Equivariant cohomology

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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 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 (that is, the group cohomology of when G is finite.) If G acts freely on X, then the canonical map is a homotopy equivalence and so one gets:

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 so-called Cartan model (see equivariant differential forms.)

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[citation needed], 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[edit]

The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of by its -action) 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 bundle EGBG 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 EGBG. This bundle XXGBG is called the Borel fibration.

An example of a homotopy quotient[edit]

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 , 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 (since connections are integrable for dimension reason). is an infinite-dimensional complex affine space and is therefore contractible.

Let be the group of all automorphisms of (i.e., gauge group.) Then the homotopy quotient of by classifies complex-analytic (or equivalently algebraic) 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[edit]

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 . An equivariant characteristic class of E is then an ordinary characteristic class of , which is an element of the completion of the cohomology ring . (In order to apply the 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 [1] 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[edit]

The localization theorem is one of the most powerful tools in equivariant cohomology.

See also[edit]

Notes[edit]

  1. ^ using Čech cohomology and the isomorphism given by the exponential map.

References[edit]

  • Atiyah, Michael; Bott, Raoul (1984), "The moment map and equivariant cohomology", Topology, 23
  • Michel Brion, "Equivariant cohomology and equivariant intersection theory" [2]
  • Goresky, Mark; Kottwitz, Robert; MacPherson, Robert (1998), "Equivariant cohomology, Koszul duality, and the localization theorem", Inventiones Mathematicae, 131: 25–83, doi:10.1007/s002220050197
  • Hsiang, Wu-Yi (1975). Cohomology Theory of Topological Transformation Groups. New York: Springer.
  • Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?" (PDF). Notices of the American Mathematical Society. 58 (03): 423–426.

Further reading[edit]

External links[edit]