Thurston's relative inequalities and Reeb components (Q554428)

The paper under review considers the relation between different versions of Thurston's inequalities for surfaces in foliated 3-manifolds. Let \(M\) be a closed connected oriented smooth \(3\)-manifold and \({\mathcal F}\) a transversely oriented foliation of codimension 1 in \(M\). Let \(\tau{\mathcal F}\) be the tangent bundle to \({\mathcal F}\) and \(e(\tau{\mathcal F})\in H^2(M;\mathbb Z)\) its Euler class. If \(M\not=S^2\times S^1\) and \({\mathcal F}\) has no Reeb components, then \textit{W. P. Thurston} proved in [Mem. Am. Math. Soc. 59, 99--130 (1986; Zbl 0585.57006)] that the absolute Thurston inequality \(|\langle e(\tau{\mathcal F}),[S]\rangle|\leq-\chi(S)\) holds for each compact oriented embedded surface \(S\). There are two relative versions of this inequality, which are both known to be true whenever \({\mathcal F}\) has no Reeb component. One is Thurston's relative \((+)\)-inequality: it considers oriented links \(\Gamma\) which are positively transverse to \({\mathcal F}\) and their Seifert surface \(S\) and states \(-<e(\tau{\mathcal F})_\Gamma,[S,\Gamma]>\leq-\chi(S)\) for the relative Euler class \(e(\tau{\mathcal{F}})_\Gamma\). The other one is Thurston's relative \((\pm)\)-inequality: it considers oriented transverse (not necessarily positively transverse) links and states \(|\langle e(\tau{\mathcal F})_\Gamma,[S,\Gamma]\rangle|\leq-\chi(S)\). The paper under review considers the logical relation between these 3 inequalities for foliations with Reeb components. \textit{Y. Mitsumatsu} had proved in [``Convergence of contact structures to foliations'', in: Walczak, P. (ed.) et al., Foliations 2005. Proceedings of the international conference, University of Łodź, Łodź, Poland, June 13--24, 2005. Hackensack, NJ: World Scientific. 353--371 (2006; Zbl 1222.53090)] that for spinnable foliations the absolute inequality is implied by the relative \((+)\)-inequality. However the paper under review produces examples of foliations and knots such that the relative \((+)\)-inequality but neither the absolute nor the relative \((\pm)\)-inequality holds. Moreover it is proved in the paper under review that a foliation with Reeb components satisfies the relative \((\pm)\)-inequality if and only if it is obtained from a foliation without Reeb components by a monotone modification along a positively transverse link.