Abstract A closed Riemannian three-manifold (Y, g) equipped with a torsion spin ᶜ structure determines a family of Dirac operators \DB\ parametrized by a b₁ (Y) -dimensional torus TY. In this paper, we develop techniques to study how the topology of the locus K TY corresponding to operators with non-trivial kernel (the three-dimensional analogue of the theta divisor of a Riemann surface) depends on the geometry of the metric. As a concrete example of our methods, we show that for any metric on the three-torus Y=T³ for which the spectral gap ₁^* on coexact 1 -forms is large, after a small perturbation of the family, the locus K is 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 spin ᶜ three-manifold (Y, s) with a large ₁^*. 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, s) in terms of the topology of the family of Dirac operators \DB\.
Francesco Lin (Mon,) studied this question.