Transgression map
# Transgression map

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

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

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 $n\in \mathbb {N}$,

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

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

