nLab
sphere spectrum

Contents

Idea

The sphere spectrum is the simplest nontrivial spectrum.

Definition

The sphere spectrum is the suspension spectrum of the point.

Properties

Homotopy type

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.

Truncations

homotopy colimits of simplicial diagrams of kk-truncated connective spectra for kk \in \mathbb{N} are modules over the kk-truncation τ k𝕊\tau_{\leq k}\mathbb{S} of the sphere spectrum.

As an E E_\infty-ring

The sphere spectrum is naturally an E-∞ ring and in fact is the initial object in the (∞,1)-category of ring spectra. It is the higher version of the ring \mathbb{Z} of integers.

References

Lecture notes include

The Postnikov tower of (localizations of) the sphere spectrum is discussed in

  • Karol Szumiło, Postnikov tower of the sphere spectrum, Master thesis 2009 (web)

  • Katja Hutschenreuter, On rigidity of the ring spectra P m𝕊 (p)P_m \mathbb{S}_{(p)} and kok o, (2012) (pdf)

Specifically the 1-truncation of the sphere spectrum (the free abelian 2-group on a single element) is discussed in

  • Niles Johnson, Angélica M. Osorno, Modeling Stable One-Types, Theory and Applications of Categories, Vol. 26, 2012, No. 20, pp (TAC, arXiv:1201.2686)

The 2-truncation appears for instance in section 3 of

  • Norio Iwase. L-S categories of simply-connected compact simple Lie groups of low rank (arXiv:math/0202122)
Revised on July 14, 2013 20:33:29 by Urs Schreiber (82.113.106.95)