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Geometry of contact transformations and domains: orderability versus squeezing

Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich

Geometry & Topology 10 (2006) 1635–1747

arXiv: math.SG/0511658


Gromov’s famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold. In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold. For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable. Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology. We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups.

Dedicated to Dusa McDuff on the occasion of her 60th birthday

contact manifolds, contact squeezing and orderability, Floer homology, holomorphic curves
Mathematical Subject Classification 2000
Primary: 53D10, 53D40
Secondary: 53D35, 53D50
Forward citations
Received: 12 February 2006
Accepted: 4 September 2006
Published: 28 October 2006
Proposed: Eleny Ionel
Seconded: Tomasz Mrowka, Peter Ozsváth
Correction: 1 February 2009
Yakov Eliashberg
Department of Mathematics
Stanford University
Stanford, CA 94305-2125
Sang Seon Kim
Departamento de Matemática
Instituto Superior Técnico
Av Roviso Pais
1049-001 Lisboa
Leonid Polterovich
School of Mathematical Sciences
The Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University
69978 Tel Aviv