Null-homotopic knots have Property R

TL;DR

This paper proves that nontrivial null-homotopic knots in certain 3-manifolds do not produce $S^1\times S^2$ summands after zero surgery, extending previous results to more general manifolds.

Contribution

It generalizes the Property R theorem for null-homotopic knots to a broader class of 3-manifolds beyond rational homology spheres.

Findings

Zero surgery on nontrivial null-homotopic knots lacks $S^1\times S^2$ summands.

Extends previous results from irreducible rational homology spheres to more general 3-manifolds.

Supports the conjecture that null-homotopic knots have Property R in wider contexts.

Abstract

We prove that if $K$ is a nontrivial null-homotopic knot in a closed oriented $3$--manfiold $Y$ such that $Y-K$ does not have an $S^1\times S^2$ summand, then the zero surgery on $K$ does not have an $S^1\times S^2$ summand. This generalizes a result of Hom and Lidman, who proved the case when $Y$ is an irreducible rational homology sphere.