What's a good reference e.g. textbook for the fact that the open n-ball and the open n-simplex are diffeomorphic?
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8$\begingroup$ Why did someone vote to close this question? $\endgroup$– Deane YangCommented Oct 12, 2010 at 1:33
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6$\begingroup$ My guess is that someone didn't realize the distinction between "homeomorphism" and "diffeomorphism" in this question. $\endgroup$– Andy PutmanCommented Oct 12, 2010 at 2:45
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2$\begingroup$ My thanks to those who answered Jim Stasheff's question. It arose in a discussion over at the nLab (and of course we're interested in the generalization to star-shaped open domains, addressed by Dan Ramras and Robin Chapman below). $\endgroup$– Todd TrimbleCommented Oct 12, 2010 at 10:55
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3$\begingroup$ And one should mention that so far Todd has been the only one in this discussion who actually did provide an explicit concrete and nice diffeomorphism (which really was the original question that Jim*s question was promted by). Reproduced here: ncatlab.org/nlab/show/ball $\endgroup$– Urs SchreiberCommented Oct 13, 2010 at 8:32
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$\begingroup$ See the answer here. $\endgroup$– Moishe KohanCommented Feb 9, 2022 at 1:01
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2 Answers
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If the compact simplex is
$$\Delta_n = \{ (x_0,\cdots,x_n) : x_i \geq 0, x_0+x_1+\cdots+x_n=1\} \subset \mathbb R^{n+1}$$
then consider this function $f : \Delta \to \mathbb R \cup \{\infty\}$ defined by
$$f(x_0,\cdots,x_n) = \frac{1}{x_0} + \cdots + \frac{1}{x_n}$$
This is a proper Morse function on the interior of $\Delta_n$, and there's only the one critical point at $(\frac{1}{n+1},\cdots,\frac{1}{n+1})$, so standard theorems in Morse theory give you a diffeo to the open ball.
I imagine this is simple enough that you could solve the corresponding ODEs explicitly and write the diffeo out in a closed-form but I haven't put in the work.
answered Oct 12, 2010 at 1:34
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I'd recommend taking a look at the book by Brocker and Janich, which discusses diffeomorphisms between star-shaped domains by defining a flow along rays from the star point. This might be an exercise in the book, rather than a theorem. (I think I have the book on my desk at school, and I'll try to give a more precise reference tomorrow.)
answered Oct 12, 2010 at 2:53
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4$\begingroup$ It's an exercise (no. 7 in ch. 8). $\endgroup$ Commented Oct 12, 2010 at 6:44
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1$\begingroup$ It's curiuous: that exercise asks the reader to prove what in Conlon's book "Differentiable manifolds" (in the edition of 2008(!)) is commented on (after lemma 10.5.5) as follows: "It seems that open star shaped sets are always diffeomorphic to R^n, but this is extremely difficult to prove." But a full prove has been written out, for instance by Stefan Born, which is reproduced as theorem 237 in Dirk Ferus' lecture notes math.tu-berlin.de/~ferus/ANA/Ana3.pdf However, that still does not give nice concrete diffeos in concrete cases like the n-simplex. $\endgroup$ Commented Oct 13, 2010 at 8:30