Description
Book SynopsisThese 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.
Table of ContentsIntroduction and history; scissors congruence group and homology; homology of flag complexes; translational scissors congruences; Euclidean scissors congruences; Sydler's theorem and non-commutative differential forms; spherical scissors congruences; hyperbolic scissors congruences; homology of Lie groups made discrete; invariants; simplices in spherical and hyperbolic 3-space; rigidity of Cheeger-Chern-Simons invariants; projective configurations and homology of the projective linear group; homology of indecomposable configurations; the case of PGI(3,F).
