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Can one vary smooth structures on $\mathbb R^4$ smoothly/continuously?

This question popped out of Ben's answer here.

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I assume you are aware of Taubes "Gauge theory on asymptotically periodic 4-manifolds", proving the existence of a uncountable family of smooth structures on $\mathbb{R}^4$, and subsequent work by Gompf "An Exotic Menagerie" proving the existence of a two-parameter family. What exactly would you mean by "smoothy" or "continuously" in this case? – Kelly Davis Jan 16 '11 at 18:34
@Kelly: I am aware of the first result you mention, bot not the second. You should add an answer! By smooth/continuous family I had in mind a smooth map $M\to N$ such that the fibers are non-diffeomorphic copies of $\mathbb R^4$; for continuous, I don't know what I mean, really :) – Mariano Suárez-Alvarez Jan 16 '11 at 18:38
@Mariano: Probably you mean not just a smooth map $M\to N$ but a smooth submersion (as in Ryan's answer). That gives smooth structures to the fibers. – Tom Goodwillie Jan 16 '11 at 21:37
@Mariano: I'm not sure how to construct the submersion. Maybe I am missing something obvious? – Kelly Davis Jan 17 '11 at 8:08
Looks like this thread withered on the vine, but, in case anyone is interested: The $L_t$ of Theorem 9.4.10 in Gompf and Stipsicz could be used as a one-parameter family of exotic $\mathbb{R}^4$ and the construction in Theorem 9.4.16(a) in the same book could be able to be used for a two parameter family. – Kelly Davis Jan 18 '11 at 20:00

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.

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How does that play along the fact that on $\mathbb R^5$ there is only one smooth structure? – Mariano Suárez-Alvarez Jan 16 '11 at 18:15
(By the way: my intuition regarding smooth structures is simply non-existent... I may be asking something silly!) – Mariano Suárez-Alvarez Jan 16 '11 at 18:16
@Mariano: I guess in that case the "strange" thing would be the projection itself, not the smooth structure on $\mathbb{R}^5$. (my intuition about smooth structures is nonexistent as well, though..) – Qfwfq Jan 16 '11 at 18:31
@Ryan: For the required result in the second paragraph, couldn't you use the $L_t$ of Theorem 9.4.10 of Gompf and Stipsicz? – Kelly Davis Jan 17 '11 at 8:04

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