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On flat manifold bundles and the connectivity of Haefliger’s classifying spaces
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Proc. Amer. Math. Soc. 152 (2024), 4943-4957

DOI: https://doi.org/10.1090/proc/16941

Published electronically: September 24, 2024

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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 $\operatorname {dim}(B)\leq \operatorname {dim}(M)$ 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 $\mathrm {Diff}_0(G)$.

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