Description:
Let A, A' be separable C*-algebras, B a stable s-unital C*-algebra. Our main result is the construction of the pairing [[A', A]]xExt-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'.