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.