V. Overtwisted, . Structures, . High, and . Dimensions, We also point out another result in a similar vein from [LMN18], that deals more precisely with the specific construction from

[. Aguilar, J. Luis-cisneros-molina, and M. , Characteristic classes and transversality, Topology and its Applications, vol.154, issue.7, pp.1220-1235, 2007.
DOI : 10.1016/j.topol.2005.11.015

P. Albers and H. Hofer, On the Weinstein conjecture in higher dimensions, Commentarii Mathematici Helvetici, vol.84, issue.2, pp.429-436, 2009.
DOI : 10.4171/CMH/167

[. Bowden, D. Crowley, and A. Stipsicz, Contact structures on $$M \times S^2$$ M ?? S 2, Mathematische Annalen, vol.20, issue.1-2, pp.351-359, 2014.
DOI : 10.14492/hokmj/1381413841

[. Borman, Y. Eliashberg, and E. Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Mathematica, vol.215, issue.2, pp.281-361, 2015.
DOI : 10.1007/s11511-016-0134-4

F. Bourgeois, Odd dimensional tori are contact manifolds, Int. Math. Res. Not, issue.30, pp.1571-1574, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01011013

[. Bourgeois, Contact homology and homotopy groups of the space of contact structures, Mathematical Research Letters, vol.13, issue.1, pp.71-85, 2006.
DOI : 10.4310/MRL.2006.v13.n1.a6

URL : https://hal.archives-ouvertes.fr/hal-01011005

R. Bott and L. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol.82, 1982.
DOI : 10.1007/978-1-4757-3951-0

K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and back
DOI : 10.1090/coll/059

J. Cerf, Sur les difféomorphismes de la sphère de dimension trois (? 4 = 0), Lecture Notes in Mathematics, issue.53, 1968.

K. Cieliebak, Subcritical Stein manifolds are split. ArXiv Mathematics e-prints, p.204351, 2002.

[. Casals, E. Murphy, and F. Presas, Geometric criteria for overtwistedness. ArXiv e-prints, pp.1503-06221, 2015.

[. Casals, D. M. Pancholi, and F. Presas, Almost contact 5-manifolds are contact, Annals of Mathematics, vol.182, issue.22, pp.429-490, 2015.
DOI : 10.4007/annals.2015.182.2.2

R. Casals, F. Presas, and S. Sandon, Small positive loops on overtwisted manifolds, Journal of Symplectic Geometry, vol.14, issue.4, pp.1013-1031, 2016.
DOI : 10.4310/JSG.2016.v14.n4.a2

URL : https://hal.archives-ouvertes.fr/hal-01465266

F. Ding and H. Geiges, Abstract, Compositio Mathematica, vol.26, issue.04, pp.1096-1112, 2010.
DOI : 10.5802/aif.1288

M. Dörner, H. Geiges, and K. Zehmisch, OPEN BOOKS AND THE WEINSTEIN CONJECTURE, The Quarterly Journal of Mathematics, vol.65, issue.3, p.869, 2014.
DOI : 10.1093/qmath/hat055

Y. Eliashberg and M. Fraser, Topologically trivial Legendrian knots, Journal of Symplectic Geometry, vol.7, issue.2, pp.77-127, 2009.
DOI : 10.4310/JSG.2009.v7.n2.a4

J. Etnyre and R. Furukawa, Braided embeddings of contact 3-manifolds in the standard contact 5-sphere, Journal of Topology, vol.36, issue.2, pp.412-446, 2017.
DOI : 10.1090/S0002-9939-1975-0375366-7

J. Etnyre and Y. Lekili, Embedding all contact 3-manifolds in a fixed contact 5-manifold. ArXiv e-prints, pp.1712-09642, 2017.
DOI : 10.1112/jlms.12164

URL : http://arxiv.org/pdf/1712.09642

, Classification of overtwisted contact structures on 3- manifolds, Invent. Math, vol.98, issue.3, pp.623-637, 1989.

, Filling by holomorphic discs and its applications, Geometry of low-dimensional manifolds, pp.45-67, 1989.

, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble), vol.42, issue.12, pp.165-192, 1992.

J. Etnyre and B. Ozbagci, Invariants of contact structures from open books, Transactions of the American Mathematical Society, vol.360, issue.06, pp.3133-3151, 2008.
DOI : 10.1090/S0002-9947-08-04459-0

[. Eliashberg and W. Thurston, Confoliations, volume 13 of University Lecture Series, 1998.

J. Etnyre, Contact structures on 5-manifolds, 2012.

[. Geiges, Contact structures on 1-connected 5-manifolds, Mathematika, vol.282, issue.02, pp.303-311, 1991.
DOI : 10.2307/1970702

[. Geiges, Applications of contact surgery, Topology, vol.36, issue.6, pp.1193-1220, 1997.
DOI : 10.1016/S0040-9383(97)00004-9

[. Geiges, Constructions of contact manifolds, Mathematical Proceedings of the Cambridge Philosophical Society, vol.121, issue.3, pp.455-464, 1997.
DOI : 10.1017/S0305004196001260

[. Geiges, An introduction to contact topology, volume 109 of Cambridge Studies in Advanced Mathematics, 2008.

N. Giroux and . Goodman, On the stable equivalence of open books in three-manifolds, Geometry & Topology, vol.87, issue.1, pp.97-114, 2006.
DOI : 10.2307/1970594

URL : https://hal.archives-ouvertes.fr/hal-00009039

H. Geiges and J. G. Pérez, ON THE TOPOLOGY OF THE SPACE OF CONTACT STRUCTURES ON TORUS BUNDLES, Bulletin of the London Mathematical Society, vol.36, issue.05, pp.640-646, 2004.
DOI : 10.1112/S0024609304003376

P. Griffiths and J. Harris, Principles of algebraic geometry, 1978.
DOI : 10.1002/9781118032527

. Giroux, Convexit?? en topologie de contact, Commentarii Mathematici Helvetici, vol.66, issue.1, pp.637-677, 1991.
DOI : 10.1007/BF02566670

. Giroux, Une infinit?? de structures de contact tendues sur une infinit?? de vari??t??s, Inventiones Mathematicae, vol.135, issue.3, pp.789-802, 1999.
DOI : 10.1007/s002220050301

E. Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Sur les transformations de contact au-dessus des surfaces . In Essays on geometry and related topics, pp.615-689, 2000.
DOI : 10.1007/s002220000082

URL : https://hal.archives-ouvertes.fr/hal-00009389

. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, pp.405-414, 2002.

. Giroux, The existence problem, contact geometry in high dimensions, 2012.

E. Giroux, Ideal Liouville Domains -a cool gadget ArXiv e-prints, 1708.

[. Geiges and M. Klukas, The fundamental group of the space of contact structures on the $3$-torus, Mathematical Research Letters, vol.21, issue.6, pp.1257-1262294, 2014.
DOI : 10.4310/MRL.2014.v21.n6.a3

R. Gompf, Handlebody Construction of Stein Surfaces, The Annals of Mathematics, vol.148, issue.2, pp.619-693, 1998.
DOI : 10.2307/121005

URL : http://arxiv.org/pdf/math/9803019

J. Gonzalo, Branched covers and contact structures, Proceedings of the American Mathematical Society, vol.101, issue.2, pp.347-352, 1987.
DOI : 10.1090/S0002-9939-1987-0902554-9

M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicae, vol.113, issue.108, pp.307-347, 1985.
DOI : 10.1016/B978-0-12-044850-0.50038-1

M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas, 1986.
DOI : 10.1007/978-3-662-02267-2

[. Geiges and A. Stipsicz, Contact structures on product five-manifolds and fibre sums along circles, GT98] Hansjörg Geiges and Charles Thomas. Contact topology and the structure of 5-manifolds with ? 1 = Z 2 . Ann. Inst. Fourier (Grenoble), pp.195-2101167, 1998.
DOI : 10.1007/978-3-662-10167-4

URL : http://real.mtak.hu/9962/1/0906.5242.pdf

H. Geiges and C. Thomas, Contact structures, equivariant spin bordism, and periodic fundamental groups, Mathematische Annalen, vol.320, issue.4, pp.685-708, 2001.
DOI : 10.1007/PL00004491

L. [. Hernández-corbato, F. Martín-merchán, and . Presas, Tight neighborhoods of contact submanifolds. ArXiv e-prints, 2018.

H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Inventiones Mathematicae, vol.33, issue.2, pp.515-563, 1993.
DOI : 10.1007/BF01232679

K. Honda, On the classification of tight contact structures. I. Geom, Topol, vol.4, pp.309-368, 2000.
URL : https://hal.archives-ouvertes.fr/hal-00009385

Y. Huang, On plastikstufe, bordered Legendrian open book and overtwisted contact structures, Journal of Topology, vol.17, issue.3, pp.720-743, 2017.
DOI : 10.1007/978-3-540-78279-7_4

URL : http://arxiv.org/pdf/1607.08262

I. Kolá?, P. Michor, and J. Slovák, Natural operations in differential geometry, 1993.
DOI : 10.1007/978-3-662-02950-3

K. Otto-van-koert and . Niederkrüger, Open book decompositions for contact structures on Brieskorn manifolds, Proc. Amer, pp.3679-3686, 2005.

K. Otto-van-koert and . Niederkrüger, Every contact manifolds can be given a nonfillable contact structure, Int. Math. Res. Not. IMRN, vol.22, issue.23, 2007.

S. Lang, Fundamentals of differential geometry, volume 191 of Graduate Texts in Mathematics, 1999.

[. Lazarev, Contact manifolds with flexible fillings

J. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics, vol.218, 2013.
DOI : 10.1007/978-1-4419-9982-5

E. Lerman, Contact fiber bundles, LMN18] Samuel Lisi, Aleksandra Marinkovi?, and Klaus Niederkrüger. Some properties of the Bourgeois contact structures. ArXiv e-prints, pp.52-66, 0869.
DOI : 10.1016/S0393-0440(03)00060-3

URL : http://arxiv.org/pdf/math/0301137

R. Lutz, Sur la g??om??trie des structures de contact invariantes, Annales de l???institut Fourier, vol.29, issue.1, pp.283-306, 1979.
DOI : 10.5802/aif.739

URL : http://archive.numdam.org/article/AIF_1979__29_1_283_0.pdf

S. Lanzat and F. Zapolsky, On the contact mapping class group of the contactization of the $$A_m$$ A m -Milnor fiber, Annales math??matiques du Qu??bec, vol.359, issue.1???2, pp.79-94, 2018.
DOI : 10.24033/bsmf.2082

P. Massot, Natural fibrations in contact topology, 2015.

[. Meyerson, Representing homology classes of closed orientable surfaces, Proceedings of the American Mathematical Society, vol.61, issue.1, pp.181-182, 1976.
DOI : 10.1090/S0002-9939-1976-0425967-3

P. Massot and K. Niederkrüger, Examples of Non-trivial Contact Mapping Classes in all Dimensions, International Mathematics Research Notices, vol.117, issue.15, pp.4784-4806, 2016.
DOI : 10.1007/978-3-0348-8508-9_6

URL : https://hal.archives-ouvertes.fr/hal-01146739

P. Massot, K. Niederkrüger, and C. Wendl, Weak and strong fillability of higher dimensional contact manifolds, Inventiones mathematicae, vol.151, issue.3, pp.287-373, 2013.
DOI : 10.1215/00127094-2010-001

URL : https://hal.archives-ouvertes.fr/hal-00949420

A. Mori, Reeb foliations on S 5 and contact 5-manifolds violating the Thurston-Bennequin inequality

J. Morgan, Recent progress on the Poincar?? conjecture and the classification of 3-manifolds, MP16] Aleksandra Marinkovi? and Milena Pabiniak, pp.57-78, 2005.
DOI : 10.1090/S0273-0979-04-01045-6

URL : https://www.ams.org/bull/2005-42-01/S0273-0979-04-01045-6/S0273-0979-04-01045-6.pdf

K. Niederkrüger, The plastikstufe ??? a generalization of the overtwisted disk to higher dimensions, Algebraic & Geometric Topology, vol.65, issue.5, pp.2473-2508, 2006.
DOI : 10.1007/BF02566630

, Mémoire d'habilitation à diriger des recherches, 2013.

K. Niederkrüger and F. Öztürk, Brieskorn manifolds as contact branched covers of spheres Some remarks on the size of tubular neighborhoods in contact topology and fillability, NP10] Klaus Niederkrüger and Francisco Presas, pp.85-97719, 2007.

F. Presas, A class of non-fillable contact structures, Geometry & Topology, vol.52, issue.4, pp.2203-2225, 2007.
DOI : 10.2140/agt.2006.6.2473

J. Stallings, Constructions of fibred knots and links, Algebraic and geometric topology (Proc. Sympos. Pure Math. Part 2, Proc. Sympos. Pure Math., XXXII, pp.55-60, 1976.
DOI : 10.1090/pspum/032.2/520522

. Amer, Math. Soc, 1978.

T. Vogel, On the uniqueness of the contact structure approximating a foliation, Geometry & Topology, vol.280, issue.5, pp.2439-2573, 2016.
DOI : 10.2140/gt.2011.15.41

T. Vogel, Non-loose unknots, overtwisted discs, and the contact mapping class group of S3, Geometric and Functional Analysis, vol.20, issue.5, pp.228-288, 2018.
DOI : 10.2140/gt.2016.20.2439

K. Zehmisch, The Gromov-Eliashberg tightness-theorem. Master's thesis, 2003.