## Algebraic & Geometric Topology

### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4275-4303.

Dates
Revised: 12 April 2018
Accepted: 21 June 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.agt/1545102071

Digital Object Identifier
doi:10.2140/agt.2018.18.4275

Mathematical Reviews number (MathSciNet)
MR3892245

Zentralblatt MATH identifier
07006391

Subjects
Primary: 58A30: Vector distributions (subbundles of the tangent bundles)

#### Citation

Colin, Vincent; Presas, Francisco; Vogel, Thomas. Notes on open book decompositions for Engel structures. Algebr. Geom. Topol. 18 (2018), no. 7, 4275--4303. doi:10.2140/agt.2018.18.4275. https://projecteuclid.org/euclid.agt/1545102071

#### References

• J Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923) 93–95
• R Casals, J L Pérez, A del Pino, F Presas, Existence $h$–principle for Engel structures, Invent. Math. 210 (2017) 417–451
• R Casals, A del Pino, F Presas, Loose Engel structures, preprint (2017)
• V Colin, Recollement de variétés de contact tendues, Bull. Soc. Math. France 127 (1999) 43–69
• V Colin, Livres ouverts en géométrie de contact (d'après Emmanuel Giroux), from “Séminaire Bourbaki Vol. 2006/2007”, Astérisque 317, Soc. Math. France, Paris (2008) Exposé 969, 91–117
• Y Eliashberg, Classification of overtwisted contact structures on $3$–manifolds, Invent. Math. 98 (1989) 623–637
• Y Eliashberg, Contact $3$–manifolds twenty years since J Martinet's work, Ann. Inst. Fourier $($Grenoble$)$ 42 (1992) 165–192
• F Engel, Zur Invariantentheorie der Systeme Pfaffscher Gleichungen, Berichte über d. Verh. d. Sachsischen Gesell. der Wiss. 41, Hirzel, Leipzig (1889) 157–176
• J B Etnyre, Lectures on open book decompositions and contact structures, from “Floer homology, gauge theory, and low-dimensional topology” (D A Ellwood, P S Ozsváth, A I Stipsicz, Z Szabó, editors), Clay Math. Proc. 5, Amer. Math. Soc., Providence, RI (2006) 103–141
• H Geiges, review of [montgomery?], MathSciNet
• H Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics 109, Cambridge Univ. Press (2008)
• E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637–677
• E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from “Proceedings of the International Congress of Mathematicians” (L Tatsien, editor), volume II, Higher Ed. Press, Beijing (2002) 405–414
• E Giroux, N Goodman, On the stable equivalence of open books in three-manifolds, Geom. Topol. 10 (2006) 97–114
• F Laudenbach, V Poénaru, A note on $4$–dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337–344
• T Lawson, Open book decompositions for odd dimensional manifolds, Topology 17 (1978) 189–192
• D McDuff, Applications of convex integration to symplectic and contact geometry, Ann. Inst. Fourier $($Grenoble$)$ 37 (1987) 107–133
• D McDuff, D Salamon, Introduction to symplectic topology, 2nd edition, Oxford Univ. Press (1998)
• R Montgomery, Engel deformations and contact structures, from “Northern California symplectic geometry seminar” (Y Eliashberg, D Fuchs, T Ratiu, A Weinstein, editors), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc., Providence, RI (1999) 103–117
• S P Novikov, Pontrjagin classes, the fundamental group and some problems of stable algebra, from “Essays on topology and related topics” (A Haefliger, R Narasimhan, editors), Springer (1970) 147–155
• F Presas, Geometric decompositions of almost contact manifolds, from “Contact and symplectic topology” (F Bourgeois, V Colin, A Stipsicz, editors), Bolyai Soc. Math. Stud. 26, János Bolyai Math. Soc., Budapest (2014) 137–172
• F Quinn, Open book decompositions, and the bordism of automorphisms, Topology 18 (1979) 55–73
• W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345–347
• T Vogel, Existence of Engel structures, Ann. of Math. 169 (2009) 79–137
• H E Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973) 45–51