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El. knyga: Introduction to l2-invariants

  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2247
  • Išleidimo metai: 29-Oct-2019
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030282974
Kitos knygos pagal šią temą:
Introduction to l2-invariantsDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2247
  • Išleidimo metai: 29-Oct-2019
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030282974
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This book introduces the reader to the most important concepts and problems in the field of l²-invariants. After some foundational material on group von Neumann algebras, l²-Betti numbers are defined and their use is illustrated by several examples. The text continues with Atiyah's question on possible values of l²-Betti numbers and the relation to Kaplansky's zero divisor conjecture. The general definition of l²-Betti numbers allows for applications in group theory. A whole chapter is dedicated to Lück's approximation theorem and its generalizations. The final chapter deals with l²-torsion, twisted variants and the conjectures relating them to torsion growth in homology.
 
The text provides a self-contained treatment that constructs the required specialized concepts from scratch. It comes with numerous exercises and examples, so that both graduate students and researchers will find it useful for self-study or as a basis for an advanced lecture course.

Recenzijos

This is an excellent introductory book, to be recommended to readers looking for an introduction to the field, as well as those that want to have an overview of recent developments. (Joan Porti, Mathematical Reviews, September, 2020)

1 Introduction
1(8)
2 Hilbert Modules and von Neumann Dimension
9(26)
2.1 Hilbert Spaces
9(7)
2.2 Operators and Operator Algebras
16(8)
2.3 Trace and Dimension
24(11)
3 l2-Betti Numbers of CW Complexes
35(32)
3.1 G-CW Complexes
35(3)
3.2 The l2-Completion of the Cellular Chain Complex
38(7)
3.3 l2-Betti Numbers and How to Compute Them
45(8)
3.4 Cohomological l2-Betti Numbers
53(4)
3.5 Atiyah's Question and Kaplansky's Conjecture
57(4)
3.6 l2-Betti Numbers as Obstructions
61(6)
3.6.1 l2-Betti Numbers Obstruct Nontrivial Self-coverings
62(1)
3.6.2 l2-Betti Numbers Obstruct Mapping Torus Structures
63(1)
3.6.3 l2-Betti Numbers Obstruct Circle Actions
64(3)
4 l2-Betti Numbers of Groups
67(20)
4.1 Classifying Spaces for Families
67(4)
4.2 Extended von Neumann Dimension
71(4)
4.3 l2-Betti Numbers of G-Spaces
75(2)
4.4 l2-Betti Numbers of Groups and How to Compute Them
77(3)
4.5 Applications of l2-Betti Numbers to Group Theory
80(7)
4.5.1 Detecting Finitely Co-Hopfian Groups
80(1)
4.5.2 Bounding the Deficiency of Finitely Presented Groups
81(2)
4.5.3 One Relator Groups and the Atiyah Conjecture
83(1)
4.5.4 The Zeroth l2-Betti Number
83(2)
4.5.5 l2-Betti Numbers of Locally Compact Groups
85(2)
5 Luck's Approximation Theorem
87(40)
5.1 The Statement
87(2)
5.2 Functional Calculus and the Spectral Theorem
89(7)
5.3 The Proof
96(4)
5.4 Extensions
100(8)
5.4.1 Infinite Type G-CW Complexes
101(1)
5.4.2 Proper G-CW Complexes
101(2)
5.4.3 Non-normal Subgroups
103(2)
5.4.4 Nontrivial Total Intersection
105(1)
5.4.5 Non-nested and Infinite Index Subgroups
106(1)
5.4.6 Further Variants
106(2)
5.5 Rank Gradient and Cost
108(8)
5.6 Approximation, Determinant, and Atiyah Conjecture
116(11)
6 Torsion Invariants
127(38)
6.1 Reidemeister Torsion
127(5)
6.2 l2-Torsion of CW Complexes
132(3)
6.3 l2-Torsion of Groups
135(2)
6.4 l2-Alexander Torsion
137(7)
6.5 Torsion in Homology
144(7)
6.6 Torsion in Twisted Homology
151(5)
6.7 Profiniteness Questions
156(9)
References 165(8)
List of Notation 173(4)
Index 177
Holger Kammeyer studied Mathematics at Göttingen and Berkeley. After a postdoc position in Bonn he is now based at Karlsruhe Institute of Technology. His research interests range around algebraic topology and group theory. The application of  ²-invariants forms a recurrent theme in his work. He has given introductory courses on the matter on various occasions.
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