Homotopical Stable Ranks for Certain $C^{\ast }$-algebras Associated to Groups

We study the general and connected stable ranks for $C^{\ast }$-algebras. We estimate these ranks for certain $C(X)$-algebras, and use that to do the same for certain group $C^{\ast }$-algebras. Furthermore, we also give estimates for the ranks of crossed product $C^{\ast }$-algebras by finite group actions with the Rokhlin property.