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Unstable $K\sb 1$-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 $SL\sb nA/E\sb nA$ is nilpotent of class at most $d$+ 1 when $A = F\sp X$ with $F$ = ${\bold R}$ or ${\bold C}$. 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 $\pi(X,G) = G\sp X/G\sp X\sb 0$ of all homotopy classes of continuous maps from $X$ to $G$ is nilpotent of class at most $d+1$.

19B14Stability for linear groups ($K_1$)
19L99Topological $K$-theory
20D15Nilpotent finite groups, $p$-groups