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球面のホモトピー群は下記
https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.
They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry.
Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

The n-dimensional unit sphere ? called the n-sphere for brevity, and denoted as Sn ? generalizes the familiar circle (S1) and the ordinary sphere (S2).
The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin.
The i-th homotopy group πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn.
This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized.
An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.
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