Selected Title s i n Thi s Serie s

90 Joh n Roe, Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke, Tigh t closur e an d it s applications , 199 6

87 Joh n Erik Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa, Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o Jimbo and Tetsuji Miwa, Algebrai c analysi s o f solvabl e lattic e modeis , 199 4

84 Hug h L. Montgomery, Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d

harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

Problems, 199 4

82 Susa n Montgomery, Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz , Geometri e analysi s an d funetio n Spaces , 199 3

80 Vaugha n F. R. Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Björn Jawerth, an d Guido Weiss, Littlewood-Pale y theor y an d th e stud y

of funetio n Spaces , 199 1

78 Edwar d Formanek, Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l Operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrell and L. Edwin Jones, Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A. Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y Dym , J contractiv e matri x funetions , reproducin g kerne l Huber t Space s an d

interpolation, 198 9

70 Richar d F. Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7

68 J . Willia m Heiton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analyti c funetions , matrices , an d electrica i engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , Operato r theory , an d quantu m mechanics ,

1987

66 G . Andrews , q-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physic s an d Compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6

64 Donal d S . Passman, Grou p rings , crosse d produet s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construetion s o f funetion s holomorphi c i n th e uni t bal l o f C n , 198 6

62 Bel a Bollobäs , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n Symmetri e Spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r Operator s an d geometr y o f Banac h Spaces , 198 6

59 Roge r How e and Allen Moy, Harish-Chandr a homomorphism s fo r p -adic groups , 198 5

58 H . Blaine Lawson, Jr., Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L . Kazdan, Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici, Cipria n Foia§, and Carl Pearcy, Dua l algebra s wit h application s t o invarian t

subspaces an d dilatio n theory , 198 5

55 Willia m Arveson, Te n lecture s o n Operato r algebras , 198 4 (Continued in the back of this publication)

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Index Theoiy, Coarse Geometiy, and Topology of Manifolds

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Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Serie s in Mathematic s

Number 9 0

Index Theoiy, Coarse Geometiy, an d Topology of Manifolds

John Ro e

Published fo r th e Conference Boar d of the Mathematica l Science s

by th e ^ g % America n Mathematica l Societ y 0 «£j

"™"i!\\! Providence , Rhod e Islan d with suppor t fro m th e

3 5̂§f̂ Nationa l Sc ienc e Foundatio n °**7^ x<

http://dx.doi.org/10.1090/cbms/090

CBMS Conferenc e o n Inde x Theory , Coars e Geometry , and Topolog y o f Manifold s

held a t th e Universit y o f Colorado , August 199 5

Research partiall y supporte d b y th e National Scienc e Foundatio n

2000 Mathematics Subject Classification. P r i m a r y 58-XX , 19K56 , 46L80 , 57R67 , 20 -XX .

Library o f Congres s Cataloging- in-Publ icat io n D a t a

Index theory , coars e geometry , an d topolog y o f manifold s / Joh n Roe . p. cm . — (Regiona l Conferenc e serie s i n mathematics , ISS N 0160-7642 ; no . 90 )

Lecture note s fro m th e Conferenc e hel d Aug . 199 5 i n Boulder , Colo . Includes bibliographica l reference s an d index . ISBN 0-8218-0413- 8 1. Inde x theor y (Mathematics)—Congresses . 2 . K-theory—Congresses . 3 . Operato r alge -

bras—Congresses. 4 . Surger y (Topology)—Congresses . I . Roe , John , 1959- . II . Series .

QA1.R33 no . 9 0 [QA614.92] 510 s—dc20 96-2205 8 [514'. 74] CI P

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofl t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

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except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

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10 9 8 7 6 5 4 3 2 1 7 1 6 1 5 1 4 1 3 1 2

Contents

Preface i x

Chapter 1 . Inde x Theor y 1 Analysis o f Dirac Operator s 1 Index theor y 4 Index theor y an d coarsenin g 6

Chapter 2 . Coars e Geometr y 9 The coars e categor y 9 Coarse algebrai c topology 1 3

Chapter 3 . C*-Algebra s 1 7 The coars e inde x 2 0 Relative inde x theory 2 2 Appendix: ÜT-theor y for C*-algebra s 2 4

Chapter 4 . A n exampl e o f a highe r inde x theorem 2 7 Odd Operator s 2 7 Partitions o f a noncompac t manifol d 2 8 The if-theor y o f C*(|R|) . 3 2

Chapter 5 . Assembl y 3 5 Kasparov's if-homolog y 3 5 The assembl y ma p 3 8 Equivariant assembl y 4 1

Chapter 6 . Surger y 4 5 Manifold structure s 4 5 Novikov's conjectur e 5 1 Bounded surger y 5 3 Positive scala r curvatur e 5 4

Chapter 7 . Mappin g surger y t o analysi s 5 7

Vlll CONTENTS

Invariants o f positive scala r curvatur e metric s 5 7 Poincare complexe s 6 1 A manifol d structur e invarian t 6 3 Bounded structure s 66

Chapter 8 . Th e coars e Baum-Connes conjectur e 6 9 The coars e assembl y ma p 6 9 The principl e o f descent 7 1

Chapter 9 . Method s o f computation 7 7 A coars e Mayer-Vietori s sequenc e 7 7 Coarse homotop y 8 0 Scaleable Space s 8 3 Coarse Baum-Conne s fo r hyperboli c Space s 8 4

Chapter 10 . Coars e structure s an d boundarie s 8 7 Abstract coars e structures 8 7 The assembl y ma p fo r continuou s contro l 8 9

References 9 3

Index 99

Preface

These are the lecture notes from the Conference 'Inde x Theory, Coarse Geometry , and Topolog y o f Manifolds ' whic h wa s hel d i n Boulde r i n Augus t 1995 . I hav e adhered fairl y closel y t o th e origina l schem e o f the lectures , althoug h th e note s naturally contai n rathe r mor e detai l an d i n a few places (especiall y i n lecture 7 ) describe construction s tha t wer e finalized onl y afte r th e lecture s wer e delivered .

It i s o f th e natur e o f thi s subjec t t o requir e a rathe r divers e background . I attempted to deal with this in the lectures by including a number of mini-surveys, and i n this boo k mos t o f lectures 1 and 6 and larg e part s o f lectures 2 and 3 are devoted t o suc h surveys . I hope tha t thes e will be useful .

The wor k reporte d o n her e ha s develope d ove r a doze n year s o r s o an d during tha t tim e I hav e ha d th e pleasur e o f learnin g fro m a grea t numbe r o f colleagues. Amon g the m I woul d mentio n Alai n Connes , Mish a Gromov , Stev e Hurder, Jonatha n Rosenberg , Stepha n Stolz , an d Guolian g Yu . Mor e recentl y I have benefite d fro m th e advic e o f Stev e Ferry , Eri k Pedersen , Andre w Ranicki , and Shmue l Weinberger , al l o f who m hav e (amon g othe r things ) answere d th e numerous question s o f a surger y neophyt e wit h exemplar y patience . Finally , a major deb t o f gratitude i s owed to Nige l Higson, with whom I have collaborate d on man y o f the project s tha t ar e summarize d i n thi s book .

The Conference was made possible by the organizational effort s o f Jeffrey Fox , Guoliang Yu , an d Carl a Farsi , an d b y th e financial suppor t o f th e Conferenc e Board fo r the Mathematica l Science s and th e National Scienc e Foundation. Th e manuscript wa s completed while I was Ulam Visiting Professor a t th e Universit y of Colorado , Boulder . I would lik e to expres s m y thanks t o al l these peopl e an d organizations fo r thei r generou s support .

John Roe , Octobe r 199 5

i x

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Index

a-approximation theorem , 67 , 8 4 assembly map , 38-44 , 50 , 52 , 64 , 69, 74 ,

80, 8 8 Baum-Connes, 43 , 7 5 coarse, 6 9 equivariant, 4 3

Baum-Connes conjecture , 3 4 coarse, see coars e Baum-Conne s con -

jecture bordism, 30 , 31 , 45, 49, 56 , 63 , 66, 6 7

spin, 5 5 Bott periodicity , 25 , 3 4 boundary

Gromov, 8 5 Higson, 8 9 map, 21 , 28, 41 , 63 of a cone-lik e end , 5 8 of a n Operator , see Operato r of compactification , 8 8

bounded geometry , 13 , 34 , 61 , 62 , 70 , 84 , 88

C*-algebra, 17 , 72 , 8 9 crossed product , 3 3 of a coars e space , see C* (X) of a group , 1 8

C*(X), 1 9 chopping function , 4 , 20 , 3 8

exact, 5 8 classifying space , 13 , 31 , 53, 71 , 75 coarse

homology, 14-16 , 6 9 map, 9 , 19 , 32 , 34 , 39 , 63 , 80 , 8 8 structure, 10 , 13 , 40 , 8 7

bounded, 8 7 continuously controlled , 8 8

coarse Baum-Conne s conjecture , 70 , 71 , 80, 8 2

with coefficients , 7 5 coarsely equivalent , 1 0

coarsening, 14 , 6 9 coarsening sequenc e o f Covers , 15 , 7 0 cobordism, see bordis m compactification, 8 8

coarse, 8 8 Higson, 89 , 9 2

concordant, 55 , 6 0

D*{X), 3 8 descent principle , 71 , 86, 9 1 Dranishnikov space , 7 0

Eilenberg swindle , 79 , 8 3 elliptic regularity , 3 , 2 0 end, 5 8 e-propagation, 20 , 8 4 excision, 37 , 8 9

strong, 8 0 excisive

decomposition, 7 7 functor, 7 2

finite propagatio n speed , 20 , 3 8 flasque, 78 , 8 1 Fredholm module , 3 5 functional calculus , 3 , 20 , 6 2

grading, 4 , 21 , 28, 38 , 6 2 Gromov-Lawson invariant , 5 9

/i-cobordism, 45 , 55 , 6 5 higher .4-genus , 32 , 4 4 higher inde x theorem , see inde x theorem ,

higher higher signature , 32 , 53 , 66 Hilbert-Poincare space , 6 1 homotopy fixe d set , 7 1 hyperbolic

differential equation , 2 0 space, 9 , 8 5

99

100 INDEX

index, 4 , 21 , 29, 38 , 5 8 coarse, 21 , 34 families, 7 4 relative, 2 2

index theorem , 6- 7 for R n -bounded manifold , 34 , 6 7 for partitione d manifold , 30-32 , 4 4 higher, 8 , 27-34 , 7 4 relative, 2 3

JC-homology, 15 , 28 , 35-38 , 70 , 8 9 K-theory, 19 , 24-2 6 Kasparov's lemma , 36 , 59 , 83 , 9 0

Lichnerowicz theorem , 6 , 22 , 55 , 5 8 Lipschitz

homotopy, 80-8 3 map, 11 , 79

Mayer-Vietoris sequence , 25 , 7 2 for if-homology , 3 7 for coarsely excisive decomposition, 7 7

microlocal analysis , 6 7 module, 3 , 18 , 5 8

adequate, 19 , 8 9 covariant, 4 1

T-adequate, 4 2 Fredholm, see Fredhol m modul e Hubert, 7 5

multiplier algebra , 20 , 4 1

noncommutative geometry , 17-1 8 normal invariant , 47-4 9 normal map , 48 , 6 5 Novikov conjecture , 43 , 51-53 , 66 , 71 , 8 3

open cone , 11 , 58, 79 , 8 3 Operator

boundary of , 27-2 8 compact, 3 , 2 5 Dirac

and assembly , 4 0 defines homolog y dass , 38 , 5 7 generalized, 2 spinor, 5 , 2 7

finite propagation , 19 , 27 , 38 , 39 , 8 8 Fredholm, 4 locally compact , 3 , 18 , 3 6 pseudodifferential, 3 6 pseudolocal, 3 6 signature, 5 , 27 , 6 1 supported nea r a set , 22 , 59 , 7 8 unbounded, 1

partition o f a manifold , 2 9 Pimsner-Voiculescu exac t sequence , 3 3 (7r-7r)-theorem, 50 , 5 9 Poincare duality , 5 , 38 , 47 , 61 , 62, 71 , 74 Pontrjagin classes , 7 , 5 2

topological invariance , 6 6 positive scala r curvature , 6 , 22 , 54-6 1 proper action , 14 , 41 , 75

rough geometry , 16 , 3 4

scaleable, 8 3 s-cobordism theorem , 12 , 4 6 shrinking, 8 2 signature, 5 , 32 , 52 , 63 , 6 7

higher, see highe r signatur e Hilbert-Poincare, 6 2 Operator, see Operator , signatur e

structure algebra , 43 , 5 7 structure invarian t

of homotop y equivalence , 6 4 of positiv e scala r curvatur e metric , 5 8

structure set , 46 , 6 5 surgery, 6 , 45-5 6

bounded, 53 , 66 , 6 9 exact sequenc e

analytic, 43 , 58 , 7 2 geometric, 4 7 transformation fro m geometri c to an -

alytic, 6 5 obstruction groups , 49-51 , 5 4

trace, 2 8

uniformly contractible , 13 , 16 , 70 , 8 8 universal cover , 10 , 14 , 43 , 6 3

Selected Title s i n Thi s Serie s (Continued from the front of this publication)

54 Willia m Fulton, Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wilhel m Klingenberg , Close d geodesic s o n Riemannia n manifolds , 198 3

52 Tsit-Yue n Lam, Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eells and Luc Lemaire, Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M. Franks , Homolog y an d dynamica l Systems , 198 2

48 W . Stephe n Wilson, Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K. Haie , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L. Graham, Rudiment s o f Ramse y theory , 198 1

44 Philli p A. Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Willia m Jaco, Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne, Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J. Newman , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin, Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finit e Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affin e spaces , 197 8

36 Car l M. Pearcy , Som e recen t development s i n Operato r theory , 197 8

35 R . Bowen, O n Axio m A diffeomorphisms , 197 8

34 L . Auslander, Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finit e groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g Operators ,

1977

29 A . Weinstein , Lecture s o n symplecti c manifolds , 197 7

28 T . A . Chapman , Lecture s o n Huber t cub e manifolds , 197 6

27 H . Blain e Lawson, Jr., Th e quantitativ e theor y o f foliations , 197 7

26 I . Reiner , Clas s group s an d Picar d group s o f grou p ring s an d Orders , 197 6

25 K . W . Gruenberg , Relatio n module s o f finit e groups , 197 6

24 M . Höchster , Topic s i n th e homologica l theor y o f module s ove r commutativ e rings , 197 5

23 M . E . Rudin, Lecture s o n se t theoreti c topology , 197 5

22 O . T. O'Meara , Lecture s o n linea r groups , 197 4

21 W . Stoll , Holomorphi c function s o f finite orde r i n severa l comple x variables , 197 4

20 H . Bass , Introductio n t o som e method s o f algebrai c if-theory , 197 4

19 B . Sz.-Nagy , Unitar y dilation s o f Huber t spac e Operator s an d relate d topics , 197 4

18 A . Friedman, Differentia l games , 197 4

17 L . Nirenberg, Lecture s o n linea r partia l differentia l equations , 197 3

16 J . L . Taylor, Measur e algebras , 197 3

15 R . G . Douglas , Banac h algebr a technique s i n th e theor y o f Toeplit z Operators , 197 3

14 S . Helgason , Analysi s o n Li e group s an d homogeneou s spaces , 197 2

13 M . Rabin , Automat a o n infinit e object s an d Church' s problem , 197 2

(See th e AM S catalo g fo r earlie r titles )