Shape theory and extensions of C*-algebras

Let $A$, $A^{\mathrm{\prime }}$ be separable $C^{\ast }$-algebras, $B$ a stable $\sigma$-unital $C^{\ast }$-algebra. Our main result is the construction of the pairing $[[A^{\mathrm{\prime }},A]]\times {\mathrm{Ext}}^{-1\mathrm{/}2}(A,B)\to {\mathrm{Ext}}^{-1\mathrm{/}2}(A^{\mathrm{\prime }},B)$, where $[[A^{\mathrm{\prime }},A]]$ denotes the set of homotopy classes of asymptotic homomorphisms from $A^{\mathrm{\prime }}$ to $A$ and ${\mathrm{Ext}}^{-1\mathrm{/}2}(A,B)$ 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 $A^{\mathrm{\prime }}$ that provides semi-invertibility of all extensions of $A^{\mathrm{\prime }}$ by $B$. This holds, in particular, if $A$ and $A^{\mathrm{\prime }}$ are shape equivalent. A similar condition implies that if ${\mathrm{Ext}}^{-1\mathrm{/}2}$ coincides with $E$-theory (via the Connes-Higson map) for $A$ then the same holds for $A^{\mathrm{\prime }}$.