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old: Talk:manifold/old, Talk:manifold/rewrite/freezer.

Impressed with this article[edit]

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 (talkcontribs) 20:36, 2006 November 12.

Animated GIF[edit]

I've removed the animated GIF of "boy's surface" because of the distraction it causes. This action is consistent with MOS:ACCESS but counter to the wishes of User:Slawekb : (The image and its caption accompany the text of the lead. If you don't like this particular image of Boy's surface, then find another one.)

Any opinions on this apart from the two of us?

-- Catskul (talk) 23:09, 14 November 2013 (UTC)

Presumably the onus is on you to make a more suitable image and convert the gif to video, per the guideline. The manual of style should not be used to dictate what kinds of informative content to have in articles. This image has informative value. Sławomir Biały (talk) 00:35, 15 November 2013 (UTC)
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
As I understand it there is no onus on editors to replace offending content. If, for example, a statement is unsourced, an editor is not obligated to find a source which negates the content before removing. While the content I am attempting to remove has value, it is neither critical to the article nor mentioned anywhere in the text.
Despite my belief that replacement is not a requirement, consensus is needed. So in an attempt to achieve consensus, I suggest replacement of the original with the following:
-- Catskul (talk) 16:30, 15 November 2013 (UTC)
Aesthetically, that image is not really an improvement over what is there now. I will work on a better replacement when I get the time. Sławomir Biały (talk) 00:08, 16 November 2013 (UTC)
I'm going to add in an opinion here. The point that some 2d manifolds cannot be embedded in 3d space without self-intersection is not central to the concept of a manifold; it is an essentially topological result that applies only for an arbitrarily constrained choice of embedding space for a some manifolds. I'd say that a far more significant (or more generally applicable) points topologically are that manifolds can be closed or can be non-orientable, which are not even mentioned in the lead. I'd think that it would be sufficient to mention one or two illustrative cases such a the Klein bottle without mentioning properties of selected embeddings. —Quondum 13:01, 16 November 2013 (UTC)

I think the idea that there are 2-manifolds that are not realizable as surfaces in Euclidean space is actually quite important for understanding why there is a mathematical notion of "manifold" at all. Certainly, one can study manifolds without worrying about their embedding properties (although there are people who build their careers entirely on the latter), but to someone with no idea what a manifold even is, I think it is very important to realize that they do represent a significant generalization of the elementary notion of a surface. Sławomir Biały (talk) 13:31, 17 November 2013 (UTC)

The topological aspects of manifolds are important, I agree, but is only one of what they are useful for. The same point about embedding can be illustrated within a Klein bottle without challenging the reader nearly as much. Emphasizing the topological aspects at the expense of the geometric aspects is also not ideal. For example, the real projective plane also represents elliptic geometry, which does not come through at all. All these issues are complex enough that more than a mention in the lead can hint to the reader that this article is hard work to understand. —Quondum 21:32, 17 November 2013 (UTC)
Meanwhile I would second the removal of any animated gif until a suitable alternative can be found. These things do my head in really quite seriously, and I am sure I am not alone. Sometimes I can rely on my browser settings, but far from always. — Cheers, Steelpillow (Talk) 15:32, 12 January 2015 (UTC)

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

Not strictly true: the south pole (for example) can be depicted as a circle. — Preceding unsigned comment added by (talkcontribs) 2014-10-16T17:39:48

Read the rest of the sentence. That circle is an example of "duplication of coverage". — Cheers, Steelpillow (Talk) 15:27, 12 January 2015 (UTC)


The lead says that the Klein bottle and real projective plane cannot be realized in three dimensions. Where does this term "realized" come from? Sure they cannot be embedded/bijected, but they sure can be injected. In the theory of abstract polytopes the idea of "realization" describes all such injections into real space, however degenerate. Does topology differ, or should the lead be amended accordingly? — Cheers, Steelpillow (Talk) 15:23, 12 January 2015 (UTC)

I agree with Steelpillow. Unless "realized" has some technical meaning that I don't know of, it is a vague statement without any clear meaning. Also, what does "in three dimensions" mean? Presumably it means in a three-dimensional real space, but it doesn't say so, and both the Klein bottle and the real projective plane can be embedded in other three-dimensional real manifolds (even if we restrict "three dimensions" to refer only to three dimensions over the real field). Consequently, I have re-worded the passage in the article. The editor who uses the pseudonym "JamesBWatson" (talk) 14:12, 29 May 2015 (UTC)
To clarify for anyone who reads this, the passage that Steelpillow mentions in the article abstract polytope says "a realization of a regular abstract polytope is a collection of points in space ... together with the face structure induced on it by the polytope, which is at least as symmetrical as the original abstract polytope..." (My emphasis.) As far as I can see, that must include injections which are not one-to-one and introduce more symmetry than in the abstract polytope. The editor who uses the pseudonym "JamesBWatson" (talk) 14:21, 29 May 2015 (UTC)
I am not sure if that quoted passage is quite correct. A few years ago "realization" was shaping up to mean any injection into some parent real space, while preservation of symmetry, number of dimensions and suchlike required the realization to be "faithful". I may have a source somewhere, I'll try to find time to check. — Cheers, Steelpillow (Talk) 18:37, 29 May 2015 (UTC)
There is a notion of a geometric realization as a functor that maps simplicial sets and incidence relations to a topological space formed from suitably glued-together geometric simplices. This is discussed in nLab at geometric realization and mentioned in the article section Simplicial set#Geometric realization. The realization in abstract polytopes that Steelpillow mentions seems an example of this more general concept, although that specialization includes constraints on symmetry. I don't have a source for this, but regarding a Klein bottle as a simplicial set, the article is stating that one cannot map the set to geometric simplices in three dimensional real space that satisfy the incidence relations. This is a fun article on Klein bottle realizations. --Mark viking (talk) 00:44, 30 May 2015 (UTC)
My favourite realization of the Klein bottle is in four-space. We make a Möbius band by taking a two-dimensional strip, twisting one end over in a third dimension and butt-joining the ends to form a twisted cylinder. Take a piece of rod instead of a flat strip. This is already 3D so we twist it over in a fourth dimension before joining the ends to form a twisted hoop. Just as a Möbius band has a definite interior surface between its edges, so too does the hoop. When either is squashed down a dimension it must self-intersect, creating a singularity where not only the boundary self-intersects but also the interior crosses over itself. The interior of the Möbius band is usually obvious to us. Once we understand that a Klein bottle is just the boundary of a solid hoop, its interior becomes equally clear. It is only the intuitive reluctance to introduce a crossover ring singularity in 3-space, when there is no such singularity in the surface, which confuses us into the mistaken idea that it is a "bottle". — Cheers, Steelpillow (Talk) 08:11, 30 May 2015 (UTC)

Betti numbers and torsion coefficients[edit]

The topological characteristics of a manifold are captured in its Betti numbers and torsion coefficients. Only the first of these is mentioned in this article. I have started a discussion at Talk:Homology (mathematics)#Betti numbers and torsion coefficients and would be grateful for any contributions. — Cheers, Steelpillow (Talk) 15:37, 12 January 2015 (UTC)