is the last odd dimensional one – the only ones are , , , and . The motivic cofiber of method is so far the most efficient method at the prime 2. The class is a map between motivic spheres. The Gheorghe–Wang–Xu theorem identifies the motivic Adams spectral sequence for the cofiber of as the algebraic Novikov spectral sequence for , which allows one to deduce motivic Adams differentials for the cofiber of from purely algebraic data. One can then pullback these motivic Adams differentials to the motivic sphere, and then use the Betti realization functor to push forward them to the classical sphere. Using this method, computes up to the 90-stem. The computation of the homotopy groups of has been reduced to a combinatorial group theory question. identify these homotopy groups as certain quotients of the Brunnian braid groups of . Under this correspondence, every nontrivial element in for may be represented by a Brunnian braid over that is not Brunnian over the disk . For example, the Hopf map corresponds to the Borromean rings. Applications The winding number (corresponding to an integer of can be used to prove the fundamental theorem of algebra, which states that every non-constant complex polynomial has a zero. The fact that implies the Brouwer fixed point theorem that every continuous map from the -dimensional ball to itself has a fixed point. The stable homotopy groups of spheres are important in singularity theory, which studies the structure of singular points of smooth maps or algebraic varieties. Such singularities arise as critical points of smooth maps from to . The geometry near a critical point of such a map can be described by an element of , by considering the way in which a small sphere around the critical point maps into a topological sphere around the critical value. The fact that the third stable homotopy group of spheres is cyclic of order 24, first proved by Vladimir Rokhlin, implies Rokhlin's theorem that the signature of a compact smooth spin 4-manifold is divisible by 16. Stable homotopy groups of spheres are used to describe the group of h-cobordism classes of oriented homotopy -spheres (for , this is the group of smooth structures on -spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by exotic spheres). More precisely, there is an injective map where is the cyclic subgroup represented by homotopy spheres