2024/01/05 by Francesco Lin, Lin, Francesco · 1 citation
Mathematics · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2401.02724
openalex publication_date 2024/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The interaction between spin geometry and positive scalar curvature has been extensively explored. In this paper, we instead focus on Dirac operators on Riemannian three-manifolds for which the spectral gap λ1^* of the Hodge Laplacian on coexact 1-forms is large compared to the curvature. As a concrete application, we show that for any spectrally large metric on the three-torus T3, the locus in the torus of flat U(1)-connections where (a small generic pertubation of) the corresponding twisted Dirac operator has kernel is diffeomorphic to a two-sphere. While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spinc three-manifold (Y,\mathfraks) with a large spectral gap λ1^*. When b1>0, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of (Y,\mathfraks) in terms of the topology of the family of Dirac operators parametrized by the torus of flat U(1)-connections on Y.