Proof. Let \(N = \mathbb{Z}[\mathcal{G}^{(0)} \times \Gamma] \otimes_{\mathcal{G}\times_\varphi \Gamma} (M \times_\varphi \Gamma)\). By definition, \(N\) is spanned by \[\{(f \otimes 1_g) \otimes (m \otimes 1_h) \colon f \in \mathbb{Z}[\mathcal{G}^{(0)}], m \in M, g,h \in \Gamma\}.\] For the first claim, it suffices to show that \((f \otimes 1_g) \otimes (m \otimes 1_h) = 0\) when \(g \neq h\). To see this, use nondegeneracy to take an idempotent \(e \in \mathbb{Z}[\mathcal{G}^{(0)}]\) such that \(em=m\). Then, by Equation 1 , \[(f \otimes 1_g) \otimes (m \otimes 1_h) = (f \otimes 1_g) \otimes ((e \otimes 1_h)(m \otimes 1_h)) = ((f \otimes 1_g)(e \otimes 1_h)) \otimes (m \otimes 1_h) =0,\] so that the first claim holds. To see the second claim, it suffices to show that the bilinear map \[(f \otimes 1_g, m \otimes 1_h) \mapsto (f \otimes 1_{kg}) \otimes (f \otimes 1_{kh})\] is balanced. This follows similarly. ◻

Proof. First, the action \(\mathcal{G}\times_\varphi \Gamma \curvearrowright X \times \Gamma\) is defined by \[(\gamma,g)(x,g\varphi(\gamma)) = (\gamma x, g),\] where \(g \in \mathcal{G}, x \in X, g \in \Gamma\) and \((\gamma,x)\) is a composable pair.

It is clear that the action \(\mathcal{G}\times_\varphi \Gamma \curvearrowright X \times \Gamma\) is free, and the anchor map \(X \times \Gamma \to \mathcal{G}^{(0)} \times \Gamma\) is a local homeomorphism. Hence, to show that action is basic, it suffices to show that the quotient map \(q \colon X \times \Gamma \to (\mathcal{G}\times_\varphi \Gamma) \backslash (X \times \Gamma)\) is a local homeomorphism. It is always open, so it suffices to show that \(q\) is locally injective. Let \(\bar{q} \colon X \to \mathcal{G}\backslash X\) is the quotient map, which is a local homeomorphism by assumption. We have the following commutative diagram: \[\begin{tikzcd} X \times \Gamma \ar[r,"q"] \ar[d] & \mathcal{G}\times_\varphi \Gamma \backslash X \times \Gamma \ar[d]\\ X \ar[r,"\bar{q}"] & \mathcal{G}\backslash X\makebox[0pt][l]{\,.} \end{tikzcd}\] The projection \(X \times \Gamma \to X\) and \(\bar{q}\) are both local homeomorphisms, so that their composition is locally injective. Hence \(q\) is locally injective. ◻

Proof. By deifinition, the right action is free and the anchor map \(\Omega \times_\tau \Gamma \to \mathcal{H}^{(0)} \times \Lambda\) is a local homeomorphism. Moreover, there is a homeomorphism \[(\Omega \times_\tau \Gamma) / (\mathcal{H}\times_\rho \Lambda) \to (\Omega / \mathcal{H}) \times \Gamma,\;[\omega,g]_{\mathcal{H}\times_\rho \Lambda} \mapsto ([\omega]_\mathcal{H},g)\] which makes the following diagram commute: \[\begin{tikzcd} & \Omega \times_\tau \Gamma \ar[ld,"q"] \ar[rd, "\bar{q} \times {\rm id}"]& \\ (\Omega \times_\tau \Gamma) / (\mathcal{H}\times_\rho \Lambda) \ar[rr,"\sim"] & &(\Omega / \mathcal{H}) \times \Gamma\makebox[0pt][l]{\,,} \end{tikzcd}\] where \(q\) and \(\bar{q}\) are quotient maps. Hence, \((\Omega \times_\tau \Gamma) / (\mathcal{H}\times_\rho \Lambda)\) is Hausdorff and \(q\) is a local homeomorphism, which implies that the right action is proper. Finally, suppose \(\Omega\) is proper, i.e., \(f\colon \Omega/ \mathcal{H}\to \mathcal{G}^{(0)}\) is a proper map. Then, the map \[(\Omega \times_\tau \Gamma) / (\mathcal{H}\times_\rho \Lambda) \cong (\Omega / \mathcal{H}) \times \Gamma \to \mathcal{G}^{(0)} \times \Gamma\] is equal to \(f \times {\rm id}\), which is a proper map. Hence \(\Omega \times_\tau \Gamma\) is proper. ◻

Proof. For \(g \in \Gamma\), define a bilinear map \[\Phi_g \colon \mathbb{Z}\Omega \times M \to \mathrm{Ind}_{\Omega \times_\tau \Gamma} (M \times_\rho \Lambda),\;(f, m) \mapsto (f \otimes 1_g) \otimes (m \otimes 1_{j(g)h})\] for \(f \in \mathbb{Z}\Omega_h, m \in M, h \in \Lambda\). Then, for \(\xi \in \mathbb{Z}\mathcal{H}\) and \(k \in \Lambda\) with \(\mathop{\mathrm{supp}}\xi \subseteq \rho^{-1}(k)\), we have \(f\xi \in \mathbb{Z}\Omega_{hk}\), so that \[\begin{align} &\Phi_g(f\xi, m) = (f \xi \otimes 1_g) \otimes (m \otimes 1_{j(g)hk}) = ((f \otimes 1_g)(\xi \otimes 1_{j(g)h})) \otimes (m \otimes 1_{j(g)hk}),and \\ &\Phi_g (f, \xi m) = (f \otimes 1_g) \otimes (\xi m \otimes 1_{j(g)h}) = (f \otimes 1_g) \otimes ((\xi \otimes 1_{j(g)h})( m \otimes 1_{j(g)hk})). \end{align}\] Hence, the map \(\Phi_g \colon \mathbb{Z}\Omega \otimes_\mathcal{H}M \to \mathrm{Ind}_{\Omega \times_\tau \Gamma} (M \times_\rho \Lambda)\) is well-defined. Now, let \[\Phi \colon (\mathrm{Ind}_\Omega M) \times_\varphi \Gamma \to \mathrm{Ind}_{\Omega \times_\tau \Gamma} (M \times_\rho \Lambda).\] be the direct sum of the maps induced from the maps \(\Phi_g\) for all \(g\in\Gamma\). A calculation shows that \(\Phi\) is a \(\mathcal{G}\times_\varphi \Gamma\)-module map. A similar argument shows that the bilinear map \[(\mathbb{Z}\Omega \otimes \mathbb{Z}\Gamma) \times (M \otimes \mathbb{Z}\Lambda) \to (\mathrm{Ind}_\Omega M) \times_\varphi \Gamma, \quad (f \otimes 1_g, m \otimes 1_l) \mapsto \delta_{j(g)h,l} (f \otimes m) \otimes 1_g\] for \(f \in \mathbb{Z}\Omega_h\) is balanced, which induces the inverse of \(\Phi\). ◻

Proof. By definition, we have \(s_{\Omega \times_\tau \Gamma}(\omega, g) = (s_\Omega(\omega), j(g)\tau(\omega))\) for \((\omega, g) \in \Omega \times_\tau \Gamma\). Hence, for \(g\in \Gamma, h \in \Lambda\) and \(\xi \in \mathbb{Z}\Omega_h\), we have \[(s_{\Omega \times_\tau \Gamma})_*(\xi \otimes 1_g) = (s_\Omega)_*(\xi) \otimes 1_{j(g)h}.\] By Lemma 1, the \(\mathbb{Z}\)-module \(\mathbb{Z}[\mathcal{G}^{(0)} \times \Gamma] \otimes_{\mathcal{G}\times_\varphi \Gamma} \mathbb{Z}[\Omega \times_\tau \Gamma]\) is spanned by \[\{(f \otimes 1_g) \otimes (\xi \otimes 1_g) \colon f \in \mathbb{Z}[\mathcal{G}^{(0)}], \xi \in \mathbb{Z}\Omega_h, g \in \Gamma, h \in \Lambda\}.\] By definition of \(\delta\), we have \[\label{eqn:delta} \delta((f \otimes 1_{kg}) \otimes (\xi \otimes 1_{kg})) = (s_{\Omega \times_\tau \Gamma})_*((f|_{\varphi^{-1}(e)}\xi) \otimes 1_{kg}) = (s_\Omega)_*(f\xi) \otimes 1_{j(k)j(g)h}\tag{3}\]

for \(k \in \Gamma\), since \(f\xi \in \mathbb{Z}\Omega_h\). Thus we have equivariance. ◻

Proof. First, we show that \(\Phi\) is well-defined. Fix \(g \in \Gamma\). Then, the bilinear map \[\Phi_g \colon \mathbb{Z}[\mathcal{G}^{(0)}] \times M \to \mathbb{Z}[\mathcal{G}^{(0)} \times \Gamma] \otimes_{\mathcal{G}\times_\varphi \Gamma} (M \times_\varphi \Gamma), \quad (f, m) \mapsto (f \otimes 1_{gh^{-1}}) \otimes (m \otimes 1_{gh^{-1}})\] for \(f\in \mathbb{Z}[\mathcal{G}^{(0)}]\), \(h\in\Gamma\), and \(m\in M_h\) is well-defined because \(M = \bigoplus_h M_h\). Let \(h,k \in \Gamma\), \(f \in \mathbb{Z}[\mathcal{G}^{(0)}]\), \(m \in M_h\), and \(\eta \in \mathbb{Z}[\mathcal{G}]\) with \(\mathop{\mathrm{supp}}\eta \subseteq \varphi^{-1}(k)\). Then, by Equation 1 , \[\begin{align} &\Phi_g(f\eta, m) = (f\eta \otimes 1_{gh^{-1}}) \otimes (m \otimes 1_{gh^{-1}}) = ((f \otimes 1_{gh^{-1}k^{-1}})(\eta \otimes 1_{gh^{-1}k^{-1}})) \otimes (m \otimes 1_{gh^{-1}}),and \\ &\Phi_g(f, \eta m) = (f \otimes 1_{gh^{-1}k^{-1}}) \otimes (\eta m \otimes 1_{gh^{-1}k^{-1}}) =(f \otimes 1_{gh^{-1}k^{-1}}) \otimes ((\eta \otimes 1_{gh^{-1}k^{-1}})(m \otimes 1_{gh^{-1}})), \end{align}\] which implies that \(\Phi_g\) is balanced and thus induces a map from \(\mathbb{Z}\Gamma \otimes (\mathbb{Z}[\mathcal{G}^{(0)}] \otimes_{\mathcal{G}} M)\). Hence, \(\Phi\) is the direct sum of the maps induced from \(\Phi_g\) for all \(g\in\Gamma\). Exactly the same argument shows that the bilinear map \[\mathbb{Z}[\mathcal{G}^{(0)} \times \Gamma] \times (M \times_\varphi \Gamma) \to \mathbb{Z}\Gamma \otimes (\mathbb{Z}[\mathcal{G}^{(0)}] \otimes_{\mathcal{G}} M), \quad (f \otimes 1_g, m \otimes 1_{g'}) \mapsto \delta_{g,g'} 1_{gh} \otimes f \otimes m\] for \(f\in \mathbb{Z}[\mathcal{G}^{(0)}]\), \(g,g',h\in\Gamma\), and \(m \in M_h\) is balanced, so that it induces a map from the tensor product \(\mathbb{Z}[\mathcal{G}^{(0)} \times \Gamma] \otimes_{\mathcal{G}\times_\varphi \Gamma} M \times_\varphi \Gamma\), which is the inverse of \(\Phi\). The latter claim follows from [33]. ◻

Proof. The LHS type spectral sequence is obtained by applying [33] to \(C_\bullet\) and \(C_\bullet'\). Since the map \(\bar{\kappa}\) from 2 is \((\Gamma,\Lambda)\)-equivariant, we see from the proof of this proposition that \(\bar{\kappa}\) induces a map between the spectral sequences in [33] for \(C_\bullet\) and \(C_\bullet'\). Moreover, the maps \(\mathrm{H}_q(C_\bullet)\to \mathrm{H}_q(C_\bullet')\) between coefficient modules of the homology groups on the \(E^2\)-page are induced from \(\bar{\kappa}\). Since \(\mathrm{H}_*(\Omega \times_\tau \Gamma)\) is induced from \(\bar{\kappa}\), this shows (i).

It remains to show that the map \(\mathrm{H}_*((C_\bullet)_\Gamma) \to \mathrm{H}_*((C'_\bullet)_\Lambda)\) induced by \(\bar{\kappa}\) is identified with \(\mathrm{H}_*(\Omega)\) via the natural transformation in Lemma 6. To see this, it suffices to show the map \[(\mathrm{Ind}_\Omega Q_\bullet)_\mathcal{G}\to (Q_\bullet)_\mathcal{H}\] induced by \({\rm id}_{\mathbb{Z}} \otimes \delta \otimes {\rm id}_{Q_\bullet\times_\rho\Lambda}\) is equal to \(\delta_\Omega\otimes{\rm id}_{Q_\bullet}\). It is the composition \[\begin{align} &(\mathrm{Ind}_\Omega Q_\bullet)_\mathcal{G}\xrightarrow{Lemma~\ref{lem:coinv}} (((\mathrm{Ind}_\Omega Q_\bullet) \times_\varphi \Gamma)_{\mathcal{G}\times_\varphi \Gamma})_\Gamma \xrightarrow{Lemma~\ref{lem:natural}} ((\mathrm{Ind}_{\Omega\times_\tau\Gamma}(Q_\bullet \times_\rho \Lambda))_{\mathcal{G}\times_\varphi \Gamma})_\Gamma \\ &\xrightarrow{{\rm id}\otimes \delta \otimes {\rm id}} ((Q_\bullet \times_\rho \Lambda)_{\mathcal{H}\times_\rho \Lambda})_\Lambda \xrightarrow{Lemma~\ref{lem:coinv}} (Q_\bullet)_\mathcal{H}. \end{align}\] For \(f \in \mathbb{Z}[\mathcal{G}^{(0)}]\), \(\xi \in \mathbb{Z}\Omega_h\), \(h \in \Lambda\), and \(m \in Q_\bullet\), we have \[\begin{align} &f \otimes (\xi \otimes m) \mapsto 1 \otimes (f \otimes 1_e) \otimes ((\xi \otimes m) \otimes 1_e) \mapsto 1 \otimes (f \otimes 1_e) \otimes ((\xi \otimes 1_e) \otimes (m \otimes 1_{h}) \\ &\mapsto 1 \otimes ((s_\Omega)_*(f \xi) \otimes 1_{h}) \otimes (m \otimes 1_{h}) \mapsto (s_\Omega)_*(f \xi) \otimes m, \end{align}\] where we have used Equation 3 . Hence, this map is equal to \(\delta_\Omega\otimes{\rm id}_{Q_\bullet}\). ◻

Proof. Let \(\iota_1 \colon \Gamma' \ltimes X \to (\Gamma \ltimes X) \times_\varphi \Gamma''\) and \(\iota_2 \colon \Lambda' \ltimes Y \to (\Lambda \ltimes Y) \times_\rho \Lambda''\) be the inclusion maps. Let \(\Omega_{\iota_1}\) and \(\Omega_{\iota_2}\) denote the correspondences associated to the étale maps \(\iota_1\) and \(\iota_2\), respectively. By [36], it suffices to show \(\Omega_{\iota_2} \circ \Omega' = (\Omega \times_\tau \Gamma'') \circ \Omega_{\iota_1}\). We can directly see that the both sides are isomorphic to the correspondence \(\Lambda \ltimes Y \colon \Gamma' \ltimes X \to (\Lambda \ltimes Y) \times_\rho \Lambda''\). ◻

Proof. Let \(\mathcal{H}_k := \Gamma_k\ltimes X_k\) and let \(\Phi_k \colon \mathbb{Z}\mathcal{H}_k \to \mathbb{Z}\mathcal{H}_{k+1}\) be the \(\mathbb{Z}\)-module map defined by \(\xi \mapsto \xi1_{X_{k+1}}\). Here, the product is defined via the left action \(\mathbb{Z}\mathcal{H}_k \curvearrowright\mathbb{Z}\mathcal{H}_{k+1} = \mathbb{Z}\Omega_k\). Concretely, for \(\xi \in \mathbb{Z}\mathcal{H}_k\) and \((h,y) \in \mathcal{H}_{k+1}\), we have \[(\Phi_k(\xi))(h,y) = \begin{cases} \xi(g,\pi_{k+1,k}(y)) & if h=j(g) for some g \in \Gamma_{k}, \\ 0 & otherwise. \end{cases}\] Here, we used equivariance in the sense of Definition 5. Let \(C_\bullet^k = \mathbb{Z}[\mathcal{H}_k^\bullet] = \mathbb{Z}[\mathcal{H}_k] \otimes_{X_k}\dots \otimes_{X_k} \mathbb{Z}[\mathcal{H}_k]\) be the chain complex obtained by taking coinvariants of the bar resolution. Then, we have \(\mathrm{H}_*(\mathcal{H}_k)=\mathrm{H}_*(C_\bullet^k)\). We first claim that \(\mathrm{H}_*(\Omega_k) \colon \mathrm{H}_*(\mathcal{H}_k) \to \mathrm{H}_*(\mathcal{H}_{k+1})\) is induced from the chain map \[\Phi_k^{\otimes n} =\Phi_k \otimes \dots \otimes \Phi_k \colon \mathbb{Z}[\mathcal{H}_k^n] \to \mathbb{Z}[\mathcal{H}_{k+1}^n]\] for each \(n\). Fix \(k \in \mathbb{N}\) and let \(j=j_{k,k+1}\), \(\pi=\pi_{k+1,k}\). Let \(\mathcal{K}=\mathcal{H}_k \ltimes X_{k+1}\) be the action groupoid, where the action \(\mathcal{H}_k \curvearrowright X_{k+1}\) is defined by \[(g,\pi(y))\cdot y = j(g)y\] for \(y \in X_{k+1}\) and \(g \in \Gamma_k\). Note that this groupoid action is well defined because of the equivariance condition from Definition 5. We identify \(\mathcal{K}\cong \Gamma_k \ltimes X_{k+1}\), where the partial action \(\Gamma_k \curvearrowright X_{k+1}\) is defined using \(j\), via the isomorphism given by \[\mathcal{H}_k \ltimes X_{k+1} \to \Gamma_k \ltimes X_{k+1},\quad (g,\pi(y),y) \mapsto (g,y).\] Let \(\varphi \colon \mathcal{K}\to \mathcal{H}_{k+1}\) be the étale homomorphism defined by \((g,y) \mapsto (j(g),y)\) for \((g,y) \in \Gamma_k \ltimes X_{k+1}\). Let \(\Omega_\varphi\) be the étale correspondence associated to \(\varphi\). Then, we can see that \(\Omega_\varphi \circ \mathcal{K}= \Omega_k\), where \(\mathcal{K}\) is considered as an étale correspondence \(\mathcal{H}_k \to \mathcal{K}\). By [36], \(\mathrm{H}_*(\mathcal{K})\) is induced from \(\tau_n^* \colon \mathbb{Z}[\mathcal{H}_k^n] \to \mathbb{Z}[\mathcal{K}^n]\), where \[\tau_n \colon \mathcal{K}^n \to \mathcal{H}_k^n,\quad( (g_1,j(g_2\cdots g_n) y), \dots, (g_n,y)) \mapsto ((g_1,g_2\cdots g_n \pi(y)), \dots, (g_n,\pi(y))).\] On the other hand, by [36], \(H_*(\Omega_\varphi)\) is induced from \(\varphi_*^n \colon \mathbb{Z}[\mathcal{K}^n] \to \mathbb{Z}[\mathcal{H}_{k+1}^{n}]\) defined by \[\varphi_*^n(f)((h_1,h_2\cdots h_n y), \dots, (h_n,y)) = \begin{cases} f((h_1,h_2\cdots h_n y), \dots, (h_n,y)) & if h_i \in j(\Gamma_k) for all i, \\ 0 & otherwise. \end{cases}\] Then, we can see that \(\varphi_*^n \circ \tau_n^* = \Phi_k^{\otimes n}\), so that \(\mathrm{H}_*(\Omega_k)\) is induced from \(\Phi_k^{\otimes \bullet}\). Note that \(\Phi_k\) is multiplicative, and we can directly see that \(\Phi_k^{\otimes \bullet}\) is indeed a well-defined chain map by using this property.

Next, we claim that \(\mathbb{Z}\mathcal{H}=\varinjlim (\mathbb{Z}\mathcal{H}_k, \Phi_k)\), where the inductive limit is taken in the category of abelian groups. To see this, it suffices to show that \(\mathbb{Z}\mathcal{H}\) is spanned by \(\{\eta1_\mathcal{H}: \eta\in \mathbb{Z}\mathcal{H}_k\}\). Clearly \(\mathbb{Z}\mathcal{H}\) is spanned by the functions supported on \(\{j_k(g)\} \times X\) for some \(k\) and \(g \in \Gamma_k\), where \(j_k\colon\Gamma_k\to\Gamma\) is the canonical map. Let \(\xi \in \mathbb{Z}[\mathcal{H}]\) be such a function. Then, there exist \(k\in \mathbb{N}\), \(g \in \Gamma_k\), and \(\tilde{\xi} \in \mathbb{Z}[X]\) such that \[\xi(h,x) = \begin{cases} \tilde{\xi}(x) &ifh = j_k(g), \\ 0 &ifh \neq j_k(g). \end{cases}\] Then, there exist \(k' \in \mathbb{N}\) and \(\tilde{\eta} \in X_{k'}\) such that \(\tilde{\xi} = \tilde{\eta} \circ \pi_k\). We may assume that \(k=k'\). Let \(\eta \mathrel{\vcenter{:}}= 1_{g} \times \tilde{\eta} \in \mathbb{Z}\mathcal{H}_k\), and then we have \(\xi = \eta 1_\mathcal{H}\).

Let \(C_\bullet = \mathbb{Z}[\mathcal{H}^\bullet]\). Then, we have \(C_\bullet = \varinjlim (C_\bullet^k, \Phi_k^{\otimes \bullet})\) because tensor products preserve inductive limits. Therefore, continuity for homology of chain complexes yields the desired property. ◻

Proof. Clearly, we have \(A\subseteq \mathscr{A}\cap\overline{A}\). Let \(a\in \mathscr{A}\cap\overline{A}\). Then, there exists \(r\in S\) and \(b\in A\) such that \(a=\tilde{\sigma}_r^{-1}(b)\). Hence, \(b=\sigma_r(a)\in\sigma_r\overline{A}=\overline{\sigma_rA}\) (where \(\overline{\sigma_rA}\) denotes the closure of \(\sigma_rA\)). Thus, \(b\in A\cap \overline{\sigma_rA}=\sigma_rA\). Now we can find \(c\in A\) such that \(b=\sigma_r(c)\), which gives \(a=c\in A\). ◻

Proof. Consider the inclusion \(\mathscr{A}/A\to\overline{\mathscr{A}}/\overline{A}\) induced from the inclusion \(\mathscr{A}\to\overline{\mathscr{A}}\). It has dense image because \(\mathscr{A}\) is dense in \(\overline{\mathscr{A}}\). Since \(\overline{A}\) is compact open in \(\overline{\mathscr{A}}\), the quotient \(\overline{\mathscr{A}}/\overline{A}\) is discrete. Hence, \(\mathscr{A}/A\to\overline{\mathscr{A}}/\overline{A}\) is surjective; this map injective by Lemma 8 and is thus an isomorphism. The claim now follows. ◻

Proof. By Lemma 9, \(\overline{A}\) is a full subset of the unit space of \(\mathscr{A}\ltimes\overline{\mathscr{A}}\). Thus, it suffices to show that \(A\ltimes\overline{A}=(\mathscr{A}\ltimes\overline{\mathscr{A}})_{\overline{A}}^{\overline{A}}\), which follows from Lemma 8. The second claim follows from Proposition 2 combined with [9] (see also [44]). ◻

Proof. Let \({\rm res}^{A}_{\sigma_sA}\colon\mathrm{H}_*(A)\to\mathrm{H}_*(\sigma_sA)\) be the transfer map (see, e.g., [33]). By 5 , we have \[\mathrm{H}_*(\mathrm{incl},\mathrm{ind}_s\vert^{M_s})\circ\mathrm{H}_*(\sigma_s\vert^{\sigma_sA},{\rm id})\circ\rho_{1,s}=\mathrm{H}_*({\rm id},\pi_{1,s}^*)=\mathrm{H}_*(\mathrm{incl},\mathrm{ind}_s\vert^{M_s})\circ{\rm res}^{A}_{\sigma_sA},\] where the second equality is from the proof of [5]. Since \(\mathrm{H}_*(\mathrm{incl},\mathrm{ind}_s\vert^{M_s})\) is invertible, the above equation implies that \[\label{eqn:rho} \mathrm{H}_*(\sigma_s\vert^{\sigma_sA},{\rm id})\circ\rho_{1,s}={\rm res}^{A}_{\sigma_sA}.\tag{8}\]

We let \(\mathrm{cor}_{\sigma_sA}^{A}\colon\mathrm{H}_*(\sigma_sA)\to\mathrm{H}_*(A)\) denote the corestriction map, i.e., the map induced from the inclusion \(\sigma_sA\to A\). We have \(\mathrm{cor}_{\sigma_sA}^{A}\circ \mathrm{H}_*(\sigma_s\vert^{\sigma_sA},{\rm id})=\mathrm{H}_*(\sigma_s,{\rm id})\), so applying \(\mathrm{cor}_{\sigma_sA}^{A}\) to both sides of 8 and using [33] gives us \(\mathrm{H}_*(\sigma_s,{\rm id})\circ\rho_{1,s}=[A:\sigma_sA]{\rm id}_{\mathrm{H}_*(A)}\). ◻

Proof. Since tensor products commute with direct limits, we have a canonical isomorphism \[\mathrm{H}_*(A,\mathbb{Z}\overline{A}) \otimes\mathbb{Z}[\mathbb{I}^{-1}]\cong\varinjlim_{s\in S}\left\{\mathrm{H}_*(A)\otimes\mathbb{Z}[\mathbb{I}^{-1}];\tilde{\rho}_{s,t}\right\}.\] By Lemma 10, \(\mathrm{H}_*(\sigma_s, {\rm id})\otimes{\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}\) and \(\tilde{\rho}_{1,s}\) are invertible with \[\label{eqn:invertrho} \tilde{\rho}_{1,s}^{-1}=\frac{1}{[A:\sigma_sA]}\mathrm{H}_*(\sigma_s, {\rm id})\otimes{\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}\tag{9}\] for every \(s\in S\). Hence, every connecting map \(\tilde{\rho}_{s,t}\) is invertible, so that \(\tilde{\rho}_{1,\infty}\) must be invertible. ◻

Proof of Theorem 9. Let \(\iota\mathrel{\vcenter{:}}= \mathrm{H}_*(\mathrm{incl},\mathrm{incl})\colon \mathrm{H}_*(A,\mathbb{Z}\overline{A})\to \mathrm{H}_*(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\) be the the isomorphism from Proposition 8. The canonical inclusion \(\mathrm{H}_*(A)\hookrightarrow \mathrm{H}_*(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\) is given by the composition \[{\textstyle\bigwedge}^*(A) \cong \mathrm{H}_*(A)\xrightarrow{\rho_{1,\infty}}\mathrm{H}_*(A,\mathbb{Z}\overline{A})\xrightarrow{\iota} \mathrm{H}_*(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}}),\] where the first isomorphism is from [33].

Using Proposition 11, we obtain a canonical inclusion \[\begin{align} \Upsilon \colon \mathrm{H}_*(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\xrightarrow{\iota^{-1}}\mathrm{H}_*(A,\mathbb{Z}\overline{A})& \xrightarrow{\mathrm{incl}}\mathrm{H}_*(A,\mathbb{Z}\overline{A}) \otimes\mathbb{Z}[\mathbb{I}^{-1}] \xrightarrow{\tilde{\rho}_{1,\infty}^{-1}} \mathrm{H}_*(A) \otimes\mathbb{Z}[\mathbb{I}^{-1}]\cong {\textstyle\bigwedge}^*(A) \otimes\mathbb{Z}[\mathbb{I}^{-1}]. \end{align}\]

It remains to compute the \(S\)-action on \(\mathrm{H}_*(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\) given by \(\theta_s = \mathrm{H}_*(\tilde{\sigma}_s,\alpha_s)\) for \(s \in S\). For each \(s\in S\), we have the following commutative diagram: \[\begin{tikzcd} \mathrm{H}_*(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\otimes\mathbb{Z}[\mathbb{I}^{-1}]\arrow{rrr}{\mathrm{H}_*(\tilde{\sigma}_s,\alpha_s)\otimes{\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}} & & & \mathrm{H}_*(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\otimes\mathbb{Z}[\mathbb{I}^{-1}] \\ \mathrm{H}_*(A,\mathbb{Z}\overline{A})\otimes\mathbb{Z}[\mathbb{I}^{-1}]\arrow[u,"{\iota\otimes {\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}}"] & & & \mathrm{H}_*(A,\mathbb{Z}\overline{A})\otimes\mathbb{Z}[\mathbb{I}^{-1}] \arrow[u,"{\iota\otimes{\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}}"]\\ \mathrm{H}_*(A)\otimes\mathbb{Z}[\mathbb{I}^{-1}]\arrow{u}{\tilde{\rho}_{1,\infty}}\arrow{rrr}{\theta_s\otimes{\rm id}} & & & \mathrm{H}_*(A)\otimes\mathbb{Z}[\mathbb{I}^{-1}]\arrow{u}{\tilde{\rho}_{1,\infty}}\makebox[0pt][l]{\,.} \end{tikzcd}\] Observe that \[\mathrm{H}_*(\tilde{\sigma}_s, \alpha_s) \circ \iota \circ \mathrm{H}_*(\sigma_1, \mathrm{ind}_1) = \iota \circ \mathrm{H}_*(\sigma_s, \mathrm{ind}_s).\] Since \(\rho_{t,\infty}=\mathrm{H}_*(\sigma_t, \mathrm{ind}_t)\) for all \(t\in S\) by 7 , we have \[\begin{align} \theta_s\otimes{\rm id}&=\tilde{\rho}_{1,\infty}^{-1}\circ(\iota\otimes {\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]})^{-1}\circ\mathrm{H}_*(\tilde{\sigma}_s,\alpha_s)\otimes{\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}\circ (\iota\otimes {\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]})\circ \tilde{\rho}_{1,\infty}\\ &=(\tilde{\rho}_{s,\infty}\circ\tilde{\rho}_{1,s})^{-1}\circ (\iota\otimes {\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]})^{-1}\circ (\iota \circ \mathrm{H}_*(\sigma_s, \mathrm{ind}_s)\otimes {\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}) \\ &=(\tilde{\rho}_{s,\infty}\circ\tilde{\rho}_{1,s})^{-1}\circ\tilde{\rho}_{s,\infty}=\tilde{\rho}_{1,s}^{-1}=\frac{1}{[A:\sigma_sA]}\mathrm{H}_*(\sigma_s,{\rm id})\otimes{\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]}. \end{align}\] For the last equality, we have used Equation 9 . It remains to observe that under the identification \(\mathrm{H}_*(A)\cong {\textstyle\bigwedge}^*(A)\), \(\mathrm{H}_*(\sigma_s,{\rm id})\) is taken to \({\textstyle\bigwedge}^*(\sigma_s)\), see [33]. ◻

Proof. The equality \(\mathrm{H}_q(A,\mathbb{Z}\overline{A})=\mathrm{H}_q(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\) follows from Proposition 8.

(i): We have \([A:nA]=n^d\) and \({\textstyle\bigwedge}^q(n)=n^q\), so by Theorem 9, we have \(\theta_s \otimes{\rm id}_{\mathbb{Z}[\mathbb{I}^{-1}]} = n^{q-d}\), so that \(\theta_s = n^{q-d}\). This implies that \(\mathrm{H}_q(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\) is divisible by \(n\) since \(q<d\), so that we have \({\textstyle\bigwedge}^qA\otimes\mathbb{Z}[1/n]\leq \mathrm{H}_q(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\).

(ii) is immediate because \({\textstyle\bigwedge}^q A = 0\). (iii): By Lemma 10, we have \(\rho_{1,n}=n^{d-d}{\rm id}_{\mathrm{H}_d(A)}={\rm id}_{\mathrm{H}_d(A)}\) for all \(n\in S\cap \mathbb{Z}_{>0}\). Thus, \(\mathrm{H}_d(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\cong \varinjlim_{s\in S}\{\mathrm{H}_d(A);\rho_{s,t}\}\cong\varinjlim_{n\in S\cap \mathbb{Z}_{>0}}\{\mathrm{H}_d(A);\rho_{n,m}\} =\mathrm{H}_d(A)=\mathbb{Z}\). ◻

Proof. For \(a\in {\rm End}\,_\mathbb{Z}(A)\cong \mathrm{M}_d(\mathbb{Z})\) with \(\det(a)\neq 0\), let \(\tilde{a}\in{\rm End}\,_\mathbb{Z}(A)\) be such that \(a\tilde{a}=\tilde{a}a=|\det(a)|\).

Put \(T\mathrel{\vcenter{:}}= \mathscr{S}\cap{\rm End}\,_\mathbb{Z}(A)\). First, we claim that for any \(a \in T\), we have \(\tilde{a} \in T\). To see this, we see that \(\tilde{a} \in {\rm End}\,_\mathbb{Z}(A)\) by definition, and we have \(\tilde{a} = a^{-1}|\det(a)| \in \mathscr{S}\) by assumption. In particular, \(T\) is a reversible monoid: Given \(a,b\in T\), we have \(|\det(a)| \in aT\) and \(|\det(b)| \in bT\), so that \(|\det(ab)| \in aT \cap bT\). Similarly, we can show \(|\det(ab)| \in Ta \cap Tb\).

Since \(|\det(a)| \in Ta\) for any \(a \in T\), the set of natural numbers \(\{|\det(t)| : t\in T\}\) is cofinal in \(T\), so that \(\mathbb{Z}_{>0}\) is cofinal in \(T\). Consider the following diagram: \[\begin{tikzcd} \varinjlim_{s\in S}\{\mathrm{H}_q(A); \rho_{s,t}\} \arrow[rr]& & \varinjlim_{s\in T}\{\mathrm{H}_q(A); \rho_{s,t}\} \\ &\varinjlim_{n\in \mathbb{Z}_{>0}}\{\mathrm{H}_q(A); \rho_{n,m}\}\makebox[0pt][l]{\,.}\arrow[lu]\arrow[ru] & \end{tikzcd}\] The arrow on the lower right is an isomorphism by cofinality of \(\mathbb{Z}_{>0}\) in \(T\). Since \(\mathbb{Z}_{>0}\subseteq S\subseteq T\), \(S\) is cofinal in \(T\), so that the top arrow is also an isomorphism. ◻

Proof. First, suppose \(0\leq q<d\). Choose \(n\in\mathscr{S}\cap\mathbb{Z}_{>1}\). Then, \(\theta_n(x)=n^{q-d}x\) for all \(x\in{\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A}\). Note that \(0<n^{q-d}<1\) because \(q<d\). We have \[\mathrm{H}_k(\langle n\rangle,{\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A})= \begin{cases} ({\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A})_{\langle n\rangle}& \text{ if } k=0,\\ ({\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A})^{\langle n\rangle}& \text{ if } k=1,\\ 0 & \text{ if } k>1, \end{cases}\] (see, for instance, [34]). Here, \(\langle n\rangle=n^\mathbb{Z}\). The group \[({\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A})_{\langle n\rangle}={\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A}/\langle(1-n^{k(q-d)})x : x\in {\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A}, k\in\mathbb{Z}\rangle\] is trivial because \({\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A}\) is divisible and \((1-n^{q-d}){\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A}\leq \langle(1-n^{k(q-d)})x : x\in {\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A}, k\in\mathbb{Z}\rangle\). The group \[({\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A})^{\langle n\rangle}=\{x\in {\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A}: \theta_n(x)=x\}\] is trivial because \(\theta_n\) has no fixed points except \(0\). Since \(\langle n\rangle\leq \mathscr{S}\) is a normal subgroup, the Lyndon–Hochschild–Serre spectral sequence now implies that \(\mathrm{H}_*(\mathscr{S},{\textstyle\bigwedge}_\mathbb{Q}^q\mathscr{A})=0\) for all \(0\leq q<d\).

By Theorem 14, we have \(\mathrm{H}_d(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})={\textstyle\bigwedge}_\mathbb{Z}^d\mathscr{A}=\mathbb{Z}\) and \(\theta_a=\mathrm{sign}(a)\). ◻

Proof. By [39], we have \(N_\mathbb{Q}(a)=\prod_{w\in V_K}w(a)\), where \(V_K\) is the set of field embeddings of \(K\) into \(\mathbb{C}\). Let \(V_{K,\mathbb{C}}\) denote the set of field embeddings \(w\) of \(K\) into \(\mathbb{C}\) such that \(w(K)\not\subseteq\mathbb{R}\). Now we have \[\frac{N_\mathbb{Q}(a)}{|N_\mathbb{Q}(a)|}=\prod_{w\in V_{K,\mathbb{C}}}\frac{w(a)}{|w(a)|} \prod_{w\in V_{K,\mathbb{R}}}\frac{w(a)}{|w(a)|}=\prod_{w\in V_{K,\mathbb{R}}}\frac{w(a)}{|w(a)|},\] where we define the empty product to be \(1\). Here, we used that for each complex embedding \(w\in V_{K,\mathbb{C}}\), there is a unique conjugate embedding \(\overline{w}\in V_{K,\mathbb{C}}\) satisfying \(w(a)\overline{w}(a)=|w(a)|^2\) for all \(a\in K\). ◻

Proof. Pick \(w \in V_{K,\mathbb{R}}\). By [39], there is \(a \in K^*\) such that \(w(a) <0\) and \(w'(a) >0\) for any \(w' \in V_{K,\mathbb{R}}\). By Lemma 11, this means that \(\mathrm{sign}(a) = -1\), and hence \(\mathrm{sign}\) is surjective. Recall that the multiplicative group \(K^*\) of \(K\) is isomorphic to the direct product \(\mu \times \Gamma\), where \(\mu\) is the group of roots of unity and \(\Gamma\) is a free abelian group (Subsection 2.4). If \(|V_{K,\mathbb{R}}|\) is even, then \(\mathrm{sign}(\mu)=\{ + 1\}\), and hence \(\mathrm{sign}|_{\Gamma}\) is surjective. If \(|V_{K,\mathbb{R}}|\) is odd, in which case \(\mathrm{sign}(\mu) = \{ \pm 1\}\), we may assume that \(\mathrm{sign}|_{\Gamma}\) is surjective by replacing \(\Gamma = \langle a_1,a_2,a_3,\cdots \rangle\) with \(\Gamma ' \mathrel{\vcenter{:}}= \langle -a_1,a_2,a_3,\cdots \rangle\) if necessary.

Pick \(b \in \Gamma\) such that \(\mathrm{sign}(b) = -1\). Then, since \(\Gamma\) is free abelian, we can take the primitive part \(a \in \Gamma\) of \(b\), i.e., the unique element \(a \in \Gamma\) such that \(b = m a\) for some integer \(m \ge 1\) and \(a\) is not divisible by any integer greater than \(1\). Then \(\mathrm{sign}(a)\) must be \(-1\), and \(a^{\mathbb{Z}}\) is a direct summand. ◻

Proof. First, suppose \(K\) is totally imaginary. In this case, the action \(K^* \curvearrowright\mathbb{Z}_\mathrm{sign}\) is the trivial action by Lemma 11. Since \(K^*/\mu\) is free abelian, we have \(K^*\cong \mu\times (K^*/\mu)\) and \(\mathrm{H}_*(K^*/\mu)\cong\bigwedge^*(K^*/\mu)\), where the homology group here denotes group homology with \(\mathbb{Z}\)-coefficients (see [33] for the latter claim). Further, note that \(K^*/\mu\) is abstractly isomorphic to \(\mathbb{Z}^{\oplus\infty}\), see Section 2.4. By the Künneth formula (Theorem 1), we have \[\begin{align} \mathrm{H}_n(K^*, \mathbb{Z}_\mathrm{sign}) &= \mathrm{H}_n(\mu \times (K^*/\mu))=\bigoplus_{p+q=n} \mathrm{H}_p(\mu) \otimes \mathrm{H}_q(K^*/\mu) \\ &= \mathrm{H}_n(K^*/\mu) \oplus \bigoplus_{p \colonodd} (\mathbb{Z}/|\mu|\mathbb{Z}) \otimes \mathrm{H}_{n-p}(K^*/\mu), \end{align}\] so that the claim (i) holds by Corollary 1. Next, suppose the number of real embeddings of \(K\) is odd. In this case, \(\mu=\{\pm 1\}\) and \(\mathrm{sign}(-1)=-1\). Let \(\Gamma = \ker \mathrm{sign}\subseteq K^*\). Then, \(\Gamma \cong K^*/\mu\) and \(K^*=\mu \times \Gamma\). By Künneth’s formula (Theorem 1), we have \[\mathrm{H}_n(K^*, \mathbb{Z}_\mathrm{sign}) = \bigoplus_{p+q=n} \mathrm{H}_p(\mu, \mathbb{Z}_\mathrm{sign}) \otimes \mathrm{H}_q(K^*/\mu).\] By [34], we can see that \[\mathrm{H}_p(\mu, \mathbb{Z}_\mathrm{sign}) = \begin{cases} \mathbb{Z}/2\mathbb{Z}& if p is even, \\ 0 & if p is odd, \end{cases}\] so that the claim (ii) is true in this case by Corollary 1. Finally, suppose the number of real embeddings of \(K\) is even and nonzero. In this case, \(\mu=\{\pm 1\}\) and \(\mathrm{sign}(-1)=1\). By Lemma 12, there exists \(a \in K^*\) with \(\mathrm{sign}(a)=-1\) such that \(a^\mathbb{Z}\) is a summand of \(K^*\). Let \(\Lambda = \{\pm 1\} \times a^\mathbb{Z}\). Then, there exists a free abelian subgroup of infinite rank \(\Gamma \subseteq K^*\) such that \(K^* = \Lambda \times \Gamma\). We can assume that \(\mathrm{sign}(\Gamma)=1\). To see this, let \(\{a_i\}_{i=1}^\infty\) be a basis of \(\Gamma\). For each \(i\), let \(b_i = a_i\) if \(\mathrm{sign}(a_i)=1\), and let \(b_i = aa_i\) if \(\mathrm{sign}(a_i) = -1\). Then, \(\{b_i\}_{i=1}^\infty\) still generates a free abelian subgroup \(\tilde{\Gamma}\) such that \(K^*=\Lambda \times \tilde{\Gamma}\), and we have \(\mathrm{sign}(\tilde{\Gamma})=1\). We calculate \(\mathrm{H}_p(\Lambda, \mathbb{Z}_\mathrm{sign})\). By the Künneth formula (Theorem 1), we have the following short exact sequence: \[0 \to \bigoplus_{s+t=p} \mathrm{H}_s(\mu) \otimes \mathrm{H}_t(a^\mathbb{Z}, \mathbb{Z}_\mathrm{sign}) \to \mathrm{H}_p(\Lambda,\mathbb{Z}_\mathrm{sign}) \to \bigoplus_{s+t=p-1} \mathrm{Tor}(\mathrm{H}_s(\mu), \mathrm{H}_t(a^\mathbb{Z}, \mathbb{Z}_\mathrm{sign})) \to 0.\] Since \[\begin{align} \mathrm{H}_t(a^\mathbb{Z}, \mathbb{Z}_\mathrm{sign}) \cong \begin{cases} \mathbb{Z}/2\mathbb{Z}&if t=0, \\ 0 &ift>0, \end{cases} \label{eqn:Z46sign46homology} \end{align}\tag{10}\] we have \[\mathrm{H}_p(\Lambda,\mathbb{Z}_\mathrm{sign}) \cong \begin{cases} \mathrm{H}_0(\mu) \otimes \mathbb{Z}/2\mathbb{Z}& if p=0, \\ \mathrm{H}_p(\mu) \otimes \mathbb{Z}/2\mathbb{Z}& if p is odd, \\ \mathrm{Tor}(\mathrm{H}_{p-1}(\mu), \mathbb{Z}/2\mathbb{Z}) & if p is even and p>0. \end{cases}\] Thus, \(\mathrm{H}_p(\Lambda,\mathbb{Z}_\mathrm{sign}) \cong \mathbb{Z}/2\mathbb{Z}\) for all \(p \geq 0\). Since \[\mathrm{H}_n(K^*, \mathbb{Z}_\mathrm{sign}) = \bigoplus_{p+q=n} \mathrm{H}_p(\Lambda, \mathbb{Z}_\mathrm{sign}) \otimes \mathrm{H}_q(K^*/\Lambda)\] by Künneth’s formula (Theorem 1), the claim (ii) is true in this case by Corollary 1. ◻

Proof. By Corollary 1, we have \(\mathrm{H}_n(\mathcal{G}_{R^\times\curvearrowright R}) = 0\) for \(0\leq n<d\) and \[\begin{align} &\mathrm{H}_d(\mathcal{G}_{R^\times\curvearrowright R}) = \mathrm{H}_0(K^*, \mathbb{Z}) = \mathbb{Z}/ \langle 1- \mathrm{sign}(a) \colon a \in K^*\rangle \\ &= \begin{cases} \mathbb{Z}& if K is totally imaginary, \\ \mathbb{Z}/2\mathbb{Z}& if K has a real embedding. \end{cases} \end{align}\] By [18], the groupoid \(\mathcal{G}_{R^\times\curvearrowright R}\) is purely infinite in the sense of [16], so in particular, \(\mathcal{G}_{R^\times\curvearrowright R}\) has comparison. Now the claim follows from Theorem 16 and [15].

By [18], the groupoid \(\mathcal{G}_{R^\times\curvearrowright R}\) is minimal. Since \(\mathcal{G}_{R^\times\curvearrowright R}\) is purely infinite and minimal, the commutator subgroup of \([\![\mathcal{G}_{R^\times\curvearrowright R}]\!]\) is simple by [16], which yields the second claim. ◻

Proof. As explained in the proof of Theorem 17, the groupoid \(\mathcal{G}_{R^\times\curvearrowright R}\) is minimal and has comparison. The statement now follows from Theorem 16 and [15]. ◻

Proof. By the finite dimensionality of a model of \(E\Gamma\), we have \(E^2_{pq} \cong 0\) for \(p > \dim E\Gamma\). Thus, the morphisms \(i_2^{pq} \colon \mathrm{DK}^2_{pq} \to \mathrm{DK}^2_{p+1,q-1}\) and \(i_2^{pq} \colon \widehat{\mathrm{DH}}{}^2_{pq} \to \widehat{\mathrm{DH}}{}^2_{p+1,q-1}\) of the derived exact couples associated to 13 are isomorphisms if \(p \geq \dim X\). Hence, by [50], they converge to \(\mathrm{KK}_{*}^{G } (X \times \underline{E}G , Y)\) and \(\widehat{\mathrm{HH}}{}^{*}_{G} (X \times \underline{E}G , Y)\) respectively. ◻

Proof. In the proof, let the pair \((\mathrm{F}^G, R)\) be either \((\mathrm{KK}_n^G, \mathbb{Z})\) or \((\widehat{\mathrm{HH}}{}_G^n ,\mathbb{C})\). Let \(\mu_{g_0,\cdots,g_k} \colon \Gamma \to \Gamma^{k+1}\) denote the \(\Gamma\)-equivariant map \(g \mapsto (gg_0,g_1,\cdots,g_n)\). Then the map \[\begin{align} \mathrm{F}^{N}(X,Y) \otimes R[\Gamma^{n+1}] \to {}& \mathrm{F}^{G} (X \times \Gamma^{k+1} , Y), \\ \delta_{g_0,\cdots,g_k} \otimes \xi \mapsto {}& \mathrm{F}^G({\rm id}_X \times \mu_{g_0,\cdots,g_k}, Y) (i_{N}^{G} \xi), \end{align}\] factors through the quotient \(\mathrm{F}^{N}(X,Y) \otimes_{R[\Gamma]} R[\Gamma^{k+1}]\) since \[\mathrm{F}^{G}(\mu_{gg_0,g_1,\cdots,g_k} \times {\rm id}_X ,Y) (i_{N}^G\xi) = \mathrm{F}^{G}(\mu_{g_0,g_1,\cdots,g_k} \times {\rm id}_X,Y) \circ \mathrm{F}^{G}(\lambda_g,Y) (i_{N}^G \xi).\] The induced map decomposes to the direct product of \[\mathrm{F}^G(X \times \Gamma , Y) \otimes_{R[\Gamma]} R[\mathrm{Im}\mu_{g_0,\cdots,g_k}] \to \mathrm{F}^G(X \times \mathrm{Im}\mu_{g_0,\cdots,g_k},Y),\] each of which is an isomorphism by definition. Moreover, if \(f \colon \Gamma^{n+1} \to \Gamma^{m+1}\) is an \(\Gamma\)-equivariant map, then \[\mathrm{F}^G({\rm id}_X \times f ,Y ) \circ \mathrm{F}^{G}({\rm id}_X \times \mu_{(g_0,g_1,\cdots,g_n)},Y) (i_{N}^G\xi) = \mathrm{F}^{G}({\rm id}_X \times \mu_{f(g_0,g_1,\cdots,g_n)},Y)(i_{N}^G\xi),\] which means that \(\mathrm{F}^G(f \times {\rm id}_X,Y)\) is identified with \(f_* \otimes_{R[\Gamma]} {\rm id}\). Finally, by Lemma 36, the induced maps are compatible with Raven’s Chern character. For either \(\mathrm{F} = \mathrm{KK}\) and \(R=\mathbb{Z}\) or \(\mathrm{HH}\) and \(R=\mathbb{C}\), the isomorphisms \[\mathrm{F}^G(X \times \Gamma^{n+1},Y) \cong \mathrm{F}^{N} (X \times \Gamma^n, Y) \cong \mathrm{F}^{N} (X, Y) \otimes R[\Gamma^{n}] \cong \mathrm{F}^{N} (X, Y) \otimes_{R[\Gamma]} R[\Gamma^{n+1}]\] follow from Lemma 36 and ?? . The functoriality with respect to an \(\Gamma\)-map follows from ?? . This identification makes the diagram \[\xymatrix{ \mathrm{F}^G(X \times \Gamma^{n+1} , Y) \ar[r] \ar[d]^{r_G^{N}} & \mathrm{F}^{N} (X,Y) \otimes_{R[\Gamma]} R[\Gamma^{n+1}] \\ \mathrm{F}^{N} (X \times \Gamma^{n+1} , Y) \ar[r]^{\cong } & \mathrm{F}^{N}(X,Y) \otimes_R R[\Gamma^{n+1}] \ar@{->>}[u] }\] commutes, where \(r_G^{N}\) denotes the forgetful map of the action of \(\Gamma\). Now, the maps \(\mathrm{F}^G(f,Y)\) and \({\rm id}\otimes f_*\) are identified with respect to this isomorphism since \(\mathrm{F}^{N}(f,Y) = {\rm id}\otimes_R f_*\). ◻

Proof. Let \(K_p\mathrel{\vcenter{:}}= \{ (t_0,\cdots, t_p) \in \Delta_p : t_i \geq 1/(2p+2) \text{ for any i} \} \cong \Delta_p\) and let \(U_p \mathrel{\vcenter{:}}= E_{p}\Gamma \setminus (\Gamma^{p+1} \times K_p)\). By 12 , we have that \(\overline{U}_p\) is homotopy equivalent to \(E_{p-1}\Gamma\). By the excision axiom for the pair \((E_p\Gamma \times \underline{E}G , \overline{U}_p \times \underline{E}G)\) and an open subset \(U_p \times \underline{E}G \subset \overline{U}_p \times \underline{E}G\), we have that \[\begin{align} \begin{aligned} \widehat{\mathrm{EH}}{}^1_{pq} \cong {}& \widehat{\mathrm{HH}}{}^{-p-q}_{G} (\underline{E}G \times \Gamma^{p+1} \times (K_p, \partial K_p), Y )\\ \cong {}& \widehat{\mathrm{HH}}{}^{-q}_{N \rtimes \Gamma} (\underline{E}G \times \Gamma^{p+1}, Y )\\ \cong {}& \widehat{\mathrm{HH}}{}^{-q}_{N} (\underline{E}G, Y ) \otimes_{\mathbb{C}[\Gamma]} \mathbb{C}[\Gamma^{p+1}] . \end{aligned}\label{eqn:E1page46tensor} \end{align}\tag{15}\] The derivation \(d^1_{pq} \colon \widehat{\mathrm{EH}}{}^1_{p,q} \to \widehat{\mathrm{EH}}{}^1_{p-1,q}\) is the boundary map \(\delta\) of the long exact sequence for the triple \((E_{p}\Gamma,E_{p-1}\Gamma, E_{p-2}\Gamma)\) (see e.g. [51]). By definition of \(E_n\Gamma\), the following diagram commutes; \[\xymatrix@C=8ex{ \widehat{\mathrm{HH}}{}^{-p-q}_{G } (\underline{E}G \times (E_p\Gamma , E_{p-1}\Gamma) , Y) \ar[r]^\delta \ar[d]^\cong & \widehat{\mathrm{HH}}{}^{-p-q+1}_{G } (\underline{E}G \times (E_{p-1}\Gamma , E_{p-2}\Gamma) , Y) \ar[d]^\cong \\ \widehat{\mathrm{HH}}{}^{-q}_{G } (\underline{E}G \times \Gamma^{p+1}, Y) \ar[r]^{\sum_i (-1)^i (({d}_i \times {\rm id}_{\underline{E}G})_*) } \ar[d]^\cong & \widehat{\mathrm{HH}}{}^{-q}_{G } (\underline{E}G \times \Gamma^{p}, Y) \ar[d]^\cong \\ \widehat{\mathrm{HH}}{}^{-q}_{N} (\underline{E}G, Y) \otimes_{\mathbb{C}[\Gamma]} \mathbb{C}[\Gamma^{p+1}] \ar[r]^{\sum_i (-1)^i ({d}_i)_* \otimes {\rm id}) } & \widehat{\mathrm{HH}}{}^{-q}_{N} (\underline{E}G, Y) \otimes_{\mathbb{C}[\Gamma]} \mathbb{C}[\Gamma^{p+1}] . }\] Indeed, by a standard argument in cellular homology theory (see for example [52]), the restriction of the boundary map to the each component, namely \[\begin{align} \delta \colon \widehat{\mathrm{HH}}{}_G^{-p-q}(\underline{E}G \times{}&{} (\Gamma \cdot (g_0,\cdots,g_n)) \times (\Delta_{p},\partial \Delta_p), Y) \\ {}&{}\to \widehat{\mathrm{HH}}{}_G^{-p-q+1}(\underline{E}G \times (\Gamma \cdot (g_0,\cdots,\hat{g_i},\cdots,g_n)) \times (\Delta_{p-1}, \partial \Delta_{p-1}), Y), \end{align}\] is given by \((-1)^i\) under a canonical identification of the domain and the range with \(\widehat{\mathrm{HH}}{}_G^{-q}(\underline{E}G,Y)\), which verifies the commutativity of the top square. The horizontal map below is nothing else than the differential of the group chain complex with coefficient in \(\widehat{\mathrm{HH}}{}^{-q}_{N} (\underline{E}G, Y)\). ◻

Proof. Recall Remark 21, in which it is proved that the action \(\mathscr{S} \curvearrowright \mathrm{HH}_{\mathscr{A}}^{*}(\underline{E}(\mathscr{A} \times \mathscr{S}) , \overline{\mathscr{A}})\) given in Lemma 14 agrees with that on \(\mathrm{H}_*(\mathscr{A} \rtimes \overline{\mathscr{A}} ;\mathbb{C})\) considered in Theorem 9, under the identification \(\mathrm{H}_*(\mathscr{A} \rtimes \overline{\mathscr{A}} ;\mathbb{C}) \cong \mathrm{H}_*(\mathscr{A} ; \mathbb{C}\overline{\mathscr{A}})\) (Proposition 2). By Proposition 15 and Lemma 15, the spectral sequence \(\widehat{\mathrm{EH}}{}_{pq}^r\) collapses at the \(E^2\)-page. Since Raven’s Chern character 14 induces a morphism of spectral sequence, a diagram chasing argument shows that this collapsing inherits to \(\mathrm{EK}_{pq}^r \otimes \mathbb{C}\). ◻

Proof. First, by \(\mathscr{A} + \overline{A} = \overline{\mathscr{A}}\) and \(\mathscr{A} \cap \overline{A} =A\) (Lemmas 8 and 9), we obtain an isomorphism of \(\mathscr{A}\)-spaces \(\mathscr{A} \times_A \overline{A} \to \overline{\mathscr{A}}\) given by the addition. Now we can apply Lemma 37 to get the commutative diagram \[\xymatrix{ \mathrm{KK}_q^{\mathscr{A}}(\underline{E}\mathscr{A},\overline{\mathscr{A}}) \ar[r]^{\mathrm{ch}_R^e} \ar[d]^\cong_{\text{Lemma \ref{lem:induction462}}} & \bigoplus_k \mathrm{HH}_{\mathscr{A}}^{-q+2k}(\underline{E}\mathscr{A}, \overline{\mathscr{A}}) \ar[d]^\cong_{\text{Lemma \ref{lem:induction462}}} \\ \mathrm{KK}_q^{A}(\underline{E}\mathscr{A},\overline{A}) \ar[r]^{\mathrm{ch}_R^e} & \bigoplus_k \mathrm{HH}_{A}^{-q+2k}(\underline{E}\mathscr{A},\overline{A}) . }\] Since \(\underline{E}\mathscr{A}\) models \(\underline{E}A\) as \(A\)-space (Remark 19 (iv)), this identifies Raven’s Chern character under consideration with \(\mathop{\mathrm{ch}_R^e} \colon \mathrm{KK}^{A}_q(\underline{E}A,\overline{A}) \to \bigoplus_{k \in \mathbb{Z}}\mathrm{HH}_{A}^{-q+2k}(\underline{E}A,\overline{A})\). Note that \(\mathbb{R}^d\) models \(\underline{E}A\) for \(A\cong \mathbb{Z}^d\). Next, by the commutativity of the diagrams \[\begin{gather} \xymatrix{ \varinjlim_s \mathrm{KK}^A_*(\underline{E}A,A/\sigma_s(A)) \ar[r]^\cong \ar[d] & \varinjlim_s \mathrm{K}_*(C(A/\sigma_s(A)) \rtimes A) \ar[d]^\cong \\ \mathrm{KK}^A_*(\underline{E}A,\overline{A}) \ar[r]^\cong & \mathrm{K}_*(C(\overline{A}) \rtimes A) } \\ \xymatrix{ \varinjlim_s \mathrm{KK}_*^A(\underline{E}A,A/\sigma_s(A)) \ar[r]^{\mathrm{ch}_R^e} \ar[d] & \varinjlim_s \bigoplus_{k}\mathrm{HH}^{-*+2k}_A(\underline{E}A,A/\sigma_s(A)) \ar[d] \\ \mathrm{KK}_*^A(\underline{E}A,\overline{A}) \ar[r]^{\mathrm{ch}_R^e} &\bigoplus_k \mathrm{HH}^{-*+2k}_A(\underline{E}A,\overline{A}), } \\ \xymatrix{ \varinjlim_s \mathrm{HH}^{-*}_A(\underline{E}A,A/\sigma_s(A)) \ar[r]^\cong \ar[d] & \varinjlim_s \mathrm{H}_*(A,\mathbb{C}[A/\sigma_s(A)]) \ar[d]^\cong \\ \mathrm{HH}^{-*}_A(\underline{E}A,\overline{A}) \ar[r]^\cong & \mathrm{H}_*(A,\mathbb{C}[\overline{A}]), } \end{gather}\] each of which is given in [53], Remark 44, and Remark 21 respectively, it turns out that Raven’s Chern character is identified with the inductive limit of \[\mathop{\mathrm{ch}_R^e} \colon \mathrm{KK}^{A}_q(\underline{E}A,A/\sigma_s(A)) \to \bigoplus_{k \in \mathbb{Z}} \mathrm{HH}_{A}^{-q+2k}(\underline{E}A,A/\sigma_s(A)).\] Again by Lemma 37, there is a commutative diagram \[\xymatrix{ \mathrm{KK}^{A}_*(\underline{E}A,A/\sigma_s(A)) \ar[r]^{\mathrm{ch}^e_R} \ar[d]^\cong_{\text{Lemma \ref{lem:induction462}}} & \bigoplus_k \mathrm{HH}^{*+2k}_A(\underline{E}A, A/\sigma_s(A)) \ar[d]^\cong_{\text{Lemma \ref{lem:induction462}}} &\\ \mathrm{KK}^{\sigma_s(A)}_*(\underline{E}A, \mathrm{pt}) \ar[r]^{\mathrm{ch}^e_R} & \bigoplus_k \mathrm{HH}^{*+2k}_{\sigma_s(A)}(\underline{E}A, \mathrm{pt}) . }\] The above discussion, and the fact that \(\underline{E}A\) models \(\underline{E}\sigma_s(A)\), reduces the problem to proving that \[\mathrm{Im}\Big( \mathop{\mathrm{ch}_R^e} \colon \mathrm{KK}^{N}_*(\underline{E}N,\mathrm{pt}) \to \bigoplus_k \mathrm{HH}_{N}^{*+2k}(\underline{E}N , \mathrm{pt})\Big) = \bigoplus_k \mathrm{H}_{-*+2k}(N , \mathbb{Z})\] via the isomorphism 11 when \(N=\mathbb{Z}^d\). To see this, as in Remark 21, consider the spectral sequence 13 , for \(\Gamma = N\) and trivial normal subgroup, for both KK- and HH-theories, and compare them via Raven’s Chern character. Then, by Lemma 14, the image of \(\mathop{\mathrm{ch}_R^e} \colon \mathrm{EK}{}^1_{p0} \to \widehat{\mathrm{EH}}{}_{p0}^1\) is identified with \[\mathbb{Z} \otimes_{\mathbb{Z}[N]}\mathbb{Z}[N^{p+1}] \subset \mathbb{C} \otimes_{\mathbb{C}[N]} \mathbb{C}[N^{p+1}],\] which agrees with the standard chain complex \(C_{\bullet}(N, \mathbb{Z})\) computing the group homology of \(N\). By Lemma 15, which states that the \(d^1\)-differential also agrees with the standard bar differential, we also obtain \(\mathrm{Im}(\mathop{\mathrm{ch}_R^e} \colon \mathrm{EK}{}^2_{p0} \to \widehat{\mathrm{EH}}{}^2_{p0}) = \mathrm{H}_p(N,\mathbb{Z})\) as subgroups of \(\widehat{\mathrm{EH}}{}^2_{p0} \cong \mathrm{H}_p(N,\mathbb{C})\). Since both \(\mathrm{EK}{}^1_{p0}\) and \(\widehat{\mathrm{EH}}{}_{p0}^1\) concentrates on the \(q=0\) row, these spectral sequences collapse at the \(E^2\) page and no extension issues arise. Finally, we obtain that \[\mathrm{Im}(\mathop{\mathrm{ch}_R^e} \colon \mathrm{EK}{}^2_{p0} \to \widehat{\mathrm{EH}}{}^2_{p0}) = \mathrm{Im}(\mathop{\mathrm{ch}_R^e} \colon \mathrm{EK}{}^\infty_{p0} \to \widehat{\mathrm{EH}}{}^\infty_{p0}) = \mathrm{Im}\Big( \mathop{\mathrm{ch}_R^e} \colon \mathrm{KK}{}^{N}_p(\underline{E}N,\mathrm{pt}) \to \bigoplus_k\widehat{\mathrm{HH}}{}^{-p+2k}_{N}(\underline{E}N,\mathrm{pt})\Big)\] is canonically identified with \(\bigoplus_k \mathrm{H}_{p+2k}(N,\mathbb{Z})\) as desired. ◻

Proof. By Lemma 17 and Proposition 15, the \(E^2\)-page \(\mathrm{EK}_{pq}^2\) is determined as \[\begin{align} \mathrm{EK}^2_{pq} \cong{}& \mathrm{H}_p(\mathscr{S} ; \mathrm{K}_{q}(C_0(\overline{\mathscr{A}}) \rtimes \mathscr{A})) \cong \mathrm{H}_p\Big( \mathscr{S} ; \bigoplus_{k \in \mathbb{Z}}\mathrm{H}_{q+2k}(\mathscr{A} \ltimes \overline{\mathscr{A}} ;\mathbb{Z}) \Big)\\ \cong {}& \begin{cases} \mathrm{H}_p(\mathscr{S}; \mathbb{Z}_{\mathrm{sign}}) & \text{ if q-d is even,}\\ 0 & \text{ if q-d is odd.} \end{cases} \end{align}\] By the assumption of \(\mathrm{sign}(S) = \{+1\}\), the group \(\mathscr{S}\) acts on the coefficient \(\mathbb{Z}_\mathrm{sign}\) trivially, and hence the right hand side becomes the free abelian group \(\mathrm{H}_{p}(\mathscr{S};\mathbb{Z}) \cong {\textstyle\bigwedge}^p \mathscr{S}\). This implies that the map of spectral sequences \(\mathrm{EK}^2_{pq} \to \mathrm{EK}^2_{pq} \otimes \mathbb{C}\) is injective. Since the spectral sequence \(\mathrm{EK}^r_{pq} \otimes \mathbb{C}\) collapses at the \(E^2\)-page by Theorem 16, an iterated diagram chase argument shows that \(d^r_{pq} =0\) for \(\mathrm{EK}^r_{pq}\). Moreover, since each component \(\mathrm{EK}_{pq}^2\) is a free \(\mathbb{Z}\)-module, there is no extension problem. This shows the desired isomorphism \[\mathrm{K}_n(C^*_r(\mathscr{A} \rtimes \mathscr{S}) \ltimes \overline{\mathscr{A}}) \cong \bigoplus_{p+q=n} \mathrm{EK}_{pq}^2 \cong \bigoplus_{k \in \mathbb{Z}}{\textstyle\bigwedge}^{n-d+2k} \mathscr{S}. \qedhere\] ◻

Proof of Theorem 23. The remaining part is the same as [22]. The case of \(\mathrm{sign}(S) =\{ +1 \}\) is already proved in Lemma 18. In the case of \(\mathrm{sign}(S) =\{ \pm 1 \}\), pick a basis \(\{ a_1, a_2, \cdots, \}\) of \(\mathscr{S}\) in the way that \(\mathrm{sign}(a_1) = -1\) and \(\mathscr{S}^\perp\mathrel{\vcenter{:}}= \langle a_2, a_3, \cdots \rangle \subset \mathscr{S}\). Set \(\mathscr{S}'\mathrel{\vcenter{:}}= \langle a_1^2, a_2,\cdots \rangle\). Apply Lemma 18 to the dynamical system \((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}}\) and\((\mathscr{A} \rtimes \mathscr{S}') \ltimes \overline{\mathscr{A}}\). Then we obtain that \[\begin{align} \mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}})) &{} \cong \bigoplus_{k\in \mathbb{Z}}\mathrm{H}_{n-d+2k}(\mathscr{S}^\perp ; \mathbb{Z}), \\ \mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}') \ltimes \overline{\mathscr{A}})) &{} \cong \bigoplus_{k \in \mathbb{Z}}\mathrm{H}_{n-d+2k}(\mathscr{S}' ; \mathbb{Z}). \end{align}\] In particular, since the morphism \(\bigoplus_{k\in \mathbb{Z}}\mathrm{H}_{n-d+2k}(\mathscr{S}^\perp ; \mathbb{Z}) \to \bigoplus_{k\in \mathbb{Z}}\mathrm{H}_{n-d+2k}(\mathscr{S}' ; \mathbb{Z})\) is injective, the morphism \(\mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}})) \to \mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}') \ltimes \overline{\mathscr{A}}))\) induced from the inclusion is injective (recall that a filtered map inducing the injective map of the subquotients is itself injective). Now, the Pimsner–Voiculescu exact sequence for the restriction of \(\alpha \colon a_1^\mathbb{Z}\curvearrowright C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}}))\) to \(a_1^{2 \mathbb{Z}}\) is \[\xymatrix{ \mathrm{K}_0(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}})) \ar[r]^{{\rm id}- \alpha^2_* } & \mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}})) \ar@{^{(}->}[r] & \mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}') \ltimes \overline{\mathscr{A}})) \ar[d] \\ \mathrm{K}_1(C^*_r((\mathscr{A} \rtimes \mathscr{S}') \ltimes \overline{\mathscr{A}})) \ar[u] & \mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp)) \ltimes \overline{\mathscr{A}}) \ar@{_{(}->}[l] & \mathrm{K}_n(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp)) \ltimes \overline{\mathscr{A}}) \ar[l]_{{\rm id}-\alpha^2_*}. }\] Hence \({\rm id}- \alpha^2_* =0\), that is, \(a_1^2\) acts on \(\mathrm{K}_*(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}}))\) by the identity. In other words, the action of \(a_1^\mathbb{Z}\) on \(\mathrm{K}_*(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}}))\) factors through \(\mathbb{Z}/2\mathbb{Z}\). Moreover, the isomorphism \[\mathrm{Gr}(\mathrm{K}_*(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}})) \cong E_{pq}^\infty,\] where the left hand side is the graded module associated to the filtration of \(\mathrm{K}_*(C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}}) \cong \mathrm{KK}_*^{\mathscr{A} \rtimes \mathscr{S}^\perp}(\underline{E}(\mathscr{A} \rtimes \mathscr{S}^\perp), \overline{\mathscr{A}})\) coming from the exact couple 13 , is equivariant under the \(a_1^{\mathbb{Z}/2 \mathbb{Z}}\)-action, and the right hand side is the direct sum of copies of \(\mathbb{Z}_{\mathrm{sign}}\) as \(\mathbb{Z}[\mathbb{Z}/2\mathbb{Z}]\)-modules. Since any extension of \(\mathbb{Z}_{\mathrm{sign}}\) by \(\mathbb{Z}_{\mathrm{sign}}\) is trivial, i.e., \(\mathop{\mathrm{Ext}}^1_{\mathbb{Z}[\mathbb{Z}/2\mathbb{Z}]}(\mathbb{Z}_{\mathrm{sign}},\mathbb{Z}_{\mathrm{sign}}) =0\), we obtain that \(\alpha_* =-{\rm id}\). Finally, the Pimsner–Voiculescu exact sequence for \(\alpha \colon a_1^\mathbb{Z}\curvearrowright C^*_r((\mathscr{A} \rtimes \mathscr{S}^\perp) \ltimes \overline{\mathscr{A}}))\) shows the desired isomorphism. ◻

Proof. By the right equation of 11 , we have the isomorphism \[\widehat{\mathrm{HH}}{}_{G}^{*} (\underline{E}G, \overline{\mathscr{A}} ) \cong \mathrm{H}_{-*} \Big( G , \mathbb{C}\Big[ \widehat{\overline{\mathscr{A}}} \Big] \Big) \cong \bigoplus_{[\gamma] \in \mathcal{C}(G_{\mathrm{tor}})} \mathrm{H}_{-*}(Z(\gamma), \mathbb{C}[\overline{\mathscr{A}}{}^\gamma]),\] where \(\widehat{\overline{\mathscr{A}}} \mathrel{\vcenter{:}}= \{ (g,a) \in G_{\mathrm{tor}} \times \overline{\mathscr{A} } : ga=a \}\). In particular, the ‘localized’ component corresponding to \([e] \in \mathcal{C}(G_{\mathrm{tor}})\) is isomorphic to the groupoid homology \(\mathrm{H}_{-*}(G , \mathbb{C}[\overline{\mathscr{A}}]) \cong \mathrm{H}_{-*}(\overline{\mathscr{A}} \rtimes G ; \mathbb{C})\).

Any torsion element \(g = a \cdot s\) in \(\mathscr{A} \rtimes \mu\) or \(\mathscr{A} \rtimes \mathscr{S}\) is conjugate to some \(\zeta \in \mu\) since the element \(\zeta = (-b) \cdot g \cdot b\), where \(b = a(1-s)^{-1} \in \mathscr{A}\), fixes \(0 \in \mathscr{A}\). The centralizer subgroup \(Z(\zeta)\) of \(\zeta \in \mu \setminus \{e\}\) is \(\mu\) since a multipricative group \(\mu\) of an integral domain acts on \(A \setminus \{0\}\) freely. By the same reason, the fixed point set of the action of \(\zeta\) on \(\overline{\mathscr{A}}\) is \(\overline{\mathscr{A}}{}^\zeta = \{ 0 \}\). In conclusion, we have \[\begin{align} \bigoplus_{[\gamma] \in \mathcal{C}(G_{\mathrm{tor}}) \setminus \{ [e] \}} \mathrm{H}_{-*}(Z(\gamma), \mathbb{C}[\overline{\mathscr{A}}{}^\gamma]) ={}&{} \bigoplus_{\zeta \in \mu \setminus \{e\}} \mathrm{H}_*(Z(\zeta) , \mathbb{C}[\overline{\mathscr{A}}{}^\zeta] ) \cong \bigoplus_{\zeta \in \mu \setminus \{ e\} } \mathrm{H}_*(\mu, \mathbb{C})\\ \cong {}&{} \begin{cases} \bigoplus_{\zeta \in \mu \setminus \{e\}}\mathbb{C} & \text{ if *=0, } \\ 0 & \text{otherwise.} \end{cases} \end{align}\] This finishes the proof. ◻

Proof. The LHS spectral sequence for the semidirect product \((\mathscr{A} \ltimes \overline{\mathscr{A}} ) \rtimes \mu\) converges to \(\mathrm{H}_n((\mathscr{A} \rtimes \mu ) \ltimes \overline{\mathscr{A}} ; \mathbb{Z})\) and its \(E^2\)-page is \[E^2_{pq} \cong \mathrm{H}_p(\mu, \mathrm{H}_q(\mathscr{A} \ltimes \overline{\mathscr{A}} ;\mathbb{C} )) \cong \begin{cases} \mathrm{H}_q(\mathscr{A} \ltimes \overline{\mathscr{A}} ;\mathbb{C} )_\mu & \text{ if p=0}, \\ 0 & \text{ otherwise.} \end{cases}\] This, together with Theorem 14, shows the lemma. ◻

Proof. By Lemmas 15, 19, 20, and Remark 25, the \(E^2\)-page of this spectral sequence is \[\begin{align} \widehat{\mathrm{EH}}{}^2_{pq} \cong \mathrm{H}_p(\Gamma, \widehat{\mathrm{HH}}{}^{-q}_{\mathscr{A} \rtimes \mu}(\underline{E}G , \overline{\mathscr{A}})) \cong {}&{} \mathrm{H}_p (\Gamma, \mathrm{H}_{q}(\overline{\mathscr{A}} \rtimes (\mu \ltimes \mathscr{A}), \mathbb{C})) \oplus \begin{cases} \mathrm{H}_{p}(\Gamma , \mathbb{C}[\mu \setminus \{e\}]) & \text{ if q=0, } \\ 0 & \text{ otherwise}. \end{cases} \\ \cong {}&{}\mathrm{H}_p (\Gamma, ({\textstyle\bigwedge}_{\mathbb{C}}^q (A \otimes \mathbb{C}))_\mu ) \oplus \begin{cases} \mathbb{C}[\mu \setminus \{e\}] \otimes {\textstyle\bigwedge}_{\mathbb{C}}^p (\Gamma \otimes \mathbb{C}) & \text{ if q=0, } \\ 0 & \text{ otherwise}. \end{cases} \end{align}\] The first direct summand is the complexification of the calculation in Proposition 15, and hence is isomorphic to \(\mathrm{H}_p(\Gamma , \mathbb{C}_{\mathrm{sign}})\) if \(q=d\) and to \(0\) otherwise. If \(\mathrm{sign} \colon \Gamma \to \{ \pm 1\}\) is non-trivial, then \(\mathrm{H}_p(\Gamma , \mathbb{C}_{\mathrm{sign}}) \cong 0\). Indeed, by decomposing \(\Gamma\) to \(s^{\mathbb{Z}} \oplus \Gamma^\perp\) in the way that \(\mathrm{sign}|_{\Gamma^\perp}\), we obtain the Künneth isomorphism \[\mathrm{H}_p(\Gamma , \mathbb{C}_{\mathrm{sign}}) \cong \bigoplus_{k}\mathrm{H}_k(s^{\mathbb{Z}} , \mathbb{C}_{\mathrm{sign}}) \otimes \mathrm{H}_{p-k}(\Gamma^\perp, \mathbb{C}) \cong 0,\] where the last isomorphism follows from \(\mathrm{H}_*(s^{\mathbb{Z}} , \mathbb{C}_{\mathrm{sign}}) \cong 0\) that is verified in 10 . In summary, the first direct summand can be non-trivial only if \(q=d\) and \(\mathrm{sign} \colon \Gamma \to \{ \pm 1\}\) is trivial, in which case it is isomorphic to \(\mathrm{H}_p(\Gamma , \mathbb{C})\). Finally, since the exact couple \((\widehat{\mathrm{DH}}, \widehat{\mathrm{EH}})\) respects the direct sum decomposition over \(\mathcal{C}(G_{\mathrm{tor}})\), no non-trivial higher differentials are possible. ◻

Proof. As is considered in [54], the induced morphism in K-theory by the inclusion \(k \colon C^*_r(A \ltimes \overline{A} ) \to C^*_r((A \rtimes \mu) \ltimes \overline{A})\) factors through the coinvariant as \[\bar{k}_* \colon \mathrm{K}_0 (C^*_r(A \ltimes \overline{A}))_\mu \to \mathrm{K}_0 (C^*_r((A \rtimes \mu) \ltimes \overline{A})).\] Indeed, for any \([p] \in \mathrm{K}_0(C^*_r(A \ltimes \overline{A}))\) and \(s \in \mu\), we have \(k_*[p]=k_*(s_*[p])\) since the two projections \(s \cdot p\) and \(p\) are conjugate by the unitary \(u_s\) in \(C^*_r((A \rtimes \mu) \ltimes \overline{A})\). By Morita invariance of C*-algebra K-theory, 11 , Lemma 17, and Theorem 14, we have \[\mathrm{K}_0(C^*_r(A \ltimes \overline{A})) \cong\mathrm{K}_0(C^*_r(\mathscr{A} \ltimes \overline{\mathscr{A}})) \cong \bigoplus_{k \in \mathbb{Z}} \mathrm{H}_{2k}(\mathscr{A}, \mathbb{Z}\overline{\mathscr{A}}) \cong \bigoplus_{0 \leq 2k <d}{\textstyle\bigwedge}^{2k}_{\mathbb{Q}}(\mathscr{A} \otimes_{\mathbb{Z}} \mathbb{Q} ) \oplus \begin{cases} {\textstyle\bigwedge}^d_{\mathbb{Z}} A & \text{ if d is even,}\\ 0 & \text{ if d is odd.} \end{cases}\] By Remark 21, 22, and compatibility of the \(\mathscr{S}\)-actions via \(\widehat{\operatorname{ch}}_R\) (a remark above Lemma 14), the above isomorphism is \(\mathscr{S}\)-equivariant, and the \(\mathscr{S}\)-action on the right group is given in Theorem 14.

An element of the first direct summand \(\bigoplus_{0 \leq 2k <d}{\textstyle\bigwedge}^{2k}_{\mathbb{Q}}(A \otimes_{\mathbb{Z}} \mathbb{Q} )\), which is divisible, is sent to \(K_{\mathrm{div}}\) by \(\bar{k}_*\). By definition of \(K_{\mathrm{div}}\), it is surjective. It is proved in [23] that it is also injective after taking coinvariant. In conclusion, we have \(K_{\mathrm{div}} \cong \bigoplus_{0 \leq 2k <d}{\textstyle\bigwedge}^{2k}_{\mathbb{Q}}(A \otimes_{\mathbb{Z}} \mathbb{Q} )_\mu\), not only as group but also as \(\Gamma\)-modules. Now, by (the proof of) Proposition 15, we obtain \(\mathrm{H}^p(\Gamma,\mathrm{K}_{\mathrm{div}}) \cong 0\). The long exact sequence of group cohomology ([33]) yields the second claim. ◻

Proof. Note that, since \(|V_{K,\mathbb{R}}|\) is assumed to be even, we have that \(\mathrm{sign}(\mu) = \{ +1\}\) (discussed in the proof of Theorem 16), and \(d=[K:\mathbb{Q}] = |V_{K,\mathbb{R}}| +2|V_{K,\mathbb{C}}|\) is even. Let us fix a basis \(A = \langle a_1,\cdots, a_d\rangle\) and identify \(A\) with \(\mathbb{Z}^d\). Take a \(\mu\)-invariant inner product \(\langle \cdot, \cdot \rangle \colon \mathbb{Z}^d \times \mathbb{Z}^d \to \mathbb{R}\) and let \((g^{ij})_{1 \leq i,j \leq d}\) denote the inverse matrix of \((\langle a_i,a_j \rangle)_{i,j}\). Consider unbounded self-adjoint operators \[X_j \colon \ell^2\mathbb{Z}^d \to \ell^2 \mathbb{Z}^d, \quad (X_j\varphi)(x_1,\cdots,x_d) = x_j \varphi(x_1,\cdots,x_d).\] Passing through the Fourier transform \(\ell^2A \to L^2\hat{A}\), each \(X_j\) is identified with \[i \frac{\partial}{ \partial \xi_j} \colon L^2(\mathbb{T}^d) \to L^2(\mathbb{T}^d),\] where \(\hat{A}\cong\mathbb{T}^d\mathrel{\vcenter{:}}=\mathbb{R}^d/(2\pi\mathbb{Z})^d\) via the basis \({a_1,\dots,a_d}\), and \(\xi_1,\dots,\xi_d\) denote the standard coordinates on \(\mathbb{T}^d\). Note that the inner product \(\langle \cdot,\cdot \rangle\) gives \(\mu\)-equivariant identifications \[\hat{A} \cong {\rm Hom}\,(A,\mathbb{R})/{\rm Hom}\,(A,2 \pi \mathbb{Z}) \cong A_{\mathbb{R}} / \check{A}, \quad \text{where} \; A_{\mathbb{R}} \mathrel{\vcenter{:}}= A \otimes_{\mathbb{Z}} \mathbb{R}, \;\check{A} \mathrel{\vcenter{:}}= \{ b \in A_{\mathbb{R}} \mid \langle b,A\rangle \subset 2\pi \mathbb{Z} \}.\] The Riemannian metric on \(A_{\mathbb{R}}\) given by \(\langle \cdot, \cdot \rangle\) induces that on \(\hat{A}\), which is preserved by the \(\mu\)-action. Since \(\mathrm{sign}(\mu)=1\), this \(\mu\)-action does not change the orientation of \(\hat{A}\cong \mathbb{T}^d\).

Let \(\mathbb{C}\ell_A\) be the Clifford algebra, i.e., the C*-algebra generated by elements \(\mathfrak{c}(a)\) for \(a \in A\) with the relations \(\mathfrak{c}(a) \mathfrak{c}(b) + \mathfrak{c}(b) \mathfrak{c}(a) =-2 \langle a,b \rangle\) and \(\mathfrak{c}(a)^* = - \mathfrak{c}(a)\) for any \(a,b \in A\). Since \(d\) is even, there is a \(2^{d/2}\)-dimensional Hilbert space \(S_A\) such that \(\mathbb{C}\ell_A \cong {\rm End}\,(S_A)\). This \(S\) is equipped with the \(\mathbb{Z}/2\)-grading so that \(\mathfrak{c}(a)\) acts by an odd operator. The \(\mu\)-action on \(\mathbb{C}\ell_A\) given by \(\mu \cdot \mathfrak{c}(a) \mathrel{\vcenter{:}}= \mathfrak{c}(\mu \cdot a)\) is a priori implemented by a projective representation \(\lambda \colon \mu \to \mathcal{U}(S_A)/\mathbb{T}\). Since \(\mathrm{sign}(\mu)=\{+1\}\), the \(\mu\)-action on \(A_{\mathbb{R}}\) preserves the orientation, and hence \(\lambda(\mu)\) preserves \(\mathbb{Z}/2\)-grading. Moreover, since \(\mu\) is a finite cyclic group, and hence satisfies \(\mathrm{H}^2(\mu , \mathbb{T}) \cong 0\), this \(\lambda\) lifts to a genuine unitary representation preserving the \(\mathbb{Z}/2\)-grading.

The Dirac operator on \(\hat{A} \cong \mathbb{T}^d\) with respect to the above Riemannian metric is an odd \(\mu\)-invariant unbounded self-adjoint operator defined by \[D \mathrel{\vcenter{:}}= \sum_{j,k =1}^d g^{jk} \cdot \mathfrak{c}(a_j) \cdot \frac{\partial}{\partial \xi_k} \colon L^2(\mathbb{T}^d, S_A) \to L^2(\mathbb{T}^d, S_A).\] This is an elliptic operator on a compact manifold, and hence is a Fredholm operator. Going back to the Fourier dual \(\ell^2A \otimes \mathbb{C}\ell_A\), this \(D\) corresponds to the odd self-adjoint operator \[C \mathrel{\vcenter{:}}= \sum_{j,k=1}^dg^{jk} \cdot \mathfrak{c}(a_j) \cdot iX_k \colon \ell^2(\mathbb{Z}^d ,S_A) \to \ell^2(\mathbb{Z}^d , S_A).\] We consider the \(\ast\)-representation \(\pi \colon C(\overline{A}) \rtimes A \to B(\ell^2A)\) arising from the covariant pair consisting of the multiplication representation of \(C(\overline{A}) \subset C_b(A)\) on \(\ell^2A\) and the left regular representation of \(A\) on \(\ell^2A\). Then, the bounded transform \(C(1+C^2)^{-1/2}\) commutes with \(C^*_r(A)\) modulo compact operators, and commutes with \(C(\overline{A})\). That is, we obtain an element of Kasparov’s KK-group \[\alpha \mathrel{\vcenter{:}}= [\ell^2(A, S_A), \pi, C(1+C^2)^{-1/2}] \in \mathrm{KK}^\mu (C(\overline{A}) \rtimes A, \mathbb{C}).\] The composition with \(\iota\) in the statement of the lemma is \[[\iota] \otimes_{C(\overline{A}) \rtimes A } \alpha = [L^2(U,S_A), m \circ \iota, PD(1+D^2)^{-1/2}P] \in \mathrm{KK}^\mu(C_0(U),\mathbb{C}),\] where \(m \colon C(\hat{A}) \to B(L^2(\hat{A}))\) denotes the multiplication representation and \(P\) denotes the projection onto \(L^2(U,S_A)\). By [55], this KK-class is identical with the one aligned with the Dirac operator on \(U\), which is nothing else than the equivariant Bott class on \(U \cong \mathbb{R}^d\). Hence, by Kasparov’s equivariant Bott periodicity theorem ([56]), it is a \(\mu\)-equivariant KK-equivalence. Applying Kasparov’s descent functor ([26]), which we write \(\cdot \rtimes \mu\) in this paper, we obtain non-equivariant KK-classes \[\begin{gather} \alpha \rtimes \mu \in \mathrm{KK}((C(\overline{A}) \rtimes A) \rtimes \mu, \mathbb{C} \rtimes \mu ) = \mathrm{KK}(C(\overline{A}) \rtimes (A \rtimes \mu) , C^*_r(\mu )),\\ [\iota] \rtimes \mu = [\iota \rtimes \mu] \in \mathrm{KK}(C_0(U) \rtimes \mu, (C(\overline{A}) \rtimes A) \rtimes \mu ) . \end{gather}\] By the compatibility of the descent with the Kasparov product, we have that \[([\iota] \rtimes \mu ) \otimes_{(C(\overline{A}) \rtimes A) \rtimes \mu}(\alpha \rtimes \mu) = ([\iota] \otimes_{C(\overline{A}) \rtimes A} \alpha) \rtimes \mu\] is a KK-equivalence, with the KK-inverse \(\beta \rtimes \mu\), where \(\beta \in \mathrm{KK}^\mu(\mathbb{C},C_0(U))\) is the \(\mathrm{KK}^\mu\)-inverse of \([\iota] \otimes_{C(\overline{A}) \rtimes A} \alpha\). Now, the Kasparov product with \((\alpha \otimes_{\mathbb{C}} \beta) \rtimes \mu\) induces a homomorphism \(\mathrm{K}_0(C(\overline{A}) \rtimes (A \rtimes \mu)) \to \mathrm{K}_0( C_0(U) \rtimes \mu)\), which is a right inverse of \((\iota \rtimes \mu)_*\). ◻

Proof. Let \(s \in \Gamma\). We prove that \(\theta_s\) preserves the subgroup \(\mathrm{Im}(\iota \rtimes \mu)_*\), and determine the action. Following Lemma 23, we fix a \(\mu\)-invariant inner product on \(A\), identify the Pontrjagin dual \(\hat{A}\) with \(A_{\mathbb{R}}/\check{A}\), and choose an open ball \(U =B_r(0) \subset A_{\mathbb{R}}\) with respect to the metric \(\langle \cdot ,\cdot \rangle\) centered at the origin, which is automatically \(\mu\)-invariant. The radius \(r>0\) is chosen sufficiently small so that the restriction of the quotient map, \(U \to \hat{A}\), is injective. For \(s \in \Gamma\), we write \(\sigma_s^\prime\) for the contragredient of \(\sigma_s\) with respect to the \(\mu\)-invariant inner product \(\langle \cdot, \cdot \rangle\) on \(A\). Since \(\langle A, \sigma_s^\prime (\check{A})\rangle = \langle \sigma_s(A), \check{A}\rangle \subset 2\pi \mathbb{Z}\), \(\sigma_s^\prime\) restricts to an endomorphism on \(\check{A}\), and hence induces a covering map \(\sigma_s^\prime \colon \hat{A} \to \hat{A}\). By definition, the pull-back \(\sigma_s^{\prime *} \colon C(\hat{A}) \to C(\hat{A})\) is identified with \(\sigma_s \colon C^*_r(A) \to C^*_r(A)\) via the Pontrjagin duality. By taking \(r >0\) sufficiently small, we may assume that the restriction of the quotient \(\sigma_s^\prime (U) \to \hat{A}\) is also injective. Note that \(\sigma_s^\prime (U)\) is also \(\mu\)-invariant since \(\sigma_s^\prime\) commutes with the \(\mu\)-action. Now, the following diagram commutes \(\mu\)-equivariantly; \[\xymatrix{ C_0(A_{\mathbb{R}}) \ar[d]^{\sigma_s^{\prime *} } & \ar[l] C_0(U) \ar[r] \ar[d]^{\sigma_s^{\prime*}} & C(\hat{A}) \ar@{=}[r] \ar[d]^{\sigma_s^{\prime*}} & C^*_rA \ar[r] \ar[d]^{\sigma_s} & C(\overline{A}) \rtimes A \ar[d]^{\alpha_s \rtimes \sigma_s} \\ C_0(A_{\mathbb{R}}) & \ar[l] C_0(\sigma_s^\prime (U)) \ar[r] & C(\hat{A}) \ar@{=}[r] & C^*_r(A) \ar[r] & C(\overline{A}) \rtimes (A) , }\] which induces the following morphisms of K-theory: \[\begin{align} \begin{aligned} \xymatrix{ \mathrm{K}_0(C_0(A_{\mathbb{R}}) \rtimes \mu) \ar[d]^{(\sigma_s^{\prime *} \rtimes \mu)_* } & \ar[l]_\cong \mathrm{K}_0(C_0(U) \rtimes \mu) \ar[r]^{(\iota \rtimes \mu)_*} \ar[d]^{(\sigma_s^{\prime*} \rtimes \mu)_*} & \mathrm{K}_0(C(\overline{A}) \rtimes (A \rtimes \mu)) \ar[d]^{\theta_s = (\alpha_s \rtimes (\sigma_s \rtimes {\rm id}_\mu))_*} \\ \mathrm{K}_0(C_0(A_{\mathbb{R}}) \rtimes \mu) & \ar[l]_\cong \mathrm{K}_0(C_0(\sigma_s^\prime (U)) \rtimes \mu) \ar[r]^{(\iota \rtimes \mu)_*} & \mathrm{K}_0(C(\overline{A}) \rtimes (A \rtimes \mu)) . } \end{aligned}\label{eqn:diagram46Kfree} \end{align}\tag{18}\] By Remark 26, the commutativity of the right square shows that \(\theta_s\) preserves the subgroup \(\mathrm{Im}(\iota \rtimes \mu)_*\). Moreover, by the same reason as Remark 26, the left horizontal maps are isomorphisms. Thus, the left vertical morphism \((\sigma_s^{\prime *} \rtimes \mu)_*\) is identified with \(\theta_s|_{\mathrm{Im}(\iota \rtimes \mu)_*}\). At the left vertical map, \(\sigma_s^{\prime} \colon A_{\mathbb{R}} \to A_{\mathbb{R}}\) is a linear automorphism that commutes with the representation of \(\mu\). Therefore, it is homotopic to the identity through \(\mu\)-equivariant linear isomorphisms if and only if its restriction to each eigenspace of \(\mu\)-representation has positive determinant.

Based on the preparations so far, we prove (i). If \(K\) is totally imaginary, then \(A_{\mathbb{R}}\) is decomposed to the direct sum of its complex embeddings \(\bigoplus_{w \in V_{K,\mathbb{C}}} \mathbb{C}\) ([39]). On each complex embedding \(\mathbb{C}\), the multiplicative group \(\Gamma \times \mu\) of \(K\) acts by a multiplication by complex numbers. Thus, the determinant as an \(\mathbb{R}\)-linear map is positive, and hence is homotopic to the identity.

We then prove (ii). By Lemma 23 and 18 , the kernel and the cokernel of \(\theta_s|_{K_\mathrm{free}} -{\rm id}\) in turn coincide with those of \((\sigma_s^{\prime *} \rtimes {\rm id}_\mu)_*-{\rm id}\). To determine these groups, we apply the computation in [25]. Since \(\mu = \{ \pm 1\}\) in this case, which acts on \(A_\mathbb{R}\) by multiplication, there is no need to care about the \(\mu\)-eigenspace decomposition. In this case, \(\det (\sigma_s^{\prime}) = \det (\sigma_s) = \operatorname{sign}(s) = -1\). Therefore, \(\sigma_s^{\prime}\) is linearly (and hence \(\mu\)-equivariantly) homotopic to \(\mathrm{diag}(1,\cdots ,1 , -1)\). This enables us to apply [25], to get \(\ker ( (\sigma_s^{\prime *} \rtimes {\rm id}_\mu)_* -{\rm id}) \cong \mathbb{Z}\) and \(\mathop{\mathrm{coker}}( (\sigma_s^{\prime*} \rtimes {\rm id}_\mu )_* -{\rm id}) \cong \mathbb{Z}\). ◻

Proof. By Lemma 22, it suffices to compute the group \(\mathrm{H}_*(\Gamma , K_{\mathrm{free}})\). Pick \(s \in \Gamma\) so that \(s^{\mathbb{Z}}\) forms a direct summand and \(\mathrm{sign}(s) =-1\) (Lemma 12), As is discussed in the proof of Lemma 22, \(\theta_s\) acts on \({\textstyle\bigwedge}^q_{\mathbb{Q}}(\mathscr{A} \otimes \mathbb{Q}) \subset K_{\mathrm{div}}\) by \(|N_{\mathbb{Q}}(s)|^{-1}{\textstyle\bigwedge}^q(s)\). Similarly, for \(m \in \mathbb{Z}_{<1}\), \(\theta_{ms}\) acts on \({\textstyle\bigwedge}^q_{\mathbb{Q}}(\mathscr{A} \otimes\mathbb{Q})\) by \(m^{q-d}|N_{\mathbb{Q}}(s)|^{-1}{\textstyle\bigwedge}^q(s)\) (cf.the first line of the proof of Proposition 15). Thus, by replacing \(s\) with \(m \cdot s\) for some \(m \in \mathbb{Z}_{>1}\) if necessary, we may assume that \(\theta_s - {\rm id}\) acts on \(K_{\mathrm{div}}\) by an automorphism. If \(m\) is chosen so that the support of \(m\) and \(s\) are disjoint in the fractional ideal group \(J_K\), then \((ms)^{\mathbb{Z}}\) is also a direct summand. and a complement \(\Gamma^\perp\) of \(s^{\mathbb{Z}}\) in \(\Gamma\) such that \(\mathrm{sign}(\Gamma^{\perp}) =\{+1\}\). Then, by Lemma 24 (ii), any \(t \in \Gamma^\perp\) acts on \(K_{\mathrm{free}}\) by the identity, and we have \[\mathrm{H}_n(s^{\mathbb{Z}} , K_{\mathrm{free}}) \cong \begin{cases} \ker (\theta_s - {\rm id}) \cong \mathbb{Z}, & \text{ if n=0, }\\ \operatorname{coker} (\theta_s - {\rm id}) \cong \mathbb{Z}, & \text{ if n=1, }\\ 0 & \text{ otherwise. } \end{cases}\] In particular, this group is torsion-free. Since \(\mathrm{H}_p(\Gamma^\perp,\mathbb{Z}) \cong {\textstyle\bigwedge}_{\mathbb{Z}}^p \Gamma\) is also torsion-free, and \(\Gamma^\perp\) acts on \(K_{\mathrm{free}}\) trivially, the Künneth theorem (see e.g. [33]) shows that \[\mathrm{H}_n (\Gamma^\perp \times s^{\mathbb{Z}} , K_{\mathrm{free}} ) \cong \bigoplus_{p+q=n}\mathrm{H}_p(\Gamma^\perp, \mathbb{Z}) \otimes \mathrm{H}_q(s^{\mathbb{Z}}, K_{\mathrm{free}} ) \cong {\textstyle\bigwedge}^{p}_{\mathbb{Z}} \Gamma^\perp \oplus {\textstyle\bigwedge}^{p-1}_{\mathbb{Z}} \Gamma^\perp \cong {\textstyle\bigwedge}^{p}_{\mathbb{Z}}\Gamma. \qedhere\] ◻

Proof of Theorem 24. Here we consider the spectral sequence 13 associated to the exact couple \[\begin{align} \begin{aligned} \mathrm{DK}_{pq}^1 \mathrel{\vcenter{:}}= & \mathrm{KK}_{p+q}^{G} (\underline{E}G \times E_p\Gamma , \overline{\mathscr{A}}),\\ \mathrm{EK}_{pq}^1 \mathrel{\vcenter{:}}= & \mathrm{KK}_{p+q}^G( \underline{E}G \times ( E_p\Gamma, E_{p-1}\Gamma) , \overline{\mathscr{A}}) \cong \mathrm{KK}^{\mathscr{A} \rtimes \mu}_{q}(\underline{E}G, \overline{\mathscr{A}}) \otimes_{\mathbb{Z}[\Gamma]}\mathbb{Z}[\Gamma^{p+1}] . \end{aligned} \end{align}\] Here, the isomorphism in the second line is verified in the same way as 15 , by Lemma 14. By Lemma 37 for \(\overline{\mathscr{A}}=\mathscr{A} \times_A \overline{A}\), Remark 19 (iv) claiming \(\underline{E}G = \underline{E}A\), and the Baum-Connes isomorphism 11 for \(A \rtimes \mu\) and \(\overline{A}\), we have the isomorphism \[\mathrm{KK}^{\mathscr{A} \rtimes \mu}_{q}(\underline{E}G, \overline{\mathscr{A}}) \cong \mathrm{KK}^{A \rtimes \mu}(\underline{E}A,\overline{A}) \cong \mathrm{K}_0(C^*_r(A \rtimes \mu) \ltimes \overline{A}) \cong K_{\mathrm{fin}}^\mu \oplus K_{\mathrm{inf}}\] is torsion-free. Thus, Raven’s Chern character gives an injective map \[\mathop{\widehat{\mathrm{ch}}_R} \colon \mathrm{KK}^{\mathscr{A} \rtimes \mu}_{q}(\underline{E}G, \overline{\mathscr{A}}) \otimes_{\mathbb{Z}[\Gamma]} \mathbb{Z}[\Gamma^{p+1}] \to \widehat{\mathrm{HH}}{}_{\mathscr{A} \rtimes \mu}^{-q}( \underline{E}G, \overline{\mathscr{A}}) \otimes_{\mathbb{C}[\Gamma]}\mathbb{C}[\Gamma^{p+1}].\] Since the differentials \(d^1_{pq}\) of the spectral sequences \(\mathrm{EK}\) and \(\widehat{\mathrm{EH}}\) are given by the boundary map of the equivariant homology theories \(\mathrm{KK}_{p+q}^G(\underline{E}G \times \cdot, \overline{\mathscr{A}})\) and \(\mathrm{KK}_{p+q}^G(\underline{E}G \times \cdot, \overline{\mathscr{A}})\), they are compatible with respect to Raven’s Chern character: \(d^1_{pq} \circ \mathop{\widehat{\mathrm{ch}}_R} = \mathop{\widehat{\mathrm{ch}}_R} \circ d_{pq}^1\). Therefore, by comparing the two spectral sequence, it turns out that \(d^1_{pq}\) for \(\mathrm{EK}\), which is the restriction of that for \(\widehat{\mathrm{EH}}\), is identical with that of group cohomology with coefficient in \(\mathrm{KK}^{\mathscr{A} \rtimes \mu}_{q}(\underline{E}G, \overline{\mathscr{A}})\). Finally, by Remark 22, we obtain that \[\mathrm{EK}^2_{pq} \cong \mathrm{H}^p(\Gamma, \mathrm{KK}_q^N(\underline{E}G, \overline{\mathscr{A}})) \cong \mathrm{H}^p(\Gamma, \mathrm{K}_q(C^*_r((A \rtimes \mu ) \ltimes \overline{A})).\]

By 17 and \(\mathrm{H}^p(\Gamma ; K_{\mathrm{div}}) \cong 0\), we have \(\mathrm{EK}_{p0}^2 \cong \mathrm{H}^p(\Gamma , K_{\mathrm{free}})\) and \(\mathrm{EK}_{p1}^2 \cong 0\). If \(K\) is totally imaginary, by Lemma 24 (1) we have \[\mathrm{EK}_{pq}^2 \cong \mathrm{H}^p(\Gamma,K_{\mathrm{free}}) \cong \begin{cases} {\textstyle\bigwedge}_{\mathbb{Z}}^p \Gamma \oplus \widetilde{R}(\mu) \otimes_{\mathbb{Z}} {\textstyle\bigwedge}_{\mathbb{Z}}^p \Gamma = R(\mu) \otimes_{\mathbb{Z}} {\textstyle\bigwedge}_{\mathbb{Z}}^p \Gamma & \text{ if q \equiv 0 \equiv d mod 2, }\\ 0 & \text{ otherwise.} \end{cases}\] If \(|V_{K,\mathbb{R}}|\) is odd, \(\Gamma\) acts on \(K_{\mathrm{fin}}^\mu\) trivially as is remarked above, and hence \[\mathrm{EK}_{pq}^2 \cong \begin{cases} \widetilde{R}(\mu) \otimes_{\mathbb{Z}} {\textstyle\bigwedge}_{\mathbb{Z}}^p \Gamma = {\textstyle\bigwedge}_{\mathbb{Z}}^p \Gamma & \text{ if q\equiv d mod 2, }\\ 0 & \text{ otherwise.} \end{cases}\] If \(|V_{L,\mathbb{R}}|\) is nonzero even, then by Lemma 25 we have \[\mathrm{EK}^2_{pq} \cong \begin{cases} {\textstyle\bigwedge}_{\mathbb{Z}}^p \Gamma & \text{ if q\equiv d mod 2, }\\ 0 & \text{ otherwise.} \end{cases}\]

In any cases, the groups \(\mathrm{EK}_{pq}^2\) are torsion-free, and hence is sent to \(\bigoplus_{k \in \mathbb{Z}}\widehat{\mathrm{EH}}{}_{p,q+2k }^2\) injectively by Raven’s Chern character \(\widehat{\mathrm{ch}}_R\). Now Lemma 21 concludes that \(\mathrm{EK}_{pq}^r\) collapses at the \(E^2\)-page. Since there is no extension problem, we obtain the desired isomorphism. ◻

Proof. Let \(x \in \Lambda^\infty\). Then, we have \(x(0,b)x(b,a)=x(0,a)\), so that \(P_b(x)x(b,a)=P_a(x)\). Hence, we have \(P_b(x) + s^bx(b,a) = P_a(x)\) in \(\mathbb{Z}\), which means that \(P_b(x)\) coincides with the image of \(P_a(x)\) under the canonical projection \(\mathbb{Z}/s^a\mathbb{Z}\to \mathbb{Z}/s^b\mathbb{Z}\). ◻

Proof. It suffices to show the desired equation only for \(x \in \mathbb{N}\) because \(\mathbb{N}\) is dense in \(\Delta\). Take \(l \in \mathbb{N}^\Sigma\) sufficiently large so that \(x \leq s^l\). Then, we have \(P_m(x) =x\) for \(m \geq l\). From the equation \(x(0,a)x(a,l+a) =x(0,l+a)\), we have \[P_a(x)+s^a(P_l(\sigma^ax)) = P_{l+a}(x) = x,\] so that we have \[P_l(\sigma^ax) = \frac{x-P_a(x)}{s^a}.\] The right hand side does not depend on \(l\). Hence, we have \(\sigma^ax \in \mathbb{N}\) and \(\sigma^ax = P_l(\sigma^ax)\), which completes the proof. ◻

Proof. Since \(\ker\rho=\mathbb{Z}\ltimes\Delta\), we have \(\mathrm{H}_0(\ker\rho)=\mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta)=\mathbb{Z}[1/n]\) (see Equation 19 ), where \(n\mathrel{\vcenter{:}}= s_1\cdots s_N\). For \(k\in\{0,...,n-1\}\), we have \((k,0,0)\in\ker\varphi\) because \(k-0=P_n(k)-P_n(0)\), so that \([1_{k+n\Delta}]_0=[1_{n\Delta}]_0\) in \(\mathrm{H}_0(\ker\varphi)\). Hence, \([1_\Delta]_0=\sum_{k=0}^{n-1}[1_{k+n\Delta}]_0=n[1_{n\Delta}]_0\), so that there is a homomorphism \(\mathbb{Z}[1/n]\to \mathrm{H}_0(\ker\varphi)\) such that \(1/n\mapsto [1_\Delta]_0\). It is easy to see that this is the inverse of \(\mathrm{H}_0(\beta)\).

In order to show the second claim, it suffices to show that \((\ker \rho)^{(0)}=\Delta \times \{e\}\) is full in \(\mathcal{G}_{S\curvearrowright\mathbb{Z}} \times_\rho \mathscr{S}\) in the sense that \(\Delta \times \{e\}\) intersects every \((\mathcal{G}_{S\curvearrowright\mathbb{Z}} \times_\rho \mathscr{S})\)-orbit. This can be seen directly. ◻

Proof. Remark 28 implies that the LHS type spectral sequence \[E_{pq}^2(\mathcal{G}_\mathrm{tor},\varphi)=\mathrm{H}_p(\mathscr{S},\mathrm{H}_q(\mathcal{G}_\mathrm{tor}\times_\varphi\mathscr{S}))\Longrightarrow \mathrm{H}_{p+q}(\mathcal{G}_\mathrm{tor}),\] collapses at the \(E^2\)-page, so that we have \[E^2_{p,0}(\mathcal{G}_\mathrm{tor},\varphi) = E^\infty_{p,0} = \mathrm{H}_p(\mathscr{S}, \mathrm{H}_0(\mathcal{G}_\mathrm{tor}\times_\varphi \mathscr{S})) = \mathrm{H}_p(\mathscr{S}, \mathrm{H}_0(\ker \varphi))\] On the other hand, \(E^2\)-page of the LHS type spectral sequence \[E_{pq}^2(\mathcal{G}_{S\curvearrowright\mathbb{Z}},\rho)=\mathrm{H}_p(\mathscr{S},\mathrm{H}_q(\mathcal{G}_{S\curvearrowright\mathbb{Z}}\times_\rho\mathscr{S}))\Longrightarrow \mathrm{H}_{p+q}(\mathcal{G}_{S\curvearrowright\mathbb{Z}})\] is \[E^2_{pq}(\mathcal{G}_{S\curvearrowright\mathbb{Z}},\rho) = \mathrm{H}_p(\mathscr{S}, \mathrm{H}_q(\mathcal{G}_{S\curvearrowright\mathbb{Z}} \times_\rho \mathscr{S})) = \mathrm{H}_p (\mathscr{S}, \mathrm{H}_q(\mathbb{Z}\ltimes \Delta))\] for \(q=0,1\), and is zero if \(q > 1\) by Lemma 28. By Theorem 4 and Example 1, \(\alpha\) induces a morphism \(E^*(\mathcal{G}_\mathrm{tor},\varphi) \to E^*(\mathcal{G}_{S\curvearrowright\mathbb{Z}}, \rho)\) of spectral sequences, which gives rise to a morphism between long exact sequences as follows: \[\begin{tikzcd} E^2_{p+1,0} \ar[r, "0"] \ar[d] & E^2_{p-1,1}(\mathcal{G}_\mathrm{tor}, \varphi) \ar[r] \ar[d] & \mathrm{H}_p(\mathcal{G}_\mathrm{tor}) \ar[r] \ar[d]& E^2_{p,0}(\mathcal{G}_\mathrm{tor}, \varphi) \ar[r, "0"] \ar[d] & E^2_{p-2,1} \ar[d] \\ E^2_{p+1,0} \ar[r, "d"] & E^2_{p-1,1}(\mathcal{G}_{S\curvearrowright\mathbb{Z}}, \rho) \ar[r] & \mathrm{H}_p(\mathcal{G}_{S\curvearrowright\mathbb{Z}}) \ar[r]& E^2_{p,0}(\mathcal{G}_{S\curvearrowright\mathbb{Z}}, \rho) \ar[r, "d"] & E^2_{p-2,1}\makebox[0pt][l]{\,.} \end{tikzcd}\] By Lemma 28, the above diagram becomes \[\begin{tikzcd} 0 \ar[r] & 0 \ar[r] \ar[d] & \mathrm{H}_p(\mathcal{G}_\mathrm{tor}) \ar[r, "\cong"] \ar[d]& \mathrm{H}_p (\mathscr{S}, \mathrm{H}_0(\ker \varphi)) \ar[r] \ar[d, "\cong"]& 0 \\ \dots \ar[r, "d"] & \mathrm{H}_{p-1} (\mathscr{S}, \mathrm{H}_1(\mathbb{Z}\ltimes \Delta)) \ar[r] & \mathrm{H}_p(\mathcal{G}_{S\curvearrowright\mathbb{Z}}) \ar[r]& \mathrm{H}_p (\mathscr{S}, \mathrm{H}_0(\mathbb{Z}\ltimes \Delta)) \ar[r, "d"]& \dots \makebox[0pt][l]{\,.} \end{tikzcd}\] In particular, the differential \(d \colon E^2_{p,0}(\mathcal{G}_{S\curvearrowright\mathbb{Z}}, \rho) \to E^2_{p-2,1}(\mathcal{G}_{S\curvearrowright\mathbb{Z}}, \rho)\) is zero, so that we obtain the split exact sequence \[\label{eqn:ses5} 0 \to \mathrm{H}_{p-1}(\mathscr{S}) \to \mathrm{H}_p(\mathcal{G}_{S\curvearrowright\mathbb{Z}}) \to \mathrm{H}_p(\mathscr{S}, \mathscr{A}) \to 0.\tag{23}\] Here, we have used that \(\mathrm{H}_1(\mathbb{Z}\ltimes \Delta)=\mathrm{H}_1(\mathscr{A},\mathbb{Z}\overline{\mathscr{A}})\) by Proposition 8, the action of \(\mathscr{S}\) on \(\mathrm{H}_1(\mathbb{Z}\ltimes \Delta)\) is trivial by Theorem 9 because \({\textstyle\bigwedge}^1(s)\otimes \frac{1}{[\mathbb{Z}: s\mathbb{Z}]}=1\) for all \(s\in S\), and that \(\mathbb{Z}\cong \mathrm{H}_1(\mathbb{Z}\ltimes \Delta)\) by Proposition 12(iii). Now the claim follows from Equation 21 . ◻

Proof. Let \(Q_\bullet \mathrel{\vcenter{:}}= \bigotimes_{i=1}^{N-1} P_\bullet(s_i)\) and \(P_\bullet \mathrel{\vcenter{:}}= P_\bullet(s_N)\). Then, since \(\mathrm{H}_0(P_{\bullet}) \cong \mathbb{Z}/(s_N-1)\mathbb{Z}\) and \(\mathrm{H}_1(P_{\bullet})=0\), the Künneth formula for chain complexes yields the following Künneth exact sequence \[0 \to \mathrm{H}_p(Q_\bullet) \otimes_{\mathbb{Z}} \mathbb{Z}/(s_N-1)\mathbb{Z} \to \mathrm{H}_p (Q_\bullet \otimes P_\bullet) \to \mathrm{Tor}^{\mathbb{Z}}_1(\mathrm{H}_{p-1}(Q_\bullet), \mathbb{Z}/(s_N-1)\mathbb{Z}) \to 0.\] This exact sequence splits because each \(P_n\) is a free \(\mathbb{Z}\)-module. By induction, we see that \[\mathrm{H}_p(Q_\bullet) \otimes \mathbb{Z}/(s_N-1)\mathbb{Z}\cong (\mathbb{Z}/g_{\Sigma}\mathbb{Z})^{\oplus \binom{N-2}{p}}and\mathrm{Tor}(\mathrm{H}_{p-1}(Q_\bullet), \mathbb{Z}/(s_N-1)\mathbb{Z}) \cong (\mathbb{Z}/g_{\Sigma}\mathbb{Z})^{\oplus \binom{N-2}{p-1}},\] and hence \(\mathrm{H}_p(\bigotimes_{s \in \Sigma} P_{\bullet }(s) ) \cong (\mathbb{Z}/g\mathbb{Z})^{\binom{n-1}{p}}\). ◻

Proof. Fix a finite subset \(\Sigma_0 \subseteq \Sigma\) such that \(g_{\Sigma_0} = g_{\Sigma}\). Let \(\Sigma_k \subseteq \Sigma\) be an increasing sequence of finite subsets of \(\Sigma\), starting with \(\Sigma_0\), such that \(\Sigma_{k+1} = \Sigma_k \sqcup \{s_k\}\) and \(\bigcup_k \Sigma_k = \Sigma\). Then, we have \(g_{\Sigma_k} = g_\Sigma\) for all \(k \geq 0\). For each \(k\), let \(\mathscr{A}_k\), \(S_k\), \(\mathscr{S}_k\), \(\Delta_k\), \(\mathcal{G}_k\mathrel{\vcenter{:}}=\mathcal{G}_{S_k\curvearrowright\mathbb{Z}}\), and \(\mathcal{P}_k\) be associated with \(\Sigma_k\) as in Section 6.1. Let \(\pi_k\colon\Delta_{k+1}\to \Delta_k\) be the canonical projection map and \(j_k\colon \mathscr{A}_k\rtimes \mathscr{S}_k\to \mathscr{A}_{k+1}\rtimes \mathscr{S}_{k+1}\) the canonical inclusion map. By Theorem 4, Lemma 7, and Proposition 2, we have the following morphism between the exact sequences from Equation 23 : \[\begin{tikzcd} 0 \arrow[r] & \mathrm{H}_{p-1}(\mathscr{S}_{k+1},\mathrm{H}_1(\mathbb{Z},\mathbb{Z}\Delta_{k+1})) \arrow[r] & \mathrm{H}_p(\mathcal{G}_{k+1}) \arrow[r] & \mathrm{H}_p(\mathscr{S}_{k+1}, \mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_{k+1})) \arrow[r] & 0 \\ 0 \arrow[r] & \mathrm{H}_{p-1}(\mathscr{S}_k,\mathrm{H}_1(\mathbb{Z},\mathbb{Z}\Delta_k)) \arrow[r]\arrow[u,"\mathrm{H}_{p-1}(j_k\text{,}\,\mathrm{H}_1({\rm id}\text{,}\,\pi_k^*))"] & \mathrm{H}_p(\mathcal{G}_k) \arrow[r]\arrow[u,"\mathrm{H}_p(\Omega_k)"] & \mathrm{H}_p(\mathscr{S}_k, \mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_k)) \arrow[r]\arrow[u,"\mathrm{H}_p(j_k\text{,}\,\mathrm{H}_0({\rm id}\text{,}\,\pi_k^*))"] & 0\makebox[0pt][l]{\,,} \end{tikzcd}\] where \(\Omega_k\colon \mathcal{G}_k\to\mathcal{G}_{k+1}\) is the étale correspondence associated with \((j_k,\pi_k)\) as in Lemma 7. To see that the equivariance condition in Definition 5 is satisfied, notice that \(\mathscr{S}_k\) acts on \(\prod_{p\in\mathcal{P}_{k+1}\setminus\mathcal{P}_k}\mathbb{Z}_p\) by homeomorphisms.

Next, we need to check that, under the identification as in Equation 19 , \(\mathrm{H}_0({\rm id},\pi_k^*)\) is identified with \({\rm id}_\mathbb{Z}\) and \(\mathrm{H}_1({\rm id},\pi_k^*)\) is identified with the inclusion map \(\iota_k \colon \mathscr{A}_k \hookrightarrow \mathscr{A}_{k+1}\). For \(q=1\), we have the following commutative diagram \[\begin{tikzcd} \mathrm{H}_1(\mathbb{Z})\arrow[r,"\rho_{1,\infty}"] \arrow[d,"{\rm id}"]& \mathrm{H}_1(\mathbb{Z},\mathbb{Z}\Delta_k)\arrow[d,"\mathrm{H}_1({{\rm id},\pi_k^*})"] \\ \mathrm{H}_1(\mathbb{Z}) \arrow[r,"\rho_{1,\infty}"]& \mathrm{H}_1(\mathbb{Z},\mathbb{Z}\Delta_{k+1})\makebox[0pt][l]{\,,} \end{tikzcd}\] where the horizontal maps are from 7 . They are isomorphisms by Proposition 12. For \(q=0\), consider the diagram \[\begin{tikzcd} \mathrm{H}_0(\mathbb{Z})\otimes \mathscr{A}_k\arrow[d,"{\rm id}"] \arrow[r,"\tilde{\rho}_{1,\infty}"] & \mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_k) \otimes\mathscr{A}_k=\mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_k)\arrow[d,"\mathrm{H}_0({{\rm id},\pi_k^*})"] \\ \mathrm{H}_0(\mathbb{Z})\otimes \mathscr{A}_k \arrow[d,"\mathrm{incl}"]\arrow[r,"\rho_{1,\infty}\otimes{\rm id}_{\mathscr{A}_k}"]& \mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_{k+1}) \otimes\mathscr{A}_k=\mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_{k+1})\arrow[d,"{\rm id}"] \\ \mathrm{H}_0(\mathbb{Z})\otimes \mathscr{A}_{k+1} \arrow[r,"\tilde{\rho}_{1,\infty}"]& \mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_{k+1})\otimes\mathscr{A}_{k+1}=\mathrm{H}_0(\mathbb{Z},\mathbb{Z}\Delta_{k+1})\makebox[0pt][l]{\,,} \end{tikzcd}\] where the top and bottom horizontal maps are the isomorphisms from Proposition 11, and \(\rho_{1,\infty}\) is the map from 7 . Proposition 12 gives us the three equalities on the right hand side. It is easy to see that each of the smaller squares commutes, so the larger square also commutes.

Consequently, we have the following morphism of short exact sequences: \[\begin{tikzcd} 0 \arrow[r] & \mathrm{H}_{p-1}(\mathscr{S}_{k+1}, \mathbb{Z}) \arrow[r] & \mathrm{H}_p(\mathcal{G}_{k+1}) \arrow[r] & \mathrm{H}_p(\mathscr{S}_{k+1}, \mathscr{A}_{k+1}) \arrow[r] & 0 \\ 0 \arrow[r] & \mathrm{H}_{p-1}(\mathscr{S}_k, \mathbb{Z}) \arrow[r] \arrow[u,"\mathrm{H}_p({j_k,{\rm id}})"] & \mathrm{H}_p(\mathcal{G}_k) \arrow[r] \arrow[u,"\mathrm{H}_p(\Omega_k)"] & \mathrm{H}_p(\mathscr{S}_k, \mathscr{A}_k) \arrow[r] \arrow[u,"\mathrm{H}_p({j_k,\iota_k})"] & 0\makebox[0pt][l]{\,.} \end{tikzcd}\]

By continuity of group homology ([33] and continuity of Tor) combined with continuity of groupoid homology (Proposition 5), we obtain the following split exact sequence: \[0 \to \mathrm{H}_{p-1}(\mathscr{S}, \mathbb{Z}) \to \mathrm{H}_p(\mathcal{G}_{S\curvearrowright\mathbb{Z}}) \to \mathrm{H}_p(\mathscr{S}, \mathscr{A}) \to 0.\]

We have \(\mathrm{H}_{p-1} (\mathscr{S}, \mathbb{Z}) = {\textstyle\bigwedge}^{p-1} \mathscr{S}\cong \mathbb{Z}^{\oplus\infty}\). In addition, we also have \[\begin{align} \mathrm{H}_p(\mathscr{S}_k,\mathscr{A}_{\sigma}) \cong {}& \mathrm{H}_p(\mathscr{S}_k,(\mathscr{A}_k)_{\sigma}) \otimes_{\mathscr{A}_k} \mathscr{A} \oplus \mathrm{Tor}^{\mathscr{A}_k}_1(\mathrm{H}_p(\mathscr{S}_k,(\mathscr{A}_k )_{\sigma}), \mathscr{A})\\ \cong {}& (\mathbb{Z}/g_{\Sigma}\mathbb{Z})^{\binom{|\Sigma_k|-1}{p-1}} \otimes_{\mathscr{A}_k} \mathscr{A} \oplus \mathrm{Tor}^{\mathscr{A}_k}_1((\mathbb{Z}/g_{\Sigma}\mathbb{Z})^{\binom{|\Sigma_k|-1}{p-1}}, \mathscr{A})\\ \cong {}&(\mathbb{Z}/g_{\Sigma}\mathbb{Z})^{\binom{|\Sigma_k|-1}{p-1}}, \end{align}\] by the universal coefficient theorem (here \(\mathscr{A}_\sigma\) denotes the \(\mathscr{A}\)-module \(\mathscr{A}\) on which \(\mathscr{S}\) acts nontrivially by \(\sigma\)) and hence \[\mathrm{H}_p(\mathscr{S},\mathscr{A}_{\sigma}) \cong \varinjlim_{k} \mathrm{H}_p(\mathscr{S}_k,\mathscr{A}_\sigma) \cong \varinjlim_{k} (\mathbb{Z}/g_{\Sigma}\mathbb{Z})^{\binom{n_k-1}{p-1}} \cong (\mathbb{Z}/g_{\Sigma}\mathbb{Z})^{\infty}. \qedhere\] ◻

Proof. We write \(s_I \mathrel{\vcenter{:}}= \prod_{i \in I} s_i\) for \(I \subset \{1,\dots,N\}\). Recall that \(B\) is an AF algebra with \(\mathrm{K}_0(B) \cong \mathbb{Z}[\mathfrak{s}^{-1}]\). We fix this isomorphism so that a rank \(1\) projection of \(B_l\) corresponds to \(1 \in \mathbb{Z}[\mathfrak{s}^{-1}]\). Under this identification, for each \(I\), the \(\mathrm{K}_0\)-group of the matrix subalgebra \(D_{\beta_I} \cong \mathbb{K}(\ell^2\Delta_{\mathfrak{s}^l / s_I}) \subset B_l \subset B\) is identified with its image in \(\mathrm{K}_0(B)\), namely the subgroup \(s_I \mathbb{Z} \subset \mathbb{Z}[\mathfrak{s}^{-1}]\). Moreover, since each partial \(\ast\)-homomorphism \(\beta_i \colon D_{\beta_{I}} \to D_{\beta_{I \setminus \{i\}}}\) sends a rank \(1\) projection to a rank \(s_i\) one, it induces multiplication by \(s_i^{-1}\) on \(\mathrm{K}_0\). This also implies that \(\beta_{i,*} = \mathrm{K}_0(B) \to \mathrm{K}_0(B)\) is also given by multiplication by \(s_i^{-1}\). In particular, \(\beta_{i,*}\) are isomorphisms. This shows \(\mathrm{K}_0(B) \cong \mathrm{K}_0(\widetilde{B})\), on which \(\mathbb{Z}^N\) acts by \(s_i^{-1}\).

Consider the AH spectral sequence \(E^{pq}_r\) associated to \(\widetilde{\mathcal{B}}\). Then we have \(E_1^{p,2q+1}\cong 0\), and by the above discussion and Remark 33, the cochain complex \((E_1^{p,2q},d_{1}^{p,2q})\) is decomposed into the tensor product of length \(2\) complexes as \[\begin{align} \begin{aligned} E_1^{*,2q} \cong {}&{} \bigg( 0 \to \bigoplus_{|I_0|=0}\mathbb{Z}[\mathfrak{s}^{-1}] \cdot x_{I_0} \xrightarrow{\partial^0} \bigoplus_{|I_1|=1} \mathbb{Z}[\mathfrak{s}^{-1}] \cdot x_{I_1} \xrightarrow{\partial^1} \cdots \xrightarrow{\partial^{N-1}} \bigoplus_{|I_N|=N} \mathbb{Z}[\mathfrak{s}^{-1}]\cdot x_{I_N} \to 0 \bigg) \\ \cong {}&{} \bigotimes_{s \in \Sigma} (0 \to \mathbb{Z}[\mathfrak{s}^{-1}] \xrightarrow{s^{-1} -1} \mathbb{Z}[\mathfrak{s}^{-1}] \to 0) \\ \cong {}&{} \bigotimes_{s \in \Sigma} (0 \to \mathbb{Z}[\mathfrak{s}^{-1}] \xrightarrow{s-1} \mathbb{Z}[\mathfrak{s}^{-1}] \to 0), \end{aligned}\label{eqn:E146B1} \end{align}\tag{25}\] where each \(I_j\) denotes a subset of \(\{1,\cdots,N\}\), \(x_I\) denotes the generator of \(\mathrm{K}_0(C_0((0,1)^I, \widetilde{\mathcal{B}}) )\) corresponding to \(1 \in \mathbb{Z}[\mathfrak{s}^{-1}]\) under the above identification, and \[\partial^j =\bigoplus_{I_{j+1}=I_j \cup \{s_i\} } \Big( \sigma(I_j,s_i) \cdot (s_i^{-1}-1) \colon \mathbb{Z}[\mathfrak{s}^{-1}] \cdot x_{I_j} \to \mathbb{Z}[\mathfrak{s}^{-1}] \cdot x_{I_{j+1}}\Big).\] Here, \(\sigma(I_j,s_i) \mathrel{\vcenter{:}}= (-1)^{m-1}\) if \(s_i\) is the \(m\)-th element when the elements of \(I_j\cup\{s_i\}\) are arranged in increasing order. The second isomorphism is given by identifying the direct summand \(\mathbb{Z}[\mathfrak{s}^{-1}] \cdot x_{I_j}\) with the tensor factor in which, for each \(s \in \Sigma\), one takes the left copy of \(\mathbb{Z}[\mathfrak{s}^{-1}]\) if \(s \notin I_j\) and the right copy if \(s \in I_j\). The third isomorphism is given by the commutative diagram below: \[\xymatrix{ \mathbb{Z}[\mathfrak{s}^{-1}] \ar[r]^{s^{-1}-1} \ar[d]^{-s^{-1}} & \mathbb{Z}[\mathfrak{s}^{-1}] \ar@{=}[d] \\ \mathbb{Z}[\mathfrak{s}^{-1}] \ar[r]^{s-1} & \mathbb{Z}[\mathfrak{s}^{-1}]. }\] By inverting the degree, this chain complex \((E_1^{N-p,2q},d_1^{N-p,2q})\) is almost the same thing as the one considered in Lemma 29, the only difference being that \(\mathbb{Z}\) is replaced with \(\mathbb{Z}[\mathfrak{s}^{-1}]\). Therefore, by a similar computation as Lemma 29, the cohomology group \(E_2^{p,2q}\) is isomorphic to \((\mathbb{Z}/g \mathbb{Z})^{\binom{N-1}{N-p}} = (\mathbb{Z}/g \mathbb{Z})^{\binom{N-1}{p-1}}\). This shows the first half of the claim.

Again by Remark 33, the AH spectral sequence \(\bar{E}_1^{p,2q}\) associated to \(\mathcal{B}_l\) satisfies \(\bar{E}_{2}^{p,2q+1} \cong 0\), and the cochain complex \((\bar{E}_1^{p,2q},d_1^{p,2q})\) a similar decomposition as \[\begin{align} \bar{E}_1^{*,2q} \cong \bigotimes_{s \in \Sigma} (0 \to s\mathbb{Z} \xrightarrow{s^{-1} -1} \mathbb{Z} \to 0) \cong \bigotimes_{s \in \Sigma} (0 \to \mathbb{Z} \xrightarrow{s-1} \mathbb{Z} \to 0), \end{align}\] where the right isomorphism is given by a restriction of the above commutative diagram. This is exactly the same as the complex considered in Lemma 29, and hence \(\bar{E}_2^{p0}\) is isomorphic to \((\mathbb{Z}/g \mathbb{Z})^{\binom{N-1}{N-p}} = (\mathbb{Z}/g \mathbb{Z})^{\binom{N-1}{p-1}}\) as well.

The above computation also shows that the map \(\bar{E}_2^{p,2q} \to E_2^{p,2q}\) induced by the inclusion \(\iota \colon \mathcal{B}_l \to \mathcal{B}\) is an isomorphism. By comparing the differentials \(d_{r}^{pq}\) inductively, it implies that the induced maps \(\bar{E}_2^{pq} \to E_2^{pq}\) are isomorphisms on every page \(E_r\) for \(r\ge 2\). ◻

Proof. Let us decompose \(B_1=\mathbb{K}(\ell^2\Delta_{\mathfrak{s}})\) into the tensor product \(\bigotimes_{s \in \Sigma} \mathbb{K}(\ell^2\Delta_{s})\) by the Chinese remainder theorem. Since \(\mathbb{Z}/\mathfrak{s}\mathbb{Z} \cong \prod_{s \in \Sigma}(\mathbb{Z}/s\mathbb{Z})\) is a ring isomorphism, the multiplication by \(s\) acts on each tensor component individually. For \(s, u \in \Sigma\), we write \(\delta_{s}^{u}\) for the partial \(\ast\)-automorphism induced from multiplication by \(s\) on \(\Delta_{u}\). If \(s \neq u\), then \(\delta_{s}^u\) is a \(\ast\)-automorphism. If \(s = u\), the domain and range are \(D_{\delta_u^u} = \mathbb{C}\cdot 1\) and \(R_{\delta_u^u} = \mathbb{C}\cdot e_u\), where \(e_u\) denotes the projection onto \(\ell^2(u\Delta_{u}) \cong \mathbb{C}\). Set \[\mathcal{D}_{u}:=\mathcal{MT}(\mathbb{K}(\ell^2\Delta_{u}), \{ \delta_{s}^{u}\}_{s \in \Sigma}).\] Then, \(\mathcal{D}_{u}|_{\mathbb{T}^{\Sigma \setminus \{u\}}}\) is the matrix bundle associated to the \(\mathbb{Z}^{\Sigma\setminus \{u\}}\)-action \(\{ \delta_{s}^{u}\}_{s \in \Sigma \setminus \{ u \} }\) by automorphisms. Notice that the tuple \(\{\delta_{s}^{u}\}_{s \in \Sigma \setminus \{u\} }\) is implemented by a mutually commuting tuple of unitaries \(v_{s}^u\) permuting the standard basis. These unitaries satisfy \(e_uv_{s}^u e_u=e_u\). Therefore, we get a continuous section \[\begin{align} e_u' \mathrel{\vcenter{:}}= e_u \otimes 1_{C([0,1]^{\Sigma \setminus \{ u \} })} \in C(\mathbb{T}^{\Sigma \setminus \{ u \}}, \mathcal{MT}(\mathbb{K}(\ell^2\Delta_u) , \{ \delta_s^u\}_{s \in \Sigma }) ) \subset C([0,1]^{\Sigma} ,\mathbb{K}(\ell^2\Delta_u)) \label{eqn:section46D46e} \end{align}\tag{26}\] of \(\mathcal{D}_u\).

By the simultaneous diagonalization, we obtain a homotopy \(\tilde{\delta}_s^u\) of \(\ast\)-automorphisms on \(\mathbb{K}(\ell^2\Delta_u)\) connecting \(\delta_s^u\) and the identity, i.e., \(\tilde{\delta}_s^u \colon \mathbb{K}(\ell^2\Delta_u) \to C([0,1]) \otimes \mathbb{K}(\ell^2\Delta_u)\) such that \(\mathrm{ev}_0 \circ \tilde{\delta}_s^u = \delta_s^u\) and \(\mathrm{ev}_1 \circ \tilde{\delta}_s^u ={\rm id}\), such that \(\mathrm{ev}_{t_1}\circ \tilde{\delta}_{s_1}^u\) and \(\mathrm{ev}_{t_2} \circ \delta_{s_2}^u\) commutes for any \(t_1,t_2 \in [0,1]\) and \(s_1,s_2 \in \Sigma\). Moreover, we may choose such \(\tilde{\delta}_s^u\) so that \(\tilde{\delta}_s^u(e_u) = e_u \otimes 1\). Now, the bundle map \[\Psi \colon \mathcal{D}_u |_{\mathbb{T}^{\Sigma \setminus \{ u \} }} \to \mathbb{T}^{\Sigma \setminus \{u\}} \times \mathbb{K}(\ell^2\Delta_u)\] given by \[\Psi( \{ t_s \}_{s \in \Sigma \setminus \{u\}} ,d) =\big( \{ t_s \}_{s \in \Sigma \setminus \{u\}} , (\mathrm{ev}_{t_{s_N}} \circ\tilde{\delta}_{s_N}^u) \circ \cdots \circ (\mathrm{ev}_{t_{s_1}} \circ\tilde{\delta}_{s_1}^u)(d)\big)\] is well-defined, gives a trivialization of \(\mathcal{D}_u|_{\mathbb{T}^{\Sigma \setminus \{u \} }}\), and sends \(e_u'\) in 26 to the constant section \(e_u \otimes 1 \in \mathbb{K}(\ell^2\Delta_u) \otimes C(\mathbb{T}^{\Sigma \setminus \{ u\}})\). \(\delta_s^u(t)\), , i.e., an action \(\tilde{\delta}^u \colon \mathbb{Z}^{\Sigma \setminus \{ u \}} \curvearrowright C([0,1]) \otimes \mathbb{K}(\ell^2\Delta_u)\) that induces the identity on \(C([0,1]) = \mathcal{Z}(C([0,1]) \otimes \mathbb{K}(\ell^2\Delta_u))\) such that \(\mathrm{ev}_0 \circ \tilde{\delta}_s^u = \delta_s^u \circ \mathrm{ev}_0\) and \(\mathrm{ev}_1 \circ \tilde{\delta}_s^u = \mathrm{ev}_1\), that fixes \(e_u\), i.e., \(\tilde{\delta}_{s}^u (e_u) = e_u \otimes 1\). Now, \[C(\mathbb{T}^{\Sigma \setminus \{ u \}} , \mathcal{MT}(C([0,1]) \otimes \mathbb{K}(\ell^2\Delta_u), \{ \tilde{\delta}^u_s \}_{s \in \Sigma} ))\] as the continuous section algebra of a locally trivial bundle over \(\mathbb{T}^{\Sigma \setminus \{u \}}\), which gives a homotopy of \(\mathcal{D}_u\) and the trivial bundle \(\mathbb{T}^{\Sigma \setminus \{u \}} \times \mathbb{K}(\ell^2\Delta_u)\). Recall that a matrix bundle corresponds, in 1-to-1 up to isomorphism, to the associated principal \(PU(u)\)-bundle. Moreover, since we now have a global section of projections \(e_u \otimes 1\), the structure group is reduced to \(P(U(u-1) \times U(1)) \cong U(u-1)\). By a standard argument (see, e.g., [66]), a homotopy connecting two principal \(U(u-1)\)-bundles induces an isomorphism between them. Going back to the associated matrix bundles, we obtain an isomorphism \(\mathcal{D}_u|_{\mathbb{T}^{\Sigma \setminus \{u\}}} \cong \mathbb{T}^{\Sigma \setminus \{u \}} \times \mathbb{K}(\ell^2\Delta_u)\) that sends the global section \(e_u \otimes 1\) to the constant section \(e_u \otimes 1\).

Now, since \(e_u'\) is identified with a constant function in the above trivialization of \(\mathcal{D}_u\), we have \[u[e_u \otimes 1_{C([0,1])^{\Sigma\setminus \{u\} }}] =[1_{C(\mathbb{T}^{\Sigma \setminus \{ u \} },\mathcal{D}_u)}] \in \mathrm{K}_0(C(\mathbb{T}^{\Sigma \setminus \{u\}},\mathcal{D}_{u})) \cong \mathrm{K}_0(C(\mathbb{T}^{\Sigma \setminus \{u\}}) \otimes \mathbb{K}(\ell^2\Delta_{u})).\] Hence there is a continuous path of projections \(\tilde{e}_u(t) \in C(\mathbb{T}^{\Sigma \setminus \{u\} },\mathcal{D}_{u} \otimes \mathcal{O}_{u} \otimes \mathbb{K})\) connecting \(\tilde{e}_u(0)=e_u \otimes 1_{\mathcal{O}_u} \otimes p\) and \(\tilde{e}_u(1)=1_{\mathcal{O}_u} \otimes p\), where \(p\) denotes a fixed rank \(1\) projection in \(\mathbb{K}\). Since \(\delta_u^u(1_{\mathbb{K}(\ell^2\Delta_u)})=e_u\), the element \(\tilde{e}_u\) is indeed contained in the subalgebra \[\begin{align} C(\mathbb{T}^{\Sigma}, \mathcal{D}_u \otimes \mathcal{O}_u \otimes \mathbb{K}) \cong {}&{} \{ b \in C([0,1], C(\mathbb{T}^{\Sigma \setminus \{u\}} , (\mathcal{D}_u \otimes \mathcal{O}_u \otimes \mathbb{K})|_{\mathbb{T}^{\Sigma \setminus \{u\}}})) : \delta_u^u(b(1)) = b(0) \}. \end{align}\]

Since \((B_1, \beta) \cong \bigotimes_{s \in \Sigma} (\mathbb{K}(\ell^2\Delta_s) , \delta^u)\), the unital inclusions \(\mathbb{K}(\ell^2\Delta_s) \to B_1\) induce \(\ast\)-homomorphisms \(\varphi_s \colon C(\mathbb{T}^N,\mathcal{D}_s) \to C(\mathbb{T}^N,\mathcal{B}_1)\) whose images are mutually commutative, and hence \[\Phi \mathrel{\vcenter{:}}= \bigotimes_{s \in \Sigma} (\varphi_s \otimes {\rm id}_{\mathcal{O}_{s} \otimes \mathbb{K}} )\colon \bigotimes_{s \in \Sigma}C(\mathbb{T}^N,\mathcal{D}_{s}\otimes \mathcal{O}_{s} \otimes \mathbb{K}) \to C(\mathbb{T}^N,\mathcal{B}_1\otimes (\mathcal{O}_{s_1} \otimes \cdots \otimes \mathcal{O}_{s_N}) \otimes \mathbb{K}).\] we have an isomorphism of continuous fields \[\begin{align} \mathcal{B}_1 \cong\mathcal{D}_{s_1} \otimes_{\mathbb{T}^N} \cdots \otimes_{\mathbb{T}^N} \mathcal{D}_{s_N} \label{eqn:bundle46assemble} \end{align}\tag{27}\] over \(\mathbb{T}^N\). Here, for continuous fields \(\mathcal{D}_1, \mathcal{D}_2\) of C*-algebras over a locally compact Hausdorff space \(X\), their tensor product \(\mathcal{D}_1 \otimes_X \mathcal{D}_2\) over \(X\) (cf.Blanchard2?) is defined by \[\mathcal{D}_1 \otimes_X \mathcal{D}_2\mathrel{\vcenter{:}}= \bigsqcup_{p \in X} \mathcal{D}_{1,p} \otimes \mathcal{D}_{2,p},\] equipped with the set of continuous sections \[C_0(X,\mathcal{D}_1) \otimes_X C_0(X,\mathcal{D}_2) \mathrel{\vcenter{:}}= C_0(X,\mathcal{D}_1) \otimes C_0(X,\mathcal{D}_2) /I_{X},\] where \(I_X\) the ideal generated by \(af \otimes b - a \otimes fb\), where \(f \in C_0(X)\), \(a \in C_0(X,\mathcal{D}_1)\) and \(b \in C_0(X,\mathcal{D}_2)\). We remark that \(\mathcal{D}_1 \otimes_X \mathcal{D}_2\) is in general an upper semi-continuous field, and it is a continuous field if either \(C_0(X,\mathcal{D}_1)\) or \(C_0(X_2,\mathcal{D}_2)\) is exact (due to McConnell? and Blanchard2?). In our setting, we can verify that \(\mathcal{D}_{s_1}\otimes_{\mathbb{T}^N}\otimes \cdots \otimes_{\mathbb{T}^N} \mathcal{D}_{s_N}\) forms a continuous field directly from an explicit isomorphism 27 . By the isomorphism 27 , This defines a projection \[\tilde{e} \mathrel{\vcenter{:}}= \Phi(\tilde{e}_{s_1} \otimes \cdots \otimes \tilde{e}_{s_N}) = \varphi_{s_1}(\tilde{e}_{s_1})\cdots \varphi_{s_N}(\tilde{e}_{s_N}) \in C(\mathbb{T}^N, \mathcal{B}_1 \otimes (\mathcal{O}_{s_1} \otimes \cdots \otimes \mathcal{O}_{s_N}) \otimes \mathbb{K}).\] For any \(t \in \mathbb{T}^N\), we have \[[\tilde{e}(t)]=[\tilde{e}(1,\cdots,1)]=[1_{\mathbb{K}(\ell^2\Delta_s) \otimes \mathcal{O}_{s_1} \otimes \cdots \otimes \mathcal{O}_{s_N}} \otimes p] \in \mathrm{K}_0(\mathbb{K}(\ell^2\Delta_{\mathfrak{s}}) \otimes \mathcal{O}_{s_1} \otimes \cdots \otimes \mathcal{O}_{s_N} \otimes \mathbb{K}).\] Note that the unit \(1\) and the rank one projection in \(\mathbb{K}(\ell^2\Delta_{\mathfrak{s}})\) represent the same \(g\)-torsion element in this K-group. Now, by choosing a unital embedding \(\iota \colon \mathcal{O}_{s_1} \otimes \cdots \otimes \mathcal{O}_{s_N} \to \mathcal{O}_{g+1}\) by means of the Kirchberg–Phillips theorem, we get the desired projection \(e:=\iota(\tilde{e})\) of \(C(\mathbb{T}^N,\mathcal{B}_1 \otimes \mathcal{O}_{g+1})\). ◻

Proof. By the Künneth theorem, the the cochain complex \((E_1^{p0},d_1^{p0})\) for the bundle \(\mathcal{B} \otimes \mathcal{O}_{g+1} \otimes \mathbb{K}\) is obtained from 25 by tensoring with \(\mathbb{Z}/g\mathbb{Z}\). Since \(s_i -1\) acts on \(\mathbb{Z}/g\mathbb{Z}\) by the identity, we have \(d_1^{pq}=0\). From this, and the Künneth theorem of cochain complexes, we see that the \(E_2\)-page takes the form described in the statement, and moreover that the inclusion \(\mathcal{B} \to \mathcal{B} \otimes \mathcal{O}_{g+1} \otimes \mathbb{K}\) induces an injection on the \(E_2\)-page.

The remaining task is to show the convergence at the \(E_2\)-page. By Lemma 32, \(\iota(f) \mathrel{\vcenter{:}}= e\cdot f\) for \(f \in C(\mathbb{T}^N)\) defines a \(\ast\)-homomorphism \(\iota \colon C(\mathbb{T}^N) \to C(\mathbb{T}^N, \mathcal{B} \otimes \mathcal{O}_{g+1} \otimes \mathbb{K})\) that respects the \(C(\mathbb{T}^N)\)-C*-algebra structure. This induces the map of AH spectral sequences. Meanwhile, the AH spectral sequence for \(\mathrm{K}^0(\mathbb{T}^N)\), say \(\tilde{E}_2^{pq}\), has the torsion-free \(E_1\)-page \(\tilde{E}_1^{p,2q} \cong \mathbb{Z}^{\binom{N}{p}}\) and \(\tilde{E}_1^{p,2q+1} \cong 0\). By comparing their ranks with those of \(\mathrm{K}^0(\mathbb{T}^N) \cong \mathrm{K}^1(\mathbb{T}^N) \cong \mathbb{Z}^{2^{N-1}}\), it follows that \(\tilde{E}_r^{pq}\) collapses at \(E_1\)-page. Moreover, since \([e(t)]\) generates \(\mathrm{K}_0(B \otimes \mathcal{O}_{g+1} \otimes \mathbb{K}) \cong \mathbb{Z}/g\mathbb{Z}\) for any \(t \in \mathbb{T}^N\), one has that \[\iota_* \colon \widetilde{E}_1^{pq} \cong \mathrm{K}_{-p}(C_0(\mathbb{T}_p^N \setminus \mathbb{T}_{p-1}^N)) \to \mathrm{K}_{-p}(C_0(\mathbb{T}_p^N \setminus \mathbb{T}_{p-1}^N , \mathcal{B} \otimes \mathcal{O}_{g+1} \otimes \mathbb{K})) \cong E_1^{pq}\] is surjective.

They show the vanishing of the higher differentials by induction on \(r\). Indeed, by the vanishing of \(d_l^{pq}\) for \(l \leq r-1\), we obtain the commutative diagram \[\xymatrix{ \tilde{E}_{r}^{p,q} \ar@{}[r]|\cong &\mathbb{Z}^{\binom{N}{p}} \ar@{->>}[r] \ar[d]^{\tilde{d}_r^{p,q} =0} & \mathbb{Z}/g\mathbb{Z}^{\binom{N}{p}} \ar@{}[r]|\cong \ar[d]^{d_r^{p,q}} & E_r^{p,q} \\ \tilde{E}_{r}^{p+r,q-r+1} \ar@{}[r]|\cong &\mathbb{Z}^{\binom{N}{p}} \ar@{->>}[r] & \mathbb{Z}/g\mathbb{Z}^{\binom{N}{p}} \ar@{}[r]|\cong & E_r^{p+r,q-r+1}, }\] which shows the vanishing of \(d_r^{pq}\). ◻

Proof. By Lemma 33, the remaining task is to show that \(\mathrm{K}_*(\mathsf{MT}(B,\beta) \otimes \mathcal{O}_{g+1})\) is isomorphic to the direct sum of \(E_\infty\)-terms, that is, there is no nontrivial extension. To see this, notice that an extension \(M\) of \((\mathbb{Z}/g\mathbb{Z})^m\) by \((\mathbb{Z}/g\mathbb{Z})^l\) is not trivial if and only if there is \(x \in M\) such that \(g x \neq 0\). Besides that, the Künneth theorem shows that \(\mathrm{K}_*(C(\mathbb{T}^N,\mathcal{B}) \otimes \mathcal{O}_{g+1})\) is isomorphic to a direct sum of abelian groups of the form \(M_1 \otimes (\mathbb{Z}/g\mathbb{Z})\) and \(\mathrm{Tor}^{\mathbb{Z}}_1 (M_2 , \mathbb{Z}/g\mathbb{Z})\), where \(M_1\) and \(M_2\) are finitely generated. Thus, any element of this group is \(g\)-torsion. This finishes the proof. ◻

Proof. First, we have \(|G \otimes_{\mathbb{Z}} (\mathbb{Z}/g\mathbb{Z})| = |G/\mathrm{Im}(g \colon G \to G) | \leq |G|\) and \(|\operatorname{Tor}^{\mathbb{Z}}_1(G,\mathbb{Z}/g\mathbb{Z})| = |\operatorname{Ker} (g \colon G \to G) | \leq |G|\). This means \(|G \otimes_{\mathbb{Z}} (\mathbb{Z}/g\mathbb{Z}) \oplus \operatorname{Tor}^{\mathbb{Z}}_1(G,\mathbb{Z}/g\mathbb{Z})| \leq |G|^2\) and the equality holds if and only if every element of \(G\) is \(g\)-torsion. This proves the “if part”. To see the “only if part”, assume that every element of \(G\) is \(g\)-torsion. Then, the exact sequence \(0 \to G_{i-1} \to G_{i} \to G_{i}/G_{i-1} \to 0\) splits. Indeed, choosing lifts of a basis of \(G_i/G_{i-1}\cong (\mathbb{Z}/g\mathbb{Z})^{m_i}\) defines a group homomorphism \(G_i/G_{i-1} \to G_i\) splitting the quotient map. This inductively shows that \(G\) is a direct sum of copies of \(\mathbb{Z}/g\mathbb{Z}\). ◻

Proof of Theorem 32. We first consider the case that \(|\Sigma| <\infty\). We first show that \[\mathrm{K}_*(C^*_r(\mathcal{G}_{\mathrm{tor}})) = \mathrm{K}_{*}(\widetilde{B} \rtimes \mathbb{Z}^N) \cong \mathrm{K}_{*-N}(C(\mathbb{T}^N , \mathcal{B})),\] where the isomorphism given in Lemma 30, is isomorphic to \[\mathrm{K}_*\Big(\bigotimes_{s \in \Sigma }\mathcal{O}_{s}\Big) \cong \bigoplus_{p \in * + 2\mathbb{Z}}(\mathbb{Z}/g\mathbb{Z})^{\binom{|\Sigma|-1}{p-1} }.\] Note that, since we have assumed \(N=|\Sigma| \geq 2\), the number of copies of \(\mathbb{Z}/g\mathbb{Z}\) of the \(\mathrm{K}_0\) and \(\mathrm{K}_1\) groups are the same, thus there is no need to care for the degree shift by \(-N\). We have already observed in Lemma 31 that the \(E_2\)-page of the AH spectral sequence for \(\mathrm{K}_*(C(\mathbb{T}^N,\mathcal{B}_1))\) is isomorphic to \((\mathbb{Z}/g\mathbb{Z})^{\binom{|\Sigma|-1}{p-1}}\) for even \(q\) and \(0\) for odd \(q\). Hence it suffices to show that this spectral sequence collapses at the \(E_2\)-page and there is no extension problem. They follow from Lemmas 33 and 34 in the following way.

As is shown in Lemma 33, the inclusion \(C(\mathbb{T}^N,\mathcal{B}_1) \to C(\mathbb{T}^N,\mathcal{B}_1 \otimes \mathcal{O}_{g+1} \otimes \mathbb{K})\) induces injections of \(E_2\)-pages, and the AH spectral sequence for \(\mathcal{B}_1 \otimes \mathcal{O}_{g+1} \otimes \mathbb{K}\) collapses at the \(E_2\)-page. Thus, a diagram chase shows that the higher derivatives \(d_r^{pq}\) for the AH spectral sequence for \(\mathcal{B}_1\) also vanish. Suppose that \(\mathrm{K}_*(C(\mathbb{T}^N,\mathcal{B}_1))\) is a nontrivial extension of \(E_\infty\)-pages. Then, by Lemma 35 and the Künneth theorem in C*-algebra K-theory, the number of elements of \(\mathrm{K}_*(C(\mathbb{T}^N,\mathcal{B}_1) \otimes \mathcal{O}_{g+1})\) is strictly smaller than the square of the number of elements of \(\mathrm{K}_*(C(\mathbb{T}^N,\mathcal{B}_1))\). This contradicts Lemma 34.

By the Kirchberg–Phillips theorem, the remaining task is to show that \([1] \in \mathrm{K}_0(C^*_r(\mathcal{G}_{\mathrm{tor}}))\) satisfies that \(n[1]=0\) if and only if \(g \mid n\). Under the isomorphism of K-theory in Lemma 30, \([1_B]\) corresponds to the image of the Bott generator \(\beta \in \mathrm{K}_N(C_0(\mathbb{R}^N) )\) via the inclusion \[C_0(\mathbb{R}^N) \cong C_0(\mathbb{R}^N) \otimes \mathbb{C}1_B \hookrightarrow C_0(\mathbb{R}^N) \otimes \widetilde{B} \hookrightarrow (C_0(\mathbb{R}^N) \otimes \widetilde{B}) \rtimes \mathbb{Z}^N.\] Up to homotopy and Morita equivalence, this \(\ast\)-homomorphism is identified with the inclusion \[C_0((0,1)^N) \otimes \mathbb{C}1_B \hookrightarrow C(\mathbb{T}^N,\mathcal{B}).\] Hence the Bott generator \(\beta \in \mathrm{K}_N(C_0(\mathbb{R}^N) \otimes \mathbb{C}1_B)\) is sent to the generator of \[E_2^{N0} = \mathop{\mathrm{Coker}}(\bigoplus \beta_s -\iota \colon \bigoplus_s \mathrm{K}_0(D_{\beta_s}) \to \mathrm{K}_0(B)) \cong \mathbb{Z}/g\mathbb{Z},\] which is a subgroup of \(\mathrm{K}_N(C(\mathbb{T}^N,\mathcal{B}))\).

Finally, we consider the case that \(|\Sigma|=\infty\). Even in this case, we have the same K-theory computation \(\mathrm{K}_*(C^*_r(\mathcal{G}_{\mathrm{tor}})) \cong (\mathbb{Z}/g\mathbb{Z})^{\infty}\). To see this, let us choose an increasing sequence of finite subsets \(\Sigma_1 \subset \Sigma_2 \subset \cdots \subset \Sigma\) such that \(\bigcup\Sigma_n =\Sigma\), \(N_n\mathrel{\vcenter{:}}= |\Sigma_n|\) is even and \(\gcd (\{ s-1 : s \in \Sigma_1\}) = g\). Then we have \[\widetilde{B}_{\Sigma} =\varinjlim_{s^a \in \mathbb{N}^{\Sigma}} \bigotimes_{s \in \Sigma} \mathbb{M}_{s^\infty} \cong \varinjlim_{n \to \infty} \bigg( \varinjlim_{s^a \in \mathbb{N}^{\Sigma_n}} \bigotimes_{s \in \Sigma} \mathbb{M}_{s^\infty} \bigg) \cong \varinjlim_{n \to \infty} \bigg( \varinjlim_{s^a \in \mathbb{N}^{\Sigma_n}} \bigotimes_{s \in \Sigma_n} \mathbb{M}_{s^\infty} \bigg) = \varinjlim_{n \to \infty} \widetilde{B}_{\Sigma_n}\] and hence \[C_r^*(\mathcal{G}_{\mathrm{tor}}) \sim_{\mathrm{Morita}} \widetilde{B}_{\Sigma} \rtimes \mathbb{Z}^\Sigma = \varinjlim_n \widetilde{B}_{\Sigma_n} \rtimes \mathbb{Z}^{\Sigma_n} \sim_{\mathrm{Morita}} \varinjlim_n C^*_r(\mathcal{G}_{n,\mathrm{tor}}),\] where \(\mathcal{G}_{n,\mathrm{tor}}\) denotes the torsion subgroupoid associated to \(\Sigma_n\). Through the K-theory isomorphism from Lemma 30, the inclusion \(\widetilde{B}_{\Sigma_n} \rtimes \mathbb{Z}^{N_n} \to \widetilde{B}_{\Sigma_{n+1}} \rtimes \mathbb{Z}^{N_{n+1}}\) corresponds to the inclusion \[\begin{align} \mathsf{MT}(\widetilde{B}_{\Sigma_n},\{ \beta_{s} \}_{s \in \Sigma_n}) \otimes C_0(\mathbb{R}^{N_{n+1}-N_n}) \to \big( \mathsf{MT}(\widetilde{B}_{\Sigma_{n+1}}, \{\beta_{s}\}_{s \in \Sigma_{n}}) \otimes C_0(\mathbb{R}^{N_{n+1}-N_n}) \big) \rtimes \mathbb{Z}^{N_{n+1} - N_n}, \end{align}\] where the left hand side is KK-equivalent to \(\mathsf{MT}(\widetilde{B}_{\Sigma_n},\beta_{n})\) and the right hand side is Morita equivalent to \(\mathsf{MT}(\widetilde{B}_{\Sigma_{n+1}},\beta_{n+1})\). This inclusion is homotopic to \[\begin{align} {}&{}C(\mathbb{T}^{N_{n}}, \mathcal{MT}(\widetilde{B}_{\Sigma_n},\{ \beta_s\}_{s \in \Sigma_n})) \otimes C_0((0,1)^{N_{n+1}-N_n}) \\ \to {}&{} C(\mathbb{T}^{N_{n}}, \mathcal{MT}(\widetilde{B}_{\Sigma_{n+1}},\{\beta\}_{s \in \Sigma_n})) \otimes C_0((0,1)^{N_{n+1}-N_n})) \cong C_0(\mathbb{T}^{N_{n+1}} \setminus \mathbb{T}^{N_n}, \mathcal{MT}(\widetilde{B}_{\Sigma_{n+1}},\{\beta\}_{s \in \Sigma_{n+1}}))\\ \subset {}&{} C(\mathbb{T}^{N_{n+1}} , \mathcal{MT}(\widetilde{B}_{\Sigma_{n+1}},\beta_{n+1})) . \end{align}\] By Lemmas 29 and (the proof of) 31, the induced map in \(E_2\)-pages is \[\mathrm{H}_{N_n-*}\Big( \bigotimes_{s \in \Sigma_n} P(s) \Big) \to \mathrm{H}_{N_{n+1}-*}\Big( \bigotimes_{s \in \Sigma_{n+1}} P(s) \Big),\] induced from the tensor product of the inclusion of chain complexes \((0 \to 0 \to \mathbb{Z} \to 0) \to (0 \to \mathbb{Z} \xrightarrow{s-1} \mathbb{Z} \to 0)\). By the Künneth theorem, this inclusion induces the injective maps to a direct summand in the \(E_2\)-terms. Thus, the above proof of Theorem 32 for the case where \(\Sigma\) is finite shows that \[\mathrm{K}_*(C^*_r(\mathcal{G}_{\mathrm{tor}})) \cong \varinjlim_n \mathrm{K}_*(C^*_r(\mathcal{G}_{n,\mathrm{tor}})) \cong \varinjlim_{n \to \infty} (\mathbb{Z}/g\mathbb{Z})^{\binom{|\Sigma_n|-1}{p-1}} \cong (\mathbb{Z}/g\mathbb{Z})^{\infty}.\] For any \(n\), the unit of \(C^*_r(\mathcal{G}_{n,\mathrm{tor}})\) satisfies \(m[1] =0 \in \mathrm{K}_0(C^*_r(\mathcal{G}_{n,\mathrm{tor}}))\) if and only if \(g \mid m\). Thus so does \([1] \in \mathrm{K}_0(C^*_r(\mathcal{G}_{\mathrm{tor}}))\). This finishes the proof. ◻

Proof. Following [32], let \[\begin{align} i_H^G \colon & \mathrm{KK}_*^H((X,A),Y) \to \mathrm{KK}_*^G(G \times_H (X,A), Y), && i_H^G[M,f,\xi] = [G \times_H M, {\rm id}_G \times_H f , \mathrm{Ind}_H^G \xi] , \\ i_H^G \colon & \mathrm{HH}_H^*((X,A),Y) \to \mathrm{HH}_G^*(G \times_H (X,A), Y), && i_H^G (\varphi) = ({\rm id}_{\mathbb{C}[G]} \otimes_{\mathbb{C}[H]} \varphi) \circ \mu_Y. \end{align}\] Here, the induction map \[\operatorname{Ind}_H^G \colon \mathrm{K}_0^H(M \times Y,\partial M \times Y) \to \mathrm{K}_0^G(G \times _H (M \times Y) , G \times_H (\partial M \times Y) )\] is given by sending \([E,F,u] \in \mathrm{K}_0^H(M \times Y, \partial M \times Y)\), where \(E,F\) are \(H\)-vector bundles over \(M \times Y\) and \(u \colon E|_{\partial M \times Y} \to F|_{\partial M \times Y}\) is an \(H\)-equivariant bundle isomorphism, to \([G \times_H E, G \times_H F, {\rm id}_G \times_H u]\). Note that, since \(Y\) is assumed to be a \(G\)-space, we have an isomorphism of \(G\)-spaces \[G \times _H (M \times Y) \cong (G \times_H M ) \times Y, \quad [g,(m,y)] \mapsto ([g,m],gy).\] We emphasize once again, as already noted in Remark 43, that the composition symbol for HH-classes is taken in the opposite direction from that for morphisms. As in [32], for a \(H\)-sheaf \(F\) on \(X \setminus A\), we canonically regard \(\mathbb{C}[G] \otimes_{\mathbb{C}[H]} F\) as a sheaf on \(G \times_H X\). This gives the induction functor between the categories of nonnegative cochain complexes of \(G\)-equivariant sheaves on \(X \setminus A\) as \[\mathbb{C}[G] \otimes_{\mathbb{C}[H]} \cdot \colon \mathbf{K}_H^+(X \setminus A, \mathbb{C}) \to \mathbf{K}_G^+(G \times_H (X \setminus A) ; \mathbb{C}).\] Since it preserves the \(c\)-softness, the tensor product morphism \({\rm id}_{\mathbb{C}[G]} \otimes_{\mathbb{C}[H]} \varphi\) is regarded as an element of \(\mathrm{HH}_G^*(G \times _H (X,A), G \times_H Y)\). The element \(\mu_Y \in \mathrm{HH}_G^*(G \times_H Y,Y)\) is given by \[\mu_Y \colon R\Gamma_c\mathbb{C}_{G \times_H Y} \cong \mathbb{C}[G] \otimes_{\mathbb{C}[H]} R\Gamma_c\mathbb{C}_Y \to R\Gamma_c \mathbb{C}_Y, \quad \mu_Y(g,s) = g \cdot s \quad \text{ for s \in R\Gamma_c\mathbb{C}_Y. }\] The inverse is given by \[\begin{align} r_G^H \colon & \mathrm{KK}_*^G(G \times _H (X,A), Y) \to \mathrm{KK}_*^H((X,A),Y), && r_G^H[M,f,\xi] = [f^{-1}(H\times_H X),f,\mathrm{Res}_G^H \xi] , \\ r_G^H \colon & \mathrm{HH}_G^*(G \times_H (X,A), Y) \to \mathrm{HH}_H^*((X,A),Y), && r_G^H \psi =\eta_{X\setminus A} \circ \mathrm{Res}_G^H \psi, \end{align}\] where \(\mathrm{Res}_G^H \colon \mathrm{K}^0_G(M \times Y, \partial M \times Y) \to \mathrm{K}^0_H(M \times Y, \partial M \times Y)\) and \(\mathrm{Res}_G^H \colon \mathbf{D}^+(\mathbb{C}[G]) \to \mathbf{D}^+(\mathbb{C}[H])\) are the restriction functors, and \[\eta_{X\setminus A} \in {\rm Hom}\,_{\mathbf{D}^+(\mathbb{C}[H])}(R\Gamma_c\mathbb{C}_{X\setminus A} , R\Gamma_c\mathbb{C}_{G \times_H (X \setminus A)})\] is given by \[\eta_{X\setminus A} \colon R\Gamma_c \mathbb{C}_{X\setminus A} \to R\Gamma_c\mathbb{C}_{G \times_H (X \setminus A)} \cong \mathbb{C}[G] \otimes_{\mathbb{C}[H]} R\Gamma_c \mathbb{C}_{X\setminus A}, \quad \eta_{X \setminus A}(s)= \delta_e \otimes s.\] They are inverses to each other. Indeed, for the \(\mathrm{KK}^G_*\)-functors, both \(r_G^H \circ i_H^G = {\rm id}\) and \(i_H^G \circ r_G^H={\rm id}\) are clear from the definition. To see the same for the \(\mathrm{HH}_G^*\)-functors, let us take a projective resolution \(P^\bullet\) of \(R\Gamma_c\mathbb{C}_{X \setminus A}\) and \(Q^\bullet\) of \(R\Gamma_c\mathbb{C}_Y\). Note that \(\mathbb{C}[G] \otimes_{\mathbb{C}[H]} P^\bullet\) is a projective resolution of \(R\Gamma_c\mathbb{C}_{G \times_H X} \cong \mathbb{C}[G] \otimes_{\mathbb{C}[H]} R\Gamma_c\mathbb{C}_X\). Now, for \(\varphi\in \mathrm{HH}_H^n((X,A),Y) = {\rm Hom}\,_{\mathbf{K}^+(\mathbb{C}[H])} (P,Q[n])\) and \(\psi \in \mathrm{HH}_G^n(G \times_H (X,A),Y) = {\rm Hom}\,_{\mathbf{K}^+(\mathbb{C}[G])}(\mathbb{C}[G] \otimes_{\mathbb{C}[H]} P,Q[n])\), we have \[\begin{gather} (\eta_{X \setminus A} \circ ({\rm id}_{\mathbb{C}[G]} \otimes_{\mathbb{C}[H]}\varphi) \circ \mu_Y) \cdot s = \mu_Y (e \otimes \varphi(s)) = \varphi(s),\\ \mu_Y \circ ({\rm id}_{\mathbb{C}[G]} \otimes_{\mathbb{C}[H]} (\psi \circ \eta_{X \setminus A}))\cdot (g \otimes s) = \mu_Y(g \otimes \psi(e \otimes s)) = g \cdot (\psi (e \otimes s)) = \psi(g \otimes s), \end{gather}\] for any \(s \in P^m\). They prove \(r_G^H \circ i_H^G = {\rm id}\) and \(i_H^G \circ r_G^H={\rm id}\). Here, for the latter equality, we have used the fact that \(\psi\) is by definition a \(\mathbb{C}[G]\)-module map.

The isomorphism of the delocalized parts of the bivariant homology theory is proved from this case. Indeed, for \(\gamma \in G_{\mathrm{tor}}\), we have \[\begin{align} (G \times_H X)^\gamma = {}&{} \{ [g,p] \in G \times_H X : g^{-1} \gamma g =: \gamma' \in H, \gamma' \cdot p=p \}\\ = {}&{} \bigsqcup_{ \substack{[\gamma'] \in \mathcal{C}(H_{\mathrm{tor}})\\ \gamma '=g^{-1}\gamma g}} gZ_G(\gamma') \times_{Z_H(\gamma')} X^{\gamma'} \end{align}\] as \(Z_G(\gamma)\)-spaces (here \(Z_H(\gamma)\) and \(Z_G(\gamma)\) denotes the centralizer subgroups in \(H\) and \(G\) respectively). This shows that the following composition is bijective: \[\begin{align} \bigoplus_{[\gamma] \in \mathcal{C}(G_{\mathrm{tor}})}\mathrm{HH}_{Z_G(\gamma) }^*((G \times_H X)^\gamma , Y^\gamma ) \cong {}&{} \bigoplus_{[\gamma] \in \mathcal{C}(G_{\mathrm{tor}})} \bigoplus_{ \substack{[\gamma'] \in \mathcal{C}(H_{\mathrm{tor}})\\ \gamma '=g^{-1}\gamma g}} \mathrm{HH}_{Z_G(\gamma)}^*(gZ_G(\gamma') \times_{Z_H(\gamma')}X^{\gamma'}, Y^{\gamma})\\ \cong {}&{} \bigoplus_{[\gamma] \in \mathcal{C}(G_{\mathrm{tor}})} \bigoplus_{ \substack{[\gamma'] \in \mathcal{C}(H_{\mathrm{tor}})\\ \gamma '=g^{-1}\gamma g}} \mathrm{HH}_{Z_G(\gamma')}^*(Z_G(\gamma') \times_{Z_H(\gamma')}X^{\gamma'}, Y^{\gamma'})\\ \xrightarrow{\oplus r^{Z_H(\gamma')}_{Z_G(\gamma')}}{}&{} \bigoplus_{[\gamma'] \in \mathcal{C}(H_{\mathrm{tor}})} \mathrm{HH}^*_{Z_H(\gamma')}(X^{\gamma'},Y^{\gamma'}). \end{align}\]

Finally, we verify that Raven’s Chern character commutes with \(r_G^H\). According to ?? , for \(\gamma \in H_{\mathrm{tor}}\), Raven’s Chern character localized at \(\gamma\) of an induced KK-cycle \(i_H^G[M,f,\xi]\), namely \[\mathrm{ch}_R^\gamma \circ i_H^G[ M, f, \xi]) = \mathrm{ch}_R^\gamma([G \times_H M,{\rm id}_G \times_H f, \mathrm{Ind}_H^G\xi]),\] is given by the composition of pull-backs, exterior products with characteristic forms, and the integration, as \[\begin{align} R\Gamma_c\mathbb{C}_{(G \times_H (X \setminus A))^\gamma} \xrightarrow{f^* }{}&{} R\Gamma_c\mathbb{C}_{(G \times_H M)^\gamma} = \Omega_c^\bullet ((G \times _H M)^\gamma ) \xrightarrow{\pi_{M^\gamma} ^* } \Omega_c^\bullet ((G \times_H M)^\gamma ) \otimes \mathbb{C}[Y^\gamma] \\ \xrightarrow{ \mathrm{ch}_G^\gamma (\mathrm{Ind}_H^G \xi)} {}&{} \Omega_c^{\bullet}((G \times_H M)^\gamma ) \otimes \mathbb{C}[Y^\gamma] \xrightarrow{\mathrm{Td}_G^\gamma((G \times_H M)^\gamma ) \otimes {\rm id}} \Omega_c^\bullet((G \times_H M)^\gamma) \otimes \mathbb{C}[Y^\gamma] \\ \xrightarrow{\mathrm{ch}_G^\gamma (\beta_{T(G \times M)^\natural \oplus n}) }{}&{} \Omega_c^{\bullet}((G \times_H (TM \oplus n))^\gamma ) \otimes \mathbb{C}[Y^\gamma] \xrightarrow{\int_{(TM \oplus n)^\gamma } \otimes {\rm id}} \mathbb{C}[Y^\gamma] . \end{align}\] Applying \(r_G^H\) to this composition is nothing but restricting it to the identity component \((eH \times_H X)^\gamma = X^\gamma\). Meanwhile, the restrictions of the characteristic forms \(\mathrm{ch}_G^\gamma(\mathrm{Ind}_H^G \xi)\), \(\mathrm{Td}_G^\gamma(G \times_H M)\), and \(\mathrm{ch}_G^\gamma (\beta_{T(G \times M^\natural) \oplus n})\) to \(M^\gamma\) are nothing but those defined from \([M,f,\xi]\), that is, \(\mathrm{ch}_H^\gamma(\xi)\), \(\mathrm{Td}_H^\gamma(M)\), and \(\mathrm{ch}_H^\gamma (\beta_{TM^\natural \oplus n})\). This concludes that \[r_G^H \circ \mathrm{ch}_{R,G}^\gamma \circ i_H^G [M,f,\xi] = \mathrm{ch}_{R,H}^\gamma [M,f,\xi],\] where \(\mathrm{ch}_{R,G}^\gamma\) and \(\mathrm{ch}_{R,G}^\gamma\) denote Raven’s \(G\)- and \(H\)- equivariant Chern character.

Finally, we show the naturality of \(i_H^G\). If one has a continuous \(H\)-map \(F \colon (X_1,A_1) \to (X_2,A_2)\), which induces \({\rm id}_G \times_H F \colon G \times_H (X_1,A_1) \to G \times_H (X_2,A_2)\), then \[\begin{align} \mathrm{KK}^G_*({\rm id}_G \times_H F,Y) \circ i_H^G[M,f,\xi] ={}&{} [G \times_H M,({\rm id}_G \times_H F) \circ ({\rm id}_G \times_H f),\operatorname{Ind}_H^G\xi] = i_H^G \circ \mathrm{KK}^G_*(F,Y)[M,f,\xi], \\ \mathrm{HH}_G^*({\rm id}_G \times_H F,Y) \circ i_H^G(\varphi) ={}&{} ({\rm id}_{\mathbb{C}[G]} \otimes_{\mathbb{C}[H]} [F]) \circ ({\rm id}_{\mathbb{C}[G]} \otimes_{\mathbb{C}[H]}\varphi) \circ \mu_Y = i_H^G \circ \mathrm{HH}_H^*(F,Y)(\varphi). \end{align}\] This shows that \(i_H^G\) are natural transformations. We also have \[\begin{align} i_H^G [[0,1] \times M, {\rm id}_{[0,1] } \times f,\xi] ={}&{} [[0,1] \times G \times_HM,{\rm id}_{[0,1]} \times {\rm id}_G \times_H f,1_{[0,1]} \otimes \operatorname{Ind}_H^G \xi] = i_H^G[M,f,\xi], \\ i_H^G (\varphi \otimes \alpha)={}&{} ({\rm id}_{\mathbb{C}[G]} \otimes_{\mathbb{C}[H]}( \varphi\otimes \alpha )) \circ (\mu_Y \otimes {\rm id}_{(0,1)}) = i_H^G(\varphi) \otimes \alpha, \end{align}\] which shows that \(i_H^G\) are compatible with the suspension isomorphisms, and hence with the boundary maps \(\delta\) by Remark 42. ◻

Proof. For the definition of the isomorphisms, consider the diagram \[\xymatrix{ \mathrm{KK}^{H}_*((X,A),Y) \ar[r]^{\mathrm{ch}_R^e} \ar[d]^{j_*} & \bigoplus_k \mathrm{HH}_{H}^{-*+2k}((X,A),Y) \ar[d]^{j_* } \\ \mathrm{KK}^{H}_*((X,A),G \times_H Y) \ar[r]^{\mathrm{ch}_R^e} \ar[d]^{i_H^G} & \bigoplus_k \mathrm{HH}_{H}^{-*+2k}((X,A),G \times_H Y) \ar[d]^{i_H^G} \\ \mathrm{KK}^{G}_*(G \times_H (X,A) , G \times_H Y) \ar[r]^{\mathrm{ch}_R^e} \ar[d]^{ \mathrm{KK}_*^G(\alpha,G \times_HY)} & \bigoplus_k \mathrm{HH}_{G}^{-*+2k}(G \times_H (X,A) , G \times_H Y) \ar[d]^{\mathrm{HH}_G^*(\alpha,G \times_HY)}\\ \mathrm{KK}^{G}_*((X,A) , G \times_H Y) \ar[r]^{\mathrm{ch}_R^e} & \bigoplus_k \mathrm{HH}_{G}^{-*+2k}((X,A) , G \times_H Y). }\] Here, \(\alpha \colon G \times_H (X,A) \to (X,A)\) is the \(G\)-action on \(X\) and \(j \colon Y \to G \times _H Y\) denotes the open embedding to \(eH \times _H Y \subset G \times_H Y\). For \(\mathrm{F}^H_* = \mathrm{KK}^H_*\) and \(\bigoplus_{k}\mathrm{HH}_H^{-*+2k}\), this \(j\) induces the morphisms of KK- and HH- groups as the restriction \[j_* \mathrel{\vcenter{:}}= \big( \mathrm{F}^H_*((X,A),J) \colon \mathrm{F}_*^H((X,A), Y^+) \to \mathrm{F}_*^H((X,A), (G \times _H Y)^+) \big)\big|_{\mathrm{F}_*^H((X,A),Y)},\] where \(J \colon (G \times_H Y)^+ \to Y^+\) is the continuous \(H\)-map such that \(J|_{j(Y)} =j^{-1}\) and \(J|_{j(Y)^c}\) is the constant map to the basepoint \(\mathrm{pt} \in Y^+\). More explicitly, this \(j_*\) is described as \(j_*[M,f,\xi] \mathrel{\vcenter{:}}= [M,f,j_*\xi]\) for \([M,f,\xi] \in \mathrm{KK}_*^H((X,A),Y)\), where \[j_* \colon \mathrm{K}^0_H(M \times Y, \partial M \times Y) \to \mathrm{K}_H^0(M \times G \times _H Y, \partial M \times G \times _H Y)\] denotes the open embedding map in equivariant K-theory. Similarly, \(j\) induces the morphism of HH-group as \(\varphi \mapsto \varphi \circ j_*\) and \(\varphi \in {\rm Hom}\,_{\mathbf{D}^+(\mathbb{C}[H])}(R\Gamma_c\mathbb{C}_X, R\Gamma_c \mathbb{C}_Y)\), where \(j_* \in {\rm Hom}\,_{\mathbf{D}^+(\mathbb{C}[H])}(R\Gamma_c\mathbb{C}_Y, \mathbb{C}[G] \otimes_{\mathbb{C}[H]}R\Gamma_c \mathbb{C}_Y)\) denotes the inclusion to \(\mathbb{C}[eH]\otimes_{\mathbb{C}[H]}R\Gamma_c \mathbb{C}_Y\). The middle square commutes by Lemma 36, and the top and bottom squares commute by the naturality of Raven’s Chern character ([32] and Remark 44). Moreover, for any a \(G\)-map \(f \colon (X_1,A_1) \to (X_2, A_2)\), we have \[\begin{align} {}&{}\mathrm{F}_*^G(\alpha, G \times_H Y) \circ i_H^G \circ \mathrm{F}_*^H((X,A),J) \circ \mathrm{F}_*^H(f,Y) \\ ={}&{} \mathrm{F}_*^G(\alpha, G \times_H Y) \circ i_H^G \circ \mathrm{F}_*^H(f,G \times_H Y) \circ \mathrm{F}_*^H((X,A),J) \\ ={}&{}\mathrm{F}_*^G(\alpha, G \times_H Y) \circ \mathrm{F}_*^G ( {\rm id}_G \times_H f,G \times_H Y) \circ i_H^G \circ \mathrm{F}_*^H((X,A),J) \\ = {}&{} \mathrm{F}_*^G(f,G \times_H Y) \circ \mathrm{F}_*^G(\alpha, G \times_H Y) \circ i_H^G \circ \mathrm{F}_*^H((X,A),J), \end{align}\] for both \(\mathrm{F}_*^H=\mathrm{KK}^H_*\) and \(\bigoplus_k \mathrm{HH}^{*+2k}_H\). This proves the naturality of the isomorphism. Here, the first equality follows from Remark 40; the second, from the naturality of \(i_H^G\) proved in Lemma 36; and the third, from the identity \(f \circ \alpha = \alpha \circ ({\rm id}_G \times_H f)\), which holds by \(G\)-equivariance of \(f\). Similarly, since the suspension isomorphism \(\mathrm{F}^H_*(X,Y) \to \mathrm{F}^H_{*+1}(\Sigma X,Y)\) is natural with respect to morphisms in both the first and second variables (Remarks 41 and 42), as well as with respect to \(i_H^G\) (Lemma 36), the above isomorphism intertwines the suspension isomorphism, and hence \(\delta\), as well.

We then show that the two vertical compositions are isomorphisms. The proof is the same as the one for Kasparov KK-theory, given in [68]. First, by a standard Mayer–Vietoris argument comparing the long exact sequences, we reduce the isomorphism to the case that \((X,A)=(G,\emptyset)\). We first consider the case that \(X\) be a free proper finite \(G\)-CW-complex and \(A=\emptyset\). This \(X\) is obtained from \(X_0=\emptyset\) by iterating finitely many times the operation of attaching a \(G\)-cell \(D_n = G \times D^{k_n}\) to \(X_n\) along its boundary to form \(X_{n+1}\). Since \(\mathrm{F}_* \mathrel{\vcenter{:}}= \mathrm{KK}^H_*(\cdot, Y)\) and \(\mathrm{F}_*' \mathrel{\vcenter{:}}= \mathrm{KK}^G_*(\cdot , G \times_H Y)\) (resp.\(\mathrm{F}_* \mathrel{\vcenter{:}}= \mathrm{HH}_H^{-*}(\cdot, Y)\) and \(\mathrm{F}_*' \mathrel{\vcenter{:}}= \mathrm{HH}_G^{-*}(\cdot , G \times_H Y)\)) are \(G\)-equivariant homology functors (cf.Remark 39), it intertwines the boundary homomorphisms \(\delta\), and hence the diagram \[\xymatrix{ \mathrm{F}_{*+1}(D^{k_n},\partial D^{k_n}) \ar[d]^\cong \ar[r]^{\delta} & \mathrm{F}_*(X_n) \ar[r] \ar[d]^\cong & \mathrm{F}_*(X_{n+1}) \ar[r] \ar[d] & \mathrm{F}_*(D^{k_n},\partial D^{k_n}) \ar[r]^\delta \ar[d]^\cong & \mathrm{F}_{*-1}(X_n) \ar[d]^\cong & \\ \mathrm{F}_{*+1}'(D^{k_n},\partial D^{k_n}) \ar[r]^\delta & \mathrm{F}_*'(X_n) \ar[r] & \mathrm{F}_*'(X_{n+1}) \ar[r] & \mathrm{F}_*'(D^{k_n},\partial D^{k_n}) \ar[r]^\delta & \mathrm{F}_{*-1}'(X_n) & }\] commutes by taking as the vertical maps those constructed above. An iterative use of the five lemma proves the isomorphism. For the case where \(A\) is non-empty, applying the same comparison to the long exact sequence for the pair \((X,A)\) yields the desired isomorphism. Finally, for possibly infinite proper \(G\)-CW-pairs \((X,A)\), the groups \(\mathrm{F}(X,A)\) and \(\mathrm{F}'(X,A)\) are both defined as limits (Remark 45), thus the isomorphism established above extends immediately.

In the case of \((X,A)=(G,\emptyset)\), by Lemma 36 we have \[\begin{gather} \mathrm{F}^H(G,Y) \cong \bigoplus_{Hg \in H \backslash G} \mathrm{F}^H(Hg, Y) \cong \bigoplus_{H g \in H \backslash G} \mathrm{F}(\mathrm{pt}, Y), \\ \mathrm{F}^G(G , G \times_H Y) \cong \mathrm{F}(\mathrm{pt},G \times_H Y) \cong \bigoplus_{gH \in G/H} \mathrm{F}(\mathrm{pt}, Y), \end{gather}\] for \(\mathrm{F}^G=\mathrm{KK}_*^G\) or \(\mathrm{HH}^*_G\). We now prove that \(\mathrm{F}^G(\alpha,G \times_H Y) \circ i_H^G \circ j_* \colon \mathrm{F}^H(G,Y) \to \mathrm{F}^G(G ,G \times_H Y)\) is identified, under the above isomorphisms, with the direct sum of the identity maps on the respective summands, where the left and the right coset corresponds by \(Hg \mapsto g^{-1}H\).

We first prove the claim for KK-theory. Let \([M,\mathrm{pr},\xi] \in \mathrm{KK}(\mathrm{pt},Y)\), i.e., \(M\) is a closed \(\mathrm{Spin}^c\)-manifold and \(\xi \in \mathrm{K}^0(M \times Y^+,M \times \mathrm{pt})\). The corresponding element in \(\mathrm{KK}^H(Hg,Y)\) is \([Hg \times M, \mathrm{pr}_{Hg}, \bar{\xi}]\), where \(\bar{\xi} \in \mathrm{K}^0_H(Hg \times M \times Y^+, Hg \times M \times \mathrm{pt})\) is \(H\)-invariant and \(\bar{\xi}|_{\{g\} \times M \times Y} = \mathrm{pr}_{M \times Y}^*\xi\). This element is sent to \[\begin{align} {}&{}\mathrm{KK}^G(\alpha,G \times_H Y) \circ i_H^G \circ \mathrm{KK}^H(\mathrm{pt},J) [Hg \times M, \mathrm{pr}_{Hg}, \bar{\xi}] \\ ={}&{}\mathrm{KK}^G(\alpha,G \times_H Y) \circ i_H^G [Hg \times M, \mathrm{pr}_{Hg}, J^*\bar{\xi}] \\ ={}&{} \mathrm{KK}^G(\alpha,G \times_H Y) [G \times_H Hg \times M, \mathrm{pr}_{G \times_H Hg}, \mathrm{Ind}_H^G \circ ({\rm id}_M \times J)^*\bar{\xi}] \\ ={}&{} [G \times_H Hg \times M, \alpha \circ \mathrm{pr}_{G \times_H Hg}, \mathrm{Ind}_H^G \circ ({\rm id}_M \times J)^*\bar{\xi}] \\ ={}&{} [G \times M, \mathrm{pr}_{G} , (\alpha^{-1} \times {\rm id}_{M} \times {\rm id}_{Y})^* \circ \mathrm{Ind}_H^G \circ ({\rm id}_M \times J)^*\bar{\xi}]. \end{align}\] The last equality is given by the identification of the topological KK-cycles given by the diffeomorphism \(\alpha \times {\rm id}_M\). Now, the element \[(\alpha^{-1} \times {\rm id}_{M} \times {\rm id}_{Y})^* \circ \mathrm{Ind}_H^G \circ ({\rm id}_M \otimes J)^*(\bar{\xi}) \in \mathrm{K}^0_G(G \times M \times (G \times_H Y^+), G \times M \times (G \times_H \mathrm{pt}))\] is \(G\)-invariant, is supported on \[(\alpha \times {\rm id}_M \times {\rm id}_Y) \big( G \cdot ((H \times_H Hg) \times M \times j(Y)) =G \cdot (Hg \times M \times j(Y)),\] and restricts to \(\xi\) on \(\{g\} \times M \times j(Y)\). By the \(G\)-equivariance, its restriction to \(\{e \} \times M \times (G \times _H Y)\) is supported on \(\{e \} \times M \times g^{-1}j(Y)\) and the corresponding restriction is \(g^*\xi \in \mathrm{K}^0(M \times g^{-1}j(Y))\). That is, \[r_G^{\{e\}} \circ \mathrm{KK}^G(\alpha,G \times_H Y) \circ i_H^G \circ j_*[Hg \times M, \mathrm{pr}_{Hg}, \bar{\xi}] = [\{e \} \times M, \mathrm{pr}, g^*\xi] \in \mathrm{KK}(\mathrm{pt}, g^{-1}H \times_H Y).\] By the pull-back via \(g^{-1} \colon gH \times_H Y \to H \times_H Y \cong Y\), we go back to the original element \([M,\mathrm{pr},\xi] \in \mathrm{KK}(\mathrm{pt},Y)\).

We then show the claim for HH-theory. Let \(S_Y^\bullet\) be a \(c\)-soft resolution of the \(G\)-sheaf \(\mathbb{C}_Y\), and let \(\varphi \in {\rm Hom}\,_{\mathbf{K}^+(\mathbb{C})} (\mathbb{C}, \Gamma_cS_Y[n] ) \cong \mathrm{HH} (\mathrm{pt},Y)\) (recall that \(\mathbb{C}_{X}\) is a c-soft resolution as itself if \(X\) is discrete). The corresponding element \(\bar{\varphi} \in {\rm Hom}\,_{\mathbf{K}^+(\mathbb{C}[H])} (\mathbb{C}[Hg], \Gamma_cS_Y[n] ) \cong \mathrm{HH}^H(Hg,Y)\) is characterized by its \(H\)-invariance and \(\bar{\varphi}(\delta_g) = \varphi (1)\). Then \[\begin{align} \begin{aligned} \mathrm{HH}_G^n(\alpha,G \times_H Y) \circ i_H^G \circ j_*(\bar{\varphi}) ={}&{}\mathrm{HH}^G(\alpha,G \times_H Y) \circ i_H^G( \bar{\varphi} \circ [J]) \\ ={}&{} \mathrm{HH}^G(\alpha,G \times_H Y) ( ({\rm id}\otimes \bar{\varphi} \circ [J]) \circ \mu_{G \times_H Y}) \\ ={}&{}[\alpha] \circ ({\rm id}\otimes \bar{\varphi} \circ j_*) \circ \mu_{G \times_H Y}. \end{aligned}\label{eqn:induction246HH} \end{align}\tag{30}\] The right hand side, which is explicitly given as \[\begin{align} \mathbb{C}[G] \xrightarrow{[\alpha]} \mathbb{C}[G \times_H Hg] ={}&{}\mathbb{C}[G] \otimes_{\mathbb{C}[H]}\mathbb{C}[Hg] \xrightarrow{{\rm id}\otimes \bar{\varphi}} \mathbb{C}[G]\otimes_{\mathbb{C}[H]}\Gamma_cS_Y[n] \\ {}&{}\xrightarrow{{\rm id}\otimes j_*}\mathbb{C}[G] \otimes_{\mathbb{C}[H]} \big( \mathbb{C}[G] \otimes_{\mathbb{C}[H]} \Gamma_cS_Y[n] \big) \xrightarrow{\mu_{G \times_HY}} \mathbb{C}[G] \otimes_{\mathbb{C}[H]} \Gamma_cS_Y[n], \end{align}\] sends \(\delta_e \in \mathbb{C}[G]\) as \[\delta_e \mapsto \delta_{g^{-1}} \otimes \delta_g \mapsto \delta_{g^{-1}} \otimes \bar{\varphi}(\delta_g) = \delta_{g^{-1}} \otimes \varphi(1) \mapsto \delta_{g^{-1}} \otimes \delta_e \otimes \varphi(1) \mapsto \delta_{g^{-1}} \otimes g^{-1}\varphi(1).\] This shows that \[\Big( r_G^{\{e\}} \circ \mathrm{HH}_G^n(\alpha,G \times_H Y) \circ i_H^G \circ j_*(\bar{\varphi})\Big) (1) = \delta_{g^{-1}} \otimes g^{-1}\cdot \varphi(1) \in {\rm Hom}\,_{\mathbf{K}^+(\mathbb{C})} (\mathbb{C}, \mathbb{C}[g^{-1}H] \otimes_{\mathbb{C}[H]}\Gamma_cS_Y[n]).\] By the pull-back via \(g^{-1} \colon gH \times_H Y \to H \times_H Y \cong Y\), corresponding to \(g \colon \mathbb{C}[g^{-1}H] \otimes_{\mathbb{C}[H]}\Gamma_cS_Y[n] \to \mathbb{C}[H] \otimes_{\mathbb{C}[H]} \Gamma_cS_Y[n]\), we go back to the original element \(\varphi\). ◻