Vaserstein, L. N. Reduction of a matrix depending on parameters to a diagonal form by addition operations. (English) Zbl 0657.55005 Proc. Am. Math. Soc. 103, No. 3, 741-746 (1988). Let \(A\) be the ring of all continuous real- or complex-valued functions on a normal topological space \(X\). The author shows that an element from \(\mathrm{SL}(n,A)\) is contained in \(E(n,A)\) if and only if its corresponding homotopy class is trivial, provided \(n\neq 2\) for real-valued functions. In the exceptional case the same result is true if and only if \(X\) is pseudocompact. Moreover, in the non-exceptional cases, it is shown that \(E(n,A)\) has bounded word-length, i.e., every element from \(E(n,A)\) is expressible as a product of a bounded number of elementary matrices, where the bound depends only on \(n\) and the dimension of \(X\). Reviewer: M.Kolster Cited in 2 ReviewsCited in 19 Documents MSC: 55N15 Topological \(K\)-theory 55Q05 Homotopy groups, general; sets of homotopy classes 18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Keywords:continuous functions; ring of continuous functions on a normal topological space; word-length; elementary matrices PDFBibTeX XMLCite \textit{L. N. Vaserstein}, Proc. Am. Math. Soc. 103, No. 3, 741--746 (1988; Zbl 0657.55005) Full Text: DOI References: [1] Allan Calder and Jerrold Siegel, Homotopy and uniform homotopy, Trans. Amer. Math. Soc. 235 (1978), 245 – 270. · Zbl 0391.55012 [2] Allan Calder and Jerrold Siegel, Homotopy and uniform homotopy. II, Proc. Amer. Math. Soc. 78 (1980), no. 2, 288 – 290. · Zbl 0452.55007 [3] R. K. Dennis and L. N. Vaserstein, On a question of M. Newman on the number of commutators, J. Algebra (submitted). · Zbl 0649.20048 [4] W. Thurston and L. Vaserstein, On \?\(_{1}\)-theory of the Euclidean space, Topology Appl. 23 (1986), no. 2, 145 – 148. · Zbl 0611.18007 [5] Wilberd van der Kallen, \?\?\(_{3}\)(\?[\?]) does not have bounded word length, Algebraic \?-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 357 – 361. · Zbl 0935.20501 [6] L. N. Vaserstein, On \?\(_{1}\)-theory of topological spaces, Applications of algebraic \?-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 729 – 740. [7] L. N. Vaseršteĭn, The stable range of rings and the dimension of topological spaces, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 17 – 27 (Russian). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.