Rational surgery exact triangles in Heegaard Floer homology

Gheehyun Nahm

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arXiv · 2601.06654

Rational surgery exact triangles in Heegaard Floer homology

Gheehyun Nahm

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Abstract

We construct a new family of surgery exact triangles in Heegaard Floer theory over the field with two elements. This family generalizes both Ozsv\'{a}th and Szab\'{o}'s $n$- and $1/n$-surgery exact triangles for positive integers $n$ and the author's recent 2-surgery exact triangle to all positive rational slopes. The construction reduces to a combinatorial problem that involves triangle and quadrilateral counting maps in a genus 1 Heegaard diagram. The main contribution of this paper is solving this combinatorial problem, which is particularly tricky for slopes $r\neq n,1/n$; one key idea is to use an involution that is closely related to the ${\rm Spin}^{c}$ conjugation symmetry.

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BibTeX RIS

Gheehyun Nahm. 2026-01-10. Rational surgery exact triangles in Heegaard Floer homology. https://arxiv.org/abs/2601.06654

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