Every orientable 3-manifold is a BΓ

الأصل · EN

We show that every orientable 3-manifold is a classifying space BΓwhere Γis a groupoid of germs of homeomorphisms of R. This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. The F we construct can be taken to be C¹ but not C². The existence of such an 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 = BΓfor some C∞ groupoid Γ.