Engel groups and universal surgery models

TL;DR

This paper introduces a new class of 4-dimensional surgery problems called 1/2-$pi_1$-null problems, showing their universality and linking their solutions to the topological surgery conjecture for all fundamental groups.

Contribution

It defines 1/2-$pi_1$-null surgery problems, proves their universality, and connects their solvability to the topological surgery conjecture, using group-theoretic and geometric methods.

Findings

1/2-$pi_1$-null problems are universal for 4D topological surgery.

Solving these problems is equivalent to solving surgery for all fundamental groups.

A weaker $pi_1$-null disk lemma suffices for surgery and s-cobordism theorems.

Abstract

We introduce a collection of 1/2-$\pi_1$-null 4-dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the $\pi_1$-null kernels which are known to admit a solution in the topological category. Using geometric applications of the group-theoretic 2-Engel relation, we show that the 1/2-$\pi_1$-null surgery problems are universal, in the sense that solving them is equivalent to establishing 4-dimensional topological surgery for all fundamental groups. As another application of these methods, we formulate a weaker version of the $\pi_1$-null disk lemma and show that it is sufficient for proofs of topological surgery and s-cobordism theorems for good groups.