Smooth homotopy sphere

A smooth homotopy $n$-sphere is a closed connected smooth $n$-manifold $\mathrm{\Sigma }$ equipped with a homotopy equivalence $\mathrm{\Sigma }\simeq S^n$. When the homotopy equivalence is only asserted to exist, $\mathrm{\Sigma }$ is still called a homotopy $n$-sphere; when a particular equivalence is used in an argument, that equivalence is part of the supplied data. An oriented smooth homotopy $n$-sphere is a smooth homotopy $n$-sphere together with a chosen orientation of $\mathrm{\Sigma }$.

No topological Poincaré assertion is built into the definition: a homotopy $n$-sphere is not assumed homeomorphic to $S^n$, and in dimension seven the exotic examples of this page are homotopy spheres whose homeomorphism with $S^n$ is a theorem proved separately, while their diffeomorphism failure is the exotic phenomenon. The homotopy type supplies the homology and fundamental-group data used later, and the chosen orientation is what the oriented connected-sum operation of the following items uses.