Some examples of nontrivial homotopy groups of modules

The concept of the homotopy theory of modules was discovered by Peter Hilton as a result of his trip in 1955 to Warsaw, Poland, to work with Karol Borsuk, and to Zurich, Switzerland, to work with Beno Eckmann. The idea was to produce an analog of homotopy theory in topology. Yet, unlike homotopy theory in topology, there are two homotopy theories of modules, the injective theory, π ¯ n ( A , B ) , and the projective theory, π ¯ n ( A , B ) . They are dual, but not isomorphic. In this paper, we deliver and carry out the precise calculation of the first known nontrivial examples of absolute homotopy groups of modules, namely, π ¯ n ( ℚ / ℤ , ℚ / ℤ ) , π ¯ n ( ℤ , ℚ / ℤ ) , and π ¯ n ( ℤ , ℤ ) , where ℚ / ℤ and ℤ are regarded as ℤ C k -modules with trivial action. One interesting phenomenon of the results is the periodicity of these homotopy groups, just as for the Ext groups.