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Transactions of the American Mathematical Society

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Repairing embeddings of $ 3$-cells with monotone maps of $ E\sp{3}$


Author: William S. Boyd
Journal: Trans. Amer. Math. Soc. 161 (1971), 123-144
MSC: Primary 54.78
DOI: https://doi.org/10.1090/S0002-9947-1971-0282352-5
MathSciNet review: 0282352
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Abstract: If $ {S_1}$ is a 2-sphere topologically embedded in Euclidean 3-space $ {E^3}$ and $ {S_2}$ is the unit sphere about the origin, then there may not be a homeomorphism of $ {E^3}$ onto itself carrying $ {S_1}$ onto $ {S_2}$. We show here how to construct a map f of $ {E^3}$ onto itself such that $ f\vert{S_1}$ is a homeomorphism of $ {S_1}$ onto $ {S_2}$, $ f({E^3} - {S_1}) = {E^3} - {S_2}$ and $ {f^{ - 1}}(x)$ is a compact continuum for each point x in $ {E^3}$. Similar theorems are obtained for 3-cells and disks topologically embedded in $ {E^3}$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0282352-5
Keywords: Wild sphere, tame sphere, monotone map, upper semicontinuous decomposition, crumpled cube, repairing embeddings
Article copyright: © Copyright 1971 American Mathematical Society

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