Transgression map

{{Short description|Concept in algebraic topology}} 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|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>G/N</math> acts on

::<math>A^N = \{ a \in A : na = a \text{ for all } n \in N\}.</math>

Then the inflation-restriction exact sequence is:

::<math>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>H^1(N, A)^{G/N} \to H^2(G/N, A^N)</math>.

Transgression is defined for general <math>n\in \N</math>,

:<math>H^n(N, A)^{G/N} \to H^{n+1}(G/N, A^N)</math>,

only if <math>H^i(N, A)^{G/N} = 0</math> for <math>i\le n-1</math>.<ref name=GS67>Gille & Szamuely (2006) p.67</ref>

==Notes== {{reflist}}

==References== * {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=Cambridge University Press | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }} * {{cite book | page=[https://archive.org/details/handbookofalgebr0003unse/page/282 282] | title=Handbook of Algebra, Volume 1 | first=Michiel | last=Hazewinkel |author-link=Michiel Hazewinkel | publisher=Elsevier | year=1995 | isbn=0444822127 | url=https://archive.org/details/handbookofalgebr0003unse/page/282 }} * {{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=Springer-Verlag | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st }} * {{cite book | pages=112–113 | title=Cohomology of Number Fields | volume=323 | series=Grundlehren der Mathematischen Wissenschaften | first1=Jürgen | last1=Neukirch | authorlink1=Jürgen Neukirch | first2=Alexander | last2=Schmidt | first3=Kay | last3=Wingberg | edition=2nd | publisher=Springer-Verlag | year=2008 | isbn=978-3-540-37888-4 | zbl=1136.11001 }} * {{cite book | page=214 | title=The Solution of The K(GV) Problem | volume=4 | series=Advanced Texts in Mathematics| first=Peter | last=Schmid | publisher=Imperial College Press | year=2007 | isbn=978-1860949708 }} * {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local Fields | translator-link=Marvin Greenberg|translator-first=Marvin Jay|translator-last=Greenberg | series=Graduate Texts in Mathematics | volume=67 | publisher=Springer-Verlag | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | pages=117–118 }}

== External links == *{{nlab|id=transgression}}

Category:Homological algebra Category:Algebraic topology