Scissors Congruences, Group Homology and Characteristic Classes (Nankai Tracts in Mathematics, V. 1.)

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. Ch. 1. Introduction And History -- Ch. 2. Scissors Congruence Group And Homology -- Ch. 3. Homology Of Flag Complexes -- Ch. 4. Translational Scissors Congruences -- Ch. 5. Euclidean Scissors Congruences -- Ch. 6. Sydler's Theorem And Non-commutative Differential Forms -- Ch. 7. Spherical Scissors Congruences -- Ch. 8. Hyperbolic Scissors Congruence -- Ch. 9. Homology Of Lie Groups Made Discrete -- Ch. 10. Invariants -- Ch. 11. Simplices In Spherical And Hyperbolic 3-space -- Ch. 12. Rigidity Of Cheeger-chern-simons Invariants -- Ch. 13. Projective Configurations And Homology Of The Projective Linear Group -- Ch. 14. Homology Of Indecomposable Configurations -- Ch. 15. The Case Of Pgl(3,f) -- App. A. Spectral Sequences And Bicomplexes. Johan L. Dupont. Includes Bibliographical References (p. 159-165) And Index. Annotation Notes based on a series of lectures delivered at the Nankai Institute of Mathematics (Tianjin, China) in fall 1998 on various aspects of Hilbert's Third Problem, in particular its relation to homological algebra and algebraic K-theory. Based largely on the work of Dupont (Aarhus U., Denmark) and the late Chih-Han Sah, the work starts from Dehn's classical solution of the problem: Are two Euclidean polyhedra with the same volume "scissors-congruent," that is, can they be subdivided into a finite number of pairwise congruent pieces? Includes previously unpublished material. c. Book News Inc It is elementary and well-known that two polygons P and P' in the Euclidean plane have the same area if and only if they are scissors congruent (s.c.), i.e. if they can be subdivided into finitely many pieces such that each piece in P is congruent to exactly one piece in P'.