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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. [22] For n ≤ k+1, the groups are called the unstable homotopy groups of spheres. [citation needed]
It is open whether non-trivial smooth homotopy spheres exist in dimension 4. Homotopy spheres form an abelian group known as Kervaire–Milnor group . Its composition is the connected sum and its neutral element is the the sphere, while inversion is given by opposite orientation .
In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnor [1] pg 14 was trying to understand the structure of ()-connected manifolds of dimension (since -connected -manifolds are homeomorphic to spheres, this is the first non-trivial case after) and found an example of a space which is homotopy equivalent to a sphere, but was not explicitly diffeomorphic.
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 spectral sequence has the advantage that the input is previously calculated homotopy groups. It was used by Oda (1977) to calculate the first 31 stable homotopy groups of spheres. For arbitrary primes one uses some fibrations found by Toda (1962):
A way out of these difficulties has been found by defining higher homotopy groupoids of filtered spaces and of n-cubes of spaces. These are related to relative homotopy groups and to n-adic homotopy groups respectively. A higher homotopy van Kampen theorem then enables one to derive some new information on homotopy groups and even on homotopy ...
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.
A constructive proof of this theorem can be found here, [6] ... and applications for calculating the homotopy groups of spheres. ...