nLab
Haefliger groupoid

Definition

For $n\in \mathbb{N}$, the Haefliger groupoid ${\Gamma }^n$ is the groupoid whose set of objects is the Cartesian space ${ℝ}^n$ and for which a morphism $x\to y$ is a germ of a diffeomorphism $({ℝ}^n,x)\to ({ℝ}^n,y)$.

Properties

Geometric structure

The Haefliger groupoid is naturally a topological groupoid. As such it is an étale groupoid.

Classification of foliations

The Haefliger groupoid classifies foliations. See at Haefliger theorem.

References

Original articles include

• André Haefliger, Groupoïdes d’holonomie et espaces classiants , Astérisque 116 (1984), 70-97

• Raoul Bott, Lectures on characteristic classes and foliations , Springer LNM 279, 1-94

A textbook account is in

• Ieke Moerdijk, Janez Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics 91, 2003. x+173 pp. ISBN: 0-521-83197-0

• Wikipedia, Haefliger structure