Scissors Congruences, Group Homology and Characteristic Classes

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World Scientific, 2001 - Mathematics - 168 pages
These lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume ?scissors-congruent?, i.e. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of groups, algebraic K-theory, characteristic classes for flat bundles, and invariants for hyperbolic manifolds. Some of the material, particularly in the chapters on projective configurations, is published here for the first time.
 

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Contents

Chapter 1 Introduction and History
1
Chapter 2 Scissors congruence group and homology
9
Chapter 3 Homology of flag complexes
17
Chapter 4 Translational scissors congruences
27
Chapter 5 Euclidean scissors congruences
37
Chapter 6 Sydlers theorem and noncommutative differential forms
45
Chapter 7 Spherical scissors congruences
53
Chapter 8 Hyperbolic scissors congruence
63
Chapter 11 Simplices in spherical and hyperbolic 3space
107
Chapter 12 Rigidity of CheegerChernSimons invariants
119
Chapter 13 Projective configurations and homology of the projective linear group
125
Chapter 14 Homology of indecomposable configurations
135
Chapter 15 The case of PGl3F
145
Appendix A Spectral sequences and bicomplexes
151
Bibliography
159
Index
167

Chapter 9 Homology of Lie groups made discrete
77
Chapter 10 Invariants
91

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