How is the monopole Floer homology of three-manifolds related to the geometry of Riemann surfaces? This paper describes this relationship, focusing on automorphisms of compact 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 . The study investigates the relationship between monopole Floer homology and Riemann surfaces. The authors 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. The result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations. This research provides a deeper understanding of the interplay between topology and geometry, with implications for both theoretical mathematics and mathematical physics.
Published in the Journal of the London Mathematical Society, this article aligns with the journal’s focus on pure mathematics. By exploring the relationship between monopole Floer homology and Riemann surfaces, the paper contributes to advanced mathematical theory, a primary area of interest for the journal.