Classifying foliations of 3-manifolds via branched surfaces (Q860483)

The author uses branched surfaces to define a pseudo-metric on the set of \(C^1\) codimension-one foliations on any closed manifold that are transverse to a nonsingular flow. She then defines two foliations to be \(b\)-equivalent if their distance measured with this pseudo-metric is zero. In this way, she obtains a discrete metric on the set of equivalence classes which has the property that foliations that are sufficiently close (up to \(b\)-equivalence) often share important topological properties, such as the existence of a compact leaf, tautness and so on. The techniques used in this paper are based on a construction that is described in an unpublished preprint by J. Christy and S. Goodman. This construction is reviewed in this paper. The construction gives a branched surface associated to a foliation \(F\), a nonsingular flow \(\phi\), and a set of compact surfaces embedded in leaves of \(F\) that satisfy certain general position requirements with respect to \(\phi\). The author in fact uses an amended version of Christy and Goodman's construction.