# Notes on bordered Floer homology

@article{Lipshitz2012NotesOB, title={Notes on bordered Floer homology}, author={Robert Lipshitz and Peter S. Ozsv'ath and Dylan Thurston}, journal={arXiv: Geometric Topology}, year={2012} }

This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.

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#### References

SHOWING 1-10 OF 57 REFERENCES

HF=HM, I : Heegaard Floer homology and
Seiberg–Witten Floer homology

- Mathematics
- Geometry & Topology
- 2020

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the… Expand

On rational blow-downs in Heegaard-Floer homology

- Mathematics
- 2007

Motivated by a result of L.P. Roberts on rational blow-downs in Heegaard-Floer homology, we study such operations along 3-manifolds that arise as branched double covers of $S^{3}$ along several… Expand

An algorithm for computing some Heegaard Floer homologies

- Mathematics
- 2006

In this paper, we give an algorithm to compute the hat version of Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtration coming from a… Expand

The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II

- Mathematics
- 2012

This paper is the sequel to "The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I" and is devoted to proving some of the technical parts of the… Expand

HF=HM, V : Seiberg–Witten Floer homology
and handle additions

- Mathematics
- 2012

This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.

Tour of bordered Floer theory

- Mathematics, Medicine
- Proceedings of the National Academy of Sciences
- 2011

This survey explains the formal structure and construction of bordered Floer homology and sketches how it can be used to compute some aspects of Heegaard Floer theory. Expand

Knot doubling operators and bordered Heegaard Floer homology

- Mathematics
- 2012

We use bordered Heegaard Floer homology to compute the τ invariant of a family of satellite knots obtained via twisted infection along two components of the Borromean rings, a generalization of… Expand

Heegaard Floer homology as morphism spaces

- Mathematics
- 2011

In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present… Expand

Fukaya categories of symmetric products and bordered Heegaard-Floer homology

- Mathematics
- 2010

The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and… Expand

Heegaard Floer homology and fibred 3-manifolds

- Mathematics
- 2007

<abstract abstract-type="TeX"><p>Given a closed $3$-manifold $Y$, we show that the Heegaard Floer homology determines whether $Y$ fibres over the circle with a fibre of negative Euler characteristic.… Expand