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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 \(C^2\).
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
58H05 Pseudogroups and differentiable groupoids
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