Bordered Floer homology and existence of incompressible tori in homology spheres

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 Poicaré sphere $\Sigma (2,3,5)$, or it is hyperbolic.

References in corpus (5)

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• Ricci Flow and the Poincare Conjecture
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Cited by in corpus (7)

• Branched covers of quasipositive links and L-spaces
• Toroidal homology spheres and SU(2)-representations
• Rank inequalities for the Heegaard Floer homology of branched covers
• The Poincare homology sphere, lens space surgeries, and some knots with tunnel number two
• Correction to the article: Floer homology and splicing knot complements
• Knot surgery formulae for instanton Floer homology I: the main theorem
• Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture