Tomiyama's K-commutative diagrams of minimal dynamical systems

Let K be the Cantor space and S-2n be an even-dimensional sphere. By applying a result of the existence of minimal skew products, we show that, associated with any Cantor minimal system (K ,alpha), there is a class Script capital R-0((alpha) over tilde) of minimal skew products on K x S-2n, such that for any two rigid homeomorphisms alpha is an element of R-0(alpha) and beta is an element of R-0((beta) over tilde), the notions of approximate K-conjugacy and C*-strongly approximate conjugacy coincide, which are also equivalent to a K-version of Tomiyama's commutative diagram. In fact, this is also the case if S-2n is replaced by any (infinite) connected finite CW-complex with torsion free K-0-group, vanished K-1-group and the so-called Lipschitz-minimal-property.