Homotopy sphere

In algebraic topology, a branch of mathematics, a homotopy sphere is a topological space homotopy equivalent to a sphere. It thus has the same homotopy groups, and a fortiori the same homology groups, as a sphere. So every homotopy sphere is an homology sphere.

The generalized Poincaré conjecture is that any homotopy sphere is a sphere; see the article Poincaré conjecture for more details.

See also:

• Homology sphere
• Homotopy groups of spheres
• Poincaré conjecture