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
The trangression 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 AN = { a 
A : na = a for all n N}. Then the inflation-restriction exact sequence is:
- 0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A)
The transgression map is the map H 1(N, A)G/N → H 2(G/N, AN)
Transgression is defined for general n
- Hn(N, A)G/N → Hn+1(G/N, AN)
only if Hi(N, A)G/N = 0 for i ≤ n-1.[1]
References
- ↑ Gille & Szamuely (2006) p.67
- Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 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. 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 323 (2nd ed.). Springer-Verlag. pp. 112–113. ISBN 3-540-37888-X. Zbl 1136.11001.
- Schmid, Peter (2007). The Solution of The K(GV) Problem. Advanced Texts in Mathematics 4. Imperial College Press. p. 214. ISBN 1860949703.
- Serre, Jean-Pierre (1979). Local fields. Graduate Texts in Mathematics 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. pp. 117–118. ISBN 0-387-90424-7. Zbl 0423.12016.