# Heegaard Floer homology and integer surgeries on links

“Heegaard Floer homology and integer surgeries on links” (2025), by Ciprian Manolescu et al. Mathematics research paper. Original abstract below, unchanged. A free PDF is available.

- Authors: Ciprian Manolescu, Peter Ozsváth
- Published: September 22, 2025
- Field: Mathematics
- DOI: https://doi.org/10.2140/gt.2025.29.2783
- Free PDF: https://arxiv.org/pdf/1011.1317

## Abstract

Let L be a link in an integral homology three-sphere.We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete system of hyperboxes consists of chain complexes for (some versions of) the link Floer homology of L and all its sublinks, together with several chain maps between these complexes.Further, we introduce a way of presenting closed four-manifolds with b C 2 2 by four-colored framed links in the three-sphere.Given a link presentation of this kind for a four-manifold X, we then describe the Ozsváth-Szabó mixed invariants of X in terms of a complete system of hyperboxes for the link.Finally, we explain how a grid diagram produces a particular complete system of hyperboxes for the corresponding link.57R58; 57R57 1. Introduction 2784 2. Conventions 2789 3. Generalized Heegaard Floer complexes for links 2790 4.An overview of the link surgery formula 2807 5. Hyperboxes of chain complexes and compression 2827 6. Quasistabilizations 2842 7. Index zero/three link stabilizations 2859 8. Hyperboxes of Heegaard diagrams 2875 9. Statement of the surgery theorem for link-minimal complete systems 2900 10.Truncation 2908 11.A general surgery exact sequence 2929 12. Proof of the surgery theorem for link-minimal complete systems 2962 13.The general link surgery formula 2980 14.Beyond the surgery theorem 3024 15.The surgery theorem applied to grid diagrams with free markings 3037

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