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

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We show that if A is a separable, nuclear, Oᵢnfty-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 Mₖ(C₀(G,v)), where G is a finite graph (and C₀(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, O₂-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 Mₖ(C₀(G,v)) (and D is O₂-absorbing and homotopic to zero in an ideal-system preserving way).