---
tag: HMRP135M6K
kind: concept
name: Holonomy groupoid of a foliation
msc: [57R30, 22A22, 58H05]
specializes: [HNVW0E732J]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_groupoid
---

## Summary

The holonomy groupoid of a [[foliation|98HTB7MSPX]] is a [[Lie groupoid|HNVW0E732J]] 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.

## Explanations

### L1 human

A foliation cuts a manifold into leaves, like the pages of a book, but the leaves can wind around in complicated ways.
To compare two points of the same leaf one walks from one to the other inside the leaf.
Walking along a path also drags nearby leaves along, and the way they are moved is the holonomy of the path.
The holonomy groupoid has an arrow for each way of walking from one point to another.
Two walks count as the same when they move nearby leaves in the same way.
For the foliation of the plane by horizontal lines, there is exactly one arrow between two points on the same line and none between points on different lines.