Shrinking without doing much at all

Freedman, Michael¹ ; Starbird, Michael²

¹Station Q, Microsoft, Santa Barbara, CA, United States, Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, United States
²Department of Mathematics, The University of Texas at Austin, Austin, TX, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2099-2114

Résumé

In 1952 Bing astonished the mathematical world with his wild involution on S3. It has been among the most seminal examples in topology. The example depends on finding shrinking homeomorphisms of Bing’s decomposition of S3 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.

Détail

DOI : 10.2140/agt.2025.25.2099

Keywords: Bing involution, shrinking, modulus of continuity

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