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Foliations on 3-Manifolds

1969; Princeton University; Volume: 89; Issue: 2 Linguagem: Inglês

10.2307/1970673

1939-8980

John W. Wood,

Homotopy and Cohomology in Algebraic Topology

Let M be a smooth manifold with tangent bundle TM. A k-plane field (or k-distribution) on M is a k-dimensional subbundle a of TM. Equivalently let a denote the section of the Grassmann bundle Gk(M) of k-planes associated to TM whose value at e M is the k-plane ax c TM,. Two k-plane fields are homotopic if they are homotopic as sections of Gk(M). Homotopic k-plane fields are equivalent as k-plane bundles over M, but not conversely. If L is an injectively immersed, smooth submanifold of M such that TLX = ax c TMx for all e L, L is called an integral submanifold of a. A kplane field a is called completely integrable if the following three equivalent conditions are satisfied. A. M is covered by open sets U with local coordinates x1, * , x such that the submanifolds defined by xk+l = constant, * * *, xm = constant are integral submanifolds of a. B. a is smooth and through every point e M there is an integral submanifold L of a. C. a is smooth and if X and Y are vector fields on M with Xx, Yx C ax for all e M then the bracket [X, Y]x e ax. The equivalence of these conditions is the Frobenius theorem. An integrable k-plane field is also called a foliation (this is equivalent to other definitions) and the maximal connected integral submanifolds are called leaves. The leaves of a foliation partition the manifold. The existence theorem for ordinary differential equations says that smooth line-fields are always integrable. In general for k > 1 the set of kplane fields which are not integrable is open and dense in the space of sections of Gk(M). G. Reeb [16] has asked if the existence of a k-plane field on a manifold implies the existence of a foliation. He has given an example of a foliation of codimension one on S3. W. B. R. Lickorish [10] and, independently, S. P. Novikov and H. Zeischang have exhibited foliations of codimension one on any closed, orientable 3-manifold. In ? 4 we consider the unorientable