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 acts on AN = { a
The transgression map is the map H 1(N, A)G/N → H 2(G/N, AN)
Transgression is defined for general n
Hn(N, A)G/N → Hn+1(G/N, AN)only if Hi(N, A)G/N = 0 for i ≤ n-1.