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 $R \geq 1$. Specifically we prove if $R \geq 1$ 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.