Mapping spaces and homology isomorphisms

Let Map(K, X) denote the space of pointed continuous maps from a finite cell complex K to a space X. Let E* be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on K and X, Map(K, X) will send an E*-isomorphism in either variable to a map that is monic in E* homology. Interesting examples arise by letting E* be K-theory, the finite complex K be a sphere, and the map in the X variable be an exotic unstable Adams map between Moore spaces.