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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.

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On almost one-to-one maps
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by Alexander Blokh, Lex Oversteegen and E. D. Tymchatyn PDF
Trans. Amer. Math. Soc. 358 (2006), 5003-5014 Request permission

Abstract:

A continuous map $f:X\to Y$ of topological spaces $X, Y$ is said to be almost $1$-to-$1$ if the set of the points $x\in X$ such that $f^{-1}(f(x))=\{x\}$ is dense in $X$; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and $\sigma$-compact spaces (e.g., $n$-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if $f$ is a minimal self-mapping of a 2-manifold $M$, then point preimages under $f$ are tree-like continua and either $M$ is a union of 2-tori, or $M$ is a union of Klein bottles permuted by $f$.
References
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Additional Information
  • Alexander Blokh
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
  • MR Author ID: 196866
  • Email: ablokh@math.uab.edu
  • Lex Oversteegen
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
  • MR Author ID: 134850
  • Email: overstee@math.uab.edu
  • E. D. Tymchatyn
  • Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6
  • MR Author ID: 175580
  • Email: tymchat@snoopy.math.usask.ca
  • Received by editor(s): February 29, 2004
  • Received by editor(s) in revised form: October 21, 2004
  • Published electronically: June 13, 2006
  • Additional Notes: The first author was partially supported by NSF Grant DMS-0140349
    The second author was partially supported by NSF Grant DMS-0072626
    The third author was partially supported by NSERC grant OGP005616
  • © Copyright 2006 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 5003-5014
  • MSC (2000): Primary 57N35, 54C10; Secondary 37B45
  • DOI: https://doi.org/10.1090/S0002-9947-06-03922-5
  • MathSciNet review: 2231882