# Is there a "geometric" language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions.

Here, let me describe this groupoid. Take a foliated manifold, which for my question might as well have constant rank. It defines various different groupoids with object-set the manifold. An important one, but not the one who's hanging out in my office, is the one whose morphisms are homotopy classes of paths that are tangent to the foliation. I believe a result of Crainic and Fernandes holds that this groupoid can be equipped with a structure of a Lie groupoid.

My groupoid is conceptually similar, but a result like Crainic and Fernandes' fails. Namely, in my groupoid is nothing more than an equivalence relation: the objects are again the points of the manifold, and two objects are isomorphic iff they are in the same leaf of the foliation, and then isomorphic in a unique way.

I believe that for general foliations, my groupoid cannot be equipped with a Lie structure. Indeed, I'm having trouble even topologizing it, although I guess I can topologize it as a quotient of the Crainic-Fernandes groupoid (which is a "source- simply connected cover" of my groupoid). But I want even more: I'd like a good language that describes things like its smooth structure, and I'd much rather refer to existing (presumably more general) literature than derive just what I need ad hoc.

I've seen things like "Lie groupoids where the morphisms space may not be Hausdorff" (sometimes called "differentiable groupoids"), and also "Lie groupoids where the morphism space may be a stack rather than a manifold (sometimes called "Weinstein groupoids"). Since I don't myself have much understanding of my groupoid except in terms of its underlying sets, I don't know whether either of these notions is useful. Any help would be appreciated.

• I like the fact you have groupoids in your office. :) I wish I had a couch for my groupoids to sit on! Jan 13, 2011 at 6:21
• You may have a locally Lie groupoid, and such will usually have an associated holonomy Lie groupoid. See the papers I refer to below. Jan 26, 2012 at 14:08
• I don't know whether this helps: you can find more information on this groupoid in the book by Moore and Schochet "Global Analysis of Foliated Spaces". The "equivalence relation groupoid" is an example of a Borel groupoid. Mar 30, 2012 at 13:10
• For a foliation F, I think the usual groupoid associated to a foliation is $[x,y,\alpha]$ where $\alpha$ is aF- tangent curvr from x to y. The equivalent relation is that $[x,y,\alpha\sim [x,y,\beta]$ iff $\alpha,\beta$ induce the same holonomy.@DavidRoberts But before we topologize it, we give an atlas of chart for this groupoid then it get topology automotically(using sum or weak topology). I think this is the standard groupoid associated to a foliation which is always haussdorf provided the foliation is real analytic. Nov 17, 2020 at 20:27
• I recommend this reference and reference 11 of this paper ehu.eus/~mtwmastm/JMS.pdf Nov 17, 2020 at 20:31

I'm being thick, but surely you can topologize the space of arrows as a subspace of $M\times M$ where $M$ is the space of objects? It may be a horrible space, but it exists. As far as naming the thing goes, I would call it a presentation of the stack-replacement of the space of leaves or similar. If $L = M/\sim$ is the space of leaves (which admittedly is 'bad', so let us assume it is 'good' for now), then your groupoid is the Cech groupoid of the canonical map $M \to L$. It is morally equivalent to the space of leaves if they are considered as stacks (leaving aside the question of whether your groupoid actually presents a stack for now). As far as a reference goes, how about the 1989 paper of Jean Pradines 'Morphisms between spaces of leaves viewed as fractions' Numdam/arXiv (the latter is an update of the original published version). The only drawback to that paper is that it uses language inherited from the Ehresmann school of category theory, which is quite ideosyncratic (there is a bit of a dictionary provided in appendix A or the arXiv version). Perhaps though for your purposes only section 1 will be necessary.

• Well, yes, I can topologize it that way. Thanks for the reference! Jan 14, 2011 at 4:43
• Ah, so one of the reasons I don't like that topologization is that when a leaf wraps densely around something, I'd rather the corresponding subset Hom(x,-) in the space of morphisms to nevertheless be a manifold. The example is a foliation on a torus by irrational lines; then the "leaf space" groupoid is the Lie groupoid for the R-action, so I'd like the source fibers to be lines, whereas topologizing as a subset of the pair space makes the open sets on each fiber complicated. Jan 16, 2011 at 23:35
• Hmm, yes. Pradines makes some point about focusing on foliation groupoids that satisfy some conditions (regular or simple) but I don't know what these entail or imply. Jan 17, 2011 at 6:40

I do not exactly understand your problem but it is possible that some developments and expositions of other ideas of Pradines might help, namely

R. Brown and M. E.-S. A.-F. Aof, The holonomy groupoid of a locally topological groupoid'', Top. and its Appl., 47 (1992) 97-113.

R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid'', Cah. Top. G\'eom. Diff. Cat 36 (1995) 345-369.

R. Brown and O. Mucuk, Foliations, locally Lie groupoids, and holonomy'', Cah. Top. G\'eom. Diff. Cat. , 37 (1996) 61-71.

These papers are related to

Pradines, Jean, "Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales." C. R. Acad. Sci. Paris Sér. A-B 263 1966 A907–A910.

where the first paper explains Théoreme 1, and the second paper explains Théoreme 2, of that Note, following explanations given to me by Jean in 1981 in Toulouse. We have other papers on these ideas, e.g. on local subgroupoids.

I thought these ideas of Jean were great and needed a full exposition. The first paper shows how a holonomy groupoid arises from an "iteration of local procedures", the local procedures being defined by a locally topological, or smooth, groupoid. Also this holonomy groupoid satisfies a universal property.

If your foliation is "regular", which roughly means the leaves have constant dimension, then every chart of the manifold has the form $$L \times T$$, where $$L$$ is the longitudinal (leafwise) direction and $$T$$ the transeversal direction. Moreover, the change of coordinates is of the form $$(x,y) \mapsto (f(x,y),h(y))$$. (This is thanks to the Frobenius theorem!). I mean the second coordinate depends only on the transversal direction. This particular map $$h : T_1 \to T_2$$ is called a holonomy. (It's a "small" holonomy, meaning it works for points close enough to each other.)

So now take two points $$p=(x_1,y_1)$$ and $$q=(x_2,y_2)$$ in the same leaf $$L$$, so that the pair $$(p,q)$$ is an arrow of your groupoid. For simplicity, say $$p$$ and $$q$$ are close to each other. Take $$T_1$$ and $$T_2$$ transversals to $$L$$, at the points $$p$$ and $$q$$ respectively. So the pair $$(p,q)$$ has coordinates $$L \times T_1 \times T_2 \times L$$.

But actually $$T_2$$ in these coordinates is redundant, because $$y_1$$ and $$y_2$$ are identified via the "small" holonomy diffeomorphism $$h$$ (that's why we assumed $$p$$ and $$q$$ are close to each other, so that there is a change of coordinates map).

Now, if $$p$$ and $$q$$ are far from each other, then you just do the classical trick: First you connect them with a smooth path $$\gamma$$ which stays on the leaf $$L$$. Then, $$\gamma$$ being compact, you cover it with a finite number of foliation charts. So you get a "big" holonomy $$h_1 \circ \ldots h_k$$.

So a charts of your groupoid at $$(p,q)$$ is of the form $$L \times T \times L$$. There you have it, your groupoid is a Lie groupoid, and its dimension is twice the dimension of the leaf plus the dimension of the transversal.

Your groupoid is really the "graph" of the foliation (the equivalence relation as you say).

If your foliation is singular, which means that the dimension of the leaves drops (in a semi-continuous way), then the graph is just a topological groupoid. It is a quotient of the holonomy groupoid of the foliation, which is also a topological groupoid in general. In fact, both of these groupoids carry a diffeological structure, so you can still do differential geometry with them. This structure is defined by the notion of bisubmersion.

• Hi Lakovos, Thanks. What you write is of course completely correct. It's hard for me to remember what I knew and didn't know 10 years ago, but I think this was things past-me knew. What I was trying to ask about in my question was exactly the non-regular case. Oct 11, 2021 at 11:27
• Theo In your question I read "constant rank" so I thought it was about regular foliations. Maybe by "constant rank" you mean the "almost regular" case. The easiest way to see this, is to start from the vector fields.That's a $C^{\infty}(M)$-submodule $F$ of which is projective. So $F$ is the sections of a Lie algebroid (just think of the Serre-Swan theorem), with anchor map injective on an open dense subset. (e.g. action of $\C^{\ast}$ on $\R^2$). For such foliations the holonomy Lie groupoid integrates F. Its smooth structure is in Debord's work. The graph may not be a Lie groupoid. Oct 24, 2021 at 11:24
• A "truly" singular foliation is a non-projective $F$. (e.g action of $GL(2,\R)$ on $\R^2$.) The holonomy groupoid in this case is NEVER a Lie groupoid, just a topological groupoid. There is also the graph, a quotient of this groupoid. I suppose the standard reference is my paper "The holonomy groupoid of a singular foliation". Oct 24, 2021 at 11:27
• What all this says is: If you want to have a groupoid which: a) is a Lie groupoid b) orbits are the leaves c) is "minimal" (meaning smallest isotropy) - then the only case is for an almost regular foliation. If your foliation is "too singular", then you are forced to pass to topological groupoids. The point here is that "too singular" has nothing to do with the dimension of the leaves, as the previous examples show. This is because, in the truly singular case, there is no longer a bijective correspondence between the partition to leaves and the module of vector fields target to them. Oct 24, 2021 at 11:29
• Sorry, in the last sentence I wanted to write "...tangent to them", not "target to them". So, if you take the partition of $\R^2$ to the origin and its complement (2 leaves), and you consider all the vector fields tangent to these leaves, you obtain the infinitesimal generators of the $GL(2)$ action I mentioned before. But there are other modules as well, for instance the inf. generators of the action of $\C^*$, which is a projective module. Another one is the inf generators of the $SL(2,\R)$ action. It's not a projective module, but "less" singular than the $GL(2)$ action (smaller isotropy). Oct 24, 2021 at 11:47