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1. APPLICATIONS OF INVOLUTIVE HEEGAARD FLOER HOMOLOGY

   https://doi.org/10.1017/S147474801900015X  

   Hendricks, Kristen; Hom, Jennifer; Lidman, Tye (April 2019, Journal of the Institute of Mathematics of Jussieu)

   We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms $$\bar{d}$$ and $$\text{}\underline{d}$$ for certain families of three-manifolds. 

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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. 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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4. 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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5. Instanton and the depth of taut foliations

   Li, Zhenkun (July 2020, ArXivorg)

   Sutured instanton Floer homology was introduced by Kronheimer and Mrowka. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the sutured instanton Floer homology provides a bound on the minimal depth of all possible taut foliations on that balanced sutured manifold. The same argument can be adapted to the monopole and even the Heegaard Floer settings, which gives a partial answer to one of Juhasz's conjectures. Using the nature of instanton Floer homology, on knot complements, we can construct a taut foliation with bounded depth, given some information on the representation varieties of the knot fundamental groups. This indicates a mystery relation between the representation varieties and some small depth taut foliations on knot complements, and gives a partial answer to one of Kronheimer and Mrowka's conjecture. 

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