# 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 (as of February 2019) whether or not there are non-trivial smooth homotopy spheres in dimension 4.

## References

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

## See also

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