Classifying foliations

TL;DR

This paper surveys methods for classifying foliations, focusing on secondary classes and dynamical properties, and explores how these aspects relate to each other in the structure of foliated spaces.

Contribution

It provides a comprehensive overview of classification approaches, linking secondary classes with dynamical properties and space decomposition.

Findings

Secondary classes relate to dynamical properties

Foliated space decomposes into components by dynamics

Comparison of dynamical structure with classification

Abstract

We give a survey of the approaches to classifying foliations, starting with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of foliations are introduced and discussed, including expansion rate, local entropy, and orbit growth rates. This leads to a decomposition of the foliated space into Borel or measurable components with these various dynamical types. The dynamical structure is compared with the classification via secondary classes.