In algebraic topology, a transgression map is a way to transfer

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

classes. It occurs, for example in the inflation-restriction exact sequence in

group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...

, and in

integration in fiber In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where ''m'' is the dimension of the fiber, via " integration". It is also called the fiber integration. Definition Let \pi: E \to B be a fiber bundle over ...

s. It also naturally arises in many

spectral sequence In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they h ...

s; see spectral sequence#Edge maps and transgressions.

Inflation-restriction exact sequence

The transgression map appears in the inflation-restriction exact sequence, an

exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definit ...

occurring in

. Let ''G'' be a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

, ''N'' a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

, and ''A'' an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

which is equipped with an action of ''G'', i.e., a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

from ''G'' to the

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

of ''A''. The quotient group

G/N

acts on ::

A^N = \.

Then the inflation-restriction exact sequence is: ::

0 \to H^1(G/N, A^N) \to H^1(G, A) \to H^1(N, A)^ \to H^2(G/N, A^N) \to H^2(G, A).

The transgression map is the map

H^1(N, A)^ \to H^2(G/N, A^N)

. Transgression is defined for general

n\in \N

, :

H^n(N, A)^ \to H^(G/N, A^N)

, only if

H^i(N, A)^ = 0

for

i\le  n-1

.Gille & Szamuely (2006) p.67

Notes

References

* * * * * *

External links

* Homological algebra Algebraic topology {{topology-stub