Rabinowitz Floer homology and symplectic homology

Cieliebak, Kai; Frauenfelder, Urs; Oancea, Alexandru

doi:10.24033/asens.2137

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, Série 4, Tome 43 (2010) no. 6, pp. 957-1015

Abstract (VO)
Résumé

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 $π_{1}$ .

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 $dim L \geq 4$ .

MR Zbl

DOI : 10.24033/asens.2137

Classification : 53D35, 53D40
Keywords: symplectic homology, Rabinowitz Floer homology, contact embeddings, free loop space
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},
     pages = {957--1015},
     year = {2010},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {6},
     doi = {10.24033/asens.2137},
     mrnumber = {2778453},
     zbl = {1213.53105},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2137/}
}

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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

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