Recent progress on the Poincaré conjecture and the classification of 3-manifolds

Author:
John W. Morgan

Journal:
Bull. Amer. Math. Soc. **42** (2005), 57-78

MSC (2000):
Primary 57M50, 57M27, 58J35

DOI:
https://doi.org/10.1090/S0273-0979-04-01045-6

Published electronically:
October 29, 2004

MathSciNet review:
2115067

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References | Similar Articles | Additional Information

**1.**Y. Burago, M. Gromov and G. Perelman, ``A. D. Aleksandrov spaces with curvature bounded below," Russian Math. Surveys**47**(1992), 1-58. MR**1185284 (93m:53035)****2.**J. Cheegar and M. Gromov, ``Collapsing Riemannian manifolds while keeping their curvatures bounded, I, II." J. Differential Geom.**23**(1986), 309-346;**32**(1990), 269-298. MR**0852159 (87k:53087)**; MR**1064875 (92a:53066)****3.**T. Colding and W. Minicozzi, ``Estimates for the extinction time for the Ricci flow on certain -manifolds and a question of Perelman," arXiv.math.DG/0308090, August 25, 2003.**4.**D. Cooper, C. Hodgson, S. Kerchoff,*-dimensional orbifolds and cone-manifolds,*Math. Soc. Japan Memoirs, Vol. 5, Tokyo, 2000. MR**1778789 (2002c:57027)****5.**R. Hamilton, ``Three-manifolds with positive Ricci curvature," J. Differential Geom.**17**(1982), 255-306. MR**0664497 (84a:53050)****6.**R. Hamilton, ``The Harnack estimate for the Ricci flow," J. Differential. Geom.**37**(1993), 225-243. MR**1198607 (93k:58052)****7.**R. Hamilton, ``Formation of singularities in the Ricci flow,"*Surveys in Differential Geometry***2**, 7-136, International Press, 1995. MR**1375255 (97e:53075)****8.**R. Hamilton, ``Four-manifolds with positive isotropic curvature," Comm. Anal. Geom.**5**(1997), 1-92. MR**1456308 (99e:53049)****9.**R. Hamilton, ``Non-singular solutions of the Ricci flow on three-manifolds," Comm. Analysis and Geometry**7**(1999), 695-729. MR**1714939 (2000g:53034)****10.**J. Hempel,*-manifolds,*Ann. of Math. Studies, Vol. 86, Princeton University Press, 1976. MR**0415619 (54:3702)****11.**H. Kneser, ``Geschlossene Flächen in drei-dimesnionalen Mannigfaltigkeiten," Jahresber. Deutsch. Math. Verbein**38**(1929), 248-260.**12.**J. Milnor, ``Groups which act on without fixed points," Amer. Journal of Math.**79**(1957), 623-630. MR**0090056 (19:761d)****13.**J. Milnor, ``Towards the Poincaré Conjecture and the classification of -manifolds," Notices AMS**50**(2003), 1226-1233. MR**2009455 (2004h:57022)****14.**G. Perelman, ``Spaces with curvature bounded below," Proceedings of ICM 1994, 517-525, Birkhäuser, 1995. MR**1403952 (97g:53055)****15.**G. Perelman, ``The entropy formula for the Ricci flow and its geometric applications," arXiv.math.DG/0211159, November 11, 2002.**16.**G. Perelman, ``Ricci flow with surgery on three-manifolds," arXiv.math.DG/0303109, March 10, 2003.**17.**G. Perelman, ``Finite extinction time for the solutions to the Ricci flow on certain three-manifolds," arXiv.math.DG/0307245, July 17, 2003.**18.**H. Poincaré, ``Cinquième complément à l'analysis situs," Rend. Circ. Mat. Palermo**18**(1904), 45-110. (See*Oeuvres,*Tome VI, Paris, 1953, p. 498.) MR**1401792 (98m:01041)****19.**P. Scott, ``The geometries of -manifolds," Bull. London Math. Soc.**15**(1983), 401-487. MR**0705527 (84m:57009)****20.**H. Seifert, ``Topologie dreidimensionaler gefaserter Räume," Acta Math.**60**(1933), 147-288.**21.**W-X. Shi, ``Deforming the metric on complete riemannian manifolds,'' J. Differential Geom.**30**(1989), 223-301. MR**1001277 (90i:58202)****22.**T. Shioya and T. Yamaguchi, ``Collapsing three-manifolds under a lower curvature bound," J. Differential Geom.**56**(2000), 1-66. MR**1863020 (2002k:53074)****23.**T. Shioya and T. Yamaguchi, ``Volume collapsed three-manifolds with a lower curvature bound," arXiv.math.DG/0304472, April 15, 2003.**24.**W. Thurston,*Three-dimensional Geometry and Topology, Vol. 1*(Silvio Levy, ed.) Princeton Math. Ser., vol. 35, Princeton Univ. Press, 1997. MR**1435975 (97m:57016)**

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Additional Information

**John W. Morgan**

Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027

Email:
jm@math.columbia.edu

DOI:
https://doi.org/10.1090/S0273-0979-04-01045-6

Received by editor(s):
June 11, 2004

Received by editor(s) in revised form:
September 1, 2004

Published electronically:
October 29, 2004

Additional Notes:
Written version of a talk presented on January 9, 2004, in the “Current Events in Mathematics" session at the AMS national meeting in Phoenix, AZ

Article copyright:
© Copyright 2004
American Mathematical Society