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Given an orbifold $\mathcal{O}$, I have seen many ways to define the orbifold fundamental group:

  • Thinking of $\mathcal{O}$ as a groupoid $\mathcal{G}$, $\pi_1^{orb}(\mathcal{O})$ can be defined as the fundamental group of the classifying space $B\mathcal{G}$
  • Thinking of $\mathcal{O}$ as a Riemannian orbifold, one can take the frame bundle $Fr(\mathcal{O})$, which is a manifold with an almost free $O(n)$-action, and then one can define $\pi_1^{orb}(\mathcal{O})=\pi_1(Fr(\mathcal{O})\times_{O(n)}EO(n))$ (this, I am pretty sure, is the same as the previous definition, but as a Riemannian geometer I prefer this definition)
  • Letting $X$ denote the disjoint union of manifold charts of $\mathcal{O}$, with an action of a pseudogroup $\mathcal{H}$, one can define $\pi_1^{orb}(\mathcal{O})$ in terms of ``$\mathcal{H}$-loops'' on $X$ (i.e. union of paths in $X$, connected into a loop via the pseudogroup $\mathcal{H}$) modulo a natural notion of homotopy.

There are more definitions (e.g. in terms of deck transformations) but I want to focus on these three, in view of generalizing this to higher homotopy groups.

When $n\geq 2$, I have seen the definition of $\pi_n^{orb}(\mathcal{O})$ in the literature only in terms of $\pi_n(B\mathcal{G})$ or as $\pi_n(Fr(\mathcal{O})\times_{O(n)}EO(n))$ (which again, I am pretty sure are litterally the same thing) but never in terms of $\mathcal{H}$-maps $(D^n,\partial D^n)\to X$ modulo homotopy.

Question: does anyone have any reference, where I can find the definition of $\pi_n^{orb}(\mathcal{O})$ in this last way, possibly with a proof that this is equal to the other definitions? I imagine this is nothing too hard, but it is somewhat technical and I would like to avoid having to reinvent the wheel.

Thank you in advance!

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  • $\begingroup$ In my thesis I defined the fundamental bigroupoid of a topological groupoid, and orbifolds can be seen as special cases of these. From this definition one can extract $\pi_2$ of the groupoid, as well as the fundamental group, in the manner of the third definition you give above. I proved that for certain groupoids this gives the expected answer, but not for arbitrary orbifolds, but I expect that proving the classifying space model to be doable. $\endgroup$
    – David Roberts ♦
    Commented Nov 9, 2021 at 10:03
  • $\begingroup$ Thank you David for your answer, I will look into your thesis then. Did you also publish your result in a journal? That would be great for citation purposes. $\endgroup$
    – CuriousUser
    Commented Nov 9, 2021 at 13:21
  • $\begingroup$ No, unfortunately, I hadn't gotten around to that bit. You can cite my thesis, though... $\endgroup$
    – David Roberts ♦
    Commented Nov 9, 2021 at 20:33

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