One-connectivity and finiteness of Hamiltonian $S^1$-manifolds with minimal fixed sets

Abstract

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M,\ddot{I}‰)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim (X)+\dim (Y)+2=\dim (M)$. We first show that $X$, $Y$ and $M$ are simply connected. Then we show that, up to $S^1$-equivariant diffeomorphism, there are finitely many such manifolds in each dimension. Moreover, we show that in low dimensions, the manifold is unique in a certain category. We use techniques from both areas of symplectic geometry and geometric topology.