Notes On Open Book Decompositions For Engel Structures

TL;DR

This paper explores the relationship between open book decompositions of 4-manifolds and Engel structures, demonstrating how certain decompositions induce Engel structures with specific properties, including the existence of loose structures.

Contribution

It establishes a method to construct Engel structures from open book decompositions with toric binding and contact monodromy, expanding understanding of Engel structures on 4-manifolds.

Findings

Constructed Engel structures from open books with toric binding.

Showed that supported Engel structures include loose structures.

Proved that every open book with toric binding on a parallelizable 4-manifold supports an Engel structure.

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 En-gel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding. In particular the pages are contact man-ifolds and the monodromy is a 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 amongst the supported Engel structures we construct, there is a class of loose Engel structures.