Contact non-squeezing at large scale via generating functions

TL;DR

The paper proves a large-scale contact non-squeezing result by developing a $\mathbb{Z}_k$-equivariant generating function homology for domains in $(\mathbb{R}^{2n} \times S^1, \xi_0)$ and its contact analog via translated $k$-chains. The approach mirrors symplectic generating function techniques, using a category-theoretic construction to define canonical equivariant homologies for isotopies, and introduces translated chains to replace fixed points in the contact setting. A key outcome is the relation $G_{\mathbb{Z}_k,*}^{(a,b]}(\widehat{\mathcal{U}}) \cong G_{\mathbb{Z}_k,*}^{(a,b]}(\mathcal{U}) \otimes H_* (S^1)$, together with explicit calculations for balls that force a contradiction to any contact squeezing when $1 \leq \pi R_2^2 \leq \pi R_1^2$. This yields a purely generating-function-based proof of the desired non-squeezing phenomenon and suggests a path to defining equivariant capacities in this setting.

Abstract

Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $πR_2^2 \leq K \leq πR_1^2$ for some integer $K$ then there is no contact squeezing in $\mathbb{R}^{2n} \times S^1$ of the prequantization of the ball of radius $R_1$ into the prequantization of the ball of radius $R_2$. This result was extended to the case of balls of radius $R_1$ and $R_2$ with $1 \leq πR_2^2 \leq πR_1^2$ by Chiu (2017) and the first author (2016), using respectively microlocal sheaves and SFT. In the present article we recover this general contact non-squeezing theorem using generating functions, a classical method based on finite dimensional Morse theory. More precisely, we develop an equivariant version, with respect to a certain action of a finite cyclic group, of the generating function homology for domains of $\mathbb{R}^{2n} \times S^1$ defined by the second author (2011). A key role in the construction is played by translated chains of contactomorphisms, a generalization of translated points.

Figures (4)

• Figure 1: The homology groups $G_{2nl}^{(a,\infty]}(B^{2n}(R))$.
• Figure 2: If $p$ is a $7$-periodic point of $\varphi$ then all $\varphi^{j}(p)$, $j = 1,\dots,6$, are $7$-periodic points of $\varphi$. The group $\mathbb{Z}_7$ acts on the set $\{\varphi^{j}(p) \;\lvert\; j=0,\dots,6$}.
• Figure 3: If the triple $(p_1,p_2,p_3)$ is a translated $3$-chain of $\phi$ then $(p_2, p_3, p_1)$ and $(p_3, p_1, p_2)$ are translated $3$-chains of $\phi$. The group $\mathbb{Z}_3$ acts on the set $\{\, (p_1, p_2, p_3) \,,\, (p_2, p_3, p_1) \,,\, (p_3, p_1, p_2)\,\}$.
• Figure 4: The homology groups $G_{\mathbb{Z}_5,2nl}^{(a,\infty]}(B^{2n}(R))$ for $l<5$.

Theorems & Definitions (53)

• Theorem 1.1: Contact non-squeezing at large scale
• Remark 1
• Proposition 1
• Lemma 1
• proof
• Remark 2
• Proposition 2
• Proposition 3
• proof
• Proposition 4