Rational surgery exact triangles in Heegaard Floer homology

TL;DR

This paper introduces a new family of surgery exact triangles in Heegaard Floer homology that generalize previous results to all positive rational slopes, using combinatorial methods and symmetry considerations.

Contribution

It constructs a comprehensive set of surgery exact triangles for all positive rational slopes in Heegaard Floer theory, extending prior specific cases.

Findings

Unified framework for rational surgeries in Heegaard Floer homology

Solved complex combinatorial counting problem for general slopes

Utilized involution related to Spin^c conjugation symmetry

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.