## Scissors Congruences, Group Homology and Characteristic ClassesThese 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 |

167 | |

### Other editions - View all

Scissors Congruences, Group Homology and Characteristic Classes Johan L. Dupont Limited preview - 2001 |

Scissors Congruences, Group Homology and Characteristic Classes Johan L. Dupont No preview available - 2001 |

### Common terms and phrases

2-torsion 3-dimensional 3rd Problem abelian group algebraically closed bicomplexes boundary map center kills chain complex chain homotopy chain map clearly follows configuration corollary defined Dehn invariant denote diagram dimension Dupont Dupont-Sah Euclidean exact sequence filtration finite formula Gal(C/Q Galois geodesic geometry given Gl(n Gram matrix group homology Hence Hilbert's homology groups hyperbolic simplex hyperhomology hyperhomology spectral sequence hyperplane inclusion induces an isomorphism intentionally left blank isometries Jessen Künneth theorem left blank CHAPTER n-simplex natural isomorphism notation Notice numbers orthoscheme particular points polyhedra polytope proof of theorem prove Pt(S Pt(U rational lune rational vector space Remark scissors congruence scissors congruence group sequence in theorem similarly simplices span spectral sequence spherical or hyperbolic St(H St(S St(X subgroup subspaces surjective theorem 1.7 torsion vector space vertices volume zero