Lawrence Conlon

Every surface is a leaf

FOR EACH integer n≥3, there are open, connected n-manifolds that cannot be realized as leaves in any compact, C0-foliated (n+1)-manifold [8] (for the C2 case, cf. [10]). For n=2, the corresponding question has been thought to be hard. Here we give a proof, announced in [7], that all open surfaces are realizable. This answers a basic question posed in [14]. In the following, L denotes an open, connected 2-manifold, M a closed, connected 3-manifold.

The dynamics of open, foliated manifolds and a vanishing theorem for the Godbillon-Vey class

Let (M,.F) be a C3-foliated n-manifold with leaves of dimension n 1. Let gv(F) E H3(M, IR) denote the characeristic class introduced by Godbillon and Vey in [G-V]. Considerable effort has been made to discover the geometric significance of this invariant (recall Thurston’s “helical wobble” [Th], which can be interpreted via the formulas in [R-W]). In addition to celebrated examples of foliations for which gv(F) does not vanish [G-V, Th, Br], the list of foliations for which it does has been growing steadily. By work of Herman [Her] and Wallet [WI, gv(Sr) = 0 for foliated circle bundles and interval bundles over T2. This result has interesting consequences that are far from being obvious. Thus, Morita and Tsuboi have deduced the vanishing of gv(X) for closed, foliated manifolds without holonomy [M-T] and Mizutani, Morita, and Tsuboi have extended this to foliations almost without holonomy [M-M-T]. In all of the above vanishing theorems, the leaves of the foliations have polynomial growth. Other classes of foliations, with all leaves polynomial and gv(,F) = 0, were found by Nishimori [N], Sergiescu [Ser], and Tsuchiya [Ts]. In a preprint [C-C2], that became obsolete within a few months of its distribution, the present authors proved the vanishing theorem for all closed, foliated manifolds in which every leaf has polynomial growth. Indeed, we proved that, if every leaf is nonresilient [L] and if no leaf lies at infinite level, then gv(sT) = 0.

An interesting class of C1 foliations

Abstract A class of depth two foliations is described in which certain growth conditions allow C 1 -smoothability, but obstruct higher order smoothability. Similarly, such conditions can obstruct even C 1 -smoothability. There result uncountably many examples of C 1 foliations that are not homeomorphic to C 2 foliations, as well as uncountably many C 0 foliations that are not homeomorphic to C 1 foliations.

LEAVES OF MARKOV LOCAL MINIMAL SETS IN FOLIATIONS OF CODIMENSION ONE

The authors continue their study of exceptional local minimal sets with holonomy modeled on symbolic dynamics (called Markov LMS [C-C 1]). Here, an unpublished theorem of G. Duminy, on the topology of semiproper exceptional leaves, is extended to every leaf, semiproper or not, of a Markov LMS. Other topological results on these leaves are also obtained.

Depth of knots

Abstract The authors show that for every k ⩾/0 there are knots intrinsically of depth k , i.e., whose complements admit smooth, taut foliations of depth k but do not admit C 2 , taut foliations of any depth less than k . They also give examples of knots whose complements do not admit taut, C 2 foliations, with leaves meeting the boundary of the knot complement in circles, of any finite depth.

Closed transversals and the genus of closed leaves in foliated 3-manifolds

Abstract The classical local theory of integrable 2-plane fields in 3-space leads to interesting qualitative questions about the global properties of solutions surface (i.e., leaves of a foliation) on 3-manifolds. It is now known that foliations admitting a closed leaf of suitably high genus abound on all closed or orientable 3-manifolds that are not rational homology spheres (S. Goodman, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 4414–4415), and this leads to natural questions about the “positions” of such leaves relative to the rest of the foliation. One such question, suggested by Goodmans theorem on closed transversals (S. Goodman, ibid.), is considered here.

Erratum to “Transversally parallelizable foliations of codimension two”

The proof given in [1] for Lemma (7.6) is incorrect. A correct proof, which is also much simpler, is easily obtained by using Corollary (8.2) in the same paper. Of course, (8.2) does not depend on (7.6). We want to show that Tr1(M, A) is not cyclic, so we assume the contrary. The two cases to consider continue to be those in which FZ does or does not admit a compact leaf. The first of these cases was correctly handled in [1], but the second was not. If the leaves of FZ are noncompact, then (8.2) implies that Fz has no limit cycles. The natural surjection