# Simple homotopy types of even dimensional manifolds

Csaba Nagy  School of Mathematics and Statistics, University of Glasgow,
United Kingdom<EMAIL_ADDRESS>,  John Nicholson  School of
Mathematics and Statistics, University of Glasgow, United Kingdom
<EMAIL_ADDRESS>and  Mark Powell  School of Mathematics and
Statistics, University of Glasgow, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We study the difference between simple homotopy equivalence and homotopy
equivalence for closed manifolds of dimension $n\geq 4$. Given a closed
$n$-manifold, we characterise the set of simple homotopy types of
$n$-manifolds within its homotopy type in terms of algebraic $K$-theory, the
surgery obstruction map, and the homotopy automorphisms of the manifold. We
use this to construct the first examples, for all $n\geq 4$ even, of closed
$n$-manifolds that are homotopy equivalent but not simple homotopy equivalent.
In fact, we construct infinite families with pairwise the same properties, and
our examples can be taken to be smooth for $n\geq 6$.

We also show that, for $n\geq 4$ even, orientable examples with fundamental
group $C_{\infty}\times C_{m}$ exist if and only if $m=4,8,9,12,15,16,18$ or
$\geq 20$. The proof involves analysing the obstructions which arise using
integral representation theory and class numbers of cyclotomic fields. More
generally, we consider the classification of the fundamental groups $G$ for
which such examples exist. For $n\geq 12$ even, we show that examples exist
for any finitely presented $G$ such that the involution on the Whitehead group
$\operatorname{Wh}(G)$ is nontrivial. The key ingredient of the proof is a
formula for the Whitehead torsion of a homotopy equivalence between doubles of
thickenings.

###### Key words and phrases:

Manifolds, simple homotopy equivalence.

###### 2020 Mathematics Subject Classification:

Primary 57N65, 57Q10; Secondary 19A31, 19B28, 19J10.

###  1\. Introduction

####  1.1. Background and main results

One of the earliest triumphs in manifold topology was the classification of
3-dimensional lens spaces up to homotopy equivalence, and up to homeomorphism,
due to Seifert-Threlfall, Reidemeister, Whitehead, and Moise [Threlfall-
Seifert, Reidemeister-lens-spaces, Whitehead-incidence, Moise]; see [Co73] for
a self-contained treatment and [Volkert-manifold-atlas] for a detailed
history. The two classifications do not coincide e.g. $L(7,1)$ and $L(7,2)$
are famously homotopy equivalent but not homeomorphic.

The homeomorphism classification made use of Reidemeister torsion to
distinguish homotopy equivalent lens spaces. While trying to understand this
more deeply, J.H.C. Whitehead defined the notion of _simple homotopy
equivalence_ [Whitehead-simplicial-spaces, Whitehead-incidence, Whitehead-
combinatorial, Whitehead-she]. A homotopy equivalence $f\colon X\to Y$ between
CW complexes is said to be _simple_ if it is homotopic to the composition of a
sequence of elementary expansions and collapses; see Section 2 for the precise
definition. Chapman [Ch74] showed that every homeomorphism $f\colon X\to Y$
between compact CW complexes is a simple homotopy equivalence, and so in the
hierarchy of equivalence relations:

$\mbox{homeomorphism}\Rightarrow\mbox{simple homotopy
equivalence}\Rightarrow\mbox{homotopy equivalence}.$

Whitehead showed that the homeomorphism classification of 3-dimensional lens
spaces coincides with the classification up to simple homotopy equivalence, so
there are many examples of homotopy but not simple homotopy equivalent lens
spaces, e.g. the aforementioned $L(7,1)$ and $L(7,2)$. Higher dimensional lens
spaces give rise to similar examples in all odd dimensions $2k-1\geq 5$
[Co73], and infinite such families of odd-dimensional manifolds were produced
by Jahren-Kwasik [JK15].

This article constructs the first examples of closed manifolds, in all even
dimensions $2k\geq 4$, that are homotopy but not simple homotopy equivalent.
In fact, we produce infinite familes.

From now on, $n\geq 4$ will be an integer and an _$n$ -manifold_ will be a
compact, connected, $\operatorname{CAT}$ $n$-manifold where
$\operatorname{CAT}\in\\{\operatorname{Diff},\operatorname{PL},\operatorname{TOP}\\}$
is the category of either smooth, piecewise linear, or topological manifolds.
Manifolds will also be assumed closed unless otherwise specified. We will also
frequently make use of the following hypothesis.

######  Hypothesis 1.1.

If $n=4$, then we assume $\operatorname{CAT}=\operatorname{TOP}$ and we
restrict to manifolds whose fundamental group is good in the sense of Freedman
(see, for example, [FQ, Freedman-book-goodgroups]).

######  Theorem A.

Let $n\geq 4$ be even and let $\operatorname{CAT}$ be as in 1.1. Then there
exists an infinite collection of orientable $\operatorname{CAT}$ $n$-manifolds
that are all homotopy equivalent but are pairwise not simple homotopy
equivalent.

We write $\simeq$ for homotopy equivalence and $\simeq_{s}$ for simple
homotopy equivalence. We note that if $M_{1}\simeq N_{1}$ and $M_{2}\simeq
N_{2}$ are odd dimensional manifolds, then $M_{1}\times
M_{2}\simeq_{s}N_{1}\times N_{2}$ (Corollary 2.32), so even dimensional
examples cannot be constructed in such a straightforward way from odd
dimensional examples.

It is currently open whether all topological 4-manifolds are homeomorphic to
CW complexes. Thus, the definition of simple homotopy equivalence given
previously does not apply when $n=4$ and
$\operatorname{CAT}=\operatorname{TOP}$. For a more general definition that
does apply in this case, see Definition 2.3.

In order to state our more detailed results, we introduce the simple homotopy
manifold set

$\mathcal{M}^{h}_{s}(M)\coloneqq\\{\operatorname{CAT}\text{ $n$-manifolds
}N\mid N\simeq M\\}/\simeq_{s}$

for a $\operatorname{CAT}$ $n$-manifold $M$. This is the set of manifolds
homotopy equivalent to $M$ up to simple homotopy equivalence. We are
interested in finding manifolds with nontrival, and ideally large, simple
homotopy manifold sets. More specifically, we will consider two questions in
this paper.

######  Question 1.

For $n\geq 4$ even, how large can $\mathcal{M}^{h}_{s}(M)$ be for a
$\operatorname{CAT}$ $n$-manifold $M$?

We prove that the size of $\mathcal{M}^{h}_{s}(M)$ can be both arbitrarily
large finite and infinite if $M=S^{1}\times L$ for some lens space $L$, which
in particular implies Theorem A. For an integer $m\geq 2$, let $C_{m}$ denote
the cyclic group of order $m$ and let $C_{\infty}$ denote the infinite cyclic
group.

######  Theorem B.

Let $n\geq 4$ be even and let $\operatorname{CAT}$ be as in 1.1. Let
$M^{n}_{m}=S^{1}\times L$, where $L$ is an $(n-1)$-dimensional lens space with
$\pi_{1}(L)\cong C_{m}$, $m\geq 2$. Then

  1. (a)

$|\mathcal{M}^{h}_{s}(M^{n}_{m})|$ only depends on $n$ and $m$, and is
independent of the choice of $L$ or $\operatorname{CAT}$;

  2. (b)

$|\mathcal{M}^{h}_{s}(M^{n}_{m})|=1$ if and only if
$m\in\\{2,3,5,6,7,10,11,13,14,17,19\\}$;

  3. (c)

$|\mathcal{M}^{h}_{s}(M^{n}_{m})|=\infty$ if and only if $m$ is not square-
free;

  4. (d)

$|\mathcal{M}^{h}_{s}(M^{n}_{m})|\to\infty$ as $m\to\infty$, uniformly in $n$.

The ingredients of the proof will be discussed in Sections 1.2, 1.3, and 1.7.
They rely on an analysis of the _Whitehead group_ [Whitehead-she]. For a group
$G$, Whitehead defined this group, $\operatorname{Wh}(G)$, and associated to a
homotopy equivalence $f\colon X\rightarrow Y$ its Whitehead torsion
$\tau(f)\in\operatorname{Wh}(\pi_{1}(Y))$. He proved the fundamental result
that $\tau(f)=0$ if and only if $f$ is simple. So to understand simple
homotopy manifold sets, one has to understand the Whitehead torsions of
homotopy equivalences between manifolds.

Here is our second motivating question.

######  Question 2.

Fix $n\geq 4$ (not necessarily even) and a category $\operatorname{CAT}$. For
which pairs $(G,w)$ is there a $\operatorname{CAT}$ $n$-manifold $M$ with
fundamental group $G$ and orientation character $w\colon G\to\\{\pm 1\\}$ such
that $|\mathcal{M}^{h}_{s}(M)|>1$?

Of course, it is necessary that $\operatorname{Wh}(G)\neq 0$, otherwise all
homotopy equivalences between spaces with fundamental group $G$ are simple. As
we only consider homotopy equivalences between $n$-manifolds, a stronger
condition can be given in terms of $\operatorname{Wh}(G,w)$, the Whitehead
group $\operatorname{Wh}(G)$ equipped with the canonical involution
$x\mapsto\overline{x}$ determined by $w$ (see Section 2.2). Namely, it is
necessary that $\mathcal{J}_{n}(G,w)\neq 0$ (see Proposition 2.34), where we
use the notation

$\displaystyle\mathcal{J}_{n}(G,w)$
$\displaystyle=\left\\{y\in\operatorname{Wh}(G,w)\mid
y=-(-1)^{n}\overline{y}\right\\}\leq\operatorname{Wh}(G,w),$
$\displaystyle\mathcal{I}_{n}(G,w)$
$\displaystyle=\left\\{x-(-1)^{n}\overline{x}\mid
x\in\operatorname{Wh}(G,w)\right\\}\leq\mathcal{J}_{n}(G,w).$

Our main result in this direction is that in high dimensions a slightly
stronger condition, the nonvanishing of $\mathcal{I}_{n}(G,w)$, is sufficient.
In fact we have the following.

######  Theorem C.

Let $n=9$ or $n\geq 11$, fix a category $\operatorname{CAT}$, let $G$ be a
finitely presented group and let $w\colon G\rightarrow\left\\{\pm 1\right\\}$
be a homomorphism. Then there is a $\operatorname{CAT}$ $n$-manifold $M$ with
fundamental group $G$ and orientation character $w$ such that
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$ if and only if
$\mathcal{I}_{n}(G,w)\neq 0$.

Here, $\mathcal{M}^{\operatorname{hCob}}_{s}(M)$ denotes the set of manifolds
$h$-cobordant to $M$ up to simple homotopy equivalence. Recall that a
cobordism $(W;M,N)$ of closed manifolds is an _$h$ -cobordism_ if the
inclusion maps $i_{M}\colon M\to W$ and $i_{N}\colon N\to W$ are homotopy
equivalences. There is a natural inclusion
$\mathcal{M}^{\operatorname{hCob}}_{s}(M)\hookrightarrow\mathcal{M}^{h}_{s}(M)$,
and so $|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$ implies that
$|\mathcal{M}^{h}_{s}(M)|>1$. Thus Theorem C can be used to construct a pair
of manifolds that are $h$-cobordant but not simple homotopy equivalent. This
will be complemented by Theorem 1.7, which gives a sufficient (but possibly
not necessary) condition for the existence of manifolds that are homotopy
equivalent, but neither simple homotopy equivalent, nor $h$-cobordant.

The notion of an $h$-cobordism is central to manifold topology. Smale’s
$h$-cobordism theorem [smale-pe, Smale-h-cob, Milnor-h-cob], together with its
extensions to other categories and dimension 4 in [Stallings-Tata, Kirby-
Siebenmann:1977-1, FQ], states that under 1.1, every simply-connected
$h$-cobordism is $\operatorname{CAT}$-equivalent to the product $M\times I$.
To generalise the $h$-cobordism theorem to non-simply connected manifolds, one
also considers $s$-cobordisms. An _$s$ -cobordism_ $(W;M,N)$ is a cobordism
for which the inclusions $M\to W$ and $N\to W$ are simple homotopy
equivalences. The $\operatorname{CAT}$ $s$-cobordism theorem [Barden,
Mazur:1963-1, Stallings-Tata, Kirby-Siebenmann:1977-1, FQ] states that under
1.1, $\operatorname{CAT}$ equivalence classes of $h$-cobordisms based on $M$
are in bijection with $\operatorname{Wh}(\pi_{1}(M))$, with the bijection
given by taking the Whitehead torsion of the inclusion $M\to W$ (Theorem 3.1).
In particular every $s$-cobordism is a product. This theorem underpins
manifold classification in dimension at least $4$. So while studying the
difference between homotopy and simple homotopy equivalence for manifolds, it
is natural to also consider the rôle of $h$-cobordisms.

For the remainder of the introduction, we will discuss our results and the
contents of this article in greater detail. In Section 1.2, we give a
characterisation of $\mathcal{M}^{h}_{s}(M)$ (Theorem D). We apply this in
Section 1.3 to the manifold $S^{1}\times L$ to obtain Theorem B, as well as
other detailed numerical results (Theorems E and F). In Section 1.4, we
establish a formula for the Whitehead torsion of a homotopy equivalences
between doubles (Theorem G). This is applied in Section 1.5 to obtain a simple
homotopy rigidity result for sphere bundles (Theorem H), leading to Theorem C.
Using the results on $S^{1}\times L$ and doubles, we revisit 2 in Section 1.6.
In Section 1.7 we conclude by discussing the involution on
$\widetilde{K}_{0}(\mathbb{Z}G)$ and our approach to its computation using
methods from algebraic number theory. This is the basis for the numerical
results in Theorems B, E, and F.

####  1.2. Characterisation of simple homotopy manifold sets

In Theorem D below, we characterise the simple homotopy manifold sets
$\mathcal{M}^{h}_{s}(M)$, for a $\operatorname{CAT}$ $n$-manifold $M$, in
terms of the Whitehead group, the homotopy automorphisms of $M$, and the
surgery obstruction map. It will be helpful to consider the following
variations of $\mathcal{M}^{h}_{s}(M)$:

$\displaystyle\mathcal{M}^{\operatorname{hCob}}_{s}(M)$
$\displaystyle\coloneqq\left\\{\operatorname{CAT}\text{ $n$-manifolds }N\mid
N\text{ is $h$-cobordant to }M\right\\}/\simeq_{s}$
$\displaystyle\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$
$\displaystyle\coloneqq\left\\{\operatorname{CAT}\text{ $n$-manifolds }N\mid
N\simeq M\right\\}/\langle\simeq_{s},\operatorname{hCob}\rangle$

where $\langle\simeq_{s},\operatorname{hCob}\rangle$ denotes the equivalence
relation generated by simple homotopy equivalence and $h$-cobordism. All our
main theorems arise by applying these characterisations to specific manifolds
$M$, for which we can compute the objects that appear in Theorem D.

Recall the definition of $\mathcal{J}_{n}(G,w)$ and $\mathcal{I}_{n}(G,w)$.
The Tate cohomology group $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ is
canonically identified with $\mathcal{J}_{n}(G,w)/\mathcal{I}_{n}(G,w)$ (see
Proposition 14.5), and we will denote the quotient map by

$\pi\colon\mathcal{J}_{n}(G,w)\rightarrow\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w)).$

If $w\equiv 1$ is the trivial orientation character, then we omit it from the
notation.

There is an action of $\operatorname{hAut}(M)$ on $\operatorname{Wh}(G,w)$ (as
a set) such that if $f\colon N\rightarrow M$ is a homotopy equivalence and
$g\in\operatorname{hAut}(M)$, then $\tau(f)^{g}=\tau(g\circ f)$. Let
$q\colon\operatorname{Wh}(G,w)\rightarrow\operatorname{Wh}(G,w)/\operatorname{hAut}(M)$
denote the quotient map, let
$\varrho\colon\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))\rightarrow
L_{n}^{s}(\mathbb{Z}G,w)$ be the homomorphism from the exact sequence (3.1),
and let $\sigma_{s}\colon\mathcal{N}(M)\rightarrow L_{n}^{s}(\mathbb{Z}G,w)$
be the surgery obstruction map (see Section 3.2). The following theorem is the
basis of our main results (see Theorems 4.11 and 4.16).

######  Theorem D.

Let $M$ be a $\operatorname{CAT}$ $n$-manifold with fundamental group $G$ and
orientation character $w\colon G\rightarrow\left\\{\pm 1\right\\}$, satisfying
1.1. There is a commutative diagram

$\textstyle{\mathcal{M}^{\operatorname{hCob}}_{s}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{M}^{h}_{s}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{q(\mathcal{I}_{n}(G,w))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\varrho\circ\pi)^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varrho^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)}$

where each row is a short exact sequence of pointed sets, and each vertical
arrow is a bijection.

The subset $\mathcal{J}_{n}(G,w)$ is invariant under the action of
$\operatorname{hAut}(M)$ on $\operatorname{Wh}(G,w)$, and there is an induced
action on $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$. In the statement
above, the subsets

$(\varrho\circ\pi)^{-1}(\operatorname{Im}\sigma_{s})\subseteq\mathcal{J}_{n}(G,w)\quad\text{
and
}\quad\varrho^{-1}(\operatorname{Im}\sigma_{s})\subseteq\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$

are both invariant under the induced actions of $\operatorname{hAut}(M)$. Note
that if the subset $\mathcal{I}(G,w)$ is invariant under the action of
$\operatorname{hAut}(M)$ on $\operatorname{Wh}(G,w)$, then
$q(\mathcal{I}_{n}(G,w))=\mathcal{I}_{n}(G,w)/\operatorname{hAut}(M)$.

It follows from the exact sequence (3.1) that the image of the homomorphism
$\psi\colon
L_{n+1}^{h}(\mathbb{Z}G,w)\rightarrow\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
is contained in $\varrho^{-1}(\operatorname{Im}\sigma_{s})$ and so we obtain
the following corollary.

######  Corollary 1.2.

Let $M$ be a $\operatorname{CAT}$ $n$-manifold with fundamental group $G$ and
orientation character $w\colon G\rightarrow\left\\{\pm 1\right\\}$, satisfying
1.1. If $\psi\colon
L_{n+1}^{h}(\mathbb{Z}G,w)\rightarrow\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
is surjective, then
$\mathcal{M}^{h}_{s}(M)\cong\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)$ and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)\cong\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))/\operatorname{hAut}(M)$.

The advantage of Corollary 1.2 is that $\mathcal{J}_{n}(G,w)$ and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ (and, more generally,
$\operatorname{Im}\psi$) only depend on $G$ and $w$, while
$\operatorname{Im}\sigma_{s}$ depends on $M$ a priori. This will allow us to
apply Corollary 1.2 by separately analysing the involution on
$\operatorname{Wh}(G,w)$ and the homotopy automorphisms of $M$. Note also
that, in cases where $M$ can be regarded as a manifold in multiple categories,
the sets $q(\mathcal{I}_{n}(G,w))$,
$\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)$ and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))/\operatorname{hAut}(M)$ do
not depend on the choice of $\operatorname{CAT}$.

To prove Theorem D, first we observe that if $N$ is a manifold homotopy
equivalent to $M$, then the set of Whitehead torsions of all possible homotopy
equivalences $f\colon N\to M$ forms an orbit of the action of
$\operatorname{hAut}(M)$ on $\operatorname{Wh}(G,w)$. By considering the
different restrictions and equivalence relations on these manifolds $N$, we
obtain maps from the various simple homotopy manifold sets of $M$ to
subsets/subquotients of $\operatorname{Wh}(G,w)/\operatorname{hAut}(M)$. We
then verify that these maps are injective, and the remaining task is to
determine their images.

In the case of $\mathcal{M}^{\operatorname{hCob}}_{s}(M)$ we use the
$s$-cobordism theorem. For every $x\in\operatorname{Wh}(G,w)$ there is an
$h$-cobordism $(W;M,N)$ with Whitehead torsion $x$, and then
$\tau(f)=-x+(-1)^{n}\overline{x}$ for the induced homotopy equivalence
$f\colon N\rightarrow M$ (see Proposition 2.37). This shows that every element
of $\mathcal{I}_{n}(G,w)$ can be realised as the Whitehead torsion of a
homotopy equivalence $N\rightarrow M$ for some $N$ that is $h$-cobordant to
$M$. For $\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$, we use the surgery
exact sequence combined with the Ranicki-Rothenberg exact sequence [Shaneson-
GxZ], [Ranicki-80-I, §9] to describe the set of vaues of $\pi(\tau(f))$ for
all homotopy equivalences $f\colon N\to M$ (see Proposition 3.6). Finally, the
characterisation of $\mathcal{M}^{h}_{s}(M)$ is obtained by combining the
results on $\mathcal{M}^{\operatorname{hCob}}_{s}(M)$ and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$.

####  1.3. The simple homotopy manifold set of $S^{1}\times L$

Recall that a lens space $L\coloneqq S^{2k-1}/C_{m}$ of dimension $2k-1\geq 3$
is a quotient of $S^{2k-1}$ by a free action of a finite cyclic group $C_{m}$
for some $m\geq 2$. The action is determined by a $k$-tuple of integers
$(q_{1},\dots,q_{k})$ with $q_{j}$ coprime to $m$ for all $j$. The
fundamamental group is $\pi_{1}(L)\cong C_{m}$.

In Part II we will prove Theorem B (and hence Theorem A) by applying Corollary
1.2 to $S^{1}\times L$. This is possible, because the map $\psi\colon
L_{n+1}^{h}(\mathbb{Z}[C_{\infty}\times
C_{m}])\rightarrow\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))$ is surjective for $n=2k$ (see Proposition 3.12), and we get that
$\mathcal{M}^{h}_{s}(S^{1}\times L)\cong\mathcal{J}_{n}(C_{\infty}\times
C_{m})/\operatorname{hAut}(S^{1}\times L)$. For the homotopy automorphisms of
$S^{1}\times L$ we have the following (see Theorems 6.6 and 6.5).

######  Theorem 1.3.

  1. (a)

Every homotopy automorphism $f\colon S^{1}\times L\to S^{1}\times L$ is
simple.

  2. (b)

If $\pi_{1}\colon\operatorname{hAut}(S^{1}\times
L)\rightarrow\operatorname{Aut}(C_{\infty}\times C_{m})$ is the map given by
taking the induced automorphism on the fundamental group, then
$\operatorname{Im}(\pi_{1})=\left\\{\left(\begin{smallmatrix}a&b\\\
0&c\end{smallmatrix}\right)\in\operatorname{Aut}(C_{\infty}\times C_{m})\mid
c^{k}\equiv\pm 1\text{ mod }m\right\\}$.

Part (a) implies that the action of $\operatorname{hAut}(S^{1}\times L)$ on
$\operatorname{Wh}(C_{\infty}\times C_{m})$ factors through the action of
$\operatorname{Aut}(C_{\infty}\times C_{m})$, which acts on
$\operatorname{Wh}(C_{\infty}\times C_{m})$ via automorphisms, using the
functoriality of $\operatorname{Wh}$ (see Definitions 4.3 and 4.9). Therefore
every orbit has cardinality at most $|\operatorname{Aut}(C_{\infty}\times
C_{m})|<2m^{2}$, and $|\mathcal{J}_{n}(C_{\infty}\times
C_{m})/\operatorname{hAut}(S^{1}\times L)|=1$ if and only if
$\mathcal{J}_{n}(C_{\infty}\times C_{m})=0$. Moreover, since it follows from
part (b) that $\operatorname{Im}(\pi_{1}\colon\operatorname{hAut}(S^{1}\times
L)\rightarrow\operatorname{Aut}(C_{\infty}\times C_{m}))$ is independent of
the choice of the integers $(q_{1},\dots,q_{k})$, the same is true for
$\mathcal{J}_{n}(C_{\infty}\times C_{m})/\operatorname{hAut}(S^{1}\times L)$.

So to prove Theorem B about $|\mathcal{M}^{h}_{s}(S^{1}\times L)|$, it remains
to study the involution on $\operatorname{Wh}(C_{\infty}\times C_{m})$ and
prove the corresponding statements about $|\mathcal{J}_{n}(C_{\infty}\times
C_{m})|$. First, the fundamental theorem for
$K_{1}(\mathbb{Z}C_{m}[t,t^{-1}])$ gives rise to a direct sum decomposition
(see Theorem 5.9):

$\operatorname{Wh}(C_{m}\times
C_{\infty})\cong\operatorname{Wh}(C_{m})\oplus\widetilde{K}_{0}(\mathbb{Z}C_{m})\oplus
NK_{1}(\mathbb{Z}C_{m})^{2}$

where $\widetilde{K}_{0}$ is the reduced projective class group, and $NK_{1}$
is the so-called Nil group. All summands have natural involutions, which are
compatible with this isomorphism. Using that $\mathcal{J}_{n}(C_{m})=0$ (see
Proposition 5.13), we obtain the following decomposition for
$\mathcal{J}_{n}(C_{\infty}\times C_{m})$ (see Proposition 5.10):

$\mathcal{J}_{n}(C_{\infty}\times
C_{m})\cong\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid
x=-\overline{x}\\}\oplus NK_{1}(\mathbb{Z}C_{m}).$

In this decomposition the first component is always finite (see Lemma 5.12),
so part (c) of Theorem B is proved by the following (see Theorems 5.8 and
5.7).

######  Theorem 1.4 (Bass-Murthy, Martin, Weibel, Farrell).

If $m$ is square-free then $NK_{1}(\mathbb{Z}C_{m})=0$. Otherwise
$NK_{1}(\mathbb{Z}C_{m})$ is infinite.

For the remaining parts (b) and (d) of Theorem B, we need to study the
involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$ and prove the analogous
statements about $\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid
x=-\overline{x}\\}$ in the case where $m$ is square-free. This will be
achieved using methods from algebraic number theory, and discussed further in
Section 1.7 (see Theorem 1.13).

We can analyse $\mathcal{I}_{n}(C_{\infty}\times C_{m})$ similarly to
$\mathcal{J}_{n}(C_{\infty}\times C_{m})$. It has the following decomposition
(see Proposition 5.10):

$\mathcal{I}_{n}(C_{\infty}\times C_{m})\cong\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}\oplus NK_{1}(\mathbb{Z}C_{m}).$

It follows from Theorem 1.3 that $\mathcal{I}_{n}(C_{\infty}\times C_{m})$ is
invariant under the action of $\operatorname{hAut}(S^{1}\times L)$, so
$q(\mathcal{I}_{n}(C_{\infty}\times C_{m}))$ can be expressed as
$\mathcal{I}_{n}(C_{\infty}\times C_{m})/\operatorname{hAut}(S^{1}\times L)$.
Thus by Theorem D we have $\mathcal{M}^{\operatorname{hCob}}_{s}(S^{1}\times
L)\cong\mathcal{I}_{n}(C_{\infty}\times C_{m})/\operatorname{hAut}(S^{1}\times
L)$. By Theorems 1.3, 1.4, and 1.14, this leads to the following theorem.

######  Theorem E.

Let $n\geq 4$ be even and let $\operatorname{CAT}$ be as in 1.1. Let
$M^{n}_{m}=S^{1}\times L$, where $L$ is an $(n-1)$-dimensional lens space with
$\pi_{1}(L)\cong C_{m}$, $m\geq 2$. Then

  1. (a)

$|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|$ only depends on $n$ and
$m$, but it is independent of the choice of $L$ or $\operatorname{CAT}$;

  2. (b)

$|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|=1$ if and only if
$m\in\\{2,3,5,6,7,10,11,13,14,15,17,19,29\\}$;

  3. (c)

$|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|=\infty$ if and only if $m$
is not square-free;

  4. (d)

$|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|\to\infty$ as $m\to\infty$
uniformly in $n$.

Finally, we will consider $\mathcal{M}^{h}_{s,\operatorname{hCob}}(S^{1}\times
L)$. It follows from Corollary 1.2 that this is isomorphic to
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))/\operatorname{hAut}(S^{1}\times L)$. The decomposition of
$\operatorname{Wh}(C_{\infty}\times C_{m})$ induces an isomorphism
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))\cong\widehat{H}^{n+1}(C_{2};\widetilde{K}_{0}(\mathbb{Z}C_{m}))$.
Hence, as above, it remains to consider the involution on
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$, but the behaviour of
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))$ is
different from $\mathcal{J}_{n}(C_{\infty}\times C_{m})$ and
$\mathcal{I}_{n}(C_{\infty}\times C_{m})$, in particular it is always finite.

######  Theorem F.

Let $n\geq 4$ be even and let $\operatorname{CAT}$ be as in 1.1. Let
$M^{n}_{m}=S^{1}\times L$, where $L$ is an $(n-1)$-dimensional lens space with
$\pi_{1}(L)\cong C_{m}$, $m\geq 2$. Then

  1. (a)

$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|$ only depends on $n$ and
$m$, but it is independent of the choice of $L$ or $\operatorname{CAT}$;

  2. (b)

$\displaystyle\liminf_{m\to\infty}\Bigl{(}\sup_{n}|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|\Bigr{)}=1$;

  3. (c)

$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|<\infty$ for all $n$ and
$m$;

  4. (d)

$\displaystyle\limsup_{m\to\infty}\Bigl{(}\inf_{n}|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|\Bigr{)}=\infty$.

Equivalently, part (b) says that there are infinitely many $m$ such that
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|=1$ for every $n$, while
part (d) says that
$\inf_{n}|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|$ is unbounded in
$m$.

####  1.4. Homotopy equivalences of doubles

Next we consider 2. By Theorem D, to show that $\mathcal{M}^{h}_{s}(M)$ (resp.
$\mathcal{M}^{\operatorname{hCob}}_{s}(M)$,
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$) is nontrivial for some $M$, we
need to understand the action of $\operatorname{hAut}(M)$ on the Whitehead
group. Therefore our aim is to develop a systematic method for constructing
manifolds $M$, with arbitrary predetermined fundamental group $G$ and
orientation character $w$, such that we have control over the Whitehead
torsion of homotopy automorphisms of $M$. To achieve this, we will study
doubles and homotopy equivalences between them. Our central result is a
formula for the Whitehead torsion of such a homotopy equivalence (Theorem G),
which we will discuss below. By applying this formula to certain doubles, we
get sufficient conditions for the existence of an $M$ with nontrivial
$\mathcal{M}^{\operatorname{hCob}}_{s}(M)$ or
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$ (and hence
$\mathcal{M}^{h}_{s}(M)$), expressed in terms of the involution on
$\operatorname{Wh}(G,w)$. These, and a few further applications of Theorem G,
will be discussed in Section 1.5.

We start by introducing the doubles that we will study. Fix positive integers
$n,k$ such that $n\geq\max(6,2k+2)$. Let $K$ be a $k$-complex, i.e. a finite
$k$-dimensional CW complex, and let $T$ be an $n$-dimensional thickening of
$K$, i.e. an $n$-manifold with boundary together with a simple homotopy
equivalence $f_{T}\colon K\rightarrow T$ (see Section 9). Then a double over
$K$ is a manifold of the form $M=T\cup_{\operatorname{Id}_{\partial T}}T$ (a
trivial double), $M=T\cup_{g}T$ where $g\colon\partial T\rightarrow\partial T$
is a diffeomorphism (a twisted double), or $M=T\cup W\cup T$ where $W$ is an
$h$-cobordism between two copies of $\partial T$ (a generalised double). In
each case $M$ comes equipped with a canonical map $\varphi\colon K\rightarrow
M$, which is the composition of $f_{T}$ and the inclusion of the first
component $T\rightarrow M$.

If $M$ is an $n$-manifold and $K$ is a $k$-complex, then we will call a map
$\varphi\colon K\rightarrow M$ a _polarisation_ of $M$, and the pair
$(M,\varphi)$ a _polarised manifold_. We will say that $(M,\varphi)$ has a
_trivial/twisted/generalised double structure_ if, for some thickening $T$ of
$K$, $M$ has a decomposition as above such that the canonical map
$K\rightarrow M$ is homotopic to $\varphi$.

We will obtain the following recognition criterion for generalised doubles
(see Proposition 10.2).

######  Proposition 1.5.

Let $(M,\varphi)$ be a polarised manifold. Then $(M,\varphi)$ has a
generalised double structure if and only if $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected.

It follows that the class of generalised doubles is closed under homotopy
equivalence, i.e. if $(M,\varphi)$ has a generalised double structure and
$M\simeq N$, then $(N,\psi)$ also has a generalised double structure for some
$\psi$. Note that the same is not true for the classes of twisted and trivial
doubles.

Given a polarised manifold $(M,\varphi)$ such that $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected, we can define an invariant
$\tau(M,\varphi)$, called the Whitehead torsion of $(M,\varphi)$, as follows.
There is a thickening $f_{T}\colon K\rightarrow T$ of $K$ and an embedding
$i\colon T\rightarrow M$ such that $i\circ f_{T}\simeq\varphi$. Let
$C=M\setminus i(\operatorname{Int}T)$, then $\varphi$ is homotopic to a
homotopy equivalence $\varphi^{\prime}\colon K\rightarrow C$ regarded as a map
$K\rightarrow M$. The inclusion $C\rightarrow M$ induces an isomorphism
$\pi_{1}(C)\cong\pi_{1}(M)$, so we can identify
$\operatorname{Wh}(\pi_{1}(C))$ with $\operatorname{Wh}(\pi_{1}(M))$.

######  Definition 1.6.

Let $\tau(M,\varphi)=\tau(\varphi^{\prime})\in\operatorname{Wh}(\pi_{1}(M))$.

This invariant will show up in the formula of Theorem G as an error term. We
will also prove in Corollary 10.8 that if $(M,\varphi)$ has a trivial double
structure, then $\tau(M,\varphi)=0$. The converse does not hold, but if
$\tau(M,\varphi)=0$, then $(M,\varphi)$ has a twisted double structure, and
more generally see Proposition 10.10 for a result characterising precisely
when a twisted double structure exists.

Next we consider homotopy equivalences between doubles. If $(M,\varphi)$ is a
polarised manifold and $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected, then we can find a CW
decomposition of $M$ such that $\varphi$ is the embedding of its $k$-skeleton.
Suppose that $N$ is another $n$-manifold with an
$\left\lfloor\frac{n}{2}\right\rfloor$-connected polarisation $\psi\colon
L\rightarrow N$ for a $k$-dimensional CW complex $L$. Then by cellular
approximation any homotopy equivalence $f\colon M\rightarrow N$ restricts to a
map $\alpha\colon K\rightarrow L$. This $\alpha$ is also a homotopy
equivalence if $(M,\varphi)$ and $(N,\psi)$ satisfy some mild restrictions on
their dimensions or double structures; in this case we call them _split
polarised_ (SP) manifolds (see Definition 10.12). The following key theorem
then allows us to compute the Whitehead torsion of $f$ from that of $\alpha$
(see Theorem 11.5).

######  Theorem G.

Suppose that $(M,\varphi)$ and $(N,\psi)$ are split polarised manifolds and
$f\colon M\rightarrow N$ is a homotopy equivalence. Then

$\tau(f)=\tau(\alpha)-(-1)^{n}\overline{\tau(\alpha)}+\tau(N,\psi)-f_{*}(\tau(M,\varphi))\in\operatorname{Wh}(\pi_{1}(N),w_{N})$

where $\alpha\colon K\rightarrow L$ is the restriction of $f$,
$\operatorname{Wh}(\pi_{1}(L))$ is identified with
$\operatorname{Wh}(\pi_{1}(N))$ via $\psi_{*}$ and $w_{N}$ is the orientation
character of $N$.

####  1.5. Applications of Theorem G

In the main applications of Theorem G we restrict to special types of doubles
where we have control over the right hand side of the formula. In particular,
when $\tau(M,\varphi)=0$ (which can be ensured by taking $M$ to be a trivial
double), then we have $\tau(f)\in\mathcal{I}_{n}(G,w)$ for every homotopy
automorphism $f$ of $M$. From this we get that $0$ is a fixed point of the
action of $\operatorname{hAut}(M)$ on
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$. By Theorem D we have the
following.

######  Theorem 1.7.

Let $n\geq 5$, let $G$ be a finitely presented group and let $w\colon
G\rightarrow\left\\{\pm 1\right\\}$ be such that $\psi\colon
L_{n+1}^{h}(\mathbb{Z}G,w)\rightarrow\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
is nontrivial. Then there exists a $\operatorname{CAT}$ $n$-manifold $M$ with
fundamental group $G$ and orientation character $w$ such that
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|>1$.

######  Remark 1.8.

For $n\geq 6$, this can also be deduced from results in Hausmann’s unpublished
preprint [hausmann-mwmdh, Sections 9–10]. He gave applications in the case
where $n$ is odd [hausmann-mwmdh, Example 10.2].

When we also have some control over the possible values of $\tau(\alpha)$,
Theorem G gives even stricter restrictions on the Whitehead torsions of
homotopy equivalences and automorphisms. For example, if $K$ and $L$ are
$k$-manifolds with orientation character $w$, then
$\tau(\alpha)\in\mathcal{J}_{k}(G,w)$ (see Proposition 2.34). We obtain the
following simple homotopy rigidity theorem for sphere bundles.

######  Theorem H.

Suppose that $j>k$ are positive integers and $j$ is odd. Let $K$ and $L$ be
$k$-manifolds, and let $S^{j}\to M\to K$ and $S^{j}\to N\to L$ be orientable
$($linear$)$ sphere bundles. Then every homotopy equivalence $f\colon M\to N$
is simple.

######  Remark 1.9.

If we think of the Whitehead torsion as an invarant analogous to the Euler
characteristic (cf. Lemma 2.17), then Theorem H can be regarded as the
analogue of the fact that odd dimensional manifolds have vanishing Euler
characteristic.

Combined with Theorem D, Theorem H leads to a proof of Theorem C (see Theorem
12.9). At the same time, it produces a class of manifolds, with arbitrary
fundamental groups, within which two manifolds are homotopy equivalent if and
only if they are simple homotopy equivalent.

Next we consider doubles over certain $2$-complexes $X$ with fundamental group
$C_{\infty}\times C_{m}$, for which Metzler showed that
$\tau\colon\operatorname{hAut}(X)\to\operatorname{Wh}(C_{\infty}\times C_{m})$
is not surjective [Me79, Theorem 1]. Using these complexes $X$, improved
constraints on the set $\tau(\operatorname{hAut}(X))$, and Theorem D, we
obtain:

######  Theorem 1.10.

Let $n\geq 5$ and let $m\geq 2$ be such that $\\{x-(-1)^{n}\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}\neq 0$. Then there is an orientable
$\operatorname{CAT}$ $n$-manifold $M$ with fundamental group $C_{\infty}\times
C_{m}$ such that $|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$.

######  Remark 1.11.

If $n$ is even, then examples where $\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}\neq 0$ are given in Theorem 1.14
(i). If $n$ is odd then, by Lemma 15.2 (i) and a similar argument to the one
used in Lemma 16.4, we have $|\\{x+\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|\geq h_{m}^{+}$ where $h_{m}^{+}$
denotes the plus part of the $m$th cyclotomic class number (see Section 1.7).
For example, we have that $h_{136}^{+}\neq 1$ [Wa97, p. 421].

Theorem 1.10 leads to examples of doubles $M$ with fundamental group
$C_{\infty}\times C_{m}$ such that
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$. When $n$ is even, these serve
as alternatives to the examples constructed in Section 1.3.

Finally we note that $\tau(M,\varphi)$ can be used to define an invariant for
certain unpolarised manifolds. We say that $M$ is a _split manifold_ if there
is a $\varphi\colon K\rightarrow M$ such that $(M,\varphi)$ is an SP manifold.
For such an $M$ we define

$\tau(M):=\pi(\tau(M,\varphi))\in\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(\pi_{1}(M),w))$

where $w\colon\pi_{1}(M)\rightarrow\left\\{\pm 1\right\\}$ is the orientation
character of $M$. It can be shown using Theorem G that this is well-defined
(see Section 12.4). If a split manifold $M$ is also a manifold without middle
dimensional handles in the sense of Hausmann [hausmann-mwmdh], then $\tau(M)$
recovers the “torsion invariant” defined in [hausmann-mwmdh, Section 9], which
was shown to be invariant under simple homotopy equivalences and homotopy
equivalences induced by $h$-cobordisms. We prove the following (see Theorem
12.21).

######  Theorem 1.12.

On the class of split manifolds, $\tau(M)$ is a complete invariant for the
equivalence relation generated by simple homotopy equivalence and
$h$-cobordism.

####  1.6. Summary for 2

We now give a brief overview of how the results presented so far can be used
to characterise pairs $(G,w)$ with an affirmative answer to 2, and the
analogous questions for $\mathcal{M}^{\operatorname{hCob}}_{s}$ and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}$.

First observe that when $G=C_{\infty}\times C_{m}$, $w=1$ and $n$ is even, we
can use the results on $S^{1}\times L$ to get a complete answer for
$\mathcal{M}^{h}_{s}$ and $\mathcal{M}^{\operatorname{hCob}}_{s}$. That is
because during the proof of Theorem B (b) (resp. Theorem E (b)) we show that
$\mathcal{J}_{n}(C_{\infty}\times C_{m})=0$ (resp.
$\mathcal{I}_{n}(C_{\infty}\times C_{m})=0$) for the listed values of $m$,
which excludes the existence of an $M$ with nontrivial
$\mathcal{M}^{h}_{s}(M)$ (resp. $\mathcal{M}^{\operatorname{hCob}}_{s}(M)$) by
Theorem D. Furthermore, by comparing the two lists, we see that $m=15$ and
$29$ are the only integers for which
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|=1$ but
$|\mathcal{M}^{h}_{s}(M^{n}_{m})|>1$ (for every $n$). These give examples of
manifolds $M^{n}_{m}$ such that every manifold $h$-cobordant to $M^{n}_{m}$ is
simple homotopy equivalent to it, but being homotopy equivalent to $M^{n}_{m}$
does not imply being simple homotopy equivalent to it.

In course of the proof of Theorem F we similarly show that a manifold $M$ with
nontrivial $\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$ exists if and only if
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))\cong\mathcal{J}_{n}(C_{\infty}\times
C_{m})/\mathcal{I}_{n}(C_{\infty}\times C_{m})$ is nontrivial. The question
whether $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))$
is trivial or not is open in general. However, the answer is known for the
previously mentioned special values of $m$ by the results on
$\mathcal{J}_{n}(C_{\infty}\times C_{m})$ and
$\mathcal{I}_{n}(C_{\infty}\times C_{m})$, and the proof of Theorem 16.3 shows
that for each case there are infinitely many $m$ for which that case occurs.
The results on $C_{\infty}\times C_{m}$ are summarised in Table 1.

$m$ | $\mathcal{M}^{h}_{s}$ | $\mathcal{M}^{\operatorname{hCob}}_{s}$ | $\mathcal{M}^{h}_{s,\operatorname{hCob}}$
---|---|---|---
$2,3,5,6,7,10,11,13,14,17,19$ | No | No | No
$15,29$ | Yes | No | Yes
otherwise | Yes | Yes | Open
Table 1. Is there an even-dimensional orientable $M$ with fundamental group
$C_{\infty}\times C_{m}$ such that $|\mathcal{M}^{h}_{s}(M)|>1$ (resp.
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$,
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|>1$)?

Next we consider the case of an arbitrary finitely presented group $G$ with a
homomorphism $w\colon G\to\\{\pm 1\\}$, using results from Section 1.5.

First, Theorem C characterises pairs $(G,w)$ such that
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$ for some $M$ with fundamental
group $G$ and orientation character $w$ in terms of the (non-)vanishing of
$\mathcal{I}_{n}(G,w)$.

For $\mathcal{M}^{h}_{s,\operatorname{hCob}}$, Theorems D and 1.7 show that
the nontriviality of $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ is
necessary and the nonvanishing of $\psi=\psi^{n+1}_{(G,w)}\colon
L_{n+1}^{h}(\mathbb{Z}G,w)\rightarrow\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
is sufficient. The case when
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))\neq 0$ but
$\psi^{n+1}_{(G,w)}=0$ is open in general.

Finally, Theorem D shows that $\mathcal{M}^{h}_{s}(M)$ is nontrivial if and
only if at least one of $\mathcal{M}^{\operatorname{hCob}}_{s}(M)$ and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$ is nontrivial. Thus the existence
of an $M$ with $|\mathcal{M}^{h}_{s}(M)|>1$ can be decided in all cases except
when $\mathcal{I}_{n}(G,w)=0$,
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))\neq 0$ and
$\psi^{n+1}_{(G,w)}=0$. We note that this case is nonempty: for example, for
$n$ even, the group $C_{4}\times C_{4}$, with $w=1$, falls into this category.
Details will be postponed for future work.

These results are summarised in Table 2, where we also indicate the
restrictions on $n$ needed for the theorems to apply.

$\mathcal{I}_{n}(G,w)$ | $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ | $\psi^{n+1}_{(G,w)}$ | $\mathcal{M}^{h}_{s}$ | $\mathcal{M}^{\operatorname{hCob}}_{s}$ | $\mathcal{M}^{h}_{s,\operatorname{hCob}}$
---|---|---|---|---|---
$=0$ | $=0$ | $=0$ | No | $n\geq 4$ | No | $n\geq 4$ | No | $n\geq 4$
$=0$ | $\neq 0$ | $\neq 0$ | Yes | $n\geq 5$ | No | $n\geq 4$ | Yes | $n\geq 5$
$=0$ | $\neq 0$ | $=0$ | Open | - | No | $n\geq 4$ | Open | -
$\neq 0$ | $=0$ | $=0$ | Yes | $n=9,\geq 11$ | Yes | $n=9,\geq 11$ | No | $n\geq 4$
$\neq 0$ | $\neq 0$ | $\neq 0$ | Yes | $n\geq 5$ | Yes | $n=9,\geq 11$ | Yes | $n\geq 5$
$\neq 0$ | $\neq 0$ | $=0$ | Yes | $n=9,\geq 11$ | Yes | $n=9,\geq 11$ | Open | -
Table 2. Is there an $n$-manifold $M$ with fundamental group $G$ and
orientation character $w$ such that $|\mathcal{M}^{h}_{s}(M)|>1$ (resp.
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$,
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|>1$)?

####  1.7. The involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$

In light of the discussion in Section 1.3, detailed information regarding the
involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$ is required in order to
complete the proofs of parts (b) and (d) in each of Theorems B, E, LABEL:, and
F. The purpose of Part IV is to explore this involution in detail and to
establish the following three theorems which are the key algebraic ingredients
behind the proofs of Theorems B, E, LABEL:, and F respectively.

######  Theorem 1.13.

Let $m\geq 2$ be a square-free integer. Then

  1. (i)

$|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}|=1$ if and
only if $m\in\\{2,3,5,6,7,10,11,13,14,17,19\\}$; and

  2. (ii)

$|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}|\to\infty$
super-exponentially in $m$.

######  Theorem 1.14.

Let $m\geq 2$ be a square-free integer. Then

  1. (i)

$|\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|=1$ if and
only if $m\in\\{2,3,5,6,7,10,11,13,14,15,17,19,29\\}$; and

  2. (ii)

$|\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|\to\infty$
super-exponentially in $m$.

######  Theorem 1.15.

Let $m\geq 2$ be an integer. Then

  1. (i)

$|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}/\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|=1$ for infinitely many $m$; and

  2. (ii)

$\displaystyle\sup_{n\leq
m}|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}/\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|\to\infty$ exponentially in $m$.

We will now explain the strategy of proof of these three theorems, as well as
some of the key ingredients. The first thing to note is that, whilst
$\widetilde{K}_{0}(R)$ is difficult to compute for an arbitrary ring, it is
often computable in the case where $R=\mathbb{Z}G$ for a finite group $G$
since finitely generated projective $\mathbb{Z}G$-modules $P$ are all locally
free, i.e. $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}P$ is a free $\mathbb{Z}_{p}G$
module for all primes $p$, where $\mathbb{Z}_{p}$ denotes the $p$-adic
integers. In particular, $\widetilde{K}_{0}(\mathbb{Z}G)\cong C(\mathbb{Z}G)$
where $C(\cdot)$ denotes the locally free class group (see Section 13.1). The
general strategy for determining the involution on
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$ is to note that, since
$\widetilde{K}_{0}(\mathbb{Z}C_{m})\cong C(\mathbb{Z}C_{m})$, we can fit
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$ into a short exact sequence of abelian
groups:

$0\to
D(\mathbb{Z}C_{m})\to\widetilde{K}_{0}(\mathbb{Z}C_{m})\to\bigoplus_{d\mid
m}C(\mathbb{Z}[\zeta_{d}])\to 0$

where $D(\mathbb{Z}C_{m})$ denotes the kernel group of $\mathbb{Z}C_{m}$ (see
Section 13.2) and $C(\mathbb{Z}[\zeta_{d}])$ denotes the ideal class group of
$\mathbb{Z}[\zeta_{d}]$, where $\zeta_{d}$ denotes a primitive $d$th root of
unity. The standard involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$
restricts to $D(\mathbb{Z}C_{m})$ and induces the involution given by
conjugation on each $C(\mathbb{Z}[\zeta_{d}])$ (see Section 15.1).

The proofs of Theorems 1.13 and 1.14 are intertwined and can be found in
Section 16.1. The approach we will take is to use Lemma 14.3, and its
consequence Lemma 16.4, which allow us to obtain information about the orders
$|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}|$ and
$|\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|$ from the
orders of the corresponding groups $|\\{x\in A\mid\overline{x}=-x\\}|$ and
$|\\{x-\overline{x}\mid x\in A\\}|$ in the cases where $A=D(\mathbb{Z}C_{m})$
or $C(\mathbb{Z}[\zeta_{d}]))$ for some $d\mid m$.

The key lemma which makes this approach possible is the following, which is a
part of Proposition 15.4. Let $h_{m}=|C(\mathbb{Z}[\zeta_{m}])|$ denotes the
class number of the $m$th cyclotomic field, and recall that it splits as a
product $h_{m}=h_{m}^{+}h_{m}^{-}$ for integers
$h_{m}^{+}=|C(\mathbb{Z}[\zeta_{m}+\zeta_{m}^{-1}])|$ and $h_{m}^{-}$ known as
the plus and minus parts of the class number. For an integer $m$, we let
$\operatorname{odd}(m)$ denote the odd part of $m$, i.e. the unique odd
integer $r$ such that $m=2^{k}r$ for some $k$.

######  Lemma 1.16.

We have that $\operatorname{odd}(h_{m}^{-})\leq|\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}|$.

It was shown by Horie [Ho89], using results from Iwasawa theory [Fr82], that
there exists finitely many $m$ for which $\operatorname{odd}(h_{m}^{-})=1$ and
$\operatorname{odd}(h_{m}^{-})\to\infty$ (see Proposition 15.8). Since Lemma
1.16 also gives a lower bound on $|\\{x\in
C(\mathbb{Z}[\zeta_{m}])\mid\overline{x}=-x\\}|$, this is enough to prove part
(ii) of Theorems 1.13 and 1.14 and to reduce the proof of part (i) to checking
finitely many cases. These cases are dealt with via a variety of methods such
as analysing the group structure on $C(\mathbb{Z}[\zeta_{m}])$ (see the proof
of Proposition 16.7) and the studying the involution on $D(\mathbb{Z}C_{m})$
by relating it to maps between units groups (see Sections 15.4 and 16.1).

The proof of Theorem 1.15 can be found in Section 16.2. Our approach will be
based around the fact that the group we are trying to compute can be expressed
as

$\frac{\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}}{\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}}\cong\widehat{H}^{1}(C_{2};\widetilde{K}_{0}(\mathbb{Z}C_{m}))$

where $\widetilde{K}_{0}(\mathbb{Z}C_{m})$ is viewed as a
$\mathbb{Z}C_{2}$-module with the $C_{2}$-action given by the involution (see
Proposition 14.5). The short exact sequence above induces a 6-periodic exact
sequence on Tate cohomology (Proposition 14.7). This, combined with the fact
that $\widehat{H}^{n}(C_{2};A)$ for $A$ finite only depends on its $2$-Sylow
subgroup $A_{(2)}$ (Proposition 14.11) gives the following, which is a part of
Lemma 16.14.

######  Lemma 1.17.

If $h_{m}$ is odd, then
$\widehat{H}^{1}(C_{2};\widetilde{K}_{0}(\mathbb{Z}C_{m}))\cong\widehat{H}^{1}(C_{2};D(\mathbb{Z}C_{m}))$.
In particular, if $h_{m}$ and $|D(\mathbb{Z}C_{m})|$ are both odd, then
$\widehat{H}^{1}(C_{2};\widetilde{K}_{0}(\mathbb{Z}C_{m}))=0$.

Using Iwasawa theory, it was shown by Washington [Wa75] that $h_{3^{n}}$ is
odd for all $n\geq 1$ (see also the work of Ichimura–Nakajima [IN12] which
extends this results to more primes $p\leq 509$). Since $C_{3^{n}}$ is a
$3$-group, $|D(\mathbb{Z}C_{3^{n}})|$ is an abelian $3$-group and so is odd
[CR87, Theorem 50.18]. By Lemma 1.17, this implies that
$|\widehat{H}^{1}(C_{2};\widetilde{K}_{0}(\mathbb{Z}C_{3^{n}}))|=1$ for all
$n\geq 1$ which gives Theorem 1.15 (i).

On the other hand, by Weber’s theorem, $h_{2^{n}}$ is odd for all $n\geq 1$
(see Lemma 15.11). Using a result of Kervaire–Murthy [KM77] which relates
$D(\mathbb{Z}C_{2^{n}})$ with $D(\mathbb{Z}C_{2^{n+1}})$ (Proposition 15.20),
and the 6-periodic exact sequence on Tate cohomology (Lemma 16.14), we show
that $|\widehat{H}^{1}(C_{2};D(\mathbb{Z}C_{2^{n}}))|$ is unbounded as
$n\to\infty$. By Lemma 1.17, this implies Theorem 1.15 (ii).

#### Organisation of the paper

The paper will be structured into four parts that approximately mirror the
structure of Sections 1.2 – 1.7 above, which can be viewed as introductions
for the parts to which they correspond. Part I follows on from Section 1.2: we
develop the necessary background on simple homotopy equivalence, Whitehead
torsion and $h$-cobordisms. We will then prove Theorem D, which is our main
general result. Part II follows on from Section 1.3: we study the manifolds
$L\times S^{1}$, leading to the proofs of Theorems B, E, LABEL:, and F subject
to results about $\widetilde{K}_{0}(\mathbb{Z}C_{m})$. Part III follows on
from Sections 1.4 – 1.5: we develop a theory of homotopy equivalences of
doubles which we apply to prove Theorems G and H. Part IV follows on from
Section 1.7: we develop the necessary background on integral representation
theory and algebraic number theory. We then study the involution of
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$, leading to proofs of Theorems 1.13-1.15.

#### Conventions

The following conventions will be in place throughout this article, unless
otherwise specified. As above, $n\geq 4$ will be an integer and an
$n$-manifold will be a compact connected $\operatorname{CAT}$ $n$-manifold
where
$\operatorname{CAT}\in\\{\operatorname{Diff},\operatorname{PL},\operatorname{TOP}\\}$.
They will be assumed closed except where it is clear from the context that
they are not, e.g. thickenings and cobordisms. We will frequently assume 1.1
but will state this as needed. Groups will be assumed to be finitely
presented. Rings $R$ will be assumed to have a multiplicative identity, and
$R$-modules will be assumed to be finitely generated.

#### Acknowledgements

We are grateful to Daniel Kasprowski and Arunima Ray for discussions on the
$s$-cobordism theorem, to Henri Johnston, Rachel Newton, and Jack Shotton for
advice on ideal class groups, and to Scott Schmieding for discussions on Nil
groups.

CsN was supported by EPSRC New Investigator grant EP/T028335/2. JN was
supported by the Heilbronn Institute for Mathematical Research. MP was
partially supported by EPSRC New Investigator grant EP/T028335/2 and EPSRC New
Horizons grant EP/V04821X/2.

###### Contents

  1. 1 Introduction
    1. 1.1 Background and main results
    2. 1.2 Characterisation of simple homotopy manifold sets
    3. 1.3 The simple homotopy manifold set of $S^{1}\times L$
    4. 1.4 Homotopy equivalences of doubles
    5. 1.5 Applications of Theorem G
    6. 1.6 Summary for 2
    7. 1.7 The involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$
  2. I General results
    1. 2 Preliminaries
      1. 2.1 Simple homotopy equivalence
      2. 2.2 The Whitehead group
      3. 2.3 The Whitehead torsion of a chain homotopy equivalence
      4. 2.4 The Whitehead torsion of a homotopy equivalence
    2. 3 Realising elements of the Whitehead group by maps between manifolds
      1. 3.1 Realising via $h$-cobordisms
      2. 3.2 Realising via the surgery exact sequence
    3. 4 Simple homotopy manifold sets
      1. 4.1 CW complexes
      2. 4.2 Manifolds
  3. II The simple homotopy manifold set of $S^{1}\times L$
    1. 5 The involution on $\operatorname{Wh}(C_{\infty}\times C_{m})$
      1. 5.1 The $\widetilde{K}_{0}$ and $NK_{1}$ groups
      2. 5.2 Bass-Heller-Swan decomposition of $\operatorname{Wh}(C_{\infty}\times G)$
      3. 5.3 The groups $\mathcal{J}_{n}(C_{\infty}\times C_{m})$, $\mathcal{I}_{n}(C_{\infty}\times C_{m})$, and $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))$
    2. 6 Homotopy automorphisms of $S^{1}\times L$
      1. 6.1 The automorphism induced on the fundamental group
      2. 6.2 The Whitehead torsion
    3. 7 The proof of Theorems B, E, , and F
  4. III Homotopy equivalences of doubles
    1. 8 Split chain complexes
    2. 9 Thickenings
    3. 10 Polarised doubles
      1. 10.1 Double structures on polarised manifolds
      2. 10.2 The Whitehead torsion of a polarised double
      3. 10.3 SP manifolds
    4. 11 The Whitehead torsion of homotopy equivalences of doubles
    5. 12 Applications
      1. 12.1 Simple doubles
      2. 12.2 Doubles over manifolds
      3. 12.3 Doubles over certain $2$-complexes
      4. 12.4 The dependence of $\tau(M,\varphi)$ on $\varphi$
      5. 12.5 An invariant of unpolarised manifolds
  5. IV The involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$
    1. 13 Locally free class groups
      1. 13.1 Definitions and properties
      2. 13.2 Kernel groups
      3. 13.3 The idèlic approach to locally free class groups
      4. 13.4 Involutions on locally free class groups
    2. 14 Tate cohomology and $\mathbb{Z}C_{2}$-modules
      1. 14.1 Tate cohomology
      2. 14.2 Tate cohomology and the structure of $\mathbb{Z}C_{2}$-modules
    3. 15 Computing the involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$
      1. 15.1 The induced involution on the maximal order
      2. 15.2 Ideal class groups of cyclotomic fields
      3. 15.3 Divisibility of class numbers of cyclotomic fields
      4. 15.4 Kernel groups of $\mathbb{Z}C_{m}$
      5. 15.5 Divisibility and lower bounds for kernel groups
    4. 16 Proof of main results on the involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$
      1. 16.1 Proof of Theorems 16.1 and 16.2
      2. 16.2 Proof of Theorem 16.3

##  Part I General results

In this part we will establish general results regarding simple homotopy
equivalence. This will be the basis for our applications in Parts II and III.
In Section 2, we will recall the basic theory of simple homotopy equivalence,
culminating in constraints on the Whitehead torsion of homotopy equivalences
between manifolds. Section 3 concerns the two methods which we will use for
constructing manifolds: via $h$-cobordisms (Section 3.1) and via the surgery
exact sequence (Section 3.2). In Section 4, which follows on from Section 1.2
in a natural way, we study the simple homotopy manifolds sets and prove
Theorem D.

###  2\. Preliminaries

In this section we recall the definition of simple homotopy equivalence, the
Whitehead group and the Whitehead torsion, as well as some of their basic
properties. Our main sources are Milnor [Mi66], Cohen [Co73], and Davis-Kirk
[DK01].

####  2.1. Simple homotopy equivalence

Let $X$ be a CW complex and let $\phi\colon D^{n}\to X$ be a cellular map.
Divide the boundary of the closed $(n+1)$-cell $e^{n+1}$ into two $n$-discs,
$\partial e^{n+1}\cong D^{n}\cup_{S^{n-1}}D^{n}$, gluing along the first copy
of $D^{n}\subseteq\partial e^{n+1}$. Then the inclusion $X\to
X\cup_{\phi}e^{n+1}$ is called an _elementary expansion_. There is a
deformation retract $X\cup_{\phi}e^{n+1}\to X$ in the other direction, and
this is called an _elementary collapse_.

######  Definition 2.1.

A homotopy equivalence $f\colon X\to Y$ between finite CW complexes is
_simple_ if $f$ is homotopic to a map that is a composition of finitely many
elementary expansions and collapses

$X=X_{0}\to X_{1}\to X_{2}\to\cdots\to X_{k}=Y.$

For a homotopy equivalence $f\colon X\to Y$, Whitehead introduced an invariant
$\tau(f)$, the _Whitehead torsion_ of $f$, which lies in the _Whitehead group_
$\operatorname{Wh}(\pi_{1}(Y))$ of $\pi_{1}(Y)$. We shall define both the
Whitehead group and the Whitehead torsion shortly. The motivation for the
Whitehead torsion is the following beautiful result, which completely
characterises whether or not a homotopy equivalence is simple.

######  Theorem 2.2 (Whitehead [Whitehead-she]).

A homotopy equivalence $f\colon X\to Y$ between CW complexes $X$ and $Y$ is
simple if and only if $\tau(f)=0\in\operatorname{Wh}(\pi_{1}(Y))$.

In dimensions $n\neq 4$, every closed $n$-manifold admits an $n$-dimensional
CW structure; see Kirby-Siebenmann [Kirby-Siebenmann:1977-1, III.2.2] for
$n\geq 5$, [Ra26] for $n=2$ and [Moise] for $n=3$. In dimension 4, it is an
open question whether this holds. Certainly every smooth or PL $n$-manifold
admits an $n$-dimensional triangulation, and hence a CW structure.

We explain how the notion of simple homotopy equivalence makes sense for
4-manifolds, even those for which we do not know whether they admit a CW
structure. The procedure, which is due to Kirby-Siebenmann [Kirby-
Siebenmann:1977-1, III, §4] works for any dimension, so we work in this
generality.

Let $M$ be a topological $n$-manifold. Embed $M$ in high-dimensional Euclidean
space. By [Kirby-Siebenmann:1977-1]*III, §4, there is a normal disc bundle
$D(M)\to M$ that admits a triangulation and hence a CW structure. The
inclusion map $z_{M}\colon M\to D(M)$ of the $0$-section is a homotopy
equivalence. Let $z_{M}^{-1}$ denote the homotopy inverse of $z_{M}$.

######  Definition 2.3.

We say that a homotopy equivalence $f\colon M\to N$ between topological
manifolds is _simple_ if the composition $z_{N}\circ f\circ z_{M}^{-1}\colon
D(M)\to D(N)$ is simple. Kirby-Siebenmann [Kirby-Siebenmann:1977-1]*III, §4
showed that whether or not this composition is simple does not depend on the
choice of normal disc bundle nor on the choice of triangulation.

If $M$ and $N$ are smooth or PL, then we can ask whether $f$ is simple using
the canonical class of triangulations of $M$ and $N$, or by forgetting the
smooth/PL structures and using the Kirby-Siebenmann method from Definition
2.3.

######  Proposition 2.4 (Kirby-Siebenmann [Kirby-Siebenmann:1977-1, III.5.1]).

For $\operatorname{CAT}\in\\{\operatorname{Diff},\operatorname{PL}\\}$, a
homotopy equivalence $f\colon M\to N$ between $\operatorname{CAT}$ manifolds
$M$ and $N$ is simple with respect to their canonical class of triangulations
if and only if it is simple with respect to Definition 2.3.

Hence we have a coherent notion of simple homotopy equivalence in all three
manifold categories. We remark that Proposition 2.4 can also be proven using
the following theorem of Chapman.

######  Theorem 2.5 (Chapman [Ch74]).

Let $f\colon X\to Y$ be a homeomorphism between compact, connected CW
complexes. Then $f$ is a simple homotopy equivalence.

Thus for closed manifolds $M$ and $N$ that admit a CW structure, one can
deduce from Chapman’s Theorem 2.5 that the question of whether $f\colon
M\xrightarrow{\simeq}N$ is simple does not depend on the CW structures. Thus
the extra work using disc bundles in Definition 2.3 is only required for non-
smoothable topological 4-manifolds.

####  2.2. The Whitehead group

######  Definition 2.6.

For a ring $R$, let
$\operatorname{GL}(R)=\operatorname{colim}_{n}\operatorname{GL}_{n}(R)$, where
we take the colimit with respect to the inclusions
$\operatorname{GL}_{n}(R)\hookrightarrow\operatorname{GL}_{n+1}(R)$,
$A\mapsto\left(\begin{smallmatrix}A&0\\\ 0&1\end{smallmatrix}\right)$. Define

$K_{1}(R)\coloneqq\operatorname{GL}(R)^{\text{ab}}.$

The Whitehead lemma [Mi66, Lemma 1.1] states that the commutator subgroup of
$\operatorname{GL}(R)$ is equal to the subgroup $E(R)$ generated by elementary
matrices (i.e. the matrices $E$ such that $A\mapsto EA$ is an elementary row
operation). It follows that we can also write

$K_{1}(R)=\operatorname{GL}(R)/E(R).$

######  Definition 2.7.

Define the _Whitehead group_ of a group $G$ to be
$\operatorname{Wh}(G)=K_{1}(\mathbb{Z}G)/{\pm G}$, where the map $\pm G\to
K_{1}(\mathbb{Z}G)$ is the composition $\pm G\subseteq
GL_{1}(\mathbb{Z}G)\subseteq\operatorname{GL}(\mathbb{Z}G)\to
K_{1}(\mathbb{Z}G)$.

Note that $\operatorname{Wh}$ is a functor from the category of groups to the
category of abelian groups. The map induced by a homomorphism $\theta\colon
G\rightarrow H$ will be denoted by
$\theta_{*}\colon\operatorname{Wh}(G)\rightarrow\operatorname{Wh}(H)$.
Similarly, for a continuous map $f\colon X\rightarrow Y$ between topological
spaces, the map induced by $\pi_{1}(f)\colon\pi_{1}(X)\rightarrow\pi_{1}(Y)$
will be denoted by
$f_{*}\colon\operatorname{Wh}(\pi_{1}(X))\rightarrow\operatorname{Wh}(\pi_{1}(Y))$.

The isomorphism type of $\operatorname{Wh}(G)$ is known in some cases. For us
the following examples will be relevant.

######  Proposition 2.8 (Stallings [St65]).

If $G$ is a finitely generated free group, then $\operatorname{Wh}(G)=0$.

######  Proposition 2.9 ([Co73, (11.5)]).

If $C_{m}$ is the finite cyclic group of order $m$, then

$\operatorname{Wh}(C_{m})\cong\mathbb{Z}^{\left\lfloor
m/2\right\rfloor+1-\delta(m)}$

where $\delta(m)$ is the number of positive integers dividing $m$.

The Whitehead group $\operatorname{Wh}(G)$ of a group $G$ is equipped with a
natural involution, i.e. an automorphism $x\mapsto\overline{x}$ such that
$\overline{\overline{x}}=x$. Equivalently, it has a $\mathbb{Z}C_{2}$-module
structure, where the generator of $C_{2}$ acts by the involution. We will
describe this involution below.

Let $R$ be a ring with involution, i.e. a ring equipped with a map
$x\mapsto\overline{x}$ that is an involution on $R$ as an abelian group, and
satisfies $\overline{xy}=\overline{y}\cdot\overline{x}$. This induces an
involution on $\operatorname{GL}(R)$, sending
$A=(A_{ij})\in\operatorname{GL}_{n}(R)$ to
$\overline{A}=(\overline{A_{ji}})\in\operatorname{GL}_{n}(R)$, the conjugate
transpose of $A$. Note that this is perhaps non-standard notation for the
conjugate transpose. We use this convention so that the notation for
involutions is consistent. This involution preserves the subgroup
$E(R)\subseteq\operatorname{GL}(R)$ and so induces an involution $K_{1}(R)$.

If $G$ is a group and $w\colon G\to\\{\pm 1\\}$ is an orientation character,
i.e. a homomorphism from $G$ to $\\{\pm 1\\}$, then the integral group ring
$\mathbb{Z}G$ has an involution given by
$\sum_{i=1}^{k}n_{i}g_{i}\mapsto\sum_{i=1}^{k}w(g_{i})n_{i}g_{i}^{-1}$ for
$n_{i}\in\mathbb{Z}$ and $g_{i}\in G$. The resulting involution on
$K_{1}(\mathbb{Z}G)$ preserves $\pm G$ and so induces an involution on
$\operatorname{Wh}(G)$.

######  Definition 2.10.

We will write $\operatorname{Wh}(G,w)$ for the abelian group
$\operatorname{Wh}(G)$ equipped with the involution determined by the
orientation character $w\colon G\to\\{\pm 1\\}$. This can equivalently be
regarded as a $\mathbb{Z}C_{2}$-module. When $w\equiv 1$ is the trivial
homomorphism, we will omit it from the notation and write
$\operatorname{Wh}(G)$ for $\operatorname{Wh}(G,1)$.

Note that the choice of $w$ only affects the involution, and
$\operatorname{Wh}(G,w)$ is equal to $\operatorname{Wh}(G)$ as an abelian
group. Next we define two subgroups of $\operatorname{Wh}(G,w)$ which will
play an important role in the rest of this article.

######  Definition 2.11.

For a group $G$ and a homomorphism $w\colon G\to\\{\pm 1\\}$, define:

$\displaystyle\mathcal{J}_{n}(G,w)$
$\displaystyle\coloneqq\\{y\in\operatorname{Wh}(G,w)\mid
y=-(-1)^{n}\overline{y}\\}\leq\operatorname{Wh}(G,w)$
$\displaystyle\mathcal{I}_{n}(G,w)$
$\displaystyle\coloneqq\\{x-(-1)^{n}\overline{x}\mid
x\in\operatorname{Wh}(G,w)\\}\leq\operatorname{Wh}(G,w)$

We have $\mathcal{I}_{n}(G,w)\leq\mathcal{J}_{n}(G,w)$, and it follows from
the definition of the Tate cohomology groups (see Proposition 14.5) that there
is a canonical isomorphism

$\mathcal{J}_{n}(G,w)/\mathcal{I}_{n}(G,w)\cong\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w)).$

We will denote the quotient map by

$\pi\colon\mathcal{J}_{n}(G,w)\rightarrow\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w)).$

####  2.3. The Whitehead torsion of a chain homotopy equivalence

Now we define the Whitehead torsion in the algebraic setting. Let $G$ be a
group and let $f\colon C_{*}\to D_{*}$ be a chain homotopy equivalence between
finitely generated, free, based (left) $\mathbb{Z}G$-module chain complexes.
Consider the algebraic mapping cone $(\mathscr{C}(f),\partial_{*})$, which is
also a finitely generated, free, based $\mathbb{Z}G$-module chain complex.
Since $f$ is a chain homotopy equivalence, $\mathscr{C}(f)$ is chain
contractible; see e.g. [Ranicki-AGS, Proposition 3.14]. Choose a chain
contraction $s\colon\mathscr{C}(f)_{*}\to\mathscr{C}(f)_{*+1}$. Now consider
the modules

$\mathscr{C}(f)_{\operatorname{odd}}\coloneqq\bigoplus_{i=0}^{\infty}\mathscr{C}(f)_{2i+1}\quad\text{
and
}\quad\mathscr{C}(f)_{\operatorname{even}}\coloneqq\bigoplus_{i=0}^{\infty}\mathscr{C}(f)_{2i}.$

These are finitely generated, free, based $\mathbb{Z}G$-modules. Since
$\mathscr{C}(f)$ is contractible, its Euler characteristic vanishes, and so
these modules are of equal, finite rank. The collection of boundary maps
$\oplus_{i=0}^{\infty}\partial_{2i+1}$ with odd degree domain, and the
collection of the maps in the chain contraction
$\oplus_{i=0}^{\infty}s_{2i+1}$ with odd degree domain, both give rise to
homomorphisms from $\mathscr{C}(f)_{\operatorname{odd}}$ to
$\mathscr{C}(f)_{\operatorname{even}}$. Using the given bases, their sum is an
element of $\operatorname{GL}(\mathbb{Z}G)$ [Co73, (15.1)], and hence
represents an element of the Whitehead group.

######  Definition 2.12.

The _Whitehead torsion_ of $f$ is the equivalence class

$\tau(f)\coloneqq\Bigl{[}\bigoplus_{i=0}^{\infty}\bigl{(}\partial_{2i+1}+s_{2i+1}\bigr{)}\colon\mathscr{C}(f)_{\operatorname{odd}}\to\mathscr{C}(f)_{\operatorname{even}}\Bigr{]}\in\operatorname{Wh}(G).$

This equivalence class is independent of the choice of the chain contraction
$s$ [Co73, (15.3)].

######  Remark 2.13.

Since $\mathscr{C}(f)_{\operatorname{odd}}$ and
$\mathscr{C}(f)_{\operatorname{even}}$ are finitely generated, the sum in the
definition of $\tau(f)$ is a finite sum.

The equivalence class $\tau(f)$ remains invariant under permutations of the
bases of $C_{*}$ and $D_{*}$, or if a basis element is multiplied by $(-1)$ or
an element of $G$ [Co73, (15.2) and (10.3)].

Now we list some useful facts about $\tau$. We fix the group $G$, and all
chain complexes will be assumed to be finitely generated, free, based, left
$\mathbb{Z}G$-module chain complexes.

######  Proposition 2.14 ([DK01, Theorem 11.27]).

Let $f,g\colon C_{*}\to D_{*}$ be homotopic chain homotopy equivalences. Then
$\tau(f)=\tau(g)$.

######  Lemma 2.15 ([DK01, Theorem 11.28]).

Let $f\colon C_{*}\rightarrow D_{*}$ and $g\colon D_{*}\rightarrow E_{*}$ be
chain homotopy equivalences. Then $\tau(g\circ f)=\tau(f)+\tau(g)$. In
particular $\tau(\operatorname{Id})=0$.

We say that a short exact sequence $0\rightarrow C^{\prime}_{*}\rightarrow
C_{*}\rightarrow C^{\prime\prime}_{*}\rightarrow 0$ of chain complexes is
based if the basis of $C_{*}$ consists of the image of the basis of
$C^{\prime}_{*}$ and an element from the preimage of each basis element of
$C^{\prime\prime}_{*}$.

######  Lemma 2.16.

Let $0\rightarrow C^{\prime}_{*}\rightarrow C_{*}\rightarrow
C^{\prime\prime}_{*}\rightarrow 0$ and $0\rightarrow D^{\prime}_{*}\rightarrow
D_{*}\rightarrow D^{\prime\prime}_{*}\rightarrow 0$ be based short exact
sequences of chain complexes, and let $(f^{\prime},f,f^{\prime\prime})$ be a
morphism between them, where $f^{\prime}\colon C^{\prime}_{*}\rightarrow
D^{\prime}_{*}$, $f\colon C_{*}\rightarrow D_{*}$, and $f^{\prime\prime}\colon
C^{\prime\prime}_{*}\rightarrow D^{\prime\prime}_{*}$ are chain homotopy
equivalences. Then $\tau(f)=\tau(f^{\prime})+\tau(f^{\prime\prime})$.

###### Proof.

The given data determines a based short exact sequence
$0\rightarrow\mathscr{C}(f^{\prime})\rightarrow\mathscr{C}(f)\rightarrow\mathscr{C^{\prime\prime}}(f)\rightarrow
0$ of mapping cones, so the statement follows from [DK01, Theorem 11.23]. ∎

The following lemma gives a useful way to compute Whitehead torsion in
favourable special cases. We will apply it in Section 6.

######  Lemma 2.17.

Let $f\colon C_{*}\to D_{*}$ be a chain map such that $f_{i}\colon C_{i}\to
D_{i}$ is an isomorphism for each $i$. Then $[f_{i}]\in\operatorname{Wh}(G)$
for each $i$ and $\tau(f)=\sum_{i=0}^{\infty}(-1)^{i}[f_{i}]$.

###### Proof.

Let $n=\max\left\\{i\mid C_{i}\neq 0\right\\}$, we will prove the statement by
induction on $n$. If $n=0$, then
$\mathscr{C}(f)_{\operatorname{odd}}=\mathscr{C}(f)_{1}\cong C_{0}$,
$\mathscr{C}(f)_{\operatorname{even}}=\mathscr{C}(f)_{0}\cong D_{0}$, and
$\partial_{1}=f_{0}$, so $\tau(f)=[f_{0}]$ by Definition 2.12.

For the induction step define $C^{\prime}_{*}$ by $C^{\prime}_{i}=C_{i}$ if
$i<n$ and $C_{i}=0$ otherwise, and define $C^{\prime\prime}_{*}$ by
$C^{\prime\prime}_{n}=C_{n}$ and $C_{i}=0$ if $i\neq n$. Define
$D^{\prime}_{*}$, $D^{\prime\prime}_{*}$, $f^{\prime}\colon
C^{\prime}_{*}\rightarrow D^{\prime}_{*}$, and $f^{\prime\prime}\colon
C^{\prime\prime}_{*}\rightarrow D^{\prime\prime}_{*}$ analogously, and apply
Lemma 2.16. ∎

For a chain complex $C_{*}$ and an integer $k$ we will denote by $C_{k+*}$ the
same chain complex with shifted grading $i\mapsto C_{k+i}$. Similarly, the
cochain complex $C_{k-*}$ is defined by changing the grading to $i\mapsto
C_{k-i}$. The notation $C^{k+i}$ and $C^{k-*}$ is defined analogously for a
cochain complex $C^{*}$.

######  Lemma 2.18.

Let $f\colon C_{*}\rightarrow D_{*}$ be a chain homotopy equivalence. For
every $k\in\mathbb{Z}$, it can also be regarded as a chain homotopy
equivalence $f\colon C_{k+*}\rightarrow D_{k+*}$, and we have $\tau(f\colon
C_{k+*}\rightarrow D_{k+*})=(-1)^{k}\tau(f\colon C_{*}\rightarrow D_{*})$.

###### Proof.

If we shift the grading by an even number, then
$\mathscr{C}(f)_{\operatorname{odd}}$ and
$\mathscr{C}(f)_{\operatorname{even}}$ remain unchanged. If we shift the
grading by an odd number, then $\mathscr{C}(f)_{\operatorname{odd}}$ and
$\mathscr{C}(f)_{\operatorname{even}}$ are swapped, and by [Co73, (15.1)], we
see that $\tau(f)$ changes its sign. ∎

######  Definition 2.19.

Let $f\colon C^{*}\rightarrow D^{*}$ be a homotopy equivalence of cochain
complexes of finitely generated, free, based, left $\mathbb{Z}G$-modules. It
can be regarded as a homotopy equivalence of chain complexes $f\colon
C^{-*}\rightarrow D^{-*}$, and we define $\tau(f\colon C^{*}\rightarrow
D^{*})\coloneqq\tau(f\colon C^{-*}\rightarrow D^{-*})$.

Next we describe how we can define the dual of a left $R$-module as another
left $R$-module, if $R$ is a ring with involution.

######  Definition 2.20.

Suppose that $R$ is a ring with involution. Let $X$ and $Y$ be a left and a
right $R$-module respectively. Then we define the abelian groups

$\displaystyle Y\otimes_{R}X$
$\displaystyle={Y\otimes_{\mathbb{Z}}X}/{(yr\otimes x=y\otimes
rx)}\quad\forall r\in R,x\in X,y\in Y;$
$\displaystyle\operatorname{Hom}_{R}^{lr}(X,Y)$
$\displaystyle=\left\\{\varphi\in\operatorname{Hom}_{\mathbb{Z}}(X,Y)\mid\forall
r\in R,x\in X\mid\varphi(rx)=\varphi(x)\overline{r}\right\\}.$

If $Y$ also has a left $S$-module structure for some ring $S$ (which is
compatible with its right $R$-module structure), then $Y\otimes_{R}X$ is a
left $S$-module with $s(y\otimes x)=(sy)\otimes x$, and
$\operatorname{Hom}_{R}^{lr}(X,Y)$ is a left $S$-module with
$(s\varphi)(x)=s\varphi(x)$.

######  Remark 2.21.

$\operatorname{Hom}_{R}^{lr}(X,Y)$ is equal to
$\operatorname{Hom}_{R}^{r}(\overline{X},Y)$, the group of right $R$-module
homomorphisms $\overline{X}\rightarrow Y$, as a subgroup of
$\operatorname{Hom}_{\mathbb{Z}}(X,Y)$ (or as a left $S$-module). Here
$\overline{X}$ denotes the right $R$-module that is equal to $X$ as an abelian
group, with its multiplication given by $xr=\overline{r}\cdot x$, where
$\cdot$ denotes multiplication in $X$.

######  Definition 2.22.

Suppose that $R$ is a ring with involution. Let $X$ be a left $R$-module. Its
dual is the left $R$-module $X^{*}=\operatorname{Hom}_{R}^{lr}(X,R)$.

Now suppose that $G$ is equipped with a group homomorphism $w\colon G\to\\{\pm
1\\}$, i.e. an orientation character. This determines an involution on the
group ring $\mathbb{Z}G$, and also on the Whitehead group
$\operatorname{Wh}(G,w)$ (see Section 2.2). So if $C_{*}$ is a finitely
generated, free, based, left $\mathbb{Z}G$-module chain complex then, using
Definition 2.22, we can define the dual cochain complex $C^{*}$, which also
consists of finitely generated, free, based, left $\mathbb{Z}G$-modules.

######  Lemma 2.23.

Let $f\colon C_{*}\rightarrow D_{*}$ be a chain homotopy equivalence and let
$f^{*}\colon D^{*}\rightarrow C^{*}$ be its dual. Then
$\tau(f^{*})=\overline{\tau(f)}\in\operatorname{Wh}(G,w)$.

###### Proof.

First note that if $g\colon X\rightarrow Y$ is an isomorphism between finitely
generated, free, based, left $\mathbb{Z}G$-modules, and $g^{*}\colon
Y^{*}\rightarrow X^{*}$ is its dual, then the matrix of $g^{*}$ is the
conjugate transpose of the matrix of $g$, hence
$\tau(g^{*})=\overline{\tau(g)}\in\operatorname{Wh}(G,w)$.

Now let $(\mathscr{C}(f),\partial^{f}_{*})$ denote the mapping cone of $f$, so
that $\mathscr{C}(f)_{i}=C_{i-1}\oplus D_{i}$. We regard $f^{*}\colon
D^{*}\rightarrow C^{*}$ as a homotopy equivalence $f^{*}\colon
D^{-*}\rightarrow C^{-*}$ of chain complexes, and define its mapping cone
$(\mathscr{C}(f^{*}),\partial^{f^{*}}_{*})$. Then
$\mathscr{C}(f^{*})_{i}=(D_{-i+1})^{*}\oplus(C_{-i})^{*}\cong(\mathscr{C}(f)_{-i+1})^{*}$,
and this implies that
$\mathscr{C}(f^{*})_{\operatorname{odd}}\cong(\mathscr{C}(f)_{\operatorname{even}})^{*}$
and
$\mathscr{C}(f^{*})_{\operatorname{even}}\cong(\mathscr{C}(f)_{\operatorname{odd}})^{*}$.
Moreover,
$\partial^{f^{*}}_{i}\colon\mathscr{C}(f^{*})_{i}\rightarrow\mathscr{C}(f^{*})_{i-1}$
is the dual of
$\partial^{f}_{-i+2}\colon\mathscr{C}(f)_{-i+2}\rightarrow\mathscr{C}(f)_{-i+1}$.

Let $s_{*}\colon\mathscr{C}(f)_{*}\rightarrow\mathscr{C}(f)_{*+1}$ be a chain
contraction. We define
$s^{\prime}_{*}\colon\mathscr{C}(f^{*})_{*}\rightarrow\mathscr{C}(f^{*})_{*+1}$
by
$s^{\prime}_{i}=(s_{-i})^{*}\colon\mathscr{C}(f^{*})_{i}\cong(\mathscr{C}(f)_{-i+1})^{*}\rightarrow\mathscr{C}(f^{*})_{i+1}\cong(\mathscr{C}(f)_{-i})^{*}$.
Then it follows by dualising the formula
$s_{*}\partial^{f}+\partial^{f}s_{*}=\operatorname{Id}$ that $s^{\prime}_{*}$
is a chain contraction of $\mathscr{C}(f^{*})$.

Thus the map
$\bigoplus_{i}\bigl{(}\partial^{f^{*}}_{2i+1}+s^{\prime}_{2i+1}\bigr{)}\colon\mathscr{C}(f^{*})_{\operatorname{odd}}\to\mathscr{C}(f^{*})_{\operatorname{even}}$
is the dual of the map
$\bigoplus_{i}\bigl{(}\partial^{f}_{2i+1}+s_{2i+1}\bigr{)}\colon\mathscr{C}(f)_{\operatorname{odd}}\to\mathscr{C}(f)_{\operatorname{even}}$.
By our earlier observations in the first paragraph of the proof, this implies
that $\tau(f^{*})=\overline{\tau(f)}$. ∎

Finally we consider the effect of changing the underlying (group) ring.

######  Definition 2.24.

Let $A$ and $B$ be groups, $X$ a left $\mathbb{Z}B$-module and
$\theta\in\operatorname{Hom}(A,B)$. The left $\mathbb{Z}A$-module $X_{\theta}$
is defined as follows. The underlying abelian group of $X_{\theta}$ is the
same as that of $X$. For every $a\in A$ and $x\in X_{\theta}$ let
$ax=\theta(a)\cdot x$, where $\cdot$ denotes multiplication in $X$.

Similarly, if $Y$ is a right $\mathbb{Z}B$-module, then the right
$\mathbb{Z}A$-module $Y^{\theta}$ is equal to $Y$ as an abelian group, and
$ya=y\cdot\theta(a)$ for every $a\in A$ and $y\in Y^{\theta}$, where $\cdot$
denotes multiplication in $Y$.

######  Lemma 2.25.

Let $A$ and $B$ be groups equipped with orientation characters $w_{A}\colon
A\rightarrow\left\\{\pm 1\right\\}$ and $w_{B}\colon B\rightarrow\left\\{\pm
1\right\\}$ respectively. Let $X$ be a left $\mathbb{Z}A$-module and let
$\theta\colon A\rightarrow B$ be an isomorphism such that
$w_{B}\circ\theta=w_{A}$. Then we have the following isomorphisms of left
$\mathbb{Z}B$-modules.

  1. (a)

$\mathbb{Z}B^{\theta}\otimes_{\mathbb{Z}A}X\cong X_{\theta^{-1}}$.

  2. (b)

$\operatorname{Hom}_{\mathbb{Z}B}^{lr}(X_{\theta^{-1}},\mathbb{Z}B)=\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}B^{\theta})\cong\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}A)_{\theta^{-1}}$,
in particular $(X_{\theta^{-1}})^{*}\cong(X^{*})_{\theta^{-1}}$.

###### Proof.

(a) If $Y$ is a right $\mathbb{Z}A$-module and a left $R$-module for some ring
$R$, then it follows from the definition of the tensor product (see Definition
2.20) that
$Y\otimes_{\mathbb{Z}A}X=Y^{\theta^{-1}}\otimes_{\mathbb{Z}B}X_{\theta^{-1}}$
(as left $R$-modules). By applying this for $Y=\mathbb{Z}B^{\theta}$ and
$R=\mathbb{Z}B$, we get that
$\mathbb{Z}B^{\theta}\otimes_{\mathbb{Z}A}X=\mathbb{Z}B\otimes_{\mathbb{Z}B}X_{\theta^{-1}}$.
Of course
$X_{\theta^{-1}}\cong\mathbb{Z}B\otimes_{\mathbb{Z}B}X_{\theta^{-1}}$ via the
map $x\mapsto 1\otimes x$.

(b) (cf. [Ni23, Proposition 3.9]) For $Y$ as above, it also follows from the
definitions that
$\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,Y)=\operatorname{Hom}_{\mathbb{Z}B}^{lr}(X_{\theta^{-1}},Y^{\theta^{-1}})$,
so
$\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}B^{\theta})=\operatorname{Hom}_{\mathbb{Z}B}^{lr}(X_{\theta^{-1}},\mathbb{Z}B)=(X_{\theta^{-1}})^{*}$.

The isomorphism $\theta$ induces an isomorphism
$\mathbb{Z}\theta\colon\mathbb{Z}A\rightarrow\mathbb{Z}B$ of abelian groups
(or rings). This map is also an isomorphism
$\mathbb{Z}\theta\colon\mathbb{Z}A\rightarrow\mathbb{Z}B^{\theta}$ of right
$\mathbb{Z}A$-modules. Hence it induces an isomorphism
$(\mathbb{Z}\theta)_{*}\colon\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}A)\rightarrow\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}B^{\theta})$
of abelian groups. We can check that $(\mathbb{Z}\theta)_{*}$ is also an
isomorphism
$\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}A)_{\theta^{-1}}\rightarrow\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}B^{\theta})$
of left $\mathbb{Z}B$-modules. So we get that
$(X^{*})_{\theta^{-1}}=\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}A)_{\theta^{-1}}\cong\operatorname{Hom}_{\mathbb{Z}A}^{lr}(X,\mathbb{Z}B^{\theta})$.
∎

######  Lemma 2.26.

Let $A$ and $B$ be groups and let $\theta\colon A\rightarrow B$ be an
isomorphism. Let $f\colon C_{*}\rightarrow D_{*}$ be a chain homotopy
equivalence of finitely generated, free, based, left $\mathbb{Z}B$-module
chain complexes, which can also be regarded as a chain homotopy equivalence
$f\colon(C_{*})_{\theta}\rightarrow(D_{*})_{\theta}$ of $\mathbb{Z}A$-module
chain complexes. We have

$\tau(f\colon(C_{*})_{\theta}\rightarrow(D_{*})_{\theta})=\theta^{-1}_{*}(\tau(f\colon
C_{*}\rightarrow D_{*}))\in\operatorname{Wh}(A).$

###### Proof.

First note that if $g\colon X\rightarrow Y$ is an isomorphism between finitely
generated, free, based, left $\mathbb{Z}B$-modules, and we regard it as an
isomorphism $g\colon X_{\theta}\rightarrow Y_{\theta}$ of
$\mathbb{Z}A$-modules, then its matrix changes by applying $\theta^{-1}$ to
each entry, hence $\tau(g\colon X_{\theta}\rightarrow
Y_{\theta})=\theta^{-1}_{*}(\tau(g\colon X\rightarrow
Y))\in\operatorname{Wh}(A)$.

Now let $(\mathscr{C}(f),\partial_{*})$ denote the mapping cone of $f$ and let
$s_{*}\colon\mathscr{C}(f)_{*}\rightarrow\mathscr{C}(f)_{*+1}$ be the chain
contraction used to define $\tau(f)$ over $\mathbb{Z}B$. Then $s_{*}$ is also
a chain contraction for $(\mathscr{C}(f)_{\theta},\partial_{*})$, the mapping
cone over $\mathbb{Z}A$.

So we can compute $\tau(f\colon(C_{*})_{\theta}\rightarrow(D_{*})_{\theta})$
from
$\bigoplus_{i}\bigl{(}\partial_{2i+1}+s_{2i+1}\bigr{)}\colon(\mathscr{C}(f)_{\theta})_{\operatorname{odd}}=(\mathscr{C}(f)_{\operatorname{odd}})_{\theta}\to(\mathscr{C}(f)_{\theta})_{\operatorname{even}}=(\mathscr{C}(f)_{\operatorname{even}})_{\theta}$,
which is
$\bigoplus_{i}\bigl{(}\partial_{2i+1}+s_{2i+1}\bigr{)}\colon\mathscr{C}(f)_{\operatorname{odd}}\to\mathscr{C}(f)_{\operatorname{even}}$
regarded as an isomorphism of $\mathbb{Z}A$-modules, hence
$\tau(f\colon(C_{*})_{\theta}\rightarrow(D_{*})_{\theta})=\theta^{-1}_{*}(\tau(f\colon
C_{*}\rightarrow D_{*}))$. ∎

####  2.4. The Whitehead torsion of a homotopy equivalence

Now let $X$ and $Y$ be finite CW complexes with universal covers
$\widetilde{X}$ and $\widetilde{Y}$, and let $F:=\pi_{1}(X)$ and
$G:=\pi_{1}(Y)$. The cellular chain complex of $Y$ with $\mathbb{Z}G$
coefficients is
$C_{*}(Y;\mathbb{Z}G)=\mathbb{Z}G\otimes_{\mathbb{Z}G}C_{*}(\widetilde{Y})\cong
C_{*}(\widetilde{Y})$, which is a finitely generated, free, left
$\mathbb{Z}G$-module chain complex. Choose a lift of each cell of $Y$ in
$\widetilde{Y}$ to obtain a basis of $C_{*}(\widetilde{Y})$, which is well-
defined up to ordering, signs and multiplication by elements of $G$.
Similarly, the cellular chain complex of $X$ with $\mathbb{Z}F$ coefficients,
$C_{*}(X;\mathbb{Z}F)\cong C_{*}(\widetilde{X})$, is a finitely generated,
free, left $\mathbb{Z}F$-module chain complex with a basis well-defined up to
ordering and multiplication by elements of $\pm F$.

Let $f\colon X\to Y$ be a cellular homotopy equivalence, and let
$\theta=\pi_{1}(f)\colon F\rightarrow G$. The right $\mathbb{Z}G$-module
$\mathbb{Z}G$ corresponds to a local coefficient system on $Y$, which is
pulled back to the local coefficient system on $X$ corresponding to the right
$\mathbb{Z}F$-module $\mathbb{Z}G^{\theta}$. Therefore $f$ induces a chain
homotopy equivalence $f_{*}\colon C_{*}(X;\mathbb{Z}G^{\theta})\rightarrow
C_{*}(Y;\mathbb{Z}G)$ of left $\mathbb{Z}G$-module chain complexes. Note that
by Lemma 2.25 (a) we have

$C_{*}(X;\mathbb{Z}G^{\theta})=\mathbb{Z}G^{\theta}\otimes_{\mathbb{Z}F}C_{*}(\widetilde{X})\cong
C_{*}(\widetilde{X})_{\theta^{-1}}\cong C_{*}(X;\mathbb{Z}F)_{\theta^{-1}},$

so this is also a finitely generated, free chain complex with a basis that is
well-defined up to ordering and multiplication by elements of $\pm G$.

######  Definition 2.27.

The _Whitehead torsion_ of the cellular homotopy equivalence $f\colon X\to Y$
is $\tau(f):=\tau(f_{*})$, where $f_{*}\colon
C_{*}(X;\mathbb{Z}G^{\theta})\rightarrow C_{*}(Y;\mathbb{Z}G)$ is the induced
chain homotopy equivalence.

By Remark 2.13, it follows that $\tau(f_{*})$ is well-defined, even though the
bases of $C_{*}(X;\mathbb{Z}G^{\theta})$ and $C_{*}(Y;\mathbb{Z}G)$ are well-
defined only up to ordering and multiplication by elements of $\pm G$.

######  Proposition 2.28 ([Co73, Statement 22.1]).

Let $f,g\colon X\to Y$ be homotopic cellular homotopy equivalences between
finite CW complexes. Then $\tau(f)=\tau(g)\in\operatorname{Wh}(G)$.

###### Proof.

This follows immediately from Proposition 2.14, because homotopic homotopy
equivalences induce homotopic chain homotopy equivalences. ∎

Now we can extend the definition of Whitehead torsion to arbitrary homotopy
equivalences. If $f\colon X\to Y$ is a homotopy equivalence between finite CW
complexes, then it is homotopic to a cellular homotopy equivalence
$f^{\prime}\colon X\to Y$, and we define $\tau(f)\coloneqq\tau(f^{\prime})$.
By Proposition 2.28 this is independent of the choice of $f^{\prime}$.
Moreover, it follows that if $f,g\colon X\to Y$ are arbitrary homotopy
equivalences and $f\simeq g$, then $\tau(f)=\tau(g)$.

Now that we have defined Whitehead torsion, it is worth recalling its key
role: Theorem 2.2 states that a homotopy equivalence $f\colon X\to Y$ between
CW complexes $X$ and $Y$ is simple if and only if its Whitehead torsion
$\tau(f)=0\in\operatorname{Wh}(\pi_{1}(Y))$.

We collect a few key properties of Whitehead torsion.

######  Proposition 2.29 ([Co73, Corollary 5.1A]).

Let $X$ and $Y$ be finite CW complexes and let $f\colon X\to Y$ be a cellular
map. Let $\operatorname{Cyl}_{f}$ denote the mapping cylinder of $f$. Then the
inclusion $Y\rightarrow\operatorname{Cyl}_{f}$ is a simple homotopy
equivalence.

######  Proposition 2.30 ([Co73, Statement 22.4]).

Let $X$, $Y$, and $Z$ be finite CW complexes and let $f\colon X\to Y$ and
$g\colon Y\to Z$ be homotopy equivalences. Then

$\tau(g\circ f)=\tau(g)+g_{*}(\tau(f)).$

######  Proposition 2.31 ([Co73, Statement 23.2]).

For $i=1,2$, let $X_{i}$, $Y_{i}$ be finite CW complexes and let $f_{i}\colon
X_{i}\to Y_{i}$ be a homotopy equivalence. Let $i_{1}\colon
Y_{1}\hookrightarrow Y_{1}\times Y_{2}$ and $i_{2}\colon Y_{2}\hookrightarrow
Y_{1}\times Y_{2}$ be natural inclusion maps defined by fixing a point in
$Y_{2}$ and $Y_{1}$ respectively. For the product map $f_{1}\times f_{2}\colon
X_{1}\times X_{2}\to Y_{1}\times Y_{2}$, we have

$\tau(f_{1}\times
f_{2})=\chi(Y_{2})\cdot{i_{1}}_{*}(\tau(f_{1}))+\chi(Y_{1})\cdot{i_{2}}_{*}(\tau(f_{2}))$

######  Corollary 2.32.

Let $f_{1}\colon M_{1}\rightarrow N_{1}$ and $f_{2}\colon M_{2}\rightarrow
N_{2}$ be homotopy equivalences between odd dimensional manifolds. Then
$f_{1}\times f_{2}\colon M_{1}\times M_{2}\rightarrow N_{1}\times N_{2}$ is a
simple homotopy equivalence.

###### Proof.

We have $\tau(f_{1}\times f_{2})=0$, by Proposition 2.31 and the fact that
$\chi(N_{1})=\chi(N_{2})=0$. ∎

######  Proposition 2.33.

Let $f\colon X\to Y$ be a homeomorphism between finite CW complexes. Then
$\tau(f)=0$.

###### Proof.

By Theorem 2.5, $f$ is a simple homotopy equivalence. Therefore $\tau(f)=0$ by
Theorem 2.2. ∎

The following generalises an observation of Wall. He pointed this out in the
case where $M$, $N$ are simple Poincaré complexes [Wa74, p. 612], i.e. finite
Poincaré complexes $X$ for which the chain duality isomorphism
$C^{*}(\widetilde{X})\to C_{*}(\widetilde{X})$ is a simple chain homotopy
equivalence. At the time, it was known that smooth manifolds are simple
Poincaré complexes [Wall-SCM, Theorem 2.1] but the case of topological
manifolds was not established until the work of Kirby-Siebenmann [Kirby-
Siebenmann:1977-1, III.5.13].

######  Proposition 2.34.

Let $M$ and $N$ be closed $n$-dimensional topological manifolds. Let
$G=\pi_{1}(N)$ with orientation character $w\colon G\rightarrow\left\\{\pm
1\right\\}$, and let $f\colon M\to N$ be a homotopy equivalence. Then
$\tau(f)\in\mathcal{J}_{n}(G,w)$.

###### Proof.

Let $F=\pi_{1}(M)$ and $\theta=\pi_{1}(f)\colon F\to G$. Since $f$ is a
homotopy equivalence, $f_{*}([M])=\pm[N]$, where $[M]$ and $[N]$ denote the
(twisted) fundamental classes of $M$ and $N$ respectively. Since Poincaré
duality is given by taking cap product with the fundamental class, and the cap
product is natural, we get a diagram of left $\mathbb{Z}G$-module chain
complexes that commutes up to chain homotopy and sign:

$\textstyle{C^{n-*}(M;\mathbb{Z}G^{\theta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{PD}}$$\textstyle{C^{n-*}(N;\mathbb{Z}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C^{n-*}(f)}$$\scriptstyle{\operatorname{PD}}$$\textstyle{C_{*}(M;\mathbb{Z}G^{\theta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{*}(f)}$$\textstyle{C_{*}(N;\mathbb{Z}G)}$

Poincaré duality is a simple chain homotopy equivalence by [Kirby-
Siebenmann:1977-1, III.5.13], so we have that $\tau(\operatorname{PD}\colon
C^{n-*}(N;\mathbb{Z}G)\to C_{*}(N;\mathbb{Z}G))=0$. By Lemmas 2.25 and 2.26,
the Poincaré duality map

$\operatorname{PD}\colon C^{n-*}(M;\mathbb{Z}G^{\theta})\cong
C^{n-*}(M;\mathbb{Z}F)_{\theta^{-1}}\to C_{*}(M;\mathbb{Z}G^{\theta})\cong
C_{*}(M;\mathbb{Z}F)_{\theta^{-1}}$

is simple too.

It follows from Proposition 2.14 and Lemma 2.15 that
$\tau(C^{n-*}(f))+\tau(C_{*}(f))=0$. Therefore it is enough to prove that
$\tau(C^{n-*}(f))=(-1)^{n}\overline{\tau(C_{*}(f))}$.

By Lemma 2.18, we know that $\tau(C^{n-*}(f))=(-1)^{n}\tau(C^{-*}(f))$, and it
follows from Definition 2.19 that $\tau(C^{-*}(f))=\tau(C^{*}(f))$. Finally
Lemma 2.23 shows that $\tau(C^{*}(f))=\overline{\tau(C_{*}(f))}$. We deduce
that $\tau(C^{n-*}(f))=(-1)^{n}\overline{\tau(C_{*}(f))}$, completing the
proof. ∎

We recall the definitions of $h$\- and $s$-cobordisms.

######  Definition 2.35.

A cobordism $(W;M,N)$ of closed manifolds is an _$h$ -cobordism_ if the
inclusion maps $i_{M}\colon M\to W$ and $i_{N}\colon N\to W$ are homotopy
equivalences. If in addition $i_{M}$ and $i_{N}$ are simple homotopy
equivalences then $W$ is an _$s$ -cobordism_.

######  Definition 2.36.

Suppose that $W$ is an $h$-cobordism between closed $n$-dimensional manifolds
$M$ and $N$. Let $G=\pi_{1}(W)$ and let $w\colon G\rightarrow\left\\{\pm
1\right\\}$ be the orientation character of $W$. We will write $\tau(W,M)$ and
$\tau(W,N)$ to denote the Whitehead torsion of the inclusions $M\rightarrow W$
and $N\rightarrow W$ respectively. The composition $N\to M$ of the inclusion
$N\rightarrow W$ and the homotopy inverse of the inclusion $M\rightarrow W$
(well-defined up to homotopy) will be called _the homotopy equivalence induced
by $W$_.

######  Proposition 2.37.

Let $W$ be an $h$-cobordism between closed, $n$-dimensional manifolds $M$ and
$N$. Let $G=\pi_{1}(W)$ and let $w\colon G\rightarrow\left\\{\pm 1\right\\}$
be the orientation character of $W$.

  1. (a)

We have that
$\tau(W,N)=(-1)^{n}\overline{\tau(W,M)}\in\operatorname{Wh}(G,w)$.

  2. (b)

If $f\colon N\rightarrow M$ denotes the homotopy equivalence induced by $W$,
then we have

$\tau(f)=-\tau(W,M)+(-1)^{n}\overline{\tau(W,M)}\in\operatorname{Wh}(G,w)$

where $\pi_{1}(M)$ is identified with $G$ via the inclusion. In particular
$\tau(f)\in\mathcal{I}_{n}(G,w)$.

###### Proof.

For (a) this is the “duality theorem” [Mi66, p. 394], translated into our
conventions; see also the Remark on [Mi66, p. 398]. Then (b) follows from part
(a) and Proposition 2.30. ∎

###  3\. Realising elements of the Whitehead group by maps between manifolds

Throughout this section, fix $n\geq 4$, a finitely presented group $G$, and
$\operatorname{CAT}\in\\{\operatorname{Diff},\operatorname{PL},\operatorname{TOP}\\}$,
satisfying 1.1.

####  3.1. Realising via $h$-cobordisms

If $(W;M,N)$ is an $h$-cobordism, then we will use the isomorphism
$\pi_{1}(i_{M})$ to identify $\pi_{1}(M)$ with $\pi_{1}(W)$, and regard
$\tau(W,M)\in\operatorname{Wh}(\pi_{1}(W))$ as an element of
$\operatorname{Wh}(\pi_{1}(M))$.

Here is the complete statement of the $s$-cobordism theorem, for closed
manifolds. It is due to Smale, Barden, Mazur, Stallings, Kirby-Siebenmann, and
Freedman-Quinn [Smale-h-cob, Barden, Mazur:1963-1, Stallings-Tata, Kirby-
Siebenmann:1977-1, FQ].

######  Theorem 3.1 ($s$-cobordism theorem).

Let $M$ be a closed, $\operatorname{CAT}$ $n$-manifold with $\pi_{1}(M)\cong
G$, satisfying 1.1.

  1. (a)

Let $(W;M,M^{\prime})$ be an $h$-cobordism over $M$. Then $W$ is trivial over
$M$, i.e. $W\cong M\times[0,1]$, via a $\operatorname{CAT}$-isomorphism
restricting to the identity on $M$, if and only if its Whitehead torsion
$\tau(W,M)\in\operatorname{Wh}(G)$ vanishes.

  2. (b)

For every $x\in\operatorname{Wh}(G)$ there exists an $h$-cobordism
$(W;M,M^{\prime})$ with $\tau(W,M)=x$.

  3. (c)

The function assigning to an $h$-cobordism $(W;M,M^{\prime})$ its Whitehead
torsion $\tau(W,M)$ yields a bijection from the
$\operatorname{CAT}$-isomorphism classes relative to $M$ of $h$-cobordisms
over $M$ to the Whitehead group $\operatorname{Wh}(G)$.

Part (b) of the theorem was stated in [RS82, p. 90] in the PL category. As
observed in [KPR-survey, Remark 3.6], the proof given there has a gap in the
case $n=4$, since it is not clear that the manifold $M^{\prime}$ has the same
fundamental group as $M$: the attaching spheres for the 3-handles can be
smoothly embedded but it is not clear that these embeddings admit geometric
duals. If one restricts to the topological category and good fundamental group
(i.e. 1.1), then as explained in [KPR-survey, Theorem 3.5], the proof can be
fixed using the sphere embedding theorem [FQ, PRT-duals].

Recall that we write
$\mathcal{I}_{n}(G,w)\coloneqq\\{y+(-1)^{n+1}\overline{y}\mid
y\in\operatorname{Wh}(G,w)\\}\leq\operatorname{Wh}(G,w)$.

######  Corollary 3.2.

Let $M$ be a closed, $\operatorname{CAT}$ $n$-manifold with $\pi_{1}(M)\cong
G$ and orientation character $w\colon G\rightarrow\left\\{\pm 1\right\\}$,
satisfying 1.1. For every $x\in\mathcal{I}_{n}(G,w)$ there exists a closed,
$\operatorname{CAT}$ $n$-manifold $N$ and a homotopy equivalence $f\colon N\to
M$ induced by an $h$-cobordism between $M$ and $N$ such that $\tau(f)=x$.

###### Proof.

Let $x\in\mathcal{I}_{n}(G,w)$ and write $x=-y+(-1)^{n}\overline{y}$ for some
$y\in\operatorname{Wh}(G,w)$. Apply Theorem 3.1 to obtain an $h$-cobordism
$(W;M,N)$ from $M$ to some $n$-manifold $N$ with $\tau(W,M)=y$. If $f$ is the
homotopy equivalence induced by $W$, then $\tau(f)=-y+(-1)^{n}\overline{y}=x$
by Proposition 2.37 (b). ∎

######  Corollary 3.3.

Let $M$ and $N$ be closed, $\operatorname{CAT}$ $n$-manifolds satisfying 1.1.
Suppose $M$ has $\pi_{1}(M)\cong G$ and orientation character $w\colon
G\rightarrow\left\\{\pm 1\right\\}$. If there is a homotopy equivalence
$f\colon N\rightarrow M$ such that $\tau(f)\in\mathcal{I}_{n}(G,w)$, then
there exists a closed, $\operatorname{CAT}$ $n$-manifold $P$ that is simple
homotopy equivalent to $N$ and $h$-cobordant to $M$.

###### Proof.

By Corollary 3.2 there is an $h$-cobordism between $M$ and some $n$-manifold
$P$ such that $\tau(g)=\tau(f)$ for the induced homotopy equivalence $g\colon
P\rightarrow M$. Moreover, if $g^{-1}$ denotes the homotopy inverse of $g$,
then $g^{-1}\circ f\colon N\rightarrow P$ is a simple homotopy equivalence,
because $\tau(g^{-1}\circ
f)=\tau(g^{-1})+g^{-1}_{*}(\tau(f))=\tau(g^{-1})+g^{-1}_{*}(\tau(g))=\tau(g^{-1}\circ
g)=\tau(\operatorname{Id})=0$ by Proposition 2.30. ∎

####  3.2. Realising via the surgery exact sequence

We recall the surgery exact sequence. Let $M$ be a closed $\operatorname{CAT}$
$n$-dimensional manifold with $\pi_{1}(M)=G$ and orientation character
$w\colon G\to\\{\pm 1\\}$, for some $n\geq 4$, satisfying 1.1.

######  Definition 3.4.

The _homotopy structure set_ of $M$, denoted $\mathcal{S}^{h}(M)$, is by
definition the set of pairs $(N,f)$, where $N$ is a closed
$\operatorname{CAT}$ $n$-manifold and $f\colon N\rightarrow M$ is a homotopy
equivalence, considered up to $h$-cobordism over $M$. That is,
$[N,f]=[N^{\prime},f^{\prime}]\in\mathcal{S}^{h}(M)$ if and only if there is
an $h$-cobordism $(W;N,N^{\prime})$, with inclusion maps $i\colon N\to W$ and
$i^{\prime}\colon N^{\prime}\to W$, together with a map $F\colon W\to M$ such
that $F\circ i=f$ and $F\circ i^{\prime}=f^{\prime}$.

We can similarly define the simple homotopy structure set $\mathcal{S}^{s}(M)$
to be the set of pairs $(N,f)$, where $N$ is a closed $\operatorname{CAT}$
$n$-manifold and $f\colon N\rightarrow M$ is a simple homotopy equivalence,
considered up to $s$-cobordism over $M$.

The Browder-Novikov-Sullivan-Wall surgery exact sequence for $x\in\\{h,s\\}$
is as follows [Wall-SCM, FQ]; see also [Luck-basic-intro-surgery, Theorem
5.12], [Freedman-book-surgery-chapter].

${\mathcal{N}(M\times[0,1],M\times\\{0,1\\})}$${L_{n+1}^{x}(\mathbb{Z}G,w)}$${\mathcal{S}^{x}(M)}$${\mathcal{N}(M)}$${L_{n}^{x}(\mathbb{Z}G,w).}$$\scriptstyle{\sigma_{x}}$$\scriptstyle{W_{x}}$$\scriptstyle{\eta_{x}}$$\scriptstyle{\sigma_{x}}$

Here, $\mathcal{N}(M\times[0,1],M\times\\{0,1\\})$ and $\mathcal{N}(M)$ denote
the sets of the normal bordism classes of degree one normal maps over
$M\times[0,1]$ and $M$ respectively. These sets do not depend on the
decoration $x$.

The groups $L_{n}^{x}(\mathbb{Z}G,w)$ are the surgery obstruction groups.
Elements of $L_{n}^{h}(\mathbb{Z}G,w)$ are represented by nonsingular
Hermitian forms over finitely generated free $\mathbb{Z}G$-modules for $n$
even, and by nonsingular formations over finitely generated free
$\mathbb{Z}G$-modules for $n$ odd, with the involution on $\mathbb{Z}G$
determined by $w$. See e.g. [Ranicki-80-I]. In the case of
$L_{n}^{s}(\mathbb{Z}G,w)$ the forms/formations are also required to be based
and simple.

The maps labelled $\sigma_{x}$ are the surgery obstruction maps. For the
definition of $\sigma_{s}$, we take a degree one normal map $(f,b)\colon N\to
M$, perform surgery below the middle dimension to make the map
$[n/2]$-connected, and then produce the based, simple form or formation (for
$n$ even or odd respectively) of the surgery kernel in the middle
dimension(s), to obtain an element of $L_{n}^{s}(\mathbb{Z}G,w)$. To define
the map $\sigma_{h}$ we perform the same procedure, and then forget the data
of the bases, to obtain an element of $L_{n}^{h}(\mathbb{Z}G,w)$. One of the
main theorems of surgery [Wall-SCM],[Luck-basic-intro-surgery] is that the
maps $\sigma_{x}$, for $x\in\\{h,s\\}$, are well-defined.

The map $W_{x}$ is the Wall realisation map. Given a $z\in
L_{n+1}^{x}(\mathbb{Z}G,w)$ and $[M_{0},f_{0}]\in\mathcal{S}^{x}(M)$, Wall
realisation produces a new element $[M_{1},f_{1}]\in\mathcal{S}^{x}(M)$
together with a degree one normal bordism between $(M_{0},f_{0})$ and
$(M_{1},f_{1})$ whose surgery obstruction equals $z$. This determines an
action of $L_{n+1}^{x}(\mathbb{Z}G,w)$ on $\mathcal{S}^{x}(M)$, and the map
$W_{x}$ is defined by acting on the equivalence class of the identity map,
$[M,\operatorname{Id}_{M}]\in\mathcal{S}^{x}(M)$.

The $s$ and $h$ decorated $L$ groups are related by the exact sequence:

$\cdots\to L_{n+1}^{s}(\mathbb{Z}G,w)\to
L_{n+1}^{h}(\mathbb{Z}G,w)\xrightarrow{\psi}\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))\to
L_{n}^{s}(\mathbb{Z}G,w)\to L_{n}^{h}(\mathbb{Z}G,w)\to\cdots$ (3.1)

######  Remark 3.5.

The only proof we could find in the literature for this exact sequence is due
to Shaneson [Shaneson-GxZ, Section 4]. There, Shaneson attributed the
derivation of the sequence to Rothenberg, by a different (unpublished) proof.

Here for the definition of the Tate group, $C_{2}$ acts on
$\operatorname{Wh}(G,w)$ via the involution. Recall that
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))\cong\mathcal{J}_{n}(G,w)/\mathcal{I}_{n}(G,w)$
and
$\pi\colon\mathcal{J}_{n}(G,w)\to\mathcal{J}_{n}(G,w)/\mathcal{I}_{n}(G,w)$ is
the quotient map. Define a map

$\displaystyle\widehat{\tau}\colon\mathcal{S}^{h}(M)$
$\displaystyle\to\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
$\displaystyle[N,f]$ $\displaystyle\mapsto\pi(\tau(f)).$

We check that this map is well-defined. By Proposition 2.34, the Whitehead
torsion of a homotopy equivalence between manifolds lies in
$\mathcal{J}_{n}(G,w)$. If we change the representative of $[N,f]$ we obtain
an $h$-cobordism between $N$ and some $N^{\prime}$ over $M$, and by
Proposition 2.37 this changes the torsion by an element of
$\mathcal{I}_{n}(G,w)$.

The map $\psi$ from (3.1) determines an action of $L^{h}_{n+1}(\mathbb{Z}G,w)$
on $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$, given by
$x^{z}=x+\psi(z)$ for $x\in\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
and $z\in L^{h}_{n+1}(\mathbb{Z}G,w)$. We will now establish the following two
properties of the map $\widehat{\tau}$.

######  Proposition 3.6.

The map $\widehat{\tau}$ is $L^{h}_{n+1}(\mathbb{Z}G,w)$-equivariant. That is,
for all $z\in L^{h}_{n+1}(\mathbb{Z}G,w)$ and
$[M_{0},f_{0}]\in\mathcal{S}^{h}(M)$, we have
$\widehat{\tau}([M_{0},f_{0}]^{z})=\widehat{\tau}([M_{0},f_{0}])^{z}$.

######  Proposition 3.7.

There is a commutative diagram

$\textstyle{\mathcal{S}^{h}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\tau}}$$\scriptstyle{\eta_{h}}$$\textstyle{\mathcal{N}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{s}}$$\scriptstyle{\sigma_{h}}$$\textstyle{L^{h}_{n+1}(\mathbb{Z}G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{h}}$$\scriptstyle{\psi}$$\textstyle{L^{h}_{n}(\mathbb{Z}G,w)}$$\textstyle{\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varrho}$$\textstyle{L^{s}_{n}(\mathbb{Z}G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$

between the surgery exact sequence and the exact sequence (3.1).

######  Remark 3.8.

This commutative diagram in Proposition 3.7 is part of a braid that appeared
in Jahren and Kwasik’s preprint [arxiv-jk-inertial-hcob, Diagram (6)], but was
removed for the published version [JK18]. In the case
$\operatorname{CAT}=\operatorname{TOP}$, see also Ranicki [Ra86, p.359–360].
Neither reference contains a proof.

In the proofs of these propositions we will need to work with Whitehead
torsion in a more general setting than (chain) homotopy equivalences, so we
will use Milnor’s definition [Mi66, Section 3–4]. Given a chain complex
$C_{*}$ of free $\mathbb{Z}G$-modules with free homology $H_{*}\coloneqq
H_{*}(C_{*})$, if bases are chosen for both $C_{*}$ and $H_{*}$, then the
Whitehead torsion $\tau(C_{*})\in\operatorname{Wh}(G)$ is defined. In
particular, if $H_{*}$ is trivial, then it has a unique basis, so it is enough
to choose a basis for $C_{*}$.

Given a chain homotopy equivalence $f\colon C_{*}\rightarrow D_{*}$ between
based chain complexes, Cohen [Co73, §16] showed that the definition of
$\tau(f)$ from Section 2 coincides with Milnor’s definition of the Whitehead
torsion of the mapping cone $\mathscr{C}(f)$ of $f$, with basis determined by
the bases of $C_{*}$ and $D_{*}$ (which is the same as the Whitehead torsion
of the pair $(\operatorname{Cyl}_{f},C_{*})$).

More generally, $\tau(C_{*})$ is also defined when $H_{*}$ is only stably
free, equipped with a stable basis. A module $X$ is called _stably free_ if
$X\oplus\mathbb{Z}G^{k}$ is free for some $k\geq 0$, and a _stable basis_ of
$X$ is by definition the data of an integer $k$ and a basis of
$X\oplus\mathbb{Z}G^{k}$ for some $k$. Note that in this more general setting,
$\tau(C_{*})$ is no longer unambiguously defined when $H_{*}$ is trivial and
$C_{*}$ is based, because the stable basis of the trivial module is not
unique. But it has a canonical stable basis (given by its unique basis), so
when $H_{*}$ is trivial, we will assume that it is equipped with the canonical
stable basis, unless we specify a different one.

In the proofs of Propositions 3.6 and 3.7, every space $X$ will be equipped
with a fixed map to $M$, which allows us to take all chain complexes and
homology groups with $\mathbb{Z}G$ coefficients. That is, we will write
$C_{*}(X)$ for $C_{*}(X;\mathbb{Z}G^{\theta})$ if $f\colon X\rightarrow M$ is
the fixed map and $\theta=\pi_{1}(f)\colon\pi_{1}(X)\rightarrow\pi_{1}(M)=G$.

###### Proof of Proposition 3.6.

The case of odd $n$ was proved in Shaneson [Shaneson-GxZ, Lemma 4.2], so we
will assume that $n=2q$ is even.

An element $z\in L^{h}_{n+1}(\mathbb{Z}G,w)$ is represented by a formation
$(H;L_{0},L_{1})$, where $H$ is a $(-1)^{q}$-symmetric hyperbolic form over
$\mathbb{Z}G$ (of rank $2k$) and $L_{0}$ and $L_{1}$ are lagrangians in $H$.
Let $[M_{0},f_{0}]\in\mathcal{S}^{h}(M)$. First we recall the definition of
the action of $z$ on $[M_{0},f_{0}]$.

Consider $M_{0}\times I$ and add $k$ trival $q$-handles, then we obtain a
cobordism $W_{0}$ between $M_{0}$ and $N\coloneqq M_{0}\\#k(S^{q}\times
S^{q})$. There is an isometry $H\cong H_{q}(\\#^{k}S^{q}\times S^{q})$ such
that $L_{0}$ corresponds to the subgroup generated by the $\ast\times S^{q}$
(recall that all homology is with $\mathbb{Z}G$ coefficients by default in
this proof). Then $W_{0}$ can also be constructed from $N\times I$ by adding
$k$ $(q+1)$-handles along a basis of $L_{0}$.

Next add $k$ $(q+1)$-handles to $N\times I$ along a basis of $L_{1}$. This
yields a cobordism $W_{1}$ between $N$ and some $M_{1}$, and we define
$W=W_{0}\cup_{N}W_{1}$. It is a cobordism between $M_{0}$ and $M_{1}$ over $M$
with the map $F\colon W\to M$, which restricts to the composition of the
projection $M_{0}\times I\rightarrow M_{0}$ and $f_{0}$ on $M_{0}\times I$,
and sends the extra handles to a point. Let

$f_{1}:=F\big{|}_{M_{1}}\colon M_{1}\to M,\,\,F_{i}:=F\big{|}_{W_{i}}\colon
W_{i}\rightarrow M\,\,(i=0,1),\text{ and }g:=F\big{|}_{N}\colon N\to M.$

Then $[M_{0},f_{0}]^{z}=[M_{1},f_{1}]$.

Fix a basis of $H$. On $L_{i}$ we consider the basis which corresponds to the
gluing maps of the handles that are added to $N\times I$ to construct $W_{i}$,
this determines a dual basis on $L_{i}^{*}$. The adjoint of the intersection
form on $H$ is an isomorphism $H\rightarrow H^{*}$, and we get a split short
exact sequence

$0\rightarrow L_{i}\rightarrow H\rightarrow L_{i}^{*}\rightarrow 0$ (3.2)

A splitting determines an isomorphism $H\cong L_{i}\oplus L_{i}^{*}$, and its
Whitehead torsion, denoted $x_{i}\in\operatorname{Wh}(G,w)$, is independent of
the choice of the splitting (see [Mi66, Section 2]).

We note that changing the basis of $H$ has the same effect on $x_{0}$ and
$x_{1}$, so $x_{1}-x_{0}$ is independent of the choice of basis. In the case
when the basis is given by the homology classes corresponding to the spheres
in $H_{q}(\\#^{k}S^{q}\times S^{q})$, we have $x_{0}=0$ and
$\psi(z)=\pi(x_{1})$ by definition; see [Shaneson-GxZ, p. 312] and the
correspondence between the different descriptions of the odd-dimensional
L-groups given at the end of [Wall-SCM, Chapter 6]. Therefore we have,
independently of the choice of basis of $H$, that

$\psi(z)=\pi(x_{1}-x_{0})\in\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w)).$
(3.3)

Now consider the triple $(\operatorname{Cyl}_{F_{i}},W_{i},N)$, where
$\operatorname{Cyl}_{F_{i}}$ denotes the mapping cylinder of $F_{i}$. We
compute the relative homology groups as follows:

  1. (1)

$H_{*}(W_{i},N)\cong H_{q+1}(W_{i},N)\cong L_{i}$;

  2. (2)

$H_{*}(\operatorname{Cyl}_{F_{i}},N)\cong
H_{q+1}(\operatorname{Cyl}_{F_{i}},N)\cong\ker(H_{q}(g))\cong H$, because
$\operatorname{Cyl}_{F_{i}}\simeq M\simeq M_{0}$, and using the construction
of $N$ as $N=M_{0}\\#k(S^{q}\times S^{q})$; and

  3. (3)

$H_{*}(\operatorname{Cyl}_{F_{i}},W_{i})\cong
H_{q+1}(\operatorname{Cyl}_{F_{i}},W_{i})\cong\ker(H_{q}(F_{i}))\cong\ker(H_{q}(g))/\ker(H_{q}(N\rightarrow
W_{i}))\cong H/L_{i}\cong L_{i}^{*},$ because $W_{i}$ is constructed from
$M_{0}\times I\simeq M\simeq_{s}\operatorname{Cyl}_{F_{i}}$ by adding
$(q+1)$-handles.

So all of these homology groups are free (over $\mathbb{Z}G$), and we equip
them with the previously chosen bases. The long exact sequence

$\cdots\rightarrow H_{q+2}(\operatorname{Cyl}_{F_{i}},W_{i})\rightarrow
H_{q+1}(W_{i},N)\rightarrow H_{q+1}(\operatorname{Cyl}_{F_{i}},N)\rightarrow
H_{q+1}(\operatorname{Cyl}_{F_{i}},W_{i})\rightarrow
H_{q}(W_{i},N)\rightarrow\cdots$ (3.4)

is therefore isomorphic to the short exact sequence (3.2). If we regard it as
a based free chain complex with vanishing homology, then by the sign
conventions in the definition (see [Mi66, p. 365]) its Whitehead torsion is
$(-1)^{q}x_{i}$. The cobordism $W_{i}$ is simple homotopy equivalent to $N$
with a $(q+1)$-cell attached for each basis element of $L_{i}$, so

$C_{*}(W_{i},N)\cong C_{q+1}(W_{i},N)\cong L_{i}\cong H_{q+1}(W_{i},N)\cong
H_{*}(W_{i},N)$

with the same choice of basis, therefore, again by the definition,
$\tau(W_{i},N)=0$. Finally,
$\operatorname{Cyl}_{F_{i}}\simeq_{s}M\simeq_{s}\operatorname{Cyl}_{g}$ by
Proposition 2.29, so
$\tau(\operatorname{Cyl}_{F_{i}},N)=\tau(\operatorname{Cyl}_{g},N)$. Hence by
[Mi66, Theorem 3.2] we have

$\tau(\operatorname{Cyl}_{g},N)=\tau(\operatorname{Cyl}_{F_{i}},W_{i})+(-1)^{q}x_{i},$

which implies that

$x_{1}-x_{0}=(-1)^{q+1}(\tau(\operatorname{Cyl}_{F_{1}},W_{1})-\tau(\operatorname{Cyl}_{F_{0}},W_{0})).$

Next consider the triple $(\operatorname{Cyl}_{F_{i}},W_{i},M_{i})$. Since
$W_{i}$ can be obtained from $M_{i}\times I$ by adding $k$ trivial
$q$-handles, we have $W_{i}\simeq_{s}M_{i}\vee(\bigvee^{k}S^{q})$. We choose
the basis corresponding to the spheres in both $C_{*}(W_{i},M_{i})\cong
C_{q}(W_{i},M_{i})\cong C_{q}(\bigvee^{k}S^{q})$ and $H_{*}(W_{i},M_{i})\cong
H_{q}(W_{i},M_{i})\cong H_{q}(\bigvee^{k}S^{q})$, so that
$\tau(W_{i},M_{i})=0$. Since $f_{i}\colon M_{i}\rightarrow M$ is a homotopy
equivalence and
$\operatorname{Cyl}_{F_{i}}\simeq_{s}M\simeq_{s}\operatorname{Cyl}_{F_{i}}$,
we have $H_{*}(\operatorname{Cyl}_{F_{i}},M_{i})\cong
H_{*}(\operatorname{Cyl}_{F_{i}},M_{i})=0$ and
$\tau(\operatorname{Cyl}_{F_{i}},M_{i})=\tau(\operatorname{Cyl}_{F_{i}},M_{i})=\tau(f_{i})$.
In the homological long exact sequence of the triple
$(\operatorname{Cyl}_{F_{i}},W_{i},M_{i})$ all terms are trivial except for
the isomorphism

$0\rightarrow
H_{q+1}(\operatorname{Cyl}_{F_{i}},W_{i})\xrightarrow{\cong}H_{q}(W_{i},M_{i})\rightarrow
0$ (3.5)

We equipped $H_{q+1}(\operatorname{Cyl}_{F_{i}},W_{i})\cong L_{i}^{*}$ with
the dual of the basis of $L_{i}$. Since the handles added to $M_{i}\times I$
to obtain $W_{i}$ (determining the basis of $H_{q}(W_{i},M_{i})$) are the
duals of the handles added to $N\times I$ to obtain $W_{i}$ (corresponding to
the basis if $L_{i}$), the isomorphism (3.5) preserves the basis. Therefore
the Whitehead torsion of the long exact sequence (3.4) vanishes. So by [Mi66,
Theorem 3.2] we have

$\tau(f_{i})=\tau(\operatorname{Cyl}_{F_{i}},W_{i}).$

Therefore $x_{1}-x_{0}=(-1)^{q+1}(\tau(f_{1})-\tau(f_{0}))$, which implies
that $\pi(x_{1}-x_{0})=\pi(\tau(f_{1})-\tau(f_{0}))$. Combining with (3.3), we
have $\pi(\tau(f_{1})-\tau(f_{0}))=\pi(x_{1}-x_{0})=\psi(z)$. Hence

$\displaystyle\widehat{\tau}([M_{0},f_{0}]^{z})$
$\displaystyle=\widehat{\tau}([M_{1},f_{1}])=\pi(\tau(f_{1}))=\pi(\tau(f_{0}))+\pi(\tau(f_{1})-\tau(f_{0}))=\pi(\tau(f_{0}))+\pi(x_{1}-x_{0})$
$\displaystyle=\widehat{\tau}([M_{0},f_{0}])+\psi(z)=\widehat{\tau}([M_{0},f_{0}])^{z},$

as required. ∎

###### Proof of Proposition 3.7.

The first triangle commutes by Proposition 3.6 combining with the fact that
$\widehat{\tau}([M,\operatorname{Id}])=0$. The last triangle commutes by the
definitions of $\sigma_{s}$ and $\sigma_{h}$, see e.g. [Ra86, p.359–360]. The
map $\sigma_{h}$ is by definition the map $\sigma_{s}$ followed by the map
$F\colon L_{n}^{s}(\mathbb{Z}G,w)\to L_{n}^{h}(\mathbb{Z}G,w)$ that forgets
bases.

So we need to prove that the square commutes. Let
$[N,f]\in\mathcal{S}^{h}(M)$, then $\eta_{h}([N,f])\in\mathcal{N}(M)$ is the
normal bordism class of $f\colon N\rightarrow M$ (with an appropriate bundle
map), and $\widehat{\tau}([N,f])=\pi(\tau(f))$. We need to determine the image
of these elements in $L^{s}_{n}(\mathbb{Z}G,w)$.

First assume that $n=2q$ is even. Since $f$ is $q$-connected,
$\sigma_{s}(\eta_{h}([N,f]))$ is defined as a form on $\ker(H_{q}(f))\cong
H_{q+1}(\operatorname{Cyl}_{f},N)$, which is trivial (because $f$ is a
homotopy equivalence), and it is equipped with a certain stable basis. The
stable basis of $H_{q+1}(\operatorname{Cyl}_{f},N)=0$ is chosen such that with
this choice $\tau(\operatorname{Cyl}_{f},N)=0$ (where
$H_{i}(\operatorname{Cyl}_{f},N)=0$ is equipped with the canonical stable
basis if $i\neq q+1$). If $H_{q+1}(\operatorname{Cyl}_{f},N)$ were also
equipped with the canonical stable basis, then the Whitehead torsion of
$(\operatorname{Cyl}_{f},N)$ would be equal to $\tau(f)$. Therefore the
transition matrix between the chosen and the standard stable basis (when both
are regarded as bases of $\mathbb{Z}G^{k}$ for some $k\geq 0$) has Whitehead
torsion $(-1)^{q}\tau(f)$. In $L^{s}_{n}(\mathbb{Z}G,w)$ the same element
$\sigma_{s}(\eta_{h}([N,f]))$ is also represented by a standard hyperbolic
form with a basis such that the transition matrix between the chosen and the
standard basis has Whitehead torsion $(-1)^{q}\tau(f)$.

The image of $\pi(\tau(f))$ in $L^{s}_{n}(\mathbb{Z}G,w)$ is represented by a
standard hyperbolic form with a basis with the property that if $x$ denotes
the Whitehead torsion of the transition matrix between the chosen and the
standard basis, then $x\in\mathcal{J}_{n}(G,w)$ and $\pi(x)=\pi(\tau(f))$ (see
[Shaneson-GxZ, p. 312]). In particular, the representative of
$\sigma_{s}(\eta_{h}([N,f]))$ constructed above also represents
$\varrho(\widehat{\tau}([N,f]))$, noting that
$\pi((-1)^{q}\tau(f))=\pi(\tau(f))$, because
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ is $2$-torsion. Therefore
the square commutes if $n$ is even.

Next consider the case when $n=2q+1$ is odd. Let $U$ denote a tubular
neighbourhood of a disjoint union of embeddings $S^{q}\rightarrow N$
representing a generating set of $\ker(\pi_{q}(f))$ and let
$N_{0}=N\setminus\operatorname{Int}U$. We identify $H_{q}(\partial U)$ with
the standard hyperbolic form, then $H_{q+1}(U,\partial U)$ (more precisely,
its image under the boundary map) corresponds to the standard lagrangian. We
can assume that $f\big{|}_{U}$ is constant, so $f\big{|}_{N_{0}}$ is a map of
pairs $(N_{0},\partial U)\rightarrow(M,\ast)$. Then
$\ker(H_{q+1}(N_{0},\partial U)\rightarrow H_{q+1}(M,\ast))\cong
H_{q+2}(\operatorname{Cyl}_{f|_{N_{0}}},N_{0}\cup\operatorname{Cyl}_{f|_{\partial
U}})$ determines another lagrangian in $H_{q}(\partial U)$. We equip
$H_{q}(\partial U)$ and $H_{q+1}(U,\partial U)$ with their standard bases, and
$H_{q+2}(\operatorname{Cyl}_{f|_{N_{0}}},N_{0}\cup\operatorname{Cyl}_{f|_{\partial
U}})$ with a stable basis such that
$\tau(\operatorname{Cyl}_{f|_{N_{0}}},N_{0}\cup\operatorname{Cyl}_{f|_{\partial
U}})=0$. Then $\sigma_{s}(\eta_{h}([N,f]))$ is represented by the formation
$(H_{q}(\partial U);H_{q+1}(U,\partial
U),H_{q+2}(\operatorname{Cyl}_{f|_{N_{0}}},N_{0}\cup\operatorname{Cyl}_{f|_{\partial
U}}))$. Since $f$ is a homotopy equivalence, we can take the empty generating
set for $\ker(\pi_{q}(f))$. Then $H_{q}(\partial U)=0$, so we have the trivial
formation, with the standard basis on the ambient form and on the first
lagrangian. The stable basis on the second lagrangian,
$H_{q+2}(\operatorname{Cyl}_{f},N)$, is chosen such that
$\tau(\operatorname{Cyl}_{f},N)=0$. Therefore the transition matrix between
this stable basis and the standard one has Whitehead torsion
$(-1)^{q+1}\tau(f)$. In $L^{s}_{n}(\mathbb{Z}G,w)$ the same element
$\sigma_{s}(\eta_{h}([N,f]))$ is also represented by a formation on the
standard hyperbolic form given by the standard lagrangians, such that the
ambient form and the first lagrangian are equipped with their standard bases,
and for the second lagrangian the transition matrix between the chosen and the
standard basis has Whitehead torsion $(-1)^{q+1}\tau(f)$.

The image of $\pi(\tau(f))$ in $L^{s}_{n}(\mathbb{Z}G,w)$ is represented by a
formation on the standard hyperbolic form given by the standard lagrangians,
such that the ambient form and the first lagrangian are equipped with their
standard bases, and if $x$ denotes the Whitehead torsion of the transition
matrix between the chosen and the standard basis of the second lagrangian,
then $x\in\mathcal{J}_{n}(G,w)$ and $\pi(x)=\pi(\tau(f))$. Again we use that
we can ignore signs because $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
is $2$-torsion. So the representative of $\sigma_{s}(\eta_{h}([N,f]))$
constructed above also represents $\varrho(\widehat{\tau}([N,f]))$, showing
that the square also commutes if $n$ is odd. ∎

######  Corollary 3.9.

  1. (a)

$\operatorname{Im}\widehat{\tau}=\varrho^{-1}(\operatorname{Im}\sigma_{s})$.

  2. (b)

$\operatorname{Im}\psi\subseteq\varrho^{-1}(\operatorname{Im}\sigma_{s})$.

###### Proof.

First we prove (a). The commutativity of the square in Proposition 3.7 implies
that
$\operatorname{Im}\widehat{\tau}\subseteq\varrho^{-1}(\operatorname{Im}\sigma_{s})$.
For the other direction assume that
$x\in\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ and
$\varrho(x)=\sigma_{s}(y)$ for some $y\in\mathcal{N}(M)$. Then
$\sigma_{h}(y)=F\circ\sigma_{s}(y)=F\circ\varrho(x)=0$, so $y=\eta_{h}(v)$ for
some $v\in\mathcal{S}^{h}(M)$. Then
$\varrho(x-\widehat{\tau}(v))=\varrho(x)-\sigma_{s}(\eta_{h}(v))=\varrho(x)-\sigma_{s}(y)=0$,
so $x-\widehat{\tau}(v)=\psi(z)$ for some $z\in L^{h}_{n+1}(\mathbb{Z}G,w)$.
So by Proposition 3.6
$x=\widehat{\tau}(v)^{z}=\widehat{\tau}(v^{z})\in\operatorname{Im}\widehat{\tau}$.

For (b), since $\operatorname{Id}_{M}$ has vanishing surgery obstruction, we
have
$\operatorname{Im}\psi=\ker\varrho\subseteq\varrho^{-1}(\operatorname{Im}\sigma_{s})$.
∎

######  Corollary 3.10.

Let $x\in\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$. If
$x\in\operatorname{Im}\psi$, then there is a $\operatorname{CAT}$ $n$-manifold
$N$ and a homotopy equivalence $f\colon N\to M$ such that $\pi(\tau(f))=x$.

###### Proof.

By Corollary 3.9, $x=\widehat{\tau}([N,f])$ for some
$[N,f]\in\mathcal{S}^{h}(M)$. ∎

######  Remark 3.11.

There is a common generalisation of the two realisation techniques used in
Section 3. Let $L_{n}^{s,\tau}(\mathbb{Z}G,w)$ denote the surgery obstruction
groups defined in [Kreck-preprint, Section 4]. Similarly to
$L_{n}^{s}(\mathbb{Z}G,w)$, the elements of $L_{n}^{s,\tau}(\mathbb{Z}G,w)$
are represented by based nonsingular forms/formations, but, unlike the
elements of $L_{n}^{s}(\mathbb{Z}G,w)$, they are not assumed to be simple.
Thus there is a natural forgetful map $L_{n}^{s,\tau}(\mathbb{Z}G,w)\to
L_{n}^{h}(\mathbb{Z}G,w)$ and $L_{n}^{s}(\mathbb{Z}G,w)$ is the kernel of a
map $L_{n}^{s,\tau}(\mathbb{Z}G,w)\to\operatorname{Wh}(G,w)$ (see [Kreck-
preprint]). There is also an action of $\operatorname{Wh}(G,w)$ on
$L_{n}^{s,\tau}(\mathbb{Z}G,w)$ (which determines a map
$\operatorname{Wh}(G,w)\to L_{n}^{s,\tau}(\mathbb{Z}G,w)$ by acting on $0$)
given by changing the basis; this action is transitive on the fibers of the
$L_{n}^{s,\tau}(\mathbb{Z}G,w)\to L_{n}^{h}(\mathbb{Z}G,w)$.

Wall’s construction for realising elements of $L_{n}^{s}(\mathbb{Z}G,w)$
[Wall-SCM] can also be applied to $L_{n}^{s,\tau}$, but it will only produce
homotopy equivalences, not simple homotopy equivalences. The Whitehead torsion
of the resulting homotopy equivalence is given by the previously mentioned map
$L_{n}^{s,\tau}(\mathbb{Z}G,w)\to\operatorname{Wh}(G,w)$, and in fact the
image of this map is in $\mathcal{J}_{n}(G,w)$.

The realisation of $L_{n}^{h}(\mathbb{Z}G,w)$, and the map $W_{h}$, can be
regarded as a special case of this construction, by first choosing a basis for
the form/formation representing an element of $L_{n}^{h}(\mathbb{Z}G,w)$,
equivalently, choosing a lift in $L_{n}^{s,\tau}(\mathbb{Z}G,w)$. As the
choice of basis is not unique, the homotopy equivalence we obtain is not
unique, and its Whitehead torsion (the value of $\widehat{\tau}$) is only
well-defined in the quotient
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$. The realisation part of the
$s$-cobordism theorem is also a special case via the map
$\operatorname{Wh}(G,w)\to L_{n}^{s,\tau}(\mathbb{Z}G,w)$. The Whitehead
torsion of the homotopy equivalence induced by the $h$-cobordism we get is of
course in $\mathcal{I}_{n}(G,w)$.

In the rest of this section we will consider the orientable ($w\equiv 1$)
case, and omit $w$ from the notation.

######  Proposition 3.12.

Let $G_{m}\coloneqq C_{\infty}\times C_{m}$, with $m\geq 2$. For every even
integer $n=2k\geq 0$, the map

$\psi\colon
L_{n+1}^{h}(\mathbb{Z}[G_{m}])\to\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G_{m}))$

is surjective.

###### Proof.

We verify that the forgetful map $F\colon L^{s}_{2k}(\mathbb{Z}G_{m})\to
L^{h}_{2k}(\mathbb{Z}G_{m})$ is injective. The conclusion then follows from
the exact sequence (3.1). First we apply Shaneson splitting to the domain and
the codomain, to obtain a commutative diagram whose rows are split short exact
sequences and whose vertical maps are the forgetful maps [Ra86].

${0}$${L^{s}_{2k}(\mathbb{Z}C_{m})}$${L^{s}_{2k}(\mathbb{Z}G_{m})}$${L^{h}_{2k-1}(\mathbb{Z}C_{m})}$${0}$${0}$${L^{h}_{2k}(\mathbb{Z}C_{m})}$${L^{h}_{2k}(\mathbb{Z}G_{m})}$${L^{p}_{2k-1}(\mathbb{Z}C_{m})}$${0.}$$\scriptstyle{F}$

By the five lemma it suffices to show that the left and right vertical maps
are injective. In [Bak-computation-L-groups], Bak gave computations of
$L_{i}^{X}(\mathbb{Z}G)$ for $X\in\\{s,h,p\\}$ and $G$ a finite group whose
2-hyperelementary subgroups are abelian, which certainly holds for finite
cyclic groups. See also [Bak-odd-L-groups-of-odd-order-groups, Bak-even-L-
groups-of-odd-order-groups] First we note that, by [Bak-computation-L-groups,
Theorem 8], the forgetful map $L^{h}_{2k-1}(\mathbb{Z}C_{m})\to
L^{p}_{2k-1}(\mathbb{Z}C_{m})$ is injective. So the right vertical map is
injective.

To prove that the left vertical map is injective we also use [Bak-computation-
L-groups]. For a finite group $G$ whose 2-Sylow subgroup $G_{2}$ is normal and
abelian, Bak defined
$r_{2}:=\operatorname{rk}H^{1}(C_{2};\operatorname{Wh}(G))$. On [Bak-
computation-L-groups, p. 386], he noted that if $G_{2}$ is cyclic, as in our
case $G=C_{m}$, then $r_{2}=0$. Therefore by the sequence (3.1), as shown on
[Bak-computation-L-groups, p. 390], it follows that
$L_{2k}^{s}(\mathbb{Z}C_{m})\to L_{2k}^{h}(\mathbb{Z}C_{m})$ is injective.
Hence $F\colon L_{2k}^{s}(\mathbb{Z}G_{m})\to L_{2k}^{h}(\mathbb{Z}G_{m})$ is
injective as desired. ∎

###  4\. Simple homotopy manifold sets

In this section we prove Theorem D about the characterisation of simple
homotopy manifold sets. First we look at the analogous problem in the setting
of CW complexes, which has a simpler answer. Then we consider the case of
manifolds, which will rely on the results of Section 3.

####  4.1. CW complexes

Fix a CW complex $X$ and let $G=\pi_{1}(X)$. Our goal is to understand the set
$\mathcal{C}^{h}_{s}(X)$ defined below.

######  Definition 4.1.

Let

$\mathcal{C}^{h}_{s}(X)\coloneqq\left\\{\text{CW complexes }Y\mid Y\simeq
X\right\\}/\simeq_{s}.$

We will need some auxiliary definitions.

######  Definition 4.2.

If $Y$ is a CW complex homotopy equivalent to $X$, then let

$t_{X}(Y)=\left\\{\tau(f)\mid f\colon Y\to X\text{ is a homotopy
equivalence}\right\\}\subseteq\operatorname{Wh}(G).$

######  Definition 4.3.

For $x\in\operatorname{Wh}(G)$ and $g\in\operatorname{hAut}(X)$ let
$x^{g}=g_{*}(x)+\tau(g)$. Clearly $x^{\operatorname{Id}}=x$ and $x^{g\circ
g^{\prime}}=(x^{g^{\prime}})^{g}$ (see Proposition 2.30), so this defines an
action of $\operatorname{hAut}(X)$ on the set $\operatorname{Wh}(G)$.

With this notation we have the following theorem.

######  Theorem 4.4.

The map $t_{X}$ induces a well-defined bijection
$\widetilde{t}_{X}\colon\mathcal{C}^{h}_{s}(X)\to\operatorname{Wh}(G)/\operatorname{hAut}(X)$.

The proof will consist of the following sequence of lemmas.

######  Lemma 4.5.

If $Y$ is a CW complex homotopy equivalent to $X$, then $t_{X}(Y)$ is an orbit
of the action of $\operatorname{hAut}(X)$ on $\operatorname{Wh}(G)$.

###### Proof.

By Proposition 2.30 we have $\tau(g\circ f)=\tau(f)^{g}$ for every homotopy
equivalence $f\colon Y\to X$ and $g\in\operatorname{hAut}(X)$. If $f\colon
Y\to X$ is a homotopy equivalence and $g\in\operatorname{hAut}(X)$, then
$g\circ f$ is also a homotopy equivalence, so if $x\in t_{X}(Y)$, then
$x^{g}\in t_{X}(Y)$ for every $g$. On the other hand, if $f,f^{\prime}\colon
Y\to X$ are homotopy equivalences, then there is a
$g\in\operatorname{hAut}(X)$ such that $f^{\prime}\simeq g\circ f$, showing
that if $x,x^{\prime}\in t_{X}(Y)$, then $x^{\prime}=x^{g}$ for some $g$. ∎

######  Lemma 4.6.

If $Y$ is a CW complex homotopy equivalent to $X$ and $Y\simeq_{s}Z$, then
$t_{X}(Y)=t_{X}(Z)$.

###### Proof.

Let $h\colon Z\to Y$ be a simple homotopy equivalence. If $x\in t_{X}(Y)$,
i.e. $x=\tau(f)$ for some homotopy equivalence $f\colon Y\to X$, then $f\circ
h\colon Z\to X$ is a homotopy equivalence with $\tau(f\circ h)=\tau(f)=x$ by
Proposition 2.30. This shows that $t_{X}(Y)\subseteq t_{X}(Z)$. We get
similarly that $t_{X}(Z)\subseteq t_{X}(Y)$, therefore $t_{X}(Y)=t_{X}(Z)$. ∎

######  Lemma 4.7.

If $Y$ and $Z$ are CW complexes homotopy equivalent to $X$ and
$t_{X}(Y)=t_{X}(Z)$, then $Y\simeq_{s}Z$.

###### Proof.

Let $x\in t_{X}(Y)=t_{X}(Z)$ be an arbitrary element, then there are homotopy
equivalences $f\colon Y\to X$ and $f^{\prime}\colon Z\to X$ with
$\tau(f)=\tau(f^{\prime})=x$. Let $f^{-1}\colon X\to Y$ denote the homotopy
inverse of $f$, then by Proposition 2.30
$0=\tau(\operatorname{Id}_{Y})=\tau(f^{-1})+f^{-1}_{*}(\tau(f))=\tau(f^{-1})+f^{-1}_{*}(x)$.
Hence we have $\tau(f^{-1}\circ
f^{\prime})=\tau(f^{-1})+f^{-1}_{*}(\tau(f^{\prime}))=\tau(f^{-1})+f^{-1}_{*}(x)=0$,
showing that $f^{-1}\circ f^{\prime}\colon Z\to Y$ is a simple homotopy
equivalence. ∎

######  Lemma 4.8 ([Co73, (24.1)]).

For every $x\in\operatorname{Wh}(G)$ there is a CW complex $Y$ and a homotopy
equivalence $f\colon Y\rightarrow X$ such that $\tau(f)=x$. ∎

###### Proof of Theorem 4.4.

By Lemma 4.5, $t_{X}$ takes values in
$\operatorname{Wh}(G)/\operatorname{hAut}(X)$ and by Lemma 4.6 it induces a
well-defined map $\widetilde{t}_{X}$ on $\mathcal{C}^{h}_{s}(X)$. Lemmas 4.7
and 4.8 imply that $\widetilde{t}_{X}$ is injective and surjective,
respectively. ∎

######  Remark 4.9.

There are two special cases when the action of $\operatorname{hAut}(X)$ on
$\operatorname{Wh}(G)$ has a simpler description.

First, assume that $\pi_{1}(g)=\operatorname{Id}_{G}$ for every
$g\in\operatorname{hAut}(X)$. Then $\tau(g\circ
g^{\prime})=\tau(g)+\tau(g^{\prime})$ for every
$g,g^{\prime}\in\operatorname{hAut}(X)$ and $x^{g}=x+\tau(g)$ for every
$x\in\operatorname{Wh}(G)$ and $g\in\operatorname{hAut}(X)$. This implies that
$\left\\{\tau(g)\mid g\in\operatorname{hAut}(X)\right\\}$ is a subgroup of
$\operatorname{Wh}(G)$ and $\operatorname{Wh}(G)/\operatorname{hAut}(X)$ is
the corresponding quotient group.

Second, assume that $\tau(g)=0$ for every $g\in\operatorname{hAut}(X)$. Then
$x^{g}=g_{*}(x)$ for every $x\in\operatorname{Wh}(G)$ and
$g\in\operatorname{hAut}(X)$. This means that the action of
$\operatorname{hAut}(X)$ factors through the map
$\pi_{1}\colon\operatorname{hAut}(X)\to\operatorname{Aut}(G)$, in particular
$\operatorname{hAut}(X)$ acts via automorphisms of the group
$\operatorname{Wh}(G)$.

####  4.2. Manifolds

Now we consider the problem in the manifold setting. Fix a closed connected
$\operatorname{CAT}$ $n$-manifold $M$ and let $G=\pi_{1}(M)$ with orientation
character $w\colon G\rightarrow\left\\{\pm 1\right\\}$. Then we can consider
either all $n$-manifolds that are homotopy equivalent to $M$, or those that