Equivariant cohomology

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 $G$.

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

$X_{hG}$ of $G$

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

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

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.\$

Borel construction of the homotopy orbit space

The homotopy quotient, also called homotopy orbit space or Borel construction, 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 $EG$ has a free $G$-action. Then the product $X\times EG$—which is homotopy equivalent to $X$ since $EG$ 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 $EG$. So we define the homotopy orbit space to be the orbit space of this $G$-action.

This construction is denoted by

$X_{hG} = X\times_G EG.$