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Bordered Floer homology for manifolds with torus boundary via immersed curves

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Rasmussen, Jacob 
Hanselman, Jonathan 
Watson, Liam 


This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M is such a manifold, we show that the type D structure CFD(M) may be viewed as a set of immersed curves decorated with local systems in ∂M. These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of HF decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of HF. In particular, it follows that a prime rational homology sphere Y with HF (Y ) < 5 must be geometric. Other results include a new proof of Eftekhary’s theorem that L-space homology spheres are atoroidal; a complete characterisation of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.



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Journal of the American Mathematical Society

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American Mathematical Society

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Engineering and Physical Sciences Research Council (EP/M000648/1)
Part of this work was completed while the authors were visiting the Newton Institute