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.
See main article: 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
AN=\{a\inA:na=aforalln\inN\}.
Then the inflation-restriction exact sequence is:
0\toH1(G/N,AN)\toH1(G,A)\toH1(N,A)G/N\toH2(G/N,AN)\toH2(G,A).
The transgression map is the map
H1(N,A)G/N\toH2(G/N,AN)
Transgression is defined for general
n\in\N
Hn(N,A)G/N\toHn+1(G/N,AN)
only if
Hi(N,A)G/N=0
i\len-1
. Jean-Pierre Serre . . Marvin Greenberg. Marvin Jay. Greenberg . . 67 . . 1979 . 0-387-90424-7 . 0423.12016 . 117–118 .