On the 3-torsion in \(K_4(\mathbb{Z})\)
The goal of the paper under review is to investigate the first homology group \(H_1(SL_4(\mathbb Z);St_4)\) of the group \(SL_4(\mathbb Z)\) of \((4 \times 4)\) integral matrices with determinant \(1\) with coefficients in the Steinberg module \(St_4\), i.e., in the top homology of the Tits building of \(SL_4\) over \(\mathbb Q\). The main theorem asserts that \(H_1(SL_4(\mathbb Z);St_4)\) is a finite abelian \(2\)-group. Its proof is based on the construction of a spectral sequence converging towards \(H_*(SL_n(\mathbb Z);St_n)\) and, in the case \(n=4\), on the calculation of the \(E_1\)-term of that spectral sequence modulo the Serre class \(\mathcal S\) of all finite abelian groups of order a power of \(2\). In the author's Ph.D. thesis [\textit{C. Soulé}, ``Groupes arithmétiques et K-théorie des anneaux d'entiers des corps de nombres'', Thèse, Université de Paris VII (1979)], this theorem was shown to imply that the fourth algebraic \(K\)-group of the ring of integers \(\mathbb Z\) satisfies: \(K_4(\mathbb Z)=0\) or \(\mathbb Z/3\) modulo \(\mathcal S\). In the recent proof of the vanishing of \(K_4(\mathbb Z)\) by \textit{J. Rognes} [``\(K_4(\mathbb Z)\) is the trivial group'', Topology 39, No. 2, 267-281 (2000; Zbl 0937.19005)], the above theorem is used in order to deduce that the \(3\)-torsion of \(K_4(\mathbb Z)\) is trivial.
scientific article; zbMATH DE number 1384247