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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Every orientable 3–manifold is a $\mathrm{B}\Gamma$

Danny Calegari

Algebraic & Geometric Topology 2 (2002) 433–447

arXiv: math.GT/0206066

Abstract

We show that every orientable 3–manifold is a classifying space BΓ where Γ is a groupoid of germs of homeomorphisms of . This follows by showing that every orientable 3–manifold M admits a codimension one foliation such that the holonomy cover of every leaf is contractible. The we construct can be taken to be C1 but not C2. The existence of such an 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.

Keywords
foliation, classifying space, groupoid, germs of homeomorphisms
Mathematical Subject Classification 2000
Primary: 57R32
Secondary: 58H05
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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