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created ball
Does theorem 237 from the notes (in German) by Ferus prove that for a star-shaped region $C \subset \mathbb{R}^n$ that $C$ is diffeomorphic to that $\mathbb{R}^n$? Because at ball we have two theorems:
Theorem In dimension $d \in \mathbb{N}$ for $d \neq 4$ we have: every open subset of $\mathbb{R}^d$ which is homeomorphic to $\mathbb{B}^d$ is also diffeomorphic to it.
Remark In dimension 4 the analog statement fails due to the existence of exotic smooth structures on $\mathbb{R}^4$. See De Michelis-Freedman.
Theorem Let $C \subset \mathbb{R}^n$ be a star-shaped open subset of a Cartesian space. Then $C$ is diffeomorphic to $\mathbb{R}^n$.
An open subset homeomorphic to $\mathbb{R}^n$ is star-shaped, so at the very least, we have to modify the second statement.
Ideally it would be good to get a theorem that told us we can get a differentiably good open cover of fake $\mathbb{R}^4$s, where the open sets are diffeomorphic to a standard $\mathbb{R}^4$. At present I don’t think we do (or perhaps I just don’t see it).
Does theorem 237 from the notes (in German) by Ferus prove that for a star-shaped region $C \subset \mathbb{R}^n$ that $C$ is diffeomorphic to that $\mathbb{R}^n$?
Yes, what is meant is $\mathbb{R}^n$ with its standard smooth structure.
An open subset homeomorphic to $\mathbb{R}^n$ is star-shaped
Not necessarily. Just take one that is and deform it continuously to make it have a “nose”.
at the very least, we have to modify the second statement.
This is a thorny subject, in particular since it seems so simple and is so subtle. So I might have to think about it harder again. But right now I am not sure what you say must be changed.
Ah, overlooked the chance of a ’nose’. I was thinking in terms of balls-shaped regions
at open balll the definition for a general metric space wasn’t stated, so I added it. It seems I also created a stub metric topology.
Wrote down the inverse to the diffeomorphism from $n$-dimensional cartesian space to the open n-ball. I used the vector notation implicitly, if that is okay.
That looks good to me.
Thanks, that’s neat. Could you add it to the entry?
I think it’s okay. But it’s quite an elementary argument, so I’d want to mull over it more. Maybe I’d enjoy a few more words on “surjective by elementary properties of flow lines”.
@Dmitri
regarding your comments on MO about differentiably good open covers, it occurred to me that one might not know every point has a geodesically convex neighbourhood. I bring this up here, since I gather you are working on a proof of existence of such good open covers.
The nLab software seems to have problems with some standard TeX commands and characters such as \bf, <, >. Is there any chance that the software can be patched to fix this?
I think the answer to this would be “it is not broken, you are just speaking the wrong dialect”.
What you need is \mathbf
instead of \bb
, \lt
for <
, \gt
for >
.
(you can pick up the “proper” dialect by seeing what is used to render similar things in the file being edited)
I made these substitutions. Does this look right now?
Suppose $T$ is a star-shaped open subset of ${\mathbb {R}}^n$ centered at the origin. Theorem 2.29 in Lee proves that there is a function $f$ on ${\mathbb{R}}^n$ such that $f\gt 0$ on $T$ and $f$ vanishes on the complement of $T$. By applying bump functions we can assume that $f\le1$ everywhere and $f=1$ in an open $\epsilon$-neighborhood of the origin, by rescaling the ambient space we can assume $\epsilon=2$.
The smooth vector field $V\colon x\mapsto f(x)\cdot x/\|x\|$ is defined on the complement of the origin in $T$. Multiply $V$ by a smooth bump function $0\le b\le1$ such that $b=1$ for $\|x\|\gt 1/2$ and $b=0$ in a neighborhood of 0. The new vector field $V$ extends smoothly to the origin and defines a smooth global flow $F\colon R\times T\to T$. (The domain of the flow is $R$ and not some interval $(-\infty,A)$ because the norm of $V$ is bounded by 1.) Observe that for $1/2\lt \|x\|\lt 2$ the vector field $V$ equals $x\mapsto x/\|x\|$. Also, all flow lines of $V$ are radial rays.
Now define the flow map $p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2}$ as $x\mapsto F(\|x\|-1,x/\|x\|)$ for $\|x\|\gt 1/2$. (The subscript $\gt 1/2$ removes the closed ball of radius $1/2$.) The flow map is the composition of two diffeomorphisms: ${\mathbb{R}}^n_{\gt 1/2}\to(-1/2,\infty)\times S^n\to T_{\gt 1/2}$, hence itself is a diffeomorphism. (The latter map is surjective by elementary properties of flow lines.)
Finally, define the desired diffeomorphism $d\colon{\mathbb{R}}^n\to T$ as the gluing of the identity map for $\|x\|\lt 2$ and as $p$ for $\|x\|\gt 1/2$. The map $g$ is smooth because for $1/2\lt \|x\|\lt 2$ both definitions give the same value.
The reason why <
and >
don’t work is because they have special meaning in HTML.
I made some further edits. Just a note that by typing $x/{\|x\|}$, with braces, one gets
$x/{\|x\|}$
whereas without braces, one gets
$x/\|x\|$
Sometimes I think readability is enhanced by using $\frac{x}{{\|x\|}}$ instead of $x/{\|x\|}$. There were other small adjustments made.
The spacing might be a bug in MathML that iTeX (the proper name of the program that does our typesetting) has just omitted to fix, I don’t know.
iTeX is never going to support all of (La)TeX because it doesn’t actually run TeX underneath, it just translates directly to MathML (that’s what enables it to avoid the nasty retypsetting problems that afflict mathjax). In particular, I expect TeX-isms like {\bf ...}
are unlikely to be feasible; the LaTeX syntax \mathbf{...}
is probably much easier for a program other than TeX to parse. However, I don’t really understand why it couldn’t support < and >, since the source goes through iTeX before it gets interpreted as HTML.
I meant “TeX-ism” as opposed to HTML, not as opposed to LaTeX. I suppose one could hardcode a syntax like {\bf ... }
, but I think it would be a bit misleading to support that but not all of the effects of TeX’s grouping constructs.
Dmitri, I quite share your disappointment with the software. Ultimately it is my fault: when I created the $n$Lab a few years back didn’t look around much which options there are. We have since long ago talked about migrating to a better system, such as for instance WorkingWiki, which does all we could ask for (MathML, true LaTeX including xypic and all, proper talk pages, and much more…) but nobody as yet had the time and energy to make it happen.
@Urs: I see that transition has already been discussed extensively, e.g., at http://nforum.mathforge.org/discussion/6093/changes-in-nlab-organization/.
What is the current status of this and what remains to be done?
Does the main problem lie in changing the syntax of nLab pages to something that can be understood by WorkingWiki?
Myself, I want to write a small program to convert nLab pages into a properly hyperlinked TeX (the existing converter is not quite good enough, in particular the double bracket links don’t work at all) so that I can compile the nLab using TeX and view it offline when I need it. Such a parser could presumably be adapted to change the current syntax to something else.
@Dmitri - thanks. I think the nLab should then make a more modern citation than a passing remark than Milnor, and certainly not back to Whitehead’s original paper, which is pretty scary. I will add something to this effect.
@Dmitri,
this is the status: a few months back Adeel and myself had an extensive exchange with Lee Worden on setting up a WorkingWiki installation for the nLab. (See for instance here at Research Math Typesetting and Publishing in WorkingWiki). Lee was effectively ready to go for it, and we had some first scripts to test transporting of $n$Lab pages. But then we ran out of steam (personally I ran out of more than just steam, and I suppose Adeel is all busy himself, too).
What it would take, I suppose, would be somebody dedicating himself to looking into the issue of writing scripts that systematically and truthfully transport the $n$Lab entry database to that WorkingWiki platform.
If you are interested, or if you know somebody who might be interested, let me know and we should get back into contact with Lee Worden.
@dmitri 23
Which makes me wonder why everybody who implements a math typesetting system for web (iTeX, MathJax) elects to write an ersatz-TeX instead of modifying TeX itself so that it outputs the resulting compiled math list (an internal TeX concept; a bit similar to what MathML presents).
Writing a new subroutine (or rather modifying the existing one that converts a math list to an hlist, another internal TeX concept)
While TeX and derivatives like LaTeX have enormous success as a way to write academic papers, it is a fairly flawed system for representing text in a form that can be manipulated. TeX is basically a macro language however it has been extended to be a Turing Complete programming language. Manipulating pure macros is easy, manipulating fully general programs is impossible and you have to deal with issues like halting.
That is why many people are interested in subsets like iTeX that are much simpler and purer. Things like graphics languages written in TeX, while useful, are pretty much abominations when viewed from outside.
@dmitri 26
Myself, I want to write a small program to convert nLab pages into a properly hyperlinked TeX (the existing converter is not quite good enough, in particular the double bracket links don’t work at all) so that I can compile the nLab using TeX and view it offline when I need it.
Why? You can save nLab pages on your computer and view them offline. If you want to send the pages to a place like FedEx Kinko’s to have them printed then it is best to convert them to PDF (Kinko’s can’t understand all formats) but there are many ways to do this.
Getting back to the mathematics of the article ball for just a second: I’m not sure who wrote the addendum after theorem 1, but the suggested argument there doesn’t seem quite right to me. I’m not saying the statement there (that closures of open starlike regions are homeomorphic to disks if they are compact) is false, just that the suggested modified argument for it doesn’t seem to work, unless I’m missing something.
What the proof of theorem 1 does in the third bullet point is show that no two boundary points can be positive multiples of each other. The suggested argument purports to prove that this still holds under the hypotheses of the more general statement. But it doesn’t. Consider the open region
$U = \{(x, y) \in \mathbb{R}^2: (x^2 + y^2 \lt 1 \; \wedge \; y \geq 0) \; \vee \; (x^2 + y^2 \lt 4 \; \wedge \; y \lt 0)\}$This $U$ is the interior of a closed region, consisting of the union $C$ of a closed northern half-disk centered at the origin, placed atop a larger closed southern half-disk. $U$ is starlike about the origin. The closure is the closed region $C$ itself. Notice that $(1, 0)$ and $(2, 0)$ are both boundary points of $C$.
This is the type of example that had bothered me before hitting on the idea presented in the proof of the theorem (under the more restrictive convexity hypothesis).
While we are at it, I would like to point out that the article good open cover could be improved by amending the proof of Proposition 1 with the remark that geodesically convex subsets are mapped diffeomorphically to a star-shaped open subset of R^n via the exponential map based at some interior point.
Indeed, the Gauss lemma shows that the tangent map of the exponential map is invertible and the exponential map is injective by definition of geodesic convexity (two different points x,y∈T_p(M) such that exp_p(x)=exp_p(y)=q give two different geodesics from p to q). Hence the exponential map is a diffeomorphism, as desired.
With this remark one can remove Proposition 2 and its somewhat convoluted proof with injectivity and convexity estimates.
Which addendum? The discussion added in rev 3?
Dmitri, feel invited to add your alternative argument. Maybe instead of removing the previous one, you could keep it. Feel invited to advertize it as “convoluted”, if you wish to.
Urs #32: By addendum, I meant the paragraph after the proof of theorem 1, where it says that by slightly modifying the proof, the argument can be extended to open star-shaped regions with compact closure. It’s not in revision 3.
This is very odd. The offending paragraph seems to have been introduced in rev 9, and the revision indicates that it was written by me! I find it incredible that I could write that, as the argument doesn’t even make sense to me (mathematically, and even almost grammatically). I just don’t recognize myself at all in it.
Re #33: I agree.
@Dmitri - and I even told you on MO that there is a proof in do Carmo! Memory…
@Urs: The proposed proof is a strict subset of the existing proof of Proposition 2; I merely observe that some constructions are unnecessary.
With this in mind, I wonder if it might look weird to have two proofs of the same proposition, where the second proof contains the first proof.
@RodMcGuire:
it is a fairly flawed system for representing text in a form that can be manipulated.
Manipulated for which purpose?
TeX is basically a macro language however it has been extended to be a Turing Complete programming language.
True, but the same can be said about PostScript, HTML+CSS (without JavaScript), PDF, all of which are Turing-complete despite being much more primitive languages than TeX. And Markdown is simply too primitive to be compared to TeX.
At the level of convenience provided by TeX it is impossible not to be Turing-complete.
That is why many people are interested in subsets like iTeX that are much simpler and purer. Things like graphics languages written in TeX, while useful, are pretty much abominations when viewed from outside.
I don’t think this is the reason for the existence of iTeX. Rather, a software capable of translating TeX formulas into reasonably-looking HTML was needed, hence iTeX was created because existing translators were incapable of producing (decent-looking) HTML.
Why? You can save nLab pages on your computer and view them offline. If you want to send the pages to a place like FedEx Kinko’s to have them printed then it is best to convert them to PDF (Kinko’s can’t understand all formats) but there are many ways to do this.
I already have a local copy of the entire nLab repository on my hard drive. What is desirable, however, is a way to view nLab with decent-looking formulas (i.e., not iTeX). Also, the only way to create PDF with good-looking formulas is via TeX.
I added the new proof to good open cover. There are now two proofs of the same statement there.
Todd, of course in principle anyone could sign with your name. I only now realize the following strange behaviour of the software: on each current page it shows the IP address of the last author, after the name that the author gave (on the very bottom of the entry), but on the pages in the history it shows only the name, not the IP.
That’s amusing. I think I lean somewhat more toward believing I wrote that crap over believing someone was impersonating me!
Additionally, I seem to have been the one who introduced the statement (in revision 7, I think) that my doppelgänger was trying to prove. More weirdness. I suppose no one will object if I simply remove all of it.
HTML+CSS is Turing complete? How does that work?
Well, I suppose that depends on what you mean by “Turing complete”. Apparently you can implement rule 110 in HTML+CSS, but only on a finite space and for finitely many steps. By any strict definition of Turing completeness, this would not count – it’s obviously a finite state machine! – but the same criticism would apply to any genuine computer programming language.
@Mike Shulman: Here is an implementation of a universal computer in CSS: http://eli.fox-epste.in/rule110-full.html.
Well, I suppose that depends on what you mean by “Turing complete”. Apparently you can implement rule 110 in HTML+CSS, but only on a finite space and for finitely many steps. By any strict definition of Turing completeness, this would not count – it’s obviously a finite state machine!
As I understand it you can use CSS to compute the next state in a cellular automata - not compute the entire sequence of states. It is impossible to give a browser CSS that doesn’t terminate while a TeX program may not terminate.
That’s cute, but I don’t think it’s Turing-completeness. The difference between primitive recursive functions and general recursive functions is precisely the existence of unbounded search. Computing the next state of a Turing machine is a primitive recursive function, but that doesn’t mean that primitive-recursion is Turing complete. To be Turing-complete, a language needs to be able to perform the unbounded search itself, without a user present to say “now try the next number”, “now try the next number”, “now try the next number”, etc. This might seem like a trivial distinction, but I think it’s necessary if “Turing complete” is to have any nonvacuous meaning.
For a real-world example, consider a total functional programming language like Gallina (the vernacular language of Coq). If “Turing complete” is to have any meaning, then Gallina must not be Turing complete, since every function you can write in it is total. But you can do the same thing with it that was done there with CSS, encoding a general recursive “function” and allowing the user to say “now evaluate the next step”, “now evaluate the next step”, “now evaluate the next step”, until an answer is found or the user gets bored.
Zhen is right that any physical implementation of a programming language must be bounded in time and space. But that just means that Turing-completeness is a property of a mathematical abstraction of a programming language rather than a physical implementation of it. Even a mathematical abstraction of CSS could not perform an unbounded search on its own.
My eyes have just run across what looks to be a nice explicit online proof of the fact that open star-shaped domains are diffeomorphic to Euclidean space. But I’m having difficulty figuring out how to incorporate it into the article ball, particularly whether this proof should supplant the one already there, or how prominently it should be mentioned.
Or just as an alternative proof in a new section. Cutting and pasting the MO answer should be ok under the Creative Commons license as long as a clear attribution is given.
I have added pointer to that MO proof to the entry, here.
@Dmitri - link to the Q?
Thanks. Here are online links:
Weil: Sur les théorèmes de de Rham. Demailly: Lemma IV.6.9 (page 214)
The article currently claims that Guillemin–Haine prove that open star-shaped regions are diffeomorphic to a ball.
In fact, Guillemin–Haine never state this fact. They only state a weaker fact, namely, convex open subsets of R^n are diffeomorphic to R^n, as Proposition 5.3.3, which is left without proof.
They do prove that any smooth manifold admits a differentiable good open cover, up to Proposition 5.3.3.
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