Equivariant cohomology

(Redirected from Borel construction)

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 ${\displaystyle \Lambda }$ of the homotopy quotient ${\displaystyle EG\times _{G}X}$:

${\displaystyle H_{G}^{*}(X;\Lambda )=H^{*}(EG\times _{G}X;\Lambda ).}$

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

If X is a manifold, G a compact Lie group and ${\displaystyle \Lambda }$ 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[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

The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of ${\displaystyle X}$ by its ${\displaystyle G}$-action) in which ${\displaystyle X}$ 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 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

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

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

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

Equivariant characteristic classes

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

Localization theorem

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

Notes

1. ^ using Čech cohomology and the isomorphism ${\displaystyle H^{1}(M;\mathbb {C} ^{*})\simeq H^{2}(M;\mathbb {Z} )}$ given by the exponential map.

References

• 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.