Talk:Manifold/rewrite/freezer

message: Should we also merge atlas (topology) into this article? Markus Schmaus 6 July 2005 12:13 (UTC)
: I would think that would make the article more complicated. I guess one can mention about atlasses, but refer to atlas (topology) for more details. Either way, importing from atlas (topology) wording of the form:
::By definition, a smooth differentiable structure (or differential structure) on a manifold M is such a maximal atlas of charts, all related by smooth coordinate changes on the overlaps.
makes me a bit nervous. :) Oleg Alexandrov 6 July 2005 15:10 (UTC)
Actually I wasn't thinking about moving content, but redirecting atlas (topology) to manifold and describing and defining "atlas" and "manifold" in the same article. Is there anything about an atlas which should not be in an article on manifolds? Markus Schmaus 6 July 2005 16:50 (UTC)
:I don't see anything in atlas which is not about manifolds. Please also note that I have added a section on atlases to the rewrite of differentiable manifold. I suggest you write first and then if atlas turns out to be superfluous we can redirect.--MarSch 6 July 2005 17:09 (UTC)
: I am fine with merging contents, as long as things don't get way too technical. It was sort of agreed that manifold should be an entry article for differential geometry, so the really heavy math should go elsewhere.
: So, again, you are welcome to merge the atlas here, as long as the contents is still somewhat readable by nonprofessional differential geometers. Oleg Alexandrov 7 July 2005 02:18 (UTC)
While we are on the topic of atlases, I have a question: what is the precise relationship between a sheaf taking values in topological spaces and an atlas?  If we require the sheaf maps to be homeomorphisms on the open sets, do we obtain an atlas?  Note that abstract varieties and schemes use sheaves, not atlases, for their "gluing" constructions.  I could imagine defining a category of topological spaces in such a way as to induce the correct restriction maps.  - Gauge 22:19, 10 July 2005 (UTC)
replies: text: Agree. I mentioned the same thing above. Oleg Alexandrov 21:13, 11 July 2005 (UTC)

text: "The continuous real-valued functions form a sheaf. An alternative definition of a topological manifold is a topological space with a sheaf of continuous functions locally isomorphic to Euclidean space with its sheaf of continuous functions."
All charts are in the sheaf of continuous functions and every homeomorphism is a chart (possibly require the range to be Rn). I'm not sure this is a sheaf, although it probably is. What happens to homeomorphism that coincide on some common domain? They couldn't be glued, cause the gluing would not be injective if the ranges are required to be Rn, but does this situation exist. Imagine an atlas for the cylinder consisting of two charts glued at both ends where they coincide such that they can be glued to form a homeomorphism of the entire cylinder to an annulus. The first map maps to the square ]-1, 1[^2. Lets stretch it in vertical direction with (x, y) |-> (x, tan(y (1 - |x|/2) \pi/2) ) with inverse (x, y) |-> (x, arctan(y)/(1 - |x|/2) \pi/2). Then these charts cannot be glued to an annulus. They must intersect somewhere and are thus non-injective and thus cannot be homeo's. --MarSch 17:32, 16 July 2005 (UTC)
thread_title:  Merging [[atlas (topology)]]? 

message: I made some changes, but I'm not so sure whether it's a good idea, so I'm solliciting opinions:
* I moved the circle example to above the excellent section on atlases etc. The idea is that that section is rather abstract and it might help the reader if he has a concrete example in mind (however, we are writing an encyclopedia).
* I added some formulas to the example. I think the formulas are easy enough to be understood, though the one for the transition map needs a bit more explanation. On the other hand, it is often said that after every formula costs you half the readers in a popular science text.
* I made the picture that I actually had in mind when writing the example (though it can probably use some annotations). However, Oleg has already uploaded a picture some days ago, which is now included in the article, so I did not upload mine. You can see both pictures at . I honestly don't know which to prefer: Oleg's is much simpler, but mine has some more information.
Awaiting your comments, Jitse Niesen (talk) 22:04, 14 July 2005 (UTC)
replies: text: I think your picture explains better than mine what charts are all about. So I have no problem if you replace my picture with yours. Oleg Alexandrov 22:09, 14 July 2005 (UTC)
replies: text: Actually I think the pictures compliment eachother. I would like both in the article for maximum clarity. Of course their colors need to be coordinated.--MarSch 17:35, 16 July 2005 (UTC)
thread_title: Jitse's edits of 14 July 2005

message: I think we should say for example "homeomorphic to Euclidean space" and not "homeomorphic to the Euclidean space". There is a Euclidean space for each natural number. Similarly "the" should be removed from other sentences.--MarSch 17:40, 16 July 2005 (UTC)
thread_title:  (the?) Euclidean space 

message: I've started a section on the construction of manifolds. So far I presented the constructions "zeros of a function" and "gluing charts", but there are at least three more constructions I'd like to include, "gluing manifolds with bounderies", such as gluing handles or moebius strips to spheres, "quotient manifolds", such as R2/Z2, the torus, and the "carthesian product" of manifolds. Markus Schmaus 19:47, 19 July 2005 (UTC)
replies: text: Excellent! I was just thinking about writing on constructing the circle with gluing, so I can do something else now. However, I am foreseeing one problem with the article: all the examples are in fact differentiable manifolds, so we have to be careful that it doesn't end up being an article about differentiable manifolds (or perhaps this is not such a bad thing). Are these constructions also valid for more general manifolds? Unfortunately, the little I do know about manifolds is all differentiable manifolds ... -- Jitse Niesen (talk) 20:31, 19 July 2005 (UTC)
thread_title:  Construction 

message: I removed subsection status on some of the headings just to make the table of contents a bit smaller. In particular, not every example requires its own subsection. I also implemented the "types of manifolds" idea to reduce clutter, and indented a couple of the examples to make them stand out from the rest of that section. - Gauge 04:32, 20 July 2005 (UTC)
replies: text: I don't like using only boldface to indicate a subheading, as this removes structure. This article is much from finished and I don't think we should care about the length of the table of contents now. Markus Schmaus 16:16, 23 July 2005 (UTC)
thread_title: Large table of contents

message: I am concerned that the article lists varieties and schemes as examples of manifolds.  Although the definition of a manifold is not explicitly given in the article, most definitions that I've seen make use of an atlas.  What is the corresponding atlas that defines a scheme or variety as a manifold? I noticed that Planetmath has a sheaf-theoretic definition of a manifold here.  - Gauge 20:28, 21 July 2005 (UTC)
thread_title: Varieties and Schemes

message: "Similarly, a cone is not a manifold because the tip of the cone looks different".  This sentence seems to end too early but I couldn't find a way to fix it.  The cone may not be the best example because of the ambiguity between bounded cones (with a base) and unbounded cones (like a light-cone in relativity).  I would like to say something like "it looks different from the rest of the space", but with bounded cones you have this corner at the base to deal with. - Gauge 05:12, 25 July 2005 (UTC)
replies: text: I agree. I had to finish quickly. The other problem is that the (unbounded) cone is a topological manifold (I think), but not a differentiable manifold. My reasoning was twofold (but not manifold): we should also give some counterexamples, otherwise it seems that everything is a manifold, and I did not like the counterexample of the cylinder precisely because of the ambiguity between bounded cylinders (which are only a manifold with boundary) and unbounded cylinders (which are bona fide manifolds). -- Jitse Niesen (talk) 09:53, 25 July 2005 (UTC)
thread_title: Counterexamples

message: Just for clarification, those edits, Gauge refered to, weren't mine. They were done by Jitse, but I liked them a lot. Markus Schmaus 12:22, 25 July 2005 (UTC)
replies: text: Whoops. Thanks for clearing that up. - Gauge 23:38, 25 July 2005 (UTC)
thread_title:  "Markus' edits"