During the past six years relationships between spaces of equivariant self-maps of spheres and ordinary stable homotopy theory have been obtained by several different authors using somewhat different techniques [1, 2, 4, 6, 7, 9, 16]. Closer examination shows that such results fall into two classes: (i) Results relating equivariant stable homotopy theory to ordinary stable homotopy as in work of G. Segal, C. Kosniowski, T. tom Dieck; and H. Hauschild [4, 6, 7, 9, 16]. (ii) Results relating spaces of unpointed equivariant self-equivalences to homotopy theory as in the work of J. C. Becker and the author [1, 2].

This paper has two objectives. The first is to provide a natural relationship between the above classes of results, and the second is to apply this relationship to a question left open in [1, 2] - describing the composition product on the spaces FG studied in those papers and describing (in principle, at least) how the Pontrjagin ring structure on H⋆(FG) may be calculated. To be precise, we shall no use the homotopy equivalences constructed in [1, 2], but instead we shall replace them with more convenient maps defined in the same spirit.

It turns out that the systems studied in (i) and (ii) more or less fit together as halves of a larger object; this is developed in Section 1.