sphere spectrum



The sphere spectrum is the suspension spectrum of the point.

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


Homotopy type

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


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.


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