OPENING CLOSED LEAVES OF FOLIATIONS J. C. HARRISON Novikov proved that every C l codimension one foliation of S 3 has a closed leaf ([5, Theorems 6.1 and 7.1]). In higher codimension, the situation is quite different. According to Schweitzer, if a manifold M has a C foliation of codimension q ^ 3 with 0 ^ r ^ oo, then it possesses a foliation with no closed leaves ([6, Theorem D]). To get the same result in codimension q — 2, Schweitzer uses the celebrated Denjoy CMoroidal flow containing a proper minimal set with no closed trajectories (see [1]). Since that phenomenon cannot occur in a C 2 flow on a surface (see [1]) his methods give only C 1 results when q = 2. In [3] the author topologically embeds the Denjoy C 1 vector field in a C 2 vector field defined on a punctured, thickened torus, N = (T 2 \D 2 ) x / to obtain a C 2 "flow plug". Much as in Schweitzer, but with an alternate exposition, this flow plug can be modified and used to open closed leaves of C 2 foliations. Let M be a C 00 smooth, paracompact manifold without boundary of dimension k ^ 3, and ^ a C r foliation of M. A leaf of $F is closed, if it is closed as a subset of M. THEOREM A. If there exists a C foliation ^ 0 of M of codimension two, r = 0, 1 or 2, then there exists such a foliation $F X with no closed leaves. In order to prove Theorem A, we reduce the problem to the case where the closed leaves of #" 0 have a locally finite family of disjoint neighborhoods in M. To do this, we use the following lemma and corollary. LEMMA 1 (Fuller [2]). There exists a C°° non-singular vector field X l defined on a neighborhood of the closed unit cube P in IR 3 satisfying (i) X^p) = —d/dzfor p in a neighborhood of (ii) X x has exactly one periodic trajectory; (iii) every trajectory of X x starting in some open subset of the top face of J 3 enters Int (/ 3 ) and never exits. Sketch proof. Let Y x be a vector field on the annulus A = S 1 x/ 1 such that S l x {|} is its only periodic trajectory. Let Y = Y l x 0 be the trivial product vector field on the thickened annulus Ax I. Smoothly embed Ax I in Int(/ 3 ) so that A x {t} c I 2 x {%t}. Let Z = — d/dz and suitably average Y and Z to obtain X^ satisfying (i)-(iii). Received 15 October, 1982. Bull. London Math. Soc, 15 (1983), 218-220