Floer Simple Manifolds and L-Space Intervals

TL;DR

This paper introduces an invariant for Floer simple manifolds with torus boundary that determines L-space filling intervals from torsion data, providing new proofs and partial solutions to classification and gluing conjectures.

Contribution

It constructs an invariant to compute L-space intervals from torsion and slope data, advancing understanding of Floer simple manifolds and their L-space fillings.

Findings

Constructed an invariant for Floer simple manifolds

Provided a new proof of Seifert fibered L-space classification

Proved a case of a conjecture on L-spaces from gluing three-manifolds

Abstract

An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call "Floer simple," we construct an invariant which computes the interval of L-space filling slopes from the Turaev torsion and a given slope from the interval's interior. As applications, we give a new proof of the classification of Seifert fibered L-spaces over $S^2$, and prove a special case of a conjecture of Boyer and Clay about L-spaces formed by gluing three-manifolds along a torus.