THE EULER CLASS AS AN OBSTRUCTION IN THE THEORY OF FOLIATIONS

Abstract

Let X denote an oriented differentiable manifold of dimension n and ? an oriented foliation' of dimension p < n. We ask whether there exists an oriented foliation ?' of X by leaves of dimension p + 1 which is concordant with ? in the sense that each p-leaf of ? lies on a single (p + 1)-leaf of ?'. This is clearly a basic question in the theory of foliations, and we shall refer to it as the first extension problem. Let us suppose, for the moment, that ? constitutes an actual fibering of X, so that there exists a base manifold M of dimension n - p and a projection map r: X -- M Let X denote an oriented differentiable manifold of dimension n and ? an oriented foliation' of dimension p < n. We ask whether there exists an oriented foliation ?' of X by leaves of dimension p + 1 which is concordant with ? in the sense that each p-leaf of ? lies on a single (p + 1)-leaf of ?'. This is clearly a basic question in the theory of foliations, and we shall refer to it as the first extension problem. Let us suppose, for the moment, that ? constitutes an actual fibering of X, so that there exists a base manifold M of dimension n - p and a projection map r: X -- M with ? = { ~-l(x) :x e M}. It is apparent that an oriented foliation ?' of X by (p + 1)-leaves concordant to ? is now equivalent to a direction field on M, in which case the topological obstruction to the first extension problem is none other than the Euler class of the tangent bundle to M. For an arbitrary foliation ?, on the other hand, the leaf-space A (which results from the identification of all points lying on a single leaf of ?) is usually not a manifold and quite unsuitable to give us an Euler class, since it carries no natural vector bundle. Let us recall at this point that in a previous paper2 we have introduced a category e which extends the category C of topological spaces and continuous maps, and we have shown that a foliation S canonically determines an object 9c in e which generalizes the notion of a base manifold. In particular, whenever ? does admit a classical base manifold M, C is naturally isomorphic to M in C. In general, S occupies an intermediate position between X and A (there exist natural projections X - 9C -- A), and appears to be a more suitable candidate for the role of a base manifold than the classical leaf-space A. This raises the question whether by a suitable development of these ideas it may not be possible to define a notion of object tangent bundle to 9C and a corresponding Euler class which will constitute an obstruction for the first extension