Every orientable 3–manifold is a BΓ

We show that every orientable $3$ –manifold is a classifying space $B Γ$ where $Γ$ is 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 $C^{1}$ but not $C^{2}$ . 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 $M = B Γ$ for some $C^{\infty}$ groupoid $Γ$ .

Keywords

foliation, classifying space, groupoid, germs of homeomorphisms

Mathematical Subject Classification 2000

Primary: 57R32

Secondary: 58H05

References

Forward citations

Publication

Received: 25 March 2002

Accepted: 28 May 2002

Published: 29 May 2002

Authors

Danny Calegari
Department of Mathematics Harvard University Cambridge, MA 02138 USA

Volume 2, issue 1 (2002)

Danny Calegari