Lifting theorems for completely positive maps

We prove lifting theorems for completely positive maps going out of exact C*-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if X is a second countable topological space, A and B are separable, nuclear C -algebras over X, and the action of X on A is continuous, then E.XI A; B/ Š KK.XI A; B/ naturally. As an application, we show that a separable, nuclear, strongly purely infinite C* -algebra A absorbs a strongly self-absorbing C* -algebra D if and only if I and I ⊗ D are KK-equivalent for every two-sided, closed ideal I in A. In particular, if A is separable, nuclear, and strongly purely infinite, then A ⊗ O₂ Š A if and only if every two-sided, closed ideal in A is KK-equivalent to zero.