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6
votes
2answers
772 views

A simple minded Poincare duality for orbifolds?

Suppose $X^n$ is an orientable compact orbifold (without boundary) with stabilisers in codimension 2, and $\bar X^n$ is the underlying topological space. We can assume, moreover, that $X^n$ is a ...
13
votes
10answers
2k views

Orbifold fundamental group in terms of loops?

In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...
1
vote
0answers
108 views

Can infinitely many orbifolds be “added up” to form a fractal space?

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it. Intuitively, an orbifold, from what I understand, is a "...
1
vote
0answers
87 views

Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.
8
votes
0answers
124 views

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
2answers
785 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
votes
0answers
56 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 $H_1,H_2\subset G$ ...
24
votes
4answers
3k views

What is meant by smooth orbifold?

There seems to be some confusion over what the tangent space to a singular point of an orbifold is. On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-...
10
votes
1answer
664 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 $C_2$ ...
7
votes
3answers
398 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
2answers
645 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
2answers
273 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 ...
5
votes
0answers
236 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
1answer
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 $H^d_{orb}(X)=\bigoplus_{(g) \in T} H^{d-2\iota_{(g)}}(X_{(g)})$ ...
0
votes
1answer
81 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
1answer
270 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 ...
16
votes
2answers
544 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 ...
0
votes
1answer
119 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 ...
6
votes
1answer
190 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
0answers
80 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 I\...
2
votes
1answer
172 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
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 $T^{2}/Z_{k}\times Z_{k}$ ...
1
vote
0answers
212 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
2answers
423 views

Branched coverings over orbifolds with reflector lines

It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum_i^r\frac{a_i-1}{...
1
vote
0answers
158 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
1answer
379 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
1answer
207 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
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
2answers
464 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 ...
2
votes
0answers
344 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 ...
1
vote
0answers
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 R^{1,4m-3}...
3
votes
2answers
562 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, ...
23
votes
2answers
1k 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 ...
8
votes
1answer
460 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 ...
21
votes
3answers
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 ...
3
votes
0answers
317 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
541 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 ...
18
votes
5answers
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 ...
1
vote
2answers
243 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 ...
5
votes
2answers
378 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 ...
5
votes
1answer
818 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 ...
1
vote
1answer
218 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
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 ...
4
votes
1answer
430 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 ...
2
votes
1answer
124 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 Z\to\operatorname{Aut}(\...
13
votes
1answer
920 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
223 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
644 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
5answers
413 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
312 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 ...