Timeline for What should be taught in a 1st course on smooth manifolds?
Current License: CC BY-SA 2.5
11 events
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| Mar 21, 2011 at 0:28 | comment | added | Jack Huizenga | Tom's comments sum up my thoughts nicely. There really isn't an issue here--either set of definitions really does give the same thing. | |
| Mar 20, 2011 at 13:59 | comment | added | R. Andrew Hicks | Jack - Spivak's "cube-sphere", as you so elegantly put it, does come equipped with a metric. So it has geometry, which makes it isometric as a metric space to a cube. | |
| Mar 20, 2011 at 3:54 | comment | added | Tom Goodwillie | Every smooth manifold in $\mathbb R^n$ in the sense of the Milnor book ("concrete manifold") is canonically a smooth manifold in the abstract sense. If $M$ and $N$ are concrete manifolds, then the smooth maps between them in the concrete sense are precisely the smooth maps between them in the abstract sense. That is, we have a full and faithful functor from the one category to the other. In fact, it is an equivalence of categories -- that is, every abstract manifold is diffeomorphic to some concrete manifold -- that is, every abstract manifold can be smoothly embedded in some $\mathbb R^n$. | |
| Mar 20, 2011 at 3:30 | comment | added | Jack Huizenga | If you take a cube, identify it with a sphere, and give it the differentiable structure from the sphere, then it can't be smoothly embedded in R^3 as a cube, and I would thus be reluctant to call it a cube anymore. It can be embedded as a sphere, however, and the smooth functions on it will all locally extend to smooth functions on the ambient space. In this sense Spivak's "cube-sphere" is also a manifold in G&P. Once you forget the embedding in G&P, the categories are the same. | |
| Mar 20, 2011 at 3:24 | comment | added | R. Andrew Hicks | @Jack: My last statement "the two categories of differentiable manifolds obtained from G&P and Spivak are very different." is not correct. | |
| Mar 20, 2011 at 3:02 | comment | added | R. Andrew Hicks | @Jack: Given the G&P definitions the cube is simply not a differentiable manifold, as you point out, so of course it isn't diffeomorphic to the usual round $S^2$. They are however, homeomorphic. From the viewpoint of the Spivak Vol. 1 definitions you would say "they are homeomorphic and AFTER you put a (any in fact) differentiable structure on the cube then it is a theorem that it is diffeomorphic to $S^2$." If this is right then the usage is really different. (Another way to say this(?): the two categories of differentiable manifolds obtained from G&P and Spivak are very different.) | |
| Mar 19, 2011 at 22:21 | comment | added | Jack Huizenga | Another way of putting it is this: every smooth manifold has an embedding in some R^n, in such a way that the smooth functions on the manifold are (pullbacks of) those functions on the image which can be locally extended to smooth functions on open neighborhoods. | |
| Mar 19, 2011 at 20:24 | comment | added | Jack Huizenga | No, I don't believe it is a different notion. Perhaps the confusion is that a cube in the sense of G&P is not a smooth manifold--thinking of a sphere as homeomorphically embedded in R^n as a cube does not put a smooth manifold structure on the sphere. | |
| Mar 19, 2011 at 16:16 | comment | added | R. Andrew Hicks | Thanks for that clarification. So it really is a different definition of diffeomorphism, and so perhaps is deserving of a different name, like 'ambient diffeomorphism'. | |
| Mar 19, 2011 at 12:24 | comment | added | Tom Goodwillie | If you take the point of view, as in these books, that smooth manifolds are certain subsets of $\mathbb R^n$ and inherit their smooth structure from $\mathbb R^n$, then there is no such thing as two smooth structures on the same set. But there is such a thing, obviously, as two smooth manifolds related by a homeomorphism that is not a diffeomorphism. And there is also such a thing, not obviously, as two smooth manifolds related by a homeomorphism but not by any diffeomorphism. | |
| Mar 19, 2011 at 1:13 | history | answered | R. Andrew Hicks | CC BY-SA 2.5 |