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 (the original Poincaré conjecture) 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 () whether or not there are non-trivial smooth homotopy spheres in dimension 4.