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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 acts on a space , 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 is unsatisfactory because for a nonfree action, the topology of the quotient can be quite bad. A solution is to find a contractible space on which acts freely, for then will have the same homotopy type as and the group will act freely on via the diagonal action. It is well known that such a space is the total space of the universal -bundle , whose base space is the classifying space of . The homotopy theorists have defined the homotopy quotient of by to be the quotient space , and the equivariant cohomology to be the ordinary cohomology of its homotopy quotient .

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

Since equivariant cohomology is a functor of -spaces, the constant map induces a homomorphism . Thus, the equivariant cohomology has the structure of a module over .

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

When is a manifold, there is a push-forward map , akin to integration along the fiber.

Suppose a torus acts on a compact manifold with fixed point set , and is an equivariantly closed class. Let be the connected components of and let be the inclusion map, the normal bundle of in , and the equivariant Euler class of . In [82] Atiyah and Bott proved a localization theorem for the equivariant cohomology with real coefficients:

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 of dimension of :

1)
If a torus action on preserves the symplectic form and has a moment map , then the push-forward of the symplectic volume under the moment map is piecewise polynomial.
2)
Under the same hypotheses, the stationary phase approximation for the integral

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.

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