Rabinowitz Floer homology and symplectic homology
By Kai Cieliebak, Urs Frauenfelder, A. Oancea. 2009
135 citations recorded by the source corpus.
The Rabinowitz-Floer homology groups $RFH_*(M,W)$ are associated to an exact embedding of a contact manifold $(M,\xi)$ into a symplectic manifold $(W,\omega)$. They 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 an exact contact embedding of $ST^*L$ (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided $\dim L\ge 4$ and the embedding induces an injection on $\pi_1$. In particular, $ST^*L$ does not admit an exact contact embedding into a subcritical Stein manifold if $L$ is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8ec33bb85bd2f03d74c2066392dac71efeb88655
How researchers see this paper
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In Geometry and Topology
- Alexander F. Ritter
- M. Abouzaid
- A. Oancea
- P. Seidel
- Kai Cieliebak
- Otto van Koert
- I. Smith
- Frederick J. Bourgeois III
In Mathematical Physics
- Alexander F. Ritter
- A. Oancea
- M. Abouzaid
- P. Seidel
- Kai Cieliebak
- Otto van Koert
- Frederick J. Bourgeois III
- I. Smith
Citations in the corpus
- Contact geometry of the restricted three‐body problem (2010, methodology): “Since the contact structures are of restricted contact type we can apply the theorem by Cieliebak-Frauenfelder-Oancea [6] and obtain that the Rabinowitz Floer homology of the energy hypersurface as defined by Cieliebak-Frauenfelder [5] is isomorphic to the direct sum of symplectic homology and…”
- Altering symplectic manifolds by homologous recombination (2010, background): “Besides the original papers [58, 57], a few references are [12, 42, 50, 7, 15, 45].”
- Symplectic homology and the Eilenberg-Steenrod axioms (2015, methodology): “The following result follows in the index-positive case (ii) from the arguments of [15], as remarked in [29, 18].”
- Leaf-wise intersections and Rabinowitz Floer homology (2008, unknown): “See [CFO09] for further examples.”
- Topological quantum field theory structure on symplectic cohomology (2010, methodology): “By exploiting recent literature on the Rabinowitz-Floer cohomology RFH(M) [11, 12], and after proving that RFH(M) = 0 iff SH(M) = 0, we will deduce: Corollary.”