ar X iv : m at h / 03 09 06 8 v 1 [ m at h . D G ] 4 S ep 2 00 3 GENERALIZING THE LOCALIZATION FORMULA IN EQUIVARIANT COHOMOLOGY

We give two generalizations of the Atiyah-Bott-Berline-Vergne lo-calization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calculating the Gysin homomorphism in ordinary co-homology of an equivariant map. As an example, we recover a formula of Akyildiz-Carrell for the Gysin homomorphism of flag manifolds. Suppose M is a compact oriented manifold on which a torus T acts. The Atiyah-Bott-Berline-Vergne localization formula calculates the integral of an equivariant cohomology class on M in terms of an integral over the fixed point set M T. This formula has found many applications, for example, in analysis, topology, symplec-tic geometry, and algebraic geometry (see [2], [10], [12], [16]). Similar, but not entirely analogous, formulas exist in K-theory ([3]), cobordism theory ([14]), and algebraic geometry ([11]). In this article we consider the problem of generalizing the localization formula in equivariant cohomology in two different directions: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the T-manifold M by a T-equivariant map f : M − → N. To begin, we give an example to show why there does not exist a localization formula for a compact connected Lie group G in terms of the fixed point set of the group. Taking cues from the work of Atiyah and Segal in K-theory [3], we state and prove a localization formula for a compact connected Lie group in terms of the fixed point set of a conjugacy class in the group. This is followed by a localization formula for a T-equivariant map. Both formulas generalize the ABBV localization formula. As an application, both of our formulas can be used to calculate the Gysin homomorphism in ordinary cohomology of an equivariant map. For a compact connected Lie group G with maximal torus T and a closed subgroup H containing T , we work out as an example the Gysin homomorphism of the canonical projection f : G/T − → G/H, a formula first obtained by Akyildiz and Carrell [1]. The application to the Gysin map in this article complements that of [16]. The previous article [16] shows how to use the ABBV localization formula to calculate the Gysin map of a fiber bundle. This article shows how to use the relative localization formula …