# 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.^{[1]}

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 open whether non-trivial smooth homotopy spheres exist in dimension 4.

## See also

[edit]## References

[edit]**^**A., Kosinski, Antoni (1993).*Differential manifolds*. Academic Press. ISBN 0-12-421850-4. OCLC 875287946.`{{cite book}}`

: CS1 maint: multiple names: authors list (link)

## External links

[edit]- Hedegaard, Rasmus. "Homotopy sphere".
*MathWorld*.