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.

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 {\displaystyle G/N} acts on

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

Then the inflation-restriction exact sequence is:

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

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

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

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

• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
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• Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. ISBN 3-540-63003-1. Zbl 0819.11044.
• 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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