Are there any manifolds that are $C^{r-1}$-diffeomorphic but not $C^r$-diffeomorphic?

Question

Things like fake-$\mathbb R^4$s and $\mathbb R^4$ are homeomorphic but not diffeomorphic. But are there any examples of some $C^r$-manifolds (smooth would be better) that are $C^{r-1}$-diffeomorphic but not $C^r$-diffeomorphic?

Answer 1

If $M$ and $N$ are $C^{r}$-manifolds which admit a $C^{s}$-diffeomorphism for some $1\leq s\leq r\leq\infty$, then there also exists a $C^{r}$-diffeomorphism. The most general proof of this I know very roughly goes as follows. One first defines a reasonable topology on the set of $C^s$-maps $C^{s}(M,N)$, $1\leq s\leq r\leq\infty$, and then shows that $C^{r}(M,N)$ is dense in $C^{s}(M,N)$ in a way such that also the set of $C^{r}$-diffeomorphisms is dense in the set of $C^{s}$-diffeomorphisms. In particular, there exists a $C^{s}$-diffeomorphism if and only if there exists a $C^{r}$-diffeomorphism.

See, for example, M. Hirsch's book "Differential Topology", Chapter 2, for more details.