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:
- "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]
- "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]