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 220.127.116.11 (talk • contribs) 20:36, 2006 November 12.
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)
I'm sure I've misunderstand the following statement from the introduction:
> Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane which cannot.
It seems to suggest Klein bottles cannot be represented in three dimensional space but then links to the article on Klein bottles with several real world models of them. Is:
the Klein bottle model(s) inaccurate?
the statement is incorrect?
the statement is correct and I've misinterpreted it?
A three-dimensional model (or representation) of the Klein bottle is the image of a differentiable map from the (abstract) Klein bottle into R3, which satisfies some regularity condition. Typically this condition is that every point of the Klein bottle has a neighborhood such that the restriction of the map to this neighborhood is a diffeomorphism of the neighborhood onto its image. It can be proved that such a three-dimensional model cannot be a manifold, that is the model is a surface, which has crossing points, where the surface is not locally homeomorphic to R2. Probably, you have been confused by the fact that there are several kinds of surfaces, and those that have crossing points or other singularities are not manifolds. D.Lazard (talk) 09:12, 5 April 2016 (UTC)
In other words, although the Klein bottle can be immersed in ordinary space, such an immersion will always self-intersect - it cannot be embedded without self-intersection. The statement you quote does seem to be a little ambiguous - your scenario 3 seems to be the case, so I'll try to clarify it. — Cheers, Steelpillow (Talk) 09:20, 5 April 2016 (UTC)
A possible source of confusion is that, in Surface, it was asserted that a surface is necessarily non-singular. I have fixed this by rewriting the lead, which was also tagged as too technical. D.Lazard (talk) 10:26, 5 April 2016 (UTC)
Convention for domain and codomain of transition maps?
The subsection "Transition Maps" and its neighbors (especially "Atlases") need a bit of work, but for now I'm just curious: does a transition map send a region of R^n to another region of R^n, or does a transition map send a part of the manifold to another part of the manifold? In this article, the first convention is used. But in the Wikipedia article Atlas (topology) the other convention is used - with a picture!
I don't know what to believe anymore. -Norbornene (talk) 15:38, 4 January 2017 (UTC)
Please, read Atlas (topology) more carefully: in both articles transition maps map open subsets of R^n to open subset of R^n. D.Lazard (talk) 16:33, 4 January 2017 (UTC)
As I understand it, a transition map describes how to convert from the coordinate system used by one chart to the coordinate system used by an overlapping chart. All are expressed in terms of R^n. A simple example would be converting (transitioning) between a map of Britain using the Greenwich meridian and a map of France using the Paris meridian, when visiting the Channel Islands. — Cheers, Steelpillow (Talk) 17:24, 4 January 2017 (UTC)