In mathematics, equivariant cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring
Contents
- Homotopy quotient
- An example of a homotopy quotient
- Equivariant characteristic classes
- Localization theorem
- References
If
If X is a manifold, G a compact Lie group and
The construction should not be confused as a more naive 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.
The 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 (the quotient of
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.
An example of a homotopy quotient
The following example is Proposition 1 of [1].
Let X be a complex projective algebraic curve. We identify X as a topological space with the set of the complex points
Let
One can define the moduli stack of principal bundles
Equivariant characteristic classes
Let E be an equivariant vector bundle on a G-manifold M. It gives rise to a vector bundle
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
Localization theorem
The localization theorem is one of the most powerful tools in equivariant cohomology.