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 M -bundle over a manifold B B where dim ⁡ ( B ) ≤ dim ⁡ ( M ) \operatorname {dim}(B)\leq \operatorname {dim}(M) is cobordant to a flat M M -bundle. In particular, we study the bordism class of flat M M -bundles over low dimensional manifolds, comparing a finite dimensional Lie group G G with D i f f 0 ( G ) \mathrm {Diff}_0(G) .