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

Main article: 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 ${\displaystyle G/N}$ 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]

Notes

1. Gille & Szamuely (2006) p.67

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