Orbifold fundamental group in terms of loops?

Question

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 "Later we shall interpret $\pi_1(O)$ in terms of loops on $O$, but this interpretation doesn't seem to appear in his notes.

My question is, well, what is this interpretation, precisely?

Here are my thoughts so far:

The example I'm currently interested in is the 1-D orbifold S¹/ℤ₂. Its universal cover is ℝ. Deck transformations are generated by a translation by 2π, which I'll call T and a reflection about the origin R. There's relations $R^2=1$ and $TR=RT^{-1}$. If I haven't messed up, this is the same as the fundamental group of the Klein bottle as well (if someone can explain how to construct the Klein bottle from S¹/ℤ₂, I would greatly appreciate it as well!). How can I relate paths on S¹/ℤ₂ to loops in the Klein bottle? Oops, I mixed up something in my head. I'll have another question on this in the future, perhaps.

I think my main trouble is making all of these observations precise, so a good reference with standard terminology / theorems (with lots of examples like the one I've been thinking about) would be appreciated as well.

Answer 1

Most of the standard intro sources on orbifolds discuss their fundamental groups in terms of coverings. One exception is Ratcliffe's book "Foundations of Hyperbolic Manifolds", chapter 13 of which contains a discussion of the fundamental group of an orbifold defined via loops.

Answer 2

I guess that the easiest way to think of this is to consider a disc with a singular point $x$ of index $2$. Take a loop $\gamma$ that turns once around $x$. Then in the universal cover of the disc, $\gamma$ lifts to a path $\tilde\gamma$ whose endpoints do not coincide (they are the two lifts $y_1,y_2$ of $\gamma(0)$). You cannot homotope it to a constant while keeping its endpoints over $\gamma(0)$ (which you would need to get an homotopy of $\gamma$ with fixed base point). Therefore this loop is not trivial.

Consider now $2\gamma$. Then in the universal cover, it lifts to a loop (that goes from $y_1$ to $y_2$, then from $y_2$ to $y_1$, continuing to turn around the (unique) lift of $x$. But now this universal cover is a disc (without singular point) and you can homotope your loop to a constant (based on $y_1$ with the above notations). This homotopy projects to an homotopy of $2\gamma$, which is zero in the fundamental group.

Answer 3

To understand the difficulties inherent in forming a "path-based" definition of the orbifold fundamental group, it is good to ponder Serre's definition of a fundamental group of a graph of groups, given in his book "Trees".

Answer 4

If you want to think in terms of loops, then choose a local lift of each loop in every chart of some atlas on such a way that they agree by some of transition maps. Then think about homotopy with such liftings.

In case your orbifold is spin and oriented and dim $\ge 3$, you may also pass to the double cover of frame bundle and think there.

Answer 5

Here are two more references where the fundamental group of an orbifold is defined in terms of loops:

The chapter by Haefliger on "orbi-espaces" in this book:

E. Ghys et P. de la Harpe. Sur les groupes hyperboliques d'après Mikhael Gromov, Progr. Math., 83, Birkhäuser Boston, Boston, MA, 1990.

and Chapter 2 of the book

M. Boileau, S. Maillot, and J. Porti. Three-dimensional orbifolds and their geometric structures. Panoramas et Synthèses, 15. Société Mathématique de France, Paris, 2003

where the "local lift" approach mentioned in Anton Petrunin's answer is implemented.

Answer 6

Given a pointed orbifold $(X,x)$ with $x$ in the smooth locus of $X$, and its pointed universal cover $(\widetilde{X},\widetilde{x})$, there is a natural correspondence between homotopy classes of loops based at $x$ and deck transformations of $\widetilde{X}$. Given a choice of loop in a homotopy class, there is a unique lift to a path in $\widetilde{X}$, and the endpoint is the image of $\widetilde{x}$ under a unique deck transformation. The endpoint is independent of homotopy class representative, and uniqueness of the deck transformation uses the smooth locus condition. Given a deck transformation, a homotopy class of paths from $\widetilde{x}$ to its image maps to a homotopy class of loops on $X$ based at $x$.

I think the main difficulty in translating the correspondence from the topological space picture comes from internalizing the definition of path on an orbifold.

Answer 7

If you think about the orbifold as being presented by a Lie groupoid (which is true in the example you mentioned, $S^1/\mathbb{Z}_2$, and any cases of an almost free action of a Lie group - almost free meaning the stabilisers are finite), then you can calculate the fundamental group using an approach going back, in essence, to Haefliger, but more recently by Moerdijk-Mrcun (Lie groupoids, sheaves and cohomology, in LMS Lecture Notes 323, 2005) and Colman (On the homotopy 1-type of Lie groupoids [arXiv:math/0612257]).

Essentially, the interval in replaced by a family of objects which are resolutions of the interval, and take into account the fact that arrows in a Lie groupoid are, from the point of view of the homotopy type of the groupoid, on an equal footing with paths in the object space.

Answer 8

There is a ambiguity about orbifold homotopy, depending of the category you consider. Regarded as a diffeological space — see Orbifold as Diffeology — and considering the general homotopy theory of diffeological spaces. This will give you, for the classical example ${\cal O}_m = {\bf R}^2/{\bf Z}_m$, the trivial homotopy, since ${\cal O}_m$ is smoothly contractible. But you can get a refinement by looking at the principal orbit of ${\rm Diff}({\cal O}_m)$ acting on ${\cal O}_m$ itself, which is ${\cal O}_m - \{0\}$. Contrarily to manifolds where the group of diffeomorphisms is transitive, the lack of transitivity of the group of diffeomorphisms for orbifolds (the structure groups of an orbifold are diffeological invariants) gives you a family of homotopy groups in each degree, one for each orbit, and especially for the principal orbit. I don't really looked how the homotopy for diffeological orbifolds, or the homotopy of the orbits of the group of diffeomorphisms, and the structure groups of the orbifolds (here ${\bf Z}_m$ at the origin) are related, but it has its own logic and it is a diffeological invariant (I consider only what is called "effective orbifolds"). I'm not sure I answer the question, but it may give some directions to dig in.

Note that, the singularities of a diffeological space are defined by the orbits of the group of diffeomorphisms, so this homotopy described above for orbifolds is a specialization of the general case.

Answer 9

There is also a natural interpretation of the orbifold fundamental group in terms of loops, using an extended version of a Wirtinger presentation. Let's start out with the closed case and mention the cusped case at the end.

As a warm-up, consider an orbifold that has singular locus a link with underlying space $S^3$. To compute the fundamental group of the orbifold can be computed first from the Wirtinger presentation of the link and then by introducing torsion relations for each meridian for example if the cone angle along the link is $2\pi/3$ everywhere, then each meridian $\mu_i$ should have the relation $\mu_i^3$. A loop around a link component does not bound a (smooth immersed) disk in the orbifold, instead it bounds a disk with a cone point of order 3. However $\mu_i^3$ does bound a disk in the orbifold, and is trivial.

To make this interpretation general, we need to consider 3-orbifolds more generally. A geometric 3-orbifold is really a trivalent graph with edges decorated by torsion (or cone angles, depending on taste) embedded in a 3-manifold, and so the underlying manifold, the embedding, and the trivalent points need to be accounted for.

In reverse order, the trivalent points are introduce relations abc=1 (compare to finite subgroups of SO(3) which are actually the isotropy subgroups that fix these types of points). The next two conditions really have to be considered together. Really what needs to be computed is a Wirtinger presentation of the complement of a trivalent graph in the underlying space. Then quotient out by the torsion relations and relations coming from the trivalent points.

For cusped (geometric) manifolds, there are extra ways to decorate the graph, namely there can be trivalent points where the orders of torsion along edges incident to the these vertices are (2,4,4), (2,3,6) or (3,3,3) and there could be 4-valent points where the torsion orders of the edges are (2,2,2,2).

It should be pointed out that such a computation can also be done by Damian Heard's ORB, which is extremely useful if a large example needs to be considered.

Answer 10

This path approach is completely explained in the book by Bridson and Haefliger "Metric spaces of nonpositive curvature" published in Springer Verlag. Maybe the language of groupoid has to be learned. Other references by Haefliger are in French unfortunately.