Homotopy groups with coefficients

This paper has two goals. It is an expository paper on homotopy groups with coefficients in an abelian group and it contains new results which correct old errors and omissions in low dimensions. The homotopy groups with coefficients are functors on the homotopy category of pointed spaces. They satisfy a universal coefficient theorem, give long exact sequences when applied to fibrations, and have Hurewicz maps into homology groups with coefficients. When the coefficient group is finitely generated, homotopy group functors are corepresentable as homotopy classes of maps out of a Peterson space. A Peterson space is a space with exactly one nonzero integral reduced cohomology group which is the coefficient group. Kan and Whitehead (1961) showed that Peterson spaces do not exist for the coefficient group of the rational numbers. Of course, rational homotopy groups can be defined by tensoring the classical homotopy groups with the rationals. For many nonfinitely generated groups, homotopy groups can be defined by a direct limit of homotopy groups with coefficients from finitely generated subgroups. This depends on having sufficient functoriality in the Peterson spaces of the coefficient subgroups. Coefficient group functoriality fails in the presence of 2-torsion.