Talk:Topological manifold

Disjoint Union of manifolds

The article asserts that the disjoint union of n manifolds of the same dimension is a manifold, which is false in general (consider the line (x,y) with x = 0 and the line (x,y) with x > 0, y = 0, their union is certainly not a manifold). I've removed this line from the article. I'm not sure of all the necessary conditions though. I do know that being compact, disjoint and same of the dimension are sufficient conditions though.

It has to be the topological coproduct, and not the set coproduct. This is an example of a coproduct in set which is not a coproduct in top. — Preceding unsigned comment added by 216.186.225.19 (talk) 23:59, 1 May 2022 (UTC)

Attention needed

MarSch created this as a cut&paste from manifold, and that's still where we are, an incoherent mess. Any work on this article is much appreciated. Oleg Alexandrov (talk) 16:59, 22 December 2005 (UTC)

"A topological manifold is a manifold that is glued together from Euclidean spaces"? A piece of paper is a Euclidian space, therefore if I glue a bunch of papers together I get a topological manifold? Serious need of attention, yes, granted. FelisSchrödingeris 14:45, 7 April 2006 (UTC)

I'm wondering where you get that quote from. It certainly isn't in the article of the date of your comment and I can't find it in the early versions. I do kind of recall saying something like that somewhere though. --MarSch 17:56, 15 April 2006 (UTC)

As a matter of fact, it's the first line of the article in it's present form. The analogy used in my comment perhaps wasn't quite relevant (although it was the first thing that popped into my head while reading it), but the idea behind it still holds...

The concept, I gather, is a basic one in the field of topology. But that sentence alone pre-supposes that the reader knows what a "manifold" is. It also uses the concept of "gluing of Euclidian spaces", which, for a non-topologist like me, doesn't make much sense. A good introduction - a gentle slope into the concepts - is mainly what's missing, I guess. FelisSchrödingeris 14:36, 18 April 2006 (UTC)

Manifold is supposed to explain the gluing. Then this article only needs to specify that the what which is being glued are the topologies of Euclidean spaces. That's why things are as they are. There are so many different kinds of manifold that it is a huge duplication of effort to explain the gluing in each one separately.--MarSch 08:59, 19 April 2006 (UTC)

How about creating glueing, which is in Wikipedia:Missing science topics/Maths4. Oleg Alexandrov (talk) 17:04, 19 April 2006 (UTC)

Interesting suggestion. What would manifold be for if we create glueing? --MarSch 11:43, 20 April 2006 (UTC)

This is now a stale thread, but topology does a great deal of gluing (correct spelling) that is not suitable for manifolds. For a manifold we must have overlapping open sets, but in the more general concept we can glue two circles together at a single point, creating a "figure-8", or glue edges of a polygon together to create an orbifold. Other fun (and useful) objects we can create in this way include a bouquet of spheres and a CW complex. --KSmrq^T 07:50, 21 November 2006 (UTC)

All the gluing examples you mention would result in what I would informally call manifolds. If there is a better term to describe all of these things I'd like to know. --MarSch 11:41, 11 March 2007 (UTC)

I would never call most of these manifolds, not even informally. I don't have the details from Talk:manifold, but I believe even authors (Marsden, for example) who allow the dimension of a manifold to vary from one connected component to another — which is already an uncommon use — do not allow dimension to vary within a component. After all, a primary motivation for manifolds is that locally they are uncomplicated, so we can pretend we're in Rⁿ for some fixed n, and carry over massive amounts of helpful machinery. By demanding open overlaps, we enforce consistent dimension and uniform simplicity. If we drop that requirement, we can glue together any two topological spaces in ever more creative ways, and the only all-encompassing term is "topological space"; sorry! --KSmrq^T 12:35, 11 March 2007 (UTC)

Separation from manifold entry

Can anybody explain to me briefly what should be in this entry in distinction to the manifold entry. Then I may be able to clean up a little bit. Hottiger 11:03, 14 April 2006 (UTC)

Besides: The example fo the hausdorff condition should clarify the topology imho. Hottiger 11:03, 14 April 2006 (UTC)

Gladly. Long ago some mathematician editors decided that manifold was supposed to be about the general concept of manifold exemplified by such concrete entities as topological manifolds, differentiable manifolds, complex manifolds and arguably algebraic varieties and schemes. The common thing being that you go from something well understood to something new by gluing well understood objects together. Because these editors thought that this was actually explainable to non-mathematicians it was decided to designate manifold as the article in which to do this and not focus on exact definitions (which don't exist for this concept) or technicalities or whatever else might stand in the way of comprehension. So manifold is supposed to explain only this process (of gluing). Of course the gory details need to be discussed somewhere too, and this article is supposed to contain them. I hope this clarifies. --MarSch 18:07, 15 April 2006 (UTC)

That is, what seemed to be the initil intention. The result is that the initial explaining image in the manifold entry explains what is a 2-dimensional Riemannian manifold (there are ankles shown). In my opinion, the entry isn't good neither for non-mathematicians nor for mathematicians. By the way, one should not even try to explain what a scheme is in a manifold entry. Hottiger 17:43, 4 October 2006 (UTC)

I like the idea that the manifolds entry is a general, non-technical article. It should cover, in broad terms, many classes of manifold: topological, smooth, infinite-dimensional, even non-Hausdorff. The present article, on the other hand, is one of its most basic important technical incarnations: finite dimensional Hausdorff second countable continuous manifolds. Although every wikipedia article should begin in as elementary a way as possible, this article should lead quite quickly to careful definitions and examples. It should be a prototype for articles on other classes of manifolds, and a springboard for discussing such issues as the classification problem for manifolds, including the piecewise linear and smooth categories, and the algebraic and differential topology of manifolds. At present it doesn't do that. Also, the examples section should focus on examples with topological significance (such as projective space, flag manifolds and Stieffel manifolds), with a more comprehensive list being moved to an article on smooth manifolds. I've commented about this on several other pages. Let me know what you think, as these ideas are all interlinked. Geometry guy 21:50, 8 February 2007 (UTC)

Merging to manifold

See Talk:Manifold#Merging from topological manifold Oleg Alexandrov (talk) 15:26, 2 May 2006 (UTC)

composition link

Can someone clarify what exactly is meant by the composition link? There are three separate definitions for mathematics-related composition on the disambig page, and so the link as it is not very helpful. All we need to know is what definition the author was using for the word. SingCal 06:58, 27 July 2006 (UTC)

Function composition. I disamb'ed the link; thanks for your note. -- Jitse Niesen (talk) 09:34, 27 July 2006 (UTC)

Thanks!! SingCal 21:42, 27 July 2006 (UTC)

5-manifold?

I had just seen an article: 5-manifold. I wondered if it should be listed under, "See Also" If so, it looks like it needs a bit of work.Brian Pearson 07:15, 21 November 2006 (UTC)

Cleanup definition

A topological space X is locally Euclidean if every point in X has a neighborhood which is homeomorphic to an open subset of Euclidean space Rⁿ. The integer n, called the dimension of X, must be the same for all points of X.

The second phrase is problematic in at least these ways:

1. Quantifiers which appear after the objects they quantify are ambiguous and bad style.

2. The way this phrase is written makes it seem that it is obvious that the dimension of X is well-defined. But this is a highly non-trivial theorem and I don't know if it is true for locally Euclidean spaces as opposed to topological manifolds.

Requiring every point to have a neighborhood homeomorphic to Rⁿ itself is mathematically clearer and probably what the author had in mind.

--80.238.205.119 23:10, 22 July 2007 (UTC)

Why not second countability?

Second countability (along with the Hausdorff and locally Euclidean condition), allow manifolds to be imbedded in finite-dimensional Euclidean space (this is useful in applications). In many branches of mathematics, spaces are assumed to be paracompact (these branches do deal with manifolds). Therefore, spaces like the Long line should not be manifolds and second countability should be a condition (since this implies paracompactness, normality and other such useful topological properties). Even if the main definition in this article does not include second countability as an assumption for manifolds, there should be a detailed discussion of manifolds for which second countability is an assumption.

Topology Expert (talk) 13:29, 18 September 2008 (UTC)

A clarification to the article would be to simply define a topological space to be an n-manifold if it is Hausdorff, second countable and locally homeomorphic to R^n. This would probably meet the needs of most readers with a sidenote that the definition may vary slightly this being one of the most commonly used considering eg. applications.

80.221.7.131 (talk) 18:59, 10 January 2009 (UTC) AnvlL

Old information

In the section "classification of manifolds", it says

The 3-dimensional case may be solved. Thurston's geometrization conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman sketched a proof of this conjecture in 2003 which (as of 2006) appears to be essentially correct.

Is this a joke? I thought wikipedia was supposed to be on the current affairs and I come up here to find an more detailed explanation whenever my book doesn't have enough detail. But 2006??? This is only a minor problem because everyone knows Perelman's proof is correct as of 2009, but I've seen lot's of "as of year" stretching all the way back to 2003. Breath of the Dying (talk) 22:08, 4 October 2009 (UTC)

Dead link

http://www.maths.ed.ac.uk/~aar/haupt/ks76.pdf (in the references) is a dead link (404). —Preceding unsigned comment added by 77.187.29.157 (talk) 20:31, 12 February 2010 (UTC)

Formal Definition seems incorrect

I think it needs to at least say "homeomorphic to a connected open subset of Euclidean space Rⁿ." Or perhaps to an open ball in Rⁿ. Or perhaps it should simply use the usual definition and say "homeomorphic to Rⁿ (or even better to Eⁿ)" as in Hocking and Young (p278). If I've missed a point here, please correct me, otherwise I'll change it in a few days to read "homeomorphic to Euclidean space Eⁿ, or equivalently, to a connected open set or open ball in Eⁿ," or something like that. Eleuther (talk) 10:58, 13 March 2012 (UTC)

I think connectedness is not actually required: if the homeomorphism is to an arbitrary open set, we can always take an open ball in this set, and a ball is homeomorphic to Rⁿ. This should confirm what I'm saying. --Nicelbole (talk) 09:21, 21 January 2015 (UTC)

However, the now-used definition "homeomorphic to real n-space Rⁿ" clashes with the first sentence of "Coordinate charts", which says "By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of $\mathbb {R} ^{n}$." Maybe it should be noted somewhere that both definitions are equivalent. IttalracS (talk) 15:11, 16 November 2020 (UTC)