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 open whether non-trivial smooth homotopy spheres exist in dimension 4.
Homotopy spheres form an abelian group known as Kervaire–Milnor group. Its composition is the connected sum and its neutral element is the sphere, while inversion is given by opposite orientation.
See also
- Homology sphere
- Homotopy groups of spheres
- Poincaré conjecture
References
External links
- Hedegaard, Rasmus. "Homotopy sphere". MathWorld.
