Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Extending cell-like maps on manifolds
HTML articles powered by AMS MathViewer

by B. J. Ball and R. B. Sher PDF
Trans. Amer. Math. Soc. 186 (1973), 229-246 Request permission

Abstract:

Let X be a closed subset of a manifold M and ${G_0}$ be a cell-like upper semicontinuous decomposition of X. We consider the problem of extending ${G_0}$ to a cell-like upper semicontinuous decomposition G of M such that $M/G \approx M$. Under fairly weak restrictions (which vanish if $M = {E^n}$ or ${S^n}$ and $n \ne 4$ we show that such a G exists if and only if the trivial extension of ${G_0}$, obtained by adjoining to ${G_0}$ the singletons of $M - X$, has the desired property. In particular, the nondegenerate elements of Bing’s dogbone decomposition of ${E^3}$ are not elements of any cell-like upper semicontinuous decomposition G of ${E^3}$ such that ${E^3}/G \approx {E^3}$. Call a cell-like upper semicontinuous decomposition G of a metric space X simple if $X/G \approx X$ and say that the closed set Y is simply embedded in X if each simple decomposition of Y extends trivially to a simple decomposition of X. We show that tame manifolds in ${E^3}$ are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat k-manifold in an m-manifold $(k,m \ne 4)$. Examples are given of an everywhere wild simply embedded simple closed curve in ${E^3}$ and of a compact absolute retract which embeds in ${E^3}$ yet has no simple embedding in ${E^3}$.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 57A60
  • Retrieve articles in all journals with MSC: 57A60
Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 186 (1973), 229-246
  • MSC: Primary 57A60
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0328950-3
  • MathSciNet review: 0328950