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

TL;DR

This paper proves that Hamiltonian $S^1$-manifolds with exactly two fixed components and minimal fixed set dimensions are simply connected, finite in number up to equivariant diffeomorphism, and often unique in low dimensions.

Contribution

It establishes finiteness and simple connectivity of such manifolds, and demonstrates uniqueness in low-dimensional cases, combining symplectic and topological methods.

Findings

Manifolds are simply connected.

Finitely many such manifolds exist in each dimension.

In low dimensions, the manifold is unique.

Abstract

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. 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.