SECONDARY CHARACTERISTIC CLASSES OF TRANSVERSELY HOMOGENEOUS FOLIATIONS

Abstract

Let \(G\) be a simple Lie group of real rank one, and \(S_{\infty}^{q}\) the ideal boundary of the corresponding hyperbolic symmetric space of noncompact type (\(H_{\mathbb{R}}^{n}\), \(H_{\mathbb{C}}^{n}\) , \(H_{\mathbb{H}}^{n}\) or \(H_{\mathbb{O}}^{2}\)). 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 \(S_{\infty}^{q}\), except the case of transversely conformally flat foliations of even codimension \(q\). For this exceptional case, we construct examples of foliations on a manifold which break the finiteness and show a weaker form of the finiteness result. 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 secondary characteristic classes of certain foliated bundles, and then obtain a rigidity result on transversely homogeneous foliations on the unit tangent sphere bundles of hyperbolic manifolds.