Rabinowitz Floer homology and symplectic homology

Kai Cieliebak; Urs Frauenfelder; Alexandru Oancea

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 6, page 957-1015
  • ISSN: 0012-9593

Abstract

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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 ≥ 4 and the embedding induces an injection on π 1 .

How to cite

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Cieliebak, Kai, Frauenfelder, Urs, and Oancea, Alexandru. "Rabinowitz Floer homology and symplectic homology." Annales scientifiques de l'École Normale Supérieure 43.6 (2010): 957-1015. <http://eudml.org/doc/272182>.

@article{Cieliebak2010,
abstract = {The first two authors have recently defined Rabinowitz Floer homology groups $RFH_*(M,W)$ associated to a separating exact embedding of a contact manifold $(M,\xi )$ 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 $RFH_*(M,W)$, 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 $\dim \,L\ge 4$ and the embedding induces an injection on $\pi _1$.},
author = {Cieliebak, Kai, Frauenfelder, Urs, Oancea, Alexandru},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {symplectic homology; Rabinowitz Floer homology; contact embeddings; free loop space},
language = {eng},
number = {6},
pages = {957-1015},
publisher = {Société mathématique de France},
title = {Rabinowitz Floer homology and symplectic homology},
url = {http://eudml.org/doc/272182},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Cieliebak, Kai
AU - Frauenfelder, Urs
AU - Oancea, Alexandru
TI - Rabinowitz Floer homology and symplectic homology
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 6
SP - 957
EP - 1015
AB - The first two authors have recently defined Rabinowitz Floer homology groups $RFH_*(M,W)$ associated to a separating exact embedding of a contact manifold $(M,\xi )$ 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 $RFH_*(M,W)$, 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 $\dim \,L\ge 4$ and the embedding induces an injection on $\pi _1$.
LA - eng
KW - symplectic homology; Rabinowitz Floer homology; contact embeddings; free loop space
UR - http://eudml.org/doc/272182
ER -

References

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