In
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, 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 () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. 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 the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
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
.
See also
*
Chromatic homotopy theory In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. ...
*
Adams-Novikov spectral sequence
*
Framed cobordism
References
*
Algebraic topology
Homotopy theory
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