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8
votes
1answer
318 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
1k 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
563 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, ...
4
votes
4answers
858 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 ...
5
votes
1answer
401 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 ...
10
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 ...
23
votes
2answers
2k 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 ...
8
votes
2answers
1k 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
182 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
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 ...
4
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 ...
2
votes
0answers
277 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
372 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
329 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 $\gamma_x$...
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 $\...
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 "...
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 $G$-...
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}{...