Alberto Candel Lawrence Conlon · Foliations I / Alberto Candel, Lawrence Conlon. p. cm. —...
Transcript of Alberto Candel Lawrence Conlon · Foliations I / Alberto Candel, Lawrence Conlon. p. cm. —...
Alberto CandelLawrence Conlon
American Mathematical Society
Graduate Studies in Mathematics
Volume 60
Foliations II
Foliations II
http://dx.doi.org/10.1090/gsm/060
Alberto CandelLawrence Conlon
Foliations II
American Mathematical SocietyProvidence, Rhode Island
Graduate Studies in Mathematics
Volume 60
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
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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–
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10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03
To our wives,
Juana and Jackie
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
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
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
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
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
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
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.
Bibliography
1. J. W. Alexander, The combinatorial theory of complexes, Ann. of Math. 31 (1930),292–320.
2. , A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. 9 (1923),93–95.
3. J. A. Alvarez Lopez and A. Candel, Generic geometry of leaves, Preprint, 1999.(Revised version in preparation.)
4. , Equicontinuous foliated spaces, Preprint, 2001.
5. , Topological description of Riemannian foliations with dense leaves, Preprint,2002.
6. T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampere Equations, Grundlehrender Mathematische Wissenschaften, Vol. 252, Springer-Verlag, New York, NY, 1982.
7. M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d’une Variete Riemannienne,Lecture Notes in Mathematics, vol. 194, Springer-Verlag, Berlin, 1971.
8. S. A. Bleiler and A. J. Casson, Automorphisms of Surfaces after Nielsen andThurston, Cambridge Univ. Press, Cambridge, 1988.
9. R. Bott, Lectures on characteristic classes and foliations, Lecture Notes in Math.,279, Springer-Verlag, New York, NY, 1972, pp. 1–94.
10. , On some formulas for the characteristic classes of group–actions, LectureNotes in Math., 652, Springer-Verlag, New York, NY, 1978, pp. 25–56.
11. R. Bott and L. Tu, Differential Forms in Algebraic Topology, Springer–Verlag, NewYork, NY, 1982.
12. N. Bourbaki, Elements de mathematique. Fasc. XIII. Livre VI: Integration. Chapitres1, 2, 3 et 4: Inegalites de convexite, Espaces de Riesz, Mesures sur les espaceslocalement compacts, Prolongement d’une mesure, Espaces Lp, Deuxieme edition,revue et augmentee. Actualites Scientifiques et Industrielles, No. 1175, Hermann,Paris, 1965.
13. R. Bowen, Weak mixing and unique ergodicity of homogeneous spaces, Israel J. ofMath. 23 (1979), 337–342.
14. L. G. Brown, P. Green, and M. A. Rieffel, Stable isomorphism and strong Moritaequivalence of C∗-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363.
527
528 Bibliography
15. B. Burde and H. Zieschang, Knots, Studies in Mathematics 5, de Gruyter, New York,NY, 1985.
16. A. Candel, Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. 26(1993), 489–516.
17. , C∗-algebras of proper foliations, Proc. Amer. Math. Soc. 124 (1996), no. 3,899–905.
18. , The harmonic measures of Lucy Garnett, To appear in Adv. Math.
19. J. Cantwell and L. Conlon, Growth of leaves, Comment. Math. Helv. 53 (1978),93–111.
20. , The Dynamics of Open Foliated Manifolds and a Vanishing Theorem for theGodbillon–Vey Class, Advances in Math. 53 (1984), 1–27.
21. , Every surface is a leaf, Topology 26 (1987), 265–285.
22. , Foliations and subshifts, Tohoku Math J. 40 (1988), 165–187.
23. , Leaves of Markov local minimal sets in foliations of codimension one, Pub-licacions Matematiques 33 (1989), 461–484.
24. , Generic leaves, Comment. Math. Helv. 73 (1998), 306–336.
25. , Isotopies of foliated 3–manifolds without holonomy, Adv. in Math. 144(1999), 13–49.
26. , Endsets of exceptional leaves; a theorem of G. Duminy, Proceedings ofthe Euroworkshop on Foliations, Geometry and Dynamics, World Scientific, 2002,pp. 225–261.
27. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Orlando, FL, 1984,Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk.
28. , Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives,Cambridge University Press, Cambridge, 2001.
29. S.Y. Cheng, P. Li, and S.-T. Yau, On the upper estimate of the heat hernel of acomplete Riemannian manifold, Amer. J. Math. 103 (1981), no. 5, 1021–1063.
30. G. Choquet, Lectures on Analysis. Vol. II: Representation Theory, W. A. Benjamin,Inc., New York-Amsterdam, 1969.
31. K. L. Chung, Lectures from Markov Processes to Brownian Motion, Grundlehren derMathematische Wissenschaften, Vol. 249, Springer-Verlag, New York, NY, 1982.
32. L. Conlon, Differentiable Manifolds; Second Edition, Birkhauser, Boston, Mas-sachusetts, 2001.
33. A. Connes, The von Neumann algebra of a foliation, Mathematical problems in the-oretical physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), Lecture Notes inPhysics, no. 80, Springer, Berlin, 1978, pp. 145–151.
34. , Sur la theorie non commutative de l’integration, Algebres d’operateurs(Sem., Les Plans-sur-Bex, 1978), Springer–Verlag, Berlin, 1979, pp. 19–143.
35. , Feuilletages et algebres d’operateurs, Lecture Notes in Math., no. 842,Springer-Verlag, Berlin, 1981, pp. 139–155.
36. , A survey of foliations and operator algebras, Proc. Symp. Pure Math.,vol. 38, Amer. Math. Soc., 1982, pp. 521–628.
37. , Cyclic cohomology and the transverse fundamental class of a foliation, Geo-metric methods in operator algebras (Kyoto, 1983), Longman Sci. Tech., Harlow,1986, pp. 52–144.
38. , Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994.
Bibliography 529
39. M. Crainic and I. Moerdijk, A remark on sheaf theory for non-Hausdorff manifolds,Tech. Report 1119, Utrecht University, 1999.
40. , A homology theory for etale groupoids, J. Reine Angew. Math. 521 (2000),25–46.
41. R. Crowell, Genus of alternating link types, Ann. of Math. 69 (1959), 258–275.
42. F. Alcalde Cuesta, G. Hector, and P. Schweitzer, Sur l’existence de feuilles compactsen codimension 1, in preparation.
43. K. R. Davidson, C∗-Algebras by Example, Fields Institute Monographs, vol. 6, Amer-ican Mathematical Society, Providence, R.I., 1996.
44. G. de Rham, Varietes Differentiables, Hermann, Paris, 1960.
45. A. Debiard, B. Gaveau, and E. Mazet, Theoremes de comparaison en geometrieriemannienne, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 2, 391–425.
46. M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135–206, Trans-lated and reprinted in “Papers on Group Theory and Topology by Max Dehn”,Springer–Verlag, 1987, pp. 256–362.
47. J. Dixmier, C∗-Algebras, North-Holland Publishing Co., Amsterdam, 1977, Trans-lated from the French by Francis Jellett, North-Holland Mathematical Library, Vol.15.
48. , Von Neumann Algebras, North-Holland Publishing Co., Amsterdam, 1981,With a preface by E. C. Lance, Translated from the second French edition by F.Jellett.
49. J. Dixmier and A. Douady, Champs continus d’espaces hilbertiens et de C∗-algebres,Bull. Soc. Math. France 91 (1963), 227–284.
50. J. Dodziuk, Maximum principle for parabolic equations and the heat flow on openmanifolds, Indiana Univ. Math. J. 32 (1983), 703–716.
51. R. Dudley, Real Analysis and Probability, Wadsworth & Brooks/Cole AdvancedBooks & Software, Pacific Grove, CA, 1989.
52. G. Duminy, L’invariant de Godbillon–Vey d’un feuilletage se localise dans les feuillesressort, preprint (1982).
53. N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, JohnWiley & Sons, Inc., New York, NY, 1988, With the assistance of William G. Badeand Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication.
54. A. H. Durfee, Foliations of odd dimensional spheres, Ann. of Math. 97 (1973), 407–411.
55. A. H. Durfee and H. B. Lawson, Fibered knots and foliations of highly connectedmanifolds, Inv. Math. 17 (1972), 203–215.
56. E. Dynkin, Theory of Markov Processes, Prentice–Hall, Englewood Cliffs, NJ, 1961,Translated from the Russian by D. E. Brown; edited by T. Kovary.
57. , Markov Processes. Vols. I, II, Translated with the authorization and as-sistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. DieGrundlehren der Mathematischen Wissenschaften, Bande 121, Vol. 122, AcademicPress Inc., Publishers, New York, NY, 1965.
58. E. G. Effros and F. Hahn, Locally compact transformation groups and C∗-algebras,Memoirs of the American Mathematical Society, no. 75, American MathematicalSociety, Providence, RI, 1967.
59. C. Ehresmann, Structures feuilletees, Proc. 5th Canadian Math. Congress, Univ. ofToronto Press, 1963, pp. 109–172.
530 Bibliography
60. D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, CompositioMath. 22 (1970), 165–173.
61. , Periodic flows on 3–manifolds, Ann. of Math. 95 (1972), 66–82.
62. T. Fack, Quelques remarques sur le spectre des C∗-algebres de feuilletages, Bull. Soc.Math. Belg. Ser. B 36 (1984), no. 1, 113–129.
63. T. Fack and G. Skandalis, Sur les representations et ideaux de la C∗-algebre d’unfeuilletage, J. Operator Theory 8 (1982), 95–129.
64. T. Fack and X. Wang. The C-algebras of Reeb foliations are not AF-embeddable,Proc. Amer. Math. Soc. 108 (1990), 941–946.
65. P. A. Fillmore, A User’s Guide to Operator Algebras, John Wiley & Sons, Inc., NewYork, NY, 1996, A Wiley-Interscience Publication.
66. G. B. Folland, Introduction to Partial Differential Equations, Princeton UniversityPress, Princeton, NJ, 1976, Preliminary informal notes of university courses andseminars in mathematics, Mathematical Notes.
67. , Real Analysis. Modern Techniques and their Applications, second ed., JohnWiley & Sons Inc., New York, NY, 1999, A Wiley-Interscience Publication.
68. H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,Princeton University Press, Princeton, NJ, 1981.
69. S. A. Gaal, Linear Analysis and Representation Theory, Die Grundlehren der Math-ematischen Wissenschaften, Band 198, Springer-Verlag, New York, NY, 1973.
70. D. Gabai, Foliations and the topology of 3–manifolds, J. Diff. Geom. 18 (1983), 445–503.
71. , Foliations and genera of links, Topology 23 (1984), 381–394.
72. , Genera of the arborescent links, Mem. Amer. Math. Soc. 59 (1986), 1–98.
73. , Foliations and the topology of 3–manifolds II, J. Diff. Geom. 26 (1987),461–478.
74. , Foliations and the topology of 3–manifolds III, J. Diff. Geom. 26 (1987),479–536.
75. , Combinatorial volume preserving flows and taut foliations, Comment. Math.Helv. 75 (2000), 109–124.
76. D. Gabai and W. Kazez, Homotopy, isotopy and genuine laminations of 3-manifolds,Geometric Topology (William Kazez, ed.) 1 (1997), 123–138.
77. L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal.51 (1983), no. 3, 285–311.
78. E. Ghys, Une variete qui n’est pas une feuille, Topology 24 (1985), 67–73.
79. , Classe d’Euler et minimal exceptionnel, Topology 26 (1987), 57–73.
80. , Topologie des feuilles generiques, Ann. of Math. 141 (1995), 387–422.
81. E. Ghys and V. Sergiescu, Sur un groupe remarquable de diffeomorphismes du cercle,Comment. Math. Helv. 62 (1987), 185–239.
82. J. Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885–911.
83. E. C. Gootman and J. Rosenberg, The structure of crossed product C∗-algebras: aproof of the generalized Effros-Hahn conjecture, Invent. Math. 52 (1979), no. 3, 283–298.
84. M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol.2 (Sussex, 1991) (Graham A. Niblo and Martin A. Roller, eds.), London Math. Soc.Lecture Note Ser., no. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295.
Bibliography 531
85. A. Haefliger, Feuilletages sur les varietes ouvertes, Topology 9 (1970), 183–194.
86. , Homotopy and integrability, Lecture Notes in Math., 197, Springer–Verlag,1971.
87. A. Haefliger and G. Reeb, Varietes (non separees) a une dimension et structuresfeuilletees du plan, Enseignement Math. (2) 3 (1957), 107–125.
88. P. Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc.242 (1978), 35–72.
89. J. Hass, Minimal surfaces in foliated manifolds, Comment. Math. Helv. 61 (1986),1–32.
90. J. Hass and P. Scott, The existence of least area surfaces in 3–manifolds, Trans.Amer. Math. Soc. 310 (1988), 87–114.
91. G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Part B, Viewegand Sohn, Braunschweig, 1983.
92. J. Heitsch and S. Hurder, Secondary classes, Weil operators and the geometry offoliations, J. Diff. Geom. 20 (1984), 291–309.
93. J. Hemple, 3–Manifolds, Princeton Univ. Press and Univ. of Tokyo Press, Princeton,NJ, and Tokyo, 1976.
94. M. Herman, Sur la conjugaison differentiable des diffeomorphismes du cercle a desrotations, Publ. Math. I.H.E.S. 49 (1979), 5–233.
95. M. Herman and F. Sergeraert, Sur un theoreme d’Arnold et Kolmogorov, C. R. Acad.Sci. Paris 273 (1971), A409–A411.
96. E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, American Mathe-matical Society Colloquium Publications, vol. XXXI, American Mathematical Soci-ety, Providence, RI, 1957.
97. M. Hilsum and G. Skandalis, Stabilite des C∗-algebres de feuilletages, Ann. Inst.Fourier (Grenoble) 33 (1983), no. 3, 201–208.
98. M. W. Hirsch, A stable analytic foliation with only exceptional minimal sets, Dynam-ical Systems, Warwick, 1974, Lecture Notes in Math., 468, Springer–Verlag, 1975,pp. 9–10.
99. , Differential Topology, Springer-Verlag, New York, NY, 1976.
100. H. Hopf, Enden offener Raume und unendliche diskontinuerliche Gruppen, Comm.Math. Helv. 16 (1944), 81–100.
101. S. Hurder, Vanishing of secondary classes for compact foliations, J. London Math.Soc. 28 (1983), 175–183.
102. , The Godbillon measure of amenable foliations, J. Diff. Geom. 23 (1986),347–365.
103. S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Ann. ofMath. 126 (1987), 221–275.
104. S. Hurder and R. Langevin, Dynamics and the Godbillon-Vey class of C1 foliations,preprint, 2000.
105. D. Husemoller, Fibre Bundles, McGraw-Hill, New York, NY, 1966.
106. K. Ito and Jr. H. P. McKean, Diffusion Processes and their Sample Paths, Springer–Verlag, Berlin, 1974, Second printing, corrected, Die Grundlehren der Mathematis-chen Wissenschaften, Band 125.
107. W. Jaco, Lectures on Three–Manifold Topology, Regional Conference Series in Math-ematics, vol. 43, American Mathematical Society, Providence, RI, 1980.
532 Bibliography
108. M. Kac, On the notion of recurrence in discrete stochastic processes, Bulletin of theAmer. Math. Soc. 53 (1947), 1002–1010.
109. W. Kaplan, Regular curve families filling the plane I, Duke Math. J. 7 (1940), 154–185.
110. , Regular curve families filling the plane II, Duke Math. J. 8 (1941), 11–46.
111. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Sys-tems, Cambridge University Press, Cambridge, England, 1995.
112. M. Kellum, Uniformly quasi–isometric foliations, Ergodic Theory Dynam. Systems13 (1993), 101–122.
113. J. R. Kinney, Continuity properties of sample functions of Markov processes, Trans.Amer. Math. Soc. 74 (1953), 280–302.
114. A. A. Kirillov, Elements of the Theory of Representations, Springer-Verlag, Berlin,1976, Translated from the Russian by Edwin Hewitt, Grundlehren der Mathematis-chen Wissenschaften, Band 220.
115. W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976),no. 1, 19–70.
116. N. Kryloff and N. Bogoliuboff, La theorie generale de la mesure dans son applicationa l’etude des systemes dynamiques de la mecanique non lineaire, Annals of Math. 38(1937), no. 2, 65–113.
117. F. Laudenbach and S. Blank, Isotopie de formes fermees en dimension trois, Inv.Math. 54 (1979), 103–177.
118. H. B. Lawson, Codimension one foliations of spheres, Ann. of Math. 94 (1971),494–503.
119. G. Levitt, Feuilletages des varietes de dimension 3 qui sont des fibres en cercles,Comment. Math. Helv. 53 (1978), 572–594.
120. P. Li and S. T. Yau, On the parabolic heat kernel of the Schrodinger operator, ActaMath. 156 (1986), 153–201.
121. W. B. R. Lickorish, A representation of orientable combinatorial 3–manifolds, Ann.of Math. 76 (1962), 531–538.
122. , Homeomorphisms of nonorientable two–manifolds, Proc. Camb. Phil. Soc.59 (1963), 307–317.
123. , A foliation for 3–manifolds, Ann. of Math. 82 (1965), 414–420.
124. G. W. Mackey, Ergodic theory and virtual groups, Math. Ann. 166 (1966), 187–207.
125. W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, NewYork, NY, 1967.
126. J. Mather, Commutators of diffeomorphisms, Comment. Math. Helv. 49 (1974), 512–528.
127. , Commutators of diffeomorphisms. II, Comment. Math. Helv. 50 (1975),33–40.
128. K. Millett, Generic properties of proper foliations, Fund. Math. 128 (1987), 131-138.
129. J. Milnor, On the existence of a connection with curvature zero, Comment. Math.Helv. 32 (1957), 215–223.
130. J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton University Press,Princeton, NJ, 1974.
131. S. Minakshisundaram, Eigenfunctions on Riemannian manifolds, J. Indian Math.Soc. (N.S.) 17 (1953), 159–165.
Bibliography 533
132. P. Molino, Riemannian Foliations, Birkhauser, Boston, MA, 1988, Translated from
the French by Grant Cairns, With appendices by Cairns, Y. Carriere, E. Ghys, E.Salem and V. Sergiescu.
133. C. C. Moore, Ergodic theory and von Neumann algebras, Operator algebras andapplications, Part 2 (Kingston, Ont., 1980) (Richard V. Kadison, ed.), ProceedingsSymposia in Pure Mathematics, vol. 38, Amer. Math. Soc., Providence, RI, 1982,pp. 179–226.
134. C. C. Moore and C. Schochet, Global Analysis on Foliated Spaces, MathematicalSciences Research Institute, vol. 9, Springer-Verlag, New York, NY, 1988.
135. R. Mosher and M. Tangora, Cohomology Operations and Applications in HomotopyTheory, Harper and Row, New York, NY, 1968.
136. R. Moussu and F. Pelletier, Sur le theoreme de Poincare–Bendixson, Ann. Inst.Fourier, Grenoble 24 (1974), 131–148.
137. J. Munkres, Elementary Differential Topology, Princeton University Press, Princeton,NJ, 1966.
138. K. Murasugi, On the genus of the alternating knot, I, II, J. Math. Soc. Japan 10(1958), 94–105, 235–248.
139. M. A. Naımark, Normed Rings, English ed., Wolters-Noordhoff Publishing, Gronin-gen, 1970, Translated from the first Russian edition by Leo F. Boron.
140. T. Natsume, The C∗-algebras of codimension one foliations without holonomy, Math.Scand. 56 (1985), no. 1, 96–104.
141. S. P. Novikov, Topology of foliations, Trans. Moscow Math. Soc. 14 (1965), 268–305.
142. R. Palais, Extending diffeomorphisms, Proc. Amer. Math. Soc. 11 (1960), 274–277.
143. F. Palmeira, Open manifolds foliated by planes, Ann. of Math. 107 (1978), 109–131.
144. J. Pasternack, Topological obstructions to integrability and Riemannian geometry offoliations, Ph.D. thesis, Princeton University, 1970.
145. G. K. Pedersen, C∗-algebras and their Automorphism Groups, Academic Press Inc.[Harcourt Brace Jovanovich Publishers], London, 1979.
146. , Analysis Now, Springer-Verlag, New York, NY, 1989.
147. K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 2,Cambridge University Press, Cambridge, 1983.
148. P. Peterson, Riemannian Geometry, Graduate Texts in Mathematics, vol. 171,Springer–Verlag, New York, NY, 1998.
149. R. R. Phelps, Lectures on Choquet’s Theorem, second ed., Lecture Notes in Mathe-matics, no. 1757, Springer-Verlag, Berlin, 2001.
150. D. Pixton, Nonsmoothable, nonstable group actions, Trans. Amer. Math. Soc. 229(1977), 259–268.
151. V. Poenaru, Homotopy theory and differentiable singularities, Lecture Notes inMath., 197, Springer-Verlag, New York, NY, 1971, pp. 106–132.
152. S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Prob-ability and Mathematical Statistics, Academic Press [Harcourt Brace JovanovichPublishers], New York, NY, 1978.
153. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,Prentice-Hall, Inc., Englewood Cliffs, NJ, 1967.
154. N. V. Que and R. Roussarie, Sur l’isotopie des formes fermees en dimension 3, Inv.Math. 64 (1981), 69–87.
534 Bibliography
155. J. Renault, A Groupoid Approach to C∗-Algebras, Lecture Notes in Mathematics,no. 793, Springer-Verlag, Berlin, 1980.
156. M. A. Rieffel, Morita equivalence for C∗-algebras and W ∗-algebras, J. Pure Appl.Algebra 5 (1974), 51–96.
157. J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds,Mem. Amer. Math. Soc. 104 (1993), no. 497, x+90.
158. D. Rolfsen, Knots and Links, Publish or Perish, Inc., Berkeley, CA, 1976.
159. H. Rosenberg, Foliations by planes, Topology 7 (1968), 131–138.
160. R. Roussarie, Plongements dans les varietes feuilletees et classification de feuilletagessans holonomie, I.H.E.S. Publ. Math. 43 (1973), 101–142.
161. W. Rudin, Real and Complex Analysis, 2nd ed., McGraw–Hill, New York, NY, 1974.
162. J.-L. Sauvageot, Ideaux primitifs de certains produits croises, Math. Ann. 231(1977/78), no. 1, 61–76.
163. P. Schweitzer, Some problems in foliation theory and related areas, Lecture Notes inMath., 652, Springer–Verlag, 1978, pp. 240–252.
164. L. C. Siebenmann, Deformations of homeomorphisms on stratified sets, Comment.Math. Helv. 47 (1972), 123–163.
165. V. V. Solodov, Components of topological foliations, Mat. Sb. (N.S.) 119 (1982),340–354, Russian; English transl., Math. USSR Sb. 47 (1984), 329-343.
166. H. Takai, C∗-algebras of Anosov foliations, Operator Algebras and their Connectionswith Topology and Ergodic Theory (Busteni, 1983), Lecture Notes in Math., 1132,Springer–Verlag, 1985, pp. 509–516.
167. M. Takamura T. Inaba, T. Nishimori and N. Tsuchiya, Open manifolds which arenon–realizable as leaves, Kodai Math. J. 8 (1985), 112–119.
168. M. Takesaki, Theory of Operator Algebras. I, Springer-Verlag, New York, NY, 1979.
169. , Structure of factors and automorphism groups, CBMS Regional ConferenceSeries in Mathematics, vol. 51, Published for the Conference Board of the Mathe-matical Sciences, Washington, D.C., by the Amer. Math. Soc., Providence, RI, 1983.
170. I. Tamura, Every odd dimensional homotopy sphere has a foliation of codimensionone, Comment. Math. Helv. 47 (1972), 164–170.
171. , Spinnable structures on differentiable manifolds, Proc. Japan. Acad. 48(1972), 293–296.
172. W. Thurston, The theory of foliations in codimension greater than one, Comment.Math. Helv. 49 (1974), 214–231.
173. , A local construction of foliations for three–manifolds, Proc. Sympos. PureMath., vol. XXVII, Amer. Math. Soc., 1975, pp. 315–319.
174. , Existence of codimension–one foliations, Ann. of Math. 104 (1976), 249–268.
175. , A norm for the homology of three–manifolds, Mem. Amer. Math. Soc. 59(1986), No. 339, pp. 99–130.
176. A. M. Torpe, K-theory for the leaf space of foliations by Reeb components, J. Funct.Anal. 61 (1985), no. 1, 15–71.
177. T. Tsuboi, On the foliated products of class C1, Ann. of Math. 130 (1989), 227–271.
178. B. L. van der Waerden, Modern Algebra, Frederick Unger, New York, NY, 1970.
179. J. W. Vick, Homology Theory, Springer–Verlag, New York, NY, 1982.
180. A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960),503–528.
Bibliography 535
181. X. Wang, On the C∗-Algebras of Foliations in the Plane, Lecture Notes in Mathe-matics, no. 1257, Springer-Verlag, Berlin, 1987.
182. A. Weil, Les familles de courbes sur le tore, Collected Papers, vol. I, Springer-Verlag,1979, pp. 113–115.
183. R. L. Wheeden and A. Zygmund, Measure and Integral, Pure and Applied Mathe-matics, vol. 43, Marcel Dekker Inc., New York, NY, 1977.
184. D. P. Williams, The topology on the primitive ideal space of transformation groupC∗-algebras and C.C.R. transformation group C∗-algebras, Trans. Amer. Math. Soc.266 (1981), no. 2, 335–359.
185. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. 79 (1973),45–51.
186. , The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75.
187. , The number of ends of the universal leaf of a Riemannian foliation, Dif-ferential geometry (College Park, MD, 1981/1982), Birkhauser, Boston, MA, 1983,pp. 247–254.
188. J. Wood, Foliations on 3–manifolds, Ann. of Math. 89 (1969), 336–358.
189. , Bundles with totally disconnected structure group, Comment. Math. Helv.46 (1971), 257–273.
190. F. Wright, Mean least recurrence time, J. London Math. Soc. 36 (1961), 382–384.
191. K. Yosida, Functional Analysis, Sixth ed., Springer-Verlag, New York, NY, 1980, DieGrundlehren der Mathematischen Wissenschaften, Band 123.
192. G. Zeller-Meier, Produits croises d’une C∗-algebre par un groupe d’automorphismes,J. Math. Pures Appl. (9) 47 (1968), 101–239.
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
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)
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
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
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
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
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)
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
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
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/.
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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