Secondary classes and transverse measure theory of a foliation

Abstract

1. The purpose of this note is to announce several theorems showing how the secondary classes of a foliation J of a compact manifold X depend upon the measure theoretic properties of the equivalence relation determined by the foliation. The relevant properties are: (i) amenability [14], which is equivalent to hyperfiniteness by ConnesFeldman-Weiss [3]; and (ii) the Murray-von Neumann type. A set B C X is saturated if it is the union of leaves of 7. The equivalence relation 7 has type I if there is a measurable subset of X which intersects almost every leaf exactly once; type II if it admits an invariant measure, finite or infinite in the given measure class but does not have an essential saturated set of type I; and type III if it does not have any essential saturated sets of types I or II. Every equivalence relation can be decomposed into parts of types I, II, and III. These types correspond to certain algebraic properties of the von Neumann algebra M(X, 7) associated with the equivalence relation [1, 13]. Let X be a compact manifold without boundary and J a C 2 , codimensionn foliation of X. The secondary classes are given by a map A* : H*(WOn) —• H*(X; R) with image spanned by the classes of the form A*(yicj). Here, yi is a basis element for the relative cohomology if*(gln, On), and cj is a Chern form of degree at most 2n. If degree cj = 2n, we say the class is residual. The Godbillon-Vey classes are those of the form A*(yicj) € # 2 n + 1 ( X ; R ) , with 2/1 G if(gln ,On) , the normalized basis element. The generalized GodbillonVey classes are those of the form A*(yiy/Cj), where yi = 1 is permitted. (For a convenient reference, see [11].) The residual secondary classes have the unusual property that they localize to the measurable saturated subsets of X: for each such B c X and residual yicj e # p (WO n ) , the restriction A*(2//Cj)|£ G H (X) is well defined [5]. The following theorems are stated for the secondary classes of 7 on X, but corresponding theorems also hold for the localized classes A*(yiCj)\B of the restriction 7\B.