Let A, A′ be separable C*-algebras, and B be a stable σ-unital C*-algebra. Our main result is the construction of the pairing [[A′, A]] × Ext −1/2(A, B) → Ext −1/2(A′, B), where [[A′, A]] denotes the set of homotopy classes of asymptotic homomorphisms from A′ to A and Ext −1/2(A, B) is the group of semi-invertible extensions of A by B. Assume that all extensions of A by B are semi-invertible. Then this pairing allows us to give a condition on A′ that provides semi-invertibility of all extensions of A′ by B. This holds, in particular, if A and A′ are shape equivalent. A similar condition implies that if Ext −1/2 coincides with E-theory (via the Connes–Higson map) for A, then the same holds for A′.
