Poincaré conjecture

Solved, 2003TopologyGeometryMillennium

Statement

Every simply connected, closed $3$ -manifold is homeomorphic to the $3$ -sphere $S^3$ . Equivalently, if $M$ is a compact $3$ -dimensional topological manifold without boundary whose fundamental group $\pi_1(M)$ is trivial, then $M \cong S^3$ .

Between November 2002 and July 2003, Grigori Perelman posted three concise preprints to the arXiv (`math/0211159`, `math/0303109`, `math/0307245`) completing Richard Hamilton's Ricci flow programme by introducing monotonic entropy functionals, a no-local-collapsing theorem, a controlled surgery procedure past neck singularities, and finite extinction time for manifolds without aspherical prime factors. This established William Thurston's full geometrization conjecture for closed $3$ -manifolds and the Poincaré conjecture as a special case. After three independent teams (Bruce Kleiner and John Lott; John Morgan and Gang Tian; Huai-Dong Cao and Xi-Ping Zhu) wrote detailed expositions confirming that the proof was complete and correct, Perelman was awarded the Fields Medal in August 2006 and the 1,000,000 USD Clay Millennium Prize in March 2010 — and declined both honours, stating regarding the Millennium Prize that his contribution to resolving the conjecture was no greater than Hamilton's.

Posed by, 1904Henri Poincaré Solved by, 2003Grigori Perelman

In three dimensions, the topological, piecewise-linear, and smooth categories coincide (by theorems of Edwin Moise and Jean Cerf), so Perelman's smooth proof settles all three versions for $S^3$ . In higher dimensions, however, the topological and smooth questions diverge dramatically: John Milnor discovered exotic $7$ -spheres in 1956 — smooth manifolds homeomorphic but not diffeomorphic to $S^7$ — and while the topological Poincaré conjecture holds in every dimension $n \ge 1$ , the smooth four-dimensional Poincaré conjecture (whether every smooth $4$ -manifold homeomorphic to $S^4$ is diffeomorphic to $S^4$ ) remains one of the greatest open problems in geometric topology.

References

Grisha Perelman (2002). The entropy formula for the Ricci flow and its geometric applications · arXiv:math/0211159v1 [preprint, not peer-reviewed]
Grisha Perelman (2003). Ricci flow with surgery on three-manifolds · arXiv:math/0303109v1 [preprint, not peer-reviewed]
Grisha Perelman (2003). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds · arXiv:math/0307245v1 [preprint, not peer-reviewed]
John Morgan, Gang Tian (2007). Ricci Flow and the Poincaré Conjecture

History of the problem

1900Henri Poincaré initially asserts that any closed $3$ -manifold with the same homology groups as $S^3$ must be homeomorphic to $S^3$ .Henri Poincaré
1904In his Cinquième complément à l'Analysis situs, Poincaré constructs a counterexample to his homology statement — the Poincaré homology sphere, whose fundamental group is the binary icosahedral group of order $120$ — and poses the refined question of whether trivial fundamental group $\pi_1(M) = 1$ suffices to characterise $S^3$ .Henri Poincaré
1961Stephen Smale proves the generalised Poincaré conjecture in all dimensions $n \ge 5$ using the $h$ -cobordism theorem and Morse theory, showing that high dimensions allow enough room to untangle handles.
1982Michael Freedman proves the $4$ -dimensional topological Poincaré conjecture using Casson handles; the same year, William Thurston formulates the geometrization conjecture for $3$ -manifolds and Richard Hamilton introduces the Ricci flow $\partial_t g_{ij} = -2 R_{ij}$ on Riemannian metrics.
2000The Clay Mathematics Institute names the Poincaré conjecture one of the seven Millennium Prize Problems.
2002In November, Grigori Perelman posts the first of three preprints (`math/0211159`) on the arXiv, introducing the $\mathcal{W}$ -entropy and $\mathcal{L}$ -length functionals for Ricci flow and proving the no-local-collapsing theorem.Grigori Perelman
2003Perelman posts the second and third preprints (`math/0303109` in March and `math/0307245` in July), constructing Ricci flow with surgery on $3$ -manifolds and proving finite extinction time in the simply connected case, completing the proof of the Poincaré and geometrization conjectures.Grigori Perelman
2006Following verification by independent exposition teams, Grigori Perelman is awarded the Fields Medal at the International Congress of Mathematicians in Madrid and becomes the first mathematician to decline it.Grigori Perelman
2010The Clay Mathematics Institute officially awards the first Millennium Prize for the resolution of the Poincaré conjecture to Perelman, who declines the 1,000,000 USD prize.Grigori Perelman

Prizes

Clay Millennium Prize Problem, US$1,000,000 — awarded