The classification of $2$-connected $7$-manifolds
We present a classification theorem for closed smooth spin $2$-connected $7$-manifolds $M$. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is a generalisation of the Eells--Kuiper invariant for any closed spin $7$-manifold, regardless of whether the spin characteristic class $p_M \in H^4(M)$ is torsion. In addition we determine the inertia group of $2$-connected $M$ -- equivalently the number of oriented smooth structures on the underlying topological manifold -- in terms of $p_M$ and the torsion linking form.
2c7m-v1.tex
% 2014-06-07
% $Id: eek.tex,v 1.44 2014/06/07 13:32:13 jnordstr Exp $

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%\title{Generalised Eells--Kuiper invariant}
\title{The classification of $2$-connected $7$-manifolds}
\author[D. Crowley]{Diarmuid Crowley}
\author[J. Nordström]{Johannes Nordström}

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\begin{document}


\begin{abstract}
We present a classification theorem for closed smooth spin $2$-connected
$7$-manifolds $M$. This builds on the almost-smooth classification from the
first author's thesis. The main additional ingredient is a generalisation of the
Eells--Kuiper invariant for any closed spin $7$-manifold, regardless of whether
the spin characteristic class $p_M \in H^4(M)$ is torsion. 
In addition we determine the inertia group of $2$-connected $M$ --
equivalently the number of oriented smooth structures on the underlying topological
manifold -- in terms of $p_M$ and the torsion linking form.
\end{abstract}

\maketitle

%\ifthenelse{\boolean{\showdate}}{\vspace{-0.8\baselineskip}}{}

% (fold)
\section{Introduction} \label{sec:introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Throughout this paper $M$ will be a closed smooth spin $7$-manifold and 
all homeomorphisms and diffeomorphisms are assumed to preserve spin structures,
unless otherwise stated.
We shall present a smooth classification of $2$-connected spin $7$-manifolds.
This requires an generalisation of the Eells-Kuiper invariant of $M$, classically
defined only when the spin characteristic class $p_M$ is torsion.
%which is one of the main innovations of this paper.

\subsection{The classification} \label{subsec:classification}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To any closed smooth spin 7-manifold $M$ we shall associate the following
algebraic invariants: 
%of which the first three are standard and the last two we explain below:
\begin{itemize}
\item The integral cohomology group $H^4(M)$, which is a finitely generated abelian
group.
\item The spin characteristic class $p_M$, which is an even element of $H^4(M)$.
It is a homeomorphism invariant (see Remark \ref{rmk:topological_p_M})
which is related to the first Pontrjagin class by $2p_M = p_1(M)$.
\item The torsion linking form $b_M : TH^4(M) \times TH^4(M) \to \Q/\Z$, which
is a torsion form on the torsion subgroup $TH^4(M) \subseteq H^4(M)$
(by this we mean that $b_M$ is symmetric, bilinear and nonsingular).
\item The \emph{quadratic linking family} $q^\circ_M$, 
a family of quadratic refinements of $(H^4(M), b_M, p_M)$.
\item The \emph{generalised Eells-Kuiper invariant} $\EK_M$, 
a mod 28 Gauss refinement of $(H^4(M), q_M^\circ, p_M)$.
\end{itemize}
For the moment we merely indicate the type of the last two invariants.

A quadratic refinement of $b_M$ is a function $q \colon TH^4(M) \to \Q/\Z$ 
which satisfies the equation
$q(x+y) = q(x) + q(y) + b_M(x, y)$, and we denote the set of such $q$ by
$\cal{Q}(b_M)$. The homogeneity defect of $q \in \cal{Q}(b_M)$ is the unique
element $\beta \in 2TH^4(M)$ such that $q(x) - q(-x) = b_M(x, \beta)$.
Let
\[ \divs_2 := \{ h \in H^4(M) \, | \, p_M - 2h \textrm{ is torsion} \} . \]
%
That $q^\circ_M$ is a \emph{family of quadratic refinements} of $(H^4(M), b_M, p_M)$
means that it is a function
%
\[ q_M^\circ \colon \divs_2 \to \cal{Q}(b_M), \quad h \mapsto q_M^h, \]
%
such that $q_M^h$ has homogeneity defect $\beta_h := p_M - 2h$, and satisfies
a transformation rule (see Definition \ref{def:foqr}).
Let $d_\pi$ be the greatest integer dividing $p_M$ modulo torsion
(or $d_\pi := 0$ if $p_M$ is a torsion element),
$\td_\pi : = \lcm \left(4, d_\pi \right)$ and
$\dM : = \gcd \bigl( \frac{\td_\pi}{4}, 28 \bigr)$. Let
%
\[ \divsp := \{ k \in H^4(M) \, | \, p_M - d_\pi k \textrm{ is torsion} \} \]
%
and set $\beta_k := p_M - d_\pi k$ for each $k \in S_{d_\pi}$.
By saying that the generalised Eells-Kuiper invariant of $M$ is a
\emph{mod 28 Gauss refinement} of $(H^4(M), q_M^\circ, p_M)$ we mean that it is a
function
%
\[ \EK_M \colon \divsp \to \Q/\dM\Z,  \]
%
such that $\EK_M(k) = \Arf(q^{\frac{d_\pi}{2} k}) \mmod \Z$,
%where $\Arf$ if the Arf invariant of a quadratic refinement defined in \eqref{eq:arf_def},
(where $\Arf$ is the Arf invariant of a quadratic refinement, computed in terms of a Gauss sum 
in \eqref{eq:arf_def})
and such the transformation rule 
\begin{equation}
\label{eq:ek_rule_28}
\EK_M(k+t) - \EK_M(k) = \frac{d_\pi^2 b_M(t, t) - 2d_\pi b_M(\beta_k, \, t)}{8}
\mod \gcd \left( 28, \textstyle \frac{d_\pi}{4} \right),
\end{equation}
holds for all $k \in \divsp$ and $t \in T$.
For the purposes of this introduction, the main significance of these
conditions is that a Gauss refinements is defined by its value at a single
element in $\divsp$, and that the difference between two Gauss refinements of
$(H^4(M), q_M^\circ, p_M)$ is just a constant in $\Z/\dM\Z$. 

\begin{rmk} \label{rmk:original_ek}
If $p_M$ is a torsion element then $d_\pi = 0$ and $\dM = 28$, while
$\divsp = T$ contains the distinguished element $0$.
The value $\frac{1}{28}\EK_M(0) \in \Q/\Z$ recovers the original
Eells-Kuiper invariant.
\end{rmk}

If $G$ is a finitely generated abelian group, $p \in 2G$ and $b$ is a torsion
form on $T$ then we call $(G,b,p)$ a \emph{base}. If $q$ is a family of
quadratic refinements of $(G,b,p)$ then we call $(G,q,p)$ a \emph{refinement};
we suppress $b$ since it can be recovered from $q$. If $\EK$ is
a mod 28 Gauss refinement of $(G,q^\circ\!,p)$ then we call the quadruple
$(G,q^\circ\!,\EK,p)$ a \emph{mod 28 distillation}.
If $F : G' \to G$ is a group isomorphism then we can
define another mod 28 distillation $(G', F^\#q, F^\#\EK, F^\#p)$ by pulling back:
$F^\#(p) := F^{-1}(p)$, $(F^\#q)^h(x) := q^{F(h)}(F(x))$,
and $F^\#\EK := \EK \circ F$.

The mod 28 distillation $(H^4(M), q_M^\circ, \EK_M, p_M)$ of $M$ is an invariant of
spin diffeomorphisms: if $f : M \to M'$ is a spin diffeomorphism
then $f^* : H^4(M') \to H^4(M)$ is an isomorphism, and
$(q_{M'}, \EK_{M'}, p_{M'}) = ((f^*)^\#q, (f^*)^\#\EK, (f^*)^\#p)$.
In fact, only $\EK_M$ depends on the smooth structure, and the refinement
$(H^4(M), q_M^\circ, p_M)$ is also invariant under spin almost diffeomorphisms.

An almost diffeomorphism $f \colon M_0 \acong M_1$ is a homeomorphism which is 
smooth except perhaps at a finite number of points.
It follows from results of the first author's thesis, see Lemma \ref{lem:2_types_of_qlf}, 
that \emph{2-connected} 7-manifolds
are classified up to homeomorphism and almost diffeomorphism by their refinements.

\begin{thm} \label{thm:almost_classification}
%Every refinement $(G, q^\circ\!, p)$ is isomorphic to
%$(H^4(M), q_M^\circ, p_M)$ for some
%closed smooth $2$-connected $7$-manifold $M$,
%$2$-connected $M$.
%and the refinement of $-M$ is $(H^4(M), -q_M^\circ, p_M)$.
%Moreover, for any pair of $2$-connected $7$-manifolds
%$M_0$ and $M_1$, an isomorphism $F \colon H^4(M_1) \to H^4(M_0)$,
%is realised by an almost diffeomorphism $f \colon M_0 \acong M_1$ such that $F = f^*$
%if and only if $(q_{M_1}^\circ, p_{M_1}) = F^\#(q_{M_0}^\circ, p_{M_0})$.
%
Every refinement $(G, q^\circ\!, p)$ is isomorphic to
$(H^4(M), q_M^\circ, p_M)$ for some $2$-connected $M$.
Moreover, if $M_0$ and $M_1$ are $2$-connected, then an isomorphism $F \colon H^4(M_1) \to H^4(M_0)$,
is realised by an almost diffeomorphism $f \colon M_0 \acong M_1$ such that $F = f^*$
if and only if $(q_{M_1}^\circ, p_{M_1}) = F^\#(q_{M_0}^\circ, p_{M_0})$.

The same statement holds when ``almost diffeomorphism'' is replaced by ``homeomorphism''.
\end{thm}

%
%We write
%%
%\[ F \colon (G_0, q_0, \EK_0, p_0) \cong (G_1, q_1, \EK_1, p_1) \]
%% 
%if there is an equality $(q_0, \EK_0, p_0) = (F^\#q_1, F^\#\EK_1, F^\#p_1)$.
%%$(G_0, \EK_0, q_0, p_0) = (G_0, F^\#\EK_1, F^\#q_1, F^\#p_1)$.

In this paper we prove that the generalised Eells-Kuiper invariant is precisely what
needs to be added to Theorem \ref{thm:almost_classification} to obtain a smooth classification of 2-connected 7-manifolds.


\begin{thm} \label{thm:classification}
%Every mod 28 distillation $(G, q, \EK, p)$ is isomorphic to
%$(H^4(M), q_M^\circ, \EK_M, p_M)$ for some 
%closed smooth $2$-connected $7$-manifold $M$,
%$2$-connected $M$.
%and the distillation of $-M$ is $(H^4(M), -\EK_M, -q_M^\circ, p_M)$.
%Moreover, for any pair of $2$-connected $7$-manifolds
%$M_0$ and $M_1$, an isomorphism $F \colon H^4(M_1) \to H^4(M_0)$,
%is realised by a diffeomorphism $f \colon M_0 \cong M_1$ such that $F = f^*$
%if and only if
%$(q_{M_1}, \EK_{M_1}, p_{M_1}) = F^\#(q_{M_0}, \EK_{M_0}, p_{M_0})$.
%
Every mod 28 distillation $(G, q^\circ\!, \EK, p)$ is isomorphic to
$(H^4(M), q_M^\circ, \EK_M, p_M)$ for some $2$-connected $M$.
%and the distillation of $-M$ is $(H^4(M), -\EK_M, -q_M^\circ, p_M)$.
Moreover, for any pair of $2$-connected $7$-manifolds $M_0$ and $M_1$,
an isomorphism $F \colon H^4(M_1) \to H^4(M_0)$,
is realised by a diffeomorphism $f \colon M_0 \cong M_1$ such that $F = f^*$
if and only if
$(q^\circ_{M_1}, \EK_{M_1}, p_{M_1}) = F^\#(q^\circ_{M_0}, \EK_{M_0}, p_{M_0})$.
\end{thm}

%\begin{rmk} \label{rmk:topological_classification}
%In Definition \ref{def:ql} we show that the refinement $q_M^\circ$ is a
%homeomorphism invariant.  This gives a simpler proof of the homeomorphism
%classification.
%It follows that Theorem \ref{thm:almost_classification} holds when ``almost diffeomorphism'' is replaced by ``homeomorphism''.
%\end{rmk}

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


\subsection{Elaboration of the classification} \label{subsec:elaboration}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If we are simply interested in whether $M_0$ and $M_1$ are diffeomorphic,
we can consider the following simpler invariant.
Let $TH^4(M) \subset H^4(M)$ be the torsion subgroup, let $\aut(b_M)$ be 
the group of automorphisms of the linking form $b_M$ and recall that 
$\beta_k = p_M - d_\pi k \in T$ for each $k \in S_{d_\pi}$.
We define the {\em reduced splitting function
of $M$} (the terminology is explained in Section \ref{subsec:splitting_functions}) to be the function
%
\[ \ol{q^\circ}_{\!\!\!\!\!\:M} \colon S_{d_\pi} \to \bigl(2 TH^4(M)/\!\aut(b) \bigr) \times \Q/\dM \Z, 
\quad k \mapsto \bigl([\beta_k], \mu(k) \bigr). \]

\begin{thm} \label{thm:classification_minimal}
Let $M_0$ and $M_1$ be $2$-connected with 
$(H^4(M_0), b_{M_0}, p_{M_0}) \cong (H^4(M_1), b_{M_0}, p_{M_1})$
and reduced smooth splitting functions $\ol{q_0^\circ}$ and $\ol{q_1^\circ}$.
The following are equivalent:
\begin{enumerate}
\item $M_0$ is diffeomorphic to $M_1$;
\item
$\ol{q^\circ_{0}}(S_{d_\pi}) = \ol{q^\circ_{1}}(S_{d_\pi})$;
\item
$\ol{q^\circ_{0}}(S_{d_\pi}) \cap \;\! \ol{q^\circ_{1}}(S_{d_\pi}) \neq \phi$.
\end{enumerate}
\end{thm}
%
\noindent The corresponding result for almost diffeomorphisms is given in
Corollary~\ref{cor:minima_almost_classification}.

We can also reformulate the above classification in categorical language as follows.
Let $\catek$ denote the category of mod 28 distillations
$(G, q^\circ\!, \EK, p)$ with morphisms isomorphisms:
%
\[ {\rm Ob}(\catek) = \{ (G, q^\circ\!, \EK, p) \}. \]
%
Let $\cal{M}_{7,2}^{spin}$ denote the category of 2-connected spin 7-manifolds with 
morphisms diffeomorphisms:
%
\[ {\rm Ob}(\cal{M}_{7, 2}^{spin}) = \{ M \, | \, \pi_1(M) = 0 = \pi_2(M) \}. \]
%
Given a diffeomorphism $f \colon M_0 \cong M_1$, write $f^* \colon H^4(M_1) \cong H^4(M_0)$ for the induced action on cohomology.
Hence we obtain the contravariant functor
%
\[ \cal{Q} \colon \cal{M}_{7, 2}^{spin} \to \catek,  \quad 
\left\{ \begin{array}{ccc} M & \mapsto & (H^4(M), q_M^\circ, \EK_M, p_M), \\
f \colon M_0 \cong M_1 & \mapsto & f^*. \end{array} \right. \]
%
The operations of connected sum and reversing orientation in $\cal{M}_{7,2}^{spin}$
are mirrored by corresponding operations in $\cal{Q}_\EK$.
The orthogonal sum of two distillations $(G_i, q^\circ_i, \EK_i, p_i)$, $i = 0, 1$, is defined by 
%
\[ (G_0, q^\circ_0, \EK_0, p_0) \oplus (G_1, q^\circ_1, \EK_1, p_1) := 
(G_0 \oplus G_1, q^\circ_0 \oplus q^\circ_1, \EK_0 \oplus \EK_1, p_0 \oplus p_1) .\]
%
For the refinements, we note that $S_2(G_0 \oplus G_1) = S_2(G_0) \times S_2(G_1)$, 
and for refinements $q_i^{h_i}$ of $b_i$, $q_0^{h_0} \oplus q_1^{h_1}$ refines $b_0 \oplus b_1$.  
For the mod $28$ distillations, 
%For the sum of distillations, $(G_i, q_i, \EK_i, p_i)$, $i = 0, 1$, 
we have that $c_i d_{\pi_i} = d_{\pi_0 \oplus \pi_1}$ for some integer $c_i$.
In this case $c_0S_{d_{\pi_0}} \times c_1S_{d_{\pi_1}} \subseteq S_{d_{\pi_0 \oplus \pi_1}}$ and we
set
%
\[ (\EK_0 \oplus \EK_1)(c_0 k_0 + c_1 k_1) = \EK_0(k_0) + \EK_1(k_1) \mod 
\gcd \left( 28, \textstyle \frac{\tilde d_{\pi_0 \oplus \pi_1}}{4} \right). \]
%
Since $\EK_0 \oplus \EK_1$ is determined by its value on a single $k \in S_{d_{\pi_0 \oplus \pi_1}}$,
this suffices to define the sum of distillations.  We define the negative of a distillation by
%
\[  -(G, q^\circ\!, \EK, p) = (G, -q^\circ\!, -\EK, p).  \]
%
%\[ (G_0, q_0, \EK_0, p_0) \oplus (G_1, q_1, \EK_1, p_1) := 
%(G_0 \oplus G_1, q_0 \oplus q_1, \EK_0 \oplus \EK_1 p_0 \oplus p_1) .\]

%Firstly, for
%refinements $(G_0, q_0, p_0)$ and $(G_1, q_1, p_1)$ we define the connected sum
%in the obvious way
%
%\[ (G_0, q_0, p_0) \oplus (G_1, q_1, p_1) := (G_0 \oplus G_1, q_0 \oplus q_1, p_0 \oplus p_1) .\]



\begin{thm} \label{thm:classification_categorical}
The functor $\cal{Q} \colon \cal{M}_{7, 2}^{spin} \to \catek$ is surjective
and faithful.  Moreover
\begin{enumerate}
\item $\cal{Q}(M_0 \sharp M_1) = \cal{Q}(M_0) \oplus \cal{Q}(M_1)$,
\item $\cal{Q}(-M) = -\cal{Q}(M)$.
\end{enumerate}
\end{thm}

\begin{rmk} \label{rmk:homotopy_classification}
A homotopy classification of $2$-connected $M$ is given in \cite[Theorem 6.11]{crowley02}.
\end{rmk}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The generalised Eells-Kuiper invariant} \label{subsec:intro_eek}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since the smooth classification of $2$-connected rational homology spheres and
the classification of general $2$-connected $M$ up to connected sum with
homotopy spheres was already given in \cite{crowley02} (with some cosmetic
differences in the description of the quadratic linking families,
\cf Remark \ref{rmk:projections}), the novelty of Theorem
\ref{thm:classification_categorical} lies in the smooth classification when
$H^4(M)$ is infinite.
The key ingredient is the generalisation of the Eells-Kuiper invariant.

Let $X$ be a closed spin $8$-manifold.  By the index theorem \cite[Theorem 5.3]{atiyah68}
$\wh A(X)$, the $\wh A$-genus of $X$, is equal to the index of the Dirac
operator on $X$, and so an integer.  
The classical Eells-Kuiper invariant is derived from on the relation
\begin{equation}
\label{eq:ahat}
p_X^2 - \sign(X) = 224\ahat(X),
\end{equation}
%which holds for any smooth closed spin 8-manifold $X$ with 
where $X$ has signature $\sign(X)$ and spin characteristic class $p_X$: the latter is 
defined in Section \ref{subsec:basic_invariants}.
By the index theorem $\ahat(X)$ is an
integer.
If $M$ is a closed 7-manifold such that $p_M$ is a \emph{torsion} class
(so rationally trivial) and $W$ is a spin coboundary of $M$, then $p_W^2$
has a well-defined integral over $W$ (it might in general take values in $\Q$
and not just $\Z$), and  \eqref{eq:ahat} implies that
\begin{equation}
\label{eq:classical}
\EK(M):= \frac{p_W^2 - \sign(W)}{8} \in \Q/28\Z
\end{equation}
is independent of the choice of coboundary $W$. This defines the classical
Eells-Kuiper invariant, modulo normalisation by a factor of 28.

To define an analogue when $p_M$ is not a torsion class we have to let it take
values not modulo 28 but modulo the integer $\dM = \gcd(\frac{\tilde d_\pi}{4}, 28)$, depending on the
divisibility of $p_M$ modulo torsion as above. Moreover, the
generalisation is not simply a constant in $\Q/\dM\Z$ but a function.

Now suppose that $W$ is a spin coboundary of $M$, and that there exists
$n \in H^4(W)$ such that the image of $p_W - d_\pi n$ under the restriction
map $H^4(W) \to H^4(M)$ is a torsion class; equivalently $jn \in \divsp$.
(If $W$ is 3-connected then such $n$ exist, and any spin $M$ has 3-connected
coboundaries: see the start of Section \ref{subsec:coboundaries}).
We define
\begin{equation} \label{eq:grW}
 \gr_W(jn) := \frac{(p_W {-} d_\pi n)^2 - \sign(W)}{8} \in \Q/\tdf \Z . 
\end{equation}
and extend $\gr_W$ to a function $\divsp \to \Q/\tdf \Z$ by the transformation
rule \eqref{eq:ek_rule_28}. Then $\gr_W$ is independent of the choices of $n$.
The following lemma (\cf \eqref{eq:compare}) implies that
$\EK_M := \gr_W \mmod \dM$ is independent of the choice of $W$, and functorial.

\begin{lem}
\label{lem:ek_defined}
Let $f : \partial W_0 \to \partial W_1$ be a spin diffeomorphism and
$X := (-W_0) \cup_f W_1$. Then
\[ \gr_{W_1} - (f^*)^\# \gr_{W_0} = 28 \ahat(X) \mmod \tdf. \]
\end{lem}

The idea of the definition is that the simplest way to change
\eqref{eq:classical} to something that is well-defined when the restriction of
$p_W$ to the boundary is rationally non-trivial is to compensate by subtracting
from $p_W$ a class that is divisible by $d_\pi$ and has the same rational image
in $H^4(M)$. The essentially different ways of doing that are parametrised by
$\divsp$, and that is why we end up with the generalised Eells-Kuiper invariant,
$\EK_M \colon \divsp \to \Q/\dM \Z$, being a function defined on $\divsp$.


The definition of the $s_1$ invariant by Kreck and Stolz \cite{kreck91}
provides as a byproduct a way to compute the classical Eells-Kuiper invariant
in terms of coboundaries that are not spin, but merely \spc. Proposition
\ref{prop:ek_via_spc} gives a similar way to compute the generalised Eells-Kuiper
invariant via \spc coboundaries. This turns out to be useful in an application
to distinguishing between closed 7-manifolds with holonomy $G_2$ that are
homeomorphic but not diffeomorphic.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Inertia and reactivity} \label{subsec:i_and_r}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\Theta_7 = \{ \Sigma \,|\, \Sigma \simeq S^7 \}$ be the group of spin diffeomorphism
classes of homotopy $7$-spheres $\Sigma$.  Since homotopy spheres are simply connected,
this is equivalent to the standard definition of $\Theta_7$ in \cite{kervaire63}.
By \cite{kervaire63}, $\Theta_7$
is an abelian group under connected sum and $\Theta_7 \cong \Z/28$.
%We can equally well regard $\Theta_7$ as
%the group of {\em spin} diffeomorphism classes of homotopy $7$-spheres,
%and we do so.
We define the {\em inertia group} of $M$ to be the following subgroup of $\Theta_7$:
%
\[ I(M) : = \{ \Sigma \, | \, M \sharp \Sigma \cong M \}.\]
%

\begin{rmk} \label{rmk:inertia}
Let $M_+$ denote the oriented manifold underlying $M$.  If $M$ is simply connected then 
$I(M) = I(M_+)$, 
where $I(M_+)$ is the usual inertia group of $M_+$, which is 
defined using orientation preserving diffeomorphisms $f_+ \colon M_+ \sharp \Sigma_+ \cong M_+$. 
%where we use the subscript $+$ the emphasise that $f$ need not preserve spin structures.  
%We use the notation $I(M)$ since we work spin category throughout this paper.
%and
%since we focus on $2$-connected $M$ where the spin inertia group and the usual
%inertia group coincide.  %Indeed, we know of no example where the spin inertia
%group differs from the usual inertia group.
\end{rmk}

%If $M$ is $2$-connected, then by Remark \ref{rmk:topological_classification},
%computing the spin inertia group $I(M)$ is equivalent to counting the number of 
%oriented diffeomorphism classes on smooth structures on a given almost-smooth $7$-manifold.

It turns out that even with Theorem \ref{thm:classification_categorical} in
hand, the determination of $I(M)$ can be a delicate problem.
The reason is that $\EK_M$ is not a constant but rather a function and so it is
possible for almost diffeomorphisms of $M$ to act non-trivially on $\EK_M$.
Equivalently, the automorphism group of a refinement $(G,q^\circ\!,p)$ can act
non-trivially on the set of mod 28 Gauss refinements.

The inertia group is closely related to what we (therefore) call the
\emph{reactivity} of $M$. % which an integer valued
%almost diffeomorphism invariant of $M$ defined as follows.
Let $\ADiff_{}(M)$ denote the group of spin almost diffeomorphisms
of $M$.  Given 
%a spin almost diffeomorphism 
$f \in \ADiff_{}(M)$,
the mapping torus $T_f$ of $f$ has
as well-defined spin characteristic class $p_{T_f} \in H^4(T_f)$ and we define the integer
$p^2(f) : = \an{p_{T_f}^2, [T_f]}$.  This defines a homomorphism 
%
\[ p^2 \colon \ADiff_{}(M) \to \Z, \quad f \mapsto p^2(f),\]
%
and the reactivity of $M$ is the non-negative integer $R(M)$ defined by
%
\begin{equation}
\label{eq:defR}
p^2(\ADiff_{}(M)) = R(M) \Z .
\end{equation}
%
Clearly $R(M)$ is an almost diffeomorphism invariant of $M$.  Since $T_f$ has zero signature
and $p_{T_f}$ is characteristic for the intersection form of $T_f$ we have $R(M) \in 8 \Z$.
%In Lemma \ref{lem:mapping_torus} we show that $f \in \ADiff_{}(M)$ is homotopic to a diffeo iff
%$p^2(f)$ is divisible by 224 and consequently, see Proposition \ref{prop:role_of_reactivity}, it
%follows that
It is well understood that $f \in \ADiff_{}(M)$ is pseudo-isotopic to a diffeomorphism if and only if
$p^2(f)$ is divisible by 224 (see Lemma \ref{lem:mapping_torus}) and consequently
% 
\begin{equation} \label{eq:i_and_r_intro}
I^{}(M) = \frac{R(M)}{8} \Theta_7.
\end{equation}
%
%see Proposition \ref{prop:role_of_reactivity}.

To determine $R(M)$ when $M$ is 2-connected, we first determine values of
$p^2(f)$ that
are realised for $H^*(f) = {\rm Id}$ (see Proposition \ref{prop:p2null}).
This reduces the determination of $R(M)$ to understanding the action of
the automorphism group $\aut_{q^\circ}(H^4(M))$ of $(H^4(M), q_M^\circ, p_M)$ on
mod 28 Gauss refinements. That can in turn be reduced to understanding
the automorphism group $\aut_b(H^4(M))$ of $(H^4(M), b_M, p_M)$, which is much
easier to deal with in practice. In fact, $R(M)$ is 
almost completely determined just 
%up to a factor of two by 
using the following `intermediate' notion of the divisibility of
$p_M$, whose significance was pointed out by
Wilkens \cite[Conjecture p.\,548]{wilkens74}:
%
\[ \mdiv := \left \{ \begin{array}{cl} 0 & \text{if $p_M$ is torsion,} \\
{\rm Max} \bigl\{ s \,|\,s, \mpl \in \Z, \; s\mpl^2 \text{~divides~} \mpl p_M
\bigr\} & \text{otherwise.} \end{array} \right.  \]
%
Corollary \ref{cor:p2image} and \eqref{eq:i_and_r_intro} give the next theorem, where
for a fraction $\frac{a}{b}$ in lowest terms $\NumB{\frac{a}{b}} = a$. 

\begin{thm} \label{thm:i_and_r}
Let $M$ be $2$-connected and let $\mdiv = \mdiv(M)$.
There is an integer $r \in \{0, 1, 2\}$ depending only on the base $(H^4(M), b_M, p_M)$, such
that 
%
\[ R(M) = \lcm(8,2^{r} \mdiv) .\]
%
Consequently
%
%\[ I(M) = \NumB{2^{r-3} \mdiv} \Theta_7 .\]
\[ I(M) = \NumB{\frac{2^{r} \mdiv}{8}} \Theta_7 .\]
%
%Moreover, if $b_M$ contains no split summand of $2$-primary order, in
%particular if $TH^4(M)$ is of odd order or if $b_M$ is hyperbolic or
%pseudo-hyperbolic, then $r = 1$.
If $TH^4(M)$ has odd order then $r = 1$.
%However, each $r \in \{0, 1, 2\}$ is realised by some base $(G, b, p)$.
%
\end{thm}

If $H^4(M)$ does have some 2-torsion then in general one needs to look
at the torsion linking form in detail to determine $r$. 
Wilkens' conjecture
\cite[Conjecture p.\,548]{wilkens74} for the inertia group is equivalent to 
supposing that $r = 1$ always, which is not true.  The invariant $r = r(G, b, p)$ is
defined in Definition \ref{def:r} and while we do not have a closed formula for $r$,
it is feasible to compute $r$ for any given example. 
For examples where $r = 0, 1$ or $2$, see Example \ref{ex:r012}.

We next discuss some consequences of Theorem \ref{thm:i_and_r} and its proof.
If $N$ is a close smooth manifold, let $n_+(N)$ denote 
the number of oriented diffeomorphism classes 
of smooth structures on the topological manifold underlying $M$.
From Theorems \ref{thm:almost_classification} and \ref{thm:i_and_r} and Remark \ref{rmk:inertia} 
we deduce

\begin{cor} \label{cor:n_+}
If $M$ is $2$-connected then $n_+(M) = \gcd\left( \Num(2^{r-3}\mdiv), 28 \right)$.
%\[ n_+(M) = \gcd\left( \frac{\lcm(8, 2^r\mdiv)}{8}, 28 \right).\]
\end{cor}
%
We call a homotopy equivalence $f \colon N_0 \simeq N_1$ of smooth manifolds if 
there is a stable vector bundle isomorphism $f^*\tau_{N_1} \cong \tau_{N_0}$ where $\tau_{N_i}$ 
is the stable tangent bundle of $N_i$.  In Lemma \ref{lem:the1} we show that 
a homotopy equivalence $f \colon M_0 \simeq M_1$ of 
$2$-connected $7$-manifolds with
$(f^*)^\sharp p_{M_1} = p_{M_0}$ is tangential.  
Together with Theorem \ref{thm:i_and_r} this entails
%
\begin{cor} \label{cor:I(M)_is_the}
Let $M_0$ and $M_1$ be $2$-connected and let $f \colon M_0 \simeq M_1$ be a tangential homotopy equivalence.
Then $I(M_0) = I(M_1)$.
\end{cor}
%
One may wonder if Corollary \ref{cor:I(M)_is_the} is true because tangentially homotopy equivalent
2-connected 7-manifolds are almost diffeomorphic (equivalently homeomorphic by Theorem \ref{thm:almost_classification}).  However this was shown not to be the case in \cite[p.\,114]{crowley02}, contradicting results of Madsen, Taylor and Williams
\cite[Theorem C and Theorem 5.10]{madsen80}: see Proposition \ref{prop:the} and Remark \ref{rmk:the}.

The computation of the reactivity of $M$ also has applications in $G_2$-topology.
We define the {\em smooth reactivity of $M$}, $R^\Diff(M)$, using the equation 
$p^2(\Diff(M)) = R^\Diff(M)\Z$.
%so that 
%
%\[ R^\Diff(M) = \lcm(R(M), 224). \]
%
By Corollary \ref{cor:p2image}, 
%$R^\Diff(M) = \lcm(224, 2^r\mdiv)$ 
%
\[ R^\Diff(M) = \lcm(2^r\mdiv, 224), \]
%
and in \cite[Section 6]{nu} we show that $R^\Diff(M)$
determines the number of \gtstr s on $M$ modulo homotopies and diffeomorphisms.


The proof of Theorem \ref{thm:i_and_r} leads to information about the mapping class groups of $M$.  
Let $I_H(M) \subseteq I(M)$ be the subgroup of the inertia
group of $M$ consisting of homotopy spheres $\Sigma$ such that there is a diffeomorphism
$f \colon M \sharp \Sigma \cong M$ where $H^*(f) = {\rm Id}$, considering $M \sharp \Sigma$ and $M$
as the same topological space.  Using the delicate algebra in Section \ref{subsec:aut_q_on_gauss_functions}
we construct a surjective homomorphism 
%
\[ \wh P \colon \aut_{q^\circ}(H^4(M)) \to I(M)/I_H(M), \]
%
such that $F \in \aut_{q^\circ}(H^4(M))$ is realised by a diffeomorphism of $M$ if and only if $\wh P(F) = 0$.
Now by Theorem \ref{thm:almost_classification}, every $F \in \aut_{q^\circ}(H^4(M))$ is realised
by an almost diffeomorphism of $M$ and in Proposition \ref{prop:pairs_of_inertia_groups} we prove 
that every subquotient of $\Theta_7$ can be realised as the pair $I(M)/I_H(M)$ for some $2$-connected $M$.
As consequence we have 
%the following theorem (see Example \ref{ex:mapping_inertia}).

\begin{thm} \label{thm:hatP_short}
There exist $2$-connected $M$ with automorphisms $F \in \aut_{q^\circ}(H^4(M))$ which are not realised by any diffeomorphism of $M$.
Necessarily every such $F$ is realised by an almost diffeomorphism of $M$.
\end{thm}


\subsection{Organisation} \label{subsec:organisation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The rest of this paper is organised as follows. In Section \ref{sec:invariants} we define the
invariants used in Theorems \ref{thm:almost_classification} and \ref{thm:classification}.
In particular, families of refinements, Gauss refinements and the generalised Eells-Kuiper invariant are defined 
in Sections \ref{subsec:families}, \ref{subsec:gr} and \ref{subsec:eek} respectively.
In Section \ref{sec:classification} we prove our main classification results
%Theorems \ref{thm:almost_classification}, \ref{thm:classification}, \ref{thm:classification_minimal}
%and \ref{thm:classification_categorical}, 
and we discuss the connected sum splitting of
$2$-connected $7$-manifolds in Theorems \ref{thm:almost_split1} and \ref{thm:split}.
Section \ref{sec:auto} is an algebraic section in which we analyse the automorphisms of refinements
and bases and the action of these automorphisms on Gauss refinements.
This section contains the proof of Theorem \ref{thm:i_and_r}
which follows from the computation of the reactivity of $M$ in Corollary \ref{cor:p2image}.
%Aside from the definitions of refinements and distillations from Section \ref{sec:invariants},
%the only input to the proof of Corollary \ref{cor:p2image} from outside Section \ref{sec:auto}
%is Proposition \ref{prop:p2null} where we construct almost diffeomorphisms $f \colon M \acong M$
%with $H^*(f) = {\rm Id}$ and which realise certain values of $p^2(f)$.

In Section \ref{sec:examples} we illustrate our classification of $2$-connected $M$ with examples:
we consider the total spaces of $3$-sphere bundles over $S^4$, examples admitting rare
metrics and we give a pair of manifolds $M_0$ and $M_1$ which are tangentially homotopy equivalent 
but not homeomorphic.  We also we present a refinement of Wilkesn' identification of the set of indecomposable generators 
for the monoid of almost diffeomorphism classes of $2$-connected $7$-manifolds under the operation of connected sum: 
see Theorem \ref{thm:indecomposable}.
In Section \ref{sec:mcg_and_inertia} we investigate the relationship between
the inertia groups of $M$ and the mapping class groups of $M$ and prove Theorem \ref{thm:hatP_short}.

\smallskip
\noindent
{\bf Acknowledgements:}  We would like to thank Sebastian Goette for helpful discussions.
DC thanks the Mathematics Department at Imperial College for its hospitality 
and acknowledges support from EPSRC Mathematics Platform grant EP/I019111/1.
DC also acknowledges the support of the Leibniz Prize of Wolfgang L\"{u}ck, granted by the Deutsche Forschungsgemeinschaft.
% section introduction (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% (fold)
\section{Invariants} \label{sec:invariants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we define the invariants needed to classify $2$-connected spin $7$-manifolds
$M$. In Section \ref{subsec:basic_invariants} we introduce the linking form of $b_M$ of $M$
and the spin characteristic class $p_M \in 2H^4(M)$. 
%This spin characteristic class is defined
%for all topological spin manifolds and in particular for spin $8$-manifolds $Y$.  An important 
%fact is that $p_Y \in H^4(Y)$ is characteristic for the intersection form of $Y$; 
%\ie~$p_Y$ determines the mod $2$ intersection form of $Y$.  

In Section \ref{subsec:coboundaries}
we recall that every $M$ has a $3$-connected spin coboundary $W$ which has an
intersection form
%
\[ \lambda_W \colon FH^4(W, \del W) \times FH^4(W, \del W) \to \Z, \]
%
where $H^4(W, \del W) = H^4(W, \del W)$ and $FH^4(W, \del W)$ is the free quotient of $H^4(W, \del W)$. 
A key fact is that $p_W$ is characteristic for $\lambda_W$, \ie $\lambda_W(x, x) = x \cup p_W$ 
mod $2$ for all $x \in H^4(W, \del W)$.  
We call a triple $(H^4(W, \del W), \lambda_W, p_W)$ a {\em characteristic form} and identify it as the salient algebraic model for $W$.

In sections
Sections \ref{subsec:tfs_and_Gauss_sums}, \ref{subsec:families} and \ref{subsec:gr} we progressively
build algebraic ``boundary invariants'' of characteristic forms.
Section \ref{subsec:tfs_and_Gauss_sums} recalls the theory of refinements of torsion forms and
Section \ref{subsec:families} shows how a characteristic form defines a family of refinements
on its boundary.  In Section \ref{subsec:gr} we define the generalised
Eells-Kuiper invariant $\EK_M$ of $M$ using the characteristic form $(FH^4(W, \del W), \lambda_W, p_W)$ 
of a spin coboundary $W$ and Hirzebruch's characteristic class formulae for the $\wh A$-genus and the $L$-genus.
The generalised Eells-Kuiper invariant is a reduced defect invariant of the $\wh A$-genus.
Finally in Section \ref{subsec:spin_c} we show how $\mu_M$ can be computed via a coboundary $W$
which is \spc rather than spin.
%The original Eells-Kuiper invariant is a define invariant based on the
%fact that the $\wh A$-genus of closed spin $8$-manifold $X$ is an integer and can be expressed
%as a rational polynomial of Pontrjagin numbers.  For spin $8$-manifolds with $W$ boundary $M$ where
%$p_M = 0 \in H^4(M; \Q)$ on can define effectively define $\mu(W) = \wh A(W) \in \Q$ and hence 
%$\mu(M) : = \mu(W) \in \Q/\Z$.  However, when $p_M \neq \in H^4(M; \Q)$ a new method must be found.
%Our idea is to define $\mu_W $



\subsection{Basic invariants} \label{subsec:basic_invariants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To any closed spin 7-manifold we can associate its integral cohomology group
$H^4(M)$ and torsion linking form
%
\[ b_M : TH^4(M) \times TH^4(M) \to \Q/\Z. \]
%
defined on the torsion subgroup of $H^4(M)$.
The linking form $b_M$ is a nonsingular symmetric bilinear form.
%torsion form as defined in \S \ref{subsec:tfs_and_Gauss_sums}.  
We also have the spin characteristic class $p_M \in 2H^4(M)$
whose definition we recall below.  
%In the categorical language of the introduction, $(H^4(M), b_M, p_M) \in \catb$.

The classifying space $BSpin$ is $3$-connected and
$\pi_4(BSpin) \cong \Z$.
It follows that $H^4(BSpin) \cong \Z$ is infinite cyclic.  A generator is
denoted $\pm \frac{p_1}{2}$ and the notation is justified since for the
canonical map $\pi \colon BSpin \to BSO$ we have $\pi^*p_1 = 2 \frac{p_1}{2}$
where $p_1$ is the first Pontrjagin class.
Given a spin manifold $N$ we write
%
\[ p_{N} : = \frac{p_1}{2}(N) \in H^4(N) .\]
%
The triple $(H^4(M), b_M, p_M)$ is
called the base of $M$ and for later use we introduce the category $\catb$ consisting of bases
with morphisms isomorphisms
%
\[ {\rm Ob}(\catb) = \{ (G, b , p) \}. \]

\begin{rmk}
\label{rmk:topological_p_M}
In order to prove the topological invariance of invariants we define in the
later subsections, we consider $p_Y$ for general topological spin
manifolds $Y$. 
We let $BTop$ denote the classifying space for
stable topological microbundles, see \cite[Essay IV, Proposition 8.1]{kirby77},
and $BTop\an{4}$ its $3$-connected cover.
Equivalently, $BTop\an{4}$ is the classifying space for stable spin topological microbundles.
By \cite[(3)]{hsiang69} there is a split short exact sequence
%
\[ 0 \to \pi_4(BSpin) \to \pi_4(BTop\an{4}) \to \Z/2 \to 0. \]
%
It follows that the canonical homomorphism $H^4(BTop\an{4}) \to H^4(BSpin)$ is an isomorphism 
and so $p_N \in H^4(N)$ is defined for any topological spin manifold.
\end{rmk}

By \cite[Lemma 6.5]{kreck91}, the mod 2 reduction of $\pm$ is the $4th$
Stiefel-Whitney class $w_4$. That has the following consequences for the parity
of the characteristic class of a closed spin manifold.

%In the following lemma $Y$ will be compact and we let $H^*_{cpt}(Y) : =
%H^*_{cpt}(Y - \del Y)$ denote the compactly support cohomology $Y$ minus its
%boundary.  Thus there is a natural isomorphism
%$H^*_{cpt}(Y) \cong H^*(Y, \del Y)$.

\begin{lem}  \label{lem:p_M}
%%%%%%
\hfill
\begin{enumerate}
\item \label{it:closed7}
Let $M$ be a closed spin $7$-manifold. Then $p_M \in 2 H^4(M)$.
\item \label{it:closed8}
Let $X$ be a closed spin $8$-manifold.
For all $x \in H^4(X; \Z/2)$
%
\[ x^2 = x \cup p_Y \in H^8(X;\Z/2). \]
\item \label{it:boundary}
Let $W$ be a compact spin $8$-manifold with boundary $M$.
For all $x \in H^4(W, M; \Z/2)$
%
\[ x^2 = x \cup p_W \in H^8(W,M; \Z/2). \]
%
\end{enumerate}
\end{lem}

\begin{proof}
%%%%%%%
By Wu's formula, see \eg
\cite[Theorem 11.14]{milnor74}, $w_4 = v_4$ for any closed spin manifold 
since the first three Wu classes of a spin manifold vanish.

\begin{enumerate}
\item Now $v_4(M) = 0$ since $M$ is $7$-dimensional, the Wu class satisfies
$v_4(M) \cup x = Sq^4(x)$ for all $x \in H^3(M; \Z_2)$,
and $Sq^4$ vanishes on degree three classes.

\item $x^2 = Sq^4(x) = x \cup v_4(X) = x \cup p_X$.

\item Let $X := W \cup_{\Id_M} (-W)$. The push-forward
$i_* : H^*(W,M) \to H^*(X)$ of the inclusion $i : W \into X$ is dual under the
Poincar\'e pairing to the restriction $i^* : H^4(X) \to H^4(W)$.
Since $i^*p_X = p_W$, (ii) gives
\[ x^2 = (i_*x)^2 = i_*x \cup p_X = x \cup p_Y , \]
where the equalities take place in
$\Z/2 \cong H^8(X; \Z/2) \cong H^8(W,M; \Z/2)$. \qedhere
\end{enumerate}
\end{proof}


\subsection{Algebraic models of coboundaries} \label{subsec:coboundaries}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $M$ be a closed spin 7-manifold. Since the bordism group $\Omega_7^{Spin}$
vanishes by \cite{milnor63}, there is a compact spin
8-manifold $W$ such that $\partial W = M$.  Applying surgery below the middle dimension to $W$ \cite[Theorem 3]{milnor61}, we can assume that $W$
is 3-connected.  
Since $W$ is $3$-connected the relative cohomology sequence of $(W, M)$ gives exactness of
%
\begin{equation} \label{eq:M_and_W}
FH^4(W, \del W) \to H^4(W) \to H^4(M) \to 0 
\end{equation}
%
where $H^4(W)$ is torsion-free and $FH^4(W, \del W) := H^4(W, \del W)/{\rm Tors}(H^4(W, \del W))$
is the free quotient of $H^4(W, \del W)$ (by Poincar\'{e} duality, the
torsion subgroup of $H^4(W, \del W)$ is isomorphic to the torsion subgroup of
$H^3(M)$.)
%The push-forward map $\hlambda_W$ can be thought of as the map
The first homomorphism in \eqref{eq:M_and_W} can be thought of as the adjoint
homomorphism
$\hlambda_W : FH^4(W, \del W) \to FH^4(W, \del W)^*$ of the intersection pairing
$\lambda_W$ on $FH^4(W, \del W)$. The principle is to regard
$(FH^4(W, \del W), \lambda_W)$ as a ``model'' for a coboundary~$W$.

Let us set up some terminology to deal with these models.
%For a finitely generated abelian group $G$, let $T$ denote the torsion
%subgroup, and let $\pi : G \to G/T$ be the projection to free quotient.
We say that $(H,\lambda)$ is an integral form if $H$ is a finitely generated
free abelian group and $\lambda : H \times H \to \Z$ is symmetric and bilinear.
Let $\hlambda$ denote the adjoint map $H \to H^*$. The ``boundary''
of $(H,\lambda)$ is $G := \coker \hlambda$. We say that an element
$\alpha \in H^*$ is \emph{characteristic} for $\lambda$ if
$\lambda(x,x) = \alpha(x) \mmod 2$ for all $x \in H$. We then call
$(H,\lambda,\alpha)$ a \emph{characteristic form}.

If $W$ is a 3-connected coboundary of $M$ then what we have said so far is that 
$(FH^4(W, \del W),\lambda_W)$ is an integral form with boundary $H^4(M)$.
By Lemma \ref{lem:p_M}\ref{it:boundary},
$(FH^4(W, \del W), \lambda_W, p_W)$ is characteristic.
(If $M$ is $2$-connected then Wall's classification of $3$-connected $8$-manifolds \cite{wall62}
ensures that the characteristic form of $W$ is a
complete invariant of diffeomorphisms: see \cite[Corollary 2.5]{crowley02}.)
In the next subsections we study the structures that an integral or
characteristic form induces on its boundary. By applying this to the
algebraic model of a 3-connected coboundary of $M$ we obtain the desired
algebraic invariants of $M$. To prove that they are independent of the
choice of $W$ we will combine a splitting result for the algebraic constructions
with the following lemma whose proof is a simple application of
the Mayer-Vietoris theorem.

\begin{lem}
\label{lem:split_mfd}
Let $W_i$ be compact 3-connected spin 8-manifolds,
$f : \partial W_0 \to \partial W_1$ a homeomorphism,
and $X := (-W_0) \cup_f W_1$ (a closed spin topological manifold).
Then $H^4_{}(W_i, \del W_i) \into H^4(X)$, and the images are orthogonal to each
other with respect to the intersection form $\lambda_X$ of $X$.
Further the restriction map $H^4(X) \to H^4(M)$ is surjective, with kernel
$H^4_{}(W_0, \del W_0) + H^4_{}(W_1, \del W_1)$. \qed
\end{lem}

\begin{rmk} \label{rmk:characteristic_gluing}
We note that by Lemma \ref{lem:p_M} \ref{it:closed8}, the triple
$(H^4(X), \lambda_X, p_X)$ of the manifold $X$ in Lemma \ref{lem:split_mfd} is
a (non-singular) characteristic form.
Moreover, the image of $p_X$ under the restriction map $H^4(X) \to H^4(W_i)$ is of course $p_{W_i}$.
\end{rmk}
%By Lemma \ref{lem:topological_p_M}, $(H^4(X), \lambda_X, p_X)$ is a characteristic form.
%Moreover, the image of $p_X$ under the restriction map $H^4(X) \to H^4(W_i)$ is $p_{W_i}$.

%If in addition $f$ is almost-smooth, then $X$ is an
%almost-smooth spin manifold and has a well-defined spin characteristic class
%$p_X \in H^4(X)$. So $(H^4(X), \lambda_X, p_X)$ is a characteristic form,
%and the image of $p_X$ under the restriction map $H^4(X) \to H^4(W_i)$ is
%$p_{W_i}$.


% (fold)
\subsection{Torsion forms and quadratic refinements on finite groups}
\label{subsec:tfs_and_Gauss_sums}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Throughout this subsection $T$ will be a \emph{finite} abelian group.
We say that $b \colon T \times T \to \Q/\Z$ is a \emph{torsion form} on $T$
if it is symmetric, bilinear, and nonsingular in the sense that
the induced map $T \to \Hom(T, \Q/\Z)$ is an isomorphism.
We call a function $q \colon T \to \Q/\Z$ a \emph{quadratic refinement} of $b$
if
%for all $x, y \in G$
%
\[ q(x+y) = q(x) + q(y) + b(x, y), ~~ \forall\,x, y \in T . \]
%
The \emph{homogeneity defect} of $q$ is the element $\beta \in 2 T$ such that
$q(x)-q(-x) = b(x, \beta)$ for all $x \in G$.
If $\beta = 0$ then $q(x) = q(-x)$ and $q$ is called \emph{homogeneous}.
We define
%
\[ \cal{Q}(b) := \{ q \,|\, \text{$q$ is quadratic refinement of $b$} \}, \]
%
and we let $\cal{Q}^0(b) \subset \cal{Q}(b)$ be the set of homogeneous quadratic 
refinements of $b$.
In this subsection we consider the problem of classifying the refinements in
$\cal{Q}(b)$ up to isomorphism.  For $\cal{Q}^0(b)$ this problem was solved by
Nikulin \cite{nikulin79} and the general solution was given independently by
the first author \cite{crowley02} and Deloup and Massuyeau \cite{deloup05}.

The first basic result \cite[Lemma 2.30, Lemma 2.31]{crowley02} is that
$\cal{Q}(b)$ and $\cal{Q}^0(b)$ are both non-empty and that $G$ acts 
freely and transitively on $\cal{Q}(b)$ via the action
%
\[  \cal{Q}(b) \times G \to \cal{Q}(b), \quad (q, a) \mapsto q_a,\]
%
where for all $x \in G$, 
%
\[ q_a(x) := q(x) + b(x, a) = q(x+a) - q(a). \]
%  

\begin{ex} \label{ex:cyclic_b_and_q}
If $T \cong \Z/r\Z$ is cyclic then all torsion forms and refinements on $T$
are given by the following examples.
Given $\theta \in \Z/r$ coprime to $r$, let
$\anb{\theta}{r}$ denote $T$ equipped with the
torsion form
\[ b(x,y) := \frac{\theta xy}{r} \in \Q/\Z . \]
Given $\theta \in \Z/2r$ coprime to $r$ and $\gamma \in \Z/r$
(so that $2 \gamma \in \Z/2r$), we define a quadratic refinement
$\anq{\theta}{2r}_\gamma$ of $\anb{\theta}{r}$ by
%
%\[ \anq{\theta/2r}_\gamma \colon \Z/r \to \Q/\Z, \quad
%x \longmapsto \theta \frac{x^2 + 2\gamma x}{2r}. \]
\[ q(x) := \theta \left(\frac{ x^2 + 2\gamma x}{2r}\right) \in \Q/\Z. \]
%

\end{ex}

Beyond the homogeneity defect,
we introduce two further equivalent invariants of $q$.
The first of these is the {\em Gauss sum of $q$} which is the complex number
%
\[ GS(q) := \sum_{x \in T} e^{2\pi i q(x)} \in \C,  \]
%
where $i = \sqrt{-1}$ and $e$ is Euler's number.
From the fact that $q_a(x) = q(x+a)-q(a)$
one easily obtains the following useful

\begin{lem}[{\cite[(4.1)]{deloup05}}] \label{lem:Gauss_sum_shift}
$GS(q_a) = e^{-2\pi iq(a)} GS(q)$. \qed
\end{lem}

It is a theorem of Milgram \cite[Theorem, p.\,127, Appendix 4]{milnor73} that if $q$ is homogeneous, then $GS(q)$
is a non-zero complex number with modulus $\sqrt{|T|}$: by Lemma \ref{lem:Gauss_sum_shift}, this
holds for all $q \in \cal{Q}(b)$.
We define the {\em Arf invariant} of $q$ to be the number $\Arf(q) \in \Q/\Z$ which is the argument of $GS(q)$ divided by $2 \pi$.  That is
%
\begin{equation} \label{eq:arf_def}
 GS(q) = \sqrt{|T|}e^{2\pi i \Arf(q)} \in \C. 
\end{equation}
%
Then Lemma \ref{lem:Gauss_sum_shift} is equivalent to
\begin{equation}
\label{eq:arf_shift}
\Arf(q_a) = \Arf(q) - q(a) .
\end{equation}

Before giving the classification theorems for $\cal{Q}(b)$, we review how elements of $\cal{Q}(b)$
can be presented as the boundaries of nondegenerate characteristic forms
$(H, \lambda, \alpha)$ and how $\Arf(q)$ is determined by
$(H, \lambda, \alpha)$ in this situation.
If $\lambda$ is nondegenerate then the boundary $T := \coker (\hlambda)$
of $(H, \lambda)$ fits into the short exact sequence
%
\[ 0 \to H \xra{~\hlambda~} H^* \xra{~\ares~} T \to 0. \]
%
It follows that $\hlambda$ becomes
an isomorphism when tensored with the rationals, $\hlambda_\Q \colon H \otimes \Q \cong H^* \otimes \Q$.
We can use the inverse $(\hlambda_\Q)^{-1}$ to pullback the form $\lambda_\Q$ on $H \otimes \Q$ induced by
$\lambda$ and we obtain a rational symmetric bilinear form on $H^* \otimes \Q$.  Restricting to $H^* \subset H^* \otimes \Q$,
we obtain the form
%
\[ \lambda^{-1} : = (\hlambda_\Q^{-1})^*(\lambda_\Q)|_{H^* \times H^*} \colon H^* \times H^* \to \Q.\]
%
Given a characteristic form $(H, \lambda, \alpha)$ and $x \in T$, let $\bar x \in H^*$ be such that $\ares(\bar x) = x$.
We define the torsion form
%
\[ b_\lambda \colon T \times T \to \Q/\Z, \quad (x, y) \mapsto \lambda^{-1}(\bar x, \bar y)~~\text{mod}~\Z, \]
%
and the quadratic refinement of $b$
%
\begin{equation} \label{eq:bndry_of_char_form}
q_{\lambda, \alpha} \colon T \to \Q/\Z, \quad x \mapsto \frac{\lambda^{-1}(\bar x, \bar x) + \lambda^{-1}(\bar x, \alpha)}{2}~~\text{mod}~\Z.
\end{equation}
The homogeneity defect of $q_{\lambda, \alpha}$ is exactly $\ares(\alpha)$.
%\dccomm{Notice the algebraic signs are unchanged - in the topology, we will introduce a minus sign, unlike in \cite{crowley02}.}

We have following fundamental result of Wall:

\begin{thm}[{\cite[Theorem 6]{wall63}}]
For all torsion forms $b$ and for every $q \in \cal{Q}^0(b)$, there is an even nondegenerate
form $(H, \lambda)$ and an isomorphism $q \cong \del(H, \lambda, 0)$.
\end{thm}

We now state Milgram's theorem on the Gauss sums of homogeneous quadratic torsion forms.

\begin{thm}[Milgram {\cite[Theorem, p.\,127, Appendix 4]{milnor73}}] \label{thm:Milgram}
Let $q \in \cal{Q}^0(b)$.
\begin{enumerate}
\item $8\Arf(q) \in \Z$.  
\item If $(H, \lambda)$ is an even nondegenerate integral form
then $8 \Arf(q_{\lambda, 0}) \equiv \sigma(\lambda)$~mod~$8$.
\end{enumerate}
%
\end{thm}

Following Milgram's theorem, we can restate Nikulin's classification of homogeneous quadratic
refinements of $b$ as follows.

\begin{thm}[{\cite[Theorem 1.11.3]{nikulin79}}] \label{thm:classification_of_homogeneous_q}
If $q_0, q_1 \in \cal{Q}^0(b)$ then $q_0$ is isomorphic to $q_1$ if and only if $\Arf(q_0) = \Arf(q_1)$.
\end{thm}

%For general quadratic refinements, we also define the Eells-Kuiper invariant of $(H, \lambda, \alpha)$ by
%
%\[ \mu(H, \lambda, \alpha) : = \frac{\lambda^{-1}(\alpha, \alpha) - \sigma(\lambda)}{8} \in \Q/\Z .\]
%

For general quadratic refinements of $b$ we have the following results.

\begin{prop}[{\cite[Proposition 5.19]{crowley02}}]
\label{prop:arf}
Let $q \in \cal{Q}(b)$ and suppose that $(G, q, b) = \del(H, \lambda, \alpha)$.  Then
%
\[ -\Arf(q) = \frac{\lambda^{-1}(\alpha, \alpha) - \sigma(\lambda)}{8} \in \Q/\Z . \]
\end{prop}

\begin{thm}[{\cite[Theorem 5.22]{crowley02}}, {\cite[Theorem 4.1]{deloup05}}] \label{thm:classification_of_q}
Two torsion forms $q_0, q_1 \in \cal{Q}(b)$ are isomorphic if and only if the following hold:
\begin{enumerate}
\item there is an automorphism $f \colon T \cong T$ of $b$ such that $f(\beta_0) = \beta_1$,
\item $\Arf(q_0) = \Arf(q_1)$.
\end{enumerate}
\end{thm}

\begin{rmk}
The proof of Theorem \ref{thm:classification_of_homogeneous_q} in \cite{nikulin79} 
and the proof of Theorem \ref{thm:classification_of_q} 
in \cite{crowley02} both apply classification results for torsion forms and case by case checking.
In contrast, the proof of Theorem \ref{thm:classification_of_q} in \cite{deloup05}
is short and general, with one elegant argument covering all cases.
\end{rmk}

% subsection   (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Families of quadratic refinements}
\label{subsec:families}

Let $G$ be a finitely generated abelian group, $p$ an element of $2G$,
and $b$ a torsion form on the torsion subgroup $T$, \ie $(G,b,p)$ belongs to
the category $\catb$.
Define
\[ \divs_2 := \{ h \in G : p_M - 2h \in T \} , \]
and for $h \in \divs_2$ write $\beta_h := p_M - 2h$. Note that $T$ acts
simply transitively on $\divs_2$ by addition.

\begin{defn} \label{def:foqr}
A \emph{family of refinements} of $(G,b,p)$ is a function
$q^\circ \colon \divs_2 \to \cal{Q}(b), \; h \mapsto q^h$, such that
\begin{enumerate}
\item \label{it:q_hom-defect}
the homogeneity defect of $q^h$ is $\beta_h$, and
\item \label{it:q_trans}
$q^{h+t} = q^h_{-t}$ for any $t \in T$.
\end{enumerate}
\end{defn}
%
An isomorphism $F : G \to G'$ obviously maps $\divs_2 \to \divs'_2$,
and $F$ pulls back a family of quadratic refinements $q'^\circ$ on
$G'$ to one on $G$ by setting
\begin{equation}
\label{eq:pull_q}
(F^\# q')^h := q'^{F(h)} \circ F_{|T} .
\end{equation}
In this case $q^\circ$ and $q'^\circ$ are isomorphic via $F$.  
The orthogonal sum of two families of refinements $(G_0, b_0, p_0)$ and $(G_1, b_1, p_1)$ 
is the obvious family of refinements of $(G_0 \oplus G_1, b_0 \oplus b_1, p_0 \oplus p_1)$
as defined in Section \ref{subsec:elaboration} of the introduction. 
The negative of a family of refinements $q^\circ$ is the family of refinements $-q^\circ$ of $(Q, -b, p)$
defined by $(-q)^h = -q^h$.  
For later use we introduce the category $\catq$ consisting of refinements
with morphisms isomorphisms
%
\[ {\rm Ob}(\catq) = \{ (G, q^\circ\!, p) \}. \]

Families of refinements as in Definition \ref{def:foqr} 
are defined naturally on the boundaries of characteristic forms
$(H, \lambda, \alpha)$ when $\lambda$ is allowed to be degenerate.
First define $(G, b, p)$ as follows.
Let $G := \coker(\hlambda)$ and $p := \ares(\alpha)$; that $p$ is even
should perhaps be seen as a consequence of existence of a characteristic
element for any $\Z_2$-valued bilinear form, degenerate or not.
Let $\rad$ be the radical $\ker(\lambda) \subset H$, and
$\res \subseteq H^*$ the annihilator of $\rad$.
% {\rm Ann}(\rad)
%Then $\res$ is rationally spanned by $\im \hl$, and $T \cong R / \im \hl$.
The form $\lambda$ descends to a nondegenerate form on $H/\rad$,
and $\res \cong (H/K)^*$. Now $R / \im(\hlambda) \cong T$, so we obtain
a torsion form $b$ on $T$ as in Section \ref{subsec:tfs_and_Gauss_sums}.

Next we define the induced family of quadratic refinements. For any
$h \in \divs_2$, pick $m \in H^*$ such that $\ares m = h$ and
set $\alpha_m = \alpha - 2m$.
Then $\ares(\alpha_m) = \beta_h$, so $\alpha_m \in \res$.
Now $\alpha_m$ is characteristic for $(H/\rad, \lambda)$, and we let
$q^h = q_{\lambda, \alpha_m}$ be the quadratic refinement of $b$ defined in \eqref{eq:bndry_of_char_form} 
in the previous subsection, 
\ie if $y \in R$ and $\ares y = x$ then
%
\begin{equation}
\label{eq:alg_qdef}
q^h(y) := \frac{\lambda^{-1}(y, y) + \lambda^{-1}(\alpha_m, y)}{2}
= \frac{\gen{\hlambda^{-1}(y), y + \alpha_m}}{2} \in \Q/\Z.
\end{equation}
%
This is independent of the choice of $m$, since if $m' = m + \hlambda r$ then
%
\[ \lambda^{-1}(2m', y) - \lambda^{-1}(2m, y) = 2\gen{r,y} \in 2\Z . \]
That (i) of Definition \ref{def:foqr} is satisfied is immediate from $\ares \alpha_m = \beta_h$.
Meanwhile, if $h' = h + t$ for some $t \in T$ then
$\ares(m' - m) = t$, so 
\[ q^{h'}(x) - q^h(x) = \gen{\hlambda^{-1}(y), m - m'}
= -b(x, t) , \]
which shows that (ii) of Definition \ref{def:foqr} holds.  We denote this family of refinements $q^\circ$ by
$\del(H, \lambda, \alpha)$ and often refer to it as the {\em boundary} of $(H, \lambda, \alpha)$.
It is clear that an isomorphism of characteristic
forms $E \colon (H_0, \lambda_0, \alpha_0) \cong (H_1, \lambda_1, \alpha_1)$
induces an isomorphism $\del E \colon \del(H_0, \lambda_0, \alpha_0) \cong (H_1, \lambda_1, \alpha_1)$. 
It is also clear that the boundary of an orthogonal sum of characteristic forms is
the orthogonal sum of the boundaries and that $\del(H, -\lambda, \alpha) = -\del(H, \lambda, \alpha)$.

%We say that a characteristic form $(H,\lambda,\alpha)$ is nonsingular
%if $\lambda$ is, \ie if the adjoint $\hlambda : H \to H^*$ is an isomorphism.
%Since $\coker(\hlambda) = 0$ if and only if $\lambda$ is nonsingular, it follows
%that $(H, \lambda, \alpha)$ is nonsingular if and only if $\del(H, \lambda, \alpha) = 0$,
%the zero quadratic refinement.
%We say that two characteristic forms are \emph{stably isomorphic} if they
%become isomorphic after addition of nonsingular characteristic forms.  It is clear from
%the discussion above that the boundaries of stably isomorphic characteristic forms are

A characteristic form $(H,\lambda,\alpha)$ is nonsingular
if $\lambda$ is, \ie if the adjoint $\hlambda : H \to H^*$ is an isomorphism.
Suppose that $(H,\lambda, \alpha)$ is a nonsingular characteristic form
and that $H_0$ is some primitive subgroup of~$H$, $H_1$ the $\lambda$-orthogonal subspace
to $H_0$, and $\lambda_i$, $\alpha_i$ the restrictions of $\lambda$ and
$\alpha$ to $H_i$.  
In this case we say that $(H_0, -\lambda_0, \alpha_0)$ and $(H_1, \lambda_1, 
\alpha_1)$ are {\em orthogonal in $(H, \lambda, \alpha)$}.
For the groups $G_i = \coker(\wh \lambda_i)$ of the boundaries of 
$(H_i, \lambda_i, \alpha_i)$, the restriction maps $H^* \to H_i^*$ 
and the isomorphism $\wh \lambda \colon H \cong H^*$ give rise to homomorphisms
$H \to H^* \to H_i^* \to G_i$
which induce isomorphisms 
%
\[ \Pi_i \colon H/(H_0 \oplus H_1) \cong G_i.\]
% 
Hence we have the canonical isomorphism
%
\begin{equation} \label{eq:canonical_iso}
 F_\lambda : = \Pi_1 \circ \Pi_0^{-1} \colon G_0 \cong G_1. 	
\end{equation}
%  
The following lemma is a routine calculation using \eqref{eq:alg_qdef} and
the fact that $(H, \lambda, \alpha)$ is nonsingular: see \cite[Lemma 3.10]{crowley02}
for the case where $(H_i, \lambda_i, \alpha_i)$ are nondegenerate.

\begin{lem} \label{lem:splitq}
Let $(H_0, -\lambda_0, \alpha_0)$ and $(H_1, \lambda_1, \alpha_1)$ 
be orthogonal characteristic forms in the nonsingular
characteristic form $(H, \lambda, \alpha)$.  
The canonical isomorphism $F_\lambda$ of \eqref{eq:canonical_iso} 
induces an isomorphism of the boundaries of $(H_0, -\lambda_0, \alpha_0)$
and $(H_1, \lambda_1, \alpha_1)$:
\[
F_\lambda^\# q^\circ_1 = -q^\circ_0 .  \hfill\qed
\]
\end{lem}

%\begin{rmk} \label{rmk:gluing}
%Given a pair of characteristic forms $(H_i, \lambda_i, \alpha_i)$
%and an isomorphism $q^\circ_0 \cong - q^\circ_1$, we can glue to form a nonsingular
%characteristic form.  This is a key step in showing that 
%\end{rmk}

As pointed out before, if $W$ is a 3-connected coboundary of a closed spin 7-manifold $M$ then $(FH^4(W, \del W), \lambda_W, p_W)$ is a characteristic form.
The associated boundary in $\catb$ is precisely $(H^4(M), -b_M, p_M)$.

\begin{defn}
\label{def:ql}
The \emph{quadratic linking family} $q^\circ_M$ of $M$ is the family of quadratic
refinements of $(H^4(M), b_M, p_M)$ defined by the characteristic form
$(FH^4(W, \del W), -\lambda_W, p_W)$ of $-W$ for any 3-connected coboundary $W$ or $M$.

That $q^\circ_M$ is independent of the choice of $W$ and natural under homeomorphisms
(in the sense that $(f^*)^\# q_{M_0}^\circ = q_{M_1}^\circ$ for any
homeomorphism $f : M_0 \to M_1$) follows from Remark \ref{rmk:topological_p_M}
and \ref{lem:split_mfd} and Lemmas  \ref{lem:splitq}:
see Remark \ref{rmk:characteristic_gluing}.
\end{defn}

\begin{rmk} \label{rmk:sign_of_q}
The sign in Definition \ref{def:ql} is introduced so that the linking form of $q^\circ_M$ is
the usual linking form $b_M$.  That this is the correct sign is shown in the proof of
\cite[Theorem 2.1]{alexander76}.  The sign differs from \cite[Definition 2.50]{crowley02}
where the wrong sign was used.
\end{rmk}

\begin{rmk} \label{rmk:t_invariant}
If $M$ is a rational homology sphere, then $0 \in S_2$ is a preferred element and 
we have the refinement $q_M := q^0_M$.  It follows from \cite[Defintion 1.4 and Theorem 2.4]{crowley13} 
that $q_M$ can be defined analytically using the eta invariant of the Dirac operator
on $M$, twisted by appropriate quaternionic line bundles.
%We could therefore define $q_M^\circ$ intrinsically
%by using splitting function of $M$ from Theorem \ref{thm:almost_split1}, and working with the resulting rational homotopy spheres 
%where $q_M$ is intrinsically defined.  
As of the time of writing, we do not know
of an intrinsic definition of the linking family $q^\circ_M$ when $M$ is not a rational homotopy sphere.
%which is defined without first using a connect sum splitting of $M$.
\end{rmk}

\begin{rmk} \label{rmk:topological_classification}
The proof in Definition \ref{def:ql} that $q^\circ_M$ is a homeomorphism invariant
relies on Remark \ref{rmk:topological_p_M} and Lemma \ref{lem:p_M}.
It is simpler than the proof given 
in \cite[Theorem 6.1]{crowley02} which used the full apparatus of smoothing theory.

Notice, however, that smoothing theory and Theorem \ref{thm:almost_classification} imply
that every $2$-connected $M$ with $H^2(M; \Z/2) \neq 0$ admits exotic self-homeomorphisms,
by which we meant homeomorphisms which are not isotopic to piecewise linear homeomorphisms.
Self-homotopy equivalences which are homotopic to exotic self-homeomorphisms were defined for rational
homotopy spheres in \cite[\S 2.b]{crowley13}, see \cite[Lemma 2.17]{crowley13}.
\end{rmk}

\begin{rmk}
\label{rmk:projections}
%Given a projection $\tau : G \to T$, the kernel of $\tau$ is isomorphic to the
%free part of $G$, and there are unique choices of
%$h \in \divs_2 \cap \ker \tau$ and $k \in \divsp \in \ker \tau$.
%We can therefore define the family of quadratic refinements as a
%as function on the set of projections instead of $\divsp$:
%$q^\tau := q^h$.

%Given a projection $\tau : G \to T$, the kernel of $\tau$ is isomorphic to the
%free part of $G$, and there is a unique choice of
%$k \in \divsp \in \ker \tau$.
%We can therefore define the family of quadratic refinements as a
%as function on the set of projections instead of $\divsp$:
%$q^\tau := q^{\frac{d_\pi}{2}k}$.

Given a section $\sigma : G/T \to G$ of the projection $\pi \colon G \to G/T$,
the image of $\sigma$ is isomorphic to the free part of $G$, and there is a
unique $k(\sigma) \in \divsp \cap \im(\sigma)$.
We can therefore define the family of quadratic refinements as a
function on the set of sections $\sect(\pi)$ of $\pi$ so that
% instead of on $\divsp$:
$q^\bullet \colon \sect(\pi) \to \cal{Q}(b),~q^{\sigma} := q^{\frac{d_\pi}{2}k(\sigma)}$.
This presentation is relevant for considering connected-sum
splittings of $M$ and is discussed further
in Section \ref{subsec:alsmost_classification}.

We could go in the direction of functions on elements on $G$ and
consider the restriction of a refinement $q^\circ$ to the subset
$\bar \divs_2 := \{ h \in G : p_M = 2h \}$;
for $h \in \bar \divs_2$, $q^h$ is a 
%genuine finite 
homogeneous refinement of $b$.
%quadratic form rather than a quadratic function.
We can define the quadratic linking families without
mentioning quadratic linking functions, provided that we can always find a spin
coboundary $W$ with even intersection form (\ie $p_W$ even).
The values of $h \mapsto q^h$ on all of $\divs_2$ can be
recovered from the values on $\bar \divs_2$ because of the transformation law,
so the restriction to $\bar \divs_2$ contains all the information of the previous
notion of a linking family.
\end{rmk}

\subsection{Gauss refinements}
\label{subsec:gr}

%Let us introduce some terminology to accommodate $g_W$ and $\EK_M$.
We can associate a further boundary invariant to a characteristic form
which we refer to as a Gauss refinement of the family of quadratic refinements.
Let $(G,p,b) \in \catb$, \ie a finitely generated abelian group $G$,
an element $p \in 2G$ and $b : T \times T \to \Q/\Z$ a torsion form.
Let $d_\pi$ denote the greatest integer dividing $p$ (or $d_\pi := 0$ if
$p$ is a torsion element), and let
\begin{equation}
\divsp := \{ k \in G : p_M - d_\pi k \in T \} .
\end{equation}
Given $k \in \divsp$ write $\beta_k := p_M - d_\pi k$
and note that $T$ acts simply transitively on $\divsp$ by addition.
We call a function
$\lf : \divsp \to \Q/\frac{d_\pi}{4} \Z$ \emph{$(b,p)$-linked} if
for all $k \in \divsp$ and $t \in T$
\begin{equation}
\label{eq:lf_def}
\lf(k + t) = \lf(k) + \frac{\Delta(k, t)}{8} ,
\end{equation}
where
\begin{equation} \label{eq:Delta}
\Delta(k, t) := d_\pi^2 b(t, t) - 2d_\pi b(\beta_k, \, t) \in \Q/2d_\pi \Z . 	
\end{equation} 
Further, if $(G,q^\circ\!,p) \in \catq$, \ie if $q$ is a family of quadratic
refinements of $(G,b, p)$, then we call a function
$\gr \colon \divsp \to \Q/\tdf \Z$ a \emph{Gauss refinement} of $q$ if
its mod $\frac{d_\pi}{4} \Z$ reduction is $(b,p)$-linked and in addition
\begin{equation}
\label{eq:gr_def}
\gr(k) = \Arf(q^{\frac{d_\pi}{2}k}) \mod \bbz.
\end{equation}
(Note that if $k \in \divsp$ then $\frac{d_\pi}{2}k \in \divs_2$,
and $q^{\frac{d_\pi}{2}k}$ is a well-defined quadratic refinement of $b$.)
%With this terminology, Lemmas \ref{lem:gs} and \ref{lem:delta} say precisely
%that $\gr_W$ is a Gauss refinement of $q_M$. Similarly we say that $\EK_M$
%is a mod 28 Gauss refinement of $q_M$.

Now suppose that $(H, \lambda, \alpha)$ is a characteristic form.
Given $k \in \divsp$, pick $n \in H^*$ such that $jn = k$,
and set $\alpha_n = p_W - d_\pi n$.
Note that $j\alpha_n = \beta_k$, and that $\alpha_n \in R \cong (H/K)^*$ is a
characteristic element for the intersection form on $H/K$. Let
\begin{equation} \label{eq:grchat}
 \gr_H(k) := \frac{\lambda^{-1}(\alpha_n, \alpha_n) - \sign(W)}{8}
\in \Q/\tdf \Z . 
\end{equation}

\begin{lem}
\label{lem:gr_well}
$\gr_H$ is well-defined, independent of the choices of $n$.
\end{lem}

\begin{proof}
Replacing $n$ by $n' := n + \hlambda r$,
so $\alpha_{n'} = \alpha_n  -d_\pi \hlambda r$,
changes the value of $\gr_H(k)$ by
\[ \frac{-2d_\pi \lambda^{-1}(\alpha, \hlambda r) +
d_\pi^2 \lambda^{-1}(\hlambda r, \hlambda r)}{8} = \frac{d_\pi}{4}
\left(-\gen{r, \alpha_n} + \frac{d_\pi}{2}\lambda(r, r)\right) . \]
The last factor is an integer, and it is even when $d_\pi$ is not divisible
by 4 (\ie when $\td_\pi = 2d_\pi$) because $\alpha_n$ is characteristic.
\end{proof}

\begin{lem}
\label{lem:gs}
$\gr_H(k) = -\Arf(q^{\frac{d_\pi}{2}k}) \mod \Z$.
\end{lem}

\begin{proof}
The $\alpha_n$ used in the definition of $\gr_H(k)$ co-incides with the $\alpha_m$
used in the definition of $g^{\frac{d_\pi}{2}k}$ in \eqref{eq:alg_qdef}.
Since $\alpha_n$ is characteristic for $\lambda$, the lemma is immediate from
Proposition \ref{prop:arf}.
\end{proof}

Finally we check the transformation law.

\begin{lem}
\label{lem:delta}
\[
\gr_H(k+t) - \gr_H(k) \; = \; \frac{\Delta(k, t)}{8}
\mod {\textstyle \frac{d_\pi}{4}} .
\]
\end{lem}

\begin{proof}
$\alpha_{n'} - \alpha_n = -d_\pi(n'-n)$, and $j(n'-n) = t$. Hence
\begin{equation*}
\lambda^{-1}(\alpha_{n'}, \alpha+{n'}) - \lambda^{-1}(\alpha_n, \alpha_n) =
-2d_\pi \lambda^{-1}(\alpha_n, n' {-} n) + d_\pi^2 \lambda^{-1}(n' {-} n, n'{-} n)
= \Delta(k, t) \mod 2d_\pi . \qedhere
\end{equation*}
\end{proof}

Lemma \ref{lem:gs} and \ref{lem:delta} say precisely that $\gr_H$
is a Gauss refinement of $q$ in the sense defined above.

\begin{rmk}
We could make an analogy with factors of automorphy of automorphic forms and
think of $\Delta$ as a ``term of automorphy''. For any linked functions to
exist is clearly equivalent to the cocycle condition
\begin{equation}
\label{eq:cocycle}
\Delta(k, s+t) = \Delta(k+s, t) + \Delta(k, s) ,
\end{equation}
which can be checked directly from the definition in \eqref{eq:lf_def}. 
The difference of two
functions with the same term of automorphy is invariant under the $T$ action;
since $T$ acts transitively on $\divsp$ that simply means that the difference
between two $(p,b)$-linked functions is a constant in $\Q/2d_\pi\Z$.
\end{rmk}

We define $\tdelta(k, t) \in \Q/2\td_\pi \Z$ by
\begin{subequations}
\begin{align}
\label{eq:delta_pi}
\tdelta(k, t) &= \Delta(k, t) \mod 2d_\pi , \\
\label{eq:delta_8}
\tdelta(k, t) &= 8 q^{\frac{d_\pi}{2}k} (\textstyle\frac{d_\pi}{2}t) \mod 8 .
\end{align}
\end{subequations}
If $d_\pi$ is not divisible by 4 then in order for $\tdelta$ to be well-defined
we need to check that the mod 4 reductions agree, which follows from
\[ 2q^{\frac{d_\pi}{2}k}(x) = b(x, x) + b(\beta_k, \, x) . \]
Similarly, for $4 \mid d_\pi$ we deduce that \eqref{eq:delta_pi} implies
\eqref{eq:delta_8} from
\[ q^{\frac{d_\pi}{2}k}(2x) = 2b(x, x) + b(\beta_k, \, x) . \]
Now any Gauss refinement $\gr$ of $q^\circ$ satisfies the transformation rule
\begin{equation} \label{eq:gr_rule}
\gr(k+t) - \gr(k) \; = \; \frac{\tdelta(k, t)}{8} ,
\end{equation}
since $\Arf(q^{\frac{d_\pi}{2}(k+t)}) - \Arf(q^{\frac{d_\pi}{2}k})
= q^{\frac{d_\pi}{2}k}(\frac{d_\pi}{2}t) \in \Q/\Z$ by \eqref{eq:arf_shift}.

The difference between two Gauss refinements of the same linking family is
simply a constant in $\Z/\tdf \Z$.
The difference between Gauss refinements of different linking families need not
be constant mod $\Z$ if $d_\pi$ is not divisible by 4 (since then $\tdelta$ can
depend on $q$), and in any case need not take integer values.

An isomorphism $F : G \to G'$ with $F(p) = p'$ maps $\divsp \to \divsp'$.
If $F^\# b' = b$ then we have $\Delta'(F(k), F(t)) = \Delta(k, t)$ for all
$k \in \divsp$ and $t \in T$, so if $\lf : \divsp' \to \Q/\frac{d_\pi}{4}\Z$
is a linked function then so is
\begin{equation}
\label{eq:pull_gauss}
F^\# \lf := \lf \circ F .
\end{equation}
Similarly, if $\gr$ is a Gauss refinement of a linking family $q^\circ$ on $G'$, then
$F^\#\gr$ is a Gauss refinement of the linking family $F^\# q^\circ$ on $G$.

We next consider the Gauss refinements of orthogonal characteristic forms.
We recall that if $(H, \lambda, \alpha)$ is nonsingular and $H_0 \subset H$
is primitive with orthogonal complement $H_1$, then for the characteristic forms
$(H_i, \lambda_i, \alpha_i)$ defined by restriction from $(\lambda, \alpha)$,
there is a canonical isomorphism 
$F_\lambda \colon \del(H_0, -\lambda_0, \alpha_0) \cong \del(H_1, \lambda_1, \alpha_1)$.  

\begin{lem}
\label{lem:alg_compare}
%Under the hypotheses of Lemma \ref{lem:splitq}
Let $(H_0, -\lambda_0, \alpha_0)$ and $(H_1, \lambda_1, \alpha_1)$ 
be orthogonal characteristic forms in the nonsingular
characteristic form $(H, \lambda, \alpha)$.  
The canonical isomorphism $F_\lambda$ of \eqref{eq:canonical_iso} 
induces an isomorphism

\[  F_\lambda^\#\gr_{H_1} - \gr_{H_0} \; = \;
\frac{\lambda(\alpha,\alpha) - \sign(\lambda)}{8} \mod \tdf. \]
\end{lem}

\begin{proof}
Note that since $(H, \lambda)$ is nonsingular, $(\wh \lambda)^{-1} \colon H^* \cong H$
is an isomorphism from $\lambda^{-1}$ to $\lambda$.
Also, we have homomorphisms $H^* \to H_i^* \to G_i$ where we recall that
$G_i = \coker(\wh \lambda_i)$.
Pick a $k \in \divsp(G_0)$, and then pick $n \in H^*$ whose image in
$G_0$ equals $k$ (the set-up means that the image of $n$ in $G_1$ is $F_\lambda(k)$).
Let $n_i$ be the image of $n$ in $H_i^*$, and set
$\alpha_{n_i} := \alpha_i - d_\pi n_i \in R_i$
%, $\alpha' := p_{W'} - d_\pi n' \in R'$
as in the definition of~$g_{H_i}$. We need to show that
\begin{equation}
\lambda^{-1}(\alpha,\alpha) =
\lambda^{-1}_1(\alpha_{n_1}, \alpha_{n_1})
- \lambda^{-1}_0(\alpha_{n_0}, \alpha_{n_0})
\mod 2\td_\pi .
\end{equation}
%(Bit of confusion here about whether to implicitly identify $H \cong H^*$).

The image of $\alpha - d_\pi n$ in $H^* \otimes \Q$ can be written as a sum
$\wh \lambda_0(\gamma_0) + \wh \lambda_1(\gamma_1)$
where $\gamma_i \in H_i \otimes \Q$ and $\hlambda_i(\gamma_i) = \alpha_{n_i}$.
Thus
%\[ \alpha^2
%= \gamma_1^2 + \gamma_0^2 + 2d_\pi n (\alpha - d_\pi n) + d_\pi^2 n^2
%= \lambda^{-1}_1(\alpha_1, \alpha_1) - \lambda^{-1}_0(\alpha_0, \alpha_0)
\begin{multline*} \lambda^{-1}(\alpha, \alpha)
= \lambda_1(\gamma_1, \gamma_1) - \lambda_0(\gamma_0, \gamma_0) + 2d_\pi \lambda^{-1}(n, \alpha - d_\pi n) 
+ d_\pi^2 \lambda^{-1}(n, n) \\
= \lambda^{-1}_1(\alpha_{n_1}, \alpha_{n_1})
- \lambda^{-1}_0(\alpha_{n_0}, \alpha_{n_0})
\mod 2\td_\pi ; 
\end{multline*}
that equality holds mod $4d_\pi$ when $d_\pi$ is not divisible by 4 follows
from $\alpha - d_\pi n$ being a characteristic element for $\lambda$.
%(Alternatively, use that to argue that RHS of \eqref{eq:compare}
%is an integer, as the LHS is a priori.)
\end{proof}

\begin{rmk} \label{rmk:neutral_forms}
We call a characteristic form $(H,\lambda,\alpha)$ \emph{neutral} if it is
nonsingular and $\lambda(\alpha,\alpha) = \sign(\lambda)$. Say that two
characteristic forms are \emph{neutrally isomorphic} if they become isomorphic
after addition of neutral forms (so this is a sharper condition than stable
isomorphism).
Lemma \ref{lem:alg_compare} implies that Gauss refinements are invariant
under neutral isomorphism. 
The gluing and splitting arguments for characteristic forms reviewed in Section 
\ref{subsec:alsmost_classification}, in particular Theorem \ref{thm:stable_iso_and_qlfs},
can be used to show that characteristic forms are classified to neutral isomorphism.
by their boundary distillations $(G, q^\circ\!, \gr, p)$ classify
%In fact, the distillations $(G,q,p,g)$ classify characteristic forms up to neutral isomorphism.
\end{rmk}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{The generalised Eells--Kuiper invariant} \label{subsec:eek}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $M$ be a spin 7-manifold and $W$ a 3-connected coboundary of $M$.
Let $\gr_W$ be the Gauss refinement of $(H^4(M), q_M^\circ, p_M)$ defined by
the characteristic form $(H^4(W, \del W), \lambda_W, p_W)$: notice that
we {\em did not use $-W$} as we did in defining the linking family of $M$.
If 
%\begin{lem} \label{lem:compare} Let
$f : M_0 \to M_1$ is a spin homeomorphism 
%then since the $W_i$ are 3-connected coboundaries for $M_i$ 
the $X := (-W_0) \cup_f W_1$ a closed
is a closed topological spin 8-manifold and Lemmas \ref{lem:split_mfd} and \ref{lem:alg_compare} 
imply
\begin{equation}
\label{eq:compare}
 \gr_{W_1} - (f^*)^\# \gr_{W_0}  = \frac{p_X^2 - \sign(X)}{8} \mod \tdf \Z .
\end{equation}
%\end{lem}
If $f$ is a diffeomorphism then $X$ is smooth, the RHS of \eqref{eq:compare}
equals $28\ahat(X)$, and $\ahat(X)$ is an integer. Letting
\begin{equation}
%\dM := \gcd \left( \textstyle \tilde d_\pi/4, 28 \right)
\dM := \gcd \left( \textstyle \frac{\wt d_\pi}{4}, 28 \right)
%\dM := \gcd \left( \textstyle \frac{\tilde d_\pi}{4}, 28 \right)
\end{equation}
as in the introduction, it follows that
\begin{equation}
\label{eq:EK_of_M}
\begin{aligned}
\EK_M : \divsp \to \Q/\dM \Z, \\
\EK_M := \gr_W \mmod \dM
\end{aligned}
\end{equation}
is a independent of the choice of $W$, and natural under diffeomorphisms:
%\begin{thm}
%\label{thm:eek_defined}
If $f : M_0 \to M_1$ is a spin diffeomorphism then
$(f^*)^\# \EK_{M_0} = \EK_{M_1}$.
%\end{thm}

From this definition of $\EK_M$ via the characteristic form of a
3-connected coboundary we recover the description in the introduction.
%That the latter description is valid also when $W$ is not 3-connected
%can be seen as a special case of Proposition \ref{prop:ek_via_spc} below.
This description is valid also when $W$ is not 3-connected, although $\gr_W$ might
not be defined on all of $S_{d_\pi}$ in this case.
This point can be seen as a special case of Proposition \ref{prop:ek_via_spc} below.

\begin{rmk} \label{rmk:splitting_definition_of_eek}
Analogously to Remark \ref{rmk:projections},
we can define Gauss refinements (and $\EK_M$) as functions of sections $\sigma \colon G/T \to G$
rather than on $\divsp$,
$\gr_W(\sigma) := \gr_W(k(\sigma))$.
Then $\gr_W(\sigma) = \Arf(q^\sigma) \mmod \bbz$, and the transformation
rule \eqref{eq:lf_def} can also be rewritten in these terms.
%On the other hand, it seems hard to do anything similar with $\EK$, since
%$\bar \divs_{d_\pi}$ may well be empty. Further, if we don't know what $q^h$ is
%for all $h \in \divs_2$ then we can't make sense of Lemma \ref{lem:gs}.
\end{rmk}


\begin{rmk} \label{rmk:analytic_eek}
Recall Remark \eqref{rmk:original_ek} saying that if $p_M$ is torsion then
$\divsp = T$ contains the distinguished element $0$ and 
$\frac{1}{28}\EK_M(0) \in \Q/\Z$ recovers the original Eells-Kuiper invariant $\mu(M)$.

Although defined extrinsically using spin co-boundaries, the original Eells-Kuiper invariant $\mu(M)$ 
was shown by Donnelly \cite[Theorem 4.2]{donnelly75} to have an intrinsic definition in terms of 
the eta invariant of the Dirac operator of $M$.  
%The generalised Eells-Kuiper invariant
%can be defined intrinsically using the smooth splitting function of
%$M$ from Theorem \ref{thm:split}, and working with the resulting rational homotopy spheres 
%where $\mu(M)$ is intrinsically defined.  
As of the time of writing, we do not know
of an intrinsic definition of the generalised Eells-Kuiper invariant when $p_M \neq 0 \in H^4(M; \Q)$.
%which is defined without first using a connect sum splitting of $M$.

For further information about the role of eta invariants in the classification of $7$-manifolds,
we refer the reader to \cite[\S 4]{goette12}. 
\end{rmk}

\begin{rmk} \label{rmk:neutral_forms_mod_224}
In \cite[\S 4.4]{crowley02}, a pair of characteristic forms
$(H, \lambda, \alpha)$ are called \emph{smoothly equivalent}
if they become isomorphic after addition of non-singular forms with  
$\lambda(\alpha,\alpha) \equiv \sign(\lambda)$~mod~$224$
(so this is a weakening of the notion of neutral equivalence from
Remark \ref{rmk:neutral_forms}).
In algebraic terms, the definition of the generalised Eells-Kuiper invariant
can be used to show that the mod $28$ Gauss refinement of $M$, $(H^4(M), q^\circ_M, \EK_M, p_M)$,
is a complete invariant of the smooth equivalence class of the characteristic
form $(H^4(W, \del W), \lambda_W, p_W)$ of a $3$-connected coboundary for $M$.
Hence Theorem \ref{thm:classification} is a development  
\cite[Theorem 4.9]{crowley02} in dimension $7$ where $2$-connected
$7$-manifolds are classified up to diffeomorphism by the smooth equivalence
class of the characteristic form of a $3$-connected coboundary.
\end{rmk}


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


% (fold)
\subsection{The computation of $\EK_M$ via \spc coboundaries} \label{subsec:spin_c}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Inspired by the $s_1$ invariant of Kreck-Stolz \cite{kreck91}, we derive
an expression for $\EK_M$ in terms of coboundaries that are not
necessarily spin (never mind 3-connected) but just \spc.

A principal \spc bundle is equivalent to a real vector bundle $E$ together
with a complex line bundle $L$ such that $c_1(L) = w_2(E) \mmod 2$.
We can associate to this the characteristic classes
\begin{align*}
z & := c_1(L) , \\
\pd & := p(E \oplus L) , \\
\pc & := \pd + z^2 .
\end{align*}
So $2\pd = p_1(E \oplus L) = p_1(E) - z^2$, and $2\pc = p_1(E) + z^2$.
%
Recall that any $U$-bundle has a natural \spc structure: if $E$ is a complex
vector bundle we may take $L := \det E$.

\begin{lem}\hfill \label{lem:pz}
\begin{enumerate}
\item \label{it:pz_basis}
$\pd$ and $z^2$ form a basis for $H^4(B\spcg)$.
\item \label{it:cx}
$\pd(E) = -c_2(E)$ for any complex bundle $E$.
%\[ \pc(E) = -c_2(E) + c_1(E)^2 . \]
\item \label{it:w}
$\pd(E) = w_4(E) \mmod 2 $ for any \spc bundle $E$.
%\[ \pc(E) = w_4(E) + w_2(E)^2 \mmod 2 . \]
\end{enumerate}
\end{lem}

\begin{proof}
Observe that $\spcg/U \cong Spin/SU$ 5-connected implies
$\pi^* : H^4(B\spcg) \to H^4(BU)$ and
$\pi^* : H^2(B\spcg) \to H^2(BU)$ are isomorphisms (with both $\bbz$ and
$\bbz_2$ coefficients). The image of $z$ is patently $c_1$.

We know that $H^4(BU)$ has basis $c_2$, $c_1^2$. Because there is
no 2-torsion,
$2\pi^* \pd  = p_1 - (\pi^*z)^2 = (-2c_2 + c_1^2) - c_1^2$
implies $\pi^* \pd = -c_2$, proving \ref{it:pz_basis} and \ref{it:cx}.

The isomorphism on $H^4(-;\bbz_2)$ implies that it suffices to check that
\ref{it:w} holds when $E$ is complex. But that follows from \ref{it:cx}.
\end{proof}

\begin{cor}
If $X$ is a compact \spc 8-manifold then
$\pc_X$ is characteristic for the intersection form $\lambda_X$ of $X$.
\end{cor}

\begin{proof}
Lemma \ref{lem:pz} gives
\[ \pc = w_4 + w_2^2 \mod 2 . \]
If $X$ is closed, then Wu's formula implies that for any closed orientable
manifold $X$ the fourth Wu class is $v_4(X) = w_4(X) + w_2(X)^2$.
The compact case follows from the closed case, as in the proof of
Lemma \ref{lem:p_M}\ref{it:boundary}.
\end{proof}

\begin{lem}
\label{lem:ind_L}
If $X$ is a closed \spc 8-manifold then the Dirac operator
of the fundamental complex spinor bundle has
\begin{equation}
\label{eq:ind_L}
28 \ind \dirac^+
\; = \; \frac{\pc_X^2 - \sign(X)}{8} - \frac{5z^2\pc_X}{12} + \frac{z^4}{4} .
\end{equation}
\end{lem}

\begin{proof}
\cite[Theorem D.15]{lawson89} expresses $\ind \dirac^+$ as
the integral of $\exp\left(\frac{z}{2}\right) \ahat(X)$, whose degree 8 part expands to
\[ \frac{-4p_2 + 7p_1^2}{2^7.45} - \frac{z^2 p_1}{24.8} + \frac{z^4}{24.16}
%\; = \; \frac{(\frac{p_1}{2})^2 - L}{2^5.7} - \frac{z^2}{2^6.3} +
\; = \; \frac{p_1^2}{2^7.7}  - \frac{L}{2^5.7} - \frac{z^2 p_1}{2^6.3} +
\frac{z^4}{2^7.3} . \]
Then substitute $p_1 = 2\pc - z^2$ to obtain \eqref{eq:ind_L}.
\end{proof}

%\begin{rmk}
%In particular, the RHS of \eqref{eq:ind_L} is an integer for a smooth \spc
%manifold $X$, and $3z^4 - 5z^2\pc$ is divisible by 12. This property is
%unchanged under cutting out an 8-disc and regluing by an exotic diffeomorphism
%of $S^7$, so also holds for almost-smooth \spc manifolds.
%Indeed, modulo~3 this is a special case of Wu's result relating $p_1 \mmod 3$
%of a closed oriented $n$-manifold $X$ to the Steenrod power operation
%$P^1 : H^{n-4}(X;\bbz_3) \to H^n(X;\bbz_3)$.
%\end{rmk}

Now suppose $M$ is a spin 7-manifold and $W$ a \spc coboundary,
such that the restriction of $z \in H^2(W)$ to $M$ is trivial. Then $z$ has a
pre-image $\bar z \in H^2_{}(W, M)$, and $\bar z^2 \in H^4(W, M)$ is
independent of the choice of $\bar z$.

\begin{defn}
\label{def:grc}
Given $k \in \divsp$, suppose there is $n \in H^*$ such that
$jn = k$. Let $\wh\alpha_n := \pc_W - d_\pi n$, and
\[ \gr^c_W(k) :=
\frac{\lambda^{-1}(\wh\alpha_n, \wh\alpha_n) - \sign(W)}{8}
- \frac{5\bar z^2 \pc_W}{12} + \frac{z^4}{4} \in \Q/\tdf \Z . \]
\end{defn}

If $z = 0$ then of course $\gr^c_W = \gr_W$. The proof that $\gr^c_W(k)$ does
not depend on the choice of $n$ is analogous to Lemma \ref{lem:gr_well},
using that $\wh\alpha$ is characteristic for intersection form $\lambda_W$.

\begin{prop}
\label{prop:ek_via_spc}
Let $(W_1,z_1)$ be a \spc coboundary of $M$ and $j_1 \colon H^4(W_1) \to H^4(M)$
the natural homomorphism.
For all $k, k' \in S_{d_\pi} \cap j_1(H^4(W_1))$ we have:
\begin{enumerate}
\item \label{it:grc}
%$\gr^c_{W_1}$ is defined and
$\gr^c_{W_1}(k) = \EK_M(k)\mmod 28$ for all $k\in S_{d_\pi} \cap j_1(H^4(W_1))$;
%reduction equal to $\EK_M$.
\item \label{it:grc_transform}
The defined values of $\gr^c_{W_1}$ satisfy the transformation rule
\eqref{eq:gr_rule}, \ie if $k, k' \in S_{d_\pi} \cap j_1(H^4(W_1))$ then
$\gr^c_{W_1}(k') = \gr^c_{W_1}(k) + \frac{\tdelta(k'{-}k,t)}{8}$.
%If $t \in H^4(M)$ is such that $k + t \in S_{d_\pi} \cap j_1(H^4(W_1))$,
%then $(k, t)$ satisfies
\end{enumerate}
\end{prop}

\begin{proof}
For part \ref{it:grc}, take $W_0$ to be a 3-connected coboundary for $M$, and let
$X := (-W_0) \cup_{\Id_M} W_1$. Then $X$ is a smooth \spc manifold, possibly
with more than one choice of $z \in H^2(X)$ restricting to $z_1$ on $W_1$ and
$0$ on $W_0$.
 While we do not trouble ourselves with
separating the algebra from the topology in this case, we essentially
adapt the proof of Lemma \ref{lem:alg_compare} to show
\begin{equation}
\label{eq:compare^c}
\gr^c_{W_1} - \gr_{W_0} =
\frac{\pc_X^2 - \sign(X)}{8} - \frac{5z^2\pc_X}{12} + \frac{z^4}{4}
\mod \tdf \Z .
\end{equation}
Since the RHS equals $28 \ind \dirac^+$ by
Lemma \ref{lem:ind_L}, while
$\gr_W = \EK_M \mmod \gcd(28,\tdf)$ by definition, the result then follows.

Pick some $n_1 \in H^4(W_1)$ such that $j_1 n_1 \in \divsp$ as in Definition
\ref{def:grc}. Because $W_0$ is 3-connected there is some $n \in H^4(X)$ whose
restriction to $W_1$ equals $n_1$.
Then $\pc_X - d_\pi n$ is a sum
of push-forwards of $\gamma_i \in H^4_{}(W_i, M;\Q)$, and
$\hlambda_{W_i} \gamma_i = \wh\alpha_i$. Meanwhile, note that regardless of
the choice of $z$, $z^2 \in H^4(X)$ is the push-forward of
$\bar z_1^2 \in H^4_{}(W_1, M)$. Hence
\begin{align*}
 \pc_X^2 - \frac{10 z^2 \pc_X}{3} + 2z^4 
\; & = \; \gamma_1^2 + \gamma_0^2 + 2d_\pi n (\pc_X - d_\pi n) + d_\pi^2 n^2
- \frac{10}{3}\bar z_1^2 \pc_{W_1} +2z_1^4 \\
& = \; \lambda_{W_1}^{-1}(\wh\alpha_1, \wh\alpha_1)
- \frac{10}{3}\bar z_1^2 \pc_{W_1} + 2z_1^4
- \lambda_{W_0}^{-1}(\wh\alpha_0, \wh\alpha_0) 
\mod 2\td_\pi .
\end{align*}
The fact that the equality holds $\mmod 4d_\pi$ when $d_\pi$ is not divisible
by 4 is due to $\pc_X - d_\pi n$ being a characteristic element for the
intersection form on $X$.

Part \ref{it:grc_transform} follows from \eqref{eq:compare^c} since $\gr_{W_0}$
satisfies \eqref{eq:gr_rule} and the RHS of \eqref{eq:compare^c} is constant.
\end{proof}

The consequence of \ref{it:grc_transform} is that we can extend
$\gr^c_W$ to a well-defined Gauss refinement as long as
$S_{d_\pi} \cap j(H^4(W))$ is non-empty.

% subsection spin_c (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% section invariants (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% (fold)
\section{The classification of $2$-connected $7$-manifolds} \label{sec:classification}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we classify closed smooth spin $2$-connected
$7$-manifolds $M$ up to spin diffeomorphism.
Recall that a homotopy $7$-sphere $\Sigma$ is a spin manifold which is homotopy equivalent to $S^7$.
In Section \ref{subsec:almost_diffeomorphisms} we recall that by definition an 
almost diffeomorphism $f \colon M \acong N$ defines a diffeomorphism $f \colon M \cong N \sharp \Sigma$,
for some homotopy sphere $\Sigma$.
In Section \ref{subsec:alsmost_classification} we relate the algebra of Section \ref{sec:invariants} 
to the algebra used in \cite{crowley02} and so give the almost diffeomorphism classification of 
$2$-connected $M$ in terms of their linking families $(H^4(M), q^\circ_M, p_M)$.

With the almost diffeomorphism classification in hand, we are lead to consider the inertia group
of $M$.  This is the group of homotopy spheres $\Sigma$ such that $M \sharp \Sigma \cong M$.
In Section \ref{subsec:inertia_and_reactivity2} we establish basic
facts about the inertia group of $M$, the reactivity of $M$ and the relationship between
reactivity and inertia.  We also construct an important family of almost diffeomorphisms $f \colon M \acong M$
in Proposition \ref{prop:p2null}.  
The almost diffeomorphisms of Proposition \ref{prop:p2null} allow us to show that 
the generalised Eells-Kuiper invariant of $M$, $\mu_M$, precisely measures the gap between
the almost diffeomorphism classification and the diffeomorphism classification.  
In Section \ref{subsec:classification_theorem} we prove that
the mod $28$ Gauss refinement of $M$, $(H^4(M), q^\circ_M, \EK_M, p_M)$, is a complete invariant diffeomorphisms.


% (fold)
\subsection{Almost diffeomorphisms} 
\label{subsec:almost_diffeomorphisms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this subsection we briefly review the almost diffeomorphism category in dimension $7$.
We shall return to consider almost diffeomorphisms in more detail in
Section \ref{subsec:inertia_and_reactivity2}.
An almost diffeomorphism $f \colon M_0 \acong M_1$
%
%\[ f \colon M_0 \acong M_1 \]
%
is a spin homeomorphism which is smooth except perhaps at a finite number of singular points 
$\{m_0, \dots, m_a\} \subset M_0$.  Notice that {\em we do not require $f$ to be non-smooth}
at $m_i$, but we rather {\em allow} it.
In this case $M_0$ and $M_1$ are called almost
diffeomorphic.  
If $M = M_0 = M_1$, then we can compose almost diffeomorphisms
and we define
%
\[ \ADiff_{}(M) := \{ f \, | \, f \colon M \acong M \}, \]
%
to be the group of almost diffeomorphisms of $M$.

Let $f_i$, $i = 0, 1$, be almost diffeomorphisms with singular
points $\{m_{1}, \dots, m_{a_0}\}$ and $\{n_{1}, \dots, n_{a_1}\}$.
A pseudo-isotopy between almost diffeomorphisms is a spin homeomorphism
%
\[ F \colon M_0 \times I \acong M_1 \times I \]
%
such that $F|_{{M_0} \times \{i\}} = f_i$ which is smooth except perhaps along 
$X \subset M_0 \times I$, where $X$ is a smoothly embedded
$1$-manifold with boundary $\del X = \{n_1, \dots, n_{a_0}\} \sqcup \{m_1, \dots, m_{a_1}\}$
and precisely $(a_0 + a_1)/2$ connected components.  In particular, $X$ has no closed components
in the interior of $M_0 \times I$ and $a_0$ and $a_1$ have the same parity.
%Let $f_i$, $i = 0, 1$, be almost diffeomorphisms.
%points $\{m_{1}, \dots, m_{a_0}\}$ and $\{n_{1}, \dots, n_{a_1}\}$.
%A pseudo-isotopy between almost diffeomorphisms is a spin homeomorphism
%
%\[ F \colon M_0 \times I \acong M_1 \times I \]
%
%such that $F|_{{M_0} \times \{i\}} = f_i$ and which can be written as a
%product 
%
%\[ F = \Pi_{i=1}^{2j}F_i \]
%
%where each $F_{2k} \colon M_0 \times I \cong M_1 \times I$ is a diffeomorphism
%and each $F_{2k+1}$ is a product homeomorphism $F_{2k+1} = g_{2k+1} \times {\rm Id}_I$ 
%with $g_{2k+1} \colon M_0 \acong M_1$ an almost diffeomorphism.
Pseudo-isotopy defines an equivalence relation on almost diffeomorphisms which is
compatible with composition and hence if $M = M_0 = M_1$ we define
%
\[ \wt\pi_0 \ADiff_{}(M) := \{ f \, | \, f \colon M \acong M \}, \]
%
to be the group of pseudo-isotopy classes of almost diffeomorphisms of $M$.

Later we shall wish to take the connected sum of two almost 
diffeomorphisms if $f \colon M_0 \acong M_1$ and $g \colon N_0 \acong N_1$.
By the disc theorem \cite[Theorem 5.5]{Palais59}, we can assume after isotopy
that there is are embeddings $\iota_M \colon D^7 \to M_0$ 
and $\iota_N \colon D^7 \to N_0$ with images disjoint from the singular points of $f$ and $g$ 
and with 
$f|_{\iota_M(D^7)} = g|_{\iota_N(D^7)}$.  Taking the connected sum of 
$M_0$ and $N_0$ along $\iota_M$ and $\iota_N$ we can then extend $f$ and $g$ to
form an almost diffeomorphism
%
\[ f \sharp g \colon M_0 \sharp N_0 \acong M_1 \sharp N_1. \]
%

% subsection almost diffeomorphisms (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{The almost diffeomorphism classification}
% of $2$-connected $7$-manifolds} 
\label{subsec:alsmost_classification}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this subsection we show how Theorem \ref{thm:almost_classification}
follows from the classification results of \cite{crowley02}.
The almost diffeomorphism classification given in \cite{crowley02} used a different but closely
related definition of a quadratic linking family.  We begin by explaining
the relationship between the two definitions of linking family and showing that
Theorem \ref{thm:almost_classification} is equivalent to \cite[Theorem B]{crowley02}.  
We then describe the main ideas of the proof of \cite[Theorem B]{crowley02} and interpret linking families
in terms of connected sum splittings.
Throughout this subsection $M$ is $2$-connected, $G = H^4(M)$ with torsion subgroup $T \subset G$
and free quotient $F = G/T$.

Let us start with some elementary algebra for the group $G$.
Let $\iota \colon T \to G$ be the inclusion of the torsion subgroup and 
let $\pi \colon G \to F$ be the canonical projection.
We let $\sect(\pi) := \{ \sigma \colon F \to G\}$ be the set of sections $\pi \colon G \to F$ and 
we let $\proj(\iota) := \{ \tau \colon G \to T \}$ be the set of projections over $\iota$:
$\tau \circ \iota = {\rm Id}_T$.
The sets $\sect(\pi)$ and $\proj(\iota)$ are in bijection by mapping $\sigma \mapsto \tau_\sigma$
where $\im(\sigma) = \ker(\tau_\sigma)$.  Both sets admit simple transitive actions of
$\hom(F, T)$ via addition of functions: 
For $\phi \in \hom(F, T)$ $f \in F$ and $g \in G$ we have
%
\[ (\sigma + \phi)(f) = \sigma(f) + \phi(f) \quad \text{and} \quad
(\tau + \phi)(g) = \tau(g) + \phi(\pi(g)).   \]
%
Notice that $\tau_{\sigma + \phi} = \tau_\sigma - \phi$.

\begin{rmk}
The action of $\hom(F, T)$ on $\sect(\pi)$ used above differs by a sign from the corresponding
action in \cite[p.\,39]{crowley02}.
\end{rmk}

Let $(G, b, p)$ be a base so that $b$ is a torsion form on $T$ and $p \in 2G$.
Recall that $\cal{Q}(b)$ is the set of refinements of $b$ and given $q \in \cal{Q}(b)$,
let us write $\beta(q)$ for the homogeneity defect of $q$: see Section \ref{subsec:tfs_and_Gauss_sums}.
%In the first author's thesis 
In \cite[Definition 2.39]{crowley02} a quadratic linking
family on a base $(G, b, p)$ was defined as a function
%
\[ q^\bullet \colon \sect(\pi) \to \cal{Q}(b)\]
%
such that for all $\sigma \in \sect(\pi)$ and for all $\phi \in \hom(G, T)$,
%
\[  q^{\sigma + \phi} = q^\sigma_{-\phi(\pi(p))} \quad \text{and} \quad
\beta(q^\sigma) = \tau_\sigma(p). \]
%
We explain the topological significance of these conditions below,
focussing for now on the algebra.
%Equivalently, we have a function
%
%\[ q_\proj \colon \proj(\iota) \to \cal{Q}(b) \]
%
%such that $\beta_{q_\proj(\tau)} = \tau(p)$ and which satisfies
%$q_\proj(\tau + \phi) = q_\proj(\tau)_{\phi(\pi(p))}$.  

In this paper we work with refinements which are functions on $S_2$. 
To relate $q^\bullet$ to the linking family $q^\circ \colon S_2 \to \cal{Q}(b)$ 
defined in \ref{def:foqr} we start from the set $S_{d_\pi} \subseteq G$.
Multiplication by $\frac{d_\pi}{2}$ gives a map $S_{d_\pi} \to S_2$ and we define
$\wh S_2 = \frac{d_\pi}{2}S_{d_\pi} \subset S_2$.  There is a function
%
\[ \sect(\pi) \to S_{d_\pi}, \quad \sigma \mapsto k(\sigma) \in {\rm Im}(\sigma) \cap S_{d_\pi} \]
%
and given a refinement $q^\circ \colon S_2 \to \cal{Q}(b)$ we define 
%
\begin{equation} \label{eq:q_bullet_and_q_sect}
q^\bullet \colon \sect(\pi) \to \cal{Q}(b), \quad q^\sigma := q^{\frac{d_\pi}{2}k(\sigma)} .
\end{equation}
%
Conversely, given $q^\bullet \colon \sect(\pi) \to \cal{Q}(b)$ we define
%
\begin{equation} \label{eq:q_sect_and_q_bullet}
q^\circ \colon \wh S_2 \to \cal{Q}(b), \quad q^{\frac{d_\pi}{2}k(\sigma)} := q^\sigma,
\end{equation}
%
and extend to all of $S_2$ by the transformation rule of Definition \ref{def:foqr} \ref{it:q_trans}.
Indeed the transformation rule ensures
that a refinement $q^\circ$ is determined by its value, $q^h$, for a single value of $h \in S_2$
and hence we have

\begin{lem} \label{lem:2_types_of_qlf}
The mappings $q^\circ \mapsto q^\bullet$ and $q^\bullet \mapsto q^\circ$ of 
\eqref{eq:q_bullet_and_q_sect} and \eqref{eq:q_sect_and_q_bullet} define inverse equivalences 
of categories between linking families defined on $S_2$ and linking families defined on $\sect(\pi)$. \qed
\end{lem}

\begin{proof}[Proof of Theorem \ref{thm:almost_classification}]
Let $M$ be $2$-connected, let $q_M^\circ$ be the linking family of $M$ as defined in Definition \ref{def:ql}
and let $-q^\bullet_M$ be the linking family of $M$ as defined in \cite[Definition 2.39]{crowley02}
(we have introduced the sign to correct the mistake in \cite[Definition 2.50]{crowley02}: see 
Remark \ref{rmk:sign_of_q}.)
Comparing these definitions, we see that for each $\sigma \in \sect(\pi)$
%
\begin{equation} \label{eq:qs_related}
q^{\frac{d_\pi}{2}k(\sigma)}_M = q^\sigma_{M}.
\end{equation}
%
Now \cite[Theorem B]{crowley02} states that
all linking families defined on $\sect(\pi)$ arise as the quadratic linking families of $2$-connected $M$ and 
that any isomorphism of linking families defined on $\sect(\pi)$ is realised by an almost diffeomorphism.  
Hence Theorem \ref{thm:almost_classification} follows by combining \cite[Theorem B]{crowley02}, \eqref{eq:qs_related}
and Lemma \ref{lem:2_types_of_qlf}.
\end{proof}

We now explain the proof of \cite[Theorem B]{crowley02}.  Recall that every $2$-connected $M$ is the boundary
of a $3$-connected $W$ and that the characteristic form of $W$, $(H^4(W, \del W), \lambda_W, \alpha_W)$
is a complete diffeomorphism invariant by \cite[Corollary 2.5]{crowley02}.  
A foundational theorem of Wilkens \cite[Theorem 3.2]{wilkens71} (see \cite[Theorem 2.24]{crowley02}),
states that for any diffeomorphism $f \colon \del W_0 \cong \del W_1$, there are $W_2$ and $W_3$ with
$\del W_2 = \del W_3 = S^7$ and a diffeomorphism $F \colon W_0 \natural W_2 \cong W_1 \natural W_3$ extending
$f$: here $\natural$ denotes boundary connected sum.
Now the boundary
of $W$ is a homotopy sphere, if and only if $\lambda_W$ is nonsingular which by definition means
that $(H^4(W, \del W), \lambda_W, \alpha_W)$ is nonsingular.  Hence we 
say that two characteristic forms are \emph{stably isomorphic} if they
become isomorphic after addition of nonsingular characteristic forms.  The above discussion shows that
classifying $2$-connected $7$-manifolds up to almost diffeomorphism is equivalent to classifying
characteristic forms up to stable isomorphism.  
This was achieved in \cite[Theorem 3.4]{crowley02} by extending ideas of Wall 
%from the proof of 
\cite[Theorem p.\,156]{wall72} from the setting of even
forms to the setting of characteristic forms.  The point is that an isomorphism 
$F \colon \del(H_0, \lambda_0, \alpha_0) \cong \del(H_1, \lambda_1, \alpha_1)$ of the boundaries of characteristic
forms can be used to glue them together to obtain a nonsingular characteristic form
%
\[ (H_0, -\lambda_0, \alpha_0) \cup_F (H_1, \lambda_1, \alpha_1). \]
%
It is then possible to explicitly write down an isomorphism of characteristic forms
%
\[ E \colon (H_0, \lambda_0, \alpha_0) \oplus \bigl( (H_0, -\lambda_0, \alpha_0) \cup_F (H_1, \lambda_1, \alpha_1) \bigr)
 \cong (H_1, \lambda_1, \alpha_1) \oplus \bigl( (H_0, -\lambda_0, \alpha_0) \cup_{\rm Id} (H_0, \lambda_0, \alpha_0) \bigr), \]
 %
such that $\del E = F$.  Combined with Lemma \ref{lem:splitq}, these methods give the following theorem, which 
is a refinement of a special case of \cite[Theorem 3.4]{crowley02}.

\begin{thm}[\cf\,{\cite[Theorem 3.4]{crowley02}}] \label{thm:stable_iso_and_qlfs}
Let $(H_i, \lambda_i, \alpha_i)$, $i = 0, 1$, be two characteristic forms.
The following are equivalent:
\begin{enumerate}
\item 
There is an isomorphism of linking families $F \colon \del(H_0, \lambda_0, \alpha_0) \cong \del(H_1, \lambda_1, \alpha_1)$;
\item
There are nonsingular characteristic forms $(H_j, \lambda_j, \alpha_j)$, $j = 2, 3$, and
an isomorphism 
%
\[ E \colon (H_0, \lambda_0, \alpha_0) \oplus (H_2, \lambda_2, \alpha_2) \cong (H_1, \lambda_1, \alpha_1) \oplus
(H_3, \lambda_3, \alpha_3) \]
%
such that $\del E  = F$;
\item
There is a nonsingular characteristic form $(H, \lambda, \alpha)$ containing $(H_0, -\lambda_0, \alpha_0)$
and $(H_1, \lambda_1, \alpha_1)$ are orthogonal summands with canonical isomorphism 
$F_\lambda = F \colon \del(H_0, \lambda_0, \alpha_0) \cong \del (H_1, \lambda_1, \alpha_1)$.
\end{enumerate}
\end{thm}

\begin{rmk}
By Lemma \ref{lem:2_types_of_qlf} the statement of Theorem \ref{thm:stable_iso_and_qlfs} and the discussion before it 
applies equally well to linking families defined over $\sect(\pi)$ and linking families defined over $S_2$.
\end{rmk}

\begin{proof}[Proof of Theorem \ref{thm:stable_iso_and_qlfs}]
This follows from \cite[Lemma 3.12]{crowley02} and the proof of \cite[Theorem 3.4]{crowley02}.
\end{proof}

We now explain the topological significance of $q^\circ_M$.
By Theorem \ref{thm:almost_classification}, $2$-connected
rational homotopy spheres $M$ with torsion linking for $(T, b)$ are classified up to almost diffeomorphism 
by the refinement $q_M \in \cal{Q}(b)$.  We shall write $M(q)$ for any rational homotopy sphere with refinement $q$.  
In contrast to rational homotopy spheres where $G = T$, the canonical examples of $2$-connected 
$M$ with $H^4(M)$ torsion free are the manifolds
%
\begin{equation} \label{eq:MF}
M(F, d_\pi) := (S^3 \tilde \times_{d_\pi} S^4) \sharp_{b-1} (S^3 \times S^4),
\end{equation}
%
where $F \cong \Z^b$, $S^3 \tilde \times_{d_\pi} S^4$ is the $S^3$-bundle over $S^4$ with trivial Euler class and
first Pontrjagin class $d_\pi$ times a generator of $H^4(S^4)$.
In the general case we shall be interested in connected sum almost splittings of $M$ which are almost diffeomorphisms
%
\[ f \colon M \acong M(q^f) \sharp M(F, d_\pi), \]
%
where $q^f$ is a certain refinement of $b_M$.
We call two connected sum almost splittings $f_0$ and $f_1$
{\em $H^*$-equivalent} if there is an almost diffeomorphism
$g \colon M \acong M$ with $H^*(g) = \Id$ and if there are almost diffeomorphisms $g_T \colon M(q_0) \acong M(q_1)$ 
and $g_F \colon M(F, d_\pi) \acong M(F, d_\pi)$ such that the following diagram commutes up to pseudo-isotopy. 
%
\[ \xymatrix{ M \ar[d]^g  \ar[r]^(0.25){f_0} & M(q^{f_0}) \sharp M(F, d_\pi) \ar[d]^{g_T \sharp g_F} \\
M \ar[r]^(0.25){f_1} & M(q^{f_1}) \sharp M(F, d_\pi). }  \]
%
We define $\Asplt(M) := \{ [f] \}$ to be the set of $H^*$-equivalence classes of almost splittings of
$M$ and note that there is a well-defined map
%
\[ \Asplt(M) \mapsto \sect(\pi), \quad [f] \mapsto \sigma(f), \]
%
where $\im(\sigma(f)) = f^*H^4(M(F, d_\pi))$.  The following theorem is implicit in \cite[Definition 2.50]{crowley02}. 

\begin{thm} \label{thm:almost_split1}
Let $M$ have linking family $q^\bullet_M \colon \sect(\pi) \to \cal{Q}(b)$.
For each $\sigma \in \sect(\pi)$ there is a unique $H^*$-equivalence class of almost splitting 
%
\[ f_\sigma \colon M \acong M(q_M^\sigma) \sharp M(F, d_\pi). \]
%
Consequently the map $\Asplt(M) \to \sect(\pi)$ is a bijection.
%
%The mapping $\Asplt(M) \to \sect(\pi), [f] \mapsto \sigma(f)$ defines a bijection such 
%that 
%
%\[ q(f) = (q_\sect)_M(\sigma(f)). \]
%There is an almost diffeomorphism $M \acong M(q) \sharp M(F, d_\pi)$ if and only if $q \in q_\sect(\sect(\pi))$.
%Moreover, 
%for each section $\sigma \in \sect(\pi)$, there is a corresponding almost diffeomorphism 
%
%$ f_\sigma \colon M \acong M(q(\sigma)) \sharp M(F, d_\pi)$
% 
%which defines the splitting $H^4(M) \cong T \oplus {\rm Im}(\sigma)$.
\end{thm}

\begin{proof}
Let $W$ be a $3$-connected coboundary of $M$ with characteristic form 
$(H^4(W, \del W), \lambda_W, \alpha_W) = (H, \lambda, \alpha)$.
We recall from the definition of the linking family defined by $W$ in
\eqref{eq:alg_qdef}, that there are orthogonal splittings of $(H, \lambda, \alpha)$
%
\[ \psi \colon (H, \lambda, \alpha) \cong (R, \lambda_R, \alpha_\psi) \oplus (F, 0, \alpha|_F), \]
%
where $(R, \lambda_R, \alpha_\psi)$ is nondegenerate.  For every such splitting $\psi$,
the classification of $3$-connected coboundaries (see \cite[Corollary 2.5]{crowley02}) implies that 
there is a corresponding boundary connected sum splitting $g_\psi \colon W \cong W_\psi \natural W_F$.
There is also a corresponding section $\sigma = \sigma(\psi) \in \sect(\pi)$ where $\im(\sigma) = j(H^4(W_f))$,
$j$ the natural homomorphism $H^4(W, M) \to H^4(M)$.
By definition, see \cite[Definition 2.50]{crowley02},
%
\[ q_M^{\sigma} = \del(R, -\lambda_R, \alpha_\psi), \]
%
%(where we have given the correct sign, unlike in \cite[Definition 2.50]{crowley02}, see
%Remark \ref{rmk:sign_of_q}).
and we define $f_\sigma \colon M \cong M(q_M^\sigma) \sharp M(F, d_\pi)$ to be the diffeomorphism on the boundary induced by the
the splitting $g_\psi$.  
This shows that $\Asplt(M) \to \sect(\pi)$ is onto.  

Suppose that $f_0$ and $f_1$ are two splittings of $M$ defining the same section $\sigma$.
Then the $H^*$-equivalence class of $f_i$ is determined by the almost diffeomorphism type
of $M(q^{f_i})$.
Now the Poincar\'{e} dual of $\im(\sigma)$ is a finitely generate free abelian group $\wh F \subset H_3(M)$.
We choose a basis $\{x_1, \dots x_b\}$ for $\wh F$ and this is represented by a set of disjoint embeddings 
$\phi \colon \sqcup_{i=1}^b \colon D^4 \times S^3 \subset M$.
We let $M_\phi$ be the outcome of surgery on $\phi$.
Clearly there are choices $\phi_0$ and $\phi_1$ for $\phi$ so that $M_{\phi_i} \acong M(q^{f_i})$.
We claim that the almost diffeomorphism type of $M_\phi$ is independent of the 
choice of $\phi$ and this 
%Assuming the claim, we have that each $f_i$ is equivalent to $M \cong M_\phi \sharp M(F, d_\pi)$ and
%so $f_0$ and $f_1$ are almost $H^*$-equivalent, 
implies that $\sigma \colon \Asplt(M) \to \sect(\pi)$ in injective.

To prove the claim, let $W_\phi$ be the trace of surgeries on $\phi$ and let 
$W_1 : = W \cup_M W_\phi$ be the union of $W_\phi$ and our original $3$-connected coboundary.
By construction, we see that there is a fixed $\alpha_\sigma \in R^*$ such that the characteristic form of 
$W_1$ is isomorphic to $(R, \lambda_R, \alpha_\sigma) \oplus (H_1, \lambda_1, \alpha_1)$ where 
$(H_1, \lambda_1, \alpha_1)$ is nonsingular.  It follows that the almost diffeomorphism type of
$M_\phi$ is well-defined.
%
\end{proof}

We conclude this subsection by identifying a simpler complete almost diffeomorphism invariant
of $2$-connected $M$.  
Recall that $\aut(b)$, the group of automorphisms of $b$, acts on $\cal{Q}(b)$, the set of refinements of $b$.
We define the reduced refinement of $M$ to be the function
%
\[ \wt q^{\; \circ}_M \colon S_2 \to \cal{Q}(b)/\!\aut(b), \quad h \mapsto [q^h] .\]
%
For the following simple corollary to Theorem \ref{thm:almost_split1}, recall that 
$\wh S_2 = \frac{d_\pi}{2}S_{d_\pi} \subseteq S_2$.

\begin{cor} \label{cor:minima_almost_classification}
Let $M_0$ and $M_1$ have the same base $(G, b, p)$ and reduced refinements $\wt q^{\; \circ}_0$ and 
$\wt q^{\; \circ}_1$.  
The following are equivalent:
\begin{enumerate}
\item \label{it:ad1}
$M_0$ is almost diffeomorphic to $M_1$;
\item \label{it:ad2}
$\wt q_0^{\; \circ}(\wh S_2) = \wt q_1^{\; \circ}(\wh S_2)$;
\item \label{it:ad3}
$\wt q_0^{\; \circ}(\wh S_2) \cap \wt q_1^{\; \circ}(\wh S_2) \neq \phi$. \qed
\end{enumerate}
\end{cor}

%%%%%%% Old stuff %%%%%%
\begin{comment}
We shall consider our classification results from the point of view of 
connected sum splittings of $2$-connected $M$ of the form
%
\[ M \cong M(T) \sharp M(F), \]
%
where $H^4(M(T)) \cong T$ and $H^4(M(F)) \cong F$.  Such connected sum splittings motivated 
the first definition of quadratic linking families in \cite{crowley02}.  We begin by
recalling quadratic linking families as defined in \cite{crowley02} and explaining their
relationship to connected sum splittings as above.  

Now let $S_2' : = \frac{d_\pi}{2}S_{d_\pi} \subset S_2$.  The following proposition
follows immediately from Lemma \ref{prop:almost_split1} and the proof of Theorem \ref{thm:almost_classification}
above.

\begin{thm} \label{thm:almost_splitting}
We fix a base $(G, b, p)$ and consider $2$-connected $M$ with this base.  
\begin{enumerate}
\item \label{it:almost_splitting_function}
For each $h \in S_2'$ there are almost diffeomorphisms
%
$ f_h \colon M \cong M(q^h) \sharp M(F, d_\pi)$
% 
with corresponding splittings $H^4(M) \cong T \oplus {\rm Im}(\sigma)$ where 
$h = \frac{d_\pi}{2} k$ and $k = k(\sigma)$.
\item \label{it:almost_splitting_criterion}
There is an almost diffeomorphism $M \acong M(q) \sharp M(F, d_\pi)$ if and only if $q \in q_M(S_2')$.
\item \label{it:almost_splitting_image}
$M_0$ and $M_1$ are almost diffeomorphic if and only if 
$q_{M_0}(S_2') = q_{M_1}(S_2') \subseteq \cal{Q}(b)$. \qed
\end{enumerate}
\end{thm}

%\begin{proof}
%\ref{it:almost_splitting_criterion}
%The refinement $q$ is defined using a $3$-connected co-boundary $W$.  Since $M$ is $2$-connected,
%$W$ is classified up to diffeomorphism by its generalised intersection form.
%The definition of $q^{\frac{d_\pi}{2}k}$ is obtained by decomposing $W = W_0 \natural W_1$ where the boundary of 
%$W_0$ is diffeomorphic to $M(F, d_\pi)$ and where the boundary of $W_1$ is diffeomorphic to $M(q^{\frac{d_\pi}{2}k})$.

%\ref{it:almost_splitting_image}  The set $q_M(\frac{d_\pi}{2}S_{d_\pi})$ is clearly an almost diffeomorphism invariant.
%The converse follows from part \ref{it:almost_splitting_criterion}.
%\end{proof}
\end{comment}
% subsection almost diffeo (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% (fold)
\subsection{Inertia and reactivity in more detail} \label{subsec:inertia_and_reactivity2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to proceed from the almost diffeomorphism classification of 2-connected 7-manifolds
to the full diffeomorphism classification we need a better understanding of  
the inertia group of $M$
%
\[ I^{}(M) = \{ \Sigma \,|\, M \sharp \Sigma \cong_{} M \}.\]
%
%\begin{rmk} \label{rmk:spin_inertia}
%If $M$ is simply connected then $I^{Spin}(M) = I(M)$, the usual inertia group of $M$.	
%\end{rmk}
%
We therefore begin with some general properties of the inertia group in dimension 7
and {\em do not assume that $M$ is $2$-connected.}
Computing $I^{}(M)$ is a delicate problem.  An important and simpler
first step is to compute the {\em (integral) (co)homology inertia group} of $M$,
%
\[ I_H(M) \subseteq I(M), \]
%
where we recall that a homotopy sphere $\Sigma \in I_H(M)$ if and only if there is a diffeomorphism 
$f \colon M \sharp \Sigma \cong M$
such that $H^*(f) = \Id$, regarding $M \sharp \Sigma$ and $M$ as the same topological space.
The main result of this subsection, Proposition \ref{prop:p2null}, establishes
%we establish basic tools for computing $I(M)$ and we establish a first
a lower bound on $I_H(M)$ for $2$-connected $M$: see Remark \ref{rmk:p2null}.

%%%%%%%%  old presentation of Diff and ADiff %%%%%%
\begin{comment}
Let $X$ be a compact spin manifold, possibly with boundary.  Recall that a pseudo-isotopy
between spin diffeomorphisms $f_0, f_1 \colon X \cong X$ is a diffeomorphism
$F \colon X \times I \cong X \times I$ which restricts to $f_i$ on $X \times \{i\}$.
Pseudo-isotopy defines an equivalence relation on the group of spin diffeomorphisms of $X$
and we define
%
\[ \wt \pi_0\Diff_{}(X) = \{ [f] \, | \, f \colon X \cong X \} \]
%
to be the group of pseudo-isotopy classes of spin diffeomorphisms of $X$.  We shall be
interested in the cases where $X = M$ and $X = M^\bullet : = M-{\rm Int}(D^7)$ and
we define
%
\[ \wt \pi_0\ADiff_{}(M) := \wt \pi_0\Diff_{}(M^\bullet), \]
%
the group of pseudo-isotopy classes of almost diffeomorphisms of $M$. 
%
%\begin{rmk} \label{rmk:pesudo-isotopy}
%Whilst there are several possible definitions of the group $\ADiff_{}(M)$, they all agree at the
%level of pseudo-isotopy.
%\end{rmk}
\end{comment}

To begin, let $\Diff_{}(M)$ be the group of diffeomorphisms of $M$.
Recall that a pseudo-isotopy
between diffeomorphisms $f_0, f_1 \colon M \cong M$ is a diffeomorphism
$F \colon M \times I \cong M \times I$ which restricts to $f_i$ on $M \times \{i\}$.
Pseudo-isotopy defines an equivalence relation on the group of diffeomorphisms of $M$
and for future use we define 
%
\[ \wt \pi_0\Diff_{}(M) = \{ [f] \, | \, f \colon M \cong M \} \]
%
to be the group of the pseudo-isotopy classes of spin diffeomorphisms of $M$.  Here we remind 
the reader that $M$ is spin and all the diffeomorphisms used to define $\pi_\Diff(M)$ are spin
diffeomorphisms.

We now consider the problem of when an almost diffeomorphism is pseudo-isotopic
to a diffeomorphism.
Let $f \colon M_0 \acong M_1$ be an almost diffeomorphism with singular set 
$\{m_1, \dots, m_a\}$.  We now associate a diffeomorphism class of
homotopy $7$-sphere to each singular point $m_i$.  Let $D^7_i \ni m_i$ be a small
disc containing $m_i$ and no other singular points of $f$.  The manifold $f(D^7_i) \subset M_1$
is a co-dimension zero submanifold of $M_1$ and so inherits a smooth structure from $M_1$
such that
%
\[ \wh f_i : = f|_{D^7_i - \{m_i\}} \colon D^7_i - \{m_i\} \cong f(D^7_i - \{m_i\})  \]
%
is a diffeomorphism.  We can therefore define the smooth homotopy $7$-sphere
%
\[ \Sigma_i : = D^7_i \cup_{\wh f_i} (-f(D^7_i)) \]
%
by gluing $D^7_i$ and $-f(D^7_i)$ together along $\wh f_i$.  From the definition
of pseudo-isotopy we deduce that the diffeomorphism class of the homotopy sphere
%
\[ \Sigma_f : = \Sigma_1 \sharp \Sigma_2 \sharp \dots \sharp \Sigma_a \]
%
is invariant under pseudo-isotopies.  Moreover, it is clear that $f$
defines a diffeomorphism
%
\begin{equation} \label{eq:Sigma_f}
f \colon M \cong M \sharp \Sigma_f,	
\end{equation}
that $f$ is pseudo-isotopic to a diffeomorphism if and only if 
$\Sigma_f \cong S^7$ and that $\Sigma_{f \circ g} = \Sigma_f \sharp \Sigma_g$.
%
%\begin{equation} \label{eq:smoothing_criterion}
%\Sigma_1 \sharp \Sigma_2 \sharp \dots \sharp \Sigma_a = S^7 \in \Theta_7. 
%\end{equation}
%
%The relevance of the above for the inertia group $I(M)$ is the following.  By fundamental
%results of Smale and Cerf \cite{smale61, cerf70}, there is an isomorphism
%
%\[ \Sigma \colon \wt \pi_0\Diff_{}(S^6) \cong \Theta_7, \quad [g] \mapsto D^7 \cup_g (-D^7).\]
%
%Restricting a diffeomorphism $f \colon M^\bullet \cong M^\bullet$ to the boundary
%gives a well-defined homomorphism $\wt \pi_0\ADiff_{}(M) \to \wt \pi_0\Diff_{}(S^6)$,
%and composing with the isomorphism $\Sigma$ above we obtain homomorphisms
%
It follows that there is a {\em singularity homomorphism}
%
\[ \del \colon \wt \pi_0\ADiff(M) \to \Theta_7 \quad [f] \mapsto \Sigma_f, \]
%
with kernel isomorphic to the image of $\wt \pi_0 \Diff(M)$ in $\wt \pi_0\ADiff(M)$.
%
%D^7 \cup_{\del f} (-D^7), \]
%\Sigma([\del f]) \]
%\[ \del \colon \wt \pi_0\ADiff_{}(M) \to \wt \pi_0\Diff_{}(S^6), \quad [f] \mapsto [\del f] \]
%
%and
%\[ \del_H \colon \wt \pi_0\ADiff_{H}(M) \to \Theta_7, \quad [f] \mapsto \Sigma_f, \]
%D^7 \cup_{\del f} (-D^7), \]
% \Sigma ([\del f]), \]
%\[ \del_H \colon \wt \pi_0\ADiff_{H}(M) \to \wt \pi_0\Diff_{}(S^6), \quad [f] \mapsto [\del f], \]
%
If $\wt \pi_0\ADiff_H(M) \subset \wt \pi_0\ADiff_{}(M)$ is the subgroup
of pseudo-isotopy classes inducing the identity on $H^*(M)$, the we define 
the singularity homomorphism
$\del_H \colon \wt \pi_0\ADiff_{H}(M) \to \Theta_7$ to be the restriction of $\del$.
From \eqref{eq:Sigma_f} we see that
%
\begin{equation} \label{eq:I_and_del}
 I_H(M) = \im(\del_H) \quad \text{and} \quad I(M) = \im(\del).
\end{equation}
%\[ I_H(M) = \del(\wt \pi_0 \ADiff_{H}(M)) \quad \text{and} \quad
%I(M) = \del(\wt \pi_0 \ADiff_{}(M)) .\]

%
Given an almost diffeomorphism $f \colon M \acong M$ with singular
points $\{m_1, \dots, m_a\}$, we now show how to determine 
$\Sigma_f \in \Theta_7$ using the mapping torus of $f$.
The mapping torus of $f$ is the almost smooth manifold constructed from
the cylinder $M \times I$ by using $f$ to identify points at either end:
%
\[ T_f : = (M \times [0, 1])/ (m,0) \sim (f(m),1). \]
%
Since $f$ is an almost diffeomorphism, the closed $8$-manifold $T_f$ admits a smooth
structure except perhaps at points $\{\ol m_1, \dots, \ol m_a\}$ corresponding
to the singular points of $f$.  Indeed if $B^8_i \ni \ol m_i$ are small open
balls containing precisely one singular point each of $T_f$ then
%
\begin{equation} \label{eq:W_f}
 W_f : = T_f - \sqcup_{i=1}^a B^8_i 
\end{equation}
%
is a compact smooth manifold with boundary
%
\[ \del W_f \cong \sqcup_{i=1}^a \Sigma_i. \]
%
We choose a spin structure on $T_f$ and denote the corresponding $8$-dimensional almost smooth 
spin manifold by $T_f$ also: 
no confusion shall arise since we are interested only in the characteristic number
%
\[  p^2(f) : = \langle p_{T_f}^2, [T_f] \rangle \in \Z, \]
% 
which depends only on the oriented almost diffeomorphism type of $T_f$ since
$2p_{T_f} = p_1(T_f)$ and $H^8(T_f) \cong \Z$ (in fact $p_{T_f}$ is independent
of the choice of spin structure by \cite[p.\,170]{cadek08}).
%Since $p^2(f)$ depends only on the oriented diffeomorphism type of $T_f$ it
It follows that $p^2(f)$ is an invariant of the pseudo-isotopy class of $f$.
For the statement of the next lemma, we recall the renormalised Eells-Kuiper invariant
of a homotopy sphere $\Sigma$, $\mu(\Sigma) \in \Z/28$ defined in \eqref{eq:classical}.
By \cite[(13)]{eells62}, $\mu(\Sigma_1) = \mu(\Sigma_2)$ if and only if $\Sigma_1 \cong \Sigma_2$.

\begin{lem} \label{lem:mapping_torus}
For every almost diffeomorphism $f \colon M \acong M$ the following hold:
\begin{enumerate}
\item \label{lem:mapping_torus:8}
$p^2(f) \in 8 \Z$,
\item \label{lem:mapping_torus:mu}
$\mu(\Sigma_f) = \frac{p^2(f)}{8} \in \Z/28$,
\item \label{lem:mapping_torus:smooth}
$f$ is pseudo-isotopic to a diffeomorphism if and only if $p^2(f) \in 224 \Z$.
\end{enumerate}
\end{lem}

\begin{proof}
\ref{lem:mapping_torus:8} This follows since by
Lemma \ref{lem:p_M}\ref{it:boundary}, $p_{T_f}$ is characteristic for
the intersection form of $T_f$.  Hence by \cite[Lemma 5.2, \S 5]{milnor73},
$p^2(f) \equiv \sigma(T_f)$~mod~$8$.  But by Novikov additivity, the signature of
$T_f$ is zero.

\noindent
\ref{lem:mapping_torus:mu} This follows since $W_f$ defined in \eqref{eq:W_f} above
is a smooth spin coboundary for $\Sigma_f$ and so can be used to compute $\mu(\Sigma_f)$.
Since $\sigma(W_f) = \sigma(T_f) = 0$, applying \eqref{eq:grW} gives the result.

\noindent
\ref{lem:mapping_torus:smooth} The almost diffeomorphism $f$ is pseudo-isotopic to a diffeomorphism if and only 
if $\Sigma_f \cong S^7$.  Hence \ref{lem:mapping_torus:smooth} follows directly from \ref{lem:mapping_torus:mu}.
\end{proof}


In the light of Lemma \ref{lem:mapping_torus}, we define the function
%
\[ p^2 \colon \wt \pi_0 \ADiff_{}(M) \to \Z, \quad [f] \mapsto p^2(f). \]
%
Since the image of $p^2$ plays a key role, we define non-negative integers called
the {\em reactivity of $M$}, $R(M)$, and the \emph{(co)homologically fixed reactivity}
of $M$, $R_H(M)$, by the following equations
%to be the smallest positive value of $p^2(f) \in \Z$
%for $f \colon M \cong M$ a spin isomorphism of $M$, so that
%There are many versions of reactivity
%depending upon what sort of isomorphism $f$ is: i.e.~diffeomorphism, almost diffeomorphism, 
%PL Homeomorphism, inducing the identity on $H^*$ etc.  The most basic version of reactivity
%is defined by
%
\[ p^2(\ADiff_{}(M)) = R(M) \Z \quad \text{and} \quad
p^2(\ADiff_{H}(M)) = R_H(M) \Z .\]
%
%Note that if $\im(p^2) = 0$, then we set $R(M) = 0$.
%We similarly define the \emph{(integral) (co)homologically fixed reactivity}
%of $M$, $R_H(M)$, by
%
%\[ p^2(\ADiff_{H}(M)) = R_H(M) \Z .\]
%
By Lemma~\ref{lem:mapping_torus}~\ref{lem:mapping_torus:8}, $R(M)$ and $R_H(M)$ are
both divisible by 8. 
By Lemma \ref{lem:mapping_torus}~\ref{lem:mapping_torus:mu} and the definition
of reactivity we have
\begin{prop} \label{prop:role_of_reactivity} \hfill
%$I_H(M) = \frac{R_H(M)}{8} \Theta_7$ and $I(M) = \frac{R(M)}{8} \Theta_7$. \qed
\begin{enumerate}
\item \label{it:r_of_r}
$I(M) = \frac{R(M)}{8} \Theta_7$,
\item $I_H(M) = \frac{R_H(M)}{8} \Theta_7$. \qed
\end{enumerate}
\end{prop}

For other problems, for example counting the number of deformation equivalence classes of $G_2$-structures
on $M$ as in \cite{nu}, it important to know the value of $p^2$ for diffeomorphisms.
Hence we define the {\em smooth reactivity of $M$}, $R^\Diff(M)$, and the
{\em smooth (co)homologically fixed reactivity of $M$}, $R^\Diff_H(M)$ by the equations
%
\[ p^2(\Diff_{}(M)) = R^\Diff(M) \Z
\quad \text{and} \quad 
 p^2(\Diff_{H}(M)) = R^\Diff_H(M) \Z. \]
%
%By definition $R(M)$ divides $R_H(M)$, and $R^\Diff(M)$ divides $R^\Diff_H(M)$.
%and by Lemma~\ref{lem:mapping_torus}~\ref{lem:mapping_torus:8} 
%$R(M), R_H(M), R^\Diff(M), R^\Diff_H(M) \in 8 \Z$.
By Lemma~\ref{lem:mapping_torus}~\ref{lem:mapping_torus:smooth} we have

\begin{lem} \label{lem:smooth_reactivity} \hfill
\begin{enumerate}
\item $R^\Diff(M) = \lcm(R(M), 224)$,
\item $R^\Diff_H(M) = \lcm(R_H(M), 224)$. \qed
%\item $\GTD(M) = 24 \cdot \Num{\frac{R(M)}{224}}$.
\end{enumerate}
\end{lem}
%By $8$ divides $R(M)$ and that $p^2(\ADiff_{}(M)) = p^2(\PLHomeo_{}(M))$.
%Then we can define
%
%\[ p^2(\Diff_{}(M)) = R^{\rm Diff}(M) \Z .\]
%
%Note that $R^{\rm Diff}(M) = \lcm(R^{}(M), 224)$ and we may not ever need to use $R^{\rm Diff}(M)$.
%We can state all theorems about the inertia group and counting $G_2$-structures using
%$R^{}(M)$.
%And similarly for $R^{}_H(M)$ and $R_H^{\rm Diff}(M)$.

We now turn to the construction of almost diffeomorphisms $f \in \ADiff_{}(M)$ on $2$-connected $M$
with $p^2(f) \neq 0$.
Recall that $d_\pi$ is the divisibility of $\pi(p_M) \in H^4(M)/TH^4(M)$ and
that $\tilde d_\pi = \lcm(4, d_\pi)$.

\begin{prop} \label{prop:p2null}
If $M$ is 2-connected then $R_H(M)\,|\, 2\tilde d_\pi$; \ie~ $2\td_\pi\Z \subseteq p^2(\ADiff_H(M))$.
%If $M$ is 2-connected then the image of $\ADiff_H(M)$ under $p^2$ contains
%$2\td_\pi\Z$.
\end{prop}

\begin{rmk} \label{rmk:p2null}
If $M$ is $2$-connected, Propositions \ref{prop:role_of_reactivity} and \ref{prop:p2null} together give
$\frac{\tilde d_\pi}{4} \Theta_7 \subseteq I_H(M)$.
In Corollary \ref{cor:p2image}~\ref{it:p2null} below we will show that $R_H(M) = 2 \tilde d_\pi$
and hence $\frac{\tilde d_\pi}{4} \Theta_7 = I_H(M)$.
\end{rmk}

For the proof of Proposition \ref{prop:p2null} it will be useful to compute the
characteristic number $p^2(T_f)$ using a co-bounding spin $8$-manifold $W$.
%By \cite{milnor63} $\Omega_7^{Spin} = 0$, such a $W$ always exists and 
We define
the closed almost smooth $8$-manifold
%
\[ X_f : = W \cup_f -W. \]
%
%where $W$ is any spin coboundary for $M$.  
%By \cite[Theorem 8]{wall67} such
%coboundaries always exist for $2$-connected $M$, but we remark that every spin $7$-manifold
%admits a 3-connected spin coboundary.

\begin{lem} \label{lem:T_and_X}
With the notation above, $p^2(f) = \an{p_{X_f}^2, [X_f]}$.
\end{lem}

\begin{proof}
Since $p^2(f) = \an{p_{T_f}^2, [T_f]}$ and $\an{p_{X_f}^2, [X_f]}$ are characteristic numbers,
if suffices to prove that $T_f$ is oriented bordant to $X_f$.
Consider the manifolds $M \times I$ and $W \sqcup -W$.  Both have boundary $M \sqcup -M$,
$T_f$ is formed from $M \times I$ by gluing $M$ to $-M$ via $f$	and $X_f$ if formed
from $W \sqcup -W$ by gluing $M$ to $-M$ via $f$.  It therefore suffices to prove
that $W \sqcup -W$ is bordant relative to the boundary to $M \times I$.  But the manifold
$W \times I$ is a rel.~boundary bordism from $W \sqcup -W$ to $M \times I$, and
we are done.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:p2null}]
%For convenience we abbreviate $\dpi{M} = d$. 
We assume $d_\pi \not= 0$, since otherwise there is nothing to prove.
By Theorem \ref{thm:almost_split1} or by \cite[Theorem 1]{wilkens72},
we may decompose $M$ as a connected sum of spin manifolds
%
\[ M \cong_{} M_1 \sharp M_2 \]
% 
where $M_1 = M(\Z, d_\pi) = S^3 \tilde \times_{d_\pi} S^4$ 
is the total space of a $3$-sphere bundle over $S^4$ defined in \eqref{eq:MF}.
%is the total space of a certain $3$-sphere
%bundle over $S^4$ with Euler class zero.
% as in Example \ref{ex:divisibilities}.
%Specifically, there is a linear $D^3$-bundle with characteristic map
%$\alpha \in \pi_3(SO(3))$ such that $M_1 = S^3 \tilde \times_{S \alpha} S^4$ is
%the total space of the sphere bundle of the stabilisation of $\alpha$,
%$S\alpha \in \pi_3(SO(4))$ and $p_{M_1} = p(\alpha) = d \cdot z$ where $z$ is a
%generator of  $H^4(M_1)$. 
%and $p_{M_1}$ is torsion.  
%Recall that $d \in 2 \cdot \Z$.  
We shall produce the required almost diffeomorphisms on the manifold $M_1$ and
then extend by the identity to $M$.  Let 
%
\[ M_1^\bullet : = M_1 - \Int(D^7) \]
%
be $M_1$ minus a small open disc.  Since $M_1$ is the total space of an $S^3$-bundle over $S^4$ there is a diffeomorphism
%
\[ M_1^\bullet \cong (D^3 \tilde \times_{d_\pi} S^4) \cup_{S^2 \times D^4} (D^3 \times D^4) \]
%
%\refcomm{make this more explicit}
where $D^3 \tilde \times_{d_\pi} S^4$ is a tubular neighbourhood of a section of $M_1 \to S^4$ and $D^3 \times D^4$ is a $3$-handle added to $D^3 \tilde \times_{d_\pi} S^4$ along the tubular neighbourhood of a fibre $2$-sphere,
$S^2 \times D^4 \subset S^2 \tilde \times_{d_\pi} S^4$.

By \cite[p.\,171 (2)]{wall62} we may identify $\pi_3(SO(4))$ as the group of
pairs of integers $(n, p)$ where $n \equiv p \mmod 2$, so that the
corresponding $S^3$-bundle over $S^4$ has Euler class $n \in H^4(S^4) = \Z$ and
first Pontrjagin class $2p$.
Let $\gamma_{n, p} \colon (D^3, S^2) \to (SO(4), {\rm Id})$ be a smooth
function representing $(n, p)$.  We define a diffeomorphism
%
\[ f^\bullet_{n, p} \colon M^\bullet_1 \cong M^\bullet_1 \]
%
where $f^\bullet_{n, p}|_{D^3 \tilde \times_\alpha S^4}$ is the identity and on
the $3$-handle we use the $D^3$ co-ordinate to twist the $D^4$-coordinate
using $\gamma_{n, p}$.  To be explicit:
%
\[ f^\bullet_{n, p}|_{D^3 \times D^4}(u, v) = (u, \gamma_{n, p}(u)(v)) .\]
%
We observe that there is a copy of $S^3 \vee S^4 \subset M^\bullet_1$
%\refcomm{prefers $S^4 \vee S^3$} such that
the restriction $f^\bullet_{n, p}|_{S^3 \vee S^4}$ is the identity and such
that $M^\bullet_1$ deformation retracts to $S^3 \vee S^4$.
It follows that $f^\bullet_{n, p}$ acts trivially on cohomology. 

By the Alexander trick, we may extend $f^\bullet_{n, p}$ to 
an almost diffeomorphism of $M_1$ and we
denote any choice of extension by $f_{n, p}$.
Since $f^\bullet_{n, p}$ acts trivially on cohomology, so does $f_{n, p}$.
Since $M_1$ admits a unique spin structure for each
orientation and since $f_{n, p}$ is orientation preserving, $f_{n, p}$ is a
spin almost diffeomorphism.
Up to pseudo-isotopy, we may assume that $f_{n, p}$ is the identity on a disc
and hence we may we extend $f_{n, p}$ to $M$ by taking the connected
sum with the identity on~$M_2$. Thus we define the almost diffeomorphism
%
\[ g_{n, p} := f_{n, p} \sharp \Id_{M_2} \colon M \cong M, \]
%
and note that $g_{n, p}$ acts trivially on cohomology and is a spin almost diffeomorphism.
%and hence the mapping torus of
%$g_{n, p}$, $T_{g_{n, p}}$, admits a spin structure. 
We claim that 
%
\begin{equation} \label{eq:spin}
p^2(g_{n, p}) = p^2(f_{n, p}) = d_\pi(2p -nd_\pi).
\end{equation}
%
The manifold $M_1 \cong S^3 \tilde \times_{d_\pi} S^4$ bounds the
$8$-dimensional $D^4$-bundle $W_1 : = D^4 \tilde \times_{d_\pi} S^4$,
and we let $W_2$ be any spin coboundary for $M_2$.  
We form the twisted doubles $X_{f_{n, p}} := W_1 \cup_{f_{n, p}} (-W_1)$
and
\begin{equation} \label{eq:Xg}
 X_{g_{n, p}} := (W_1 \natural W_2) \cup_{g_{n, p}} (-W_1 \natural -W_2) \cong  
X_{f_{n, p}} \sharp \bigl( W_2 \cup_{{\rm id}} (-W_2) \bigr). 
\end{equation}
Applying Lemma \ref{lem:T_and_X} we have,
%
\[ p^2(g_{n, p}) = \an{p^2(X_{g_{n, p}}), [X_{g_{n, p}}]} = 
\an{p^2(X_{f_{n, p}}), [X_{f_{n, p}}]} = p^2(f_{n, p}),\]
%
where the second equality holds by \eqref{eq:Xg} since the characteristic number $p^2$
is a bordism invariant which is additive for connected sums and $W_2 \cup (-W_2) = \del (W_2 \times I)$.
Writing $X_{n, p} := X_{f_{n, p}}$, it therefore remains to compute $\an{p^2(X_{{n, p}}), [X_{{n, p}}]}$.
From the construction of $X_{n, p}$ we see that $H_4(X_{n, p}) \cong \Z(x) \oplus \Z(y)$ 
where $x$ is represented by the zero section of $W_1$ and $y = [D^4 \cup D^4]$ is represented 
by an embedded $4$-sphere obtained by gluing two fibres of the $D^4$-bundle $W_1$ together, 
one from each copy of $W_0$.  
By construction, the normal bundle of the $4$-sphere $D^4 \cup D^4$ has characteristic function 
$\gamma_{n, p}$ and hence Euler number $n$.   It follows that the intersection form of $X_{n, p}$ 
with respect to the basis $\{x, y\} $ is given by the following matrix:
%
\[ \left( \begin{array}{cc} 0 & 1 \\ 1 & n  \end{array} \right) \]
%
Moreover since $x$ is represented by an embedded $4$-sphere with tubular neighbourhood diffeomorphic
to $D^4 \tilde \times_{d_\pi} S^4$
and since $y$ is represented by an embedded $4$-sphere with normal bundle $\gamma_{n, p}$, 
we have $p_{X_{n, p}}(x) = d_\pi$ and $p_{X_{n,p}}(y) = p$.  We conclude that the Poincar\'{e} 
dual of $p_{X_{n, p}}$ is given by
%
\[ PD(p_{X_{n, p}}) = (p - nd_\pi) x + d_\pi y. \]
%
It follows that 
$\an{p_{X_{n, p}}^2, [X_{n, p}]}= 2d_\pi(p-nd_\pi) + n d_\pi^2 = d_\pi(2p - n d_\pi)$, 
and the claim \eqref{eq:spin} is proven.

Finally we need to choose $n$ and $p$ so that $d_\pi(2p-nd_\pi) = 2\td_\pi$.
Recall that we may choose $n$ and $p$ freely subject to the constraint that 
$n \equiv p$~mod~$2$.  If $d_\pi = 4k+2$, then $2 \tilde d_\pi = 4 d_\pi$ and we choose 
$(n, p) = (0, 2)$.  If $d_\pi = 4k$, then $2 \tilde d_\pi = 2d_\pi$ and we set 
$(n, p) = (1, 2k+1)$.
\end{proof}
% subsection inertia and reactivity (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{The proof of the main classification theorem} \label{subsec:classification_theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The {\em refined quadratic linking family} of $M$ is the pair $(H^4(M), q_M^\circ, \EK_M, p_M)$
The mod 28 distillation of $M$ is the quadruple $(H^4(M), q_M^\circ, \EK_M, p_M)$
where $q^\circ_M$ is the quadratic linking family of $M$ as in
Definition \ref{def:ql} and the generalised Eells-Kuiper invariant
%
\[ \EK_M \colon \divsp \to \Q/ \dM \Z \]
%
is the mod 28 Gauss refinement of $q^\circ_M$ defined by \eqref{eq:EK_of_M}.  
In this subsection we prove Theorem \ref{thm:classification} which states that
mod 28 distillations give a complete invariant of diffeomorphisms of $2$-connected
$M$.  For the remainder of the subsection $M$ is $2$-connected.

Recall that the (renormalised) classical Eells-Kuiper invariant, as defined
by \eqref{eq:classical}, gives a group isomorphism
%
\[ \Theta_7 \cong \Z/28 \Z, \quad \Sigma \mapsto \mu(\Sigma) : = \mu_\Sigma(0) .\]
%
The following lemma is obvious from the definitions of $q^\circ_M$ and $\EK_M$.
%$\rho_{\dM} \colon \Q/28\Z \to \Q/\dM\Z$ denotes reduction mod~$\dM$.

\begin{lem} \label{lem:g_M_sharp_Sigma}
For all $\Sigma \in \Theta_7$, $q^\circ_{M \sharp \Sigma} = q^\circ_M$ and
$\EK_{M \sharp \Sigma} = \EK_M + [\mu(\Sigma)]$, where $[\mu(\Sigma)]$ is the mod~$\dM$
reduction of $\mu(\Sigma)$. \qed
\end{lem}

%Now suppose that $M_0$ and $M_1$ are $2$-connected $7$-manifolds,
%and let $F : H^4(M_1) \to H^4(M_1)$ be a group isomorphism.
%with mod 28 distillations $(G_0, q_0, \EK_0, p_0)$ and
%$(G_1, q_1, \EK_1, p_1)$, and let $F \colon G_1 \cong G_0$ be an isomorphism.
%We have defined the pull-backs $F^\#q_{M_0}$ and $F^\#\EK_{M_0}$
%in \eqref{eq:pull_q} and \eqref{eq:pull_gauss} (and $F^\#p_{M_0}$ is simply
%$F^{-1}(p_{M_0})$).

%\begin{thm}
%\label{thm:classify}
%There is a diffeomorphism $f \colon M_0 \cong M_1$ with
%$H^*(f) = F \colon H^4(M_1) \cong H^4(M_0)$ if and only if
%$F^\#(q_{M_0}, \EK_{M_0}, p_{M_0}) = (q_{M_1}, \EK_{M_1}, p_{M_1})$.
%\end{thm}

\begin{proof}[Proof of Theorem \ref{thm:classification}]
%To complete the proof of Theorem \ref{thm:classification}, we merely remark
%that the existence of a smooth $M$ with mod 28 distillation isomorphic
%to a prescribed $(G,q,\EK,p) \in \catek$ follows from the corresponding
%existence statement in Theorem \ref{thm:almost_classification}, since
%Lemma \ref{lem:g_M_sharp_Sigma} lets us freely change the Eells-Kuiper
%invariant of a manifold with a prescribed refinement $(G,q,p) \in \catq$.
The existence of a smooth $M$ with mod 28 distillation isomorphic
to a prescribed $(G,q^\circ,\EK,p) \in \catek$ follows from the corresponding
existence statement in Theorem \ref{thm:almost_classification}, since
Lemma \ref{lem:g_M_sharp_Sigma} lets us freely change the Eells-Kuiper
invariant of a manifold with a prescribed refinement $(G,q^\circ,p) \in \catq$.

By Lemma \ref{lem:ek_defined}, which is proven in \eqref{eq:compare},
the generalised Eells-Kuiper invariant is a diffeomorphism invariant.
Now we suppose
$F^\#(q^\circ_{M_0}, \EK_{M_0}, p_{M_0}) = (q^\circ_{M_1}, \EK_{M_1}, p_{M_1})$.
As explained in \eqref{eq:Sigma_f}, Theorem \ref{thm:almost_classification}
means that there is a homotopy sphere $\Sigma$ and a diffeomorphism 
%
\[ f \colon M_0 \sharp \Sigma \cong M_1 \]
%
such that  $H^*(f) = F \colon H^4(M_1) \cong H^4(M_0)$.
It remains to show that $\Sigma \in I_H(M_0)$.  For if so, there is a diffeomorphism 
%
\[ h \colon M_0 \cong M_0 \sharp \Sigma \]
%
with $H^*(h) = \Id$ and then $f \circ h \colon M_0 \cong M_1$ is a diffeomorphism 
with $H^*(f \circ h) = H^*(f) = F$.

Since $f$ is a diffeomorphism it preserves the mod~28 Gauss refinements.  
Applying Lemma \ref{lem:g_M_sharp_Sigma} we have
%
\[ \EK_{M_1} = F^\#(\EK_{M_0} + \mu(\Sigma)) = F^\#(\EK_{M_0}) + \mu(\Sigma) .\]
%
On the other hand, our assumption is that $F^\#(\EK_{M_0}) = \EK_{M_1}$.  Since $d_{M_0} = d_{M_1}$,
%
\[
\mu(\Sigma) = \EK_{M_1} - F^\#(\EK_{M_0}) = 0 \in \Z/d_{M_1} \Z = \Z/d_{M_0} \Z.
\]
%
By Remark \ref{rmk:p2null}, $\Sigma \in I_H(M_0)$ and this completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:classification_categorical}]
That the functor $\cal{Q} \colon \cal{M}_{7,2}^{spin} \to \cal{Q}_\EK$ is surjective and faithful
is a restatement of Theorem \ref{thm:classification}.
To see that $\cal{Q}$ is additive and compatible with orientation reversal,
let $i = 0, 1$, and let $M_i = \del W_i$ where $W_i$ has characteristic from $(H_i, \lambda_i, \alpha_i)$
with boundary refinement $q^\circ_i$.
The mod $28$ Gauss refinement of $M_i$ is $(q^\circ_i, \del g_{W_i})$
where we set $\del g_{W_i} := g_{W_i}$~mod~$\dM$ as in \eqref{eq:EK_of_M}.
Since the characteristic form of $-W_i$ is $(H_i, -\lambda_i, \alpha_i)$ and
the characteristic form of the boundary connected sum $W_0 \natural W_1$ is $(H_0, \lambda_0, \alpha_0) \oplus (H_1, \lambda_1, \alpha_1)$,
it follows that mod $28$ Gauss refinements of $-M_i$ and $M_0 \sharp M_1$ are
$(-q^\circ_i, -\del g_{W_i})$ and $(q_1^\circ, \del g_{W_0}) \oplus (q_1^\circ, \del g_{W_1})$ respectively.
\end{proof}

% subsection classification theorem (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{Smooth splitting functions} \label{subsec:splitting_functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this subsection we consider connected sum splittings of $2$-connected $M$
in the smooth category and we prove a smooth analogue of Theorem \ref{thm:almost_split1}.
We also prove Theorem \ref{thm:classification_minimal} which is the smooth analogue
of Corollary \ref{cor:minima_almost_classification}. 
Throughout the subsection $M$ is 2-connected.

Let $(T, b)$ be a torsion form and let $d$ be an even integer.  We define the set
%
\[ \wh{\cal{Q}}_{d}(b) := \{(q, s)\} \subset \cal{Q}(b) \times \Q/d \Z, \]
%
which consists of pairs of quadratic refinements $q$ of $b$ and rational residues mod $d$ where 
$A(q) \equiv s$~mod $\Z$.
By Theorem \ref{thm:classification}, a rational homotopy sphere $M$ with linking form isomorphic to $(T, b)$
is classified up to diffeomorphism by the pair $(q_{M}, \EK(M)) \in \wh{\cal{Q}}_{28}(b)$.  
We denote this rational homotopy sphere by
%
\[ M = M(q, s), \]
%
where $s = \EK(M)$.
Suppose that we are given a base $(G, b, p)$ with $F = G/T \cong \Z^b$ and $\pi(p) \in F$
of divisibility $d_\pi$.   
%Note that $M(F, d_\pi)$ has distillation $(F, 0, 0, d_\pi x)$ where $x \in F$ is primitive.
By Theorem \ref{thm:almost_split1}, if $M$ has base $(H^4(M), p_M, b_M) \cong (G, b, p)$,
there is a connected sum splitting
%
\[ f \colon M \cong M(q^f, s(f)) \sharp M(F, d_\pi), \]
%
for some rational homotopy sphere $M(q^f, s(f))$.  We recall that $\sigma(f) \in \sect(\pi)$ is
the section defined by $\im(\sigma) = f^*(H^4(M(F, d_\pi)))$.  In considering the
uniqueness of $M(q, s)$ in such a splitting, we note that 
by Theorem \ref{thm:classification} $I(M(F, d_\pi)) = \dM\Theta_7$ 
(see also Remark \ref{rmk:p2null}).
%and
%we observe that if $\Sigma$ is a homotopy sphere in $\dM\Theta_7$,
%then we can obtain a smooth connected sum splitting
%
%\[ f_\Sigma \colon M \cong M \sharp \Sigma \sharp (-\Sigma) 
%\cong \bigl( M(q^f, s(f)) \sharp \Sigma \bigr) \sharp \bigl(M(F, d_\pi) \sharp (-\Sigma) \bigr)
%\cong \bigl( M(q^f, s(f)) \sharp \Sigma \bigr) \sharp M(F, d_\pi),
%\]
%
%with $\sigma(f_\Sigma) = \sigma(f)$.  
As a consequence, we see that if $(q_i, s_i) \in \cal{Q}_{28}(b)$, $i = 0, 1$, have
$s_0 \equiv s_1$~mod~$\dM\Z$, then there is a diffeomorphism
%
\begin{equation} \label{eq:splitting_inertia}
g \colon M(q, s_0) \sharp M(F, d_\pi) \cong M(q, s_1) \sharp M(F, d_\pi) 
\end{equation} 
%
such that $H^*(g)$ preserves the induced splittings of $H^4$.  
Hence we define two splittings $f_0$ and
$f_1$ to be $H^*$-equivalent if there is a diffeomorphism
$g \colon M \cong M$ with $H^*(g) = \Id$ and if there are almost diffeomorphisms $g_T \colon M(q_0) \acong M(q_1)$ 
and $g_F \colon M(F, d_\pi) \acong M(F, d_\pi)$ with $\Sigma_{g_T} = -\Sigma_{g_F} \in \dM\Theta_7$
such that the following diagram commutes up to pseudo-isotopy:
%
\[ \xymatrix{ M \ar[d]^g  \ar[r]^(0.25){f_0} & M(q^{f_0}, s(f_0)) \sharp M(F, d_\pi) \ar[d]^{g_T \sharp g_F} \\
M \ar[r]^(0.25){f_1} & M(q^{f_1}, s(f_1)) \sharp M(F, d_\pi) }  \]
%
We define $\splt(M) := \{ [f] \}$ to be the set of $H^*$-equivalence classes of splittings of
$M$.  We also define the {\em smooth splitting function} of $M$ 
%
\[ \wh q^{\; \bullet}_M \colon \sect(\pi) \to \wh{\cal{Q}}_{\dM}(b), 
\quad \sigma \longmapsto \wh q_M^{\;\sigma} := \bigl( q_M^{k(\sigma)}, \EK_M(k(\sigma)) \bigr), \]
%
where we recall that $k(\sigma) \in S_{d_\pi}$ is defined by $k(\sigma) \in S_{d_\pi} \cap \im(\sigma)$.
From the diffeomorphism in \eqref{eq:splitting_inertia}, we see that 
$M\bigl( q^{k(\sigma)}, \EK(k(\sigma)) \bigr) \sharp M(F, d_\pi)$ 
gives a well-defined diffeomorphism type for each section $\sigma$.

\begin{thm} \label{thm:split}
Let $M$ have smooth splitting function $\wh q^{\; \bullet}_M \colon \sect(\pi) \to \wh{\cal{Q}}_{\dM}(b)$.
For each $\sigma \in \sect(\pi)$ there is a unique $H^*$-equivalence class of splitting 
%
\[ f_\sigma \colon M \cong M(\wh q^{\; \sigma}_M) \sharp M(F, d_\pi). \]
%
Consequently the map $\splt(M) \to \sect(\pi), [f] \mapsto \sigma(f)$ is a bijection.
\end{thm}

\begin{proof}
The proof is a refined version of the proof Theorem~\ref{thm:almost_split1}
and we adopt the notation of that proof so that $M$ has $3$-connected coboundary $W$.
Specifically, the proof of the existence of $f_\sigma$ is verbally the same,
except that now by \cite[Definition 2.50]{crowley02} we have
%
\[ \wh q_M^{\; \sigma} = \left(\del(R, -\lambda_R, \alpha_\psi), 
\frac{\lambda^{-1}_R\bigl( \alpha_\psi, \alpha_\psi \bigr) - \sigma(\lambda_R) }{8} \right). \]
%
Hence the splitting $W \cong W_\psi \natural W_F$ defines the splitting 
$f_\sigma \colon M \cong M(\wh q_M^{\; \sigma}) \sharp M(F, d_\pi)$.	

To show that splittings $f_0$ and $f_1$ defining the same section $\sigma$ are
$H^*$-equivalent, we consider the nonsingular characteristic form $(H_1, \lambda_1, \alpha_1)$.
The symmetric form $(H_1, \lambda_1)$ has a Lagrangian $L \subset H_1$ corresponding to $H^3(M)$ and 
hence $\alpha_1(L) = d_\pi \Z$.  The proof of Proposition \ref{prop:p2null} now shows that
$\frac{\tilde d_\pi}{4}$ divides $(\lambda_1^{-1}(\alpha_1, \alpha_1) - \sigma(\lambda_1))/8$.
Consequently the diffeomorphism type of $M_\phi$ is determined up to connected sum with
$\Sigma \in \dM \Theta_7$, and this shows that $f_0$ and $f_1$ are $H^*$-equivalent splittings.
%
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:classification_minimal}]
Let $\aut(b)$ act on $\wh{\cal{Q}}_{\dM}(b)$ by $F \cdot (q, s) = (q \circ F, s)$
and let $[q, s]$ denote the $\aut(b)$ equivalence class of $(q, s)$.
We define the map
%
\[ \beta \colon \wh{\cal{Q}}_{\dM}(b) \to 2T/\!\aut(b) \times \Q/\dM\Z, \quad (q, s) \mapsto ([\beta_q], s).  \]
%
Since $A(q) = s$, it follows from Theorem~\ref{thm:classification_of_q} that $[q_0, s_0] = [q_1, s_1]$
if and only if $\beta(q_0, s_0) = \beta(q_1, s_1)$.  

Now let $M_0$ and $M_1$ have smooth splitting functions 
$\wh q_i^{\; \bullet} \colon \sect(\pi) \to \wh{\cal{Q}}_{\dM}(b)$, $i = 0,1$.
By definition, these splitting functions descend to splitting functions
%
\[ \wh q_i^{\; \circ} \colon S_{d_\pi} \to \wh{\cal{Q}}_{\dM}(b). \]
%
By Theorem \ref{thm:split},
$M_0$ and $M_1$ are diffeomorphic if and only if there are sections $\sigma$ and $\sigma'$, a homotopy sphere
$\Sigma \in \dM\Theta$ and a diffeomorphism $M(\wh q_0^{\; \sigma}) \cong M(\wh q_1^{\; \sigma'}) \sharp \Sigma$.
But this happens if and only if there is an isomorphism $F \colon q_0^{\sigma} \cong q_1^{\sigma'}$ and
$\mu_0(k(\sigma)) = \mu_1(k(\sigma'))$; \ie if and only if
$\beta(\wh q_0^{\; \sigma}) = \beta(\wh q_1^{\; \sigma'})$.
The reduced splitting function of $M_i$, defined
in the introduction just prior to the statement of Theorem \ref{thm:classification_minimal}, is
the function,
%
\[ \ol{q^\circ}_{\!\!\!\!\!\;i} \colon S_{d_\pi} \to \bigl(2 T/\!\aut(b) \bigr) \times \Q/\dM \Z, 
\quad k \mapsto \beta(\wh q_i^{\;k}). \]
%
The above shows that $M_0$ and $M_1$ are diffeomorphic if and only if the images of $\ol{q^\circ}_{\!\!\!\!\!\;0}$
and $\ol{q^\circ}_{\!\!\!\!\!\:1}$ intersect and consequently this happens if and only if the images
of $\ol{q^\circ}_{\!\!\!\!\!\;0}$ and $\ol{q^\circ}_{\!\!\!\!\!\:1}$ coincide.
%
\end{proof}

%%%%%% old stuff %%%%%%
\begin{comment}
\[ \wh q^{\; \circ}_M \colon S_{d_\pi} \to \wh{\cal{Q}}_{\dM}(b), \quad k \mapsto (q^{\frac{d_\pi}{2}k}, \EK(k)). \]

\begin{proof}
\ref{it:splitting_function}  The proof is a refined version of the proof of
Lemma \ref{prop:almost_split1}.
The distillation of $M$ is defined via a $3$-connected coboundary $W$ of $M$
which is classified up to diffeomorphism by its
generalised intersection form $(H^4(W, \del W), \lambda_W, \alpha_W) = (H, \lambda, \alpha)$.  
In the definition of $\EK_M(k)$, a splitting 
$\psi \colon (H, \lambda, \alpha) \cong (R, \lambda_R, \alpha_\psi) \oplus (F, 0, \alpha|_F)$
corresponds to taking a splitting of $W = W_{(R, \psi)} \natural W_{(F, \psi)}$ where the boundary of 
$W_{(F, \psi)}$ is diffeomorphic to $M(F, d_\pi)$ and where the boundary of $W_{(R, \psi)}$ 
is diffeomorphic to $M(q, s)$ modulo $I_H(M)$ and where $q = q^{\frac{d_\pi}{2}k}$ and $s \in \Q/28 \Z$ reduces to $\EK(k) \in \Q/\dM \Z$.

\ref{it:splitting_criterion}  This follows from Theorem \ref{thm:classification}.

\ref{it:splitting_image}
The set $\wh q_M(S_{d_\pi}) \subset \wh{\cal{Q}}_{\dM}(b)$ is clearly a diffeomorphism invariant.
Conversely, suppose that $q_{M_0}(S_{d_\pi}) = q_{M_1}(S_{d_\pi})$.  Then by part \ref{it:splitting_criterion}
there is a diffeomorphism
%
\[ M_0 \cong M(F, d_\pi) \sharp M(q, s) \cong M_1  \]
%
for every $(q, s)$ such that $(q, \rho_{\wh d})(s) \in \cal{Q}_{\wh d}(b)$.
\end{proof}


Given a mod $28$ distillation $(q^\circ\!, \EK)$ of $(G, b, p)$ we have the {\em splitting function} of $(q^\circ\!, \EK)$,
%
\[ \wh q \colon S_{d_\pi} \to \wh{\cal{Q}}_{\dM}(b), \quad k \mapsto (q^{\frac{d_\pi}{2}k}, \EK(k)), \]
%
where we recall that $\dM = \gcd(\frac{\tilde d_\pi}{4}, 28)$.
%
\begin{thm} \label{thm:slitting}
We fix a base $(G, b, p)$ and consider $2$-connected $M$ with this base.  
\begin{enumerate}
\item \label{it:splitting_function}
For each $k \in S_{d_\pi}$ there are diffeomorphisms 
$f_k \colon M \cong M(q^{\frac{d_\pi}{2}k}, \EK_M(k)) \sharp M(F, d_\pi)$ 
with corresponding splittings $H^4(M) \cong T \oplus {\rm Im}(\sigma)$ where $k = k(\sigma)$ and $\sigma \in \sect(\pi)$.
\item \label{it:splitting_criterion}
There is a diffeomorphism $M \cong M(F, d_\pi) \sharp M(q, s)$ if and only if $(q, \rho_{\dM}(s)) \in \wh q_M(S_{d_\pi})$.
\item \label{it:splitting_image}
$M_0$ and $M_1$ are diffeomorphic if and only if 
$\wh q_{M_0}(S_{d_\pi}) = \wh q_{M_1}(S_{d_\pi}) \subseteq \wh{\cal{Q}}_{\dM}(b)$.
\end{enumerate}
\end{thm}
	
%In order to characterise the image of splitting functions, we define the following equivalence 
%relation on the mod $\wh d$ distillations of $(T, b)$.  This is based on the transformation
%rules for $(b, p)$-linked functions, \eqref{eq:lf_def}, and for 
%Gauss refinements \eqref{eq:delta_pi} and \eqref{eq:delta_8}.  We observe that both $\Delta(k, t)$
%and $\wt \Delta(k, t)$ depend only on the value of $q^{\frac{d_\pi}{2}k}$ and hence we can
%equally well define $\Delta(q, t)$ and $\wt \Delta(q, t)$.
%
%We shall say that two distillations $(q_0, s_0), (q_1, s_1) \in \cal{Q}_{\wh d}(b)$ are $d_\pi$-equivalent if 
%there is a $t \in T$ such that
%
%\begin{enumerate}
%\item $q_1 = (q_0)_{\frac{-d_\pi}{2}t}$,
%\item $s_1 = s_0 + \wt \Delta(q_0, t)$.
%\end{enumerate}
%
%From the definition, it is not obvious that $d_\pi$-equivalence is an equivalence relation.  However,
%we observe that $(q_0, s_0)$ is $d_\pi$-equivalent to $(q_1, s_1)$ if and only if there is a distillation
%$(q, \EK)$ such that $(q_0, s_0), (q_1, s_1) \in {\rm Im}(\wh q)$ and this shows that
%we indeed have an equivalence relation.  We define
%
%\[ [q, \EK] = {\rm Im}(\wh q) \subseteq \wh{\cal{Q}}_{\wh d}(b) \]
%
%to be the $d_\pi$-equivalence class defined by the image of $\wh q$.
%
%We observe that the diffeomorphism classes of 
%rational homotopy spheres $M(q, s)$ which appear in such a connected sum splitting of 
%$M$ form an equivalence class.  Namely, we define the equivalence relation 
%$(q_0, s_0) \simeq_{d_\pi} (q_1, s_1)$ if and only if there is a diffeomorphism
%
%\[ M(q_0, s_0) \sharp M(F, d_\pi) \cong M(q_1, s_1) \sharp M(F, d_\pi). \]
%
%We now describe how the distillation of $M$ determines this equivalence
%class of rational homotopy spheres.
%
%Let $\rho_{\dM} \colon \Q/28 \Z \to \Q/\dM \Z$ be reduction mod $\dM$.  By Remark \ref{rmk:p2null},
%$I_H(M(F, d_\pi)) \subset \dM \Theta_7$ and so if $(q, s)$ and $(q, s') \in \cal{Q}_{28}(b)$ are such that
%$\rho_{\dM}(s) = \rho_{\dM}(s')$, then there is a diffeomorphism $M(q, s) \sharp \Sigma \cong M(q, s')$
%where $\Sigma \in I_H(M(F, d_\pi))$ and hence a diffeomorphism
%
%\[ M(q, s) \sharp M(F, d_\pi) \cong M(q, s') \sharp M(F, d_\pi).\]
%
%It follows for $(q, s) \in \cal{Q}_{\dM}(b)$ that $M(q, s) \sharp M(F, d_\pi)$ gives a well-defined
%diffeomorphism type.  

\end{comment}
% subsection splitting functions (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\begin{comment}
\subsection{The homotopy classification}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We conclude this section by giving the classification of $2$-connected $7$-manifolds
up to orientation preserving homotopy equivalence from \cite[Theorem 6.11]{crowley02}.
We shall summarise the main ideas and refer the reader to \cite[\S 6]{crowley02} for details.

Evaluation with the universal spin characteristic class $p \in H^4(BSpin)$ defines an isomorphism
$p \colon \pi_4(BSpin) \cong \Z$.  To pass to the homotopy category, we must pass
from stable spin vector bundles which are classified by maps to $BSpin$ to stable spin
spherical fibrations which are classified by maps to $BG\an{4}$, the $3$-connective
cover of $BG$, where $BG$ is the classifying space for stable spherical fibrations.
Passing from a stable spin vector bundle to the underlying stable spin spherical
fibration defines a map $J \colon BSpin \to BG\an{4}$.
It is well known -cite that $\pi_{4}(BG\an{4}) \cong \Z/24$ and that the induced map
%
\[ J_* \colon \pi_4(BSpin) \to \pi_4(BG\an{4}) \]
%
is onto.  It follows that $H^4(BG\an{4}; \Z/24) \cong \Z/24$ and we denote the generator
which maps to the mod $24$ reduction of $p$ under $J^*$ by $p^J$.  
Evaluation with $p^J$ defines an isomorphism $\pi_4(BG\an{4}; \Z/24) \cong \Z/24$.
For a spin manifold
$Y$ with $p_Y \in H^4(Y)$, we let $p_Y^J \in H^4(Y; \Z/24)$ be the mod $24$ reduction of $p_Y$.

Better to introduce the Spivak normal fibration and spin Poincar\'{e} complexes.  In this
way $(q_M^J)^\circ$ will be a well defined homotopy invariant for all spin $7$-dimensional
Poincar\'{e} complexes.

%Now given $W_0$ and $W_1$, two 3-connected spin $8$-manifolds with two-connected boundaries
%$M_0$ and $M_1$, we can ask when there is a homotopy equivalence of pairs
%
%\[ f \colon (W_0, M_0) \simeq (W_1, M_1).\]
%
%The intersection form is clearly preserved under a homotopy equivalence of pairs, but this is
%also true for $p_{W_i}^J$ -cite{wall, crowley}.  Hence we define the $J$-characteristic
%form of $W$ to be the triple $(H^4(W, \del W), \lambda_W, p_W^J)$.

%\begin{lem} \label{lem:homotopy_handlebody}
%There is an isomorphism $F \colon H^4_{cpt}(W_1) \cong H^4_{cpt}(W_0)$ preserving the
%$J$-characteristic forms of $W_0$ and $W_1$ if and only if there is a homotopy equivalence
%of pairs $f \colon (W_0, M_0) \simeq (W_1, M_1)$ such that $F = f^*$.
%\end{lem}
\end{comment}
% subsection homotopy classification (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% section classification (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% (fold)
\section{Automorphisms of $H^4(M)$} \label{sec:auto}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The smooth classification Theorem \ref{thm:classification} implies that the number
of different smooth structures on the same 2-connected almost-smooth 7-manifold
corresponds to the number of different mod 28 Gauss refinements
of the linking family $(H^4(M), q_M^\circ, p_M)$.
The first estimate of the number of smooth structures on $M$, $\dM = \gcd(\frac{\td_\pi}{4}, 28)$, 
only counts smooth structures on $M$ modulo almost diffeomorphisms that act
trivially on $H^4(M)$. To get the full picture, we need to understand how
automorphisms of the quadratic linking family $q^\circ_M$ act on the Gauss refinements.
We begin this process in Section \ref{subsec:aut_b_on_gauss_functions}.

Conveniently enough, it turns out that this
problem can be reduced to understanding how automorphisms of the base
$(H^4(M), b_M, p_M)$ act on linked function: see Proposition \ref{prop:imtp} in
Section \ref{subsec:aut_q_on_gauss_functions}.
While we do not have a complete
description of this action in general we still have control up to a 
factor $2^r$ where $r \in {0, 1, 2}$ is explicitly defined in \eqref{eq:rdef}.  
Moreover it is feasible to understand it for explicit examples:
see Examples \ref{ex:r0}, \ref{ex:r2}, \ref{ex:split_vs_hyperbolics} and \ref{ex:r_for_split_sums}.
With Proposition \ref{prop:imtp} in hand, we proceed in Section \ref{subsec:computation_of_reactiviy}
to determine the reactivity of $2$-connected $M$ in terms of $r$ and the integer
$d_\kappa$ defined in Section \ref{subsec:auto_notation}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{Notation} \label{subsec:auto_notation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We begin by setting up some terminology.
Given a finitely generated abelian group $G$, $p \in 2G$, and
$b : T \times T \to \Q/\Z$ a torsion form,
let $\aut_b$ denote the group of isomorphisms $F : G \to G$ preserving $p$
and $b$. %, \ie $F(p) = p$ and $F_{|T}$ is an automorphism of $(T,b)$.
If $q^\circ$ is a family of quadratic refinements of $(G,b,p)$, let
$\aut_{q^\circ}$ be the subgroup of $\aut_b$ that preserves $q$ too.

Let $\pi \colon G \to G/T$ be the projection to the free quotient of $G$.  
Let $\shearb \subseteq \aut_b$ be the subgroup of ``pure shears'', \ie $F$
acting trivially on $T$ and $G/T$. In other words,
$F = \Id_G + \rho \circ \pi$ for some homomorphism
$\rho : G/T \to T$ such that $\rho(\frac{\pi(p)}{d_\pi})$ is $d_\pi$-torsion
(the last condition is equivalent to $F(p) = p$); so actually $\shearb$
does not depend on $b$ at all.
Similarly let $\shearq = \shearb \cap \aut_{q^\circ}$, the subgroup of shears in
$\aut_{q^\circ}$. For $h \in \divs_2$,
$(F^\#q)^h = q^{F(h)} = q^{h + \rho(\frac{\pi(p)}{2})} = q^h_{-\rho(\frac{\pi(p)}{2})}$, 
%$(F^\#q)^h = q^{F(h)} = q^{h + \rho(\pi(p)/2)} = q^h_{-\rho(\pi(p)/2)}$, 
so for $F$ to preserve $q$ we need
$\rho(\frac{\pi(p)}{2}) = 0$. 
%$\rho(\pi(p)/2) = 0$
Hence $\shearq$ simply corresponds to
homomorphisms $\rho : G/T \to T$ such that 
$\rho(\frac{\pi(p)}{d_\pi})$
%$\rho(\pi(p)/d_\pi)$
is $\frac{d_\pi}{2}$-torsion. In particular: $\shearq$ actually depends
on neither $q^\circ$ nor $b$ but only on $p$, and if $F \in \shearb$
then $F^2 \in \shearq$.

We say that a cyclic subgroup $C \subseteq T$ is a \emph{split summand}
if $T$ is a direct sum of $C$ and its $b$-orthogonal complement.
We call $x \in T$ \emph{split} if it generates a split summand; this is equivalent to
\[ b(x,x) \not= 0 \mod \frac{1}{n} , \]
where $n$ is the order of $x$. (Then $x$ cannot be divisible by any integer
dividing $n$.)

%Recall that a torsion form $b$ on a finite group $T$ can always be decomposed
%as an orthogonal direct sum of its $s$-primary parts $T_s$, summing over all
%primes $s$. If $s$ is an odd prime then $T_s$ is an orthogonal direct sum of
%cyclic subgroups, but a splitting of 

Given the element $p \in G$ we consider the following notions of its
divisibility (if $p$ is a torsion element we set all three integers to be 0):
\begin{align*}
%
d_{\phantom \pi} & :=
{\rm Max} \{s \in \Z \, | \, r \textrm{ divides } p \in G \}, \\
d_\pi & :=
{\rm Max} \{s \in \Z \, | \, r \textrm{ divides } \pi(p) \in G/T \}, \\
\mdiv & :=
{\rm Max} \{ s \,|\,s, \mpl \in \Z, \; r\mpl^2 \text{~divides~} m p \in G \} .
\end{align*}
We have an obvious chain of divisibilities
%
\[ 2 \mid d \mid \mdiv \mid d_\pi. \]
%
Further $d = d_\pi$ if and only if $\mdiv = d_\pi$, since the latter
implies that the maximum in the definition of $\mdiv$ is attained with
$\mpl = 1$.
%

%
%\noindent
%For a prime $q$ and $s$ an integer, let $\ord_qs$ be the exponent of $q$ in
%the prime factorisation of $s$; \eg~$\ord_q q^j = j$.
For an integer $s$, let $\ord_2 s$ be the exponent of $2$ in
the prime factorisation of $s$; \eg~$\ord_2 2^j = j$.

\begin{defn} \label{def:extremal_exponent}
%Let $q$ be a prime.
%A non-negative integer $j$ is an {\em $q$-extremal exponent} for $(G, p)$ if
%for some $N$ such that $\mdiv N^2$ divides $Np$, $\ord_qN = j$.
A non-negative integer $e$ is a {\em $2$-extremal exponent} for $(G, p)$ if
for some $\mpl$ such that $\mdiv \mpl^2$ divides $\mpl p$, $\ord_2 \mpl = e$.
\end{defn}

\begin{ex}
Let $p = (2^a, 2^c) \in \bbz \times \bbz/2^b\Z$ with $a,  b \geq c \geq 1$.
Then $d_\pi = 2^a$, $\mdiv = \max(2^c, 2^{a-b+c})$ and $d = 2^c$.
The 2-extremal exponents are 0 for $a \leq b$, and $b-c$ for $a \geq b$.
\end{ex}


%\begin{ex} \label{ex:divisibilities}
%%%%%%
%Let $\alpha  \in \pi_3(SO(3)) \cong \Z$, let $S\alpha \in \pi_3(SO(4))$ be
%its stabilisation and let $M = S^3 \tilde \times_{S\alpha} S^4$ be the total
%space of the sphere bundle associated to $S\alpha$.  Then by \cite{milnor56}
%$(H^4(M), p_M) \cong (\Z, 2\alpha)$ and so
%$d(p_M) = \dpi{M} = \dinfty{M} = 2 \alpha$.
%
%\end{ex}

% subsection divisibilities (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% (fold)
\subsection{The action of $\aut_b$ on linked functions} \label{subsec:aut_b_on_gauss_functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Given $F \in \aut_b$ and any $k \in \divsp$, set 
$t := F(k) -k \in T$ (not necessarily $d_\pi$-torsion, unless $F_{|T}$
is the identity) and $\beta_k := p - d_\pi k$, and let
\begin{equation}
\label{eq:pnaive}
P(F) := d_\pi^2 b(t, t) - 2d_\pi b(\beta_k, t) \in \Q/2d_\pi\Z .
\end{equation}
In other words, $P(F) = \Delta(k, t)$ from \eqref{eq:Delta}.
Equivalently, we can define $P(F)$ by
\begin{equation}
\label{eq:pdiff}
F^\# \lf = \lf + \frac{P(F)}{8} \mod \frac{d_\pi}{4}\Z
\end{equation}
for any linked function $\lf$ (use that $(F^\# \lf)(k) = \lf(F(k)) =
\lf(k + t) = g(k) + \frac{\Delta(k,t)}{8}$ by the condition \eqref{eq:lf_def}
for $\lf$ to be a linked function.)
The first definition, \eqref{eq:pnaive}, is independent of $\lf$ and the
second, \eqref{eq:pdiff}, of $k$, so in fact $P$ depends on neither.
If $F$ preserves a family of quadratic refinements, then
taking $\lf$ to be a Gauss refinement of that family shows that $P$
takes values in $8\Z/2d_\pi\Z$ (in the next subsection we study a corresponding
$8\Z/2\td_\pi\Z$-valued function $\tp$).
Even if $F$ does not preserve a family of quadratic refinements, the
fact that the Arf invariant $\!\!\mmod\! \quart \Z$ of a quadratic refinement
of $(b,p)$ depends only on $(b,p)$ itself shows that $P$ takes values in $2\Z$. It is also clear
from \eqref{eq:pdiff}, or from \eqref{eq:pnaive} together with
\eqref{eq:cocycle}, that $P$ is a homomorphism
$\aut_b \to 2\Z/2d_\pi\Z$.

Let $j_\pi = \ord_2 d_\pi$, and $j_\msub = \ord_2 \mdiv$.

\begin{lem}
\label{lem:autb}
$P(\aut_b) \subseteq \mdiv \Z/2d_\pi \Z$.
If $b$ lacks a split $2^{2e+j_\msub}$ summand for some $2$-extremal
exponent $e$, then $P(\aut_b) \subseteq 2\mdiv \Z/2d_\pi \Z$.
\end{lem}

\begin{proof}
Pick some $y \in G$ such that $\mpl^2\mdiv y = m p$. Then $s := Fy - y$ is
an $\mpl^2\mdiv$-torsion element. It suffices to show that
\begin{equation}
\label{eq:pdinfty}
P(F) = \mpl^2\mdiv^2 b(s, s) \mod 2\mdiv ,
\end{equation}
because the RHS is $\mdiv$ if the 2-primary part of $s$ is split, and $0$
otherwise.

Note that $\pio := \frac{d_\pi}{\mdiv}$ and $\frac{\mpl}{\pio}$ are
integers. Let $k := \frac{\mpl}{\pio}y$. Then
$k \in \divsp$, and $\pio t = \mpl s$, so \eqref{eq:pnaive} implies 
\[ P(F) = \pio^2 \mdiv^2 b(t, t) - 2 \pio \mdiv b(\beta_k, t)
= \mdiv^2(\mpl s, \mpl s) - 2\mdiv(\beta_k, \mpl s) %= \mdiv^2m^2b(s,s)
\mod 2\mdiv . \]
Since $\beta_k = p - d_\pi k$ is $\mpl$-torsion, \eqref{eq:pdinfty} and the
result follows.
\end{proof}

%Let $\shearb \subseteq \aut_b$ be the subgroup of ``pure shears'', \ie $F$
%acting trivially on $T$ and $G/T$. In other words,
%$F = \Id_G + \rho \circ \pi$ for some homomorphism
%$\rho : G/T \to T$ such that $\rho(\frac{\pi(p)}{d_\pi})$ is $d_\pi$-torsion
%(the last condition is equivalent to $F(p) = p$); so actually $\shearb$
%does not depend on $b$ at all.

If $F \in \shearb \subseteq \aut_b$, \ie $F = \Id_G + \rho \circ \pi$ for some
homomorphism $\rho : G/T \to T$, then
$t = F(k) - k = \rho(\frac{\pi(p)}{d_\pi})$
independent of the choice of $k \in \divsp$.
Since $\frac{\pi(p)}{d_\pi} \in G/T$ is
a primitive element of a free abelian group, we can prescribe its image
under a homomorphism $\rho$ arbitrarily. Determining the image
$P(\shearb)$ therefore amounts to computing the RHS of \eqref{eq:pnaive}
for all $d_\pi$ torsion elements $t \in T$.

\begin{lem}
\label{lem:aut0} 
$4\mdiv \Z/2d_\pi \Z \subseteq P(\shearb)$. If $j_\pi \not = j_\msub + 1$
or if $b$ has no split $2^{j_\pi}$ summand, then
$2\mdiv \Z/2d_\pi \Z \subseteq P(\shearb)$.
\end{lem}

\begin{proof}
The key claim is that there exists a $d_\pi$-torsion
element $t$ such that $b(\beta_k, t) = \frac{1}{\pio}$, where
$d_\pi = \pio \mdiv$.
By the non-degeneracy of $b$, this is equivalent to $\beta_k$ having order at
least $\pio$, and not being divisible by more than $\mdiv$. That any divisor of
$\beta_k$ also divides $\mdiv$ is obvious, and if $\mpl\beta_k = 0$ then
$\mpl p$ is divisible by $\mpl d_\pi$, which indeed implies $\pio \mid \mpl$ by
the definition of $\mdiv$.

Let $\rho$ be any homomorphism $G/T \to T$ mapping
$\frac{\pi(p)}{d_\pi} \mapsto t$, and
$F := \Id_G + \rho \circ \pi \in \shearb$.
If the 2-primary part of $t$ does
not generate a split $2^{j_\pi}$ summand then $d_\pi^2 b(t, t)$ is divisible by
$2d_\pi$, so 
\[ P(F) = d_\pi^2b(t,t) - 2d_\pi b(\beta_k, t) = 2\mdiv \mod 2d_\pi , \]
and we are done. Otherwise
$P(F) = d_\pi - 2\mdiv = (\pio-2)\mdiv \mmod 2d_\pi$. The subgroup this
generates is precisely $n\mdiv\Z/2d_\pi\Z$, where $n = \gcd(\pio{-}2, 2\pio) =
\gcd(\pio{-}2,4)$. Clearly $n$ is 1 or 2 except when $j_\pi = j_\msub + 1$,
in which case $n = 4$.
\end{proof}

Lemmas \ref{lem:autb} and \ref{lem:aut0} imply that the following is well-defined.

\begin{defn} \label{def:r}
Define $r(G,p,b) \in \{0,1,2\}$ by
\begin{equation}
\label{eq:rdef}
\im P = 2^r \mdiv \Z/2d_\pi \Z .
\end{equation}
\end{defn}

\begin{rmk} \label{rmk:values_of_r}
Lemmas \ref{lem:autb} and \ref{lem:aut0} provide necessary conditions for $r = 0$ or $r = 2$.
In particular, if $G$ has no 2-torsion then $r = 1$.  
%Examples \ref{ex:r0} and \ref{ex:r2}
The next examples show that there are bases with $r = 0$ and bases with $r=2$.
\end{rmk}


\begin{ex}
\label{ex:r0}
Let $G = \Z \oplus \Z/2^j$, $b = \an{\frac{1}{2^j}}$ and $p = (2^j, 0)$
(so $d_\pi = \mdiv = 2^j$).
Then the shear $F : (x,y) \mapsto (x, x+y)$ has $P(F) = 2^j \mmod 2^{j+1}$,
\ie $P(F) = \mdiv \mmod 2d_\pi$. Thus $r = 0$.
\end{ex}

\begin{ex}
\label{ex:r2}
Let $G = \Z \oplus \Z/2^j $, $b = \an{\frac{1}{2^j}}$ and
$p = (2^j, 2^{j-1})$ (so $d_\pi = 2^j$,
while $\mdiv = 2^{j-1}$). Now any $t \in T$ has
$d_\pi^2 b(t,t) + 2d_\pi(\beta_k, t) = 0 \mmod 2^{j+1}$, so $r = 2$.
\end{ex}

%\begin{rmk}\label{rmk:r=2}
%The situation when $P(\shearb) = 4\mdiv \Z/2d_\pi\Z$ is rather special;
%the condition that
%$\frac{d_\pi}{2}b(t,t) = b(\beta_k, t) \in \frac{1}{2}\Z/\Z$ for every
%$d_\pi$-torsion element $t$ means that one may then choose a splitting
%$\sigma$ of $G$ (see Remark \ref{rmk:projections} so that in a splitting of
%$T$ into elementary pieces, $\beta_{k(\sigma)}$ has component
%$2^{j_\pi {-} 1}$ in every diagonal $\Z/2^{j_\pi}\Z$ summand and $0$ in all
%other 2-torsion components.
%%It would be good to understand whether it can happen that
%%$2\mdiv \subseteq P(\aut_b)$ in this situation.
%\end{rmk}

% Failed example
%
%\begin{ex}
%Let $G = \Z \oplus \Z/16 \Z \oplus \Z/4\Z$
%with $p = (16, 8, 2)$ (so $d_\pi = 16$, while $\mdiv = 8$),
%and linking form $(\diag(\frac{1}{16}, \frac{\epsilon}{4})$.
%Any $d_\pi$-torsion element $t$ has
%$d_\pi^2 b(t,t) + 2d_\pi(\tau p, t) = 0 \mmod 32$, so $P(\shearb)$ is
%trivial (and thus equal to $4\mdiv\Z/2d_\pi\Z$).
%If $\epsilon = -1$ we can define an automorphism $A$ of $(T,b)$ by
%\[ \begin{pmatrix} 3 & 3 & \\ 4 & 1 \end{pmatrix}. \]
%...
%\end{ex}
% subsection aut_b_on_gauss_functions (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{The action of $\aut_{q^\circ}$ on Gauss refinements} \label{subsec:aut_q_on_gauss_functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now let $q^\circ$ be a family of quadratic refinements of the base $(G, b, p)$, and let
$\aut_{q^\circ}$ denote its group of automorphisms.
%and $\shearq$ the subgroup acting trivially on $T$, \ie pure shears
%such that $\rho(\frac{\pi(p)}{d_\pi})$ is $\frac{d_\pi}{2}$-torsion.
For an automorphism $F \in \aut_{q^\circ}$ we define $\tp(F) \in 8\Z/2\td_\pi \Z$ by
\begin{gather*}
\tp(F) = P(F) \mod 2d_\pi,
\\ \tp(F) = 0 \mod 8 .
\end{gather*}
Equivalently, $\tp(F) = \tdelta(k, t)$ for any $k \in \divsp$ and
$t := F(k) - k$. Now $\tp$ is a homomorphism
$\wt P \colon \aut_{q^\circ} \to 8\Z/2\td_\pi\Z$, such that
%
\begin{equation} \label{eq:P_tilde}
F^\# \gr = \gr + \frac{\tp(F)}{8} \mod \frac{\td_\pi}{4}\Z 
\end{equation}
for any Gauss refinement $\gr$ of $q^\circ$.

%\begin{rmk}
%Even though the first definition of $\tp(F)$ does not involve $q$ directly,
%it only makes sense if $F$ preserves some $q$, since otherwise
%there is no guarantee that $P(F) = 0 \mod \gcd(2d_\pi, 8)$. On the other hand,
%if one tries to define $\tp(F)$ as $\tdelta(k, t)$ or $8(F^\# \gr - \gr)$
%relative to some arbitrary $q$, then there is no guarantee that the result is
%independent of $k$ or $g$ respectively.
%
%A related issue is: How many different linking families might $F$ preserve?
%Two linking families refining the same $(p,b)$ are related by a 2-torsion
%element, so the answer would seem to depend on how $F$ permutes those.
%For example, if $F$ is a pure shear %(acting trivially on $T$)
%then it preserves some $q$ if and only if it preserves all of them.
%\end{rmk}

%Let $\shearq \subseteq \aut_{q^\circ}$ be the subgroup acting trivially on $T$ and
%$G/T$, \ie pure shears $F = \Id_G + \rho \circ \pi$. For $h \in \divs_2$,
%$(F^\#q)^h = q^{F(h)} = q^{h + \rho(\frac{\pi(p)}{2})}
%= q^h_{-\rho(\frac{\pi(p)}{2})}$, so for $F$ to preserve $q$ we need
%$\rho(\frac{\pi(p)}{2}) = 0$. Hence $\shearq$ simply corresponds to
%homomorphisms $\rho : G/T \to T$ such that $\rho(\frac{\pi(p)}{d_\pi})$
%is $\frac{d_\pi}{2}$-torsion. In particular: $\shearq$ actually depends
%on neither $q$ nor $b$ but only on $p$, and if $F \in \shearb$
%then $F^2 \in \shearq$.

We can get some control on the image of the shear subgroup
$\shearq \subseteq \aut_{q^\circ}$ just from the observation that $F^2 \in \shearq$ for any $F \in \shearb$.
 
\begin{lem}
\label{lem:imtp0}
$\tp(\shearq) \supseteq 4\mdiv\Z/2\td_\pi\Z$.
\end{lem}

\begin{proof}
The proof of Lemma \ref{lem:aut0} showed that we can achieve
$P(F) = 2\mdiv$ or $P(F) = 2\mdiv + d_\pi$ for some $F \in \shearb$.
Then $F^2 \in \shearq$ has $P(F^2) = 4\mdiv$.
\end{proof}

\begin{prop}
\label{prop:imtp}
$\im \tp = \{ n \in 8\Z/2\td_\pi\Z : n \mmod 2d_\pi \in \im P \}
= \lcm(8, 2^r\mdiv)\Z/2\td_\pi \Z$.
\end{prop}

\begin{proof}
If $d_\pi$ is not divisible by 4 then the result follows from
Lemma \ref{lem:imtp0}.
If $4 \mid d_\pi$ and $F \in \aut_b$, then $P(F) = n \in 8\Z/2d_\pi\Z$ implies
that for any $k \in \divsp$, $t := F(k)- k$ and $h := \frac{d_\pi}{2}k$,
we get $q^{h}(\frac{d_\pi}{2}t) = 0$.
Thus $(F^\# q)^h$ and $q^h$, which have equal inhomogeneity $\beta_h = p - 2h$
by definition, also have equal Arf invariant by \eqref{eq:arf_shift}.
Therefore by Theorem \ref{thm:classification_of_q} there is an
automorphism $F_T$ of $(T,b)$ such that $(F^\# q)^h \circ F_T = q^h$
(necessarily $F_T$ fixes $\beta_h$).

Now suppose that $\sigma$ is a section of $\pi$, and
$k \in \im \sigma \cap \divsp$ (\cf Remark \ref{rmk:projections}).
Then $G \cong \im \sigma \oplus T$, and we may define
$\Id_{\im \sigma} + F_T \in \aut_b$. This fixes $k$ and $h$, so the composition
$F' := F \circ (\Id_{\im \sigma} + F_T)$ has
\[ (F'^\#q)^h = \left((\Id_{\im(\sigma)} + F_T)^\#F^\#q\right)^h
= (F^\#q)^h \circ F_T = q^h . \]
Hence $F' \in \aut_{q^\circ}$, and $F'(k) = F(k)$ implies $P(F') = P(F)$.
\end{proof}

\begin{ex}
\label{ex:r0'}
For the base $(\Z \oplus \Z/2^j, \an{\frac{1}{2^j}}, (2^j, 0))$ of Example \ref{ex:r0}
let $q^\circ$ be the refinement with $q^{(2^{j-1}, 0)} = \anq{1}{2^{j+1}}$.
The isomorphism $F$ of the base in Example \ref{ex:r0} does not preserve $q^\circ$: 
if $j > 1$ then $F$ alters the homogeneity defect of $q^{(2^{j-1}, 0)}$ and if $j=1$ 
$F$ alters the Arf invariant.  However, if $j \geq 3$ then 
$F' : (x,y) \mapsto (x, \; x + (2^{j-1} {+} 1) y)$ is an
isomorphism of $q^\circ$ with $\tp(F') = P(F) = 2^m \mmod 2^{j+1}$.
\end{ex}

\noindent
The following examples illustrate that $r$, and hence $\im \tp$, can depend
on $b$ as well as $(G,p)$.

\begin{ex} \label{ex:split_vs_hyperbolics}
Let $G = \Z \oplus (\Z/2^j)^2$ and $p = (2^j, 0, 0)$
(so $d_\pi = \mdiv = 2^j$).
Choosing the torsion form
$b_0 = \an{\frac{1}{2^j}} \oplus \an{\frac{1}{2^j}}$ on $TG$,
using Example \ref{ex:r0} shows that $r = 0$.

%Let $G_1 = \Z \oplus (\Z/2^j \Z)^2$
Let $b_1$ be the hyperbolic torsion form on $TG$ with matrix
%
\[ \begin{pmatrix} 0 & 2^{-j} \\ 2^{-j} & 0 \end{pmatrix} . \]
%
%and $p_1 = (2^j, 0, 0)$ (so $d_\pi = \mdiv = 2^j$).
Since $d_\pi = \mdiv$, it follows that $r_1 = 0$ or $1$.  But $b_1$ contains no split
cyclic summands, and so by Lemma \ref{lem:aut0}, we conclude that $r_1 = 1$.
\end{ex}

The next example shows that $r$ cannot be determined merely from the type
of splitting of~$b$ (cyclics versus hyperbolics), but can depend on the
%precise values of the coefficients.
isomorphism classes of split cyclic summands.
 
\begin{ex} \label{ex:r_for_split_sums}
Let $G = \Z \oplus \Z/8 \oplus \Z/64 \oplus \Z/512$ with torsion form
$\anb{1}{8}\oplus \anb{1}{64} \oplus \anb{\epsilon}{512}$ 
($\epsilon = \pm 1$),
and $p = (64, 0, 8, 0)$. Then $d_\pi = 64$ and $\mdiv = 8$,
so $r$ is 0 or 1. The 2-extremal exponents are 0 and~3. If $F \in \aut_b$ then
by \eqref{eq:pdinfty}
\[ P(F) = \mdiv \!\mmod 2\mdiv
\Leftrightarrow (\Id-F)(1,0,0,0) \textrm{ split 512-torsion}
\Leftrightarrow (\Id-F)(8,0,1,0) \textrm{ split 8-torsion} . \]
Thus if $r = 0$ there must be some automorphism $f$ of $(T,b)$ such that
$(\Id - f)(0,1,0)$ plus a split 8-torsion element is divisible by 8, \ie
$f(0,1,0) = (a, 8b + 1, 8c)$ with $a$ odd. If $\epsilon = +1$ then the would-be
image has norm $\frac{17}{64}$ for any $a, b, c$, so there can be no such $f$;
hence $r = 1$. On the other hand, if $\epsilon = -1$ we can define
such an $f$ by the matrix
\[ \begin{pmatrix} 1 & -8 & 0 \\ 1 & 1 & 8 \\ 1 & 1 & 1\end{pmatrix}. \]
%Setting $F = \Id_\Z + \rho + f$ with $\rho(1,0,0,0) = (0, 0, -1) \in T$
Setting $F = \Id_\Z + \rho + f$ with $\rho : \Z \to T, n \mapsto (0, 0, -n)$
makes $\Id - F$ map $(8,0,1,0) \in G$ to the split 8-torsion element
$(1,0,0) \in T$ (and $(1,0,0,0)$ to $(0,0,-1)$),
so $P(F) = \mdiv \mmod 2\mdiv$, and $r = 0$.
\end{ex}

\begin{cor} \label{cor:counting_Gauss_refinements}
Modulo the action of $\aut_{q^\circ}$, the number of possible Gauss refinements of
$(G,q^\circ\!,p)$ is
\[ \NumB{\frac{2^r \mdiv}{8}} , \]
and the number of possible mod 28 Gauss refinements is
\[ \gcd\left(28, \NumB{\frac{2^r \mdiv}{8}} \right) . \]
\end{cor}

\begin{rmk}
Notice that Corollary \ref{cor:counting_Gauss_refinements} combined with 
Theorems \ref{thm:almost_classification} and \ref{thm:classification} gives the computation of
the inertia group $I(M)$ for $2$-connected $M$ from Theorem \ref{thm:i_and_r}.  
\end{rmk}

% subsection aut_q_on_gauss_functions (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{The computation of reactivity} \label{subsec:computation_of_reactiviy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this subsection we use Proposition \ref{prop:imtp} to prove lower bounds on the
reactivity of every spin $7$-manifold $M$.  When $M$ is $2$-connected we also prove
that this lower bound is sharp and so compute the reactivity of $2$-connected $M$.
%
Recall from Section \ref{subsec:inertia_and_reactivity2} that if $f$ is a self-almost diffeomorphism of $M$, 
then the mapping torus $T_f$ is
almost smooth, the spin characteristic class $p_{T_f} \in H^4(T_f)$ is well defined and so is the integer
%
\[ p^2(f) = \an{p_{T_f}^2, [T_f]} \in 8\Z.\]
%
%$p^2(f)$ is divisible by 8 a priori because $p_{T_f}$ is characteristic and
%$\sign(T_f) = 0$.
%
The bridge between the algebraic arguments of
\S \ref{subsec:aut_b_on_gauss_functions}-\ref{subsec:aut_q_on_gauss_functions}
and the computation of reactivity of $M$, as defined in \eqref{eq:defR}, is the
following
%\noindent
%Then we establish that the residue of $p^2(f) \mmod 2\td_\pi$
%is determined by the action of $f^*$ on $H^4(M)$.
%Together with Proposition \ref{prop:imtp} this completely determines
%the image of $\ADiff_{}$ under $p^2$.

\begin{lem} \label{lem:p2mod}
$\tp(f^*) = p^2(f) \mmod 2\td_\pi$ for any self-almost diffeomorphism of $M$.
\end{lem}

\begin{proof}
Let $f \colon M \acong M$ be an almost diffeomorphism and let $W$ be a $3$-connected 
spin coboundary for $M$.  By definition we can use $W$ to compute the generalised
Eells-Kuiper invariant of $M$: $\EK_M = \gr_W$~mod~$\dM$.
We can also use $f$ to glue two copies of $W$ together along $M$
and form the almost smooth spin manifold
$X : = (-W) \cup_f W$.  By Lemma \ref{lem:T_and_X}, $p^2(f) = p^2_X$.  From
the definition of $\wt P(f)$ in \eqref{eq:P_tilde} and from the comparison of Gauss refinements
in \eqref{eq:compare} we have
%
\[ \wt P(f^*) \equiv 8(\gr_W - (f^*)^\sharp g_W) \equiv p^2_X - \sigma(X) \equiv p^2_X \equiv p^2(f) \mod 2 \tilde d_\pi ,\]
%
where $\sigma(X) = \sigma(W) - \sigma(W) = 0$ by Novikov additivity, and are all equivalences are mod $2 \tilde d_\pi$.
\end{proof}

%Together with Proposition \ref{prop:imtp}, Lemma \ref{lem:p2mod} completely determines
%the image of $\ADiff_{}(M)$ under $p^2$.
%Recall that $\Diff_H \subseteq \Diff_{}$ and $\ADiff_H \subseteq \ADiff_{}$
%are the subgroups acting trivially on cohomology.

\begin{cor}
\label{cor:p2image}
For any closed spin 7-manifold $M$ we have that:
\begin{enumerate}
\item \label{it:p2null-H} 
$R_H(M)$ is divisible by $2\td_\pi$;
\item \label{it:p2null}
$R(M)$ is divisible by $\lcm(8, 2^r \mdiv)$;
\item \label{it:p2null-DiffH}
$R^{\Diff}_H(M)$ is divisible by $\lcm(224, 2\td_\pi)$; \vspace{0.75 mm}
\item \label{it:p2null-Diff}
$R^{\Diff}(M)$ is divisible by $\lcm(224, 2^r \mdiv)$.
\end{enumerate}
If $M$ is 2-connected then equality holds in each case.
%\begin{enumerate}
%\item \label{it:p2null} $\im p^2(\ADiff_H(M)) = 2\td_\pi \Z$.
%\item $\im p^2(\Diff_H(M)) = \lcm(224, 2\td_\pi) \Z$.
%\item $\im p^2(\ADiff_{}(M)) = \lcm(8, 2^r \mdiv) \Z$.
%\item $\im p^2(\Diff_{}(M)) = \lcm(224, 2^r \mdiv) \Z$.
%\end{enumerate}
\end{cor}

\begin{proof}
%The first two are immediate from Proposition \ref{prop:p2null} and
%Lemma \ref{lem:p2mod}, while the last two also rely on Proposition \ref{prop:imtp}.
For part \ref{it:p2null-H},
Lemma \ref{lem:p2mod} shows that $R_H(M) \mid 2 \td_\pi$, while 
Proposition \ref{prop:p2null} shows that
$R_H(M) \mid  2 \td_\pi$ if $M$ is 2-connected.
For part \ref{it:p2null}, let $M$ have refinement $(G, q^\circ, p)$.
Proposition \ref{prop:imtp} computes the image ${\rm Im} \wt P = \lcm(8, 2^r\mdiv) \Z/2\td_\pi$, so Lemma \ref{lem:p2mod} gives $\lcm(8, 2^r\mdiv) \mid R(M)$.
On the other hand, if $M$ is 2-connected then
Theorem \ref{thm:almost_classification} states that every automorphism of
$(G, q^\circ, p)$ is realised by an almost diffeomorphism $f \colon M \acong M$,
and so part \ref{it:p2null-H} and Proposition \ref{prop:imtp} imply the equality.
Parts \ref{it:p2null-DiffH} and \ref{it:p2null-Diff} follow from parts \ref{it:p2null-H} and \ref{it:p2null} 
and Lemma~\ref{lem:smooth_reactivity}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:i_and_r}]
The computation of $R(M) = \lcm(8, 2^r\mdiv)$ is given in Corollary \ref{cor:p2image} \ref{it:p2null}.
Then $I(M) = \Num\big(\frac{2^r\mdiv}{8}\big)\Theta_7$
by Proposition \ref{prop:role_of_reactivity}~\ref{it:r_of_r}.
By Remark \ref{rmk:values_of_r}, $r=1$ if $TH^4(M)$ is of odd order.  
%From the existence part of Theorem \ref{thm:classification}
%applied to the examples of bases in Example \ref{ex:r0} and \ref{ex:r2}, we conclude that there are $2$-connected
%$M$ where $r = 0$ and $2$-connected $M$ where $r = 2$.
\end{proof}

% subsection p^2 (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% section auto (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\section{Examples} \label{sec:examples}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ever since Milnor's discovery of exotic 7-spheres \cite{milnor56}, $2$-connected
$7$-manifolds have provided interesting examples in topology and geometry.
In this section we discuss a number of examples of $2$-connected $7$-manioflds.
In Section \ref{subsec:sphere_bundles} we consider the total spaces of $3$-sphere bundles over $S^4$ 
and their connected sums.
In Section \ref{subsec:geometry} we mention some examples admitting interesting metrics.
In Section \ref{subsec:the} we give examples which are
tangentially homotopy equivalent but not homeomorphic.
Finally in Section \ref{subsec:monoid} we present a refinement of Wilkens' list \cite[Theorem 1]{wilkens72} 
of the indecomposable generators
for the monoid of almost diffeomorphism classes of $2$-connected $7$-manifolds.

%We recall that $2$-connected every rational homotopy sphere
%$M$ is diffeomorphic to $M(q, s)$ where $s \in \Q/28\Z$ and $A(q) \equiv s$~mod $\Z$.
%In order to more precisely describe certain $M(q, s)$, we establish notation for refinements on cyclic groups.  
%Let $T := \Z/r\Z$. Given $\theta \in \Z/r\Z$ coprime to $r$, let
%$\anb{\theta}{r}$ denote $T$ equipped with the
%torsion form
%\[ b(x,y) := \frac{\theta xy}{r} \in \Q/\Z . \]
%Given $\theta \in \Z/2r$ coprime to $r$ and $\gamma \in \Z/r$
%(so that $2 \gamma \in \Z/2r$), we define a quadratic refinement
%$\anq{\theta}{2r}_\gamma$ of $\anb{\theta}{r}$ by
%
%\[ \anq{\theta/2r}_\gamma \colon \Z/r \to \Q/\Z, \quad
%x \longmapsto \theta \frac{x^2 + 2\gamma x}{2r}. \]
%\[ q(x) := \theta \frac{x^2 + 2\gamma x}{2r} \in \Q/\Z. \]
%

% (fold)
\subsection{$3$-sphere bundles over $S^4$ and their connected sums} \label{subsec:sphere_bundles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Following the notation of \cite{crowley13}, let $(n, p)$ be integers with same parity
and let $M_{n, p} := S(\xi_{n, p})$ denote the total space of the $3$-sphere bundle 
over $S^4$ which there corresponding vector bundle $\xi_{n, p}$ has Euler class $e(\xi_{n, p}) = n \in H^4(S^4)$
and spin characteristic class $\frac{p_1}{2}(\xi_{n, p}) = p \in H^4(S^4)$.  
By definition, we $M_{0, p} = M(\Z, p)$.  Using \eqref{eq:alg_qdef} and \eqref{eq:grchat}
and recalling the notation of Example \ref{ex:cyclic_b_and_q}, we compute that for $n \neq 0$ there is a diffeomorphism
%
\[ M_{n, p} \cong M\!\left( \biganq{-1}{2n}_{-p}, \; \frac{p^2-|n|}{8n} \right). \]
%
\begin{ex} \label{ex:Milnor_sphere}
The Milnor sphere, $\Sigma_{\rm M} : = M_{1, 3}$, is homeomorphic to $S^7$ be not diffeomorphic 
to $S^7$ since $\mu(\Sigma_{\rm M}) = 1 \neq 0$~mod $28$: see \cite{milnor56} and \cite{eells62}.
\end{ex}
\noindent
In \cite{crowley03} the total spaces of $3$-sphere bundles over $S^4$ were classified up to
homotopy homeomorphism and diffeomorphism.

We now give an example which illustrates the subtleties of the inertia group.
Building on Examples \ref{ex:r0} and \ref{ex:r2}, Theorem \ref{thm:i_and_r} 
gives to the following

\begin{ex} \label{ex:r012}
The connected sums
%
\[ M_0 : = M_{-8, 0} \sharp M_{0, 8}, \quad M_1 : = M_{-8, 2} \sharp M_{0, 8} 
\quad \text{and} \quad M_2 : = M_{-8, 4} \sharp M_{0, 8}, \]
%
have $r(M_i) = i$.  In each case $d_\pi(M_i) = 8$, whereas
$\mdiv(M_0) = 8$, $\mdiv(M_1) = 2$ and $\mdiv(M_2) = 4$.  From Theorem \ref{thm:i_and_r}
we have $I(M_0) \cong I(M_1) \cong \Theta_7$ and $I(M_2) \cong 2\Theta_7$.

Notice that when $r = 1$ the \cite[Conjecture p.\,548]{wilkens74} correctly predicts  $I(M_1) = \Theta_7$.
However when $r \neq 1$, \cite[Conjecture p.\,548]{wilkens74} incorrectly predicts that $I(M_0)$ is $2 \Theta_7$ and
that $I(M_2)$ is $\Theta_7$.
\end{ex}

\begin{ex} \label{ex:pre_exotic_G2}
While \cite[Theorem 1]{wilkens74} and Theorem \ref{thm:i_and_r} give 
$I(M(\Z^b, d)) = \NumB{\frac{d}{4}}\Theta_7$, 
the classical Eells-Kuiper invariant is not defined for $M(\Z^b, d)$ when $d_\pi = d \neq 0$.
Using \eqref{eq:grchat} we compute that
%
\[ \mu \bigl( M(\Z^b, d) \bigr) \sharp \Sigma) = [\mu(\Sigma)] \in \Z/\dM \Z . \]
%
Hence for the Minor sphere $\Sigma_{\rm M}$ we have
$\mu(M(\Z^b, 8) \sharp \Sigma_{\rm M}) = 1 \in \Z/2\Z$, whereas
$\mu(M(\Z^b, 8)) = 0$, 
and the generalised Eells-Kuiper 
invariant distinguishes the diffeomorphism types of $M(\Z^b, 8)$ and
$M(\Z^b, 8) \sharp \Sigma_{\rm M}$.  

We can also deduce from Theorem \ref{thm:classification_categorical} that, for example, $M(\Z^b, 8) \sharp \Sigma_{\rm M}$
admits an orientation reversing diffeomorphism, whereas $M(\Z^b, 16) \sharp \Sigma_{\rm M}$ does not.
\end{ex}

\begin{ex} \label{ex:circle_actions}
If $N$ is a simply connected oriented $6$-manifold with $\pi_2(N) \cong \Z$ and $S^1 \to M \to N$
is a principle $S^1$ bundle with primitive first Chern class, then $M$ is $2$-connected with a preferred
orientation and hence spin structure.  Conversely, by \cite[Lemma 2.1]{jiang13}, every free
$S^1$ action on $M$ is equivalent to such a principle bundle action.  In \cite[Theorem 1.3]{jiang13}
Yi Jiang identifies
the homeomorphism types and diffeomorphism types of all $2$-connected $M$ which admit free circle actions.
In particular, by \cite[Theorem 1.3]{jiang13} every such 
$M$ is almost diffeomorphic to a connected sum $M_{bk, b(k+12m)} \sharp_{2r}M_{0, 0}$ for $b \in \{1, 2\}$,
$r \in \Z^{\geq 0}$ and $m, k \in \Z$.
\end{ex}
% subsection sphere bundles (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{Examples from geometry} \label{subsec:geometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There are a number of examples of $2$-connected $7$-manifolds admitting metrics
with interesting geometric properties.  Indeed by \cite[Theorem B]{crowley14}, every $2$-connected
$7$-manifold admits a metric with positive Ricci curvature.

The smooth manifold underlying the Gromoll-Meyer sphere $\Sigma_{GM}$ \cite{gromoll74} 
is an exotic $7$-sphere admitting a metric of non-negative sectional curvature.  
If $Sp(n)$ denotes the $n$-dimensional
symplectic group of orthogonal $n \times n$ quaternionic matrices, then $\Sigma_{GM}$
is a certain quotient of $Sp(2) \times Sp(1)$ by $Sp(1) \times Sp(1)$.  By \cite[Theorem 1]{gromoll74},
$\Sigma_{GM} \cong M_{-1, -5} \cong 3 \Sigma_{\rm M}$.

The smooth manifold underlying the Berger space $B$ is a homogeneous
space $B = SO(5)/SO(3)$ (where $SO(3) \into SO(5)$ by the adjoint
representation) that admits a metric of positive sectional curvature.
The Berger space is 2-connected with $H^4(B) \cong \Z/10$,
and Goette, Kitchloo and Shankar \cite[Corollary 2]{goette04} proved there is a spin diffeomorphism
%
\[ B \cong M_{10, 8}. \]
%\cong M(q(\an{-1/20}, -8), 27/40). \]
%

More recently Grove, Verdiani and Ziller \cite[Theorem A]{grove11} constructed
a metric of positive sectional curvature  on a 2-connected manifold $P_2$ with
$H^4(P_2) \cong \Z/2$.  Applying \cite[Theorem A]{crowley02},
they deduced that there is an almost diffeomorphism $P_2 \acong S(TS^4)$, where $S(TS^4) = M_{2, 0}$ 
is the unit tangent sphere bundle of $S^4$.  
In \cite{goette11} Goette proved that there is a diffeomorphism
%
\[ P_2 \cong M_{2, 2} \sharp (-\Sigma_{\rm M}). \]
%\cong M \bigl( q(\an{-1/4}, 0 \bigr), -7/8). \]
%
%where $\mu(\Sigma_{\rm M}) = \frac{1}{28}$.

In \cite{g2m} Corti, Haskins, the second author and Pacini constructed 
a very large class of examples of simply connected manifolds with $G_2$ holonomy
metrics. Many of these examples are $2$-connected with $H^4(M)$ torsion-free.
For instance, \cite[Table 3]{g2m} gives 7 explicit ways to
construct holonomy $G_2$ metrics on
%
$M(\Z^{85}, 2)$.
%
%by starting from Fano 3-folds of Picard rank 1
By Theorem \ref{thm:almost_classification}, the underlying topological manifold
admits a unique smooth structure.
On the other hand, the same table lists 2 different ways to construct holonomy $G_2$ metrics with underlying topological manifold $M(\Z^{99}, 8)$.
In this case $I(M) = 2\Theta(7)$, so $M$ admits two inequivalent smooth
structures, and to determine whether those two \gtmfd s are diffeomorphic one
needs to evaluate their generalised Eells-Kuiper invariants.

% subsection geometric examples (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{Tangentially homotopy equivalent manifolds} \label{subsec:the}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $N_0$ and $N_1$ be closed smooth manifolds of the same dimension and let $\tau_{N_i}$
denote the stable tangent bundle of $N_i$.  
A tangential homotopy equivalence $f \colon N_0 \to N_1$ is a homotopy equivalence
such that there is a stable vector bundle isomorphism $f^*\tau_{N_1} \cong \tau_{N_2}$.
It is natural to ask under what conditions tangentially homotopy equivalent manifolds
are necessarily homeomorphic, and this question was studied in detail 
by Madsen, Taylor and Williams in \cite{madsen80}.

In \cite[p.\,144]{crowley02} it was proven the $2$-connected manifolds give rise to examples of
non-homeomorphic tangentially homotopy equivalent manifolds.  We present a simplified
version of the proof here, which starts with the following

\begin{lem} \label{lem:the1}
Let $M_0$ and $M_1$ be $2$-connected and 
let $f \colon M_0 \simeq M_1$ be a homotopy equivalence such that $(f^*)^{\sharp}p_{M_1} = p_{M_0}$.
Then $f$ is tangential.
\end{lem}

\begin{proof}
Let $\wt{KO}$ denote reduced real $K$-theory.  By \cite{kervaire59} the spin characteristic 
class $p$ defines an isomorphism $p \colon \wt{KO}(S^4) \cong H^4(S^4)$.
The Atiyah-Hirzebruch spectral sequence for $\wt{KO}(M_0)$ then shows that $p$ defines
an isomorphism $p \colon \wt{KO}(M) \cong H^4(M)$.  By definition, 
$p_{M_i} = p(\tau_{M_i})$ and so the assumption $(f^*)^{\sharp}p_{M_1} = p_{M_0}$
gives $f^*\tau_{M_1} \cong \tau_{M_0}$, which is to say that $f$ is tangential.
\end{proof}

\begin{prop}[\cf~{\cite[p.\,114]{crowley02}}] \label{prop:the}
The manifolds $M_{8, 1}$ and $M_{8, 25}$ are tangentially homotopy equivalent
but not homeomorphic.	
\end{prop}

\begin{proof}
We first show that $M_{8, 1}$ and $M_{8, 25}$ are homotopy equivalent.
Let $SG(4)$ denote the topological monoid of orientation preserving self-homotopy equivalences
of $S^3$.  Elements of $SG(4)$ are clutching functions for spherical fibrations over $S^4$ with
fibre $S^3$.  The natural map $i \colon SO(4) \in SG(4)$ induces a map on homotopy groups
$i_* \colon \pi_3(SO(4)) \to \pi_3(SG(4))$.  In \cite[p.\,337]{crowley03} it is shown that
$\pi_3(SO(4)) \cong \Z \oplus \Z/12$ and that
%
\[ i_*(\xi_{n, p}) = \left(n, \left[\frac{p-n}{2}\right]\right) \in \Z \oplus \Z/12, \]
%
where $[\frac{p-n}{2}] \in \Z/12$ is the mod $12$ reduction of $[\frac{p-n}{2}]$.  
Hence the bundles $\xi_{8, 1}$ and $\xi_{8, 25}$
are fibre homotopy equivalent and so there is a fibre homotopy equivalence $f \colon M_{8, 1} \simeq M_{8, 25}$
between their sphere bundles.  

To see that $f$ is tangential, let $M_0 = M_{8, 1}$, $M_1 = M_{8, 25}$, 
let $x \in H^4(S^4)$ be a fixed generator and let $x_i \in H^4(M_i)$ be the pull back of
$x$ to under the bundle map $M_i \to S^4$.  By definition $p(M_i) = x_i$.
Since $f$ is a fibre homotopy equivalence $(f^*)^\sharp x_1 = x_0$.
Hence $(f^*)^\sharp p_{M_1} = p_{M_0}$ and so $f$ is tangential by Lemma \ref{lem:the1}.

To see that $M_0 := M_{8, 1}$ and $M_1 : = M_{8, 25}$ are not homeomorphic, we use Proposition \ref{prop:arf}
to compute that $A(q_{M_0}) = -7/64$ mod $\Z$ but $A(q_{M_1}) = -23/64$ mod $\Z$.  
By Theorem \ref{thm:almost_classification},
the refinement $q$ is a homeomorphism invariant, and hence $M_{8, 1}$ and $M_{8, 25}$ are not homeomorphic.
\end{proof}

\begin{rmk} \label{rmk:the}
Proposition \ref{prop:the} contradicts \cite[Theorem C and Theorem 5.10]{madsen80} 
where it is stated, amongst other things, that all tangentially homotopy equivalent 
$2$-connected $7$-manifolds
are homeomorphic.  The source of the mistake in the arguments of \cite{madsen80} can be
found in \cite[Theorem 3.12]{madsen80} which is not correct.
It is claimed that a certain cohomology class 
%
\[ f^*\pi^*(l_n) \in H^{4n}(S^2\Omega^2(SG[3, \infty]); \Z_{(2)}) \]
% 
vanishes.
Here $SG[3, \infty]$ is the $2$-connected cover of $SG$, $S^2\Omega^2$
denotes the double suspension of the double loop space, the coefficient group $\Z_{(2)}$
is the integers localised at $2$ and we shall not define the maps $f$ or $\pi$.
However the argument given for the proof of \cite[Theorem 3.12]{madsen80}
only shows that $f^*\pi^*(l_n) = 2 x$ for some $x \in H^{4n}(S^2\Omega^2(SG[3, \infty]); \Z_{(2)})$
and not that $f^*\pi^*(l_n) = 0$.
To the best of our knowledge, this is the only
flaw in the arguments of \cite{madsen80}.
\end{rmk}
% subsection the (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (fold)
\subsection{Generators for the monoid of $2$-connected $7$-manifolds} \label{subsec:monoid}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The connected sum operation gives the set of spin diffeomorphism classes of 
$2$-connected $7$-manifolds the structure of a commutative monoid with unit $S^7$.
Owing to the existence of homotopy $7$-spheres, every $M$ has non-trivial connected sum splittings
%
\[ M \cong (M \sharp \Sigma) \sharp (-\Sigma) \]
%
for each $\Sigma \in \Theta_7$.  Hence we call $M$ {\em topologically decomposable} if
there is a diffeomorphism
%
\[ M \cong M_0 \sharp M_1 \]
% 
where neither $M_0$ nor $M_1$ is a homotopy sphere, and we call 
$M$ {\em topologically indecomposable} if $M$ is not decomposable.

By Theorem \ref{thm:classification_categorical}, every connected sum splitting of $M$
gives rise to an orthogonal splitting of the refinement of $M$,
and by Theorem \ref{thm:classification} every orthogonal splitting of the
refinement of $M$ is realised by a connected sum splitting of $M$.
Hence we call a refinement $(G, q^\circ\!, p)$ or a base $(G, b, p)$ decomposable 
if it can be written as a non-trivial orthogonal sum and indecomposable otherwise.  
It is clear from the definitions that a refinement is indecomposable if and only if its base if indecomposable.  
Moreover, the indecomposable bases are of the form $(\Z, 0, p)$ and $(T, b, p)$ where $b$ is an
indecomposable torsion form; \ie~$b$ cannot be written as a non-trivial orthogonal sum. In this case 
we also call $q$ indecomposable.  A list of all isomorphism classes of indecomposable torsion forms 
was give by Wall \cite[Theorem 4]{wall63} and torsion forms were then classified by 
Kawauchi and Kojima \cite[Theorem 4.1]{kawauchi80}.  We do 
not go into details but note that if $(T, b, p)$ is indecomposable then $T \cong \Z/p^k$ for a prime $p$
or $T \cong (\Z/2^k)^2$.  Summarising the above discussion we have the following refinement of 
a theorem of Wilkens.

\begin{thm}[\cf~{\cite[Theorem 1]{wilkens72}}] \label{thm:indecomposable}
Every $2$-connected $M$ is diffeomorphism to a connected sum of topologically indecomposable 
manifolds $M_i:$
%
\[ M \cong \sharp_{i=1}^n M_i. \]
Moreover, $M$ is topologically indecomposable if and only if it is almost diffeomorphic to 
a manifold of one of the following forms
%
\[ S^7, \quad M(\Z, d), \quad M(q, s), \]
%
where in the final case, $q$ is a prime refinement and hence $H^4(M(q, s)) \cong \Z/p^k$ for
$p$ a prime or $H^4(M(q, s)) \cong (\Z/2^k)^2$. \qed
\end{thm}

Even up to almost diffeomorphism, the splitting $M \cong \sharp_{i=1}^n M_i$ 
of Theorem \ref{thm:indecomposable} is in general far from being unique.  For
example the manifolds $M_{4, 0}$ and $M_{4, 2}$ have non-isomorphic bases 
$(\Z/4, \an{\frac{-1}{4}}, 0)$ and $(\Z/4, \an{\frac{-1}{4}}, 2)$ respectively, but 
$M_{4, 0} \sharp M_{0, 2}$ and $M_{4, 2} \sharp M_{0, 2}$ are diffeomorphic.
Even when $H^4$ is torsion, there are many examples of torsion forms
where $b_0 \oplus b_2 \cong b_1 \oplus b_3$ but $b_0$ is not isomorphic to 
$b_1$ or $b_3$ and the same holds for $b_2$: see for example \cite[\S 3]{kawauchi80}.  
This leads to non-uniqueness
of connected sum splittings for manifolds with linking form isomorphic to
$b_0 \oplus b_2$.
% subsection monoid generators (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



% section examples (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




% (fold)
\section{Mapping class groups and inertia} \label{sec:mcg_and_inertia}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we point out some
implications of our classification results for mapping class groups of $2$-connected $M$.  
Throughout this section $M$ will be $2$-connected. 

Recall that $\Diff_{}(M)$ are $\ADiff_{}(M)$ denote the groups
of spin diffeomorphisms of $M$ and spin almost diffeomorphisms of $M$ respectively.
Recall also that $\Diff_H(M)$ and $\ADiff_H(M)$ denote subgroups of the above 
automorphism groups whose elements induce the identity on $H^*(M)$.
For brevity, let $\aut_\EK(H^4(M))$ denote
the group of automorphisms of the mod 28 distillation
$(H^4(M), q_M^\circ, \EK_M, p_M)$ and let
$\aut_{q^\circ}(H^4(M))$ denote the group of automorphisms of the classifying triple
$(H^4(M), q_M^\circ, p_M)$.  As an immediate consequence of 
Theorem \ref{thm:classification} we obtain
%
\begin{prop} \label{prop:mcg1}
For each $2$-connected $M$, there is a short exact sequence
\[
0 \to \wt \pi_0 \Diff_H(M) \to \wt \pi_0\Diff_{}(M) \to \aut_\EK(H^4(M)) \to 0. \qed 
\] 
\end{prop}
%
\begin{rmk} \label{rmk:mcg}
The exact sequence of Proposition \ref{prop:mcg1} serves as a starting point
to study the mapping class groups $\wt \pi_0\Diff_{}(M)$.  
The determination of 
$\wt \pi_0 \Diff_H(M)$ and the extension in the sequence of Proposition \ref{prop:mcg1}
lie outside the scope of this paper.
For example, we do not currently know if $\wt \pi_0 \Diff_H(M)$ is abelian in general.
\end{rmk}
  
We now consider $\wt\pi_0\ADiff(M)$, the mapping class group of almost diffeomorphisms of $M$.
We recall the homomorphism $\wt P \colon \aut_{q^\circ}(H^4(M)) \to \lcm(8, 2^r\mdiv)\Z/2 \tilde d_\pi \Z$ from 
Section \ref{subsec:aut_q_on_gauss_functions}: see \eqref{eq:P_tilde} and Proposition \ref{prop:imtp}.
Recalling that $\dM = \gcd(\frac{\tilde d_\pi}{4}, 28)$, we then define
%
\[ \wh P \colon \aut_{q^\circ}(H^4(M)) \to \NumB{2^{r-3}\mdiv}\Z/\dM \Z, \quad F \mapsto \frac{\wt P(F)}{8} \mod \dM, \]
% \rho_{\dM} \circ \frac{1}{8}\wt P
%
to be the mod $\dM$ reduction of $\wt P$ divided by $8$.  Notice that by Theorem \ref{thm:i_and_r} and
Remark \ref{rmk:p2null} we have $I(M)/I_H(M) \cong \NumB{2^{r-3}\mdiv}\Z/\dM \Z$ and so we can equally regard
$\wh P$ as a homomorphism $\wh P \colon \aut_{q^\circ}(H^4(M)) \to I(M)/I_H(M)$.

\begin{thm} \label{thm:mcg_all}
For each $2$-connected $M$ there is a commutative diagram of group homomorphisms with short exact sequences for rows
and with exact columns:
% where $\del_H, \del$ and $\wh P$ are onto:
%
\[
\xymatrix{  0 \ar[r] & \wt \pi_0\Diff_H(M) \ar[d] \ar[r] & \wt \pi_0 \Diff_{}(M) \ar[d] \ar[r] & 
\aut_\EK(H^4(M)) \ar[d] \ar[r] & 0 \\
0 \ar[r] & \wt \pi_0 \ADiff_H(M) \ar[d]^{\del_H} \ar[r] & \wt \pi_0 \ADiff_{}(M) \ar[d]^{\del} \ar[r] & 
\aut_{q^\circ}(H^4(M)) \ar[d]^{\wh P_{}} \ar[r] & 0 \\
0 \ar[r] & I_H(M) \ar[d] \ar[r] & I(M) \ar[d] \ar[r] & I(M)/I_H(M) \ar[d]  \ar[r] & 0 \\
& 0 & 0 & 0 } 
\]
In particular $F \in \aut_{q^\circ}(H^4(M))$ is realised by a diffeomorphism of $M$ if and only if $\wh P(F) = 0$.
%\begin{equation} \label{eq:mcg_all}
%\xymatrix{  0 \ar[r] & \wt \pi_0\Diff_H(M) \ar[d] \ar[r] & \wt \pi_0 \Diff_{}(M) \ar[d] \ar[r] & 
%\aut_\EK(H^4(M)) \ar[d] \ar[r] & 0 \\
%0 \ar[r] & \wt \pi_0 \ADiff_H(M) \ar[d] \ar[r] & \wt \pi_0 \ADiff_{}(M) \ar[d] \ar[r] & 
%\aut_{q^\circ}(H^4(M)) \ar[d] \ar[r] & 0 \\
%0 \ar[r] & I_H(M) \ar[r] & I(M) \ar[r] & I(M)/I_H(M)  \ar[r] & 0 } 
%\end{equation}
%A feature of diagram \eqref{eq:mcg_all} ...
\end{thm}

\begin{proof}
The top row is the exact sequence of Proposition \ref{prop:mcg1}.
The exactness of the second row follows from Theorem \ref{thm:almost_classification}.
The first two columns are exact by the discussion at the beginning of Section \ref{subsec:inertia_and_reactivity2},
and in particular \eqref{eq:I_and_del}, and the third column is exact by the definition of $\wh P_{}$.
The only part of the commutativity of the diagram which needs comment is the bottom right hand square
where the commutativity follows from from Lemma \ref{lem:mapping_torus}~\ref{lem:mapping_torus:mu}
and Lemma~\ref{lem:p2mod}.  The final statement follows from the exactness of final row
and the top column.
\end{proof}

We shall call an almost diffeomorphisms {\em exotic} if it is not pseudo-isotopic to a diffeomorphism,
A feature of the diagram in Proposition \ref{thm:mcg_all} is that when $I(M)/I_H(M) \neq 0$, $M$ admits
exotic almost diffeomorphisms which are detected by their action on $H^4(M)$.  
Specifically, if $f \colon M \acong M$ is an almost diffeomorphism,
then $\wh P_{}(f^*)$ is the obstruction to $f^* \colon H^4(M) \cong H^4(M)$ being induced by {\em any diffeomorphism}
of $M$.  Since $\wh P$ is onto, it is enough to find cases where $I(M)/I_H(M)$ is non-zero to
show that $\wh P$ is non-zero.

\begin{prop} \label{prop:pairs_of_inertia_groups}
Any pair of subgroups $I_0 \subseteq I_1 \subseteq \Theta_7$ can arise as the pair of inertia
groups $(I_0, I_1) = (I_H(M), I(M))$ for some $2$-connected $M$.
\end{prop}

\begin{proof}
There are three pairs of subgroups $(T_0, T_1)$ in $\Z/7$ and six pairs of subgroups $(T_0, T_1)$ in $\Z/4$,
leaving 18 cases to realise.
By Theorem \ref{thm:i_and_r} and Remark \ref{rmk:p2null}, $I(M)$ and $I_H(M)$ depend only on the
base $(G, b, p)$.  We list manifolds, their bases and the pairs of inertia groups they realise
in the following table, where it is helpful to note that $102 = 7 \times 16$:
%
\begin{center}
\begin{tabular}{c|c|c|c}
$M$  & $(G, b, p)$  & $(I_0, I_1)$ & $I_1/I_0 $\\ \hline
$S^7$& $(0, 0, 0)$ & $(0, 0)$ & 0 \\
$M_{0, 56}$& $(\Z, 0, 56)$ & $(\Z/2, \Z/2)$ & $0$ \\
$M_{0, 28}$& $(\Z, 0, 28)$ & $(\Z/4, \Z/4)$ & $0$ \\
$M_{0, 16}$& $(\Z, 0, 16)$ & $(\Z/7, \Z/7)$ & $0$ \\
$M_{0, 8}$& $(\Z, 0, 8)$ & $(\Z/14, \Z/14)$ & $0$ \\
$M_{0, 4}$& $(\Z, 0, 4)$ & $(\Z/28, \Z/28)$ & $0$ \\
$M_{-16, 0} \sharp M_{0, 102} $ & $\bigl(\Z/16 \oplus \Z, \an{1/16}, (0, 102) \bigr)$ & $(0, \Z/2)$ & $\Z/2$ \\
$M_{-8,0 } \sharp M_{0, 56}$ & $\bigl(\Z/8 \oplus \Z, \an{1/8}, (0, 56) \bigr)$ & $(\Z/2, \Z/4)$ & $\Z/2$ \\
$M_{-16, 2} \sharp M_{0, 102}$ & $\bigl(\Z \oplus \Z/16, \an{1/16}, (102, 2) \bigr)$ & $(0, \Z/4)$ & $\Z/4$ \\
$M_{-16, 0} \sharp M_{0, 16}$ & $(\Z/16 \oplus \Z, \an{1/16}, (0, 16) \bigr)$ & $(\Z/7, \Z/14)$ & $\Z/2$ \\
$M_{-8, 0} \sharp M_{0, 8}$ & $\bigl(\Z/8 \oplus \Z, \an{1/8}, (0, 8) \bigr)$ & $(\Z/14, \Z/28)$ & $\Z/2$ \\
$M_{-16, 2} \sharp M_{0, 16}$ & $\bigl(\Z/16 \oplus \Z, \an{1/16}, (2, 16) \bigr)$ & $(\Z/7, \Z/28)$ & $\Z/4$ \\
$M_{-7, 1} \sharp M_{0, 102}$ & $\bigl(\Z/7 \oplus \Z, \an{1/7}, (1, 102) \bigr)$ & $(0, \Z/7)$ & $\Z/7$ \\
$M_{-7, 1} \sharp M_{0, 56}$ & $\bigl(\Z/7 \oplus \Z, \an{1/7}, (1, 56) \bigr)$ & $(\Z/2, \Z/14)$ & $\Z/7$ \\
$M_{-0, 14} \sharp M_{0, 14}$ & $\bigl(\Z/7 \oplus \Z, \an{1/7}, (1, 14) \bigr)$ & $(\Z/4, \Z/28)$ & $\Z/7$ \\
$M_{-7, 1} \sharp M_{102, 16}$ & $\bigl(\Z/102 \oplus \Z, \an{1/102}, (16, 102) \bigr)$ & $(0, \Z/14)$ & $\Z/14$ \\
$M_{-56, 0} \sharp M_{0, 56}$ & $\bigl(\Z/56 \oplus \Z, \an{1/56}, (8, 56) \bigr)$ & $(\Z/2, \Z/28)$ & $\Z/14$\\
$M_{-102, 2} \sharp M_{0, 102}$ & $\bigl(\Z/102 \oplus \Z, \an{1/102}, (2, 102) \bigr)$ & $(0, \Z/28)$ & $\Z/28$\\
\end{tabular}
\end{center}
%%%%%% old argument %%%%%
\begin{comment}
We indicate how to list the necessary bases,
using Theorem \ref{thm:i_and_r} and Remark \ref{rmk:p2null} to compute $I(M)$ and $I_H(M)$.
At the prime $7$, the three pairs $(0, 0)$, $(\Z/7, \Z/7)$ and $(0, \Z/7)$ are realised respectively 
by the bases
%
\[ \bigl(0, 0, 0 \bigr), \quad \bigl( \Z, 0, 2 \bigr),
\quad (\Z \oplus \Z/7, \an{1/7}, (98, 1) \bigr).\]
%
%where $(\Z/7, \an{\frac{1}{7}})$ is the linking form on $\Z/7$ with $b(x, x) = \frac{1}{7}$ for some $x \in \Z/7$.
At the prime $2$, the six pairs $(0, 0)$, $(\Z/2, \Z/2)$, $(\Z/4, \Z/4)$ and $(0, \Z/2)$, $(\Z/2, \Z/4)$ and
$(0, \Z/4)$ are realised respectively by the bases
%
\[ \bigl(0, 0, 0 \bigr), \quad \bigl(\Z, 0, 8 \bigr), \quad \bigl(\Z, 0, 4 \bigr), \]
%
and 
%
\[ (\Z \oplus \Z/16, \an{1/16}, (16, 0) \bigr), \quad (\Z \oplus \Z/8, \an{1/8}, (8, 0) \bigr), 
\quad \bigl(\Z \oplus \Z/16, \an{1/16}, (16, 2) \bigr). \]
%
%\left(\Z \oplus (\Z/16)^2, \left( \begin{array}{cc} 0 & 1/16 \\ 1/16 & 0  \end{array} \right) , 
%(16, 2, 2) \right).\]
%
The fourth and fifth cases at the prime $2$ are from Example \ref{ex:r0} where $r=0$.
In the sixth case at the prime $2$, $\mdiv = 1$ whereas $d_\pi = 16$ and we
we do not need to know $r$. 
We leave the reader to adapt the bases in the lists above to realise all 18 cases.
\end{comment}
\end{proof}

Theorem \ref{thm:hatP_short} follows immediately from Theorem \ref{thm:mcg_all} and
Proposition \ref{prop:pairs_of_inertia_groups}.  We now give an explicit example.

\begin{ex} \label{ex:mapping_inertia}
Let $M = M_{-102, 2} \sharp M_{0, 102}$ so that $H^4(M) = \Z/102 \oplus \Z$ and consider
the automorphism of $(H^4(M), q^\circ_M, p_M)$ defined by
%
\[ F = \begin{pmatrix} 1 & [1] \\ 0 & 1 \end{pmatrix} \colon \Z/102 \oplus \Z \cong \Z/102 \oplus \Z. \]
%
In this case $\dM = 28$ and from the proof of Proposition \ref{prop:pairs_of_inertia_groups}, we see that
$M$ admits an almost diffeomorphism $f \colon M \acong M$ with $f^* = F$, and $\wh P(f) = 1 \in \Z/28 \Z$.
By Theorem \ref{thm:mcg_all}, $F^n$ is realised by a diffeomorphism of $M$ if and only  if $n \equiv 0$~mod~28.
%
\end{ex}
% section mapping class groups (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\bibliographystyle{amsinitial}
\bibliography{g2geom}
\bigskip
\noindent
\emph{Diarmuid Crowley}\\
  {\small
  \begin{tabular}{@{\qquad}l}
    Max Planck Institute for Mathematics\\
    Vivatsgasse 7, 53111 Bonn, Germany\\
    \textsf{diarmuidc23@gmail.com}\\
	%\textsf{http://www.dcrowley.net}\\
  \end{tabular}

\bigskip

\noindent
\emph{Johannes Nordstr\"{o}m}\\
  {\small
  \begin{tabular}{@{\qquad}l}
   Department of Mathematical Sciences\\
   University of Bath\\
   Claverton Down, Bath BA2 7AY, United Kingdom\\
    \textsf{j.nordstrom@bath.ac.uk}\\
    %\textsf{http://people.bath.ac.uk/jlpn20/}
  \end{tabular}


\end{document}

2c7m-v1.bbl
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% \MRhref is called by the amsart/book/proc definition of \MR.
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}
\providecommand{\href}[2]{#2}
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