# Questions tagged [orbifolds]

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure.

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### Equivalence of Lie groupoids $\phi: H \rightarrow G$ induces an equivalence of categories $\phi^*: G\text{-spaces} \rightarrow H\text{-spaces}$.

In Orbifolds as Groupoids there is the notion of an equivalence $\phi: H \rightarrow G$ between Lie groupoids (2.4) and of $G$-spaces (5.1). Given a smooth functor $\phi: H \rightarrow G$ we can ...
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### number of orbit of a group

is there any general algorithem to find number of orbits of a permutation group under permuting the coordinates action? for example if g=(1,2,3) is an ellemnt of a group G and x=(4,7,8) in $R^{3}$ we ...
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### Why the teardrop is a bad orbifold?

I found this The teardrop and the spindle are bad orbifolds, but I do not totally understand the explanation. Could you help me, please?
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### Orbifold of the three-sphere

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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### What is a singular space?

A book I am reading on orbifolds uses the word singular space but doesn't say what it means. The book is Orbifolds and Stringy Topology by ALR the quote is "Orbifolds are singular spaces that are ...
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### Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
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### A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n$

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...
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### Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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### points which are fixed points of a finite group action

consider an open set $\tilde{U}\subset\mathbb{R}^n$ and a finite Lie-group $G$, which acts smoothly on $\tilde{U}$, i.e. we have a smooth map $G\times \tilde{U}\rightarrow\tilde{U}$. Suppose further, ...
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### Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
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### Prove existence of local orbifold chart

Let $(U, G, \phi)$ be a $n$-dimensional complex orbifold chart over $x \in X$, i.e., $x \in \phi(U)$ I want to know if there is a a subset $V$ of $U$ such that $(V, G_x, \phi|_V)$ is also an orbifold ...
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### Is the Groupoid of germs associated to an orbifold a Hausdorff proper Lie groupoid?

I was studying the book Introduction to Foliations and Lie Groupoids by I. Moerdijk and J. Mrcun and I have a doubt. On page 140 they give the following definition for a proper Lie groupoid. A Lie ...
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### Orbifold chart.

i'm trying to define an orbifold chart for the teardrop $R^2/Z_2$, where $Z_2$ acts via rotations. My advisor gave a tip: to use stereographic projections. But I'm a little stuck. Any help?
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### Weighed projective space charts

I'm studying weighed projective spaces and I found this reference http://arxiv.org/pdf/math/0510331v1.pdf* where it describes its orbifold charts (starts at page 53 of the PDF). My doubt is very ...
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### Set of non fixed points of an automorphism

I am trying to prove the following "For an orbifold chart $(\tilde{U},G,\phi)$ the set of non fixed point of $g : \tilde{U} \rightarrow \tilde{U}$ where $1 \neq g \ \in G$ is dense in $\tilde {U}$"...
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### Does Stokes' Theorem hold on spaces with singular points?

I have come across the question whether Stokes' theorem holds also on orbifolds. Let us take the simple case of $T^2/Z_2$ with a one-form $A$, then the question becomes: For a region $\Gamma$ with ...
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### Singular locus of orientable 3-orbifolds

Any reference (or any hints if the proof is easy) for the proof that the singular locus of a 3-dimensional orientable orbifold is a trivalent graph with each edge labelled by integers $a,b,c>1$ and ...
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### Quotient of Poincare dodecahedral space-example of spherical orbifold

Let $\mathcal{O}$ the orbifold with underlying space $S^3$ and singular locus the trefoil knot with local groups of order five. Then how can we see that it is the quotient space of the Poincare ...
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### General introduction to orbifolds?

Where should I go to learn about orbifolds? I am interested in a general introduction that gives precise definitions and clear explanations. I have a fair background in topological and smooth ...
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### Reference - Riemannian Orbifolds

I am looking for papers or textbooks talking about the various analog theorems of Riemannian Geometry of Manifolds to Riemannian Orbifolds like Toponogovs Theorem, Bonnet-Myers, Gauss Bonnet etc. So ...
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### Does this orbifold embed into $\mathbb{R}^3$?

Let $X$ be the space obtained by gluing together two congruent equilateral triangles along corresponding edges. Note that $X$ has the structure of a Riemannian manifold except at the three cone ...
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### Orbifolds and singular points

If I understand correctly, we roughly define the singular locus $\Sigma_O$ of an orbifold $O$ to be the set of points where the orbifold fails to be a manifold. In particular, if $x \in O$ has a ...
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### Connected sum while keeping curvature bounded.

Is it possible to perform a connected sum of two Riemannian Manifolds or Orbifolds while keeping curvature bounded from below? More explicitly, If $M_1$ and $M_2$ are two Riemannian manifolds (or ...
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### What is this orbifold?

Consider the non-negative real line with identification $x\sim 2x$. Is there a special name for this quotient space; what's known about it?
Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...