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 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 [math]\displaystyle{ G/N }[/math] acts on
- [math]\displaystyle{ A^N = \{ a \in A : na = a \text{ for all } n \in N\}. }[/math]
Then the inflation-restriction exact sequence is:
- [math]\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). }[/math]
The transgression map is the map [math]\displaystyle{ H^1(N, A)^{G/N} \to H^2(G/N, A^N) }[/math].
Transgression is defined for general [math]\displaystyle{ n\in \N }[/math],
- [math]\displaystyle{ H^n(N, A)^{G/N} \to H^{n+1}(G/N, A^N) }[/math],
only if [math]\displaystyle{ H^i(N, A)^{G/N} = 0 }[/math] for [math]\displaystyle{ i\le n-1 }[/math].[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.
- Hazewinkel, Michiel (1995). Handbook of Algebra, Volume 1. Elsevier. p. 282. ISBN 0444822127. https://archive.org/details/handbookofalgebr0003unse/page/282.
- Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci.. 62 (2nd printing of 1st ed.). Springer-Verlag. ISBN 3-540-63003-1.
- 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.
- 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. Springer-Verlag. pp. 117–118. ISBN 0-387-90424-7.
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