nLab
sphere spectrum

Contents

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 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

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

The 2-truncation appears for instance in section 3 of