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Mazur-type manifolds with \(L\)-space boundary. (English) Zbl 1441.57021

A Mazur-type manifold \(M(n)\) is a \(4\)-manifold obtained by the following construction. Start with \(B^4\), attach a \(1\)-handle and then a \(2\)-handle along a knot \(K\) (with framing \(n\)) that has intersection number \(1\) with the co-core of the \(1\)-handle. The paper under review proves: if \(Y=\partial M(n)\) is an irreducible \(L\)-space, then it is a \(3\)-sphere and \(M(n)\cong B^4\). (Recall that an \(L\)-space is a rational homology \(3\)-sphere \(Y\) whose Heegaard-Floer homology is the simplest possible, that is, \(\widehat{HF}(Y)\) is a free abelian group of \(\sharp H_1(Y)\) generators.)
The proof roughly proceeds as follows. Let \(K^\prime\) be a meridian of \(K\), that is, the boundary of the co-core of the \(1\)-handle. The authors use a result of S. Akbulut and Ç. Karakurt [Proc. Am. Math. Soc. 142, No. 11, 4001–4013 (2014; Zbl 1305.57049)] to show that \(\pm 1\)-surgeries at \(K^\prime\) yield \(L\)-spaces again. In particular, both \(K^\prime\) and its mirror \(-K^\prime\) are \(L\)-space knots. (A knot is called an \(L\)-space knot if some positive surgery yields an \(L\)-space.) Further the authors show that the complement of \(K^\prime\) in \(Y\) is irreducible. It follows from known results that \(L\)-space knots with irreducible complement are fibered and (thus) support a tight contact structure. K. Honda et al. [Invent. Math. 169, No. 2, 427–449 (2007; Zbl 1167.57008)] have shown that the supporting open book decompositions of tight contact structures have right-veering monodromy. Since the monodromy associated to \(-K^\prime\) is the inverse of the monodromy associated to \(K^\prime\), this means that the monodromy must be trivial – otherwise the inverse of right-veering diffeomorphism could not be right-veering. Since \(Y\) is a rational homology sphere, this implies that the page of the open book is a disk, that \(K^\prime\) is the unknot, and that \(Y=S^3\). Finally, the authors show that \(M(n)\) is a minimal strong symplectic filling of the contact structure on \(Y=S^3\), which by a result of Gromov and McDuff implies that \(M(n)\) is diffeomorphic to \(B^4\).
As an application, the authors give a new proof of property R. Recall that Gabai’s property R theorem asserts that \(0\)-surgery at a knot \(K^\prime\subset S^3\) can yield \(S^1\times S^2\) only if \(K^\prime\) is the unknot. The argument is that the surgery yields a \(4\)-dimensional cobordism between \(Y\) and \(S^1\times S^2\), which turns out to be a Mazur-type manifold such that \(K^\prime\) is isotopic to the boundary of the co-core of the \(2\)-handle. Then the proof of the main theorem implies that \(K^\prime\) is the unknot. The authors’ proof works more generally for knots in irreducible integer homology sphere \(L\)-spaces, in particular for knots in the Poincaré homology sphere.

MSC:

57K40 General topology of 4-manifolds
57M50 General geometric structures on low-dimensional manifolds
53D10 Contact manifolds (general theory)
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