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1. A remark on the geography problem in Heegaard Floer homology

   https://doi.org/10.1090/pspum/102/08  

   Hanselman, Jonathan; Kutluhan, Çagatay; Lidman, Tye (July 2019, Proceedings of symposia in pure mathematics)

   Gay, David; Wu, Weiwei (Ed.)

   We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsváth, Stipsicz, and Szabó to argue that there is no algebraic obstruction to the existence of knots with trivial epsilon invariant and non-trivial upsilon invariant cannot be realized as the knot Floer complex of a knot. 

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2. Naturality and Mapping Class Groups in Heegaard Floer Homology

   https://doi.org/10.1090/memo/1338  

   Juhász, András; Thurston, Dylan; Zemke, Ian (September 2021, Memoirs of the American Mathematical Society)

   We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds. 

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3. Homology cobordism and triangulations

   https://doi.org/10.1142/9789813272880_0092  

   Manolescu, Ciprian (January 2018, Proceedings of the International Congress of Mathematicians)

   null (Ed.)

   The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin.2/- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology. 

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4. Floer Homology Beyond Borders

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

   Gross, David; Yao, Andrew Chi-Chih; Yau, Shing-Tung (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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5. 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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