A remark on the contactomorphism group of overtwisted contact spheres

Fernández, Eduardo; Gironella, Fabio

Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 189-196.

On prouve l’existence d’éléments d’ordre infini dans certains groupes d’homotopie du groupe des contactomorphismes des sphères vrillées. En particulier, il s’en suit que le groupe des contactomorphismes de certaines sphères vrillées n’est pas homotopiquement équivalent à un groupe de Lie de dimension finie.

We show the existence of elements of infinite order in some homotopy groups of the contactomorphism group of overtwisted spheres. It follows in particular that the contactomorphism group of some high dimensional overtwisted spheres is not homotopically equivalent to a finite dimensional Lie group.

[1] Antonelli, Peter L.; Burghelea, Dan; Kahn, Peter J. The non-finite homotopy type of some diffeomorphism groups, Topology, Volume 11 (1972), pp. 1-49 | DOI | MR | Zbl

[2] Borman, Matthew Strom; Eliashberg, Yakov; Murphy, Emmy Existence and classification of overtwisted contact structures in all dimensions, Acta Math., Volume 215 (2015) no. 2, pp. 281-361 | DOI | MR | Zbl

[3] Bott, Raoul The stable homotopy of the classical groups, Ann. Math., Volume 70 (1959), pp. 313-337 | DOI | MR | Zbl

[4] Casals, Roger; Keating, Ailsa; Smith, Ivan Symplectomorphisms of exotic discs, J. Éc. Polytech., Math., Volume 5 (2018), pp. 289-316 (With an appendix by Sylvain Courte) | DOI | MR | Zbl

[5] Casals, Roger; del Pino, Álvaro; Presas, Francisco Loose Engel structures, Compos. Math., Volume 156 (2020) no. 2, pp. 412-434 | DOI | MR | Zbl

[6] Casals, Roger; Presas, Francisco A remark on the Reeb flow for spheres, J. Symplectic Geom., Volume 12 (2014) no. 4, pp. 657-671 | DOI | MR | Zbl

[7] Casals, Roger; Spáčil, Oldřich Chern-Weil theory and the group of strict contactomorphisms, J. Topol. Anal., Volume 8 (2016) no. 1, pp. 59-87 | DOI | MR | Zbl

[8] Ding, Fan; Geiges, Hansjörg The diffeotopy group of $S^1\times S^2$ via contact topology, Compos. Math., Volume 146 (2010) no. 4, pp. 1096-1112 | DOI | MR | Zbl

[9] Dymara, Katarzyna Legendrian knots in overtwisted contact structures on $S^3$, Ann. Global Anal. Geom., Volume 19 (2001) no. 3, pp. 293-305 | DOI | MR | Zbl

[10] Eliashberg, Yakov Classification of overtwisted contact structures on $3$-manifolds, Invent. Math., Volume 98 (1989) no. 3, pp. 623-637 | DOI | MR | Zbl

[11] Eliashberg, Yakov Contact $3$-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier, Volume 42 (1992) no. 1-2, pp. 165-192 | DOI | MR | Zbl

[12] Farrell, F. Thomas; Hsiang, Wu Chung On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Stanford, Calif., 1976), Part 1 (Proceedings of Symposia in Pure Mathematics), Volume 32, American Mathematical Society, 1978, pp. 325-337 | DOI | MR | Zbl

[13] Félix, Yves; Oprea, John; Tanré, Daniel Algebraic models in geometry, Oxford Graduate Texts in Mathematics, 17, Oxford University Press, 2008, xxii+460 pages | MR | Zbl

[14] Geiges, Hansjörg An introduction to contact topology, Cambridge Studies in Advanced Mathematics, 109, Cambridge University Press, 2008, xvi+440 pages | DOI | MR | Zbl

[15] Geiges, Hansjörg; Gonzalo Perez, Jesús On the topology of the space of contact structures on torus bundles, Bull. Lond. Math. Soc., Volume 36 (2004) no. 5, pp. 640-646 | DOI | MR | Zbl

[16] Gironella, Fabio Examples of contact mapping classes of infinite order in all dimensions (2018) (https://arxiv.org/abs/1809.07762, to appear in Math. Res. Lett.) | Zbl

[17] Gironella, Fabio Examples of nontrivial contact mapping classes for overtwisted contact manifolds in all dimensions, Algebr. Geom. Topol., Volume 19 (2019) no. 3, pp. 1207-1227 | DOI | MR | Zbl

[18] Giroux, Emmanuel Sur les transformations de contact au-dessus des surfaces, Essays on geometry and related topics, Vol. 1, 2 (Monographies de l’Enseignement Mathématique), Volume 38, L’Enseignement Mathématique, 2001, pp. 329-350 | MR | Zbl

[19] Giroux, Emmanuel; Massot, Patrick On the contact mapping class group of Legendrian circle bundles, Compos. Math., Volume 153 (2017) no. 2, pp. 294-312 | DOI | MR | Zbl

[20] Harris, Bruno Some calculations of homotopy groups of symmetric spaces, Trans. Am. Math. Soc., Volume 106 (1963), pp. 174-184 | DOI | MR | Zbl

[21] Hatcher, Allen E. A proof of the Smale conjecture, $\mathrm{Diff}\left(S^3\right)\simeq \mathrm{O}\left(4\right)$, Ann. Math., Volume 117 (1983) no. 3, pp. 553-607 | DOI | MR | Zbl

[22] Kachi, Hideyuki Homotopy groups of symmetric spaces ${\Gamma }_n$, J. Fac. Sci., Shinshu Univ., Volume 13 (1978) no. 2, pp. 103-120 | MR | Zbl

[23] Lanzat, Sergei; Zapolsky, Frol On the contact mapping class group of the contactization of the $A_m$-Milnor fiber, Ann. Math. Qué., Volume 42 (2018) no. 1, pp. 79-94 | DOI | MR | Zbl

[24] Massey, William S. Obstructions to the existence of almost complex structures, Bull. Am. Math. Soc., Volume 67 (1961), pp. 559-564 | DOI | MR | Zbl

[25] Massot, Patrick Natural fibrations in contact topology, 2015 (Available at https://www.math.u-psud.fr/~pmassot/files/exposition/fibrations.pdf)

[26] Massot, Patrick; Niederkrüger, Klaus Examples of non-trivial contact mapping classes in all dimensions, Int. Math. Res. Not., Volume 2016 (2016) no. 15, pp. 4784-4806 | DOI | MR | Zbl

[27] Mukai, Juno Remarks on homotopy groups of symmetric spaces, Math. J. Okayama Univ., Volume 32 (1990), pp. 159-164 | MR | Zbl

[28] Vogel, Thomas Non-loose unknots, overtwisted discs, and the contact mapping class group of $S^3$, Geom. Funct. Anal., Volume 28 (2018) no. 1, pp. 228-288 | DOI | MR | Zbl

[29] Weiss, Michael; Williams, Bruce Automorphisms of manifolds, Surveys on surgery theory, Vol. 2 (Annals of Mathematics Studies), Volume 149, Princeton University Press, 2001, pp. 165-220 | MR | Zbl