Why study orbifolds? [closed]

Question

Question is as in the title.

Why study orbifolds?

I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. Groupoid) $\mathcal{G}$ and a homeomorphism $|\mathcal{G}|\rightarrow X$, where $|\mathcal{G}|$ is orbit space of the groupoid $\mathcal{G}$.

Let $G$ be a Lie group acting smoothly on a manifold $M$. If one further assumes the action is proper and free then, the quotient space $M/G$ has a manifold structure.

One reason why I find orbifolds interesting is, loosely, if I ignore the condition that the group $G$ (compact Lie group) freely and impose that it only acts almost freely, then there is still some interesting structure on the quotient space $M/G$, i.e., of orbifold.

I don’t know any other places where one would see Orbifolds and how it would be interesting.

Correct me if I have a wrong understand and share your opinion of how to see an orbifold.

Edit : Wikipedia page does not say anything about groupoid way of looking at orbifolds. I did not mean to ask for references for orbifolds or groupoids. It is about how do you explain others what orbifolds are and how they occur naturally and what tools do we use to study thei geometry.

Answer 1

I can not say why one studies orbifolds (or e.g. why one studies math at all). However, I can try the approach which might convince your funding agency: There are tons of interesting examples of how orbifolds arise in "applications" (= mathematics):

1. The quotient spaces appearing in symplectic reduction are not always manifolds. If they fail to be manifolds, then one can (also not always but often enough) give them the structure of an orbifold (and then hope to do differential geometry on them), see Ana Cannas da Silvas's notes for some interesting examples: da Silvas - Lectures on symplectic geometry.

2. If one is interested in shape analysis (see Bauer, Bruveris, and Michor - Overview of the geometries of shape spaces and diffeomorphism groups), one wants to study Riemannian geometry on quotients of the form $\operatorname{Imm}([0,1] , \mathbb{R}^n) / \operatorname{Diff}_+([0,1])$ (Immersions of the interval into $\mathbb{R}^n$ mod the orientation preserving diffeomorphisms of the unit interval). Unfortunately this is not a manifold as there are immersions which may have a finite stabiliser subgroup under the reparametrisation action of $\operatorname{Diff}_+ ([0,1])$ (this is a result of P.W. Michor and collaborators, see V. Cervera, F. Mascaro, and P. W. Michor. The action of the diffeomorphism group on the space of immersions. Differential Geom. Appl., 1(4):391–401, 1991 (MSN)). So in essence, these spaces are "infinite-dimensional orbifolds" (in shape analysis this is immediately disregarded as one then concentrates on the open subset of elements with trivial stabiliser).

3. Orbifolds appear naturally in questions connected to foliation theory (see, e.g., Moerdijk/Mrčun: Introduction to Foliations and Lie Groupoids (MSN)).

4. Thurston studied them in his work on geometrisation of $3$-dimensional manifolds (see Thurston - Geometry and topology of 3-manifolds)

Though this is by no means an exhaustive list, I hope that there is some example you find interesting enough to justify interest in orbifolds.

Answer 2

I would like to propose an answer to this question, since 15 year ago I was asking it to myself and was thinking that orbiolds are useless. I read your question (maybe wrongly) as a question in mathematical psychology (or just in psychology). At the present moment I use orbifolds very often. And there was a turning point for me, when I understood one particular statement about orbifolds that made geometric sense to me.

Statement. An orbifold of negative curvature is a global quotient.

For some reason, once I got this statement, I decided that orbifolds are cool. But I know that this is very personal, so I don't expect that this particular statement will necessarily be interesting for you.

Also, there is a very nice example of an orbifold mentioned Thurston's notes on 3-manifolds - the Barber shop.