Closed transversals and the genus of closed leaves in foliated 3-manifolds

Abstract

The classical local theory of integrable 2-plane fields in 3-space leads to interesting qualitative questions about the global properties of solutions surface (i.e., leaves of a foliation) on 3-manifolds. It is now known that foliations admitting a closed leaf of suitably high genus abound on all closed or orientable 3-manifolds that are not rational homology spheres (S. Goodman, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 4414–4415), and this leads to natural questions about the “positions” of such leaves relative to the rest of the foliation. One such question, suggested by Goodman's theorem on closed transversals (S. Goodman, ibid.), is considered here.