Topology of the Dirac Equation on Spectrally Large Three-Manifolds

A closed Riemannian three-manifold $(Y,g)$ equipped with a torsion spin ᶜ structure determines a family of Dirac operators \ 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 λ 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 λ 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 \ .