# 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 the set of points $$x$$ for which $$G_x \neq 1$$, and these points are called singular points. Non-singular points are called regular. The singular locus is a closed, nowhere dense subset of $$X$$, and the set of regular points $$X^r$$ is open and dense in $$X$$.

The orbifold $$X$$ has a stratificiation into strata of points of equal type, i.e. $$X$$ is the disjoint union of connected components along which the isotropy group is constant (up to isomorphism). In particular, the strata of codimension $$0$$ form the regular points, and the strata of codimension $$\geq 1$$ form the singular locus.

Now, my question: is the set of regular points $$X^r$$ connected?

I've done a bit of background research, but I seem to find contradictory statements. Moreover, the original definition of orbifolds (i.e. Satake's V-manifolds) assumed that there were no codimension $$1$$ strata, in which case the statement is clearly true. To add to the confusion, some authors still use this definition.

So, what have I found so far:

Suggesting "Yes":

• "This stratum is a connected manifold ..." [Riemannian orbifolds with non-negative curvature, Dmytro Yeroshkin]
• "The regular points of an orbifold form the top-dimensional stratum. It is open, dense and path-connected." [The Topology of locally volume collapsed 3-Orbifolds, Daniel Faessler]
• "Observe that the set of regular points is a dense connected open subset of the topological space underlying a connected orbifold" [Seifert Fibred 3-Orbifolds, Bonahon & Siebenmann]
• "Observe that the set of regular points is a dense connected open subset of $$X$$" [Lusternik-Schnirelmann category of Orbifolds, Hellen Colman]
• "The regular part of an orbifold is connected." (proof given, which uses that the regular part is locally connected) [Orbifolds from a metric viewpoint, Christian Lange]

Suggesting "No":

• "Having done this, the set of special points has codimension at least 2. The set of regular points is therefore a connected manifold." [Differential Topology, Foliations, and Group Actions, Paul A. Schweitzer]
• "...we require that the fixed-point set is of codimension at least two. [...] This requirement has the consequence that the non-fixed-point set is locally connected." [Stringy geometry and topology of orbifolds, Yongbin Ruan]
• The first bullet point that you have labelled 'Suggesting "No"' does not suggest no, instead it explicitly says yes. – Lee Mosher Apr 12 '18 at 12:13
• Also, in an $n$-dimensional orbifold $X$, the set of regular points forms an open set which is an $n$-manifold, and the union of the set of regular points and codimension 1 strata forms an open set which is an $n$-manifold with boundary; connectivity of the regular points of $X$ follows quickly as a consequence of connectivity of $X$ itself. – Lee Mosher Apr 12 '18 at 12:23
• @LeeMosher The first point states that connectedness is a consequence of not having codimension one strata, which (to me) sounds like the regular points may not be connected if codimension one strata are present. – Ezueneok Apr 12 '18 at 12:26
• Usually when special cases are proved, there is no implied suggestion that other special cases will have an opposite conclusion. – Lee Mosher Apr 12 '18 at 12:29
• @LeeMosher I'll think about the manifold with boundary for a bit. Perhaps you should post it as an answer, so I can accept it once I've convinced myself :) – Ezueneok Apr 12 '18 at 12:31

Part of the reason that people consider orbifolds whose singular points have codimension $\ge 2$ is because they are restricting their attention to oriented orbifolds. For example, if you are working with $n$-orbifolds over the complex numbers $\mathbb{C}$ then each element of each isotropy group is a holomorphic transformation, and every holomorphic transformation preserves orientation; in that situation, the singular points have codimension $\ge 2$.
If you do not restrict to oriented orbifolds, there is only one way that a codimension 1 stratum can arise: the isotropy group is the order 2 cyclic group $\mathbb{Z}/2\mathbb Z$, acting by reflection across some codimension 1 plane in Euclidean space. From this, it follows that for each point $p$ in a codimension 1 stratum, there is an orbifold neighborhood of $p$ homeomorphic to $$\{x=(x_1,...,x_n) \in \mathbb{R}^n \mid \|x\| < 1, x_n \ge 0\}$$ such that each point of that neighborhood corresponding to $x_n=0$ is on a codimension 1 stratum with isotropy group $\mathbb{Z}/2\mathbb{\mathbb Z}$, and each point of that neighborhood corresponding to $x_n>0$ is a regular point. Thus, the union of the regular points and the codimension 1 strata has the structure of an $n$-manifold with boundary, the codimension 1 strata are precisely the boundary points of the manifold, the regular points are precisely the interior of the manifold, and all the rest of the singular locus has codimension $\ge 2$.