### 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_{∞} module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A_{∞} 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.

Original language | English (US) |
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Pages (from-to) | 1-294 |

Number of pages | 294 |

Journal | Memoirs of the American Mathematical Society |

Volume | 254 |

Issue number | 1216 |

DOIs | |

State | Published - Jul 2018 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Extended topological field theory
- Heegaard floer homology
- Holomorphic curves
- Low-dimensional topology
- Three-manifold topology

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## Cite this

*Memoirs of the American Mathematical Society*,

*254*(1216), 1-294. https://doi.org/10.1090/memo/1216