Summary: We study semiprojective, subhomogeneous $C^*$-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous mbox$C^*$-algebras: one in terms of their primitive ideal spaces and one by means of special direct limit structures over one-dimensional NCCW complexes. These results are obtained by working out several new permanence results for semiprojectivity, including a complete description of its behavior with respect to extensions by homogeneous mbox$C^*$-algebras.

46L05, 46L80, 46L85, 54C55, 54F50

C^*-algebras, semiprojectivity, subhomogeneous, quantum permutation algebras