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Notes on Bordered Floer Homology

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Contact and Symplectic Topology

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 26))

Abstract

Bordered Heegaard Floer homology is an extension of Ozsváth-Szabós Heegaard Floer homology to 3-manifolds with boundary, enjoying good properties with respect to gluings. In these notes we will introduce the key features of bordered Heegaard Floer homology: its formal structure, a precise definition of the invariants of surfaces, a sketch of the definitions of the 3-manifold invariants, and some hints at the analysis underlying the theory. We also talk about bordered Heegaard Floer homology as a computational tool, both in theory and practice.

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Notes

  1. 1.

    Strictly speaking, in the original definition the manifolds were only totally-real, not Lagrangian. It was shown in [54] that a Kähler form can be chosen making the relevant submanifolds Lagrangian.

  2. 2.

    The ground ring for bordered Floer homology is \(\mathbb {Z}/{2}\mathbb {Z}\); hence the signs usually appearing in the differential graded Leibniz rule become irrelevant.

  3. 3.

    This I(S) was denoted I D (S) in [27], where I(S) was used for I A (S) introduced in Section 4.4.

  4. 4.

    The discussion in this section is taken from [27, Section 11.2].

  5. 5.

    As usual, we will suppress the fact that one needs to perturb the almost-complex structure in order to achieve transversality from the discussion.

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Acknowledgements

These are notes from a series of lectures by the first author during the CaST conference in Budapest in the summer of 2012. We thank Jennifer Hom for many helpful comments on and corrections to earlier drafts of these notes, and the participants in the CaST summer school for many further corrections. We also thank the Rényi Institute for providing a stimulating environment without which these notes would not have been written, and the Simons Center for providing a peaceful one, without which these notes would not have been revised.

RL was supported by NSF Grant number DMS-0905796 and a Sloan Research Fellowship. PSO was supported by NSF grant number DMS-0804121. DPT was supported by NSF grant number DMS-1008049.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Robert Lipshitz

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Peter Ozsváth

  3. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Dylan P. Thurston

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  1. Robert Lipshitz
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  2. Peter Ozsváth
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  3. Dylan P. Thurston
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Correspondence to Robert Lipshitz .

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Editors and Affiliations

  1. Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Orsay, France

    Frédéric Bourgeois

  2. Lab. de Mathématiques Jean Leray UMR6629, University of Nantes, Nantes, France

    Vincent Colin

  3. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    András Stipsicz

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Lipshitz, R., Ozsváth, P., Thurston, D.P. (2014). Notes on Bordered Floer Homology. In: Bourgeois, F., Colin, V., Stipsicz, A. (eds) Contact and Symplectic Topology. Bolyai Society Mathematical Studies, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-02036-5_7

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