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1. Floer homology beyond borders

   Lipshitz, Robert; Ozsváth, Peter; Thurston, Dylan P (September 2024, International Press)

   Gross, D; Yao, A C-C; Yau, S-T (Ed.)

   Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zoltán Szabó and the second author. In addition to its conceptual interest, bordered Floer homology also provides powerful computational tools. This survey outlines the theory, focusing on recent developments and applications. 

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2. Involutive bordered Floer homology

   https://doi.org/10.1090/tran/7557  

   Hendricks, Kristen; Lipshitz, Robert (April 2019, Transactions of the American Mathematical Society)

   We give a bordered extension of involutive HF-hat and use it to give an algorithm to compute involutive HF-hat for general 3-manifolds. We also explain how the mapping class group action on HF-hat can be computed using bordered Floer homology. As applications, we prove that involutive HF-hat satisfies a surgery exact triangle and compute HFI-hat of the branched double covers of all 10-crossing knots. 

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3. Bordered Floer homology and contact structures

   https://doi.org/10.1017/fms.2023.19  

   Alishahi, Akram; Földvári, Viktória; Hendricks, Kristen; Licata, Joan; Petkova, Ina; Vértesi, Vera (January 2023, Forum of Mathematics, Sigma)

   Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $$(M, \xi , \mathcal {F})$$ whose convex boundary is equipped with a signed singular foliation $$\mathcal {F}$$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $$c_D$$ and $$c_A$$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $$\mathcal {F}$$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić. 

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4. Splicing knot complements and bordered Floer homology

   https://doi.org/10.1515/crelle-2014-0064  

   Hedden, Matthew; Levine, Adam Simon (January 2016, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology. 

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5. A bordered HF^- algebra for the torus

   https://doi.org/10.4310/JSG.250403021401  

   Lipshitz, Robert; Ozsváth, Peter; Thurston, Dylan P (January 2025, Journal of Symplectic Geometry)

   Auroux, Denis; Biran, Paul; Donaldson, Simon; Mrowka, Tomasz (Ed.)

   We describe a weighted A-infinity algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work. 

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