Editing Homotopy groups of spheres (section)

==General theory==

As noted already, when {{mvar|i}} is less than {{mvar|n}}, {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>) {{=}} 0}}, the [[trivial group]]. The reason is that a continuous mapping from an {{mvar|i}}-sphere to an {{mvar|n}}-sphere with {{math|''i'' < ''n''}} can always be deformed so that it is not [[surjective]]. Consequently, its image is contained in {{math|''S''<sup>''n''</sup>}} with a point removed; this is a [[contractible space]], and any mapping to such a space can be deformed into a one-point mapping.{{sfn|Hatcher|2002|p=349}}

When {{math|''i'' {{=}} ''n''}}, {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} Z}}, the [[infinite cyclic group]], generated by the identity map from the {{mvar|n}}-sphere to itself. It follows from the definition of homotopy groups that the identity map and its multiples are elements of {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>)}}. That these are the only elements can be shown using the [[Freudenthal suspension theorem]], which relates the homotopy groups of a space and its suspension. In the case of spheres, the suspension of an {{mvar|n}}-sphere is an {{math|(''n''+1)}}-sphere, and the suspension theorem states that there is a group homomorphism {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>) → π<sub>''i''+1</sub>(''S''<sup>''n''+1</sup>)}} which is an isomorphism for all {{math|''i'' < 2''n''-1}} and is surjective for {{math|''i'' {{=}} 2''n''-1}}. This implies that there is a sequence of group homomorphisms
:<math>\pi_1(S^1) \to \pi_2(S^2) \to \pi_3(S^3) \to \cdots</math>
in which the first homomorphism is a surjection and the rest are isomorphisms. As noted already, {{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z}}, and {{math|π<sub>2</sub>(''S''<sup>2</sup>)}} contains a copy of {{math|Z}} generated by the identity map, so the fact that there is a surjective homomorphism from {{math|π<sub>1</sub>(''S''<sup>1</sup>)}} to {{math|π<sub>2</sub>(''S''<sup>2</sup>)}} implies that {{math|π<sub>2</sub>(''S''<sup>2</sup>) {{=}} Z}}. The rest of the homomorphisms in the sequence are isomorphisms, so {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} Z}} for all {{mvar|n}}.{{sfn|Hatcher|2002|p=361}}

The homology groups {{math|''H''<sub>''i''</sub>(''S''<sup>''n''</sup>)}}, with {{math|''i'' > ''n''}}, are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general.{{cn|date=January 2025}} This is the case that is of real importance: the higher homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>)}}, for {{math|''i'' > ''n''}}, are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics.{{cn|date=February 2022}}

===Table===
The following table gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either a) the [[trivial group]] 0, the infinite cyclic group {{math|Z}}, b) the finite [[cyclic group]]s of order {{mvar|n}} (written as {{math|Z<sub>''n''</sub>}}), or c) the [[direct product of groups|direct products]] of such groups (written, for example, as {{math|Z<sub>24</sub>×Z<sub>3</sub>}} or {{math|1=Z{{su|lh=1|b=2|p=2}} = Z<sub>2</sub>×Z<sub>2</sub>}}). Extended tables of homotopy groups of spheres are given [[#Table of homotopy groups|at the end of the article]].

{| class="wikitable" style="text-align:center"
|-
!
!style="width:4em"| π<sub>1</sub>
!style="width:4em"| π<sub>2</sub>
!style="width:4em"| π<sub>3</sub>
!style="width:4em"| π<sub>4</sub>
!style="width:4em"| π<sub>5</sub>
!style="width:4em"| π<sub>6</sub>
!style="width:4em"| π<sub>7</sub>
!style="width:4em"| π<sub>8</sub>
!style="width:4em"| π<sub>9</sub>
!style="width:4em"| π<sub>10</sub>
!style="width:4em"| π<sub>11</sub>
!style="width:4em"| π<sub>12</sub>
!style="width:4em"| π<sub>13</sub>
!style="width:4em"| π<sub>14</sub>
!style="width:4em"| π<sub>15</sub>
|-

!style="height: 2.5em"| ''S''<sup>1</sup>
|style="background:white"| Z
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|style="background:white"| 0
|-
!style="height: 2.5em"| ''S''<sup>2</sup>
|style="background:#FFDDDD; border-top: solid black 2px"| 0
|style="background:#DDDDFF; border-top: solid black 2px; border-right: solid black 2px"| Z
|style="background:#FFFFCC"| Z
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>12</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>3</sub>
|style="background:white"| Z<sub>15</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z<sub>12</sub>×Z<sub>2</sub>
|style="background:white"| Z<sub>84</sub>×Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|-
!style="height: 2.5em"| ''S''<sup>3</sup>
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF; border-top: solid black 2px"| Z
|style="background:#DDFFDD; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>12</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>3</sub>
|style="background:white"| Z<sub>15</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z<sub>12</sub>×Z<sub>2</sub>
|style="background:white"| Z<sub>84</sub>×Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|-
!style="height: 2.5em"| ''S''<sup>4</sup>
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD; border-top: solid black 2px"| Z<sub>2</sub>
|style="background:#FFDDDD; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>2</sub>
|style="background:#FFFFCC"| Z×Z<sub>12</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z{{su|lh=1|b=2|p=2}}
|style="background:white"| Z<sub>24</sub>×Z<sub>3</sub>
|style="background:white"| Z<sub>15</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|style="background:white"| {{su|p=Z<sub>120</sub>×|b=Z<sub>12</sub>×Z<sub>2</sub>}}
|style="background:white"| Z<sub>84</sub>×Z{{su|lh=1|b=2|p=5}}
|-
!style="height: 2.5em"| ''S''<sup>5</sup>
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD; border-top: solid black 2px"| Z<sub>2</sub>
|style="background:#DDDDFF; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>24</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>30</sub>
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|style="background:white"| Z<sub>72</sub>×Z<sub>2</sub>
|-
!style="height: 2.5em"| ''S''<sup>6</sup>
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD"| Z<sub>2</sub>
|style="background:#DDDDFF; border-top: solid black 2px"| Z<sub>24</sub>
|style="background:#DDFFDD; border-top: solid black 2px; border-right: solid black 2px"| 0
|style="background:#FFFFCC"| Z
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>60</sub>
|style="background:white"| Z<sub>24</sub>×Z<sub>2</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|-
!style="height: 2.5em"| ''S''<sup>7</sup>
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD"| Z<sub>2</sub>
|style="background:#DDDDFF"| Z<sub>24</sub>
|style="background:#DDFFDD; border-top: solid black 2px"| 0
|style="background:#FFDDDD; border-top: solid black 2px; border-right: solid black 2px"| 0
|style="background:white"| Z<sub>2</sub>
|style="background:white"| Z<sub>120</sub>
|style="background:white"| Z{{su|lh=1|b=2|p=3}}
|-
!style="height: 2.5em"| ''S''<sup>8</sup>
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| 0
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD"| 0
|style="background:#DDDDFF"| Z
|style="background:#DDFFDD"| Z<sub>2</sub>
|style="background:#FFDDDD"| Z<sub>2</sub>
|style="background:#DDDDFF"| Z<sub>24</sub>
|style="background:#DDFFDD"| 0
|style="background:#FFDDDD; border-top: solid black 2px"| 0
|style="background:#DDDDFF; border-top: solid black 2px; border-right: solid black 2px"| Z<sub>2</sub>
|style="background:#FFFFCC; border-bottom: solid black 2px"| Z×Z<sub>120</sub>
|}

The first row of this table is straightforward. The homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>1</sup>)}} of the 1-sphere are trivial for {{math|''i'' > 1}}, because the universal [[covering space]], <math>\mathbb{R}</math>, which has the same higher homotopy groups, is contractible.{{sfn|Hatcher|2002|p=342}}

Beyond the first row, the higher homotopy groups ({{math|''i'' > ''n''}}) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle.
* The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring).
* Most of the groups are finite.  The only infinite groups are either on the main diagonal or immediately above the jagged line (highlighted in yellow).
* The second and third rows of the table are the same starting in the third column (i.e., {{math|π<sub>''i''</sub>(''S<sup>2</sup>'') {{=}} π<sub>''i''</sub>(''S<sup>3</sup>'')}} for {{math| ''i'' ≥ 3}}).  This isomorphism is induced by the Hopf fibration {{math| ''S''<sup>3</sup> → ''S''<sup>2</sup>}}.
* For {{math|1=''n'' = 2, 3, 4, 5}} and {{math|''i'' ≥ ''n''}} the homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>)}} do not vanish. However, {{math|1=π<sub>''n''+4</sub>(''S''<sup>''n''</sup>) = 0}} for {{math|''n'' ≥ 6}}.
These patterns follow from many different theoretical results.{{cn|date=February 2022}}

===Stable and unstable groups===
The fact that the groups below the jagged line in the table above are constant along the diagonals is explained by the [[Freudenthal suspension theorem|suspension theorem]] of [[Hans Freudenthal]], which implies that the suspension homomorphism from  {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} to  {{math|π<sub>''n''+''k''+1</sub>(''S''<sup>''n''+1</sup>)}} is an isomorphism for {{math|''n'' &gt; ''k'' + 1}}.  The groups {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} with {{math|''n'' > ''k'' + 1}} are called the ''stable homotopy groups of spheres'', and are denoted {{math|π{{su|lh=1|b=''k''|p=''S''}}}}: they are finite abelian groups for {{math|''k'' ≠ 0}}, and have been computed in numerous cases, although the general pattern is still elusive.{{sfn|Hatcher|2002|loc=Stable homotopy groups, pp. 385–393}} For {{math|''n'' ≤ ''k''+1}}, the groups are called the ''unstable homotopy groups of spheres''.{{cn|date=February 2022}}

===Hopf fibrations===
The classical [[Hopf fibration]] is a [[fiber bundle]]:

:<math>S^1\hookrightarrow S^3\rightarrow S^2.</math>

The general theory of fiber bundles {{math|''F'' → ''E'' → ''B''}} shows that there is a [[homotopy group#Long exact sequence of a fibration|long exact sequence]] of homotopy groups

:<math> \cdots \to \pi_i(F) \to \pi_i(E) \to \pi_i(B) \to \pi_{i-1}(F) \to \cdots.</math>

For this specific bundle, each group homomorphism {{math|π<sub>''i''</sub>(''S''<sup>1</sup>) → π<sub>''i''</sub>(''S''<sup>3</sup>)}}, induced by the inclusion {{math|''S''<sup>1</sup> → ''S''<sup>3</sup>}}, maps all of {{math|π<sub>''i''</sub>(''S''<sup>1</sup>)}} to zero, since the lower-dimensional sphere {{math|''S''<sup>1</sup>}} can be deformed to a point inside the higher-dimensional one {{math|''S''<sup>3</sup>}}. This corresponds to the vanishing of {{math|π<sub>1</sub>(''S''<sup>3</sup>)}}. Thus the long exact sequence breaks into [[short exact sequence]]s,

:<math>0\rightarrow \pi_i(S^3)\rightarrow \pi_i(S^2)\rightarrow \pi_{i-1}(S^1)\rightarrow 0 .</math>

Since {{math|''S''<sup>''n''+1</sup>}} is a [[suspension (topology)|suspension]] of {{math|''S''<sup>''n''</sup>}}, these sequences are [[splitting lemma|split]] by the [[Freudenthal suspension theorem|suspension homomorphism]] {{math|π<sub>''i''−1</sub>(''S''<sup>1</sup>) → π<sub>''i''</sub>(''S''<sup>2</sup>)}}, giving isomorphisms

:<math>\pi_i(S^2)= \pi_i(S^3)\oplus \pi_{i-1}(S^1) .</math>

Since {{math|π<sub>''i''−1</sub>(''S''<sup>1</sup>)}} vanishes for {{mvar|i}} at least 3, the first row shows that {{math|π<sub>''i''</sub>(''S''<sup>2</sup>)}} and {{math|π<sub>''i''</sub>(''S''<sup>3</sup>)}} are isomorphic whenever {{mvar|i}} is at least 3, as observed above.

The Hopf fibration may be constructed as follows: pairs of complex numbers {{math|(''z''<sub>0</sub>,''z''<sub>1</sub>)}} with {{math|{{abs|''z''<sub>0</sub>}}<sup>2</sup> + {{abs|''z''<sub>1</sub>}}<sup>2</sup> {{=}} 1}} form a 3-sphere, and their ratios {{math|{{sfrac|''z''<sub>0</sub>|''z''<sub>1</sub>}}}} cover the [[Riemann sphere|complex plane plus infinity]], a 2-sphere. The Hopf map {{math|''S''<sup>3</sup> → ''S''<sup>2</sup>}} sends any such pair to its ratio.{{cn|date=February 2022}}

Similarly (in addition to the Hopf fibration <math>S^0\hookrightarrow S^1\rightarrow S^1</math>, where the bundle projection is a double covering), there are [[Hopf fibration#Generalizations|generalized Hopf fibrations]]

:<math>S^3\hookrightarrow S^7\rightarrow S^4</math>

:<math>S^7\hookrightarrow S^{15}\rightarrow S^8</math>

constructed using pairs of [[quaternion]]s or [[octonion]]s instead of complex numbers.{{sfn|Hatcher|2002}} Here, too, {{math|π<sub>3</sub>(''S''<sup>7</sup>)}} and {{math|π<sub>7</sub>(''S''<sup>15</sup>)}} are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations.

:<math>\pi_i(S^4)= \pi_i(S^7)\oplus \pi_{i-1}(S^3) ,</math>

:<math>\pi_i(S^8)= \pi_i(S^{15})\oplus \pi_{i-1}(S^7) .</math>

The three fibrations have base space {{math|''S''<sup>''n''</sup>}} with {{math|''n'' {{=}} 2<sup>''m''</sup>}}, for {{math|''m'' {{=}} 1, 2, 3}}. A fibration does exist for {{math|''S''<sup>1</sup>}} ({{math|''m'' {{=}} 0}}) as mentioned above, but not for {{math|''S''<sup>16</sup>}} ({{math|''m'' {{=}} 4}}) and beyond. Although generalizations of the relations to {{math|''S''<sup>16</sup>}} are often true, they sometimes fail; for example,

:<math>\pi_{30}(S^{16})\neq \pi_{30}(S^{31})\oplus \pi_{29}(S^{15}) .</math>

Thus there can be no fibration

:<math>S^{15}\hookrightarrow S^{31}\rightarrow S^{16} ,</math>

the first non-trivial case of the [[Hopf invariant]] one problem, because such a fibration would imply that the failed relation is true.{{cn|date=February 2022}}

===Framed cobordism===

Homotopy groups of spheres are closely related to [[cobordism]] classes of manifolds.
In 1938 [[Lev Pontryagin]] established an isomorphism between the homotopy group {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} and the group {{math|Ω{{su|lh=1|b=''k''|p=framed}}(''S''<sup>''n''+''k''</sup>)}} of cobordism classes of [[Differentiable manifold|differentiable]] {{mvar|k}}-submanifolds of {{math|''S''<sup>''n''+''k''</sup>}}  which are "framed", i.e. have a trivialized [[normal bundle]].  Every map {{math|''f'' : ''S''<sup>''n''+''k''</sup> → ''S''<sup>''n''</sup>}} is homotopic to a differentiable map with {{math|1=''M''<sup>''k''</sup> = ''f''<sup>&minus;1</sup>(1, 0, ..., 0) ⊂ ''S''<sup>''n''+''k''</sup>}} a framed {{mvar|k}}-dimensional submanifold. For example, {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} Z}} is the cobordism group of framed 0-dimensional submanifolds of {{math|''S''<sup>''n''</sup>}}, computed by the algebraic sum of their points, corresponding to the [[degree of a map|degree]] of maps {{math|''f'' : ''S''<sup>''n''</sup> → ''S''<sup>''n''</sup>}}. The projection of the [[Hopf fibration]] {{math|''S''<sup>3</sup> → ''S''<sup>2</sup>}} represents a generator of {{math|π<sub>3</sub>(''S''<sup>2</sup>) {{=}} Ω{{su|lh=1|b=1|p=framed}}(''S''<sup>3</sup>) {{=}} Z}} which corresponds to the framed 1-dimensional submanifold of {{math|''S''<sup>3</sup>}} defined by the standard embedding {{math|''S''<sup>1</sup> ⊂ ''S''<sup>3</sup>}} with a nonstandard trivialization of the normal 2-plane bundle. Until the advent of more sophisticated algebraic methods in the early 1950s (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 group]]s of spaces and [[Spectrum (homotopy theory)|spectra]]. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups.{{sfn|Scorpan|2005}}

===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}}

===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}}

===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}}