On modular cohomotopy groups

Let p be a prime and let πn(X; ℤ/pr) = [X, Mn(ℤ/pr)] be the set of homotopy classes of based maps from CW-complexes X into the mod pr Moore spaces Mn(ℤ/pr) of degree n, where ℤ/pr denotes the integers mod pr. In this paper we firstly determine the modular cohomotopy groups πn(X; ℤ/pr) up to extensions by classical methods of primary cohomology operations and give conditions for the splitness of the extensions. Secondly we utilize some unstable homotopy theory of Moore spaces to study the modular cohomotopy groups; especially, the group π3(X; ℤ(2)) with dim(X) ≤ 6 is determined.