Haefliger structure

Summary

A Haefliger structure is a generalisation of a foliation in which the local defining maps need not be submersions. It is given by local maps to $\R^q$ glued by germs 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 $B\Gamma_q$. This homotopy theory is the basis of the existence theory of foliations.