Existence of Engel structures

Abstract

We develop a construction of Engel structures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M1,M2 satisfying a certain condition on the characteristic foliations, there is an Engel structure on M1#M2#(S2 × S2) which is closely related to the original Engel structures. © Copyright 2008. Annals of Mathematics. All rights reserved.