Monopole Floer homology and SOLV geometry

Francesco Lin

doi:10.5802/ahl.56

Abstract

We study the monopole Floer homology of a SOLV rational homology sphere Y from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable SOLV metrics on Y, small regular perturbations of the Seiberg-Witten equations do not admit irreducible solutions; in particular, this provides a geometric proof that Y is an L-space.

Highlights

Among the three-dimensional model geometries, Solv, i.e. R3 equipped with the metric e2zdx2 + e−2zdy2 + dz[2], is the least symmetric one [Sco83]
Within the classification scheme of Thurston’s geometrization theorem; they can be characterized as the geometric manifolds which are neither Seifert nor hyperbolic. Their importance stems from the fact that many Solv manifolds arise as cusps of Hirzebruch modular surfaces [Hir73]; and the understanding of their signature defect was the main motivation behind the discovery of the Atiyah–Patodi–Singer index theorem for manifolds with boundary [APS75], see [ADS83]
In this paper we study the monopole Floer homology of a Solv rational homology sphere Y from a geometric viewpoint

Summary

Introduction

Among the three-dimensional model geometries, Solv, i.e. R3 equipped with the metric e2zdx2 + e−2zdy2 + dz[2], is the least symmetric one [Sco83]. The main ingredient in the proof of Theorem 1.1 is the following relation, for a rational homology sphere, between the existence of irreducible solutions to the Seiberg–Witten equations and the first eigenvalue λ∗1 of the Hodge Laplacian on coexact 1-forms (which improves on the main result of [Lin17]). As these metrics have λ∗1 is exactly 1, they lie in the borderline case of Theorem 1.3, and transversality is a quite subtle issue.

Compact Solvmanifolds and their Fourier analysis

The spectrum on coexact 1-forms

Transversality