K-HOMOLOGY, ASYMPTOTIC REPRESENTATIONS, AND UNSUSPENDED E-THEORY

Connes and Higson defined a bivariant homology theory E(A, B) for sepa rable C-algebras. The elements of E(A, B) are taken to be homotopy cl asses of asymptotic momorphisms from SA X K to SB X K. In symbols E(A, B)congruent to[[SA, SB X K]]. If A is K-nuclear then E-theory agrees with Kasparov's bivariant K-theory. We show that, in many cases, one n eed not take suspensions to calculate the E-theory group E(A, B). For many A, we show E(A, B)congruent to[[A, B X K]] for all B. Among the A for which this is true are C-0(X\{pt}) for X with the homotopy type o f a finite, connected CW complex. This gives a concrete realization of K-homology, related to the Brown-Douglas-Fillmore description. For ex ample, K-0(X) arises as asymptotic representations, K-0(X)congruent to [[C-0(X), K]]. Other A for which our isomorphic holds include the nonu nital dimension-drop intervals. In this case, there is no distinction between -homomorphisms and asymptotic morphisms so we have succeeded in classifying all -homomorphisms from a dimension-drop interval to a stable C-algebra. This was subsequently used by Elliott (J. Reine An gew. Math. 443 (1993), 179-219) in the classification of certain C-al gebras. The dimension-drop interval may also be used to describe K(X; Z/n) in terms of paths of asymptotic representations of C-0(X). (C) 1 994 Academic Press, Inc.