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 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.