Mazur-type manifolds with $L$-space boundaries

TL;DR

This paper proves that Mazur-type 4-manifolds with L-space boundaries are trivial, specifically the 4-ball with a 3-sphere boundary, providing a new proof of Gabai's Property R.

Contribution

It establishes a rigidity result linking L-space boundaries to the trivial 4-ball, offering a novel proof of a classical topological property.

Findings

Mazur-type manifolds with L-space boundaries are the 4-ball.

The boundary of such manifolds must be the 3-sphere.

Provides a new proof of Gabai's Property R.

Abstract

In this note, we prove that if the boundary of a Mazur-type $4$-manifold is an irreducible Heegaard Floer homology $L$-space, then the manifold must be the $4$-ball, and the boundary must be the $3$-sphere. We use this to give a new proof of Gabai's Property R.