Page contents not supported in other languages.
From Wikipedia, the free encyclopedia
WikiProject iconMathematics B‑class Low‑priority
WikiProject iconThis article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
BThis article has been rated as B-class on Wikipedia's content assessment scale.
 Low This article has been rated as Low-priority on the project's priority scale.


The stub is mathematically incorrect. What it describes is the so-called "good" orbifold, but there are bad ones also.

Be bold; why don't you add the info about the good and the bad ones? Dysprosia 10:31, 28 Sep 2003 (UTC)

Differentiable manifolds?[edit]

Are we talking about differentiable manifolds here? Also, I'm not sure whether the word "discrete" adds anything to the definition. AxelBoldt 19:52, 18 Jan 2004 (UTC)

Hyperbolic orbifolds[edit]

There's a whole topic of hyperbolic orbifolds (replace R^n with H^n, etc.); I know just enough to mention this might be important, but don't feel qualified enough to offer a presentation. Just wanted to throw the idea out there.


Should, "finite quations of Rn," not be "finite quotients of Rn?"

Linear actions[edit]

The formal definition talks about "linear actions of a finite group on an open subset of Rn". What's the meaning of "linear" here? Also, I find it surprising that the compatibility condition does not refer to the group actions at all. AxelBoldt 17:39, 23 Jun 2004 (UTC)

I changed it to linear transformation. On the group action, I think it is still correct, the compatibility condition for the group actions follow from it (let me know if I'm wrong). Tosha 19:42, 23 Jun 2004 (UTC)
So we're talking about a finite group of linear transformations acting on an open subset U of Rn? That seems a bit strange to me: there aren't many such actions, because most linear transformations won't map U bijectively onto U. Maybe we should allow a finite group of diffeomorphisms? Is there a good book to read up on these things, maybe something by Thurston? Thanks, AxelBoldt 09:55, 25 Jun 2004 (UTC)
sure, there aren't many such actions for general U, but given an action one can choose an invariant U, I do not see a problem here. If you want to change def. to more stadard(?) it is fine, but I like this one... If you change to diffeomorphsm it will give the same orbifold, but the def will become mor complecated. I do not know a good ref. I'm not sure but it seems that Thurston was mostly interested in 3-d case. I took a def from some paper (now I do not remember which) and modifyed it a bit Tosha 12:09, 5 Jul 2004 (UTC)

Homogeneous space[edit]

Isn't it the case that the base space (Euclidean space modulo group of linear xforms) is just an example of a homogeneous space? Perhaps I'm missing something, but I'm having trouble figureing out why euclidean space is singled out in this article, instead of having a more general definition for a homogeneous space. In fact, the "string theory" definition seems to be trying to say this, without actually blurting out that M/G must be a homogeneous space in order to have the usual properties. linas 16:37, 19 September 2006 (UTC)Reply[reply]

Fundamental group and Euler characteristic[edit]

I stopped short of putting in the normal (non-orbifold) characteristics for S2 by way of contrast, opting instead for the links. I just found myself wondering as I read it "Ok but what are they usually if they don't have to be the same" Zero sharp 04:44, 21 July 2007 (UTC)Reply[reply]

Definition of orbifold[edit]

The definition given in the article does not seem to agree with the definitions of either Satake or Thurston. It is the definition given in the short note of Haefliger in the Ghys-de la Harpe seminar of the late 1980s, which seems incomplete in that it does not give a manifestly invariant definition of isotropy groups (up to isomorphism). The later work of Haefliger on orbihedra (Trieste notes) takes a definition equivalent to the definitions of Satake and Thurston in the category of simplicial complexes. In order to make Haefliger's first definition of orbifold work, some theorems are needed (eg at least versions of Newman's theorem, possibly more). Standard references on the application to Seifert fibered 3-manifolds such as the BLMS article of Peter Scott use Thurston's definition. So, after checking the definitions currently in use in this mainstream part of the mathematical literature (part of the 2,000 odd orbifold articles on mathscinet), it would seem a better idea to give Thurston's definition, indicating the precise relation to other definitions. The book of Bridson and Haefliger does not discuss orbihedra, only the underlying complexes of groups. This was undoubtedly a deliberate decision of the authors, but does not help to resolve the slightly problematic definition of orbifold that Haefliger gave, that he gave once more in the coda of his book with Bridson and that is repeated in this WP article. --Mathsci (talk) 21:45, 21 November 2007 (UTC)Reply[reply]

I have now made these changes and will later add some more references. --Mathsci (talk) 13:17, 22 November 2007 (UTC)Reply[reply]

Use of orbifolds in the geometrization of 3-manifolds[edit]

Could somebody please add material on this, explaining why orbifolds on 2-manifolds arise in Seifert fibrations, etc, etc.? This would definitely improve the article. Thanks in advance, Mathsci (talk) 13:17, 22 November 2007 (UTC)Reply[reply]

properly discontinuous[edit]

Would you mind commenting on the sentence,

The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a finite group;

at Talk:Properly_discontinuous_action#is there an isotropy subgroup for a properly discontinuous action?? -- JanCK (talk) 22:21, 19 December 2007 (UTC)Reply[reply]

I have replied at the place you linked to. But note, WP is not a forum like sci.math (in which I participate from time to time as rusty). Mathsci (talk) 22:48, 19 December 2007 (UTC)Reply[reply]

Triangles of groups and Complexes of groups[edit]

I propose to fork out these long sections to separate articles and use the summary style here. Reasons:

  • Sufficiently notable to merit their own articles.
  • Not quite in line with other types of orbifolds that can be discussed here.
  • Already long and with potential of further expansion.

At the moment, there aren't even redirects to these terms (and the built-in search function isn't very helpful in finding them), but I think that in the long run, separate articles is the way to go. Arcfrk (talk) 04:24, 29 February 2008 (UTC)Reply[reply]

I do not quite understand what you mean by "not in line with other types of orbifolds discussed here". Are you making a judgement on the large body of work by Gromov, Haefliger, Bridson, Ballmann, Stallings and others? The article does not at present mention orbifolds in three dimensions in any detail (apart from what I have added myself). Unless you are extremely careful, you will produce an inconsistent article by forking. The constructions of orbifolds, orbispaces, etc, are absolutely essential for understanding triangles/complexes of groups. Do you have any paricular expertise on these topics or the extensive recent literature? The original definition of orbifold in this article (Haefliger's) was not quite right and Thurston's definition was substituted. At the moment I am not particularly in favour of forking. I would have a different view if the original article had been better written and not simply a not particularly informative list. Your statements to me seem to reflect a personal point of view, not borne out by recent mathematical literature. WP mathematics articles should reflect current mainstream knowledge, not prejudices or possible gaps in knowledge of editors. Gromov's work in the 80's changed the viewpoint on orbifolds and that is reflected in the current article. Mathsci (talk) 19:02, 1 March 2008 (UTC)Reply[reply]

This article kicks ass. Nothing needs to be removed or forked. (talk) 03:28, 4 March 2008 (UTC)Reply[reply]

I do not quite see the relevance of your statement

I would have a different view if the original article had been better written and not simply a not particularly informative list

to the question that I raised of whether or not triangles of groups and complexes of groups deserve their own articles. The definition of a triangle of groups, in particular, does not require the full strength of the notion of orbifold given in this article, and much of the theory can be desribed in a self-contained, elementary fashion. On the other hand, regardless of the level of generality ultimately adopted, it may be preferable to organize material in smaller parcels from the point of view of readibility. For example, while the notion of a connection in a vector or principal bundle clearly relies on the notion of the bundle, on Wikipedia this material is (in my opinion, rightly) distributed over several articles. Moreover, including a nontechnical description of triangles and complexes of groups, either as part of a "summary style" section here, or in the lead or introduction of a separate dedicated article, will greatly increase the benefit for potential readers (Gromov and Thurston already know the definition of orbifold, and Stallings and Haefliger already know what triangles and complexes of groups are, so we do not have to impress them with the text here). Arcfrk (talk) 04:41, 4 March 2008 (UTC)Reply[reply]


Incidentally, I note that you've chosen a personal attack over a substantative answer to points 1 and 3 (I address your response to point 2 in more detail in a separate section below).

Wikipedia is a collaborative project.

I thought I should mention it, since I have repeatedly seen you react in a hostile way to any editors who have made any contributions interfering with your own. The anonymity of Wikipedia and certain laxness in enforcing the rules of civil behaviour gives you a lot of opportunities to harrass and put down other editors and generally behave in a way not befitting someone implicating an editing license from Gromov and Stallings. However, this is an illusory freedom, and you would be well advised to pay attention to the goals of Wikipedia and to conform to the standards of behavior here (which are not very much different from common standards of behavior for responsible adults). I hope that you understand my concern for maintaining friendly environment conductive to work on collabaratively expanding and improving Wikipedia. Arcfrk (talk) 04:41, 4 March 2008 (UTC)Reply[reply]

What is an orbifold?[edit]

MathSci has insinuated above that the main place of orbifolds within "current mainstream knowledge" and "borne out by recent mathematical literature" is confined to "large body of work by Gromov, Haefliger, Bridson, Ballmann, Stallings and others" based on "Gromov work in the 80's". Recent mathematical and physical literature certainly includes other occurrences of orbifolds, with rather different flavor, and some of them have been already mentioned in the lead: orbifolds in symplectic geometry, algebraic geometry, and string theory immediately come to mind. As a matter of fact, on March 3, 2008, MathSciNet had the following distribution of articles with the word "Orbifold" occuring in the title according to their primary MSC numbers:

  • 81 (Quantum theory): 600
  • 57 (Manifolds and cell complexes): 130
  • 83 (Relativity and gravitational theory): 53
  • 14 (Algebraic geometry): 52
  • 58 (Global analysis, analysis on manifolds): 49
  • 53 (Differential geometry): 46
  • 32 ( Several complex variables and analytic spaces): 31

and even lower numbers in other subject areas. While this can only be considered a rough guide to the relative significance of orbifolds for various disciplines, it does point out towards a clear trend in recent literature to predominantly associate orbifolds with string theory and related subjects, not with topology and combinatorial group theory (which may have been the case in the 1970 and 1980s). Arcfrk (talk) 04:41, 4 March 2008 (UTC)Reply[reply]

Please avoid the use of uncivil words like "insinuate". You have given a completely incorrect interpretation of what I wrote, which seems unhelpful. Most importantly you have not produced extra definitions/applications of orbifold(s). If you are referring to the Deligne-Mumford quotient X//G in algebraic geometry/symplectic geometry (eg from the work of Kirwan, Jeffrey, Woodward, Meinrecken et al), why not say so? This is related to orbifolds, but only tangentially; it includes the theory of the moment map (symplectic quotient or "Marsden-Weinstein reduction"). But if you don't even make reference to locally closed orbits, stability/semistability, symplectic reduction, or actions of algebraic/reductive complex Lie groups, how can anybody have any idea at all what material you are discussing?
It was you that wanted to remove the orbifold material of Gromov and his school from the current article by making an unwarranted fork: I objected because there was no mathematical justification. There is no need to used emotive terms here (elsewhere you used emotive words like "snide" and "condescending") instead of discussing specific mathematical issues. Have you checked the articles on the moment map and the symplectic quotient? They have the failing that they do not mention the link between the symplectic reduction (for the compact group) and the algebraic geometric quotient (for its complexification). I'm not at all sure that this article on orbifolds is the place to address that particular problem, although obviously, as I have written below, the necessary material should be written somewhere on wikipedia (starting with Francis Kirwan's thesis for example) and amplified on here with wikilinks and examples. When unaccompanied by any mathematical discussion, lists from mathscinet are unhelpful and make it look as if your trying to prove some (non-mathematical) WP:POINT. If you have other applications in mind, other than what I have already mentioned here and below, please make concrete suggestions, or better still just add material to the WP elsewhere and/or here . That is exactly how I came to add a significant amount of material on examples to the article on buildings: that required a lot of hard work (in that case actions of p-adic reductive groups were involved). Mathsci (talk) 09:36, 11 May 2008 (UTC)Reply[reply]
The concept in physics as described in the article in the section of the article on CFT (originally fixed points of vertex algebras, now widely used in D-branes, M-theory, "swampland", etc) is different from the mathematical one. That concept accounts for the vast majority of references. As for the rest, it is surely deprecated to add vague statements in the lede undiscussed in the main article. Add the detailed content (yes it takes time!) and then record it in the lede, which is only intended to be a summary of the material in the main part of the article. If there is extra material, by all means it should be unearthed and included in new sections. Thus finding new material to be added (eg on Seifert fibre spaces or symplectic geometry) is only to be encouraged. However, making disparaging remarks about sourced material on orbifolds seems to serve no purpose at all. The subject classifications of mathscinet give no hint of which part of the mathematical aspects of orbifold theory are being studied. I put in orbifold as a search term "anywhere" on mathscinet and came up with 3072 citations. However 600 + 130 + 53 + 52 + 49 + 45 + 31 = 960 is considerably less than 3072. There are 1064 articles with orbifold in the title and 93 references just to V-manifold (did Arcfrk look for these?). Then there is all the work on compactifications of moduli space (Borel, Serre, etc), which might not explicitly use the word orbifold, but often involves orbifolds. I think mathematicians should add any scholarly content that reflects current knowledge. (I'm not sure that moduli spaces or geometric invariant theory are properly covered at the moment on the wikipedia, but the geometric quotient of G.I.T. is clearly a topic that could be referred to in the current article.) Arcfrk himself made a value judgement, which did not seem to reflect recent mathematical activity, as represented eg in the latest ICMs, another reasonable way to gauge progress; his comments did not seem particularly helpful. Any additional material on topics such as Borel-Serre compactification, the relation to Poincare's work, Seifert fibre spaces, Satake's index theorem, etc, would be fantastic. Adding this is a more profitable exercise than (a) compiling statistical lists of debatable relevance or (b) making complaints on the WikiProject Mathematics page. It seems advisable to concentrate on adding actual detailed mathematical content outside the lede, only mentioning it in the lede when it's been written (or if there is some plan to write it). Mathsci (talk) 03:37, 11 May 2008 (UTC)Reply[reply]
On Mathscinet, I just tried anywhere=orbifold and MSC primary=14 (algebraic geometry) and got 192 entries. 52 of these have orbifold in the title. As he/she wrote, Arcfrk just looked for orbifold in the title. Making a search on the title alone in the post 1940 database of all published mathematical articles clearly will miss the vast majority of articles and not provide helpful statistics. Please learn to use mathscinet more efficiently, Arcfrk :) Mathsci (talk) 03:54, 11 May 2008 (UTC)Reply[reply]

Focus on basic orbifold stuff[edit]

This page is a mess. Its main problem is that it has forgotten that it is but one article in an encyclopedia, and it is trying to do too much. We have stuff about orbihedra, orbispaces, triangles of groups, etc. but very little on the basic orbifold concept, as originally set forth by Thurston. Indeed, all these topics I mentioned come before some simple 2-dimensional examples are explained. Imagine how ridiculous it would be if the manifold article was mostly about more general theories of manifold type objects and explained at the very end some basic examples of manifolds. I think this page should focus on explaining Thurston's definition of orbifold (which means the string theory stuff should get its own page too), and go through some basic examples in low dimensions. This way people reading articles on various topics such as wallpaper groups or geometrization theorem can come here and get a readable introduction to the basic topics. The more advanced topics would be well-suited to separate articles. --C S (talk) 03:33, 11 July 2008 (UTC)Reply[reply]

There is alas no such thing as "basic orbifold stuff", since it is a ragbag of different topics by different groups of mathematicians and physicists. I completely agree however that, if other articles/content can be added to WP on (a) orbifold index theorems and (b) the use of orbifolds in the geometrization programme, then this article could be transformed into a short summary of the themes leading onto 5 or more other articles. Please go ahead and add content on 3-manifolds. I already appealed on this page for editors to do exactly that in Novermber 2007 here. As far as I recall, I had to include a brief description of the results of Michel Boileau at al, even though this is far from my expertise. (I do in fact know the CFT stuff quite well, although more from the point of view of mathematical physics.) I would suggest a separate article on 3-dimensional orbifolds. The present material does contain several separate themes which could usefully be broken up into separate articles (index theorems for orbifolds, 2-dimensional orbifolds (made into an article rather than a list), negatively curved orbihedra, orbifolds in string theory and 3-dimensional orbifolds) with this article as a linking article, giving basic definitions (orbifold and orbifold funcdamental group) and very brief summaries of the other themes. At this stage it would be probably a good idea if someone actually wrote the "3-dimensional orbifolds" article. After that, the future of the whole article could be rediscussed and a proper linking article made. The material on orbihedra is coherent and explains the whole circle of ideas to do with "triangles of groups" from the late 1980s and early 1990s. There is also at present no article on index theorems for orbifolds: it would not be hard to write a short article (it is related to later localisation theorems in equivariant K-theory and is part of my own expertise). So please add the extra content here and/or elsewhere. That seems the first priority. Cheers, Mathsci (talk) 12:32, 11 July 2008 (UTC)Reply[reply]
I agree with C S that "This page is a mess." I came here to find out what an orbifold is, in a general way. The introduction is a congeries of details that belong in the body of the article. I'm a mathematician and I can't understand a word, even though I know what a manifold is, what a group action is, etc. I wish I could do a rewrite but I can't extract enough meaning from this article. Please, someone who knows, try to write a real introduction! It would repay the effort to find an intuitive way to explain what an orbifold is, without technical details, and hold the precise statements, including the exact differences between various definitions, for afterwards, in new sections. I'm sure it can't be non-technical, but can't it be less specialized? Thank you. Zaslav (talk) 05:16, 1 April 2011 (UTC)Reply[reply]

is a manifold then a "trivial" or special case of orbifold?[edit]

That is, if you quotient an open subset U of a manifold M by the identity action, you get U again, so I guess M is an orbifold? At least that's my current (mis?)understanding. But I've often thought I've understtod things. If it is indeed true, it should be stated. Although article does say orbifold is generalization of manifold, stating the "converse" would help me and possibly others learn it more easily. I don't think the article needs to be an IQ test, please consider emphasizing main things for me and possibly others to remember and think about. Thanks, Rich Peterson199.33.32.40 (talk) 01:23, 9 December 2010 (UTC)Reply[reply]

'Ad' notation[edit]

Does anyone know what the 'Ad' s that appear in some of the formulas are supposed to mean? Based on context I'm guessing it's the adjoint of a matrix, but not everyone would think of that immediately, given that matrices are referred to as 'members of a finite group acting linearly on R^n'. If anyone can confirm this, it should be mentioned in the article. Cyrapas (talk) 14:05, 17 April 2013 (UTC)Reply[reply]

Some things that I and possibly other readers would want to know, that may be too obvious for[edit]

the knowledgeable people who wrote the article to realize it's not obvious to others(After all, if a person is thinking about something all the time, he may forget it's not second nature to someone else): 1) What is a "point" in that topological space that is called an orbifold? Is it an equivalence class of points from the original manifold? Like the orbit of a point under the action of the group G? 2) Supposing the original manifold to have a distance defined on it, does the resulting orbifold have a distance defined on it that is naturally related to the original distance? 3) Can projective spaces be thought of as orbifolds of R^n with the group of nonzero scalar n by n matrices,? If not, is it only because the group of scalar matices is infinite? What about if the string theory of orbifold is used instead? 4) Is there a functor that could be called "orbifoldization" on pairs of (M,G) where G is the group and M is the manifold resulting in the orbifold O(M,G)? What would the adjoint functor be? 5) Or am i just way off and don't get it at all? 6)When, if ever, is an orbifold still a manifold if G is nontrivial?... With sincere thanks, Rich Peterson76.218.104.120 (talk) 04:59, 20 August 2013 (UTC)Reply[reply]

The orbifold begins by dismissing the formalism of the n-tuple and its signature a priori. But why - because you cannot think how to define a distinct point-wise limit?

Since the orbifold is a manifold with an induced structure, you must show first that you are not talking about a structure on R induced as Borel sets by the point-wise limit of the n-tuple ring signature.

The absurdity is clear in the application to music: What kind of topology can be defined by points that are not distinct?

For example, the guitar tuning is a collection of intervals that define a Borel set, such as (0 5 5 5 4 5). The tuning is a precise point. Now, can you show that the "orbifold" is not simply a Borel measure induced in the octave metric space by the guitar tuning ring signaure whose lattice is the union and intersection of the guitar strings? The orbifold is just an induced Borel set on the Baire metric space.

Better idea: The affine sets in music cover the projective space completely.

You should eliminate the section on music. If you are implying music is a torus, that is absurd since the system fundamental cannot have 2 independent generators (in spite of Euler's tone net assertion). A torus cannot shrink to a point but music topology must.


I was surprised that stacks are never mentioned in the article. I thought it was now generally understood that this is the correct language to work with orbifolds. — Preceding unsigned comment added by 2A02:1810:9529:FA00:4A9:3E0F:F05C:B7F7 (talk) 21:32, 24 March 2018 (UTC)Reply[reply]