Timeline for What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
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| Nov 7, 2009 at 6:58 | comment | added | Tom Church | Yes. There is a subset U of R^4 which is an open submanifold and, with the induced topology, homeomorphic to R^4; however, the induced smooth structure on U coming from the standard smooth structure on R^4 makes U not diffeomorphic to the standard R^4. Such spaces are called "small" exotic R^4's, see e.g. en.wikipedia.org/wiki/Exotic_R4. | |
| Nov 7, 2009 at 4:33 | comment | added | Oliver | Ok, but there is something I don't really understand : is the exotic smooth R^4 open subset of R^4 you consider provided with the differential structure induced by the standard R^4 ? Recall that I consider in my question only one differential structure (the standard one) on U and R^4. | |
| Nov 7, 2009 at 3:57 | comment | added | Ryan Budney | Nope. As I mentioned earlier, there are exotic smooth R^4's that embed smoothly in R^4. So such open subsets of R^4 are contractible and simply connected at infinity (since they're homeomorphic to R^4), but not diffeomorphic to the standard R^4 as that's what "exotic smooth R^4" means. | |
| Nov 7, 2009 at 3:48 | history | answered | Oliver | CC BY-SA 2.5 |