Questions tagged [orbifolds]
The orbifolds tag has no usage guidance.
128
questions
4
votes
1
answer
83
views
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 ...
1
vote
0
answers
186
views
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....
4
votes
0
answers
83
views
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 $...
3
votes
1
answer
100
views
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 $...
5
votes
1
answer
153
views
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\ :=\ \...
6
votes
1
answer
233
views
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 ...
4
votes
0
answers
108
views
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$ ...
5
votes
0
answers
119
views
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 ...
2
votes
1
answer
117
views
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 ...
2
votes
0
answers
66
views
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\...
3
votes
0
answers
66
views
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^{...
8
votes
0
answers
119
views
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= ...
8
votes
2
answers
222
views
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 ...
4
votes
0
answers
121
views
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 ...
6
votes
0
answers
151
views
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 ...
1
vote
0
answers
129
views
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$?
3
votes
0
answers
123
views
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 ...
9
votes
1
answer
915
views
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 ...
1
vote
0
answers
49
views
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 ...
4
votes
1
answer
143
views
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 ...
3
votes
0
answers
42
views
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 ...
5
votes
0
answers
178
views
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 ...
2
votes
0
answers
75
views
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 ...
1
vote
0
answers
47
views
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\}, $$
where $G_x$ is the isotropy group.
For $u\in K_v(TX)$, as ...
5
votes
0
answers
116
views
A fiber bundle of the Euclidean space over an orbifold
Consider a fiber bundle $p: F\hookrightarrow
E \to B$, where $E$ and $F$ are smooth manifolds and $B$ is a smooth orbifold. More precisely, each point $b \in B$ has an orbifold chart $U=\tilde U/\...
1
vote
0
answers
60
views
Quotient of Euclidean space with maximal volume growth
Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.
If there exists a point $p \in O$ such that
$$
\lim_{r \to \infty}\frac{\text{...
4
votes
0
answers
138
views
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 ...
1
vote
0
answers
129
views
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 ...
4
votes
1
answer
1k
views
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: ...
8
votes
0
answers
152
views
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"...
6
votes
1
answer
362
views
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 ...
3
votes
0
answers
445
views
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/(...
9
votes
3
answers
784
views
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 ...
6
votes
0
answers
133
views
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 ...
4
votes
1
answer
236
views
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 ...
11
votes
0
answers
198
views
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 (...
2
votes
1
answer
203
views
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 ...
8
votes
1
answer
238
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 $...
12
votes
0
answers
242
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 ...
7
votes
0
answers
336
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,\...
1
vote
0
answers
168
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 ...
3
votes
1
answer
273
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\...
0
votes
2
answers
692
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. ...
3
votes
2
answers
271
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 $\mathcal{H}$...
3
votes
1
answer
275
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 $\Sigma X$ is ...
2
votes
1
answer
459
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 ...
4
votes
1
answer
132
views
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 $p$ ...
2
votes
0
answers
45
views
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 ...
6
votes
1
answer
347
views
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 ...
3
votes
1
answer
166
views
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