Algebraic and Geometric Topology 2 (2002), paper no. 21, pages 433-447.

Every orientable 3-manifold is a B\Gamma

Danny Calegari

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 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\Gamma for some C^\infty groupoid \Gamma .

Keywords. Foliation, classifying space, groupoid, germs of homeomorphisms

AMS subject classification. Primary: 57R32. Secondary: 58H05.

DOI: 10.2140/agt.2002.2.433

E-print: arXiv:math.GT/0206066

Submitted: 25 March 2002. Accepted: 28 May 2002. Published: 29 May 2002.

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

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