Equivariant cohomology

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In mathematics, equivariant cohomology is a theory from algebraic topology which applies to spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory.

Specifically, given a group $G$ (discrete or not), a topological space $X$ and an action

$G\times X\rightarrow X,$

equivariant cohomology determines a graded ring

$H^*_GX,$

the equivariant cohomology ring. If $G$ is the trivial group, this is just the ordinary cohomology ring of $X$ , whereas if $X$ is contractible, it reduces to the group cohomology of $G$ .

[edit] Outline construction

Equivariant cohomology can be constructed as the ordinary cohomology of a suitable space determined by $X$ and $G$ , called the homotopy orbit space

X h G

G

on $X$ . (The 'h' distinguishes it from the ordinary orbit space $X G$ .)

If $G$ is the trivial group this space $X h G$ will turn out to be just $X$ itself, whereas if $X$ is contractible the space will be a classifying space for $G$ .

[edit] Properties of the homotopy orbit space

If $G\times X\rightarrow X$ is a free action then $X_{hG}\sim X_G$ .
If $G\times X\rightarrow X$ is a trivial action then $X_{hG}\sim X\times BG$ .
In particular (as a special case of either of the above) if $G$ is trivial then $X_{hG}\sim X$ .

[edit] Construction of the homotopy orbit space

The homotopy orbit space is a “homotopically correct” version of the orbit space (the quotient of $X$ by its $G$ -action) in which $X$ is first replaced by a larger but homotopy equivalent space so that the action is guaranteed to be free.

To this end, construct the universal bundle $EG\rightarrow BG$ for $G$ and recall that $E G$ has a free $G$ -action. Then the product $X\times EG$ —which is homotopy equivalent to $X$ since $E G$ is contractible—has a “diagonal” $G$ -action defined by taking the $G$ -action on each factor: moreover, this action is free since it is free on $E G$ . So we define the homotopy orbit space to be the orbit space of this $G$ -action.