Exercise 14, § 6

Let G be a group, H a subgroup of finite index of G. Put $n = (G : H)$. a) Let Q, M be two $\mathbf{Z}^{(G)}$-modules. Now prove that there exists a $\mathbf{Z}$-homomorphism

such that, for every system of representatives $g_1, \ldots, g_n$ of the classes of G modulo H, If $j : \mathrm{Hom}_{\mathbf{Z}(G)}(Q, M) \to \mathrm{Hom}_{\mathbf{Z}(H)}(Q, M)$ is the canonical injection, one has $t \circ j = n.\mathrm{Id}$. b) Let N be a right $\mathbf{Z}^{(G)}$-module. With the notations of a), now prove that there exists a $\mathbf{Z}$-homomorphism $\theta : N \otimes_{\mathbf{Z}(G)} Q \to N \otimes_{\mathbf{Z}(H)} Q$ such that If $p : N \otimes_{\mathbf{Z}(H)} Q \to N \otimes_{\mathbf{Z}(G)} Q$ is the canonical mapping, one has $p \circ \theta = n.\mathrm{Id}$. c) Let P be a projective resolution of the $\mathbf{Z}^{(G)}$-module $\mathbf{Z}$, q an integer. Applying a) and b) with $Q = P_q$, define homomorphisms $t^q : H^q(H, M) \to H^q(G, M)$ and $\theta_q : H_q(G, N) \to H_q(H, N)$. If $i$ denotes the canonical injection of H into G, one has $t^q \circ i^q = n.\mathrm{Id}$ and $i_q \circ \theta_q = n.\mathrm{Id}$. d) Consider the $\mathbf{Z}^{(G)}$-modules $\hat{M} = \mathrm{Hom}_{\mathbf{Z}(H)}(\mathbf{Z}^{(G)}, M)$ and $\tilde{N} = N \otimes_{\mathbf{Z}(H)} \mathbf{Z}^{(G)}$ (exercise 13). Taking $Q = \mathbf{Z}^{(G)}$, one obtains from the homomorphisms t and $\theta$ of a) and b) homomorphisms $t_M : \hat{M} \to M$ and $\theta_N : N \to \tilde{N}$. Now prove that these homomorphisms are $\mathbf{Z}^{(G)}$-linear. Now prove that the composed homomorphism (resp. $H_q(G, N) \xrightarrow{H_q(G, \theta_N)} H_q(G, \tilde{N}) \xrightarrow{(\tilde{i}_q)^{-1}} H_q(H, N)$) is equal to $t^q$ (resp. $\theta_q$).

e) Let $(g_1, \ldots, g_n)$ be a system of representatives of the classes of G modulo H, and let $x \in H^0(H, M)$. Now prove that one has $t^0(x) = \sum_{i=1}^n g_i x$. Describe $t^1$. f) If one identifies the groups $H_1(G, \mathbf{Z})$ and $H_1(H, \mathbf{Z})$ with $G/(G, G)$ and $H/(H, H)$ respectively, the homomorphism $t_1$ is identified with a homomorphism ($\text{transfer}$) $V : G/(G, G) \to H/(H, H)$. If $\tilde{s} : H/G \to G$ is a section of the canonical mapping, now prove that V is obtained by passing to the quotient from the mapping $V_s : G \to H$ such that