Talk:Atlas (topology)

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The definition of a differential structure by means of a maximal atlas is the one I learned first. I recall some other definition, as a sheaf cohomology H1 class; with respect to some sheaf built from diffeomorphisms. This does seem to have advantages - perhaps I'm wrong. Anyway, I'd quite like to see this idea in Wikipedia.

Charles Matthews 15:42, 14 Jun 2004 (UTC)

manifold atlas?[edit]

Does atlas have any meaning besides the one in the definition of a manifold? If not perhaps we should integrate it into manifold. Both articles would benefit from some more context. --MarSch 14:30, 2 Jun 2005 (UTC)

One can talk about G-atlases on fiber bundles. Anyway, I think it's probably better to keep this page separate. Even for manifolds, we can have different kinds of atlases depending on what kind of manifold we have (topological, Ck, smooth, real-analytic, complex-analytic, etc.) -- Fropuff 14:57, 2005 Jun 2 (UTC)
The technical details about atlases are I think better discussed in a stand-alone article, as this one. The manifold article is already long and in some places quite complicated, no need to make it even more so. Oleg Alexandrov 22:27, 2 Jun 2005 (UTC)
perhaps manifold and fiber bundle both need a section on atlases? There is currently one article for topological manifolds and differentiable manifolds. Maybe we should split that up. --MarSch 10:57, 3 Jun 2005 (UTC)
It is good that manifold starts with a discussion of topological manifolds before going to differential manifolds. Being a topological manifold is a big part of what a differential manifold is about, so you can't really avoid it. Splitting things up will make the concept of differential manifold harder to understtand. That's my take. Oleg Alexandrov 15:49, 3 Jun 2005 (UTC)
Yeah, well I don't really want to split them up, since I like to merge articles. I am a bit surprised that you don't want to split. Just defaulting to disagreeing with me ;) ? --MarSch 16:23, 3 Jun 2005 (UTC)
If you mean that've been bugging you too much lately, then you are right. I will try to take it in a more relaxed way. Oleg Alexandrov 04:34, 4 Jun 2005 (UTC)
PS But I still stand by my opinon above. Oleg Alexandrov 04:34, 4 Jun 2005 (UTC)
Yes, charts and atlases are also used for fiber bundles. Should the page be renamed to, e.g., Atlas (finite-dimensional manifolds in Topology)? If not, should it be expanded to cover fiber bundles and infinite-dimensional ( Banach, Fréchet) manifolds?
Note that while it is possible to give definitions encompassing both manifolds and fiber bundles, that would be OR. Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:34, 29 May 2015 (UTC)

Can anybody fix this sentence?[edit]

"For each point in the complicated space, a neighborhood of that point that is homeomorphic to a simple space. The homeomorphism is called a chart."

"If for each point in the complicated space, a neighborhood of that point that is homeomorphic to a simple space, then the homeomorphism is called a chart."

Did I change the meaning? Be gentle, I'm new at this. I'm not sure about the comma. (talk) 06:31, 1 November 2011 (UTC)

Does it mean this: "If for each point in the topological space a (subspace) including that point is homeomorphic to a Euclidian space then the homeomorphism is called a chart."? JLewis98856 (talk) 06:55, 1 November 2011 (UTC)
What is the origin of this sentence? It's not in the article. Are you just looking for general clarification? Or is this something taken from the "simple English Wikipedia" (or whatever it's called)? Either way, you're looking for something like...
"Each point in the complicated space has at least one neighborhood that is homeomorphic to a simple space, and each pairing of a neighborhood with its homeomorphism is called a chart."
In other words, an atlas is a book of maps ("charts") that can overlap. Each map appears to be a two-dimensional surface where distances along the surface of the earth appear to be straight-line Euclidean distances. However, the book in total represents positions on the surface of the spherical planet. Note that an atlas of the Earth must then have more than two maps in it. If it only had one map covering the entire surface of the earth, open edges of that map would omit a line of points from the earth. Furthermore, closed edges of that map would give the impression that the two far edges were very far apart even though they were the same line. Thus, atlases must be open covers. —TedPavlic (talk/contrib/@) 14:10, 1 November 2011 (UTC)

Local patches[edit]

The term local patch in Metric tensor (general relativity) redirected to Chart (topology) which redirected to here. Is the term a synonym for something in the article? modify 19:28, 24 April 2006 (UTC)

Yep. Basically the coordinate system for a local patch is defined by a chart (a map from part of Rn to the local patch), the coordinate system for Rn is mapped to give coordinates for the local patch. An atlas is the set of all the charts. See Manifold for a more extensive discussion aimed at a general audience. --Salix alba (talk) 23:31, 24 April 2006 (UTC)
Thank you, Salix alba. That was helpful. modify 06:19, 25 April 2006 (UTC)

Maximal atlas without Zorn's lemma[edit]

The article claims the existence of a maximal atlas without using Zorn's lemma. But the only given reference uses it and I only know proofs of it's existence making use of Zorn's lemma. Could somebody give a reference to a source, that shows existence without it? (talk) 22:17, 7 May 2008 (UTC)

The article should be rewritten[edit]

Please see the page, User talk: Waltpohl, where I have made some comments regarding why this page should be rewritten.

Topology Expert (talk) 09:16, 12 August 2008 (UTC)

In fact I might as well include what I said here:

  • The article refers to a manifold as a 'complicated space'; this word is not mathematical and doesn't explain what a manifold really is
  • Additional to this, the article claims that a manifold is made up of 'simple spaces'. According to this, a 'simple space' could be an open interval in R. But any open interval in R is also a manifold (being homeomorphic to R) which means that according to a space can be 'simple' and 'compicated' at the same time

There is no such concept of a 'simple space' in mathematics anyhow. There is such thing as a simple function (often used in measure theory), however.

  • Worst of all, the article has not included the actual definition of an atlas
  • The article has been written in a non-formal manner

Topology Expert (talk) 09:20, 12 August 2008 (UTC)

A separate topic on coordinate chart should be added[edit]

It seems Wikipedia does not have a topic on coordinate chart (which by the way redirects to atlas). However coordinate chart, by its own rights, is a quite vast subject. One should also talk about the flaws with charts (like singularities) which on the first place leads to the need of atlases. —Preceding unsigned comment added by (talk) 17:22, 3 February 2011 (UTC)

The definition of a transition map is incorrect. The transition map is defined on the image of U_{\alpha} \intersect U_{\beta} under \phi_{\alpha}. — Preceding unsigned comment added by (talk) 01:28, 10 August 2011 (UTC)

Local Frame[edit]

… redirects to this article about atlases, but isn't mentioned anywhere in it. — Preceding unsigned comment added by (talk) 11:34, 9 August 2012 (UTC)

Yep. We also have articles on smooth frames and on smooth structures, which do not define this either. The smooth frame article is about frames on the tangent bundle; it fails to generalize to general vector bundles. It also fails to mention atlases. This article defines the notion of "local" but never uses the word "local" in regards to that. It also never defines a frame' istead it hand-waves though the section "more structure". Sigh. This article needs a major rewrite. (talk) 04:51, 24 October 2016 (UTC)

sheafs and ringed spaces[edit]

These days, its popular to define manifolds with sheaves and ringed spaces. At least some mention of this should be made here ... (talk) 03:22, 2 October 2016 (UTC)