Editing Homotopy groups of spheres (section)

===Ring structure===

The [[Direct sum of groups|direct sum]]
:<math>\pi_{\ast}^S=\bigoplus_{k\ge 0}\pi_k^S</math>
of the stable homotopy groups of spheres is a [[supercommutative ring|supercommutative]] [[graded ring]], where multiplication is given by composition of representing maps, and any element of non-zero degree is [[nilpotent]];{{sfn|Nishida|1973}} the [[nilpotence theorem]] on [[complex cobordism]] implies Nishida's theorem.{{cn|date=February 2022}}

Example: If {{mvar|η}} is the generator of {{math|π{{su|lh=1|b=1|p=S}}}} (of order 2),
then {{math|''η''<sup>2</sup>}} is nonzero and generates {{math|π{{su|lh=1|b=2|p=S}}}}, and {{math|''η''<sup>3</sup>}} is nonzero and 12 times a generator of {{math|π{{su|lh=1|b=3|p=S}}}}, while {{math|''η''<sup>4</sup>}} is zero because the group  {{math|π{{su|lh=1|b=4|p=S}}}} is trivial.{{cn|date=February 2022}}

If {{mvar|f}} and {{mvar|g}} and {{mvar|h}} are elements of {{math|π{{su|lh=1|b=*|p=S}}}} with {{math|''f'' ''g'' {{=}} 0}} and {{math|''g''⋅''h'' {{=}} 0}}, there is a [[Toda bracket]] {{math|{{angle bracket|''f'', ''g'', ''h''}}}} of these elements.{{sfn|Toda|1962}} The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of products of certain other elements. [[Hiroshi Toda]] used the composition product and Toda brackets to label many of the elements of homotopy groups. There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of [[Massey product]]s in [[cohomology]].{{cn|date=February 2022}}
Every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.{{sfn|Cohen|1968}}