L 2 -Invariants: theory and applications to geometry and K-theory
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin ; Heidelberg
Springer
[2002]
|
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete
Folge 3 ; 44 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009897979&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke (2010) |
| Umfang: | xv, 595 Seiten |
| ISBN: | 3540435662 9783540435662 9783642078101 |
Internformat
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| 020 | |a 9783642078101 |c softcover |9 978-3-642-07810-1 | ||
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| 100 | 1 | |a Lück, Wolfgang |d 1957- |e Verfasser |0 (DE-588)133867072 |4 aut | |
| 245 | 1 | 0 | |a L 2 -Invariants |b theory and applications to geometry and K-theory |c Wolfgang Lück |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [2002] | |
| 264 | 4 | |c © 2002 | |
| 300 | |a xv, 595 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 |v 44 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke (2010) | ||
| 650 | 4 | |a Invariante - Mannigfaltigkeit - Komplex <Topologie> - Selbstadjungierter Operator - K-Theorie | |
| 650 | 0 | 7 | |a K-Theorie |0 (DE-588)4033335-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mannigfaltigkeit |0 (DE-588)4037379-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Komplex |g Topologie |0 (DE-588)4164883-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Selbstadjungierter Operator |0 (DE-588)4180810-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Invariante |0 (DE-588)4128781-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Invariante |0 (DE-588)4128781-2 |D s |
| 689 | 0 | 1 | |a Mannigfaltigkeit |0 (DE-588)4037379-4 |D s |
| 689 | 0 | 2 | |a Komplex |g Topologie |0 (DE-588)4164883-3 |D s |
| 689 | 0 | 3 | |a Selbstadjungierter Operator |0 (DE-588)4180810-1 |D s |
| 689 | 0 | 4 | |a K-Theorie |0 (DE-588)4033335-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-662-04687-6 |
| 830 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete |v Folge 3 ; 44 |w (DE-604)BV000899194 |9 44 | |
| 856 | 4 | 2 | |m SWB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009897979&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-009897979 | |
Datensatz im Suchindex
| _version_ | 1819302499763355648 |
|---|---|
| adam_text | CONTENTS 0. INTRODUCTION ..............................................
1 0.1 WHAT ARE L 2 -INVARIANTS? . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 1 0.2 SOME APPLICATIONS OF L 2 -INVARIANTS . . .
. . . . . . . . . . . . . . . . . . . . 2 0.3 SOME OPEN PROBLEMS
CONCERNING L 2 -INVARIANTS . . . . . . . . . . . . 3 0.4 L 2 -INVARIANTS
AND HEAT KERNELS . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.5 L 2 -INVARIANTS AND CELLULAR CHAIN COMPLEXES . . . . . . . . . . . .
. . . 6 0.6 L 2 -BETTI NUMBERS AND BETTI NUMBERS . . . . . . . . . . . .
. . . . . . . . . 7 0.7 L 2 -INVARIANTS AND RING-THEORY . . . . . . . .
. . . . . . . . . . . . . . . . . . . 8 0.8 L 2 -INVARIANTS AND K
-THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 0.9
L 2 -INVARIANTS AND ASPHERICAL MANIFOLDS . . . . . . . . . . . . . . . .
. . . 11 0.10 L 2 -INVARIANTS AND GROUPS . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 12 1. L 2 -BETTI NUMBERS
........................................ 13 INTRODUCTION. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 13 1.1 GROUP VON NEUMANN ALGEBRAS AND HILBERT MODULES. . . . . .
. . . 14 1.1.1 GROUP VON NEUMANN ALGEBRAS . . . . . . . . . . . . . . .
. . . . . . 14 1.1.2 HILBERT MODULES. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 16 1.1.3 DIMENSION THEORY . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 16 1.1.4 HILBERT CHAIN
COMPLEXES . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.1.5
INDUCTION FOR GROUP VON NEUMANN ALGEBRAS . . . . . . . . . 29 1.2
CELLULAR L 2 -BETTI NUMBERS. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 30 1.2.1 SURVEY ON G - CW -COMPLEXES . . . . . . . . . .
. . . . . . . . . . . . 31 1.2.2 THE CELLULAR L 2 -CHAIN COMPLEX . . . .
. . . . . . . . . . . . . . . . 33 1.2.3 BASIC PROPERTIES OF CELLULAR L
2 -BETTI NUMBERS . . . . . . . 37 1.2.4 L 2 -BETTI NUMBERS AND
ASPHERICAL SPACES . . . . . . . . . . . . 45 1.3 ANALYTIC L 2 -BETTI
NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.3.1 THE CLASSICAL HODGE-DE RHAM THEOREM . . . . . . . . . . . . . 49
1.3.2 ANALYTIC DEFINITION OF L 2 -BETTI NUMBERS . . . . . . . . . . . .
52 1.4 COMPARISON OF ANALYTIC AND CELLULAR L 2 -BETTI NUMBERS . . . . .
54 1.4.1 SURVEY ON UNBOUNDED OPERATORS AND SPECTRAL FAMILIES 54 1.4.2 L
2 -HODGE-DE RHAM THEOREM . . . . . . . . . . . . . . . . . . . . . . 58
1.5 L 2 -BETTI NUMBERS OF MANIFOLDS WITH BOUNDARY . . . . . . . . . . .
. . 63 1.6 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 65 EXERCISES . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 68 X CONTENTS 2. NOVIKOV-SHUBIN INVARIANTS
................................ 71 INTRODUCTION. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71 2.1 SPECTRAL DENSITY FUNCTIONS . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 72 2.1.1 SPECTRAL DENSITY FUNCTIONS OF MORPHISMS .
. . . . . . . . . . . 72 2.1.2 SPECTRAL DENSITY FUNCTIONS OF HILBERT
CHAIN COMPLEXES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 81 2.1.3 PRODUCT FORMULA FOR NOVIKOV-SHUBIN INVARIANTS . . . .
. . 86 2.1.4 THE LAPLACIAN IN DIMENSION ZERO. . . . . . . . . . . . . .
. . . . . 93 2.2 CELLULAR NOVIKOV-SHUBIN INVARIANTS . . . . . . . . . .
. . . . . . . . . . . . . 96 2.3 ANALYTIC NOVIKOV-SHUBIN INVARIANTS . .
. . . . . . . . . . . . . . . . . . . . . 103 2.4 COMPARISON OF ANALYTIC
AND CELLULAR NOVIKOV-SHUBIN INVARIANTS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.5 ON THE
POSITIVITY AND RATIONALITY OF THE NOVIKOV-SHUBIN INVARIANTS . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 112 2.6 NOVIKOV-SHUBIN INVARIANTS OF MANIFOLDS WITH BOUNDARY. . .
. . . 114 2.7 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 115 EXERCISES . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 117 3. L 2 -TORSION
................................................ 119 INTRODUCTION. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 119 3.1 SURVEY ON TORSION INVARIANTS . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 120 3.1.1 WHITEHEAD GROUPS . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.1.2
WHITEHEAD TORSION . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 121 3.1.3 REIDEMEISTER TORSION . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 123 3.2 FUGLEDE-KADISON DETERMINANT . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 126 3.3 L 2 -TORSION OF
HILBERT CHAIN COMPLEXES . . . . . . . . . . . . . . . . . . . 139 3.3.1
BASIC DEFINITIONS AND PROPERTIES OF L 2 -TORSION . . . . . . . 140 3.3.2
L 2 -TORSION AND CHAIN CONTRACTIONS . . . . . . . . . . . . . . . . .
144 3.3.3 PROOFS OF THE BASIC PROPERTIES OF L 2 -TORSION . . . . . . . .
. 148 3.4 CELLULAR L 2 -TORSION . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 160 3.4.1 CELLULAR L 2 -TORSION IN THE
WEAKLY-ACYCLIC CASE . . . . . . 160 3.4.2 CELLULAR L 2 -TORSION IN THE
WEAKLY-ACYCLIC AND ASPHERICAL CASE . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 171 3.4.3 TOPOLOGICAL L 2 -TORSION FOR
RIEMANNIAN MANIFOLDS . . . . 176 3.5 ANALYTIC L 2 -TORSION . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.5.1
DEFINITION OF ANALYTIC L 2 -TORSION . . . . . . . . . . . . . . . . . .
. 178 3.5.2 THE LAPLACE TRANSFORM OF A DENSITY FUNCTION . . . . . . . .
181 3.5.3 COMPARISON OF TOPOLOGICAL AND ANALYTIC L 2 -TORSION . . 186
3.5.4 ANALYTIC L 2 -TORSION FOR HYPERBOLIC MANIFOLDS . . . . . . . . 186
3.6 L 2 -TORSION OF MANIFOLDS WITH BOUNDARY . . . . . . . . . . . . . .
. . . . . 189 3.7 COMBINATORIAL COMPUTATIONS OF L 2 -INVARIANTS . . . .
. . . . . . . . . . 193 3.8 MISCELLANEOUS . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 200 EXERCISES . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 205 CONTENTS XI 4. L 2 -INVARIANTS OF
3-MANIFOLDS .............................. 211 INTRODUCTION. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 211 4.1 SURVEY ON 3-MANIFOLDS . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 211 4.2 L 2 -INVARIANTS OF
3-MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214 4.3 L 2 -INVARIANTS OF KNOT COMPLEMENTS . . . . . . . . . . . . . .
. . . . . . . . 217 4.4 MISCELLANEOUS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 218 EXERCISES . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 220 5. L 2 -INVARIANTS OF SYMMETRIC SPACES
........................ 223 INTRODUCTION. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.1
SURVEY ON SYMMETRIC SPACES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 223 5.2 L 2 -INVARIANTS OF SYMMETRIC SPACES OF NON-COMPACT
TYPE . . . 227 5.3 L 2 -INVARIANTS OF SYMMETRIC SPACES. . . . . . . . .
. . . . . . . . . . . . . . . 229 5.4 MISCELLANEOUS . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 232 6. L 2 -INVARIANTS FOR GENERAL
SPACES WITH GROUP ACTION ...... 235 INTRODUCTION. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
235 6.1 DIMENSION THEORY FOR ARBITRARY MODULES . . . . . . . . . . . . .
. . . . . 237 6.2 COMPARISON OF MODULES AND HILBERT MODULES . . . . . .
. . . . . . . . 246 6.3 INDUCTION AND THE EXTENDED VON NEUMANN
DIMENSION. . . . . . . 253 6.4 THE EXTENDED DIMENSION FUNCTION AND
AMENABLE GROUPS . . . 255 6.4.1 SURVEY ON AMENABLE GROUPS . . . . . . .
. . . . . . . . . . . . . . . . 256 6.4.2 AMENABILITY AND THE
COINVARIANTS OF THE GROUP VON NEUMANN ALGEBRA . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 258 6.4.3 AMENABILITY AND FLATNESS
PROPERTIES OF THE GROUP VON NEUMANN ALGEBRA OVER THE GROUP RING . . . .
. . . . . 258 6.5 L 2 -BETTI NUMBERS FOR GENERAL SPACES WITH GROUP
ACTIONS . . . 263 6.6 L 2 -EULER CHARACTERISTIC . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 276 6.6.1 DEFINITION AND BASIC
PROPERTIES OF L 2 -EULER CHARACTERISTIC . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 277 6.6.2 L 2 -EULER
CHARACTERISTIC, EQUIVARIANT EULER CHARACTERISTIC AND THE BURNSIDE GROUP
. . . . . . . . . . . . . . 280 6.7 FINITELY PRESENTED TORSION MODULES
AND NOVIKOV-SHUBIN INVARIANTS . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 285 6.8 MISCELLANEOUS .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 287 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 289 7. APPLICATIONS
TO GROUPS ................................... 293 INTRODUCTION. . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 293 7.1 GROUPS WITH VANISHING L 2 -BETTI NUMBERS . . . . .
. . . . . . . . . . . . 293 7.1.1 GENERAL CRITERIONS FOR THE VANISHING
OF THE L 2 -BETTI NUMBERS OF A GROUP . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 294 7.1.2 THE VANISHING OF THE L 2 -BETTI
NUMBERS OF THOMPSON*S GROUP . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 297 XII CONTENTS 7.2 EULER CHARACTERISTICS OF GROUPS .
. . . . . . . . . . . . . . . . . . . . . . . . . 298 7.3 DEFICIENCY OF
GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 299 7.3.1 SURVEY ON DEFICIENCY OF GROUPS . . . . . . . . . . . . .
. . . . . . . 299 7.3.2 APPLICATIONS OF L 2 -BETTI NUMBERS TO DEFICIENCY
AND TO SIGNATURES OF 4-MANIFOLDS . . . . . . . . . . . . . . . . . . . .
. . . 303 7.4 GROUP AUTOMORPHISMS AND L 2 -TORSION . . . . . . . . . . .
. . . . . . . . . 304 7.4.1 AUTOMORPHISMS OF GROUPS G WITH FINITE MODELS
FOR BG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 304 7.4.2 AUTOMORPHISMS OF SURFACES . . . . . . . . .
. . . . . . . . . . . . . . . 307 7.4.3 A COMBINATORIAL APPROACH FOR THE
L 2 -TORSION OF AN AUTOMORPHISM OF A FINITELY GENERATED FREE GROUP . . .
308 7.4.4 GENERALIZATIONS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 310 7.5 MISCELLANEOUS . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 EXERCISES .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 314 8. THE ALGEBRA OF AFFILIATED OPERATORS
...................... 317 INTRODUCTION. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 8.1
THE ALGEBRA OF AFFILIATED OPERATORS . . . . . . . . . . . . . . . . . .
. . . . . 318 8.2 BASIC PROPERTIES OF THE ALGEBRA OF AFFILIATED
OPERATORS . . . . . . 323 8.2.1 SURVEY ON ORE LOCALIZATION . . . . . . .
. . . . . . . . . . . . . . . . . 323 8.2.2 SURVEY ON VON NEUMANN
REGULAR RINGS . . . . . . . . . . . . . 325 8.2.3 BASIC PROPERTIES OF
THE ALGEBRA OF AFFILIATED OPERATORS 327 8.3 DIMENSION THEORY AND L 2
-BETTI NUMBERS OVER THE ALGEBRA OF AFFILIATED OPERATORS . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.4 VARIOUS
NOTIONS OF TORSION MODULES OVER A GROUP VON NEUMANN ALGEBRA . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.5
MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 333 EXERCISES . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
9. MIDDLE ALGEBRAIC K -THEORY AND L -THEORY OF VON NEUMANN ALGEBRAS
.................................................. 335 INTRODUCTION. . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 335 9.1 SURVEY ON VON NEUMANN ALGEBRAS . . . . . . . .
. . . . . . . . . . . . . . . . 336 9.1.1 DEFINITION OF A VON NEUMANN
ALGEBRA . . . . . . . . . . . . . . . 336 9.1.2 TYPES AND THE
DECOMPOSITION OF VON NEUMANN ALGEBRAS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 337 9.1.3 FINITE VON
NEUMANN ALGEBRAS AND TRACES . . . . . . . . . . . 338 9.1.4 EXTENDING
RESULTS FOR GROUP VON NEUMANN ALGEBRAS TO FINITE VON NEUMANN ALGEBRAS .
. . . . . . . . . . . . . . . . . . 340 9.2 MIDDLE K -THEORY OF A
NEUMANN ALGEBRA . . . . . . . . . . . . . . . . . . 341 9.2.1 K 0 OF A
VON NEUMANN ALGEBRA . . . . . . . . . . . . . . . . . . . . . 341 9.2.2
K 1 OF A VON NEUMANN ALGEBRA . . . . . . . . . . . . . . . . . . . . .
343 9.3 MIDDLE K -THEORY OF THE ALGEBRA OF AFFILIATED OPERATORS . . . .
. 345 9.4 L -THEORY OF A VON NEUMANN ALGEBRA AND THE ALGEBRA OF
AFFILIATED OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 347 CONTENTS XIII 9.5 APPLICATION TO MIDDLE K - AND G
-THEORY OF GROUP RINGS . . . . 354 9.5.1 DETECTING ELEMENTS IN K 1 OF A
COMPLEX GROUP RING . . 354 9.5.2 SURVEY ON THE ISOMORPHISM CONJECTURE
FOR K 0 OF COMPLEX GROUP RINGS, THE BASS CONJECTURE AND THE
HATTORI-STALLINGS RANK . . . . . . . . . . . . . . . . . . . . . 357
9.5.3 G -THEORY OF COMPLEX GROUP RINGS . . . . . . . . . . . . . . . . .
363 9.6 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 365 EXERCISES . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 366 10. THE ATIYAH CONJECTURE ...................................
369 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 369 10.1 SURVEY ON THE ATIYAH
CONJECTURE . . . . . . . . . . . . . . . . . . . . . . . . . 370 10.1.1
VARIOUS FORMULATIONS OF THE ATIYAH CONJECTURE . . . . . . . 370 10.1.2
RELATIONS OF THE ATIYAH CONJECTURE TO OTHER CONJECTURES. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 10.1.3
SURVEY ON POSITIVE RESULTS ABOUT THE ATIYAH CONJECTURE . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.1.4 A
COUNTEREXAMPLE TO THE STRONG ATIYAH CONJECTURE . . 379 10.2 A STRATEGY
FOR THE PROOF OF THE ATIYAH CONJECTURE . . . . . . . . . . 381 10.2.1
THE GENERAL CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 381 10.2.2 SURVEY ON UNIVERSAL LOCALIZATIONS AND DIVISION AND
RATIONAL CLOSURE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 385 10.2.3 THE STRATEGY FOR THE PROOF OF LINNELL*S THEOREM . .
. . . . 387 10.3 THE PROOF OF LINNELL*S THEOREM . . . . . . . . . . . .
. . . . . . . . . . . . . . 391 10.3.1 THE PROOF OF ATIYAH*S CONJECTURE
FOR FREE GROUPS . . . . 391 10.3.2 SURVEY ON CROSSED PRODUCTS . . . . .
. . . . . . . . . . . . . . . . . . 396 10.3.3 PROPERTY (R) ASCENDS TO
FINITE EXTENSIONS . . . . . . . . . . 399 10.3.4 PROPERTY (R) ASCENDS TO
EXTENSIONS BY INFINITE CYCLIC GROUPS. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 399 10.3.5 PROPERTY (K)
AND EXTENSIONS BY VIRTUALLY FINITELY GENERATED ABELIAN GROUPS . . . . .
. . . . . . . . . . . . . . . . . . . 402 10.3.6 PROPERTY (K) HOLDS FOR
VIRTUALLY FREE GROUPS . . . . . . . . 405 10.3.7 THE INDUCTION STEP FOR
DIRECTED UNIONS . . . . . . . . . . . . . 408 10.4 MISCELLANEOUS . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 410 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 413 11. THE SINGER
CONJECTURE .................................... 417 INTRODUCTION. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 417 11.1 SURVEY ON POSITIVE RESULTS ABOUT THE SINGER
CONJECTURE . . . . . 418 11.1.1 LOW-DIMENSIONAL MANIFOLDS. . . . . . . .
. . . . . . . . . . . . . . . . 418 11.1.2 PINCHED CURVATURE . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 420 11.1.3 ASPHERICAL
MANIFOLDS AND LOCALLY SYMMETRIC SPACES . . 421 XIV CONTENTS 11.2 THE
SINGER CONJECTURE FOR K¨ AHLER MANIFOLDS . . . . . . . . . . . . . . .
421 11.2.1 HODGE THEORY ON K¨ AHLER MANIFOLDS . . . . . . . . . . . . .
. . . . 424 11.2.2 THE L 2 -LEFSCHETZ THEOREM . . . . . . . . . . . . .
. . . . . . . . . . . 426 11.2.3 NOVIKOV-SHUBIN INVARIANTS FOR K¨ AHLER
HYPERBOLIC MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 429 11.2.4 NON-VANISHING OF THE MIDDLE L 2
-BETTI NUMBER FOR K¨ AHLER HYPERBOLIC MANIFOLDS. . . . . . . . . . . . .
. . . . . . . . . . 431 11.3 MISCELLANEOUS . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 432 EXERCISES . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 434 12. THE ZERO-IN-THE-SPECTRUM CONJECTURE
.................... 437 INTRODUCTION. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 12.1 AN
ALGEBRAIC FORMULATION OF THE ZERO-IN-THE-SPECTRUM CONJECTURE . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 438 12.2 SURVEY ON POSITIVE RESULTS ABOUT THE ZERO-IN-THE-SPECTRUM
CONJECTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 440 12.2.1 THE ZERO-IN-THE-SPECTRUM CONJECTURE
FOR LOW-DIMENSIONAL MANIFOLDS. . . . . . . . . . . . . . . . . . . . . .
. . 440 12.2.2 THE ZERO-IN-THE-SPECTRUM CONJECTURE FOR LOCALLY SYMMETRIC
SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
441 12.2.3 THE ZERO-IN-THE-SPECTRUM CONJECTURE FOR K¨ AHLER HYPERBOLIC
MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
12.2.4 THE ZERO-IN-THE-SPECTRUM CONJECTURE FOR HYPEREUCLIDEAN MANIFOLDS
. . . . . . . . . . . . . . . . . . . . . . . . . 442 12.2.5 THE
ZERO-IN-THE-SPECTRUM CONJECTURE AND THE STRONG NOVIKOV CONJECTURE . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 12.2.6 THE
ZERO-IN-THE-SPECTRUM CONJECTURE AND FINITE ASYMPTOTIC DIMENSION . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 443 12.3 COUNTEREXAMPLES
TO THE ZERO-IN-THE-SPECTRUM CONJECTURE IN THE NON-ASPHERICAL CASE . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 12.4
MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 449 EXERCISES . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
13. THE APPROXIMATION CONJECTURE AND THE DETERMINANT CONJECTURE
............................................... 453 INTRODUCTION. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 453 13.1 SURVEY ON THE APPROXIMATION CONJECTURE AND
DETERMINANT CONJECTURE . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 454 13.1.1 SURVEY ON POSITIVE
RESULTS ABOUT THE APPROXIMATION CONJECTURE AND THE DETERMINANT
CONJECTURE . . . . . . . . . 454 13.1.2 RELATIONS TO OTHER CONJECTURES.
. . . . . . . . . . . . . . . . . . . . 455 13.1.3 A CLASS OF GROUPS . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 13.2 THE
PROOF OF THE APPROXIMATION CONJECTURE AND THE DETERMINANT CONJECTURE IN
SPECIAL CASES . . . . . . . . . . . . . . . . . . 459 13.2.1 THE GENERAL
STRATEGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
CONTENTS XV 13.2.2 LIMITS OF INVERSE SYSTEMS . . . . . . . . . . . . . .
. . . . . . . . . . . 465 13.2.3 COLIMITS OF DIRECTED SYSTEMS . . . . .
. . . . . . . . . . . . . . . . . 467 13.2.4 AMENABLE EXTENSIONS . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 469 13.2.5 QUOTIENTS
WITH FINITE KERNELS . . . . . . . . . . . . . . . . . . . . . . 473 13.3
VARIATIONS OF THE APPROXIMATION RESULTS . . . . . . . . . . . . . . . .
. . 475 13.4 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 482 EXERCISES . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 483 14. L 2 -INVARIANTS AND THE SIMPLICIAL VOLUME
.................. 485 INTRODUCTION. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 14.1
SURVEY ON SIMPLICIAL VOLUME . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 485 14.1.1 BASIC DEFINITIONS . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 485 14.1.2 ELEMENTARY PROPERTIES . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 487 14.1.3 BOUNDED
COHOMOLOGY AND AMENABLE GROUPS . . . . . . . . 489 14.1.4 THE SIMPLICIAL
VOLUME OF HYPERBOLIC AND LOW-DIMENSIONAL MANIFOLDS. . . . . . . . . . .
. . . . . . . . . . . . . 489 14.1.5 VOLUME AND SIMPLICIAL VOLUME . . .
. . . . . . . . . . . . . . . . . 491 14.1.6 SIMPLICIAL VOLUME AND BETTI
NUMBERS . . . . . . . . . . . . . . . 494 14.1.7 SIMPLICIAL VOLUME AND S
1 -ACTIONS . . . . . . . . . . . . . . . . . . 496 14.1.8 SOME
INFORMATION ABOUT THE SECOND BOUNDED COHOMOLOGY OF GROUPS . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 496 14.1.9 FURTHER PROPERTIES
OF THE SIMPLICIAL VOLUME . . . . . . . . . 496 14.2 SIMPLICIAL VOLUME
AND L 2 -INVARIANTS OF UNIVERSAL COVERINGS OF CLOSED MANIFOLDS . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
14.2.1 HYPERBOLIC MANIFOLDS AND 3-MANIFOLDS . . . . . . . . . . . . . .
497 14.2.2 S 1 -ACTIONS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 498 14.2.3 AMENABLE FUNDAMENTAL GROUPS . . .
. . . . . . . . . . . . . . . . . 498 14.2.4 SELFMAPS OF DEGREE
DIFFERENT FROM * 1, 0 AND 1 . . . . . . . 499 14.2.5 NEGATIVE SECTIONAL
CURVATURE AND LOCALLY SYMMETRIC SPACES . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 501 14.2.6 SIMPLICIAL
VOLUME AND L 2 -INVARIANTS . . . . . . . . . . . . . . . . 501 14.3
MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 502 EXERCISES . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
15. SURVEY ON OTHER TOPICS RELATED TO L 2 -INVARIANTS ......... 507
INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 507 15.1 L 2 -INDEX THEOREMS . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
15.2 L P -COHOMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 508 15.3 INTERSECTION COHOMOLOGY . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 509 15.4 KNOT
CONCORDANCE AND L 2 -SIGNATURE . . . . . . . . . . . . . . . . . . . . .
. 509 16. SOLUTIONS OF THE EXERCISES .................................
511 REFERENCES .................................................... 559
NOTATION ...................................................... 583
INDEX ......................................................... 587
|
| any_adam_object | 1 |
| author | Lück, Wolfgang 1957- |
| author_GND | (DE-588)133867072 |
| author_facet | Lück, Wolfgang 1957- |
| author_role | aut |
| author_sort | Lück, Wolfgang 1957- |
| author_variant | w l wl |
| building | Verbundindex |
| bvnumber | BV014555413 |
| classification_rvk | SK 300 SK 350 SK 370 |
| ctrlnum | (OCoLC)248566129 (DE-599)BVBBV014555413 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV014555413 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T11:04:56Z |
| institution | BVB |
| isbn | 3540435662 9783540435662 9783642078101 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009897979 |
| oclc_num | 248566129 |
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| owner | DE-384 DE-355 DE-BY-UBR DE-20 DE-19 DE-BY-UBM DE-29T DE-824 DE-703 DE-11 DE-188 DE-83 |
| owner_facet | DE-384 DE-355 DE-BY-UBR DE-20 DE-19 DE-BY-UBM DE-29T DE-824 DE-703 DE-11 DE-188 DE-83 |
| physical | xv, 595 Seiten |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Springer |
| record_format | marc |
| series | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 |
| spellingShingle | Lück, Wolfgang 1957- L 2 -Invariants theory and applications to geometry and K-theory Ergebnisse der Mathematik und ihrer Grenzgebiete Invariante - Mannigfaltigkeit - Komplex <Topologie> - Selbstadjungierter Operator - K-Theorie K-Theorie (DE-588)4033335-8 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Komplex Topologie (DE-588)4164883-3 gnd Selbstadjungierter Operator (DE-588)4180810-1 gnd Invariante (DE-588)4128781-2 gnd |
| subject_GND | (DE-588)4033335-8 (DE-588)4037379-4 (DE-588)4164883-3 (DE-588)4180810-1 (DE-588)4128781-2 |
| title | L 2 -Invariants theory and applications to geometry and K-theory |
| title_auth | L 2 -Invariants theory and applications to geometry and K-theory |
| title_exact_search | L 2 -Invariants theory and applications to geometry and K-theory |
| title_full | L 2 -Invariants theory and applications to geometry and K-theory Wolfgang Lück |
| title_fullStr | L 2 -Invariants theory and applications to geometry and K-theory Wolfgang Lück |
| title_full_unstemmed | L 2 -Invariants theory and applications to geometry and K-theory Wolfgang Lück |
| title_short | L 2 -Invariants |
| title_sort | l 2 invariants theory and applications to geometry and k theory |
| title_sub | theory and applications to geometry and K-theory |
| topic | Invariante - Mannigfaltigkeit - Komplex <Topologie> - Selbstadjungierter Operator - K-Theorie K-Theorie (DE-588)4033335-8 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Komplex Topologie (DE-588)4164883-3 gnd Selbstadjungierter Operator (DE-588)4180810-1 gnd Invariante (DE-588)4128781-2 gnd |
| topic_facet | Invariante - Mannigfaltigkeit - Komplex <Topologie> - Selbstadjungierter Operator - K-Theorie K-Theorie Mannigfaltigkeit Komplex Topologie Selbstadjungierter Operator Invariante |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009897979&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000899194 |
| work_keys_str_mv | AT luckwolfgang l2invariantstheoryandapplicationstogeometryandktheory |