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Solution of the Poincaré conjecture

A one-dimensional sphere is a circle, which can be thought of as the set of points, ( x, y ), in two dimensions that satisfy the equation x 2 y 2 = r 2, where r is the radius. A two-dimensional sphere is the surface of a globe, or the set of points, ( x, y, z ) in three dimensions that satisfy the equation x 2 y 2 z 2 = r 2. And a three-dimensional sphere is the set of points in four dimensions, ( x, y, z, w ), that satisfy the equation x 2 y 2 z 2 w 2 = r 2.
Image:CIRCLE 1.svg Circle is a one dimensional sphere
Image:Sphere-wireframe.png The boundary of a common ball is the 2-dimensional sphere.
Image:Hypersphere coord.PNG The [3-sphere] is one dimension higher, the boundary of a four-dimensional ball
What is a manifold ?

A manifold is a surface created by taking another surface -- for example, a piece of paper -- and warping it. A cylinder is a manifold since it can be formed by attaching the two opposite sides of the paper to each other. The cylinder can be deformed into another manifold by attaching the two circles at each end of the cylinder, to get a torus (ie. donut).

A manifold is a space created by gluing together pieces of Euclidean space, called [charts] . For example you could take two 2-dimensional disks and curve them around two hemispheres and then glue them together to make a 2-dimensional sphere.

center   thumb  Hemispheres mapped together to form a Sphere

A torus (the surface of a donut) could be built using a rectangular chart as seen in this image.

center  400px  thumb  Progressive build of a torus from a rectangle

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