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Bordered Floer homology and existence of incompressible tori in homology spheres
- Part of
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 6 / June 2018
- Published online by Cambridge University Press:
- 18 May 2018, pp. 1222-1268
- Print publication:
- June 2018
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- Article
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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.
