Scissors Congruences, Group Homology And Characteristic Classes

By: Johan L. Dupont, Weiping Zhang (Editor)

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Paperback | 1 March 2001

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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", ie. 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, characteristics 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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Non-Fiction Mathematics Geometry Euclidean Geometry Topology Analytic Geometry Booktopia Publisher Services World Scientific Publishing

Preface	p. v
Introduction and History	p. 1
Scissors congruence group and homology	p. 9
Homology of flag complexes	p. 17
Translational scissors congruences	p. 27
Euclidean scissors congruences	p. 37
Sydler's theorem and non-commutative differential forms	p. 45
Spherical scissors congruences	p. 53
Hyperbolic scissors congruence	p. 63
Homology of Lie groups made discrete	p. 77
Invariants	p. 91
Simplices in spherical and hyperbolic 3-space	p. 107
Rigidity of Cheeger-Chern-Simons invariants	p. 119
Projective configurations and homology of the projective linear group	p. 125
Homology of indecomposable configurations	p. 135
The case of PGl(3,F)	p. 145
Spectral sequences and bicomplexes	p. 151
Bibliography	p. 159
Index	p. 167
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Scissors Congruences, Group Homology And Characteristic Classes

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