Secondary characteristic classes of transversely homogeneous foliations
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.
Keywords
Cite
@article{arxiv.1205.3375,
title = {Secondary characteristic classes of transversely homogeneous foliations},
author = {Jesús A. Álvarez López and Hiraku Nozawa},
journal= {arXiv preprint arXiv:1205.3375},
year = {2015}
}
Comments
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