We show that there exists a family of smooth orientable circle bundles over closed orientable 3-manifolds each of which has a codimension-one foliation transverse to the fibres of class C0 but has none of class C3. There arises a necessary condition induced from the Milnor-Wood inequality for the existence of a foliation transverse to the fibres of an orientable circle bundle over a closed orientable 3-manifold. We show that with some exceptions this necessary condition is also sufficient for the existence of a smooth transverse foliation if the base space is a closed Seifert fibred manifold.
Cite this article
Shigeaki Miyoshi, On foliated circle bundles over closed orientable 3-manifolds. Comment. Math. Helv. 72 (1997), no. 3, pp. 400–410DOI 10.1007/S000140050024