A remark on taut foliations and Floer homology

TL;DR

This paper strengthens the connection between taut foliations and Floer homology by showing the homology contains a direct summand, which could help identify counterexamples to the L-space conjecture through Floer theory.

Contribution

It proves that the reduced Floer homology of rational homology spheres with taut foliations has a direct summand as an $F$-vector space, advancing the understanding of their structure.

Findings

Reduced Floer homology has a direct $F$-summand.

Potential to detect counterexamples to the L-space conjecture.

Strengthens the link between taut foliations and Floer homology.

Abstract

It is well-known that the reduced Floer homology of a rational homology sphere admitting a taut foliation does not vanish. We strengthen this by showing that (when thought of as an $\mathbb{F}[U]$-module) it also admits a direct $\mathbb{F}$-summand. Hence, one could potentially detect counterexamples to the foliation part of the $L$-space conjecture via purely Floer theoretic computations.