Equivariant cohomology

If ${\displaystyle G}$ acts freely, then the equivariant cohomology ring is just the singular cohomology ring of the quotient space ${\displaystyle X/G}$. In particular, if ${\displaystyle G}$ is the trivial group, then the equivariant cohomology ring is the ordinary cohomology ring of ${\displaystyle X}$. If ${\displaystyle X}$ is contractible, it reduces to the cohomology ring of the classifying space ${\displaystyle BG}$ (that is, the group cohomology of ${\displaystyle G}$ when ${\displaystyle G}$ is finite.)

Definitions

Let ${\displaystyle X}$ be a topological space, with a (left) action by a group ${\displaystyle G}$. Let ${\displaystyle EG}$ be a contractible space with a free (right) action by the same group. Define ${\displaystyle EG{\times }_{G}X}$ to be the quotient space ${\displaystyle (EG\times X)/\sim }$, where the equivalence relation ${\displaystyle \sim }$ identifies ${\displaystyle (e\cdot g,x)}$ with ${\displaystyle (e,g\cdot x)}$ for all ${\displaystyle e\in EG,x\in X,g\in G}$. The equivariant cohomology ring of ${\displaystyle X}$ (with respect to the group ${\displaystyle G}$) is then defined as the singular cohomology ring of this space:$${\displaystyle H_G^{\ast }(X):=H^{\ast }(EG{\times }_{G}X).}$$The space ${\displaystyle EG{\times }_{G}X}$ is known as the Borel construction, or homotopy quotient. Projection onto the first factor gives it the structure of a fibre bundle over the classifying space ${\displaystyle BG}$. This is the associated ${\displaystyle X}$-bundle obtained from the action of ${\displaystyle G}$ on ${\displaystyle X}$ and the principal bundle ${\displaystyle EG\to BG}$. The bundle ${\displaystyle X\to EG{\times }_{G}X\to BG}$ is called the Borel fibration.

The group cohomology ring of ${\displaystyle G}$ can be recovered by taking ${\displaystyle X}$ to be a point, as in this case the homotopy quotient is just the classifying space of ${\displaystyle G}$. If ${\displaystyle G}$ acts freely on ${\displaystyle X}$, then the canonical map ${\displaystyle EG{\times }_{G}X\to X/G}$ is a homotopy equivalence and so one gets $H_G^{\ast }(X)=H^{\ast }(X/G).$

It is also possible to define the equivariant cohomology ${\displaystyle H_G^{\ast }(X;A)}$ of ${\displaystyle X}$ with coefficients in a ${\displaystyle G}$-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 coefficients are taken in 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.

Relation with groupoid cohomology

For a Lie groupoid ${\displaystyle \mathfrak{X}=[X_1\rightrightarrows X_0]}$ equivariant cohomology of a smooth manifold is a special example of the groupoid cohomology of a Lie groupoid. This is because given a ${\displaystyle G}$-space ${\displaystyle X}$ for a compact Lie group ${\displaystyle G}$, there is an associated groupoid

${\displaystyle {\mathfrak{X}}_G=[G\times X\rightrightarrows X]}$

whose equivariant cohomology groups can be computed using the Cartan complex ${\displaystyle {\mathrm{\Omega }}_G^{\bullet }(X)}$ which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex are

${\displaystyle {\mathrm{\Omega }}_G^n(X)=\underset{2k+i=n}{⨁}({\text{Sym}}^k({\mathfrak{g}}^{\vee })\otimes {\mathrm{\Omega }}^i(X){)}^G}$

where ${\displaystyle {\text{Sym}}^{\bullet }({\mathfrak{g}}^{\vee })}$ is the symmetric algebra of the dual Lie algebra from the Lie group ${\displaystyle G}$, and ${\displaystyle (-{)}^G}$ corresponds to the ${\displaystyle G}$-invariant forms. This is a particularly useful tool for computing the cohomology of ${\displaystyle BG}$ for a compact Lie group ${\displaystyle G}$ since this can be computed as the cohomology of

${\displaystyle [G\rightrightarrows \ast ]}$

where the action is trivial on a point. Then,

${\displaystyle H_{dR}^{\ast }(BG)=\underset{k\ge 0}{⨁}{\text{Sym}}^{2k}({\mathfrak{g}}^{\vee }{)}^G}$

For example,

${\displaystyle \begin{array}{cc}{H}_{dR}^{\ast }(BU(1))& =\underset{k=0}{⨁}{\text{Sym}}^{2k}({\mathbb{R}}^{\vee })\\ & \cong \mathbb{R}[t]\\ & \text{ where }\mathrm{deg}(t)=2\end{array}}$

since the ${\displaystyle U(1)}$-action on the dual Lie algebra is trivial.