Mathematical Sciences Research Institute 9 Publications Editors S.S. Chern I. Kaplansky C.c. Moore I.M. Singer Calvin C. Moore Claude Schochet Global Analysis on Foliated Spaces With 16 Illustrations Springer-Verlag New Yo rk Berlin Heidelberg London Paris Tokyo Calvin C. Moore Claude Schochet Department of Mathematics Department of Mathematics University of Califomia Wayne State University Berkeley, CA 94720, USA Detroit, MI 48202, USA Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, CA 94720, USA AMS Classification: 19K56, 28D15, 46L05, 46L50, 57R30, 58FIl, 58GIO, 58GII, 58GI5 Library of Congress Cataloging-in-Publication Data Moore, C. C. (Calvin c.) Global analysis on foliated spaces, (Mathematical Sciences Research Institute publications : 9) Bibliography: p, I. Global analysis. 2. Foliations (Mathematics) I. Schochet, Claude. 11. Title. 111. Series. QA614.M65 1988 514'.74 87-28631 © 1988 by Springer-Verlag New York Inc. Softcover reprint of (he hardcover 1st edition 1988 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York. NY 10010. USA). except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software. or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names. trademarks. eIe. in this publication. even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Acl. may accordingly be used freely by anyone. Camera-ready copy provided by the authors. 9 8 7 6 5 4 3 2 1 I SBN-13: 978-1-4613-9594-2 e-ISBN -13: 978-1-4613-9592-8 DOI: 10.1007/978-1-4613-9592-8 PRBFACB 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 [C03] , a theorem which asserts the equality of the analytic and the topological index (two real numbers) which are associated to a tanaentially elliptic operator. The exposition, we believe, serves an additional purpose of preparing the way towards the more general index theorem of Connes . and Skandalis [CS]. This index theorem describes the abstract index ,.. dass in KO(Cr(G(M))), the index group of the C -algebra of the foUated space, and is neceasarily substantially more abstract, while the tools used here are relatively elementary and strairhtforward, 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 exiat had we not been so stimulated by his results to try to understand them better. Table of Contents Introduetion 1 I. Loeally Traeeable Operators 16 11. Foliated Spaees 38 111. Tangential Cohomology 68 IV. Transverse ~easures 92 V. Charaeteristie Classes 137 VI. Operator Algebras 163 VII. Pseudodifferential Operators 207 VIII. The Index Theorem 260 A: The ä operator 279 by S. Hurder B: L2 Harmonie Forms on Non-Compaet ~anifolds 308 by Calvin C. ~oore, Claude Sehoehet, and Robert J. Zimmer C: Positive Sealar Curvature Along the Leaves 316 by Robert J. Zimmer Referenees 322 INTRODUCTION 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 classicallv in the Gauss-Bonnet theorem and its generalizations. which culminate in the Ativah-Singer Index Theorem [ASI] which 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 topologie al properties of the manifold. The Ativah-Singer Index Theorem has been generalized in several directions. notably by Atiyah-Singer to an index theorem for families [AS4]. The typical setting here is given by a family of = elliptic operators P (Pb) on the total space of a fibre bundle F_M_B. where Pb is defined on the Hilbert space L21p -llbl.dvollFll. In this case there is an abstract index class indlPI E ROIBI. Once the problem is properly formulated it turns out that no further deep analvtic information is needed in order to identify the class. These theorems and their equivariant counterparts have been enormously useful in topology. geometry. physics. and in representation theory. A smooth manifold Mn with an integrable p-dimensional subbundle F of its tangent bundle TM may be partitioned into p-dimensional manifolds called 1 ea. v es such that the restriction of F to the leaf is .iust the tangent bundle of the leaf. This structure is called a f 01 i at ion of M. Localiy a foliation has the form IRPXN. with leaves of the form IRPX(n). Locally. then. a foliation is a fibre bundle. However the same leaf may pass through a given coordinate patch infinitely often. So globally the situation is much more complicated. Foliations arise in the study of flows and dynamies. in group representations. automorphic forms. groups acting on spaces Icontinuously or even measurablyl. and in situations not easily modeled in classical algebraic topology. For instance. a diffeomorphism acting ergodicallv on a manifold M yields al-dimensional foliation on MXZIR with each leaf dense. The space of leaves of a foliation in these cases is not decent topologically levery point is dense in the example above) or even measure-theoretically (the space may not be a standard Borel spacel. Foliations carry interesting differential operators, such as signature operators along the leaves. Fo11owing the Atiyah-Singer pattern. one might hope that there would be an index c1ass of the type = indlP) Average indlP x), There are two difficulties. First of a11 , leaves of compact foliations need not be compact, so an elliptic operator on a leaf may weH have infinite dimensional kernel or cokernel. and thus "indIPx)" makes no sense. This problem aside, the fact that the space of leaves may not be even a standard Borel space suggests strongly that there is no way to average over it. There was thus no analytic index to try to compute for foliations. Alain Connes saw his way through these difficulties. He realized that the "space of leaves" of a foliation should be a • * non-commutative space -- that iso aC-algebra Cr(G(M». In the case of a foliated fibre bundle this algebra is stably isomorphic to the algebra of continuous functions on the base space. This suggests * KO(CrIGIM») as a home for an abstract index indlP) for tangentially elliptic operators. [Subsequentlv Connes and Skandalis proved [CS] an abstract index theorem which identifies this class.] Next Connes realized that in the fibre bundle case there is an invariant transverse measure v which corresponds to the volume measure on B. So we must assume given some invariant transverse measure in general. [These may not exist. If one exists it may not be unique up to scale.] An invariant transverse measure v gives rise to a trace t6 v on C *r IG(M)) and thus areal number = ind vIP) t6 v(ind(P» E lJi! which Connes declared to be the anallltic index. [Actually we are cheating here: the most basic definition of the analytic index is in terms of locally traceable operators as we sha11 explain below and in Chapters J and IV.] With an analytic index to compute, Connes computed it. Connes Index Theorem. Let M be a compact smooth manifold with an oriented foliation and let )I be an invariant transverse measure with associated RueUe-Sullivan current C)I' Let P be a tangentially elliptic pseudodifferential operator. Then Connes' theorem is very satisfying and its proof involves a tour of many areas of modern mathematics. The authors decided to expose this theorem and to use it as a centerpiece to discuss this region of mathematics. Along the way we realized that the setting of Jo li at ed spaces (local picture IRPXN with N not necessarily Euclidean) was at once simpler pedagogicallv and yielded a somewhat more general theorem, since foliated spaces which are not manifolds occur with some frequency. The local picture of a foliated space is simply aspace of the form IRPXN, where we regard sets of the form IRPXCn) as leaves and N is a transversal. N To such aspace is canonically associated a p-plane vector bundle I F RPXN with F (t,n) E! T(IRP). The global picture of a foliated space X is somewhat more complex. We stipulate that X be a separable metrizable space with coordinate patches Ux :!! IRPXNx and continuous change of coordinate maps of the form 3 t' = CPlt.nl n' = ",In) . which are smooth along the leaves, in the sense that a set in Ux of the form IRPXn is sent to a set of the form IRPX",lnl by a smooth map. This guarantees that the leaves in each coordinate patch coalesce to form leaves i. in X which are smooth p-manifolds. The I bundles F U. coalesce to form a p-plane bundle F over X such that 1 F Ii. ::! Tli.) for each leaf i.. Any foliated manifold is a foliated space. There are interesting examples of foliated spaces which are not foliated manifolds. For instance. a solenoid is a foliated space with leaves of dimension 1 and with Ni homeomorphic to Cantor sets. If Mn is a manifold which is foliated by leaves of dimension p and if N is a transversal of Mn then any subset of N determines a foliated subspace of M simply by ta king those leaves of Mn which meet the subset. This includes the laminat ions of much current interest in low dimensional topology. Finally, X may weIl be infinite dimensional: take n7s1 foliated by lines corresponding to algebraically independent irrational rotations. Then ( l) X n;sl is a transversal! If X is a foliated space then C;IXI is the ring of continuous functions on X which are smooth in the leaf directions. If E -!!... X is a foliated bundle ILe .. E is also foliated. 7f takes leaves to leaves. and 7f is smooth on each leaf) then r r(E) :: r rIX.E) denotes = continuous tangentiaIly smooth sections of E. We let n~IX) rrIAkF*) and define the ta.noentia.l cohomoloOll groups of a foliated space by where d: nrk( X) - n~ + 1 (XI is the analogue of the de R,ham differential obtained by differentiating in the leaf directions. Similar Ibut not the same) groups have been studied by many authors. Tangential cohomology groups are based upon forms which are 4 cont i nuous transversely (even if X is a foliated manifold.) It turns out that this small point has some major consequences. The groups may be described as where CX'T is the sheaf of germs of continuous functions which are constant along leaves. The tangential cohomology groups are functors from foliated spaces and leaf-preserving tangentially smooth maps to graded commutative IR-algebras. They vanish for k > p. There is the usual apparatus of long exact sequences, suspension isomorphisms, and a Thom isomorphism for oriented k-plane bundles. * The groups H'T(X) have a natural topology and are not necessarily Hausdorff; we let ii~(X) = H~(X)/CO} denote the maximal Hausdorff quotient. For example, if X is the irrational flow on the torus then H~(X) has infinite dimension but ii~(X) :!! IR. The parallel between de Rham theory and tangential cohomology theory extends to the existence of characteristic classes. Given a tangentially smooth vector bundle E _ X we construct tangential connections. curvature forms, and Chern classes. This leads to a tangential Chern character, a tangential Todd genus and hence a topological index where • denotes the tangential Thom isomorphism. Next we recall the construction of the groupoid of a foliated space; the idea is due to Ehresmann, Thom and Reeb and was elaborated upon by Winkelnkemper. If X is a foliated space then there is a natural equivalence relation: x y if and only if x and y N are on the same leaf. The resulting space CX(X) C X X X is not a well-behaved topological space. The holonomy groupoid G(X) of a foliated space is designed to by-pass this difficulty. It contains holonomy data not given by CX(X); holonom.v is essential for diffeomorphism and structural questions about the foliated space. The holonoml/ arou1)oid G(X) consists of tripIes (x,y,[a]) where x 5