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Solution of the Poincaré conjecture

center   thumb  Neither of the colored loops on this torus can be contracted to a point without leaving the surface.

What does homeomorphic mean?

Generally, two shapes are homeomorphic if you can deform one into the other without a break or discontinuity. A homeomorphism is a continuous function mapping points from one object to another. This means that if two points are close to each other in the first object, they will be close together when the points are mapped onto the second object. The surface of a square and the surface of a sphere are not homeomorphic, since the square has edges and the sphere doesn't, so the mapping function has to jump somewhere, and at that point it won't be continuous.

A [homeomorphism] between two spaces, A and B, is a correspondence between the points of A and B, such that each point of A corresponds to exactly one point of B and vice versa, which is a [continuous function] both when viewed from A to B and from B to A. Intuitively, this means that if two points are close to each other in A, the corresponding points in B are also close to each other, and vice versa. Two spaces are called homeomorphic if a homeomorphism between them exists.

For example, a 2-dimensional sphere is homeomorphic to the surface of a cube; similarly, a 3-sphere is homeomorphic to the 3-dimensional [boundary] of a 4-dimensional [hypercube] .

Putting all these terms together, we can now understand the statement of the Poincaré conjecture:

The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected must be homeomorphic to a 3-sphere.

Perelman's proof based on Hamilton's Ricci flow

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