Questions tagged [orbifolds]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
0answers
94 views

Metropolis-Hastings sampling as a group action

Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...
7
votes
1answer
659 views

Condensed / pyknotic approach to orbifolds?

Does condensed / pyknotic mathematics afford an (yet!) another approach to orbifold theory? Let me admit up-front that I don't know much about either condensed / pyknotic mathematics or about orbifold ...
11
votes
2answers
608 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
44 views

Clarification about the process of naturally endowing a space with a Riemann orbifold structure supported on a sphere

I am having some difficulties understanding an argument in a proof. Here is an excerpt from Lyubich–Peters - Classification of invariant Fatou components for dissipative Henon maps, first geometric ...
4
votes
1answer
132 views

Absolute and relative tilings of the hyperbolic plane

In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it). The ...
3
votes
0answers
32 views

A canonical map from a Euclidean cone-manifold $M^3$ to $\mathbb{E}^3/\mathrm{Hol}(M)$

Suppose we have a 3-dimensional Euclidean cone-manifold $M$—in my book that just means $M$ is a manifold whose geometry is constructed by gluing it out of Euclidean tetrahedra, with faces paired by ...
5
votes
0answers
147 views

Pseudoreflection groups in affine varieties

Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result: (C-S-T): Let $G$ be a ...
2
votes
0answers
72 views

Equivalence between integrals over a reduced space

Context: I have been trying to understand this paper from Y. Cho and K. Kim. More precisely, a specific argument in Lemma 2.2 where they say the ABBV localization formula on an integral over a ...
11
votes
4answers
2k views

What is the “right” definition of the homology(cohomology) of an orbifold?

What is the "right" analog in the orbifold case of a singular homology of a topological space? We can not just take the homology of the underlying space, because it does not contain much information. ...
11
votes
1answer
2k views

What is the intuition behind the 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
0answers
47 views

About the orientation of index formula on orbifold

Let $X$ be a closed oriented orbifold with singularity $\Sigma X$. The singularity is defined as $$ \Sigma X=\{(x)|~x\in X,~G_x\neq1\}, $$ where $G_x$ is the isotropy group. For $u\in K_v(TX)$, as ...
4
votes
0answers
93 views

A fiber bundle of the Euclidean space over an orbifold

Consider a fiber bundle $p: F\hookrightarrow E \to B$, where $E$ and $F$ are smooth manifolds and $B$ is a smooth orbifold. More precisely, each point $b \in B$ has an orbifold chart $U=\tilde U/\...
1
vote
0answers
56 views

Quotient of Euclidean space with maximal volume growth

Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold. If there exists a point $p \in O$ such that $$ \lim_{r \to \infty}\frac{\text{...
4
votes
0answers
123 views

The Fock space in Costello's paper “Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products”

Let $X$ be a smooth projective variety. In this Annals paper, Costello expressed the descendent genus $g$ Gromov-Witten (GW) invariants of $X$ in terms of genus zero GW invariants of the symmetric ...
1
vote
0answers
124 views

Berglund-Hübsch-Hori-Vafa mirror symmetry is a ring isomorphism?

Let $W = \sum_{i=1}^{m} a_i \prod_{j=1}^{n} x_j^{b_{ij}}$ be a homogeneous polynomial of degree $d$ in $n$ variables. I focus on the $m=n$ case (invertible polynomial in the Berglund-Hübsch ...
4
votes
1answer
932 views

Is a free and discrete group action on the plane a covering space action?

Let $\mathbb{R}^2$ be the plane, and let a group $G$ act on it with orientation preserving homeomorphisms, and assume that every orbit of $G$ is a discrete subset in $\mathbb{R}^2$ $G$ acts freely: ...
7
votes
0answers
141 views

Smooth sub-orbifolds in the language of stacks

In most geometric categories, "monomorphism" is too general to describe useful notions of "embedding". This is the case e.g. for schemes, complex manifolds, and differentiable manifolds. So "embedding"...
6
votes
1answer
287 views

What are orbifolds with corners?

What is the geometric definition of orbifolds with corners? Here “geometric" means that there is a definition in chapter 8 of the draft of Dominic Joyce's book D-manifolds and d-orbifolds: a theory of ...
9
votes
3answers
740 views

What is the official definition of $\mathcal{M}_g$ as an orbifold, and how much can I ignore it?

There is a well-known description of $\mathcal{M}_g$ as $\mathcal{T}_g/\Gamma$ where $\mathcal{T}_g$ is the Teichmuller space and $\Gamma$ is the mapping class group. Teichmuller space is homeomorphic ...
2
votes
0answers
363 views

Geometry of the irrational torus

One of the motivations of diffeology is to study singular spaces such as the irrational torus. The irrational torus $T_α$ of slope $α∈R∖Q $ as a diffeological space is given by the quotient space $ R/(...
0
votes
2answers
634 views

Why study orbifolds? [closed]

Question is as in the title. Why study orbifolds? I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. ...
6
votes
0answers
116 views

How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?

My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...
3
votes
1answer
255 views

Proper and etale groupoid is locally a translation groupoid

I am reading Orbifolds as Groupoids: an Introduction by Ieke Moerdijk. In page $8$ when explaining local charts, it says the following : Let $\mathcal{G}$ be a Lie groupoid. For an open set $U\...
4
votes
1answer
194 views

Geodesic representatives in the orbifold fundamental group

Does every element in the orbifold fundamental group $\pi_1^{orb}(X,x)$ of a closed hyperbolic 2-orbifold $X$ admit a unique geodesic arc representing it? Does every free homotopy class in $X$ admit ...
10
votes
0answers
144 views

Flat spherical orbifolds

What is known about existence and classification of flat spherical orbifolds?. Here I mean orbifolds that admit a flat Riemannian metric (Euclidean orbifolds) and whose underlying topological space (...
2
votes
1answer
161 views

Global symplectic reduction

Let $M$ be a symplectic manifold equipped with a hamiltonian action of a compact Lie group $G$ with moment map $\mu\colon M\to \mathfrak g^*$. Assume $c\in \mathfrak g^*$. Then the symplectic ...
8
votes
1answer
227 views

Does geometrization of Alexandrov 3-spaces follow from that of 3-orbifolds?

Galaz-Garcia and Guijarro proved the geometrization of closed (compact, boundaryless) Alexandrov 3-spaces. Part of the strategy was to use the so-called ramified double cover $\tilde{X}$ of the space $...
12
votes
0answers
224 views

If two group actions lead to the same orbifold, are they conjugate?

In The Geometry and Topology of Three-Manifolds, Thurston says: "In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we ...
7
votes
0answers
311 views

Equivariant De Rham theorem for orbifolds

Recall that the "classical" equivariant De Rham theorem states that, for a compact Lie group $G$ acting on a compact smooth manifold $M$, $$H_G^*(M,\mathbb{R})\cong H^*(\Omega_G(M)),$$ where $H_G^*(M,\...
3
votes
2answers
243 views

Necessity/Motivation for generalised homomorpisms

I am reading Ieke Moerdijk's article "Orbifolds as Groupoids : an Introduction". In that notes author defines a notion of generalized map between Lie groupoids. Let $\mathcal{G}$ and $\mathcal{H}$...
1
vote
0answers
138 views

Idea behind definition of classifying space over an orbifold

Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction. Definition : Let $X$ be a locally compact ...
3
votes
1answer
231 views

Do regular points of an orbifold form a connected set?

First, a bit of background on orbifolds: Let $X$ be a connected (effective) orbifold. To every point $x \in X$, we associated a group $G_x$ called the isotropy group. The singular locus $\Sigma X$ is ...
0
votes
2answers
163 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 ...
2
votes
1answer
381 views

How singular is the metric on an orbifold

I am reading some stuff on orbifolds. I am particularly interested in the metrics on orbifolds. The famous example of one orbifold is the "American football", which is $\mathbb{S}^2$ quotient by the ...
4
votes
1answer
124 views

Can we perturb a surface away from an orbifold point?

Let $X$ be a smooth, compact, orbifold of dimension $4$, where the stabilisers are only allowed to be cyclic groups. Let $p \in X$ be an isolated orbifold point (i.e. the orbifold chart about $p$ ...
1
vote
0answers
51 views

Orbifolds with maximal diameter

Orbifolds with positive curvature and maximal diameter are investigated in this article, by J. Borzellino. Theorem 1 of the article states: Let $\mathcal{O}$ be a complete $n$-dimensional ...
2
votes
0answers
42 views

Minimization of the volume of the image of space-filling convex polytopes under similarities

Suppose $A:\mathbb{R}^n \to \mathbb{R}^n$ is a similarity, given by $A(x) = \lambda Ox$, where $\lambda > 1$ and $O$ is an orthogonal matrix (i.e., $A$ is a particular loxodromic repelling ...
6
votes
1answer
337 views

Homotopy groups of smooth part of moduli space

Let $M_g$ be the moduli space of Riemann surfaces, as described for example in the book of Harris and Morrison - Moduli of curves. As a topological space, or better as orbifold, it has smooth points ...
2
votes
1answer
164 views

Is there a pair of pants decomposition analogue for orbifolds?

The pair of pants decomposition is a useful tool is surface theory. Is there an analogous decomposition for orbifolds? Thanks
1
vote
0answers
54 views

Integer valued signature of $4n$ dimensional orbifolds

Let $M^{4n}$ be a smooth oriented $4n$-dimensional manifold without boundary. Then we have an intersection form in $H^{2n}(M^{4n},\mathbb R)$ and such a form has signature $(n_+, n_-)$. Question. I ...
5
votes
1answer
264 views

Maps to the universal punctured elliptic curve

I have just started reading Hain's paper On the Universal Elliptic KZB Connection. I am a bit confused about a comment made there about base points on orbifolds. I am still very new to the idea of ...
8
votes
1answer
504 views

Smoothing of a Kähler orbifold metric on a complex surface

Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb ...
2
votes
1answer
163 views

Cobordism/bordism group based on orbifolds with corners

We define a geometric homology group of a topological space $X$ as follows: the chain complex $C_{\bullet}$is freely generated by the maps $f$ from a compact oriented orbifold with corners $P$ to $X$, ...
23
votes
2answers
1k views

Is every rational realized as the Euler characteristic of some manifold or orbifold?

Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds. Then, if the answer is No, one can remove various conditions on the dimension, and allow non-compact ...
2
votes
0answers
92 views

Does any smooth oriented closed orbifold have a fundamental class

This thread:triangulation of orbifolds has shown that any smooth closed orbifold has a triangulation. My further question is: if the difference of any two triangulations $P$ and $Q$ is a boundary of a ...
3
votes
1answer
340 views

Does the torus $T^d$ 2-fold cover an orbifold $Q^d$ with underlying space $S^d$?

I asked the same question on math.stackexchange recently (https://math.stackexchange.com/questions/2134978/is-it-possible-to-orbifold-torus-td-into-a-sphere-sd-using-mathbbz-2), but it didn't receive ...
3
votes
0answers
200 views

What is a proper n-etale morphism?

Let $Y$ be a complex algebraic variety, and let $n\in \mathbb Z_{\geq 1}\cup \{\infty\}$. How do I think about a proper $n$-etale morphism $X\to Y$? If $n=1$, I think this should be a finite etale ...
4
votes
0answers
303 views

Symplectic orbifolds

I will start by saying that I am not a symplectic topology. However, in my research I now have on my hands a symplectic 4-orbifold, which I would like to understand better. Certain results for ...
3
votes
2answers
350 views

Orbifold of the three-sphere (and lens spaces)

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&=...
7
votes
3answers
1k views

Is the Čech cohomology of an orbifold isomorphic to its singular cohomology?

Let $\mathcal{O}$ be a finite-dimensional, paracompact, Hausdorff, smooth (and compact, if it helps) orbifold. Is there an isomorphism between the real Čech cohomology and singular cohomology of the ...