Localization of equivariant cohomology rings of real Grassmannians

Abstract

We use localization method to understand the rational equivariant cohomology rings of real Grassmannians and oriented Grassmannians, then relate this to the Leray-Borel description which says the ring generators are equivariant Pontryagin classes, Euler classes in even dimension, and one more new type of classes in odd dimension, as stated by Casian and Kodama. We give additive basis in terms of equivariant characteristic polynomials and equivariant Schubert/canonical classes. We also calculate Poincaré series, equivariant Littlewood-Richardson coefficients and equivariant characteristic numbers. Since all these Grassmannians with torus actions are equivariantly formal, many results for equivariant cohomology have similar statements for ordinary cohomology.

Updates in V3: chose a more precise title and made some small corrections

References in corpus (4)

• Classification and equivariant cohomology of circle actions on 3d manifolds
• Grassmannians and the equivariant cohomology of isotropy actions
• Towards Generalizing Schubert Calculus in the Symplectic Category
• Localization of certain odd-dimensional manifolds with torus actions