Preface For this second volume of Foliations, we have selected three special topics: analysis on foliated spaces, characteristic classes of foliations, and foliated 3-manifolds. Each of these is an example of deep interaction between foli- ation theory and some other highly developed area of mathematics. In all cases, our aim is to give useful, in-depth introductions. In Part 1 we treat C∗-algebras of foliated spaces and generalize heat flow and Brownian motion in Riemannian manifolds to such spaces. The first of these topics is essential for the “noncommutative geometry” of these spaces, a deep theory originated and pursued by A. Connes. The second is due to L. Garnett. While the heat equation varies continuously from leaf to leaf, its solutions have an essentially global character, making them hard to compare on different leaves. We will show, however, that leafwise heat diffusion defines a continuous, 1-parameter semigroup of operators on the Banach space C(M) and, following Garnett [77], we will construct prob- ability measures on M that are invariant under this semiflow. These are called harmonic measures, and they lead to a powerful ergodic theory for foliated spaces. This theory has profound topological applications (cf. The- orem 3.1.4), but its analytic and probabilistic foundations have made access difficult for many topologists. For this reason, we have added two survey appendices, one on heat diffusion in Riemannian manifolds and one on the associated Brownian flow. For similar reasons, we have added an appendix on the basics of C∗-algebras. We hope that these will serve as helpful guides through the analytic foundations of Part 1. Part 2 is devoted to characteristic classes and foliations. Following R. Bott [9], we give a Chern-Weil type construction of the exotic classes based on the Bott vanishing theorem (Theorem 6.1.1). The resulting theory xi
Previous Page Next Page