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

Let $R^2$ be the plane, and let G act on it with orientation preserving homeomorphisms, and assume that every orbit of G is a discrete subset in $R^2$ G acts freely: $(\forall g \in G, g \neq e)$, $...
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What is an orbifold with corners

Can one have a formal definition of orbifold with corners? note that it is not parallel to the definition of manifold with corners, as a manifold with boundary is already an orbifold(not with boundry)....
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298 views

Is the torus an orbifold?

Reading the book Geometry of Surfaces by Stillwell I notice that he defines an orbifold as the orbit space $S_\Pi = S/\Gamma$ where $S = \mathbb{C}, S^2, \mathbb{H}^2$ and $\Gamma$ is a group generate ...
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Orbifold Subchart Definition

I am currently reading Zvonkine's "An Introduction to Moduli Spaces of Curves and Their Intersection Theory" and I am hoping that someone here would be willing to clarify some aspects of his ...
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Is every quotient by a finite group an orbifold?

It is required, in order to be an orbifold, to be locally like $\mathbb{R}^n/\Gamma$ where $\Gamma$ is a finite subgroup of $GL(n,\mathbb{R})$ and that the fixed points of the action of $\Gamma$ have ...
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About the definition of orbifolds

I am new to orbifolds. By reading the definition (classical ones, not in terms of stacks or groupoids), I am wondering why only finite group action is allowed in the definition of local charts. I am ...
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102 views

pull back of Groupoid spaces

I am reading Orbifolds as Groupoids: an Introduction https://arxiv.org/abs/math/0203100 Let $\mathcal{G}$ be a Lie groupoid. A right $\mathcal{G}$ space is a smoooth manifold $E$ equipped with an ...
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Finite surface covers of orbifolds

Suppose that we have an orbifold, for example $S^2(3,5,7)$. There are many references that this is a good orbifold and so finitely covered by a surface. By Riemann-Hurwitz, the surface would have ...
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Motivation for orbifold base points

I have been reading Hain's notes Lectures on Moduli Spaces of Elliptic Curves, and would like some "philosophical" intuition on the definition of orbifold basepoints. Let $X$ be a simply connected ...
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Global quotient orbifold

I am a little bit confused about global quotient orbifolds. I dont know if there are any conditions that must be satisfied by the group action on the manifold. This is what I thought could be done: ...
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Is there any version of triangulation of orbifolds

Every closed manifold admits a triangulation, but is it true for closed orbifold? The simplex in the triangulation should be replaced by some orbifold versions, otherwise it must be a manifold. Thanks ...
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Is it possible to orbifold torus $T^d$ into a sphere $S^d$ using $\mathbb{Z}_2$?

A related question for $d=2$ has already been raised (and positively answered) at Math StackExchange here: Is it possible to obtain a sphere from a quotient of a torus? It is also trivially true for $...
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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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219 views

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?
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How do we check if a covering of an orbifold is a manifold?

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 ...
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What's the difference between an orbifold with a conical singularity and a conifold?

In Becker, Becker, Schwarz's book 'String Theory and M-Theory: A Modern Introduction', page 360 they explain how an orbifold of $\mathbb{C}/\mathbb{Z}_{2}$ (which is equivalent to $\mathbb{R}_{2}/\...
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“[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)”, its orbifold notation?

I am trying to understand this frieze pattern $D_{\infty h}$ aka its orbifold notation. This describes spider's silk. The authors call it a space group, some sort of generalization from orbifolds. ...
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Magic theorem for cylinders? Symmetry classes according to Conway's notation?

My teacher Kirsi of Mat-1.3000 in Aalto University stated 17 symmetry classes for planes and 14 for spherical things (some lecture slides here). She used Conway Thurston's notation to classify ...
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796 views

Quotient Riemann surfaces

Let $\mathbb{H}$ be an upper half plane (this is a Riemann surface), then $PSL(2,\mathbb{Z})$ acts on $\mathbb{H}$ and it is well-know that $$ \mathbb{H}/PSL(2,\mathbb{Z})\cong \mathbb{C} $$ is again ...
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Etale groupoid and Morita equivalence

Let $\mathcal{G}=(G_{1}\rightrightarrows G_0)$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $\mathcal{G}$ is called etale if both the source and target ...
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Definition of Homology for an orbifold

I am moving to the study of moduli of curves and I am looking at these notes where $\mathcal{M}_{g,n}$ is described as an orbifold. In order to define the tautological ring the notion of homology and ...
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The teardrop and the spindle are bad orbifolds

I should prove that the teardop $S^2(p)$ (the orbifold with underlying surface $S^2$ and a single cone point of order $p>1$) and the spindle $S^2(p,q)$ (the orbifold with underlying surface $S^2$ and ...