Engel groups and universal surgery models

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.