# Talk:Manifold

Manifold is a former featured article candidate. Please view the links under Article milestones below to see why the nomination failed. For older candidates, please check the archive.
Article milestones
DateProcessResult
November 18, 2005Peer reviewReviewed
January 31, 2006Featured article candidateNot promoted
Current status: Former featured article candidate
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Field:  Geometry
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## Untitled

Since I usually nitpick on discussion pages, I just wanted to say that this article is very coherently constructed, provides good examples, and covers the topic well for a wide range of readers. Thank you to all who contributed to it. —The preceding unsigned comment was added by 24.205.231.209 (talkcontribs) 20:36, 2006 November 12.

## Equivalence

The article states: Two atlases are said to be equivalent if their union is also an atlas. I cannot imagine what it means. In the first place is according to the definition of an atlas any extension of an atlas with a chart again an atlas. And couldn't it be the case that compatibility is meant? So, something is missing here. Madyno (talk) 21:24, 25 September 2017 (UTC)

In the smooth category, the union of two atlases need not be an atlas, since nothing forces the composite ${\displaystyle \phi \circ \psi ^{-1}}$ to be smooth if ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are just homeomorphisms that define different smooth structures. However, in the topological category, any two atlases are equivalent. Sławomir Biały (talk) 23:18, 25 September 2017 (UTC)
As I understand it, it means that if two atlases both describe the same topological manifold then their union - the bound volume of both sets of charts - will also be an atlas for that manifold. Perhaps I am being too simple-minded? — Cheers, Steelpillow (Talk) 08:23, 26 September 2017 (UTC)
Correct. But also the union of two topological atlases will always be a topological atlas, because the composite of homeomorphisms is a homeomorphism. So the definition of equivalence is fairly vacuous in the topological case. Sławomir Biały (talk) 10:56, 26 September 2017 (UTC)

As far as I can see, nothing in the definitions puts a condition on the transition maps. Madyno (talk) 12:55, 26 September 2017 (UTC)

For a smooth chart, the transition maps are assumed to be smooth. (For a topological chart, the transition maps are automatically continuous, so this is not needed as an extra hypothesis.) Sławomir Biały (talk) 14:38, 26 September 2017 (UTC)

Quite strange, you don't get my point. I know what you're saying. The point is, it isn't mentioned in the definitions. Madyno (talk) 13:27, 27 September 2017 (UTC)

What words would you propose adding to the article, then? Perhaps that will help us to see what you mean. — Cheers, Steelpillow (Talk) 14:04, 27 September 2017 (UTC)
A chart is assumed to "preserve the structure". Usually though, one takes a chart as what defines the structure. For example, a differentiable manifold is a topological space covered by an atlas where the transition functions are differentiable. You can give a different atlas which is differentiable in the sense that its transition maps with itself are differentiable, but for which the transition maps with the other atlas are not. So these are not equivalent atlases. They define two different differentiable structures. Sławomir Biały (talk) 14:50, 27 September 2017 (UTC)

## Chart

"The surface of the Earth requires (at least) two charts to include every point"

Not true, see, for example "Mercator_projection". There's a bit more precision (or hand waving) required in this explanation. — Preceding unsigned comment added by 125.239.100.117 (talk) 08:50, 2 November 2017 (UTC)

The Mercator projection leaves out the poles, and also fails to be continuous on one of the meridians. Sławomir Biały (talk) 10:37, 2 November 2017 (UTC)
Any chart on a sphere must leave out at least one point. Any single wrapping which covers it completely must introduce other discontinuities and so is not a chart. — Cheers, Steelpillow (Talk) 11:26, 2 November 2017 (UTC)

## Too technical, especially the summary

The summary of this article does not comply with wikipedia guidelines for technical articles.

Before removing the technical tag, the issue should be addressed. If an editor believes the subject matter is too technical for a simple explanation, that should be explained in the talk page first before removing the Template:Technical.

There's already a pretty simple explanation as the second paragraph. Consider expanding this and swapping with the current lead sentence. — Preceding unsigned comment added by 24.148.30.174 (talkcontribs) 20:05, 28 November 2018 (UTC)