On the Haefliger-Thurston conjecture

Abstract

The classifying space for the framed Haefliger structures of codimension $q$ and class $C^r$ is $(2q-1)$-connected, for $1\le r\le \infty$. 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.

This preprint is withdrawn, for an error occurs at the first line of page 10, while applying Lemma 1.1 to the microfoliation $\mathcal{M}$ of ${\mathcal{G}}_{\alpha }$. There is no global open neighborhood $U$ of the zero section in the total space of the normal bundle, such that $\mathcal{M}|U$ would be $C^r$ over the complement of $\alpha \times 0$, as asked in Lemma 1.1 (page 7, line 8)