Contact non-squeezing at large scale in ${\mathbb{R}}^{2n}\times S^1$

Abstract

We define a ${\mathbb{Z}}_k$-equivariant version of the cylindrical contact homology used by Eliashberg-Kim-Polterovich (2006) to prove contact non-squeezing for prequantized integer-capacity balls $B(R)\times S^1\subset {\mathbb{R}}^{2n}\times S^1$, $R\in \mathbb{N}$ and we use it to extend their result to all $R\ge 1$. Specifically we prove if $R\ge 1$ there is no $\psi \in \text{Cont}({\mathbb{R}}^{2n}\times S^1)$, the group of compactly supported contactomorphisms of ${\mathbb{R}}^{2n}\times S^1$ which squeezes $\hat{B}(R)=B(R)\times S^1$ into itself, i.e. maps the closure of $\hat{B}(R)$ into $\hat{B}(R)$. A sheaf theoretic proof of non-existence of corresponding $\psi \in {\text{Cont}}_0({\mathbb{R}}^{2n}\times S^1)$, the identity component of $\text{Cont}({\mathbb{R}}^{2n}\times S^1)$, is due to Chiu (2014); 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 ${\mathbb{Z}}_k$-action generated by a vertical $1/k$-shift but can also be related, for prequantized balls, to the ${\mathbb{Z}}_k$-equivariant contact homology developed by Milin (2008) in her proof of orderability of lens spaces.

Minor changes: corrected typos, improved wording in Intro and added more technical detail in construction

References in corpus (2)

• Non-squeezing property of contact balls
• Automatic transversality in contact homology I: Regularity

Cited by in corpus (1)

• Non-squeezing property of contact balls