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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2209.07630 |
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| _version_ | 1866916906396549120 |
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| author | Freedman, Michael Starbird, Michael |
| author_facet | Freedman, Michael Starbird, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07630 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Shrinking Without Doing Much At All Freedman, Michael Starbird, Michael Geometric Topology 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. |
| title | Shrinking Without Doing Much At All |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2209.07630 |