Taiwanese Journal of Mathematics


Wen-Hsiung Lin

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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

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

First available in Project Euclid: 20 July 2017

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Zentralblatt MATH identifier

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

surfaces 3-manifolds


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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  • M. T. Anderson, Geometrization of $3$-Manifolds via the Ricci Flow, Notices of the AMS, 51(2) (2004), 184-193.
  • W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1973.
  • R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255-306.
  • R. Hamilton, The Harnack estimate for the Ricci flow, J. Differential. Geom. 37 (1993), 225-243.
  • R. Hamilton, Formation of singularities in the Ricci flow, Surveys in Differential Geometry 2, International Press, 1995, pp. 7-136.
  • R. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom.5 (1997), 1-92.
  • R. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Analysis and Geometry 7 (1999), 695-729.
  • John W. Morgan, Recent progress on the Poincaré Conjecture and the classification of $3$-manifolds, Bull. AMS. 42(1) (2004), 57-78.
  • W. S. Massey, Algebraic Topology: An Introduction, Harcourt-Brace, New York, 1967.
  • John W. Milnor, Topology from the differentiable viewpoint, University of Virginia Press, Charlottesville, 1965.
  • John W. Milnor, Towards the Poincaré Conjecture and the Classification of $3$-Manifolds, Notices of the AMS, 50(10)) (2003), 1226-1233.
  • G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.math.DG/0211159, November 11, 2002.
  • G. Perelman, Ricci flow with surgery on three-manifolds, arXiv.math.DG/0303109, March 10, 2003.
  • G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv.math.DG/0307245, July 17, 2003.
  • W. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, Princeton University Press, 1997.
  • Jeffrey Weeks, The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations, Notices of the AMS, 51(6) (2004), 610-619.
  • J. A. Wolf, Spaces of constant curvature, Publish & Perish, Boston, 1984.