Bordered Floer homology for manifolds with torus boundary via immersed curves

By J. Hanselman, Jacob Rasmussen, Liam Watson. 2016

101 citations recorded by the source corpus.

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 $\widehat{\mathit{CFD}}$ may be viewed as a set of immersed curves decorated with local systems in $\partial M$. 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 $\widehat{\mathit{HF}}$ 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 $\widehat{\mathit{HF}}$. In particular, it follows that a prime rational homology sphere $Y$ with $\widehat{\mathit{HF}}(Y)<5$ 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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/d7bf54f26a811b0ec20a92687f47e4b6ba75f137

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Auslander orders over nodal stacky curves and partially wrapped Fukaya categories (2017, background): “1]: it appears as the algebra associated to the punctured torus (see also [11]).”