  Let $A$, $B$ be C*-algebras; $A$ separable, $B$ $\sigma$-unital and stable.
We introduce a notion of translation invariance for asymptotic homomorphisms
from $SA=C_0(\mathbb R)\otimes A$ to $B$ and show that the Connes-Higson
construction applied to any extension of $A$ by $B$ is homotopic to a
translation invariant asymptotic homomorphism. In the other direction we give a
construction which produces extensions of $A$ by $B$ out of such a translation
invariant asymptotic homomorphism. This leads to our main result; that the
homotopy classes of extensions coincide with the homotopy classes of
translation invariant asymptotic homomorphisms.
