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 {\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} .
Notes
References 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. Hazewinkel, Michiel (1995). Handbook of Algebra, Volume 1. Elsevier. p. 282. ISBN 0444822127. 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. Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. Vol. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. pp. 117–118. ISBN 0-387-90424-7. Zbl 0423.12016.
External links transgression at the nLab