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We define a $\mathbb{Z}_k$-equivariant version of the cylindrical contacthomology used by Eliashberg-Kim-Polterovich (2006) to prove contactnon-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 theirresult 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 compactlysupported 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 ofcorresponding $\psi \in \text{Cont}_0(\mathbb{R}^{2n} \times S^1)$, theidentity 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 theadvantage of retaining the contact homological viewpoint ofEliashberg-Kim-Polterovich and its potential for application inprequantizations of other Liouville manifolds. It makes use of the$\mathbb{Z}_k$-action generated by a vertical $1/k$-shift but can also berelated, for prequantized balls, to the $\mathbb{Z}_k$-equivariant contacthomology developed by Milin (2008) in her proof of orderability of lens spaces.

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