Bordered Floer homology and existence of incompressible tori in homology spheres

TL;DR

This paper links bordered Floer homology of knot complements in homology spheres to knot Floer complexes, revealing that homology spheres with trivial Heegaard Floer homology lack incompressible tori and classifying certain L-spaces.

Contribution

It computes the bordered Floer complex of knot complements in homology spheres using knot Floer complexes, providing new insights into the topology of these manifolds.

Findings

Homology spheres with the same Heegaard Floer homology as S^3 contain no incompressible tori.

Irreducible homology sphere L-spaces are either S^3, Poincaré sphere, or hyperbolic.

The paper establishes a relationship between bordered Floer homology and the existence of incompressible tori.

Abstract

Let $K$ denote a knot inside the homology sphere $Y$. The zero-framed longitude of $K$ gives the complement of $K$ in $Y$ the structure of a bordered three-manifold, which may be denoted by $Y(K)$. We compute the quasi-isomorphism type of the bordered Floer complex of $Y(K)$ in terms of the knot Floer complex associated with $K$. As a corollary, we show that if a homology sphere has the same Heegaard Floer homology as $S^3$ it does not contain any incompressible tori. Consequently, if $Y$ is an irreducible homology sphere $L$-space then $Y$ is either $S^3$, or the Poicar\'e sphere $\Sigma(2,3,5)$, or it is hyperbolic.