In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.
Author(s): Wolfgang Lück
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Edition: Reprint
Publisher: Springer
Year: 2004
Language: English
Pages: 608
What are L2-Invariants?......Page 14
Some Applications of L2-Invariants......Page 15
Some Open Problems Concerning L2-Invariants......Page 16
L2-Invariants and Heat Kernels......Page 18
L2-Invariants and Cellular Chain Complexes......Page 19
L2-Betti Numbers and Betti Numbers......Page 20
L2-Invariants and Ring-Theory......Page 21
L2-Invariants and K-Theory......Page 23
L2-Invariants and Aspherical Manifolds......Page 24
L2-Invariants and Groups......Page 25
Introduction......Page 26
Group von Neumann Algebras......Page 27
Dimension Theory......Page 29
Hilbert Chain Complexes......Page 37
Induction for Group von Neumann Algebras......Page 42
Cellular L2-Betti Numbers......Page 43
Survey on G-CW-Complexes......Page 44
The Cellular L2-Chain Complex......Page 46
Basic Properties of Cellular L2-Betti Numbers......Page 50
L2-Betti Numbers and Aspherical Spaces......Page 58
The Classical Hodge-de Rham Theorem......Page 62
Analytic Definition of L2-Betti Numbers......Page 65
Survey on Unbounded Operators and Spectral Families......Page 67
L2-Hodge-de Rham Theorem......Page 71
L2-Betti Numbers of Manifolds with Boundary......Page 76
Miscellaneous......Page 78
Exercises......Page 81
Introduction......Page 84
Spectral Density Functions of Morphisms......Page 85
Spectral Density Functions of HilbertChain Complexes......Page 94
Product Formula for Novikov-Shubin Invariants......Page 99
The Laplacian in Dimension Zero......Page 106
Cellular Novikov-Shubin Invariants......Page 109
Analytic Novikov-Shubin Invariants......Page 116
Comparison of Analytic and Cellular Novikov-Shubin Invariants......Page 119
On the Positivity and Rationality of the Novikov-ShubinInvariants......Page 125
Novikov-Shubin Invariants of Manifolds with Boundary......Page 127
Miscellaneous......Page 128
Exercises......Page 130
Introduction......Page 132
Whitehead Groups......Page 133
Whitehead Torsion......Page 134
Reidemeister Torsion......Page 136
Fuglede-Kadison Determinant......Page 139
L2-Torsion of Hilbert Chain Complexes......Page 152
Basic Definitions and Properties of L2-Torsion......Page 153
L2-Torsion and Chain Contractions......Page 157
Proofs of the Basic Properties of L2-Torsion......Page 161
Cellular L2-Torsion in the Weakly-Acyclic Case......Page 173
Cellular L2-Torsion in the Weakly-Acyclic and Aspherical Case......Page 184
Topological L2-Torsion for Riemannian Manifolds......Page 189
Definition of Analytic L2-Torsion......Page 191
The Laplace Transform of a Density Function......Page 194
Analytic L2-Torsion for Hyperbolic Manifolds......Page 199
L2-Torsion of Manifolds with Boundary......Page 202
Combinatorial Computations of L2-Invariants......Page 206
Miscellaneous......Page 213
Exercises......Page 218
Survey on 3-Manifolds......Page 224
L2-Invariants of 3-Manifolds......Page 227
L2-Invariants of Knot Complements......Page 230
Miscellaneous......Page 231
Exercises......Page 233
Survey on Symmetric Spaces......Page 236
L2-Invariants of Symmetric Spaces of Non-Compact Type......Page 240
L2-Invariants of Symmetric Spaces......Page 242
Miscellaneous......Page 244
Exercises......Page 245
Introduction......Page 248
Dimension Theory for Arbitrary Modules......Page 250
Comparison of Modules and Hilbert Modules......Page 259
Induction and the Extended von Neumann Dimension......Page 266
The Extended Dimension Function and Amenable Groups......Page 268
Survey on Amenable Groups......Page 269
Amenability and Flatness Properties of the Group von Neumann Algebra over the Group Ring......Page 271
L2-Betti Numbers for General Spaces with Group Actions......Page 276
L2-Euler Characteristic......Page 289
Definition and Basic Properties of L2-Euler Characteristic......Page 290
L2-Euler Characteristic, Equivariant Euler Characteristic and the Burnside Group......Page 293
Finitely Presented Torsion Modules and Novikov-Shubin Invariants......Page 298
Miscellaneous......Page 300
Exercises......Page 302
Groups with Vanishing L2-Betti Numbers......Page 306
General Criterions for the Vanishing of the L2-Betti Numbers of a Group......Page 307
The Vanishing of the L2-Betti Numbers of Thompson's Group......Page 310
Euler Characteristics of Groups......Page 311
Survey on Deficiency of Groups......Page 312
Applications of L2-Betti Numbers to Deficiency and to Signatures of 4-Manifolds......Page 316
Automorphisms of Groups G with Finite Modelsfor BG......Page 317
Automorphisms of Surfaces......Page 320
A Combinatorial Approach for the L2-Torsion of an Automorphism of a Finitely Generated Free Group......Page 321
Generalizations......Page 323
Miscellaneous......Page 324
Exercises......Page 327
Introduction......Page 330
The Algebra of Affiliated Operators......Page 331
Survey on Ore Localization......Page 336
Survey on von Neumann Regular Rings......Page 338
Basic Properties of the Algebra of Affiliated Operators......Page 340
Dimension Theory and L2-Betti Numbers over the Algebra of Affiliated Operators......Page 342
Various Notions of Torsion Modules over a Groupvon Neumann Algebra......Page 344
Exercises......Page 346
Introduction......Page 348
Definition of a von Neumann Algebra......Page 349
Types and the Decomposition of von Neumann Algebras......Page 350
Finite von Neumann Algebras and Traces......Page 351
Extending Results for Group von Neumann Algebras to Finite von Neumann Algebras......Page 353
K0 of a von Neumann Algebra......Page 354
K1 of a von Neumann Algebra......Page 356
Middle K-Theory of the Algebra of Affiliated Operators......Page 358
L-Theory of a von Neumann Algebra and the Algebra of Affiliated Operators......Page 360
Detecting Elements in K1 of a Complex Group Ring......Page 367
Survey on the Isomorphism Conjecture for K0 of Complex Group Rings, the Bass Conjecture and the Hattori-Stallings Rank......Page 370
G-Theory of Complex Group Rings......Page 376
Miscellaneous......Page 378
Exercises......Page 379
Introduction......Page 382
Various Formulations of the Atiyah Conjecture......Page 383
Relations of the Atiyah Conjecture to Other Conjectures......Page 389
Survey on Positive Results about the Atiyah Conjecture......Page 391
A Counterexample to the Strong Atiyah Conjecture......Page 392
The General Case......Page 394
Survey on Universal Localizations and Division and Rational Closure......Page 398
The Strategy for the Proof of Linnell's Theorem......Page 400
The Proof of Atiyah's Conjecture for Free Groups......Page 404
Survey on Crossed Products......Page 409
Property (R) Ascends to Extensions by Infinite Cyclic Groups......Page 412
Property (K) and Extensions by Virtually Finitely Generated Abelian Groups......Page 415
Property (K) holds for Virtually Free Groups......Page 418
The Induction Step for Directed Unions......Page 421
Miscellaneous......Page 423
Exercises......Page 426
Introduction......Page 430
Low-Dimensional Manifolds......Page 431
Pinched Curvature......Page 433
The Singer Conjecture for Kähler Manifolds......Page 434
Hodge Theory on Kähler manifolds......Page 437
The L2-Lefschetz Theorem......Page 439
Novikov-Shubin Invariants for Kähler Hyperbolic Manifolds......Page 442
Non-Vanishing of the Middle L2-Betti Number for Kähler Hyperbolic Manifolds......Page 444
Miscellaneous......Page 445
Exercises......Page 447
Introduction......Page 450
An Algebraic Formulation of the Zero-in-the-Spectrum Conjecture......Page 451
The Zero-in-the-Spectrum Conjecture for Low-Dimensional Manifolds......Page 453
The Zero-in-the-Spectrum Conjecture for Locally Symmetric Spaces......Page 454
The Zero-in-the-Spectrum Conjecture for HyperEuclidean Manifolds......Page 455
The Zero-in-the-Spectrum Conjecture and Finite Asymptotic Dimension......Page 456
Counterexamples to the Zero-in-the-Spectrum Conjecture in the Non-Aspherical Case......Page 457
Miscellaneous......Page 462
Exercises......Page 463
Introduction......Page 466
Survey on Positive Results about the Approximation Conjecture and the Determinant Conjecture......Page 467
Relations to Other Conjectures......Page 468
A Class of Groups......Page 469
The Proof of the Approximation Conjecture and the Determinant Conjecture in Special Cases......Page 472
The General Strategy......Page 473
Limits of Inverse Systems......Page 478
Colimits of Directed Systems......Page 480
Amenable Extensions......Page 482
Quotients with Finite Kernels......Page 486
Variations of the Approximation Results......Page 488
Miscellaneous......Page 495
Exercises......Page 496
Basic Definitions......Page 498
Elementary Properties......Page 500
The Simplicial Volume of Hyperbolic and Low-Dimensional Manifolds......Page 502
Volume and Simplicial Volume......Page 504
Simplicial Volume and Betti Numbers......Page 507
Further Properties of the Simplicial Volume......Page 509
Hyperbolic Manifolds and 3-Manifolds......Page 510
Amenable Fundamental Groups......Page 511
Selfmaps of Degree Different from -1, 0 and 1......Page 512
Simplicial Volume and L2-Invariants......Page 514
Miscellaneous......Page 515
Exercises......Page 517
L2-Index Theorems......Page 520
Lp-Cohomology......Page 521
Knot Concordance and L2-signature......Page 522
Solutions of the Exercises......Page 524
References......Page 537
Index......Page 596
