Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A characterization of manifolds


Author: Louis F. McAuley
Journal: Trans. Amer. Math. Soc. 209 (1975), 101-107
MSC: Primary 57A15
MathSciNet review: 0391099
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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\vert(X - C),f\vert{f^{ - 1}}[0,1/2)$, and $ f\vert{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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57A15

Retrieve articles in all journals with MSC: 57A15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0391099-X
PII: S 0002-9947(1975)0391099-X
Article copyright: © Copyright 1975 American Mathematical Society