Scissors congruences, group homology and characteristic classes
As the author writes in the introduction, this book shows the relationship between Hilbert's 3rd problem and its generalizations, including the non-Euclidean cases, and questions in homological algebra and topology. The author generalizes the notion of scissor congruence to that of \(G\)-scissor congruence, \(G\)-s.c., where \(G\) is a group of isometries, and two polytopes \(P\) and \(P'\) are said to be \(G\)-s.c. if there exist finite subdivisions \(\{ P_i\}\) and \(\{ P'_i\}\), \(i =1, \dots k\), respectively of \(P\) and \(P'\), such that \(P'_i = g_i P_i\) for some \(g_i \in G\), \(i =1, \dots,k\). NEWLINENEWLINENEWLINEThe book is divided into 15 chapters and it also includes an appendix on spectral sequences and bicomplexes. The first chapter provides a historical introduction to the subject, with bibliographic references to the main results. In chapter 2, the author gives a homological description of the scissor congruence group. Chapter 3 deals with Steinberg and polytope modules, establishing some exact sequences relating these submodules in different dimensions. Focussing on the Euclidean case, the author states necessary and sufficient conditions for translational scissor congruence -- the Hadwiger invariants -- in chapter 4, studying the s.c. group \(P(E^n,G)\) when \(G\) is a group of isometries of the Euclidean \(n\)-space \(E^n\) in the next chapter. In chapter 6, a direct homological proof that \(H_2(SO(3), \mathbb{R}^3) = 0\) is given, thus proving Sydler's theorem in 3 and 4 dimensions. Chapter 7 deals with scissor congruence group in spherical space, showing that \(P(S^{2n})\) is isomorphic to \(P(S^{2n-1})\) and identifying \(P(S^{n})\) for all \(n\geq 0\) by means of exact sequences. In chapter 8, focussing on the hyperbolic geometry, the author describes the situation of s.c. groups in different dimensions, pointing out that there is no easy way to relate \(P(H^{2n})\) with \(P(H^{2n-1})\), so the problem is much more difficult than in the previous cases. Next, in chapter 9, and aiming to understand \(P(H^3)\), the author looks at the isometry groups of 3-geometries and their homological properties as discrete Lie groups. NEWLINENEWLINENEWLINEChapters 10, 11 and 12 deal with Cheeger-Chern-Simons classes for flat \(Sl(2,C)\)-bundles, obtaining some invariants for s.c. in dimension 3 by twisting these invariants with a field automorphism of \(C\). Following this approach further, the remaining chapters deal with the homology of the projective linear group \(\text{PGL}(n+1, F)\) in terms of configurations of points in projective \(n\)-space \(P^n(F)\), where \(F\) is a field of characteristic \(0\).
scientific article