In
stable homotopy theory, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
''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 isom ...
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
*
Adams-Novikov spectral sequence
*
Framed cobordism
References
*
Algebraic topology
Homotopy theory
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