Mathematics – Symplectic Geometry
Scientific paper
2010-03-09
Mathematics
Symplectic Geometry
59 pages (39 + appendix), 17 figures. Version 4: shorter revised version, the symplectic cohomology and wrapped Floer cohomolo
Scientific paper
We construct the TQFT on twisted symplectic cohomology and twisted wrapped Floer cohomology, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic cohomology. These twisted constructions yield new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. In the Appendix we construct in detail the untwisted TQFT theory, which was missing in the literature. This leads to a new result: if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary of M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian lying in the boundary. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology. Finally, we use the TQFT to prove that symplectic cohomology vanishes iff Rabinowitz Floer cohomology vanishes.
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