#### Volume 2, issue 1 (2002)

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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 $B\Gamma$ 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 $\mathsc{ℱ}$ such that the holonomy cover of every leaf is contractible. The $\mathsc{ℱ}$ we construct can be taken to be ${C}^{1}$ but not ${C}^{2}$. The existence of such an $\mathsc{ℱ}$ 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
Primary: 57R32
Secondary: 58H05