Purely infinite C*-algebras: ideal-preserving zero homotopies

We show that if A is a separable, nuclear, O_infty-absorbing (or strongly purely infinite) C*-algebra, which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form M_k(C_0(G,v)), where G is a finite graph (and C_0(G,v) is the algebra of continuous functions on G that vanish at a distinguished point v in G). We show further that any separable, nuclear, stable, O_2-absorbing C*-algebra is isomorphic to a crossed product of a C*-algebra D with the integers by an action alpha, where D is an inductive limit of C*-algebras of the form M_k(C_0(G,v)) (and D is O_2-absorbing and homotopic to zero in an ideal-system preserving way).