On flat manifold bundles and the connectivity of Haefliger's classifying spaces.

We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston's conjecture predicts that every M-bundle over a manifold B where \operatorname {dim}(B)\leq \operatorname {dim}(M) is cobordant to a flat M-bundle. In particular, we study the bordism class of flat M-bundles over low dimensional manifolds, comparing a finite dimensional Lie group G with \mathrm {Diff}_0(G). [ABSTRACT FROM AUTHOR]