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 ::A^N = \{ a \in A : na = a \text{ for all } n \in N\}. Then the inflation-restriction exact sequence is: ::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} \to H^2(G/N, A^N). Transgression is defined for general n\in \N, :H^n(N, A)^{G/N} \to H^{n+1}(G/N, A^N), only if H^i(N, A)^{G/N} = 0 for i\le n-1. ==Notes==