Every orientable 3–manifold is a $\mathrm{B}\Gamma$

Abstract

We show that every orientable $3$–manifold is a classifying space $B\Gamma$ where $\Gamma$ is a groupoid of germs of homeomorphisms of $ℝ$. This follows by showing that every orientable $3$–manifold $M$ admits a codimension one foliation $\mathcal{ℱ}$ such that the holonomy cover of every leaf is contractible. The $\mathcal{ℱ}$ we construct can be taken to be $C^1$ but not $C^2$. The existence of such an $\mathcal{ℱ}$ answers positively a question posed by Tsuboi [Classifying spaces for groupoid structures, notes from minicourse at PUC, Rio de Janeiro (2001)], but leaves open the question of whether $M=B\Gamma$ for some $C^{\infty }$ groupoid $\Gamma$.