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
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
, which is an element of the completion of the cohomology ring
(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.