Contact Homology, Capacity and Non-Squeezing in ${ℝ}^{2n}\times S^1$ via Generating Functions

Sandon, Sheila

Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 145-185.

Inspirés par le travail de Bhupal, nous étendons à la géométrie de contact la notion de capacité de Viterbo ainsi que la construction, dûe à Traynor, d’homologie symplectique. Comme application, nous obtenons une démonstration alternative du Théorème de Non-Tassement d’Eliashberg, Kim et Polterovitch.

Starting from the work of Bhupal we extend to the contact case the Viterbo capacity and Traynor’s construction of symplectic homology. As an application we get a new proof of the Non-Squeezing Theorem of Eliashberg, Kim and Polterovich.

Classification : 53D35
Keywords: Contact non-squeezing, contact capacity, contact homology, orderability of contact manifolds, generating functions
Mot clés : non tassement de contact, capacité de contact, homologie de contact, ordonnabilité des varitétés de contact, fonctions génératrices

[1] Bhupal, M. Legendrian intersections in the 1-jet bundle, University of Warwick (1998) (Ph. D. Thesis)

[2] Bhupal, M. A partial order on the group of contactomorphisms of ${ℝ}^{2n+1}$ via generating functions, Turkish J. Math., Volume 25 (2001), pp. 125-135 | MR | Zbl

[3] Biran, P.; Polterovich, L.; Salamon, D. Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J., Volume 119 (2003), pp. 65-118 | DOI | MR | Zbl

[4] Chaperon, M. Une idée du type “géodésiques brisées” pour les systémes hamiltoniens, C. R. Acad. Sci. Paris, Sér. I Math., Volume 298 (1984), pp. 293-296 | MR | Zbl

[5] Chaperon, M.; H. Hofer et al. On generating families, The Floer Memorial Volume (Progr. Math.), Volume 133, Birkhauser, Basel, 1995, pp. 283-296 | MR | Zbl

[6] Chekanov, Y. Critical points of quasi-functions and generating families of Legendrian manifolds, Funct. Anal. Appl., Volume 30 (1996), pp. 118-128 | DOI | MR | Zbl

[7] Chekanov, Y.; van Koert, O.; Schlenk, F. Minimal atlases of closed contact manifolds, arXiv:0807.3047

[8] Chekanov, Y.; Pushkar, P. Combinatorics of fronts of Legendrian links, and Arnold’s 4-conjectures, Russian Math. Surveys, Volume 60 (2005), pp. 95-149 | DOI | MR | Zbl

[9] Chernov, V.; Nemirovski, S. Legendrian links, causality, and the Low conjecture, arXiv:0810.5091v2

[10] Chernov, V.; Nemirovski, S. Non-negative Legendrian isotopy in $S{T}^{*}M$, arXiv:0905.0983

[11] Cieliebak, K.; Ginzburg, V.; Kerman, E. Symplectic homology and periodic orbits near symplectic submanifolds, Comment. Math. Helv., Volume 79 (2004), pp. 554-581 | DOI | MR | Zbl

[12] Colin, V.; Ferrand, E.; Pushkar, P. Positive loops of Legendrian embeddings (2007) (preprint)

[13] Eiseman, P.; Lima, J.; Sabloff, J.; Traynor, L. A partial ordering on slices of planar Lagrangians, J. Fixed Point Theory Appl., Volume 3 (2008), pp. 431-447 | DOI | MR | Zbl

[14] Eliashberg, Y. New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc., Volume 4 (1991), pp. 513-520 | DOI | MR | Zbl

[15] Eliashberg, Y.; Gromov, M. Lagrangian intersection theory : finite-dimensional approach, Geometry of differential equations (Amer. Math. Soc. Transl. Ser. 2), Volume 186, Amer. Math. Soc., Providence, RI, 1998 p. 27–118, see also : Lagrangian intersections and the stable Morse theory Boll. Un. Mat. Ital., B(7) 11 suppl. (1997), p. 289–326 | MR | Zbl

[16] Eliashberg, Y.; Kim, S. S.; Polterovich, L. Geometry of contact transformations and domains: orderability vs squeezing, Geom. and Topol., Volume 10 (2006), pp. 1635-1747 | DOI | MR | Zbl

[17] Eliashberg, Y.; Polterovich, L. Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal., Volume 10 (2000), pp. 1448-1476 | DOI | MR | Zbl

[18] Ferrand, E.; Pushkar, P. Morse theory and global coexistence of singularities on wave fronts, J. London Math. Soc., Volume 74 (2006), pp. 527-544 | DOI | MR | Zbl

[19] Fuchs, D.; Rutherford, D. Generating families and Legendrian contact homology in the standard contact space, arXiv:0807.4277

[20] Geiges, H. An Introduction to Contact Topology, Cambridge University Press, 2008 | MR | Zbl

[21] Ginzburg, V.; Gürel, B. Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles, Duke Math. J., Volume 123 (2004), pp. 1-47 | DOI | MR | Zbl

[22] Givental, A. Nonlinear generalization of the Maslov index, Theory of singularities and its applications (Adv. Soviet Math.), Volume 1, Amer. Math. Soc., Providence, RI, 1990, pp. 71-103 | MR | Zbl

[23] Givental, A.; H. Hofer et al. A symplectic fixed point theorem for toric manifolds, The Floer Memorial Volume (Progr. Math.), Volume 133, Birkhauser, Basel, 1995, pp. 445-481 | MR | Zbl

[24] Gromov, M. Pseudoholomorphic curves in symplectic manifolds, Invent.Math., Volume 82 (1985), pp. 307-347 | DOI | MR | Zbl

[25] Hermann, D. Inner and outer Hamiltonian capacities, Bull. Soc. Math. France, Volume 132 (2004), pp. 509-541 | Numdam | MR | Zbl

[26] Hofer, H.; Zehnder, E. Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994 | MR | Zbl

[27] Hörmander, L. Fourier integral operators I, Acta Math., Volume 127 (1971), pp. 17-183 | DOI | MR | Zbl

[28] Jordan, J.; Traynor, L. Generating family invariants for Legendrian links of unknots, Algebr. Geom. Topol., Volume 6 (2006), pp. 895-933 | DOI | MR | Zbl

[29] Laudenbach, F.; Sikorav, J. C. Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibre cotangent, Invent. Math., Volume 82 (1985), pp. 349-357 | DOI | MR | Zbl

[30] McDuff, D.; Salamon, D. Introduction to Symplectic Topology, Oxford University Press, 1998 | MR | Zbl

[31] Milinković, D. Morse homology for generating functions of Lagrangian submanifolds, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 3953-3974 | DOI | MR | Zbl

[32] Sikorav, J. C. Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale, C.R. Acad. Sci. Paris, Sér. I Math., Volume 302 (1986), pp. 119-122 | MR | Zbl

[33] Sikorav, J. C. Problemes d’intersections et de points fixes en géométrie hamiltonienne, Comment. Math. Helv., Volume 62 (1987), pp. 62-73 | DOI | MR | Zbl

[34] Théret, D. Utilisation des fonctions génératrices en géométrie symplectique globale, Université Denis Diderot (Paris 7) (1995) (Ph. D. Thesis)

[35] Théret, D. Rotation numbers of Hamiltonian isotopies in complex projective spaces, Duke Math. J., Volume 94 (1998), pp. 13-27 | DOI | MR | Zbl

[36] Théret, D. A complete proof of Viterbo’s uniqueness theorem on generating functions, Topology Appl., Volume 96 (1999), pp. 249-266 | DOI | MR | Zbl

[37] Traynor, L. Symplectic Homology via generating functions, Geom. Funct. Anal., Volume 4 (1994), pp. 718-748 | DOI | MR | Zbl

[38] Traynor, L. Generating Function Polynomials for Legendrian Links, Geom. and Topol., Volume 5 (2001), pp. 719-760 | DOI | MR | Zbl

[39] Viterbo, C. Functors and computations in Floer homology with applications, Part II (Preprint) | Zbl

[40] Viterbo, C. Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systémes hamiltoniens, Bull. Soc. Math. France, Volume 115 (1987), pp. 361-390 | Numdam | MR | Zbl

[41] Viterbo, C. Symplectic topology as the geometry of generating functions, Math. Ann., Volume 292 (1992), pp. 685-710 | DOI | MR | Zbl