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On $K\sb 1$-theory of the Euclidean space. (English) Zbl 0611.18007
It is shown that any $n$ by $n$ matrix of determinant 1, $n\ge 3$, whose entries are continuous real functions of $N=3$ variables, can be reduced to the identity matrix by addition operations (with continuous coefficients). In the case $n=2$, this is not true. The case of $N=1$ was done in a previous paper by a different method. The case of arbitrary $N$ is finished by a different method in a paper to appear in Proc. Am. Math. Soc. The similar result for polynomial functions (n$\ge 3)$ is due to A. Suslin.

MSC:
18F25Algebraic $K$-theory and $L$-theory
46L80$K$-theory and operator algebras
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References:
[1] H. Bass, K-theory and stable algebra, Publ. Math. IHES, 22. · Zbl 0248.18025
[2] Vaserstein, L. N.: The stable range of rings and the dimension of topological spaces. Funnk. an. Pril. 5, No. 2, 17-27 (1971)
[3] L.N. Vaserstein, On K1-theory of topological spaces, A.M.S. Cont. Math., to appear.
[4] Vaserstein, L. N.: Vector bundles and projective modules. Trans. amer. Math. soc. 294, 749-755 (1986) · Zbl 0592.55012