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Is there any source where the basic facts about orbifolds are written and proved in full detail? I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more complete and modern source.

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9 Answers 9

There is a very nice short paper by Andre Henriques: http://arxiv.org/abs/math/0112006 He explains different possible definitions of orbifolds and some relations between them. He also gives many good example.

Ieke Moerdijk also has a nice paper, but it is a bit longer and has less examples http://arxiv.org/abs/math/0203100

One should note that there are a few ways of thinking about orbifolds:

  1. As spaces which are almost like manifolds (ie instead of locally being R^n, they are locally R^n/G, G a finite group acting linearly).

  2. As a special kind of differentiable stack, equivalently they are Lie groupoids in which every point has a finite isotropy group.

The second way of thinking is the more modern approach and my references above are more in this line of thought.

In either way of thinking, they often arise as quotients X/G where G is a compact Lie group acting on a manifold X, with G acting locally freely (all stabilizers finite).

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A great place to start is Peter Scott's beautiful paper "The geometries of 3-manifolds".

It goes through tons of examples of 1- and 2-dimensional orbifolds; this includes those coming from spherical/Euclidean/hyperbolic triangle groups, and he gives a complete classification of 2-dimensional Euclidean and spherical orbifolds. Along the way he covers all the usual tools like orbifold fundamental group, van Kampen's theorem, and Euler characteristic. Finally he uses orbifolds heavily, in the context of Seifert fibered manifolds, to explain which manifolds admit one of Thurston's eight geometries, and how. The other topics touched upon in the paper (Lie groups, connections and holonomy, group actions, foliations, group extensions, etc.) are all well worth knowing, and presented very explicitly and clearly here.

(A scan is available from Scott's webpage, but it is oriented sideways and the filesize is large. Anyone with access should get the paper from the journal directly.)

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One standard and beautiful source is chapter 13 of Thurston's notes on 3-manifolds, which you may find here.

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MSRI updated website. Link no longer works. Any other place to find these notes? –  Max M Nov 5 '10 at 15:26
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Never mind. library.msri.org/books/gt3m –  Max M Nov 5 '10 at 15:28
    
Thanks, Max M.${}$ –  j.c. Nov 5 '10 at 15:33
    
Thurston worked just with effective orbifolds isn't it ? –  Zoran Skoda Nov 5 '10 at 19:50

There are some recent lectures by Michael Davis called "lectures on orbifolds and reflection groups." They are available on his webpage here

They seem good.

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The link seems to send one to here and not to Davis's webpage. ??? –  José Figueroa-O'Farrill Nov 27 '09 at 5:49
    
Thank you José. I've fixed it now. –  Grétar Amazeen Nov 27 '09 at 12:40

The reference I would use is Three-Dimensional Orbifolds and Their Geometric Structures by Joan Porti, Michel Boileau, Sylvain Maillot, Panorama et Syntheses 15 (2003). It contains references for what it doesn't prove, and is extremely readable. Another choice (which I like less) is Three-dimensional orbifolds and cone-manifolds by D. Cooper, C.D. Hodgson, abd S.P. Kerckhoff, MSJ Memoirs, 5. (2000).

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Since Ilya mentioned orbifolds in Physics, the earliest reference I am aware of is the 1985 paper Strings on orbifolds by Dixon, Harvey, Vafa and Witten. This is one of the landmark papers of the so-called "first superstring revolution" because of the realisation that one could get a "realistic compactification" (meaning a model with 3 generations of quarks and leptons) out of a $\mathbb{Z}_3$ orbifold of a six-torus. This orbifold admits a Calabi-Yau resolution.

It should be mentioned in that most uses of the word "orbifold" in string theory refer to global orbifolds, so riemannian manifolds of the form $M/G$ where $M$ is a riemannian manifold and $G$ a finite subgroup of isometries. In most applications, $M$ is actually a euclidean space or a torus.

The philosophy behind the use of orbifolds in string theory is that if one knows how to describe string propagation on $M$ then one knows how to describe it on $M/G$, but there are very few $M$ for which one knows how to do this. As the above $T^6/\mathbb{Z}_3$ example shows, smooth Calabi-Yau (and not just CY) manifolds may have orbifold points in their moduli space and it is at those points that one can obtain information from string theory calculations.

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Eh, it depends on what you need to know about orbifolds.

Here are the basic facts you certainly need to know: orbifold is a smooth manifold X together with a very good finite group G action — it should be similar to the free action, meaning that it may have some isolated fixed points, but no more than that.

The idea of orbifold is that it allows to do computations with singular X/G as if it was smooth. E.g. you can define a differential form on an orbifold — that would be a G-invariant form on X, you can define Euler characteristics, etc.

Now I'm not sure where you wan to go from there. Since orbifolds first appeared in physics (there's a more abstract mathematical notion of stack) you might want to read some physical papers to learn why they are relevant to string theory.

Alternatively, you can search for orbifold on arxiv and find many math papers that prove some specific and sometimes generally interesting things about orbifolds.

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The definition you've given certainly gives some orbifolds, but normally one allows more general things. In the normal definition, an orbifold needn't be a global quotient of a smooth manifold, rather each singular point should have a neighbourhood which is homeomorphic to the quotient of R^n by the linear action of a finite group. Also one normally allows non-isolated fixed points. –  Joel Fine Oct 23 '09 at 8:52
    
Yes, the ones I gave are the simplest ones, and I think this preserves the basic idea --- orbifolds are things that are sufficiently close to smooth manifolds. –  Ilya Nikokoshev Oct 23 '09 at 15:56
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I don't think that orbifolds appeared first in Physics, even though they perhaps have become popular because of Physics. Satake's paper certainly predates any use of orbifolds in Physics that I'm aware of. Orbifolds entered the Physics collective consciousness in the mid 1980s, when it was realised that string theory, unlike QFT, can be consistently defined on orbifolds. This gave rise to the abstract notion of an "orbifold conformal field theory". –  José Figueroa-O'Farrill Nov 26 '09 at 12:40
    
Yes, they started appearing significantly in physics in mid 1980-s with orbifold models in CFT and also with seminal papers of Vafa et al. on strings on orbifolds. –  Zoran Skoda Nov 5 '10 at 19:52

In terms of introductory material, Peter Scott's paper is pretty good, although as noted above, it is less modern in its exposition. There is also the first part of the book "Orbifolds and Stringy Topology" by Adem, Leida and Ruan which covers both approaches to some degree. It also covers the orbifold cohomology ring, which depending on your interests, might be of use.

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A word of warning: the "paths" on an orbifold are subtle.

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Indeed, I asked a question about this... mathoverflow.net/questions/20511/… –  j.c. May 29 '10 at 1:02
    
More generally, the maps between orbifolds are subtle. This is related to the fact that orbifolds form a 2-category (there are arrows between arrows). –  André Henriques Sep 7 '10 at 13:48

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