# Transgression map

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{{Short description|Concept in algebraic topology}}
In algebraic topology, a '''transgression map''' is a way to transfer [cohomology](/source/cohomology) classes.
It occurs, for example in the [inflation-restriction exact sequence](/source/inflation-restriction_exact_sequence) in [group cohomology](/source/group_cohomology), and in [integration in fiber](/source/integration_in_fiber)s. It also naturally arises in many [spectral sequence](/source/spectral_sequence)s; see [spectral sequence#Edge maps and transgressions](/source/spectral_sequence).

== Inflation-restriction exact sequence ==
{{main|Inflation-restriction exact sequence}}

The transgression map appears in the [inflation-restriction exact sequence](/source/inflation-restriction_exact_sequence), an [exact sequence](/source/exact_sequence) occurring in [group cohomology](/source/group_cohomology).  Let ''G'' be a [group](/source/group_(mathematics)), ''N'' a [normal subgroup](/source/normal_subgroup), and ''A'' an [abelian group](/source/abelian_group) which is equipped with an action of ''G'', i.e., a [homomorphism](/source/homomorphism) from ''G'' to the [automorphism group](/source/automorphism) 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](/source/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](/source/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](/source/Springer_Science%2BBusiness_Media) | 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](/source/Local_Fields) | translator-link=Marvin Greenberg|translator-first=Marvin Jay|translator-last=Greenberg | series=[Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics) | volume=67 | publisher=[Springer-Verlag](/source/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

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