Semi-invertible extensions and asymptotic homomorphisms

We consider the semigroup $Ext(A,B)$ of extensions of a separable C*-algebra $A$ by a stable C*-algebra $B$ modulo unitary equivalence and modulo asymptotically split extensions. This semigroup contains the group $Ex{t}^{-1\mathrm{/}2}(A,B)$ of invertible elements (i.e. of semi-invertible extensions). We show that the functor $Ex{t}^{1\mathrm{/}2}(A,B)$ is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from $C(\mathbb{T})\otimes A$ to $M(B)$ that map $SA\subseteq C(\mathbb{T})\otimes A$ into $B$.