nLab sphere spectrum

Definition

The sphere spectrum is the suspension spectrum of the point.

As a symmetric spectrum, see Schwede 12, example I.2.1

Properties

Homotopy type

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

Truncations

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

As an $E_\infty$-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.

Related concepts

• sphere

• spherical fibration

• equivariant sphere spectrum

• 2-periodic sphere spectrum

• Cohomotopy

  • stable Cohomotopy

  • twisted Cohomotopy

  • equivariant Cohomotopy

  • differential Cohomotopy

• superalgebra

• The Music of the Spheres

References

Lecture notes include

• John Rognes, The sphere spectrum (4 pages) (2004) (pdf)

• Stefan Schwede, Example I.2.1 in Symmetric spectra, 2012 (pdf)

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

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

• Katja Hutschenreuter, On rigidity of the ring spectra $P_m \mathbb{S}_{(p)}$ and $k o$, (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)