|| 16256 || 2 || 16 || 16 || 523264 || 24 |- ! | 1 || 1 || 1 || 1 || 1 || 1 || 28 || 1 || 2 || 1 || 992 || 1 || 1 || 1 || 8128 || 1 || 2 || 1 || 261632 || 1 |- ! | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24 |- ! | 1 || 2 || 1 || 1 || 1 || 2 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2×2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24 |- !index | – || 2 || – || – || – || 2 || – || – || – || – || – || – || – || 2 || – || – || – || – || – || – |} Note that for dim , then are , , , and . Further entries in this table can be computed from the information above together with the table of stable homotopy groups of spheres. By computations of stable homotopy groups of spheres, proves that the sphere has a unique smooth structure, and that it is the last odd-dimensional sphere with this property – the only ones are , , , and . Explicit examples of exotic spheres Milnor's construction One of the first examples of an exotic sphere found by was the following. Let be the unit ball in , and let be its boundary—a 3-sphere which we identify with the group of unit quaternions. Now take two copies of , each with boundary , and glue them together by identifying in the first boundary with in the second boundary. The resulting manifold has a natural smooth structure and is homeomorphic to , but is not diffeomorphic to . Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.