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Generalized Poincaré conjecture

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1 Generalized Poincaré conjecture on Tue Dec 14, 2010 8:20 am

In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a category of manifolds: topological (Top), differentiable (Diff), or piecewise linear (PL). Then the statement is
Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) is isomorphic to the n-sphere in the chosen category, i.e. homeomorphic, diffeomorphic, or PL-isomorphic.
The name derives from the Poincaré conjecture, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected. The Generalized Poincaré conjecture is known to be true or false in a number of instances, due to the work of many distinguished topologists, including the Fields medal recipients John Milnor, Steve Smale, Michael Freedman and Grigori Perelman.[1]

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