Engel groups and universal surgery models

Abstract

We introduce a collection of 1 2 - π 1 -null four-dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the π 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 - π 1 -null surgery problems are universal, in the sense that solving them is equivalent to establishing four-dimensional topological surgery for all fundamental groups. As another application of these methods, we formulate a weaker version of the π 1 -null disk lemma and show that it is sufficient for proofs of topological surgery and s-cobordism theorems for good groups.