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5
votes
1answer
159 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 ...
2
votes
0answers
61 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
1answer
99 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
1answer
94 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
2answers
164 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 $T^{2}/Z_{k}\times Z_{k}$ ...
1
vote
0answers
130 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 ...
7
votes
1answer
217 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 ...
0
votes
0answers
138 views

Weighted projective space, its varieties, the universal hyperplane section description and the Segre embedding

We will view here the weighted projective space as an orbifold. Let $S=k[x_0,...,x_n]$ be a positively graded ring (I don't assume $S$ to be finitely generated in degree 1). To the grading ...
4
votes
1answer
164 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
1answer
164 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 $$ X^{+} = \beta\alpha' p^{+}\tau $$ $$ p^{+} = \frac{2\pi}{\beta} P^{\tau +} $$ ...
6
votes
2answers
289 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 ...
1
vote
0answers
77 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
2answers
520 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 ...
19
votes
3answers
819 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 ...
4
votes
1answer
186 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 ...
3
votes
0answers
246 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 ...
7
votes
2answers
434 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 ...
3
votes
1answer
369 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 ...
4
votes
2answers
317 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
1answer
174 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
1answer
194 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 ...
21
votes
2answers
724 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 ...
2
votes
1answer
117 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 ...
1
vote
0answers
322 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 ...
17
votes
5answers
1k 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 ...
13
votes
1answer
697 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 ...
1
vote
1answer
201 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
1answer
515 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 ...
6
votes
5answers
382 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 ...
6
votes
2answers
288 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 ...
2
votes
0answers
429 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
1answer
319 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
0answers
190 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
1answer
798 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 ...
1
vote
2answers
224 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 ...
8
votes
1answer
567 views

Orbifold fundamental group and configuration space

Hi, 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 ...
6
votes
1answer
393 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
vote
0answers
195 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
1answer
532 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 ...
7
votes
1answer
294 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 ...
9
votes
3answers
967 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
votes
2answers
531 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, ...
2
votes
4answers
688 views

Quotient Surface of A Hyperelliptic Involution

Let $X$ be a hyperelliptic Riemann surface, and let $J$ be the hyperelliptic involution. Then consider the quotient surface $X/ < J > ,$ my question is whether $X/ < J > $ is a Riemann ...
4
votes
1answer
341 views

Ramification formula for orbifolds

It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this ...
8
votes
1answer
1k views

Why isn't the orbifold cohomology of $pt/G$ equal to the cohomology of $BG$?

The classifying space of a group $G$ is given by taking a contractible space $E$ equipped with a free $G$-action, and looking at the quotient, which we dub $BG$. The homotopy type of this space (and ...
20
votes
2answers
1k views

Is there a Chern-Gauss-Bonnet theorem for orbifolds?

There's a Gauss-Bonnet theorem for compact 2-orbifolds(due to Satake, I think), which gives a relation between the curvature of a Riemannian orbifold and the orbifold topology(i.e. taking into account ...
7
votes
2answers
892 views

Euler characteristic of orbifolds

Hello, Suppose $M$ is a compact oriented smooth manifold and $G$ is a finite group acting on it. Then it is well-known, although I have yet to find a proof or derivation of it, that the (normal ...
1
vote
1answer
178 views

When do maps of ineffective orbifolds descend to their effective part?

If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...
4
votes
1answer
405 views

Intrinsic Characterization of when an orbifold (or more general stack) is effective?

Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie ...
3
votes
2answers
1k views

Are orbifold singularities canonical?

This is a direct consequence of my previous question: Extending group actions on varieties In his answer, inkspot said that group actions can be extended if the variety has ample canonical class and ...