Monday, January 01, 2007

Poincaré conjecture

In mathematics, the Poincaré conjecture is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. The conjecture concerns a space that locally looks like ordinary three dimensional space but is finite in size and lacks any boundary (a closed 3-manifold). The conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is just a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincare, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

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Weekend Edition Saturday, July 29, 2006

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