#### Volume 19, issue 3 (2015)

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“Slicing” the Hopf link

### 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
##### Publication
Received: 25 February 2014
Revised: 5 August 2014
Accepted: 3 September 2014
Published: 21 May 2015
Proposed: Robion Kirby
Seconded: Michael Freedman, Walter Neumann
##### Authors
 Vyacheslav Krushkal Department of Mathematics University of Virginia Charlottesville, VA 22904 USA http://www.math.virginia.edu/~vk6e