# Questions tagged [orbifolds]

The orbifolds tag has no usage guidance.

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### Geodesic representatives in the orbifold fundamental group

Does every element in the orbifold fundamental group $\pi_1^{orb}(X,x)$ of a closed hyperbolic 2-orbifold $X$ admit a unique geodesic arc representing it?
Does every free homotopy class in $X$ admit ...

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### Flat spherical orbifolds

What is known about existence and classification of flat spherical orbifolds?.
Here I mean orbifolds that admit a flat Riemannian metric (Euclidean orbifolds) and whose underlying topological space (...

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### Global symplectic reduction

Let $M$ be a symplectic manifold equipped with a hamiltonian action of a compact Lie group $G$ with moment map $\mu\colon M\to \mathfrak g^*$. Assume $c\in \mathfrak g^*$. Then the symplectic ...

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201 views

### Does geometrization of Alexandrov 3-spaces follow from that of 3-orbifolds?

Galaz-Garcia and Guijarro proved the geometrization of closed (compact, boundaryless) Alexandrov 3-spaces. Part of the strategy was to use the so-called ramified double cover $\tilde{X}$ of the space $...

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212 views

### If two group actions lead to the same orbifold, are they conjugate?

In The Geometry and Topology of Three-Manifolds, Thurston says: "In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we ...

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144 views

### Equivariant De Rham theorem for orbifolds

Recall that the "classical" equivariant De Rham theorem states that, for a compact Lie group $G$ acting on a compact smooth manifold $M$,
$$H_G^*(M,\mathbb{R})\cong H^*(\Omega_G(M)),$$
where $H_G^*(M,\...

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129 views

### Cohomological description of gerbes over stacks (orbifolds)

When understanding about gerbe over a manifold $X$ from Hitchin - Lectures on special Lagrangian submanifolds it is said that
We are basically in gerbe territory (for smooth manifolds) if any one ...

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86 views

### Idea behind definition of classifying space over an orbifold

Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction.
Definition : Let $X$ be a locally compact ...

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147 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\...

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427 views

### Why study orbifolds? [closed]

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. ...

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190 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 $\mathcal{H}$...

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108 views

### gerbes over orbifolds

I am reading Orbifolds from Ieke Moerdijk’s paper https://arxiv.org/abs/math/0203100
In that they define orbifold as some sort of Lie groupoids. In that the author concludes the paper by defining ...

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188 views

### Do regular points of an orbifold form a connected set?

First, a bit of background on orbifolds:
Let $X$ be a connected (effective) orbifold. To every point $x \in X$, we associated a group $G_x$ called the isotropy group. The singular locus $\Sigma X$ ...

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### Description and counting non-crystallographic discrete subgroups of Euclidean isometries

A few days ago I have posted this question on Math.Stackexchange there was thankfully a response from YCor and he recommended reposting it here in order to get more details
An n-dimensional ...

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246 views

### How singular is the metric on an orbifold

I am reading some stuff on orbifolds. I am particularly interested in the metrics on orbifolds. The famous example of one orbifold is the "American football", which is $\mathbb{S}^2$ quotient by the ...

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### Can we perturb a surface away from an orbifold point?

Let $X$ be a smooth, compact, orbifold of dimension $4$, where the stabilisers are only allowed to be cyclic groups. Let $p \in X$ be an isolated orbifold point (i.e. the orbifold chart about $p$ ...

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### Orbifolds with maximal diameter

Orbifolds with positive curvature and maximal diameter are investigated in this article, by J. Borzellino. Theorem 1 of the article states:
Let $\mathcal{O}$ be a complete $n$-dimensional ...

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### Minimization of the volume of the image of space-filling convex polytopes under similarities

Suppose $A:\mathbb{R}^n \to \mathbb{R}^n$ is a similarity, given by $A(x) = \lambda Ox$, where $\lambda > 1$ and $O$ is an orthogonal matrix (i.e., $A$ is a particular loxodromic repelling ...

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### Homotopy groups of smooth part of moduli space

Let $M_g$ be the moduli space of Riemann surfaces, as described for example in the book of Harris and Morrison - Moduli of curves. As a topological space, or better as orbifold, it has smooth points ...

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### Is there a pair of pants decomposition analogue for orbifolds?

The pair of pants decomposition is a useful tool is surface theory. Is there an analogous decomposition for orbifolds?
Thanks

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### Integer valued signature of $4n$ dimensional orbifolds

Let $M^{4n}$ be a smooth oriented $4n$-dimensional manifold without boundary. Then we have an intersection form in $H^{2n}(M^{4n},\mathbb R)$ and such a form has signature $(n_+, n_-)$.
Question. I ...

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207 views

### Maps to the universal punctured elliptic curve

I have just started reading Hain's paper On the Universal Elliptic KZB Connection. I am a bit confused about a comment made there about base points on orbifolds. I am still very new to the idea of ...

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### Cobordism/bordism group based on orbifolds with corners

We define a geometric homology group of a topological space $X$ as follows: the chain complex $C_{\bullet}$is freely generated by the maps $f$ from a compact oriented orbifold with corners $P$ to $X$, ...

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### Smoothing of a Kähler orbifold metric on a complex surface

Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb ...

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### 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 ...

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### Does any smooth oriented closed orbifold have a fundamental class

This thread:triangulation of orbifolds
has shown that any smooth closed orbifold has a triangulation. My further question is: if the difference of any two triangulations $P$ and $Q$ is a boundary of a ...

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242 views

### Does the torus $T^d$ 2-fold cover an orbifold $Q^d$ with underlying space $S^d$?

I asked the same question on math.stackexchange recently (https://math.stackexchange.com/questions/2134978/is-it-possible-to-orbifold-torus-td-into-a-sphere-sd-using-mathbbz-2), but it didn't receive ...

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176 views

### What is a proper n-etale morphism?

Let $Y$ be a complex algebraic variety, and let $n\in \mathbb Z_{\geq 1}\cup \{\infty\}$. How do I think about a proper $n$-etale morphism $X\to Y$?
If $n=1$, I think this should be a finite etale ...

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### Symplectic orbifolds

I will start by saying that I am not a symplectic topology. However, in my research I now have on my hands a symplectic 4-orbifold, which I would like to understand better. Certain results for ...

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### 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 ...

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### Orbifold of the three-sphere (and lens spaces)

Think of the three-sphere as given by $\lbrace|z|^2+|w|^2=1, \;z,w\in \mathbb{C}^2\rbrace$. We can regard it in terms of Hopf coordinates
\begin{align*}
z&= \cos(\theta/2)e^{i(\phi+\psi)}\\
w&=...

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### Einstein's field equation on orbifolds

I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).
Here, by an orbifold I mean the "stacky" quotient of, ...

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### Embedding of Gorenstein orbifold as a hypersurface

I am trying to understand if three complex dimensional orbifold singularity $\mathbb{C}^3 / \Gamma$ can be embedded as hypersurface in $\mathbb{C}^4$. The condition of being Gorenstein and having ...

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### Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.

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### Equivariant and orbifold Chern classes

Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...

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### 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 ...

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### Submanifolds invariant under subgroups with identical quotients

given a smooth manifold $M^n$ and a finite group $G$ acting smoothly and effectively, let's consider two (embedded) $k$-dimensional submanifolds $N_1,N_2\subset M$ and two subgroups $H_1,H_2\subset G$ ...

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### 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 ...

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695 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.
( ...

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### manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...

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### Rationally graded singular cohomology

In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the orbifold cohomology group of an orbifold $X$ by
$H^d_{orb}(X)=\bigoplus_{(g) \in T} H^{d-2\iota_{(g)}}(X_{(g)})$
...

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### Orbifold singularities over a smooth map

I recently starded studying the book "Orbifolds and Stringy Topology" by Adem, Leida and Ruan and I'm trying to see if there is a relation between the singularites of two orbifolds when there is a ...

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346 views

### Is there a Riemann existence theorem for orbifolds?

For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...

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### Orbit spaces of crystallographic groups

In their paper "On Three-Dimensional Space Groups", Conway et al. write
Although this paper was inspired by the orbifold concept, we did not need to consider the 219 orbifolds of space groups ...

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### Properties of the induced map between inertia stacks

Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to I\...

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156 views

### étalé space of sheaves on a differentiable stack

If $F$ is a sheaf on a topological space $X$, the well-known étalé space
contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is
isomorphic to the sheaf of sections of $\Gamma F$.
On ...

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286 views

### Real weight modular forms as sections of a line bundle

Background: I've been trying to read
Baily, Walter L., Jr.
The decomposition theorem for V-manifolds.
Amer. J. Math. 78 1956
and I have problems with the language used in the paper. Firstly I am ...

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### Topological invariants of toroidal orbifolds

Which are the most powerful topological invariants of toroidal orbifolds?
In particular I am looking for topological invariants of two-dimensional toroidal orbifolds such as $T^{2}/Z_{k}\times Z_{k}$ ...

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### Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on $\...

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### 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 ...