Shrinking Without Doing Much At All

Michael Freedman; Michael Starbird

doi:10.2140/agt.2025.25.2099

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arXiv · 2209.07630

Shrinking Without Doing Much At All

Michael Freedman·Michael Starbird

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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.

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Michael Freedman, Michael Starbird. 2022-09-15. Shrinking Without Doing Much At All. https://doi.org/10.2140/agt.2025.25.2099

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