Monopole Floer homology and invariant theta characteristics - Lin, Francesco (Journal of the London Mathematical Society, 2024)
We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism φ$φ$ of a compact Riemann surface Σ$Σ$ with quotient P1$\mathbb {P}^1$, there is a natural correspondence between theta characteristics L$L$ on Σ$Σ$ which are invariant under φ$φ$ and self‐conjugate spinc${\text{spin}}^c$ structures sL$\mathfrak {s}_L$ on the mapping torus Mφ$M_{φ }$ of φ$φ$. We show that the monopole Floer homology groups of (Mφ,sL)$(M_{φ },\mathfrak {s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of φ$φ$ on H0(L)$H^0(L)$, the space of holomorphic sections of L$L$, and discuss several consequences of this description. Our result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations.