Combinatorial formulas for products of Thom classes

Abstract

Let G be a torus of dimension n > 1 and M 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 $\{ W_p^+ ; p \in M^G\}$ 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 $\SS(\fg^*)$ and, in particular, $$τ_p^+ τ_q^+ = \sum c_{pq}^r τ_r^+$$ with $c_{pq}^r \in \SS(\fg^*)$. For manifolds of GKM type we obtain a combinatorial description of these $τ_p^+$'s and, from this description, a combinatorial formula for $c_{pq}^r$.