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The groups π n+k (S n) with n > k + 1 are called the stable homotopy groups of spheres, and are denoted π S k: they are finite abelian groups for k ≠ 0, and have been computed in numerous cases, although the general pattern is still elusive. [21] For n ≤ k+1, the groups are called the unstable homotopy groups of spheres. [citation needed]
One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions.
Note that this implies for =, this computes the -torsion of the homotopy groups of the sphere spectrum, i.e. the stable homotopy groups of the spheres. Also, because for any CW-complex Y {\displaystyle Y} we can consider the suspension spectrum Σ ∞ Y {\displaystyle \Sigma ^{\infty }Y} , this gives the statement of the previous formulation as ...
Bott periodicity computes the homotopy of the stable unitary group and stable orthogonal group. The Whitehead group of a ring (the first K-group) can be defined in terms of (). Stable homotopy groups of spheres are the stable groups associated with the suspension functor.
The direct sum = of the stable homotopy groups of spheres is a supercommutative graded ring, where multiplication (called composition product) is given by composition of representing maps, and any element of non-zero degree is nilpotent (Nishida 1973).
The stable homotopy category, or homotopy category of (CW) spectra is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.
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 S 0 , i.e., a set of two points. Explicitly, the n th space in the sphere spectrum is the n -dimensional sphere S n , and the structure maps from the suspension of S n to S n +1 are the ...
In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n -sphere, and so every homotopy sphere is necessarily a homology sphere .