## Algebraic & Geometric Topology

### Every orientable 3–manifold is a $\mathrm{B}\Gamma$

Danny Calegari

#### Abstract

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 $Γ$.

#### Article information

Source
Algebr. Geom. Topol., Volume 2, Number 1 (2002), 433-447.

Dates
Accepted: 28 May 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882701

Digital Object Identifier
doi:10.2140/agt.2002.2.433

Mathematical Reviews number (MathSciNet)
MR1917061

Zentralblatt MATH identifier
0991.57028

#### Citation

Calegari, Danny. Every orientable 3–manifold is a $\mathrm{B}\Gamma$. Algebr. Geom. Topol. 2 (2002), no. 1, 433--447. doi:10.2140/agt.2002.2.433. https://projecteuclid.org/euclid.agt/1513882701

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