Shrinking Without Doing Much At All

TL;DR

This paper presents a novel method to shrink Bing's decomposition of the 3-sphere with minimal geometric disturbance, advancing understanding of topological involutions and their conjugacy classes.

Contribution

It introduces a counterintuitive construction that shrinks Bing's decomposition with minimal lengthening or rotation, expanding the geometric understanding of Bing's involutions.

Findings

Successfully shrinks Bing's decomposition with minimal geometric change

Demonstrates new techniques for controlling topological involutions

Provides insights into the conjugacy class of Bing's involutions

Abstract

In 1952 Bing astonished the mathematical world with his wild involution on $S^3$. It has been among the most seminal examples in topology. The example depends on finding shrinking homeomorphisms of Bing's decomposition of $S^3$ into points and arcs. If Bing's original homeomorphisms are varied, Bing's original wild involution changes by conjugation, which preserves some analytic properties \cite{fs22} while altering others. In 1988, Bing published a second paper "Shrinking Without Lengthening," answering a question that one of the present authors posed to him in an effort to understand the geometry of the entire conjugacy class. In this paper we produce a counterintuitive construction, namely, a method to shrink the Bing decomposition doing almost nothing at all--neither lengthening much nor rotating much.