Topology of the Dirac equation on spectrally large three-manifolds
Francesco Lin
TL;DR
This work investigates Dirac operators on closed orientable 3-manifolds with a large spectral gap, defined by the first coexact-form eigenvalue $\lambda_1^*$, and connects linear spectral data to non-linear Seiberg–Witten theory and monopole Floer homology. On the 3-torus with spectrally large metrics, the locus $\mathsf{K}$ of flat connections $[B]$ for which the twisted Dirac operator has nontrivial kernel is shown to be diffeomorphic to $S^2$, a result proven using perturbations of the Chern–Simons–Dirac functional and a detailed obstruction theory for transversality. The authors develop a framework for perturbing the Seiberg–Witten equations to achieve transversality while maintaining a detailed handle on the solution spaces, defining a notion of simplest type manifolds (type $1$) and deriving explicit descriptions of $\widehat{HM}_*(Y,\mathfrak{s})$ via a spectral sequence whose $E^1$-page encodes the topology of the Jacobian and the triple cup product $\cup_Y^3$. Concrete computations for $S^1\times\Sigma_h$ ($h=2,3$) illustrate the method, linking to theta divisors and yielding explicit Floer groups, while the main results for $T^3$ provide a precise Floer-theoretic invariant and bridge to Heegaard-type computations. Overall, the paper illuminates how spectral geometry constraints on $\lambda_1^*$ shape monopole Floer theory and yield explicit topological invariants for spectrally large 3-manifolds.
Abstract
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 $λ_1^*$ 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 $T^3$, 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 spin$^c$ three-manifold $(Y,\mathfrak{s})$ with a large spectral gap $λ_1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak{s})$ in terms of the topology of the family of Dirac operators parametrized by the torus of flat $U(1)$-connections on $Y$.