Holonomy groupoid of a foliation

Remarks

The holonomy germ does not depend on the choice of transversals up to the natural identifications, so the equivalence relation is well defined. Homotopic leafwise paths with fixed endpoints have the same holonomy, so each arrow is a union of leafwise homotopy classes of paths. The holonomy groupoid has a natural structure of a Lie groupoid of dimension $2n - q$, whose manifold of arrows need not be Hausdorff. Its source fibre over $x$ is the holonomy cover of the leaf through $x$.

Example: foliate $\R^2$ by the horizontal lines $y = c$. All holonomy germs are identities, so there is one arrow from $(a, c)$ to $(b, c)$ for all $a, b, c$, and no arrow between points with different second coordinates. The holonomy groupoid is the equivalence relation $\set{((b, c), (a, c))}$, a manifold of dimension $3 = 2 \cdot 2 - 1$.

Non-example: the monodromy groupoid, whose arrows are leafwise homotopy classes of leafwise paths, is in general different. For the foliation of $S^1 \times \R$ by the circles $S^1 \times \set{c}$, a loop going once around a circle has trivial holonomy but is not null-homotopic in its leaf. So it gives the identity arrow of the holonomy groupoid but a nontrivial arrow of the monodromy groupoid. In the Reeb foliation, the compact leaf has nontrivial holonomy, so some loops in it give nontrivial arrows in both.