that the homotopy groups are independent of for . These are called the stable homotopy groups of spheres and have been computed for values of 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 ) are more erratic; nevertheless, they have been tabulated for . 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. 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. -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: 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, , for reasons given below. The same idea applies for any dimension ; the equation produces the -sphere as a geometric object in ()-dimensional space. For example, the 1-sphere is a circle. Disk with collapsed rim: written in topology as This construction moves from geometry to pure topology. The disk is the region contained by a circle, described by the inequality , and its rim (or "boundary") is the circle , described by the equality . 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, produces . For example, is a line segment, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an -dimensional disk is glued to a point, producing a CW complex. Suspension of equator: written in topology as This construction, though simple, is of great theoretical importance. Take the circle to be the equator, and sweep each