Combinatorial Formulas for Products of Thom Classes

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 pg r .