Bordered Floer homology and existence of incompressible tori in homology spheres

Author name not available (Why is that?)

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

S i g m a (2, 3, 5)

, or it is hyperbolic.

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