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Every orientable $$3$$-manifold is a $$B\Gamma$$. (English) Zbl 0991.57028
The author proves that a smooth closed orientable 3-manifold $$M$$ admits a $$C^1$$ surface foliation $$\mathcal F$$, each leaf of which has a contractible cover. To obtain $$\mathcal F$$, he uses an open book decomposition $$M=K\coprod N$$ by R. Myers [Proc. Am. Math. Soc. 72, 397-402 (1978; Zbl 0395.57002)] where $$K$$ is a tubular neighborhood of a knot, and $$N$$ is a surface bundle over $$S^1$$ with fiber the punctured surface $$\Sigma$$ such that the boundary curve $$\partial\Sigma$$ is not a meridian for $$K$$. For $$I=[0, 1]$$, he constructs a suitable (i.e. generic and well-tapered) representaton $$\rho: \pi_1(\Sigma)\to \text{Homeo}(I)$$ which is $$C^{\infty}$$ on $$\text{int}(I)$$, and then he applies an argument by D. Pixton [Trans. Am. Math. Soc. 229, 259-268 (1977; Zbl 0361.58005)], on a representation of $${\mathbb Z}\oplus{\mathbb Z}$$ in the group of $$C^1$$ homeomorphisms of $$I$$. By A. Haefliger [Astérisque 116, 70-97 (1984; Zbl 0562.57012)], $$M$$ is a classifying space $$B\Gamma$$ of the holonomy groupoid $$\Gamma$$ of $$\mathcal F$$. The author notes that $$\mathcal F$$ is $$C^1$$ but not $$C^2$$.
##### MSC:
 57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology 58H05 Pseudogroups and differentiable groupoids
##### Keywords:
foliation; classifying space; groupoid; germs of homeomorphisms
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