Rabinowitz Floer homology and symplectic homology
[Homologie de Rabinowitz-Floer et homologie symplectique]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 6, pp. 957-1015.

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 RFH * (M,W). Ceux-ci dépendent uniquement de la composante bornée V de WM. 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 RFH * (M,W), qui finalement est envoyée vers la cohomologie symplectique de V. Nous calculons RFH * (ST * L,T * L) pour le fibré cotangent unitaire ST * L d’une variété compacte sans bord L. Nous démontrons que l’image d’un plongement exact et séparant de ST * 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 dimL4.

The first two authors have recently defined Rabinowitz Floer homology groups RFH * (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 WM. 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 RFH * (M,W), which then maps to symplectic cohomology of V. We compute RFH * (ST * L,T * L), where ST * 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 ST * L cannot be displaced away from itself by a Hamiltonian isotopy, provided dimL4 and the embedding induces an injection on π 1 .

DOI : 10.24033/asens.2137
Classification : 53D35, 53D40
Keywords: symplectic homology, Rabinowitz Floer homology, contact embeddings, free loop space
Mot clés : homologie symplectique, homologie de Rabinowitz Floer, plongements de contact, espaces de lacets libres
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     title = {Rabinowitz {Floer} homology and symplectic homology},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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Cieliebak, Kai; Frauenfelder, Urs; Oancea, Alexandru. Rabinowitz Floer homology and symplectic homology. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 6, pp. 957-1015. doi : 10.24033/asens.2137. http://archive.numdam.org/articles/10.24033/asens.2137/

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

[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

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

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

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

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

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