"Slicing" the Hopf link

TL;DR

This paper introduces a new 4-manifold M to distinguish between the sliceness of the Borromean rings and the Hopf link, providing insights into the A-B slice problem and link theory in 4-manifolds.

Contribution

It constructs a specific 4-manifold M that differentiates the sliceness properties of the Borromean rings and the Hopf link, advancing understanding of link slicing in 4-manifolds.

Findings

Borromean rings are not M-slice in the constructed manifold

Hopf link is M-slice in the constructed manifold

Provides an obstruction related to the A-B slice problem

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 "the most non-slice" 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.