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Notes on open book decompositions for Engel structures

Vincent Colin, Francisco Presas and Thomas Vogel

Algebraic & Geometric Topology 18 (2018) 4275–4303

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.

open book decomposition, Engel structures, contact structure
Mathematical Subject Classification 2010
Primary: 58A30
Received: 21 February 2018
Revised: 12 April 2018
Accepted: 21 June 2018
Published: 11 December 2018
Vincent Colin
Laboratoire de Mathématiques Jean Leray
Université de Nantes
Francisco Presas
Instituto de Ciencias Matemáticas
Thomas Vogel
Mathematisches Institut der Ludwig-Maximilians-Universität
Universität München