In this chapter we put everything together to state and prove the Connes Index Theorem for compact foliated spaces.

Let X be a compact foliated space with leaves of dimension p and foliation bundle F which we assume oriented and equipped with a tangentially smooth oriented tangential Riemannian structure g. Let G= G(X) denote the associated holonomy groupoid of the foliated space. Let D be a tangential, tangentially elliptic pseudodifferential operator on (bundles over) X. For each leaf 𝓁 the spaces KerD𝓁 and KerD*𝓁 are well defined by Proposition 7.23 and are locally finite-dimensional with local index measure.

In this chapter we discuss certain cohomology groups associated with a foliated space, which we shall call tangential cohomology groups. It will be in these groups that invariants connected with the index theorem will live. Similar groups have been considered, for instance, in [Kamber and Tondeur 1975; Molino 1973; Vaisman 1973; Sarkaria 1978; Heitsch 1975; El Kacimi-Alaoui 1983; Haefliger 1980] (we discuss Haefliger’s work at the end of chapter IV). The similarities and differences between the three situations are easy to describe; all involve differential forms which are smooth in the tangential direction of the foliation. The difference comes in the assumptions on the transverse behavior: for foliated manifolds (Kamber, Tondeur, and others), forms are C∞ in the transverse direction; for foliated spaces (the present treatment), the forms are to be continuous in the transverse directions, since that is all that makes sense; and finally for foliated measure spaces, the forms are to be measurable in the transverse direction, for again that is all that makes sense.

Thus, we let X be a metrizable foliated space with foliation tangent bundle FX→ X, as defined in Chapter II. The quickest and simplest way to introduce the tangential cohomology is via sheaf theory and sheaf cohomology, but for those readers who are not familiar with such notions we show how to define the groups via a de Rham complex and also show in an appendix how to give a completely algebraic definition. For details concerning sheaves and their cohomology, consult [Godement 1973] and [Wells 1973].

We consider the sheaf ℛτ on X of germs of continuous real-valued tangentially locally constant functions. Specifically, this sheaf assigns to each open set U of X the set of continuous real-valued functions on U that are locally constant in the tangential direction on the foliated space U (given the induced foliation from X). This is obviously a presheaf and it is immediate that the additional conditions defining a sheaf [Godement 1973, p. 109] are satisfied.

A lot has happened in the realm of foliated spaces and their operator algebras since 1988, when this book first appeared. We are pleased that, as we had hoped, this book has served as an introduction to the subject and a reference for researchers and students.

Our colleagues have convinced us that there is merit in issuing a second edition of our work, so that a new generation of students may have access to its contents. Cambridge University Press was amenable to the idea, so we (slowly) went to work.

We have taken the opportunity of a new edition to make a number of changes and additions to the book:

• (1) We have corrected a few minor errors, filled some gaps, and made many changes to improve the exposition.

• (2) We have added updates at the end of each chapter as well as occasional footnotes in which we discuss some of the relevant mathematical developments since 1988. This discussion is understandably brief. We do try to point the reader to the papers where the results themselves appear.

• (3) We have enlarged the bibliography correspondingly.

• (4) We have added a new appendix; it is a reprint of a Mathematical Reviews Featured Review by the second author on the Gap Labeling Theorem. We felt this was appropriate since it illustrates a very interesting and important application of the Index Theorem.

• (5) We have added an index to the book.

• (6) MSRI has provided for the resetting of the book in LATEX and for the redrafting of all the art.

As originally formulated, the Connes’ Index Theorem [1979] applied to foliated manifolds. The version presented here is valid for foliated spaces, a category that is strictly larger than foliated manifolds and laminations obtained from manifolds. It turns out that this extra generality is crucial for some of the applications of the Index Theorem in the past few years. For instance, the Gap Labeling results discussed in Appendix D require this extra generality. We discuss this in some detail at the end of Chapter VIII.

We acknowledge with gratitude the help that we have received from Jean Bellissard, Alberto Candel, Larry Conlon, Steve Hurder, Jerry Kaminker, Ma-soud Khalkhali, Paul Muhly, and especially our friend and editor par excellence Silvio Levy in the preparation of this edition. The second author is grateful to Baruch Solel and the faculty of the Technion for a sabbatical year at a critical time. We are grateful to the editors of Mathematical Reviews for permission to reproduce the Featured Review on the Gap Labeling Theorem as Appendix D of this work.

If M is a compact oriented manifold then the Hodge theorem supplies a unique harmonic form associated to each de Rham cohomology class of M. If the compactness assumption is dropped the situation becomes considerably more sensitive. In this appendix we demonstrate how to use the index theorem for foliated spaces to produce L2 harmonic forms on the leaves of certain foliated spaces.

We begin by recalling the Hirzebruch signature theorem. If M is a com-pact oriented manifold of dimension 4r then its signature is defined to be the signature of the bilinear form on H2r(M)

Recall that there is a signature operator A (Chapter VIII), and the signature of the manifold, SignM, is the Fredholm index of this operator. If M has positive signature then H2r(M) must be nontrivial and must contain classes represented by harmonic forms, by Hodge theory.

This is a review of three articles: [Bellissard et al. 2005] (Jean Bel-lissard, Riccardo Benedetti and Jean-Marc Gambaudo, “Spaces of tilings, finite telescopic approximations, and gap-labeling”, to appear in Communications in Mathematical Physics), [Benameur and Oyono-Oyono 2003] (“Gap-labelling for quasi-crystals”, pp. 11–22 in Operator algebras and mathematical physics, Theta Foundation, Bucharest, 2003), and [Kaminker and Putnam 2003] (“A proof of the gap labeling conjecture”, Michigan Mathematical Journal 51 (2003), 537–546). It first appeared as a Featured Review in Mathematical Reviews, and is reprinted here by permission, with slight modifications. The three reviewed articles are herein referred to as BBG, BO and KP.

The Gap Labeling Theorem was originally conjectured in [Bellissard et al. 2000]. The problem arises in a mathematical version of solid state physics in the context of aperiodic tilings. Its three proofs, discovered independently by the authors above, all lie in K-theory. Here is the core result of these papers.

In this chapter we introduce the basic definitions and elementary properties of foliated spaces.

The local picture of a foliated space is a topological space of the form L×N, where L is a copy of ℝp and N is a separable metric space, not necessarily a manifold. A tangentially smoothfunction

f: L×N → ℝ

is a continuous function with the following properties:

• (1) For each n ∈ N, the function f(·, n) L→ ℝ is smooth.

• (2) All partial derivatives of f in the L directions are continuous on L × N.

In this chapter we concentrate upon the measure theoretic aspects of foliated spaces, including especially the notion of transverse measures.

We begin with a general study of groupoids, first in the measurable and later in the topological context. Our examples come from the holonomy groupoid of a foliated space (2.20) and a discrete version corresponding to a complete transversal. We introduce transverse measures ν with a given modulus and discuss when these are invariant.

Next we look in the tangential direction, defining a tangential measure λ to be a collection of measures λ = { λx} (one for each leaf in the case of a foliated space) which satisfies certain invariance and smoothness conditions. For instance, a tangential, tangentially elliptic operator D yields a tangential measure lD as follows. Restrict D to a leaf l. It follows from Chapter I that KerD` and Ker D*lare locally finite-dimensional and hence the local index.

Next we specialize to topological groupoids and continuous Radon tangential measures. In the case of a foliated space we recount the Ruelle–Sullivan construction of a current associated to a transverse measure and we show that the current is a cycle if and only if the transverse measure is invariant.

This book grew out of lectures and the lecture notes generated therefrom by the first named author at UC Berkeley in 1980 and by the second named author at UCLA, also in 1980. We were motivated to develop these notes more fully by the urgings of our colleagues and friends and by the desire to make the general subject and the work of Alain Connes in particular more readily accessible to the mathematical public. The book develops a variety of aspects of analysis and geometry on foliated spaces which should be useful in many contexts. These strands are then brought together to provide a context and to expose Connes’ index theorem for foliated spaces [Connes 1979], a theorem which asserts the equality of the analytic and the topological index (two real numbers) which are associated to a tangentially elliptic operator. The exposition, we believe, serves an additional purpose of preparing the way towards the more general index theorem of Connes and Skandalis [1981; 1984]. This index theorem describes the abstract index class in K0(C*r(G(M))), the index group of the C*-algebra of the foliated space, and is necessarily substantially more abstract, while the tools used here are relatively elementary and straightforward, and are based on the heat equation method.

We must thank several people who have aided us in the preparation of this book. The origins of this book are embedded in lectures and seminars at Berkeley and UCLA (respectively) and we wish to acknowledge the patience and assistance of our colleagues there, particularly Bill Arveson, Ed Effros, Marc Rieffel and Masamichi Takesaki. More recently, we have benefitted from conversations and help from Ron Douglas, Peter Gilkey, Jane Hawkins, Steve Hurder, Jerry Kaminker, John Roe, Jon Rosenberg, Bert Schreiber, George Skandalis, Michael Taylor, and Bob Zimmer

We owe a profound debt to Alain Connes, whose work on the index theorem aroused our own interest in the subject. This work would not exist had we not been so stimulated by his results to try to understand them better.

Global analysis has as its primary focus the interplay between the local analysis and the global geometry and topology of a manifold. This is seen classically in the Gauss–Bonnet theorem and its generalizations, which culminate in the Atiyah–Singer Index Theorem [Atiyah and Singer 1968a]. This places constraints on the solutions of elliptic systems of partial differential equations in terms of the Fredholm index of the associated elliptic operator and characteristic differential forms which are related to global topological properties of the manifold.