Surgery obstructions and Heegaard Floer homology

TL;DR

This paper introduces a new obstruction derived from Heegaard Floer homology that determines whether a homology sphere can be obtained by surgery on a knot, leading to the construction of numerous small Seifert fibered examples.

Contribution

It provides the first Heegaard Floer homology-based obstruction to knot surgery on homology spheres, expanding the toolkit for classifying 3-manifolds.

Findings

Constructed infinitely many small Seifert fibered homology spheres not arising from knot surgery.

Established a new obstruction criterion using Heegaard Floer homology.

Extended understanding of the relationship between surgery and manifold topology.

Abstract

Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.