The set of foliations carried by the same branched surface as a foliation f includes, but is not restricted to, the set of foliations sufficiently close to f. We show that a branched surface may be constructed to ensure that no foliation carried by it has more dead-end components than f. In particular, if f is taut, all foliations carried by the branched surface will be taut. We use the result to classify the foliations of closed 3-manifolds having the fewest number of dead-end components. We then develop an algorithm to determine a lower bound on the depth of a foliation in a given class. Introduction.The topology of certain orientable Riemannian 3-manifolds has recently been studied by constructing transversely orientable foliations of these manifolds (see [4]-[7]). For this, it useful to work with the simplest foliations of the manifolds. These we consider to be the foliations with the least number of "dead-end components" (described in [13, p. 104] and [9]) that are at the lowest "depth" ([1]).Branched surfaces were used in [16] to define an equivalence relation on codimension one orientable C 1 foliations of a compact 3-manifold M that are transverse to a flow φ. For the definition, we used the simplest branched surface that can be associated with each foliation. Such a branched surface gives a skeletal outline of the foliation and equivalent foliations correspond to the same branched surface. Under this relation, foliations in the same class need not be topologically equivalent but do share certain topological properties such as the existence of a compact leaf of a particular genus ([15]) or a covering by a trivial product of hyperplanes ([14]). In this paper we show that all foliations in the same class as a foliation f will have the same number of dead-end components as f. In particular, this relation classifies foliations of M that are transverse to φ and have the fewest number of dead-end components. The simplest branched surfaces associated with these foliations provide a combinatorial classification.One of the consequences of the above mentioned result is that all foliations in the same equivalence class as a taut foliation are taut. (Tautness rules 305
We give sufficient conditions for an R-covered codimension one foliation F of a closed 3-manifold to be carried by an Rcovered branched surface; that is, a branched surface carrying only foliations with the R-covered property. These conditions can be readily verified for many examples. In cases where the branched surface is generated by disks, the R-covered property is stable for F in the sense that all nearby foliations are also R-covered.
We study the topology of codimension one taut foliations of closed orientable 3-manifolds which are smooth along the leaves. In particular, we focus on the lifts of these foliations to the universal cover, specifically when any set of leaves corresponding to nonseparable points in the leaf space can be totally ordered. We use the structure of branching in the lifted foliation to find conditions that ensure two nonseparable leaves are left invariant under the same covering translation. We also determine when the set of leaves nonseparable from a given leaf is finite up to the action of covering translations. The hypotheses for the results are satisfied by all Anosov foliations.Introduction.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.