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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 $n\ne 2$ 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

55N15$K$-theory (algebraic topology)
55Q05General theory of homotopy groups
18F25Algebraic $K$-theory and $L$-theory
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