Stable Homotopy theory
The sphere spectrum is the suspension spectrum of the point.
As a symmetric spectrum, see Schwede 12, example I.2.1
The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
homotopy colimits of simplicial diagrams of -truncated connective spectra for are modules over the -truncation of the sphere spectrum.
As an -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 of integers.
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 and , (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