Bordered Heegaard Floer homology: Invariance and pairing

Abstract

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

277 pages, 59 figures; v6: published version, with additional small corrections and one Erratum for a missing holomorphic curve

References in corpus (13)

• Floer homology and knot complements
• Holomorphic disks and link invariants
• Floer homology and surface decompositions
• Legendrian knots, transverse knots and combinatorial Floer homology
• Bimodules in bordered Heegaard Floer homology
• Fukaya categories and deformations
• Contact homology and virtual fundamental cycles
• Cables of thin knots and bordered Heegaard Floer homology
• Bordered Floer homology for sutured manifolds
• Computing HF^ by factoring mapping classes
• Knot doubling operators and bordered Heegaard Floer homology
• Combinatorial cobordism maps in hat Heegaard Floer theory
• Floer homology and splicing knot complements

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