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

Sandon, Sheila

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

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.

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.

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