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Hello,

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :

For example :

1) Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

2) Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs

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Huh, at first this looked suspiciously like a homework problem, but I see from the comments below that it's for real! –  Scott Morrison Nov 7 '09 at 6:37

6 Answers 6

Ad question 1): Yes, all open star-shaped subsets of $\mathbb{R}^n$ are diffeomorphic to $\mathbb{R}^n$.

This is surprisingly little-known and there is a proof due to Stefan Born. You can find this (fairly complicated) proof in Dirk Ferus's course notes

http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf

page 154, Satz 237 [The notes are alas in German]

Added December 30, 2009: My excellent colleague Erwann Aubry informs me that this result is also proved more simply on page 60 of Gonnord & Tosel's book "Calcul Différentiel", ellipses,1998.

[This book is in French, and moreover published by "ellipses" a valiant little publisher, completely unknown outside of France because it caters to the idiosyncratic French academic system]

Kudos to any reference in honest English, rather than exotic foreign languages :)

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Why is this so complicated? –  Kevin H. Lin Jan 5 '10 at 10:17

You can certainly have a set diffeomorphic to R^n but not star-shaped. For example, for n=2, the Riemann mapping theorem implies that any simply connected open set is diffeomorphic to the plane. More concretely, you can take a ball and just deform it a little bit so it's very badly not convex (in particular, not star-convex) but still diffeormorphic to the ball. For example, you a thickened letter M in two dimensions.

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There are several characterizations of manifolds diffeomorphic to R^n when n>4, e.g. an open manifold that is simply-connected at infinity (Stallings), or the image of a degree one proper map from R^n (Siebenmann), but looks like this is not what you want. Surely tons of subsets of R^n that are diffeomorphic to R^n can be constructed by attaching "fingers" to a ball.

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No, not really. In dimension 4, for example, an open subset of R^4 can be homeomorphic to R^4 but not diffeomorphic, as there are exotic smooth R^4's that embed smoothly in R^4.

But in dimensions different from 4, R^n admits a unique smooth structure. So your neccessary and sufficient condition can be that the open subset is homeomorphic to R^n. That's probably not what you want to hear?

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Following up on Ryan Budney's response, there's some discussion of subsets of R^n which are homeomorphic to R^n here: math.niu.edu/~rusin/known-math/95/contractible . Contractibility is not enough, but I don't think any full necessary and sufficient conditions are given in that thread. –  j.c. Nov 7 '09 at 1:00
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en.wikipedia.org/wiki/Simply_connected_at_infinity claims that contractibility and simple connectedness at infinity are equivalent to being homeomorphic to R^n. So I guess the results described in the page I linked to above go both ways after all. –  j.c. Nov 7 '09 at 1:11
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Yeah, I think the argument goes like this: if its simply-connected at infinity, you apply Larry Siebenmann's dissertation to find a manifold compactification. Contractibility tells you this compactification is a topological n-ball. This argument requires a dimension restriction to n >= 6 though. –  Ryan Budney Nov 7 '09 at 1:19

The answer for 2) is no. Think of an annulus in R^2 with a radius removed.

1) seems much less trivial. It is true in 2 dimension, but the easiest way I can think of is to use the fact that star-shaped implies simply connected and use the Riemann mapping theorem. So complex analysis here yields a purely topological conclusion.

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Thank you for your answers,

So I guess that if $n\neq 4$, then the necessary and sufficient condition is precisely "contractible and simply connected at infinity". Here, there is only one possible differential structure.

In dimension 4, you have an infinity of possible differential structures. Is is true then that:

Question : If $U$ is an open contractible simply connected at infinity subset of $\mathbb{R}^4$ on which we consider the standard differential structure. Then is $U$ diffeomorphic to $\mathbb{R}^4$ (with its standard differential structure) ?

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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. –  Ryan Budney Nov 7 '09 at 3:57
    
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. –  Oliver Nov 7 '09 at 4:33
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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. –  Tom Church Nov 7 '09 at 6:58

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