Editing Homotopy groups of spheres (section)

===The J-homomorphism===
{{main|J-homomorphism}}

An important subgroup of {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}}, for {{math|''k'' ≥ 2}}, is the image of the [[J-homomorphism]]
{{math|''J'' : π<sub>''k''</sub>(SO(''n'')) → π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}}, where {{math|SO(''n'')}} denotes the [[special orthogonal group]].{{sfn|Adams|1966}} In the stable range {{math|''n'' ≥ ''k'' + 2}}, the homotopy groups  {{math|π<sub>''k''</sub>(SO(''n''))}} only depend on {{math|''k'' (mod 8)}}. This period 8 pattern is known as [[Bott periodicity]], and it is reflected in the stable homotopy groups of spheres via the image of the {{mvar|J}}-homomorphism which is:
* a cyclic group of order 2 if {{mvar|k}} is [[congruence relation|congruent]] to 0 or 1 [[modular arithmetic|modulo]] 8;
* trivial if {{mvar|k}} is congruent to 2, 4, 5, or 6 modulo 8; and
* a cyclic group of order equal to the denominator of {{math|{{sfrac|''B''<sub>2''m''</sub>|4''m''}}}}, where {{math|''B''<sub>2''m''</sub>}} is a [[Bernoulli number]], if {{math|''k'' {{=}} 4''m'' − 1 ≡ 3 (mod 4)}}.
This last case accounts for the elements of unusually large finite order in {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} for such values of {{mvar|k}}. For example, the stable groups {{math|π<sub>''n''+11</sub>(''S''<sup>''n''</sup>)}} have a cyclic subgroup of order 504, the denominator of {{math|{{sfrac|''B''<sub>6</sub>|12}} {{=}} {{sfrac|1|504}}}}.{{cn|date=February 2022}}

The stable homotopy groups of spheres are the direct sum of the image of the {{mvar|J}}-homomorphism, and the kernel of the Adams {{mvar|e}}-invariant, a homomorphism from these groups to {{math|<math>\mathbb{Q} / \mathbb{Z}</math>}}. Roughly speaking, the image of the {{mvar|J}}-homomorphism is the subgroup of "well understood" or "easy" elements of the stable homotopy groups. These well understood elements account for most elements of the stable homotopy groups of spheres in small dimensions. The quotient of {{math|π{{su|lh=1|b=''n''|p=S}}}} by the image of the {{mvar|J}}-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres {{harv|Adams|1966}}. (Adams also introduced certain order 2 elements {{math|μ<sub>''n''</sub>}} of {{math|π{{su|lh=1|b=''n''|p=S}}}} for {{math|''n'' &equiv; 1 or 2 (mod 8)}}, and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part {{math|im(''J'')}} to save space.{{cn|date=February 2022}}