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 \Lambda of the homotopy quotient EG \times_G X:

H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda).

If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when G is finite.) If G acts freely on X, then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^*(X; \Lambda) = H^*(X/G; \Lambda).

If X is a manifold, G a compact Lie group and \Lambda 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, 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 X by its G-action) in which X 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 G. 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 X(\mathbb{C}), 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 BG is 2-connected and X has real dimension 2. Fix some smooth G-bundle P_\text{sm} on X. Then any principal G-bundle on X is isomorphic to P_\text{sm}. In other words, the set \Omega 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 P_\text{sm} or equivalently the set of holomorphic connections on X (since connections are integrable for dimension reason). \Omega is an infinite-dimensional complex affine space and is therefore contractible.

Let \mathcal{G} be the group of all automorphisms of P_\text{sm} (i.e., gauge group.) Then the homotopy quotient of \Omega by \mathcal{G} classifies complex-analytic (or equivalently algebraic) principal G-bundles on X; i.e., it is precisely the classifying space B\mathcal{G} of the discrete group \mathcal{G}.

One can define the moduli stack of principal bundles \operatorname{Bun}_G(X) as the quotient stack [\Omega/\mathcal{G}] and then the homotopy quotient B\mathcal{G} is, by definition, the homotopy type of \operatorname{Bun}_G(X).

Equivariant characteristic classes[edit]

Let E be an equivariant vector bundle on a G-manifold M. It gives rise to a vector bundle \widetilde{E} on the homotopy quotient EG \times_G M so that it pulls-back to the bundle EG \times E \to EG \times M. An equivariant characteristic class of E is then an ordinary characteristic class of \widetilde{E}, which is an element of the completion of the cohomology ring H^*(EG \times_G M) = H^*_G(M). (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 H^2(M; \mathbb{Z}).[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 H^2_G(M; \mathbb{Z}).

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 H^1(M; \mathbb{C}^*) \simeq H^2(M; \mathbb{Z}) given by the exponential map.

References[edit]

  • M. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984).
  • Michel Brion, Equivariant cohomology and equivariant intersection theory [2]
  • M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131 (1998).
  • Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?". AMS Notices 58 (03): 423–426. 

Further reading[edit]

External links[edit]