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Alberto Candel Lawrence Conlon American Mathematical Society Graduate Studies in Mathematics Volume 60 Foliations II

Transcript of Alberto Candel Lawrence Conlon · Foliations I / Alberto Candel, Lawrence Conlon. p. cm. —...

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Alberto CandelLawrence Conlon

American Mathematical Society

Graduate Studies in Mathematics

Volume 60

Foliations II

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Foliations II

http://dx.doi.org/10.1090/gsm/060

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Alberto CandelLawrence Conlon

Foliations II

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 60

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EDITORIAL COMMITTEE

Walter CraigNikolai Ivanov

Steven G. KrantzDavid Saltman (Chair)

2000 Mathematics Subject Classification. Primary 57R30.

The first author was supported by NSF Grants DMS-9973086and DMS 0049077

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-60

Library of Congress Cataloging-in-Publication Data

Candel, Alberto, 1961–Foliations I / Alberto Candel, Lawrence Conlon.

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 23)Includes bibliographical references and index.ISBN 0-8218-0809-5 (alk. paper)1. Foliations (Mathematics) I. Title. II. Series. III. Conlon, Lawrence, 1933–

QA613.62.C37 1999514′.72 21—dc21 99-045694

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To our wives,

Juana and Jackie

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Contents

Preface xi

Part 1. Analysis and Geometry on Foliated Spaces

Foreword to Part 1 3

Chapter 1. The C∗-Algebra of a Foliated Space 5

§1.1. Twisted Forms and Densities 6

§1.2. Functions on Non-Hausdorff Spaces 8

§1.3. The Graph of a Foliated Space 11

§1.4. The C∗-algebra of a Foliated Space 18

§1.5. The Basic Examples 27

§1.6. Quasi-invariant Currents 37

§1.7. Representations of the Foliation C∗-algebra 48

§1.8. Minimal Foliations and their C∗-algebras 54

Chapter 2. Harmonic Measures for Foliated Spaces 61

§2.1. Existence of Harmonic Measures 62

§2.2. The Diffusion Semigroup 68

§2.3. The Markov Process 80

§2.4. Characterizations of Harmonic Measures 86

§2.5. The Ergodic Theorem 96

§2.6. Ergodic Decomposition of Harmonic Measures 99

§2.7. Recurrence 112

vii

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viii Contents

Chapter 3. Generic Leaves 119

§3.1. The Main Results and Examples 119

§3.2. The Holonomy Graph 122

§3.3. Proof of the Theorems 128

§3.4. Generic Geometry of Leaves 131

Part 2. Characteristic Classes and Foliations

Foreword to Part 2 139

Chapter 4. The Euler Class of Circle Bundles 141

§4.1. Generalities about Bundles 142

§4.2. Cell Complexes 144

§4.3. The First Obstruction 148

§4.4. The Euler Class 155

§4.5. Foliated Circle Bundles 164

§4.6. Further Developments 174

Chapter 5. The Chern-Weil Construction 177

§5.1. The Chern-Weil Homomorphism 178

§5.2. The Structure of I∗n(K) 181

§5.3. Chern Classes and Pontryagin Classes 184

Chapter 6. Characteristic Classes and Integrability 187

§6.1. The Bott Vanishing Theorem 187

§6.2. The Godbillon-Vey Class in Arbitrary Codimension 192

§6.3. Construction of the Exotic Classes 194

§6.4. Haefliger Structures and Classifying Spaces 200

Chapter 7. The Godbillon-Vey Classes 209

§7.1. The Godbillon Class and Measure Theory 209

§7.2. Proper Foliations 232

§7.3. Codimension One 234

§7.4. Quasi-polynomial Leaves 239

Part 3. Foliated 3-Manifolds

Foreword to Part 3 251

Chapter 8. Constructing Foliations 253

§8.1. Orientable 3-Manifolds 254

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Contents ix

§8.2. Open Book Decompositions 261

§8.3. Nonorientable 3-Manifolds 262

§8.4. Raymond’s Theorem 265

§8.5. Thurston’s Construction 274

Chapter 9. Reebless Foliations 285

§9.1. Statements of Results 286

§9.2. Poincare-Bendixson Theory and Vanishing Cycles 290

§9.3. Novikov’s Exploding Disk 300

§9.4. Completion of the Proofs of Novikov’s Theorems 307

§9.5. The Roussarie-Thurston Theorems 312

Chapter 10. Foliations and the Thurston Norm 325

§10.1. Compact Leaves of Reebless Foliations 326

§10.2. Knots, Links, and Genus 333

§10.3. The Norm on Real Homology 340

§10.4. The Unit Ball in the Thurston Norm 345

§10.5. Foliations without Holonomy 355

Chapter 11. Disk Decomposition and Foliations of Link Complements 361

§11.1. A Basic Example 361

§11.2. Sutured Manifolds 364

§11.3. Operations on Sutured Manifolds 367

§11.4. The Main Theorem 376

§11.5. Applications 385

§11.6. Higher Depth 397

Appendix A. C∗-Algebras 399

§A.1. Bounded Operators 399

§A.2. Measures on Hausdorff Spaces 400

§A.3. Hilbert Spaces 403

§A.4. Topological Spaces and Algebras 406

§A.5. C∗-Algebras 408

§A.6. Representations of Algebras 410

§A.7. The Algebra of Compact Operators 415

§A.8. Representations of C0(X) 418

§A.9. Tensor Products 420

§A.10. Von Neumann Algebras 422

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x Contents

Appendix B. Riemannian Geometry and Heat Diffusion 425

§B.1. Geometric Concepts and Formulas 425

§B.2. Estimates of Geometric Quantities 428

§B.3. Basic Function Theory 432

§B.4. Regularity Theorems 433

§B.5. The Heat Equation 436

§B.6. Construction of the Heat Kernel 438

§B.7. Estimates for the Heat Kernel 445

§B.8. The Green Function 447

§B.9. Dirichlet Problem and Harmonic Measure 449

§B.10. Diffusion and Resolvent 453

Appendix C. Brownian Motion 461

§C.1. Probabilistic Concepts 461

§C.2. Construction of Brownian Motion 465

§C.3. The Markov Process 469

§C.4. Continuity of Brownian Paths 474

§C.5. Stopping Times 478

§C.6. Some Consequences of the Markov Property 481

§C.7. The Discrete Dirichlet and Poisson Problems 483

§C.8. Dynkin’s Formula 486

§C.9. Local Estimates of Exit Times 492

Appendix D. Planar Foliations 497

§D.1. The Space of Leaves 497

§D.2. Basic Isotopies 501

§D.3. The Hausdorff Case 506

§D.4. Decomposing the Foliation 510

§D.5. Construction of the Diffeomorphism 513

Bibliography 527

Index 537

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Preface

For this second volume of Foliations, we have selected three special topics:analysis on foliated spaces, characteristic classes of foliations, and foliated3-manifolds. Each of these is an example of deep interaction between foli-ation theory and some other highly developed area of mathematics. In allcases, our aim is to give useful, in-depth introductions.

In Part 1 we treat C∗-algebras of foliated spaces and generalize heatflow and Brownian motion in Riemannian manifolds to such spaces. Thefirst of these topics is essential for the “noncommutative geometry” of thesespaces, a deep theory originated and pursued by A. Connes. The second isdue to L. Garnett. While the heat equation varies continuously from leafto leaf, its solutions have an essentially global character, making them hardto compare on different leaves. We will show, however, that leafwise heatdiffusion defines a continuous, 1-parameter semigroup of operators on theBanach space C(M) and, following Garnett [77], we will construct prob-ability measures on M that are invariant under this semiflow. These arecalled harmonic measures, and they lead to a powerful ergodic theory forfoliated spaces. This theory has profound topological applications (cf. The-orem 3.1.4), but its analytic and probabilistic foundations have made accessdifficult for many topologists. For this reason, we have added two surveyappendices, one on heat diffusion in Riemannian manifolds and one on theassociated Brownian flow. For similar reasons, we have added an appendixon the basics of C∗-algebras. We hope that these will serve as helpful guidesthrough the analytic foundations of Part 1.

Part 2 is devoted to characteristic classes and foliations. FollowingR. Bott [9], we give a Chern-Weil type construction of the exotic classesbased on the Bott vanishing theorem (Theorem 6.1.1). The resulting theory

xi

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xii Preface

can be viewed either as a topic in algebraic topology, motivated by folia-tion theory, or as a deep application of algebraic topology to the study offoliations. We take the latter viewpoint, emphasizing qualitative aspectssuch as G. Duminy’s celebrated vanishing theorem for the Godbillon-Veyclass (unpublished) and S. Hurder’s analogous theorems for the generalizedGodbillon-Vey classes in higher codimension [102]. We begin Part 2 witha chapter on the “grandfather” of all characteristic classes, the Euler classof oriented circle bundles, giving complete proofs of the applications, due toJ. Milnor [129] and J. Wood [189], concerning obstructions to the existenceof foliations transverse to the fibers of circle bundles over surfaces.

In Part 3, we study compact 3-manifolds foliated by surfaces, a topicthat has been popular since the advent of the Reeb foliation of S3. The spe-cial methods of 3-manifold topology yield existence theorems and qualitativeproperties unique to dimension three. The theorem of S. P. Novikov [141]on the existence of Reeb components has the consequence that “Reebless fo-liations” carry important topological information about the ambient 3-man-ifold. Together with a theorem of W. Thurston [175] on compact leavesof Reebless foliations, this led to D. Gabai’s groundbreaking work in whichtaut foliations are used as powerful tools for studying 3-manifold topology.We develop this theory up to Gabai’s constructions of taut, finite depthfoliations on certain sutured 3-manifolds, giving details only in the disk de-composable case (depth one). This will bring the reader to the threshold ofthe “modern age” of essential laminations. These laminations are general-izations simultaneously of taut foliations and incompressible surfaces, andare the object of much current research. Essential laminations, however,need a book of their own and we hope that one or more of the specialistswill provide such.

Appendix D pertains to Part 3, being a detailed account of Palmeira’stheorem that the only simply connected n-manifold foliated by leaves dif-feomorphic to Rn−1 is Rn. In fact, if n ≥ 3, the foliated manifold is diffeo-morphic to R2 × Rn−2 in such a way that the foliation is the product of afoliation of R2 by the space Rn−2. Although valid in all dimensions n ≥ 3,this result has important applications to Reebless-foliated 3-manifolds.

The bibliography is not intended to be a comprehensive list of all pub-lications on these areas of foliation theory. Only references explicitly citedin the text are included, with the result that many important papers andbooks are omitted (with apologies to the authors).

The three parts of this book can be read independently. One minorexception to this is that certain standard properties of the Euler class, provenin Part 2, are needed in Part 3. Of course, all parts depend in various ways

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Preface xiii

on material in Volume I. All references to that volume will be of the form[I,. . . ].

Finally, the first named author expresses his sincere thanks and appre-ciation to the second for his invitation to join in this journey through thetheory of foliations, and for seeing that it got to an end.

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Index

∼−, 499

∼+, 499

A∗�(M), 224

A∗�(M,�), 224

�′(U), 9

absolutely continuous, 88

abutment, 125

abuts, 125

adequate neighborhood, 314

adjoint operator, see also operator(s)

admissible imbedding, see also imbedding

admissible surface, see also surface

Alcalde Cuesta, F., 286

Alexander, J., 261, 289

algebra, 406

anticommutative, 177

Banach, 406

involutive, 408

C∗-, see also C∗-algebraChern, 178, 181, 184

graded commutative, 177

tensor product, 196

Pontryagin, 178, 181, 185

representations of, 410–415

containment of, 411

cyclic, 411

definition of, 410

direct integral of, 411

direct sum of, 411

faithful, 411

involutive, 410

irreducible, 412

nondegenerate, 410

topologically irreducible, 412

unitarily equivalent, 411

weak containment of, 413

truncated polynomial, 196

von Neumann, 52, 422

Alvarez Lopez, J. A., 131, 494

ambient isotopy, 502

Anosov diffeomorphism, 120

anticommutative algebra, see also algebra

approximate unit, 409

Bkg , 126

(H), 407

(X,Y ), 400

Ballantine ale rings, 337

Banach algebra, see also algebra

Banach space, 399

barycenter, 259

barycentric subdivision, see also subdivision

basic connection, see also connection

Bauer maximum principle, 106

bee, 483

Bishop’s comparison theorem, 430

Blank, S., 326

blow up nicely, 385

Blumenthal’s zero-one law, 473

Bogoliuboff, N., 67, 108

Borel

σ-field, 461

map, 124

measure, see also measure(s)

set, 123

transversal, 42, 47

Borromean rings, 337, 347

Bott connection, see also connection

Bott vanishing theorem, 187–192

statement, 188

Bott, R., 7, 139, 187

boundary of subcomplex, 126

bounded domain, see also domain

bounded geometry, 428

bounded operator, see also operator(s)

Bourbaki, N., 400

branch point, 499

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538 Index

Brownian

expectation, 478

motion, 461–496

construction, 465–469

process, 471

particle, 461, 474

paths, 474–477

bundle, 142–144

2-plane, 141

circle, 141–175

foliated, 164–173

dual, 156

normal, 188, 206

of densities, 7

pullback, 142

homotopy invariance of, 143

universal, 202

C(D), 425

C0(D), 425

C0(X), 402

C0, 29

Cc(X), 35, 400

C∞c (M), 8

CJ , 502

Ck(D), 425

Ck(D), 425

Ck(D), 426

C∗-algebra, 408–410definition of, 408

noncommutative, 5

nuclear, 421

of a foliated space, 3, 18–27

definition of, 24

full, 26

reduced, 24

of a minimal foliation, 54–59

positive linear functional on, 410

primitive, 58

simple, 413

C∗-norm, 408

Candel, A., 131

Cantwell, J., 134

Cantwell-Conlon, 121

Cayley graph, 122

Cech cohomology, 201

cell, 144

cell complex, see also complex

cellular

approximation theorem, 145

map, 145

center tangency, see also tangency

characteristic class, see also class

Cheng, S. Y., 445

Chern

algebra, see also algebra

class, see also class

Chern∗(E), 181

Chern-Weil construction, 139

Chern-Weil homomorphism, 178–181

definition of, 180

Choquet’s theorem, 107

Choquet, G., 107

circle bundle, see also bundle

circle tangency, see also tangency

class

characteristic, 139

Chern-Weil construction of, 177–186

for vector bundles, 177–186

Chern, 141, 184

construction of, 184–186

first, 156

total, 185

Euler, 139, 155–164

definition of, 155

relative, 155

exotic, 139, 191

construction, 194–200

definition, 198

Godbillon, 210–212

definition of, 210

Godbillon-Vey, 139

arbitrary codimension, 192–194

definition of, 192

generalized, 199

Pontryagin, 184

construction of, 184–186

total, 185

secondary, 191

Whitney, 201

classifying space, 200–208

for vector bundles, 201–202

Haefliger, 139, 206–208

Milnor, 201–202

cobble, 509

cocycle

ε-tempered, 230

Γq , 203–206

Gq, 200–201

Haefliger

definition of, 204

of a foliation, 204

holonomy, 122, 203

integrable, 221

measurable, 221

obstruction, 150–153

on a groupoid, 221

pull-back, 201

structure, 178

coherent, 200, 204

commutant, 422

compact operator, see also operator(s)

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Index 539

completely invariant harmonic measure, seealso measure(s)

complexcell, 144–147

homology of, 144regular, 146skeleton of, 144subcomplex of, 144weakly regular, 146

CW, 144

Conlon, L., 134connection

basic, 189–190existence, 190

Bott, 139, 189–190existence, 190

form, 178Connes, A., xi, 3, 5, 6, 10, 11, 52, 54, 422containment of representations of an alge-

bra, see also algebracontinuity of diffusion, 73

convergencestrong, 400weak, 400

convolutionon Γc(G,∞), 22

on Γc(G,�1/2)in the non-Hausdorff case, 21

in the Hausdorff case, 19convolution of a family of operators, see also

operator(s)counter-orientation, 276, 280counter-oriented triangulation, 280Crainic, M., 8crystalline subdivision, see also subdivisioncurrent, 38

equivalent, 38invariant, 40quasi-invariant, 37–48

definition of, 40

curvature, 178form, 178

CW complex, see also complexcycle

vanishing, 285definition of, 287simple, 302

cyclic vector, 411cyclic representation of an algebra, see also

algebracylinder sets, 82cylindrical collar, 517

DM , 269∂τM -incompressible, 313Davidson, K. R., 399Debiard, A., 494

decomposable operator, see also operator(s)Dehn twist, 254Dehn’s Lemma, 329Dehn, M., 254, 329density, 7

α-, 7half-, 7

Hilbert space of, 8square integrable, 7

order of, 7positive, 7

density point, 111diagonalizable operator, see also operator(s)

DiffJ , 502diffused measure, see also measure(s)

diffusion operator, see also operator(s)diffusion semigroup, 68–80

definition of, 69Dirac’s bra-ket, 416direct integral of representations of an alge-

bra, see also algebradirect sum of representations of an algebra,

see also algebraDirichlet problem, 449

discrete, 483

discrete homogeneous space, 132discrete Poisson problem, 486disk decomposable surface, see also surfacedisk decomposable sutured manifold, see also

sutured manifolddisk decomposition, 361–398distribution, 433distribution solution, 434divergence, 426

divergence theorem, 427relative, 427

Dixmier, J., 34, 399, 422Dixmier-Douady invariant, 34domain, 425

bounded, 425regular, 425

Douady, A., 34double of M , 269

double points, 301dual bundle, see also bundledual norm, see also normDuminy

decomposition, 235vanishing theorem for gv(�), 139, 234

Duminy, G., 54, 131, 209, 210, 214, 234, 273Durfee, A. H., 261

Dynkin’s formula, 486–492general version, 487simplest version, 486

Dynkin, E., 83, 453, 474, 486

ε-tempering, 239

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540 Index

edgepath, 125Effros, E. G., 59Ehresmann, C., 11elliptic regularity theorem, 435EmbJ , 502end of a graph, 123

Epstein, D. B. A., 232, 274Epstein-Millett filtration, 232equivalent currents, see also currentergodic, 121ergodic component, 121ergodic decomposition of harmonic measures,

see also measure(s)

ergodic measure, see also measure(s)ergodic theorem, 96, 98essential loop, see also loopEuler class, see also classexceptional minimal sets, 265–274

generic leaf of, 131

exotic class, see also classexpectation, 462

conditional, 463expected value, 462exploding

annulus, 321disk, 287, 300–307

plateau, 300extreme point, 103

�-flat, 297�◦, 302Fack, T., 5, 6, 54, 55, 59

faithful representation of an algebra, see alsoalgebra

fibered face, 356fibered ray, see also rayfield, 461

σ-field, 461Borel, 461

generated by S, 461field of operators, see also operator(s)Fillmore, P. A., 399finite rank operator, see also operator(s)first exit time, 478first obstruction, see also obstruction

flat, 297flat connection, 165foliated circle bundle, see also bundlefoliated face, 356foliated ray, see also rayfoliated space, 5

graph of, 11–18

definition of, 11transitive, 58

foliation cone, 356foliation(s)

constructions of, 253–283

planar, 497space of leaves of, 497–501

Reebless, 285–323Riemannian, 191

taut, 322without holonomy, 355–359

Folland, G. B., 400frog, 484fundamental family of measurable vector fields,

404

fundamental solution of the heat equation,437

G, 503Gn, 503GA, 12GB , 12

GBA , 12

Γc, 7Γq cocycle, see also cocycleΓq-structure, 203

Gq cocycle, see also cocycleGabai, D., 251, 323, 325, 361, 364, 501Garnett, L., xi, 3, 74Gaveau, B., 494Gelfand-Fuks cohomology, 187

Gelfand-Naımark-Segal representation, 415general position, 277

loop, 301Roussarie, 314Thurston, 322

genus of knots and links, 333–339definition of, 335

Ghys’s Proposition Fondamentale, 118Ghys, E., 3, 112, 113, 117, 119, 121, 134,

135, 174Godbillon class, see also class

Godbillon measure, 212–214definition of, 214

Godbillon-Vey class, see also classGodement resolution, 9good Borel set, 117

Goodman, S., 287, 308Gootman, E. C., 59graded commutative algebra, see also alge-

bragradient, 426

Green function, 448Green operator, see also operator(s)Green’s formula, 428Green-Gauss-Stokes-Ostrogradski formula, 427Gromov, M., 206

groupoid, 217of germs, 203

H(�X), 222H′(�X), 222

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Index 541

H∗(M ;�), 210

H(�X), 222

H′(�X), 222

Haefliger classifying space, see also classify-ing space

Haefliger cocycle, see also cocycle

Haefliger structure, 200–208

definition of, 204

homotopy of, 205

of a foliation, 204

Haefliger, A., 203, 277, 285

Hahn, F., 59

Hahn-Banach theorem, 66

handlebody, 255

harmonic function, 432

harmonic measure, see also measure(s)

harmonic measure one, 108

harmonic measure zero, 108

Harnack principle, 451

Harnack’s theorem, 452

Hass, J., 323

heat equation, 436

heat kernel, 437

existence and uniqueness, 442, 444

Hector, G., 131, 274, 286

Heegaard splitting, 258

Heitsch, J., 54, 225

Herman number, 275

Herman, M., 274

Hilbert integral, 405

Hilbert space(s), 403

dimension of, 403

field of, 404

direct integral of, 405

measurable, 404

separable, 403

tensor product of, 404

Hilbert sum, 404

Hille, E., 69

Hille-Yosida theorem, 74

Hilsum, M., 37

Hirsch example, 95

Hirsch, M. W., 119

holonomy

covering, 11

graph, 122

group, 11

groupoid, 11–18

definition of, 11

pseudogroup, 122

representation, 11

holonomy cocycle, see also cocycle

homology of a cell complex, see also complex

homotopy extension theorem, 145

Hopf fibration, 157

Hopf, H., 119

Hurder, S., 54, 139, 192, 209, 214, 220, 225,226, 229, 232, 239

imbeddingadmissible, 313reduced, 314

incompressible surface, see also surfaceindex of a vector field at a singularity, 161index sum of a vector field, 161index theorem

Atiyah-Singer, 6foliation, 6

inductive limit topology, 401inessential loop, see also loopinfinitesimal generator, 70, 456

infinitesimal holonomy, 214, 217inflate, 282injectivity radius, 428integrable cocycle, see also cocycle

integral linear functional, 350integral norm, see also normintegration along the fiber, 195interior of subcomplex, 126

intrinsic domain, 454invariant current, see also currentinvolution, 408

on Γc(G,�1/2)in the Hausdorff case, 19

in the non-Hausdorff case, 21involutive Banach algebra, see also algebrainvolutive representation of an algebra, see

also algebrairreducible, 288

irreducible representation of an algebra, seealso algebra

irregular point, 479isometry of Hilbert spaces, 403isotopy respecting �′, 315Ito, K., 474

jiggle a triangulation, 276, 277juncture, 387

Kg(x), 125Kg(x)c, 125

(H), 416Kac’s recurrence theorem, 113Kac, M., 113, 114Kaplan, W., 500

Katok, A., 54, 232, 239Kazez, W., 501Kellum, M., 135Kinney, J. R., 83, 474Kirillov, A. A., 399

knot, 333alternating, 392–397complement, 334

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542 Index

Kolmogoroff, A. N., 474Krein-Milman theorem, 106Krieger, W., 54Kryloff, N., 67, 108

Langevin, R., 209

Laplace operator, see also operator(s)Laplacian, 426

leafwise, 62Laudenbach, F., 326Lawson, H. B., 261

leafgeneric, 119, 128π1-injective, 287totally recurrent, 122

leafwise Riemannian measure, see also mea-

sure(s)leafwise Stokes’ theorem, 225Lebesgue

current, 40density theorem, 218

level, 520level-preserving

diffeotopy, 502embedding, 502map, 502

Levitt, G., 174Levy, P., 474Li, P., 445Lickorish, W., 253, 254link, 333

alternating, 392–397complement, 334Whitehead, 337, 346, 370Whitehead-like, 338

link complement, 363, 386

link exterior, 386linking number, 346longitude, 386loop

essential, 256, 286

inessential, 286nullhomotopic, 286

loop in general position, see also general po-sition

Loop Theorem, 329

M◦, 302Mackey, G. W., 35Markov

process, 470on a foliated space, 80–86

property, 471, 478

for functions, 471strong, 480

Mather, J., 274maximum principle, 432

boundary, 433

Mazet, E., 494

McKean, H. P., 474

meager, 128

measurable cocycle, see also cocycle

measurable space, 461

measure class, 219

measure zero (with respect to a current), 39

measure(s)

Borel, 401

diffused, 109

ergodic, 99, 121

harmonic, 61–120, 450

characterizations of, 86–96

completely invariant, 91

definition of, 63

ergodic decomposition of, 108

existence of, 67, 90

holonomy-invariant, 120

leafwise Riemannian, 63

Lebesgue, 120

mutually singular, 104

push-forward, 401

Radon, 62, 401

smooth, 88

visual, 91

Wiener, 461

meridian, 386

metric

on a graph, 122

on a leaf L, 123

Millett, K., 232

Milnor classifying space, see also classifyingspace

Milnor, J., 139, 141, 164, 202

Minakshisundaram, S., 439

modular function, 40

Moerdijk, I., 8

monotone class theorem, 464

Moore, C., 6, 52

Morita equivalence, 36

Morse tangency, see also tangency

Moussou, R., 192

Murasugi sum, 388–390

definition of, 388

Murray, F. J., 52

mutually singular measures, see also mea-sure(s)

Natsume, T., 37

naturality, 186

Naımark, M. A., 399

negative saddle tangency, see also tangency

non-Hausdorff spaces

functions on, 8–11

noncommutative geometry, 6

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Index 543

nondegenerate representation of an algebra,see also algebra

nonwandering set, 112norm

dual, 340, 345integral, 350

Thurston, 325–359of a homology class, 326of a surface, 326on real homology, 340–345

normal bundle, see also bundlenormal decomposition, 510normal plug, 507

bases of, 507wall of, 507

Novikov, S. P., 253, 285, 287, 288nuclear C∗-algebra, see also C∗-algebranullhomotopic loop, see also loop

Ω(L), 82Ω(M), 81Ω(X), 461obstruction

cochain, 151cocycle, 153

relative, 155first, 148–155

definition of, 151open book decomposition, 261–262

definition of, 261of nonorientable manifolds, 265

operator(s)

adjoint, 409bounded, 399–400

between Hilbert spaces, 403definition of, 399extension of, 400

compact, 415–418definition of, 416

decomposable, 406diagonalizable, 406diffusion, 453family

convolution of, 73strongly continuous, 72

field of, 405measurable, 405

Green, 459Laplace, 407norm, 400of finite rank, 416projection, 423

resolvent, 459ring of, 422tensor product of, 404topology, 400unitary, 403

orthonormal system, 403

π1-injective, 285Px, 461, 467pair of pants, 346Palais, R., 502

Palmeira, F., 286, 289, 497, 501Papakyriakopoulos, C. D., 329parallel normal fields, 189parametrix, 439

Pasternack, J., 192Pedersen, G. K., 399, 400Pelletier, F., 192Petersen, K., 114Phillips, A., 206

Phillips, R. S., 69pinched annulus, 296Pixton, D., 398planar foliation, see also foliation(s)

plumbing, 389Poenaru, V., 206Poincare lemma, 195Poincare-Hopf theorem, 161

for surfaces with boundary, 163Poincare’s recurrence theorem, 114Poincare, H., 114Poisson problem, 453

infinite domains, 490

probabilistic solution, 490Pont∗(E), 181Pontryagin

algebra, see also algebra

class, see also classpositive linear functional, 410positive saddle tangency, see also tangencyprobability space, 462

product decomposition, 376projection operator, see also operator(s)properly imbedded surface, see also surfaceproperty P, 388property R, 386–388

definition of, 388pseudo-analytic, 14pseudogroup (holonomy), 122pullback bundle, see also bundle

quantitative theory, 200

quasi-invariant current, see also currentquasi-isometry type, 131

coarse, 131quasi-polynomial growth, 241

quasi-regular point, 108quasi-regular set, 109quasi-symmetric, 133

r-graph, 123Radon measure, see also measure(s)

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544 Index

Radon-Nikodym theorem, 463range map, 12ray

fibered, 356foliated, 355

Raymond, B., 253, 265

recurrence in foliated spaces, 112–118reduced imbedding, see also imbeddingReeb

component, 285foliation, see also foliation(s)modification, 253

stability (local), 127Reeb, G., 253Reebless foliations, see also foliation(s)regular cell complex, see also complexregular domain, see also domainregular foliated atlas, 17regular points, 112

regular representation of a group, 414representable functor, 202, 334representation

holonomy, 11of the foliation C∗-algebra, 48–54of the graph, 48

regular, 48, 50trivial, 48, 50

representation of an algebra, see also alge-bra

residual, 119resolvent operator, see also operator(s)Rieffel, M. A., 36Riesz representation theorem, 402

ring of subsets, 461Rosenberg, H., 286, 288Rosenberg, J., 59Roussarie general position, see also general

positionRoussarie, R., 312, 314Rudin, W., 400

σ-cylinder, 466σg(x), 125Σg , 147saddle tangency, see also tangency

Saint-Venant equation, 490Sauvageot, J.-L., 59Schochet, C., 6Schweitzer, P., 274, 286secondary class, see also classSeifert

algorithm, 391–392

circle, 391surface, see also surface

semigroup of operators, 69Sergeraert, F., 274Sergiescu, V., 174

sheaf topology, 203Siebenmann, L. C., 286simple C∗-algebra, see also C∗-algebrasimple vanishing cycle, see also cycleSkandalis, G., 5, 6, 37, 54, 55skeleton of a cell complex, see also complex

smooth measure, see also measure(s)Solodov, V., 286source, 203source map, 12space of leaves, 5spanning surface, see also surface

spinning, 253, 262spiral ramps, 282Stallings, J., 329star of vertex, 126state, 410stochastic process, 470stopping time, 478

hitting time, 479strong convergence, see also convergencestrongly continuous family of operators, see

also operator(s)subcomplex, see also complexsubdivision

barycentric, 276, 278

crystalline, 276, 277Sullivan, D., 192, 323surface

admissible, 313disk decomposable, 370incompressible, 313properly imbedded, 312

∂τM -incompressible, 313incompressible, 313

Seifert, 335, 365spanning, 334

surgery, 254suture, 364

sutured manifold, 361, 363–367definition of, 364disk decomposable, 369hierarchy, 397taut, 367

symbolic dynamics, 273symmetric polynomial, 181

Takesaki, M., 399, 420, 422Tamura, I., 261tangency

center, 315circle, 314

Morse, 290saddle, 314

negative, 332positive, 331

target, 203

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Index 545

thickness, 235thin, 234three-link chain, 339, 348Thurston ball, 345–355

Thurston cone, 356Thurston general position, see also general

positionThurston norm, see also norm

on real homology, see also norm

Thurston, W., 174, 191, 207, 251, 253, 274,312, 325

topologically irreducible representation of analgebra, see also algebra

Torpe, A. M., 37total, 403

transitive point, 111transversality, 277trefoil knot, 335Tsuboi, T., 191

Tu, L., 7turbulization, 253twisted

density, 6–8

form, 6–8definition of, 6

unit, 407unitarily equivalent representations of an al-

gebra, see also algebra

unitary operator, see also operator(s)universal bundle, see also bundleunknot, 335

vanishing cycle, see also cycleVectq(X), 201

visual measure, see also measure(s)vol, 63von Neumann algebra, see also algebravon Neumann, J., 52

Wallace, A. H., 254wandering

leaf, 112point, 112

set, 112Wang, X., 37weak convergence, see also convergenceweakly regular cell complex, see also com-

plex

weak∗ topology, 89Weil, A., 68Whitehead

double, 335link, see also link

Whitehead-like link, see also linkWhitney class, see also classWhitney duality, 185

Wiener measures, see also measure(s)Wiener, N., 461, 474Williams, D. P., 59Winkelnkemper, E., 11, 261Wood, J., 139, 141, 165, 253, 262, 274Wright, F., 114

Yau, S.-T., 445Yosida, K., 69, 108, 399, 453

Zkg , 126, 128

Z(r), 123Z×r, 123Zeller-Meier, G., 59Zieschang, H., 253

Page 35: Alberto Candel Lawrence Conlon · Foliations I / Alberto Candel, Lawrence Conlon. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; ... we study compact 3-manifolds foliated

Titles in This Series

60 Alberto Candel and Lawrence Conlon, Foliations II, 2003

59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic,2003

58 Cedric Villani, Topics in optimal transportation, 2003

57 Robert Plato, Concise numerical mathematics, 2003

56 E. B. Vinberg, A course in algebra, 2003

55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003

54 Alexander Barvinok, A course in convexity, 2002

53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002

52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis,geometry and physics, 2002

51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002

50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002

49 John R. Harper, Secondary cohomology operations, 2002

48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002

47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation,2002

46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry andLie groups, 2002

45 Inder K. Rana, An introduction to measure and integration, second edition, 2002

44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002

43 N. V. Krylov, Introduction to the theory of random processes, 2002

42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002

41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002

40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable,2002

39 Larry C. Grove, Classical groups and geometric algebra, 2002

38 Elton P. Hsu, Stochastic analysis on manifolds, 2002

37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modulargroup, 2001

36 Martin Schechter, Principles of functional analysis, second edition, 2002

35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001

34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001

33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001

32 Robert G. Bartle, A modern theory of integration, 2001

31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methodsof financial mathematics, 2001

30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001

29 Javier Duoandikoetxea, Fourier analysis, 2001

28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000

27 Thierry Aubin, A course in differential geometry, 2001

26 Rolf Berndt, An introduction to symplectic geometry, 2001

25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000

24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/.

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www.ams.orgAMS on the WebGSM/60

This is the second of two volumes on foliations (the fi rst is Volume 23 of this series). In this volume, three specialized topics are treated: analysis on foliated spaces, characteristic classes of foliations, and foliated three-manifolds. Each of these topics represents deep interaction between foliation theory and another highly developed area of mathematics. In each case, the goal is to provide students and other interested people with a substantial introduction to the topic leading to further study using the extensive available literature.

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