# Questions tagged [orbifolds]

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140
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### Direct limit of the orbifolds $\mathbb{R}^n/S_n$? as $n \to \infty$?

While studying the particle interchange symmetry of the Bosons in physics, I have arrived at the notion of $\mathbb{R}^n$ with coordinate interchange symmetry.
That is, I take the quotient of $\mathbb{...

3
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1
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### The difference between two lifts along an orbifold chart

The motivation for this question comes from the study of lifts of an orbifold chart, which is simplified as the following:
Suppose that $ U $ is an open connected subset of $ \mathbb{R}^n $ and $ G $ ...

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### Is there a relation between symplectic toric orbifolds and semi-toric systems?

So recently I have been studying semi-toric systems which are a generalization of toric symplectic manifolds and allow for the presence of focus-focus fibers. These were proved to be classified by $5$ ...

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### Is there a contractible hyperbolic 3-orbifold of finite volume?

Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...

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### Difference between affine quotient variety and a global quotient orbifold

Given a smooth affine variety $X$ and a finite group $G$ acting by automorphisms on $X$, the quotient space $X/G$ has the structure of an affine variety which is in general not smooth. However, in the ...

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### Can a compact good orbifold be realized as a global quotient of a compact manifold?

Let $\mathcal{O}$ be a compact good orbifold, where we understand a good orbifold to be an orbifold obtained as a global quotient $M/G$, where $M$ is a manifold and $G$ is a discrete group. Are there ...

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votes

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### In what sense is Bass-Serre theory the one-dimensional version of orbifold theory

The Wikipedia article on Bass-Serre theory claims that graphs of groups (in the context of Bass-Serre theory) "can be viewed as one dimensional versions of orbifolds." I hazily see a ...

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### Is there an extension of Ogg's results to surfaces of Genus 1

The first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero iff p is ...

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### 2-orbifolds that I expect to be hyperbolic, but they're nonnegatively curved

I'm considering some complex 1-dimensional/real 2-dimensional orbifolds that I expect to be hyperbolic. However, some of them seem to be Euclidean or spherical. Any thoughts what's going on here? Here ...

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### Isometric embedding of a 2-dimensional orbifold with constant curvature and three cone points

There are classical surfaces of revolution, shaped like footballs, that have constant positive curvature, except for their two cone points. How about such a surface with three cone points?
To give ...

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### Gluing of orbifolds

Suppose that $P$ and $Q$ are $n$-dimensional orbifolds, with boundaries. Suppose also that there is an isomorphism $f \colon \partial P \rightarrow \partial Q$ (as orbifolds). Is there a way to glue $...

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### Reference for foliation on orbifolds

Can anyone recommend a good reference for foliation on orbifolds ? Thanks!

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### Extension of an orbifold structure from punctured balls to balls

Let $\hat{D} := D \backslash \{0\}$ be a ball in $R^n$ with the origin $\{0\}$ removed. Assume that $\hat{D}$ has a structure as an orbifold (may be distinct from its standard manifold structure). Is ...

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### In what sense is an orbifold a DM stack?

My advisor mentioned in passing that orbifolds are Deligne-Mumford stacks, and I'd like to know in which sense this is true. The only reference I can find is this article (https://arxiv.org/abs/0806....

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### About the construction of the associated complex vector bundle of an orbifold one

My question has to do with the general construction that associates to each complex orbifold vector bundle $\mathscr E\rightarrow\mathscr X$ over an orbifold Riemann surface, a complex vector bundle $...

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### Almost free Lie group action

It's known that if a compact Lie group $G$ acts freely on a compact manifold $M$, then the orbit space $M/G$ is a manifold. If we only assume that $G$ acts almost freely (i.e. $G_x$ is finite for any $...

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### Existence of orbifold vertex algebras – current status?

Let the finite group $G$ act on a vertex algebra $V$. It is expected that there are certain vector spaces $V_g$ (with the structure of $g$ twisted $V$ modules), with $V_1=V^G$, and
$$V/G\ :=\ \...

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### When is a compact orbifold Riemann surface a global quotient of a Riemann surface

While reading the paper Seifert Fibred Homology 3-Spheres and the Yang-Mills Equations on Riemann Surfaces with Marked points by M. Furuta and B. Steer, I stumbled upon the following statement:
Any ...

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### An orbifold atlas of linear charts

It is well-known that, if $M$ is a smooth manifold and $G$ is a finite group of diffeomorphisms of $M$, then for each $x\in M$ fixed by the action of $G$ there exists a coordinate system in which $G$ ...

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### Higher homotopy groups of an orbifold

Given an orbifold $\mathcal{O}$, I have seen many ways to define the orbifold fundamental group:
Thinking of $\mathcal{O}$ as a groupoid $\mathcal{G}$, $\pi_1^{orb}(\mathcal{O})$ can be defined as ...

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votes

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### HNN-extension of the orbifold fundamental group

Let $\Gamma$ be a discrete subgroup of $\operatorname{PSL}_2(\mathbb{R})$. I know that $\mathbb{H}^2/\Gamma$ has an orbifold structure. Let $\gamma$ be a non-separating closed curve. If $\Gamma$ has ...

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### Product of good orbifolds [closed]

Let $M,N$ be two (very)good orbifold,(i.e. (finitely)covered by a manifold). The local coordinate are given by $U/ \Gamma_M$ and $V/\Gamma_N$
Then the coordinate charts $U\times V/\Gamma_M\times\...

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### Fundamental group of hyperbolic 2-orbifold

Suppose $\Gamma$ is a cocompact lattice of $PSL_2(\mathbb{R})$. Then $\mathbb{H}^2/\Gamma$ has a natural structure of orbifold. My questions are:
What is $\pi_1(\mathbb{H}^2/\Gamma)$?
What is $\pi_1^{...

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### Configuration space of 4 points as an orbifold

Setup: Consider the braid group $B_n$. One way to define this is as the fundamental group of the unordered configuration space $UC_n(\mathbb{C}) = \{\{z_1,\dotsc,z_n\}\subset \mathbb{C} \mid z_i \not= ...

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### Smooth rank one foliations with closed leaves

Let $F$ be a smooth rank one foliation on a manifold $M$. Suppose that all leaves of $F$ are compact (that is, circles). Then its leaf space (edit: when additional assumptions are taken) is an ...

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### How to judge whether an orbifold is good

My own case comes from dynamic system on compact complex manifolds. To be precise, let $M$ be a compact complex 3-dimentional manifold, $W^c$ a holomorphic foliation of M with 1-dimentional uniformly ...

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### Find an explicit quasi-smooth embedding $X_{38} \subset \mathbb P(5, 6, 8, 19)$

This question is not quite about research-level mathematics, so I apologize for bringing it here. I asked it in Math.SE first, but I got no answers, and only a suggestion to ask it here.
Consider the ...

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145
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### Orbifold vs étale fundamental group of complex ball quotient

Let $X$ be a quotient of the complex ball by an arithmetic group. How does the orbifold fundamental group of the complex points of $X$ compare to the étale fundamental group of $X$?

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### Metropolis-Hastings sampling as a group action

Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...

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### Condensed / pyknotic approach to orbifolds?

Does condensed / pyknotic mathematics afford an (yet!) another approach to orbifold theory?
Let me admit up-front that I don't know much about either condensed / pyknotic mathematics or about orbifold ...

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### Clarification about the process of naturally endowing a space with a Riemann orbifold structure supported on a sphere

I am having some difficulties understanding an argument in a proof. Here is an excerpt from Lyubich–Peters - Classification of invariant Fatou components for dissipative Henon maps, first geometric ...

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### Absolute and relative tilings of the hyperbolic plane

In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it).
The ...

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### A canonical map from a Euclidean cone-manifold $M^3$ to $\mathbb{E}^3/\mathrm{Hol}(M)$

Suppose we have a 3-dimensional Euclidean cone-manifold $M$—in my book that just means $M$ is a manifold whose geometry is constructed by gluing it out of Euclidean tetrahedra, with faces paired by ...

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### Pseudoreflection groups in affine varieties

Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result:
(C-S-T): Let $G$ be a ...

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### Equivalence between integrals over a reduced space

Context: I have been trying to understand this paper from Y. Cho and K. Kim. More precisely, a specific argument in Lemma 2.2 where they say the ABBV localization formula on an integral over a ...

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### About the orientation of index formula on orbifold

Let $X$ be a closed oriented orbifold with singularity $\Sigma X$. The singularity is defined as
$$ \Sigma X=\{(x)|~x\in X,~G_x\neq1\}, $$
where $G_x$ is the isotropy group.
For $u\in K_v(TX)$, as ...

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### A fiber bundle of the Euclidean space over an orbifold

Consider a fiber bundle $p: F\hookrightarrow
E \to B$, where $E$ and $F$ are smooth manifolds and $B$ is a smooth orbifold. More precisely, each point $b \in B$ has an orbifold chart $U=\tilde U/\...

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### Quotient of Euclidean space with maximal volume growth

Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.
If there exists a point $p \in O$ such that
$$
\lim_{r \to \infty}\frac{\text{...

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### The Fock space in Costello's paper "Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products"

Let $X$ be a smooth projective variety. In this Annals paper, Costello expressed the descendent genus $g$ Gromov-Witten (GW) invariants of $X$ in terms of genus zero GW invariants of the symmetric ...

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### Berglund-Hübsch-Hori-Vafa mirror symmetry is a ring isomorphism?

Let $W = \sum_{i=1}^{m} a_i \prod_{j=1}^{n} x_j^{b_{ij}}$ be a homogeneous polynomial of degree $d$ in $n$ variables. I focus on the $m=n$ case (invertible polynomial in the Berglund-Hübsch ...

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### Is a free and discrete group action on the plane a covering space action?

Let $\mathbb{R}^2$ be the plane, and let a group $G$ act on it with orientation preserving homeomorphisms, and assume that
every orbit of $G$ is a discrete subset in $\mathbb{R}^2$
$G$ acts freely: ...

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### Smooth sub-orbifolds in the language of stacks

In most geometric categories, "monomorphism" is too general to describe useful notions of "embedding". This is the case e.g. for schemes, complex manifolds, and differentiable manifolds.
So "embedding"...

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### What are orbifolds with corners?

What is the geometric definition of orbifolds with corners? Here “geometric" means that there is a definition in chapter 8 of the draft of Dominic Joyce's book D-manifolds and d-orbifolds: a theory of ...

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### Geometry of the irrational torus

One of the motivations of diffeology is to study singular spaces such as the irrational torus. The irrational torus $T_α$ of slope $α∈R∖Q $ as a diffeological space is given by the quotient space $ R/(...

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### What is the official definition of $\mathcal{M}_g$ as an orbifold, and how much can I ignore it?

There is a well-known description of $\mathcal{M}_g$ as $\mathcal{T}_g/\Gamma$ where $\mathcal{T}_g$ is the Teichmuller space and $\Gamma$ is the mapping class group. Teichmuller space is homeomorphic ...

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### How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?

My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...

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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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### 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 $...