“Slicing” the Hopf link

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.