Combinatorial Formulas for Products of Thom Classes

Abstract

Let G be a torus of dimension n>1 and M be a compact Hamiltonian G-manifold with M ^G finite. A circle S ¹ in G is generic if M ^G=M ^{S 1}. For such a circle the moment map associated with its action on M is a perfect Morse function. Let {W ⁺ _p ;p∈ M ^G} be the Morse-Whitney stratification of M associated with this function and let τ ⁺ _p be the equivariant Thom class dual to W ⁺ _p . These classes form a basis of H ^* _G (M) as a module over $\mathbb{S}(\mathfrak{g}*)$ and, in particular,

$$ \tau _p^ + \tau _q^ + = \sum {c_{pq}^r \tau _r^ + }$$

with $c_{pq}^r \in \mathbb{S}(\mathfrak{g}*)$. For a large class of manifolds of this type we obtain a combinatorial description of these τ ⁺ _p s and, from this description, a combinatorial formula for c ^r _pg .