Looking for an introduction to orbifolds

Question

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.

Answer 1

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).

Answer 2

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.)

Answer 3

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

Answer 4

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.

Answer 5

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).

Answer 6

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.

Answer 7

Since there is still room for some additional references I would like to mention the following list:

1. P. Scott -The geometries of 3-manifolds
2. W. P. Thurston - The Geometry and Topology of Three-Manifolds
3. J.E. Borzellino - PhD Thesis
4. I. Satake - On a generalization of the notion of manifold
5. J. Ratcliffe - Foundations of Hyperbolic Manifolds
6. M. Boileau, S. Maillot, J. Porti - Three-Dimensional Orbifolds and their Geometric Structures
7. B. Kleiner, J.Lott - Geometrization of Three-Dimensional Orbifolds via Ricci Flow
8. D. Cooper, C.D. Hodgson, S.P. Kerckhoff - Three-dimensional Orbifolds and Cone-Manifolds

Reference 1 provides an overview of the topic and is a complete (first) introduction to orbifolds (mostly topological). References 3, 5, 8 provide supplementary material especially in terms of the Riemannian Geometry of Orbifolds (more geometric approach).

Note: Some of the references were already mentioned in other answers but I include them also here for completeness and convenience.

Answer 8

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.

Answer 9

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

Answer 10

There's another reference I'd like to promote:

Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach, by Coufal, Pronk, Rovi, Scull, and Thatcher, in Women in topology: collaborations in homotopy theory, 135–166,Contemp. Math., 641, Amer. Math. Soc., Providence, RI, 2015.

Many of the other references mentioned either (a) take a naive view of orbifolds, which do not allow one to define "maps" between orbifolds in a sensible or (b) dig into different possible definitions, substantially complicating the picture.

If you define an orbispace as a proper étale topological groupoid, the definitions are all pleasant, and in fact familiar to topologists via thinking about smooth manifolds as defined by charts. You do need to make peace with the category being fundamentally a 2-category. This reference straightforwardly lays out that point of view, with clear illustrations and examples.

Answer 11

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.

Answer 12

Eugene Lerman’s Orbifolds as stacks? discuss about orbifolds from point of view of Lie groupoids/Differentiable stacks.

The user dvitek mentioned the same paper in a comment to some answer.