Volume 19, issue 3 (2015)

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Vyacheslav Krushkal

Geometry & Topology 19 (2015) 1657–1683
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 $M\phantom{\rule{0.3em}{0ex}}$–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 $M\phantom{\rule{0.3em}{0ex}}$–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.

Keywords
Slice links, the Milnor group, the A-B slice problem
Mathematical Subject Classification 2010
Primary: 57N13
Secondary: 57M25, 57M27