Rabinowitz Floer homology and symplectic homology

Cieliebak, Kai; Frauenfelder, Urs; Oancea, Alexandru

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 $(M,\xi )$ dans une variété symplectique $(W,\omega )$, les deux premiers auteurs ont défini des groupes d’homologie dits de Rabinowitz Floer $RF{H}_{*}(M,W)$. Ceux-ci dépendent uniquement de la composante bornée $V$ de $W\setminus M$. 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 $RF{H}_{*}(M,W)$, qui finalement est envoyée vers la cohomologie symplectique de $V$. Nous calculons $RF{H}_{*}(S{T}^{*}L,T^{*}L)$ 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 $dimL\ge 4$.

The first two authors have recently defined Rabinowitz Floer homology groups $RF{H}_{*}(M,W)$ associated to a separating exact embedding of a contact manifold $(M,\xi )$ into a symplectic manifold $(W,\omega )$. These depend only on the bounded component $V$ of $W\setminus M$. 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 $RF{H}_{*}(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RF{H}_{*}(S{T}^{*}L,T^{*}L)$, 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 $dimL\ge 4$ and the embedding induces an injection on ${\pi }_1$.

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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