Coinflation functor on homology

Definition

Suppose $G$ is a group and $N$ is a normal subgroup of $G$. Suppose $A$ is an abelian group and $\varphi :G\to \operatorname {Aut} (A)$ is a homomorphism of groups, making $A$ into a $G$-module. Denote by $A_{N}$ the quotient of $A$ by the subgroup of elements of the form $\varphi (g)a-a$, with $g\in N$.

Then, the coinflation homomorphism $\operatorname {coinf} :H_{*}(G,A)\to H_{*}(G/N,A_{N})$ is defined as the composite:

$H_{*}(G,A)\to H_{*}(G,A/N){\stackrel {\operatorname {cor} }{\to }}H_{*}(G/N,A_{N})$

where the first map is the natural map obtained by viewing cohomology as a covariant functor in its second coordinate, applied to the surjection $A\to A_{N}$ of $G$-modules, and the second map is obtained by applying the corestriction functor on homology to the quotient map $G\to G/N$.

Related notions

• Restriction functor on cohomology
• Corestriction functor on homology
• Coinflation functor on homology