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