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

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

2010-10-12

arxiv.org/abs/1010.2505v2

Mathematics

Symplectic Geometry

Scientific paper

Let the circle act 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, under some minor dimension restrictions, up to $S^1$-equivariant diffeomorphism, there are finitely many such manifolds in each dimension. Moreover, we show that, up to non-equivariant diffeomorphism, there are finitely many such manifolds in each dimension. In low dimensions, we have uniqueness up to equivariant diffeomorphism or homeomorphism. We use techniques from both areas of symplectic geometry and geometric topology; our main technique for the proof of finiteness is surgery theory.

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