L DUPONT J (EN) — SCISSORS CONGRUENCES, GROUP HOMOLOGY & CHARACTERISTIC CLASSES

SCISSORS CONGRUENCES, GROUP HOMOLOGY & CHARACTERISTIC CLASSES краткое содержание

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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”, 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.Contents:Introduction and HistoryScissors Congruence Group and HomologyHomology of Flag ComplexesTranslational Scissors CongruencesEuclidean Scissors CongruencesSydler's Theorem and Non-Commutative Differential FormsSpherical Scissors CongruencesHyperbolic Scissors CongruenceHomology of Lie Groups Made DiscreteInvariantsSimplices in Spherical and Hyperbolic 3-SpaceRigidity of Cheeger-Chern-Simons InvariantsProjective Configurations and Homology of the Projective Linear GroupHomology of Indecomposable ConfigurationsThe Case of PGl(3,F)Readership: Graduate students and researchers in geometry and topology.Key Features:The emphasis and originality of this book is to envisage the emergence of a knowledge selection era and its associated risk, and therefore the need to establish a wise risk strategy