are $h$-cobordant to $M$, up to simple homotopy equivalence, or manifolds
homotopy equivalent to $M$ up to the equivalence relation generated by simple
homotopy equivalence and $h$-cobordism.

######  Definition 4.10.

Let

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

As before, $\operatorname{hAut}(M)$ acts on the set $\operatorname{Wh}(G,w)$,
and a subset $t_{M}(N)\subseteq\operatorname{Wh}(G,w)$ is defined for every
manifold $N$ that is homotopy equivalent to $M$. By Proposition 2.34 if
$f\colon N\rightarrow M$ is a homotopy equivalence between manifolds, then
$\tau(f)\in\mathcal{J}_{n}(G,w)$, so we can also define

$u_{M}(N)=\pi(t_{M}(N))=\left\\{\pi(\tau(f))\mid f\colon N\to M\text{ is a
homotopy
equivalence}\right\\}\subseteq\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w)).$

######  Theorem 4.11.

  1. (a)

The subset $\mathcal{J}_{n}(G,w)\subseteq\operatorname{Wh}(G,w)$ is invariant
under the action of $\operatorname{hAut}(M)$. The action of
$\operatorname{hAut}(M)$ on $\mathcal{J}_{n}(G,w)$ induces an action on the
set $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$.

  2. (b)

The map $t_{M}$ induces well-defined injective maps

$\widetilde{t}_{M}\colon\mathcal{M}^{h}_{s}(M)\to\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)\text{
and
}\widetilde{t}^{\prime}_{M}\colon\mathcal{M}^{\operatorname{hCob}}_{s}(M)\to
q(\mathcal{I}_{n}(G,w)),$

where
$q\colon\mathcal{J}_{n}(G,w)\to\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)$
denotes the quotient map, so that

$q(\mathcal{I}_{n}(G,w))\subseteq\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M).$

Similarly, the map $u_{M}$ induces a well-defined map

$\widetilde{u}_{M}\colon\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)\to\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))/\operatorname{hAut}(M).$

  3. (c)

There is a commutative diagram

$\textstyle{\mathcal{M}^{\operatorname{hCob}}_{s}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{t}^{\prime}_{M}}$$\textstyle{\mathcal{M}^{h}_{s}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{t}_{M}}$$\textstyle{\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{u}_{M}}$$\textstyle{q(\mathcal{I}_{n}(G,w))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))/\operatorname{hAut}(M)}$

In each row the first map is injective and the second map is surjective. In
the top row the composition of the maps is trivial, while the bottom row is an
exact sequence of pointed sets.

  4. (d)

If 1.1 are satisfied, then $\widetilde{t}^{\prime}_{M}$ is surjective,
$\widetilde{u}_{M}$ is injective, and the top row is an exact sequence of
pointed sets.

  5. (e)

If 1.1 are satisfied, then, using the notation from Section 3.2, the subsets

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

are invariant under the action of $\operatorname{hAut}(M)$, and we have

$\operatorname{Im}\widetilde{t}_{M}=(\varrho\circ\pi)^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)\text{
and
}\operatorname{Im}\widetilde{u}_{M}=\varrho^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M).$

This implies Theorem D (see also Theorem 4.16 below). Note that parts (a), (b)
and (c) do not require 1.1 and so apply to smooth/PL 4-manifolds as well as
topological 4-manifolds with arbitrary fundamental group.

We will use the following lemmas in the proof of Theorem 4.11.

######  Lemma 4.12.

Let $N$ and $P$ be manifolds homotopy equivalent to $M$. If $N\simeq_{s}P$ or
$N$ is $h$-cobordant to $P$, then $u_{M}(N)=u_{M}(P)$.

###### Proof.

If $N\simeq_{s}P$, then $t_{M}(N)=t_{M}(P)$ by Lemma 4.6, so
$u_{M}(N)=\pi(t_{M}(N))=\pi(t_{M}(P))=u_{M}(P)$.

If $N$ is $h$-cobordant to $P$ and $h\colon P\rightarrow N$ is a homotopy
equivalence induced by an $h$-cobordism, then $\pi(\tau(f\circ
h))=\pi(\tau(f))+\pi(f_{*}(\tau(h)))=\pi(\tau(f))$ for every homotopy
equivalence $f\colon N\rightarrow M$ (because
$f_{*}(\tau(h))\in\mathcal{I}_{n}(G,w)$ by Proposition 2.37). Hence
$u_{M}(N)\subseteq u_{M}(P)$. We get similarly that $u_{M}(P)\subseteq
u_{M}(N)$, therefore $u_{M}(N)=u_{M}(P)$. ∎

######  Lemma 4.13.

Suppose that 1.1 are satisfied, and $N$ and $P$ are manifolds homotopy
equivalent to $M$. If $u_{M}(N)=u_{M}(P)$, then there is a manifold $Q$ that
is simple homotopy equivalent to $N$ and $h$-cobordant to $P$.

###### Proof.

Since $u_{M}(N)=u_{M}(P)$, there are homotopy equivalences $f\colon
N\rightarrow M$ and $g\colon P\rightarrow M$ such that
$\pi(\tau(f))=\pi(\tau(g))$, equivalently,
$\tau(f)-\tau(g)\in\mathcal{I}_{n}(G,w)$. If $g^{-1}$ denotes the homotopy
inverse of $g$, then $\tau(g^{-1}\circ
f)=\tau(g^{-1})+g^{-1}_{*}(\tau(f))=\tau(g^{-1})+g^{-1}_{*}(\tau(g))+g^{-1}_{*}(\tau(f)-\tau(g))=\tau(g^{-1}\circ
g)+g^{-1}_{*}(\tau(f)-\tau(g))=g^{-1}_{*}(\tau(f)-\tau(g))$ by Proposition
2.30. So we can apply Corollary 3.3 to the homotopy equivalence $g^{-1}\circ
f\colon N\rightarrow P$. ∎

######  Lemma 4.14.

Suppose that $N$ and $P$ are manifolds satisfying 1.1. If there is a manifold
$Q$ that is simple homotopy equivalent to $N$ and $h$-cobordant to $P$, then
there is a manifold $R$ that is $h$-cobordant to $N$ and simple homotopy
equivalent to $P$.

###### Proof.

Apply Corollary 3.3 to the composition of a homotopy equivalence $P\rightarrow
Q$ induced by an $h$-cobordism and a simple homotopy equivalence $Q\rightarrow
N$. ∎

######  Proposition 4.15.

Suppose that $N$ and $P$ are manifolds satisfying 1.1. Then the following are
equivalent.

  * (a)

The manifolds $N$ and $P$ are equivalent under the equivalence relation
generated by simple homotopy equivalence and $h$-cobordism.

  * (a)

There is a manifold $Q$ that is simple homotopy equivalent to $N$ and
$h$-cobordant to $P$.

###### Proof.

A chain between $N$ and $P$ of alternating simple homotopy equivalences and
$h$-cobordisms can be reduced to a chain of length $2$ using Lemma 4.14. ∎

Now we are ready to begin the proof of Theorem 4.11.

###### Proof of Theorem 4.11.

(a) If $g\in\operatorname{hAut}(M)$, then $\tau(g)\in\mathcal{J}_{n}(G,w)$ by
Proposition 2.34. Moreover, $\pi_{1}(g)\circ w=w$, hence
$g_{*}\colon\operatorname{Wh}(G,w)\rightarrow\operatorname{Wh}(G,w)$ is
compatible with the involution, so if $x=-(-1)^{n}\overline{x}$, then
$g_{*}(x)=-(-1)^{n}\overline{g_{*}(x)}$. Therefore if
$x\in\mathcal{J}_{n}(G,w)$, then $x^{g}\in\mathcal{J}_{n}(G,w)$, showing that
$\operatorname{hAut}(M)$ acts on $\mathcal{J}_{n}(G,w)$.

If $x,y\in\mathcal{J}_{n}(G,w)$ and $x-y\in\mathcal{I}_{n}(G,w)$, then
$x^{g}-y^{g}=(g_{*}(x)+\tau(g))-(g_{*}(y)+\tau(g))=g_{*}(x-y)\in\mathcal{I}_{n}(G,w)$
(because $g_{*}$ is compatible with the involution). Therefore there is an
induced action on $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$.

(b) If $f\colon N\rightarrow M$ is a homotopy equivalence for some
$n$-manifold $N$, then $\tau(f)\in\mathcal{J}_{n}(G,w)$ by Proposition 2.34.
Hence $t_{M}(N)\subseteq\mathcal{J}_{n}(G,w)$ for every manifold $N$ that is
homotopy equivalent to $M$. Lemmas 4.5, 4.6, LABEL:, and 4.7 can be used again
to show that $t_{M}$ induces well-defined injective maps
$\widetilde{t}_{M}\colon\mathcal{M}^{h}_{s}(M)\to\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)$
and
$\widetilde{t}^{\prime}_{M}\colon\mathcal{M}^{\operatorname{hCob}}_{s}(M)\to\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)$.
If $N$ is $h$-cobordant to $M$ and $f\colon N\rightarrow M$ is a homotopy
equivalence induced by an $h$-cobordism, then $\tau(f)\in\mathcal{I}_{n}(G,w)$
by Proposition 2.37, showing that $t_{M}(N)\in q(\mathcal{I}_{n}(G,w))$.

The analogue of Lemma 4.5 shows that $u_{M}(N)$ is an orbit of the action of
$\operatorname{hAut}(M)$ on $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
(using that $\pi(\tau(g\circ f))=\pi(\tau(f)^{g})=\pi(\tau(f))^{g}$ for every
homotopy equivalence $f\colon N\to M$ and $g\in\operatorname{hAut}(M)$). Lemma
4.12 shows that $u_{M}$ induces a well-defined map on
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$.

(c) This follows immediately from the definitions.

(d) The surjectivity of $\widetilde{t}^{\prime}_{M}$ and injectivity of
$\widetilde{u}_{M}$ follow from Corollary 3.2 and Lemma 4.13, respectively.

For the exactness of the top row, assume that $N$ is equivalent to $M$ under
the equivalence relation generated by simple homotopy equivalence and
$h$-cobordism. By Proposition 4.15 there is a manifold $P$ that is simple
homotopy equivalent to $N$ and $h$-cobordant to $M$. This $P$ represents an
element of $\mathcal{M}^{\operatorname{hCob}}_{s}(M)$, which is mapped to the
equivalence class of $N$ in $\mathcal{M}^{h}_{s}(M)$.

(e) First consider $\varrho^{-1}(\operatorname{Im}\sigma_{s})$, which is equal
to $\operatorname{Im}\widehat{\tau}$ by Corollary 3.9. This is
$\operatorname{hAut}(M)$-invariant, because for any
$[N,f]\in\mathcal{S}^{h}(M)$ and $g\in\operatorname{hAut}(M)$ we have
$\pi(\tau(f))^{g}=\pi(\tau(g\circ f))=\widehat{\tau}([N,g\circ f])$. Since the
action of $\operatorname{hAut}(M)$ on
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ is induced by its action on
$\mathcal{J}_{n}(G,w)$, and $\varrho^{-1}(\operatorname{Im}\sigma_{s})$ is
$\operatorname{hAut}(M)$-invariant,
$\pi^{-1}(\varrho^{-1}(\operatorname{Im}\sigma_{s}))$ is
$\operatorname{hAut}(M)$-invariant too.

Now suppose that $[N]\in\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$. Then
there is a homotopy equivalence $f\colon N\rightarrow M$ and
$\widetilde{u}_{M}([N])$ is the orbit of
$\pi(\tau(f))=\widehat{\tau}([N,f])\in\operatorname{Im}\widehat{\tau}=\varrho^{-1}(\operatorname{Im}\sigma_{s})$.
This shows that
$\operatorname{Im}\widetilde{u}_{M}\subseteq\varrho^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)$.
On the other hand, if
$x\in\varrho^{-1}(\operatorname{Im}\sigma_{s})=\operatorname{Im}\widehat{\tau}$,
then $x=\pi(\tau(f))$ for some homotopy equivalence $f\colon N\rightarrow M$,
and then $[N]\in\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$ and
$\widetilde{u}_{M}([N])$ is the orbit of $x$. Therefore
$\operatorname{Im}\widetilde{u}_{M}\supseteq\varrho^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)$.

By the commutativity of the diagram in part (c),
$\operatorname{Im}\widetilde{t}_{M}$ is contained in the inverse image of
$\operatorname{Im}\widetilde{u}_{M}=\varrho^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)$,
which is
$\pi^{-1}(\varrho^{-1}(\operatorname{Im}\sigma_{s}))/\operatorname{hAut}(M)$.
In the other direction, if $x\in\mathcal{J}_{n}(G,w)$ and
$\pi(x)\in\varrho^{-1}(\operatorname{Im}\sigma_{s})=\operatorname{Im}\widehat{\tau}$,
then there exists a manifold $N_{1}$ and a homotopy equivalence $f_{1}\colon
N_{1}\to M$ with $\pi(\tau(f_{1}))=\pi(x)$. It follows that $x=\tau(f_{1})+y$
for some $y\in\mathcal{I}_{n}(G,w)$. Use $(f_{1})_{*}$ to identify
$\pi_{1}(N_{1})$ with $G=\pi_{1}(M)$ (since $f_{1}$ is a homotopy equivalence,
the orientation character of $N$ is also $w$). By Corollary 3.2 there exists a
manifold $N_{2}$ and a homotopy equivalence $f_{2}\colon N_{2}\to N_{1}$ with
$\tau(f_{2})=y$. By Proposition 2.30 we have $\tau(f_{1}\circ
f_{2})=\tau(f_{1})+(f_{1})_{*}(\tau(f_{2}))=\tau(f_{1})+\tau(f_{2})=\tau(f_{1})+y=x$
(where $(f_{1})_{*}=\operatorname{Id}$ because we used $f_{1}$ to identify
$\pi_{1}(N_{1})$ with $G$). So for $[N_{2}]\in\mathcal{M}^{h}_{s}(M)$ we get
that $\widetilde{t}_{M}([N_{2}])$ is the orbit of $x$, showing that
$\operatorname{Im}\widetilde{t}_{M}\supseteq(\varrho\circ\pi)^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)$.
∎

In particular we proved the following, which is immediate from Theorem 4.11.

######  Theorem 4.16.

If 1.1 are satisfied, then there are bijections

$\displaystyle\widetilde{t}^{\prime}_{M}$
$\displaystyle\colon\mathcal{M}^{\operatorname{hCob}}_{s}(M)\xrightarrow{\cong}q(\mathcal{I}_{n}(G,w))$
$\displaystyle\widetilde{t}_{M}$
$\displaystyle\colon\mathcal{M}^{h}_{s}(M)\xrightarrow{\cong}(\varrho\circ\pi)^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)$
$\displaystyle\widetilde{u}_{M}$
$\displaystyle\colon\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)\xrightarrow{\cong}\varrho^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M).$

With extra input from the map $\psi$, the latter two statements become
cleaner.

######  Corollary 4.17.

If 1.1 are satisfied and $\psi$ is surjective, then there are bijections
$\mathcal{M}^{h}_{s}(M)\cong\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)$ and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)\cong\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))/\operatorname{hAut}(M)$.

###### Proof.

By Corollary 3.9 (b) we have
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))=\operatorname{Im}\psi\subseteq\varrho^{-1}(\operatorname{Im}\sigma_{s})$.
Therefore
$\varrho^{-1}(\operatorname{Im}\sigma_{s})=\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
and
$(\varrho\circ\pi)^{-1}(\operatorname{Im}\sigma_{s})=\mathcal{J}_{n}(G,w)$. ∎

######  Remark 4.18.

If $\tau(g)\in\mathcal{I}_{n}(G,w)$ for every $g\in\operatorname{hAut}(M)$,
then the proof of Theorem 4.11 (a) shows that $\mathcal{I}_{n}(G,w)$ is also
an invariant subset under the action of $\operatorname{hAut}(M)$, and
$q(\mathcal{I}_{n}(G,w))=\mathcal{I}_{n}(G,w)/\operatorname{hAut}(M)$. It also
means that $\pi(\tau(g))=0$ for every $g$, so $\operatorname{hAut}(M)$ acts on
the group $\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$ via automorphisms.

As in Remark 4.9, if $\tau(g)=0$ for every $g$, then $\operatorname{hAut}(M)$
acts on the groups $\mathcal{I}_{n}(G,w)$ and $\mathcal{J}_{n}(G,w)$ via
automorphisms. And if $\pi_{1}(g)=\operatorname{Id}_{G}$ for every $g$, then
$\mathcal{J}_{n}(G,w)/\operatorname{hAut}(M)$ and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))/\operatorname{hAut}(M)$ are
quotient groups of $\mathcal{J}_{n}(G,w)$ and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$.

Now we consider some corollaries of Theorems 4.11 and 4.16.

######  Definition 4.19.

Let $M$ be a closed, $\operatorname{CAT}$ $n$-manifold with $\pi_{1}(M)\cong
G$ and orientation character $w\colon G\rightarrow\left\\{\pm 1\right\\}$. We
define the subsets

$\displaystyle T(M)$ $\displaystyle=\left\\{\tau(g)\mid
g\in\operatorname{hAut}(M)\right\\}\subseteq\mathcal{J}_{n}(G,w)$
$\displaystyle U(M)$ $\displaystyle=\left\\{\pi(\tau(g))\mid
g\in\operatorname{hAut}(M)\right\\}\subseteq\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$

######  Proposition 4.20.

Let $M$ be a closed, $\operatorname{CAT}$ $n$-manifold satisfying 1.1. Let
$G=\pi_{1}(M)$ with orientation character $w\colon G\rightarrow\left\\{\pm
1\right\\}$. Then $|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$ if and only
if $\mathcal{I}_{n}(G,w)\setminus T(M)$ is nonempty.

###### Proof.

Note that $T(M)=t_{M}(M)\subseteq\mathcal{J}_{n}(G,w)$ is the orbit of $0$
under the action of $\operatorname{hAut}(M)$. So $q(\mathcal{I}_{n}(G,w))$
contains more than one element if and only if $\mathcal{I}_{n}(G,w)\setminus
T(M)$ is nonempty. By Theorem 4.16 this is equivalent to
$\mathcal{M}^{\operatorname{hCob}}_{s}(M)$ containing more than one element. ∎

######  Proposition 4.21.

Let $M$ be a closed, $\operatorname{CAT}$ $n$-manifold satisfying 1.1. Let
$G=\pi_{1}(M)$ with orientation character $w\colon G\rightarrow\left\\{\pm
1\right\\}$. Then $|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|>1$ if and only
if $\varrho^{-1}(\operatorname{Im}\sigma_{s})\setminus U(M)$ is nonempty. In
particular, it is a sufficient condition that
$\operatorname{Im}(\psi)\setminus U(M)$ is nonempty.

###### Proof.

Again $U(M)=u_{M}(M)\subseteq\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w))$
is the orbit of $0$ under the action of $\operatorname{hAut}(M)$. So
$\varrho^{-1}(\operatorname{Im}\sigma_{s})/\operatorname{hAut}(M)$ contains
more than one element if and only if
$\varrho^{-1}(\operatorname{Im}\sigma_{s})\setminus U(M)$ is nonempty. By
Theorem 4.16 this is equivalent to
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$ containing more than one element.
Finally, by Corollary 3.9(b) we have
$\varrho^{-1}(\operatorname{Im}\sigma_{s})\setminus
U(M)\supseteq\operatorname{Im}(\psi)\setminus U(M)$. ∎

######  Proposition 4.22.

Let $M$ be a closed, $\operatorname{CAT}$ $n$-manifold satisfying 1.1. Then
the following are equivalent:

  1. (i)

$|\mathcal{M}^{h}_{s}(M)|>1$.

  2. (ii)

Either $|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$ or
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|>1$.

  3. (iii)

$(\varrho\circ\pi)^{-1}(\operatorname{Im}\sigma_{s})\setminus T(M)$ is
nonempty.

###### Proof.

$\eqref{item:prop-he-not-she-equiv-i}\Leftrightarrow\eqref{item:prop-he-not-
she-equiv-ii}$. By Theorem 4.11 (c) and (d) we have
$|\mathcal{M}^{h}_{s}(M)|=1$ if and only if
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|=1$ and
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|=1$.

$\eqref{item:prop-he-not-she-equiv-i}\Leftrightarrow\eqref{item:prop-he-not-
she-equiv-iii}$. Follows from the fact that $T(M)$ is the orbit of $0$ and
Theorem 4.16. ∎

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

In this part we consider $S^{1}\times L$, the product of the circle with a
lens space $L$. We will prove Theorems B, E, LABEL:, and F about the simple
homotopy manifold sets of $S^{1}\times L$. As described in Sections 1.2 and
1.3, we need to study the involution on $\operatorname{Wh}(C_{\infty}\times
C_{m})$ and the group $\operatorname{hAut}(S^{1}\times L)$. Section 5 contains
our results about the groups $\mathcal{J}_{n}(C_{\infty}\times C_{m})$,
$\mathcal{I}_{n}(C_{\infty}\times C_{m})$ and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))$ (for $n$
even), which rely on the results of Part IV. In Section 6, we prove that all
homotopy automorphisms of $S^{1}\times L$ are simple, and determine the
automorphisms of $C_{\infty}\times C_{m}$ induced by them. In Section 7, we
combine these results to prove Theorems B, E, LABEL:, and F.

###  5\. The involution on $\operatorname{Wh}(C_{\infty}\times C_{m})$

In order to understand its involution, we first consider a direct sum
decomposition of the Whitehead group $\operatorname{Wh}(C_{\infty}\times G)$.
This decomposition is derived from the fundamental theorem for
$K_{1}(\mathbb{Z}G[t,t^{-1}])$ [Ba68]; see also [Ra86], [We13, III.3.6].
Whilst this theorem appeared for the first time in Bass’ book, the paper of
Bass-Heller-Swan [BHS64, Theorem 2’] is often mentioned in conjunction with
it, as an early version which contained several key ideas, and the theorem for
left regular rings, appeared there. See [Ba68, pXV] for further discussion.

We will use the version given by Ranicki [Ra86]. We start by defining the
terms appearing in the decomposition in Section 5.1. Section 5.2 contains the
decomposition of $\operatorname{Wh}(C_{\infty}\times C_{m})$ and the induced
decompositions of $\mathcal{J}_{n}(C_{\infty}\times C_{m})$,
$\mathcal{I}_{n}(C_{\infty}\times C_{m})$ and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))$. These
will be combined with the results of Part IV in Section 5.3, to prove Theorems
5.14, 5.15, LABEL:, and 5.17, which will be the key algebraic ingredients in
the proofs of Theorems B, E, LABEL:, and F respectively.

####  5.1. The $\widetilde{K}_{0}$ and $NK_{1}$ groups

First we consider the $K_{0}$ groups. For a ring $R$, we will define the
abelian group $K_{0}(R)$, and, for a group $G$, the reduced group
$\widetilde{K}_{0}(\mathbb{Z}G)$. A convenient reference for this material is
[We13]. In Part IV, we will study the involution on
$\widetilde{K}_{0}(\mathbb{Z}G)$ in more detail when $G$ is a finite cyclic
group.

######  Definition 5.1.

For a ring $R$ let $P(R)$ denote the set of isomorphism classes of (finitely
generated left) projective $R$-modules, which is a monoid under direct sum.
Define $K_{0}(R)$ to be the Grothendieck group of this monoid, i.e. the
abelian group generated by symbols $[P]$, for every $P\in P(R)$, subject to
the relations $[P_{1}\oplus P_{2}]=[P_{1}]+[P_{2}]$ for $P_{1},P_{2}\in P(R)$.

The assignment $R\mapsto K_{0}(R)$ defines a functor from the category of
rings to the category of abelian groups. If $f\colon R\to S$ is a ring
homomorphism, then $K_{0}(f)\colon K_{0}(R)\to K_{0}(S)$ is the map induced by
extension of scalars $P\mapsto f_{\\#}(P):=S\otimes_{R}P$ (see Definition
2.20) for $P\in P(R)$.

######  Definition 5.2.

For a group $G$, let

$\widetilde{K}_{0}(\mathbb{Z}G)\coloneqq
K_{0}(\mathbb{Z}G)/K_{0}(\mathbb{Z}),$

where the map $K_{0}(\mathbb{Z})\to K_{0}(\mathbb{Z}G)$ is induced by the
inclusion $i\colon\mathbb{Z}\rightarrow\mathbb{Z}G$.

######  Definition 5.3.

For a group $G$ and a (finitely generated) projective $\mathbb{Z}G$-module
$P$, the rank of $P$ is defined to be

$\operatorname{rk}(P):=\operatorname{rk}_{\mathbb{Z}}(\varepsilon_{\\#}(P))$

where $\varepsilon\colon\mathbb{Z}G\to\mathbb{Z}$ denotes the augmentation map
and $\operatorname{rk}_{\mathbb{Z}}$ denotes the rank of a free
$\mathbb{Z}$-module.

We have that $K_{0}(\mathbb{Z})\cong\mathbb{Z}$ and, for $P\in
P(\mathbb{Z}G)$, the rank $\operatorname{rk}(P)$ coincides the the image of
$[P]$ under the composition $K_{0}(\mathbb{Z}G)\to
K_{0}(\mathbb{Z})\cong\mathbb{Z}$ induced by $\varepsilon$. Since
$\varepsilon\circ i=\operatorname{Id}_{\mathbb{Z}}$, we have a splitting of
abelian groups
$K_{0}(\mathbb{Z}G)\cong\mathbb{Z}\oplus\widetilde{K}_{0}(\mathbb{Z}G)$ given
by $[P]\mapsto(\operatorname{rk}(P),[P])$. In particular,
$\widetilde{K}_{0}(\mathbb{Z}G)$ is the set of equivalence classes of
projective $\mathbb{Z}G$-modules $P$ where $P\sim Q$ if
$P\oplus\mathbb{Z}G^{i}\cong Q\oplus\mathbb{Z}G^{j}$ for some $i,j\geq 0$.

For $R$ a ring with involution, we can define a natural involution on
$K_{0}(R)$. If $P\in P(R)$, then $P^{*}\in P(R)$ (see Definition 2.22) since
$P\oplus Q\cong R^{n}$ implies that $P^{*}\oplus Q^{*}\cong R^{n}$.

######  Definition 5.4.

The standard involution on $K_{0}(R)$ is given by $[P]\mapsto-[P^{*}]$. This
map preserves the rank of projective $\mathbb{Z}G$-modules and so induces an
involution on $\widetilde{K}_{0}(\mathbb{Z}G)$.

Next we define the Nil groups $NK_{1}(R)$ and recall some of their properties.

######  Definition 5.5.

For a ring $R$, let

$NK_{1}(R)\coloneqq\operatorname{coker}(K_{1}(R)\rightarrow K_{1}(R[t])),$

where the map $K_{1}(R)\rightarrow K_{1}(R[t])$ is induced by the inclusion
$R\rightarrow R[t]$.

Note that $NK_{1}$ is a functor from the category of rings to the category of
abelian groups.

If $R$ (and hence $R[t]$) is a ring with involution, then $K_{1}(R[t])$ also
has a natural involution (see Section 2.2). This involution preserves the
image of $K_{1}(R)\rightarrow K_{1}(R[t])$, and hence induces an involution on
$NK_{1}(R)$, which we will denote by $x\mapsto\overline{x}$.

######  Definition 5.6.

We equip $NK_{1}(R)^{2}=NK_{1}(R)\oplus NK_{1}(R)$ with the involution
$(x,y)\mapsto(\overline{y},\overline{x})$.

We will need the following result of Farrell [Fa77] (note that $\mathrm{Nil}$,
which is also written as $\mathrm{Nil}_{0}$, coincides with $NK_{1}$ by [Ba68,
Chapter XII; 7.4(a)]).

######  Theorem 5.7 (Farrell).

For any ring $R$, if $NK_{1}(R)\neq 0$, then $NK_{1}(R)$ is not finitely
generated.

The following can be proven by combining results of Bass-Murthy [BM67], Martin
[Ma75], and Weibel [We09].

######  Theorem 5.8.

Let $m\geq 1$. Then $NK_{1}(\mathbb{Z}C_{m})=0$ if and only if $m$ is square-
free.

###### Proof.

If $m$ is square-free, then $NK_{1}(\mathbb{Z}C_{m})=0$ by [BM67, Theorem 10.8
(d)]. If $n\mid m$ and $NK_{1}(\mathbb{Z}C_{n})\neq 0$, then
$NK_{1}(\mathbb{Z}C_{m})\neq 0$ (by, for example, [Ma75, Theorem 3.6]). It
remains to show that $NK_{1}(\mathbb{Z}C_{p^{2}})\neq 0$ for all primes $p$.
This was achieved in the case where $p$ is odd in [Ma75, Theorem B] and in the
case $p=2$ in [We09, Theorem 1.4]. ∎

####  5.2. Bass-Heller-Swan decomposition of
$\operatorname{Wh}(C_{\infty}\times G)$

By [Ra86, p.329, p.357] we have the following theorem.

######  Theorem 5.9 (Bass-Heller-Swan decomposition).

Let $G$ be a group. Then there is an isomorphism of $\mathbb{Z}C_{2}$-modules,
which is natural in $G$:

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

where the $\mathbb{Z}C_{2}$-module structure of each component is determined
by the involution defined in Section 2.2, Definition 5.4 and Definition 5.6
respectively.

######  Proposition 5.10.

Let $G$ be a group. The decomposition of Theorem 5.9 restricts to the
following isomorphisms:

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

where $NK_{1}(\mathbb{Z}G)$ is embedded into $NK_{1}(\mathbb{Z}G)^{2}$ by the
map $x\mapsto(x,-(-1)^{n}\overline{x})$.

###### Proof.

Since Theorem 5.9 gives a decomposition of $\mathbb{Z}C_{2}$-modules, both
$\mathcal{I}_{n}(C_{\infty}\times G)$ and $\mathcal{J}_{n}(C_{\infty}\times
G)$ are decomposed into the corresponding subgroups of the components on the
right-hand side. If $(x,y)\in NK_{1}(\mathbb{Z}G)^{2}$, then
$\overline{(x,y)}=(\overline{y},\overline{x})$, so
$(x,y)=-(-1)^{n}\overline{(x,y)}$ holds if and only if
$y=-(-1)^{n}\overline{x}$, so

$\\{(x,y)\in
NK_{1}(\mathbb{Z}G)^{2}\mid(x,y)=-(-1)^{n}\overline{(x,y)}\\}=\\{(x,-(-1)^{n}\overline{x})\mid
x\in NK_{1}(\mathbb{Z}G)\\}\cong NK_{1}(\mathbb{Z}G).$

Similarly, we have
$(x,y)-(-1)^{n}\overline{(x,y)}=(x-(-1)^{n}\overline{y},-(-1)^{n}\overline{x-(-1)^{n}\overline{y}})$,
therefore

$\\{(x,y)-(-1)^{n}\overline{(x,y)}\mid(x,y)\in
NK_{1}(\mathbb{Z}G)^{2}\\}=\\{(x,-(-1)^{n}\overline{x})\mid x\in
NK_{1}(\mathbb{Z}G)\\}\cong NK_{1}(\mathbb{Z}G).\qed$

We immediately get the following corollary; see also [Ra86, p.358].

######  Corollary 5.11.

Let $G$ be a group. The decomposition of Theorem 5.9 induces an isomorphism

$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
G))\cong\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))\oplus\widehat{H}^{n+1}(C_{2};\widetilde{K}_{0}(\mathbb{Z}G)).$

####  5.3. The groups $\mathcal{J}_{n}(C_{\infty}\times C_{m})$,
$\mathcal{I}_{n}(C_{\infty}\times C_{m})$, and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))$

In this section we prove, using results from Part IV, our main results about
the groups $\mathcal{J}_{n}(C_{\infty}\times C_{m})$,
$\mathcal{I}_{n}(C_{\infty}\times C_{m})$, and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))$ for $n$
even.

First note that by Proposition 5.10 and Theorem 5.7, for any group $G$ and any
$n$, if $NK_{1}(\mathbb{Z}G)\neq 0$, then $\mathcal{J}_{n}(C_{\infty}\times
G)$ and $\mathcal{I}_{n}(C_{\infty}\times G)$ are not finitely generated. If
$G$ is finite and $n$ is even, then we also have the following lemma.

######  Lemma 5.12.

Suppose that $n$ is even and $G$ is a finite group. Then the following are
equivalent:

  1. (i)

$|\mathcal{J}_{n}(C_{\infty}\times G)|<\infty$

  2. (ii)

$|\mathcal{I}_{n}(C_{\infty}\times G)|<\infty$

  3. (iii)

$NK_{1}(\mathbb{Z}G)=0$.

###### Proof.

Let $SK_{1}(\mathbb{Z}G)=\ker(K_{1}(\mathbb{Z}G)\to K_{1}(\mathbb{Q}G))$ where
the map is induced by inclusion $\mathbb{Z}G\subseteq\mathbb{Q}G$. It was
shown by Wall [Wa74, Proposition 6.5] (see also [Ol88, Theorem 7.4]) that
$SK_{1}(\mathbb{Z}G)$ is isomorphic to the torsion subgroup of
$\operatorname{Wh}(G)$. Let
$\operatorname{Wh}^{\prime}(G)=\operatorname{Wh}(G)/SK_{1}(\mathbb{Z}G)$ be
the free part. The standard involution on $\operatorname{Wh}(G)$ induces an
involution on $\operatorname{Wh}^{\prime}(G)$, and it was shown by Wall (see
[Ol88, Corollary 7.5]) that this induced involution is the identity.

It follows that if $x=-\overline{x}\in\operatorname{Wh}(G)$, then $x$ maps to
$0\in\operatorname{Wh}^{\prime}(G)$, so $x\in SK_{1}(\mathbb{Z}G)$. Hence
$\mathcal{I}_{n}(G)\leq\mathcal{J}_{n}(G)\leq SK_{1}(\mathbb{Z}G)$. If $G$ is
finite, then $SK_{1}(\mathbb{Z}G)$ is finite [Wa74] and
$\widetilde{K}_{0}(\mathbb{Z}G)$ is finite [Sw60] (see also Proposition 13.5
(ii)). Therefore the lemma follows from Proposition 5.10 and Theorem 5.7. ∎

######  Proposition 5.13.

Suppose that $n$ is even. Then for every $m\geq 2$ we have
$\mathcal{J}_{n}(C_{m})=0$, and hence
$\mathcal{I}_{n}(C_{m})\cong\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{m}))=0$.

###### Proof.

By Proposition 2.9, $\operatorname{Wh}(C_{m})$ is a finitely generated free
abelian group. By [Ba74, Proposition 4.2], the involution acts trivially on
$\operatorname{Wh}(G)$ for every finite abelian group $G$. This implies that
$\mathcal{J}_{n}(C_{m})=\\{x\in\operatorname{Wh}(C_{m})\mid x=-x\\}=0$. ∎

The next three theorems are the main results of Section 5. They will be
established as a consequence of Theorems 16.1, 16.2 and 16.3, which will be
proven in Section 16 (which is in Part IV). These results were stated in the
introduction as Theorems 1.13, 1.14 and 1.15 respectively.

######  Theorem 5.14.

Suppose that $n$ is even and $m\geq 2$ is an integer.

  1. (i)

$|\mathcal{J}_{n}(C_{\infty}\times C_{m})|=1$ if and only if
$m\in\\{2,3,5,6,7,10,11,13,14,17,19\\}$;

  2. (ii)

$|\mathcal{J}_{n}(C_{\infty}\times C_{m})|=\infty$ if and only if $m$ is not
square-free;

  3. (iii)

$|\mathcal{J}_{n}(C_{\infty}\times C_{m})|\to\infty$ super-exponentially in
$m$.

######  Theorem 5.15.

Suppose that $n$ is even and $m\geq 2$ is an integer.

  1. (i)

$|\mathcal{I}_{n}(C_{\infty}\times C_{m})|=1$ if and only if
$m\in\\{2,3,5,6,7,10,11,13,14,15,17,19,29\\}$;

  2. (ii)

$|\mathcal{I}_{n}(C_{\infty}\times C_{m})|=\infty$ if and only if $m$ is not
square-free;

  3. (iii)

$|\mathcal{I}_{n}(C_{\infty}\times C_{m})|\to\infty$ super-exponentially in
$m$.

######  Remark 5.16.

To write Theorem 5.14 (i) another way, the $m\geq 2$ for which
$|\mathcal{J}_{n}(C_{\infty}\times C_{m})|=1$ are as follows:

$m=\begin{cases}\text{$p$},&\text{where $p\leq 19$ is prime}\\\
\text{$2p$},&\text{where $p\leq 7$ is an odd prime}.\end{cases}$

Furthermore, Theorem 5.15 (i) states that $|\mathcal{I}_{n}(C_{\infty}\times
C_{m})|=1$ if and only if $|\mathcal{J}_{n}(C_{\infty}\times C_{m})|=1$ or
$m\in\\{15,29\\}$.

###### Proof of Theorems 5.14 and 5.15.

First note that Theorem 5.14 (ii) and Theorem 5.15 (ii) immediately follow
from Theorem 5.8 and Lemma 5.12. For the proofs of (i) and (iii), we can
therefore restrict to the case where $m$ is square-free, when
$NK_{1}(\mathbb{Z}C_{m})=0$. By Propositions 5.10 and 5.13 we then have:

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

So the results follow from Theorems 16.1 and 16.2, which will be proved in
Section 16. ∎

######  Theorem 5.17.

Suppose that $n$ is even and $m\geq 2$ is an integer.

  1. (i)

$|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))|=1$ for
infinitely many $m$.

  2. (ii)

$|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times C_{m}))|<\infty$
for every $m$.

  3. (iii)

$\displaystyle\sup_{l\leq
m}|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{l}))|\to\infty$ exponentially in $m$.

Note that (i) and (iii) imply that

$\liminf_{m\to\infty}|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))|=1,\quad\limsup_{m\to\infty}|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))|=\infty.$

###### Proof.

It follows from Corollary 5.11 and Proposition 5.13 that

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

Hence parts (i) and (iii) follows from Theorem 16.3, which will be proved in
Section 16. Part (ii) follows from the fact that
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$ is finite [Sw60] (see also Proposition
13.5 (ii)). ∎

###  6\. Homotopy automorphisms of $S^{1}\times L$

Fix $k,m\geq 2$ and $q_{1},\ldots,q_{k}$ such that $\gcd(m,q_{j})=1$. Let
$L=L_{2k-1}(m;q_{1},\ldots,q_{k})$ be a $(2k-1)$-dimensional lens space with
$\pi_{1}(L)\cong C_{m}$. We will be interested in the product $S^{1}\times L$
and its homotopy automorphisms. Fix basepoints of $S^{1}$ and $L$ and let
$i_{1}\colon S^{1}\rightarrow S^{1}\times L$ and $i_{2}\colon L\rightarrow
S^{1}\times L$ denote the standard embeddings, we will identify $S^{1}$ and
$L$ with their images in $S^{1}\times L$. Let $G=C_{\infty}\times
C_{m}\cong\pi_{1}(S^{1}\times L)$.

####  6.1. The automorphism induced on the fundamental group

Our first goal is to determine which automorphisms of $G$ can be realised by
homotopy automorphisms of $S^{1}\times L$, i.e. to describe the image of the
map $\pi_{1}\colon\operatorname{hAut}(S^{1}\times
L)\rightarrow\operatorname{Aut}(G)$. Note that the automorphisms of
$G=C_{\infty}\times C_{m}$ can be expressed as matrices of the form
$\left(\begin{smallmatrix}a&b\\\ 0&c\end{smallmatrix}\right)$, where
$a\in\\{\pm 1\\}$, $b\in C_{m}$ and $c\in
C_{m}^{\times}\cong\operatorname{Aut}(C_{m})$.

######  Lemma 6.1 ([Co73, Statement 29.5]).

Let $c\in C_{m}^{\times}$. There is a homotopy automorphism $f\colon
L\rightarrow L$ such that $\pi_{1}(f)=c$ if and only if $c^{k}\equiv\pm
1\,\,\text{mod}\,m$.

######  Lemma 6.2.

  1. (a)

There is a diffeomorphism $f\in\operatorname{Diff}(S^{1}\times L)$ such that
$\pi_{1}(f)=\left(\begin{smallmatrix}-1&0\\\ 0&1\end{smallmatrix}\right)$.

  2. (b)

There is a diffeomorphism $f\in\operatorname{Diff}(S^{1}\times L)$ such that
$\pi_{1}(f)=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$.

###### Proof.

(a) Take $f=r\times\operatorname{Id}_{L}$ for an orientation-reversing
diffeomorphism $r\colon S^{1}\rightarrow S^{1}$.

(b) Let
$S^{2k-1}=\\{(z_{1},\ldots,z_{k})\mid\sum|z_{j}|^{2}=1\\}\subseteq\mathbb{C}^{k}$.
Recall that $L=L_{2k-1}(m;q_{1},\ldots,q_{k})$ is the quotient of $S^{2k-1}$
by the $C_{m}$-action generated by
$(z_{1},\ldots,z_{k})\mapsto(\zeta^{q_{1}}z_{1},\ldots,\zeta^{q_{k}}z_{k})$,
where $\zeta=e^{2\pi i/m}$. Define $\phi\colon S^{1}\to\operatorname{Diff}(L)$
by $e^{2\pi
it}\mapsto([z_{1},\ldots,z_{k}]\mapsto[\zeta^{q_{1}t}z_{1},\ldots,\zeta^{q_{k}t}z_{k}])$.
Then we can define a suitable diffeomorphism $f$ by $f(x,y)=(x,\phi(x)(y))$. ∎

######  Lemma 6.3.

For every homotopy automorphism $f\colon S^{1}\times L\rightarrow S^{1}\times
L$ there is a homotopy automorphism $\bar{f}_{2}\colon L\rightarrow L$ such
that $f\circ i_{2}\simeq i_{2}\circ\bar{f}_{2}\colon L\rightarrow S^{1}\times
L$.

###### Proof.

Since $\pi_{1}(L)\cong C_{m}$ is the torsion subgroup of $\pi_{1}(S^{1}\times
L)\cong C_{\infty}\times C_{m}$, the restriction of $\pi_{1}(f)$ is an
isomorphism $\pi_{1}(L)\rightarrow\pi_{1}(L)$, and hence
$\operatorname{Im}\pi_{1}(f\circ i_{2})=\operatorname{Im}\pi_{1}(\rho)$ where
the covering $\rho\colon\mathbb{R}\times L\rightarrow S^{1}\times L$ is the
product of the universal covering of $S^{1}$ and $\operatorname{Id}_{L}$. By
the lifting criterion [Ha02, Proposition 1.33], $f\circ i_{2}$ has a lift
$\widetilde{f}_{2}\colon L\rightarrow\mathbb{R}\times L$ such that $f\circ
i_{2}=\rho\circ\widetilde{f}_{2}$. Let $\pi\colon\mathbb{R}\times L\rightarrow
L$ denote the projection. Since $\mathbb{R}$ is contractible, the universal
covering map $\mathbb{R}\to S^{1}$ is null-homotopic. It follows that
$\rho\simeq i_{2}\circ\pi$. Thus

$f\circ i_{2}=\rho\circ\widetilde{f}_{2}\simeq
i_{2}\circ\pi\circ\widetilde{f}_{2}=i_{2}\circ\bar{f}_{2},$

where by definition $\bar{f}_{2}=\pi\circ\widetilde{f}_{2}\colon L\rightarrow
L$.

This implies that $\pi_{1}(\bar{f}_{2})$ is the restriction of $\pi_{1}(f)$,
so it is an isomorphism. Since $\pi_{j}(i_{2})$ is an isomorphism for $j>1$
and $f$ is a homotopy automorphism (hence $\pi_{j}(f)$ is an isomorphism),
$\pi_{j}(\bar{f}_{2})$ is also an isomorphism. Therefore $\bar{f}_{2}$ is a
homotopy automorphism too. ∎

######  Definition 6.4.

Let $A_{2k}(m)\leq\operatorname{Aut}(G)$ denote the subgroup of matrices
$\left(\begin{smallmatrix}a&b\\\ 0&c\end{smallmatrix}\right)$ such that
$c^{k}\equiv\pm 1$ mod $m$.

######  Theorem 6.5.

We have $\operatorname{Im}(\pi_{1}\colon\operatorname{hAut}(S^{1}\times
L)\rightarrow\operatorname{Aut}(G))=A_{2k}(m)$.

###### Proof.

Suppose that $f\colon S^{1}\times L\rightarrow S^{1}\times L$ is a homotopy
automorphism with $\pi_{1}(f)=\left(\begin{smallmatrix}a&b\\\
0&c\end{smallmatrix}\right)$. By Lemma 6.3 there is a homotopy automorphism
$\bar{f}_{2}\colon L\rightarrow L$ such that $f\circ i_{2}\simeq
i_{2}\circ\bar{f}_{2}$. This implies that $\pi_{1}(\bar{f}_{2})=c$, so by
Lemma 6.1 $c^{k}\equiv\pm 1$ mod $m$. Therefore $\pi_{1}(f)\in A_{2k}(m)$.

Let $c\in C_{m}^{\times}$ be such that $c^{k}\equiv\pm 1$ mod $m$. By Lemma
6.1 there is a homotopy automorphism $f\colon L\rightarrow L$ such that
$\pi_{1}(f)=c$. Then $\operatorname{Id}_{S^{1}}\times f$ is a homotopy
automorphism of $S^{1}\times L$ such that
$\pi_{1}(\operatorname{Id}_{S^{1}}\times f)=\left(\begin{smallmatrix}1&0\\\
0&c\end{smallmatrix}\right)$. Matrices of this form, together with the
matrices realised in Lemma 6.2, generate $A_{2k}(m)$. Therefore
$A_{2k}(m)\subseteq\operatorname{Im}(\pi_{1}\colon\operatorname{hAut}(S^{1}\times
L)\rightarrow\operatorname{Aut}(G))$. ∎

####  6.2. The Whitehead torsion

Next we prove that every homotopy automorphism of $S^{1}\times L$ is simple.

######  Theorem 6.6.

If $f\in\operatorname{hAut}(S^{1}\times L)$, then $\tau(f)=0$.

######  Remark 6.7.

When $m$ is odd, this also follows from results of Khan and Hsiang-Jahren. By
[Kh17, Corollary 6.2], $\operatorname{hAut}(S^{1}\times L)$ is generated by
maps of the form $f\times\operatorname{Id}_{L}$,
$\operatorname{Id}_{S^{1}}\times g$ and $r_{h}$, where
$f\in\operatorname{hAut}(S^{1})$, $g\in\operatorname{hAut}(L)$, $h\colon
S^{1}\rightarrow\operatorname{Map}(L)$ and $r_{h}(x,y)=(x,h(x)(y))$. By
Proposition 2.31, generators of the first two types have vanishing Whitehead
torsion. If $m$ is odd, then it follows from [HJ83, Proposition 3.1] that
$\tau(r_{h})=0$ (see also [Kh17, Corollary 6.4]).

The proof consists of two parts, which we will prove in the following two
lemmas. For a CW complex $X$ and an integer $n\geq 0$, let
$\operatorname{sk}_{n}(X)$ denote the $n$-skeleton of $X$. We will endow
$S^{1}\times L$ with a standard CW decomposition, that we recall during the
next proof.

######  Lemma 6.8.

If $g\colon S^{1}\times L\rightarrow S^{1}\times L$ is a homotopy automorphism
such that $g\big{|}_{L\cup\operatorname{sk}_{2k-2}(S^{1}\times
L)}=\operatorname{Id}$, then $\tau(g)=0$.

###### Proof.

We will denote the generators of the components of $G$ by $\alpha\in
C_{\infty}$ and $\beta\in C_{m}$. Let
$\Sigma:=\sum_{j=0}^{m-1}\beta^{j}\in\mathbb{Z}C_{m}\subseteq\mathbb{Z}G$ be
the norm element of $\mathbb{Z}C_{m}$.

The lens space $L=L_{2k-1}(m;q_{1},\ldots,q_{k})$ has a CW decomposition with
one cell $E_{j}$ in each dimension $0\leq j\leq 2k-1$ (see e.g. [Co73, §28]).
The corresponding cellular chain complex $(C^{L}_{j},d^{L}_{j})$ with
$\mathbb{Z}C_{m}$ coefficients has $C^{L}_{j}\cong\mathbb{Z}C_{m}$ for every
$j$ and

$d^{L}_{j}=\begin{cases}\Sigma&\text{if $j$ is even}\\\
\beta^{r_{h}}-1&\text{if $j=2h-1$}\end{cases}$

where $r_{h}q_{h}\equiv 1$ mod $m$.

The product $M=S^{1}\times L$ has a CW decomposition containing the cells of
$L$, plus a $(j+1)$-cell $E^{\prime}_{j+1}$, the product of $E_{j}$ and the
$1$-cell of $S^{1}$, for every $0\leq j\leq 2k-1$. Its chain complex
$(C^{M}_{j},d^{M}_{j})$ with $\mathbb{Z}G$ coefficients is the tensor product
of $(C^{L}_{j},d^{L}_{j})$ and the chain complex of $S^{1}$ with
$\mathbb{Z}C_{\infty}$ coefficients,
$\mathbb{Z}C_{\infty}\stackrel{{\scriptstyle\alpha-1}}{{\longrightarrow}}\mathbb{Z}C_{\infty}$.
It has $C^{M}_{0}\cong C^{M}_{2k}\cong\mathbb{Z}G$ and
$C^{M}_{j}\cong\mathbb{Z}G^{2}$ for every $1\leq j\leq 2k-1$ (with
$\mathbb{Z}G\oplus\left\\{0\right\\}$ corresponding to $E_{j}$ and
$\left\\{0\right\\}\oplus\mathbb{Z}G$ corresponding to $E^{\prime}_{j}$). The
differentials are

$\begin{gathered}d^{M}_{1}=\begin{pmatrix}\beta^{r_{1}}-1\\\
\alpha-1\end{pmatrix},\quad d^{M}_{2h-1}=\begin{pmatrix}\beta^{r_{h}}-1&0\\\
\alpha-1&\Sigma\end{pmatrix}\quad\text{for $h>1$},\\\
d^{M}_{2k}=(1-\alpha,\beta^{r_{k}}-1),\quad
d^{M}_{2h}=\begin{pmatrix}\Sigma&0\\\
1-\alpha&\beta^{r_{h}}-1\end{pmatrix}\quad\text{for $h<k$}.\end{gathered}$

We write elements of $\mathbb{Z}G^{j}$ as row vectors, so a homomorphism
$\mathbb{Z}G^{j}\rightarrow\mathbb{Z}G^{h}$ is multiplication on the right by
a $j\times h$ matix.

We can assume that $g$ is cellular, let $g_{j}\colon C^{M}_{j}\rightarrow
C^{M}_{j}$ denote the induced chain map. Since
$g\big{|}_{L\cup\operatorname{sk}_{2k-2}(L\times S^{1})}=\operatorname{Id}$,
we have $g_{j}=\operatorname{Id}$ for every $j\leq 2k-2$ and
$g_{2k-1}\big{|}_{\mathbb{Z}G\oplus\left\\{0\right\\}}=\operatorname{Id}$.

We investigate the map $g_{2k-1}\colon C_{2k-1}^{M}\to C_{2k-1}^{M}$. It is
represented by a $2\times 2$ matrix with entries in $\mathbb{Z}G$, with
respect to the basis corresponding to $E_{2k-1}$ and $E_{2k-1}^{\prime}$.
Since
$g_{2k-1}\big{|}_{\mathbb{Z}G\oplus\left\\{0\right\\}}=\operatorname{Id}$, the
matrix is $\left(\begin{smallmatrix}1&0\\\ x&y\end{smallmatrix}\right)$ for
some $x,y\in\mathbb{Z}G$. To compute $x$ and $y$, we consider the difference
between $g$ and $\operatorname{Id}$ on $E_{2k-1}^{\prime}$. Since
$g\big{|}_{\partial E^{\prime}_{2k-1}}=\operatorname{Id}$, the cell
$E^{\prime}_{2k-1}$ and $g(E^{\prime}_{2k-1})$ together form a map
$S^{2k-1}\rightarrow S^{1}\times L$. This map represents an element of
$\pi_{2k-1}(S^{1}\times L)\cong\pi_{2k-1}(\mathbb{R}\times
S^{2k-1})\cong\mathbb{Z}$, so its homotopy class is an integer multiple of
that of the inclusion $S^{2k-1}\rightarrow\mathbb{R}\times S^{2k-1}$, composed
with the projection $\mathbb{R}\times S^{2k-1}\rightarrow S^{1}\times L$. The
image of the fundamental class of $S^{2k-1}$ under this composition is
represented by the chain
$\Sigma\in\mathbb{Z}G\oplus\left\\{0\right\\}<\mathbb{Z}G^{2}\cong
C^{M}_{2k-1}$. So $E^{\prime}_{2k-1}$ and $g(E^{\prime}_{2k-1})$ differ by
some integer multiple of $\Sigma\in\mathbb{Z}G\oplus\left\\{0\right\\}$.
Therefore (by changing $g$ on $E^{\prime}_{2k-1}$ and $E^{\prime}_{2k}$, to
replace it with another cellular map which is homotopic to it) we can assume
that $y=1$ and $x=a\Sigma$ for some $a\in\mathbb{Z}$. That is,
$g_{2k-1}=\left(\begin{smallmatrix}1&0\\\ a\Sigma&1\end{smallmatrix}\right)$.

Since $(g_{j})$ is a chain map, we have $d^{M}_{2k}\circ g_{2k}=g_{2k-1}\circ
d^{M}_{2k}$. When converted to matrices this yields

$g_{2k}(1-\alpha,\beta^{r_{k}}-1)=(1-\alpha,\beta^{r_{k}}-1)\left(\begin{smallmatrix}1&0\\\
a\Sigma&1\end{smallmatrix}\right)=(1-\alpha,\beta^{r_{k}}-1),$

since $\beta^{r_{k}}\Sigma=\Sigma$. Looking at the first coordinate, we have
$g_{2k}(1-\alpha)=(1-\alpha)$. Multiplication by $(1-\alpha)$ is an injective
map $\mathbb{Z}G\rightarrow\mathbb{Z}G$, so we deduce that $g_{2k}=1$.

Therefore $g_{j}$ is an isomorphism for every $j$. By Lemma 2.17 we can
compute the torsion via the formula $\tau(g)=\sum_{j=0}^{2k}(-1)^{j}[g_{j}]$.
Since $g_{2k-1}$ is an elementary matrix and $g_{j}=\operatorname{Id}$ for
$j\neq 2k-1$, this implies that $\tau(g)=0$. ∎

######  Lemma 6.9.

For every homotopy automorphism $f\colon S^{1}\times L\rightarrow S^{1}\times
L$ there is another homotopy automorphism $g$ such that $\tau(f)=\tau(g)$ and
$g\big{|}_{L\cup\operatorname{sk}_{2k-2}(L\times S^{1})}=\operatorname{Id}$.

###### Proof.

By Lemma 6.3 there is a homotopy automorphism $\bar{f}_{2}\colon L\rightarrow
L$ such that $f\circ i_{2}\simeq i_{2}\circ\bar{f}_{2}$. Let
$\overline{h}\colon L\rightarrow L$ be the homotopy inverse of $\bar{f}_{2}$,
and let $h:=\operatorname{Id}_{S^{1}}\times\overline{h}\colon S^{1}\times
L\rightarrow S^{1}\times L$. Note that $h\circ
i_{2}=i_{2}\circ\overline{h}\colon L\to S^{1}\times L$. By Corollary 2.32
$\tau(h)=0$. Let $f^{\prime}=f\circ h$. Then by Proposition 2.30, we have that
$\tau(f^{\prime})=\tau(f)$. Moreover, combining the above observations we have

$f^{\prime}\circ i_{2}=f\circ h\circ i_{2}=f\circ i_{2}\circ\overline{h}\simeq
i_{2}\circ\bar{f}_{2}\circ\overline{h}\simeq i_{2}.$

This means that after changing $f^{\prime}$ by a homotopy we can assume that
$f^{\prime}\big{|}_{L}=\operatorname{Id}$.

The embedding $i_{1}\colon S^{1}\rightarrow S^{1}\times L$ represents
$(1,0)\in\pi_{1}(S^{1}\times L)\cong C_{\infty}\times C_{m}$. Since
$\pi_{1}(f^{\prime})$ is an automorphism of $C_{\infty}\times C_{m}$,
$\pi_{1}(f^{\prime})^{-1}(1,0)=(a,b)$ for some $a\in\left\\{\pm 1\right\\}$,
$b\in C_{m}$. It follows from Lemma 6.2 that there is a diffeomorphism
$h\in\operatorname{Diff}(S^{1}\times L)$ such that $\pi_{1}(h)(1,0)=(a,b)$.
Let $f^{\prime\prime}=f^{\prime}\circ h$. Then
$\tau(f^{\prime\prime})=\tau(f^{\prime})$ by Proposition 2.30 and Chapman’s
Theorem 2.5. Moreover $\pi_{1}(f^{\prime\prime})(1,0)=(1,0)$, i.e.
$f^{\prime\prime}\circ i_{1}\simeq i_{1}$. The diffeomorphisms constructed in
Lemma 6.2 keep $L$ pointwise fixed, so
$f^{\prime\prime}\big{|}_{L}=\operatorname{Id}$, and after applying a homotopy
we can also assume $f^{\prime\prime}\big{|}_{S^{1}}=\operatorname{Id}$.

Finally, we recursively construct maps $f^{\prime\prime}_{j}$ for every $1\leq
j\leq 2k-2$ such that $f^{\prime\prime}\simeq f^{\prime\prime}_{j}$ rel
$S^{1}\vee L$ and
$f^{\prime\prime}_{j}\big{|}_{L\cup\operatorname{sk}_{j}(L\times
S^{1})}=\operatorname{Id}$. We start with
$f^{\prime\prime}_{1}=f^{\prime\prime}$. If $f^{\prime\prime}_{j-1}$ is
already defined, we consider the single $j$-cell $E^{\prime}_{j}$ in
$S^{1}\times L-L$. Since $f^{\prime\prime}_{j-1}\big{|}_{\partial
E^{\prime}_{j}}=\operatorname{Id}$, the cell $E^{\prime}_{j}$ and
$f^{\prime\prime}_{j-1}(E^{\prime}_{j})$ together form a map $S^{j}\rightarrow
S^{1}\times L$. This map is nullhomotopic (because $\pi_{j}(S^{1}\times
L)\cong\pi_{j}(\mathbb{R}\times S^{2k-1})=0$ for $2\leq j\leq 2k-2$), so
$f^{\prime\prime}_{j-1}\big{|}_{E^{\prime}_{j}}$ is homotopic to
$\operatorname{Id}_{E^{\prime}_{j}}$ rel $\partial E^{\prime}_{j}$. We can
extend this homotopy to a homotopy between $f^{\prime\prime}_{j-1}$ and an
$f^{\prime\prime}_{j}$ with
$f^{\prime\prime}_{j}\big{|}_{L\cup\operatorname{sk}_{j}(L\times
S^{1})}=\operatorname{Id}$. Therefore $f^{\prime\prime}_{j}$ can be defined
for every $1\leq j\leq 2k-2$.

To complete the proof of Lemma 6.9, we take $g=f^{\prime\prime}_{2k-2}\colon
S^{1}\times L\to S^{1}\times L$. ∎

###### Proof of Theorem 6.6.

Consider a homotopy equivalence $f\colon S^{1}\times L\to S^{1}\times L$. By
Lemma 6.9, we can replace $f$ by a homotopy equivalence $g\colon S^{1}\times
L\to S^{1}\times L$ such that $\tau(f)=\tau(g)$ and
$g\big{|}_{L\cup\operatorname{sk}_{2k-2}(L\times S^{1})}=\operatorname{Id}$.
Then $g$ satisfies the hypotheses of Lemma 6.8, and so $\tau(g)=0$. Therefore
$\tau(f)=0$, and $f$ is a simple homotopy equivalence, as desired. ∎

###  7\. The proof of Theorems B, E, LABEL:, and F

We can now combine the results of the previous sections to prove Theorems B,
E, LABEL:, and F. Let $n=2k\geq 4$ be an even integer and fix a category
$\operatorname{CAT}$ satisfying 1.1.

Recall the definition of the sets $\mathcal{M}^{h}_{s}(M)$,
$\mathcal{M}^{\operatorname{hCob}}_{s}(M)$ and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)$ from Definition 4.10. Also recall
the definition of $A_{2k}(m)$ (Definition 6.4):
$A_{2k}(m)\leq\operatorname{Aut}(C_{\infty}\times C_{m})$ denotes the subgroup
of matrices $\left(\begin{smallmatrix}a&b\\\ 0&c\end{smallmatrix}\right)$ such
that $c^{k}\equiv\pm 1$ mod $m$. Note that $A_{2k}(m)$ acts on
$\operatorname{Wh}(C_{\infty}\times C_{m})$ and on the subgroups
$\mathcal{I}_{n}(C_{\infty}\times C_{m})$ and
$\mathcal{J}_{n}(C_{\infty}\times C_{m})$.

######  Proposition 7.1.

Let $m\geq 2$ and $q_{1},\ldots,q_{k}$ such that $\gcd(m,q_{j})=1$. Let
$L=L_{2k-1}(m;q_{1},\ldots,q_{k})$ be an $(n-1)$-dimensional lens space with
$\pi_{1}(L)\cong C_{m}$. Let $G:=C_{\infty}\times
C_{m}\cong\pi_{1}(S^{1}\times L)$. Then there are bijections of pointed sets
$\mathcal{M}^{h}_{s}(S^{1}\times L)\cong\mathcal{J}_{n}(G)/A_{2k}(m)$,
$\mathcal{M}^{\operatorname{hCob}}_{s}(S^{1}\times
L)\cong\mathcal{I}_{n}(G)/A_{2k}(m)$ and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(S^{1}\times
L)\cong\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))/A_{2k}(m)$.

###### Proof.

Firstly note that, since $C_{\infty}\times C_{m}$ is polycyclic, it is a good
group [FQ, Freedman-book-goodgroups], so 1.1 is satisfied. Moreover, since the
map $\psi$ is surjective for $G=C_{\infty}\times C_{m}$, $w\equiv 1$ and $n$
even by Proposition 3.12, the vertical maps in the diagram of Theorem 4.11 (c)
are bijections if $M=S^{1}\times L$. Therefore
$\mathcal{M}^{h}_{s}(S^{1}\times
L)\cong\mathcal{J}_{n}(G)/\operatorname{hAut}(S^{1}\times L)$,
$\mathcal{M}^{\operatorname{hCob}}_{s}(S^{1}\times L)\cong
q(\mathcal{I}_{n}(G))$, and
$\mathcal{M}^{h}_{s,\operatorname{hCob}}(S^{1}\times
L)\cong\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))/\operatorname{hAut}(S^{1}\times
L)$.

By Theorem 6.6, every homotopy automorphism of $S^{1}\times L$ is simple,
therefore the action of $\operatorname{hAut}(S^{1}\times L)$ on
$\operatorname{Wh}(G)$ factors through the action of $\operatorname{Aut}(G)$
(see Definition 4.3 and Remark 4.18). By Theorem 6.5, the image of
$\pi_{1}\colon\operatorname{hAut}(S^{1}\times
L)\rightarrow\operatorname{Aut}(G)$ is $A_{2k}(m)$. In particular, the orbits
of the action of $\operatorname{hAut}(S^{1}\times L)$ and $A_{2k}(m)$ on
$\mathcal{J}_{n}(G)$, $\mathcal{I}_{n}(G)$, and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))$ coincide. This implies that
$\mathcal{J}_{n}(G)/\operatorname{hAut}(S^{1}\times
L)=\mathcal{J}_{n}(G)/A_{2k}(m)$,
$q(\mathcal{I}_{n}(G))=\mathcal{I}_{n}(G)/A_{2k}(m)$ and
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))/\operatorname{hAut}(S^{1}\times
L)=\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))/A_{2k}(m)$, as required. ∎

In particular Proposition 7.1 implies that $|\mathcal{M}^{h}_{s}(S^{1}\times
L)|$, $|\mathcal{M}^{\operatorname{hCob}}_{s}(S^{1}\times L)|$ and
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(S^{1}\times L)|$ are independent of
the choice of the $q_{j}$ and of $\operatorname{CAT}$, and only depend on $n$
and $m$. This proves part (a) of Theorems B, E, LABEL:, and F.

From now on, we will write $M^{n}_{m}$ for $S^{1}\times L$, where $L$ is any
$(n-1)$-dimensional lens space with $\pi_{1}(L)\cong C_{m}$.

######  Lemma 7.2.

Let $n=2k\geq 4$ be an even integer, $m\geq 2$ and $G=C_{\infty}\times C_{m}$.
Then the following hold.

  1. (a)

$|\mathcal{M}^{h}_{s}(M^{n}_{m})|=1$ if and only if $\mathcal{J}_{n}(G)=0$.

  2. (b)

$|\mathcal{M}^{h}_{s}(M^{n}_{m})|=\infty$ if and only if
$|\mathcal{J}_{n}(G)|=\infty$.

  3. (c)

If $\mathcal{M}^{h}_{s}(M^{n}_{m})$ is finite, then
$\frac{|\mathcal{J}_{n}(G)|}{2m^{2}}<|\mathcal{M}^{h}_{s}(M^{n}_{m})|\leq|\mathcal{J}_{n}(G)|$.

  4. (d)

$|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|=1$ if and only if
$\mathcal{I}_{n}(G)=0$.

  5. (e)

$|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|=\infty$ if and only if
$|\mathcal{I}_{n}(G)|=\infty$.

  6. (f)

If $\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})$ is finite, then
$\frac{|\mathcal{I}_{n}(G)|}{2m^{2}}<|\mathcal{M}^{\operatorname{hCob}}_{s}(M^{n}_{m})|\leq|\mathcal{I}_{n}(G)|$.

  7. (g)

$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|=1$ if and only if
$\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))=0$.

  8. (h)

$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|=\infty$ if and only if
$|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))|=\infty$.

  9. (i)

If $\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})$ is finite, then
$\frac{|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))|}{2m^{2}}<|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|\leq|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G))|$.

###### Proof.

(a) By Proposition 7.1,
$|\mathcal{M}^{h}_{s}(M^{n}_{m})|=|\mathcal{J}_{n}(G)/A_{2k}(m)|$. The group
$A_{2k}(m)\leq\operatorname{Aut}(G)$ acts on $\operatorname{Wh}(G)$, and hence
on $\mathcal{J}_{n}(G)$, by automorphisms. So $0\in\mathcal{J}_{n}(G)$ is a
fixed point of this action, and $|\mathcal{J}_{n}(G)/A_{2k}(m)|=1$ if and only
if $\mathcal{J}_{n}(G)=0$.

(b) $\operatorname{Aut}(G)$, and hence its subgroup $A_{2k}(m)$, is finite.
Hence $|\mathcal{J}_{n}(G)/A_{2k}(m)|=\infty$ if and only if
$|\mathcal{J}_{n}(G)|=\infty$.

(c) It is easy to see that
$\frac{|\mathcal{J}_{n}(G)|}{|A_{2k}(m)|}\leq|\mathcal{J}_{n}(G)/A_{2k}(m)|\leq|\mathcal{J}_{n}(G)|$.
We have $|A_{2k}(m)|\leq|\operatorname{Aut}(G)|\leq 2m(m-1)<2m^{2}$, because
elements of $\operatorname{Aut}(G)$ can be represented as matrices of the form
$\left(\begin{smallmatrix}a&b\\\ 0&c\end{smallmatrix}\right)$, where
$a\in\\{\pm 1\\}$, $b\in C_{m}$ and $c\in C_{m}^{\times}$.

The proofs of parts (d), (e), and (f) (resp. parts (g), (h) and (i)) are
entirely analogous to those of parts (a), (b), and (c) respectively, and so
will be omitted for brevity. ∎

######  Theorem 7.3.

Let $n=2k\geq 4$ be an even integer and $m\geq 2$. Then the following hold.

  1. (a)

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

  2. (b)

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

  3. (c)

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

  4. (d)

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

  5. (e)

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

  6. (f)

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

  7. (g)

There are infinitely many $m$ such that
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|=1$ for every $n$.

  8. (h)

$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|$ is finite for every $n$
and $m$.

  9. (i)

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

###### Proof.

(a), (b), (d), (e), (g), and (h) follow immediately from Lemma 7.2 and
Theorems 5.14, 5.15, LABEL:, and 5.17.

(c) For every $m$, it follows from Lemma 7.2 (c) that
$|\mathcal{M}^{h}_{s}(M^{n}_{m})|>\frac{|\mathcal{J}_{n}(C_{\infty}\times
C_{m})|}{2m^{2}}$ for every $n$. By Theorem 5.14 (iii)
$\frac{|\mathcal{J}_{n}(C_{\infty}\times C_{m})|}{2m^{2}}\to\infty$ as
$m\to\infty$. Item (f), can be proved similarly using Theorem 5.15 (iii).

(i) For every $m$, it follows from Lemma 7.2 (i) that
$\inf_{n}|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M^{n}_{m})|>\frac{|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))|}{2m^{2}}$. By Theorem 5.17 (iii), we see that
$\frac{1}{2m^{2}}\sup_{l\leq
m}|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{l}))|\to\infty$ as $m\to\infty$. This implies that $\sup_{l\leq
m}\frac{|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{l}))|}{2l^{2}}\to\infty$ as $m\to\infty$. So the map
$m\mapsto\frac{|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))|}{2m^{2}}$ is unbounded, therefore
$\limsup_{m\to\infty}\frac{|\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(C_{\infty}\times
C_{m}))|}{2m^{2}}=\infty$. ∎

This completes the proof of the remaining parts of Theorems B, E, LABEL:, and
F, subject to the proofs of Theorems 16.1, 16.2, and 16.3 which will be
postponed until Section 16. We can now deduce:

######  Theorem 7.4.

There exists an infinite collection of orientable, $\operatorname{CAT}$
$n$-manifolds that are all homotopy equivalent to one another but are pairwise
not simple homotopy equivalent.

###### Proof.

By Theorem 7.3 (b), if $m$ is not square-free (e.g. if $m=4$), then
$\mathcal{M}^{h}_{s}(M^{n}_{m})$ is infinite. Note that $M^{n}_{m}=S^{1}\times
L$ is orientable, so the same is true for every $n$-manifold homotopy
equivalent to it. Therefore we can get a suitable infinite collection by
choosing a representative from each element of
$\mathcal{M}^{h}_{s}(M^{n}_{m})$. ∎

This completes the proof of Theorem A. Note that this does not depend on the
results postponed to Section 16. In particular, if $m$ is not square-free,
then $|NK_{1}(\mathbb{Z}C_{m})|=\infty$ (Theorem 1.4) which implies
$|\mathcal{J}_{n}(C_{\infty}\times C_{m})|=\infty$ (Lemma 5.12) and so
$|\mathcal{M}^{h}_{s}(M^{n}_{m})|=\infty$ (Lemma 7.2 (b)).

##  Part III Homotopy equivalences of doubles

In this part we prove Theorem G about the Whitehead torsion of a homotopy
equivalence between doubles, and then derive the results announced in Section
1.5 from it. Every manifold will be assumed to be smooth, compact and
connected, but not necessarily closed.

We start by considering the algebraic version of the problem in Section 8. We
consider _split chain complexes_ , which are those that split as the direct
sum of their lower and upper halves, cf. Definitions 8.1 and 8.2. Provided
that they are also based and satisfy a version of Poincaré duality, in Lemma
8.6 we compute the Whitehead torsion of a chain homotopy equivalence between
two such chain complexes in terms of the Whitehead torsion of the restriction
of the chain map to the lower half. We note that the formula contains two
error terms, which are intrinsic to the two chain complexes (given by the
Whitehead torsion of the restriction of the Poincaré duality map).

In Section 9, we recall Wall’s results on thickenings. In Section 10, we
introduce polarised doubles, SP manifolds and the invariant $\tau(M,\varphi)$.
We show that if $(M,\varphi)$ is an SP manifold, then its cellular chain
complex splits and $\tau(M,\varphi)$ is equal to the associated error term. In
Section 11, we combine the previous results to prove Theorem G. In Section 12,
we consider various applications of Theorem G, including Theorems 1.7, H, C,
1.10 and 1.12.

###  8\. Split chain complexes

In this section we define split chain complexes and prove Lemma 8.6 about
computing the Whitehead torison of a chain homotopy equivalence between two
split chain complexes.

Fix some positive integer $n$ and a group $G$ with orientation character
$w\colon G\rightarrow\left\\{\pm 1\right\\}$. Every chain complex $C_{*}$ will
be assumed to consist of finitely generated free, left $\mathbb{Z}G$-modules,
satisfying the condition $C_{i}=0$ for $i<0$ and $i>n$ (and similarly for
cochain complexes).

######  Definition 8.1.

Let $(C_{*},d_{*})$ be a chain complex. We define the chain complex
$(C^{\ell}_{*},d^{\ell}_{*})$ by $C^{\ell}_{i}=C_{i}$ and
$d^{\ell}_{i}=d_{i}\colon C_{i}\to C_{i-1}$ if $i<\frac{n}{2}$ and
$C^{\ell}_{i}=0$ if $i\geq\frac{n}{2}$. Similarly, we define
$(C^{u}_{*},d^{u}_{*})$ by $C^{u}_{i}=C_{i}$ if $i>\frac{n}{2}$,
$d^{u}_{i}=d_{i}$ if $i>\frac{n}{2}+1$, and $C^{u}_{i}=0$ if
$i\leq\frac{n}{2}$.

Let $(C^{*},d^{*})$ be a cochain complex. We define
$((C^{u})^{*},(d^{u})^{*})$ by $(C^{u})^{i}=C^{i}$ and
$(d^{u})^{i}=d^{i}\colon C^{i}\to C^{i+1}$ if $i>\frac{n}{2}$ and
$(C^{u})^{i}=0$ if $i\leq\frac{n}{2}$. Similarly, we define
$((C^{\ell})^{*},(d^{\ell})^{*})$ by $(C^{\ell})^{i}=C^{i}$ if
$i<\frac{n}{2}$, $(d^{\ell})^{i}=d^{i}$ if $i<\frac{n}{2}-1$, and
$(C^{\ell})^{i}=0$ if $i\geq\frac{n}{2}$.

######  Definition 8.2.

We say that a chain complex $(C_{*},d_{*})$ _splits_ if $C_{n/2}=0$ (if $n$ is
even) or $d_{\left\lceil n/2\right\rceil}=0$ (if $n$ is odd). We say that a
cochain complex $(C^{*},d^{*})$ _splits_ if $C^{n/2}=0$ (if $n$ is even) or
$d^{\left\lfloor n/2\right\rfloor}=0$ (if $n$ is odd).

Note that $C_{*}$ (resp. $C^{*}$) splits if and only if $C_{*}\cong
C^{\ell}_{*}\oplus C^{u}_{*}$ (resp.
$C^{*}\cong(C^{u})^{*}\oplus(C^{\ell})^{*}$).

######  Definition 8.3.

Let $C_{*}$ and $D_{*}$ be chain complexes. We define
$\operatorname{chEq}(C_{*},D_{*})\subseteq\operatorname{Hom}(C_{*},D_{*})/{\simeq}$
to be the set of chain homotopy classes of chain homotopy equivalences
$C_{*}\rightarrow D_{*}$.

######  Lemma 8.4.

Suppose that $(C_{*},d^{C}_{*})$ and $(D_{*}d^{D}_{*})$ are split chain
complexes. Then there is a bijection

$\operatorname{chEq}(C_{*},D_{*})\cong\operatorname{chEq}(C^{\ell}_{*},D^{\ell}_{*})\times\operatorname{chEq}(C^{u}_{*},D^{u}_{*}),$

where the projections
$\operatorname{chEq}(C_{*},D_{*})\rightarrow\operatorname{chEq}(C^{\ell}_{*},D^{\ell}_{*})$
and
$\operatorname{chEq}(C_{*},D_{*})\rightarrow\operatorname{chEq}(C^{u}_{*},D^{u}_{*})$
are defined by restriction.

###### Proof.

Since $C_{*}$ and $D_{*}$ split, restrictions define an isomorphism

$\operatorname{Hom}(C_{*},D_{*})\cong\operatorname{Hom}(C^{\ell}_{*},D^{\ell}_{*})\oplus\operatorname{Hom}(C^{u}_{*},D^{u}_{*}).$

Let $f,f^{\prime}\colon C_{*}\rightarrow D_{*}$ be chain maps. The
restrictions of a chain homotopy between $f$ and $f^{\prime}$ are homotopies
between $f\big{|}_{C^{\ell}_{*}}$ and $f^{\prime}\big{|}_{C^{\ell}_{*}}$ and
between $f\big{|}_{C^{u}_{*}}$ and $f^{\prime}\big{|}_{C^{u}_{*}}$,
respectively, because $d^{D}_{\left\lceil n/2\right\rceil}=0$ and
$d^{C}_{\left\lfloor n/2\right\rfloor+1}=0$. Conversely, a pair of homotopies
between $f\big{|}_{C^{\ell}_{*}}$ and $f^{\prime}\big{|}_{C^{\ell}_{*}}$ and
between $f\big{|}_{C^{u}_{*}}$ and $f^{\prime}\big{|}_{C^{u}_{*}}$ can be
combined to a homotopy between $f$ and $f^{\prime}$. Therefore there is an
induced isomorphism
$(\operatorname{Hom}(C_{*},D_{*})/{\simeq})\cong(\operatorname{Hom}(C^{\ell}_{*},D^{\ell}_{*})/{\simeq})\oplus(\operatorname{Hom}(C^{u}_{*},D^{u}_{*})/{\simeq})$
on homotopy classes.

This isomorphism is defined for every pair $(C_{*},D_{*})$ of split chain
complexes, it is compatible with the composition maps (induced by
$\operatorname{Hom}(D_{*},E_{*})\times\operatorname{Hom}(C_{*},D_{*})\to\operatorname{Hom}(C_{*},E_{*})$,
and similarly for $\operatorname{Hom}(C^{\ell}_{*},D^{\ell}_{*})$ and
$\operatorname{Hom}(C^{u}_{*},D^{u}_{*})$) and
$[\operatorname{Id}_{C_{*}}]\in(\operatorname{Hom}(C_{*},C_{*})/{\simeq})$
corresponds to
$([\operatorname{Id}_{C^{\ell}_{*}}],[\operatorname{Id}_{C^{u}_{*}}])$. Since
$\operatorname{chEq}(C_{*},D_{*})$ consists of those elements of
$(\operatorname{Hom}(C_{*},D_{*})/{\simeq})$ which have an inverse in
$(\operatorname{Hom}(D_{*},C_{*})/{\simeq})$ (and similarly for
$\operatorname{chEq}(C^{\ell}_{*},D^{\ell}_{*})$ and
$\operatorname{chEq}(C^{u}_{*},D^{u}_{*})$), this implies that the isomorphism
above restricts to a bijection
$\operatorname{chEq}(C_{*},D_{*})\cong\operatorname{chEq}(C^{\ell}_{*},D^{\ell}_{*})\times\operatorname{chEq}(C^{u}_{*},D^{u}_{*})$.
∎

Recall that the orientation character $w$ determines an involution on
$\mathbb{Z}G$, which allows us to define the dual of a left
$\mathbb{Z}G$-module as another left $\mathbb{Z}G$-module (see Definition
2.22). Note that a chain complex $C_{*}$ splits if and only if the dual
cochain complex $C^{*}$ (defined by $C^{i}=(C_{i})^{*}$) splits.

######  Definition 8.5.

Let $C_{*}$ and $D_{*}$ be chain complexes, $C^{*}$ and $D^{*}$ the dual
cochain complexes, and let $P\colon C^{n-*}\rightarrow C_{*}$ and $Q\colon
D^{n-*}\rightarrow D_{*}$ be chain homotopy equivalences. Then define
$\operatorname{chEq}(C_{*},D_{*})_{P,Q}\subseteq\operatorname{chEq}(C_{*},D_{*})$
to be the subset consisting of those chain homotopy equivalences $f\colon
C_{*}\rightarrow D_{*}$ which make the diagram

$\textstyle{C^{n-*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\textstyle{D^{n-*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\scriptstyle{Q}$$\textstyle{C_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D_{*}}$

of chain complexes commute up to chain homotopy, where $f^{*}$ denotes the
dual of $f$.

######  Lemma 8.6.

Let $C_{*}$ and $D_{*}$ be chain complexes, $C^{*}$ and $D^{*}$ the dual
cochain complexes, and let $P\colon C^{n-*}\rightarrow C_{*}$ and $Q\colon
D^{n-*}\rightarrow D_{*}$ be chain homotopy equivalences. Suppose that $C_{*}$
and $D_{*}$ split, so that $P$ and $Q$ restrict to chain homotopy equivalences
$P\big{|}_{(C^{\ell})^{n-*}}\colon(C^{\ell})^{n-*}\rightarrow C^{u}_{*}$ and
$Q\big{|}_{(D^{\ell})^{n-*}}\colon(D^{\ell})^{n-*}\rightarrow D^{u}_{*}$, and
let $\alpha=\tau(P\big{|}_{(C^{\ell})^{n-*}})$ and
$\beta=\tau(Q\big{|}_{(D^{\ell})^{n-*}})\in\operatorname{Wh}(G,w)$. Then there
is a commutative diagram

$\textstyle{\operatorname{chEq}(C_{*},D_{*})_{P,Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{chEq}(C^{\ell}_{*},D^{\ell}_{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}+\beta-\alpha}$

where the vertical map on the left is given by restriction, i.e. it is the
composition of the inclusion
$\operatorname{chEq}(C_{*},D_{*})_{P,Q}\rightarrow\operatorname{chEq}(C_{*},D_{*})$
and the projection
$\operatorname{chEq}(C_{*},D_{*})\rightarrow\operatorname{chEq}(C^{\ell}_{*},D^{\ell}_{*})$
from Lemma 8.4.

###### Proof.

Let $f\in\operatorname{chEq}(C_{*},D_{*})_{P,Q}$. Since $C_{*}$ and $D_{*}$
split, by Lemma 2.16 we have that
$\tau(f)=\tau(f\big{|}_{C^{\ell}_{*}})+\tau(f\big{|}_{C^{u}_{*}})$, so it is
enough to prove that
$\tau(f\big{|}_{C^{u}_{*}})=-(-1)^{n}\overline{\tau(f\big{|}_{C^{\ell}_{*}})}+\beta-\alpha$.

The diagram in Definition 8.5 restricts to a homotopy commutative diagram

$\textstyle{(C^{\ell})^{n-*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P|_{(C^{\ell})^{n-*}}}$$\textstyle{(D^{\ell})^{n-*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}|_{(D^{\ell})^{n-*}}}$$\scriptstyle{Q|_{(D^{\ell})^{n-*}}}$$\textstyle{C^{u}_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{C^{u}_{*}}}$$\textstyle{D^{u}_{*}}$

By Proposition 2.14 and Lemma 2.15 we have
$\tau(f^{*}\big{|}_{(D^{\ell})^{n-*}})+\alpha+\tau(f\big{|}_{C^{u}_{*}})=\beta$,
or equivalently
$\tau(f\big{|}_{C^{u}_{*}})=-\tau(f^{*}\big{|}_{(D^{\ell})^{n-*}})+\beta-\alpha$.
Therefore it suffices to prove that
$\tau(f^{*}\big{|}_{(D^{\ell})^{n-*}})=(-1)^{n}\overline{\tau(f\big{|}_{C^{\ell}_{*}})}$.

By Lemma 2.18 it follows that
$\tau(f^{*}\big{|}_{(D^{\ell})^{n-*}})=(-1)^{n}\tau(f^{*}\big{|}_{(D^{\ell})^{-*}})$,
and it follows from Definition 2.19 that
$\tau(f^{*}\big{|}_{(D^{\ell})^{-*}})=\tau(f^{*}\big{|}_{D^{*}_{l}})$. Finally
by Lemma 2.23, we have
$\tau(f^{*}\big{|}_{D^{*}_{l}})=\overline{\tau(f\big{|}_{C^{\ell}_{*}})}$,
which completes the proof. ∎

###  9\. Thickenings

We recall Wall’s definition of thickenings and some of their basic properties.
Fix positive integers $n,k$ with $n\geq 2k+1$, and let $K$ be a connected
finite CW complex of dimension (at most) $k$. Recall that we work in the
smooth category in this part.

######  Definition 9.1.

Suppose that $n\geq\max(k+3,2k+1)$. An $n$-dimensional _thickening_ of $K$ is
a simple homotopy equivalence $f_{T}\colon K\rightarrow T$, where $T$ is an
$n$-manifold with boundary such that the inclusion map $\partial
T\hookrightarrow T$ induces an isomorphism $\pi_{1}(\partial
T)\cong\pi_{1}(T)$.

Two thickenings $f_{T}\colon K\rightarrow T$ and $f_{T^{\prime}}\colon
K\rightarrow T^{\prime}$ are equivalent if there is a diffeomorphism $H\colon
T\rightarrow T^{\prime}$ such that $H\circ f_{T}\simeq f_{T^{\prime}}$.

Since $n\geq 2k+1$, we can assume that $f_{T}$ is an embedding. From now on
thickening will mean $n$-dimensional thickening, unless indicated otherwise.

######  Lemma 9.2 (Wall [Wa66, Proposition 5.1]).

Suppose that $n\geq\max(6,2k+1)$. The assignment $(f_{T}\colon K\rightarrow
T)\mapsto f_{T}^{*}(\nu_{T})$, where $\nu_{T}$ denotes the stable normal
bundle of $T$, is a bijection between the set of equivalence classes of
thickenings of $K$ and the set of isomorphism classes of stable vector bundles
over $K$. ∎

######  Remark 9.3.

The arguments of [Wa66, §5, §7 and §8] also show that if $n=5\geq 2k+1$, then
the assignment $(f_{T}\colon K\rightarrow T)\mapsto f_{T}^{*}(\nu_{T})$ is
surjective.

######  Lemma 9.4 (Wall).

Suppose that $n\geq\max(6,2k+1)$, $M$ is an $n$-manifold, and $\varphi\colon
K\rightarrow M$ is a continuous map. Then there is a thickening $f_{T}\colon
K\rightarrow T$ and an embedding $i\colon T\rightarrow\operatorname{Int}M$
such that $i\circ f_{T}\simeq\varphi$. The thickening $f_{T}$ is unique up to
equivalence.

###### Proof.

The existence part follows from Wall’s embedding theorem [Wa66]. For the
uniqueness assume that there is another thickening $f_{T^{\prime}}\colon
K\rightarrow T^{\prime}$ and an embedding $i^{\prime}\colon
T^{\prime}\rightarrow M$ such that $i^{\prime}\circ
f_{T^{\prime}}\simeq\varphi$. Then $f_{T}^{*}(\nu_{T})\cong
f_{T}^{*}(i^{*}(\nu_{M}))\cong\varphi^{*}(\nu_{M})\cong
f_{T^{\prime}}^{*}((i^{\prime})^{*}(\nu_{M}))\cong
f_{T^{\prime}}^{*}(\nu_{T^{\prime}})$, so by Lemma 9.2 it follows that $f_{T}$
and $f_{T^{\prime}}$ are equivalent. ∎

######  Lemma 9.5.

Suppose that $n\geq\max(k+3,2k+1)$ and $f_{T}\colon K\rightarrow T$ is a
thickening. Then the inclusion $\partial T\rightarrow T$ is
$(n-k-1)$-connected.

###### Proof.

By the definition of thickenings $\pi_{1}(\partial T)\rightarrow\pi_{1}(T)$ is
an isomorphism. Let $G=\pi_{1}(T)$ and consider homology with $\mathbb{Z}G$
coefficients. We have $H_{i}(T,\partial T)\cong H^{n-i}(T)\cong H^{n-i}(K)=0$
if $n-i>k$, equivalently, $i\leq n-k-1$. ∎

######  Lemma 9.6.

Suppose that $n\geq\max(6,2k+2)$ and $f_{T}\colon K\rightarrow T$ is an
$n$-dimensional thickening. Then there is a map $f_{\partial T}\colon
K\rightarrow\partial T$ such that $f_{\partial T}\simeq f_{T}$ as maps
$K\rightarrow T$. Furthermore, $f_{\partial T}$ is unique up to homotopy.

###### Proof.

By [Wa66, §5] there is an $(n-1)$-dimensional thickening $f_{V}\colon
K\rightarrow V$ and a diffeomorphism $T\approx V\times I$ such that $f_{T}$ is
homotopic to the composition $K\stackrel{{\scriptstyle
f_{V}}}{{\longrightarrow}}V\rightarrow V\times I\approx T$. Hence $\partial
T\approx V\cup_{\operatorname{Id}_{\partial V}}V$, and we can take
$f_{\partial T}$ to be the composition $K\stackrel{{\scriptstyle
f_{V}}}{{\longrightarrow}}V\rightarrow V\cup_{\operatorname{Id}_{\partial
V}}V\approx\partial T$. Since the inclusion $\partial T\rightarrow T$ is at
least $(k+1)$-connected, any two maps from $K$ to $\partial T$ that are
homotopic as $K\rightarrow T$ maps are also homotopic as $K\rightarrow\partial
T$ maps. Therefore $f_{\partial T}$ is unique up to homotopy. ∎

######  Lemma 9.7.

Suppose that $n\geq\max(6,2k+2)$. Let $M$ be a closed $n$-manifold,
$f_{T}\colon K\rightarrow T$ a thickening, $\varphi\colon K\rightarrow M$ an
$\left\lfloor\frac{n}{2}\right\rfloor$-connected map and $i\colon T\rightarrow
M$ an embedding such that $i\circ f_{T}\simeq\varphi$. Let $C=M\setminus
i(\operatorname{Int}T)$ denote the complement of $i(T)$, let $j\colon
C\rightarrow M$ be the inclusion, and let $\varphi^{\prime}$ denote the
composition $K\stackrel{{\scriptstyle f_{\partial
T}}}{{\longrightarrow}}\partial T\stackrel{{\scriptstyle
i}}{{\longrightarrow}}i(\partial T)\rightarrow C$.

  1. (a)

The map $\varphi^{\prime}\colon K\rightarrow C$ is a homotopy equivalence.

  2. (b)

A map $f\colon K\rightarrow C$ is homotopic to $\varphi^{\prime}$ if and only
if $j\circ f\simeq\varphi\colon K\rightarrow M$.

###### Proof.

(a) Since $\varphi$ is $\left\lfloor\frac{n}{2}\right\rfloor$-connected and
$f_{T}$ is a homotopy equivalence, $i$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected too. The inclusions $\partial
T\rightarrow T$ and $i$ induce isomorphisms on $\pi_{1}$, so it follows from
the Van Kampen theorem that the inclusions $i(\partial T)=\partial
C\rightarrow C$ and $j$ also induce isomorphisms on $\pi_{1}$. Let
$G=\pi_{1}(M)$, identify the fundamental groups of $T$, $\partial T$ and $C$
with $G$ via the inclusions, and consider their homology with $\mathbb{Z}G$
coefficients.

By excision and Lemma 9.5 we have $H_{r}(M,C)\cong H_{r}(T,\partial T)=0$ if
$r\leq n-k-1$. Therefore $H_{r}(j)\colon H_{r}(C)\rightarrow H_{r}(M)$ is an
isomorphism if $r\leq n-k-2$, in particular if
$r\leq\left\lceil\frac{n}{2}\right\rceil-1$. The induced homomorphism
$H_{r}(\varphi)\colon H_{r}(K)\rightarrow H_{r}(M)$ is an isomorphism for
$r\leq\left\lfloor\frac{n}{2}\right\rfloor$, because $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected and $K$ has dimension
$k<\left\lfloor\frac{n}{2}\right\rfloor$. Since $\varphi\simeq i\circ
f_{T}\simeq i\circ f_{\partial T}\simeq j\circ\varphi^{\prime}\colon
K\rightarrow M$, we get that $H_{r}(\varphi^{\prime})\colon
H_{r}(K)\rightarrow H_{r}(C)$ is an isomorphism if
$r\leq\left\lceil\frac{n}{2}\right\rceil-1$. We also have that $H_{r}(C)\cong
H^{n-r}(C,\partial C)\cong H^{n-r}(M,i(T))=0$ and $H_{r}(K)=0$ if
$r\geq\left\lceil\frac{n}{2}\right\rceil$ (hence
$n-r\leq\left\lfloor\frac{n}{2}\right\rfloor$). Therefore $\varphi^{\prime}$
induces an isomorphism on $\pi_{1}$ and all homology groups, so it is a
homotopy equivalence.

(b) We already saw that $\varphi\simeq j\circ\varphi^{\prime}$. This implies
the only if direction. Also, by part (a) this implies that $j$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected. Hence if $j\circ
f\simeq\varphi\simeq j\circ\varphi^{\prime}$ for some $f\colon K\rightarrow
C$, then $f\simeq\varphi^{\prime}$, because $K$ has dimension
$k\leq\left\lfloor\frac{n}{2}\right\rfloor-1$. ∎

The next lemma is implicit in the discussion on [Wa66, p. 77].

######  Lemma 9.8.

Let $M$ be an $n$-manifold with boundary, and let
$N\subset\operatorname{Int}M$ be a codimension $0$ submanifold. Then the
following hold.

  1. (a)

If $M\setminus\operatorname{Int}N$ is an $h$-cobordism between $\partial N$
and $\partial M$, then the inclusion $i\colon N\rightarrow M$ is a homotopy
equivalence.

  2. (b)

If the inclusions induce isomorphisms $\pi_{1}(\partial N)\cong\pi_{1}(N)$ and
$\pi_{1}(\partial M)\cong\pi_{1}(M)$, and $i\colon N\rightarrow M$ is a
homotopy equivalence, then $M\setminus\operatorname{Int}N$ is an
$h$-cobordism.

  3. (c)

If the assumptions in (b) hold, then
$\tau(i)=j_{*}(\tau(M\setminus\operatorname{Int}N,\partial
N))\in\operatorname{Wh}(\pi_{1}(M))$, where $j\colon
M\setminus\operatorname{Int}N\rightarrow M$ is the inclusion.

###### Proof.

(a) Since the inclusion $\partial N\rightarrow M\setminus\operatorname{Int}N$
is a homotopy equivalence, by applying homotopy excision [Ha02, Theorem 4.23]
to the map $(M\setminus\operatorname{Int}N,\partial N)\to(M,N)$ we deduce that
$i$ is a homotopy equivalence too.

(b) Let $G=\pi_{1}(M)$, it follows from the Van Kampen theorem and the
assumptions that the inclusions identify $\pi_{1}(N)$, $\pi_{1}(\partial N)$,
$\pi_{1}(M\setminus\operatorname{Int}N)$ and $\pi_{1}(\partial M)$ with $G$,
in particular $\pi_{1}(\partial
N)\cong\pi_{1}(M\setminus\operatorname{Int}N)\cong\pi_{1}(\partial M)$.
Consider homology with $\mathbb{Z}G$ coefficients. Since $i$ is a homotopy
equivalence, we have $H_{*}(M\setminus\operatorname{Int}N,\partial N)\cong
H_{*}(M,N)=0$ by excision. By Poincaré duality we also have
$H_{*}(M\setminus\operatorname{Int}N,\partial M)=0$, therefore the inclusions
$\partial N\rightarrow M\setminus\operatorname{Int}N$ and $\partial
M\rightarrow M\setminus\operatorname{Int}N$ are both homotopy equivalences.

(c) Fix a CW structure on $\partial N$ and extend it to a CW structure on $M$
(so that $\partial N$ is a subcomplex of $N$ and
$M\setminus\operatorname{Int}N$). Consider cellular chain complexes with
$\mathbb{Z}G$ coefficients. We have a commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{*}(\partial
N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{*}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{C_{*}(N,\partial
N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{*}(M\setminus\operatorname{Int}N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{*}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{*}(M,M\setminus\operatorname{Int}N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the rows are short exact sequences of chain compexes. The left and
middle vertical maps are chain homotopy equivalences by our assumptions and
part (b). The vertical map on the right is a chain homotopy equivalences with
vanishing Whitehead torsion, because the cells in $N\setminus\partial N$
determine the standard basis in both $C_{*}(N,\partial N)$ and
$C_{*}(M,M\setminus\operatorname{Int}N)$. So by Lemma 2.16 (additivity with
respect to short exact sequences) $\tau(i_{*})=\tau(C_{*}(\partial
N)\rightarrow C_{*}(M\setminus\operatorname{Int}N))\in\operatorname{Wh}(G)$. ∎

###  10\. Polarised doubles

Fix positive integers $n,k$ with $n\geq 2k+1$, and let $M$ be a closed
$n$-manifold, $K$ a finite CW complex of dimension (at most) $k$ and let
$\varphi\colon K\rightarrow M$ be a continuous map. We will think of (the
homotopy class of) $\varphi$ as extra structure on $M$ and call the pair
$(M,\varphi)$ a polarised manifold.

####  10.1. Double structures on polarised manifolds

We start by defining the different types of double structures that a polarised
manifold may have.

######  Definition 10.1.

Let $(M,\varphi)$ be a polarised manifold.

  * $\bullet$

A _generalised double structure_ on $(M,\varphi)$ is a diffeomorphism $h\colon
T\cup_{g_{0}}W\cup_{g_{1}}T\rightarrow M$ such that $\varphi\simeq h\circ
i_{1}\circ f_{T}$, where $f_{T}\colon K\rightarrow T$ is a thickening, $W$ is
an $h$-cobordism with $\partial W=\partial_{0}W\sqcup\partial_{1}W$,
$g_{0}\colon\partial_{0}W\rightarrow\partial T$ and $g_{1}\colon\partial
T\rightarrow\partial_{1}W$ are diffeomorphisms and $i_{1}\colon T\rightarrow
T\cup_{g_{0}}W\cup_{g_{1}}T$ is the inclusion of the first component.

  * $\bullet$

A _twisted double structure_ on $(M,\varphi)$ is a diffeomorphism $h\colon
T\cup_{g}T\rightarrow M$ such that $\varphi\simeq h\circ i_{1}\circ f_{T}$,
where $f_{T}\colon K\rightarrow T$ is a thickening, $g\colon\partial
T\rightarrow\partial T$ is a diffeomorphism, and $i_{1}\colon T\rightarrow
T\cup_{g}T$ is the inclusion of the first component.

  * $\bullet$

A _trivial double structure_ on $(M,\varphi)$ is a twisted double structure
with $g=\operatorname{Id}_{\partial T}$.

Of course, $(M,\varphi)$ has a twisted double structure if and only if it has
a generalised double structure with $W\approx\partial T\times I$. If the
manifold $M$ is of the form $T\cup_{g_{0}}W\cup_{g_{1}}T$, $T\cup_{g}T$, or
$T\cup_{\operatorname{Id}_{\partial T}}T$, then $\operatorname{Id}_{M}$ is a
generalised/twisted/trivial double structure on $(M,i_{1}\circ f_{T})$.

######  Proposition 10.2.

Suppose that $n\geq\max(6,2k+2)$. Then $(M,\varphi)$ has a generalised double
structure if and only if $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected.

###### Proof.

First assume that a generalised double structure $h$ exists as in Definition
10.1. By Lemma 9.8 (a), the inclusion $T\rightarrow T\cup W$ is a homotopy
equivalence. By Lemma 9.5 the pair $(T,\partial T)$ is $(n-k-1)$-connected,
and by homotopy excision the same holds for the pair $(T\cup W\cup T,T\cup
W)$. Therefore $i_{1}$ is $(n-k-1)$-connected too. Since $\varphi\simeq h\circ
i_{1}\circ f_{T}$ and $h$ and $f_{T}$ are homotopy equivalences, $\varphi$ is
also $(n-k-1)$-connected, in particular it is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected.

Now assume that $\varphi$ is $\left\lfloor\frac{n}{2}\right\rfloor$-connected.
By Lemma 9.4 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
$C=M\setminus i(\operatorname{Int}T)$ denote the complement of $i(T)$. By
Lemma 9.7 the composition $K\stackrel{{\scriptstyle f_{\partial
T}}}{{\longrightarrow}}\partial T\stackrel{{\scriptstyle
i}}{{\longrightarrow}}i(\partial T)\rightarrow C$, denoted by
$\varphi^{\prime}$, is a homotopy equivalence. Applying Lemma 9.4 again, there
is a thickening $f_{T^{\prime}}\colon K\rightarrow T^{\prime}$ of $K$ and an
embedding $i^{\prime}\colon T^{\prime}\rightarrow\operatorname{Int}C$ such
that $i^{\prime}\circ f_{T^{\prime}}\simeq\varphi^{\prime}$. Moreover, since
$\varphi$ and $\varphi^{\prime}$ are homotopic as $K\rightarrow M$ maps,
$f_{T}$ and $f_{T^{\prime}}$ are equivalent thickenings, i.e. there is a
diffeomorphism $H\colon T\rightarrow T^{\prime}$ such that $H\circ f_{T}\simeq
f_{T^{\prime}}$. Let $W=C\setminus i^{\prime}(\operatorname{Int}T^{\prime})$.
The embedding $i^{\prime}\colon T^{\prime}\rightarrow C$ is a homotopy
equivalence, because $\varphi^{\prime}\colon K\rightarrow C$ and
$f_{T^{\prime}}$ are both homotopy equivalences and $i^{\prime}\circ
f_{T^{\prime}}\simeq\varphi^{\prime}$. So by Lemma 9.8 (b) $W$ is an
$h$-cobordism. Therefore the decomposition $M=i(T)\cup W\cup(i^{\prime}\circ
H)(T)$ determines a generalised double structure on $(M,\varphi)$. ∎

######  Corollary 10.3.

Suppose that $n\geq\max(6,2k+2)$, and let $N$ be a closed $n$-manifold. If
$(M,\varphi)$ has a generalised double structure and $M\simeq N$, then
$(N,\psi)$ also has a generalised double structure for some $\psi\colon
K\rightarrow N$.

###### Proof.

If $h\colon M\rightarrow N$ is a homotopy equivalence, then $\psi\coloneqq
h\circ\varphi$ is $\left\lfloor\frac{n}{2}\right\rfloor$-connected. ∎

####  10.2. The Whitehead torsion of a polarised double

Next we define and study the invariant $\tau(M,\varphi)$, the torsion
associated to a polarised double. Note that this invariant gives an invariant
of a single polarised manifold, and not of a homotopy equivalence between
manifolds.

######  Definition 10.4.

Suppose that $n\geq\max(6,2k+2)$ and $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected. The _Whitehead torsion
associated to $(M,\varphi)$_, denoted by $\tau(M,\varphi)$, is defined as
follows. Let $f_{T}\colon K\rightarrow T$ be a thickening and let $i\colon
T\rightarrow M$ be an embedding such that $i\circ f_{T}\simeq\varphi$. Let
$C=M\setminus i(\operatorname{Int}T)$ denote the complement of $i(T)$ with
inclusion $j\colon C\rightarrow M$, and let $\varphi^{\prime}$ be the
composition

$\varphi^{\prime}\colon K\stackrel{{\scriptstyle f_{\partial
T}}}{{\longrightarrow}}\partial T\stackrel{{\scriptstyle
i}}{{\longrightarrow}}i(\partial T)\rightarrow C.$

Then

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

######  Proposition 10.5.

The Whitehead torsion $\tau(M,\varphi)$ is well-defined.

###### Proof.

First we check that the assumptions made in Definition 10.4 are satisfied. A
suitable thickening $f_{T}$ and embedding $i$ exist by Lemma 9.4. The
composition $\varphi^{\prime}\colon K\rightarrow C$ is a homotopy equivalence
by Lemma 9.7, so $\tau(\varphi^{\prime})\in\operatorname{Wh}(\pi_{1}(C))$ is
defined. Since $j\circ\varphi^{\prime}\simeq\varphi$ and $\pi_{1}(\varphi)$
and $\pi_{1}(\varphi^{\prime})$ are isomorphisms,
$j_{*}\colon\operatorname{Wh}(\pi_{1}(C))\rightarrow\operatorname{Wh}(\pi_{1}(M))$
is an isomorphism too.

Next we check that $\tau(M,\varphi)$ is independent of choices. By Lemma 9.4,
the thickening $f_{T}$ is well-defined (up to equivalence). Suppose that
$i_{0}\colon T\rightarrow M$ and $i_{1}\colon T\rightarrow M$ are two
embeddings with $i_{0}\circ f_{T}\simeq i_{1}\circ f_{T}\simeq\varphi$ and
corresponding $C_{0},j_{0},\varphi^{\prime}_{0}$ and
$C_{1},j_{1},\varphi^{\prime}_{1}$. We will prove that

$(j_{0})_{*}(\tau(\varphi^{\prime}_{0}))=(j_{1})_{*}(\tau(\varphi^{\prime}_{1})).$

Since $n\geq 2k+2$, we can assume that $f_{T}$ is an embedding, and since the
embeddings $i_{0}\circ f_{T}$ and $i_{1}\circ f_{T}\colon K\rightarrow M$ are
homotopic, they are isotopic [Whitney-36, Theorem 6].

By the isotopy extension theorem there is a diffeomorphism $H\colon
M\rightarrow M$, isotopic to $\operatorname{Id}_{M}$, such that $i_{0}\circ
f_{T}=H\circ i_{1}\circ f_{T}$. Let $i_{2}=H\circ i_{1}$,
$C_{2}=H(C_{1})=M\setminus i_{2}(\operatorname{Int}T)$, $j_{2}=H\circ
j_{1}\circ H^{-1}\colon C_{2}\rightarrow M$ and
$\varphi^{\prime}_{2}=H\circ\varphi^{\prime}_{1}\colon K\rightarrow C_{2}$.

Since $H$ is a diffeomorphism,
$\tau(\varphi^{\prime}_{2})=H_{*}(\tau(\varphi^{\prime}_{1}))\in\operatorname{Wh}(\pi_{1}(C_{2}))$
by Proposition 2.30 and Theorem 2.5, and hence
$(j_{2})_{*}(\tau(\varphi^{\prime}_{2}))=H_{*}\circ(j_{1})_{*}(\tau(\varphi^{\prime}_{1}))$.
We have
$H_{*}=\operatorname{Id}\colon\operatorname{Wh}(\pi_{1}(M))\rightarrow\operatorname{Wh}(\pi_{1}(M))$,
because $H\simeq\operatorname{Id}_{M}$, therefore
$(j_{2})_{*}(\tau(\varphi^{\prime}_{2}))=(j_{1})_{*}(\tau(\varphi^{\prime}_{1}))$.
So it is enough to prove that

$(j_{0})_{*}(\tau(\varphi^{\prime}_{0}))=(j_{2})_{*}(\tau(\varphi^{\prime}_{2})).$

The image of $i_{0}\circ f_{T}=i_{2}\circ f_{T}\colon K\to M$ is contained in
$i_{0}(T)\cap i_{2}(T)$, so we can apply Lemma 9.4 to get an embedding
$i_{3}\colon T\rightarrow i_{0}(T)\cap i_{2}(T)$ such that $i_{3}\circ
f_{T}\simeq i_{0}\circ f_{T}$, and define the corresponding $C_{3}:=M\setminus
i_{3}(\operatorname{Int}T)$, with inclusion $j_{3}\colon C_{3}\to M$, and
$\varphi^{\prime}_{3}\colon K\to C_{3}$. Since $i_{3}(T)\subseteq i_{0}(T)$,
we have $C_{3}\supseteq C_{0}$. Let $h\colon i_{3}(T)\rightarrow i_{0}(T)$ and
$h^{\prime}\colon C_{0}\rightarrow C_{3}$ denote the inclusions (hence
$j_{0}=j_{3}\circ h^{\prime}$).

By construction, $h\circ i_{3}\circ f_{T}\simeq i_{0}\circ f_{T}\colon
K\rightarrow i_{0}(T)$. Since $f_{T}$ and the diffeomorphisms $i_{0}\colon
T\rightarrow i_{0}(T)$ and $i_{3}\colon T\rightarrow i_{3}(T)$ are simple
homotopy equivalences, $h$ is a simple homotopy equivalence too. By Lemma 9.8
(b) and (c), we see that $i_{0}(T)\setminus i_{3}(\operatorname{Int}T)$ is an
$s$-cobordism, and this in turn implies that $h^{\prime}$ is a simple homotopy
equivalence. We have $j_{3}\circ
h^{\prime}\circ\varphi^{\prime}_{0}=j_{0}\circ\varphi^{\prime}_{0}\simeq\varphi$,
so by Lemma 9.7 (b), we have that
$h^{\prime}\circ\varphi^{\prime}_{0}\simeq\varphi^{\prime}_{3}\colon
K\rightarrow C_{3}$. Use this, Proposition 2.30, and the fact that
$\tau(h^{\prime})=0$, to deduce that

$(j_{3})_{*}(\tau(\varphi^{\prime}_{3}))=(j_{3})_{*}(\tau(h^{\prime}\circ\varphi^{\prime}_{0}))=(j_{3})_{*}(h^{\prime}_{*}(\tau(\varphi^{\prime}_{0})))=(j_{0})_{*}(\tau(\varphi^{\prime}_{0})).$

Since also $i_{3}(T)\subseteq i_{2}(T)$, we can prove similarly that
$(j_{3})_{*}(\tau(\varphi^{\prime}_{3}))=(j_{2})_{*}(\tau(\varphi^{\prime}_{2}))$.
Therefore
$(j_{0})_{*}(\tau(\varphi^{\prime}_{0}))=(j_{2})_{*}(\tau(\varphi^{\prime}_{2}))$
as required. ∎

######  Proposition 10.6.

Suppose that $n\geq\max(6,2k+2)$ and $(M,\varphi)$ has a generalised double
structure $h\colon T\cup_{g_{0}}W\cup_{g_{1}}T\rightarrow M$, and let $i_{1}$,
$i_{2}$, and $i_{3}$ denote the inclusions of the three components of
$T\cup_{g_{0}}W\cup_{g_{1}}T$.

  1. (a)

There is a homotopy automorphism $\alpha\colon K\rightarrow K$ such that
$\varphi\circ\alpha\simeq h\circ i_{3}\circ f_{T}\colon K\rightarrow M$, and
such an $\alpha$ is unique up to homotopy.

  2. (b)

This $\alpha$ satisfies
$\alpha^{*}(\varphi^{*}(\nu_{M}))\cong\varphi^{*}(\nu_{M})$.

  3. (c)

For this $\alpha$, we have $\tau(M,\varphi)=(h\circ
i_{2})_{*}(\tau(W,\partial_{1}W))-\varphi_{*}(\tau(\alpha))$.

###### Proof.

First we introduce some notation. Let $i=h\circ i_{1}\colon T\rightarrow M$,
$C=h(W\cup_{g_{1}}T)$, and $\varphi^{\prime}=h\circ i_{1}\circ f_{\partial
T}\colon K\rightarrow C$, and let $j\colon C\rightarrow M$ denote the
inclusion. Then, by Definition 10.4, we have
$\tau(M,\varphi)=j_{*}(\tau(\varphi^{\prime}))=h_{*}(\tau(\varphi^{\prime\prime}))$,
where $\varphi^{\prime\prime}=i_{1}\circ f_{\partial T}\colon K\rightarrow
W\cup_{g_{1}}T$.

(a) By Lemma 9.7, $\varphi^{\prime}$ is a homotopy equivalence, and since
$h\big{|}_{W\cup_{g_{1}}T}\colon W\cup_{g_{1}}T\rightarrow C$ is a
diffeomorphism, $\varphi^{\prime\prime}$ is a homotopy equivalence too. Let

$\alpha:=(\varphi^{\prime\prime})^{-1}\circ i_{3}\circ f_{T}\colon
K\rightarrow K,$

where $i_{3}$ is regarded as an inclusion $T\rightarrow W\cup_{g_{1}}T$ and
$(\varphi^{\prime\prime})^{-1}\colon W\cup_{g_{1}}T\rightarrow K$ is the
homotopy inverse of $\varphi^{\prime\prime}$. By Lemma 9.8 (a), $i_{3}$ is a
homotopy equivalence, so $\alpha$ is a homotopy equivalence too. Moreover, we
compute that

$\varphi\circ\alpha\simeq h\circ i_{1}\circ f_{T}\circ\alpha\simeq h\circ
i_{1}\circ f_{\partial
T}\circ\alpha=h\circ\varphi^{\prime\prime}\circ\alpha\simeq h\circ i_{3}\circ
f_{T}\colon K\rightarrow M.$

If $\alpha,\alpha^{\prime}\colon K\rightarrow K$ are two maps such that
$\varphi\circ\alpha\simeq\varphi\circ\alpha^{\prime}\colon K\rightarrow M$,
then $\alpha\simeq\alpha^{\prime}$, because $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected and $K$ has dimension
$k\leq\left\lfloor\frac{n}{2}\right\rfloor-1$.

(b) Since $\varphi\circ\alpha\simeq h\circ i_{3}\circ f_{T}$, we have
$\alpha^{*}(\varphi^{*}(\nu_{M}))\cong f_{T}^{*}((h\circ
i_{3})^{*}(\nu_{M}))\cong f_{T}^{*}(\nu_{T})$, where we used that $h\circ
i_{3}\colon T\rightarrow M$ is a codimension $0$ embedding. Since
$\varphi\simeq h\circ i_{1}\circ f_{T}$, and $h\circ i_{1}\colon T\rightarrow
M$ is a codimension $0$ embedding, we similarly obtain
$\varphi^{*}(\nu_{M})\cong f_{T}^{*}(\nu_{T})$. Hence
$\alpha^{*}(\varphi^{*}(\nu_{M}))\cong\varphi^{*}(\nu_{M})$.

(c) It follows from the definition of $\alpha$ that
$\varphi^{\prime\prime}\simeq i_{3}\circ f_{T}\circ\alpha^{-1}\colon
K\rightarrow W\cup_{g_{1}}T$. We have

$\tau(i_{3}\circ
f_{T}\circ\alpha^{-1})=\tau(i_{3})+(i_{3})_{*}(\tau(f_{T}))+(i_{3}\circ
f_{T})_{*}(\tau(\alpha^{-1}))$

by Proposition 2.30. The map $f_{T}$ is a simple homotopy equivalence, and by
Lemma 9.8 (c), we have that $\tau(i_{3})=(i_{2})_{*}(\tau(W,\partial_{1}W))$,
where $i_{2}$ is regarded as an embedding $W\rightarrow W\cup_{g_{1}}T$. We
also have
$0=\tau(\alpha^{-1}\circ\alpha)=\tau(\alpha^{-1})+\alpha^{-1}_{*}(\tau(\alpha))$,
so

$\tau(i_{3}\circ
f_{T}\circ\alpha^{-1})=(i_{2})_{*}(\tau(W,\partial_{1}W))-(i_{3}\circ
f_{T}\circ\alpha^{-1})_{*}(\tau(\alpha)).$

Since $\varphi\simeq h\circ i_{3}\circ f_{T}\circ\alpha^{-1}$, we obtain

$\tau(M,\varphi)=h_{*}(\tau(\varphi^{\prime\prime}))=h_{*}(\tau(i_{3}\circ
f_{T}\circ\alpha^{-1}))=(h\circ
i_{2})_{*}(\tau(W,\partial_{1}W))-\varphi_{*}(\tau(\alpha)).\qed$

######  Corollary 10.7.

Suppose that $n\geq\max(6,2k+2)$ and $(M,\varphi)$ has a twisted double
structure $h\colon T\cup_{g}T\rightarrow M$. Let
$\alpha\in\operatorname{hAut}(K)$ be the image of $g$ under the restriction
map $\operatorname{hAut}(\partial T)\rightarrow\operatorname{hAut}(K)$ $($see
Remark 10.13, Lemma 11.3 and Remark 11.4$)$. Then
$\tau(M,\varphi)=-\varphi_{*}(\tau(\alpha))$.

###### Proof.

Since the twisted double structure of $(M,\varphi)$ determines a generalised
double structure with $W\approx\partial T\times I$, it is enough to show that
$\varphi\circ\alpha\simeq h\circ i_{2}\circ f_{T}\colon K\rightarrow M$, where
$i_{2}\colon T\rightarrow T\cup_{g}T$ is the inclusion of the second
component. This holds, because

$\varphi\circ\alpha\simeq h\circ i_{1}\circ f_{T}\circ\alpha\simeq h\circ
i_{1}\circ f_{\partial T}\circ\alpha\simeq h\circ i_{1}\circ g\circ
f_{\partial T}=h\circ i_{2}\circ f_{\partial T}\simeq h\circ i_{2}\circ
f_{T}.\qed$

######  Corollary 10.8.

Suppose that $n\geq\max(6,2k+2)$ and $(M,\varphi)$ has a trivial double
structure. Then $\tau(M,\varphi)=0$.

###### Proof.

We can apply Corollary 10.7 with $g=\operatorname{Id}_{\partial T}$, and hence
$\alpha=\operatorname{Id}_{K}$. ∎

######  Remark 10.9.

The converse does not hold. There are even simply-connected counterexamples,
e.g. take $K$ to be a point and $M$ an exotic sphere.

######  Proposition 10.10.

Suppose that $n\geq\max(6,2k+2)$ and $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected. Then $(M,\varphi)$ has a
twisted double structure if and only if there is a homotopy automorphism
$\alpha\in\operatorname{hAut}(K)$ such that there is an isomorphism of stable
vector bundles $\alpha^{*}(\varphi^{*}(\nu_{M}))\cong\varphi^{*}(\nu_{M})$ and
$\tau(M,\varphi)=-\varphi_{*}(\tau(\alpha))\in\operatorname{Wh}(\pi_{1}(M))$.

###### Proof.

If $(M,\varphi)$ has a twisted double structure, then it follows from
Proposition 10.6 (b) and the proof of Corollary 10.7 that such an $\alpha$
exists.

Now suppose that there is an $\alpha$ with
$\alpha^{*}(\varphi^{*}(\nu_{M}))\cong\varphi^{*}(\nu_{M})$ and
$\tau(M,\varphi)=-\varphi_{*}(\tau(\alpha))$. Define $f_{T}\colon K\rightarrow
T$, $i$, $C$, $j$ and $\varphi^{\prime}$ as in Definition 10.4, so that
$\tau(M,\varphi)=j_{*}(\tau(\varphi^{\prime}))$. Let
$\psi:=\varphi^{\prime}\circ\alpha\colon K\rightarrow C$. Then

$j_{*}(\tau(\psi))=j_{*}(\tau(\varphi^{\prime})+\varphi^{\prime}_{*}(\tau(\alpha)))=\tau(M,\varphi)+(j\circ\varphi^{\prime})_{*}(\tau(\alpha))=\tau(M,\varphi)+\varphi_{*}(\tau(\alpha))=0$

by Proposition 2.30, Lemma 9.7 (b), and the hypothesis. As $j_{*}$ is an
isomorphism, this means that $\psi$ is a simple homotopy equivalence, hence
$\psi\colon K\rightarrow C$ is a thickening. Moreover,

$\psi^{*}(\nu_{C})\cong\alpha^{*}((\varphi^{\prime})^{*}(j^{*}(\nu_{M})))\cong\alpha^{*}(\varphi^{*}(\nu_{M}))\cong\varphi^{*}(\nu_{M})\cong
f_{T}^{*}(i^{*}(\nu_{M}))\cong f_{T}^{*}(\nu_{T}).$

So by Lemma 9.2 the thickenings $\psi\colon K\rightarrow C$ and $f_{T}\colon
K\rightarrow T$ are equivalent, i.e. there is a diffeomorphism $H\colon
T\rightarrow C$ such that $H\circ f_{T}\simeq\psi$. Then $(i\cup H)\colon
T\cup_{g}T\rightarrow M$ is a twisted double structure on $(M,\varphi)$, where
$g$ is the composition

$g\colon\partial T\stackrel{{\scriptstyle H}}{{\longrightarrow}}\partial
C=i(\partial T)\stackrel{{\scriptstyle i^{-1}}}{{\longrightarrow}}\partial
T.\qed$

######  Corollary 10.11.

Suppose that $n\geq\max(6,2k+2)$ and $\varphi$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected. If $\tau(M,\varphi)=0$, then
$(M,\varphi)$ has a twisted double structure.

###### Proof.

Take $\alpha=\operatorname{Id}_{K}$ and apply Proposition 10.10. ∎

####  10.3. SP manifolds

Next we show that if we impose some mild restrictions on a double
$(M,\varphi)$, then $\tau(M,\varphi)$ can be expressed in terms of the
Poincaré duality chain homotopy equivalence.

######  Definition 10.12.

The pair $(M,\varphi)$ is a _split polarised manifold_ (SP manifold for short)
if at least one of the following conditions holds:

  1. (SP1)

$n\geq\max(7,2k+2)$ and $(M,\varphi)$ has a generalised double structure;

  2. (SP2)

$n\geq\max(6,2k+2)$ and $(M,\varphi)$ has a twisted double structure; or

  3. (SP3)

$n\geq\max(k+3,2k+1)$ and $(M,\varphi)$ has a trivial double structure.

######  Remark 10.13.

If $n\geq\max(6,2k+2)$ and $f_{T}\colon K\rightarrow T$ is an $n$-dimensional
thickening, then $n-1\geq\max(k+3,2k+1)$ and the proof of Lemma 9.6 shows that
$(\partial T,f_{\partial T})$ satisfies (SP3).

We extend the definition of $\tau(M,\varphi)$ to the case when
$\max(6,2k+2)>n\geq\max(k+3,2k+1)$ and $(M,\varphi)$ has a trivial double
structure by setting $\tau(M,\varphi)=0$. Then $\tau(M,\varphi)$ is defined
for every SP manifold $(M,\varphi)$, and if (SP3) holds, then
$\tau(M,\varphi)=0$.

######  Theorem 10.14.

Suppose that $(M,\varphi)$ is an SP manifold, and let $G=\pi_{1}(M)$ with
orientation character $w\colon G\rightarrow\left\\{\pm 1\right\\}$. Then $M$
has a CW decomposition with the following properties.

  1. (a)

The $\left\lfloor\frac{n}{2}\right\rfloor$-skeleton of $M$ is identified with
$K$ via an embedding $K\rightarrow M$ homotopic to $\varphi$.

  2. (b)

Let $C_{*}(M)=C_{*}(M;\mathbb{Z}G)$ denote the cellular chain complex of $M$
with $\mathbb{Z}G$ coefficients. Then $C_{*}(M)$ splits $($see Definition
8.2$)$.

  3. (c)

Let $\operatorname{PD}\colon C^{n-*}(M)\rightarrow C_{*}(M)$ denote the chain
homotopy equivalence induced by Poincaré duality $($which is determined up to
chain homotopy by a choice of twisted fundamental class $[M]\in
H_{n}(M;\mathbb{Z}^{w})\cong\mathbb{Z}$, where $\mathbb{Z}^{w}$ is the
orientation module$)$. Then

$\tau(\operatorname{PD}\big{|}_{C^{\ell}(M)^{n-*}}\colon
C^{\ell}(M)^{n-*}\rightarrow
C^{u}_{*}(M))=\tau(M,\varphi)\in\operatorname{Wh}(G,w)\text{.}$

###### Proof.

(a) Let $h\colon T\cup_{g_{0}}W\cup_{g_{1}}T\rightarrow M$ or $h\colon
T\cup_{g}T\rightarrow M$ be a generalised, twisted or trivial double structure
on $(M,\varphi)$ if (SP1), (SP2) or (SP3) holds, respectively (in the last
case $g=\operatorname{Id}_{\partial T}$). Let $i_{1}$, $i_{2}$ and $i_{3}$
denote the embeddings of the components of $T\cup_{g_{0}}W\cup_{g_{1}}T$, or
let $i_{1}$ and $i_{2}$ denote the embeddings of the components of
$T\cup_{g}T$.

The thickening $T$ has a handlebody decomposition such that the embedding
$f_{T}\colon K\rightarrow T$ identifies $K$ with the CW complex formed by the
cores of the handles, in particular there is a bijection between the
$i$-handles and the $i$-cells of $K$ (see [Wa66, §7]). This handlebody
decomposition determines a Morse function $m_{0}\colon T\to[0,1]$ such that
index-$i$ critical points of $m_{0}$ correspond to $i$-handles and
$m_{0}^{-1}(1)=\partial T$. By the normal form lemma [Milnor-h-cob], [Luck-
basic-intro-surgery, Lemma 1.24], if $n\geq 7$, then there is a Morse function
$m_{1}\colon W\to[0,1]$ on the $h$-cobordism $W$ such that all critical points
have index $\left\lfloor\frac{n}{2}\right\rfloor+1$ or
$\left\lfloor\frac{n}{2}\right\rfloor+2$, and $m_{1}^{-1}(0)=\partial_{0}W$
and $m_{1}^{-1}(1)=\partial_{1}W$.

Now we can define a Morse function $m\colon M\to\mathbb{R}$ on $M$. In the
case of (SP1) we take $m_{0}$ on $h\circ i_{1}(T)$, $m_{1}+1$ on $h\circ
i_{2}(W)$ and $3-m_{0}$ on $h\circ i_{3}(T)$. In the case of (SP2) and (SP3)
we take $m_{0}$ on $h\circ i_{1}(T)$ and $2-m_{0}$ on $h\circ i_{2}(T)$.

The Morse function $m$ determines a handlebody decomposition of $M$, and by
[Ma02, Theorem 4.18] $M$ is homeomorphic to the CW complex formed by the cores
of the handles (having one $i$-cell for each index-$i$ critical point of $m$).
The critical points of $m$ have index at most
$k\leq\left\lfloor\frac{n}{2}\right\rfloor$ in $h\circ i_{1}(T)$,
$\left\lfloor\frac{n}{2}\right\rfloor+1$ or
$\left\lfloor\frac{n}{2}\right\rfloor+2$ in $h\circ i_{2}(W)$, and at least
$n-k\geq\left\lfloor\frac{n}{2}\right\rfloor+1$ in $h\circ i_{2}(T)$ or
$h\circ i_{3}(T)$. Therefore the
$\left\lfloor\frac{n}{2}\right\rfloor$-skeleton of $M$ consists of the cores
of the handles in $h\circ i_{1}(T)$.

Let $f_{M}=h\circ i_{1}\circ f_{T}\colon K\rightarrow M$, then by the above
$f_{M}$ identifies the $\left\lfloor\frac{n}{2}\right\rfloor$-skeleton of $M$
(which is also the $k$-skeleton) with $K$. Moreover, $f_{M}\simeq\varphi$ by
Definition 10.1.

(b) If $n\geq 2k+2$, then $k\leq\left\lfloor\frac{n}{2}\right\rfloor-1$, so
$M$ has no $\left\lfloor\frac{n}{2}\right\rfloor$-cells. This means that
$C_{\left\lfloor n/2\right\rfloor}(M)=0$, hence $C_{*}(M)$ splits.

If $n=2k+1$, then $(M,\varphi)$ satisfies (SP3), so $h^{-1}$ is a
diffeomorphism $M\rightarrow T\cup_{\operatorname{Id}_{\partial T}}T$. There
is a well-defined retraction
$\operatorname{Id}_{T}\cup\operatorname{Id}_{T}\colon
T\cup_{\operatorname{Id}_{\partial T}}T\rightarrow T$. Since the embedding
$f_{T}$ is a homotopy equivalence, $K$ is a deformation retract of $T$, and we
can compose $h^{-1}$ with the two retractions to get a retraction $r\colon
M\to K$. It induces a chain map $C_{*}(r)\colon C_{*}(M)\rightarrow C_{*}(K)$
such that the composition $C_{*}(K)\stackrel{{\scriptstyle
C_{*}(f_{M})}}{{\longrightarrow}}C_{*}(M)\stackrel{{\scriptstyle
C_{*}(r)}}{{\longrightarrow}}C_{*}(K)$ is the identity. Since $C_{i}(f_{M})$
is an isomorphism for $i\leq k$, we get that $C_{k}(r)$ is an isomorphism too,
and this implies that the differential $C_{k+1}(M)\rightarrow C_{k}(M)$
vanishes. Therefore $C_{*}(M)$ splits.

(c) Let $\overline{m}=-m\colon M\rightarrow\mathbb{R}$ be the reverse Morse
function on $M$. It has the same critical points as $m$, with index-$i$
critical points turning into index-$(n-i)$ critical points. It determines a
new CW complex homeomorphic to $M$, which we will denote by $\overline{M}$. By
cellular approximation there is a cellular map $\iota\colon
M\rightarrow\overline{M}$ homotopic to $\operatorname{Id}_{M}$.

Now let $I\colon C^{n-*}(M)\rightarrow C_{*}(\overline{M})$ denote the
isomorphism that sends the (cochain) dual of an $(n-i)$-cell of $M$ to the
corresponding $i$-cell of $\overline{M}$. Then the chain homotopy equivalence
$\operatorname{PD}\colon C^{n-*}(M)\rightarrow C_{*}(M)$ inducing Poincaré
duality can be defined (up to chain homotopy) as the composition
$C_{*}(\iota)^{-1}\circ I$, where $C_{*}(\iota)^{-1}$ denotes the homotopy
inverse of the chain homotopy equivalence $C_{*}(\iota)\colon
C_{*}(M)\rightarrow C_{*}(\overline{M})$.

The chain complex $C_{*}(\overline{M})$ splits, because $C^{n-*}(M)$ splits
and $I$ is an isomorphism. By Lemma 8.4, we have that $\operatorname{PD}$,
$I$, and $C_{*}(\iota)$ all restrict to chain homotopy equivalences between
the upper and lower halves of the chain complexes involved, and
$C_{*}(\iota)^{-1}\big{|}_{C^{\ell}_{*}(\overline{M})}=(C_{*}(\iota)\big{|}_{C^{\ell}_{*}(M)})^{-1}$
and
$C_{*}(\iota)^{-1}\big{|}_{C^{u}_{*}(\overline{M})}=(C_{*}(\iota)\big{|}_{C^{u}_{*}(M)})^{-1}$
(up to chain homotopy). Hence we have:

$\operatorname{PD}\big{|}_{C^{u}(M)^{n-*}}=(C_{*}(\iota)\big{|}_{C^{\ell}_{*}(M)})^{-1}\circ
I\big{|}_{C^{u}(M)^{n-*}},\quad\operatorname{PD}\big{|}_{C^{\ell}(M)^{n-*}}=(C_{*}(\iota)\big{|}_{C^{u}_{*}(M)})^{-1}\circ
I\big{|}_{C^{\ell}(M)^{n-*}}.$

Since the isomorphism $I$ and its restrictions preserve the standard bases,
this implies that

$\tau(\operatorname{PD}\big{|}_{C^{u}(M)^{n-*}})=-\tau(C_{*}(\iota)\big{|}_{C^{\ell}_{*}(M)})\text{
and
}\tau(\operatorname{PD}\big{|}_{C^{\ell}(M)^{n-*}})=-\tau(C_{*}(\iota)\big{|}_{C^{u}_{*}(M)}).$

Now note that
$\tau(C_{*}(\iota)\big{|}_{C^{u}_{*}(M)})+\tau(C_{*}(\iota)\big{|}_{C^{\ell}_{*}(M)})=\tau(C_{*}(\iota))=0$.
Here we used that $\iota$ is homotopic to $\operatorname{Id}_{M}$, so we can
apply Proposition 2.14 and Lemma 2.15 to deduce that $\tau(C_{*}(\iota))=0$.
Therefore
$\tau(\operatorname{PD}\big{|}_{C^{\ell}(M)^{n-*}})=-\tau(C_{*}(\iota)\big{|}_{C^{u}_{*}(M)})=\tau(C_{*}(\iota)\big{|}_{C^{\ell}_{*}(M)})$.

Let $L$ denote the $\left\lfloor\frac{n}{2}\right\rfloor$-skeleton of
$\overline{M}$ and let $C=M\setminus h\circ i_{1}(\operatorname{Int}T)$. In
$\overline{M}$ the handles corresponding to the critical points of
$\overline{m}$ of index at most $\left\lfloor\frac{n}{2}\right\rfloor$ (which
are the critical points of $m$ of index at least
$\left\lceil\frac{n}{2}\right\rceil$) together make up $C$, and $L$ consists
of the cores of these handles. By [Ma02, Theorem 4.18], it follows that $C$ is
homeomorphic to the mapping cylinder of the projection $\partial C\rightarrow
L$, so the inclusion $L\rightarrow C$ is a simple homotopy equivalence by
Proposition 2.29. The cellular map $\iota$ restricts to a map
$\iota\big{|}_{f_{M}(K)}\colon f_{M}(K)\rightarrow L$ between the
$\left\lfloor\frac{n}{2}\right\rfloor$-skeletons of $M$ and $\overline{M}$.
The inclusions determine isomorphisms $C^{\ell}_{*}(M)\cong C_{*}(f_{M}(K))$
and $C^{\ell}_{*}(\overline{M})\cong C_{*}(L)$ (preserving the standard
bases), hence
$\tau(C_{*}(\iota)\big{|}_{C^{\ell}_{*}(M)})=\tau(\iota\big{|}_{f_{M}(K)})$.

First assume that (SP1) or (SP2) holds. Let $\varphi^{\prime}$ denote the
composition $K\stackrel{{\scriptstyle f_{\partial
T}}}{{\longrightarrow}}\partial T\stackrel{{\scriptstyle
i_{1}}}{{\longrightarrow}}i_{1}(\partial T)\rightarrow C$, so that
$\tau(M,\varphi)=\tau(\varphi^{\prime})$ (when $\pi_{1}(C)$ is identified with
$\pi_{1}(M)$ via the inclusion). The composition $K\stackrel{{\scriptstyle
f_{M}}}{{\longrightarrow}}f_{M}(K)\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}L\rightarrow
C\rightarrow M$ is homotopic to $\varphi$, because $f_{M}\simeq\varphi$ and
$\iota\simeq\operatorname{Id}_{M}$. So by Lemma 9.7 (b) the composition
$K\stackrel{{\scriptstyle
f_{M}}}{{\longrightarrow}}f_{M}(K)\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}L\rightarrow
C$ is homotopic to $\varphi^{\prime}$. This shows that
$\tau(\iota\big{|}_{f_{M}(K)})=\tau(\varphi^{\prime})=\tau(M,\varphi)$,
because the homeomorphism $f_{M}\colon K\rightarrow f_{M}(K)$ and the
inclusion $L\rightarrow C$ have vanishing Whitehead torsion.

Now assume that (SP3) holds. Then $L=h\circ i_{2}\circ f_{T}(K)$ and
$r\big{|}_{L}\colon L\rightarrow K$ is a homeomorphism (by the definition of
$r$). Moreover, $r\circ\iota\circ f_{M}\simeq r\circ\operatorname{Id}_{M}\circ
f_{M}=\operatorname{Id}_{K}$. Since $f_{M}$ is also a homeomorphism, this
means that $\iota\big{|}_{f_{M}(K)}$ is homotopic to a homeomorphism, so
$\tau(\iota\big{|}_{f_{M}(K)})=0=\tau(M,\varphi)$.

Therefore $\tau(\operatorname{PD}\big{|}_{C^{\ell}(M)^{n-*}})=\tau(M,\varphi)$
in all cases. ∎

###  11\. The Whitehead torsion of homotopy equivalences of doubles

In this section we prove Theorem G. Fix positive integers $n,k$ with $n\geq
2k+1$. Let $M$ and $N$ be closed $n$-manifolds, and let $K$ and $L$ be finite
CW complexes of dimension (at most) $k$. Suppose that $\varphi\colon
K\rightarrow M$ and $\psi\colon L\rightarrow N$ are continuous maps such that
$(M,\varphi)$ and $(N,\psi)$ are SP manifolds.

Let $F:=\pi_{1}(M)$ and $G:=\pi_{1}(N)$. The maps $\varphi$ and $\psi$ are
$\left\lfloor\frac{n}{2}\right\rfloor$-connected by Proposition 10.2, so
$\pi_{1}(\varphi)$ and $\pi_{1}(\psi)$ are isomorphisms. We use these
isomorphisms to identify $\pi_{1}(K)$ with $F=\pi_{1}(M)$ and $\pi_{1}(L)$
with $G=\pi_{1}(N)$.

Let $w_{M}\colon F\rightarrow\left\\{\pm 1\right\\}$ and $w=w_{N}\colon
G\rightarrow\left\\{\pm 1\right\\}$ be the orientation characters of $M$ and
$N$ respectively. Fix twisted fundamental classes $[M]\in
H_{n}(M;\mathbb{Z}^{w_{M}})$ and $[N]\in H_{n}(N;\mathbb{Z}^{w})$. For any
homotopy equivalence $f\colon M\rightarrow N$ we have $w\circ\pi_{1}(f)=w_{M}$
and $f_{*}([M])=\varepsilon[N]$ for $\varepsilon=1$ or $-1$.

######  Definition 11.1.

For topological spaces $X$, $Y$, let $\operatorname{hEq}(X,Y)$ denote the set
of homotopy classes of homotopy equivalences $X\to Y$. For an isomorphism
$\theta\colon\pi_{1}(X)\rightarrow\pi_{1}(Y)$, let
$\operatorname{hEq}_{\theta}(X,Y)\subseteq\operatorname{hEq}(X,Y)$ denote the
subset consisting of homotopy equivalences $f$ such that $\pi_{1}(f)=\theta$.

For $\varepsilon=\pm 1$, let
$\operatorname{hEq}(M,N)_{\varepsilon}\subseteq\operatorname{hEq}(M,N)$ denote
the subset of degree-$\varepsilon$ homotopy equivalences, i.e. those that send
$[M]$ to $\varepsilon[N]$. For an isomorphism $\theta\colon F\rightarrow G$,
let
$\operatorname{hEq}_{\theta}(M,N)_{\varepsilon}=\operatorname{hEq}_{\theta}(M,N)\cap\operatorname{hEq}(M,N)_{\varepsilon}$.

######  Lemma 11.2.

Suppose that $X$ is a CW complex of dimension at most $k$, and let $f,g\colon
X\rightarrow L$ be continuous maps. If $\psi\circ f\simeq\psi\circ g\colon
X\rightarrow N$, then $f\simeq g$.

###### Proof.

By Theorem 10.14 (a), $N$ has a CW decomposition such that (up to homotopy)
$\psi$ is an embedding identifying $L$ with the
$\left\lfloor\frac{n}{2}\right\rfloor$-skeleton of $N$.

If $(N,\psi)$ satisfies (SP1) or (SP2), then $N$ has no $(k+1)$-cells.
Therefore if we make the homotopy between $\psi\circ f$ and $\psi\circ g$
cellular, we obtain a homotopy between $f$ and $g$.

If $(N,\psi)$ satisfies (SP3), then we can compose the homotopy between
$\psi\circ f$ and $\psi\circ g$ with the retraction $r\colon N\rightarrow L$
from the proof of Theorem 10.14 (b) to get a homotopy between $f$ and $g$. ∎

######  Lemma 11.3.

There is a well-defined restriction map
$\operatorname{hEq}(M,N)\rightarrow\operatorname{hEq}(K,L)$.

###### Proof.

Again we fix CW decompositions on $M$ and $N$ using Theorem 10.14 (a).

Consider a continuous map $M\rightarrow N$. After cellular approximation it
can be restricted to a map $K\rightarrow L$, and by Lemma 11.2 the
restriction’s homotopy class is independent of the choice of the
approximation. Therefore restriction defines a map $[M,N]\rightarrow[K,L]$.
Similarly we get a map $[N,M]\rightarrow[L,K]$.

Now suppose that $f\colon M\rightarrow N$ is a cellular homotopy equivalence
and $g$ is its cellular homotopy inverse. Then $f\circ
g\simeq\operatorname{Id}_{N}$ and $g\circ f\simeq\operatorname{Id}_{M}$, hence
$f\big{|}_{\varphi(K)}\circ
g\big{|}_{\psi(L)}\simeq\operatorname{Id}_{\psi(L)}\colon\psi(L)\rightarrow N$
and $g\big{|}_{\psi(L)}\circ
f\big{|}_{\varphi(K)}\simeq\operatorname{Id}_{\varphi(K)}\colon\varphi(K)\rightarrow
M$. Lemma 11.2 implies that $f\big{|}_{\varphi(K)}\circ
g\big{|}_{\psi(L)}\simeq\operatorname{Id}_{\psi(L)}\colon\psi(L)\rightarrow\psi(L)$
and $g\big{|}_{\psi(L)}\circ
f\big{|}_{\varphi(K)}\simeq\operatorname{Id}_{\varphi(K)}\colon\varphi(K)\rightarrow\varphi(K)$,
therefore $f\big{|}_{\varphi(K)}\colon\varphi(K)\rightarrow\psi(L)$ is a
homotopy equivalence. ∎

######  Remark 11.4.

The restriction $\alpha\in\operatorname{hEq}(K,L)$ of a map
$f\in\operatorname{hEq}(M,N)$ is characterised by the property that
$\psi\circ\alpha\simeq f\circ\varphi\colon K\rightarrow N$, i.e. the following
diagram is homotopy commutative:

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{N}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\varphi}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$

This implies that $\pi_{1}(\alpha)=\pi_{1}(f)\in\operatorname{Hom}(F,G)$.
Therefore for every isomorphism $\theta\colon F\rightarrow G$ the restriction
map of Lemma 11.3 restricts to a map
$\operatorname{hEq}_{\theta}(M,N)\rightarrow\operatorname{hEq}_{\theta}(K,L)$.

Finally, we will establish the following, which is an equivalent formulation
of Theorem G.

######  Theorem 11.5.

For every isomorphism $\theta\colon F\rightarrow G$ with $w\circ\theta=w_{M}$
there is a commutative diagram

$\textstyle{\operatorname{hEq}_{\theta}(M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{hEq}_{\theta}(K,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}+\tau(N,\psi)-\theta_{*}(\tau(M,\varphi))}$

where the vertical map on the left is given by restriction.

###### Proof.

Fix CW decompositions on $M$ and $N$ as in Theorem 10.14. We denote the
corresponding (split) cellular chain complexes by
$C_{*}(M)=C_{*}(M;\mathbb{Z}F)$ and $C_{*}(N)=C_{*}(N;\mathbb{Z}G)$. Let
$\operatorname{PD}^{M}\colon C^{n-*}(M)\rightarrow C_{*}(M)$ and
$\operatorname{PD}^{N}\colon C^{n-*}(N)\rightarrow C_{*}(N)$ denote the chain
homotopy equivalences given by Poincaré duality. By Theorem 10.14 (c), we have
$\tau(\operatorname{PD}^{M}\big{|}_{C^{\ell}(M)^{n-*}})=\tau(M,\varphi)$ and
$\tau(\operatorname{PD}^{N}\big{|}_{C^{\ell}(N)^{n-*}})=\tau(N,\psi)$. We will
show that there is a commutative diagram

$\textstyle{\operatorname{hEq}_{\theta}(M,N)_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{*}({-})}$$\textstyle{\operatorname{chEq}(C_{*}(M)_{\theta^{-1}},C_{*}(N))_{\operatorname{PD}^{M},\operatorname{PD}^{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{hEq}_{\theta}(K,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{*}({-})}$$\textstyle{\operatorname{chEq}(C_{*}(K)_{\theta^{-1}},C_{*}(L))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}+\tau(N,\psi)-\theta_{*}(\tau(M,\varphi))}$ (11.1)

For the notation see Definitions 2.24 and 8.5. We begin by describing the
three maps not yet defined.

Since $K$ and $L$ are the $\left\lfloor\frac{n}{2}\right\rfloor$-skeletons of
$M$ and $N$ respectively, we have $C^{\ell}_{*}(M)\cong C_{*}(K)$ (hence
$C^{\ell}_{*}(M)_{\theta^{-1}}\cong C_{*}(K)_{\theta^{-1}}$) and
$C^{\ell}_{*}(N)\cong C_{*}(L)$. By Lemma 8.4, there is a restriction map
$\operatorname{chEq}(C_{*}(M)_{\theta^{-1}},C_{*}(N))\to\operatorname{chEq}(C_{*}(K)_{\theta^{-1}},C_{*}(L))$,
and we define the vertical map in the middle to be its restriction to
$\operatorname{chEq}(C_{*}(M)_{\theta^{-1}},C_{*}(N))_{\operatorname{PD}^{M},\operatorname{PD}^{N}}$.

Suppose that $f\in\operatorname{hEq}_{\theta}(K,L)$, then $f$ induces a chain
map $C_{*}(K;\mathbb{Z}G^{\theta})\rightarrow C_{*}(L;\mathbb{Z}G)$. We have
$C_{*}(K;\mathbb{Z}G^{\theta})\cong C_{*}(K;\mathbb{Z}F)_{\theta^{-1}}$ (see
Lemma 2.25 (a)), so taking the induced chain map defines a map
$C_{*}({-})\colon\operatorname{hEq}_{\theta}(K,L)\rightarrow\operatorname{chEq}(C_{*}(K)_{\theta^{-1}},C_{*}(L))$.

Similarly, a homotopy equivalence $f\in\operatorname{hEq}_{\theta}(M,N)$
induces maps

$\displaystyle C_{*}(f)$ $\displaystyle\colon
C_{*}(M;\mathbb{Z}G^{\theta})\cong C_{*}(M)_{\theta^{-1}}\to
C_{*}(N;\mathbb{Z}G)=C_{*}(N)$ $\displaystyle C^{*}(f)$ $\displaystyle\colon
C^{*}(N;\mathbb{Z}G)=C^{*}(N)\to C^{*}(M;\mathbb{Z}G^{\theta})\cong
C^{*}(M)_{\theta^{-1}}$

where the last isomorphism follows from Lemma 2.25 (b). Moreover,
$C^{*}(M;\mathbb{Z}G^{\theta})$ is the dual of $C_{*}(M)_{\theta^{-1}}$, and
$C^{*}(f)$ is identified with the dual of $C_{*}(f)$. The chain homotopy
equivalence $C^{n-*}(M;\mathbb{Z}G^{\theta})\cong
C^{n-*}(M)_{\theta^{-1}}\rightarrow C_{*}(M;\mathbb{Z}G^{\theta})\cong
C_{*}(M)_{\theta^{-1}}$ given by Poincaré duality (with $\mathbb{Z}G^{\theta}$
coefficients) is identified with $\operatorname{PD}^{M}$ under the
identification
$\operatorname{Hom}(C^{n-*}(M)_{\theta^{-1}},C_{*}(M)_{\theta^{-1}})=\operatorname{Hom}(C^{n-*}(M),C_{*}(M))$.
If $f$ has degree $1$, i.e. $f_{*}([M])=[N]$, then it induces a homotopy
commutative diagram

$\textstyle{C^{n-*}(M)_{\theta^{-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{PD}^{M}}$$\textstyle{C^{n-*}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C^{n-*}(f)}$$\scriptstyle{\operatorname{PD}^{N}}$$\textstyle{C_{*}(M)_{\theta^{-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{*}(f)}$$\textstyle{C_{*}(N)}$

since Poincaré duality can be defined by taking cap product with the
fundamental class. Therefore we get a restricted map
$C_{*}({-})\colon\operatorname{hEq}_{\theta}(M,N)_{1}\rightarrow\operatorname{chEq}(C_{*}(M)_{\theta^{-1}},C_{*}(N))_{\operatorname{PD}^{M},\operatorname{PD}^{N}}$.

Next we verify that the diagram in (11.1) commutes. The square on the left
commutes since both downward pointing arrows are defined by restriction. The
square on the right commutes by Lemma 8.6. Note that if
$\operatorname{PD}^{M}\big{|}_{C^{\ell}(M)^{n-*}}$ is regarded as a chain
homotopy equivalence $C^{\ell}(M)^{n-*}_{\theta^{-1}}\rightarrow
C^{u}_{*}(M)_{\theta^{-1}}$ (instead of $C^{\ell}(M)^{n-*}\rightarrow
C^{u}_{*}(M)$), then its Whitehead torsion is $\theta_{*}(\tau(M,\varphi))$
(instead of $\tau(M,\varphi)$), see Lemma 2.26.

The Whitehead torsion of a homotopy equivalence is defined as the Whitehead
torsion of the induced chain homotopy equivalence, so we get a commutative
diagram

$\textstyle{\operatorname{hEq}_{\theta}(M,N)_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{hEq}_{\theta}(K,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}+\tau(N,\psi)-\theta_{*}(\tau(M,\varphi))}$

We can apply the same argument to $-N$ instead of $N$ (where $-N$ is the same
manifold $N$ with the opposite twisted fundamental class $[-N]=-[N]$). Then we
get a commutative diagram

$\textstyle{\operatorname{hEq}_{\theta}(M,N)_{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{hEq}_{\theta}(K,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}+\tau(N,\psi)-\theta_{*}(\tau(M,\varphi))}$

because
$\operatorname{hEq}_{\theta}(M,-N)_{1}=\operatorname{hEq}_{\theta}(M,N)_{-1}$
and $\tau(-N,\psi)=\tau(N,\psi)$ (because the definition of $\tau(N,\psi)$
does not depend on the choice of the twisted fundamental class).

Since
$\operatorname{hEq}_{\theta}(M,N)=\operatorname{hEq}_{\theta}(M,N)_{1}\sqcup\operatorname{hEq}_{\theta}(M,N)_{-1}$,
we can combine the two diagrams to get the diagram in the statement. ∎

###  12\. Applications

Now we consider some applications of Theorem G and prove the results announced
in Section 1.5.

####  12.1. Simple doubles

Let $(M,\varphi)$ be a polarised manifold such that $\tau(M,\varphi)$ is
defined. We say that $(M,\varphi)$ is _simple_ if $\tau(M,\varphi)=0$. Recall
from Corollaries 10.8 and 10.11 that if $(M,\varphi)$ has a trivial double
structure, then it is simple, and if $(M,\varphi)$ is simple, then it has a
twisted double structure. We now state a consequence of Theorem 11.5 in the
special case when $(M,\varphi)$ is simple, and then use it to prove Theorem
1.7.

######  Theorem 12.1.

Suppose that $M$ and $N$ are $n$-manifolds, $K$ and $L$ are CW complexes of
dimension $($at most$)$ $k$, and $\varphi\colon K\to M$ and $\psi\colon L\to
N$ are continuous maps such that $(M,\varphi)$ and $(N,\psi)$ are SP
manifolds. Let $G=\pi_{1}(N)$ with orientation character $w\colon
G\rightarrow\left\\{\pm 1\right\\}$, and identify $\pi_{1}(L)$ with $G$ via
$\psi$. Suppose that $\tau(M,\varphi)=0$. Then there is a commutative diagram

$\textstyle{\operatorname{hEq}(M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{hEq}(K,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}+\tau(N,\psi)}$

where the vertical map on the left is given by restriction.

###### Proof.

We have
$\operatorname{hEq}(M,N)=\bigsqcup_{\theta}\operatorname{hEq}_{\theta}(M,N)$,
where the union ranges over all isomorphisms
$\theta\colon\pi_{1}(M)\rightarrow G$ with $w\circ\theta=w_{M}$, where $w_{M}$
is the orientation character of $M$. For each such $\theta$ we can apply
Theorem 11.5, and since $\theta_{*}(\tau(M,\varphi))=0$, we can combine the
resulting diagrams to get the diagram in the statement. ∎

The following is obtained by further specialising.

######  Theorem 12.2.

Suppose that $M$ is an $n$-manifold, $K$ is a CW complex of dimension (at
most) $k$, and $\varphi\colon K\to M$ is a continuous map such that
$(M,\varphi)$ is an SP manifold. Let $G=\pi_{1}(M)$ with orientation character
$w\colon G\rightarrow\left\\{\pm 1\right\\}$, and identify $\pi_{1}(K)$ with
$G$ via $\varphi$. Suppose that $\tau(M,\varphi)=0$. Then there is a
commutative diagram

$\textstyle{\operatorname{hAut}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{hAut}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}}$

where the vertical map on the left is given by restriction.

###### Proof.

Apply Theorem 12.1 to the case where $M=N$, $K=L$, and $\varphi=\psi$. ∎

In particular, this implies that $\tau(g)\in\mathcal{I}_{n}(G,w)$ for every
$g\in\operatorname{hAut}(M)$, hence $T(M)\subseteq\mathcal{I}_{n}(G,w)$ and
$U(M)=\\{0\\}$ (see Definition 4.19).

We use this to show the following theorem, which implies Theorem 1.7. For the
definition of the map $\psi$, see Section 3.2.

######  Theorem 12.3 (cf. Hausmann [hausmann-mwmdh, Sections 9–10]).

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 an $n$-manifold $M$ with fundamental group
$G$ and orientation character $w$ such that
$|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|>1$.

###### Proof.

Let $K$ be a finite $2$-dimensional CW complex with $\pi_{1}(K)\cong G$ (and
we fix an isomorphism). Let $\nu$ be a stable vector bundle over $K$ with
orientation character $w$. By Lemma 9.2 and Remark 9.3 there is an
$n$-dimensional thickening $f_{T}\colon K\to T$ such that
$f_{T}^{*}(\nu_{T})\cong\nu$. Let $M=T\cup_{g}T$ be a twisted double of $T$
such that $(M,\varphi)$ is an SP manifold and $\tau(M,\varphi)=0$, where
$\varphi$ is the composition of $f_{T}$ and the inclusion $T\to M$ of the
first component (e.g. let $g=\operatorname{Id}_{\partial T}$). Then
$\pi_{1}(M)\cong G$ with orientation character $w$.

It follows from Theorem 12.2 that $U(M)=\\{0\\}$. So
$\operatorname{Im}(\psi)\setminus U(M)$ is nonempty, and by Proposition 4.21
this implies that $|\mathcal{M}^{h}_{s,\operatorname{hCob}}(M)|>1$. ∎

We saw in Proposition 3.12 that for $G=C_{\infty}\times C_{m}$, $w=0$, and $n$
even, that the map $\psi$ can be nontrivial. So Theorem 12.3 applies in this
case.

####  12.2. Doubles over manifolds

We now consider the special case of Theorem 11.5 when $K$ and $L$ are closed
$k$-manifolds. We get an especially nice statement when $n-k$ is odd. Then we
present some applications, proving Theorems H and then C.

######  Theorem 12.4.

Suppose that $M$ and $N$ are $n$-manifolds, $K$ and $L$ are $k$-manifolds and
$\varphi\colon K\to M$ and $\psi\colon L\to N$ are continuous maps such that
$(M,\varphi)$ and $(N,\psi)$ are SP manifolds. Let $F=\pi_{1}(M)$ and
$G=\pi_{1}(N)$, and identify $\pi_{1}(K)$ with $F$ and $\pi_{1}(L)$ with $G$
via $\varphi$ and $\psi$ respectively. Let $w_{M}\colon
F\rightarrow\left\\{\pm 1\right\\}$ be the orientation character of $M$ and
let $w=w_{N}\colon G\rightarrow\left\\{\pm 1\right\\}$ be the orientation
character of $N$. Assume that the orientation character of $L$ is also $w$.

  1. (a)

If $n-k$ is odd, then $\tau(f)=\tau(N,\psi)-f_{*}(\tau(M,\varphi))$ for every
homotopy equivalence $f\colon M\to N$.

  2. (b)

If $n-k$ is even, then for every isomorphism $\theta\colon F\rightarrow G$
with $w\circ\theta=w_{M}$ there is a commutative diagram

$\textstyle{\operatorname{hEq}_{\theta}(M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)}$$\textstyle{\operatorname{hEq}_{\theta}(K,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\operatorname{Wh}(G,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
2x+\tau(N,\psi)-\theta_{*}(\tau(M,\varphi))}$

###### Proof.

Let $f\colon M\to N$ be a homotopy equivalence. By Lemma 11.3 it restricts to
a homotopy equivalence $\alpha\colon K\rightarrow L$. By Theorem 11.5 we have
$\tau(f)=\tau(\alpha)-(-1)^{n}\overline{\tau(\alpha)}+\tau(N,\psi)-f_{*}(\tau(M,\varphi))\in\operatorname{Wh}(G,w)$.
Since $K$ and $L$ are $k$-manifolds and the orientation character of $L$ is
$w$, by Proposition 2.34 we see that $\tau(\alpha)\in\mathcal{J}_{k}(G,w)$.
That is,
$\tau(\alpha)=-(-1)^{k}\overline{\tau(\alpha)}\in\operatorname{Wh}(G,w)$.
Therefore

$\tau(f)=\tau(\alpha)-(-1)^{n}\overline{\tau(\alpha)}+\tau(N,\psi)-f_{*}(\tau(M,\varphi))=\tau(\alpha)+(-1)^{n-k}\tau(\alpha)+\tau(N,\psi)-f_{*}(\tau(M,\varphi)).\qed$

We will use this to prove Theorem H.

######  Theorem 12.5.

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.

###### Proof.

Let $n=j+k$ be the dimension of $M$ and $N$. It follows from the assumptions
that $j\geq\max(3,k+1)$, so $n\geq\max(k+3,2k+1)$.

The manifold $M$ is the sphere bundle of some orientable rank $(j+1)$ vector