The orbifolds tag has no wiki summary.
2
votes
2answers
149 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
120 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
169 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
103 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
145 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
147 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
236 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
73 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 ...
11
votes
1answer
235 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
792 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
170 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
214 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
401 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
259 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
307 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
145 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
191 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 ...
20
votes
2answers
613 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
113 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
312 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
621 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
187 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
489 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 ...
5
votes
5answers
370 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 ...
5
votes
2answers
276 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
412 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
296 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
180 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
740 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
210 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
525 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
372 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 ...
0
votes
0answers
185 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
503 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
266 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
892 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
519 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
634 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
323 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
933 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 ...
19
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
803 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
174 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
388 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
2answers
940 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 ...
2
votes
0answers
243 views
Quasi-projective orbifolds and algebraic line bundles
The notion of quasi-projective orbifold is generally accepted to contain at least the following: let $X$ be a (simply-connected) complex manifold, $G$ a group acting on $X$ by biholomorphisms, and ...
5
votes
1answer
321 views
Diffeomorphism groups of orbifolds
A lot is known about geometric and topological properties of diffeomorphism groups of surfaces (here, I am mainly thinking about the work of Smale and Eells-Elworthy). Is there anything known for ...
5
votes
3answers
288 views
Smoothness of frame bundle of (global) orbifolds [reference request]
Background
Let $(M,g)$ be a riemannian manifold and let $G$ be a finite group acting effectively and isometrically on $M$. Recall that this means that for all $x \in G$, the diffeomorphism ...
2
votes
4answers
4k views
Cotangent bundle of a differentiable stack
If you ever wanted to construct the tangent bundle of a differentiable stack, it's relatively simple:
First, if $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$, you could just say ...
