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Concept
Equivariant cohomology
Summary
In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring with coefficient ring of the homotopy quotient :
If is the trivial group, this is the ordinary cohomology ring of , whereas if is contractible, it reduces to the cohomology ring of the classifying space (that is, the group cohomology of when G is finite.) If G acts freely on X, then the canonical map is a homotopy equivalence and so one gets:
It is also possible to define the equivariant cohomology
of with coefficients in a
module A; these are abelian groups.
This construction is the analogue of cohomology with local coefficients.
If X is a manifold, G a compact Lie group and is the field of real numbers or the field of complex numbers (the most typical situation), then the above cohomology may be computed using the so-called Cartan model (see equivariant differential forms.)
The construction should not be confused with other cohomology theories,
such as Bredon cohomology or the cohomology of invariant differential forms: if G is a compact Lie group, then, by the averaging argument, any form may be made invariant; thus, cohomology of invariant differential forms does not yield new information.
Koszul duality is known to hold between equivariant cohomology and ordinary cohomology.
For a Lie groupoid equivariant cohomology of a smooth manifold is a special example of the groupoid cohomology of a Lie groupoid. This is because given a -space for a compact Lie group , there is an associated groupoidwhose equivariant cohomology groups can be computed using the Cartan complex which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex arewhere is the symmetric algebra of the dual Lie algebra from the Lie group , and corresponds to the -invariant forms.
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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 looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups .