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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.

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