Notation: $\mathbf{R}^4$ is a smooth manifold with underlying topology $(\mathbb{R})^4$; ${\mathbb{R}}^4$ is the standard smooth structure.

The two things I know best about $\mathbf{R}^4$ is that it is *locally* diffeomorphic to $\mathbb{R} ^4$, and that it's contractible. It's easy to see that the contraction can be acheived by a smooth map ${\mathbb{R}}^4\times I\rightarrow{\mathbb{R}}^4$.

- Do I suppose correctly that the same contraction is
*not*smooth as a map ${\mathbf{R}}^4\times I\rightarrow{\mathbf{R}}^4$? - Do the exotic smooth structures have any smooth contractions?
- If not, are there
*continuous*contractions $\mathbf{R}^4\times I\rightarrow\mathbf{R}^4$ within the smooth maps $\mathbf{R}^4\rightarrow\mathbf{R}^4$?

topologicallyirreducible w.r.t. connect-sum. – some guy on the street Apr 15 '10 at 2:38