Homotopy groups of spheres Guide, Meaning , Facts, Information and Description
In
mathematics, the
homotopy groups of spheres are the groups
- πk(Sn)
in
algebraic topology, more specifically
homotopy theory, where π
k(.) for
k ≥ 1 denotes the
homotopy group and
Sn the
n-sphere. From a geometric point of view these are fundamental invariants; on the other hand there is ample evidence from the algebraic aspect that they involve substantial complexity of structure, and intense study from around 1950 has not completely elucidated that.
The case k < n is trite: for example if n ≥ 2 the n-sphere is simply connected. The case k = n is always infinite cyclic, with mappings classified by their degree. It is the case k > n that is of real importance. Here, according to a suspension theorem of Freudenthal, the group depends only on
- k − n = i
for
k large enough; and is a finite group (abelian), denoted by
- πiS.
These are the
stable homotopy groups of spheres. They have been computed in numerous cases, but the general pattern is still elusive. There are
ad hoc methods for the cases of
i small; a systematic tool is the J-homomorphism.
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