Vaserstein, L. N. Unstable \(K_ 1\)-theory of topological spaces is nilpotent. (English) Zbl 0762.19002 Algebraic \(K\)-theory, commutative algebra, and algebraic geometry, Proc. Jt. US-Italy Semin., Santa Margherita Ligure/Italy 1989, Contemp. Math. 126, 193-196 (1992). [For the entire collection see Zbl 0742.00073.]The author proves that for an arbitrary topological space \(X\) of finite dimension \(d\) and any integer \(> 1\), the group \(\mathrm{SL}_ nA/E_ nA\) is nilpotent of class at most \(d\)+ 1 when \(A = F^ X\) with \(F = \mathbb R\) or \(\mathbb C\). More generally, Theorem 7 asserts that the group \(G(A)/E(A)\) is nilpotent of class at most \(d+1\) when \(G\) is a simply connected almost simple Chevalley group and \(E(A)\) is the subgroup of \(G(A)\) generated by all root elements. Therefore \(E(A)\) is a characteristic subgroup of \(G(A)\). Theorem 8 asserts that for any almost simple Lie group \(G\), the group \(\pi(X,G) = G^ X/G^ X_ 0\) of all homotopy classes of continuous maps from \(X\) to \(G\) is nilpotent of class at most \(d+1\). Reviewer: L. N. Vaserstein (University Park) Cited in 1 Document MSC: 19B14 Stability for linear groups 19L99 Topological \(K\)-theory 20D15 Finite nilpotent groups, \(p\)-groups Keywords:Chevalley group; almost simple Lie group Citations:Zbl 0742.00073 PDFBibTeX XMLCite \textit{L. N. Vaserstein}, Contemp. Math. 126, 193--196 (1992; Zbl 0762.19002)