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Bordered Heegaard Floer homology


About this Title

Robert Lipshitz, Peter S. Ozsvath and Dylan P. Thurston

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 254, Number 1216
ISBNs: 978-1-4704-2888-4 (print); 978-1-4704-4748-9 (online)
DOI: https://doi.org/10.1090/memo/1216
Published electronically: June 13, 2018
Keywords:Three-manifold topology, low-dimensional topology, Heegaard Floer homology, holomorphic curves, extended topological field theory

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. $\mathcal {A}_{\infty }$ structures
  • Chapter 3. The algebra associated to a pointed matched circle
  • Chapter 4. Bordered Heegaard diagrams
  • Chapter 5. Moduli spaces
  • Chapter 6. Type $D$ modules
  • Chapter 7. Type $A$ modules
  • Chapter 8. Pairing theorem via nice diagrams
  • Chapter 9. Pairing theorem via time dilation
  • Chapter 10. Gradings
  • Chapter 11. Bordered manifolds with torus boundary
  • Appendix A. Bimodules and change of framing

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 ) is a module over the algebra and the other of which (type ) is an module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the tensor product of the type module of one piece and the type module from the other piece is 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 . We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

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