Mazur-type manifolds with $L$-space boundaries

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.

Exotic Mazur manifolds and knot trace invariants

“…It has been conjectured that the only integral homology sphere L-spaces are S 3 and the Poincaré homology sphere; see [19]. The following quick verification of this conjecture among homology spheres that bound Mazur manifolds also answers our questions for L-spaces, and has been observed independently by Conway and Tosun [11]. Proof By hypothesis, W is built from S 1 × B 3 by attaching a single 2-handle along a knot K in S 1 × S 2 which generates H 1 (S…”

“…As observed by Conway and Tosun [11], this gives an alternative to Gabai's proof of Property R: Indeed, if a knot K in an L-space Y admits a surgery to S 1 ×S 2 , then the above argument shows that Y = S 3 and that the surgery dual to K is S 1 × {pt} ⊂ S 1 × S 2 . Applying Lemma 3.1, we conclude that K ⊂ S 3 is the unknot.…”

Dehn surgery and non-separating two-spheres

“…Recently, Conway and Tosun [CT18] showed that the boundary of a non-trivial Mazur manifold is not an L-space. Ni has pointed out that an alternate proof follows from [Ni13].…”