Heegaard Floer homology and Morse surgery

TL;DR

This paper develops formulas for how Heegaard Floer homology changes under p/q surgery on knots in three-manifolds, generalizing previous results and showing surgery on non-trivial knots cannot yield the three-sphere.

Contribution

It introduces new surgery formulas for filtered Heegaard Floer homology associated with rational surgeries on null-homologous knots, extending prior work by Ozsváth and Szabó.

Findings

Established surgery formulas for Heegaard Floer homology filtration changes.

Proved that non-trivial knot surgeries cannot produce S^3.

Re-proved non-vanishing results for knot Floer homology after surgery.

Abstract

We establish surgery formulas for filtration of the Heegaard Floer homology associated with p/q surgery on a null-homologous knot K in a three-manifold Y, induced by K_{p/q}. Here K_{p/q} is the core of the attached solid torus (which produces the surgery). This would generalize a result of Ozsvath and Szabo. We will re-prove that surgery on non-trivial knots can not produce S^3, as a corollary of a non-vanishing result for the knot Floer homology of K_{p/q}.