Corestriction functor on homology

Definition

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:

$\operatorname {cor} _{G_{1}}^{G_{2}}:H_{*,\varphi _{1}}(G_{1},A)\to H_{*,\varphi _{2}}(G_{2},A)$

This homomorphism is termed the corestriction homomorphism and the functor that sends the map $\alpha :G_{1}\to G_{2}$ to this homomorphism is the corestriction functor on homology.

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 $G_{2}$.

Related notions

• Restriction functor on cohomology
• Inflation functor on cohomology
• Coinflation functor on homology