Sphere spectrum

In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum of S⁰, i.e., a set of two points. Explicitly, the nth space in the sphere spectrum is the n-dimensional sphere Sⁿ, and the structure maps from the suspension of Sⁿ to Sⁿ⁺¹ are the canonical homeomorphisms. The k-th homotopy group of a sphere spectrum is the k-th stable homotopy group of spheres. The localization of the sphere spectrum at a prime number p is called the local sphere at p and is denoted by [math]\displaystyle{ S_{(p)} }[/math].

See also

• Chromatic homotopy theory
• Adams-Novikov spectral sequence
• Framed cobordism

References

• Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, 1974