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Shrinking without doing
much at all
Michael Freedman and Michael Starbird |
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Algebraic & Geometric Topology 25 (2025) 2099–2114
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Abstract |
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In 1952 Bing astonished the mathematical world with his wild involution on . It has been among the most seminal examples in topology. The example depends on finding shrinking homeomorphisms of Bing’s decomposition of into points and arcs. If Bing’s original homeomorphisms are varied, Bing’s original wild involution changes by conjugation, which preserves some analytic properties 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. Here we produce a counterintuitive construction, namely a method to shrink the Bing decomposition doing almost nothing at all: neither lengthening much nor rotating much. |
Keywords
Bing involution, shrinking, modulus of continuity
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Mathematical Subject Classification
Primary: 57K30
Secondary: 57M60
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References |
Publication
Received: 24 January 2023
Revised: 4 October 2023
Accepted: 23 October 2023
Published: 11 August 2025
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Authors |
| Michael Freedman | |
| Station Q Microsoft Santa Barbara, CA United States |
Department of Mathematics University of California, Santa Barbara Santa Barbara, CA United States |
| Michael Starbird | |
| Department of Mathematics The University of Texas at Austin Austin, TX United States |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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