Taiwanese Journal of Mathematics

TOWARD THE POINCARÉ CONJECTURE

Wen-Hsiung Lin

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Abstract

Let $n$ be a positive integer. An $n$-manifold $M$ is a Hausdorff topological space with a countable base of open sets such that $M$ is locally the Euclidean space $\mathbb{R}^n$, that is, for each $x \in M$ there exists an open neighborhood $U$ of $x$ and a homeomorphism $U \stackrel{\varphi}{\to} V$ from $U$ onto an open subset $V$ of $\mathbb{R}^n$. A compact $n$-manifold is an $n$-manifold which is compact as a topological space. It is clear that any $n$-manifold is locally path-connected and so is a path-connected space if it is a connected space. It is also clear that a compact $n$-manifold has only a finite number of path components, each of these being a compact path-connected $n$-manifold.

$\mathbb{R}^n$ and any of its non-empty open subsets are examples of $n$-manifolds. These are not compact manifolds. Let \[ S^n = \{ x = (x_1,\dotsc,x_{n+1}) \in \mathbb{R}^{n+1} |\ \|x\| = \sqrt{x^2_1+\cdots+x^2_{n+1}} = r \gt 0 \}, \] called the $n$-sphere (of radius $r$). $S^n$ is a path-connected compact $n$-manifold (note that $n \ge 1$). It is easy to see that any two $n$-spheres of different radii are homeomorphic.

Article information

Source
Taiwanese J. Math., Volume 10, Number 5 (2006), 1109-1129.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500557292

Digital Object Identifier
doi:10.11650/twjm/1500557292

Mathematical Reviews number (MathSciNet)
MR2253368

Zentralblatt MATH identifier
1108.57013

Subjects
Primary: 57M20: Two-dimensional complexes 57M40: Characterizations of $E^3$ and $S^3$ (Poincaré conjecture) [See also 57N12] 57R60: Homotopy spheres, Poincaré conjecture

Keywords
surfaces 3-manifolds

Citation

Lin, Wen-Hsiung. TOWARD THE POINCARÉ CONJECTURE. Taiwanese J. Math. 10 (2006), no. 5, 1109--1129. doi:10.11650/twjm/1500557292. https://projecteuclid.org/euclid.twjm/1500557292


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