bundle $\xi$ over $K$. Since $j+1>k$, there is a rank $j$ vector bundle
$\xi_{0}$ such that $\xi\cong\xi_{0}\oplus\varepsilon^{1}$, where
$\varepsilon^{1}$ denotes the trivial rank $1$ bundle. Let $T$ be the disc
bundle of $\xi_{0}$. Then $T$ is a thickening of $K$ (with the zero-section
$K\rightarrow T$) and $M\approx T\cup_{\operatorname{Id}_{\partial T}}T$. This
means that if $\varphi$ denotes the composition $K\rightarrow T\rightarrow M$
of the zero section and the inclusion of the first component, then
$(M,\varphi)$ has a trivial double structure, so it satisfies (SP3).
Similarly, $(N,\psi)$ satisfies (SP3) for the analogous $\psi\colon
L\rightarrow N$.

Let $G=\pi_{1}(N)$ and let $w\colon G\rightarrow\left\\{\pm 1\right\\}$ be the
orientation character of $N$. Then $\pi_{1}(\psi)$ determines an
identification $\pi_{1}(L)\cong G$, and since $\xi$ is orientable, $w$ is also
the orientation character of $L$.

Since $M$ and $N$ are trivial doubles, $\tau(M,\varphi)=0$ and
$\tau(N,\psi)=0$ by Corollary 10.8. Since $n-k=j$ is odd, Theorem 12.4 (a)
implies that $\tau(f)=0$ for every homotopy equivalence $f\colon M\to N$. ∎

We obtain the following two theorems as immediate corollaries.

######  Theorem 12.6.

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
sphere bundles. If $M$ and $N$ are homotopy equivalent, then they are simple
homotopy equivalent.

######  Theorem 12.7.

Suppose that $j>k$ are positive integers and $j$ is odd. Let $K$ be a
$k$-manifold and let $S^{j}\to M\to K$ be an orientable sphere bundle. Let
$G=\pi_{1}(M)$ with orientation character $w\colon G\rightarrow\left\\{\pm
1\right\\}$. Then $T(M)=\\{0\\}$ (see Definition 4.19).

######  Remark 12.8.

In the three theorems above, the sphere bundle $M$ could be replaced with any
twisted double $T\cup_{g}T$ of the disc bundle $T$ of an orientable rank $j$
bundle $\xi_{0}$ over $K$ such that $(M,\varphi)$ is an SP manifold and
$\tau(M,\varphi)=0$, where $\varphi$ is the composition of the zero section
$K\rightarrow T$ and the inclusion $T\to M$ of the first component (and
similarly for $N$).

We will use Theorem 12.7 to prove Theorem C.

######  Theorem 12.9.

Let $n\geq 11$ or $n=9$ and
$\operatorname{CAT}\in\\{\operatorname{TOP},\operatorname{PL},\operatorname{Diff}\\}$.
Let $G$ be a finitely presented group with an orientation character $w\colon
G\rightarrow\left\\{\pm 1\right\\}$. Then the following are equivalent.

  1. (a)

There is a closed smooth $n$-manifold $M$ with fundamental group $G$ and
orientation character $w$ such that
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$.

  2. (b)

There is a closed $\operatorname{CAT}$ $n$-manifold $M$ with fundamental group
$G$ and orientation character $w$ such that
$|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$.

  3. (c)

We have $\mathcal{I}_{n}(G,w)\neq 0$.

###### Proof.

(c) $\Rightarrow$ (a) Under the assumptions on $n$ there is an integer $k\geq
4$ such that $n-k>k$ and $n-k$ is odd. For instance, we can take $k=4$ for
$n\geq 9$ odd and $k=5$ for $n\geq 12$ even. Since $k\geq 4$, there is a
closed $k$-manifold $K$ with $\pi_{1}(K)\cong G$ and orientation character
$w$. Let $M$ be an orientable $S^{n-k}$-bundle over $K$ (e.g. $K\times
S^{n-k}$). Then $\pi_{1}(M)\cong G$ with orientation character $w$, and by
Theorem 12.7, it follows that $T(M)=\\{0\\}$. So
$\mathcal{I}_{n}(G,w)\setminus T(M)$ is nonempty, and by Proposition 4.20 this
implies that $|\mathcal{M}^{\operatorname{hCob}}_{s}(M)|>1$.

(a) $\Rightarrow$ (b) This implication holds because a smooth manifold has a
$\operatorname{CAT}$ structure.

(b) $\Rightarrow$ (c) If $f$ is the homotopy equivalence induced by an
$h$-cobordism between $n$-manifolds with fundamental group $G$ and orientation
character $w$, then $\tau(f)\in\mathcal{I}_{n}(G,w)$ by Proposition 2.37. If
the manifolds are not simple homotopy equivalent, then $\tau(f)\neq 0$, so
$\mathcal{I}_{n}(G,w)\neq 0$. ∎

####  12.3. Doubles over certain $2$-complexes

In this section we will consider specific $2$-dimensional CW complexes for
which it is known by work of Metzler [Me79] that the Whitehead torsions of
their homotopy automorphisms are contained in a certain subgroup of the
Whitehead group. We can exploit this property by applying Theorem 12.2 to
doubles over such $2$-complexes, leading to a proof of Theorem 1.10.

Recall that for any group presentation $\mathcal{P}=\langle
g_{1},\ldots,g_{s}\mid r_{1},\ldots,r_{t}\rangle$ there is a corresponding
presentation complex, denoted by $X_{\mathcal{P}}$, which consists of one
$0$-cell, one $1$-cell for each generator $g_{i}$, and one $2$-cell for each
relator $r_{j}$, with gluing map determined by $r_{j}$.

Also recall that, by Theorem 5.9, there is an isomorphism

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

The following is a generalisation of the main result of [Me79].

######  Theorem 12.10.

Let $m\geq 1$ and let $X:=X_{\mathcal{P}}$, where $\mathcal{P}=\langle x,y\mid
y^{m},[x,y]\rangle$ is the standard presentation for $C_{\infty}\times C_{m}$.
Then the composition

$\operatorname{hAut}(X)\xrightarrow[]{\tau}\operatorname{Wh}(C_{\infty}\times
C_{m})\twoheadrightarrow\widetilde{K}_{0}(\mathbb{Z}C_{m})$

is the zero map. In particular,
$\tau(\operatorname{hAut}(X))\subseteq\operatorname{Wh}(C_{m})\oplus\\{0\\}\oplus
NK_{1}(\mathbb{Z}C_{m})^{2}\subseteq\operatorname{Wh}(C_{\infty}\times
C_{m})$.

###### Proof.

Let $G=C_{\infty}\times C_{m}$ and let
$\psi\colon\mathbb{Z}C_{m}\twoheadrightarrow\mathbb{Z}C_{m}/\Sigma$ be the
quotient map, factoring out by the ideal generated by
$\Sigma:=\sum_{i=0}^{m-1}y^{i}$, the group norm in $\mathbb{Z}C_{m}$. Since
$\mathbb{Z}G\cong(\mathbb{Z}C_{m})[C_{\infty}]$ and
$\mathbb{Z}G/\Sigma\cong(\mathbb{Z}C_{m}/\Sigma)[C_{\infty}]$, $\psi$ induces
a map $\Psi\colon\mathbb{Z}G\twoheadrightarrow\mathbb{Z}G/\Sigma$. Metzler
showed [Me79, Lemma 2] that the composition

$\operatorname{hAut}(X)\xrightarrow[]{\tau}\operatorname{Wh}(G)\cong
K_{1}(\mathbb{Z}G)/{\pm
G}\xrightarrow[]{\Psi_{*}}K_{1}(\mathbb{Z}G/\Sigma)/(\mathbb{Z}G/\Sigma)^{\times}$

is the zero map. Since
$\mathbb{Z}G/\Sigma\cong(\mathbb{Z}C_{m}/\Sigma)[t,t^{-1}]$, we set
$R=\mathbb{Z}C_{m}/\Sigma$, so that $\mathbb{Z}G/\Sigma\cong R[t,t^{-1}]$. A
variant of the Bass-Heller-Swan decomposition for arbitrary Laurent polynomial
rings $R[t,t^{-1}]$ [We13, III.3.6] (cf. Section 5.2) implies that

$K_{1}(\mathbb{Z}G/\Sigma)/(\mathbb{Z}G/\Sigma)^{\times}\cong(K_{1}(\mathbb{Z}C_{m}/\Sigma)/(\mathbb{Z}C_{m}/\Sigma)^{\times})\oplus\widetilde{K}_{0}(\mathbb{Z}C_{m}/\Sigma)\oplus
NK_{1}(\mathbb{Z}C_{m}/\Sigma)^{2}.$

The map $\Psi_{*}$ respects this splitting and so we have that:

$\displaystyle\tau(\operatorname{hAut}(X))\subseteq$
$\displaystyle\ker(\psi_{*}\colon\operatorname{Wh}(C_{m})\to
K_{1}(\mathbb{Z}G/\Sigma)/(\mathbb{Z}G/\Sigma)^{\times})$
$\displaystyle\quad\oplus\ker(\psi_{*}\colon\widetilde{K}_{0}(\mathbb{Z}C_{m})\to\widetilde{K}_{0}(\mathbb{Z}C_{m}/\Sigma))\oplus\ker(\psi_{*}\colon
NK_{1}(\mathbb{Z}C_{m})\to NK_{1}(\mathbb{Z}C_{m}/\Sigma))^{2}.$

It therefore suffices to prove that
$\psi_{*}\colon\widetilde{K}_{0}(\mathbb{Z}C_{m})\to\widetilde{K}_{0}(\mathbb{Z}C_{m}/\Sigma)$
is injective. To see this, consider the following pullback square of rings

${\mathbb{Z}C_{m}}$${\mathbb{Z}C_{m}/\Sigma}$${\mathbb{Z}}$${\mathbb{Z}/m}$$\scriptstyle{\psi}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon^{\prime}}$$\scriptstyle{\psi^{\prime}}$
(12.1)

where $\psi,\psi^{\prime}$ are the quotient maps and
$\varepsilon,\varepsilon^{\prime}$ are induced by augmentation. Note that
(12.1) has the property that at least one of the maps $\psi^{\prime}$ and
$\varepsilon^{\prime}$ is surjective, i.e. (12.1) is a Milnor square. It
follows from [CR87, Theorem 42.13] that (12.1) induces a long exact sequence:

$K_{1}(\mathbb{Z})\oplus
K_{1}(\mathbb{Z}C_{m}/\Sigma)\xrightarrow[]{(\psi^{\prime}_{*},\varepsilon^{\prime}_{*})}K_{1}(\mathbb{Z}/m)\xrightarrow[]{\partial}\widetilde{K}_{0}(\mathbb{Z}C_{m})\xrightarrow[]{(\epsilon_{*},\psi_{*})}\widetilde{K}_{0}(\mathbb{Z})\oplus\widetilde{K}_{0}(\mathbb{Z}C_{m}/\Sigma).$

More specifically, by [CR87, Theorem 42.13], we obtain an exact sequence with
$\widetilde{K}_{0}$ replaced by $K_{0}$ throughout. It is clear from the
definition of $\partial$ that its image lies in
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$ and, from this, we obtain the exact
sequence above.

By [CR87, p. 343], we have
$\operatorname{Im}(\partial)=T(C_{m})\subseteq\widetilde{K}_{0}(\mathbb{Z}C_{m})$
where $T(G)\subseteq\widetilde{K}_{0}(\mathbb{Z}G)$ denotes the Swan subgroup
of a finite group $G$. By [CR87, Proposition 53.6 (iii)], we have $T(C_{m})=0$
and so $\partial=0$. Since $\mathbb{Z}$ is a PID, we have
$\widetilde{K}_{0}(\mathbb{Z})=0$. Hence
$\psi_{*}\colon\widetilde{K}_{0}(\mathbb{Z}C_{m})\to\widetilde{K}_{0}(\mathbb{Z}C_{m}/\Sigma)$
is injective. It follows that the composition in the statement of the theorem
is the zero map. ∎

This implies that the map
$\tau\colon\operatorname{hAut}(X)\to\operatorname{Wh}(C_{\infty}\times C_{m})$
is not surjective when $\widetilde{K}_{0}(\mathbb{Z}C_{m})\neq 0$, which is a
broad generalisation of [Me79, Theorem 1].

######  Corollary 12.11.

Let $n\geq 5$ and $m\geq 1$. Suppose that $M$ is an $n$-manifold and
$\varphi\colon X_{\mathcal{P}}\rightarrow M$ is a continuous map such that
$(M,\varphi)$ is an SP manifold and $\tau(M,\varphi)=0$, where
$\mathcal{P}=\langle x,y\mid y^{m},[x,y]\rangle$ is the standard presentation
for $C_{\infty}\times C_{m}$. Then the composition

$\operatorname{hAut}(M)\xrightarrow[]{\tau}\operatorname{Wh}(C_{\infty}\times
C_{m})\twoheadrightarrow\widetilde{K}_{0}(\mathbb{Z}C_{m})$

is the zero map.

###### Proof.

Consider the diagram

${\operatorname{hAut}(M)}$${\operatorname{Wh}(C_{\infty}\times
C_{m})}$${\widetilde{K}_{0}(\mathbb{Z}C_{m})}$${\operatorname{hAut}(X_{\mathcal{P}})}$${\operatorname{Wh}(C_{\infty}\times
C_{m})}$${\widetilde{K}_{0}(\mathbb{Z}C_{m})}$$\scriptstyle{\tau}$$\scriptstyle{\tau}$$\scriptstyle{x\mapsto
x-(-1)^{n}\overline{x}}$$\scriptstyle{x\mapsto x-(-1)^{n}\overline{x}}$

where the vertical map on the left is given by restriction. The first square
commutes by Theorem 12.2. The second square commutes by Theorem 5.9. Since the
composition of the maps in the bottom row vanishes by Theorem 12.10, the
commutativity of the diagram implies that the composition of the maps in the
top row vanishes too. ∎

######  Remark 12.12.

By the first square of the diagram, we also have
$T(M)\subseteq\mathcal{I}_{n}(C_{\infty}\times C_{m})$. So, by combining
Proposition 5.10 with Corollary 12.11, we get that
$T(M)\subseteq\mathcal{I}_{n}(C_{m})\oplus\\{0\\}\oplus
NK_{1}(\mathbb{Z}C_{m})$, where $NK_{1}(\mathbb{Z}G)$ is embedded into
$NK_{1}(\mathbb{Z}G)^{2}$ by the map $x\mapsto(x,-(-1)^{n}\overline{x})$.

######  Theorem 12.13.

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

###### Proof.

Let $\mathcal{P}=\langle x,y\mid y^{m},[x,y]\rangle$, and let $T$ be an
oriented thickening of $X_{\mathcal{P}}$ (e.g. a regular neighbourhood of an
embedding $X_{\mathcal{P}}\rightarrow\mathbb{R}^{n}$). 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 $\varphi\colon K\to M$ denotes the composition of
$f_{T}$ and the inclusion $T\to M$ of the first component (e.g. let
$g=\operatorname{Id}_{\partial T}$). Then
$\pi_{1}(M)\cong\pi_{1}(X_{\mathcal{P}})\cong C_{\infty}\times C_{m}$ and $M$
is orientable.

By Proposition 4.20 it is enough to show that
$\mathcal{I}_{n}(C_{\infty}\times C_{m})\setminus T(M)$ is nonempty. It
follows from Proposition 5.10 and Corollary 12.11 that
$\mathcal{I}_{n}(C_{\infty}\times C_{m})\setminus T(M)$ contains
$\\{x-(-1)^{n}\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}\setminus\\{0\\}$, which is nonempty
by our assumption. ∎

####  12.4. The dependence of $\tau(M,\varphi)$ on $\varphi$

Theorem 11.5 allows us to describe the set of possible values of
$\tau(M,\varphi)$ for a fixed $M$.

######  Proposition 12.14.

Suppose that $n\geq\max(6,2k+2)$, $M$ is a closed $n$-manifold and
$G=\pi_{1}(M)$ with orientation character $w\colon G\rightarrow\left\\{\pm
1\right\\}$. Let $\varphi\colon K\rightarrow M$ be an
$\left\lfloor\frac{n}{2}\right\rfloor$-connected map for some CW complex $K$
of dimension $k$. Then $\tau(M,\varphi)\in\mathcal{J}_{n}(G,w)$.

###### Proof.

By Proposition 10.2 there is a generalised double structure $h\colon
T\cup_{g_{0}}W\cup_{g_{1}}T\rightarrow M$ on $(M,\varphi)$. Let $i_{1}$,
$i_{2}$, and $i_{3}$ denote the inclusions of the three components of
$T\cup_{g_{0}}W\cup_{g_{1}}T$. Let
$d_{0}\colon\partial_{1}W\rightarrow\partial_{0}W$ be the composition of the
inclusion $\partial_{1}W\rightarrow W$ and the homotopy inverse of
$\partial_{0}W\rightarrow W$. By the construction of $d_{0}$, we have
$i_{2}\big{|}_{\partial_{1}W}\simeq i_{2}\big{|}_{\partial_{0}W}\circ
d_{0}\colon\partial_{1}W\rightarrow T\cup_{g_{0}}W\cup_{g_{1}}T$. Hence, with
the notation $d\coloneqq g_{0}\circ d_{0}\circ
g_{1}\in\operatorname{hAut}(\partial T)$, and using that $g_{1}$ and $g_{0}$
are the gluing maps, we have

$i_{3}\big{|}_{\partial T}=i_{2}\big{|}_{\partial_{1}W}\circ g_{1}\simeq
i_{2}\big{|}_{\partial_{0}W}\circ d_{0}\circ g_{1}=i_{1}\big{|}_{\partial
T}\circ g_{0}\circ d_{0}\circ g_{1}=i_{1}\big{|}_{\partial T}\circ
d\colon\partial T\rightarrow T\cup_{g_{0}}W\cup_{g_{1}}T.$ (12.2)

By Proposition 10.6 there is a homotopy automorphism $\alpha\colon
K\rightarrow K$ such that

$\varphi\circ\alpha\simeq h\circ i_{3}\circ f_{T}$

and

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

By Remark 10.13, $(\partial T,f_{\partial T})$ is an SP manifold, so by Lemma
11.3 there is a restriction map $\operatorname{hAut}(\partial
T)\rightarrow\operatorname{hAut}(K)$. The homotopy automorphism $\alpha$ is
the restriction of $d$. To see this recall Remark 11.4 and note that

$h\circ i_{1}\circ f_{\partial T}\circ\alpha\simeq h\circ i_{1}\circ
f_{T}\circ\alpha\simeq\varphi\circ\alpha\simeq h\circ i_{3}\circ f_{T}\simeq
h\circ i_{3}\circ f_{\partial T}\simeq h\circ i_{1}\circ d\circ f_{\partial
T}\colon K\rightarrow M,$

using Definition 10.1, the definition of $\alpha$, and (12.2). The map $h\circ
i_{1}\big{|}_{\partial T}\colon\partial T\rightarrow M$ is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected, because it is the
composition of the inclusion $\partial T\rightarrow T$ (which is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected by Lemma 9.5) and $h\circ
i_{1}\colon T\rightarrow M$ (which is also
$\left\lfloor\frac{n}{2}\right\rfloor$-connected, because $h\circ i_{1}\circ
f_{T}\simeq\varphi$ and $f_{T}$ is a homotopy equivalence). Since $K$ has
dimension $k\leq\left\lfloor\frac{n}{2}\right\rfloor-1$, we get that
$f_{\partial T}\circ\alpha\simeq d\circ f_{\partial T}\colon
K\rightarrow\partial T$, and this means that $\alpha$ is the restriction of
$d$.

Since $(\partial T,f_{\partial T})$ has a trivial double structure,
$\tau(\partial T,f_{\partial T})=0$. It follows from Theorem 11.5 that
$\tau(d)=(f_{\partial
T})_{*}(\tau(\alpha)-(-1)^{n-1}\overline{\tau(\alpha)})$. Let $j=h\circ
i_{1}\big{|}_{\partial T}\colon\partial T\rightarrow M$. Then we have

$\displaystyle j_{*}(\tau(d))$ $\displaystyle=(h\circ i_{1}\circ f_{\partial
T})_{*}(\tau(\alpha)-(-1)^{n-1}\overline{\tau(\alpha)})=(h\circ i_{1}\circ
f_{T})_{*}(\tau(\alpha)-(-1)^{n-1}\overline{\tau(\alpha)})$
$\displaystyle=\varphi_{*}(\tau(\alpha)-(-1)^{n-1}\overline{\tau(\alpha)}).$

Again we used that $h\circ i_{1}\circ f_{T}\simeq\varphi$ (Definition 10.1),
to obtain the last equality.

By Proposition 2.37 (b), we have
$\tau(d_{0})=\tau(W,\partial_{1}W)-(-1)^{n-1}\overline{\tau(W,\partial_{1}W)}$
where $\pi_{1}(\partial_{0}W)$ is identified with $\pi_{1}(W)$ via the
inclusion. Since $g_{0}$ and $g_{1}$ are diffeomorphisms, we have
$\tau(d)=(g_{0})_{*}(\tau(d_{0}))$, by Proposition 2.30 and Theorem 2.5. Hence

$\displaystyle j_{*}(\tau(d))$ $\displaystyle=(h\circ i_{1}\circ
g_{0})_{*}(\tau(d_{0}))=(h\circ i_{2})_{*}(\tau(d_{0}))$
$\displaystyle=(h\circ
i_{2})_{*}(\tau(W,\partial_{1}W))-(-1)^{n-1}\overline{(h\circ
i_{2})_{*}(\tau(W,\partial_{1}W))}.$

By combining the above formulae we obtain

$\displaystyle\tau(M,$
$\displaystyle\varphi)-(-1)^{n-1}\overline{\tau(M,\varphi)}$
$\displaystyle=(h\circ
i_{2})_{*}(\tau(W,\partial_{1}W))-(-1)^{n-1}\overline{(h\circ
i_{2})_{*}(\tau(W,\partial_{1}W))}-\varphi_{*}(\tau(\alpha))+(-1)^{n-1}\overline{\varphi_{*}(\tau(\alpha)})$
$\displaystyle=j_{*}(\tau(d))-j_{*}(\tau(d))=0.$

Therefore $\tau(M,\varphi)=-(-1)^{n}\overline{\tau(M,\varphi)}$, i.e.
$\tau(M,\varphi)\in\mathcal{J}_{n}(G,w)$. ∎

If $n<\max(6,2k+2)$ and $(M,\varphi)$ is an SP manifold, then it satisfies
(SP3), so $\tau(M,\varphi)=0$. Therefore
$\tau(M,\varphi)\in\mathcal{J}_{n}(G,w)$ for every SP manifold $(M,\varphi)$.

######  Remark 12.15.

The proof of Proposition 12.14 shows that $\tau(M,\varphi)$ can be regarded as
a secondary invariant of an inertial $h$-cobordism on $\partial T$. An
inertial $h$-cobordism on $\partial T$ is an $h$-cobordism between two copies
of $\partial T$, more precisely, an $h$-cobordism $W$ with $\partial
W=\partial_{0}W\sqcup\partial_{1}W$ together with diffeomorphisms
$g_{0}\colon\partial_{0}W\rightarrow\partial T$ and $g_{1}\colon\partial
T\rightarrow\partial_{1}W$. If $d\colon\partial T\rightarrow\partial T$ is the
homotopy equivalence induced by $W$, $g_{0}$ and $g_{1}$, then
$\tau(d)\in\mathcal{I}_{n-1}(G,w)$ for two different reasons:
$\tau(d)=\tau(W,\partial_{1}W)-(-1)^{n-1}\overline{\tau(W,\partial_{1}W)}$ by
Proposition 2.37 (b), and
$\tau(d)=\tau(\alpha)-(-1)^{n-1}\overline{\tau(\alpha)}$ by Theorem 11.5,
where $\alpha\in\operatorname{hAut}(K)$ is the restriction of $d$. In general
$\tau(W,\partial_{1}W)\neq\tau(\alpha)$ (the former does not depend on the
diffeomorphisms $g_{0}$ and $g_{1}$, but $\alpha$ does), and
$\tau(W,\partial_{1}W)-\tau(\alpha)=\tau(M,\varphi)$ for
$M=T\cup_{g_{0}}W\cup_{g_{1}}T$ and $\varphi=i_{1}\circ f_{T}$. So
$\tau(M,\varphi)$ is a secondary invariant in the sense that it equals the
difference between two reasons that $\tau(d)\in\mathcal{I}_{n-1}(G,w)$, i.e.
two cochains in the Tate cochain group
$\widehat{C}^{n-1}(C_{2};\operatorname{Wh}(G,w))$ mapping to $\tau(d)$ under
the coboundary map.

######  Proposition 12.16.

Suppose that $M$ is a closed $n$-manifold, $G=\pi_{1}(M)$ and $w\colon
G\rightarrow\left\\{\pm 1\right\\}$ is the orientation character of $M$. Let
$\varphi\colon K\rightarrow M$ and $\varphi^{\prime}\colon
K^{\prime}\rightarrow M$ be continuous maps such that $(M,\varphi)$ and
$(M,\varphi^{\prime})$ are SP manifolds. Then
$\tau(M,\varphi)-\tau(M,\varphi^{\prime})\in\mathcal{I}_{n}(G,w)$.

###### Proof.

We will apply Theorem 11.5 to $\operatorname{Id}_{M}$, regarded as a map
between the SP manifolds $(M,\varphi)$ and $(M,\varphi^{\prime})$. Then
$\operatorname{Id}_{M}\in\operatorname{hEq}_{\operatorname{Id}}(M,M)$, and it
has a restriction $f\in\operatorname{hEq}_{\operatorname{Id}}(K,K^{\prime})$,
where $\pi_{1}(K)$ and $\pi_{1}(K^{\prime})$ are identified with $G$ via
$\varphi$ and $\varphi^{\prime}$. Let $x=\tau(f)$. Since
$\operatorname{Id}_{M}$ is a diffeomorphism, $\tau(\operatorname{Id}_{M})=0$.
Therefore $0=x-(-1)^{n}\overline{x}+\tau(M,\varphi^{\prime})-\tau(M,\varphi)$.
This means that

$\tau(M,\varphi)-\tau(M,\varphi^{\prime})=x-(-1)^{n}\overline{x}\in\mathcal{I}_{n}(G,w).\qed$

######  Remark 12.17.

Suppose that $(M,\varphi)$ is an SP manifold satisfying (SP1) and $k\geq 3$.
Let $G=\pi_{1}(M)\cong\pi_{1}(K)$, and let $w\colon G\rightarrow\left\\{\pm
1\right\\}$ be the orientation character of $M$. Since $k\geq 3$, for every
$x\in\operatorname{Wh}(G)$ there is a $k$-dimensional CW complex $K^{\prime}$
and a homotopy equivalence $f\colon K^{\prime}\rightarrow K$ such that
$\tau(f)=x$. Let $\varphi^{\prime}=\varphi\circ f\colon K^{\prime}\rightarrow
M$, then by Proposition 10.2 $\varphi$, and hence $\varphi^{\prime}$, is
$\left\lfloor\frac{n}{2}\right\rfloor$-connected, so $(M,\varphi^{\prime})$
also satisfies (SP1). The proof of Proposition 12.16 shows that
$\tau(M,\varphi^{\prime})-\tau(M,\varphi)=x-(-1)^{n}\overline{x}$. Therefore
if $\psi$ varies across all maps $L\rightarrow M$ such that $(M,\psi)$ is an
SP manifold, then the set of possible values of $\tau(M,\psi)$ is precisely
the coset $\tau(M,\varphi)+\mathcal{I}_{n}(G,w)$ in $\mathcal{J}_{n}(G,w)$.

####  12.5. An invariant of unpolarised manifolds

Based on the observations of Section 12.4, we define an invariant for
manifolds that can be obtained from SP manifolds by forgetting the
polarisation.

######  Definition 12.18.

We say that an $n$-manifold $M$ is _split_ if there is a positive integer $k$,
a $k$-dimensional CW complex $K$ and a continuous map $\varphi\colon
K\rightarrow M$ such that $(M,\varphi)$ is an SP manifold.

By Proposition 10.2, if $n\geq 7$ and $M$ has a CW decomposition with no
$\left\lfloor\frac{n}{2}\right\rfloor$-cells, then it is split (as we can take
$\varphi$ to be the inclusion of its
$\left\lfloor\frac{n}{2}\right\rfloor$-skeleton).

######  Definition 12.19.

Suppose that $M$ is a split $n$-manifold, and let $G=\pi_{1}(M)$ with
orientation character $w\colon G\rightarrow\left\\{\pm 1\right\\}$. We define
the $\tau$-invariant of $M$ by

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

where $k$ is a positive integer, $K$ is a $k$-dimensional CW complex and
$\varphi\colon K\rightarrow M$ is a continuous map such that $(M,\varphi)$ is
an SP manifold.

It follows from Propositions 12.14 and 12.16 that $\tau(M)$ is well-defined.

If $M$ is a manifold without middle dimensional handles in the sense of
Hausmann [hausmann-mwmdh], then $\tau(M)$ coincides with the “torsion
invariant” defined in [hausmann-mwmdh, Section 9]. By the proof of Theorem
10.14, this is the case if $n\geq 8$ and $(M,\varphi)$ satisfies (SP1) for
some $\varphi$. For such manifolds, Hausmann showed that the $\tau$-invariant
is preserved by simple homotopy equivalences and homotopy equivalences induced
by $h$-cobordisms. Both of the two theorems below can be regarded as
strengthenings of this statement.

Firstly, in analogy with Theorem 11.5, we obtain the following formula for
$\pi(\tau(f))$ for a homotopy equivalence $f$ between split manifolds.

######  Theorem 12.20.

Suppose that $M$ and $N$ are split $n$-manifolds and $f\colon M\rightarrow N$
is a homotopy equivalence. Let $G=\pi_{1}(N)$ with orientation character
$w\colon G\rightarrow\left\\{\pm 1\right\\}$. Then

$\pi(\tau(f))=\tau(N)-f_{*}(\tau(M))\in\widehat{H}^{n+1}(C_{2};\operatorname{Wh}(G,w)).$

In particular, if $f\colon M\rightarrow N$ is a homotopy equivalence between
split manifolds, then $\pi(\tau(f))$ depends only on the induced homomorphism
$\pi_{1}(f)$.

###### Proof.

By Proposition 2.34 $\tau(f)\in\mathcal{J}_{n}(G,w)$, so $\pi(\tau(f))$ is
defined. Fix a $\varphi\colon K\rightarrow M$ and a $\psi\colon L\rightarrow
N$ such that $(M,\varphi)$ and $(N,\psi)$ are SP manifolds. Let $\alpha\colon
K\rightarrow L$ be the restriction of $f$, and let
$x=\tau(\alpha)\in\operatorname{Wh}(G)$. By Theorem 11.5
$\tau(f)=x-(-1)^{n}\overline{x}+\tau(N,\psi)-f_{*}(\tau(M,\varphi))$.
Therefore

$\displaystyle\pi(\tau(f))$
$\displaystyle=\pi(x-(-1)^{n}\overline{x})+\pi(\tau(N,\psi))-\pi(f_{*}(\tau(M,\varphi)))$
$\displaystyle=0+\pi(\tau(N,\psi))-f_{*}(\pi(\tau(M,\varphi)))=\tau(N)-f_{*}(\tau(M)).\qed$

Secondly, we show that $\tau$ is a complete invariant for the equivalence
relation generated by simple homotopy equivalence and $h$-cobordism,
restricted to split manifolds (see also Proposition 4.15).

######  Theorem 12.21.

Suppose that $M$ and $N$ are split $n$-manifolds. The following are
equivalent.

  1. (a)

There is a homotopy equivalence $f\colon M\rightarrow N$ such that
$\tau(N)=f_{*}(\tau(M))$.

  2. (b)

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

###### Proof.

We can assume that $n\geq 5$. (If $n=4$, then $(N,\psi)$ has a trivial double
structure for some $1$-dimensional $L$ and $\psi\colon L\to N$, so
$\tau(N)=0$, and similarly $\tau(M)=0$. We also get that
$\pi_{1}(N)\cong\pi_{1}(L)$ is free, so $\operatorname{Wh}(\pi_{1}(N))=0$ by
Proposition 2.8. Hence every homotopy equivalence $M\rightarrow N$ is simple.)

$\eqref{item:theorem-tau-complete-a}\Rightarrow\eqref{item:theorem-tau-
complete-b}$. Let $G=\pi_{1}(N)$ with orientation character $w\colon
G\rightarrow\left\\{\pm 1\right\\}$. By Theorem 12.20, we have
$\pi(\tau(f))=0$, so $\tau(f)\in\mathcal{I}_{n}(G,w)$ and we can apply
Corollary 3.3.

$\eqref{item:theorem-tau-complete-b}\Rightarrow\eqref{item:theorem-tau-
complete-a}$. Let $f=h\circ g$ for a simple homotopy equivalence $g\colon
M\rightarrow P$ and a homotopy equivalence $h\colon P\rightarrow N$ induced by
an $h$-cobordism. Then $\tau(f)=\tau(h)+h_{*}(\tau(g))$. Since $\tau(g)=0$,
this implies that $\tau(f)\in\mathcal{I}_{n}(G,w)$ by Proposition 2.37.
Therefore $\pi(\tau(f))=0$, so $\tau(N)=f_{*}(\tau(M))$ by Theorem 12.20. ∎

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

The aim of this part will be to prove Theorems 16.1, 16.2, LABEL:, and 16.3,
which are key ingredients in the proofs of Theorems 5.14, 5.15, and 5.17
respectively. In Sections 13 and 14, we will recall the necessary background
on class groups, Tate cohomology, and $\mathbb{Z}C_{2}$-modules. The main
technical heart of this part will be Section 15 where we will investigate the
involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$ and prove general results
which allow it to be computed. Finally, in Section 16, we make use of the
results in Section 15 to prove Theorems 16.1, 16.2, LABEL:, and 16.3.
Throughout, we will assume that all modules are left modules.

###  13\. Locally free class groups

In this section, we will recall the theory of locally free class groups for
orders in semisimple $\mathbb{Q}$-algebras. Good references for this material
are [Sw80, Section 1-3] and [CR87, Section 49A & 50E].

####  13.1. Definitions and properties

For a ring $A$, a nonzero $A$-module is simple if it contains no simple
$A$-submodules other than itself and $0$, and is semisimple if isomorphic as
$A$-modules to a direct sum of its simple $A$-submodules. We say that a ring
$A$ is simple (respectively semisimple) if $A$, viewed as an $A$-module, is
simple (respectively semisimple). Recall that, for a field $K$, a $K$-algebra
is a ring $A$ for which $K$ is a subring of the centre $Z(A)$. We say that a
$K$-algebra is finite-dimensional if it is finite-dimensional as a $K$-vector
space.

######  Definition 13.1.

Let $A$ be a finite-dimensional semisimple $\mathbb{Q}$-algebra. An order in
$A$ is a subring $\Lambda\subseteq A$ that is finitely generated as an abelian
group and which has $\mathbb{Q}\cdot\Lambda=A$.

For example, let $G$ be a finite group. Then $A=\mathbb{Q}G$ is a finite
dimensional $\mathbb{Q}$-algebra which is semisimple by Maschke’s theorem in
representation theory, and $\Lambda=\mathbb{Z}G$ is an order in $A$. For an
even simpler example, let $K/\mathbb{Q}$ is a finite field extension. Then
$A=K$ is a simple $\mathbb{Q}$-algebra and its ring of integers
$\Lambda=\mathcal{O}_{K}$ is an order in $A$.

From now on, fix an order $\Lambda$ in a finite-dimensional semisimple
$\mathbb{Q}$-algebra $A$. For a prime $p$, let
$\Lambda_{p}=\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\Lambda$ and let
$A_{p}=\mathbb{Q}_{p}\otimes_{\mathbb{Q}}A$ denote the $p$-adic completions of
$\Lambda$ and $A$.

######  Definition 13.2.

A $\Lambda$-module $M$ is locally free if
$M_{p}=\mathbb{Z}_{p}\otimes_{\mathbb{Z}}M$ is a free $\Lambda_{p}$-module for
all primes $p$.

The following is [Sw80, Lemma 2.1]. Note that the converse need not hold, i.e.
there exist orders $\Lambda$ and projective $\Lambda$-modules that are not
locally free [Sw80, p. 156].

######  Proposition 13.3.

If $M$ is a locally free $\Lambda$-module, then $M$ is projective.

We say that two locally free $\Lambda$-modules $M$ and $N$ are stably
isomorphic, written $M\cong_{\operatorname{st}}N$, if there exists $r,s\geq 0$
such that $M\oplus\Lambda^{r}\cong N\oplus\Lambda^{s}$ are isomorphic as
$\Lambda$-modules.

######  Definition 13.4.

Define the locally free class group $C(\Lambda)$ to be the set of equivalence
classes of locally free $\Lambda$-modules up to stable isomorphism. This is an
abelian group under direct sum (since the direct sum of locally free modules
is locally free).

It follows that $C(\Lambda)\leq\widetilde{K}_{0}(\Lambda)$ is a subgroup,
where $\widetilde{K}_{0}$ is the $0$th reduced algebraic $K$-group as defined
in Section 5.1. It is a consequence of the Jordan-Zassenhaus theorem that
$C(\Lambda)$ is finite [CR87, Remark 49.11 (ii)].

We will now specialise to the case where $\Lambda=\mathbb{Z}G$ for $G$ a
finite group. In contrast to the situation for general orders, we have the
following [Sw80, p. 156]. By Proposition 13.3, this implies that a
$\mathbb{Z}G$-module is projective if and only if it is locally free.

######  Proposition 13.5.

Let $G$ be a finite group.

  1. (i)

If $M$ is a projective $\mathbb{Z}G$-module, then $M$ is locally free.

  2. (ii)

There is an isomorphism of abelian groups $\widetilde{K}_{0}(\mathbb{Z}G)\cong
C(\mathbb{Z}G)$. In particular, $\widetilde{K}_{0}(\mathbb{Z}G)$ is finite.

This means that $\widetilde{K}_{0}(\mathbb{Z}G)$ is potentially computable,
since, as we shall see below, methods exist for classifying locally free class
groups (and more generally locally free modules) which do not apply to
arbitrary projective modules over orders, let alone over arbitrary rings.

Finally we note the following which relates locally free class groups to ideal
class groups (see [Re75, Section 35]).

######  Proposition 13.6.

Let $K/\mathbb{Q}$ be a finite field extension. Then $C(\mathcal{O}_{K})$
coincides with the ideal class group of $\mathcal{O}_{K}$.

####  13.2. Kernel groups

Let $A$ be a finite-dimensional semisimple $\mathbb{Q}$-algebra $A$. An order
in $A$ is said to be maximal if it is not properly contained in another other
order in $A$. Since every finite field extension of $\mathbb{Q}$ is separable,
$A$ is a separable algebra and so every order in $A$ is contained in a maximal
order [Sw70, Proposition 5.1].

Let $\Lambda$ be an order in $A$ and let $\Gamma$ be a maximal order in $A$
containing $\Lambda$. The inclusion map $i\colon\Lambda\hookrightarrow\Gamma$
induces a map $i_{*}\colon C(\Lambda)\to C(\Gamma)$ given by extension of
scalars $[M]\mapsto[\Gamma\otimes_{\Lambda}M]$ which is necessarily surjective
by [CR87, Theorem 49.25].

######  Definition 13.7.

Define the kernel group $D(\Lambda)$ to be the kernel of the map $i_{*}\colon
C(\Lambda)\to C(\Gamma)$. This is often also referred to as the defect group.

The group $C(\Gamma)$ does not depend on the choice of maximal order in $A$;
in fact, if $\Gamma_{1}$, $\Gamma_{2}$ are maximal orders in $A$, then
$C(\Gamma_{1})\cong C(\Gamma_{2})$ [CR87, Theorem 49.32]. Furthermore, the
kernel group $D(\Lambda)$ does not depend on the choice of maximal order. This
can be seen from the fact that it can be equivalently defined without
reference to a maximal order: if $M$ is a locally free $\Lambda$-module, then
$[M]\in D(\Lambda)$ if and only if there exists a finitely generated
$\Lambda$-module $X$ such that $M\oplus X\cong\Lambda^{n}\oplus X$ as
$\Lambda$-modules for some $n$ [CR87, Proposition 49.34].

In particular, we have a well-defined exact sequence of abelian groups:

$0\to D(\Lambda)\to C(\Lambda)\to C(\Gamma)\to 0$

where $\Gamma$ can be taken to be any maximal order in $A$ containing
$\Lambda$.

####  13.3. The idèlic approach to locally free class groups

Let $\Lambda$ be an order in a finite-dimensional semisimple
$\mathbb{Q}$-algebra $A$. One thing that makes the locally free class group
$C(\Lambda)$ often much easier to compute than the full projective class group
$\widetilde{K}_{0}(\Lambda)$ is that its elements can be represented by
idèles.

######  Definition 13.8.

Define the idèle group

$J(A)=\\{(\alpha_{p})\in\prod_{p}A_{p}^{\times}\mid\alpha_{p}\in\Lambda_{p}^{\times}\textit{
for all but finitely many $p$}\\}\subseteq\prod_{p}A_{p}^{\times}.$

As a subgroup of $A_{p}^{\times}$, this is independent of the choice of order
$\Lambda$ [CR87, p. 218]. Every class in $C(\Lambda)$ is represented by a
locally free $\Lambda$-module $M\subseteq A$ [CR87, p. 218]. For each $p$,
there exists $\alpha_{p}\in A_{p}$ such that
$M_{p}=\Lambda_{p}\alpha_{p}\subseteq A_{p}$. For all but finitely many $p$,
$M_{p}\cong\Lambda_{p}$ and so $\alpha_{p}\in\Lambda_{p}^{\times}$. In
particular, $\alpha=(\alpha_{p})\in J(A)$. Conversely, given an idèle
$\alpha\in J(A)$, we have that
$\Lambda\alpha=A\cap\bigcap_{p}\Lambda_{p}\alpha_{p}\subseteq A$ is a locally
free $\Lambda$-ideal. Let $\alpha,\beta\in J(A)$. Then
$\Lambda\alpha\cong\Lambda\beta$ as $\Lambda$-modules if and only if $\beta\in
U(\Lambda)\cdot\alpha\cdot A^{\times}$ where $A^{\times}\subseteq J(A)$ by
sending $a\in A^{\times}$ to $\alpha_{p}=1\otimes_{\mathbb{Q}}a$ for all $p$,
and
$U(\Lambda)=\\{(\alpha_{p})\in\prod_{p}A_{p}^{\times}\mid\alpha_{p}\in\Lambda_{p}^{\times}\text{
for all $p$}\\}\subseteq J(A)$ [CR87, 49.6]. Furthermore, we have that
$\Lambda\alpha\oplus\Lambda\beta\cong\Lambda\oplus\Lambda\alpha\beta$ [CR87,
49.8]. This leads to the following.

######  Proposition 13.9.

There is a surjective group homomorphism

$[\Lambda\,\cdot\,]\colon J(A)\twoheadrightarrow
C(\Lambda),\quad\alpha\mapsto[\Lambda\alpha].$

######  Remark 13.10.

Whilst we will not make use of it in this article, we note that this leads to
the formula

$C(\Lambda)\cong\frac{J(A)}{J_{0}(A)\cdot A^{\times}\cdot U(\Lambda)}$

for the locally free class group, where $J_{0}(A)=\\{x\in
J(A)\mid\operatorname{nr}(x)=1\\}$ and $\operatorname{nr}\colon J(A)\to
J(Z(A))$ is induced by the reduced norm map. This is due to Fröhlich [Fr75]
(see also [CR87, Theorem 49.22]).

####  13.4. Involutions on locally free class groups

Let $\Lambda$ be an order in a finite-dimensional semisimple
$\mathbb{Q}$-algebra $A$. Suppose further that $A$ is a ring equipped with an
involution $\overline{\cdot}\colon A\to A$ which restricts to $\Lambda$, i.e.
the map $\overline{\cdot}$ is an involution on $A$ as an abelian group which
satisfies $\overline{xy}=\overline{y}\cdot\overline{x}$ for all $x,y\in A$ and
$\overline{x}\in\Lambda$ for all $x\in\Lambda$. For example, if $G$ is a
finite group and $w\colon G\to\\{\pm 1\\}$ is an orientation character, then
$A=\mathbb{Q}G$ has an involution given by
$\sum_{i=1}^{k}n_{i}g_{i}\mapsto\sum_{i=1}^{k}w(g_{i})n_{i}g_{i}^{-1}$ for
$n_{i}\in\mathbb{Z}$ and $g_{i}\in G$ which restricts to $\Lambda=\mathbb{Z}G$
(see Section 2.2). Given a (left) $\Lambda$-module $M$, we now have the notion
of a dual (left) $\Lambda$-module $M^{*}$ (see Definition 2.22).

We now note the following properties of the dual of locally free modules.

######  Lemma 13.11.

Let $M$ be a locally free $\Lambda$-module. Then:

  1. (i)

$M^{*}$ is locally free;

  2. (ii)

The evaluation map $\text{ev}:M\to M^{**}$, $m\mapsto(f\mapsto f(m))$ is an
isomorphism of $\Lambda$-modules.

###### Proof.

(i) This follows from the fact that dualising commutes with $p$-adic
localisations.

(ii) A module with this property is called reflexive. By Proposition 13.3, $M$
is a projective $\Lambda$-module. Projective modules are reflexive since
finitely generated free modules are reflexive and direct summands of reflexive
modules are reflexive. ∎

We can therefore use this to define an involution on the locally free class
group.

######  Definition 13.12.

There is an involution of abelian groups:

$\ast\colon C(\Lambda)\to C(\Lambda),\quad[M]\mapsto-[M^{*}].$

That is, $\ast$ is a group homomorphism such that
$\ast^{2}=\operatorname{Id}_{C(\Lambda)}$.

In the case $\Lambda=\mathbb{Z}G$, we have that
$\widetilde{K}_{0}(\mathbb{Z}G)\cong C(\mathbb{Z}G)$ and the involution above
coincides with the standard involution on $\widetilde{K}_{0}$ as defined in
Section 5.1.

We will now explore some properties of this involution. In what follows we
will refer to [CR87, pp. 275-6]. Note that this deals only with the case
$\Lambda=\mathbb{Z}G$, though the arguments there apply to the more general
setting described above.

It is an immediate consequence of the alternative description of the kernel
group given in Section 13.2 that, if $[M]\in D(\Lambda)$, then $[M^{*}]\in
D(\Lambda)$ [CR87, p. 275]. In particular, the involution $\ast$ restricts to
$D(\Lambda)$. This implies that, if $i\colon\Lambda\hookrightarrow\Gamma$ for
$\Gamma$ a maximal order in $A$, then $\ast$ induces an involution on
$C(\Gamma)$ via the map $i_{*}\colon C(\Lambda)\twoheadrightarrow C(\Gamma)$,
and this coincides with the involution on $C(\Gamma)$ coming from the fact
that $\Gamma$ is an order in $A$.

Recall that an involution on an abelian group is the same structure as a
$\mathbb{Z}C_{2}$-module, where the $C_{2}$-action is given by the involution.
In particular, we have shown the following.

######  Proposition 13.13.

There is a short exact sequence of $\mathbb{Z}C_{2}$-modules

$0\to D(\Lambda)\to C(\Lambda)\to C(\Gamma)\to 0$

where $D(\Lambda)$, $C(\Lambda)$, and $C(\Gamma)$ are
$\mathbb{Z}C_{2}$-modules under the involutions described above.

We will conclude this section by noting that the idèlic approach to class
groups gives a different way to define an involution on $C(\Lambda)$. The
involution $\overline{\cdot}\colon A\to A$ induces involutions on $A_{p}$ for
each $p$ and so on $J(A)$. It can be shown that the involution fixes the
subgroups $A^{\times}$, $U(\Lambda)$ and $J_{0}(A)$ and so induces an
involution on $C(\Lambda)$ given by sending
$[\Lambda\alpha]\mapsto[\Lambda\overline{\alpha}]$ (see [CR87, p. 274]).

The following is proven in [CR87, p. 274].

######  Proposition 13.14.

The involution on $C(\Lambda)$ induced by the involution on $J(A)$ coincides
with the standard involution $\ast\colon C(\Lambda)\to C(\Lambda)$,
$[M]\mapsto-[M^{*}]$.

This gives an alternate way to understand the standard involution on
$C(\mathbb{Z}G)\cong\widetilde{K}_{0}(\mathbb{Z}G)$. We will make further use
of this description the following section.

###  14\. Tate cohomology and $\mathbb{Z}C_{2}$-modules

In this section, we will recall some basic facts about
$\mathbb{Z}C_{2}$-modules which will be used throughout the proofs of Theorems
16.1, 16.2, LABEL:, and 16.3. We will also explain how the methods of Tate
cohomology can be applied to $\mathbb{Z}C_{2}$-modules in preparation for the
proof of Theorem 16.3.

####  14.1. Tate cohomology

The Tate cohomology groups are defined as follows [Br94, VI.4].

######  Definition 14.1.

Given a finite group $G$ and a $\mathbb{Z}G$-module $A$, the _Tate cohomology
groups_ $\widehat{H}^{n}(G;A)$ for $n\in\mathbb{Z}$ are defined as follows.
Let $A^{G}=\\{x\in A\mid g\cdot x=x\text{ for all $g\in G$}\\}$ be the
invariants, let $A_{G}=A/\langle g\cdot x-x\mid g\in G,x\in A\rangle$ be the
coinvariants and let $N\colon A_{G}\to A^{G}$ be the norm map
$x\mapsto\sum_{g\in G}g\cdot x$ which is a well-defined homomorphism of
abelian groups. Then define:

$\widehat{H}^{n}(G;A)=\begin{cases}H^{n}(G;A),&\text{if $n\geq 1$}\\\
\operatorname{coker}(N\colon A_{G}\to A^{G}),&\text{if $n=0$}\\\ \ker(N\colon
A_{G}\to A^{G}),&\text{if $n=-1$}\\\ H_{-n-1}(G;A),&\text{if
$n\leq-2$},\end{cases}$

where $H^{n}$ and $H_{-n-1}$ denote the usual group cohomology and homology
groups.

We now recall the following basic properties. The first can be found in [Br94,
VI.5.1], the second follows from the first since functoriality means that
$\alpha_{\ast}$ is split whenever $\alpha$ is, the third is [CE56, XII.2.5],
and the fourth is [CE56, XII.2.7].

######  Proposition 14.2.

Let $G$ be a finite group.

  1. (i)

Let $0\to A\xrightarrow[]{\alpha}B\xrightarrow[]{\beta}C\to 0$ be a short
exact sequence of $\mathbb{Z}G$-modules. Then there is a long exact sequence
of Tate cohomology groups:

$\cdots\to\widehat{H}^{n-1}(G;C)\xrightarrow[]{\partial}\widehat{H}^{n}(G;A)\xrightarrow[]{\alpha_{*}}\widehat{H}^{n}(G;B)\xrightarrow[]{\beta_{*}}\widehat{H}^{n}(G;C)\xrightarrow[]{\partial}\widehat{H}^{n+1}(G;A)\to\cdots$

  2. (ii)

Let $A,B$ be $\mathbb{Z}G$-modules. Then $\widehat{H}^{n}(G;A\oplus
B)\cong\widehat{H}^{n}(G;A)\oplus\widehat{H}^{n}(G;B)$ for all
$n\in\mathbb{Z}$.

  3. (iii)

Let $A$ be a $\mathbb{Z}G$-module. Then $|G|\cdot\widehat{H}^{n}(G;A)=0$, i.e.
$|G|\cdot x=0$ for all $x\in\widehat{H}^{n}(G;A)$.

  4. (iv)

Let $A$ be a finite $\mathbb{Z}G$-module and suppose $(|G|,|A|)=1$. Then
$\widehat{H}^{n}(G;A)=0$ for all $n\in\mathbb{Z}$.

####  14.2. Tate cohomology and the structure of $\mathbb{Z}C_{2}$-modules

Let $A$ be a $\mathbb{Z}C_{2}$-module or, equivalently, an abelian group with
an involution $\overline{\cdot}\colon A\to A$. In order to prove Theorems
16.1, 16.2, and 16.3, we would like to find techniques to determine the
following groups associated to $A$:

$\\{x\in A\mid x=(-1)^{n}\overline{x}\\},\quad\\{x+(-1)^{n}\overline{x}\mid
x\in A\\},\quad\frac{\\{x\in A\mid
x=(-1)^{n}\overline{x}\\}}{\\{x+(-1)^{n}\overline{x}\mid x\in A\\}}.$

We are especially interested in the case $n$ odd (i.e. $n=1$) but we will
consider both cases. Note that, for the groups on the left, the notation
$A^{-}=\\{x\in A\mid x=-\overline{x}\\}$ and $A^{+}=\\{x\in A\mid
x=\overline{x}\\}$ is often used since they are the $(-1)$ and
$(+1)$-eigenspaces of the involution action. If $2\in A$ is invertible, then
$A\cong A^{+}\oplus A^{-}$ but this need not hold in general.

The following lemma will suffice for the study of the first two classes of
groups. Part (i) follows from the fact that $A\mapsto A^{C_{2}}$ is a left-
exact functor where $A$ is given the altered involution
$x\mapsto(-1)^{n}\overline{x}$, and part (ii) is immediate.

######  Lemma 14.3.

Let $0\to A\xrightarrow[]{\alpha}B\xrightarrow[]{\beta}C\to 0$ be an exact
sequence of $\mathbb{Z}C_{2}$-modules and let $n\in\mathbb{Z}$.

  1. (i)

Then $\alpha,\beta$ induce an exact sequence of abelian groups:

$0\to\\{x\in A\mid\overline{x}=(-1)^{n}x\\}\xrightarrow[]{\alpha}\\{x\in
B\mid\overline{x}=(-1)^{n}x\\}\xrightarrow[]{\beta}\\{x\in
C\mid\overline{x}=(-1)^{n}x\\}.$

  2. (ii)

There are injective and surjective maps induced by $\alpha,\beta$:

$\\{x+(-1)^{n}\overline{x}\mid x\in
A\\}\xhookrightarrow{\alpha}\\{x+(-1)^{n}\overline{x}\mid x\in
B\\}\xrightarrow[]{\beta}\mathrel{\mkern-14.0mu}\rightarrow\\{x+(-1)^{n}\overline{x}\mid
x\in C\\}.$

######  Remark 14.4.

For each $n$, the right map in (i) need not be surjective. For example, take
$0\to\mathbb{Z}\xrightarrow[]{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$ where
$\mathbb{Z}/2$ has the trivial involution and $\mathbb{Z}$ has the involution
$x\mapsto(-1)^{n}x$. For each $n$, the sequence (ii) need not be exact in the
middle. For example, take
$0\to\mathbb{Z}/2\to\mathbb{Z}\oplus\mathbb{Z}/2\to\mathbb{Z}\to 0$ where
$\mathbb{Z}/2$ has the trivial involution, $\mathbb{Z}\oplus\mathbb{Z}/2$ has
involution $(x,y)\mapsto((-1)^{n+1}x,x+y)$ and $\mathbb{Z}$ has the involution
$x\mapsto(-1)^{n}x$.

The third class of groups can be studied using Tate cohomology due to the
following. This is standard (see, for example, [CE56, p. 251]) but we will
include a proof here for convenience.

######  Proposition 14.5.

Let $A$ be a $\mathbb{Z}C_{2}$-module and let $n\in\mathbb{Z}$. Then

$\widehat{H}^{n}(C_{2};A)\cong\frac{\\{x\in A\mid
x=(-1)^{n}\overline{x}\\}}{\\{x+(-1)^{n}\overline{x}\mid x\in A\\}}.$

###### Proof.

Since $C_{2}$ has 2-periodic Tate cohomology group (see, for example, [Br94,
VI.9.2]), it suffices to compute $\widehat{H}^{0}(C_{2};A)$ and
$\widehat{H}^{-1}(C_{2};A)$. We have $A^{C_{2}}=\\{x\in A\mid
x=\overline{x}\\}$ and $A_{C_{2}}=A/\\{x-\overline{x}\mid x\in A\\}$ and so
the norm map is given by

$N\colon\frac{A}{\\{x-\overline{x}\mid x\in A\\}}\to\\{x\in A\mid
x=\overline{x}\\},\quad x\mapsto x+\overline{x}.$

The result then follows by reading off $\widehat{H}^{-1}(C_{2};A)=\ker(N)$ and
$\widehat{H}^{0}(C_{2};A)=\operatorname{coker}(N)$. ∎

######  Remark 14.6.

This shows that, for a group $G$ and $w\colon G\to\\{\pm 1\\}$ an orientation
character, we have

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

which was noted already in Section 2.2.

We now recall a series of special facts about the Tate cohomology of $C_{2}$.
The first, which can be found in [Se79, p. 133], is a consequence of
Proposition 14.2 (i) and the fact that finite cyclic groups have $2$-periodic
cohomology. Note that this also applies for $C_{2}$ replaced by an arbitrary
finite cyclic group.

######  Proposition 14.7.

Let $0\to A\xrightarrow[]{\alpha}B\xrightarrow[]{\beta}C\to 0$ be a short
exact sequence of $\mathbb{Z}C_{2}$-modules. Then there is a $6$-periodic
exact sequence of abelian groups:

${\widehat{H}^{1}(C_{2};A)}$${\widehat{H}^{1}(C_{2};B)}$${\widehat{H}^{1}(C_{2};C)}$${\widehat{H}^{0}(C_{2};C)}$${\widehat{H}^{0}(C_{2};B)}$${\widehat{H}^{0}(C_{2};A).}$$\scriptstyle{\alpha_{*}}$$\scriptstyle{\beta_{*}}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$$\scriptstyle{\beta_{*}}$$\scriptstyle{\alpha_{*}}$

We will next consider results which apply only in the case where $A$ is finite
$\mathbb{Z}G$-module, i.e. a $\mathbb{Z}G$-module whose underlying abelian
group is finite. This is a consequence of the theory of Herbrand quotients
(see [Se79, Chapter VIII, Section 4]) though we will include a direct proof
here for convenience.

######  Proposition 14.8.

Let $A$ be a finite $\mathbb{Z}C_{2}$-module. Then there exists $d\geq 0$ such
that

$\widehat{H}^{n}(C_{2};A)\cong(\mathbb{Z}/2)^{d}$

for all $n\in\mathbb{Z}$. In particular, $|\widehat{H}^{n}(G;A)|$ is
independent of $n\in\mathbb{Z}$.

###### Proof.

First note that, since $A$ is finite, so is $\widehat{H}^{n}(C_{2};A)$ by
Proposition 14.5. Next note that there are exact sequences of finite abelian
groups (see, for example, [Se79, p. 134]):

$0\to\\{x\in A\mid\overline{x}=x\\}\to A\xrightarrow[]{x\mapsto
x-\overline{x}}A\to\frac{A}{\\{x-\overline{x}\mid x\in A\\}}\to 0$
$0\to\widehat{H}^{1}(C_{2};A)\to\frac{A}{\\{x-\overline{x}\mid x\in
A\\}}\xrightarrow[]{x\mapsto x+\overline{x}}\\{x\in
A\mid\overline{x}=x\\}\to\widehat{H}^{0}(C_{2};A)\to 0.$

The first implies that $|\\{x\in
A\mid\overline{x}=x\\}|=|A/\\{x-\overline{x}\mid x\in A\\}|$ and, by combining
this with the second, we get that
$|\widehat{H}^{1}(C_{2};A)|=|\widehat{H}^{0}(C_{2};A)|$. Hence, since
$\widehat{H}^{n}(C_{2};A)$ is 2-periodic (see Proposition 14.5), we get that
$|\widehat{H}^{n}(C_{2};A)|$ is independent of $n\in\mathbb{Z}$.

Next recall that, by Proposition 14.2 (iii), we have
$2\cdot\widehat{H}^{n}(C_{2};A)=0$, i.e. every element in
$\widehat{H}^{n}(C_{2};A)$ has order at most two. This implies that
$\widehat{H}^{n}(C_{2};A)\cong(\mathbb{Z}/2)^{d_{n}}$ for some $d_{n}\geq 0$.
Since $|\widehat{H}^{n}(C_{2};A)|$ is independent of $n$, this implies that
$d_{n}$ is also independent of $n$. ∎

######  Remark 14.9.

In particular, by Propositions 14.5 and 14.8, there is an isomorphism of
abelian groups

$\frac{\\{x\in A\mid x=-\overline{x}\\}}{\\{x-\overline{x}\mid x\in
A\\}}\cong\frac{\\{x\in A\mid x=\overline{x}\\}}{\\{x+\overline{x}\mid x\in
A\\}}$

whenever $A$ is a finite $\mathbb{Z}C_{2}$-module.

For a finite abelian group $A$ and a prime $p$, let $A_{(p)}=\\{x\in A\mid
p^{n}\cdot x=0\text{ for some $n\geq 1$}\\}$ denote the $p$-primary component
of $A$. This coincides with the Sylow $p$-subgroup of $A$ and we use this
notation since $A_{(p)}\cong\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}}A$ where
$\mathbb{Z}_{(p)}$ denotes the localisation of $\mathbb{Z}$ at
$S=\mathbb{Z}\setminus(p)$.

######  Lemma 14.10.

Let $A$ be a finite $\mathbb{Z}C_{2}$-module. For $p$ prime, $A_{(p)}\leq A$
is a $\mathbb{Z}C_{2}$-submodule and

$A\cong\bigoplus_{p\mid|A|}A_{(p)}$

is an isomorphism of $\mathbb{Z}C_{2}$-modules.

###### Proof.

By the classification of finite abelian groups, we have
$A\cong\bigoplus_{p\mid|A|}A_{(p)}$ as abelian groups. Recall that, if $G$ and
$H$ are finite abelian groups with $(|G|,|H|)=1$, then the natural inclusion
$\operatorname{Aut}(G)\times\operatorname{Aut}(H)\hookrightarrow\operatorname{Aut}(G\times
H)$ is an isomorphism. In particular, we have

$\operatorname{Aut}(A)\cong\prod_{p\mid|A|}\operatorname{Aut}(A_{(p)}).$

A $\mathbb{Z}C_{2}$-module structure on $A$ is an involution, i.e. an element
of $\operatorname{Aut}(A)$ of order at most two. The above isomorphism implies
that the involution fixes each $A_{(p)}$ and so $A_{(p)}\leq A$ is a
$\mathbb{Z}C_{2}$-submodule. It follows that that
$A\cong\bigoplus_{p\mid|A|}A_{(p)}$ is an isomorphism of
$\mathbb{Z}C_{2}$-modules. ∎

######  Proposition 14.11.

Let $A$ be a finite $\mathbb{Z}C_{2}$-module and let $n\in\mathbb{Z}$. Then

$\widehat{H}^{n}(C_{2};A)\cong\widehat{H}^{n}(C_{2};A_{(2)}).$

###### Proof.

By Lemma 14.10 and Proposition 14.2 (ii), we get that

$\widehat{H}^{n}(C_{2};A)\cong\bigoplus_{p\mid|A|}\widehat{H}^{1}(C_{2};A_{(p)}).$

If $p\neq 2$, then $|A_{(p)}|$ is odd and so
$\widehat{H}^{1}(C_{2};A_{(p)})=0$ by Proposition 14.2 (iv). This gives that
$\widehat{H}^{n}(C_{2};A)\cong\widehat{H}^{n}(C_{2};A_{(2)})$, as required. ∎

###  15\. Computing the involution on $\widetilde{K}_{0}(\mathbb{Z}C_{m})$

The aim of this section will be to investigate the involution on
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$ in preparation for the proofs of Theorems
16.1, 16.2, LABEL:, and 16.3 in Section 16.

We will view $\widetilde{K}_{0}(\mathbb{Z}C_{m})$ as a
$\mathbb{Z}C_{2}$-module with the $C_{2}$-action coming from the standard
involution on $\widetilde{K}_{0}$ as defined in Section 5.1. We saw in Section
13.1 that $\widetilde{K}_{0}(\mathbb{Z}C_{m})\cong C(\mathbb{Z}C_{m})$ is an
isomorphism of $\mathbb{Z}C_{2}$-modules where $C(\mathbb{Z}C_{m})$ denotes
the locally free class group, and in particular it is finite. Our basic
approach for computing $C(\mathbb{Z}C_{m})$ will be to use the following short
exact sequence of $\mathbb{Z}C_{2}$-modules established in Section 13.4

$0\to D(\mathbb{Z}C_{m})\to C(\mathbb{Z}C_{m})\to C(\Gamma_{m})\to 0$

where $\Gamma_{m}$ is a maximal order in $\mathbb{Q}C_{m}$ containing
$\mathbb{Z}C_{m}$ and $D(\mathbb{Z}C_{m})$ has the induced involution.

The plan for this section is as follows. In Section 15.1, we relate the
involution on $C(\Gamma_{m})$ to the involution on $C(\mathbb{Z}[\zeta_{d}])$
induced by conjugation. In Section 15.2, we study the conjugation action on
$C(\mathbb{Z}[\zeta_{d}])$ and its relation to the class numbers
$h_{d}=|C(\mathbb{Z}[\zeta_{d}])|$. In Section 15.3, we survey results on
divisibility of class numbers as well as make some minor extensions (see
Proposition 15.8 (ii)). Finally, in Section 15.4, we investigate the
involution on $D(\mathbb{Z}C_{m})$.

####  15.1. The induced involution on the maximal order

Let $m\geq 2$ and let $C_{m}=\langle x\mid x^{m}\rangle$. Then there is an
isomorphism of $\mathbb{Q}$-algebras:

$\mathbb{Q}C_{m}\cong\prod_{d\mid m}\mathbb{Q}(\zeta_{d}),\quad
x\mapsto\prod_{d\mid m}(\zeta_{d})$

where $\zeta_{d}=e^{2\pi i/d}$ denotes a $d$th primitive root of unity. Since
$\mathcal{O}_{\mathbb{Q}(\zeta_{d})}=\mathbb{Z}[\zeta_{d}]$, it follows that
$\Gamma_{m}=\prod_{d\mid m}\mathbb{Z}[\zeta_{d}]$ is a maximal order in
$\mathbb{Q}C_{m}$. The image of $\mathbb{Z}C_{m}$ under the isomorphism above
is contained in $\Gamma_{m}$ and so $\Gamma_{m}$ contains $\mathbb{Z}C_{m}$.
In fact, $\Gamma_{m}$ is the unique maximal order in $\mathbb{Q}C_{m}$
containing $\mathbb{Z}C_{m}$ [CR87, p. 243]. This implies that there is an
isomorphism of abelian groups

$C(\Gamma_{m})\cong\bigoplus_{d\mid m}C(\mathbb{Z}[\zeta_{d}]),$

where, as noted in Proposition 13.6, the locally free class group
$C(\mathbb{Z}[\zeta_{d}])$ coincides with the ideal class group of
$\mathbb{Z}[\zeta_{d}]$.

For an integer $d\geq 1$, let $\overline{\,\cdot\,}\colon
C(\mathbb{Z}[\zeta_{d}])\to C(\mathbb{Z}[\zeta_{d}])$ denote the map induced
by conjugation, i.e. if
$\sigma\colon\mathbb{Z}[\zeta_{d}]\to\mathbb{Z}[\zeta_{d}]$ is the ring
homomorphism generated by $\zeta_{d}\mapsto\zeta_{d}^{-1}$, then
$\overline{\,\cdot\,}=\sigma_{*}$ is the induced map on
$C(\mathbb{Z}[\zeta_{d}])$. We will now compute the induced involution on
$\bigoplus_{d\mid m}C(\mathbb{Z}[\zeta_{d}])$. This was shown in the case
where $m$ is prime in [Re68] (see also [CR87, p. 275]).

######  Proposition 15.1.

Let $i\colon\mathbb{Z}C_{m}\hookrightarrow\Gamma_{m}$, $x\mapsto\prod_{d\mid
m}(\zeta_{d})$ and let $i_{*}\colon C(\mathbb{Z}C_{m})\to\bigoplus_{d\mid
m}C(\mathbb{Z}[\zeta_{d}])$ denote the induced map. Then, under $i_{*}$, the
standard involution on $C(\mathbb{Z}C_{m})$ induces the conjugation map on
each $C(\mathbb{Z}[\zeta_{d}])$.

This means that, if $x\in C(\mathbb{Z}C_{m})$ and $i_{*}(x)=\prod_{d\mid
m}x_{d}$, then $i_{*}(c(x))=\prod_{d\mid m}c_{d}(x_{d})$ where $c\colon
C(\mathbb{Z}C_{m})\to C(\mathbb{Z}C_{m})$ is the standard involution and
$c_{d}\colon C(\mathbb{Z}[\zeta_{d}])\to C(\mathbb{Z}[\zeta_{d}])$ is induced
by conjugation.

###### Proof.

For each $d\mid m$, let
$i^{(d)}\colon\mathbb{Z}C_{m}\to\mathbb{Z}[\zeta_{d}]$, $x\mapsto\zeta_{d}$.
It suffices to prove that, under the map $i^{(d)}_{*}\colon
C(\mathbb{Z}C_{m})\to C(\mathbb{Z}[\zeta_{d}])$, the involution on
$C(\mathbb{Z}C_{m})$ induces conjugation on $C(\mathbb{Z}[\zeta_{d}])$. By
Proposition 13.14, the standard involution on $C(\mathbb{Z}C_{m})$ is induced
by the involution on the idèle group $J(\mathbb{Q}C_{m})$. Note that
$i^{(d)}_{*}$ is induced by the map $J(i^{(d)})\colon J(\mathbb{Q}C_{m})\to
J(\mathbb{Q}(\zeta_{d}))$. Under the map $i^{(d)}$, the involution on
$\mathbb{Q}C_{m}$ induces conjugation on $\mathbb{Q}(\zeta_{d})$. In
particular, the involution on $C(\mathbb{Z}[\zeta_{d}])$ induced by
$i^{(d)}_{*}$ coincides with the involution induced by conjugation on
$J(\mathbb{Q}(\zeta_{d}))$. The result now follows since, if a locally free
$\mathbb{Z}[\zeta_{d}]$-ideal
$M=(x_{1},\dots,x_{n})\subseteq\mathbb{Q}(\zeta_{d})$ is represented by
$\alpha\in J(\mathbb{Q}(\zeta_{d}))$, then
$\overline{M}=(\overline{x}_{1},\dots,\overline{x}_{n})\subseteq\mathbb{Q}(\zeta_{d})$
is represented by $\overline{\alpha}\in J(\mathbb{Q}(\zeta_{d}))$. ∎

In summary, we have shown that there is a short exact sequence of
$\mathbb{Z}C_{2}$-modules

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

where $C(\mathbb{Z}C_{m})$ has the standard involution, $D(\mathbb{Z}C_{m})$
has the induced involution, and each $C(\mathbb{Z}[\zeta_{d}])$ has the
involution induced by conjugation.

####  15.2. Ideal class groups of cyclotomic fields

For every integer $m\geq 2$, let $\lambda_{m}=\zeta_{m}+\zeta_{m}^{-1}$. Let
$i\colon\mathbb{Z}[\lambda_{m}]\hookrightarrow\mathbb{Z}[\zeta_{m}]$ denote
inclusion and recall that the map $i_{*}\colon C(\mathbb{Z}[\lambda_{m}])\to
C(\mathbb{Z}[\zeta_{m}])$ is injective [La78, Theorem 4.2]. Furthermore, the
norm map gives a surjection $N\colon C(\mathbb{Z}[\zeta_{m}])\to
C(\mathbb{Z}[\lambda_{m}])$ such that the composition

$C(\mathbb{Z}[\zeta_{m}])\xrightarrow[]{N}C(\mathbb{Z}[\lambda_{m}])\xrightarrow[]{i_{*}}C(\mathbb{Z}[\zeta_{m}])$

is the map $x\mapsto x+\overline{x}$ (see, for example, [La78, pp. 83-4]). By
viewing $C(\mathbb{Z}[\zeta_{d}])$ and $C(\mathbb{Z}[\lambda_{m}])$ as
$\mathbb{Z}C_{2}$-modules under the conjugation action, the maps $i_{*}$ and
$N$ are $\mathbb{Z}C_{2}$-module homomorphisms. Note that the conjugation
action induces the identify on $C(\mathbb{Z}[\lambda_{m}])$.

This has the following useful consequences. Recall from earlier that, for $A$
a $\mathbb{Z}C_{2}$-module, we defined $A^{-}=\\{x\in A\mid
x=-\overline{x}\\}$ and $A^{+}=\\{x\in A\mid x=\overline{x}\\}$.

######  Lemma 15.2.

  1. (i)

The map $i_{*}$ induces an isomorphism
$C(\mathbb{Z}[\lambda_{m}])\cong\\{x+\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}$.

  2. (ii)

There is a short exact sequence of $\mathbb{Z}C_{2}$-modules:

$0\to C(\mathbb{Z}[\zeta_{m}])^{-}\to
C(\mathbb{Z}[\zeta_{m}])\xrightarrow[]{N}C(\mathbb{Z}[\lambda_{m}])\to 0.$

######  Remark 15.3.

Since $\mathbb{Q}(\lambda_{m})$ is the maximal real subfield of
$\mathbb{Q}(\zeta_{m})$, it is often written as $\mathbb{Q}(\zeta_{m})^{+}$.
However, whilst $C(\mathbb{Z}[\lambda_{m}])\subseteq
C(\mathbb{Z}[\zeta_{m}])^{+}$ is a subgroup, these groups are not equal in
general. For example, if $m=29$, then $C(\mathbb{Z}[\lambda_{29}])=0$ and
$C(\mathbb{Z}[\zeta_{29}])^{+}\cong(\mathbb{Z}/2)^{3}$.

###### Proof.

(i) Since $N$ is surjective, we have
$\operatorname{Im}(i_{*})=\operatorname{Im}(i_{*}\circ
N)=\\{x+\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}$.

(ii) Since $i_{*}$ is injective, we have $\ker(N)=\ker(i_{*}\circ N)=\\{x\in
C(\mathbb{Z}[\zeta_{m}])\mid x+\overline{x}=0\\}$. ∎

In order to set up later applications, we will now use Lemma 15.2 to obtain
information about each of the following groups:

$\\{x\in C(\mathbb{Z}[\zeta_{m}])\mid
x=-\overline{x}\\},\quad\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\},\quad\underbrace{\frac{\\{x\in
C(\mathbb{Z}[\zeta_{m}])\mid x=-\overline{x}\\}}{\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}}}_{\cong\widehat{H}^{1}(C_{2};C(\mathbb{Z}[\zeta_{m}]))}.$

Since $C(\mathbb{Z}[\zeta_{m}])$ is a finite abelian group (see, for example,
Section 13.1), we can define the class number of the cyclotomic integers to be
$h_{m}=|C(\mathbb{Z}[\zeta_{m}])|$. By Lemma 15.2 (ii), we have that
$h_{m}=h_{m}^{-}h_{m}^{+}$ where $h_{m}^{-}=|C(\mathbb{Z}[\zeta_{m}])^{-}|$
and $h_{m}^{+}=|C(\mathbb{Z}[\lambda_{m}])|$. We refer to $h_{m}^{-}$ as the
minus part of the class number and $h_{m}^{+}$ as the plus part of the class
number respectively.

For an integer $m$, let $\operatorname{odd}(m)$ denote the odd part of $m$,
i.e. $\operatorname{odd}(m)$ is the unique odd integer $r$ such that
$m=2^{k}r$ for some $k$.

######  Proposition 15.4.

There are subgroups:

  1. (i)

$C(\mathbb{Z}[\zeta_{m}])^{-}\leq\\{x\in C(\mathbb{Z}[\zeta_{m}])\mid
x=-\overline{x}\\}$;

  2. (ii)

$2\cdot C(\mathbb{Z}[\zeta_{m}])^{-}\leq\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}$.

In particular, $h_{m}^{-}$ divides $|\\{x\in C(\mathbb{Z}[\zeta_{m}])\mid
x=-\overline{x}\\}|$ and $\operatorname{odd}(h_{m}^{-})$ divides
$|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}|$.

###### Proof.

Parts (i) and (ii) each follows from Lemma 14.3, Lemma 15.2 (ii) and the fact
that $\overline{x}=-x$ for all $x\in C(\mathbb{Z}[\zeta_{m}])^{-}$. For the
last part note that, since $C(\mathbb{Z}[\zeta_{m}])^{-}$ is a finite abelian
group, we have that $C(\mathbb{Z}[\zeta_{m}])^{-}\cong A\oplus B$ where $|A|$
is even and $|B|=\operatorname{odd}(h_{m}^{-})$ is odd. Since $B$ is odd,
$2\cdot B=B$ and so

$B\leq 2\cdot A\oplus B=2\cdot
C(\mathbb{Z}[\zeta_{m}])^{-}\leq\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}.$

The result follows since $\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}$ has a subgroup of size
$\operatorname{odd}(h_{m}^{-})$. ∎

######  Remark 15.5.

Here we used Lemma 15.2 (ii) in order to obtain these bounds. It is also
possible to use Lemma 15.2 (i) instead. In particular, this implies that
$C(\mathbb{Z}[\zeta_{m}])$ has quotient

$\operatorname{coker}(i_{*})\cong
C(\mathbb{Z}[\zeta_{m}])/\\{x+\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}.$

If $[x]\in\operatorname{coker}(i_{*})$, then $[\overline{x}]=-[x]$. By Lemma
14.3, $\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}$ has quotient
$2\cdot\operatorname{coker}(i_{*})$. The same argument now applies since
$|\operatorname{coker}(i_{*})|=h_{m}^{-}$.

####  15.3. Divisibility of class numbers of cyclotomic fields

The aim of this section will be to survey results on the divisibility of the
class numbers $h_{m}$ and $h_{m}^{-}$. The most basic divisibility results are
that, for $n\mid m$, we have $h_{n}\mid h_{m}$ [Wa97, p. 205] and
$h_{n}^{-}\mid h_{m}^{-}$ [MM76, Lemma 5]. Motivated by Proposition 15.4, we
will now pursue divisibility results of two distinct types. We will start by
considering $\operatorname{odd}(h_{m}^{-})$, i.e. the unique odd integer $r$
such that $h_{m}^{-}=2^{k}r$ for some $k$, and we will then go on to consider
the parity of $h_{m}^{-}$.

Recall the following theorem of Masley-Montgomery [MM76] (see also [Wa97, p.
205]). In anticipation of its application in the proofs of Theorems 16.1 and
16.2, we will separate out the case where $m$ is square-free.

######  Proposition 15.6.

The complete list of $m\geq 2$ for which $h_{m}^{-}=1$ is as follows.

  1. (i)

If $m$ is square-free, then

$m=\begin{cases}\text{$p$},&\text{where $p\in\\{2,3,5,7,11,13,17,19\\}$}\\\
\text{$2p$},&\text{where $p\in\\{3,5,7,11,13,17,19\\}$}\\\ \text{$pq$ or
$2pq$},&\text{where $(p,q)\in\\{(3,5),(3,7),(3,11),(5,7)\\}$}.\end{cases}$

  2. (ii)

If $m$ is not square-free, then

$m\in\\{4,8,9,12,16,18,20,24,25,27,28,32,36,40,44,45,48,50,54,60,84,90\\}.$

Furthermore, $h_{m}^{-}=1$ if and only if $h_{m}=1$.

The proof of Proposition 15.6 is based on the fact that $h_{m}^{-}\to\infty$
as $m\to\infty$. We will now establish lower bounds on the growth rate of
$h_{m}^{-}$, and hence on $h_{m}$ since $h_{m}\geq h_{m}^{-}$. Let
$\varphi(m)$ denote Euler’s totient function.

######  Proposition 15.7.

There exists a constant $C>0$ such that, for all $m\geq 1$, we have:

$h_{m}^{-}\geq e^{C\tfrac{m\log m}{\log\log m}}.$

In particular, $h_{m}^{-}\to\infty$ super-exponentially in $m$.

###### Proof.

It is shown in [Wa97, Theorem 4.20] that $\log
h_{m}^{-}/(\frac{1}{4}\varphi(m)\log m)\to 1$ as $m\to\infty$. This implies
that $\log h_{m}^{-}\geq C_{0}\varphi(m)\log m$ for some $C_{0}>0$. The result
now follows from the fact that $\varphi(m)\geq m/2\log\log m$ for $m$
sufficiently large [HW54, Theorem 328]. ∎

The following result gives the analogue of Proposition 15.6 for
$\operatorname{odd}(h_{m}^{-})$. Note that $h_{m}^{-}=1$ implies
$\operatorname{odd}(h_{m}^{-})=1$, so we need not consider these $m$ since
they are classified in Proposition 15.6. This was established by Horie [Ho89,
Theorems 2 and 3] and builds on Friedman’s theorem from Iwasawa theory [Fr82]
and the Brauer-Siegel theorem for abelian fields [Uc71].

######  Proposition 15.8.

The complete list of $m\geq 2$ for which $\operatorname{odd}(h_{m}^{-})=1$ and
$h_{m}^{-}\neq 1$ is as follows.

  1. (i)

If $m$ is square-free, then $m\in\\{29,39,58,65,78,130\\}$.

  2. (ii)

If $m$ is not square-free, then $m\in\\{56,68,120\\}$.

Furthermore, $\operatorname{odd}(h_{m}^{-})=1$ if and only if
$\operatorname{odd}(h_{m})=1$.

In [Ho89, Theorem 1], Horie also showed that
$\operatorname{odd}(h_{m}^{-})\to\infty$ as $m\to\infty$ but gave no bound on
the growth rate. In fact, we have the following result analogous to
Proposition 15.7.

######  Proposition 15.9.

There exists a constant $C>0$ such that, for all $m\geq 1$, we have:

$\operatorname{odd}h_{m}^{-}\geq e^{C\tfrac{m\log m}{\log\log m}}.$

In particular, $\operatorname{odd}(h_{m}^{-})\to\infty$ super-exponentially in
$m$.

We will prove this by tracing through Horie’s proof of [Ho89, Theorem 1].
Recall that an abelian field $K$ is a finite Galois extension $K/\mathbb{Q}$
with $\operatorname{Gal}(K/\mathbb{Q})$ abelian and we can assume that
$K\subseteq\mathbb{C}$. For an abelian field $K$ with maximal real subfield
$K^{+}$, let $h_{K}=|C(\mathcal{O}_{K})|$,
$h_{K}^{+}=|C(\mathcal{O}_{K^{+}})|$ and $h_{K}^{-}=h_{K}/h_{K}^{+}$ (which is
an integer). Let $\operatorname{disc}(K)$ denote the discriminant of a number
field $K$.

###### Proof.

We will use that, if $L/K$ is an extension of abelian fields, then
$\operatorname{odd}(h_{K}^{-})\mid\operatorname{odd}(h_{L}^{-})$ [Ho89, Lemma
1]. For each abelian field $K$, let $K^{\prime}$ denote the maximal subfield
of $K$ with degree a power of 2. By the fundamental theorem of Galois theory
and the fact that a finite abelian group $A$ has a subgroup of order $d$ for
all $d\mid|A|$, we get that $|K^{\prime}/\mathbb{Q}|$ is the highest power of
2 dividing $|K/\mathbb{Q}|$. For an integer $n\geq 1$, let
$\mathcal{A}_{n}=\\{m\mid\operatorname{odd}(h_{m}^{-})\leq n\\}$ and
$\mathcal{B}_{n}=\\{|\mathbb{Q}(\zeta_{m})^{\prime}/\mathbb{Q}|\mid
n\in\mathcal{A}_{n}\\}$.

We will begin by finding a bound for $\sup(\mathcal{B}_{n})$. Let
$K=\mathbb{Q}(\zeta_{m})^{\prime}$ for some $m\in\mathcal{A}_{n}$. Then
$\mathbb{Q}(\zeta_{m})/K$ is an extension of abelian fields and so
$\operatorname{odd}(h_{K}^{-})\mid h_{m}^{-}$ and so
$\operatorname{odd}(h_{K}^{-})\leq n$. Since $|K/\mathbb{Q}|$ is a power of 2,
$h_{K}$ must be odd [Wa97] and so $h_{K}^{-}=\operatorname{odd}(h_{K}^{-})\leq
n$. Furthermore, note that $K$ is imaginary unless $K=\mathbb{Q}$ [Ho89, p.
468].

It follows from the proof of Theorem 1 and Proposition 1 in [Uc71] that
$\frac{|K/\mathbb{Q}|}{\log|\operatorname{disc}(K)|}$ is uniformly bounded
across imaginary abelian fields $K$, where $\operatorname{disc}(K)$ denotes
the discriminant, and that $h_{K}^{-}\geq|\operatorname{disc}(K)|^{6}$ for all
but finitely many imaginary abelian fields $K$. This implies that there exists
a constant $C>0$ such that $|K/\mathbb{Q}|\leq C\log(h_{K}^{-})$ for all
imaginary abelian fields $K$. Hence, if $K=\mathbb{Q}(\zeta_{m})^{\prime}$ for
$m\in\mathcal{A}_{n}$, then $|K/\mathbb{Q}|\leq C\log n$. This implies that
$\sup(\mathcal{B}_{n})\leq C\log n$. It also follows that there are only
finitely many fields of the form $\mathbb{Q}(\zeta_{m})^{\prime}$ for
$m\in\mathcal{A}_{n}$.

We now aim to find a bound for $\sup(\mathcal{A}_{n})$. First let $S$ denote
the set of primes which are ramified in some field
$\mathbb{Q}(\zeta_{m})^{\prime}$ for $m\in\mathcal{A}_{n}$. This is finite
since there are finitely many such fields, and coincides with the primes which
are ramified in $\mathbb{Q}(\zeta_{m})$ for some $m\in\mathcal{A}_{n}$ [Ho89,
p. 468]. Let $S=\\{p_{1},\dots,p_{s}\\}$ for distinct primes $p_{i}$. By
[Ho89, p. 469] there exists a cyclotomic field $L=\mathbb{Q}(\zeta_{\ell})$
such that $L\subseteq\mathbb{Q}(\zeta_{m})\subseteq L_{\infty}$ where
$L_{\infty}$ is the basic
$\mathbb{Z}_{p_{1}}\times\cdots\times\mathbb{Z}_{p_{s}}$-extension over $L$.
Furthermore, we have
$|\mathbb{Q}(\zeta_{m})/L|=\prod_{i=1}^{s}p_{i}^{n(p_{i})}$ for some
$n(p_{i})\geq 1$ and so, in the notation of [Fr82], we can write
$\mathbb{Q}(\zeta_{m})=L_{N}$ where $N=(n_{1},\dots,n_{s})$. Let $e_{N}^{(2)}$
denote the highest power of 2 dividing $h_{m}$. By [Fr82, Theorem B], we have
that $e_{N}^{(2)}=A\cdot n(2)+B$ for all but finitely many $N$, where $A,B\geq
0$ are integers that do not depend on $N$. This implies that there exists a
constant $C^{\prime}>0$ such that $e_{N}^{(2)}\leq C^{\prime}\cdot n(2)$ for
all $N$.

Let $K=\mathbb{Q}(\zeta_{m})^{\prime}$. Then
$|\mathbb{Q}(\zeta_{m})/\mathbb{Q}|=2^{r}t$ for some $r\geq 0$ and $t$ odd,
where $2^{r}=|K/\mathbb{Q}|\leq C\log n$ by the bound on
$\log(\mathcal{B}_{n})$. Since
$|\mathbb{Q}(\zeta_{m})/L|\mid|\mathbb{Q}(\zeta_{m})/\mathbb{Q}|$, we have
that $2^{n(2)}\leq 2^{r}\leq C\log n$. Hence
$h_{m}^{-}/\operatorname{odd}(h_{m}^{-})\leq
h_{m}/\operatorname{odd}(h_{m})=2^{e_{N}^{(2)}}\leq(C\log n)^{C^{\prime}}$.
Since $m\in\mathcal{A}_{n}$, we have $\operatorname{odd}(h_{m}^{-})\leq n$.
This gives that $h_{m}^{-}\leq a(\log n)^{b}n$ for some constants $a,b>0$ and
so, for any $\varepsilon>0$, we have that $h_{m}^{-}\leq an^{1+\varepsilon}$.
Combining this with Proposition 15.7 gives that $\log(an^{1+\varepsilon})\geq
C_{0}\frac{m\log m}{\log\log m}$ for some $C_{0}>0$, which implies that $\log
n\geq C\frac{m\log m}{\log\log m}$ for some $C>0$.

Finally, fix $m\geq 2$. Then $m\in\mathcal{A}_{n}$ where
$n=\operatorname{odd}(h_{m}^{-})$, and so gives

$\log(\operatorname{odd}(h_{m}^{-}))=\log n\geq C\frac{m\log m}{\log\log m},$

which is the required bound. ∎

We will now consider the parity of $h_{m}$ and $h_{m}^{-}$. Whilst we will not
explicitly make use of it, we will record the following basic observation
which dates back to Kummer (see, for example, [Ha52, Satz 45], [Yo98, Remark
1]).

######  Lemma 15.10.

Let $m\geq 2$. Then $h_{m}$ is odd if and only if $h_{m}^{-}$ is odd.

We will now state more detailed results in the case where $m$ is a prime
power.

######  Lemma 15.11.

Let $p$ be a prime such that $p\leq 509$ and let $n\geq 1$. Then:

  1. (i)

$h_{p}$ is odd if and only if

$p\not\in\\{29,113,163,197,239,277,311,337,349,373,397,421,463,491\\}.$

  2. (ii)

$h_{p^{n}}$ is odd if and only if $h_{p}$ is odd.

###### Proof.

(i) This was proven by Schoof [Sc98, Table 4.4].

(ii) The case $p=2$ is proven in [Ha52, Satz 36’], but the original result is
attributed to an 1886 article of Weber [We86]. See also [Yo98, p. 2590]. For
$n\geq 1$ and $p\leq 509$ an odd prime, it was shown by Ichimura–Nakajima that
$h_{p^{n}}/h_{p}$ is odd [IN12, Theorem 1 (II)]. The result follows. ∎

######  Remark 15.12.

Prior to the results of Ichimura–Nakajima, it was shown by Washington that
$h_{p^{n}}/h_{p}$ is odd for $p=3,5$ [Wa75]. Note that both results of
Washington and Ichimura–Nakajima all depend on Iwasawa theory. As far as we
are aware, this is an essential ingredient in all known proofs that there
exists an odd prime $p$ such that $h_{p^{n}}$ is odd for all $n\geq 1$.

####  15.4. Kernel groups of $\mathbb{Z}C_{m}$

The aim of this section will be to determine the involution on
$D(\mathbb{Z}C_{m})$ which is induced by the involution on
$C(\mathbb{Z}C_{m})$ (see Proposition 13.13).

We will begin with the following classical result due to Rim [Ri59, Theorem
6.24] (see also [CR87, Theorem 50.2]). Recall that, if $p$ is a prime and
$\Gamma_{p}$ is the maximal order in $\mathbb{Q}C_{p}$ containing
$\mathbb{Z}C_{p}$, then $\Gamma_{p}\cong\mathbb{Z}\times\mathbb{Z}[\zeta_{p}]$
and so $C(\Gamma_{p})\cong C(\mathbb{Z}[\zeta_{p}])$ since $\mathbb{Z}$ is a
PID.

######  Lemma 15.13.

Let $p$ be a prime. Then the map $\mathbb{Z}C_{p}\to\mathbb{Z}[\zeta_{p}]$,
$x\mapsto\zeta_{p}$ induces an isomorphism $C(\mathbb{Z}C_{p})\cong
C(\mathbb{Z}[\zeta_{p}])$. In particular, $D(\mathbb{Z}C_{p})=0$.

We will now determine $D(\mathbb{Z}C_{m})$, as well as its involution, in the
case where $m$ is square-free. This will be used in the proofs of Theorems
16.1 and 16.2. As usual, we will view an abelian group with involution as a
$\mathbb{Z}C_{2}$-module.

Let $\pi_{1}=1$, let $\pi_{p}=1-\zeta_{p}$ for a prime $p$ and, more
generally, let $\pi_{m}=\prod_{p\mid m}\pi_{p}$ for an integer $m\geq 2$. Note
that $\mathbb{Z}[\zeta_{m}]/\pi_{m}\cong\bigoplus_{p\mid
m}\mathbb{Z}[\zeta_{m}]/\pi_{p}$ since the $\pi_{p}$ are coprime (see, for
example, [CR87, p. 249]). Let
$\Psi_{m}\colon\mathbb{Z}[\zeta_{m}]^{\times}\to(\mathbb{Z}[\zeta_{m}]/\pi_{m})^{\times}$
be the natural map.

######  Definition 15.14.

For a square-free integer $m\geq 2$, define

$V_{m}=\operatorname{coker}(\Psi_{m}\colon\mathbb{Z}[\zeta_{m}]^{\times}\to(\mathbb{Z}[\zeta_{m}]/\pi_{m})^{\times}).$

We will view this as a $\mathbb{Z}C_{2}$-module with the involution induced by
the conjugation map

$\overline{\cdot}\colon(\mathbb{Z}[\zeta_{m}]/\pi_{m})^{\times}\to(\mathbb{Z}[\zeta_{m}]/\pi_{m})^{\times},\quad\zeta_{m}\mapsto\zeta_{m}^{-1}.$

Note that, if $p$ is prime, then
$\mathbb{Z}[\zeta_{p}]/\pi_{p}\cong\mathbb{F}_{p}$. We can then see that
$\Psi_{p}\colon\mathbb{Z}[\zeta_{p}]^{\times}\to\mathbb{F}_{p}^{\times}$ is
surjective by considering the cyclotomic units
$1+\zeta_{p}+\cdots+\zeta_{p}^{i-1}\in\mathbb{Z}[\zeta_{p}]^{\times}$ for
$(i,p)=1$. In particular, $V_{p}=1$. We will now show how $D(\mathbb{Z}C_{m})$
is related to $V_{d}$ for $d\mid m$.

######  Lemma 15.15.

Let $m\geq 2$ be a square-free integer. Let $d_{1},\dots,d_{n}$ be the
distinct nontrivial positive divisors of $m$, ordered such that $d_{i+1}$ has
at least as many prime factors as $d_{i}$ (so $d_{1}$ is prime and $d_{n}=m$).
Then there is a chain of $\mathbb{Z}C_{2}$-modules

$1=A_{0}\leq\cdots\leq A_{n}=D(\mathbb{Z}C_{m})$

such that $A_{i}/A_{i-1}\cong V_{d_{n-i+1}}$ for $1\leq i\leq n$.

It is proven in [CR87, Theorem 50.6] that $|D(\mathbb{Z}C_{m})|=\prod_{d\mid
n}|V_{d}|$. Our proof will involve following the argument given there, and
extending it to determine the group structure and the involution.

###### Proof.

First recall that $\Gamma=\bigoplus_{d\mid m}\mathbb{Z}[\zeta_{d}]$ is the
unique maximal order in $\mathbb{Q}C_{m}$ which contains $\mathbb{Z}C_{m}$. In
particular, this implies that there is a pullback square:

${\mathbb{Z}C_{m}}$${\Gamma}$${\mathbb{Z}C_{m}/m\Gamma}$${\Gamma/m\Gamma.}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{j_{2}}$$\scriptstyle{j_{1}}$

By [CR87, p. 246] this induces an exact sequence

$(\mathbb{Z}C_{m}/m\Gamma)^{\times}\oplus\Gamma^{\times}\xrightarrow[]{(j_{1},j_{2})}(\Gamma/m\Gamma)^{\times}\xrightarrow[]{\partial}D(\mathbb{Z}C_{m})\to
0$

where $\partial\colon u\mapsto M(u)$ where

$M(u)=\\{(x,y)\in(\mathbb{Z}C_{m}/m\Gamma)\times\Gamma\mid
j_{1}(x)=j_{2}(y)u\in\Gamma/m\Gamma\\}$

is the $\mathbb{Z}C_{m}$-module with action
$\lambda\cdot(x,y)=(i_{1}(\lambda)x,i_{2}(\lambda)y)$ for
$\lambda\in\mathbb{Z}C_{m}$. We now claim that the conjugation map on
$(\Gamma/m\Gamma)^{\times}$ induces the involution on $D(\mathbb{Z}C_{m})$.
First note that, by Proposition 13.14, the involution on $D(\mathbb{Z}C_{m})$
is induced by the natural involution $x\mapsto x^{-1}$ on the idèle group
$J(\mathbb{Q}C_{m})\subseteq\prod_{p}\mathbb{Q}_{p}C_{m}$. For all primes $p$,
we have
$\mathbb{Z}_{p}C_{m}\subseteq\Gamma_{p}\oplus(\mathbb{Z}C_{m}/m\Gamma)_{p}\subseteq\mathbb{Q}_{p}C_{m}$.
If $M(u)=\mathbb{Z}C_{m}\alpha$ for $\alpha=(\alpha_{p})\in
J(\mathbb{Q}C_{m})$, then [CR87, Exercise 53.1] implies that

$\alpha_{p}=\begin{cases}(1,1)\in\Gamma_{p}\oplus(\mathbb{Z}C_{m}/m\Gamma)_{p},&\text{if
$(p,m)=1$}\\\
(u_{p},1)\in\Gamma_{p}\oplus(\mathbb{Z}C_{m}/m\Gamma)_{p},&\text{if $(p,m)\neq
1$},\end{cases}$

where $u_{p}\in\Gamma_{p}$ is any element such that
$j_{2}(u_{p})=[u]\in(\Gamma/m\Gamma)_{p}$. By the same argument as in the
proof of Proposition 15.1, the involution on $J(\mathbb{Q}C_{m})$ induces an
involution on $J(\mathbb{Q}(\zeta_{d}))$ which coincides with the involution
induced by conjugation. In particular, the involution maps
$\alpha_{p}\mapsto(1,1)$ or $(\overline{u}_{p},1)$ where
$\overline{\,\cdot\,}\colon\Gamma_{p}\to\Gamma_{p}$ is induced by conjugation
on $\Gamma$. In particular, this coincides with the involution induced by
conjugation on $(\Gamma/m\Gamma)^{\times}$.

By [CR87, Lemma 50.7], $j_{1}$ can be replaced by the map
$\alpha\colon(\mathbb{Z}C_{m}/m\mathbb{Z}C_{m})^{\times}\to(\Gamma/m\Gamma)^{\times}$.
By [CR87, Lemma 50.8], $\operatorname{coker}(\alpha)\cong\bigoplus_{d\mid
m}(\mathbb{Z}[\zeta_{d}]/\pi_{d})^{\times}$ and it follows from the proof that
conjugation map on $(\Gamma/m\Gamma)^{\times}$ induces conjugation on
$(\mathbb{Z}[\zeta_{d}]/\pi_{d})^{\times}$ for each $d\mid m$. If
$\gamma\colon\Gamma^{\times}\to\operatorname{coker}(\alpha)$ is the map
induced by $j_{2}$, then we obtain an exact sequence

$\bigoplus_{d\mid
m}\mathbb{Z}[\zeta_{d}]^{\times}\xrightarrow[]{\gamma}\bigoplus_{d\mid
m}(\mathbb{Z}[\zeta_{d}]/\pi_{d})^{\times}\xrightarrow[]{\overline{\partial}}D(\mathbb{Z}C_{m})\to
0.$

Let $d_{1},\dots,d_{n}$ be the ordered sequence of divisors of $m$. By [CR87,
p. 250] we have that

$\gamma\mid_{\mathbb{Z}[\zeta_{d_{i}}]^{\times}}\colon\mathbb{Z}[\zeta_{d_{i}}]^{\times}\to(\mathbb{Z}[\zeta_{d_{i}}]/\pi_{d_{i}})^{\times}\oplus\bigoplus_{p\mid\frac{m}{d_{i}}}(\mathbb{Z}[\zeta_{p{d_{i}}}]/\pi_{p})^{\times}\subseteq\bigoplus_{j\geq
i}(\mathbb{Z}[\zeta_{d_{j}}]/\pi_{d_{j}})^{\times},\quad
x\mapsto(x,x^{-1},\dots,x^{-1}),$

where the last inclusion comes from the fact that
$\mathbb{Z}[\zeta_{d}]/\pi_{p}\subseteq\bigoplus_{q\mid
d}\mathbb{Z}[\zeta_{d}]/\pi_{q}\cong\mathbb{Z}[\zeta_{d}]/\pi_{d}$ for primes
$q$ since the $\pi_{q}$ are pairwise coprime in $\mathbb{Z}[\zeta_{d}]$. If
$x\in\mathbb{Z}[\zeta_{k}]^{\times}$, then
$x^{-1}\in(\mathbb{Z}[\zeta_{k}]/\pi_{p})^{\times}\subseteq(\mathbb{Z}[\zeta_{pk}]/\pi_{p})^{\times}$.

For $1\leq i\leq n$, this shows that $\gamma$ restricts to a map
$\gamma_{i}\colon\bigoplus_{j\geq
i}\mathbb{Z}[\zeta_{d_{j}}]^{\times}\to\bigoplus_{j\geq
i}(\mathbb{Z}[\zeta_{d_{j}}]/\pi_{j})^{\times}$ where $\gamma=\gamma_{1}$. By
a mild generalisation of [CR87, Exercise 50.2], this implies that there is an
exact sequence induced by the projection map:

$1\to\operatorname{coker}(\gamma_{i+1}^{\prime}\colon W\to\bigoplus_{j\geq
i+1}(\mathbb{Z}[\zeta_{d_{j}}]/\pi_{j})^{\times})\to\operatorname{coker}(\gamma_{i})\to\operatorname{coker}(\Psi_{d_{i}})\to
1,$

where $W=\\{x\in\mathbb{Z}[\zeta_{d_{i}}]^{\times}\mid x\equiv
1\,\,\text{mod}\,\pi_{d_{i}}\\}\oplus\bigoplus_{j\geq
i+1}\mathbb{Z}[\zeta_{d_{j}}]^{\times}$. By [CR87, p. 252], we have
$\operatorname{Im}(\gamma_{i+1}^{\prime})=\operatorname{Im}(\gamma_{i+1})$ and
so
$\operatorname{coker}(\gamma_{i+1}^{\prime})=\operatorname{coker}(\gamma_{i+1})$.
Let $A_{i}=\operatorname{coker}(\gamma_{n-i+1})$ for $1\leq i\leq n$ and
$A_{0}=1$. Then we have that $1=A_{0}\leq\cdots\leq
A_{n}=\operatorname{coker}(\gamma)=D(\mathbb{Z}C_{m})$such that there are
isomorphisms $A_{i}/A_{i-1}\cong\operatorname{coker}(\Psi_{d_{n-i+1}})$ as
abelian groups.

Since the involution on $D(\mathbb{Z}C_{m})$ is induced by conjugation on
$(\Gamma/m\Gamma)^{\times}$, it follows that it restricts to $A_{i}$ where it
acts via conjugation on the $\mathbb{Z}[\zeta_{d}]$. Hence, with respect to
the involution, $A_{i}\leq D(\mathbb{Z}C_{m})$ is a $\mathbb{Z}C_{2}$-modules
and the chain $A_{0}\leq\cdots\leq A_{n}$ is a chain of
$\mathbb{Z}C_{2}$-modules. Under the abelian group isomorphism

$A_{i}/A_{i-1}\cong\operatorname{coker}(\Psi_{d_{n-i+1}}\colon\mathbb{Z}[\zeta_{d_{n-i+1}}]^{\times}\to(\mathbb{Z}[\zeta_{d_{n-i+1}}]/\pi_{d_{n-i+1}})^{\times}),$

the involution on $\operatorname{coker}(\Psi_{d_{n-i+1}})$ induced by the
involution on $A_{i}$ coincides with the involution induced by conjugation on
$(\mathbb{Z}[\zeta_{d_{n-i+1}}]/\pi_{d_{n-i+1}})^{\times}$. Hence there is an
isomorphism of $\mathbb{Z}C_{2}$-modules $A_{i}/A_{i-1}\cong V_{n-i+1}$, as
required. ∎

Let $m\geq 2$ be a square-free integer. We will now give a method for
analysing the involution on $V_{m}=\operatorname{coker}(\Psi_{m})$. For a
field $\mathbb{F}$ and $\alpha_{1},\dots,\alpha_{n}\in\mathbb{C}$, we will
write $\mathbb{F}[\alpha_{1},\dots,\alpha_{n}]$ to denote
$\mathbb{F}\otimes_{\mathbb{Z}}\mathbb{Z}[\alpha_{1},\dots,\alpha_{n}]$. Note
that this may not be a field and so need not coincide with
$\mathbb{F}(\alpha_{1},\dots,\alpha_{n})$.

First note that, as described above, we have that

$\Psi_{m}\colon\mathbb{Z}[\zeta_{m}]^{\times}\to\bigoplus_{p\mid
m}(\mathbb{Z}[\zeta_{m}]/\pi_{p})^{\times},$

where $p$ ranges over the prime factors of $m$. If $p\mid m$ then, since $m$
is square-free, we have $m=pk$ where $(k,p)=1$ and so we can take
$\zeta_{m}=\zeta_{p}\cdot\zeta_{k}$. This implies that
$\mathbb{Z}[\zeta_{m}]=\mathbb{Z}[\zeta_{p},\zeta_{k}]$ and so

$\mathbb{Z}[\zeta_{m}]/\pi_{p}\cong\mathbb{Z}[\zeta_{p},\zeta_{k}]/\pi_{p}\cong\mathbb{F}_{p}[\zeta_{k}].$

Let $m=p_{1}\cdots p_{n}$ for distinct primes $p_{i}$. Then $\Psi_{m}$ can be
written as

$\Psi_{m}\colon\mathbb{Z}[\zeta_{m}]^{\times}\to\bigoplus_{i=1}^{n}\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times},$

where $\zeta_{m}=\prod_{j}\zeta_{p_{j}}$, $\zeta_{m/p_{i}}=\prod_{j\neq
i}\zeta_{p_{j}}$ and the map
$\mathbb{Z}[\zeta_{m}]^{\times}\to\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}$
is the map sending $\zeta_{p_{i}}\mapsto 1$. This motivates the following
definition.

######  Definition 15.16.

For a square-free integer $m\geq 2$, define

$\widetilde{V}_{m}\cong\operatorname{coker}\left(\Psi_{m}^{+}\colon\mathbb{Z}[\zeta_{m}]^{\times}\to\bigoplus_{i=1}^{n}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}}{\mathbb{F}_{p_{i}}[\lambda_{m/p_{i}}]^{\times}}$}\right),$

where
$\mathbb{F}_{p_{i}}[\lambda_{m/p_{i}}]^{\times}\leq\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}$
is induced by the inclusion
$\mathbb{Z}[\lambda_{m/p_{i}}]\leq\mathbb{Z}[\zeta_{m/p_{i}}]$ and
$\Psi_{m}^{+}$ is the composition of $\Psi_{m}$ with the quotient maps
$\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}\twoheadrightarrow\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}/\mathbb{F}_{p_{i}}[\lambda_{m/p_{i}}]^{\times}$.

We will view this as a $\mathbb{Z}C_{2}$-module with the involution induced by
the conjugation map

$\overline{\cdot}\colon\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}\to\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times},\quad\zeta_{m/p_{i}}\mapsto\zeta_{m/p_{i}}^{-1}.$

######  Lemma 15.17.

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

  1. (i)

There is a surjective $\mathbb{Z}C_{2}$-module homomorphism
$V_{m}\twoheadrightarrow\widetilde{V}_{m}$.

  2. (ii)

$\widetilde{V}_{m}=\widetilde{V}_{m}^{-}$, i.e. if $x\in\widetilde{V}_{m}$,
then $\overline{x}=-x\in\widetilde{V}_{m}$.

###### Proof.

(i) For $1\leq i\leq n$, we let
$f_{i}=\operatorname{Id}_{\mathbb{F}_{p_{i}}}\otimes_{\mathbb{Z}}\iota_{i}$,
where
$\iota_{i}\colon\mathbb{Z}[\lambda_{m/p_{i}}]\hookrightarrow\mathbb{Z}[\zeta_{m/p_{i}}]$
is the natural inclusion map. Then the surjective homomorphism
$V_{m}\twoheadrightarrow\widetilde{V}_{m}$ is induced by noting that

$\displaystyle\widetilde{V}_{m}$
$\displaystyle\cong\operatorname{coker}((\Psi_{m},f_{1},\dots,f_{n})\colon\mathbb{Z}[\zeta_{m}]^{\times}\oplus\bigoplus_{i=1}^{n}\mathbb{F}_{p_{i}}[\lambda_{m/p_{i}}]^{\times}\to\bigoplus_{i=1}^{n}\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times})$
$\displaystyle\cong\operatorname{coker}((f_{1},\dots,f_{n})\colon\bigoplus_{i=1}^{n}\mathbb{F}_{p_{i}}[\lambda_{m/p_{i}}]^{\times}\to
V_{m}).$

(ii) We first claim that, if $p$ is prime and $n$ is an integer, then
$\alpha\in\mathbb{F}_{p}[\zeta_{n}]$ implies
$\alpha\cdot\overline{\alpha}\in\mathbb{F}_{p}[\lambda_{n}]$. Let
$\beta=\alpha\cdot\overline{\alpha}$. Since $\mathbb{Z}[\zeta_{n}]$ has
integral basis $\\{\zeta_{n}^{i}\\}_{i=0}^{n-1}$, we can write
$\beta=\sum_{i=0}^{n-1}a_{i}\otimes\zeta_{n}^{i}$ for
$a_{i}\in\mathbb{F}_{p}$. Note that $\overline{\beta}=\beta$ which implies
that

$\sum_{i=0}^{n-1}a_{i}\otimes\zeta_{n}^{i}=\sum_{i=0}^{n-1}a_{n-i}\otimes\zeta_{n}^{i}\in\mathbb{F}_{p}[\zeta_{n}].$

Since
$\mathbb{F}_{p}[\zeta_{n}]=\mathbb{F}_{p}\otimes_{\mathbb{Z}}\mathbb{Z}[\zeta_{n}]\cong\mathbb{F}_{p}^{n}$
as an abelian group, we get that $a_{n}=a_{n-i}\in\mathbb{F}_{p}$ for all $i$
and so

$\beta=a_{0}+\sum_{i=1}^{\lfloor\frac{n-1}{2}\rfloor}a_{i}\otimes(\zeta_{n}^{i}+\zeta_{n}^{-i})+\varepsilon\in\mathbb{F}_{p}[\lambda_{n}],$

where $\varepsilon=-a_{n/2}$ for $n$ even and $\varepsilon=0$ for $n$ odd.

Finally, let $f\colon V_{m}\to\widetilde{V}_{m}$ be the map described above
and let $\alpha=[(\alpha_{1},\dots\alpha_{n})]$, where
$\alpha_{i}\in\mathbb{F}_{p_{i}}[\zeta_{d/p_{i}}]^{\times}$. We have shown
that
$\alpha_{i}\cdot\overline{\alpha}_{i}\in\mathbb{F}_{p_{i}}[\lambda_{d/p_{i}}]^{\times}$
for all $i$ and so

$f(\alpha)\cdot
f(\overline{\alpha})=[(\alpha_{1}\cdot\overline{\alpha}_{1},\dots,\alpha_{n}\cdot\overline{\alpha}_{n})]=[(1,\dots,1)]=1$

and $f(\overline{\alpha})=f(\alpha)^{-1}$. Since $f$ is surjective and induces
the involution on $\widetilde{V}_{m}$, this implies that $\overline{x}=-x$ for
all $x\in\widetilde{V}_{m}$, where we now write the inverse as $-x$ rather
than $x^{-1}$ since $\widetilde{V}_{m}$ is an abelian group. ∎

We will now deduce the following, which is the main result of this section.
Note that this is analogous to Proposition 15.4 which applied in the case of
ideal class groups. Recall that, for an integer $m$, we let
$\operatorname{odd}(m)$ denote the unique odd integer $r$ such that $m=2^{k}r$
for some $k$.

######  Proposition 15.18.

Let $m\geq 2$ be a square-free integer. Let $d_{1},\dots,d_{n}$ be the
distinct nontrivial positive divisors of $m$, ordered such that $d_{i+1}$ has
at least as many prime factors as $d_{i}$, and let $A_{i}$ be the
$\mathbb{Z}C_{2}$-modules defined in Lemma 15.15. Then there is a chain of
abelian subgroups

$1=\\{x-\overline{x}\mid x\in A_{0}\\}\leq\cdots\leq\\{x-\overline{x}\mid x\in
A_{n}\\}=\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}$

and, for each $1\leq i\leq n$, there are surjective group homomorphisms

$\\{x-\overline{x}\mid x\in A_{i}\\}/\\{x-\overline{x}\mid x\in
A_{i-1}\\}\twoheadrightarrow\\{x-\overline{x}\mid x\in
V_{d_{n-i+1}}\\}\twoheadrightarrow 2\cdot\widetilde{V}_{d_{n-i+1}}.$

In particular, $\prod_{d\mid m}\operatorname{odd}(|\widetilde{V}_{d}|)$
divides $|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|$.

###### Proof.

The chain of abelian subgroups follows direct from Lemma 15.15 and Lemma 14.3
(ii). Let $1\leq i\leq n$. By Lemma 15.15, there is a short exact sequence of
$\mathbb{Z}C_{2}$-modules

$0\to A_{i-1}\to A_{i}\to V_{d_{n-i+1}}\to 0.$

By Lemma 14.3 (ii), there are injective and surjective maps:

$\\{x-\overline{x}\mid x\in A_{i-1}\\}\hookrightarrow\\{x-\overline{x}\mid
x\in A_{i}\\}\twoheadrightarrow\\{x-\overline{x}\mid x\in V_{d_{n-i+1}}\\}.$

Since the composition is necessarily the zero map, this gives the first
surjective homomorphism. The second is a direct consequence of both parts of
Lemma 15.17 as well as Lemma 14.3 (ii) again.

For the last part, the two statements we have proved so far imply that

$|\\{x-\overline{x}\mid x\in
D(\mathbb{Z}C_{m})\\}|=\prod_{i=1}^{n}|\\{x-\overline{x}\mid x\in
A_{i}\\}/\\{x-\overline{x}\mid x\in A_{i-1}\\}|$

and $|2\cdot\widetilde{V}_{d_{n-i+1}}|$ divides $|\\{x-\overline{x}\mid x\in
A_{i}\\}/\\{x-\overline{x}\mid x\in A_{i-1}\\}|$ for all $1\leq i\leq n$. It
now suffices to note that $\operatorname{odd}(|\widetilde{V}_{d_{n-i+1}}|)$
divides $|2\cdot\widetilde{V}_{d_{n-i+1}}|$ by the same argument as in the
proof of Proposition 15.4. ∎

######  Remark 15.19.

In order to obtain a complete analogue of Proposition 15.4, it would be
desirable to also obtain bounds on $|\\{x\in D(\mathbb{Z}C_{m})\mid
x=-\overline{x}\\}|$. However, unlike Proposition 15.4, the bounds we obtain
in Proposition 15.18 are obtained by subquotients rather than just subgroups.
We therefore can’t simply apply Lemma 14.3 (i) since the final map in the
sequence need not be surjective in general (see Remark 14.4). As we shall see
in Section 16, it is possible to circumvent the need for such bounds in the
proof of Theorem 16.1.

We now conclude this section with a result which holds in the case that $m$ is
not square-free. Firstly an analogue of Lemma 15.15 holds in the case that $m$
is a prime power, by Kervaire–Murthy [KM77, Theorem 1.2]. For brevity, we will
not state this result here. We will instead make do with the following
consequence of their result in the case that $p=2$ which will be used in the
proof of Theorem 16.3. Note that $V_{2^{n+1}}$ is directly analogous to the
$\mathbb{Z}C_{2}$-modules $V_{m}$ defined in Definition 15.14 in the case that
$m$ is square-free.

######  Proposition 15.20.

Let $n\geq 1$. Then there exists an exact sequence of
$\mathbb{Z}C_{2}$-modules

$0\to V_{2^{n+1}}\to D(\mathbb{Z}C_{2^{n+1}})\to D(\mathbb{Z}C_{2^{n}})\to 0,$

where $V_{2^{n+1}}=\bigoplus_{i=1}^{n-2}(\mathbb{Z}/2^{i})^{2^{n-i-2}}$ with
the involution acting by negation.

###### Proof.

This is a consequence of results of Kervaire–Murthy [KM77]. Specifically, in
[KM77, p. 419] they show that there is an exact sequence of
$\mathbb{Z}C_{2}$-modules

$0\to
V_{2^{n+1}}\to\widetilde{K}_{0}(\mathbb{Z}C_{2^{n+1}})\xrightarrow[]{\alpha}\widetilde{K}_{0}(\mathbb{Z}C_{2^{n}})\oplus\widetilde{K}_{0}(\mathbb{Z}[\zeta_{2^{n+1}}])\to
0$

where $\alpha$ is induced by the natural map of rings
$\mathbb{Z}C_{2^{n+1}}\to\mathbb{Z}C_{2^{n}}\times\mathbb{Z}[\zeta_{2^{n+1}}]$.
Since the maximal order
$\mathbb{Z}C_{2^{n}}\subseteq\Gamma_{2^{n}}\subseteq\mathbb{Q}C_{2^{n}}$ is
given by $\Gamma_{2^{n}}=\bigoplus_{i=1}^{n}\mathbb{Z}[\zeta_{2^{i}}]$, we get
that

$\ker(\alpha)\cong\ker(\beta\colon D(\mathbb{Z}C_{2^{n+1}})\to
D(\mathbb{Z}C_{2^{n}}))$

where $\beta$ is the $\mathbb{Z}C_{2}$-module homomorphism induced by map
$\mathbb{Z}C_{2^{n+1}}\to\mathbb{Z}C_{2^{n}}$. This gives an exact sequence of
the required form. It follows from [KM77, Theorem 1.1] and the discussion
which follows that $V_{2^{n+1}}$ is as described. ∎

####  15.5. Divisibility and lower bounds for kernel groups

The aim of this section will be to establish divisibility results for
$|D(\mathbb{Z}C_{m})|$ and $\operatorname{odd}(|\widetilde{V}_{m}|)$. These
results are necessary for determining the involution on $D(\mathbb{Z}C_{m})$
in an analogous way to how divisibility results for class numbers $h_{m}$ were
necessary for determining the involution on $C(\mathbb{Z}[\zeta_{m}])$ (see
Section 15.3). The results on $\operatorname{odd}(|\widetilde{V}_{m}|)$ are
motivated by Proposition 15.18.

We begin by recalling the following result, which can be found in [CR87,
Theorem 50.18].

######  Proposition 15.21.

If $p$ is a prime and $G$ is a finite $p$-group, then $D(\mathbb{Z}G)$ is an
abelian $p$-group. In particular, if $p\neq 2$, then $|D(\mathbb{Z}G)|$ is
odd.

We will now find conditions on $m\geq 2$ square-free for which
$\operatorname{odd}(|\widetilde{V}_{m}|)\neq 1$. Our strategy is motivated by
the bounds $d$ such that $d\mid|D(\mathbb{Z}C_{m})|$ which were obtained by
Cassou-Noguès in [CN72, CN74]. In particular, our argument shows that these
bounds actually give factors of $|\widetilde{V}_{m}|$. Recall that
$\widetilde{V}_{m}=\operatorname{coker}(\Psi_{m}^{+})$, where

$\Psi_{m}^{+}\colon\mathbb{Z}[\zeta_{m}]^{\times}\to\bigoplus_{i=1}^{n}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}}{\mathbb{F}_{p_{i}}[\lambda_{m/p_{i}}]^{\times}}$}$

is the map defined in Definition 15.16.

######  Lemma 15.22.

Let $m\geq 2$ be an integer, let $p$ be a prime such that $p\nmid m$, let
$f_{p}=\operatorname{ord}_{m}(p)$ denote the order of $p$ in $\mathbb{Z}/m$
and let $g_{p}=\varphi(m)/2f_{p}$.

  1. (i)

The inclusion $\mathbb{Z}[\lambda_{m}]\subseteq\mathbb{Z}[\zeta_{m}]$ induces
inclusion $\mathbb{F}_{p}[\lambda_{m}]\subseteq\mathbb{F}_{p}[\zeta_{m}]$.

  2. (ii)

If $m\geq 3$, then
$|\mathbb{F}_{p}[\zeta_{m}]^{\times}|=(p^{f_{p}}-1)^{g_{p}}$.

  3. (iii)

If $m\geq 3$, then
$|\mathbb{F}_{p}[\lambda_{m}]^{\times}|=\begin{cases}(p^{\frac{f_{p}}{2}}-1)^{g_{p}},&\text{if
$f_{p}$ is even}\\\ (p^{f_{p}}-1)^{\frac{g_{p}}{2}},&\text{if $f_{p}$ is
odd.}\end{cases}$

If $m=2$, then $\zeta_{2},\lambda_{2}\in\mathbb{Z}$ and so
$|\mathbb{F}_{p}[\zeta_{2}]^{\times}|=|\mathbb{F}_{p}[\lambda_{2}]^{\times}|=p-1$.

###### Proof.

The proofs of (ii) and (iii) are analogous. We will prove (iii) only as it is
more complicated. First note that $\mathbb{Q}(\lambda_{m})/\mathbb{Q}$ is a
Galois extension and $p$ is unramified in $\mathbb{Q}(\lambda_{m})/\mathbb{Q}$
since it is unramified in $\mathbb{Q}(\zeta_{m})/\mathbb{Q}$ (see, for
example, [Wa97, Lemma 15.48]). This implies that
$p\cdot\mathbb{Z}[\lambda_{m}]=\mathcal{P}_{1}\cdots\mathcal{P}_{g}$ for some
$g\geq 1$, where the $\mathcal{P}_{i}\subseteq\mathbb{Z}[\lambda_{m}]$ are
distinct prime ideals. The $\mathcal{P}_{i}$ coincide under the Galois action
and the $\mathbb{Z}[\lambda_{m}]/\mathcal{P}_{i}$ all coincide with the
splitting field $\mathbb{F}_{p}(\lambda_{m})$ and so

$\mathbb{F}_{p}[\lambda_{m}]\cong\mathbb{Z}[\lambda_{m}]/(p)\cong\mathbb{F}_{p}(\lambda_{m})^{g}$

which implies that
$\mathbb{F}_{p}[\lambda_{m}]^{\times}\cong(\mathbb{F}_{p}(\lambda_{m})^{\times})^{g}$.

Let $f=[\mathbb{F}_{p}(\lambda_{m}):\mathbb{F}_{p}]$. Since
$\operatorname{Gal}(\mathbb{F}_{p}(\lambda_{m})/\mathbb{F}_{p})$ is generated
by the Frobenius element $\operatorname{Frob}_{p}\colon x\mapsto x^{p}$, we
get that $f$ is the smallest positive integer such that
$\operatorname{Frob}_{p}^{f}=\operatorname{Id}_{\mathbb{F}_{p}(\lambda_{m})}$.
Note that
$\operatorname{Frob}_{p}^{f}(\lambda_{m})=\zeta_{m}^{p^{f}}+\zeta_{m}^{-p^{f}}$.
This implies that
$\operatorname{Frob}_{p}^{f}=\operatorname{Id}_{\mathbb{F}_{p}(\lambda_{m})}$
if and only if $\zeta_{m}^{p^{f}}=\zeta_{m}^{\pm 1}$ and so $f$ is the order
of $p$ in $(\mathbb{Z}/m)^{\times}/\\{\pm 1\\}$. Since
$[\mathbb{Q}(\lambda_{m}):\mathbb{Q}]=\varphi(m)/2$, we have that
$|\mathbb{F}_{p}[\lambda_{m}]|=p^{\varphi(m)/2}$. Since
$|\mathbb{F}_{p}(\lambda_{m})^{g}|=p^{fg}$, this gives that $g=\varphi(m)/2f$.
Hence we have
$|\mathbb{F}_{p}[\lambda_{m}]^{\times}|=|\mathbb{F}_{p}(\lambda_{m})^{\times}|^{g}=(p^{f}-1)^{g}$.
Note that $f=f_{p}$ if and only if $f_{p}$ is odd, and otherwise $f=f_{p}/2$.

Finally, (i) follows by comparing the expressions for
$\mathbb{F}_{p}[\lambda_{m}]$ and $\mathbb{F}_{p}[\zeta_{m}]$ as products of
fields. ∎

We will now use Lemma 15.22 to obtain bounds on $|\widetilde{V}_{pq}|$ for
$p,q$ distinct odd primes, and for $|\widetilde{V}_{2p}|$, where $p$ is an odd
prime respectively.

######  Proposition 15.23.

Let $p$ and $q$ be distinct odd primes. Let $f_{p}=\operatorname{ord}_{q}(p)$,
$g_{p}=(q-1)/2f_{p}$, $f_{q}=\operatorname{ord}_{p}(q)$ and
$g_{q}=(p-1)/2f_{q}$. Define

$c_{pq}=\begin{cases}\frac{1}{2pq}(p^{\frac{f_{p}}{2}}+1)^{g_{p}}(q^{\frac{f_{q}}{2}}+1)^{g_{q}},&\text{if
$f_{p},f_{q}$ are even}\\\
\frac{1}{2pq}(p^{\frac{f_{p}}{2}}+1)^{g_{p}}(q^{f_{q}}-1)^{\frac{g_{q}}{2}},&\text{if
$f_{p}$ is even and $f_{q}$ is odd}\\\
\frac{1}{2pq}(p^{f_{p}}-1)^{\frac{g_{p}}{2}}(q^{\frac{f_{q}}{2}}+1)^{g_{q}},&\text{if
$f_{p}$ is odd and $f_{q}$ is even}\\\
\frac{1}{2pq}(p^{f_{p}}-1)^{\frac{g_{p}}{2}}(q^{f_{q}}-1)^{\frac{g_{q}}{2}},&\text{if
$f_{p},f_{q}$ are odd.}\end{cases}$

Then $c_{pq}\mid|\widetilde{V}_{pq}|$. In particular, if
$\operatorname{odd}(c_{pq})\neq 1$, then
$\operatorname{odd}(|\widetilde{V}_{pq}|)\neq 1$.

###### Proof.

Let
$E=\langle\zeta_{pq},\mathbb{Z}[\lambda_{pq}]^{\times}\rangle\leq\mathbb{Z}[\zeta_{pq}]^{\times}$.
By [Wa97, Corollary 4.13], $E$ has index two with
$\mathbb{Z}[\zeta_{pq}]^{\times}/E\cong\mathbb{Z}/2$ generated by
$1-\zeta_{pq}$. Let
$\psi_{p}\colon\mathbb{Z}[\zeta_{pq}]^{\times}\to\mathbb{F}_{p}[\zeta_{q}]^{\times}$
be the map sending $\zeta_{p}\mapsto 1$. If
$\alpha\in\mathbb{Z}[\lambda_{pq}]^{\times}$, then
$\alpha=\sum_{i=0}^{\frac{pq-1}{2}}a_{i}(\zeta_{pq}^{i}+\zeta_{pq}^{-i})$ for
some $a_{i}\in\mathbb{Z}$ and so

$\psi_{p}(\alpha)=\sum_{i=0}^{\frac{q-1}{2}}\widetilde{a}_{i}(\zeta_{q}^{i}+\zeta_{q}^{-i})\in\mathbb{F}_{p}[\lambda_{q}]^{\times}$

where $\widetilde{a}_{i}=\sum_{k=0}^{\frac{p-1}{2}}a_{i+kq}$. Hence the
composition
$\mathbb{Z}[\zeta_{pq}]^{\times}\xrightarrow[]{\psi_{p}}\mathbb{F}_{p}[\zeta_{q}]^{\times}\to\frac{\mathbb{F}_{p}[\zeta_{q}]^{\times}}{\mathbb{F}_{p}[\lambda_{q}]^{\times}}$
is trivial and so

$\widetilde{V}_{pq}=\operatorname{coker}(\Psi_{pq}^{+})=\operatorname{coker}\Big{(}\mathbb{Z}/pq\oplus\mathbb{Z}/2\to\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{p}[\zeta_{q}]^{\times}}{\mathbb{F}_{p}[\lambda_{q}]^{\times}}$}\oplus\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{q}[\zeta_{p}]^{\times}}{\mathbb{F}_{q}[\lambda_{p}]^{\times}}$}\Big{)}$

where $1\in\mathbb{Z}/pq$ maps to $\Psi_{pq}^{+}(\zeta_{pq})$ and
$1\in\mathbb{Z}/2$ maps to $\Psi_{pq}^{+}(1-\zeta_{pq})$. In particular, this
implies that $|\widetilde{V}_{pq}|$ is divisible by
$\frac{1}{2pq}\cdot|\frac{\mathbb{F}_{p}[\zeta_{q}]^{\times}}{\mathbb{F}_{p}[\lambda_{q}]^{\times}}|\cdot|\frac{\mathbb{F}_{q}[\zeta_{p}]^{\times}}{\mathbb{F}_{q}[\lambda_{p}]^{\times}}|$.
The result now follows from Lemma 15.22. ∎

######  Remark 15.24.

The argument of Proposition 15.23 also applies in the case $q=2$ and gives the
bound

$\frac{1}{4p}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{p}[\zeta_{2}]^{\times}}{\mathbb{F}_{p}[\lambda_{2}]^{\times}}$}\Big{|}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{2}[\zeta_{p}]^{\times}}{\mathbb{F}_{2}[\lambda_{p}]^{\times}}$}\Big{|}=\begin{cases}\frac{1}{4p}(2^{\frac{f_{2}}{2}}+1)^{g_{2}},&\text{if
$f_{2}$ is even}\\\ \frac{1}{4p}(2^{f_{2}}-1)^{\frac{g_{2}}{2}},&\text{if
$f_{2}$ is odd}\end{cases}$

for $|\widetilde{V}_{2p}|$. However, this bound in improved upon in the
proposition below.

######  Proposition 15.25.

Let $p$ be an odd prime. Let $f_{2}=\operatorname{ord}_{p}(2)$ and
$g_{2}=(p-1)/2f_{2}$. Define

$c_{2p}=\begin{cases}\frac{1}{p}(2^{\frac{f_{2}}{2}}+1)^{g_{2}},&\text{if
$f_{2}$ is even}\\\ \frac{1}{p}(2^{f_{2}}-1)^{\frac{g_{2}}{2}},&\text{if
$f_{2}$ is odd.}\end{cases}$

Then $c_{2p}\mid|\widetilde{V}_{2p}|$. In particular, if
$\operatorname{odd}(c_{2p})\neq 1$, then
$\operatorname{odd}(|\widetilde{V}_{2p}|)\neq 1$.

###### Proof.

Let
$\pi_{2p}\colon\mathbb{Z}[\zeta_{p}]^{\times}\to\mathbb{F}_{2}[\zeta_{p}]^{\times}$
be reduction mod $2$. Since $\zeta_{2p}=-\zeta_{p}$ and $\zeta_{2}=-1$, we
have
$\Psi_{2p}\colon\mathbb{Z}[\zeta_{p}]^{\times}\to\mathbb{F}_{2}[\zeta_{p}]^{\times}\oplus\mathbb{F}_{p}^{\times}$.
It is shown in [CR87, Theorem 50.14] that projection induces an isomorphism
$V_{2p}=\operatorname{coker}(\Psi_{2p})\cong\operatorname{coker}(\pi_{2p})$.
Similarly, if
$\pi_{2p}^{+}\colon\mathbb{Z}[\zeta_{p}]^{\times}\to\frac{\mathbb{F}_{2}[\zeta_{p}]^{\times}}{\mathbb{F}_{2}[\lambda_{p}]^{\times}}$,
then
$\widetilde{V}_{2p}=\operatorname{coker}(\Psi_{2p}^{+})\cong\operatorname{coker}(\pi_{2p}^{+})$.
By [Wa97, Corollary 4.13],
$\mathbb{Z}[\zeta_{p}]^{\times}=\langle\zeta_{p},\mathbb{Z}[\lambda_{p}]^{\times}\rangle$.
The same argument as Proposition 15.23 implies that
$|\operatorname{coker}(\pi_{2p}^{+})|$ is divisible by

$\mbox{\footnotesize$\displaystyle\frac{1}{p}$}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{p}[\zeta_{2}]^{\times}}{\mathbb{F}_{p}[\lambda_{2}]^{\times}}$}\Big{|}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{2}[\zeta_{p}]^{\times}}{\mathbb{F}_{2}[\lambda_{p}]^{\times}}$}\Big{|}=\mbox{\footnotesize$\displaystyle\frac{1}{p}$}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{2}[\zeta_{p}]^{\times}}{\mathbb{F}_{2}[\lambda_{p}]^{\times}}$}\Big{|}.$

The result now follows from Lemma 15.22. ∎

###  16\. Proof of main results on the involution on
$\widetilde{K}_{0}(\mathbb{Z}C_{m})$

The aim of this section will be to prove the following three theorems. As we
saw previously, these theorems are required to prove Theorems 5.14, 5.15,
LABEL:, and 5.17 respectively.

######  Theorem 16.1.

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

  1. (i)

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

  2. (ii)

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

######  Theorem 16.2.

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

  1. (i)

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

  2. (ii)

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

######  Theorem 16.3.

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

  1. (i)

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

  2. (ii)

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

####  16.1. Proof of Theorems 16.1 and 16.2

The proofs of Theorems 16.1 and 16.2 are best handled together since
$\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}\leq\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}$.
In particular, it suffices to prove the following four statements.

  1. (A1)

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

  2. (A2)

If $m\not\in\\{2,3,5,6,7,10,11,13,14,15,17,19,29\\}$, then
$|\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|\neq 1$.

  3. (A3)

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

  4. (A4)

If $m\in\\{15,29\\}$, then $|\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|=1$ and
$|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}|\neq 1$.

To see this, note that (A1) coincides with Theorem 16.2 (ii) and implies
Theorem 16.1 (ii). The forwards direction of Theorem 16.1 (i) is implied by
(A2) and (A4), and the backwards direction coincides with (A3). The forwards
direction of Theorem 16.2 (i) coincides with (A2) and the backwards direction
is implied by (A3) and (A4).

Recall from Section 13.1 that $\widetilde{K}_{0}(\mathbb{Z}C_{m})\cong
C(\mathbb{Z}C_{m})$ is an isomorphism of $\mathbb{Z}C_{2}$-modules, where
$C(\mathbb{Z}C_{m})$ denotes the locally free class group. We therefore have
the following short exact sequence of $\mathbb{Z}C_{2}$-modules established in
Section 15.1:

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

where $D(\mathbb{Z}C_{m})$ has the induced involution, and each
$C(\mathbb{Z}[\zeta_{d}])$ has the involution induced by conjugation. Each of
statements (A1)–(A4) will be proven via the following lemma.

######  Lemma 16.4.

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

  1. (i)

$|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|\leq|\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|$;

  2. (ii)

$\prod_{d\mid m}|\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{d}])\\}|\leq|\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|$;

  3. (iii)

if $|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|\neq 1$ or
$|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}|\neq 1$, then
$|\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|\neq 1$
$($and so also
$|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}|\neq 1)$;
and

  4. (iv)

if $|\\{x\in D(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}|=1$ and $\prod_{d\mid
m}|\\{x\in C(\mathbb{Z}[\zeta_{d}])\mid\overline{x}=-x\\}|=1$, then
$|\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\mid\overline{x}=-x\\}|=1$ $($and
so also $|\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|=1)$.

###### Proof.

By Lemma 14.3, there is an exact sequence

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

as well as injective and surjective maps

$\\{x-\overline{x}\mid x\in
D(\mathbb{Z}C_{m})\\}\hookrightarrow\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}\twoheadrightarrow\bigoplus_{d\mid
m}\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{d}])\\}.$

The second sequence implies (i) and (ii), with (iii) as a corollary. The first
sequence implies (iv). ∎

######  Remark 16.5.

Part (iii) might look as though it is weaker than it needs to be since, by
(ii), $|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{d}])\\}|\neq 1$ for
some $d\mid m$ also implies that $|\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{m})\\}|\neq 1$. However, by Lemma 14.3
(ii), $|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{d}])\\}|\neq 1$ for
some $d\mid m$ implies $|\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}|\neq 1$.

We will now proceed to prove each of statements (A1)-(A4). We will begin with
the following which, by Lemma 16.4 (ii), implies (A1).

######  Proposition 16.6.

We have that $|\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}|\to\infty$ super-exponentially in $m$.

###### Proof.

It was shown in Proposition 15.4 that $\operatorname{odd}(h_{m}^{-})$ divides
$|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}|$. The result now
follows from that fact that, by Proposition 15.9,
$\operatorname{odd}(h_{m}^{-})\to\infty$ super-exponentially in $m$. ∎

We will now prove (A2). Our approach will be to use Lemma 16.4 (iii). In
particular, we will begin by classifying the $m\geq 2$ square-free for which
$|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}|=1$. We will then
determine the subset of these values for which $|\\{x-\overline{x}\mid x\in
D(\mathbb{Z}C_{m})\\}|=1$.

######  Proposition 16.7.

The complete list of $m\geq 2$ square-free for which $|\\{x-\overline{x}\mid
x\in C(\mathbb{Z}[\zeta_{m}])\\}|=1$ is as follows:

$m=\begin{cases}\text{$p$},&\text{where $p\in\\{2,3,5,7,11,13,17,19,29\\}$}\\\
\text{$2p$},&\text{where $p\in\\{3,5,7,11,13,17,19,29\\}$}\\\ \text{$pq$ or
$2pq$},&\text{where
$(p,q)\in\\{(3,5),(3,7),(3,11),(5,7),(3,13)\\}$}.\end{cases}$

###### Proof.

First note that $h_{m}^{-}=1$ implies $|\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}|=1$. On the other hand, it follows from
Proposition 15.4 that $\operatorname{odd}(h_{m}^{-})\neq 1$ implies
$|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}|\neq 1$. In
Proposition 15.8 (i), it is shown that the $m\geq 2$ square-free for which
$\operatorname{odd}(h_{m}^{-})=1$ and $h_{m}^{-}\neq 1$ are precisely the
$m\in S$, where $S=\\{29,39,58,65,78,130\\}$. It remains to determine for
which $m\in S$ we have $|\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}|=1$.

Suppose $m\in S$. By [Wa97, p. 421], we have $h_{m}^{+}=1$ and so
$C(\mathbb{Z}[\zeta_{m}])=C(\mathbb{Z}[\zeta_{m}])^{-}$ by Lemma 15.2 (ii). In
particular, we have:

$\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}=2\cdot
C(\mathbb{Z}[\zeta_{m}]).$

If $m=29$ or $68$, then $C(\mathbb{Z}[\zeta_{m}])\cong(\mathbb{Z}/2)^{3}$ and
$2\cdot C(\mathbb{Z}[\zeta_{m}])=0$ [Wa97, p. 412]. If $m=39$ or $78$, then
$h_{m}=2$ and $C(\mathbb{Z}[\zeta_{m}])\cong\mathbb{Z}/2$ and $2\cdot
C(\mathbb{Z}[\zeta_{m}])=0$ [Wa97, p. 412]. If $m=65$ or $130$, then [Ho93,
Proposition 1 (iv)] gives that
$C(\mathbb{Z}[\zeta_{m}])\cong(\mathbb{Z}/2)^{2}\times(\mathbb{Z}/4)^{2}$ and
$2\cdot C(\mathbb{Z}[\zeta_{m}])\cong(\mathbb{Z}/2)^{2}$.

Hence we have shown that, for $m\geq 2$ square-free, $|\\{x-\overline{x}\mid
x\in C(\mathbb{Z}[\zeta_{m}])\\}|=1$ if and only if $h_{m}^{-}=1$ or
$m\in\\{29,39,58,78\\}$. The result now follows by Proposition 15.6 (i). ∎

We now prove the following. By Lemma 16.4 (iii), this completes the proof of
(A2).

######  Proposition 16.8.

The complete list of $m\geq 2$ square-free for which $|\\{x-\overline{x}\mid
x\in C(\mathbb{Z}[\zeta_{m}])\\}|=1$ and $|\\{x-\overline{x}\mid x\in
D(\mathbb{Z}C_{m})\\}|=1$ is as follows:

$m=\begin{cases}\text{$p$},&\text{where $p\in\\{2,3,5,7,11,13,17,19,29\\}$}\\\
\text{$2p$},&\text{where $p\in\\{3,5,7\\}$},\\\ \text{$pq$},&\text{where
$(p,q)=(3,5)$}.\end{cases}$

That is, $m\in\\{2,3,5,6,7,10,11,13,14,15,17,19,29\\}$.

Before turning to the proof, we will begin by recalling that Cassou-Noguès
determined the integers $m\geq 2$ for which $D(\mathbb{Z}C_{m})=0$ [CN74,
Theorème 1] (see also [CN72]). The following can be deduced by comparing this
with Proposition 16.7.

######  Lemma 16.9.

Let $m\geq 2$ be a square-free integer such that $|\\{x-\overline{x}\mid x\in
C(\mathbb{Z}[\zeta_{m}])\\}|=1$. Then $D(\mathbb{Z}C_{m})=0$ if and only if
$m$ is prime or $m=2p$ where $p\in\\{3,5,7\\}$.

###### Proof of Proposition 16.8.

Suppose $m\in\\{2,3,5,6,7,10,11,13,14,15,17,19,29\\}$. If $m\neq 15$, then
Lemma 16.9 implies that $D(\mathbb{Z}C_{m})=0$ and so $|\\{x-\overline{x}\mid
x\in D(\mathbb{Z}C_{m})\\}|=1$. If $m=15$, then it is shown in [CN72, p. 48]
that $|D(\mathbb{Z}C_{15})|=2$. This implies that
$D(\mathbb{Z}C_{15})\cong\mathbb{Z}/2$ has the trivial action and so
$|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{15})\\}|=1$.

By Proposition 16.7, the remaining $m\geq 2$ square-free for which
$|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}|=1$ are:

$m=\begin{cases}\text{$2p$},&\text{where $p\in\\{11,13,17,19,29\\}$}\\\
\text{$pq$},&\text{where $(p,q)\in\\{(3,7),(3,11),(5,7),(3,13)\\}$}\\\
\text{$2pq$},&\text{where
$(p,q)\in\\{(3,5),(3,7),(3,11),(5,7),(3,13)\\}$}.\end{cases}$

By Proposition 15.18, we have that $\prod_{d\mid
m}\operatorname{odd}(|\widetilde{V}_{d}|)$ divides $|\\{x-\overline{x}\mid
x\in D(\mathbb{Z}C_{m})\\}|$, where $\widetilde{V}_{d}$ is as defined in
Definition 15.16. It therefore suffices to prove that, for each $m$ listed
above, we have $\operatorname{odd}(|\widetilde{V}_{d}|)\neq 1$ for some $d\mid
m$.

In the case $m=2p$, the bound $c_{2p}$ from Proposition 15.25 is computed as
in the following table.

$p$ | 11 | 13 | 17 | 19 | 29
---|---|---|---|---|---
$c_{2p}$ | 3 | 5 | 17 | 27 | 565

In each case, $\operatorname{odd}(c_{2p})\neq 1$ and so
$\operatorname{odd}(|\widetilde{V}_{2p}|)\neq 1$. Hence
$|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|\neq 1$ for $m=2p$ where
$p\in\\{11,13,17,19,29\\}$.

In the case $m=pq$ for odd primes $p,q$, the bound $c_{pq}$ from Proposition
15.23 is computed as in the following table.

$(p,q)$ | (3,5) | (3,7) | (3,11) | (5,7) | (3,13)
---|---|---|---|---|---
$c_{pq}$ | 2 | 4 | 44 | 90 | 104

In the cases $(p,q)\in\\{(3,11),(5,7),(3,13)\\}$,
$\operatorname{odd}(c_{pq})\neq 1$ and so
$\operatorname{odd}(|\widetilde{V}_{pq}|)\neq 1$. Hence
$|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|\neq 1$ for $m=pq$ or $2pq$
where $(p,q)\in\\{(3,11),(5,7),(3,13)\\}$.

We will deal the the three remaining cases $m=21,30$ and $42$ directly from
the definition of $\widetilde{V}_{m}$:

$\widetilde{V}_{m}\cong\operatorname{coker}\Big{(}\Psi_{m}^{+}\colon\mathbb{Z}[\zeta_{m}]^{\times}\to\bigoplus_{i=1}^{n}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{p_{i}}[\zeta_{m/p_{i}}]^{\times}}{\mathbb{F}_{p_{i}}[\lambda_{m/p_{i}}]^{\times}}$}\Big{)}.$

First suppose that $m=30$. Then we have

$\Psi_{30}^{+}\colon\mathbb{Z}[\zeta_{15}]^{\times}\to\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{2}[\zeta_{15}]^{\times}}{\mathbb{F}_{2}[\lambda_{15}]^{\times}}$}\oplus\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{3}[\zeta_{5}]^{\times}}{\mathbb{F}_{3}[\lambda_{5}]^{\times}}$}\oplus\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{5}[\zeta_{3}]^{\times}}{\mathbb{F}_{5}[\lambda_{3}]^{\times}}$}$

and $E=\langle\zeta_{15},\mathbb{Z}[\lambda_{15}]^{\times}\rangle$ has index
two in $\mathbb{Z}[\zeta_{15}]^{\times}$. By the same argument as Proposition
15.23, we get that
$|\widetilde{V}_{30}|=|\operatorname{coker}(\Psi_{30}^{+})|$ is divisible by

$c_{30}=\mbox{\footnotesize$\displaystyle\frac{1}{30}$}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{2}[\zeta_{15}]^{\times}}{\mathbb{F}_{2}[\lambda_{15}]^{\times}}$}\Big{|}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{3}[\zeta_{5}]^{\times}}{\mathbb{F}_{3}[\lambda_{5}]^{\times}}$}\Big{|}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{5}[\zeta_{3}]^{\times}}{\mathbb{F}_{5}[\lambda_{3}]^{\times}}$}\Big{|}=c_{15}\cdot\Big{|}\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{2}[\zeta_{15}]^{\times}}{\mathbb{F}_{2}[\lambda_{15}]^{\times}}$}\Big{|}=2\cdot(2^{\frac{f_{2}}{2}}+1)^{g_{2}}=10$

since $f_{2}=\operatorname{ord}_{15}(2)=4$ and $g_{2}=1$. Since
$\operatorname{odd}(c_{30})\neq 1$, this implies that
$\operatorname{odd}(|\widetilde{V}_{30}|)\neq 1$. Hence
$|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{15})\\}|\neq 1$.

We will deal with the remaining cases $m=21$, $42$ by computing the involution
on $\widetilde{V}_{21}$ explicitly. This turns out to be necessary since, by
Remark 16.13, we have $|D(\mathbb{Z}C_{m})|=4$ in each case and so
$\operatorname{odd}(|\widetilde{V}_{21}|)=\operatorname{odd}(|\widetilde{V}_{42}|)=1$.
If $|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|=1$ for $m=21$ or $42$
then, by Proposition 15.18, we would have $|2\cdot\widetilde{V}_{21}|=1$.
Hence it suffices to prove that $|2\cdot\widetilde{V}_{21}|\neq 1$.

We now claim that
$\widetilde{V}_{21}=\operatorname{coker}(\Psi_{21}^{+})\cong\mathbb{Z}/4$,
which implies that $2\cdot\widetilde{V}_{21}\cong\mathbb{Z}/2$. First note
that, by the proof of Proposition 15.23, we have that

$\operatorname{coker}(\Psi_{21}^{+})\cong\operatorname{coker}\Big{(}\mathbb{Z}/21\oplus\mathbb{Z}/2\to\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{3}[\zeta_{7}]^{\times}}{\mathbb{F}_{3}[\lambda_{7}]^{\times}}$}\oplus\mbox{\footnotesize$\displaystyle\frac{\mathbb{F}_{7}[\zeta_{3}]^{\times}}{\mathbb{F}_{7}[\lambda_{3}]^{\times}}$}\Big{)}$

where $1\in\mathbb{Z}/21$ maps to $\Psi_{21}^{+}(\zeta_{21})$ and
$1\in\mathbb{Z}/2$ maps to $\Psi_{21}^{+}(1-\zeta_{21})$.

Since $g_{3}=1$, $\mathbb{F}_{3}[\zeta_{7}]\cong\mathbb{F}_{3}(\zeta_{7})$ is
a field and $\mathbb{F}_{3}[\lambda_{7}]=\mathbb{F}_{3}[\lambda_{7}]$ is a
subfield. This implies that
$\frac{\mathbb{F}_{3}[\zeta_{7}]^{\times}}{\mathbb{F}_{3}[\lambda_{7}]^{\times}}\cong\frac{\mathbb{Z}/(3^{6}-1)}{\mathbb{Z}/(3^{3}-1)}\cong\mathbb{Z}/28$.
We have $\mathbb{F}_{7}[\zeta_{3}]=\mathbb{Z}[x]/\langle
7,1+x+x^{3}\rangle=\mathbb{Z}[x]/\langle
7,(x-2)(x-4)\rangle\cong\mathbb{F}_{7}\times\mathbb{F}_{7}$ where
$\zeta_{3}\mapsto(2,4)$ and so
$\mathbb{F}_{7}[\zeta_{3}]^{\times}\cong(\mathbb{Z}/6)^{2}$. Since
$\mathbb{F}_{7}[\lambda_{3}]^{\times}=\mathbb{F}_{7}^{\times}\cong\mathbb{Z}/6$,
this implies that
$\frac{\mathbb{F}_{7}[\zeta_{3}]^{\times}}{\mathbb{F}_{7}[\lambda_{3}]^{\times}}\cong\mathbb{Z}/6$.
Hence
$D\coloneqq\frac{\mathbb{F}_{3}[\zeta_{7}]^{\times}}{\mathbb{F}_{3}[\lambda_{7}]^{\times}}\oplus\frac{\mathbb{F}_{7}[\zeta_{3}]^{\times}}{\mathbb{F}_{7}[\lambda_{3}]^{\times}}\cong\mathbb{Z}/28\oplus\mathbb{Z}/6$.

Now $\Psi_{21}^{+}(\zeta_{21})=[(\zeta_{7},\zeta_{3})]$ where
$[\zeta_{7}]\in\mathbb{Z}/28$ has order $7$ and $[\zeta_{3}]\in\mathbb{Z}/6$
has order $3$. This implies that $\operatorname{coker}(\mathbb{Z}/21\to
D)\cong\mathbb{Z}/4\oplus\mathbb{Z}/2$. Note that
$\Psi_{21}^{+}(1-\zeta_{21})=[(1-\zeta_{7},1-\zeta_{3})]$. The isomorphism
$\mathbb{F}_{7}[\zeta_{3}]^{\times}/\mathbb{F}_{7}[\lambda_{3}]^{\times}\to\mathbb{Z}/6$
sends $1-\zeta_{3}\mapsto 1-2=-1$ and so the image of
$\Psi_{21}^{+}(1-\zeta_{21})$ in $\operatorname{coker}(\mathbb{Z}/21\to
D)\cong\mathbb{Z}/4\oplus\mathbb{Z}/2$ has the form $(*,-1)$. Since it has
order two, this implies that
$\operatorname{coker}(\Psi_{21}^{+})\cong\operatorname{coker}(\mathbb{Z}/21\oplus\mathbb{Z}/2\to
D)\cong\mathbb{Z}/4$ as required. ∎

We will now prove (A3). Our approach will be to use Lemma 16.4 (iv), and will
be analogous to our proof of (A2). In particular, we will begin by classifying
the $m\geq 2$ square-free for which $\prod_{d\mid m}|\\{x\in
C(\mathbb{Z}[\zeta_{d}])\mid x=-\overline{x}\\}|=1$. We will then determine
the subset of these values for which $|\\{x\in D(\mathbb{Z}C_{m})\mid
x=-\overline{x}\\}|=1$.

######  Proposition 16.10.

Let $m\geq 2$ be square-free. Then $\prod_{d\mid m}|\\{x\in
C(\mathbb{Z}[\zeta_{d}])\mid x=-\overline{x}\\}|=1$ if and only if
$h_{m}^{-}=1$.

###### Proof.

If $h_{m}^{-}=1$, then $h_{d}^{-}=1$ for all $d\mid m$ and so $\prod_{d\mid
m}|\\{x\in C(\mathbb{Z}[\zeta_{d}])\mid x=-\overline{x}\\}|=1$. Conversely, if
$h_{m}^{-}\neq 1$ and $m\not\in\\{29,39,58,78\\}$, then Proposition 16.7
implies that $|\\{x-\overline{x}\mid x\in C(\mathbb{Z}[\zeta_{m}])\\}|\neq 1$
and so $|\\{x\in C(\mathbb{Z}[\zeta_{m}])\mid x=-\overline{x}\\}|\neq 1$. It
now suffices to show that, if $m\in\\{29,39,58,78\\}$, then $|\\{x\in
C(\mathbb{Z}[\zeta_{m}])\mid x=-\overline{x}\\}|\neq 1$.

Suppose $m\in\\{29,39,58,78\\}$. By Proposition 16.7, we have that
$C(\mathbb{Z}[\zeta_{m}])=C(\mathbb{Z}[\zeta_{m}])^{-}$ and so

$\\{x\in C(\mathbb{Z}[\zeta_{m}])\mid x=-\overline{x}\\}=\\{x\in
C(\mathbb{Z}[\zeta_{m}])\mid 2x=0\\}.$

If $m=29$ or $58$, then $C(\mathbb{Z}[\zeta_{m}])\cong(\mathbb{Z}/2)^{3}$ and
so $\\{x\in C(\mathbb{Z}[\zeta_{m}])\mid 2x=0\\}\cong(\mathbb{Z}/2)^{3}$
[Wa97, p. 412]. If $m=39$ or $78$, then
$C(\mathbb{Z}[\zeta_{m}])\cong\mathbb{Z}/2$ and so $\\{x\in
C(\mathbb{Z}[\zeta_{m}])\mid 2x=0\\}\cong\mathbb{Z}/2$ [Wa97, p. 412]. ∎

We will now prove the following. By Lemma 16.4 (iv), this completes the proof
of (A3).

######  Proposition 16.11.

The complete list of $m\geq 2$ square-free for which

$\textstyle\prod_{d\mid m}|\\{x\in C(\mathbb{Z}[\zeta_{d}])\mid
x=-\overline{x}\\}|=1\quad\text{and}\quad|\\{x\in D(\mathbb{Z}C_{m})\mid
x=-\overline{x}\\}|=1$

is as follows:

$m=\begin{cases}\text{$p$},&\text{where $p\in\\{2,3,5,7,11,13,17,19\\}$}\\\
\text{$2p$},&\text{where $p\in\\{3,5,7\\}$}.\end{cases}$

That is, $m\in\\{2,3,5,6,7,10,11,13,14,17,19\\}$.

###### Proof.

If $|\\{x\in D(\mathbb{Z}C_{m})\mid x=-\overline{x}\\}|=1$, then
$|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|=1$. By Propositions 15.6,
16.8, LABEL:, and 16.10, the $m\geq 2$ square-free for which $\prod_{d\mid
m}|\\{x\in C(\mathbb{Z}[\zeta_{d}])\mid x=-\overline{x}\\}|=1$ and
$|\\{x-\overline{x}\mid x\in D(\mathbb{Z}C_{m})\\}|=1$ are as follows:

$m=\begin{cases}\text{$p$},&\text{where $p\in\\{2,3,5,7,11,13,17,19\\}$}\\\
\text{$2p$},&\text{where $p\in\\{3,5,7\\}$},\\\ \text{$pq$},&\text{where
$(p,q)=(3,5)$}.\end{cases}$

If $m=p$ for $p\leq 17$ prime or $m=2p$ for $p\in\\{3,5,7\\}$, then Lemma 16.9
implies that $D(\mathbb{Z}C_{m})=0$ and so $|\\{x\in D(\mathbb{Z}C_{m})\mid
x=-\overline{x}\\}|=1$. If $m=15$, then it is shown in [CN72, p. 48] that
$|D(\mathbb{Z}C_{15})|=2$. This implies that
$D(\mathbb{Z}C_{15})\cong\mathbb{Z}/2$ has the trivial action and so $\\{x\in
D(\mathbb{Z}C_{m})\mid x=-\overline{x}\\}\cong\mathbb{Z}/2$. ∎

Finally, we will now prove (A4). Note that these results are implied by
computations used the proofs of Propositions 16.8 and 16.11, but we will
repeat them here for the convenience of the reader.

######  Proposition 16.12.

  1. (i)

$\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{15})\\}=0$ and
$\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{15})\mid
x=-\overline{x}\\}\cong\mathbb{Z}/2$.

  2. (ii)

$\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{29})\\}=0$ and
$\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{29})\mid
x=-\overline{x}\\}\cong(\mathbb{Z}/2)^{3}$.

###### Proof.

(i) By [Wa97, p. 412], we have $h_{15}=1$ and so
$\widetilde{K}_{0}(\mathbb{Z}C_{15})\cong D(\mathbb{Z}C_{15})$. It is shown in
[CN72, p. 48] that $|D(\mathbb{Z}C_{15})|=2$ and so
$D(\mathbb{Z}C_{15})\cong\mathbb{Z}/2$ has the trivial involution. Hence we
have $\\{x-\overline{x}\mid x\in\widetilde{K}_{0}(\mathbb{Z}C_{15})\\}=0$ and
$\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{15})\mid
x=-\overline{x}\\}\cong\mathbb{Z}/2$.

(ii) By Lemma 16.9, we have $D(\mathbb{Z}C_{29})=0$ and so
$\widetilde{K}_{0}(\mathbb{Z}C_{29})\cong C(\mathbb{Z}[\zeta_{29}])$. By
[Wa97, p. 412], we have that
$C(\mathbb{Z}[\zeta_{29}])\cong(\mathbb{Z}/2)^{3}$. We also have
$C(\mathbb{Z}[\zeta_{29}])^{+}=1$ [Wa97, p. 412] and so, by Lemma 15.2 (ii),
$C(\mathbb{Z}[\zeta_{29}])=C(\mathbb{Z}[\zeta_{29}])^{-}$ and so has the
trivial involution. Hence we have $\\{x-\overline{x}\mid
x\in\widetilde{K}_{0}(\mathbb{Z}C_{29})\\}=0$ and
$\\{x\in\widetilde{K}_{0}(\mathbb{Z}C_{29})\mid
x=-\overline{x}\\}\cong(\mathbb{Z}/2)^{3}$. ∎

This completes the proofs of statements (A1)-(A4) and so, by the discussion at
the start of this section, completes the proofs of Theorems 16.1 and 16.2.

######  Remark 16.13.

Note that statements (A3) and (A4) only require the computation of
$D(\mathbb{Z}C_{m})$ and its involution for finitely many $m$. Similarly, the
proof of (A2) involves the verification of lower bounds for
$|\widetilde{V}_{m}|$ for finitely many $m$ which in turn give lower bounds
for $|D(\mathbb{Z}C_{m})|$. It is therefore possible to do both problems
computationally. While we have not implemented an algorithm to determine the
involution on $D(\mathbb{Z}C_{m})$, we have computed $D(\mathbb{Z}C_{m})$ for
all relevant $m$ using the algorithm described in [BB06] and implemented in
Magma by Werner Bley.

We checked that $D(\mathbb{Z}C_{m})=0$ for the $m$ listed in Lemma 16.9, we
checked that all the bounds $c_{m}$ computed in the proof of Proposition 16.8
divide $|D(\mathbb{Z}C_{m})|$ and we computed
$D(\mathbb{Z}C_{21})\cong\mathbb{Z}/4$, $D(\mathbb{Z}C_{42})\cong\mathbb{Z}/2$
and $D(\mathbb{Z}C_{15})\cong\mathbb{Z}/2$ which is consistent with the
calculations above.

####  16.2. Proof of Theorem 16.3

For $m\geq 2$, let

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

where the isomorphism comes from Proposition 14.5. Similarly to the proofs of
Theorems 16.1 and 16.2, we will begin by noting that it now suffices to prove
the following two statements.

  1. (B1)

If $n\geq 1$, then $|A_{3^{n}}|=1$

  2. (B2)

If $n\geq 1$, then $|A_{2^{n}}|\cdot|A_{2^{n+1}}|\geq 2^{2^{n-2}-1}$.

To see this, note that (B1) directly implies Theorem 16.3 (i). Next, Theorem
16.3 (ii) follows from (B2) since it implies that

$\displaystyle\sup_{n\leq m}|A_{n}|$
$\displaystyle\geq\max\\{|A_{2^{\lfloor\log_{2}(m)\rfloor}}|,|A_{2^{\lfloor\log_{2}(m)\rfloor-1}}|\\}\geq\sqrt{|A_{2^{\lfloor\log_{2}(m)\rfloor}}|\cdot|A_{2^{\lfloor\log_{2}(m)\rfloor-1}}|}$
$\displaystyle\geq 2^{2^{\lfloor\log_{2}(m)\rfloor-3}-1}\geq
2^{\frac{m}{16}-1}$

which tends to infinity exponentially in $m$.

We will again make use of the following short exact sequence of
$\mathbb{Z}C_{2}$-modules established in Section 15.1:

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

where $D(\mathbb{Z}C_{m})$ has the induced involution, and each
$C(\mathbb{Z}[\zeta_{d}])$ has the involution induced by conjugation. Each of
statements (B1) and (B2) will be proven via the following lemma.

######  Lemma 16.14.

Let $m\geq 2$. Then there is a $6$-periodic exact sequence of finite abelian
groups

${\widehat{H}^{1}(C_{2};D(\mathbb{Z}C_{m})_{(2)})}$${A_{m}}$${\bigoplus_{d\mid
m}\widehat{H}^{1}(C_{2};C(\mathbb{Z}[\zeta_{d}])_{(2)})}$${\bigoplus_{d\mid
m}\widehat{H}^{0}(C_{2};C(\mathbb{Z}[\zeta_{d}])_{(2)})}$${\widehat{H}^{0}(C_{2};C(\mathbb{Z}C_{m})_{(2)})}$${\widehat{H}^{0}(C_{2};D(\mathbb{Z}C_{m})_{(2)}).}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$

Furthermore, we have that:

  1. (i)

If $h_{m}$ is odd, then
$A_{m}\cong\widehat{H}^{1}(C_{2};D(\mathbb{Z}C_{m})_{(2)})$

  2. (ii)

If $|D(\mathbb{Z}C_{m})|$ is odd, then $A_{m}\cong\bigoplus_{d\mid
m}\widehat{H}^{1}(C_{2};C(\mathbb{Z}[\zeta_{d}])_{(2)})$.

  3. (iii)

If $h_{m}$ and $|D(\mathbb{Z}C_{m})|$ are both odd, then $A_{m}=0$.

###### Proof.

By Proposition 14.11, we have that
$A_{m}\cong\widehat{H}^{1}(C_{2};C(\mathbb{Z}C_{m})_{(2)})$. The short exact
sequence stated above induces a short exact sequence on their $2$-primary
submodules:

$0\to D(\mathbb{Z}C_{m})_{(2)}\to C(\mathbb{Z}C_{m})_{(2)}\to\bigoplus_{d\mid
m}C(\mathbb{Z}[\zeta_{d}])_{(2)}\to 0.$

This follows since, for example, localisation is an exact functor. The
existence of the required 6-periodic exact sequence now follows from
Proposition 14.7 and the fact that

$\widehat{H}^{n}(C_{2};\bigoplus_{d\mid
m}C(\mathbb{Z}[\zeta_{d}])_{(2)})\cong\bigoplus_{d\mid
m}\widehat{H}^{n}(C_{2};C(\mathbb{Z}[\zeta_{d}])_{(2)})$

for $n\in\mathbb{Z}$ by Proposition 14.2 (ii).

To prove (i), suppose $h_{m}$ is odd. Then $h_{d}$ is odd for all $d\mid m$
since $h_{d}\mid h_{m}$ [Wa97, p. 205]. This implies that
$C(\mathbb{Z}[\zeta_{d}])_{(2)}=0$ for all $d\mid m$ and so $\bigoplus_{d\mid
m}\widehat{H}^{n}(C_{2};C(\mathbb{Z}[\zeta_{d}])_{(2)})=0$ for all
$n\in\mathbb{Z}$. The 6-periodic exact sequence then gives that
$A_{m}\cong\widehat{H}^{1}(C_{2};D(\mathbb{Z}C_{m})_{(2)})$.

To prove (ii), suppose that $|D(\mathbb{Z}C_{m})|$ is odd. Then
$D(\mathbb{Z}C_{m})_{(2)}=0$,
$\widehat{H}^{n}(C_{2};D(\mathbb{Z}C_{m})_{(2)})=0$ for all $n\in\mathbb{Z}$,
and so the 6-periodic exact sequence gives that $A_{m}\cong\bigoplus_{d\mid
m}\widehat{H}^{1}(C_{2};C(\mathbb{Z}[\zeta_{d}])_{(2)})$.

If both $h_{m}$ and $|D(\mathbb{Z}C_{m})|$ are odd, then
$D(\mathbb{Z}C_{m})_{(2)}=0$ and so
$A_{m}\cong\widehat{H}^{1}(C_{2};D(\mathbb{Z}C_{m})_{(2)})=0$, which proves
(iii). ∎

We will now prove the following which implies (B1) since $h_{3}=1$ is odd
(see, for example, Proposition 15.6). Note that it also implies that
$|A_{p^{n}}|=1$ for $n\geq 1$ and other primes $p\neq 3$.

######  Proposition 16.15.

Let $n\geq 1$ and let $p\leq 509$ be an odd prime such that $h_{p}$ is odd.
Then $|A_{p^{n}}|=1$.

######  Remark 16.16.

By Lemma 15.11 (i), this condition holds precisely for the odd primes $p\leq
509$ with

$p\not\in\\{29,113,163,197,239,277,311,337,349,373,397,421,463,491\\}.$

###### Proof.

Since $p$ is an odd prime, Proposition 15.21 implies that
$|D(\mathbb{Z}C_{p^{n}})|$ is odd. Since $p\leq 509$ and $h_{p}$ is odd, Lemma
15.11 (ii) implies that $h_{p^{n}}$ is odd. Hence $|A_{p^{n}}|=1$ by Lemma
16.14 (iii). ∎

We will now prove (B2). By the discussion above, this completes the proof of
Theorem 16.3.

######  Proposition 16.17.

If $n\geq 1$, then $|A_{2^{n}}|\cdot|A_{2^{n+1}}|\geq 2^{2^{n-2}-1}$.

###### Proof.

By Lemma 15.11 (ii), $h_{2^{n}}$ is odd and so Lemma 16.14 (i) implies that

$A_{2^{n}}\cong\widehat{H}^{1}(C_{2};D(\mathbb{Z}C_{2^{n}})_{(2)}).$

By Propositions 14.7 and 15.20, we have a $6$-periodic exact sequence of
finite abelian groups:

${\widehat{H}^{1}(C_{2};V_{2^{n+1}})}$${A_{2^{n+1}}}$${A_{2^{n}}}$${\widehat{H}^{0}(C_{2};D(\mathbb{Z}C_{2^{n}}))}$${\widehat{H}^{0}(C_{2};D(\mathbb{Z}C_{2^{n+1}}))}$${\widehat{H}^{0}(C_{2};V_{2^{n+1}}).}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$

Furthermore, by Proposition 14.8, we have that
$\widehat{H}^{0}(C_{2};D(\mathbb{Z}C_{2^{k}}))\cong A_{2^{k}}$ as abelian
groups for all $k\geq 1$, and
$\widehat{H}^{1}(C_{2};V_{2^{n+1}})\cong\widehat{H}^{0}(C_{2};V_{2^{n+1}})$ as
abelian groups.

Since the involution on $V_{2^{n+1}}$ acts by negation, we have that

$\widehat{H}^{1}(C_{2};V_{2^{n+1}})\cong
V_{2^{n+1}}/2V_{2^{n+1}}\cong\bigoplus_{i=1}^{n-2}(\mathbb{Z}/2)^{2^{n-i-2}}\cong(\mathbb{Z}/2)^{2^{n-2}-1}.$

This implies that the 6-periodic exact sequence restricts to:

$A_{2^{n}}\xrightarrow[]{\alpha}(\mathbb{Z}/2)^{2^{n-2}-1}\xrightarrow[]{\beta}A_{2^{n+1}}.$

By the first isomorphism theorem and exactness, we get that:

$2^{2^{n-2}-1}=|\ker(\beta)|\cdot|\operatorname{Im}(\beta)|=|\operatorname{Im}(\alpha)|\cdot|\operatorname{Im}(\beta)|\leq|A_{2^{n}}|\cdot|A_{2^{n+1}}|$

which was the required bound. ∎

### References