Secondary characteristic classes of transversely homogeneous foliations

TL;DR

This paper investigates the finiteness of secondary characteristic classes in transversely homogeneous foliations modeled on simple Lie groups, identifying exceptions and providing new examples and formulas.

Contribution

It extends finiteness results of secondary characteristic classes to broader transverse structures, with explicit constructions and formulas, and explores rigidity in hyperbolic manifold bundles.

Findings

Finiteness of secondary characteristic classes for most cases

Construction of examples breaking finiteness in specific cases

Formulas for Godbillon-Vey classes and rigidity results

Abstract

Let G be a simple Lie group of real rank one, and S the ideal boundary of the corresponding symmetric space of noncompact type (H^n_R, H^n_C, H^n_H or H^2_O). We show the finiteness of the possible values of the secondary characteristic classes of transversely homogeneous foliations on a fixed manifold whose transverse structures are modeled on the G-action on S, except the case where G=SO(n+1,1) for even n. For this exceptional case, we construct examples of foliations on a manifold which break the finiteness and show a weaker form of the finiteness. These are generalizations of a finiteness theorem of secondary characteristic classes of transversely projective foliations on a fixed manifold by Brooks-Goldman and Heitsch to other transverse structures. We also show Bott-Thurston-Heitsch type formulas to compute the Godbillon-Vey classes of certain foliated bundles, and then obtain a rigidity result on transversely homogeneous foliations on the unit tangent sphere bundles of hyperbolic manifolds.