still elusive. For , the groups are called the unstable homotopy groups of spheres. Hopf fibrations The classical Hopf fibration is a fiber bundle: The general theory of fiber bundles shows that there is a long exact sequence of homotopy groups For this specific bundle, each group homomorphism , induced by the inclusion , maps all of to zero, since the lower-dimensional sphere can be deformed to a point inside the higher-dimensional one . This corresponds to the vanishing of . Thus the long exact sequence breaks into short exact sequences, Since is a suspension of , these sequences are split by the suspension homomorphism , giving isomorphisms Since vanishes for at least 3, the first row shows that and are isomorphic whenever is at least 3, as observed above. The Hopf fibration may be constructed as follows: pairs of complex numbers with form a 3-sphere, and their ratios cover the complex plane plus infinity, a 2-sphere. The Hopf map sends any such pair to its ratio. Similarly, there are generalized Hopf fibrations constructed using pairs of quaternions or octonions instead of complex numbers. Here, too, and are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations. The three fibrations have base space with , for . A fibration does exist for (), but not for () and beyond. Although generalizations of the relations to are often true, they sometimes fail; for example, Thus there can be no fibration the first non-trivial case of the Hopf invariant one problem, because such a fibration would imply that the failed relation is true. Framed cobordism Homotopy groups of spheres are closely related to cobordism classes of manifolds. In 1938 Lev Pontryagin established an isomorphism between the homotopy group and the group of cobordism classes of differentiable -submanifolds of which are "framed", i.e. have a trivialized normal bundle. Every map is homotopic to a differentiable map with a framed -dimensional submanifold. For example, is the cobordism group of framed 0-dimensional submanifolds of , computed by the algebraic sum of their points, corresponding to the degree of maps . The projection of the Hopf fibration represents a generator of which corresponds to the framed 1-dimensional submanifold of defined by the standard embedding with a nonstandard trivialization of the normal 2-plane bundle. Until the advent of more sophisticated algebraic methods in the early 1950s