Homotopy groups of spheres

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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. 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) captures the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. It does not distinguish between mappings that can be continuously deformed into one another; its elements are therefore equivalence classes of maps under homotopy. An addition operation makes this set of equivalence...

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