Notes on open book decompositions for Engel structures

Abstract

We relate open book decompositions of a $4$–manifold $M$ with its Engel structures. Our main result is, given an open book decomposition of $M$ whose binding is a collection of $2$–tori and whose monodromy preserves a framing of a page, the construction of an Engel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding.

In particular, the pages are contact manifolds and the monodromy is a compactly supported contactomorphism. As a consequence, on a parallelizable closed $4$–manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that among the supported Engel structures we construct, there are loose Engel structures.