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 $$\text{dim}$(B)\leq $\text{dim}$(M)$iscobordanttoaflat$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 ${\text{Diff}}_0(G)$ and localizing the holonomy of flat M-bundles to be supported in a ball.