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.

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 ${\displaystyle G/N}$ acts on

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

Then the inflation-restriction exact sequence is:

${\displaystyle 0\to H^{1}(G/N,A^{N})\to H^{1}(G,A)\to H^{1}(N,A)^{G/N}\to H^{2}(G/N,A^{N})\to H^{2}(G,A).}$

The transgression map is the map ${\displaystyle H^{1}(N,A)^{G/N}\to H^{2}(G/N,A^{N})}$.

Transgression is defined for general ${\displaystyle n\in \mathbb {N} }$,

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

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

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• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. pp. 112–113. ISBN 978-3-540-37888-4. Zbl 1136.11001.
• Schmid, Peter (2007). The Solution of The K(GV) Problem. Advanced Texts in Mathematics. Vol. 4. Imperial College Press. p. 214. ISBN 978-1860949708.
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• transgression at the nLab

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