Equivariant cohomology


In mathematics, equivariant cohomology (or Borel 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 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 (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:

Definitions edit

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 (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.

Koszul duality is known to hold between equivariant cohomology and ordinary cohomology.

Relation with groupoid cohomology edit

For a Lie groupoid   equivariant cohomology of a smooth manifold[1] is a special example of the groupoid cohomology of a Lie groupoid. This is because given a  -space   for a compact Lie group  , there is an associated groupoid


whose equivariant cohomology groups can be computed using the Cartan complex   which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex are


where   is the symmetric algebra of the dual Lie algebra from the Lie group  , and   corresponds to the  -invariant forms. This is a particularly useful tool for computing the cohomology of   for a compact Lie group   since this can be computed as the cohomology of


where the action is trivial on a point. Then,


For example,


since the  -action on the dual Lie algebra is trivial.

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   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  . (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  [2] 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. ^ Behrend 2004
  2. ^ 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: 1–28, doi:10.1016/0040-9383(84)90021-1
  • Brion, M. (1998). "Equivariant cohomology and equivariant intersection theory" (PDF). Representation Theories and Algebraic Geometry. Nato ASI Series. Vol. 514. Springer. pp. 1–37. arXiv:math/9802063. doi:10.1007/978-94-015-9131-7_1. ISBN 978-94-015-9131-7. S2CID 14961018.
  • Goresky, Mark; Kottwitz, Robert; MacPherson, Robert (1998), "Equivariant cohomology, Koszul duality, and the localization theorem", Inventiones Mathematicae, 131: 25–83, CiteSeerX, doi:10.1007/s002220050197, S2CID 6006856
  • Hsiang, Wu-Yi (1975). Cohomology Theory of Topological Transformation Groups. Springer. doi:10.1007/978-3-642-66052-8. ISBN 978-3-642-66052-8.
  • Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?" (PDF). Notices of the American Mathematical Society. 58 (3): 423–6. arXiv:1305.4293.

Relation to stacks edit

  • Behrend, K. (2004). "Cohomology of stacks" (PDF). Intersection theory and moduli. ICTP Lecture Notes. Vol. 19. pp. 249–294. ISBN 9789295003286. PDF page 10 has the main result with examples.

Further reading edit

  • Guillemin, V.W.; Sternberg, S. (1999). Supersymmetry and equivariant de Rham theory. Springer. doi:10.1007/978-3-662-03992-2. ISBN 978-3-662-03992-2.
  • Vergne, M.; Paycha, S. (1998). "Cohomologie équivariante et théoreme de Stokes" (PDF). Département de Mathématiques, Université Blaise Pascal.

External links edit

  • Meinrenken, E. (2006), "Equivariant cohomology and the Cartan model" (PDF), Encyclopedia of mathematical physics, pp. 242–250, ISBN 978-0-12-512666-3 — Excellent survey article describing the basics of the theory and the main important theorems
  • "Equivariant cohomology", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Young-Hoon Kiem (2008). "Introduction to equivariant cohomology theory" (PDF). Seoul National University.
  • What is the equivariant cohomology of a group acting on itself by conjugation?