In mathematics, equivariant cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a groupaction. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring with coefficient ring of the homotopy quotient : If is the trivial group, this is the ordinary cohomology ring of, whereas if is contractible, it reduces to the cohomology ring of the classifying space If G acts freely on X, then the canonical map is a homotopy equivalence and so one gets: It is also possible to define the equivariant cohomology of with coefficients in a -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 is the field of real numbers or the field of complex numbers, then the above cohomology may be computed using the so-called Cartan model 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.
Homotopy quotient
The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space in which 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 bundleEG → 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.g = : moreover, this diagonal action is free since it is free on EG. So we define the homotopy quotient XG to be the orbit space /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 bundleEG → BG. This bundle X → XG → BG is called the Borel fibration.
An example of a homotopy quotient
The following example is Proposition 1 of . Let X be a complex projective algebraic curve. We identify X as a topological space with the set of the complex points, which is a compact Riemann surface. 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 is 2-connected and X has real dimension 2. Fix some smooth G-bundle on X. Then any principal G-bundle on is isomorphic to. In other words, the set 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 or equivalently the set of holomorphic connections on X. is an infinite-dimensional complex affine space and is therefore contractible. Let be the group of all automorphisms of Then the homotopy quotient of by classifies complex-analytic principal G-bundles on X; i.e., it is precisely the classifying space of the discrete group. One can define the moduli stack of principal bundles as the quotient stack and then the homotopy quotient is, by definition, the homotopy type of.
Equivariant characteristic classes
Let E be an equivariant vector bundle on a G-manifold M. It gives rise to a vector bundle on the homotopy quotient so that it pulls-back to the bundle over. An equivariant characteristic class of E is then an ordinary characteristic class of, which is an element of the completion of the cohomology ring. 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. 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 In the equivariant case, this translates to: the equivariant first Chern gives a bijection between the set of all isomorphism classes of equivariant complex line bundles and.
Localization theorem
The localization theorem is one of the most powerful tools in equivariant cohomology.
