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.
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.