Editing Homotopy groups of spheres (section)

===Finiteness and torsion===
In 1951, [[Jean-Pierre Serre]] showed that homotopy groups of spheres are all finite except for those of the form {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>)}} or {{math|π<sub>4''n''−1</sub>(''S''<sup>2''n''</sup>)}} (for positive {{mvar|n}}), when the group is the product of the [[infinite cyclic group]] with a finite abelian group.{{sfn|Serre|1951}} In particular the homotopy groups are determined by their {{mvar|p}}-components for all primes {{mvar|p}}. The 2-components are hardest to calculate, and in several ways behave differently from the {{mvar|p}}-components for odd primes.{{cn|date=February 2022}}

In the same paper, Serre found the first place that {{mvar|p}}-torsion occurs in the homotopy groups of {{mvar|n}} dimensional spheres, by showing that {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} has no {{mvar|p}}-[[torsion (algebra)|torsion]] if {{math|''k'' < 2''p'' − 3}}, and has a unique subgroup of order {{mvar|p}} if {{math|''n'' ≥ 3}} and {{math|''k'' {{=}} 2''p'' − 3}}.  The case of 2-dimensional spheres is slightly different: the first {{mvar|p}}-torsion occurs for {{math|''k'' {{=}} 2''p'' − 3 + 1}}. 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 {{mvar|p}} is an odd prime and {{math|''n'' {{=}} 2''i'' + 1}}, then elements of the {{mvar|p}}-[[component (group theory)|component]] of {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} have order at most {{math|''p''<sup>''i''</sup>}}.{{sfn|Cohen|Moore|Neisendorfer|1979}} This is in some sense the best possible result, as these groups are known to have elements of this order for some values of {{mvar|k}}.{{sfn|Ravenel|2003|p=4}} Furthermore, the stable range can be extended in this case: if {{mvar|n}} is odd then the double suspension from {{math|π<sub>''k''</sub>(''S''<sup>''n''</sup>)}} to {{math|π<sub>''k''+2</sub>(''S''<sup>''n''+2</sup>)}} is an isomorphism of {{mvar|p}}-components if {{math|''k'' < ''p''(''n'' + 1) − 3}}, and an epimorphism if equality holds.{{sfn|Serre|1952}} The {{mvar|p}}-torsion of the intermediate group {{math|π<sub>''k''+1</sub>(''S''<sup>''n''+1</sup>)}} can be strictly larger.{{cn|date=February 2022}}

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 {{mvar|p}} in terms of that of odd-dimensional spheres,
:<math>\pi_{2m+k}(S^{2m})(p) =  \pi_{2m+k-1}(S^{2m-1})(p)\oplus \pi_{2m+k}(S^{4m-1})(p)</math>
(where {{math|(''p'')}} means take the {{mvar|p}}-component).{{sfn|Ravenel|2003|p=25}} 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.{{cn|date=February 2022}}

For stable homotopy groups there are more precise results about {{mvar|p}}-torsion. For example,  if {{math|''k'' < 2''p''(''p'' &minus; 1) &minus; 2}} for a prime {{mvar|p}} then the {{mvar|p}}-primary component of the stable homotopy group {{math|π{{su|lh=1|b=''k''|p=S}}}} vanishes unless {{math|''k'' + 1}} is divisible by {{math|2(''p'' &minus; 1)}}, in which case it is cyclic of order {{mvar|p}}.{{sfn|Fuks|2001}}