Monopole Floer homology and invariant theta characteristics

TL;DR

This paper establishes a direct link between monopole Floer homology of certain three-manifolds and the geometry of Riemann surfaces, providing explicit computations based on automorphism actions and ramification data.

Contribution

It introduces a novel relationship connecting monopole Floer homology with invariant theta characteristics and automorphisms of Riemann surfaces, enabling explicit calculations.

Findings

Floer homology groups are determined by automorphism eigenvalues.

Dimension of holomorphic sections equals Reidemeister-Turaev torsion.

Explicit computations for automorphisms of prime order.

Abstract

We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. Consider an automorphism $\varphi$ of a compact Riemann surface $\Sigma$ with quotient $\mathbb{P}^1$. There is a natural correspondence between theta characteristics $L$ on $\Sigma$ which are invariant under $\varphi$ and self-conjugate spin$^c$ structures $\mathfrak{s}_L$ on the mapping torus $M_{\varphi}$ of $\varphi$. We show that the monopole Floer homology groups of $(M_{\varphi},\mathfrak{s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of $\varphi$ on $H^0(L)$, the space of holomorphic sections of $L$. Decategorifying our computation, we also obtain that the dimension of $H^0(L)$ equals the Reidemeister-Turaev torsion of $(M_{\varphi},\mathfrak{s}_L)$. Finally, we combine our description with the Atiyah-Bott $G$-spin theorem to provide explicit computations of the Floer homology groups for all automorphisms $\varphi$ of prime order in terms of ramification data.