Mapping the homology of a group to the K-theory of its C *-algebra

For a CW-complex X and for 0 j 2, we construct natural homomorphisms ₉^X H₉ (X;\, Z) K₉ (X) that are rationally right-inverses of the Chern character. We show that ₉^X is injective for j=0 and j=1. The case j=3 is treated using Z12-coefficients. The study of these maps is motivated by the connection with the Baum-Connes conjecture on the K-theory of group C^*-algebras.