Geodesic
Every orientable 3–manifold is a BΓ
Calegari, Danny1
1Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
Algebraic and Geometric Topology, Tome 2 (2002) no. 1, pp. 433-447
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We show that every orientable 3–manifold is a classifying space BΓ 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 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 = BΓ for some C∞ groupoid Γ.

DOI : 10.2140/agt.2002.2.433
Keywords: foliation, classifying space, groupoid, germs of homeomorphisms

Calegari, Danny  1

1 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA 
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Calegari, Danny. Every orientable 3–manifold is a BΓ. Algebraic and Geometric Topology, Tome 2 (2002) no. 1, pp. 433-447. doi: 10.2140/agt.2002.2.433

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