Graph manifolds and taut foliations
Recall that a manifold \(M\) is called a graph manifold if there is a collection \(\mathcal T\) of disjoint tori embedded in \(M\) such that the manifold obtained by splitting \(M\) open along \(\mathcal T\) is a (non-necessarily connected) Seifert-fibered space. Recall also that a foliation \({\mathcal F}\) is taut if every leaf has a closed loop passing through it which is everywhere transverse to the leaves of \(\mathcal F\). In this paper, the authors prove the following results: (1) There exist infinitely many graph manifolds which admit foliations without Reeb components but no taut foliations. (2) There exist three Seifert-fibered spaces \(M\) for which every taut foliation must have a (non-separating) torus leaf, and each space admits taut foliations. (3) There exist infinitely many graph manifolds \(M\) which admit \(C^0\) taut foliations with no compact leaf, but every \(C^2\) foliation must have a (separating) torus leaf. In particular, each manifold admits no \(C^2\) taut foliation. (4) There exists infinitely many graph manifolds \(M\) which admit \(C^0\) taut foliations with no compact leaf, but no Anosov flows. In the last part of the paper, the authors propose several interesting open questions.
scientific article; zbMATH DE number 1145700