Proposition 7

In the notation from above, the following diagram commutes:

Ex$_{\tau}(G,F)^{u^*}$ // Ex$_{\tau}(G',F)$

$\Theta_{\tau}\Theta_{\tau'}$

$H_2(G,F)^{u^*}$ // $H_2(G',F)$.

Let H be a subgroup of G of finite index. Let $s$ be a section of the canonical surjection from G to $H\backslash G$. We denote by $(g, x)\mapsto x\cdot g$ the right action of G on $H\backslash G$ induced by the right action of G on itself by multiplication. Note that for every $x\in H\backslash G$ and $g\in G$, the element $s(x)gs(x\cdot g)^{-1}$ belongs to H. For any mapping $c: H\times H\rightarrow F$, we can therefore define a mapping $\widetilde{c}_s: G\times G\rightarrow F$ by the relation

for $g_1,g_2\in G$. The mapping $c\mapsto \widetilde{c}_s$ is a group homomorphism from $F^{H\times H}$ to $F^{G\times G}$.