The Geometry of the Bing Involution

TL;DR

This paper investigates the wildness of Bing's 1952 involution on the 3-sphere, proving that any topologically conjugate involution must have a nearly exponential modulus of continuity, revealing extreme distortion properties.

Contribution

The paper establishes a lower bound on the modulus of continuity for involutions topologically conjugate to Bing's, showing it must be nearly exponential, and introduces the concept of inherent modulus of continuity.

Findings

Any conjugate involution has nearly exponential modulus of continuity.

Bing's original involution has a modulus exceeding a square-root exponential.

For any growth function, a corresponding involution with faster modulus growth exists.

Abstract

In 1952 Bing published a wild (not topologically conjugate to smooth) involution $I$ of the 3-sphere $S^3$. But exactly how wild is it, analytically? We prove that any involution $I^h$, topologically conjugate to $I$, must have a nearly exponential modulus of continuity. Specifically, given any $\alpha>0$, there exists a sequence of $\delta$'s converging to zero, $\delta > 0$, and points $x,y \in S^3$ with dist$(x,y) < \delta$, yet dist$(I^h(x), I^h(y)) > \epsilon$, where $\delta^{-1} = e^{\left(\frac{\epsilon^{-1}}{\log^{(1+\alpha)}(\epsilon^{-1})}\right)}$, and dist is the usual Riemannian distance on $S^3$. In particular, $I^h$ stretches distance much more than a Lipschitz function ($\delta^{-1} = c\epsilon^{-1}$) or a H\"{o}lder function ($\delta^{-1} = c^\prime(\epsilon^{-1})^{p}$, $1 < p < \infty$). Bing's original construction and known alternatives (see text) for $I$ have a modulus of continuity $\delta^{-1} > c \sqrt{2}^{\epsilon^{-1}}$, so the theorem is reasonably tight -- we prove the modulus must be at least exponential up to a polylog, whereas the truth may be fully exponential. Actually, the functional for $\delta^{-1}$ coming out of the proof can be chosen slightly closer to exponential than stated here (see Theorem 1). Using the same technique we analyze a large class of ``ramified'' Bing involutions and show, as a scholium, that given any function $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$, no matter how rapid its growth, we can find a corresponding involution $J$ of the 3-sphere such that any topological conjugate $J^h$ of $J$ must have a modulus of continuity $\delta^{-1}(\epsilon^{-1})$ growing faster than $f$ (near infinity). There is a literature on inherent differentiability (references in text) but as far as the authors know the subject of inherent modulus of continuity is new.