each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6-, 12-, 56- or 61-spheres. Some higher-dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an unsolved problem. Classification The monoid of smooth structures on n-spheres is the collection of oriented smooth n-manifolds which are homeomorphic to the n-sphere, taken up to orientation-preserving diffeomorphism. The monoid operation is the connected sum. Provided , this monoid is a group and is isomorphic to the group of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian. In dimension 4 almost nothing is known about the monoid of smooth spheres, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite; see the section on Gluck twists. All homotopy n-spheres are homeomorphic to the n-sphere by the generalized Poincaré conjecture, proved by Stephen Smale in dimensions bigger than 4, Michael Freedman in dimension 4, and Grigori Perelman in dimension 3. In dimension 3, Edwin E. Moise proved that every topological manifold has an essentially unique smooth structure (see Moise's theorem), so the monoid of smooth structures on the 3-sphere is trivial. Parallelizable manifolds The group has a cyclic subgroup represented by n-spheres that bound parallelizable manifolds. The structures of and the quotient are described separately in the paper , which was influential in the development of surgery theory. In fact, these calculations can be formulated in a modern language in terms of the surgery exact sequence as indicated here. The group is a cyclic group, and is trivial or order 2 except in case , in which case it can be large, with its order related to the Bernoulli numbers. It is trivial if n is even. If n is 1 mod 4 it has order 1 or 2; in particular it has order 1 if n is 1, 5, 13, 29, or 61, and proved that it has order 2 if mod 4 is not of the form . It follows from the now almost completely resolved Kervaire invariant problem that it has order 2 for all n bigger than 126; the case is still open. The order of for is where B is the numerator of , and is a Bernoulli number. (The formula in the topological literature differs slightly because topologists use a different convention