Questions tagged [orbifolds]
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35
questions with no upvoted or accepted answers
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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 ...
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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 (...
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643 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 ...
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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"...
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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,\...
6
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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 ...
6
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405 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 ...
asked Apr 20 '15 at 0:49
Mohammad Farajzadeh-Tehrani
6,3301 gold badge22 silver badges62 bronze badges
5
votes
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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 ...
4
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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/\...
4
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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 ...
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305 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 ...
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96 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 ...
3
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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 ...
3
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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 ...
3
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176 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.
3
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125 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\...
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450 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 ...
3
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371 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 ...
3
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251 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 $(M_i,...
2
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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 ...
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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/(...
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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 ...
2
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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 ...
2
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257 views
Einstein's field equation on orbifolds
I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).
Here, by an orbifold I mean the "stacky" quotient of, ...
2
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567 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 ...
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362 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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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119 views
Embedding of Gorenstein orbifold as a hypersurface
I am trying to understand if three complex dimensional orbifold singularity $\mathbb{C}^3 / \Gamma$ can be embedded as hypersurface in $\mathbb{C}^4$. The condition of being Gorenstein and having ...
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205 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 $\...
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88 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}...
