Bordered Floer homology and existence of incompressible tori in homology spheres

Abstract

Let $Y$ be a homology sphere which contains an incompressible torus. We show that $Y$ cannot be an $L$-space, i.e. the rank of $\widehat{\text{HF}}(Y)$ is greater than $1$. In fact, if the homology sphere $Y$ is an irreducible $L$-space, then $Y$ is $S^{3}$, the Poincaré sphere $\unicode[STIX]{x1D6F4}(2,3,5)$ or hyperbolic.