Contact non-squeezing at large scale in ℝ2n × S1

Abstract

We define a [Formula: see text]-equivariant version of the cylindrical contact homology used by Eliashberg–Kim–Polterovich [11] to prove contact non-squeezing for prequantized integer-capacity balls [Formula: see text], [Formula: see text] and we use it to extend their result to all [Formula: see text]. Specifically, we prove if [Formula: see text] there is no [Formula: see text], the group of compactly supported contactomorphisms of [Formula: see text] which squeezes [Formula: see text] into itself, i.e. maps the closure of [Formula: see text] into [Formula: see text]. A sheaf theoretic proof of non-existence of corresponding [Formula: see text], the identity component of [Formula: see text], is due to Chiu [7] it is not known if this is strictly weaker. Our construction has the advantage of retaining the contact homological viewpoint of Eliashberg–Kim–Polterovich and its potential for application in prequantizations of other Liouville manifolds. It makes use of the [Formula: see text]-action generated by a vertical [Formula: see text]-shift but can also be related, for prequantized balls, to the [Formula: see text]-equivariant contact homology developed by Milin [16] in her proof of orderability of lens spaces.