Holonomy groupoid of a foliation

Summary

The holonomy groupoid of a foliation is a Lie groupoid whose objects are the points of the manifold. Its arrows are paths inside leaves, where two paths count as equal when they have the same holonomy. Its orbits are the leaves, and its isotropy groups are the holonomy groups of the leaves. It replaces the leaf space, which is often a badly behaved topological space, by a smooth object. It is the starting point of the noncommutative geometry of foliations, through its convolution algebra.