Homology of tangent groups considered as discrete groups and scissors congruence
\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, \[ H_n(SU_2(\mathbb C),su_2(\mathbb C))\to H_n(SL_2(\mathbb C),sl_2(\mathbb C))^+, \] where \((\;)^+\) denotes the +1-eigenspace for complex conjugation, is bijective for \(n=0,1\) and injective for \(n=2\). Then using (2) we can deduce that \[ H_1(SU_2(\mathbb C),su_2(\mathbb C))\cong \Omega^1_\mathbb C,\quad H_2(SU_2(\mathbb C),su_2(\mathbb C))=0. \] And using the Lyndon-Hochschild-Serre spectral sequence applied on \[ 1\to \{\pm 1\} \to SU_2( \mathbb C) \to SO_3(\mathbb R) \to 1, \] we can deduce (1). The ingredients of the proof of theorem 1 (and other similar results) involve the study of the homology of tangent groups of Lie groups (always considered as discrete groups) and also the extension to dual numbers of geometric techniques developed by \textit{M. Bökstedt, M. Brun} and \textit{J. Dupont} [``Homology of \(O(n)\) and \(O^1(1,n)\) made discrete: An application of edgewise subdivision'', J. Pure Appl. Algebra 123, No. 1-3, 131-152 (1998; Zbl 0910.20026)], and \textit{C.-H. Sah} [``Homology of classical Lie groups made discrete. I: Stability theorems and Schur multipliers'', Comment. Math. Helv. 61, 308-347 (1986; Zbl 0607.57025)]. The author also proves some stability results in low degrees for the homology of tangent groups to unitary groups (section 4), and, to \(O(n)\) and \(O^1(1,n)\) (section 5).
scientific article; zbMATH DE number 1308241
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