Dirac spectral flow and Floer theory of hyperbolic three-manifolds

TL;DR

This paper links hyperbolic geometry with monopole Floer homology for certain three-manifolds, providing the first explicit computations of Floer complexes in this setting by analyzing Dirac eigenvalue crossings using advanced trace formulas.

Contribution

It introduces a geometric approach to describe monopole Floer homology for hyperbolic three-manifolds and performs the first explicit calculations of these Floer complexes.

Findings

Floer homology can be described using geodesic data.

First computations of Floer chain complexes for hyperbolic 3-manifolds.

Eigenvalue crossings analyzed via Selberg trace formulas.

Abstract

We study the interplay between hyperbolic geometry and monopole Floer homology for a closed oriented three-manifold $Y$ with $b_1=1$ equipped with a torsion spin$^c$ structure $\mathfrak{s}$. We show that, under favorable circumstances, one can completely describe the Floer theory of $(Y,\mathfrak{s})$ purely in terms of geometric data such as the lengths and holonomies of closed geodesics. In particular, we perform the first computations of monopole Floer chain complexes with non-trivial homology for hyperbolic three-manifolds. The examples we consider admit no irreducible solutions to the Seiberg-Witten equations, and the non-triviality of the Floer homology groups is a consequence of the geometry of the $1$-parameter family of Dirac operators associated to flat spin$^c$ connections. The main technical challenge is to understand explicitly how the Dirac eigenvalues with small absolute value cross the value zero in this family; we tackle this using Fourier analytic tools via the corresponding $1$-parameter family of odd Selberg trace formulas and its derivative.