The Godbillon-Vey Invariant of a Transversely Homogeneous Foliation

Abstract

ABSTRACT. A real projective foliation is a foliation 3 with a system of local coordinates transverse to 5 modelled on RP1 (so that the coordinate changes are real linear fractional transformations).Given a closed manifold M, there is but a finite set of values in f/3(M;R) which the Godbillon-Vey invariant of such foliations may assume. A bound on the possible values, in terms of the fundamental group, is computed. For M an oriented circle bundle over a surface, this finite set is explicitly computed.In [5], C. Godbillon and J. Vey introduced a characteristic class associated to a codimension-one foliation which lies in the third real cohomology group of the ambient manifold. The question was then raised whether one could assign some geometric significance to this invariant. The problem of understanding the GodbillonVey invariant in such a manner was underscored by a result of Thurston [12], who showed that for any closed 3-manifold there are families of codimension-one foliations whose Godbillon-Vey invariant varies continuously in a parameter, and which take on arbitrarily large values.It is the main goal of this paper to show that this situation changes when one restricts to foliations which are transversely homogeneous (see §1). To explain the geometric picture which emerges, it is worthwhile reviewing briefly Gromov's notion of simplicial volume [7].Given a compact manifold M, Gromov assigns to M a volume ||A7||, defined topologically, which measures the complexity of representing the fundamental class of M by singular simplices. For many manifolds, for instance spheres, it is easily seen that ||M|| = 0. However, inspired by the theorem of Milnor [9] and Wood [15], Gromov and Thurston (see [7]) were able to show that when M is a hyperbolic manifold, ||M|| is equal to the volume of M in its hyperbolic metric, up to a constant depending only on dim(M). This is done in part by comparing ||M|| to a certain volume invariant associated to representations from 7Ti(M) to the isometry group of hyperbolic space. At the heart of the argument is the elementary fact that a geodesic simplex in hyperbolic space has bounded volume.In the present case, we consider foliations of a manifold M which are transversely projective, that is, which are locally modelled on the action of PSL(2, R) on the circle 5 '. The holonomy of such a foliation is described by a homomorphism 7Ti(M) -» (SL(2,R))~, the universal covering of PSL(2,R), as is discussed in § §1and 2.

Characteristic classes of transversely homogeneous foliations

“…A stronger version of this theorem is proved by Brooks and Goldman in [3] for the special case of (PSL(2,R), 5 ^-foliations and extended to the case of (PSL(<7 + 1,R), S "^-foliations by Heitsch (see [8]). This special case is of interest since there are examples of nontrivial classes in this setting.…”

Classifying foliations

“…There are many "classic" papers on the subject: [43,45,55,75,76,106,147,171,226,227,225,297,307,308,309].…”

Chern–Simons theory, surface separability, and volumes of 3-manifolds

“…Let D be the set of all closed orientable 3-manifolds N with D(M, N ) finite for any fixed M . By Theorem 1.1 (4), SV(N ) = HV(N ) = 0 if N / ∈ D. It is known that (see [7], for example) N ∈ D if and only if N contains a prime factor Q with non-trivial geometric decomposition, or supporting an SL 2 (R) or a hyperbolic geometry. This fact combined with Theorem 1.1(2), (3), (4) implies that if vol(N, Iso e SL 2 (R)) = {0}, then necessarily a prime factor of N has a non-trivial geometric decomposition, or supports an SL 2 (R) or a hyperbolic geometry and if vol(N, PSL(2; C)) = {0}, then necessarily a prime factor of N contains some hyperbolic JSJ pieces.…”