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 *S**n* — generalizes the familiar circle (*S*1) and the ordinary sphere (*S*2). 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* π_{i}(*S**n*) summarizes the different ways in which the *i*-dimensional sphere *S**i* can be mapped continuously into the *n*-dimensional sphere *S**n*. 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.

The problem of determining π_{i}(*S**n*) falls into three regimes, depending on whether *i* is less than, equal to, or greater than *n*. For 0 < *i* < *n*, any mapping from *S**i* to *S**n* is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of *S**i* to a single point of *S**n*. When *i* = *n*, every map from *S**n* to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies π_{n}(*S**n*) with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping. However, the most interesting and surprising results occur when *i* > *n*. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere *S*3 around the usual sphere *S*2 in a non-trivial fashion, and so is not equivalent to a one-point mapping.

The question of computing the homotopy group π_{n+k}(*S**n*) for positive *k* turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups π_{n+k}(*S**n*) are independent of *n* for *n* ≥ *k* + 2. These are called the **stable homotopy groups of spheres** and have been computed for values of *k* up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for *n* < *k* + 2) are more erratic; nevertheless, they have been tabulated for *k* < 20. Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns have been established, yet much remains unknown and unexplained.

Read more about Homotopy Groups Of Spheres: Background, Low-dimensional Examples, History, General Theory, Computational Methods, Applications, Table of Homotopy Groups, Table of Stable Homotopy Groups

### Famous quotes containing the words groups and/or spheres:

“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special *groups* that constitute the community of scientists.”

—Thomas S. Kuhn (b. 1922)

“Perhaps a modern society can remain stable only by eliminating adolescence, by giving its young, from the age of ten, the skills, responsibilities, and rewards of grownups, and opportunities for action in all *spheres* of life. Adolescence should be a time of useful action, while book learning and scholarship should be a preoccupation of adults.”

—Eric Hoffer (1902–1983)