The orbifolds tag has no usage guidance.

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### Equivariant and orbifold Chern classes

Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...

**16**

votes

**2**answers

778 views

### What is an example of an orbifold which is not a topological manifold?

In Thurston's book The Geometry and Topology of Three-Manifolds it is proven that the underlying space of a two-dimensional orbifold is always a topological surface.
Are there any easy examples of ...

**0**

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**0**answers

54 views

### Submanifolds invariant under subgroups with identical quotients

given a smooth manifold $M^n$ and a finite group $G$ acting smoothly and effectively, let's consider two (embedded) $k$-dimensional submanifolds $N_1,N_2\subset M$ and two subgroups $H_1,H_2\subset G$ ...

**7**

votes

**3**answers

393 views

### Conditions for underlying space of an orbifold $\Bbb T^n/\Gamma$ to be a sphere?

Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific ...

**6**

votes

**2**answers

640 views

### Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( ...

**5**

votes

**0**answers

227 views

### manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...

**0**

votes

**1**answer

102 views

### Rationally graded singular cohomology

In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the orbifold cohomology group of an orbifold $X$ by
$H^d_{orb}(X)=\bigoplus_{(g) \in T} H^{d-2\iota_{(g)}}(X_{(g)})$
...

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votes

**1**answer

80 views

### Orbifold singularities over a smooth map

I recently starded studying the book "Orbifolds and Stringy Topology" by Adem, Leida and Ruan and I'm trying to see if there is a relation between the singularites of two orbifolds when there is a ...

**6**

votes

**1**answer

266 views

### Is there a Riemann existence theorem for orbifolds?

For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...

**6**

votes

**1**answer

187 views

### Orbit spaces of crystallographic groups

In their paper "On Three-Dimensional Space Groups", Conway et al. write
Although this paper was inspired by the orbifold concept, we did not need to consider the 219 orbifolds of space groups ...

**3**

votes

**0**answers

79 views

### Properties of the induced map between inertia stacks

Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to ...

**0**

votes

**1**answer

118 views

### étalé space of sheaves on a differentiable stack

If $F$ is a sheaf on a topological space $X$, the well-known étalé space
contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is
isomorphic to the sheaf of sections of $\Gamma F$.
On ...

**2**

votes

**1**answer

160 views

### Real weight modular forms as sections of a line bundle

Background: I've been trying to read
Baily, Walter L., Jr.
The decomposition theorem for V-manifolds.
Amer. J. Math. 78 1956
and I have problems with the language used in the paper. Firstly I am ...

**2**

votes

**2**answers

188 views

### Topological invariants of toroidal orbifolds

Which are the most powerful topological invariants of toroidal orbifolds?
In particular I am looking for topological invariants of two-dimensional toroidal orbifolds such as $T^{2}/Z_{k}\times Z_{k}$ ...

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vote

**0**answers

154 views

### Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on ...

**8**

votes

**1**answer

367 views

### Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)

A quick Question:
Is there some duality known between the quasi Hopf algebra
$D^\omega(H)$ of a finite group $H$ to an orbifold model (such as
SU(2)/$G$ or SO(3)/$G$ orbifold of some ...

**4**

votes

**1**answer

205 views

### hyperbolic orbifolds of small area

Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller ...

**3**

votes

**1**answer

191 views

### Does fixing the reparameterization invariance of the string action correspond to some kind of orbifolding?

Does fixing the reparameterization invariance of the string action, for example by choosing the light-cone gauge
$$
X^{+} = \beta\alpha' p^{+}\tau
$$
$$
p^{+} = \frac{2\pi}{\beta} P^{\tau +}
$$
...

**8**

votes

**2**answers

440 views

### Orbifolds vs. branched covers

Forgive me if this is a basic question. I'm just learning about orbifolds, and covering spaces are my happy place for thinking about group actions.
If $M$ is a manifold and $G$ is a group acting ...

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**0**answers

82 views

### Pseudo-Euclidean orbifolds

Are there any papers (reviews) devoted mainly to pseudo-Euclidean orbifolds in mathematics and physics (e.g. string theory)? A more specific question is related to orbifolds of type $\mathbb ...

**16**

votes

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540 views

### why most of the angles are right

The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which ...

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1k views

### What tools cannot work for orbifolds?

Consider all of your basic constructions/tools/theorems for manifolds: fundamental group, Euler characteristic, triangulations, orientation, smoothness, bundle structure, cobordisms, etc.. Viewing ...

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260 views

### Seifert Fibrations and their associated Spectral Sequence

In a somewhat limited setting, a Seifert Fibre Space is a 3-manifold $M$ with a "nice" decomposition into circles (http://en.wikipedia.org/wiki/Seifert_fiber_space). That is, $M$ is decomposed into ...

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311 views

### “Step-by-Step” toric resolution process?

WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial).
The classical toric ...

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**2**answers

534 views

### What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?

I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something ...

**5**

votes

**1**answer

772 views

### What is the intuition inertia orbifold (or stack)?

I am studying orbifolds with view towards Chen-Ruan cohomology. I have been struggling with inertia orbifolds but have no intuition about them at this point. I would appreciate your motivating me by ...

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votes

**2**answers

373 views

### How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...

**1**

vote

**1**answer

215 views

### Enumerativity of Gromov-Witten invariants of orbifolds

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf.
Is there some sense, or some ...

**1**

vote

**1**answer

202 views

### Link of a vertex of a 3-orbifold (link orbifold)

I know the notion of the link of a vertex of a 3-manifold. In his article Geometric structures on low-dimensional manifolds, Suhyoung Choi first defined the notion of "projective triangulation of an ...

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997 views

### Cobordism of orbifolds?

Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...

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votes

**1**answer

122 views

### Can stabilizer groups in an orbifold have global twisting?

Can stabilizer groups in an orbifold have global twisting?
For example, consider the two groups $\mathbb Z/3\times\mathbb Z$ and $\mathbb Z/3\rtimes\mathbb Z$ (where $\mathbb ...

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votes

**0**answers

340 views

### Low Dimensional Spin Manifolds

I am looking for examples of 2- and 3-dimensional flat spin manifolds with Euclidean and Lorentzian signatures, which admit parallel spinors and the dimension of the space of the parallel spinors is ...

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2k views

### How should one understand orbifold fundamental groups?

I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise ...

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**1**answer

907 views

### Homotopy theory of topological stacks/orbifolds

Motivation $\newcommand{\T}{\mathscr{T}}$
I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its ...

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vote

**1**answer

222 views

### finite generation of $G$-equivariant holomorphic maps by polynomials?

Let $V$ and $W$ be two complex vector spaces with an action of a finite group $G$. The $G$-equivariant polynomial maps from $V$ to $W$ are finitely generated as a module over the ring of $G$-invariant ...

**3**

votes

**1**answer

635 views

### What does the 'V' in 'V-manifold' stand for?

The story of how the name 'orbifold' came about is pretty well-documented, but I can't find any explanation as to why Satake originally named orbifolds 'V-manifolds'. The 'manifold' part is clear ...

**8**

votes

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410 views

### A terminological question concerning orbifolds.

The notion of orbifold is quite well established by now. I would like to ask how one should call a point of an orbifold with non-trivial stabilizer? Should one call this a singular point? Of something ...

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votes

**2**answers

309 views

### how to construct a $C^\infty$ stack from a holomorphic stack

Given a complex manifold, you can `weaken' its structure to give a smooth manifold. Is there an analogous construction that constructs a stack over the category of smooth manifolds from a stack over ...

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469 views

### Thurston's definition of an orbifold

I'm currently trying to understand the definition of an orbifold as expressed in Thurston's Geometry and topology of three manifolds (The definition is in chapter 13 p300). I'm confused about the ...

**8**

votes

**1**answer

446 views

### tangent bundles of orbifolds

Hello,
I want to take a unit $n$-ball $B_n$ and quotient it by some subgroup $G \subset \mathbb{Z}_2^n$.
Here is a sketch of the construction: take the obvious inscribed $n$-hypercube of $B_n$; we ...

**3**

votes

**0**answers

214 views

### Seek “typical examples” for the structure of spaces with two-sided Ricci bounds

By a 1990 paper of Michael Anderson, the following is true:
Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds ...

**1**

vote

**1**answer

842 views

### Penner's formula for volume of the Moduli Space

In his paper "Weil-Petersson Volumes" Penner gives the following formula for the integral of a top-dimensional cohomology class $\omega$ on the moduli space $\mathcal M_g^s$ of $s$-punctured riemann ...

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vote

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241 views

### Is the zero set of a equivariant polynomial map of minimal degree a union of linear subspaces?

Suppose that a finite group acts on two vector spaces $X$ and $Y$, and that $f:X\longrightarrow Y$ is an equivariant polynomial map which is homogeneous of degree $n$, and that there does not exist ...

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**1**answer

657 views

### Orbifold fundamental group and configuration space

I'm not very familiar with (even simple examples of) orbifolds, so my first question is:
Let $C_2$ be $\mathbb{C}$ with one cone singularity at 0 of index 2. What is the fundamental group of $C_2$ ...

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votes

**1**answer

416 views

### Stable homotopy theory of orbifolds

Is there a notion of stable homotopy, spectrum, or a stable homotopy category which corresponds to orbifolds and orbispaces, in the same way that classical stable homotopy theory corresponds to ...

**1**

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211 views

### What is known about orbifolding ordered groups and sets? Who has been involved? Links to Lee metrics?

In mathematical music theory several ordered groups are considered. Some examples contain the frequency space or Tonnetzes. Other groups (commutative and non-commutative ones) are discussed by Dawid ...

**3**

votes

**1**answer

638 views

### A simple minded Poincare duality for orbifolds?

Suppose $X^n$ is an orientable compact orbifold (without boundary) with stabilisers in codimesnion 2, and $\bar X^n$ is the underlying topological space. We can assume moreover that $X^n$ is a ...

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316 views

### 3-orbifolds with a Seifert geometry that are not actually Seifert fibered

It is well-known that Seifert fibered $3$--manifolds are geometric: they admit one of the Thurston geometries $S^2 \times R$, $R^3$, $H^2 \times R$, $S^3$, $Nil$, and $PSL(2,R)$. Furthermore, the ...

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1k views

### Groupoids vs Pseudogroups

(Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second ...

**3**

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558 views

### Spinors on orbifolds

Let $R^{n}$ be a cone over sphere $S^{n-1}$ with the metric $g = dr^2 + r^{2}g[S^{n-1}]$ ($r> 0$).
Whether it is true that the cone over $S^{n-1}/Z_{2} = RP^{n-1}$ has twice less parallel spinors, ...