Reduction of a Matrix Depending on Parameters to a Diagonal Form by Addition Operations

Abstract

ABSTRACT. It is shown that any n by n matrix with determinant 1 whose entries are real or complex continuous functions on a finite dimensional normal topological space can be reduced to a diagonal form by addition operations if and only if the corresponding homotopy class is trivial, provided that n j¿2 for real-valued functions; moreover, if this is the case, the number of operations can be bounded by a constant depending only on n and the dimension of the space. For real functions and n = 2, we describe all spaces such that every invertible matrix with trivial homotopy class can be reduced to a diagonal form by addition operations as well as all spaces such that the number of operations is bounded.

A solution of Gromov's Vaserstein problem

“…(Proof of Theorem 1 for SL 2 (C).) By a result of Vaserstein [19,Theorem 4] we have a continuous map F : X → C K for some natural number K such that f (x) = Ψ K (F (x)). Using Lemma 2.1 we see that F = (F 1 , .…”

“…In the case of continuous complex valued functions on a topological space X the problem was studied and partially solved by Thurston and Vaserstein [18] and then finally solved by Vaserstein [19,Theorem 4].…”

Generation of the special linear group by elementary matrices in some measure Banach algebras

“…For d ě 3, the answer to (Q) is 'Yes', and this is the K 1 -analogue of Serre's conjecture, which is the Suslin stability theorem [24]. ‚ Rings of continuous real-or complex-valued functions on a topological space X were considered in [25]. ‚ For the ring O X of holomorphic functions on Stein spaces in C d , the question (Q) was posed as an explicit open problem by Gromov in [11], and was solved in [13].…”

Unipotent Factorization of Vector Bundle Automorphisms

“…Replacing SL k (R) or SL k (C) with SL k (R) where R is the ring of continuous real or complex valued functions on a topological space X, we arrive at a much more subtle problem. This problem was adressed Thurston and Wasserstein [TW86] in the case where X is the Euclidean space and more generally by Wasserstein [Was88] for a finite dimensional normal topological space X. In particular, Wasserstein [Was88] proves that for any finite dimensional normal topological space X, and any continuous map F : X → SL k (R) for k ≥ 3 (and SL k (C) for k ≥ 2, respectively) which is homotopic to the constant map x → id, there are continuous maps E 1 , .…”

“…However, we do not believe the statement of the theorem is true in the non-compact setting. In particular (generalising the construction in [Was88]), we expect that the following null-homotopic maps from R to Sp 2k (R) for any positive k provide examples which can't be written as a product of unipotent factors:…”