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 -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th homotopy group summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . 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 falls into three regimes, depending on whether is less than, equal to, or greater than : For , any mapping from to is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of to a single point of . Therefore the homotopy group is the trivial group. When , every map from to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group 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. The most interesting and surprising results occur when . The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere around the usual sphere in a non-trivial fashion, and so is not equivalent to a one-point mapping. The question of computing the homotopy group for positive 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