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 classes into an abelian group whose identity element is the class of any constant map, i.e. a one that maps all of Si to a single point of Sn. The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n:
For 0 < i < n, any map from Si to Sn is homotopic (i.e., continuously deformable) to a constant map, so πi(Sn) is the trivial group. This is a consequence of the cellular approximation theorem. When i = n, every map from Sn to itself can be assigned a degree that intuitively measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group πn(Sn) with the group of integers under addition. For i > n, the groups πi(Sn) depend strongly on i and n. The historically first interesting example is the group π3(S2) which is generated by the Hopf fibration. The problem of computing the groups πn+k(Sn) for positive k is very difficult and of central importance to the field of algebraic topology, and as such has motivated the development of many fundamental tools and techniques. One early important result is the Freudenthal suspension theorem, which states that πn+k(Sn) is independent of n for n ≥ k + 2. These groups, known as the stable homotopy groups of spheres, can be studied using the tools of stable homotopy theory and are completely known for values of k up to 90.. The groups in the unstable range n < k + 2 are less accessible, but have been tabulated at least for all k < 19.
Background The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed. Algebraic topology provides the larger context, itself built on topology and abstract algebra, with homotopy groups as a basic example.
n-sphere An ordinary sphere in three-dimensional space—the surface, not the solid ball—is just one example of what a sphere means in topology. Geometry defines a sphere rigidly, as a shape. Here are some alternatives.
Implicit surface: x20 + x21 + x22 = 1 This is the set of points in 3-dimensional Euclidean space found exactly one unit away from the origin. It is called the 2-sphere, S2, for reasons given below. The same idea applies for any dimension n; the equation x20 + x21 + ⋯ + x2n = 1 produces the n-sphere as a geometric object in (n + 1)-dimensional space. For example, the 1-sphere S1 is a circle. Disk with collapsed rim: written in topology as D2/S1 This construction moves from geometry to pure topology. The disk D2 is the region contained by a circle, described by the inequality x20 + x21 ≤ 1, and its rim (or "boundary") is the circle S1, described by the equality x20 + x21 = 1. If a balloon is punctured and spread flat it produces a disk; this construction repairs the puncture, like pulling a drawstring. The slash, pronounced "modulo", means to take the topological space on the left (the disk) and in it join together as one all the points on the right (the circle). The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere. Generalized, Dn/Sn−1 produces Sn. For example, D1 is a line segment, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an n-dimensional disk is glued to a point, producing a CW complex. Suspension of equator: written in topology as ΣS1 This construction, though simple, is of great theoretical importance. Take the circle S1 to be the equator, and sweep each point on it to one point above (the North Pole), producing the northern hemisphere, and to one point below (the South Pole), producing the southern hemisphere. For each positive integer n, the n-sphere x20 + x21 + ⋯ + x2n = 1 has as equator the (n − 1)-sphere x20 + x21 + ⋯ + x2n−1 = 1, and the suspension ΣSn−1 produces Sn. Some theory requires selecting a fixed point on the sphere, calling the pair (sphere, point) a pointed sphere. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience. For spheres constructed as a repeated suspension, the point (1, 0, 0, ..., 0), which is on the equator of all the levels of suspension, works well; for the disk with collapsed rim, the point resulting from the collapse of the rim is another obvious choice.
Homotopy group
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