On foliated circle bundles over closed orientable 3-manifolds (Q1379589)

This paper is an approach to the following question. Given a circle bundle \(E\) over a closed orientable 3-manifold \(M\), when is there a codimension-one \(C^{\infty}\) foliation on \(E\) whose leaves are transverse to the fibers? If the base \(M\) is a surface, this question has an old answer due to J. Milnor (1957-58) and J.W. Wood (1971): There exists such a foliation if and only if certain inequality is satisfied, which involves the Euler number of the circle bundle and the Euler characteristic of \(M\). For dimension 3 there is a generalization of this Milnor-Wood inequality that is necessary for an affirmative answer. The author shows that, when \(M\) is a Seifert fibred manifold satisfying some mild condition, the Milnor-Wood inequality is also sufficient for the existence of such a \(C^{\infty}\) foliation. The proof is very ingenious, involving the Thurston norm on \(H_{2}(M,Z)\) and Fuchsian groups. The author also constructs a family of circle bundles over closed orientable 3-manifolds with the following property. They have codimension-one \(C^{0}\) foliations transverse to the fibers but have none of class \(C^{3}\). A key step in the non-existence of such foliations of class \(C^{3}\) is a rigidity theorem of E. Ghys: For \(r\geq 3\), certain homomorphisms of surface fundamental groups to the group of \(C^{r}\)-diffeomorphisms of the circle are conjugate to injective homomorphisms into \(PSL(2,R)\) whose images are discrete cocompact groups.