stable homotopy theory In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''⁰, i.e., a set of two points. Explicitly, the ''n''th space in the sphere spectrum is the ''n''-dimensional

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

''S''^''n'', and the structure maps from the suspension of ''S''^''n'' to ''S''^''n''+1 are the canonical

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

s. 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

S_

References

* Algebraic topology Spectra (topology) {{topology-stub

See also

References