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Reduction of a matrix depending on parameters to a diagonal form by addition operations. (English) Zbl 0657.55005

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

MSC:

55N15 Topological \(K\)-theory
55Q05 Homotopy groups, general; sets of homotopy classes
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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