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homotopy sphere
In algebraic topology, a branch of mathematics, a homotopy sphere is an ''n''-manifold that is homotopy equivalent to the ''n''-sphere. It thus has the same homotopy groups and the same homology groups as the ''n''-sphere, and so every homotopy sphere is necessarily 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, by 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 homotopy spheres in those dimensions are precisely exotic spheres. It is still an open question (as of 2014) whether or not there are non-trivial smooth homotopy spheres in dimension 4.
==References==
* A. Kosinski, ''Differential Manifolds.'' Academic Press 1993.
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