Homotopy sphere

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 a homology sphere.

The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic 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.

The resolution of the smooth Poincaré conjecture in dimensions 5 and larger implies that in such dimensions, homotopy spheres are precisely exotic spheres. It is still an open question whether or not there are non-trivial smooth homotopy spheres in dimension 4.

References[edit]

  • A. Kosinski, Differential Manifolds. Academic Press 1993.

See also[edit]