Open Saturated Sets Without Holonomy

TL;DR

This paper classifies holonomy-free saturated sets in 3-manifold foliations using a finite system of convex cones in cohomology, extending previous classifications to more complex foliation types.

Contribution

It introduces a finite cone system in cohomology that classifies holonomy-free foliations of finite depth in 3-manifolds, generalizing earlier results.

Findings

Classifies foliations via cohomology cones

Establishes a one-to-one correspondence between cones and foliation classes

Extends classification from depth one to finite depth foliations

Abstract

Open, connected, saturated sets W without holonomy in codimension one foliations play key roles as fundamental building blocks. Here, for the case of foliated 3-manifolds, we produce a finite system of closed, convex, non-overlapping polyhedral cones in the first cohomology of W with real coefficients such that the isotopy classes of possible foliations of W without holonomy, either dense leaved in W or proper, correspond one-to-one to the rays in the interiors of these cones. This generalizes our classification of depth one foliations to foliations of finite depth and more general foliations.