Secondary characteristic classes of transversely homogeneous foliations
arXiv:1205.3375
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.
52 pages. Minor corrections in v2. Lemma 8.13 in v2 was wrong and erased in v3. Theorem 1.14 and Corollary 1.15-(ii) were accordingly corrected in v3. In v4, the constant r_G in Theorem 1.7 was corrected. The constant r_{F_4(-20)} is not included anymore, because it requires more detailed computation with special techniques on F_4. The formula (1.1) in Theorem 1.5 was corrected