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.

Filter by
Sorted by
Tagged with
0
votes
1answer
25 views

Does the Collection of Graphs not Embeddable in a Fixed Orbifold have a Well-quasi-ordering?

Using the Robertson-Seymour Theorem, one can show that given a fixed surface $S$ the collection of graphs which can't be embedded in it are defined by a finite set of forbidden minors - just as the ...
3
votes
1answer
37 views

Short exact sequence of groups acting on a space

Let $X$ be a metrizable topological space, and suppose we have three groups $\Gamma$, $\Gamma'$, and $\Gamma''$, acting properly on, respectively $X$, $X$, and $X/\Gamma'$. What can we say about them ...
2
votes
0answers
58 views

Questions on the orbifold structure of complex weighted projective spaces

Let $a_0, \dots, a_n$ be positive integers with $\gcd = 1$. Consider the weighted projective space $\mathbb {CP}^n[a_0, \dots, a_n]$. I have used the following procedure to construct a coordinate ...
1
vote
1answer
72 views

How is $\Bbb{C}$ topologically the same as $\Bbb{C}/\Bbb{Z}_n$?

This article on Orbifolds says that $\Bbb{C}$ is topologically the same as $\Bbb{C}/\Bbb{Z}_n$, where $\Bbb{Z}_n$ are the $n$th roots of unity. What does it mean to say that these two spaces are ...
1
vote
0answers
26 views

Why is the Orbifold Euler characteristic of $M_{1, 1}$ equal to $\zeta(-1) $

The orbifold euler characteristic of $M_{1, 1}$= $PSL(2, \mathbb{Z}) $/$\mathbb{H}$is equal to -1/12. This is due to the fact that $M_{1,1}$ has one two-cell with $\mathbb{Z_{2}}$ automorphism, two ...
1
vote
1answer
34 views

Action torsion elements in the fundamental group of geometric orbifolds

In chapter 2 of Three-dimensional Orbifolds and Cone-Manifolds, theorem 2.26 states that complete geometric orbifolds $Q$ modeled on $(G,X)$, whith $X$ simply connected, are such that the holonomy ...
1
vote
0answers
42 views

Definition and properties of “real weighted projective spaces”

I've seen a lot of literature on Complex weighted projective spaces (sometimes defined in a flavour of algebraic geometry) which can be obtained as the orbit spaces of the action of $\mathbb{C}^*$ on $...
0
votes
0answers
5 views

How are two embeddings of orbifolds are related?

The following Lemma is used to prove that smooth embedding of orbifold charts give rise to injective homomorphism in groups acting on respective manifolds, how do I prove following: Given 2 embeddings ...
4
votes
1answer
118 views

The proof of the quotient space of a manifold with properly discontinuous action is an orbifold.

I am reading the proof of the following proposition from Thurston's the geometry and topology of 3-manifolds: My questions are: By $U_x=\tilde{U}_x/I_x$, if $\cap_{i=1}^kU_{x_i}\not=\emptyset$, then ...
3
votes
0answers
34 views

Is a trivial $T^2\times S^1$ bundle with a $\mathbb{Z}_4$ orbifold action the same as a $T^2$ bundle over $S^1$ with a $\mathbb{Z}_4$ twist?

Suppose we have a square torus (i.e. complex structure $\tau=i$) which is fibred trivially over a circle of radius $4R$. We then use the $\mathbb{Z}_4$ rotational symmetry of the torus lattice to ...
2
votes
0answers
43 views

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 ...
1
vote
1answer
198 views

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?
1
vote
1answer
93 views

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: $(\forall g \in G, g \neq e)$, $...
1
vote
1answer
613 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 ...
1
vote
0answers
31 views

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 ...
8
votes
1answer
192 views

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 ...
1
vote
1answer
136 views

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 ...
1
vote
1answer
128 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 ...
1
vote
1answer
98 views

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 ...
4
votes
1answer
191 views

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 ...
1
vote
0answers
165 views

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: ...
3
votes
0answers
64 views

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 ...
2
votes
0answers
195 views

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 $...
1
vote
0answers
66 views

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&=...
4
votes
1answer
1k views

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 ...
2
votes
0answers
77 views

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 ...
6
votes
1answer
172 views

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 $\mathbb R^n$ ...
0
votes
1answer
139 views

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 ...
0
votes
1answer
141 views

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, ...
3
votes
0answers
71 views

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 ...
2
votes
1answer
40 views

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 ...
5
votes
0answers
169 views

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 ...
1
vote
1answer
134 views

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?
1
vote
0answers
27 views

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 ...
3
votes
1answer
180 views

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 $\tilde {U}$"...
10
votes
0answers
511 views

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 ...
0
votes
1answer
119 views

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 ...
2
votes
0answers
102 views

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 ...
6
votes
1answer
432 views

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 ...
3
votes
2answers
428 views

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 ...
5
votes
2answers
164 views

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 ...
1
vote
1answer
270 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 ...
4
votes
1answer
160 views

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 ...
2
votes
1answer
142 views

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?
4
votes
0answers
145 views

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 ...
3
votes
1answer
804 views

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}/\...
2
votes
1answer
94 views

“[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. ...
7
votes
1answer
474 views

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 ...
6
votes
1answer
920 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 ...
3
votes
1answer
399 views

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 ...