Homotopy sphere

From Wikipedia, the free encyclopedia

In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups, as the n-sphere. So every homotopy sphere is an homology sphere.

The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorpic to the n-sphere; it was solved by Stephen Smale in dimensions five and higher, Michael Freedman in dimension 4, and for dimension 3 by Grigori Perelman in 2005.

[edit] See also

• Homology sphere
• Homotopy groups of spheres
• Poincaré conjecture