H-cobordism of homotopy spheres equals oriented diffeomorphism

If $f:\mathrm{\Sigma }\to {\mathrm{\Sigma }}^{\mathrm{\prime }}$ is an orientation-preserving diffeomorphism, the product $\mathrm{\Sigma }\times [0,1]$ with the two boundary identifications given by the identity and by $f$ is an oriented h-cobordism from $\mathrm{\Sigma }$ to ${\mathrm{\Sigma }}^{\mathrm{\prime }}$, so oriented diffeomorphism implies oriented h-cobordism.

Conversely, suppose $\mathrm{\Sigma },{\mathrm{\Sigma }}^{\mathrm{\prime }}$ are oriented h-cobordant and let $C$ be a compact oriented h-cobordism between them; then $C$ has dimension $n+1\ge 6$, and its boundary faces are the closed simply connected manifolds $\mathrm{\Sigma },{\mathrm{\Sigma }}^{\mathrm{\prime }}$, since both are homotopy spheres.

By [L2] and [A1] the cobordism $C$ is diffeomorphic to a product relative to the face $\mathrm{\Sigma }$; hence its other face ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is orientation-preservingly diffeomorphic to $\mathrm{\Sigma }$.

Combining steps 1.1 and 3.1, oriented h-cobordism and orientation-preserving diffeomorphism define the same equivalence relation on oriented smooth homotopy $n$-spheres, so the classes of ${\mathrm{\Theta }}_n$ are oriented diffeomorphism classes.