Existence of Reeb components and periodic transverse orbits in codimension one foliations

Paul Schweitzer S.J. (PUC-RJ)

Resumo: (Joint with F. Alcalde e G. Hector) We show that under a certain condition a homological (m - 2)-dimensional vanishing cycle in a C1 transversely oriented codimension one foliation F of a closed oriented oriented m-manifold M lies on the boundary of a homological Reeb component. This extends Novikov's famous theorem [1] to higher dimensions. A (homological) Reeb component with connected boundary is a compact foliated manifold whose interior bers over the circle with the leaves as bers and whose boundary is a single compact leaf. The Lefschetz index can be used to show that in many cases every transverse ow must have a periodic orbit.

[1] S.P. Novikov, Topology of foliations. Trans. Moscow Math. Soc., 14 (1965), 268-304..