Bökstedt, Marcel; Brun, Morten; Dupont, Johan Homology of \(O(n)\) and \(O^1(1,n)\) made discrete: An application of edgewise subdivision. (English) Zbl 0910.20026 J. Pure Appl. Algebra 123, No. 1-3, 131-152 (1998). Let \(O(n)\) be the group of real orthogonal \(n\times n\)-matrices and \(O^1(1,n)\) the group of isometries of hyperbolic \(n\)-space. It is proved that the standard injection of \(O(n)\) into \(O^1(1,n)\) induces an isomorphism on homology in degrees \(\leq n-1\), where homology means group homology for the corresponding groups made discrete. This result establishes a conjecture of C. H. Sah [Appendix A in Comment. Math. Helv. 61, 308-347 (1986; Zbl 0607.57025)]. It has a number of consequences: It shows that the statement of the Friedlander-Milnor conjecture for \(O(n)\) is equivalent to that for \(O^1(1,n)\) in the stable range. Secondly it reduces the calculation of the scissors congruence group for the 3-sphere to a well known problem in algebraic K-theory; it implies in particular that this group is a rational vector space and gives necessary and sufficient conditions for determining the scissors congruence classes of spherical polyhedra with vertices whose coordinates are algebraic integers. Among the tools is edgewise subdivision which has apparently not been used in this context before. Reviewer: J.Huebschmann (Villeneuve d’Ascq) Cited in 1 ReviewCited in 4 Documents MSC: 20G10 Cohomology theory for linear algebraic groups 20J05 Homological methods in group theory 22E99 Lie groups 52B45 Dissections and valuations (Hilbert’s third problem, etc.) 57T10 Homology and cohomology of Lie groups 55R45 Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity Keywords:homology of Lie groups made discrete; scissors congruence group; edgewise subdivision; orthogonal groups PDF BibTeX XML Cite \textit{M. Bökstedt} et al., J. Pure Appl. Algebra 123, No. 1--3, 131--152 (1998; Zbl 0910.20026) Full Text: DOI References: [1] Bökstedt, M.; Hsiang, W.C.; Madsen, I., The cyclotomic trace and algebraic K-theory of spaces, Invent. math., 111, 465-540, (1993) · Zbl 0804.55004 [2] Borel, A., Cohomologie de SL_n et valeurs de fonctions zeta aux points entiers, Ann. scuola norm. sup. Pisa cl. sci., 4, 4, 613-636, (1977) · Zbl 0382.57027 [3] Dupont, J.L., The dilogarithm as a characteristic class for flat bundles, J. pure appl. algebra, 44, 134-164, (1987) · Zbl 0624.57024 [4] Dupont, J.L.; Parry, W.; Sah, C.-H., Homology of classical Lie groups made discrete II. H_2, H3, and relations with scissors congruences, J. algebra, 113, 215-260, (1988) · Zbl 0657.55022 [5] Dupont, J.L.; Sah, C.-H., Homology of euclidean groups of motions made discrete and Euclidean scissors congruences, Acta math., 164, 1-27, (1990) · Zbl 0724.57027 [6] Eckmann, B., Cohomology of groups and transfer, Ann. of math., 58, 2, 481-493, (1953) · Zbl 0052.02002 [7] Iversen, B., Hyperbolic geometry, () · Zbl 0766.51002 [8] Lawson, H.B.; Michelsohn, M.L., Spin geometry, () · Zbl 0688.57001 [9] MacLane, S., () [10] Milnor, J., On the homology of Lie groups made discrete, Comment. math. helv., 58, 72-85, (1983) · Zbl 0528.20033 [11] Sah, C.-H., Homology of classical Lie groups made discrete, 1. stability theorems and Schur multipliers, Comment, math. helv., 61, 308-347, (1986) · Zbl 0607.57025 [12] Sah, C.-H., Homology of classical Lie groups made discrete, III, J. pure appl. algebra, 56, 269-312, (1989) · Zbl 0684.57020 [13] Sah, C.-H.; Wagoner, J.B., Second homology of Lie groups made discrete, Comm. algebra, 5, 611-642, (1977) · Zbl 0375.18006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.