# nLab sphere spectrum

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

## 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 $k$-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.

## 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 $P_m \mathbb{S}_{(p)}$ 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)

Revised on February 5, 2016 06:56:21 by Urs Schreiber (86.187.77.97)