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 ${\tau }_p^{+}$ be the equivariant Thom class dual to $W_p^{+}$. These classes form a basis of $H_G^{\ast }(M)$ as a module over $\SS(\fg^*)$ and, in particular, $${\tau }_p^{+}{\tau }_q^{+}=\sum c_{pq}^r{\tau }_r^{+}$$ with $c_{pq}^r \in \SS(\fg^*)$. For manifolds of GKM type we obtain a combinatorial description of these ${\tau }_p^{+}$'s and, from this description, a combinatorial formula for $c_{pq}^r$.

30 pages