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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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A characterization of manifolds
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by Louis F. McAuley PDF
Trans. Amer. Math. Soc. 209 (1975), 101-107 Request permission

Abstract:

The purpose of this paper is (1) to give a proof of one general theorem characterizing certain manifolds and (2) to illustrate a technique which should be useful in proving various theorems analogous to the one proved here. Theorem. Suppose that $f:X \Rightarrow [0,1]$, where $X$ is a compactum, and that $f$ has the properties: (1) for $0 \leqslant x < 1/2,{f^{ - 1}}(x) = {S^n} \cong {M_0}$, (2) ${f^{ - 1}}(1/2) \cong {S^n}$ with a tame (or flat) $k$-sphere ${S^k}$ shrunk to a point, (3) for $1/2 < x \leqslant 1,{f^{ - 1}}(x) \cong$ a compact connected $n$-manifold ${M_1} \cong {S^{n - (k + 1)}} \times {S^{k + 1}}$ (a spherical modification of ${M_0}$ of type $k$), and (4) there is a continuum $C$ in $X$ such that (letting ${C_x} = {f^{ - 1}}(x) \cap C$) (a) $0 \leqslant x < 1/2,{C_x} \cong {S^k}$, (b) ${C_{1/2}} = \{ p\}$ a point, (c) for $1/2 < x \leqslant 1$, and (d) each of $f|(X - C),f|{f^{ - 1}}[0,1/2)$, and $f|{f^{ - 1}}(1/2,1]$ is completely regular. Then $X$ is homeomorphic to a differentiable $(n + 1)$-manifold $M$ whose boundary is the disjoint union of ${\bar M_0}$ and ${\bar M_1}$ where ${M_i} = {\bar M_i},i = 0,1$.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 209 (1975), 101-107
  • MSC: Primary 57A15
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0391099-X
  • MathSciNet review: 0391099