Proof of Proposition 3.24

We need to prove that $\tau _{(-)}$ induces isomorphisms in homotopy groups. To prove surjectivity, let $a\in \mathrm {Wh}(M)$ be such that $a=(-1)^{d+n}\overline a$ and let be an h-cobordism rel boundary with $ \tau (W, M\times \Lambda _0^{n+1})=a $ . The manifold W admits a stratified structure over $\Delta ^{n+1}$ with $0$ -th horn $\Lambda _0(W)=M\times \Lambda _0^{n+1}$ and $0$ -th face $W_{\langle 1,\dots ,n+1\rangle }$ . It is not difficult to see that $\tau _W=\tau _{a}$ by Proposition 3.14 and noting that $\tau _W$ satisfies face-horn dualities for all faces (in particular, $\Delta ^{n+1}$ ).

For injectivity, let $W\in F_n(M)$ be a cycle such that $[\tau _W]=0$ in $\pi _n(F^{\mathsf {alg}}_{\bullet }(\mathrm {Wh}(M))$ . By the first step in the proof of Proposition 3.20, $\tau _W=\tau _{b+(-1)^{d+n}\overline b}$ for some $b\in \mathrm {Wh}(M)$ . Let be an h-cobordism rel boundary with torsion $\tau (V,W)=-b$ (after having identified $\pi _1(W)$ with $\pi _1(M)$ appropriately. Then $\pi _1(V)$ and $\pi _1(W')$ get identified with $\pi _1(M)$ too). We claim $W'$ is (face-preservingly) diffeomorphic to $M\times \Delta ^{n+1}$ (i.e., we have to show that $\tau (W',\Lambda _0(W'))=0$ by the s-cobordism theorem). Since $\Lambda _0(W')= \Lambda _0(W)=M\times \Lambda _0^{n+1}$ ,

$$ \begin{align*}\tau(W,\Lambda_0(W))+\tau(V,W)=\tau(W',\Lambda_0(W'))+\tau(V,W') \end{align*} $$

which, by duality, yields

$$ \begin{align*}\tau(W',\Lambda_0(W'))=b+(-1)^{d+n}\overline b-b-(-1)^{d+n+1}\overline{(-b)}=0. \end{align*} $$

Let $\phi : W'\cong M\times \Delta ^{n+1}$ be a diffeomorphism fixing $\Lambda _0(W')=M\times \Lambda _0^{n+1}$ , and consider $V':=M_\phi \circ V$ , where denotes the mapping cylinder Footnote ⁴ of $\phi $ . Then using the canonical diffeomorphism rel boundary $\Delta ^{n+1}\cong \Lambda _1^{n+2}$ , the manifold $V'$ admits a stratified structure over $\Delta ^{n+2}$ with $\partial _1V'=W$ and $\Lambda _1(V')=M\times \Lambda _1^{n+2}$ . Therefore, $V'$ provides a null-homotopy of W in $F_{\bullet }(M)$ , as desired.

Proof. Let $\widetilde {\mathcal S}^{h/s}_{\bullet }(M)$ denote the ordinary/simple block structure spaces of M [Reference QuinnQui70]. Roughly speaking, a p-simplex in the space $\widetilde {\mathcal S}^{h/s}_{\bullet }(M)$ is a pair $(W^{d+p},f)$ consisting of a manifold W stratified over $\Delta ^p$ and a face-preserving homotopy equivalence $f: W\overset {\simeq _{h/s}}\longrightarrow _{\Delta }M\times \Delta ^p$ . For $d\geq 5$ , surgery theory establishes a diagram of fibration sequences [Reference QuinnQui70, §3]

Taking homotopy fibres, we obtain an equivalence

(3.20) $$ \begin{align} \Omega^{\infty+d+1}\operatorname{hofib}\left(\mathbb{L}^s(M)\to \mathbb{L}^h(M)\right)\simeq\operatorname{hofib}\left(\widetilde{\mathcal S}^s(M)\longrightarrow\widetilde{\mathcal S}^h(M)\right). \end{align} $$

However, it is not difficult to see that there is another diagram of fibration sequences

where u is the (geometric realisation of the) forgetful map sending a p-simplex $(W^{d+p},f)$ in $\widetilde {\mathcal S}^{h/s}_{\bullet }(M)$ to $W\in \widetilde {\mathcal M}^{h/s}_p$ . Taking again homotopy fibres (in the basepoint components corresponding to M and $\mathrm {Id}_M$ ), we get a map $\overline {u}:\operatorname {hofib}(\widetilde {\mathcal S}^s(X)\longrightarrow \widetilde {\mathcal S}^h(X))\to |F_{\bullet }(M)|\simeq \Omega ^\infty (H\mathrm {Wh}(M)_{hC_2})$ which is an equivalence onto the components that are hit. For each $[a]\in H_0(C_2;\mathrm {Wh}(M))$ , write $\Omega ^\infty _{[a]}(H\mathrm {Wh}(M)_{hC_2})\subset \Omega ^\infty (H\mathrm {Wh}(M)_{hC_2})$ for the connected component corresponding to $[a]$ . There is hence a chain of equivalences

$$ \begin{align*} \Omega^{\infty+d+1}\operatorname{hofib}\left(\mathbb{L}^s(M)\to \mathbb{L}^h(M)\right)\overset{\textrm{(3.20)}}\simeq \operatorname{hofib}\left(\widetilde{\mathcal S}^s(X)\to\widetilde{\mathcal S}^h(X)\right)\simeq\hspace{-3pt} \bigsqcup_{[a]\in \mathrm{Im} (\pi_0(\overline{u}))} \Omega^{\infty}_{[a]}(H\mathrm{Wh}(M)_{hC_2}). \end{align*} $$

To establish (3.19), it remains to argue that $\mathrm {Im}(\pi _0(\overline {u}))=\widehat {H}^1(C_2;\mathrm {Wh}(M))$ . Choose a $0$ -simplex in $\operatorname {hofib}(\widetilde {\mathcal S}^s_{\bullet }(X)\longrightarrow \widetilde {\mathcal S}^h_{\bullet }(X))$ , that is, a $1$ -simplex $(W,f)\in \widetilde {\mathcal S}^h_1(M)$ such that $W_0=M$ , $f_0=\mathrm {Id}_M$ and $f_1: W_1\overset {\simeq _s}\longrightarrow M\times \{1\}$ is a simple homotopy equivalence. In particular, is an h-cobordism starting at M. By definition of $\psi _{\bullet }^{\mathrm {Wh}(M)}$ (see (3.15)),

$$ \begin{align*}\pi_0(\overline u): \pi_0\left(\operatorname{hofib}\left(\widetilde{\mathcal S}^s(X)\longrightarrow\widetilde{\mathcal S}^h(X)\right)\right)\longrightarrow H_0(C_2,\mathrm{Wh}(M)), \quad [W,f]\longmapsto [\tau(W,M)]. \end{align*} $$

Now $0=\tau (f_0)=(i_M)_*^{-1}\tau (f)+\tau (W,M)$ and, by duality, $0=(h^W_*)^{-1}\tau (f_1)=(i_M)_*^{-1}\tau (f)+(-1)^{d}\overline {\tau }(W,M)$ . Putting these two together, we obtain $\tau (W,M)=(-1)^{d}\overline {\tau }(W,M)$ , so it follows that $\mathrm {Im}(\pi _0(\overline u))\subset \widehat {H}^1(C_2;\mathrm {Wh}(M))$ . For the other inclusion, let $W\in F_0(M)$ be such that $\tau (W,M)=(-1)^{d}\overline {\tau }(W,M)$ , and pick some face-preserving homotopy equivalence $f: W\overset {\simeq _h}\longrightarrow _\Delta M\times I$ . By post-composing f with $f_0^{-1}\times I$ , for some $f_0^{-1}$ homotopy inverse to $f_0$ , we may assume that $f_0=\mathrm {Id}_M$ . Then

$$ \begin{align*}\tau(f_1)=(i_{W_1})_*^{-1}\tau(f)+(-1)^{d}h^W_*\overline{\tau}(W,M)=(i_{W_1})_*^{-1}\tau(f)+h^W_*\tau(W,M)=h^W_*\tau(f_0)=0, \end{align*} $$

so $f_1: W_1\simeq _s M\times \{1\}$ is a simple homotopy equivalence, and thus, $(W,f)$ represents a $0$ -simplex in $\operatorname {hofib}(\widetilde {\mathcal S}^s_{\bullet }(X)\longrightarrow \widetilde {\mathcal S}^h_{\bullet }(X))$ . This concludes the proof of Proposition 3.26.

Proof. The inclusion $(\subset )$ is immediate. Conversely, if $W: M\leadsto M'$ is an inertial h-cobordism with $\phi : M\cong M'$ , let $W': M\leadsto M$ denote the h-cobordism $M_{\phi ^{-1}}\circ W$ . Recall that the isomorphism $\pi _0(F_{\bullet }(M))\cong H_0(C_2;\mathrm {Wh}(M))$ sends the class represented by W to that of its torsion $\tau (W,M)=\tau (W',M)$ . As $W'$ represents a class in $\pi _1(\widetilde {\mathcal M}^h,\{M\})$ , we are done.

Proof. The basepoint change map sends an n-cycle $V^{d+n}\in B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(L)_n$ to the manifold (see Figure 1)

$$ \begin{align*}W_\#V:=V\cup_{L\times \partial\Delta^n} (W\times \partial \Delta^n). \end{align*} $$

The union is made along the boundary $\partial V=L\times \partial \Delta ^n$ . The manifold $W_\#V$ is naturally stratified over $\Delta ^n$ , and clearly represents an n-cycle in $B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(M)$ . As mentioned before, the boundary map $\partial : \pi _n(B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(L))\longrightarrow H_{n-1}(C_2;\mathrm {Wh}(L))$ sends $[V]$ to the class represented by $\tau (V,V_0)=\tau (V,L)$ in $H_{n-1}(C_2;\mathrm {Wh}(L))$ . We thus need to show that

$$ \begin{align*}\tau(W_\#V,M)\equiv h_*^W\tau(V,L)\quad \mod \{\sigma+(-1)^{d+n-1}\overline \sigma: \sigma\in \mathrm{Wh}(L)\}. \end{align*} $$

We compute $\tau (W_\#V,M)$ directly. For any subspaces $A, B\subset W_\#V$ with $A\subset B$ , write $i_A^B$ for the inclusion. If $P:=V\cup _{V_0=L}W$ (see Figure 1), we can factor the inclusion as

We compute the torsion of these three maps using the inclusion-exclusion principle (2.3):

$$ \begin{align*} \tau(W,M)&=(-1)^{d}h^W_*\overline\tau,\\ \tau(P,W)&=(i_L^W)_*\tau(V,L)+(i_W^W)_*\tau(W,W)-(i_L^W)_*\tau(L,L)\\ &=(i_L^W)_*\tau(V,L),\\ \tau(W_\#V,P)&=(i_V^P)_*\tau(W_\#V,V)+(i_W^P)_*\tau(W,W)-(i_L^P)_*\tau(W,L)\\ &=(i^P_V)_*(i_{\partial V}^{V})_*\tau(W\times\partial\Delta^n,L\times\partial\Delta^n)-(i_L^P)_*\tau\\ &=\chi(\partial\Delta^n)\cdot (i^P_{L})_*\tau-(i_L^P)_*\tau\\ &=(-1)^{n-1}(i^P_{L})_*\tau. \end{align*} $$

In the penultimate line, we have used that $i^P_V\circ i^{V}_{\partial V}\circ i_{L\times 0}^{L\times \partial \Delta ^n}=i_L^P$ and the product rule (2.4) of $\tau (-)$ , for which we need the condition $n\geq 2$ for $\partial \Delta ^n$ to be connected. By the composition rule (2.2), we get

$$ \begin{align*} \tau(W_\#V,M)&=(-1)^dh^W_*\overline\tau+(i_M^W)^{-1}_*(i_L^W)_*\tau(V,L)+(-1)^{n-1}(i_M^P)^{-1}_*(i^P_{L})_*\tau\\ &=(-1)^dh^W_*\overline\tau+h_*^W\tau(V,L)+(-1)^{n-1}h^W_*\tau\\ &=h_*^W\tau(V,L)+(-1)^{n-1}\left(h^W_*\tau+(-1)^{d+n-1}\overline{h^W_*\tau}\right), \end{align*} $$

where in the second line we have used the commutative diagram

so $(i_M^P)^{-1}_*(i^P_{L})_*=(i_M^W)^{-1}_*(i^W_{L})_*=h^W_*$ . This finishes the proof.

Figure 1 Illustration of $W_\#V$ and $P:=V\cup _{L\times \{0\}}W$ when $n=2$ .

Proof. If $\frac {I(L)}{\mathcal D(L)}=0$ then $\pi _1(B\widetilde {\mathrm {Diff}}(L))$ surjects onto $\pi _1(B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(L))$ , and hence, the latter is finite too. As $\pi _1(B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(M))\cong \pi _1(B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(L))$ surjects onto $\frac {I(M)}{\mathcal D(M)}$ , this is also finite, say of cardinality $N>1$ since $\frac {I(M)}{\mathcal D(M)}\neq 0$ . Write

$$ \begin{align*}K:=\left\{x\in \pi_1(B\widetilde{\mathrm{Diff}}{\vphantom{\mathrm{Diff}}}^{\hspace{1pt} h}(M))\cong\pi_1(B\widetilde{\mathrm{Diff}}{\vphantom{\mathrm{Diff}}}^{\hspace{1pt} h}(L)): \partial x=0\in \frac{I(M)}{\mathcal D(M)}\right\}. \end{align*} $$

We now have two extensions of finite groups

where the left vertical isomorphism is a consequence of Proposition 4.4. Even though $\partial $ is a crossed homomorphism, we still have that $N\cdot |K|=|\pi _1(B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(M))| =|\pi _1(B\widetilde {\mathrm {Diff}}{\vphantom {\mathrm {Diff}}}^{\hspace {1pt} h}(L))|$ , and therefore, the result follows.

Proof. The algebraic involution $\bar {\cdot }: \mathrm {Wh}(\pi )\to \mathrm {Wh}(\pi )$ is trivial when $\pi $ is a finite abelian group [Reference BassBas74, Prop. 4.2]. Therefore, by (4.2), the subgroups of double h-cobordisms $\mathcal D(L)$ and $\mathcal D(M)$ are trivial since L and M are odd-dimensional and orientable (so their first Stiefel–Whitney classes vanish), and $\pi _1(L)\cong \pi _1(M)\cong C_7$ is certainly finite abelian. It thus suffices to show that $I(L)=0$ and $|I(M)|=3$ .

The first assertion follows from [Reference MilnorMil66, Cor. 12.12]. We now prove that $I(M)\neq 0$ (i.e., we construct nontrivial inertial h-cobordisms starting at M). For a diffeomorphism $\phi \in \mathrm {Diff}(L)$ , write $V_\phi $ for the inertial h-cobordism

(i.e., $V_\phi =\overline {W}\cup _{\phi ^{-1}} W\in I(M)$ ). Observe that $h^{\overline W}: M\overset {\simeq }\longrightarrow L$ is homotopy inverse to $h^W: L\overset {\simeq }\longrightarrow M$ because $\overline W\circ W=D(W)$ and $W\circ \overline W=D(\overline W)$ are h-cobordant rel boundary to the trivial h-cobordisms $L\times I$ and $M\times I$ , respectively (see Figure 2). So $h^{\overline W\circ W}=h^{\overline W}\circ h^W$ is homotopic to $h^{L\times I}=\mathrm {Id}_L$ (and similarly, $h^W\circ h^{\overline W}\simeq \mathrm {Id}_M$ ). The h-cobordism $V_\phi $ then has torsion

$$ \begin{align*} \tau(V_\phi,M)&=\tau(\overline{W},M)+(h^{\overline W})_*^{-1}\tau(W\circ M_{\phi^{-1}}, L)\\ &=(-1)^{12k-1}h^W_*\overline u+h^W_*\phi_*u\\ &=h^{W}_*(\phi_*u-u), \end{align*} $$

where we have used that $\overline u=u$ by the triviality of the algebraic involution. Therefore, if we are able to find self-diffeomorphisms $\phi $ of L for which $\phi _*u\neq u$ in $\mathrm {Wh}(L)$ , then we will have achieved our task.

Figure 2 h-cobordisms rel boundary and .

Proof of Claim. In fact, by the main theorem of [Reference Hsiang and JahrenHJ83] (see (4.5) below), the natural map $\pi _0(\mathrm {Diff}(L))\to \pi _0(s\mathrm {Aut}(L))$ is surjective, so it will suffice to find simple homotopy automorphisms $f_i\in s\mathrm {Aut}(L)$ for each $i\in (\mathbb {Z}/7)^\times $ such that $\pi _1(f_i): t\mapsto t^{i}$ . By [Reference OlumOlu53, Thm. II.b & Thm. V], the natural map $\gamma :\pi _0(h\mathrm {Aut}(L))\to \mathrm {Aut}(\pi _1 L)\cong (\mathbb {Z}/7)^\times $ is injective with image

$$ \begin{align*}\mathrm{Im}\ \gamma=\{i\in \mathbb{Z}/7: i^{\hspace{1pt} 6k}\equiv \pm 1\quad \mod 7\}. \end{align*} $$

The classes $[f]$ sent to $i\in (\mathbb {Z}/7)^\times $ with $i^n\equiv +1\ \mod 7$ (resp. $i^n\equiv -1\ \mod 7$ ) are orientation-preserving (resp. orientation-reversing). But if $i\in (\mathbb {Z}/7)^\times $ , $i^6\equiv 1\ \mod 7$ and so $i^{6k}\equiv 1\ \mod 7$ too. Therefore, for each $i\in (\mathbb {Z}/7)^\times $ , there exists some (orientation-preserving) homotopy automorphism $f_i\in h\mathrm {Aut}(L)$ such that $\pi _1(f_i):t\mapsto t^{i}$ . By [Reference MilnorMil66, Lem. 12.5], $f_i$ is a simple homotopy automorphism if and only if

$$ \begin{align*}(f_i)_*\Delta(L)=\Delta(L)\in \mathbb{Q}[C_7]/\sim, \end{align*} $$

where $\Delta (L)$ denotes the R-torsion of L [Reference MilnorMil66, Lem. 12.4]. Here, two elements $x,y\in \mathbb {Q}[C_7]$ are related $x\sim y$ if and only if there exists some $g\in C_7$ such that $x=\pm g\cdot y$ . Recall also [Reference MilnorMil66, p. 406] that the R-torsion of L is

$$ \begin{align*}\Delta(L)=\prod_{j=1}^{6k}(t^{r_j}-1)=\prod_{j=1}^6(t^{\hspace{1pt}j}-1)^k, \end{align*} $$

and so

$$ \begin{align*}(f_i)_*\Delta(L)=(f_i)_*\left(\prod_{j=1}^6(t^{\hspace{1pt}j}-1)^k\right)=\prod_{i=1}^6(t^{i\cdot j}-1)^k=\Delta(L), \quad i\in (\mathbb{Z}/7)^\times. \end{align*} $$

Therefore $f_i$ is a simple homotopy equivalence for $i\in (\mathbb {Z}/7)^\times $ , as claimed.

Now it is easily checked that $(\phi _6)_*u=u$ , so $(\phi _2)_*u=(\phi _5)_*u$ and $(\phi _3)_*u=(\phi _4)_ *u$ in $\mathbb {Q}[C_7]/\sim $ . However, the three nontrivial units

$$ \begin{align*}u=2+2t-t^3-t^4-t^5, \quad (\phi_2)_*u=2+2t^2-t^6-t-t^3, \quad (\phi_3)_*u=2+2t^3-t^2-t^5-t \end{align*} $$

represent different elements in $\mathrm {Wh}(L)$ (for instance, the difference between the powers of t of the terms whose coefficient is $2$ in the three units is different mod $7$ ). By our previous argument, the h-cobordisms $V_{\phi _1}=M\times I$ , $V_{\phi _2}$ and $V_{\phi _3}$ are pairwise non-diffeomorphic inertial h-cobordisms starting at M, so $|I(M)|\geq 3$ . Note that $V_{\phi _i}\cong V_{\phi _{7-i}}$ for $i=1,\dots 6$ . Conversely, suppose that is an inertial h-cobordism (by possibly post-composing with a mapping cylinder, we may assume that the target of W is M itself). Then the inertial h-cobordism must be trivial as $I(L)=0$ , and $(h^{\overline W\circ V\circ W})^{-1}_*=(h^W)^{-1}_*(h^{V})_*^{-1}h^W_*=(\phi _i)_*$ for some $i\in (\mathbb {Z}/7)^\times $ because $\mathrm {Aut}(\pi _1L)\cong (\mathbb {Z}/7)^\times $ . Using $\tau (\overline W\circ V\circ W,L)=0$ , we get that

$$ \begin{align*}\tau(V,M)=(h^V)^{-1}_*h^W_*u-h^W_*u=h_*^W((\phi_i)_*u-u)=\tau(V_{\phi_i},M) \end{align*} $$

(i.e., V is diffeomorphic to $V_{\phi _i}$ rel M). Hence, $|I(M)|=3$ , and this finishes the proof.

Warning 4.9. The main theorem of [Reference Kwasik and SchultzKS99] states that $I(M)=0$ if M is a fake lens space – that is, the orbit space of a free (possibly non-linear) action of a finite cyclic group on a sphere. This clearly contradicts Proposition 4.7, but we believe that the proof of the result in [Reference Kwasik and SchultzKS99] is fallacious: following a trail of references [Reference Kwasik and SchultzKS99, Claim 2], [Reference Kwasik and SchultzKS92, Prop. 3.2] and [Reference KwasikKwa86, p. 353], it is eventually stated that if $\tau \in \mathrm {Wh}(M)$ is the torsion of some inertial h-cobordism of M, then $2\tau =0$ . This supposedly follows from the proof of [Reference MilnorMil66, Prop. 12.8], but in the statement of this result, it is required that the h-cobordism between special manifolds be compatible with the given identifications of the fundamental groups (i.e., that $h^W_*: \mathrm {Wh}(M)\to \mathrm {Wh}(M')$ has been trivialised beforehand). This requirement does not hold in the case of [Reference KwasikKwa86, p. 353], and is exactly what we exploit in the proof of Proposition 4.7.

Proof. According to the main result of [Reference Hsiang and JahrenHJ83], the mapping class group of L fits into an extension of groups

(4.5)

The group H is the image of $[\Sigma L_+, \mathrm {Top}/O]_*$ in $[\Sigma L_+, G/O]_*$ . By [Reference Hsiang and SharpeHS76, Thm. 1.1], the group Q also appears in an exact sequence

(4.6) $$ \begin{align} L_{2n+2}^s(\mathbb{Z}[C_7])\longrightarrow H_0(C_2;\pi_0(C(L)))\longrightarrow Q\longrightarrow L^s_{2n+1}(\mathbb{Z}[C_7]), \end{align} $$

where $\pi _0(C(L))$ is the group of (isotopy classes) of pseudoisotopies of L. By a computation of Hatcher–Wagoner [Reference Hatcher and WagonerHW73] and corrections of Igusa [Reference IgusaIgu82], it fits in yet another exact sequence

(4.7)

as long as $\dim L\geq 7$ ; this is our case. In (4.6) and (4.7), for an abelian group with involution $\pi $ ,

• • $L^s_*(\mathbb {Z}\pi )$ are the simple quadratic L-groups of the group ring $\mathbb {Z}\pi $ ,

• • $\mathrm {Wh}_2(\pi )$ is the abelian group (with involution) defined as the cokernel of the map arising from algebraic K-theory (see (B.1))

  (4.8) $$ \begin{align} \pi_2^s((B\pi)_+)\longrightarrow K_2(\mathbb{Z}\pi)\longrightarrow \mathrm{Wh}_2(\pi)\longrightarrow 0, \end{align} $$

• • $\mathrm {Wh}_1^+(\pi ;\mathbb {F}_2):=H_0(C_2; \mathbb {F}_2[\pi ])$ with a certain involution.

We now show that each of the groups in the extension (4.5) are finite. As mentioned in the proof of Proposition 4.7, $\pi _0(s\mathrm {Aut}(L))\subset \pi _0(h\mathrm {Aut}(L))\subset (\mathbb {Z}/7)^\times $ , so it is definitely finite.

By Proposition A.1, $K_2(\mathbb {Z}[C_7])$ is finite, and hence, so is $\mathrm {Wh}_2(C_7)$ by (4.8). Since $\mathbb {F}_2[C_7]$ is finite, $\mathrm {Wh}_1^+(C_7;\mathbb {F}_2)$ is so too. Moreover, the (simple) L-theory of $\mathbb {Z}\pi $ for finite groups $\pi $ of odd order is zero in odd degrees [Reference BakBak75, Thm. 1]. Therefore, $L^s_{2n+1}(\mathbb {Z}[C_7])=0$ , and thus, Q is finite by (4.6).

The finiteness of H is a consequence of the following two observations: firstly that the infinite loop space $\mathrm {Top}/O$ has finite homotopy groups in every degree (see [Reference Kirby, Siebenmann and EssayKS77, Thm. 5.5]), and secondly that if X is a (pointed) finite CW-complex and Y a (pointed) space with finite homotopy groups in every degree, then the set $[X,Y]_*$ of (pointed) maps from X to Y up to homotopy is finite – indeed, this follows easily by induction on the skeleta $\{X_k\}_{k\geq 0}$ of X by considering the cofibre sequences

where $I_k$ is a finite set (because X is a finite CW-complex). Hence, $[\Sigma L_+, \mathrm {Top}/O]_*$ is finite, and thus, so is H. This finishes the proof.