Homology of tangent groups considered as discrete groups and scissors congruence
type: Item
instance of: scholarly article
title: Homology of tangent groups considered as discrete groups and scissors congruence
zbMATH Open document ID: 0928.18006
DOI: 10.1016/S0022-4049(97)00136-9
author: Jean-Louis Cathelineau
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-03T00:00:00Z
review text: \textit{J. L. Dupont} proved [``Algebra of polytopes and homology of flag complexes'', Osaka J. Math. 19, 599-691 (1982; Zbl 0499.51014)], that Sydler's theorem, on completeness of the volume and Dehn invariants, for scissors congruences of polyhedra in Euclidean three-space, is equivalent to the following purely algebraic statement: \[ H_1(SO_3(\mathbb R),\mathbb R^3)\cong \Omega^1_\mathbb R,\quad H_2(SO_3(\mathbb R),\mathbb R^3)=0, \tag{1} \] where \(\mathbb R\) is the field of real numbers, and \(H\) means homology of Lie groups considered as discrete groups, the action of the classical orthogonal group \(SO_3(\mathbb R)\) on \(\mathbb R^3\) is the usual geometric one (which, in this case, is equivalent to the adjoint action), and \(\Omega^1_\mathbb R\) is the \(\mathbb R\)-vector space of absolute Kähler differentials. In an attempt to find a direct algebraic approach to (1), the author has shown [\textit{J.-L. Cathelineau}, ``Sur l'homologie de \(SL_2\) à coefficients dans l'action adjointe'', Math. Scand. 63, No. 1, 51-86 (1988; Zbl 0682.55013)], that for any algebraically closed field \(K\) of characteristic zero: \[ H_1(SL_2(K),sl_2(K))\cong \Omega^1_K,\quad H_2(SL_2(K),sl_2(K))=0. \tag{2} \] Here, the action of \(SL_2(K)\) on the Lie algebra \(sl_2(K)\) is the adjoint action. The initial idea of the author was to deduce (1) from (2) by a ``descent argument'' specialised to \(K=\mathbb C\), the field of complex numbers. Meanwhile, \textit{J. L. Dupont} and \textit{C.-H. Sah} [``Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences'', Acta Math. 164, No. 1/2, 1-27 (1990; Zbl 0724.57027)] were able to give a direct, but quite intricated, algebraic proof of (1). In this paper the author achieves his initial goal and gives an elegant proof of (1). In fact, he proves (theorem 1) that the natural map between homology groups of Lie groups considered as discrete groups, with coefficients in adjoint representations, \[ H_n(SU_2(\mat…
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 18G99
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 22E99
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51C05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 1308241
zbMATH Keywords: homology of Lie groups made discrete
zbMATH Keywords: scissors congruences of polyhedra
zbMATH Keywords: tangent groups
zbMATH Keywords: geometric algebra over dual numbers
zbMATH Keywords: homological stability
zbMATH Keywords: complex of configurations
zbMATH Keywords: hyperbolic space
zbMATH Keywords: orthogonal groups
zbMATH Keywords: unitary groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308241