Quantum cohomology and $S^1$-actions with isolated fixed points

Abstract

This paper studies symplectic manifolds that admit semi-free circle actions with isolated fixed points. We prove, using results on the Seidel element due to McDuff and Tolman, that the (small) quantum cohomology of a $2n$ dimensional manifold of this type is isomorphic to the (small) quantum cohomology of a product of $n$ copies of $\bb{P}^1$. This generalizes a result due to Tolman and Witsman.