The Poincaré conjecture
Prove that every closed (compact, without boundary) three-dimensional manifold in which every loop can be continuously shrunk to a point is homeomorphic to the three-dimensional sphere.
History and references
Solved in 2003 by Grigori Perelman.
Poincaré posed the question in 1904. Higher dimensional versions were proved first, by Stephen Smale for dimensions above four in 1961 and by Michael Freedman for dimension four in 1982. Richard Hamilton proposed attacking dimension three with the Ricci flow, an equation that smooths the geometry of a manifold over time, but singularities of the flow blocked his program. Grigori Perelman overcame these obstacles in papers posted to arXiv in 2002 and 2003, proving Thurston's more general geometrization conjecture. Several teams checked the work over the next few years and found only minor gaps that could be filled with his methods. Perelman declined both the 2006 Fields Medal and the Clay Millennium Prize.
References
Solutions 0
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