Abelian Group Theory and p-Maps

Abstract

The aim of this paper is to introduce to abelian group theorists a type of homomorphism, which is useful in topology, with the hope that it will be a useful tool in the study of abelian groups. These homomorphisms, or p-maps, arise in the following way: If f: X → Y is a continuous map of simply connected topological spaces such that f induces an isomorphism H_n(X,ℤ(p)) → H_n(Y,ℤ(p)), all n, of the homology groups modulo p and if f induces an isomorphism π_m(X) → π_m(Y) of the homo topy groups for all m < k, some k, then f induces a p-map on the k-th homotopy groups π_k(X) → π_k(Y). An important question arises from this setting: under what conditions is a p-map necessarily an isomorphism?