What are the properties of orbifolds and how do they differ from manifolds?

[scottbekerham]

What does orbifold means? The wikipedia article says it is a generalization of manifolds which looks like a quotient space locally . So if my understanding is correct , If R1 is the Euclidean 1-space and A is an element of R1 and we identify each A with A+2*PI we get a circle so there is an orbifold that looks locally as a circle . Is this correct?

 

[mathwonk]

a circle is a manifold. every manifold is an orbifold.

 

[scottbekerham]

Thanks , but this does not answer my question .

 

[mathwonk]

If your question was " is there an orbifold that looks locally as a circle?"

the answer is yes, and it does seem to be answered also by my previous answer.

I.e. every thing that is locally a circle is a manifold, and every manifold is an orbifold. so yes, anything that looks locally like a circle, is a manifold, and hence also an orbifold.

Indeed you have just proved by your construction that there is an orbifold, even one which is a global quotient, that looks locally like a circle.Am I still misunderstanding your question?

 

[mathwonk]

the example you gave is not what is meant by a group quotient giving an orbifold. your group has no fixed points, so you are getting a manifold. There may be some special meaning to the orbifold you defined that differs from its normal manifold structure, but to me it is just a manifold, more usually defined as a "trivial" orbifold.

The concept of orbifold is usually used to define a structure on a space which is not everywhere a manifold, by showing how the non manifold points can arise from taking a quotient of a manifold.

the construction is meant to widen the realm of spaces that can be studied using manifolds. I.e. an orbifold is locally a manifold plus a finite group action. The non manifold points are obtained when the group action has fixed points, i.e. points that are fixed by non trival group elements.

 

[Vargo]

Hello.

Let me illustrate what mathwonk said by giving an example of an orbifold that is not a manifold. Take the unit disk in R2 and let G be the group of rotations generated by a 120 degree turn. The disk modulo this group is a cone, which is not smooth at the vertex. Hence this is not a manifold. However, locally, it looks like a manifold almost everywhere, which is a typical feature of orbifolds.

 

[mathwonk]

Here is a reference for quotient singularities of surfaces, by an expert.

These are all orbifolds.

http://homepages.warwick.ac.uk/~masda/surf/more/cyclic.pdf

 

[Jamma]

Hello.

Let me illustrate what mathwonk said by giving an example of an orbifold that is not a manifold. Take the unit disk in R2 and let G be the group of rotations generated by a 120 degree turn. The disk modulo this group is a cone, which is not smooth at the vertex. Hence this is not a manifold. However, locally, it looks like a manifold almost everywhere, which is a typical feature of orbifolds.

This is a very fundamental example. Strictly what you are saying isn't quite correct - this quotient space still is a manifold, since it is locally homeomorphic to the disc everywhere (even the cone point!).

Usually in orbifold theory though, you keep track of these cone points, so you have more information than just the topology of the orbifold. I believe that keeping track of these things can give you different invariants that you associate to the orbifold which reflect its structure in a more honest way (or, I should say, more useful way for some applications). Also note that there are different ways of defining orbifolds, with many of the more modern definitions (this is stretching my knowledge at this point) fitting into some of the more modern work around groupoids, sheave theory and so on which have some very neat, if rather abstract, ways of associating invariants to these gadgets.

 

[Vargo]

I guess I should have specified that I meant smooth manifold with smooth structure somehow inherited from the disk. There is an interesting discussion at
http://mathoverflow.net/questions/19530/what-is-meant-by-smooth-orbifold
where they discuss, among other things, the subtleties of defining what would be meant by the "tangent space" to an orbifold at the cone point.

 

[Alesak]

So if I have half plane H², can I regard it as a orbifold that is localy similar to ℝ², except at points at x-axis, where it is localy similar to ℝ² with group action that flips y-axis?

 

[mathwonk]

the theory of group quotient singularities of a complex curve says that if the fixed locus has complex codimension 1 then there is actually no singularity of the quotient.

thus group quotient, i.e. orbifold, singularities do not occur until you are dealing with quotients of C^2 rather than C^1.

Even then the fixed locus (of a holomorphic action) has to be discrete.