Table Of ContentMathematical 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
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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
