Rational surgery exact triangles in Heegaard Floer homology

Abstract

We construct a new family of surgery exact triangles in Heegaard Floer theory over the field with two elements. This family generalizes both Ozsváth and Szabó'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 $r\ne n,1/n$; one key idea is to use an involution that is closely related to the ${\mathrm{Spin}}^c$ conjugation symmetry.

55 pages, 15 figures, comments welcome!