**Memoirs of the American Mathematical Society**

2018;
279 pp;
Softcover

MSC: Primary 57;

**Print ISBN: 978-1-4704-2888-4
Product Code: MEMO/254/1216**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4748-9
Product Code: MEMO/254/1216.E**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

# Bordered Heegaard Floer Homology

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*Robert Lipshitz; Peter Ozsváth; Dylan P. Thurston*

The authors 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
\(\mathcal A_\infty\) module. Both are well-defined up to chain
homotopy equivalence. For a decomposition of a 3-manifold into two
pieces, the \(\mathcal A_\infty\) tensor product of the type
\(D\) module of one piece and the type \(A\) module from
the other piece is \(\widehat{HF}\) of the glued manifold.

As a special case of the construction, the authors 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
\(\widehat{HF}\). The authors relate the bordered Floer
homology of a three-manifold with torus boundary with the knot Floer
homology of a filling.

#### Table of Contents

# Table of Contents

## Bordered Heegaard Floer Homology

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 110
- Chapter 2. 𝒜_{∞} structures 918
- Chapter 3. The algebra associated to a pointed matched circle 2938
- Chapter 4. Bordered Heegaard diagrams 4554
- Chapter 5. Moduli spaces 6170
- 5.1. Overview of the moduli spaces 6170
- 5.2. Holomorphic curves in \textalt{Σ×[0,1]×\RR}Sigma × [0,1] × R 6473
- 5.3. Holomorphic curves in \textalt{\RR×𝑍×[0,1]×\RR} R × Z × [0,1] × R 7079
- 5.4. Compactifications via holomorphic combs 7382
- 5.5. Gluing results for holomorphic combs 8291
- 5.6. Degenerations of holomorphic curves 8998
- 5.7. More on expected dimensions 99108

- Chapter 6. Type 𝐷 modules 109118
- Chapter 7. Type 𝐴 modules 145154
- Chapter 8. Pairing theorem via nice diagrams 157166
- Chapter 9. Pairing theorem via time dilation 161170
- Chapter 10. Gradings 189198
- Chapter 11. Bordered manifolds with torus boundary 205214
- 11.1. Torus algebra 205214
- 11.2. Surgery exact triangle 208217
- 11.3. Preliminaries on knot Floer homology 209218
- 11.4. From \textalt{\CFDa}CFDˆ to \textalt{\HFKm}HFK- 212221
- 11.5. From \textalt{\CFKm}CFK- to \textalt{\CFDa}CFDˆ: Statement of results 216225
- 11.6. Generalized coefficient maps and boundary degenerations 220229
- 11.7. From \textalt{\CFKm}CFK- to \textalt{\CFDa}CFDˆ: Basis-free version 225234
- 11.8. Proof of Theorem 11.26 244253
- 11.9. Satellites revisited 250259

- Appendix A. Bimodules and change of framing 255264
- Bibliography 269278
- Index of Definitions 273282
- Back Cover Back Cover1294