Rabinowitz Floer homology and symplectic homology

Cieliebak, Kai; Frauenfelder, Urs; Oancea, Alexandru

Rabinowitz Floer homology and symplectic homology
[Homologie de Rabinowitz-Floer et homologie symplectique]

Cieliebak, Kai ; Frauenfelder, Urs ; Oancea, Alexandru

Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010), p. 957-1015 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Étant donné un plongement exact et séparant d’une variété de contact $(M, ξ)$ dans une variété symplectique $(W, ω)$ , les deux premiers auteurs ont défini des groupes d’homologie dits de Rabinowitz Floer $R F H_{*} (M, W)$ . Ceux-ci dépendent uniquement de la composante bornée $V$ de $W ∖ M$ . Nous construisons une suite exacte longue dans laquelle la cohomologie symplectique de $V$ est envoyée vers l’homologie symplectique de $V$ , qui à son tour est envoyée vers l’homologie de Rabinowitz Floer $R F H_{*} (M, W)$ , qui finalement est envoyée vers la cohomologie symplectique de $V$ . Nous calculons $R F H_{*} (S T^{*} L, T^{*} L)$ pour le fibré cotangent unitaire $S T^{*} L$ d’une variété compacte sans bord $L$ . Nous démontrons que l’image d’un plongement exact et séparant de $S T^{*} L$ ne peut pas être disjointe d’elle-même par une isotopie hamiltonienne, à condition que le plongement induise une injection sur le groupe fondamental et $dim L \geq 4$ .

The first two authors have recently defined Rabinowitz Floer homology groups $R F H_{*} (M, W)$ associated to a separating exact embedding of a contact manifold $(M, ξ)$ into a symplectic manifold $(W, ω)$ . These depend only on the bounded component $V$ of $W ∖ M$ . We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$ , which in turn maps to Rabinowitz Floer homology $R F H_{*} (M, W)$ , which then maps to symplectic cohomology of $V$ . We compute $R F H_{*} (S T^{*} L, T^{*} L)$ , where $S T^{*} L$ is the unit cosphere bundle of a closed manifold $L$ . As an application, we prove that the image of a separating exact contact embedding of $S T^{*} L$ cannot be displaced away from itself by a Hamiltonian isotopy, provided $dim L \geq 4$ and the embedding induces an injection on $π_{1}$ .

Publié le : 2010-01-01

MR 2778453 | Zbl 1213.53105

DOI : https://doi.org/10.24033/asens.2137
Classification: 53D35, 53D40
Mots clés: homologie symplectique, homologie de Rabinowitz Floer, plongements de contact, espaces de lacets libres

@article{ASENS_2010_4_43_6_957_0,
     author = {Cieliebak, Kai and Frauenfelder, Urs and Oancea, Alexandru},
     title = {Rabinowitz Floer homology and symplectic homology},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {43},
     year = {2010},
     pages = {957-1015},
     doi = {10.24033/asens.2137},
     mrnumber = {2778453},
     zbl = {1213.53105},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2010_4_43_6_957_0}
}

Cieliebak, Kai; Frauenfelder, Urs; Oancea, Alexandru. Rabinowitz Floer homology and symplectic homology. Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010) pp. 957-1015. doi : 10.24033/asens.2137. http://gdmltest.u-ga.fr/item/ASENS_2010_4_43_6_957_0/

Bibliographie

[1] A. Abbondandolo & M. Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006), 254-316. | MR 2190223 | Zbl 1084.53074

[2] P. Biran, Lagrangian non-intersections, Geom. Funct. Anal. 16 (2006), 279-326. | MR 2231465 | Zbl 1099.53054

[3] P. Biran & K. Cieliebak, Lagrangian embeddings into subcritical Stein manifolds, Israel J. Math. 127 (2002), 221-244. | MR 1900700 | Zbl 1165.53378

[4] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki & E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799-888. | MR 2026549 | Zbl 1131.53312

[5] F. Bourgeois & A. Oancea, An exact sequence for contact- and symplectic homology, Invent. Math. 175 (2009), 611-680. | MR 2471597 | Zbl 1167.53071

[6] F. Bourgeois & A. Oancea, Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces, Duke Math. J. 146 (2009), 71-174. | MR 2475400 | Zbl 1158.53067

[7] G. E. Bredon, Topology and geometry, Graduate Texts in Math. 139, Springer, 1993. | MR 1224675 | Zbl 0791.55001

[8] K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur. Math. Soc. (JEMS) 4 (2002), 115-142. | MR 1911873 | Zbl 1012.53066

[9] K. Cieliebak, A. Floer, H. Hofer & K. Wysocki, Applications of symplectic homology. II. Stability of the action spectrum, Math. Z. 223 (1996), 27-45. | MR 1408861 | Zbl 0869.58013

[10] K. Cieliebak & U. A. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math. 239 (2009), 251-316. | MR 2461235 | Zbl 1221.53112

[11] K. Cieliebak & U. A. Frauenfelder, Morse homology on noncompact manifolds, preprint arXiv:0911.1805. | Zbl 1232.53076

[12] K. Cieliebak, U. A. Frauenfelder & G. P. Paternain, Symplectic topology of Mañé's critical values, Geometry & Topology 14 (2010), 1765-1870. | MR 2679582 | Zbl 1239.53110

[13] S. Eilenberg & N. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, 1952. | MR 50886 | Zbl 0047.41402

[14] H. Hofer & D. Salamon, Floer homology and Novikov rings, in The Floer memorial volume, Progr. Math. 133, Birkhäuser, 1995, 483-524. | MR 1362838 | Zbl 0842.58029

[15] M. Mclean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), 1877-1944. | MR 2497314 | Zbl 1170.53070

[16] A. Oancea, A survey of Floer homology for manifolds with contact type boundary or symplectic homology, in Symplectic geometry and Floer homology. A survey of the Floer homology for manifolds with contact type boundary or symplectic homology, Ensaios Mat. 7, Soc. Brasil. Mat., 2004, 51-91. | MR 2100955 | Zbl 1070.53056

[17] A. Oancea, The Künneth formula in Floer homology for manifolds with restricted contact type boundary, Math. Ann. 334 (2006), 65-89. | MR 2208949 | Zbl 1087.53078

[18] A. Ritter, Topological quantum field theory structure on symplectic cohomology, preprint arXiv:1003.1781. | MR 3065181 | Zbl pre06272206

[19] J. Robbin & D. Salamon, The Maslov index for paths, Topology 32 (1993), 827-844. | MR 1241874 | Zbl 0798.58018

[20] D. Salamon, Lectures on Floer homology, in Symplectic geometry and topology (Park City, UT, 1997), IAS/Park City Math. Ser. 7, Amer. Math. Soc., 1999, 143-229. | MR 1702944 | Zbl 1031.53118

[21] D. Salamon & J. Weber, Floer homology and the heat flow, Geom. Funct. Anal. 16 (2006), 1050-1138. | MR 2276534 | Zbl 1118.53056

[22] D. Salamon & E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303-1360. | MR 1181727 | Zbl 0766.58023

[23] F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv. 81 (2006), 105-121. | MR 2208800 | Zbl 1094.37031

[24] M. Schwarz, Morse homology, Progress in Math. 111, Birkhäuser, 1993. | MR 1239174 | Zbl 0806.57020

[25] P. Seidel, A biased view of symplectic cohomology, in Current developments in mathematics, 2006, Int. Press, Somerville, MA, 2008, 211-253. | MR 2459307 | Zbl 1165.57020

[26] J.-C. Sikorav, Some properties of holomorphic curves in almost complex manifolds, in Holomorphic curves in symplectic geometry, Progr. Math. 117, Birkhäuser, 1994, 165-189. | MR 1274929

[27] M. Vigué-Poirrier & D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976), 633-644. | MR 455028 | Zbl 0361.53058

[28] C. Viterbo, A new obstruction to embedding Lagrangian tori, Invent. Math. 100 (1990), 301-320. | MR 1047136 | Zbl 0727.58015

[29] C. Viterbo, Functors and computations in Floer homology with applications. I, Geom. Funct. Anal. 9 (1999), 985-1033. | MR 1726235 | Zbl 0954.57015

[30] C. Viterbo, Functors and computations in Floer homology with applications. II, preprint Université Paris-Sud no 98-15, 1998. | Zbl 0954.57015