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# Mathematical Research Letters

## Volume 27 (2020)

### Number 1

### Mazur-type manifolds with $L$-space boundary

Pages: 35 – 42

#### Authors

#### 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 $\mathrm{R}$.

The first author was partially supported by NSF grant DMS-1344991 and the second author was partially supported by the Simons Foundation grant 636841, BT.

Received 6 August 2018

Accepted 10 February 2019

Published 8 April 2020