Specifically, the equivariant cohomology ring of a space
is the trivial group, this is the ordinary cohomology ring of
is contractible, it reduces to the cohomology ring of the classifying space
If G acts freely on X, then the canonical map
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[citation needed], any form may be made invariant; thus, cohomology of invariant differential forms does not yield new information.
whose equivariant cohomology groups can be computed using the Cartan complex
which is the totalization of the de-Rham double complex of the groupoid.
-action on the dual Lie algebra is trivial.
The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of
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 → BG for G and recall that EG admits a free G-action.
Then the product EG × X —which is homotopy equivalent to X since EG is contractible—admits a “diagonal” G-action defined by (e,x).g = (eg,g−1x): moreover, this diagonal action is free since it is free on EG.
So we define the homotopy quotient XG to be the orbit space (EG × X)/G of this free G-action.
In other words, the homotopy quotient is the associated X-bundle over BG obtained from the action of G on a space X and the principal bundle EG → BG.
This bundle X → XG → BG is called the Borel fibration.
Let X be a complex projective algebraic curve.
We identify X as a topological space with the set of the complex points
Let G be a complex simply connected semisimple Lie group.
Then any principal G-bundle on X is isomorphic to a trivial bundle, since the classifying space
of all isomorphism classes of pairs consisting of a principal G-bundle on X and a complex-analytic structure on it can be identified with the set of complex-analytic structures on
is an infinite-dimensional complex affine space and is therefore contractible.
One can define the moduli stack of principal bundles
(In order to apply Chern–Weil theory, one uses a finite-dimensional approximation of EG.)
Alternatively, one can first define an equivariant Chern class and then define other characteristic classes as invariant polynomials of Chern classes as in the ordinary case; for example, the equivariant Todd class of an equivariant line bundle is the Todd function evaluated at the equivariant first Chern class of the bundle.
(An equivariant Todd class of a line bundle is a power series (not a polynomial as in the non-equivariant case) in the equivariant first Chern class; hence, it belongs to the completion of the equivariant cohomology ring.)
In the non-equivariant case, the first Chern class can be viewed as a bijection between the set of all isomorphism classes of complex line bundles on a manifold M and
The localization theorem is one of the most powerful tools in equivariant cohomology.