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I've been trying to figure out orbifolds, and in all of the sources I seem to be confused with the orbifold structure on quotient orbifolds. A quotient orbifold is defined as follows.

Let $M$ be a smooth manifold, $G$ a compact Lie group which acts smoothly (i.e. $\cdot : G \times M \rightarrow M$ is smooth), effectively (i.e. for $g\in G$, if there exists $x\in M$ such that $gx=x$ then $g=1$), and alsmost freely (i.e. for any $x\in M$ the stabilizer subgroup $G_x=\{ g\in G \, \vert \, gx=x \}$ is finite) on $M$. According to the paper by Adem, "Lectures on Orbifolds and Group Cohomology", the orbifold charts are to be given by: for $x\in M$, find a chart $\phi : \tilde{U} \rightarrow U$ with $ x \in \tilde{U} \subseteq M$ which is invariant under the action of $G_x$, $U \cong \mathbb{R^k}$. Then the chart is given by $(U, G_x, \pi : U \rightarrow U/G_x )$. This definition seems to conveniently leave out what the underlying topological space for the quotient orbifold is! The "obvious choice" being $M/G$ doesn't seem to work out though because of the following example.

Consider $M=G=\mathbb{S}^1$ with the obvious action. That action is smooth, effective and almost free (in fact free because $G_{\theta}=\{1\}$ for all $\theta \in \mathbb{S}^1$. This action is also transitive. So $M/G \cong \mathbb{R}^0$ or a point. But the charts in the above definition would look something like $\pi : U \rightarrow U/G_x = U$ where $U= \{ \theta \, \vert \, a < arg(\theta) < b\}$ for some $0< a < b < 2 \pi$. According to the strict definition of this being an orbifold chart, we would need the map $U/G_x \rightarrow M/G$ to be a homeomorphism onto its image, which it certainly is not.

So what is going on here? What is the underlying space for the charts above, or what is the correct description of the charts? Is the action supposed to have more restrictive conditions on it, like being nowhere discontinuous?

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Just a nitpick, but your definition of "effective" seems to be off - what you wrote is the same as free. Usually, "effective" means if gx = x for all x, then g = 1. –  Jason DeVito Apr 19 '11 at 2:22

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It's meant to be a slice through x for the action-sorry about the confusion. We use that a smooth action is locally smooth. You might want to look at Bredon's book on transformation groups for details. And yes, the orbit space M/G is the underlying space.

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Thanks very much for the help! –  Glen M Wilson Apr 18 '11 at 22:36

The example of the weighted projective space is illustrative of what is meant by a "quotient orbifold". It is incorrectly suggested that these quotient orbifolds are "classical orbifolds" in the introduction, especially as you have pointed out, that their charts be homeomorphisms. But later in the article, the notion is formalized. In any case, the underlying space of the orbifold is the quotient space. But it comes with additional data. In your example above $M/G$ is a point.

Try looking at the weighted projective space: the quotient of $S^3 = \{(z_0,z_1) : |z_0|^2 + |z_1|^2 = 1\}$ by the circle action where for each $\lambda \in S^1$, $\lambda.(z_0,z_1) = (\lambda z_0, \lambda^2 z_1)$. The quotient space is a sphere which the author refers to as having a teardrop shape.

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