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arXiv · 1010.2505

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

Hui Li·Martin Olbermann·Donald Stanley
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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.

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Hui Li, Martin Olbermann, Donald Stanley. 2010-10-12. One-connectivity and finiteness of Hamiltonian $S^1$-manifolds with minimal fixed sets. https://doi.org/10.1112/jlms%2Fjdv029

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