Monopole Floer homology and invariant theta characteristics

2022/05/09 by Lin, Francesco · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2205.04351

Abstract

We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. Consider an automorphism φ of a compact Riemann surface Σ with quotient ℙ¹. There is a natural correspondence between theta characteristics L on Σ which are invariant under φ and self-conjugate spin^c structures \mathfraks_L on the mapping torus M_φ of φ. We show that the monopole Floer homology groups of (M_φ,\mathfraks_L) are explicitly determined by the eigenvalues of the (lift of the) action of φ on H⁰(L), the space of holomorphic sections of L. Decategorifying our computation, we also obtain that the dimension of H⁰(L) equals the Reidemeister-Turaev torsion of (M_φ,\mathfraks_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 φ of prime order in terms of ramification data.

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