In algebraic topology, a transgression map is a way to transfer cohomology classes. It occurs, for example in the inflation-restriction exact sequence in group cohomology, and in integration in fibers. It also naturally arises in many spectral sequences; see spectral sequence#Edge maps and transgressions.

Inflation-restriction exact sequence

The transgression map appears in the inflation-restriction exact sequence, an exact sequence occurring in group cohomology. Let G be a group, N a normal subgroup, and A an abelian group which is equipped with an action of G, i.e., a homomorphism from G to the automorphism group of A. The quotient group G / N {\displaystyle G/N} acts on

A N = { a ∈ A : n a = a for all n ∈ N } . {\displaystyle A^{N}=\{a\in A:na=a{\text{ for all }}n\in N\}.}

Then the inflation-restriction exact sequence is:

0 → H 1 ( G / N , A N ) → H 1 ( G , A ) → H 1 ( N , A ) G / N → H 2 ( G / N , A N ) → H 2 ( G , A ) . {\displaystyle 0\to H^{1}(G/N,A^{N})\to H^{1}(G,A)\to H^{1}(N,A)^{G/N}\to H^{2}(G/N,A^{N})\to H^{2}(G,A).}

The transgression map is the map H 1 ( N , A ) G / N → H 2 ( G / N , A N ) {\displaystyle H^{1}(N,A)^{G/N}\to H^{2}(G/N,A^{N})}.

Transgression is defined for general n ∈ N {\displaystyle n\in \mathbb {N} },

H n ( N , A ) G / N → H n + 1 ( G / N , A N ) {\displaystyle H^{n}(N,A)^{G/N}\to H^{n+1}(G/N,A^{N})},

only if H i ( N , A ) G / N = 0 {\displaystyle H^{i}(N,A)^{G/N}=0} for i ≤ n − 1 {\displaystyle i\leq n-1}.

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