R-covered branched surfaces

Abstract

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.

The simplest branched surfaces for a foliation

“…In particular, certain structural criteria on a standard branched surface W carrying F are sufficient to guarantee that every foliation carried by W , including F, has some topological property such as tautness or the R-covered property; see [Goodman and Shields 2000;Shields 2004;1997;1996]. In such cases, we know that topological property is C 1 stable for F since all foliations sufficiently close to F are also carried by W [Shields 1991]; that is, each foliation within some ε > 0 of F, in the C 1 metric defined by Hirsch [1973], is carried by W .…”

“…So although the topological behaviors of leaves in two equivalent foliations F and G can differ substantially, there is often a branched surface W ∈ [K ] carrying both F and G. In this case, if the intersection of the set {γ : γ = π W (γ F ) for some integral curve γ F of F} with the set {γ : γ = π W (γ G ) for some integral curve γ G of G} is sufficiently large, then F and G will share many important topological properties. See [Shields 2004;1991;1997;1996].…”

“…A branched surface carrying (F, φ) often reflects the topology of F. For example, topological properties such as tautness or the R-covered property can often be detected from it [Goodman and Shields 2000;Shields 2004;1997]. Indeed, a primary motivation for using branched surfaces to study foliations is that a large number of foliations can be approximated by the same branched surface.…”