Extension theory of infinite symmetric products
Jerzy Dydak
We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension $\ExD(X)$ was introduced by A.N.Dranishnikov \cite {D$_5$} in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products $SP(L)$. One of the main ideas of the paper is to treat $\ExD(X)\leq SP(L)$ as the fundamental concept of cohomological dimension theory instead of $\dim_G(X)\leq n$. In a subsequent paper \cite{Dy$_6$} we show how properties of infinite symmetric products lead naturally to a calculus of graded groups which implies most of classical results of the cohomological dimension. The basic notion in \cite{Dy$_6$} is that of homological dimension of a graded group which allows for simultanous treatment of cohomological dimension of compacta and extension properties of CW complexes. \par We introduce cohomology of $X$ with respect to $L$ (defined as homotopy groups of the function space $SP(L)^X$). As an application of our results we characterize all countable groups $G$ so that the Moore space $M(G,n)$ is of the same extension type as the Eilenberg-MacLane space $K(G,n)$. Another application is characterization of infinite symmetric products of the same extension type as a compact (or finite-dimensional and countable) CW complex.
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\begin{document}

\title[Extension theory of infinite symmetric products]%
   {
Extension theory of infinite symmetric products}

\author{Jerzy Dydak}
\date{April 17, 2004}
\keywords{ Cohomological dimension, dimension, Eilenberg-MacLane
complexes, dimension functions, infinite symmetric product}

\subjclass{ 54F45, 55M10, 55N99, 55Q40, 55P20}

\thanks{ Research supported in part by a grant
 DMS-0072356 from the National Science Foundation.}

\begin{abstract}

We present an approach to cohomological dimension theory
based on infinite symmetric products and on the
 general theory of dimension called
the extension dimension. The notion of the extension dimension $\ExD(X)$
was introduced
by A.N.Dranishnikov \cite {D$_5$} in the context
of compact spaces and CW complexes. 
 This paper investigates extension types of
infinite symmetric products $SP(L)$.
One of the main ideas of the paper is to treat
$\ExD(X)\leq SP(L)$ as the fundamental concept of
cohomological dimension theory
instead of $\dim_G(X)\leq n$. In a subsequent paper \cite{Dy$_6$}
we show how properties of infinite symmetric products
 lead naturally
to a calculus of graded groups which implies most of classical results of
the cohomological dimension. The basic notion in \cite{Dy$_6$} is that of homological
dimension of a graded group which allows for simultanous treatment
of cohomological dimension of compacta and extension properties
of CW complexes.
\par
We introduce cohomology
 of $X$ with respect to $L$ (defined as homotopy groups of the function space
$SP(L)^X$).
 As an application of our results we characterize all countable
groups $G$ so that the Moore space $M(G,n)$ is of the same
extension type as the Eilenberg-MacLane space $K(G,n)$.
Another application is characterization of infinite symmetric
products of the same extension type as a compact (or finite-dimensional
and countable) CW complex.


\end{abstract}


\maketitle

\medskip
%Printed on \today.
\medskip
\tableofcontents


\section{Introduction}

\begin{notation} \label{XXXi.0} Throughout the paper $K$, $L$, and $M$ are
reserved for CW complexes. $X$ and $Y$ are general topological spaces
(quite often compact or compact metrizable).
We will frequently omit
coefficients in the case of
integral homology and cohomology. Thus,
 $H_n(K;\Z)$ will be shortened to $H_n(K)$ and $H^n(X;\Z)$ will be
shortened to $H^n(X)$.
\end{notation} 

Recall that $K$ is
an {\bf absolute extensor} of $X$ (this is denoted by $K\in AE(X)$)
if any map $f:A\to K$, $A$ closed in $X$,
extends over $X$.
That concept creates a partial order
on the class of CW complexes. Namely, $K\leq L$ if $K\in AE(X)$
implies $L\in AE(X)$ for every compact space $X$.
The partial order induces an equivalence relation on the class of all CW
complexes. The equivalence class $[K]$ of $K$ is called its
{\bf extension type}. 
\begin{Def} \label{XXXi.ext} A CW complex $K$ is called
the {\bf extension dimension} of a compact space $X$
(notation: $K=\ExD(X)$) if $K$ is a minimum
of the class $\{L\mid L\in AE(X)\}$.
\end{Def} 

\begin{Thm}[Dranishnikov Duality Theorem \cite {D$_5$}] \label{XXXi.1} 
Extension
dimension of compact spaces exists and each extension type
is equal to the extension dimension of some compact space.
\end{Thm} 

The concept of extension dimension generalizes both
the covering dimension $\dim(X)$ and the cohomological
dimension $\dim_G(X)$ with respect to an Abelian group $G$.
Indeed, $\dim(X)\leq n$ is equivalent to $\ExD(X)\leq S^n$
and $\dim_G(X)\leq n$ is equivalent to $\ExD(X)\leq K(G,n)$.

\par The theory of extension dimension is mostly
geometric in nature (see Section 2).
We introduce algebra to it following the basic idea of
\cite{Git}, where algebraic topology is outlined
via properties of infinite symmetric products $SP(K)$.
Thus, in this paper we show that the relation $SP(K)\leq SP(L)$
is of purely algebraic nature. We analyze it by generalizing
the connectivity index of Shchepin \cite{S} to the concept
of homological dimension of CW complexes.
To analyze relation $\ExD(X)\leq SP(L)$ we introduce
the concept of cohomology groups $H^\ast(X;L)$
of $X$ with coefficients in a CW complex $L$ (see Section 4).
Those cohomology groups have natural formulae facilitating
proofs and applications.
We show in Section 6 that the class $\{SP(L)\mid \ExD(X)\leq SP(L)\}$
has a minimum which
should be interpreted as the coefficient-free cohomological dimension
of $X$.
\par In Section 8 we dualize the connectivity index and use it
to derive algebraic implications of $\ExD(X)=\ExD(Y)$.
\par In a subsequent paper \cite{Dy$_6$} we will explain
that Bockstein Theory plays the role of homological algebra
in algebraic topology. In this paper we use Bockstein Theory
to give necessary and sufficient conditions for $SP(L)$
to have the same extension type as an Eilenberg-MacLane
space $K(G,n)$ (see Section 7).
Later on (Section 9) we characterize extension types
$[SP(L)]$ containing compact (respectively, countable and
finite-dimensional) CW complexes. That characterization
generalizes all the previously known results about
different extension types.


The last part of the paper (Section 9) is devoted to
comparison of extensional properties of $M(G,n)$ and $K(G,n)$.

\begin{Def} \label{XXXi.5} Suppose $G$ is an Abelian group and $n\ge 1$ is
an integer.
By $M(G,n)$ we will denote {\bf the Moore space}, i.e. a CW complex
$K$ so that $\tilde H_n(K;\Z)=G$ and $\tilde H_m(K;\Z)=0$ for $m\ne n$.

\end{Def} 


More precisely, Moore spaces discussed in this paper are constructed as
follows.
Choose a short exact sequence $0\to F_1\to F\to G\to 0$ so that $F$ is free
Abelian.
Let $L$ be the wedge of $n$-spheres enumerated by generators of $F$. Attach
$(n+1)$-cells to $L$ enumerated by generators of $F_1$ via characteristic maps
corresponding to $F_1\to F$. In particular, such Moore spaces are
finite-dimensional
and one has a map $M(\pi_n(K),n))\to K$ 
(provided $\pi_1(K)$ is Abelian if $n=1$) inducing isomorphism of $n$-th
homotopy groups
for any space $K$.

\par

We will use the following generalization
of the relation $K\leq L$ (see \cite{D-D$_1$}):

\begin{Def} \label{XXXi.2} Suppose $K$ and $L$ are CW complexes and $\cal C$
is a class of spaces. $K\leq_{\cal C}L$ means that if $X\in\cal C$ and
$K\in AE(X)$,
then $L\in AE(X)$. In particular $K\leq_X L$ means $K\leq_{\{X\}} L$.
\end{Def} 

The author would like to thank Akira Koyama for discussions on the subject
of the paper, and E.Shchepin for giving a series of excellent talks
(in Russian) during the workshop \lq Algebraic ideas in dimension
theory\rq \ held in Warsaw (Fall 1998). Those talks were the starting
point in author's understanding of paper \cite{S}
which eventually led to the ideas developed in this paper.
\par Some of the results of this paper were circulated
in unpublished notes \cite{Dy$_7$}.

\section{Geometry of extension theory }


The purpose of this section is to list results of extension
theory which involve no algebraic computations.

%delXXX2.1

\begin{Prop} \label{XXX2.1} (see \cite{D-D$_1$},\cite{D$_1$}) Suppose $K$
is a CW complex and $X$ is metrizable. If $K\in AE(X)$, then every map
$f:X\to \Sigma(K)$ from $X$ to the suspension of $K$ is null-homotopic.

\end{Prop} 





%delXXX2.2

\begin{Thm} \label{XXX2.2} (see \cite{D$_4$}) 
If $X=\bigcup\limits_{n=1}^\infty X_n$, $X_n$ is a closed subset of $X$ and $K\in
AE(X_n)$
for all $n$, then
$K\in AE(X)$ provided $X$ is normal and $K\in ANE(X)$.

\end{Thm} 





%delXXX2.3

\begin{Thm} \label{XXX2.3} (see \cite{Wa} and \cite{S}) 
Suppose $K$ is a CW complex. If $K\in AE(X)$, then
$K\in AE(Y)$ for every $Y\subset X$ if $X$ is metrizable.

\end{Thm} 





%delXXX2.4

\begin{Thm} \label{XXX2.4} (see \cite{Dy$_3$}) 
If $X=A\cup B$ is metrizable, and $K\in AE(A)$, $L\in AE(B)$
are CW complexes,
then $K\ast L\in AE(X)$.

\end{Thm} 





%delXXX2.5

\begin{Thm} \label{XXX2.5} (see \cite{D$_4$}) 
Suppose $K$ and $L$
are countable CW complexes.
If $K\ast L\in AE(X)$ and $X$ is a compactum,
then there is a $G_\delta$-subset $A$ of $X$ such that
  $K\in AE(A)$ and $L\in AE(X-A)$.

\end{Thm} 





%delXXX2.6

\begin{Thm} \label{XXX2.6} (see \cite{O$_1$}) 
Suppose $K$ is a countable CW complex. If $K\in AE(X)$ and $X$ is a subset
of a metric separable space $X'$,
then there is a $G_\delta$-subset $A$ of $X'$ containing $X$ such that
  $K\in AE(A)$.

\end{Thm} 


\begin{Thm} \label{XXX2.11} (see \cite{D-D$_1$}) Suppose $X$ is compact or 
metrizable and $K$ is
a pointed connected CW complex. The following conditions are
equivalent:
\par{1.} $K\in AE(X\times I)$.
\par{2.} $\Omega(K)\in AE(X)$, where $\Omega(K)$ is the loop-space
of $K$.
\par{3.} $\lbrack X/A,K\rbrack=0$ for all closed subsets $A\ne \emptyset$ of $X$.
\par{4.} $K\in AE(\Sigma(X))$.

\end{Thm} 

\begin{Thm} \label{XXX2.9} (see \cite{D$_3$}) 
Suppose $K$ is a CW complex. If $X$ is finite-dimensional and
$\prod\limits_{i=1}^{\infty}K(\pi_i(K),i)\in AE(X)$, then $K\in AE(X)$.

\end{Thm} 


\section{Transition to algebra in extension theory}


 Given a space
$X$ and $k>0$, the $k$-th {\bf symmetric product}
$SP^k(X)$ of $X$ is the  space of orbits of the action of
the symmetric group $S_k$ on $X^k$.  Points of  $SP^k(X)$ can be written
in the form $\sum\limits_{i=1}^{k} x_i$. $SP^k(X)$ is equipped
 with the quotient topology given by the natural  map $\pi :X^k\to SP^k(X)$.
If $X$ is
metrizable, then $\pi :X^k\to SP^k(X)$ is both open and closed
(see p. 255 of \cite {D-T}), so $SP^k(X)$ is metrizable, too 
(use 4.4.18 of \cite {En}).
\par\noindent If $X$ has a base point $a$, then $SP^k(X)$
has $\sum\limits_{i=1}^{k} a$ as its base point. Notice that there is
a natural inclusion $i:SP^n(X)\to SP^k(X)$ for all $n<k$.
It is given by the formula $i(\sum\limits_{i=1}^{n}
x_i)=\sum\limits_{i=1}^{n} x_i+(k-n)a$.
In this way, points of the form  $\sum\limits_{i=1}^{n} x_i$, $n<k$, can
be considered as belonging to $SP^k(X)$.
\par\noindent The direct limit of $SP^2(X)\to \ldots \to SP^n(X)\to \ldots$
is denoted by
$SP(X)$ and is called {\bf the infinite symmetric product}
(see \cite {D-T} or \cite{Git}, p.168).
\par

The main property of the infinite symmetric product is expressed
in the following result.



%delXXX3.1

\begin{Thm}[Dold-Thom Theorem \cite {D-T}] \label{XXX3.1} If $K$ is a pointed CW complex, then
the natural inclusion $i:K\to SP(K)$
induces an isomorphism $\tilde H_i(K)\to \pi_i(SP(K))$.

\end{Thm} 

 The meaning of Dold-Thom Theorem is that one can define
singular homology groups geometrically, without the apparatus of
homological algebra,
as homotopy groups of infinite symmetric products (see \cite{Git}).

Of major importance to us is the following result of Dranishnikov:
\begin{Thm} \label{XXX2.10a} (see \cite{D$_5$}) 
Suppose $K$ is a CW complex. If $X$ is compact and $K\in AE(X)$, then
$SP(K)\in AE(X)$. 
\end{Thm} 


Since $SP(K)$ is homotopy equivalent
to the weak product of Eilenberg-MacLane spaces
$K(\tilde H_i(K),i)$ (see \cite{Git}, Corollary 6.4.17 on p.223)
one has the following:
\begin{Thm} \label{XXX2.10b} (see \cite{D$_5$}) 
Suppose $K$ is a CW complex. If $X$ is compact, then
$SP(K)\in AE(X)$
is equivalent to $K(\tilde H_i(K),i)\in AE(X)$ for all $i\ge 0$. 
\end{Thm}

The following result of Dranishnikov uses a high level
of algebraic arguments and can be viewed as an analog
of the Hurewicz Theorem in Extension Theory.

\begin{Thm} \label{XXX2.10} (see \cite{D$_5$}) 
Suppose $K$ is a CW complex. If $X$ is compact
finite-dimensional, $K$ is simply connected, and
$SP(K)\in AE(X)$, then $K\in AE(X)$.
\end{Thm} 


\par Let us move to algebraic concepts associated with extension theory
by employing the connectivity index introduced by
 E.Shchepin \cite{S}. 

\begin{Def} \label{XXX1.3} (\cite{S},p.985) 
Suppose $K$ is a CW complex. Its {\bf connectivity index}
$cin(K)$ is either $\infty$ or a non-negative integer. $cin(K)=\infty$ means
that all reduced integral homology groups of $K$ are trivial.
$cin(K)=n$ means that $\tilde H_n(K;\Z)\ne 0$
and $\tilde H_k(K;\Z)=0$ for all $k<n$.
\end{Def} 

\begin{Prop} \label{XXX2.7} Suppose $K$ is a simply-connected
CW complex. The following numbers are equal:
\par\par{1.} $cin(K)$, 
\par\par{2.} the supremum of all $n\ge 0$ so that
$K\in AE(S^n)$,
\par\par{3.} the supremum of all $n\ge 0$ so that
$K$ is homotopy $k$-connected for all $k<n$.
\end{Prop} 


\begin{pf}  If $K\in AE(S^n)$, then any map $S^j\to K$, $j<n$,
extends over $S^n$ and must be null-homotopic. Hence
$H_j(K)=0$ for $j<n$.
If $H_j(K)=0$ for $j<n$, then $\pi_j(K)=0$ for $j<n$
and any map $f:A\to K$, $A$ closed in $S^n$, can be extended
over $S^n$ as follows: first extend it over a polyhedral
neighborhood $N$ of $A$, then keep extending over simplices
which are not contained in $N$ using induction over
the dimension of simplices.
 \end{pf}

\begin{Prop} \label{XXX2.8} Suppose $K$ and $L$ are pointed CW
complexes.
If $K$ is countable and $K\le_{X} L$ for all finite-dimensional compacta $X$,
then $cin(K\wedge M)\le cin(L\wedge M)$
for every CW complex $M$.
\end{Prop} 


\begin{pf}  
\par Case 1: $M$ is countable.
\par 
Since suspending a CW complex pushes its connectivity index up by 1,
it suffices to prove \ref{XXX2.8} for $M=\Sigma^2 M'$, in which case $K\wedge M$
and $L\wedge M$ are simply connected and we may use \ref{XXX2.7}.
Notice that $cin(K\wedge M)=cin(K\ast M)-1$, where $K\ast M$
is the join of $K$ and $M$.
Suppose $K\ast M\in AE(S^m)$. Express $S^m$ as $A\cup B$
so that $K\in AE(A)$, $M\in AE(B)$, and $A$ is $F_\sigma$ (see  \ref{XXX2.5}).
Now, $L\in AE(A)$ (see \ref{XXX2.2}) which implies 
$L\ast M\in AE(S^m)$ by \ref{XXX2.4}.
\par Case 2:  $M$ is arbitrary.
\par Suppose $m=cin(K\wedge M)> cin(L\wedge M)=n$.
There is an integral cycle $c$
in $H_n(L\wedge M)-\{0\}$. That cycle lies in $L\wedge M_1$ for
some finite subcomplex $M_1$ of $M$. 
Given countable subcomplex $M_i$ of $M$ construct countable subcomplex 
 $M_{i+1}\supset M_i$ of $M$ so that $H_k(K\wedge M_i)\to H_k(K\wedge M_{i+1})$
is trivial for $k<m$.
Let $M'$ be the union of all $M_i$.
By Case 1, $m\leq cin(K\wedge M')\leq cin(L\wedge M')$
which means that $c$ becomes $0$ in $H_n(L\wedge M')$,
a contradiction.
 \end{pf}

\section{Cohomology with coefficients in a complex}

\par With the hindsight of homological algebra
one has the pairing
$$H_*(K)\wedge H_*(L)\to \pi_*(SP(K\wedge L))$$
for any two pointed CW complexes $K$ and $L$.
It corresponds to the well-known K\" unneth formula for homology
and we will give it a slightly non-traditional form.



%delXXX3.2

\begin{Thm} \label{XXX3.2} If $K$ and $L$ are pointed
CW complexes, then
$$H_n(K\wedge L)\equiv \bigoplus\limits_{i+j=n}H_i(K;H_j(L)).$$

\end{Thm} 


\begin{Rem} \label{XXX3.2.5}
 Theorem \ref{XXX3.2} is a direct consequence of the Universal
Coefficient Theorem (see \cite{Sp}, p.222) and the classical
K\" unneth Theorem (see \cite{Sp}, p.235). 
\end{Rem}


Notice that
$$\bigoplus\limits_{i+j=n}H_i(K;H_j(L))\equiv
\bigoplus\limits_{i+j=n}H_i(L;H_j(K))$$
as both groups are isomorphic to $H_n(K\wedge L)$. A natural question to
ask is if
$$\bigoplus\limits_{i}H^i(K;H_{n-i}(L))$$
or
$$\bigoplus\limits_{i}H_i(L;H^{n-i}(K))$$
have similar geometrical interpretation, and if they are isomorphic.
Knowing that the smash product
$K\wedge L$ is adjoint to the function space functor
one can speculate the following:



%delXXX3.3

\begin{Idea} \label{XXX3.3}
There ought to be a dual pairing
$$H^*(X)\wedge H_*(L)\equiv \pi_*(SP(L)^X).$$

\end{Idea} 


The above pairing should correspond to K\" unneth formula for cohomology.
It turns out such pairing exists for pointed compact spaces.
\par Our first idea is to introduce cohomology of a pointed space
$X$ via homotopy groups of function spaces $SP(K)^X$.




%delXXX3.4

\begin{Def} \label{XXX3.4} Suppose $K$ is a pointed CW complex and $X$
is a pointed space. The {\bf cohomology} $H^n(X;K)$ of $X$ {\bf
with coefficients} in $K$ is defined as follows:
\par{1.} $H^n(X;K)=[X,SP(\Sigma^{n}(K)]$ if $n\ge 0$,
\par{2.} $H^n(X;K)=[\Sigma^{-n}X,SP(K)]$ if $n\leq 0$.
\end{Def} 





%delXXX3.5

\begin{Prop} \label{XXX3.5} If $K=M(G,n)$ is a Moore space, then $H^k(X;K)=H^{n+k}(X;G)$.

\end{Prop} 


\begin{pf}  Notice that $\Sigma^r(K)=M(G,n+r)$ and
$SP(M(G,r))=K(G,r)$ for $r\ge 1$ (see 3.1). As $H^m(X;G)=[X,K(G,m)]$
one gets \ref{XXX3.5} immediately from the definition of $H^m(X;K)$.
 \end{pf}



It is shown in \cite{D-T} that $SP(K)$
is homotopy equivalent to $\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(H_n(K),n)$ for every connected CW complex $K$
(see also \cite{Git}, Corollary 6.4.17 on p.223). We will need a more general
result.




%delXXX3.6

\begin{Prop} \label{XXX3.6} Suppose $K$ is a pointed CW complex
and $X$ is a pointed $k$-space. There is a weak homotopy equivalence
$i:\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(\pi_n(SP(K)^X),n)\to SP(K)^X$.

\end{Prop} 


\begin{pf}  Notice any map $f:L\to SP(K)^X$ extends
to $F:SP(L)\to SP(K)^X$ if $L$ is a CW complex.
Indeed, one can define $F$ as follows:
$F(\sum\limits_{i=1}^na_i)(x)=\sum\limits_{i=1}^nf(a_i)(x)$
for $\sum\limits_{i=1}^na_i\in SP(L)$ and $x\in X$.
Let $L$ be the wedge of $M(\pi_n(SP(K)^X),n)$, $n\ge 1$.
There is a map $f:L\to SP(K)^X$ so that
$\pi_n(f|M(\pi_n(SP(K)^X),n))$ is an isomorphism for each $n$
(see the discussion after  \ref{XXXi.5}). In particular, $H_k(L)\to \pi_k(SP(K)^X)$
is an isomorphism for all $k\ge 1$.
Pick an extension $F:SP(L)\to SP(K)^X$ of $f$.
Since $H_k(L)\to \pi_k(SP(L))$
is an isomorphism for all $k\ge 1$ and $H_k(L)\to \pi_k(SP(K)^X)$
is an isomorphism for all $k\ge 1$, $F$ is a weak homotopy
equivalence. As shown in \cite{D-T},
$SP(L)$
is homotopy equivalent to $\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(H_n(L),n)$ which is exactly the space
 $\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(\pi_n(SP(K)^X),n)$.
 \end{pf}






%delXXX3.7

\begin{Cor} \label{XXX3.7} Suppose $K$ is a pointed connected
CW complex and $X$ is a pointed $k$-space. $SP(K)$ and
the spaces $\Omega^r(SP(\Sigma^r(K)))$
are homotopy equivalent for $r\ge 1$. In particular,
 $H^n(X,\Sigma^r(K))\equiv H^{n+r}(X;K)$
for all $n\in\Z$ and all $r\ge 1$.

\end{Cor} 


\begin{pf}  Notice that $\pi_k(\Omega^r(SP(\Sigma^r(K))))=
\pi_{k+r}(SP(\Sigma^r(K)))=H_{k+r}(\Sigma^r(K))=H_k(K)$ for 
each $k\ge 0$.
By \ref{XXX3.6} (applied to $X=S^r$) there is is a weak homotopy equivalence
 $\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(H_{n}(K),n)\to \Omega^r(SP(\Sigma^r(K)))$. This map
is a homotopy equivalence as both spaces are homotopy
equivalent to CW complexes.
As in \cite{D-T}, 
$\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(H_{n}(K),n)$ is homotopically equivalent to $SP(K)$.
\par If $n\ge 0$, then
\par\noindent
$H^n(X;\Sigma^r(K))=[X,SP(\Sigma^{n+r}(K)]=H^{n+r}(X;K)$.
\par
If $n<0$, then $H^n(X;\Sigma^r(K))=[\Sigma^{-n}(X),SP(\Sigma^r(K))]$.
If $n+r\ge 0$, then
\par\noindent
$[\Sigma^{-n}(X),SP(\Sigma^r(K))]=
[X,\Omega^{-n}(SP(\Sigma^r(K)))]=[X,(SP(\Sigma^{n+r}(K)))]=
H^{n+r}(X;K)$.
If $n+r<0$, then
$[\Sigma^{-n}(X),SP(\Sigma^r(K))]=
[\Sigma^{-n-r}(X),\Omega^{r}(SP(\Sigma^r(K)))]$
\par\noindent
$=[\Sigma^{-n-r}(X),SP(K)]=
H^{n+r}(X;K)$.
 \end{pf}




The purpose of the next result is to generalize the well known
theorem of Cohen \cite{C}.



%delXXX3.8

\begin{Thm} \label{XXX3.8} Suppose $K$ is a pointed CW complex
and $X$ is compact or metrizable. The following conditions are equivalent:
\par{1.} $SP(K)\in AE(X)$.
\par{2.} $H^n(X/A;K)=0$ for all $n>0$ and
all closed subsets $A$ of $X$.
\par{3.} $H^{1}(X/A;K)=0$ for
all closed subsets $A$ of $X$.

\end{Thm} 


\begin{pf}  $H^n(X/A;K)$ was defined as $[X/A,SP(\Sigma^n(K))]$
for $n\ge 1$ (see \ref{XXX3.4}). \ref{XXX2.11} says that
$[X/A,SP(\Sigma(K))]=0$ for all closed subsets $A$ of $X$
if and only if $\Omega(SP(\Sigma(K)))\in AE(X)$. By \ref{XXX3.7},
$\Omega(SP(\Sigma(K)))$ is homotopy equivalent
to $SP(K)$ which proves 1)$\iff$ 3).
\par Clearly, 2) is stronger than 3).
\par Suppose 3) and 1) hold. If $n>1$, then (see \ref{XXX2.11})
$SP(\Sigma^n(K))\in AE(X)$
and $H^1(X/A;\Sigma^{n-1}(K))=0$ for all closed subsets $A$ of $X$
(as 1) is equivalent to 3) for all CW complexes $K$).
Now, \ref{XXX3.7} says $H^n(X/A;K)=H^1(X/A;\Sigma^{n-1}(K))$ which proves that 2) holds.
 \end{pf}


Notice that in \ref{XXX3.4} the homotopy groups of $(SP(K))^X$ correspond
to negative cohomology groups of $X$ with coefficients in $K$. It seems
unnatural
but we chose it that way in order to adhere to common practices.
However, it is time to break with tradition and achieve a better theory
that way.




%delXXX3.9

\begin{Def} \label{XXX3.9} Suppose $h^*$ is a cohomology theory.
By ${}^rh^*$ we will
denote the {\bf reversed cohomology} defined via $^rh^m(X)=h^{-m}(X)$.

\end{Def} 


The following result shows that using reversed cohomology one can
achieve similarity between homology and cohomology (compare \ref{XXX3.10} with 
 \ref{XXX3.2}).



%delXXX3.10

\begin{Thm} \label{XXX3.10} If $K$ is a pointed CW complex
and $X$ is a pointed  compact space, then
$${}^rH^n(X;K)\equiv \bigoplus\limits_{i}{}^rH^i(X;H_{n-i}(K))$$
and 
$${}^rH^n(X;K)\equiv \bigoplus\limits_{i}H_i(K;{}^rH^{n-i}(X)).$$

\end{Thm} 


\begin{pf}  If $Y$ is a pointed compact space and $L$ is a pointed
CW complex, then $[Y,SP(L)]$ is the direct
sum $\bigoplus\limits_{i}{}[Y,K(H_i(L),i)]$. Indeed,
$SP(L)$ is homotopically equivalent to 
$\bigcup\limits_{m=1}^\infty\prod\limits_{n=0}^m
K(H_{n}(L),n)$ (see \cite{D-T}) and any map from $Y$
(or any homotopy from $Y\times I$) to
$\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(H_{n}(L),n)$ is contained in $\prod\limits_{n=0}^m
K(H_{n}(L),n)$ for some $m$.
\par
If $n\ge 0$, then 
$${}^rH^{-n}(X;K)=[X,SP(\Sigma^{n}(K)]=$$
$$\bigoplus\limits_i
[X,K(H_{i-n}(K),i)]=\bigoplus\limits_{i}{}^rH^{-i}(X;H_{i-n}(K)).$$
\par
If $n< 0$, then ${}^rH^{-n}(X;K)=[\Sigma^{-n}X,SP(K)]$ is
 $\bigoplus\limits_i
[\Sigma^{-n}(X),K(H_{i}(K),i)]$ which is the same as
$\bigoplus\limits_{i}H^i(X;H_{i-n}(K))=\bigoplus\limits_{i}{}^rH^{-i}(X;H_{i-n}(
K)).$
\par By the Universal Coefficient Theorem for homology (see Theorem 14
in \cite{Sp} on p.226)
$\bigoplus\limits_{i}H_i(K;{}^rH^{n-i}(X))$ is isomorphic
to 
$$G_1=\bigoplus\limits_{i}(H_i(K)\otimes {}^rH^{n-i}(X)\oplus H_{i-1}(K)*{}^rH^{n-i}(X))$$
($G*G'$ is the torsion product of $G$ and $G'$).
By the Universal Coefficient Theorem for cohomology (see Statement 5
in \cite{K} on p.4)
$\bigoplus\limits_{i}{}^rH^i(X;H_{n-i}(K))$ is isomorphic
to 
$$G_2=\bigoplus\limits_{i}(H_{n-i}(K)\otimes {}^rH^{i}(X)\oplus H_{n-i}(K)*{}^rH^{i-1}(X)).$$
Notice that $G_1$ is isomorphic to $G_2$ (change $i$ to $n-i$ in the formula for $G_1$).
 \end{pf}


As a simple consequence of the fact that $K^Y$ is homotopy equivalent to a CW
complex if $K$ is a CW complex and $Y$ is compact, we get the following
version of K\" unneth
Formula.



%delXXX3.11

\begin{Thm} \label{XXX3.11} Suppose $K$ is a pointed CW complex.
If $X$ 
and $Y$ are pointed compact spaces, then
$${}^rH^n(X\wedge Y;K)\equiv \bigoplus\limits_{i}{}^rH^i(X;{}^rH^{n-i}(Y;K)).$$

\end{Thm} 


\begin{pf}  Suppose $n\ge 0$.
${}^rH^n(X\wedge Y;K)=[\Sigma^n(X\wedge Y),SP(K)]=
[\Sigma^n(X),SP(K)^Y]$. Since $SP(K)^Y$ is homotopy equivalent
to a CW complex, \ref{XXX3.6} says that it is homotopy equivalent to
$\bigcup\limits_{m=1}^\infty\prod\limits_{i=0}^m
K({}^rH^{i}(Y;K),i)$. Thus,
$${}^rH^n(X\wedge Y;K)=\bigoplus\limits_iH^i(\Sigma^n(X);{}^rH^{i}(Y;K))=$$

$$\bigoplus\limits_iH^{i-n}(X;{}^rH^{i}(Y;K))=\bigoplus\limits_i{}^rH^{n-i}(X;{}^r
H^{i}(Y;K)).$$
\par If $n<0$, then one reduces it to the case $n=0$ by observing
that ${}^rH^n(Z;K)={}^rH^0(Z;\Sigma^{-n}(K))$ for every pointed $k$-space
$Z$ (see \ref{XXX3.7}).
 \end{pf}





%delXXX3.12

\begin{Cor} \label{XXX3.12} If $K$ is a pointed CW complex
and $X$ is a pointed compact space, then
the following conditions are equivalent ($m$ is an integer):
\par{1.}
$H^n(X;K)=0$ for all $n\ge m$.
\par{2.}
$H_i(K;H^n(X))=0$ for all $i\leq n-m$.
\par{3.}
$H^{i}(X;H_{n}(K))=0$ for all $i\ge n+m$.

\end{Cor} 


\begin{pf}  Notice that $H^n(X;K)={}^rH^{-n}(X;K)$
is isomorphic to $\bigoplus\limits_{i}H_i(K;{}^rH^{-n-i}(X))$ (see \ref{XXX3.10})
which is the same as $\bigoplus\limits_{i}H_i(K;H^{n+i}(X))$.
That means $H^n(X;K)=0$ for all $n\ge m$ is equivalent to
$H_i(K;H^{n+i}(X))=0$ for all $n\ge m$ which is the same as saying that
$H_i(K;H^n(X))=0$ for all $i\leq n-m$.
Similarly, $H^n(X;K)={}^rH^{-n}(X;K)$
is isomorphic to $\bigoplus\limits_{i}{}^rH^i(X;H_{-n-i}(K))$ (see \ref{XXX3.10})
which is the same as $\bigoplus\limits_{i}H^{-i}(X;H_{-n-i}(K))$.
That means $H^n(X;K)=0$ for all $n\ge m$ is equivalent to
$H^{-i}(X;H_{-n-i}(K))=0$ for all $n\ge m$ which is the same as saying
$H^{i}(X;H_{n}(K))=0$ for all $i\ge n+m$.
 \end{pf}





%delXXX3.13

\begin{Cor} \label{XXX3.13} Suppose $K$ is a pointed CW complex
and $X$ is a compact space. The following conditions are equivalent:
\par{1.} $SP(K)\in AE(X)$.
\par{2.} $H_i(K;H^n(X;A))=0$ for all $i<n$ and all closed subsets $A$ of $X$.
\par{3.} $H^i(X/A;H_n(K))=0$ for all $n<i$ and all closed subsets $A$ of $X$.

\end{Cor} 


\begin{pf}  By \ref{XXX3.8}, $SP(K)\in AE(X)$ if and only if
$H^n(X/A;K)=0$ for all $n\ge 1$ and all closed subsets $A$ of $X$. 
Using \ref{XXX3.12} one gets that
$SP(K)\in AE(X)$ if and only if $H_i(K;H^n(X,A))=0$ for all $i\leq n-1$.
That proves 1)$\iff$ 2). 2)$\iff$ 3) follows from \ref{XXX3.12} applied to $m=1$.
 \end{pf}

\section{Homological dimension of CW complexes}

\ref{XXX3.13} suggests that $K\leq L$ should have
 some algebraic implications. The next result specifies the nature
of those implications.




%delXXX3.14

\begin{Thm} \label{XXX3.14} Suppose $n>0$, $G$ is an Abelian group,
$K,L$ are CW complexes, and $\cal C$ is a class of spaces
containing all finite-dimensional
compacta. If $K$ is countable, $K\leq_{\cal C}L$, and $\tilde H_i(K;G)=0$ for all $i\leq n$, then
$\tilde H_i(L;G)=0$ for all $i\leq n$.

\end{Thm} 


\begin{pf}  
Make $K$ and $L$ pointed CW complexes and switch from the reduced homology
to ordinary homology of pointed CW complexes.
By \ref{XXX3.2}, $H_i(A\wedge M(G,1))=H_{i-1}(A;G)$ for any pointed
CW complex $A$, where $M(G,1)$ is
the Moore space. Thus $cin(K\wedge M(G,1))\ge n+2$. By \ref{XXX2.8},
$cin(L\wedge M(G,1))\ge n+2$ which is the same as 
$\tilde H_i(L;G)=0$ for all $i\leq n$.
 \end{pf}

Theorem \ref{XXX3.14} suggests a new concept
of {\bf homological dimension} $\dim_G(K)$
of a pointed CW complex.

\begin{Def} \label{XXX3.15a} Suppose $K$ is a pointed CW complex and $G$
is an Abelian group. $\dim_G(K)=n < \infty$ means that $H_i(K;G)=0$ for all
$i < n$ and
$H_n(K;G)\ne 0$. $\dim_G(K)=\infty$ means that $H_i(K;G)=0$ for 
all $i$.
\end{Def} 

\begin{Rem} \label{XXX3.15b}
Notice that the above concept generalizes the concept of connectivity
index. Indeed $cin(K)=\dim_\Z(K)$ for all pointed CW complexes $K$.
\end{Rem}


Definition \ref{XXX3.15a} suggests a new partial order on the class of CW complexes:



%delXXX3.15

\begin{Def} \label{XXX3.15} Suppose $K$ and $L$ are CW complexes and $G$
is an Abelian group. $K\leq_GL$ means $\dim_G(K)\leq \dim_G(L)$.
\par
If $\cal G$
is a class of Abelian groups, then $K\leq_{\cal G}L$ means that
$K\leq_GL$ for all $G\in\cal G$. $K\sim_{\cal G}L$ means that
$K\leq_{\cal G}L$ and $L\leq_{\cal G}K$.
\par $K\leq_{Gr} L$ means that
$K\leq_GL$ for all Abelian groups $G$. $K\sim_{Gr} L$ means that
$K\leq_{Gr} L$ and $L\leq_{Gr} K$.

\end{Def} 





%delXXX3.16

\begin{Cor} \label{XXX3.16} $K\sim_{Gr} SP(K)$ for each pointed
CW complex $K$.

\end{Cor} 


\begin{pf}  
Notice that $H_n(K)$ is a direct summand of $H_n(SP(K))$ for each $n$.
 Indeed, $SP(K)$ is homotopy equivalent
to $\bigcup\limits_{m=1}^\infty\prod\limits_{n=1}^m
K(H_n(K),n)$ and each $K(H_n(K),n)$ (whose $n$-th homology group is 
$H_n(K)$) is a retract of $SP(K)$.
If $H_k(SP(K);G)=0$ for $k<n$, then it amounts to
$H_k(SP(K))\otimes G=H_{k-1}(SP(K))*G=0$ for $k<n$
(in view of the Universal Coefficient Theorem). Therefore
$H_k(K)\otimes G=H_{k-1}(K)*G=0$ for $k<n$ and $H_k(K;G)=0$ for $k<n$.
This proves $SP(K)\leq_{Gr} K$.
\par To prove
$K\leq_{Gr} SP(K)$ notice that, for $K$ countable, it follows 
from \ref{XXX3.14} and  \ref{XXX2.10a}.
Suppose $H_k(K;G)=0$ for $k\leq n$ and there is $c\in H_n(SP(K);G)-\{0\}$.
Find a finite subcomplex $K_1$ of $K$ and a countable subgroup $G_1$ of $G$
such that $c$ belongs to the image of $H_n(SP(K_1);G_1)\to H_n(SP(K);G)$.
By induction find countable subcomplexes $K_1\subset K_2\subset \ldots$
of $K$ and countable subgroups $G_1\subset G_2\subset\ldots$ of $G$
so that $H_k(K_i;G_i)\to H_k(K_{i+1};G_{i+1})$ is trivial for all $i$
and all $k\leq n$. Let $K'$ be the union of all $K_i$ and let $G'$ be the union of all $G_i$.
Notice that $H_k(K';G')=0$ for $k\leq n$. Therefore $H_n(SP(K');G')=0$
contradicting $c\ne 0$.
 \end{pf}






%delXXX3.17

\begin{Thm} \label{XXX3.17} Suppose $K$ and $L$ are connected CW
complexes and $K$ is countable.
Consider the following conditions:
\par{1.} $K\leq_{Gr} L$.
\par{2.} $SP(K)\leq_X\ SP(L)$ for all compact $X$.
\par{3.} $cin(K\wedge M)\le cin(L\wedge M)$ for each complex $M$.
\par{4.} $K\leq_X\ L$ for all finite-dimensional compacta $X$.
\par Condition 4 implies Condition 1. Conditions 1-3 are equivalent. 
If $L$ is simply connected, then Conditions 1-4 are equivalent.

\end{Thm} 


\begin{pf}  4)$\implies $1) follows from \ref{XXX3.14}.
\par
3) $\implies$ 1). Suppose $H_k(K;G)=0$ for $k<n$.
Use $M=M(G,1)$ and \ref{XXX3.2} to conclude that $H_k(K\wedge M)=0$
for $k\leq n$. Therefore $H_k(L\wedge M)=0$
for $k\leq n$ which means $H_k(L;G)=0$
for $k<n$.
\par
1) $\implies$ 3). Suppose  $H_k(K\wedge M)=0$
for $k< n$. That means (see \ref{XXX3.2}) $H_i(K;H_j(M))=0$ if $i+j<n$.
Since $K\leq_{Gr} L$, $H_i(L;H_j(M))=0$ if $i+j<n$,
i.e. $H_k(L\wedge M)=0$
for $k< n$.
\par 2)$\implies $1) follows from \ref{XXX3.14} and \ref{XXX3.16}.
\par  1)$\implies $2). Suppose $SP(K)\in AE(X)$.
\ref{XXX3.13} says that $H_n(K;H^p(X;A))=0$ for all $n<p$
and all closed subsets $A$ of $X$.
Thus, $H_n(L;H^p(X;A))=0$ for all $n<p$
and all closed subsets $A$ of $X$. Applying \ref{XXX3.13} again we get
 $SP(L)\in AE(X)$.
\par If $L$ is simply connected, then 2) implies 4) by \ref{XXX2.10}.
 \end{pf}


\begin{Cor} \label{XXX3.18} If $K$ and $L$ are countable, pointed,
connected CW
complexes, then the following conditions are equivalent:
\par{1.} $\dim_G(K)=\dim_G(L)$ for all Abelian groups $G$.
\par{2.} $\dim_Z(K\wedge M)=\dim_Z(L\wedge M)$ for all
pointed CW complexes $M$.
\end{Cor} 
\begin{pf} In view of  \ref{XXX3.16}, Condition 1)
is equivalent to $SP(K)\sim_{Gr} SP(L)$,
and that is equivalent (see \ref{XXX3.17}) to Condition 2).
\end{pf}

\begin{Rem} \label{extrarem}
\ref{XXX3.18} is dual to the well known
characterization of $\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$
($X$ and $Y$ are compact). Namely, $\dim_G(X)=\dim_G(Y)$ is equivalent
to $\dim_\Z(X\times T)=\dim_\Z(Y\times T)$ for all compact spaces $T$ (see \cite{K}).
\end{Rem}

\section{Cohomological dimension of compact spaces}

The purpose of this section is to show that, given
a compact space $X$, the class $\{SP(L)\mid \ExD(X)\leq SP(L)\}$
has a minimum which we call {\bf the cohomological
dimension} of $X$. To do so we will need
the basics of Bockstein Theory (see \cite{B} or \cite{K}).

\begin{Def} [Bockstein groups] \label{XXX6.1.0}
The set $\B$ of {\bf Bockstein groups}  is 
$$\{\Q\}\cup \bigcup\limits_{\p \ prime} 
\{\Z /\p,\Z /\p^{\infty}, \Z _{(\p)} \},$$
where $\Z /\p^{\infty}$ is the $\p$-torsion of $\Q /\Z$,
and $\Z _{(\p)}$ are the rationals whose
denominator is not divisible by $\p$.
\end{Def}

\begin{Def} [Bockstein basis] \label{XXX6.1.00}
Given an abelian group $G$
its   {\bf Bockstein basis}  $\sigma(G)$ is the subset
of $\B$ defined as follows:
\par{1.} $\Q\in \sigma(G)$ iff $\Q\otimes G\ne 0$, 
\par{2.} $\Z/\p\in \sigma(G)$ iff $(\Z/\p)\otimes G\ne 0$, 
\par{3.} $\Z_{(\p)}\in \sigma(G)$ iff $(\Z/\p^{\infty})\otimes G\ne 0$, 
\par{4.} $\Z/\p^{\infty}\in \sigma(G)$ iff 
$(\Z/\p^{\infty})*G\ne 0$ (here $H*G$ is the torsion product
of groups $H$ and $G$) or $(\Z/\p)\otimes G\ne 0$. 
\end{Def}


\begin{Rem} \label{XXX6.1.00rem}
Our definition of Bockstein basis
is slightly different from the standard ones
(see \cite{K} or  \cite{D$_8$}). Namely, if
 $\Z_{(\p)}\in \sigma(G)$ (respectively,
($\Z/\p\in \sigma(G)$), then $\Z/\p\in \sigma(G)$
(respectively, $\Z/\p^{\infty}\in \sigma(G)$. 
In traditional definitions of Bockstein basis
only one group among $\{\Z_{(\p)},\Z/\p,\Z/\p^{\infty}\}$
is admitted for any $\p$.
Since our only application of Bockstein basis is \ref{XXX6.1.000},
the change of the definition will not cause any problems
as $\dim_{\Z_{(\p)}}(X)\ge \dim_{\Z/\p}(X) \ge \dim_{\Z/\p^{\infty}}(X)$
for all primes $\p$ (see \cite{K}).
\end{Rem}


\begin{Thm} [First Bockstein Theorem] \label{XXX6.1.000}
If $X$ is compact,
then \par\noindent
\centerline {$\dim_G(X)=\max\{\dim_H(X) \mid H\in \sigma(G)\}.$}
\end{Thm}


Here is the existence of cohomological dimension:
\begin{Thm} \label{XXX6.1.0000}
If $X$ is compact,
then there is a countable CW complex $K_X$
such that $SP(K_X)\in AE(X)$
and $SP(K_X)\leq SP(L)$
for every CW complex $L$ satisfying 
$SP(L)\in AE(X)$.
\end{Thm}
\begin{pf} Consider the set $\cal B_X$
of all Bockstein groups $H$ such that $\dim_H(X)<\infty$.
Put $K_X=\bigvee\limits_{H\in\cal B_X}K(H,\dim_H(X))$.
Notice that $K_X\in AE(X)$, hence $SP(K_X)\in AE(X)$
(see \ref{XXX2.10a}). Suppose $SP(L)\in AE(X)$
for some CW complex $L$ and $SP(K_X)\in AE(Y)$
for some compact space $Y$. We need to show
that $SP(L)\in AE(Y)$.
\par Since $SP(K_X)\in AE(Y)$ is equivalent to
$H_i(K_X;H^n(Y,A))=0$ for all $i<n$ and all closed subsets $A$ of $Y$
(see  \ref{XXX3.13}), one has
$H_i(K(H,\dim_H(X));H^n(Y,A))=0$
for all $i<n$ and all closed subsets $A$ of $Y$.
This, in turn, implies $K(H,\dim_H(X))\in AE(Y)$
(see \ref{XXX3.16} and \ref{XXX3.13}) for all $H\in \cal B_X$.
Thus $\dim_H(Y)\leq \dim_H(X)$ for all Bockstein groups $H$
and, in view of \ref{XXX6.1.000}, 
$\dim_G(Y)\leq \dim_G(X)$ for all Abelian groups $G$.
Since, for any compact space $T$, $SP(L)\in AE(T)$
is equivalent to $\dim_{H_i(L)}(T)\leq i$
for all $i$, one has
$\dim_{H_i(L)}(Y)\leq \dim_{H_i(L)}(X)\leq i$ for all $i$,
i.e. $SP(L)\in AE(Y)$.
\end{pf}


\section{Extension types of Eilenberg-MacLane spaces }


The following problem was posed in \cite{D-D$_1$}.



%delXXX6.1

\begin{Problem} \label{XXX6.1} Find necessary and sufficient conditions
for two CW complexes $K$ and $L$ to be of the same extension type.

\end{Problem} 

In this section we consider \ref{XXX6.1} in case of $K$
being the infinite symmetric product and $L$ being an Eilenberg-MacLane space.
 Our characterization of infinite symmetric products having extension type of
an Eilenberg-MacLane space
involves
an enlargement of Bockstein basis.

\begin{Def} \label{XXX6.tauG}
Let $G$ be an abelian group.  $\tau(G)$ is the subset of Bockstein groups
containing $\sigma(G)$ and satisfying the following conditions
for each prime $\p$:
\par{1.} $\Z/\p\in \tau(G)$ iff $\Z/\p^{\infty}\in \sigma(G)$, 
\par{2.} $\Z_{(\p)}\in \tau(G)$ iff $\Z/\p^{\infty}\in \sigma(G)$
and $\Q\in \sigma(G)$. 
\end{Def}

\begin{Lem} \label{XXX6.tauG.1} Suppose $G$ and $F$ are non-trivial Abelian groups
and $m\ge 1$. If $\sigma(F)\setminus \tau(G)\ne\emptyset$,
then there is a compact space $X$ such that $\dim_G(X)=m$
and $\dim_F(X)=\infty$.
\end{Lem} 
\begin{pf} Define $\alpha:\B\to N$ as follows:
$\alpha(H)=\infty$ iff $H\in \B\setminus\tau(G)$,
$\alpha (H)=m+1$ iff $H\in \tau(G)\setminus \sigma(G)$,
and
$\alpha (H)=m$ iff $H\in \sigma(G)$,
\par Let us show that $\alpha$ is a Bockstein function,
i.e. the following inequalities hold for all primes $\p$:
\par 1. $\alpha(\Z/\p^\infty)\leq \alpha(\Z/\p)\leq \alpha(\Z/\p^\infty)+1$,
\par 2. $\alpha(\Z/\p)\leq \alpha(\Z_{(\p)})$,
\par 3. $\alpha(\Q)\leq \alpha(\Z_{(\p)})$,
\par 4. $\alpha(\Z_{(\p)})\leq \max(\alpha(\Q),\alpha(\Z/\p^\infty)+1)$,
\par 5. $\alpha(\Z/\p^\infty)\leq \max(\alpha(\Q),\alpha(\Z_{(\p)})-1)$.
\par Inequalities 1) can fail only if 
$\Z/\p\in\tau(G)$ and $\Z/\p^\infty\in\sigma(G)$. 
In that case $\alpha(\Z/\p^\infty)=m$ and $m\leq \alpha(\Z/\p)\leq m+1$, so Inequalities
1) hold.
Inequality 2) can fail only if 
$\Z_{(\p)}\in\tau(G)$. In that case, however, 
$\Z/\p^\infty\in\sigma(G)$ implying $\Z/\p\in \tau(G)$.
Since $\Z_{(\p)}\in\sigma(G)$ implies $\Z/\p\in \sigma(G)$,
Inequality 2) holds.
Inequality 3) can fail only if 
$\Z_{(\p)}\in\tau(G)$. In that case, however, 
$\Q\in\sigma(G)$, so
Inequality 3) holds.
Inequality 4) can fail only if $\Z/\p^\infty\in\tau(G)$
and $\Q\in\tau(G)$. That however implies 
$\Z/\p^\infty\in\sigma(G)$
and $\Q\in\sigma(G)$. Consequently, $\Z_{(\p)}\in\tau(G)$ and
$\alpha(\Z_{(\p)})\leq m+1\leq \max(\alpha(\Q),\alpha(\Z/\p^\infty)+1)$,
i.e. 4) holds.
Inequality 5) can fail only if $\Z_{(\p)}\in \tau(G)$.
Therefore $\Z/\p^\infty\in\sigma(G)$ and $\Q\in\sigma(G)$.
Hence $\alpha(\Z/\p^\infty)=m\leq \max(\alpha(\Q),\alpha(\Z_{(\p)})-1)$.
\par By Dranishnikov's Realization Theorem (see \cite{D$_1$} or \cite{D$_8$})
there is a compactum $X$
such that $\dim_H(X)=\alpha (H)$ for all $H\in \B$.
It is clear, in view of Bockstein's First Theorem \ref{XXX6.1.000}, that $X$
satisfies the desired conditions.
\end{pf}


\begin{Lem} \label{XXX6.tauG.2} Suppose $G$ and $F$ are non-trivial Abelian groups
and $m\ge 1$. If $\sigma(F)\subset \tau(G)$ and
$\sigma(F)\setminus \sigma(G)\ne\emptyset$,
then there is a compact space $X$ such that $\dim_F(X)=\dim(X)=m+1$
and $\dim_G(X)=m$.
\end{Lem} 
\begin{pf} Notice that $\sigma(F)\setminus \sigma(G)$
contains only groups of the form $\Z/\p$ or $\Z_{(\p)}$
for some $\p$.
\par
Case I: There is a prime $\q$ such that
$\Z/\q\in \sigma(F)\setminus \sigma(G)$.
Define $\alpha:\B\to N$ 
by sending $\Z/\q$ and $\Z_{(\q)}$ to $m+1$, and sending
all the other groups to $m$.
 Notice that $\alpha$ is a Bockstein function.
  By Dranishnikov's Realization Theorem (see \cite{D$_1$} or \cite{D$_8$})
there is a compactum $X$
such that $\dim_H(X)=\alpha (H)$ for all $H\in \B$
and $\dim(X)=\max(\alpha )$.
It is clear, in view of Bockstein's First Theorem, that $X$
satisfies the desired conditions.
\par Case II: There is a prime $\q$ such that
$\Z_{(\q)}\in \sigma(F)\setminus \sigma(G)$
and all groups in $\sigma(F)\setminus \sigma(G)$
are torsion-free.
Define $\alpha:\B\to N$ 
by sending $\Z_{(\q)}$ to $m+1$, and sending
all the other groups to $m$. Notice that $\alpha$ is a Bockstein function.
 By Dranishnikov's Realization Theorem (see \cite{D$_1$} or \cite{D$_8$})
there is a compactum $X$
such that $\dim_H(X)=\alpha (H)$ for all $H\in \B$
and $\dim(X)=\max(\alpha )$.
It is clear, in view of Bockstein's First Theorem, that $X$
satisfies the desired conditions.
\end{pf}

\begin{Thm} \label{XXX6.1.1} Suppose $L$ is a pointed countable
CW complex, $G$ 
is an Abelian group, and $n\ge 1$.
 $SP(L)$ is of the same extension type as $K(G,n)$
if and only if the following conditions are satisfied:
\par{a.} $H_i(L)=0$ for $i<n$.
\par{b.} $\sigma(H_n(L))=\sigma(G)$.
\par{c.} $\sigma(H_i(L))\subset \tau(G)$ for all $i\ge n$.
\end{Thm} 
\begin{pf} We can reduce the general case to that of $K(G,n)$
being a countable CW complex. Indeed,  \ref{XXX6.1.000}
implies that any $K(G,n)$ has the extension type
of $K(G',n)$ such that $G'$ is countable and $\sigma(G)=\sigma(G')$.
\par
Assume $SP(L)$ is of the same extension type as $K(G,n)$.
Since $L\sim_{Gr} K(G,n)$ (see  \ref{XXX3.16} and  \ref{XXX3.17}), a) follows.
Pick $i\ge n$ and denote $H_i(L)$ by $F$.
Suppose $\sigma(F)\setminus \tau(G)\ne \emptyset$.
By \ref{XXX6.tauG.1} there is a compactum $X$ such that $\dim_G(X)=n$
and $\dim_F(X)=\infty$. Since $SP(L)$ is of the same extension type as $K(G,n)$,
$\dim_G(X)=n$ implies $SP(L)\in AE(X)$. Consequently,
$\dim_F(X)=\dim_{H_i(L)}(X)\leq i$ (see  \ref{XXX2.10b}), a contradiction.
Thus c) holds.
\par Denote $H_n(L)$ by $F$. Thus $\sigma(F)\subset \tau(G)$.
Suppose $\sigma(F)\setminus \sigma(G)\ne \emptyset$.
By \ref{XXX6.tauG.2} there is a compact space $X$ such that 
$\dim_G(X)=\dim(X)=n+1$
and $\dim_F(X)=n$.
Since $\dim(X)=n+1$, $\dim_{H_i(L)}(X)\leq i$ for all $i\ge n$.
Consequently, see  \ref{XXX2.10b}, $SP(L)\in AE(X)$
which implies $K(G,n)\in AE(X)$ contradicting $\dim_G(X)=n+1$.
That proves $\sigma(F)\subset \sigma(G)$.

\par Suppose $\sigma(G)\setminus\sigma(F)\ne\emptyset$.
By \ref{XXX6.tauG.2} there is a compact space $X$ such that $\dim_F(X)=\dim(X)=n+1$
and $\dim_G(X)=n$. Since $SP(L)$ is of the same extension type as $K(G,n)$,
$\dim_G(X)=n$ implies $SP(L)\in AE(X)$. Consequently,
$\dim_F(X)=\dim_{H_n(L)}(X)\leq n$ (see  \ref{XXX2.10b}), a contradiction.
That proves b).
\par Suppose a), b), and c) hold.
 If $K(G,n)\in AE(X)$ (i.e., $\dim_G(X)\leq n$),
then $\dim_F(X)\leq n+1$ for all $F\in\tau(G)$ in view of Bockstein's Inequalities.
Hence, $\dim_{H_i(L)}(X)\leq i$ for all $i\ge n$
which implies $SP(L)\in AE(X)$ (see  \ref{XXX2.10b}). 
That shows $K(G,n)\leq SP(L)$.
\par If $SP(L)\in AE(X)$, then $K(H_n(L),n)\in AE(X)$ (see  \ref{XXX2.10b})
which is equivalent to $K(G,n)\in AE(X)$ in view of
$\sigma(G)=\sigma(H_n(L))$ and the First Bockstein Theorem.
\end{pf}

\section{Dimension types and the connectivity index}

\par Shchepin's connectivity index is of homological nature.
Similarly, one can introduce the {\bf homotopy connectivity index}
$hcin(K)$.

\begin{Def} \label{XXXaa3.1} Suppose $X$ is a pointed
 space. $hcin(X)$ is a non-negative integer defined as follows:
\par a. $hcin(X)=0$ means that $X$ is not path-connected.
\par b. If $\infty > r > 0$, then
$hcin(X)=r$ means that $\pi_r(X)\ne 0$
and $\pi_k(X)=0$ for all $0\leq k<r$. 
\par c. $hcin(X)=\infty$ means that $\pi_k(X)=0$ for all $0\leq k< \infty$.
\end{Def} 

Homotopy connectivity index can be easily dualized. Following
G.Whitehead \cite{Wh} (p.421--423) we introduce
the {\bf anticonnectivity index} $acin(X)$ as follows.

\begin{Def} \label{XXXaaa3.1} Suppose $X$ is a pointed
 space. $acin(X)$ is an integer greater than or equal to $-1$
or infinity defined as follows:
\par a. $acin(X)=-1$ means that $X$ is path-connected and all its
homotopy groups are trivial.
\par b.
If $0 \leq r <\infty$, then
$acin(X)=r$ means that $\pi_r(X)\ne 0$
and $\pi_k(X)=0$ for all $k>r$. 
\par c.
$acin(X)=\infty$ means that infinitely many homotopy groups of $X$
are non-trivial.
\end{Def}

\par We start with the the concept of the total function
space which is related to Shchepin's \cite{S} concept of the total
cohomology of a space.


\begin{Def} \label{XXXa3.1} Suppose $X$ is a pointed
compact space and $P$
is a pointed CW complex. The {\bf total function space} $Tot(P^X)$
is the wedge of all function spaces $P^{X/A}$, where $A$ 
is a closed subspace of $X$.
\end{Def}

Using homotopy connectivity index and the concept of the total
function space one can introduce a new relation on the class
of compact spaces: $X\sim Y$ iff $hcin(Tot(P^X))=hcin(Tot(P^Y))$
for all pointed CW complexes $P$. It turns out this relation
means exactly that $\Sigma X$ and $\Sigma Y$ are of the same
extension dimension. The first part of this section is devoted
to that fact and culminates in \ref{XXX.1}.
\par In the case of pointed CW complexes one can define
a relation $K\sim L$ to mean that $cin(K\wedge M)=cin(L\wedge M)$
for all $M$. We saw in \ref{XXX3.17} and \ref{XXX3.18}
that, in the case of countable CW complexes, that relation is identical
with equality of extension types of $SP(K)$ and $SP(L)$.
Dualizing that relation leads to $K\sim L$ iff
$acin(P^K)=acin(P^L)$ for all $P$. In the last part of this section
we investigate how extension types are connected to that relation.

\begin{Lem} \label{XXXhcin.1} Let $X$ be a compact connected space.
If $P$ is a pointed CW complex, then
$$hcin(Tot(\Omega P)^X)\ge hcin(Tot( P)^X)-1.$$
Moreover, if $hcin(Tot( P)^X)\ge 1$, then
$$hcin(Tot(\Omega P)^X)=hcin(Tot( P)^X)-1.$$
\end{Lem} 
\begin{pf} If $hcin(Tot( P)^X)=0$, then the inequality obviously
holds, so assume $hcin(Tot( P)^X)$ is at least $1$. Now, 
\ref{XXXhcin.1} follows from equality
$[S^{n-1},(\Omega P)^{X/A}]=[S^n,P^{X/A}]$
for all closed subsets $A\ne\emptyset$ of $X$ and all $n\ge 1$.
\end{pf}

\begin{Rem} \label{XXXhcin.2} If $hcin(Tot( P)^X)=0$,
then $hcin(Tot(\Omega P)^X)$ may be arbitrarily high.
For example, if $P=X=S^1$, then $Tot(\Omega P)^X$ is contractible
but $Tot( P)^X$ is not path-connected.
\end{Rem} 

\begin{Lem} \label{XXXhcin.3} Let $X$ be a compact connected space and $k > 0$.
If $P$ is a pointed CW complex, then the following conditions are equivalent:
\par a. $hcin(Tot( P)^X)\ge k$.
\par b. $\Omega^k (P)\in AE(X)$.
\end{Lem} 
\begin{pf} Notice that both $hcin(Tot( P)^X)$
and $\Omega^k (P)$ depend only on the component of the base point
of $P$, so we may reduce \ref{XXXhcin.3}
to the case of $P$ being connected.
\par
Special Case: $k=1$. Notice that
$hcin(Tot(P^X))\ge 1$ is equivalent to
$[X/A,P]=\ast$ for all closed subsets $A\ne\emptyset$ of $X$.
That statement, in view of \ref{XXX2.11}, is equivalent
to $\Omega P\in AE(X)$.
\par If $k > 1$, then $hcin(Tot(P^X))\ge k$ is equivalent
(in view of  \ref{XXXhcin.1})
to $$hcin(Tot((\Omega^{k-1}P)^X))\ge 1$$ which is equivalent,
by the Special Case, to $\Omega^k(P)\in AE(X)$.
\end{pf}

\begin{Thm} \label{XXX.1} If $X$ and $Y$ are non-empty connected
compact spaces, the following
conditions
are equivalent:
\par a. $\Sigma X$ and $\Sigma Y$ are of the same
extension dimension.
\par{b.} $\Omega(P)\in AE(X)$ is equivalent to
 $\Omega(P)\in AE(Y)$ for all pointed CW complexes $P$.
\par{c.} $hcin(Tot(P^X))=hcin(Tot(P^Y))$
for all pointed CW complexes $P$.
\end{Thm} 


\begin{pf}  a)$\implies$ b). 
Suppose $\Omega P\in AE(X)$ for some pointed CW complex $P$.
Since $\Omega P=\Omega P_0$, where $P_0$ is the component
of the base point of $P$,
we may assume $P$ is connected.
By \ref{XXX2.11}, $P\in AE(\Sigma X)$ which implies
$P\in AE(\Sigma Y$, as $\Sigma Y$ is of the same extension
dimension as $\Sigma X$.
Again, by \ref{XXX2.11}, $\Omega P\in AE(X)$.
The same argument shows that  $\Omega P\in AE(Y)$
implies  $\Omega P\in AE(X)$ for any pointed CW complex $P$.

\par
b)$\implies$ c). Suppose $hcin(Tot(P^X))\ge k$.
It suffices to show (by symmetry) that
$hcin(Tot(P^Y))\ge k$.
If $k=0$ it is obviously true, so assume $k\ge 1$.
However, in that case,
$hcin(Tot(P^T))\ge k$ is equivalent (see \ref{XXXhcin.3})
to $\Omega^k(P)\in AE(T)$ for any compact connected space $T$,
so b)$\implies $ c) follows from \ref{XXXhcin.3}
as $\Omega^k(P)\in AE(X)$ is equivalent to $\Omega^k(P)\in AE(Y)$.
\par c)$\implies$ a).
Suppose $P\in AE(\Sigma X)$. It suffices to show $P\in AE(\Sigma Y)$.
Since $\Sigma X$ is connected and contains at least two points,
$P$ must be connected (otherwise we pick $x_1\ne x_2\in \Sigma X$,
map them to two different components of $P$, and that map
cannot be extended over $\Sigma X$), so (see \ref{XXX2.11})
$P\in AE(\Sigma X)$ is equivalent to $\Omega P\in AE(X)$.
Hence $\Omega P\in AE(X)$ and, by \ref{XXX2.11},
$P\in AE(\Sigma Y)$.
 \end{pf}



 \ref{XXX.1} implies that if two compacta $X$ and $Y$ are of the same
dimension type, then the total function spaces $Tot(P^X)$ and $Tot(P^Y)$
have the same homotopy connectivity index. The rest of this section
is devoted to the dual result: if two countable complexes $K$ and
$L$ are of the same extension type, then the function spaces
$P^K$ and $P^L$ have the same anticonnectivity index for certain $P$.


\par Since some of the techniques of this section are well-known
(see \cite{D-W$_3$} and \cite{D$_8$}), we will only outline
how one translates known results in terms of trunctated cohomology
to results in terms of function spaces.
\par The following result corresponds to the Combinatorial Vietoris-Begle
Theorem of \cite{D-W$_3$} (see Lemma 2 there or Lemma 5.9 in \cite{D$_8$})
and the proof is similar.




%delXXX.2

\begin{Prop} \label{XXX.2}
Suppose $P$ is a CW complex and $f:K\to L$ is a combinatorial map from a CW
complex $K$
to a finite simplicial complex $L$.
If $P^{f\1(\Delta)}$ is weakly contractible for each simplex $\Delta$
of $L$, then $f^*:P^L\to P^K$ is a weak homotopy equivalence.

\end{Prop} 


The next two results extract essential parts of Lemma 3 in of \cite{D-W$_3$}
with proofs being similar.



%delXXX.3

\begin{Lem} \label{XXX.3}
Suppose $P$ is a CW complex. The following conditions are equivalent:
\par{1.} $[S^k,P]$ is finite for each $k\ge 0$ (unpointed spheres).
\par{2.} $[K,P]$ is finite for each finite CW complex $K$.
\par{3.} for each pair $(K,L)$ of finite CW complexes
and each map $f:L\to P$, the set
of all homotopy classes rel.$L$ of maps $g:K\to P$ such that $g|L=f$, is
finite.

\end{Lem} 





%delXXX.4

\begin{Lem} \label{XXX.4}
Suppose $P$ is a CW complex such that
 $[S^k,P]$ is finite for each $k\ge 0$ (unpointed spheres).
If $K$ is a countable CW complex and $f:K\to P$ is a map such that $f|L\sim
0$ for each finite subcomplex
$L$ of $K$, then $f\sim const$.

\end{Lem} 






%delXXX.5

\begin{Thm} \label{XXX.5}
Suppose $P$ is a CW complex and $K$ is a countable CW complex
such that $P^K$ is weakly contractible.
Suppose $L$ is a countable complex and $f:L\to P$ is a homotopically
non-trivial map.
There is a compactum $X$ and a map $g:X\to L$ so that $f\circ g$ is 
homotopically non-trivial
and $K\in AE(X)$ if one of the following conditions is satisfied:
\par{1.} $K$ is compact and $L$ is finitely dominated,
\par{2.} $[S^k,P]$ is finite for each $k\ge 0$ (unpointed spheres).

\end{Thm} 


\begin{pf}  In case of 2), one follows the same technique as described
in \cite{D$_8$} (see 5.5-5.7 there). We will outline how to use that technique
in case of both $K$ and $L$ being compact.
The inductive step consists of a homotopically non-trivial map $f':L'\to P$
and a map $h':L_0\to K$, where $L'$ is a compact simplicial complex
and $L_0$ is a subcomplex of $L'$. As in Lemma 5.6 of \cite{D$_8$}
one constructs a combinatorial map $\pi:L''\to L'$ 
so that $\pi\1(\Delta)$ is either contractible or homotopy
equivalent to $K$
for each simplex $\Delta$ of $L'$. Moreover, the composition
$\pi\1(L_0)\to L_0\to K$ extends over $L''$. 
A construction in \cite{Dy$_4$} consists of $L''$ being a subcomplex
of $L'\times K$, \cite{D$_8$} constructs $L''$ as the pull-back of
a certain diagram. \ref{XXX.2} says that the composition $L''\to L'\to P$
is homotopically non-trivial. Starting with $f:L\to P$ one can
construct inductively finite simplicial complexes $L_n$
and maps $f_n:L_{n+1}\to L_n$ so that the inverse limit $X$
of the sequence $\ldots L_{n+1}\to L_n\ldots $ satisfies
$K\in AE(X)$, $L_1$ is homotopically equivalent to $L$,
and the composition $L_n\to\ldots\L_1\to P$ is homotopically
non-trivial for all $n$ (see 5.5 and 5.9 of \cite{D$_8$}
for details).
\par If $L$ is finitely dominated, we pick $L'$ finite
and maps $u:L\to L'$, $d:L'\to L$ so that $d\circ u\approx id_L$.
Let $f'=f\circ d:L'\to P$. Notice that $f'$ is homotopically non-trivial.
By the previous case, there is compactum $X$ and a map $g':X\to L'$
so that $K\in AE(X)$ and $f'\circ g'$ is homotopically non-trivial.
Let $g=d\circ g'$. Since $f\circ g=f'\circ g'$, it is homotopically non-trivial.
 \end{pf}





%delXXX.6

\begin{Cor} \label{XXX.6} Suppose $K\leq L$ are countable pointed
complexes.
Suppose $P$ is a pointed CW complex so that $\pi_i(P^K)=0$ for all $i\ge r$,
where $r\ge 1$. Then the homotopy groups $\pi_i(P^L)$ are trivial for all $i\ge r$ if one
of the following conditions hold:
\par{1.} $\pi_i(P)$ is finite for all $i\ge r$,
\par{2.} $K$ is compact and $L$ is finitely dominated,
\par{3.} both $K$ and $L$ are finitely dominated and $r\ge 2$.

\end{Cor} 


\begin{pf}  Suppose $\pi_m(P^L)\ne 0$ for some $m\ge r$.
Put $L'=\Sigma(L)$ in cases 1) and 2), put $L'=\Sigma^2(L)$
in case 3).
Put $P'=\Omega^{m-1}(P)$ in cases 1) and 2), put $P'=\Omega^{m-2}(P)$
in case 3). Put $K'=K$ in cases 1) and 2), put $K'=\Sigma(K)$
in case 3).
Notice that $K'$ and $L'$ are homotopically equivalent to
compact CW complexes in cases 2) and 3) (see \cite{Wall}).
In all cases there is a homotopically non-trivial map $f:L'\to P'$,
so there is a map $g:X\to L'$ such that $K'\in AE(X)$
and $f\circ g$ is homotopically non-trivial (see \ref{XXX.5}). 
This contradicts \ref{XXX2.1}. Indeed, $K\leq L$ implies $\Sigma(K)\leq \Sigma(L)$
(see \cite{D-D$_1$}). That means $K'\in AE(X)$ implies that
any map $X\to L'$ is null-homotopic by \ref{XXX2.1}.
 \end{pf}



\begin{Cor} \label{XXX.6b} Suppose $K\leq L$ are countable pointed
complexes and $\p$ is a prime.
If $H^\ast(K;\Z/\p)=0$, then $H^\ast(L;\Z/\p)=0$.
\end{Cor}
\begin{pf} Suppose $H^\ast(K;\Z/\p)=0$ and $H^n(L;\Z/\p)\ne 0$
for some $n\ge 0$. If $n=0$, then $L$ must be disconnected
and $K$ is connected. Hence $K\in AE(S^1)$ and $L\notin AE(S^1)$,
a contradiction. Thus, $n\ge 1$.
Put $P=K(\Z/\p,n+1)$. Notice that $\pi_1(P^L)=H^n(L;\Z/\p)\ne 0$
and $\pi_i(P^K)=H^{n+1-i}(K;\Z/\p)=0$ for all $i\ge 1$,
a contradiction in view of \ref{XXX.6}.
\end{pf}

%delXXX.7

\begin{Thm} \label{XXX.7} Suppose $L$ is a countable, finite-dimensional CW
complex
and $\p$ is a prime number. If $H^n(L;\Z/\p)\ne 0$ for some $n\ge 1$,
then there is a non-trivial map $f:X\to\Sigma^{n+2}(L)$ from a compactum $X$
so that $\dim_{\Z[1/\p]}X=1=\dim_{\Z/\p}X$.

\end{Thm}
\begin{pf}  Let $M=M(\Z/\p,1)$ and let $L'=\Sigma^{n+2}(L)$.
In particular, $H^{2n+2}(L';\Z/\p)\ne 0$. Notice that $K(\Z,2n+3)^M$
is a $K(\Z/\p,2n+2)$ so there is a non-trivial map
$g:L'\to K(\Z,2n+3)^M$. Its adjoint $g':L'\wedge M\to K(\Z,2n+3)$
is non-trivial and we may assume that its image is contained
in a finite subcomplex $A$ of $K(\Z,2n+3)$ which is $(2n+2)$-connected.
Notice that $P=A^M$ is simply connected and its homotopy groups are
finite. Indeed, the homotopy groups of $P$ coincide with groups of
homotopy clases of maps from suspensions of $M$ to $A$.
Those sets are the same as mod $\p$ homotopy groups of $A$
(see \cite{N}). In view of Proposition 1.4 on p.3 in \cite{N} 
one has an exact sequence
$$ 0\to \pi_m(A)\otimes \Z/\p\to \pi_m(A;\Z/\p)\to \pi_{m-1}(A)\ast \Z/\p\to 0$$
and, since the homotopy groups of $A$ are finitely generated,
the homotopy groups of $P$ are finite.
By Miller's Theorem (Sullivan Conjecture - see \cite{Mi}),
$A^{K(\Z/\p,1)}$ is weakly contractible as $A$ is finite-dimensional. Therefore,
$P^{K(\Z/\p,1)}=(A^{K(\Z/\p,1)})^M$ is weakly contractible.
Since
$K(\Z[1/\p],1)\wedge M$ is contractible (compute its homology groups),
we get that $P^{K(\Z[1/\p],1)}$ is weakly contractible.
Let $K=K(\Z/\p,1)\vee K(\Z[1/\p],1)$. Notice that $P^K$
is weakly contractible. Applying \ref{XXX.5} one gets a non-trivial map
$f:X\to L'$ so that $K\in AE(X)$.
 \end{pf}


\ref{XXX.7} and \ref{XXX2.1} imply the following.



%delXXX.8

\begin{Cor} \label{XXX.8} Suppose $L$ is a countable, finite-dimensional CW
complex
and $\p$ is a prime number. If $H^n(L;\Z/\p)\ne 0$ for some $n\ge 1$,
then there is a compactum $X$
so that $\dim_{\Z[1/\p]}X=1=\dim_{\Z/\p}X$ and $L$ is not an absolute
extensor of $X$.

\end{Cor} 


\section{Extension types of infinite symmetric products }

In this section we consider Problem \ref{XXX6.1}
in the case of $K$ being an infinite symmetric space
and $L$ being a compact CW complex (respectively, a countable,
finite-dimensional CW complex).

\begin{Lem} \label{XXX6.1.5} Suppose $\p$ is a prime. The following conditions are
equivalent
for any countable connected pointed CW complex $K$:
\par{1.} $H^*(SP(K);\Z/\p)=0$.
\par{2.} $\Z/\p^\infty\notin\sigma(H_s(K))$ for all $s\ge 1$.

\end{Lem}
\begin{pf} 1)$\implies$ 2). Let $G_i=H_i(K)$ for $i\ge 1$.
If $\p\cdot G_i\ne G_i$ for some $i$,
there is a non-trivial map $K(G_i,i)\to K(G_i/\p\cdot G_i,i)$,
a contradiction as $G_i/\p\cdot G_i$ is a direct sum of copies of $\Z/\p$.
Thus $\p\cdot G_i=G_i$ for all $i$.
If $\p-\Tor(G_i)\ne 0$ for some $i$,
then $K(G_i,i)$ dominates $K(\Z/\p^\infty,i)$. To complete the proof of 
1)$\implies$ 2) it suffices to show that
$K=K(\Z/\p^\infty,i)$ has non-trivial
$\Z/\p$-cohomology. Put $L=K(\Z/\p,i+2)$
an notice that $K\leq L$ in view of Bockstein's Inequalities. 
Put $P=K(\Z/\p,i+4)$
and notice that the triviality of $\Z/\p$-cohomology of $K$
means $\pi_n(P^K)=0$ for all $n\ge 1$. \ref{XXX.6b} says that
$\pi_n(P^L)=0$ for all $n\ge 1$, a contradiction as $\pi_2(P^K)=\Z/\p$.
\par 2)$\implies$ 1). Notice that $M(\Z[1/\p],1)\leq_{Gr} K$.
Indeed, $\Z/\p^\infty\notin\sigma(H_s(K))$ for all $s\ge 1$ means
that $H_*(K)$ has no $\p$-torsion and is divisible
by $\p$. If $F\otimes \Z[1/\p]=0$ and $F$ is a Bockstein group, then 
$F$ must be either $\Z/\p$ or $\Z/\p^\infty$. Therefore
$H_*(K;F)=0$ which completes the proof of $M(\Z[1/\p],1)\leq_{Gr} K$.
Also, $H^*(K(\Z[1/\p],1);\Z/\p)=0$. If $H^*(SP(K);\Z/\p)\ne 0$,
then there is a compactum $X$ so that $K(\Z[1/\p],1)\in AE(X)$
but $SP(K)\notin AE(X)$ (see \ref{XXX.8})
which contradicts
\par\noindent $K(\Z[1/\p],1)\sim_{Gr} M(\Z[1/\p],1)\leq_{Gr}
K\sim_{Gr}SP(K)$
(see \ref{XXX3.16} and \ref{XXX3.17}).
 \end{pf}

%delXXX6.2

\begin{Thm} \label{XXX6.2} Suppose $L$ is a connected countable
CW complex and $n\ge 0$.
If $\Sigma^n(SP(L))$ has the extension type of a countable,
finite-dimensional and non-trivial CW complex,
then either $SP(L)$ is of the same extension type as $K(\Z_{(l)},1)$
for some subset $l$ of primes or 
it is of the same extension type as
$K(\Q,m)$ for some $m\ge 1$.
\end{Thm}
\begin{pf} 
Let $G_i=H_i(L)$ for $i\ge 1$.
\par Suppose $SP(L)$ is not of the same extension type as
$K(\Q,m)$ for all $m$.
There is the smallest $r$ with $\sigma(G_r)\ne\sigma(\Q)$ (see \ref{XXX6.1.1}).
There must be a prime $\p$ with $\Z/\p^\infty\in\sigma(G_r)$ which means
$H^*(SP(L);\Z/\p)\ne 0$ (see \ref{XXX6.1.5}).
Suppose $\Sigma^n(SP(L))\sim K$, $K$ is finite-dimensional.
That means $H^*(K;\Z/\p)\ne 0$ (see \ref{XXX.6b}) and there is a
compactum $X_\p$ with $\dim_{\Z[1/\p]}(X_\p)=1=\dim_{\Z/\p}(X_\p)$ so that
$K$ is not an
absolute extensor of $X_\p$ (see \ref{XXX.8}).
If $r\ge 2$, then $\dim_{G_i}(X_\p)\leq i$ for all $i$ as $\dim_\Z(X_\p)\leq 2$.
Hence $SP(L)\in AE(X_\p)$ which implies $K\in AE(X_\p)$, a
contradiction.
Thus, $r=1$. If $\sigma(G_1)\ne \sigma(\Z_{(l)})$ for all sets of primes $l$,
then $\p$ above may be chosen so that $\Z_{(p)}\notin\sigma(G_1)$
which implies $\dim_{G_1}(X_\p)=1$. Again, $SP(L)\in AE(X_\p)$ which
implies $K\in AE(X_\p)$ (see \ref{XXX3.4}), a contradiction.
 Assume $\sigma(G_1)=\sigma(\Z_{(l)})$. If $\sigma(G_s)\subset\sigma(\Z_{(l)})$
for each $s>1$, then we are done by  \ref{XXX6.1.1}.
Suppose $\sigma(G_s)\subset\sigma(\Z_{(l)})$ does not hold
for some $s>1$.
There must be a prime $\p\notin l$ so that $\Z/\p^\infty\in\sigma(G_s)$.
Again, there is a compactum $X_\p$ with
$\dim_{\Z[1/\p]}(X_\p)=1=\dim_{\Z/\p}(X_\p)$ so
that $K$ is not an
absolute extensor of $X_\p$.
This implies $\dim_{G_i}X_\p\leq 1$ for all $i$ and $SP(L)\in AE(X_p)$.
Again, $K\in AE(X_\p)$, a contradiction.
 \end{pf}





%delXXX6.3

\begin{Thm} \label{XXX6.3} Suppose $L$ is a connected countable
CW complex and $n\ge 0$.
If $\Sigma^n(SP(L))$ is
 of a compact non-trivial extension type
(i.e., there is a compact CW complex $K$ of the same extension type as
$\Sigma^n(SP(L))$),
then $SP(L)$ is of the same extension type as $S^1$.

\end{Thm} 


\begin{pf}  Let $K$ be a compact CW complex of the same extension type as
$\Sigma^n(SP(L))$.
 By \ref{XXX6.2} either
$SP(L)\sim K(\Z_{(l)},1)$ or $SP(L)\sim K(\Q,m)$ for some
$m\ge 1$.
I $l$ is not the set of all primes or $SP(L)\sim K(\Q,m)$ for some
$m\ge 1$,
then there is a prime $\p$ such that $H^*(SP(L);\Z/\p)=0$
(choose $\p\notin l$ if $SP(L)\sim K(\Z_{(l)},1)$ or
any $\p$ if $SP(L)\sim K(\Q,m)$). That implies
$H^*(K;\Z/\p)=0$ by \ref{XXX.6b}. Since $K$ is finite, $\tilde H_*(K)$ must be a torsion
graded group
and $\tilde H_*(K;\Q)=0$. Thus,
$\tilde H_*(SP(L);\Q)=0$ (see \ref{XXX3.17}), a contradiction
as $SP(L)\sim K(\Z_{(l)},1)$ or $SP(L)\sim K(\Q,m)$ for
some $m\ge 1$.
 \end{pf}


Theorem \ref{XXX6.3} generalizes all the known theorems related to the difference
between
extension types:
\par{1.} $S^n$ and $K(\Z,n)$ are of different extension types
for $n\ge 3$ (\cite{D$_2$}).
\par{2.} $S^n$ and $K(\Z,n)$ are of different extension types
for $n\ge 2$ (\cite{D-W$_3$}).
\par{3.} $M(\Z/\p,n)$ and $K(\Z/\p,n)$ are of different extension types
for $n\ge 1$
(\cite{M$_2$}).
\par{3.} $M(\Z/2,1)$ and $K(\Z/2,1)$ are of different extension types
(\cite{L}).
\par{4.} $RP^n$ and $RP^{\infty}$ are of different extension types
for $n\ge 1$
(\cite{D-R$_2$}).

\par Besides generalizing the above-mentioned results, a major reason
for \ref{XXX6.3} was our interest in pursuing ways of proving/disproving
existence of universal spaces of given cohomological dimension.
In \cite{Dy$_3$}, the author generalized a result of Shvedov which states that,
for any compact CW complex $K$, the class of compacta $X$ such that
$K$ is an absolute extensor of $X$ has a universal space.
That generalization deals with $K$ homotopy dominated by a compact
CW complex. Thus a natural way to see if there is a universal space
of a given cohomological dimension is to verify if a particular
CW complex has extension type of a compact CW complex. Theorem \ref{XXX6.3}
closes that route of proving existence of universal spaces
for any infinite symmetric product but $S^1$.




%delXXX6.4

\begin{Lem} \label{XXX6.4} $M(\Q,n)$ and $K(\Q,n)$
are of the same extension type for all $n\ge 2$.

\end{Lem} 


\begin{pf}  $M=M(\Q,n)$ can be realized
as the telescope of $f_m:S^n\to S^n$,
where $f_m$ is of degree $m!$ for $m\ge 1$.
In particular, homotopy groups of $M(\Q,n)$ are torsion groups
for $i\ge 2n$.
By Sullivan's Theorem \cite{Su}
there is a map $f:M(\Q,n)\to K$ such that
$H_i(K)=H_i(M)\otimes\Q$, $\pi_i(K)=\pi_i(M)\otimes\Q$
for all $i\ge 1$ and
$f_*:H_i(M)\to H_i(K)$ corresponds to $H_i(M)\otimes\Z\to H_i(M)\otimes \Q$
for all $i\ge 1$. Thus, $f$ is a homotopy equivalence.
In particular, $\pi_i(M)=0$ for $i\ge 2n$.
 Thus, $M(\Q,n)\sim K(\Q,n)$
(see \cite{Dy$_1$}).
 \end{pf}





%delXXX6.5

\begin{Thm} \label{XXX6.5} Suppose $G$ is a countable Abelian group. The
following conditions
are equivalent:
\par{1.} There is a Moore space $M(G,n)$ of the same extension type as
$K(G,n)$.
\par{2.} Either $n=1$ and there is a subset $l$ of primes
such that $K(G,1)$ is of the same extension type as $K(\Z_{(l)},1)$ or $n\ge 2$ and
$K(G,n)$ is of the same extension type as $K(\Q,n)$.
\end{Thm} 


\begin{pf}  2)$\implies$ 1) follows from \ref{XXX6.4} and the fact that
one can choose $M(\Q,1)$ to be $K(\Q,1)$.
\par 1)$\implies$ 2). Notice that $\Sigma(M(G,n))$ is homotopy equivalent to
a finite-dimensional CW complex and has the same extension type as
$\Sigma(K(G,n))$. Use \ref{XXX6.2}.
 \end{pf}


\begin{Rem} \label{XXX6.6bef} \ref{XXX6.5} solves the following problem posed by 
Dranishnikov (see \cite{S}, p.983):
\par Is it true that for any compactum $X$ and any countable group $G$
the conditions $M(G,1)\in AE(X)$ and $K(G,1)\in AE(X)$
are equivalent?
 
\end{Rem}






%delXXX6.6

\begin{Problem} \label{XXX6.6} Suppose the extension type of
a countable CW complex $K$ is at most the extension type of
a compact non-contractible CW complex $L$. Is there
a finite-dimensional, countable CW complex $M$ of the same
extension type as $K$?

\end{Problem} 


\begin{Rem} \label{XXX6.6b} One cannot replace compactness of $L$
by finite-dimensionality of $L$. Indeed, $K(\Z,2)\leq M(\Q,2)$
but $K(\Z,2)$ does not have the extension type
of a countable finite-dimensional CW complex (see \ref{XXX6.2}
and  \ref{XXX6.1.1}).
 
\end{Rem}
 




%delXXX6.7

\begin{Problem} \label{XXX6.7} Suppose the extension type of
a countable CW complex $K$ is at most the extension type of
a compact non-contractible CW complex $L$. Is there
a universal space among all compacta $X$ so that $K\in AE(X)$?

\end{Problem} 








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J.Dydak,
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Topology and its Appl. 64 (1995),1--7.

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V.I.Kuzminov,
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\medskip
\medskip
\medskip
\medskip
Jerzy Dydak,
Math Dept, University of Tennessee, Knoxville, TN 37996-1300, USA,
E-mail address: dydak@@math.utk.edu

\end{document}
math0404332.extracted_bib
\begin{thebibliography}{99}

%\bibitem{A$_1$}
%P.S.Alexandroff,
%{\em Dimensionstheorie, ein Beitrag zur Geometrie der abgeschlossenen Mengen},
%Math. Ann. 106 (1932),161--238.

%\bibitem{A$_2$}
%P.S.Alexandroff,
%{\em Einige Problemstellungen in der mengentheoretischen Topologie},
%Math. Sbor. 43 (1936),619--634.


\bibitem{Git}
M.Aguilar, S.Gitler, and C.Prieto,
{\em Algebraic Topology from a Homotopical Viewpoint},
Springer-Verlag (2002).


\bibitem{B}
B. F. Bockstein,
{\em Homological invariants of topological spaces, I},
Trudy Moskov. Mat. Obshch. 5 (1956),3--80.

\bibitem{C}
H. Cohen,
{\em A cohomological definition of dimension for locally compact Hausdorff spaces},
Duke Math. J. 21 (1954),209--224.

\bibitem{D-T}
A.Dold and R.Thom,
{\em Quasifaserungen und Unendliche Symmetrische Produkte},
Annals of Math. 67 (1958),239--281.

\bibitem{D$_1$}
A.N.Dranishnikov,
{\em Homological dimension theory},
Russian Math. Surveys 43(4) (1988),11--63.

\bibitem{D$_2$}
A.N.Dranishnikov,
{\em On a problem of P. S. Alexandroff},
Matem. Sbornik 135:4 (1988),551--557.

\bibitem{D$_3$}
A.N.Dranishnikov,
{\em On intersection of compacta in euclidean space II},
Proceedings of AMS 113 (1991),1149--1154.

\bibitem{D$_4$}
A.N.Dranishnikov,
{\em On the mapping intersection problem},
Pacific Journal of Mathematics 173 No.2 (1996),403--412.

\bibitem{D$_5$}
A.N.Dranishnikov,
{\em Extension of maps into CW complexes},
Math. USSR Sbornik 74 (1993),47-56.

%\bibitem{D$_6$}
%A.N.Dranishnikov,
%{\em Eilenberg-Borsuk theorem for maps into arbitrary complexes},
%Math. Sbornik 185 (1994),81--90.

%\bibitem{D$_7$}
%A.N.Dranishnikov,
%{\em On dimension of the product of two compacta and the dimension of their intersection  in general position in Euclidean  space},
%Transactions of the American Math.Soc. 352 (2000),5599--5618.

\bibitem{D$_8$}
A.N.Dranishnikov,
{\em Basic elements of the cohomological dimension theory of compact metric spaces},
 (1999).

\bibitem{D-D$_1$}
A.Dranishnikov and J.Dydak,
{\em Extension dimension and extension types},
Proceedings of the Steklov Institute of Mathematics 212 (1996),55--88.

%\bibitem{D-D$_2$}
%A.Dranishnikov and J.Dydak,
%{\em Extension theory of separable metrizable spaces  with applications to dimension theory},
%Transactions of the American Math.Soc. 353 (2000),133--156.

%\bibitem{{D-D-W}}
%A.Dranishnikov, J.Dydak and J.Walsh,
%{\em Cohomological dimension theory  with applications},
% (1992), unpublished notes.

%\bibitem{D-R$_1$}
%A.Dranishnikov and D.Repov\v s,
%{\em The Urysohn-Menger Sum Formula: An extension of the Dydak-Walsh theorem to dimension one},
%J.Austral. Math. Soc., Ser.A 59 (1995),273--282.

\bibitem{D-R$_2$}
A.N. Dranishnikov, D.Repov\v s,
{\em On Alexandroff theorem for general Abelian spaces},
Topology and its Appl. 111 (2001),343--353.

%\bibitem{D-R-S}
%A.Dranishnikov, D.Repov\v s,
%{\em On the failure of the Urysohn-Menger sum formula for cohomological dimension},
%Proc.Amer.Math.Soc. 120 (1994),1267--1270.

\bibitem{Dy$_1$}
J.Dydak,
{\em Cohomological dimension and metrizable spaces},
Transactions of the Amer.Math.Soc. 337 (1993),219--234.

%\bibitem{Dy$_2$}
%J.Dydak,
%{\em Union theorem for cohomological dimension: A simple counterexample},
%Proceedings of AMS 121 (1994),295--297.

\bibitem{Dy$_3$}
J.Dydak,
{\em Cohomological dimension and metrizable spaces II},
Trans.Amer.Math.Soc. 348 (1996),1647--1661.

\bibitem{Dy$_4$}
J.Dydak,
{\em Realizing dimension functions via homology},
Topology and its Appl. 64 (1995),1--7.

%\bibitem{Dy$_5$}
%J.Dydak,
%{\em Extension theory: The interface between set-theoretic and algebraic topology},
%Topology and its Appl. 20 (1996),1--34.

\bibitem{DHGT}
J.Dydak,
{\em Cohomological dimension theory},
 in Handbook of Geometric Topology, 423--470
(Edited by
       R.J. Daverman, 
       R.B. Sher),
Elsevier Science (2001).



\bibitem{Dy$_7$}
J.Dydak,
{\em Geometry and algebra of dimension theory},
preprint.

\bibitem{Dy$_6$}
J.Dydak,
{\em Algebra of dimension theory},
Trans.Amer.Math.Soc (to appear).


%\bibitem{D-K}
%J.Dydak and A.Koyama,
%{\em Cohomological dimension of locally connected compacta},
%Topology and its Appl. 113 (2001),39--50.

%\bibitem{D-W$_1$}
%J.Dydak and J.J.Walsh,
%{\em Spaces without cohomological dimension  preserving compactifications},
%Proceedings of the Amer.Math.Soc. 113 (1991),1155--1162.

\bibitem{D-W$_2$}
J.Dydak and J.J.Walsh,
{\em Dimension theory and the Sullivan conjecture},
 in Topology Hawaii, World Scientific, (1992),75--89.

\bibitem{D-W$_3$}
J.Dydak and J.J.Walsh,
{\em Infinite dimensional compacta having cohomological dimension two: An application of the Sullivan Conjecture},
Topology 32 (1993),93--104.

\bibitem{En}
R.Engelking,
{\em General Topology},
Heldermann Verlag Berlin (1989).

%\bibitem{H-W}
%W.Hurewicz and H.Wallman,
%{\em Dimension Theory},
%Princeton University Press (1941).

\bibitem{K}
V.I.Kuzminov,
{\em Homological dimension theory},
Russian Math. Surveys 23 (1968),1--45.

\bibitem{L}
M.Levin,
{\em Constructing compacta of different extensional dimensions},
Canad. Math. Bull. 44 (2001), no. 1, 80--86.

%\bibitem{M$_1$}
%T.Miyata,
%{\em Test spaces for cohomological dimension modulo p},
%preprint (1994).

\bibitem{M$_2$}
T.Miyata,
{\em Moore spaces and cohomological dimension},
Bulletin of the Polish Academy of Sciences 48 (2000), no. 3, 231--235.

\bibitem{Mi}
H.Miller,
{\em The Sullivan conjecture on maps from classifying spaces},
Annals of Mathematics 120 (1984),39--87.

\bibitem{N}
J.A.Neisendorfer,
{\em Primary homotopy theory},
Memoirs of AMS 232 (1980),1--67.

\bibitem{O$_1$}
W.Olszewski,
{\em Completion theorem for cohomological dimensions},
Proceedings of AMS 123 (1995),2261--2264.

%\bibitem{O$_2$}
%W.Olszewski,
%{\em Universal separable metrizable spaces for cohomological dimension},
%Topology and its Appl. 61 (1995),293--299.

%\bibitem{P}
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