Restriction functor on cohomology

General definition
Suppose $$G_1,G_2$$ are groups and $$A$$ is an abelian group. Suppose $$\varphi_1:G_1 \to \operatorname{Aut}(A)$$, $$\varphi_2: G_2 \to \operatorname{Aut}(A)$$, and $$\alpha: G_1 \to G_2$$ are homomorphisms such that $$\varphi_2 \circ \alpha = \varphi_1$$, i.e., the $$G_1$$-action and $$G_2$$-action on $$A$$ are compatible. Then, we get an induced homomorphism between the cohomology groups:

$$\operatorname{res}^{G_2}_{G_1}: H^*_{\varphi_2}(G_2,A) \to H^*_{\varphi_1}(G_1,A)$$

This homomorphism is termed the restriction homomorphism and the functor that sends the map $$\alpha:G_1 \to G_2$$ to this homomorphism is the restriction functor on cohomology.

Note that the direction of this homomorphism is reverse to the direction of the original homomorphism. The association gives a contravariant functor.

Note that it is sufficient to specify $$\varphi_2$$ and $$\alpha$$ since $$\varphi_1$$ is defined in terms of them. In other words, we can define the restriction functor in terms of $$\varphi_2:G_2 \to \operatorname{Aut}(A)$$ and $$\alpha: G_1 \to G_2$$.

Typical case
The typical case where we talk of the restriction functor is where the map $$\alpha: G_1 \to G_2$$ is injective, and we naturally identify $$G_1$$ with its image subgroup of $$G_2$$. We thus talk of restricting cohomology to a subgroup.

Related notions

 * Corestriction functor on homology
 * Inflation functor on cohomology
 * Coinflation functor on homology