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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. 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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3. A note on PL-disks and rationally slice knots

   Hendricks, Kristen; Hom, Jennifer; Stoffregen, Matthew; Zemke, Ian (April 2024, Proceedings of symposia in pure mathematics)

   We give infinitely many examples of manifold-knot pairs (Y, J) such that Y bounds an integer homology ball, J does not bound a non-locally-flat PL-disk in any integer homology ball, but J does bound a smoothly embedded disk in a rational homology ball. The proof relies on formal properties of involutive Heegaard Floer homology. 

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4. 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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5. Surface Gluing with Signs and Gradings in Decategorified Heegaard Floer Theory

   https://doi.org/10.1093/imrn/rnad238  

   Manion, Andrew (October 2023, International Mathematics Research Notices)

   Abstract A previous result about the decategorified bordered (sutured) Heegaard Floer invariants of surfaces glued together along intervals, generalizing the decategorified content of Rouquier and the author’s higher-tensor-product-based gluing theorem in cornered Heegaard Floer homology, was proved only over $${\mathbb{F}}_2$$ and without gradings. In this paper we add signs and prove a graded version of the interval gluing theorem over $${\mathbb{Z}}$$, enabling a more detailed comparison of these aspects of decategorified Heegaard Floer theory with modern work on non-semisimple 3d TQFTs in mathematics and physics. 

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