#### 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.

Notes on file formats
Danny Calegari

Department of Mathematics, Harvard University

Cambridge MA, 02138, USA

Email: dannyc@math.harvard.edu

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