## 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. |

### 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 abelian group algebraically closed bicomplexes boundary map center kills chain complex chain homotopy chain map Cheeger-Chern-Simons Cindep clearly follows configuration corollary Csing defined Dehn invariant denote dimension Dupont Dupont-Sah en+1 Euclidean exact sequence filtration formula Gal(C/Q Galois geodesic geometry given Gl(n Gram matrix group homology H₂ Hence Hilbert's 3rd Problem homology groups hyperhomology hyperhomology spectral sequence hyperplane inclusion induces an isomorphism intentionally left blank Jessen Künneth theorem left blank CHAPTER Lie groups Math module n-simplex natural isomorphism notation Notice numbers orthoscheme P(H³ P₁ particular polyhedra polytope proof of theorem proposition rational vector space Remark scissors congruence scissors congruence group sequence in theorem similarly simplices sin² Sl(n spectral sequence spherical or hyperbolic subgroup subspaces surjective Suslin theorem 1.7 vector space vertices zero