Decomposition rank of subhomogeneous $C^{\ast }$-algebras

We analyze the decomposition rank (a notion of covering dimension for nuclear $C^{\ast }$-algebras introduced by E. Kirchberg and the author) of subhomogeneous $C^{\ast }$-algebras. In particular we show that a subhomogeneous $C^{\ast }$-algebra has decomposition rank $n$ if and only if it is recursive subhomogeneous of topological dimension $n$ and that $n$ is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C. Phillips to show the following: Let $A$ be the crossed product $C^{\ast }$-algebra coming from a compact smooth manifold and a minimal diffeomorphism. Then the decomposition rank of $A$ is dominated by the covering dimension of the underlying manifold.