  Let $A$, $A'$ be separable $C^*$-algebras, $B$ a stable $\sigma$-unital
$C^*$-algebra. Our main result is the construction of the pairing
$[[A',A]]\times\operatorname{Ext}^{-1/2}(A,B)\to\operatorname{Ext}^{-1/2}(A',B)$,
where $[[A',A]]$ denotes the set of homotopy classes of asymptotic
homomorphisms from $A'$ to $A$ and $\operatorname{Ext}^{-1/2}(A,B)$ 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 $\operatorname{Ext}^{-1/2}$ coincides with $E$-theory
(via the Connes-Higson map) for $A$ then the same holds for $A'$.
