  We show that the E-theory of Connes and Higson can be formulated in terms of
C*-extensions in a way quite similar to the way in which the KK-theory of
Kasparov can. The essential difference is that the role played by split
extensions should be taken by asymptotically split extensions. We call an
extension of a C*-algebra $A$ by a stable C*-algebra $B$ asymptotically split
if there exists an asymptotic homomorphism consisting of right inverses for the
quotient map. An extension is called semi-invertible if it can be made
asymptotically split by adding another extension to it. Our main result is that
there exists a one-to-one correspondence between asymptotic homomorphisms from
$SA$ to $B$ and homotopy classes of semi-invertible extensions of $S^2A$ by
$B$.
