Can one vary smooth structures on $\mathbb R^4$ smoothly/continuously?
This question popped out of Ben's answer here.
Can one vary smooth structures on $\mathbb R^4$ smoothly/continuously? This question popped out of Ben's answer here. 


Yeah, sure. A $1$parameter family of smooth structures on a topological manifold $M$ can be taken to be a smooth structure on $M \times I$ such that the projection map $M \times I \to I$ is a submersion. Similarly for higher families. To relate it to your comments on Ben's thread, you can (apparently) find a $1$parameter family of smooth structures on $\mathbb R^4$ such that all pairs of fibres $\mathbb R^4 \times \{a\}$ and $\mathbb R^4 \times \{b\}$ for all $a \neq b$ are not diffeomorphic. That 2nd paragraph is really Larry Siebenmann talking. I don't believe I've ever seen such a construction. 

