(Serre) the Pontrjagin isomorphism was the main tool for computing the homotopy groups of spheres. In 1954 the Pontrjagin isomorphism was generalized by René Thom to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as homotopy groups of spaces and spectra. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups. Finiteness and torsion In 1951, Jean-Pierre Serre showed that homotopy groups of spheres are all finite except for those of the form or (for positive ), when the group is the product of the infinite cyclic group with a finite abelian group. In particular the homotopy groups are determined by their -components for all primes . The 2-components are hardest to calculate, and in several ways behave differently from the -components for odd primes. In the same paper, Serre found the first place that -torsion occurs in the homotopy groups of dimensional spheres, by showing that has no -torsion if , and has a unique subgroup of order if and . The case of 2-dimensional spheres is slightly different: the first -torsion occurs for . In the case of odd torsion there are more precise results; in this case there is a big difference between odd and even dimensional spheres. If is an odd prime and , then elements of the -component of have order at most . This is in some sense the best possible result, as these groups are known to have elements of this order for some values of . Furthermore, the stable range can be extended in this case: if is odd then the double suspension from to is an isomorphism of -components if , and an epimorphism if equality holds. The -torsion of the intermediate group can be strictly larger. The results above about odd torsion only hold for odd-dimensional spheres: for even-dimensional spheres, the James fibration gives the torsion at odd primes in terms of that of odd-dimensional spheres, (where means take the -component). This exact sequence is similar to the ones coming from the Hopf fibration; the difference is that it works for all even-dimensional spheres, albeit at the expense of ignoring 2-torsion. Combining the results for odd and even dimensional spheres shows that much of the odd torsion of unstable homotopy groups is determined by the odd torsion of the stable homotopy groups. For stable homotopy groups there are