Foliations and the topology of 3-manifolds

In this announcement we discuss the close relationship between the topology of 3-manifolds and the foliations that is possesses. We will introduce and state the main result, then use it and the ideas of its proof to state some geometric and topological corollaries. Details to almost all the results can be found in [G4]. Given a compact, connected, oriented 3-manifold, when does there exist a codimension-1 transversely oriented foliation 7 which is transverse to dM and has no Reeb components? If such an 7 exists dM necessarily is a (possibly empty) union of tori and M is either SxS (and 7 is the product foliation) or irreducible. The first condition follows by Euler characteristic reasons while the latter basically follows from the work of Reeb [Re] and Novikov [N] although first observed by Rosenberg [Ro]. Our main result says that such conditions are sufficient when Ü2(M, dM) # 0. If such a foliation 7 exists on M then it follows from the work of Thurston [Ti] that any compact leaf L is a Thurston norm minimizing surface [i.e., \x{L')\ < IxCni for any properly embedded T with [T] = [L] G H2{M, dM) (or H2{M) if we were discussing the norm on H2(M)), where S' denotes ^-(sphere and disc components)] for the class [L] G Ü2(M1dM). Our main result says that for a 3-manifold M satisfying the above necessary conditions any norm minimizing surface can be realized as a compact leaf of a foliation without Reeb components.