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Recent progress on the Poincaré conjecture and the classification of 3-manifolds. (English) Zbl 1100.57016
This article, written in 2004, surveys Perelman’s work on Thurston’s Geometrization Conjecture, which includes the famous Poincaré Conjecture as a very special case. After a brief overview, enough about the geometric structures and prime decomposition of \(3\)-manifolds is presented to give a clear statement of the Conjecture. Then, the Ricci flow and R. Hamilton’s initial applications of it to geometrization are presented. The second half of the article describes Perelman’s work, focusing somewhat more on the topological side than the analytic. The closing section is a fine one-page summary of the work of Hamilton and Perelman.

MSC:
57M50 General geometric structures on low-dimensional manifolds
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
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