# What is the relation between the holonomy groupoid of a foliation and the corresponding Haefliger groupoid?

Given a foliation, there is a holonomy groupoid and a classifying map to the Haefliger classifying space via the Haefliger groupid. What is the relation between these groupids?

## 1 Answer

Write $$F$$ your foliation, $$M$$ its ambiant manifold, $$q=dim(M)-dim(F)$$ its codimension. The holonomy groupoid $$H(F)$$, if I'm correct, is the set of classes of triples $$(x,\gamma,y)$$ where $$\gamma$$ is a tangential path; and $$(x,\gamma,y)~(x,\gamma',y)$$ iff $$\gamma$$ and $$\gamma'$$ have the same holonomy.

I'm not expert enough in groupoids to use the proper vocabulary, but the relation is as follows between the holonomy groupoid $$H(F)$$ of your foliation and the universal groupoid $$H(B\Gamma_q)$$. Let $$c:M\to B\Gamma_q$$ be the Haefliger classifying map of $$F$$. Then, $$F$$ is the pullback of the universal foliation on $$B\Gamma_q$$ (whatever this means) through $$c$$. In particular, there is an induced groupoid morphism $$C: H(F)\to H(B\Gamma_q)$$. Moreover, one has a partial injectivity property: for $$(x,\alpha,y)$$ and $$(x,\beta,y)$$ in $$H(F)$$ with the same endpoints, one has $$C(x,\alpha,y)=C(x,\beta,y)$$ iff $$\alpha=\beta$$. Does this help?

On the other hand, if you rather mean to compare the holonomy groupoid $$\Gamma=H(F)$$ of the given foliation with the holonomy groupoid $$H(B\Gamma)$$ of the Haefliger classifying space of $$\Gamma$$, then the continuous classifying map $$c:M\to B\Gamma$$ induces an equivalence. Precisely, $$H(B\Gamma)$$ is simple (at most 1 arrow between two units) and $$c$$ induces a bijection between the set of orbits of $$\Gamma$$ and the set of orbits of $$H(B\Gamma)$$.

• Welcome to math overflow. Try to use TeX whenever possible. Your description of the holonomy groupoid looks correct. At the end, when you say $\alpha=\beta$, you probably mean that the holonomy classes of these paths are equal? – Sebastian Goette Feb 19 '17 at 20:05
• Thank you for the welcome; yes $\alpha=\beta$ means the same holonomy class. Reading Jim's question again, I'm not sure any more if he means the universal Haefliger classifying space B\Gamma_q, or the classifying space of the groupoid (or pseudo-group) of his foliation. – Gael Meigniez Feb 19 '17 at 20:14
• Thanks to you both. I'm happy with the universal Haefliger classifying space B\Gamma_q but if `his foliation' means the one I asked about, I'd be happy with that also. – Jim Stasheff Feb 20 '17 at 21:38