On the homotopy groups of symmetric spectra

By S. Schwede. 2006

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We construct a natural, tame action of the monoid of injective self-maps of the set of natural numbers on the homotopy groups of a symmetric spectrum. This extra algebraic structure allows a conceptual and uniform understanding of various phenomena related to ‐isomorphisms, semistability and the relationship between naive and true homotopy groups for symmetric spectra. 55P42; 55U35 Symmetric spectra are an easy-to-define and convenient model for the stable homotopy category with a nice smash product. Symmetric ring spectra first showed up under the name “FSP on spheres” in the context of algebraic K ‐theory and topological Hochschild homology. Around 1993, Jeff Smith made the crucial observation that the “FSPs on spheres” are the monoids in a category of “symmetric spectra” with respect to an associative and commutative smash product, and he suspected compatible model category structures so that one obtains as homotopy categories “the” stable homotopy category (for symmetric spectra), the homotopy category of A1 ring spectra (for symmetric ring spectra), respectively the homotopy category of E1 ring spectra (for commutative symmetric ring spectra). The details of various model structures were worked out by Hovey, Shipley and Smith in [3]. Maybe the only tricky point with symmetric spectra is that the stable equivalences can not be defined by looking at stable homotopy groups (defined as the classical sequential colimit of the unstable homotopy groups of the terms in a symmetric spectrum). Formally inverting the ‐isomorphisms, ie, those morphisms which induce isomorphisms of stable homotopy groups, leaves too many homotopy types. Instead, Hovey, Shipley and Smith introduce a strictly larger class of stable equivalences, defined as the morphisms which induce isomorphisms on all cohomology theories. The difference between ‐isomorphisms and stable equivalences has previously confused at least the present author. The precise relationship between the naively defined homotopy groups and the “true” homotopy groups (morphisms from sphere in the stable homotopy category) has largely been mysterious (although Shipley’s detection functor [7, Section 3] sheds considerable light on this). In this paper we advertise and systematically exploit extra algebraic structure on the (classical) homotopy groups of a symmetric spectrum which, in the authors opinion,

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f342528ed23281137873d59188a6c321e955aff

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In Geometry and Topology

  1. Steven V. Sam
  2. Kathryn Hess
  3. Andrew Snowden
  4. John E. Harper
  5. Jan Draisma
  6. David White
  7. Andrew Putman
  8. Rohit Nagpal

In Mathematical Physics

  1. Kathryn Hess
  2. Steven V. Sam
  3. Andrew Snowden
  4. John E. Harper
  5. David White
  6. Andrew Putman

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