Robust 4-manifolds and robust embeddings

TL;DR

This paper investigates how certain 4-manifolds influence the classification of homotopically trivial links in the 4-ball, revealing embedding-dependent properties relevant to the A-B slice problem in 4-dimensional topology.

Contribution

It introduces the concept of robust 4-manifolds and demonstrates their embedding-dependent properties, providing new obstructions in the A-B slice problem.

Findings

Existence of 4-manifolds with embedding-dependent trivial link properties

New secondary obstructions in the A-B slice problem

Insights into the role of manifold embeddings in link triviality

Abstract

A link in the 3-sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4-ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links when used in place of disks in an analogous definition. We show that there are 4-manifolds for which this property depends on their embedding in the 4-ball. This work is motivated by the A-B slice problem, a reformulation of the 4-dimensional topological surgery conjecture. As a corollary this provides a new, secondary, obstruction in the A-B slice problem for a certain class of decompositions of D^4.