A remark on 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.