Questions tagged [orbifolds]

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27
votes
4answers
4k views

What is meant by smooth orbifold?

There seems to be some confusion over what the tangent space to a singular point of an orbifold is. On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-...
3
votes
1answer
255 views

Proper and etale groupoid is locally a translation groupoid

I am reading Orbifolds as Groupoids: an Introduction by Ieke Moerdijk. In page $8$ when explaining local charts, it says the following : Let $\mathcal{G}$ be a Lie groupoid. For an open set $U\...
9
votes
2answers
2k views

Euler characteristic of orbifolds

Hello, Suppose $M$ is a compact oriented smooth manifold and $G$ is a finite group acting on it. Then it is well-known, although I have yet to find a proof or derivation of it, that the (normal ...
15
votes
10answers
3k views

Orbifold fundamental group in terms of loops?

In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...
8
votes
3answers
636 views

Conditions for underlying space of an orbifold $\Bbb T^n/\Gamma$ to be a sphere?

Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific ...
22
votes
5answers
4k views

How should one understand orbifold fundamental groups?

I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise ...
24
votes
3answers
1k views

What tools cannot work for orbifolds?

Consider all of your basic constructions/tools/theorems for manifolds: fundamental group, Euler characteristic, triangulations, orientation, smoothness, bundle structure, cobordisms, etc.. Viewing ...
9
votes
1answer
598 views

Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)

A quick Question: Is there some duality known between the quasi Hopf algebra $D^\omega(H)$ of a finite group $H$ to an orbifold model (such as SU(2)/$G$ or SO(3)/$G$ orbifold of some group $G$)? What ...
17
votes
1answer
1k views

Homotopy theory of topological stacks/orbifolds

Motivation $\newcommand{\T}{\mathscr{T}}$ I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its ...
23
votes
2answers
1k views

Is every rational realized as the Euler characteristic of some manifold or orbifold?

Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds. Then, if the answer is No, one can remove various conditions on the dimension, and allow non-compact ...
20
votes
2answers
1k views

What is an example of an orbifold which is not a topological manifold?

In Thurston's book The Geometry and Topology of Three-Manifolds it is proven that the underlying space of a two-dimensional orbifold is always a topological surface. Are there any easy examples of ...
7
votes
3answers
1k views

Is the Čech cohomology of an orbifold isomorphic to its singular cohomology?

Let $\mathcal{O}$ be a finite-dimensional, paracompact, Hausdorff, smooth (and compact, if it helps) orbifold. Is there an isomorphism between the real Čech cohomology and singular cohomology of the ...
6
votes
2answers
799 views

Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true: For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections. ( ...
4
votes
1answer
488 views

Intrinsic Characterization of when an orbifold (or more general stack) is effective?

Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie ...
3
votes
2answers
243 views

Necessity/Motivation for generalised homomorpisms

I am reading Ieke Moerdijk's article "Orbifolds as Groupoids : an Introduction". In that notes author defines a notion of generalized map between Lie groupoids. Let $\mathcal{G}$ and $\mathcal{H}$...