## 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