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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.
These homotopy classes form a group, called the n-th homotopy group, (), of the given space X with base point. Topological spaces with differing homotopy groups are never homeomorphic , but topological spaces that are not homeomorphic can have the same homotopy groups.
Coefficient ring: The coefficient groups π n (S) are the stable homotopy groups of spheres, which are notoriously hard to compute or understand for n > 0. (For n < 0 they vanish, and for n = 0 the group is Z.) Stable homotopy is closely related to cobordism of framed manifolds (manifolds with a trivialization of the normal bundle).
Hiroshi Toda used the composition product and Toda brackets to label many of the elements of homotopy groups. Cohen (1968) showed that every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.
The Barratt–Priddy theorem is sometimes colloquially rephrased as saying that "the K-groups of F 1 are the stable homotopy groups of spheres". This is not a meaningful mathematical statement, but a metaphor expressing an analogy with algebraic K-theory.
These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called stable groups, though this term normally means something quite different in model theory. Certain examples of stable groups are easier to study than "unstable" groups, the groups occurring in the limit.
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 ...
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