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The Geometrization Conjecture

John Morgan and Gang Tian
Publisher: 
American Mathematical Society/Clay Mathematics Institute
Publication Date: 
2014
Number of Pages: 
291
Format: 
Hardcover
Series: 
Clay Mathematics Monographs 5
Price: 
81.00
ISBN: 
9780821852019
Category: 
Monograph
[Reviewed by
Scott Taylor
, on
01/23/2015
]

Thurston’s Geometrization Conjecture for 3-dimensional manifolds is one of the 20th century’s most spectacular insights into the nature of 3-dimensional spaces. Its resolution in 2002–2003 by Grigori Perelman is one of the 21st century’s most spectacular early achievements. As is well-known, Perelman simply posted his papers on the arXiv and declined to submit them to a journal. There followed a dramatic race to examine his results and fill out his arguments with additional proofs.

The book under review is the second volume produced by one of the teams in the race: John Morgan and Gang Tian. It builds on their first volume [5]. The two volumes together have become (along with some of the sources mentioned below) the standard resources for understanding the Geometrization Conjecture and its solution. The first volume focused on Perelman’s proof of the Poincaré Conjecture; this second volume focusses on the proof of the full Geometrization Conjecture. Although it contains expositions of much preliminary material (such as Gromov-Hausdorff Convergence and Alexandrov Spaces), it is likely completely accessible only to those with a firm foundation in modern differential geometry. For those with the background or the patience to develop it on-the-fly, the book is extremely readable and highly recommended.

Rather than giving in to the temptation to describe the Geometrization Conjecture and its importance, I’ll point to other resources which may be of interest to someone either encountering this beautiful subject for the first time or to those who want to explore other options for learning the technical aspects of the solution.

The personalities, drama, and informal introductions to the mathematics are (occasionally controversially) given in [6], [8], [13], and, of course, on the Wikipedia page [14]. Peter Scott’s 1983 article [12] is mathematically best place to start, as it puts Thurston’s Conjecture in context and gives lots of mathematical details on how the conjecture is connected to well-known results from 2-dimensional geometry, topology, and group theory. An exposition of Thurston’s solution for a particular class of 3-manifolds (known as “Haken”) are given in the books [3], [7]. As far as the solution to the Geometrization Conjecture goes, Perelman’s papers are on the arXiv ([9],[10], [11]). The papers [2], [4] also provide many more details for Perelman’s papers. An approach aimed at topologists and which depends on the Geometrization Conjecture for Haken manifolds is given in the book [1].


References

[1] Laurent Bessières, Gérard Besson, Sylvain Maillot, Michel Boileau, Joan Porti, Geometrisation of 3-manifolds (European Mathematical Society, 2010)

[2] Huai-Dong Cao, Xi-Ping Zhu, “A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow,” Asian J. Math., 10 (2006) 165–492.

[3] Michael Kapovich, Hyperbolic manifolds and discrete groups (Birkhäuser, 2009).

[4] Bruce Kleiner, John Lott, “Notes on Perelman’s papers” Geom. Topol., 12 (2008) 2587–2855.

[5] John Morgan, Gang Tian, Ricci flow and the Poincaré conjecture (American Mathematical Society and Clay Mathematics Institute, 2007)

[6] Sylvia Nasar, David Gruber, “Manifold Destiny” The New Yorker, August 28, 2006.

[7] Jean-Pierre Otal, “Thurston’s hyperbolization of Haken manifolds,” Surveys in differential geometry, Vol. III, 77–194, (Int. Press, 1998)

[8] Donal O’Shea, The Poincaré Conjecture: In Search of the Shape of the Universe (Walker, 2007).

[9] Grisha Perelman, “The entropy formula for the Ricci flow and its geometric applications” arXiv:math/0211159v1.

[10] Grisha Perelman, “Ricci flow with surgery on 3-manifolds” arXiv:math/0303109v1.

[11] Grisha Perelman, “Finite extinction time for the solutions to the Ricci flow on certain 3-manifolds” arXiv:math/0307245v1.

[12] Peter Scott, “The geometries of $3$-manifolds” Bull. London Math. Soc. 15 (1983) 401–487.

[13] George Szpiro, Poincaré’s Prize: The hundred-year quest to solve one of math’s greatest puzzles (Plume, 2008).

[14] Wikipedia, The Geometrization Conjecture.


Scott Taylor is a 3-manifold topologist at Colby College. He has never been given the chance to decline the Fields Medal.

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