Claim: For each abelian group A and natural number j, there is a space whose jth homotopy group is isomorphic to A and whose other homotopy groups are zero and has the remarkable property that it is a classifying space for cohomology.
Evidence: For each abelian group A and natural number j, there is a space K(A,j) whose jth homotopy group is isomorphic to A and whose other homotopy groups are zero. This space has the remarkable property that it is a classifying space for cohomology: there is a natural element u of H(K(A,j),A), and every cohomology class of degree j on every space X is the pullback of u by some continuous map X → K(A,j).