  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
$Ext^{-1/2}(A,B)$ of invertible elements (i.e. of semi-invertible extensions).
We show that the functor $Ext^{1/2}(A,B)$ 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 $B$.
