On the Haefliger-Thurston conjecture

The classifying space for the framed Haefliger structures of codimension $q$ and class $C^r$ is $(2q-1)$-connected, for $1\le r\le \mathrm{\infty }$. The corollaries deal with the existence of foliations, with the homology and the perfectness of the diffeomorphism groups, with the existence of foliated products, and of foliated bundles.