204 Foliations on Open Manifolds, I by ANTHONY PHILLIPS (Berkeley) 1. Introduction Let M be a smooth n-dimensional manifold, with tangent bundle TM. A smooth section in the bundle of p-planes of TM is called a p-planefield (also, "p-dimensional distribution") on M. A p-plane field a gives a p-dimensional subbundle of TM, with fibre over x~M equal to a(x). This bundle will also be denoted by a. Picking a Riemannian metric for M associates to a a complementary (n-p)-plane field a• a • (x) is the tangent subspace orthogonal to the p-plane a (x). The p-plane field a is called integrable if M has a smooth foliation ~ (see w 2 for this definition) such that at each x~M thep-plane a(x) is tangent to ~-. This is equi- valent to saying that each x~M has a neighborhood U with coordinates xl,..., x, such that the tangent vectors O/Oxl]y .... , O/Oxply span a(y) at each ye U. There is a classical criterion for integrability of a p-plane a, namely that a be involutive. This means that if v and w are vectorfields contained in a, i.e. such that v(x)~a(x), w(x)~a(x) at each point x, then their Poisson bracket Iv, w] is also contained in a. It is easy to see that integrable implies involutive. The converse is FROnENIUS' Theorem [4, Theorem 5.1]. From the point of view of differential topology it is natural to ask which p-plane fields are homotopic to integrable fields (see [1], p. 373). This article presents a par- tial answer to that question. THEOREM 1.1. Suppose M is open (i.e. has no compact components). A p-plane field a on M, whose complementary bundle a I is trivial, is homotopic to an integrablefield. THEOREM 1.2. Suppose M is open, and n-dimensional. Every (n- 1)-plane fieM a on M is homotopic to an integrable field. Remark. The hypothesis, that M be open, seems quite restrictive. For instance, in the case n = 3 Theorem 1.2 for compact M and orientable a has been proved by JOHN WOOD, a graduate student at Berkeley. On the other hand, it is easy to check that all the foliations constructed in this article are analytic, in the sense of [1], p. 368. In this respect, Theorem 1.2 should be compared with the theorem on p. 392 of [1] : if nl M contains only elements of finite order, then M can carry an analytic foliation of co-dimension 1 only if M is open. Proof of theorem 1.1. By assumption, the bundle a I contains a field ~ of (n-p)- frames. The theorem is an immediate consequence of Theorem B of I-3] which implies that, since M is open, ~ is homotopic to the gradient (n-p)-frame ections