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- Thread starter scottbekerham
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mathwonk

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

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Thanks , but this does not answer my question .

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mathwonk

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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?

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mathwonk

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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.

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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.

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mathwonk

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These are all orbifolds.

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

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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

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

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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.

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mathwonk

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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.

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