Homotopy sphere

A homotopy $n$-sphere is a closed topological $n$-manifold that is homotopy equivalent to the standard sphere $S^n$. A smooth homotopy sphere is a closed smooth $n$-manifold with the same homotopy type.

Every manifold homeomorphic to $S^n$ is a homotopy sphere, but the converse is the content of the generalized Poincaré problem. A smooth homotopy sphere can be topologically standard while carrying an exotic smooth structure, meaning it is not diffeomorphic to the standard smooth sphere.