On the Haefliger-Thurston conjecture

TL;DR

This paper proves that the classifying space for framed Haefliger structures of a given codimension and regularity is highly connected, leading to various results in foliation theory and diffeomorphism groups.

Contribution

It establishes the connectivity of the classifying space for framed Haefliger structures, advancing understanding of foliations and related geometric structures.

Findings

Classifying space for framed Haefliger structures is (2q-1)-connected.

Results on existence of foliations and foliated bundles.

Insights into homology and perfectness of diffeomorphism groups.

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.