Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 0282352
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54.78

Retrieve articles in all journals with MSC: 54.78

Additional Information

Keywords: Wild sphere, tame sphere, monotone map, upper semicontinuous decomposition, crumpled cube, repairing embeddings
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society