|
The conjecture that every simply connected 3-manifold is homeomorphic
to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincar¨¦
(Poincar¨¦ 1953, pp. 486 and 498), and subsequently generalized to the
conjecture that every compact -manifold is homotopy-equivalent to the
-sphere iff it is homeomorphic to the -sphere. The generalized statement
reduces to the original conjecture for .
The case of the generalized conjecture is trivial, the case is classical,
remains open, was proved by Freedman (1982) (for which he was awarded
the 1986 Fields medal), by Zeeman (1961), by Stallings (1962), and by
Smale in 1961. Smale subsequently extended his proof to include .
Compact Manifold, Homeomorphic, Homotopy, Manifold, Property P, Simply
Connected, Sphere, Thurston's Geometrization Conjecture
References
Adams, C. C. "The Poincar¨¦ Conjecture, Dehn Surgery, and the Gordon-Luecke
Theorem." ¡́9.3 in The Knot Book: An Elementary Introduction to the Mathematical
Theory of Knots. New York: W. H. Freeman, pp. 257-263, 1994.
Batterson, S. Stephen Smale: The Mathematician Who Broke the Dimension
Barrier. Providence, RI: Amer. Math. Soc., 2000. Bing, R. H. "Some Aspects
of the Topology of 3-Manifolds Related to the Poincar¨¦ Conjecture." In
Lectures on Modern Mathematics, Vol. II (Ed. T. L. Saaty). New York: Wiley,
pp. 93-128, 1964.
Birman, J. "Poincar¨¦'s Conjecture and the Homeotopy Group of a Closed,
Orientable 2-Manifold." J. Austral. Math. Soc. 17, 214-221, 1974.
Clay Mathematics Institute. "The Poincar¨¦ Conjecture." http://www.claymath.org/prize_problems/poincare.htm.
Freedman, M. H. "The Topology of Four-Differentiable Manifolds." J. Diff.
Geom. 17, 357-453, 1982.
Gabai, D. "Valentin Poenaru's Program for the Poincar¨¦ Conjecture." In
Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topol.,
VI (Ed. S.-T. Yau). Cambridge, MA: International Press, pp. 139-166, 1995.
Gillman, D. and Rolfsen, D. "The Zeeman Conjecture for Standard Spines
is Equivalent to the Poincar¨¦ Conjecture." Topology 22, 315-323, 1983.
Jakobsche, W. "The Bing-Borsuk Conjecture is Stronger than the Poincar¨¦
Conjecture." Fund. Math. 106, 127-134, 1980.
Milnor, J. "The Poincar¨¦ Conjecture." http://www.claymath.org/prize_problems/poincare.pdf.
Papakyriakopoulos, C. "A Reduction of the Poincar¨¦ Conjecture to Group
Theoretic Conjectures." Ann. Math. 77, 250-205, 1963.
Poincar¨¦, H. uvres de Henri Poincar¨¦, tome VI. Paris: Gauthier-Villars,
pp. 486 and 498, 1953.
Rourke, C. "Algorithms to Disprove the Poincar¨¦ Conjecture." Turkish J.
Math. 21, 99-110, 1997.
Stallings, J. "The Piecewise-Linear Structure of Euclidean Space." Proc.
Cambridge Philos. Soc. 58, 481-488, 1962.
Smale, S. "Generalized Poincar¨¦'s Conjecture in Dimensions Greater than
Four." Ann. Math. 74, 391-406, 1961.
Smale, S. "The Story of the Higher Dimensional Poincar¨¦ Conjecture (What
Actually Happened on the Beaches of Rio)." Math. Intell. 12, 44-51, 1990.
Smale, S. "Mathematical Problems for the Next Century." In Mathematics:
Frontiers and Perspectives 20000821820702 (Ed. V. Arnold, M. Atiyah, P.
Lax, and B. Mazur). Providence, RI: Amer. Math. Soc., 2000.
Thickstun, T. L. "Open Acyclic 3-Manifolds, a Loop Theorem, and the Poincar¨¦
Conjecture." Bull. Amer. Math. Soc. 4, 192-194, 1981.
Zeeman, E. C. "The Generalised Poincar¨¦ Conjecture." Bull. Amer. Math.
Soc. 67, 270, 1961.
Zeeman, E. C. "The Poincar¨¦ Conjecture for ." In Topology of 3-Manifolds
and Related Topics, Proceedings of the University of Georgia Institute,
1961. Englewood Cliffs, NJ: Prentice-Hall, pp. 198-204, 1961.
|