Engel relations in 4-manifold topology

TL;DR

This paper explores the application of Engel relations to 4-manifold topology, providing new insights into the 4-dimensional surgery conjecture and introducing universal surgery problems related to homotopically trivial links.

Contribution

It demonstrates how 2-Engel relations can be applied to solve the A-B slice problem and introduces a new class of universal surgery problems using homotopically trivial links.

Findings

Homotopy solution to the A-B slice problem.

Introduction of universal surgery problems based on trivial links.

Connection between n-Engel relations and higher order double points in 4-space.

Abstract

We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional topological surgery conjecture. The A-B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surgery problems, defined using ramifications of homotopically trivial links. More generally we show how n-Engel relations arise from higher order double points of surfaces in 4-space.