Foliations with Nonorientable Leaves

We prove that a codimension one foliation of an orientable paracompact manifold has at most a countable number of nonorientable leaves. We also give an example of a codimension one foliation of a compact orientable manifold with an infinite number of nonorientable leaves. Introduction. The purpose of this paper is to investigate the occurrence of nonorientable leaves in a codimension one foliation of an orientable manifold. Suppose that ?Fis a codimension one foliation of the orientable manifold M defined by the subbundle E of the tangent bundle TM of M. Let Q = TM/E be the normal bundle of CF. If Q is orientable, then so is E (since TM is orientable) and each leaf L of F must be orientable (since TL = E\L). If Q is nonorientable, then so is E and the leaves of ÍFmay or may not be orientable.1 For example, let M be the total space of the flat 5'-bundle over RP2 defined by the homomorphism cp:<7Tx(RP2) -* Diff(S") with <p(u) complex conjugation on S] c C, u nonzero in ttx(RP2). Then M is covered by S2 X Sx and the images of the sets S2 X (y), y e S1, define a codimension one foliation of M. This foliation has exactly two nonorientable leaves which correspond to the two fixed points of y(u) (see §1). On the other hand, it is not possible for all leaves of % to be nonorientable. In fact, a nonorientable leaf of a foliation of an orientable manifold is easily seen to have nontrivial leaf holonomy. It then follows from a result of Epstein, Millett, and Tischler [1] that the union of all nonorientable leaves is contained in the complement of a dense Gs. Our result is the following. Theorem 1. Let ^ be a codimension one foliation of a connected orientable paracompact manifold M. Then *§ has at most a countable number of nonorientable leaves. The proof of this result is given in §3. An example is given in §1 to show that the number of nonorientable leaves can be infinite. Received by the editors February 3, 1983. 1980 Mathematics Subject Classification. Primary 57D30; Secondary 57D99. 1 In a conversation with several of the authors, W. S. Massey asked whether an orientable manifold could have a foliation with nonorientable leaves. It was his question that motivated this paper. We are also indebted to Larry Conlon for helpful conversations on this matter. ft 1983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page