Monopole Floer homology and invariant theta characteristics

Abstract

AbstractWe describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism of a compact Riemann surface with quotient , there is a natural correspondence between theta characteristics on which are invariant under and self‐conjugate structures on the mapping torus of . We show that the monopole Floer homology groups of are explicitly determined by the eigenvalues of the (lift of the) action of on , the space of holomorphic sections of , 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.