Gromov's famous non-squeezing theorem (1985) states that the standardsymplectic ball cannot be symplectically squeezed into any cylinder of smallerradius. Does there exist an analogue of this result in contact geometry? Ourmain finding is that the answer depends on the sizes of the domains inquestion: We establish contact non-squeezing on large scales, and show that itdisappears on small scales. The algebraic counterpart of the (non)-squeezingproblem for contact domains is the question of existence of a natural partialorder on the universal cover of the contactomorphisms group of a contactmanifold. In contrast to our earlier beliefs, we show that the answer to thisquestion is very sensitive to the topology of the manifold. For instance, weprove that the standard contact sphere is non-orderable while the realprojective space is known to be orderable. Our methods include a new embeddingtechnique in contact geometry as well as a generalized Floer homology theorywhich contains both cylindrical contact homology and Hamiltonian Floerhomology. We discuss links to a number of miscellaneous topics such as topologyof free loops spaces, quantum mechanics and semigroups. An erratum is attached whose purpose is to is to correct a number ofinconsistencies in the main paper. These are related to the grading ofgeneralized Floer homology and do not affect formulations and proofs of themain results of the paper.
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