---
tag: 8GYWYYBFWE
kind: concept
name: Haefliger structure
msc: [57R32, 57R30, 58H05]
sources:
  - url: https://en.wikipedia.org/wiki/Haefliger_structure
same_as:
  wikidata: Q5638159
  wikipedia: Haefliger structure
---

## Summary

A Haefliger structure is a generalisation of a [[foliation|98HTB7MSPX]] in which the local defining maps need not be submersions.
It is given by local maps to $\R^q$ glued by [[germs|VZXF7RNFKS]] of local diffeomorphisms.
In other words, it is a cocycle with values in the groupoid of such germs.
Unlike foliations, Haefliger structures can be pulled back along any continuous map.
Up to homotopy, they are classified by maps into a [[classifying space|WSZ40H5E4T]] $B\Gamma_q$.
This homotopy theory is the basis of the existence theory of foliations.

## Explanations

### L1 human

A foliation of codimension $q$ is locally described by a submersion to $\R^q$, whose level sets are the pieces of leaves.
On overlaps the local submersions differ by local diffeomorphisms of $\R^q$.
A Haefliger structure keeps the gluing data but drops the requirement that the local maps be submersions.
A constant map to $\R^q$ already gives one, even though it does not foliate anything.
This flexibility makes Haefliger structures behave like vector bundles: they pull back along maps and have a classifying space.