Claim: 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 and has the remarkable property that it is a classifying space for cohomology, a general term for a sequence of abelian groups associated to a topological space.
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). More precisely, pulling back the class u gives a bijection for every space X with the homotopy type of a CW complex. Here [X,Y] denotes the set of homotopy classes of continuous maps from X to Y. In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.