  Suppose that A is a separable C*-algebra and that G_* is a (graded) subgroup
of K_*(A). Then there is a natural short exact sequence 0 \to G_* \to K_*(A)
\to K_*(A)/G_* \to 0.
  In this note we demonstrate how to geometrically realize this sequence at the
level of C*-algebras. As a result, we KK-theoretically decompose A as 0 \to
A\otimes \Cal K \to A_f \to SA_t \to 0 where K_*(A_t) is the torsion subgroup
of K_*(A) and K_*(A_f) is its torsionfree quotient. Then we further decompose
A_t : it is KK-equivalent to \oplus_p A_p where K_*(A_p) is the p-primary
subgroup of the torsion subgroup of K_*(A). We then apply this realization to
study the Kasparov group K^*(A) and related objects.
