Talk:Orbifold

message: 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)
thread_title: Stub

message: 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)
thread_title: Differentiable manifolds?

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title:  Linear actions

message: 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)
thread_title:  Homogeneous space 

message: 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)
thread_title: Fundamental group and Euler characteristic

message: 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)
replies: text: I have now made these changes and will later add some more references. --Mathsci (talk) 13:17, 22 November 2007 (UTC)
thread_title:  Definition of orbifold 

message: 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)
thread_title: Use of orbifolds in the geometrization of 3-manifolds

message: 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)
replies: text: 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)
thread_title:  properly discontinuous 

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title:  Triangles of groups and Complexes of groups 

message: 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)
thread_title:  Apropos 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Focus on basic orbifold stuff 

message: 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)
thread_title: is a manifold then a "trivial" or special case of orbifold?

message: 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)
thread_title:  'Ad' notation 

message: 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)
thread_title:  Some things that I and possibly other readers would want to know, that may be too obvious for 

message: 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)
thread_title:  Stacks