NOTES ON OPEN BOOK DECOMPOSITIONS FOR ENGEL STRUCTURES

Vincent Colin (1) , Francisco Presas (2) , Thomas Vogel (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Instituto de Ciencias Matematicas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Cientficas
3 LMU - Mathematisches Institut [München]

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.

Domaines: Géométrie symplectique [math.SG]

Vincent Colin, Francisco Presas, Thomas Vogel. NOTES ON OPEN BOOK DECOMPOSITIONS FOR ENGEL STRUCTURES. Algebraic and Geometric Topology, 2018, 18, pp.4275--4303. ⟨hal-01714524⟩