Isotopies of foliated 3-manifolds without holonomy
Let \(M\) be a compact, connected, oriented 3-manifold and \(\mathcal F\) a smooth, transversely oriented foliation of \(M\). There is a transverse, holonomy invariant measure \(\mu\) associated to \(\mathcal F\). Let \([\mu]\) be a foliated class in \(H^1(M;\mathbb{R})\) and \([{\mathcal F}]\) the equivalence class of foliations defined by \([\mu]\). \textit{F. Laudenbach} and \textit{S. Blank} [Invent. Math. 54, 103-177 (1979; Zbl 0435.58002)] prove the following theorem: Let \(M\) be a compact oriented 3-manifold. If \({\mathcal F}_0\) and \({\mathcal F}_1\) are smooth, transversely oriented foliations without holonomy, transverse to \(\partial M\) and such that \([{\mathcal F}_0]=[{\mathcal F}_1]\), then there is a \(C^0\) isotopy of \(M\) carrying \({\mathcal F}_0\) to \({\mathcal F}_1\). If \({\mathcal F}_0\) and \({\mathcal F}_1\) are defined by closed, nowhere vanishing 1-forms \(\omega_0\) and \(\omega_1\), respectively, this isotopy is smooth. If \(\omega_0\) and \(\omega_1\) are scaled so that \([\omega_0]=[\omega_1]\in H^1(M;\mathbb{R})\), the isotopy carries \(\omega_0\) to \(\omega_1\). The authors find a new proof for this theorem based on the Nielsen-Thurston classification of surface automorphisms. They study three cases: periodic, pseudo-Anosov and reducible. For each of these cases the proof is given. Many interesting results have been obtained. Finally, the authors consider the smoothness problems associated to pseudo-Anosov maps and flows.
scientific article; zbMATH DE number 1334986
scientific article