A general Heegaard Floer surgery formula

TL;DR

This paper introduces a new exact triangle in the Fukaya category of the torus, providing a simplified proof of Heegaard Floer surgery formulas and extending their applicability to arbitrary links and bordered manifolds.

Contribution

A new exact triangle in the Fukaya category that simplifies proofs of Heegaard Floer surgery formulas and extends their scope to more general settings.

Findings

Extended link surgery formula to arbitrary links in closed 3-manifolds.

Proved invariance of modules for bordered manifolds with torus boundaries.

Provided a simple proof of the surgery formula for knot and link Floer complexes.

Abstract

We give several new perspectives on the Heegaard Floer Dehn surgery formulas of Manolescu, Ozsv\'{a}th and Szab\'{o}. Our main result is a new exact triangle in the Fukaya category of the torus which gives a new proof of these formulas. This exact triangle is different from the one which appeared in Ozsv\'{a}th and Szab\'{o}'s original proof. This exact triangle simplifies a number of technical aspects in their proofs and also allows us to prove several new results. A first application is an extensions of the link surgery formula to arbitrary links in closed 3-manifolds, with no restrictions on the link being null-homologous. A second application is a proof that the modules for bordered manifolds with torus boundaries, defined by the author in a previous paper, are invariants. Another application is a simple proof of a version of the surgery formula which computes knot and link Floer complexes in terms of subcubes of the link surgery hypercube. As a final application, we show that the knot surgery algebra is homotopy equivalent to an endomorphism algebra of a sum of two decorated Lagrangians in the torus, mirroring a result of Auroux concerning the algebras of Lipshitz, Ozsv\'{a}th and Thurston.