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?


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 $H(B\Gamma)$.

  • $\begingroup$ 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? $\endgroup$ – Sebastian Goette Feb 19 '17 at 20:05
  • $\begingroup$ 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. $\endgroup$ – Gael Meigniez Feb 19 '17 at 20:14
  • $\begingroup$ 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. $\endgroup$ – Jim Stasheff Feb 20 '17 at 21:38

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