×

“Slicing” the Hopf link. (English) Zbl 1321.57025

Author’s abstract: A link in the 3-sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the 4-ball. More generally, given a 4-manifold \(M\) with a distinguished circle in its boundary, a link in the 3-sphere is called \(M\)-slice if its components bound in the 4-ball disjoint embedded copies of \(M\). A 4-manifold \(M\) is constructed such that the Borromean rings are not \(M\)-slice but the Hopf link is. This contrasts the classical link-slice setting where the Hopf link may be thought of as “the most nonslice” link. Further examples and an obstruction for a family of decompositions of the 4-ball are discussed in the context of the A-B slice problem.
Reviewer: Ioan Pop (Iaşi)

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M H Freedman, A geometric reformulation of \(4\)-dimensional surgery, Topology Appl. 24 (1986) 133 · Zbl 0898.57005
[2] M H Freedman, V Krushkal, Topological arbiters, J. Topol. 5 (2012) 226 · Zbl 1242.57018
[3] M H Freedman, X S Lin, On the \((A,B)\)-slice problem, Topology 28 (1989) 91 · Zbl 0845.57016
[4] M H Freedman, F Quinn, Topology of \(4\)-manifolds, Princeton Math. Series 39, Princeton Univ. Press (1990) · Zbl 0705.57001
[5] V S Krushkal, Exponential separation in \(4\)-manifolds, Geom. Topol. 4 (2000) 397 · Zbl 0957.57015
[6] V S Krushkal, A counterexample to the strong version of Freedman’s conjecture, Ann. of Math. 168 (2008) 675 · Zbl 1176.57025
[7] V S Krushkal, Robust four-manifolds and robust embeddings, Pacific J. Math. 248 (2010) 191 · Zbl 1202.57008
[8] W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience (1966) · Zbl 0138.25604
[9] J Milnor, Link groups, Ann. of Math. 59 (1954) 177 · Zbl 0055.16901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.