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Let $(M,\omega)$ be a $4$-dimensional closed symplectic manifold. Assume there exists a Hamiltonian $S^1$-action on $M$, let $\mu:M \to \mathbb{R}^*$ be its moment map and let $M_{\text{red}}=\mu^{-1}(r)/S^1$ denote the symplectic reduction/symplectic quotient, where $r$ is a regular value of $\mu$ and where we assume that $S^1$ acts freely on $\mu^{-1}(r)$. For dimensional reasons, $M_{\text{red}}$ is a closed manifold with $\dim M_{\text{red}}=2$, i.e. a closed surface.

Q: (How) can one determine the genus of $M_{\text{red}}$ from the topology of $M$ and the $S^1$-action?

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By a result of Hui Li https://arxiv.org/abs/math/0605133, the fundamental groups of $M_{red}$ and $M$ are isomorphic, so one can deduce the genus of $M_{red}$ from the fundamental group of $M$.

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