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A torus is not homeomorphic to a sphere. Perelman completed this portion of the proof. Several teams of mathematicians verified that Perelman’s proof was correct. 1,000,000 prize for the first correct solution.

Perelman was awarded the Millennium Prize on March 18, 2010. Unsourced material may be challenged and removed. 120, while the 3-sphere had a trivial fundamental group. In this way he was able to conclude that these two spaces were, indeed, different. In the same paper, PoincarĂ© wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. Consider a compact 3-dimensional manifold V without boundary.

Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere? PoincarĂ© never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the PoincarĂ© conjecture. 1930s he first claimed a proof, and then retracted it. In the 1950s and 1960s, other mathematicians were to claim proofs only to discover flaws in them.