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

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 $G/N$ 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 $i\le n-1$.

External links

• transgression at the nLab