Bimodules in bordered Heegaard Floer homology

By Robert Lipshitz, Peter S. Ozsv'ath, Dylan Thurston. 2010

122 citations recorded by the source corpus.

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of A(F). We also prove a duality theorem relating the two versions of the 3-manifold invariant. Finally, in the case of a genus one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/45d61ff8194edea30eab8d65d028bb18d99d34e2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Categorifications from planar diagrammatics (2010, background): “ectic manifold X, which is necessarily Calabi-Yau, so that one of the examples on the above list becomes relevant. Triangulated categories behind extended Donaldson-Floer and Ozsv´ath-Szab´o theories [34] are assigned to surfaces and closely related to Fukaya-Floer categories of the representation v”
Bordered Floer homology for manifolds with torus boundary via immersed curves (2016, methodology): “Then the pairing theorem of Lipshitz, Ozsváth and Thurston asserts that the homology of ĈFA(M0, α0, β0) ĈFD(M1, α1, β1) is isomorphic to the Heegaard Floer homology ĤF (M0 ∪hM1) of the closed manifold obtained from the homeomorphism h : ∂M1 → ∂M0 specified by h(α1) = β0 and h(β1) = α0 [25, Theor”
Fukaya categories and bordered Heegaard-Floer homology (2010, background): “egaarddiagramswhere both α- and β-arcsare simultaneously present; and pairing results arise from “top-to-bottom” stacking of cobordisms. On the other hand, the work of Lipshitz, Ozsv´ath and Thurston [6, 7] provides a diﬀerent construction of bimodules associated to cobordisms between decorated surf”
Floer simple manifolds and L-space intervals (2015, unknown): “It suffices to show that ĈFD(Y, μ, λ) is determined for one particular choice of μ and λ, since the invariant of any other choice can then be determined using the change of basis bimodules in [30].”