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at diffeomorphism I started listing theorems and references on statements about when the existence of a homeomorphism implies the existence of a diffeomorphism.
I dug out ancient references for the statement that in $d \neq 4$ everything homeomorphic to an open $d$-ball is also diffeomorphic to it. What would be a more modern, more canonical, more textbook-like reference for this?
And I’d also like to cite a reference for what is maybe obvious, that if that something in $d = 4$ is an open subset of $\mathbb{R}^4$ equipped with the induced smooth structure of the standard smooth structure, then the statement is also true in that dimension.
In fact, I am looking for nice/explicit/useful diffeomorphisms from the open $n$-ball onto the open $n$-simplex. I can of course fiddle around and cook up something, but I haven’t found anything that would count as nice. But probably some engineer out there working with finite elements or something does have a convenient choice.
In fact, I am looking for nice/explicit/useful diffeomorphisms from the open n-ball onto the open n-simplex.
Hey, finally found the relevant MO-discussion which gives a link to an article that from its page 154 on gives a detailed construction for every star-shaped region.
Lucky me that there is MO. And good luck for them that I will archive this information now in a way that next time it is easier to find! ;-)
Urs, I’m not exactly sure what you’re looking for, but here is what I cooked up:
Represent the n-disk as consisting of points $(x_1, \ldots, x_n)$ such that $x_i \gt 0$ and $x_1 + \ldots + x_n \lt 1$. Map this diffeomorphically onto $\mathbb{R}^n$ by the mapping
$(x_1, \ldots, x_n) \mapsto (\log(\frac{x_1}{1 - x_1 - \ldots -x_n}), \ldots, \log(\frac{x_n}{1 - x_1 - \ldots - x_n}))$and then map $\mathbb{R}^n$ onto the $n$-ball by
$(x_1, \ldots, x_n) \mapsto (\frac{x_1}{\sqrt{1 + r^2}}, \ldots, \frac{x_n}{\sqrt{1 + r^2}})$where $r^2 = x_1^2 + \ldots + x_n^2$. I feel pretty sure I can write down the smooth inverses to these maps, so we have a diffeomorphism. I haven’t checked this super-carefully, but I can’t see what would go wrong.
By the way, Urs – for whatever it’s worth, I wrote up a silly article on my personal web here called “Balls” where I prove a less general result than the one you seem to have found.
Ah, Todd, you are great, as always. That looks simple enough now that I see it. All right, let me think and type a bit…
I wrote up a silly article on my personal web here called “Balls” where I prove a less general result than the one you seem to have found.
But there you look just at homeomorphisms, no?
Yes, just homeomorphisms there, not diffeomorphisms. But still, the proof caused me some difficulty to nail down. It’s now there FWIW.
But still, the proof caused me some difficulty to nail down. It’s now there FWIW.
Right, good point. Maybe we could copy this over to the nLab page ball, where it would be easier to find?
I certainly don’t mind if you think it would be of any benefit! My only hesitation is that it might look out of place, but if you don’t think so, we could go ahead.
Hi Todd,
I think it would make very good sense to copy this over.
I think we can use the result about star-shaped regions being diffeomorphic to $\mathbb{R}^d$ to strengthen the statement about existence of good open covers on paracompact manifolds at good open cover:
it should follow that every paracompact manifold of dimension $d$ admits an open cover such that every non-empty finite intersection is diffeomorphic to $\mathbb{R}^d$.
I typed the argument into this section at good open cover.
Okay, I just transferred the argument at my article on balls over to ball. It fits a little more smoothly (no pun intended) than I had expected, since the homeomorphism/diffeomorphism distinction becomes more manifest in the case of closed balls as opposed to open balls.
Thanks, Todd!
Christoph Wockel points out to me that there is a remark below lemma 10.5.5 of the book Lawrence Conlon “Differentiable manifolds”, which says:
It seems that open, star-shaped sets $U \subset M$ are always diffeomorphic to $\mathbb{R}^n$, but this is extremely difficult to prove.
Maybe I should contact Dirk Ferus to ask if he plans to publish his proof.
Urs: Jim Stasheff took the trouble of asking the question over at MO, and he received two helpful replies. In particular, there’s a book by Brocker and Janich that shows how to construct diffeomorphisms between open star-shaped domains (in an exercise).
Thanks, Todd.
Hm, that book by Brocker and Janich, in its exercise 7 in chapter 8 asks the reader to show that every star-shaped region is diffeomorphic to a ball.
This is the statement of which Conlon in his textbook says “it seems to be true but is extremely difficult to prove” !
Are any hints given in that exercise? Certainly the proposition looks not so easy to me, although I am intrigued by Ryan Budney’s solution and the prospects that it might be generalizable to the context of that exercise.
On another topic: I am wondering about that remark about the exceptional case in dimension 4. Certainly I am aware of the result on exotic smooth structures on $\mathbb{R}^4$, but is there really an example of an open region in $\mathbb{R}^4$ which is homeomorphic but not diffeomorphic to $\mathbb{R}^4$, provided we use the smooth structure it inherits from the ambient $\mathbb{R}^4$? That would be very surprising to me.
So I have been editing the proof at good open cover, the second one right at the beginning, which claims to show that every paracompact manifold has a good open cover such that every non-empty finite intersection is diffeomorphic to an n-ball.
I guess for $n \neq 4$ this follows from very general results, but here the proof strategy is to give the more or less explicit diffeos obtained by geodesic flow and those diffeos between star-shaped regions and balls that we are talking about.
If I can assume that the $U_i$ can indeed be chosen such that geodesic flow inside them is a diffeo – and I think I can, by arguing that they contain no conjugate points since they are themselves obtained from a small geodesic flow – then this should be all right.
started a section Relation to homotopy equivalences lsiting cases in which it is known that homotopy equivalences are homotopic to diffeos.
What I am really after is finding out how close we get to the homotopy type of the diffeomorphism group if we consider homotopy equivalences slices over $B O(n)$ via the map that classifies the tangent bundle.
Maybe one can get some clue from comparing (more general) tangent microbundles.
I have meanwhile gotten a very good reply on MO.
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