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

TL;DR

This paper explores a conjecture relating flat manifold bundles and cobordism, analyzing the bordism classes of such bundles over low-dimensional manifolds and comparing different Lie group structures.

Contribution

It provides new insights into Haefliger-Thurston's conjecture by studying the bordism classes of flat bundles and comparing Lie group structures in this context.

Findings

Flat M-bundles over low-dimensional manifolds are classified up to bordism.

Comparison between finite-dimensional Lie groups G and Diff_0(G) in the context of flat bundles.

Localization of holonomy in flat M-bundles to a ball supports the conjecture.

Abstract

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