Scanning probe microscopic (SPM) image of optimum CC cinnamon oil NE (NE-4) shown in (Fig. 11) depicted a group of spheres have minimum interparticulate contact and generally optimal properties.
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.
More precisely, there is an injective map\Theta_n / bP_{n+1} \to \pi_n^S / J, \,\ where bPn+1 is the cyclic subgroup represented by homotopy spheres that bound a parallelizable manifold, πnS is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism.
For n ≤ k+1, the groups are called the unstable homotopy groups of spheres.
Homotopy groups of spheres are closely related to cobordism classes of manifolds.
Extended tables of homotopy groups of spheres are given at the end of the article.
Tables of homotopy groups of spheres sometimes omit the "easy" part Im(J) to save space.
(These tables are based on the table of homotopy groups of spheres in).
Tables of homotopy groups of spheres are most conveniently organized by showing πn+k(Sn).
The first few homotopy groups of spheres can be computed using ad hoc variations of the ideas above; beyond this point, most methods for computing homotopy groups of spheres are based on spectral sequences.
