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Localization in equivariant cohomology

Just as singular cohomology is a functor from the category of topological spaces to the category of rings, so when a group $ G$ acts on a space $ M$ , one seeks a functor that would incorporate both the topology of the space and the action of the group.

The naive construction of taking the cohomology of the quotient space $ M/G$ is unsatisfactory because for a nonfree action, the topology of the quotient can be quite bad. A solution is to find a contractible space $ EG$ on which $ G$ acts freely, for then $ EG \times M$ will have the same homotopy type as $ M$ and the group $ G$ will act freely on $ EG \times M$ via the diagonal action. It is well known that such a space is the total space of the universal $ G$ -bundle $ EG\to BG$ , whose base space is the classifying space of $ G$ . The homotopy theorists have defined the homotopy quotient $ M_G$ of $ M$ by $ G$ to be the quotient space $ (EG \times M)/G$ , and the equivariant cohomology $ H_G^*(M)$ to be the ordinary cohomology of its homotopy quotient $ M_G$ .

The equivariant cohomology of the simplest $ G$ -space, a point, is already quite interesting, for it is the ordinary cohomology of the classifying space of $ G$ :

$\displaystyle H_G^*(\operatorname{pt})= H^*((EG\times \operatorname{pt})/G )= H^*(EG/G)= H^*(BG).

Since equivariant cohomology is a functor of $ G$ -spaces, the constant map $ M \to \operatorname{pt}$ induces a homomorphism $ H_G^*(\operatorname{pt}) \to H_G^*(M)$ . Thus, the equivariant cohomology $ H_G^*(M)$ has the structure of a module over $ H^*(BG)$ .

Characteristic classes of vector bundles over $ M$ extend to equivariant characteristic classes of equivariant vector bundles.

When $ M$ is a manifold, there is a push-forward map $ \pi_*^M:
H_G^*(M) \to H_G^*(\operatorname{pt})$ , akin to integration along the fiber.

Suppose a torus $ T$ acts on a compact manifold $ M$ with fixed point set $ F$ , and $ \phi \in H_T^*(M)$ is an equivariantly closed class. Let $ P$ be the connected components of $ F$ and let $ \iota_P:P\to M$ be the inclusion map, $ \nu_P$ the normal bundle of $ P$ in $ M$ , and $ e(\nu_P)$ the equivariant Euler class of $ \nu_P$ . In [82] Atiyah and Bott proved a localization theorem for the equivariant cohomology $ H_T^*(M)$ with real coefficients:

$\displaystyle \pi_*^M \phi = \sum_P \pi_*^P \left( \dfrac{\iota_P^* \phi}{e(\nu_P)}

It should be noted that Berline and Vergne [BV] independently proved the same theorem at about the same time.

This localization theorem has as consequences the following results of Duistermaat and Heckman on a symplectic manifold $ (M,\omega)$ of dimension of $ 2n$ :

If a torus action on $ M$ preserves the symplectic form and has a moment map $ f$ , then the push-forward $ f_{*} (\omega^n)$ of the symplectic volume under the moment map is piecewise polynomial.
Under the same hypotheses, the stationary phase approximation for the integral

$\displaystyle \int_M e^{-itf} \dfrac{\omega^n}{n!}

is exact.

In case the vector field on the manifold is generated by a circle action, the localization theorem specializes to Bott's Chern number formulas [41] of the Sixties, thus providing an alternative explanation for the Chern number formulas.

next up previous
Next: Yang-Mills equations over Riemann Up: The life and works Previous: The cohomology of the
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