane spaces and the equivariant
  {S}eifert-van {K}ampen and suspension theorems}, Topology Appl. \textbf{48}
  (1992), no.~1, 25--61. \MR{MR1195124 (93i:55016)}

\bibitem{Le:Hurewicz}
\bysame, \emph{The equivariant {H}urewicz map}, Trans. Amer. Math. Soc.
  \textbf{329} (1992), no.~2, 433--472. \MR{92j:55024}

\bibitem{LMM:roghomology}
L.~G. Lewis, Jr., J.~P. May, and J.~E. McClure, \emph{Ordinary {$RO(G)$}-graded
  cohomology}, Bull. Amer. Math. Soc. (N.S.) \textbf{4} (1981), no.~2,
  208--212. \MR{MR598689 (82e:55008)}

\bibitem{LMS:eqhomotopy}
L.~G. Lewis, Jr., J.~P. May, and M.~Steinberger, \emph{Equivariant stable
  homotopy theory}, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag,
  Berlin, 1986, With contributions by J. E. McClure. \MR{MR866482 (88e:55002)}

\bibitem{Lu:transfgroups}
W.~L{\"u}ck, \emph{Transformation groups and algebraic {$K$}-theory}, Lecture
  Notes in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989, Mathematica
  Gottingensis. \MR{91g:57036}

\bibitem{Mac:categories}
S.~Mac\, Lane, \emph{Categories for the working mathematician},
  Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 5.
  \MR{MR0354798 (50 \#7275)}

\bibitem{MM:orthogonal}
M.~A. Mandell and J.~P. May, \emph{Equivariant orthogonal spectra and
  {$S$}-modules}, Mem. Amer. Math. Soc. \textbf{159} (2002), no.~755, x+108.
  \MR{2003i:55012}

\bibitem{May:foundations}
J.~P. May, \emph{Homotopical foundations of algebraic topology}, unpublished
  manuscript, 1974.

\bibitem{May:alaska}
\bysame, \emph{Equivariant homotopy and cohomology theory}, CBMS Regional
  Conference Series in Mathematics, vol.~91, Published for the Conference Board
  of the Mathematical Sciences, Washington, DC, 1996, With contributions by M.
  Cole, G. Comeza\~na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L.
  G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. \MR{MR1413302
  (97k:55016)}

\bibitem{May:concise}
\bysame, \emph{A concise course in algebraic topology}, Chicago Lectures in
  Mathematics, University of Chicago Press, Chicago, IL, 1999. \MR{MR1702278
  (2000h:55002)}

\bibitem{May:thomiso}
\bysame, \emph{Equivariant orientations and {T}hom isomorphisms}, Tel Aviv
  Topology Conference: Rothenberg Festschrift (1998), Contemp. Math., vol. 231,
  Amer. Math. Soc., Providence, RI, 1999, pp.~227--243. \MR{MR1707345
  (2000i:55042)}

\bibitem{MaySig:parametrized}
J.~P. May and J.~Sigurdsson, \emph{Parametrized homotopy theory}, Mathematical
  Surveys and Monographs, vol. 132, American Mathematical Society, Providence,
  RI, 2006. \MR{MR2271789}

\bibitem{MS:characteristicclasses}
J.~W. Milnor and J.~D. Stasheff, \emph{Characteristic classes}, Princeton
  University Press, Princeton, N. J., 1974, Annals of Mathematics Studies, No.
  76. \MR{MR0440554 (55 \#13428)}

\bibitem{Pe:pseudo}
T.~Petrie, \emph{Pseudoequivalences of {$G$}-manifolds}, Algebraic and
  geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford,
  Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc.,
  Providence, R.I., 1978, pp.~169--210. \MR{MR520505 (80e:57039)}

\bibitem{Pu:rogbordism}
W.~Pulikowski, \emph{{$RO(G)$}-graded {$G$}-bordism theory}, Bull. Acad. Polon.
  Sci. S\'er. Sci. Math. Astronom. Phys. \textbf{21} (1973), 991--995.
  \MR{MR0339234 (49 \#3996a)}

\bibitem{Str:MonadsI}
R.~Street, \emph{The formal theory of monads}, J. Pure Appl. Algebra \textbf{2}
  (1972), no.~2, 149--168. \MR{MR0299653 (45 \#8701)}

\bibitem{TW:mackey}
J.~Th{\'e}venaz and P.~J. Webb, \emph{Simple {M}ackey functors}, Proceedings of
  the Second International Group Theory Conference (Bressanone, 1989), no.~23,
  1990, pp.~299--319. \MR{MR1068370 (91g:20011)}

\bibitem{tD:transfgroups}
T.~tom Dieck, \emph{Transformation groups}, de Gruyter Studies in Mathematics,
  vol.~8, Walter de Gruyter \& Co., Berlin, 1987. \MR{MR889050 (89c:57048)}

\bibitem{Wal:surgery}
C.~T.~C. Wall, \emph{Surgery on compact manifolds}, Academic Press, London,
  1970, London Mathematical Society Monographs, No. 1. \MR{55 \#4217}

\bibitem{Wan:gcwv}
S.~Waner, \emph{{$G$-CW($V$)} complexes}, unpublished manuscript, 1979.

\bibitem{Wan:class}
\bysame, \emph{Classification of oriented equivariant spherical fibrations},
  Trans. Amer. Math. Soc. \textbf{271} (1982), no.~1, 313--324. \MR{83g:57026}

\bibitem{Wan:ROGbordism}
\bysame, \emph{Equivariant {$RO(G)$}-graded bordism theories}, Topology Appl.
  \textbf{17} (1984), no.~1, 1--26. \MR{85e:57042}

\bibitem{Was:difftopology}
Arthur~G. Wasserman, \emph{Equivariant differential topology}, Topology
  \textbf{8} (1969), 127--150. \MR{MR0250324 (40 \#3563)}

\bibitem{Wh:homotopy}
G.~W. Whitehead, \emph{Elements of homotopy theory}, Graduate Texts in
  Mathematics, vol.~61, Springer-Verlag, New York, 1978. \MR{MR516508
  (80b:55001)}

\bibitem{Wi:duality}
K.~Wirthm{\"u}ller, \emph{Equivariant homology and duality}, Manuscripta Math.
  \textbf{11} (1974), 373--390. \MR{MR0343260 (49 \#8004)}

\end{thebibliography}


\end{document}

math0310237.extracted_bib
\begin{thebibliography}{10}

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J.~F. Adams, \emph{Prerequisites (on equivariant stable homotopy) for
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G.~E. Bredon, \emph{Equivariant cohomology theories}, Lecture Notes in
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R.~Brown, \emph{Elements of modern topology}, McGraw-Hill Book Co., New York,
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M.~Clapp, \emph{Duality and transfer for parametrized spectra}, Arch. Math.
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M.~Clapp and D.~Puppe, \emph{The homotopy category of parametrized spectra},
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S.~R. Costenoble, \emph{The structure of some equivariant {T}hom spectra},
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\bibitem{Co:monoids}
\bysame, \emph{Classifying spaces of bundles of monoids}, preprint, 2002.

\bibitem{CMW:orientation}
S.~R. Costenoble, J.~P. May, and S.~Waner, \emph{Equivariant orientation
  theory}, Homology Homotopy Appl. \textbf{3} (2001), no.~2, 265--339
  (electronic), Equivariant stable homotopy theory and related areas (Stanford,
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\bibitem{CW:loopspaces}
S.~R. Costenoble and S.~Waner, \emph{Fixed set systems of equivariant infinite
  loop spaces}, Trans. Amer. Math. Soc. \textbf{326} (1991), no.~2, 485--505.
  \MR{MR1012523 (91k:55015)}

\bibitem{CW:duality}
\bysame, \emph{Equivariant {P}oincar\'e duality}, Michigan Math. J. \textbf{39}
  (1992), no.~2, 325--351. \MR{MR1162040 (93d:55007)}

\bibitem{CW:spivaknormal}
\bysame, \emph{The equivariant {S}pivak normal bundle and equivariant surgery},
  Michigan Math. J. \textbf{39} (1992), no.~3, 415--424. \MR{MR1182497
  (94d:57058)}

\bibitem{CW:thomiso}
\bysame, \emph{The equivariant {T}hom isomorphism theorem}, Pacific J. Math.
  \textbf{152} (1992), no.~1, 21--39. \MR{MR1139971 (93c:55005)}

\bibitem{CW:simpleduality}
\bysame, \emph{Equivariant simple {P}oincar\'e duality}, Michigan Math. J.
  \textbf{40} (1993), no.~3, 577--604. \MR{MR1236180 (94m:55006)}

\bibitem{CW:homologypreprint}
\bysame, \emph{Equivariant ordinary homology and cohomology}, preprint,
  arXiv:math/0310237v1 [math.AT], 2003.

\bibitem{Dold:parahomology}
A.~Dold, \emph{Chern classes in general cohomology}, Symposia Mathematica, Vol.
  V (INDAM, Rome, 1969/70), Academic Press, London, 1971, pp.~385--410.
  \MR{MR0276968 (43 \#2707)}

\bibitem{DR:surgery}
K.~H. Dovermann and M.~Rothenberg, \emph{Equivariant surgery and
  classifications of finite group actions on manifolds}, Mem. Amer. Math. Soc.
  \textbf{71} (1988), no.~379, viii+117. \MR{89h:57032}

\bibitem{Dr:reps}
A.~W.~M. Dress, \emph{Contributions to the theory of induced representations},
  Algebraic $K$-theory, II: ``Classical'' algebraic $K$-theory and connections
  with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972),
  Springer, Berlin, 1973, pp.~183--240. Lecture Notes in Math., Vol. 342.
  \MR{MR0384917 (52 \#5787)}

\bibitem{GM:eqhomotopy}
J.~P.~C. Greenlees and J.~P. May, \emph{Equivariant stable homotopy theory},
  Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp.~277--323.
  \MR{MR1361893 (96j:55013)}

\bibitem{Gr:groupefond}
A.~Grothendieck, \emph{R{\^e}vetements {{\'e}}tale et groupe fondemental
  {(SGA1)}}, Lecture Notes in Math., vol. 224, Springer-Verlag, 1971.

\bibitem{Ill:homology}
S.~Illman, \emph{Equivariant singular homology and cohomology. {I}}, Mem. Amer.
  Math. Soc. \textbf{1} (1975), no.~issue 2, 156, ii+74. \MR{MR0375286 (51
  \#11482)}

\bibitem{Ill:triangulation}
\bysame, \emph{The equivariant triangulation theorem for actions of compact
  {L}ie groups}, Math. Ann. \textbf{262} (1983), no.~4, 487--501. \MR{MR696520
  (85c:57042)}

\bibitem{Kle:KleisliCat}
H.~Kleisli, \emph{Every standard construction is induced by a pair of adjoint
  functors}, Proc. Amer. Math. Soc. \textbf{16} (1965), 544--546. \MR{MR0177024
  (31 \#1289)}

\bibitem{Ko:rogbordism}
C.~Kosniowski, \emph{A note on {$RO(G)$} graded {$G$} bordism theory}, Quart.
  J. Math. Oxford Ser. (2) \textbf{26} (1975), no.~104, 411--419. \MR{MR0388418
  (52 \#9254)}

\bibitem{Le:Greenfunctors}
L.~G. Lewis, Jr., \emph{The theory of {G}reen functors}, unpublished
  manuscript, 1981.

\bibitem{Le:emspaces}
\bysame, \emph{Equivariant {E}ilenberg-{M}ac {L}ane spaces and the equivariant
  {S}eifert-van {K}ampen and suspension theorems}, Topology Appl. \textbf{48}
  (1992), no.~1, 25--61. \MR{MR1195124 (93i:55016)}

\bibitem{Le:Hurewicz}
\bysame, \emph{The equivariant {H}urewicz map}, Trans. Amer. Math. Soc.
  \textbf{329} (1992), no.~2, 433--472. \MR{92j:55024}

\bibitem{LMM:roghomology}
L.~G. Lewis, Jr., J.~P. May, and J.~E. McClure, \emph{Ordinary {$RO(G)$}-graded
  cohomology}, Bull. Amer. Math. Soc. (N.S.) \textbf{4} (1981), no.~2,
  208--212. \MR{MR598689 (82e:55008)}

\bibitem{LMS:eqhomotopy}
L.~G. Lewis, Jr., J.~P. May, and M.~Steinberger, \emph{Equivariant stable
  homotopy theory}, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag,
  Berlin, 1986, With contributions by J. E. McClure. \MR{MR866482 (88e:55002)}

\bibitem{Lu:transfgroups}
W.~L{\"u}ck, \emph{Transformation groups and algebraic {$K$}-theory}, Lecture
  Notes in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989, Mathematica
  Gottingensis. \MR{91g:57036}

\bibitem{Mac:categories}
S.~Mac\, Lane, \emph{Categories for the working mathematician},
  Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 5.
  \MR{MR0354798 (50 \#7275)}

\bibitem{MM:orthogonal}
M.~A. Mandell and J.~P. May, \emph{Equivariant orthogonal spectra and
  {$S$}-modules}, Mem. Amer. Math. Soc. \textbf{159} (2002), no.~755, x+108.
  \MR{2003i:55012}

\bibitem{May:foundations}
J.~P. May, \emph{Homotopical foundations of algebraic topology}, unpublished
  manuscript, 1974.

\bibitem{May:alaska}
\bysame, \emph{Equivariant homotopy and cohomology theory}, CBMS Regional
  Conference Series in Mathematics, vol.~91, Published for the Conference Board
  of the Mathematical Sciences, Washington, DC, 1996, With contributions by M.
  Cole, G. Comeza\~na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L.
  G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. \MR{MR1413302
  (97k:55016)}

\bibitem{May:concise}
\bysame, \emph{A concise course in algebraic topology}, Chicago Lectures in
  Mathematics, University of Chicago Press, Chicago, IL, 1999. \MR{MR1702278
  (2000h:55002)}

\bibitem{May:thomiso}
\bysame, \emph{Equivariant orientations and {T}hom isomorphisms}, Tel Aviv
  Topology Conference: Rothenberg Festschrift (1998), Contemp. Math., vol. 231,
  Amer. Math. Soc., Providence, RI, 1999, pp.~227--243. \MR{MR1707345
  (2000i:55042)}

\bibitem{MaySig:parametrized}
J.~P. May and J.~Sigurdsson, \emph{Parametrized homotopy theory}, Mathematical
  Surveys and Monographs, vol. 132, American Mathematical Society, Providence,
  RI, 2006. \MR{MR2271789}

\bibitem{MS:characteristicclasses}
J.~W. Milnor and J.~D. Stasheff, \emph{Characteristic classes}, Princeton
  University Press, Princeton, N. J., 1974, Annals of Mathematics Studies, No.
  76. \MR{MR0440554 (55 \#13428)}

\bibitem{Pe:pseudo}
T.~Petrie, \emph{Pseudoequivalences of {$G$}-manifolds}, Algebraic and
  geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford,
  Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc.,
  Providence, R.I., 1978, pp.~169--210. \MR{MR520505 (80e:57039)}

\bibitem{Pu:rogbordism}
W.~Pulikowski, \emph{{$RO(G)$}-graded {$G$}-bordism theory}, Bull. Acad. Polon.
  Sci. S\'er. Sci. Math. Astronom. Phys. \textbf{21} (1973), 991--995.
  \MR{MR0339234 (49 \#3996a)}

\bibitem{Str:MonadsI}
R.~Street, \emph{The formal theory of monads}, J. Pure Appl. Algebra \textbf{2}
  (1972), no.~2, 149--168. \MR{MR0299653 (45 \#8701)}

\bibitem{TW:mackey}
J.~Th{\'e}venaz and P.~J. Webb, \emph{Simple {M}ackey functors}, Proceedings of
  the Second International Group Theory Conference (Bressanone, 1989), no.~23,
  1990, pp.~299--319. \MR{MR1068370 (91g:20011)}

\bibitem{tD:transfgroups}
T.~tom Dieck, \emph{Transformation groups}, de Gruyter Studies in Mathematics,
  vol.~8, Walter de Gruyter \& Co., Berlin, 1987. \MR{MR889050 (89c:57048)}

\bibitem{Wal:surgery}
C.~T.~C. Wall, \emph{Surgery on compact manifolds}, Academic Press, London,
  1970, London Mathematical Society Monographs, No. 1. \MR{55 \#4217}

\bibitem{Wan:gcwv}
S.~Waner, \emph{{$G$-CW($V$)} complexes}, unpublished manuscript, 1979.

\bibitem{Wan:class}
\bysame, \emph{Classification of oriented equivariant spherical fibrations},
  Trans. Amer. Math. Soc. \textbf{271} (1982), no.~1, 313--324. \MR{83g:57026}

\bibitem{Wan:ROGbordism}
\bysame, \emph{Equivariant {$RO(G)$}-graded bordism theories}, Topology Appl.
  \textbf{17} (1984), no.~1, 1--26. \MR{85e:57042}

\bibitem{Was:difftopology}
Arthur~G. Wasserman, \emph{Equivariant differential topology}, Topology
  \textbf{8} (1969), 127--150. \MR{MR0250324 (40 \#3563)}

\bibitem{Wh:homotopy}
G.~W. Whitehead, \emph{Elements of homotopy theory}, Graduate Texts in
  Mathematics, vol.~61, Springer-Verlag, New York, 1978. \MR{MR516508
  (80b:55001)}

\bibitem{Wi:duality}
K.~Wirthm{\"u}ller, \emph{Equivariant homology and duality}, Manuscripta Math.
  \textbf{11} (1974), 373--390. \MR{MR0343260 (49 \#8004)}

\end{thebibliography}