On the homotopy groups of symmetric spectra

Abstract

The symmetric spectra introduced by Hovey, Shipley and Smith are a convenient model for the stable homotopy category with a nice associative and commutative smash product on the point set level and a compatible Quillen closed model structure. About the only disadvantage of this model is that the stable equivalences cannot be defined by inverting those morphisms which induce isomorphisms on homotopy groups, because this would leave too many homotopy types. In this sense the naively defined homotopy groups are often `wrong`, and then their precise relationship to the `true` homotopy groups (i.e., morphisms in the stable homotopy category from sphere spectra) appears mysterious. In this paper I discuss and exploit extra algebraic structure on the naively defined homotopy groups of symmetric spectra, namely a special kind of action of the monoid of injective self maps of the natural numbers. This extra structure clarifies several issues about homotopy groups and stable equivalences and explains why the naive homotopy groups are not so wrong after all. For example, the monoid action allows a simple characterization of semistable symmetric spectra.