In mathematics, the homotopy groups of spheres are the groups

π_k(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 the suspension theorem of Hans Freudenthal, the group depends only on

k − n = i

for k large enough; and is a finite group (abelian), denoted by

π_i^S.

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 .

External link

• Stable homotopy groups of spheres, Allen Hatcher