# Monopole Floer homology and SOLV geometry

## TL;DR

This paper investigates the monopole Floer homology of SOLV rational homology spheres using spectral theory and Fourier analysis, demonstrating that certain metrics lead to no irreducible solutions, thus confirming Y as an L-space.

## Contribution

It introduces a spectral theory approach to monopole Floer homology on SOLV manifolds and proves that specific metrics prevent irreducible solutions, establishing Y as an L-space.

## Key findings

- Small regular perturbations do not admit irreducible solutions on suitable SOLV metrics.
- Provides a geometric proof that SOLV rational homology spheres are L-spaces.
- Uses Fourier analysis on solvable groups in Floer homology context.

## 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.