View source for Homotopy groups of spheres

{{Short description|How spheres of various dimensions can wrap around each other}}
[[Image:Sphere wrapped round itself.png|right|thumb|250px|Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.]]
In the [[Mathematics|mathematical]] field of [[Algebraic topology|algebraic topology]], the '''homotopy groups of spheres''' describe how spheres of various [[Physics:Dimension#In mathematics|dimension]]s can wrap around each other. They are examples of topological invariants, which reflect, in [[Abstract algebra|algebraic]] terms, the structure of spheres viewed as [[Topological space|topological space]]s, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the [[Homotopy group|homotopy group]]s are surprisingly complex and difficult to compute.

[[Image:Hopf Fibration.png|right|thumb|The [[Hopf fibration|Hopf fibration]] is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere.]]
[[Image:Hopfkeyrings.jpg|right|thumb|This picture mimics part of the Hopf fibration, an interesting mapping from the three-dimensional sphere to the two-dimensional sphere. This mapping is the generator of the third homotopy group of the 2-sphere.]]
The {{mvar|n}}-dimensional unit [[N-sphere|sphere]] — called the {{mvar|n}}-sphere for brevity, and denoted as {{math|''S''<sup>''n''</sup>}} — generalizes the familiar [[Circle|circle]] ({{math|''S''<sup>1</sup>}}) and the ordinary [[Sphere|sphere]] ({{math|''S''<sup>2</sup>}}).  The {{mvar|n}}-sphere may be defined geometrically as the set of points in a [[Euclidean space]] of dimension {{math|''n'' + 1}} located at a unit distance from the origin. The {{mvar|i}}-th ''homotopy group'' {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>)}} summarizes the different ways in which the {{mvar|i}}-dimensional sphere {{math|''S''<sup>''i''</sup>}} can be [[Map (mathematics)|mapped]] continuously into the {{nowrap|{{mvar|n}}-dimensional}} sphere {{math|''S''<sup>''n''</sup>}}. This summary does not distinguish between two mappings if one can be continuously [[Homotopy|deformed]] to the other; thus, only [[Equivalence class|equivalence class]]es of mappings are summarized.  An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an [[Abelian group|abelian group]].

The problem of determining {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>)}} falls into three regimes, depending on whether {{mvar|i}} is less than, equal to, or greater than {{mvar|n}}:
* For {{math|0 < ''i'' < ''n''}}, any mapping from {{math|''S''<sup>''i''</sup>}} to {{math|''S''<sup>''n''</sup>}} is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of {{math|''S''<sup>''i''</sup>}} to a single point of {{math|''S''<sup>''n''</sup>}}. Therefore the homotopy group is the [[Trivial group|trivial group]].
* When {{math|''i'' {{=}} ''n''}}, every map from {{math|''S''<sup>''n''</sup>}} to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>)}} with the group of [[Integer|integer]]s under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping. 
* The most interesting and surprising results occur when {{math|''i'' > ''n''}}. The first such surprise was the discovery of a mapping called the [[Hopf fibration]], which wraps the 3-sphere {{math|''S''<sup>3</sup>}} around the usual sphere {{math|''S''<sup>2</sup>}} in a non-trivial fashion, and so is not equivalent to a one-point mapping.

The question of computing the homotopy group {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} for positive {{mvar|k}} turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} are independent of {{mvar|n}} for {{math|''n'' ≥ ''k'' + 2}}. These are called the '''stable homotopy groups of spheres''' and have been computed for values of {{mvar|k}} up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for {{math|''n'' < ''k'' + 2}}) are more erratic; nevertheless, they have been tabulated for  {{math|''k'' &lt; 20}}. Most modern computations use [[Spectral sequence|spectral sequence]]s, a technique first applied to homotopy groups of spheres by [[Biography:Jean-Pierre Serre|Jean-Pierre Serre]]. Several important patterns have been established, yet much remains unknown and unexplained.

==Background==
The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed. [[Algebraic topology]] provides the larger context, itself built on [[Topology|topology]] and [[Abstract algebra|abstract algebra]], with [[Homotopy group|homotopy group]]s as a basic example.

=== {{mvar|n}}-sphere ===
An ordinary [[Sphere|sphere]] in three-dimensional space — the surface, not the solid ball — is just one example of what a sphere means in topology. [[Geometry]] defines a sphere rigidly, as a shape. Here are some alternatives.
* '''Implicit surface''': {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ''x''{{su|lh=1|b=2|p=2}} {{=}} 1}}
: This is the set of points in 3-dimensional [[Euclidean space]] found exactly one unit away from the origin. It is called the 2-sphere, {{math|''S''<sup>2</sup>}}, for reasons given below. The same idea applies for any [[Physics:Dimension|dimension]] {{mvar|n}}; the equation {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ⋯ + ''x''{{su|lh=1|b=''n''|p=2}} {{=}} 1}} produces the [[N-sphere|{{mvar|n}}-sphere]] as a geometric object in ({{math|''n'' + 1}})-dimensional space. For example, the 1-sphere {{math|''S''<sup>1</sup>}} is a [[Circle|circle]].{{sfn|Hatcher|2002|p=xii}}
* '''Disk with collapsed rim''': written in topology as {{math|''D''<sup>2</sup>/''S''<sup>1</sup>}}
: This construction moves from geometry to pure topology. The [[Unit disk|disk]] {{math|''D''<sup>2</sup>}} is the region contained by a circle, described by the inequality {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} ≤ 1}}, and its rim (or "[[Boundary (topology)|boundary]]") is the circle {{math|''S''<sup>1</sup>}}, described by the equality {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} {{=}} 1}}. If a balloon is punctured and spread flat it produces a disk; this construction repairs the puncture, like pulling a drawstring. The slash, pronounced "modulo", means to take the topological space on the left (the disk) and in it join together as one all the points on the right (the circle). The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere. Generalized, {{math|''D''<sup>''n''</sup>/''S''<sup>''n''−1</sup>}} produces {{math|''S''<sup>''n''</sup>}}. For example, {{math|''D''<sup>1</sup>}} is a [[Line segment|line segment]], and the construction joins its ends to make a circle. An equivalent description is that the boundary of an {{mvar|n}}-dimensional disk is glued to a point, producing a [[CW complex]].{{sfn|Hatcher|2002|loc=Example 0.3, p. 6}}
* '''Suspension of equator''': written in topology as {{math|Σ''S''<sup>1</sup>}}
: This construction, though simple, is of great theoretical importance. Take the circle {{math|''S''<sup>1</sup>}} to be the [[Astronomy:Equator|equator]], and sweep each point on it to one point above (the North Pole), producing the northern hemisphere, and to one point below (the South Pole), producing the southern hemisphere. For each positive integer {{mvar|n}}, the {{mvar|n}}-sphere {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ⋯ + ''x''{{su|lh=1|b=''n''|p=2}} {{=}} 1}} has as equator the ({{math|''n'' &minus; 1}})-sphere {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ⋯ + ''x''{{su|lh=1|b=''n''−1|p=2}} {{=}} 1}}, and the suspension {{math|Σ''S''<sup>''n''−1</sup>}} produces {{math|''S''<sup>''n''</sup>}}.{{sfn|Hatcher|2002|p=129}}

Some theory requires selecting a fixed point on the sphere, calling the pair {{math|(sphere, point)}} a ''[[Pointed space|pointed sphere]]''. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience.{{sfn|Hatcher|2002|p=28}} For spheres constructed as a repeated suspension, the point {{math|(1, 0, 0, ..., 0)}}, which is on the equator of all the levels of suspension, works well; for the disk with collapsed rim, the point resulting from the collapse of the rim is another obvious choice.

=== Homotopy group ===
[[Image:Homotopy of pointed circle maps.png|thumb|right|Homotopy of two circle maps keeping base point fixed]]
[[Image:Homotopy group addition.svg|thumb|right|Addition of two circle maps keeping base point fixed]]
The distinguishing feature of a [[Topological space|topological space]] is its continuity structure, formalized in terms of [[Open set|open set]]s or neighborhoods. A continuous map is a function between spaces that preserves continuity. A [[Homotopy|homotopy]] is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic.{{sfn|Hatcher|2002|p=3}} The idea common to all these concepts is to discard variations that do not affect outcomes of interest. An important practical example is the [[Residue theorem|residue theorem]] of [[Complex analysis|complex analysis]], where "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity.{{sfn|Miranda|1995|pp=123–125}}

The first homotopy group, or [[Fundamental group|fundamental group]], {{math|π<sub>1</sub>(''X'')}} of a (path connected) topological space {{mvar|X}} thus begins with continuous maps from a pointed circle {{math|(''S''<sup>1</sup>,''s'')}} to the pointed space {{math|(''X'',''x'')}}, where maps from one pair to another map {{mvar|s}} into {{mvar|x}}. These maps (or equivalently, closed [[Curve|curve]]s) are grouped together into [[Equivalence class|equivalence class]]es based on homotopy (keeping the "base point" {{mvar|x}} fixed), so that two maps are in the same class if they are homotopic. Just as one point is distinguished, so one class is distinguished: all maps (or curves) homotopic to the constant map {{math|''S''<sup>1</sup>↦''x''}} are called null homotopic. The classes become an [[Abstract algebra|abstract algebra]]ic [[Group (mathematics)|group]] with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere (here a circle) to the distinguished point, producing a "bouquet of spheres" — two pointed spheres joined at their distinguished point. The two maps to be added map the upper and lower spheres separately, agreeing on the distinguished point, and composition with the pinch gives the sum map.{{sfn|Hu|1959|p=[https://books.google.com/books?id=iVhMPU0X2G4C&pg=PA107 107]}}

More generally, the {{mvar|i}}-th homotopy group, {{math|π<sub>''i''</sub>(''X'')}} begins with the pointed {{mvar|i}}-sphere {{math|(''S''<sup>''i''</sup>, ''s'')}}, and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for {{mvar|X}} equal to {{math|''S''<sup>''n''</sup>}} (for positive {{mvar|n}}) — the homotopy groups of spheres — the groups are [[Abelian group|abelian]] and [[Finitely generated group|finitely generated]]. If for some {{mvar|i}} all maps are null homotopic, then the group {{math|π<sub>''i''</sub>}} consists of one element, and is called the [[Trivial group|trivial group]].{{cn|date=February 2022}}

A continuous map between two topological spaces induces a [[Group homomorphism|group homomorphism]] between the associated homotopy groups. In particular, if the map is a continuous [[Bijection|bijection]] (a [[Homeomorphism|homeomorphism]]), so that the two spaces have the same topology, then their {{mvar|i}}-th homotopy groups are isomorphic for all {{mvar|i}}. However, the real [[Plane (mathematics)|plane]] has exactly the same homotopy groups as a solitary point (as does a Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also make certain computations easier.{{cn|date=February 2022}}

==Low-dimensional examples==
The low-dimensional examples of homotopy groups of spheres provide a sense of the subject, because these special cases can be visualized in ordinary 3-dimensional space.{{sfn|Hatcher|2002|p=29}} However, such visualizations are not mathematical proofs, and do not capture the possible complexity of maps between spheres.{{cn|date=February 2022}}

==={{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z}}===
[[Image:Fundamental group of the circle.svg|300px|thumb|Elements of {{math|π<sub>1</sub>(''S''<sup>1</sup>)}}]]

The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a [[Engineering:Rubber band|rubber band]] around one's finger: it can be wrapped once, twice, three times and so on.  The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group {{math|π<sub>1</sub>(''S''<sup>1</sup>)}} is therefore an infinite cyclic group, and is isomorphic to the group {{math|Z}} of [[Integer|integer]]s under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the [[Winding number|winding number]] of a loop around the [[Origin (mathematics)|origin]] in the [[Plane (mathematics)|plane]].{{sfn|Hatcher|2002|p=29}}

The identification (a [[Group isomorphism|group isomorphism]]) of the homotopy group with the integers is [[Abuse of notation|often written]] as an equality: thus {{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z}}.<ref>See, e.g., {{harvnb|Homotopy type theory|2013}}, Section 8.1, "<math display=inline>\pi_1(S^1)</math>".</ref>

==={{math|π<sub>2</sub>(''S''<sup>2</sup>) {{=}} Z}}===

[[Image:Sphere wrapped round itself.png|200px|thumb|right|Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.]]

Mappings from a 2-sphere to a 2-sphere can be visualized as wrapping a plastic bag around a ball and then sealing it.  The sealed bag is topologically equivalent to a 2-sphere, as is the surface of the ball. The bag can be wrapped more than once by twisting it and wrapping it back over the ball. (There is no requirement for the continuous map to be injective and so the bag is allowed to pass through itself.) The twist can be in one of two directions and opposite twists can cancel out by deformation. The total number of twists after cancellation is an integer, called the ''degree'' of the mapping. As in the case mappings from the circle to the circle, this degree identifies the homotopy group with the group of integers, {{math|Z}}.

These two results generalize: for all {{math|''n'' > 0}},  {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} Z}} (see below).

==={{math|π<sub>1</sub>(''S''<sup>2</sup>) {{=}} 0}}===
[[Image:P1S2all.jpg|400px|thumb|A homotopy from a circle around a sphere down to a single point]]
Any continuous mapping from a circle to an ordinary sphere can be continuously deformed to a one-point mapping, and so its homotopy class is trivial. One way to visualize this is to imagine a rubber-band wrapped around a frictionless ball: the band can always be slid off the ball. The homotopy group is therefore a [[Trivial group|trivial group]], with only one element, the identity element, and so it can be identified with the [[Subgroup|subgroup]] of {{math|Z}} consisting only of the number zero. This group is often denoted by 0.  Showing this rigorously requires more care, however, due to the existence of
[[Space-filling curve|space-filling curve]]s.{{sfn|Hatcher|2002|p=348}}

This result generalizes to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if {{math|''i'' < ''n''}}, then  {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>) {{=}} 0}}.  This can be shown as a consequence of the [[Cellular approximation theorem|cellular approximation theorem]].{{sfn|Hatcher|2002|p=349}}

==={{math|π<sub>2</sub>(''S''<sup>1</sup>) {{=}} 0}}===
All the interesting cases of homotopy groups of spheres involve mappings from a higher-dimensional sphere onto one of lower dimension. Unfortunately, the only example which can easily be visualized is not interesting: there are no nontrivial mappings from the ordinary sphere to the circle. Hence, {{math|π<sub>2</sub>(''S''<sup>1</sup>) {{=}} 0}}. This is because {{math|''S''<sup>1</sup>}} has the real line as its universal cover which is contractible (it has the homotopy type of a point). In addition, because {{math|''S''<sup>2</sup>}} is simply connected, by the lifting criterion,{{sfn|Hatcher|2002|p=61}} any map from {{math|''S''<sup>2</sup>}} to {{math|''S''<sup>1</sup>}} can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space.{{cn|date=February 2022}}

==={{math|π<sub>3</sub>(''S''<sup>2</sup>) {{=}} Z}}===

[[Image:Hopf Fibration.png|right|thumb|The [[Hopf fibration|Hopf fibration]] is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere. Each colored circle maps to the corresponding point on the 2-sphere shown bottom right.]]

The first nontrivial example with {{math|''i'' > ''n''}} concerns mappings from the [[3-sphere|3-sphere]] to the ordinary 2-sphere, and was discovered by [[Biography:Heinz Hopf|Heinz Hopf]], who constructed a nontrivial map from {{math|''S''<sup>3</sup>}} to {{math|''S''<sup>2</sup>}}, now known as the [[Hopf fibration]].{{sfn|Hopf|1931}} This map [[Generating set of a group|generates]] the homotopy group {{math|π<sub>3</sub>(''S''<sup>2</sup>) {{=}} Z}}.{{sfn|Walschap|2004|p=90}}

==History==
In the late 19th century [[Biography:Camille Jordan|Camille Jordan]] introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory.{{sfn|O'Connor|Robertson|2001}} A more rigorous approach was adopted by [[Biography:Henri Poincaré|Henri Poincaré]] in his 1895 set of papers ''Analysis situs'' where the related concepts of [[Homology (mathematics)|homology]] and the [[Fundamental group|fundamental group]] were also introduced.{{sfn|O'Connor|Robertson|1996}}

Higher homotopy groups were first defined by [[Biography:Eduard Čech|Eduard Čech]] in 1932.{{sfn|Čech|1932|p=203}} (His first paper was withdrawn on the advice of [[Biography:Pavel Alexandrov|Pavel Sergeyevich Alexandrov]] and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.) Witold Hurewicz is also credited with the introduction of homotopy groups in his 1935 paper and also for the [[Hurewicz theorem]] which can be used to calculate some of the groups.{{sfn|May|1999a}}
An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was [[Biography:Hans Freudenthal|Hans Freudenthal]]'s [[Freudenthal suspension theorem|suspension theorem]], published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results.{{sfn|May|1999a}} In 1953 [[Biography:George W. Whitehead|George W. Whitehead]] showed that there is a metastable range for the homotopy groups of spheres. [[Biography:Jean-Pierre Serre|Jean-Pierre Serre]] used spectral sequences to show that most of these groups are finite, the exceptions being {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>)}} and {{math|π<sub>4''n''−1</sub>(''S''<sup>2''n''</sup>)}}. Others who worked in this area included [[Biography:José Adem|José Adem]], [[Biography:Hiroshi Toda|Hiroshi Toda]], [[Biography:Frank Adams|Frank Adams]], [[Biography:J. Peter May|J. Peter May]], [[Biography:Mark Mahowald|Mark Mahowald]], Daniel Isaksen, Guozhen Wang, and [[Biography:Zhouli Xu|Zhouli Xu]]. The stable homotopy groups {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} are known for {{mvar|k}} up to 90, and, as of 2023, unknown for larger {{mvar|k}}.{{sfn|Isaksen|Wang|Xu|2020}}

==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|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|contractible space]], and any mapping to such a space can be deformed into a one-point mapping.{{sfn|Hatcher|2002|p=349}}

The case {{math|''i'' {{=}} ''n''}} has also been noted already, and is an easy consequence of the [[Hurewicz theorem]]: this theorem links homotopy groups with homology groups, which are generally easier to calculate; in particular, it shows that for a simply-connected space ''X'', the first nonzero homotopy group {{math|π<sub>''k''</sub>(''X'')}}, with {{math|''k'' &gt; 0}}, is isomorphic to the first nonzero homology group {{math|''H''<sub>''k''</sub>(''X'')}}. For the {{mvar|n}}-sphere, this immediately implies that for {{math|''n'' ≥ 2}}, {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} ''H''<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} Z}}.{{cn|date=February 2022}}

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. 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 the [[Trivial group|trivial group]] 0, the infinite cyclic group {{math|Z}}, finite [[Cyclic group|cyclic group]]s of order {{mvar|n}} (written as {{math|Z<sub>''n''</sub>}}), or [[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 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 two rows of this table are straightforward. The homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>0</sup>)}} of the 0-dimensional sphere are trivial for {{math|''i'' > 0}}, because any base point preserving map from an {{mvar|i}}-sphere to a 0-sphere ''is'' a one-point mapping. Similarly, the homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>1</sup>)}} of the 1-sphere are trivial for {{math|''i'' > 1}}, because the [[Covering space|universal covering space]], <math>\mathbb{R}</math>, which has the same higher homotopy groups, is contractible.{{cn|date=February 2022}}

Beyond these two rows, 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 third and fourth 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 [[Biography:Hans Freudenthal|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 sequences,

:<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|quaternion]]s or [[Octonion|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|cobordism]] classes of manifolds.
In 1938 [[Biography:Lev Pontryagin|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|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 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 [[Biography:René Thom|René Thom]] to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as [[Homotopy group|homotopy group]]s of spaces and 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, [[Biography:Jean-Pierre Serre|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 [[Graded ring|graded ring]], where multiplication is given by composition of representing maps, and any element of non-zero degree is [[Nilpotent|nilpotent]];{{sfn|Nishida|1973}} the [[Nilpotence theorem|nilpotence theorem]] on [[Complex cobordism|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. [[Biography:Hiroshi Toda|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|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}}

==Computational methods==
If {{mvar|X}} is any finite simplicial complex with finite fundamental group, in particular if {{mvar|X}} is a sphere of dimension at least 2, then its homotopy groups are all [[Finitely generated abelian group|finitely generated abelian group]]s. To compute these groups, they are often factored into their [[Component (group theory)|{{mvar|p}}-components]] for each [[Prime number|prime]] {{mvar|p}}, and calculating each of these [[P-group|{{mvar|p}}-groups]] separately. The first few homotopy groups of spheres can be computed using ad hoc variations of the ideas above; beyond this point, most methods for computing homotopy groups of spheres are based on [[Spectral sequence|spectral sequence]]s.{{sfn|Ravenel|2003}} This is usually done by constructing suitable fibrations and taking the associated long exact sequences of homotopy groups; spectral sequences are a systematic way of organizing the complicated information that this process generates.{{cn|date=February 2022}}

*"The method of killing homotopy groups", due to {{harvs|txt=yes|last=Cartan|last2=Serre|year1=1952a|year2=1952b}} involves repeatedly using the [[Hurewicz theorem]] to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an [[Eilenberg–MacLane space]]. In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group.
*The [[Serre spectral sequence]] was used by Serre to prove some of the results mentioned previously. He used the fact that taking the [[Loop space|loop space]] of a well behaved space shifts all the homotopy groups down by 1, so the {{mvar|n}}th homotopy group of a space {{mvar|X}} is the first homotopy group of its ({{math|''n''−1}})-fold repeated loop space, which is equal to the first homology group of the ({{math|''n''−1}})-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of {{mvar|X}} to the calculation of homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used to calculate the homology of loop spaces. The Serre spectral sequence tends to have many non-zero differentials, which are hard to control, and too many ambiguities appear for higher homotopy groups. Consequently, it has been superseded by more powerful spectral sequences with fewer non-zero differentials, which give more information.{{cn|date=February 2022}}
* The [[EHP spectral sequence]] can be used to compute many homotopy groups of spheres; it is based on some fibrations used by Toda in his calculations of homotopy groups.{{sfn|Mahowald|2001}}{{sfn|Toda|1962}}
* The classical [[Adams spectral sequence]] has {{math|''E''<sub>2</sub>}} term given by the Ext groups {{math|Ext{{su|lh=1|b=''A''(''p'')|p=∗,∗}}(Z<sub>''p''</sub>, Z<sub>''p''</sub>)}} over the mod {{mvar|p}} [[Steenrod algebra]] {{math|''A''(''p'')}}, and converges to something closely related to the {{mvar|p}}-component of the stable homotopy groups. The initial terms of the Adams spectral sequence are themselves quite hard to compute: this is sometimes done using an auxiliary spectral sequence called the [[May spectral sequence]].{{sfn|Ravenel|2003|pp=67–74}}
*At the odd primes, the Adams–Novikov spectral sequence is a more powerful version of the Adams spectral sequence replacing ordinary cohomology mod {{mvar|p}} with a generalized cohomology theory, such as [[Complex cobordism|complex cobordism]] or, more usually, a piece of it called [[Brown–Peterson cohomology]]. The initial term is again quite hard to calculate; to do this one can use the [[Chromatic spectral sequence|chromatic spectral sequence]].{{sfn|Ravenel|2003|loc=Chapter 5}}

[[Image:BorromeanRings.svg|150px|thumb|Borromean rings]]
*A variation of this last approach uses a backwards version of the Adams–Novikov spectral sequence for Brown–Peterson cohomology: the limit is known, and the initial terms involve unknown stable homotopy groups of spheres that one is trying to find.{{sfn|Kochman|1990}}

*The motivic Adams spectral sequence converges to the motivic stable homotopy groups of spheres. By comparing the motivic one over the complex numbers with the classical one, Isaksen gives rigorous proof of computations up to the 59-stem. In particular, Isaksen computes the Coker J of the 56-stem is 0, and therefore by the work of Kervaire-Milnor, the sphere {{math|''S''<sup>56</sup>}} has a unique smooth structure.{{sfn|Isaksen|2019}}

*The Kahn–Priddy map induces a map of Adams spectral sequences from the suspension spectrum of infinite real projective space to the sphere spectrum. It is surjective on the Adams {{math|''E''<sub>2</sub>}} page on positive stems. Wang and Xu develops a method using the Kahn–Priddy map to deduce Adams differentials for the sphere spectrum inductively. They give detailed argument for several Adams differentials and compute the 60 and 61-stem. A geometric corollary of their result is the sphere {{math|''S''<sup>61</sup>}} has a unique smooth structure, and it is the last odd dimensional one – the only ones are {{math|''S''<sup>1</sup>}}, {{math|''S''<sup>3</sup>}}, {{math|''S''<sup>5</sup>}}, and {{math|''S''<sup>61</sup>}}.{{sfn|Wang|Xu|2017}}

*The motivic cofiber of {{math|''τ''}} method is so far the most efficient method at the prime 2. The class {{math|''τ''}} is a map between motivic spheres. The Gheorghe–Wang–Xu theorem identifies the motivic Adams spectral sequence for the cofiber of {{math|''τ''}} as the algebraic Novikov spectral sequence for {{math|''BP''<sub>*</sub>}}, which allows one to deduce motivic Adams differentials for the cofiber of {{math|''τ''}} from purely algebraic data. One can then pullback these motivic Adams differentials to the motivic sphere, and then use the Betti realization functor to push forward them to the classical sphere.{{sfn|Gheorghe|Wang|Xu|2021}} Using this method, {{harvtxt|Isaksen|Wang|Xu|2020}} computes up to the 90-stem.{{sfn|Isaksen|Wang|Xu|2020}}

The computation of the homotopy groups of {{math|''S''<sup>2</sup>}} has been reduced to a [[Geometric group theory|combinatorial group theory]] question. {{harvtxt|Berrick|Cohen|Wong|Wu|2006}} identify these homotopy groups as certain quotients of the Brunnian [[Braid group|braid group]]s of {{math|''S''<sup>2</sup>}}. Under this correspondence, every nontrivial element in {{math|π<sub>''n''</sub>(''S''<sup>2</sup>)}} for {{math|''n'' > 2}} may be represented by a Brunnian braid over {{math|''S''<sup>2</sup>}} that is not Brunnian over the disk {{math|''D''<sup>2</sup>}}. For example, the Hopf map {{math|''S''<sup>3</sup> → ''S''<sup>2</sup>}} corresponds to the [[Borromean rings]].{{sfn|Berrick|Cohen|Wong|Wu|2006}}

==Applications==

* The [[Winding number|winding number]] (corresponding to an integer of {{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z)}} can be used to prove the [[Fundamental theorem of algebra|fundamental theorem of algebra]], which states that every non-constant [[Complex number|complex]] [[Polynomial|polynomial]] has a zero.{{sfn|Fine|Rosenberger|1997}}
* The fact that {{math|π<sub>''n''&minus;1</sub>(''S''<sup>''n''&minus;1</sup>) {{=}} Z}} implies the Brouwer fixed point theorem that every continuous map from the {{mvar|n}}-dimensional [[Ball (mathematics)|ball]] to itself has a fixed point.{{sfn|Hatcher|2002|p=32}}
* The stable homotopy groups of spheres are important in [[Singularity theory|singularity theory]], which studies the structure of singular points of smooth maps or algebraic varieties. Such singularities arise as [[Critical point (mathematics)|critical point]]s of smooth maps from {{math|<math>\mathbb{R}</math><sup>''m''</sup>}} to {{math|<math>\mathbb{R}</math><sup>''n''</sup>}}. The geometry near a critical point of such a map can be described by an element of {{math|π<sub>''m''&minus;1</sub>(''S''<sup>''n''&minus;1</sup>)}}, by considering the way in which a small {{math|''m'' &minus; 1}} sphere around the critical point maps into a topological {{math|''n'' &minus; 1}} sphere around the [[Critical value|critical value]].{{cn|date=February 2022}}
* The fact that the third stable homotopy group of spheres is cyclic of order 24, first proved by Vladimir Rokhlin, implies [[Rokhlin's theorem]] that the [[Signature (topology)|signature]] of a compact smooth [[Spin structure|spin]] [[4-manifold|4-manifold]] is divisible by 16.{{sfn|Scorpan|2005}}
* Stable homotopy groups of spheres are used to describe the group {{math|Θ<sub>''n''</sub>}} of [[H-cobordism|h-cobordism]] classes of oriented homotopy {{mvar|n}}-spheres (for {{math|''n'' ≠ 4}}, this is the group of [[Smooth structure|smooth structure]]s on {{mvar|n}}-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by [[Exotic sphere|exotic sphere]]s). More precisely, there is an injective map
::<math>\Theta_n / bP_{n+1} \to \pi_n^S / J ,</math>
:where {{math|''bP''<sub>''n''+1</sub>}} is the cyclic subgroup represented by homotopy spheres that bound a [[Parallelizable manifold|parallelizable manifold]], {{math|π{{su|lh=1|b=''n''|p=S}}}} is the {{mvar|n}}th stable homotopy group of spheres, and {{mvar|J}} is the image of the {{mvar|J}}-homomorphism. This is an isomorphism unless {{mvar|n}} is of the form {{math|2<sup>''k''</sup> &minus; 2}}, in which case the image has index 1 or 2.{{sfn|Kervaire|Milnor|1963}}
* The groups {{math|Θ<sub>''n''</sub>}} above, and therefore the stable homotopy groups of spheres, are used in the classification of possible smooth structures on a topological or [[Piecewise linear manifold|piecewise linear manifold]].{{sfn|Scorpan|2005}}
* The Kervaire invariant problem, about the existence of manifolds of [[Kervaire invariant]] 1 in dimensions {{math|2<sup>''k''</sup> &minus; 2}}  can be reduced to a question about stable homotopy groups of spheres. For example, knowledge of stable homotopy groups of degree up to 48 has been used to settle the Kervaire invariant problem in dimension {{math|2<sup>6</sup> &minus; 2 {{=}} 62}}. (This was the smallest value of {{mvar|k}} for which the question was open at the time.){{sfn|Barratt|Jones|Mahowald|1984}}
* The [[Barratt–Priddy theorem]] says that the stable homotopy groups of the spheres can be expressed in terms of the [[Plus construction|plus construction]] applied to the [[Classifying space|classifying space]] of the [[Symmetric group|symmetric group]], leading to an identification of K-theory of the [[Field with one element|field with one element]] with stable homotopy groups.{{sfn|Deitmar|2006}}

==Table of homotopy groups==
Tables of homotopy groups of spheres are most conveniently organized by showing {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}}.

The following table shows many of the groups {{math|π<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}}. The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following conventions:<ref>These tables are based on the [http://www.math.nus.edu.sg/%7Ematwujie/homotopy_groups_sphere.html table of homotopy groups of spheres] in {{harvtxt|Toda|1962}}.</ref>
* The entry "⋅" denotes the trivial group.
* Where the entry is an [[Integer|integer]], {{mvar|m}}, the homotopy group is the [[Cyclic group|cyclic group]] of that order (generally written {{math|Z<sub>''m''</sub>}}).
* Where the entry is ∞, the homotopy group is the infinite cyclic group, {{math|Z}}.
* Where entry is a product, the homotopy group is the [[Cartesian product|cartesian product]] (equivalently, [[Direct sum of groups|direct sum]]) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when {{mvar|a}} and {{mvar|b}} have no common factor, {{math|Z<sub>''a''</sub>×Z<sub>''b''</sub>}} is isomorphic to {{math|Z<sub>''ab''</sub>}}.)

'''Example''': {{math|π<sub>19</sub>(''S''<sup>10</sup>) {{=}} π<sub>9+10</sub>(''S''<sup>10</sup>) {{=}} Z×Z<sub>2</sub>×Z<sub>2</sub>×Z<sub>2</sub>}}, which is denoted by {{math|∞⋅2<sup>3</sup>}} in the table.

{| class="toccontents" border="1" cellspacing="0" cellpadding="4" style="text-align: center"
|- style="background: #fff; text-align: center;"
! style="background: #eee" | ''S''<sup>''n''</sup> →
! style="width:4em" | ''S''<sup>0</sup>
! style="width:4em" | ''S''<sup>1</sup>
! style="width:4em" | ''S''<sup>2</sup>
! style="width:4em" | ''S''<sup>3</sup>
! style="width:4em" | ''S''<sup>4</sup>
! style="width:4em" | ''S''<sup>5</sup>
! style="width:4em" | ''S''<sup>6</sup>
! style="width:4em" | ''S''<sup>7</sup>
! style="width:4em" | ''S''<sup>8</sup>
! style="width:4em" | ''S''<sup>9</sup>
! style="width:4em" | ''S''<sup>10</sup>
! style="width:4em" | ''S''<sup>11</sup>
! style="width:4em" | ''S''<sup>12</sup>
! style="width:4em" | ''S''<sup>≥13</sup>
|- style="background: #eef"
! style="background: #fff" | π<sub><''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fff" |
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
|- style="background: #ddf"
! style="background: #eee" | π<sub>0+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fdd" | 2
| style="background: #fdd" | ∞
| ∞
| ∞
| ∞
| ∞
| ∞
| ∞
| ∞
| ∞
| ∞
| ∞
| ∞
| ∞
|- style="background: #eef"
! style="background: #fff" | π<sub>1+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fee" | ⋅
| style="background: #fee" | ⋅
| style="background: #fee" | ∞
| 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
|- style="background: #ddf"
! style="background: #eee" | π<sub>2+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fdd" | ⋅
| style="background: #fdd" | ⋅
| style="background: #fdd" | 2
| style="background: #fdd" | 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
| 2
|- style="background: #eef"
! style="background: #fff" | π<sub>3+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fee" | ⋅
| style="background: #fee" | ⋅
| style="background: #fee" | 2
| style="background: #fee" | 12
| style="background: #fee" | ∞⋅12
| 24
| 24
| 24
| 24
| 24
| 24
| 24
| 24
| 24
|- style="background: #ddf"
! style="background: #eee" | π<sub>4+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fdd" | ⋅
| style="background: #fdd" | ⋅
| style="background: #fdd" | 12
| style="background: #fdd" | 2
| style="background: #fdd" | 2<sup>2</sup>
| style="background: #fdd" | 2
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
|- style="background: #eef"
! style="background: #fff" | π<sub>5+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fee" | ⋅
| style="background: #fee" | ⋅
| style="background: #fee" | 2
| style="background: #fee" | 2
| style="background: #fee" | 2<sup>2</sup>
| style="background: #fee" | 2
| style="background: #fee" | ∞
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
|- style="background: #ddf"
! style="background: #eee" | π<sub>6+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fdd" | ⋅
| style="background: #fdd" | ⋅
| style="background: #fdd" | 2
| style="background: #fdd" | 3
| style="background: #fdd" | 24⋅3
| style="background: #fdd" | 2
| style="background: #fdd" | 2
| style="background: #fdd" | 2
| 2
| 2
| 2
| 2
| 2
| 2
|- style="background: #fee"
! style="background: #fff" | π<sub>7+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 3
| 15
| 15
| 30
| 60
| 120
| ∞⋅120
| style="background: #eef" | 240
| style="background: #eef" | 240
| style="background: #eef" | 240
| style="background: #eef" | 240
| style="background: #eef" | 240
|- style="background: #fdd"
! style="background: #eee" | π<sub>8+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 15
| 2
| 2
| 2
| 24⋅2
| 2<sup>3</sup>
| 2<sup>4</sup>
| 2<sup>3</sup>
| style="background: #ddf" | 2<sup>2</sup>
| style="background: #ddf" | 2<sup>2</sup>
| style="background: #ddf" | 2<sup>2</sup>
| style="background: #ddf" | 2<sup>2</sup>
|- style="background: #fee"
! style="background: #fff" | π<sub>9+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 2
| 2<sup>2</sup>
| 2<sup>3</sup>
| 2<sup>3</sup>
| 2<sup>3</sup>
| 2<sup>4</sup>
| 2<sup>5</sup>
| 2<sup>4</sup>
| ∞⋅2<sup>3</sup>
| style="background: #eef" | 2<sup>3</sup>
| style="background: #eef" | 2<sup>3</sup>
| style="background: #eef" | 2<sup>3</sup>
|- style="background: #fdd"
! style="background: #eee" | π<sub>10+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 2<sup>2</sup>
| 12⋅2
| {{resize|93%|120⋅12⋅2}}
| 72⋅2
| 72⋅2
| 24⋅2
| 24<sup>2</sup>⋅2
| 24⋅2
| 12⋅2
| 6⋅2
| style="background: #ddf" | 6
| style="background: #ddf" | 6
|- style="background: #fee"
! style="background: #fff" | π<sub>11+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 12⋅2
| 84⋅2<sup>2</sup>
| 84⋅2<sup>5</sup>
| 504⋅2<sup>2</sup>
| 504⋅4
| 504⋅2
| 504⋅2
| 504⋅2
| 504
| 504
| ∞⋅504
| style="background: #eef" | 504
|- style="background: #fdd"
! style="background: #eee" | π<sub>12+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 84⋅2<sup>2</sup>
| 2<sup>2</sup>
| 2<sup>6</sup>
| 2<sup>3</sup>
| 240
| ⋅
| ⋅
| ⋅
| 12
| 2
| 2<sup>2</sup>
| rowspan="8" valign="middle" align="center" style="background: #fff" | ''See<br />below''
|- style="background: #fee"
! style="background: #fff" | π<sub>13+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 2<sup>2</sup>
| 6
| 24⋅6⋅2
| 6⋅2
| 6
| 6
| 6⋅2
| 6
| 6
| 6⋅2
| 6⋅2
|- style="background: #fdd"
! style="background: #eee" | π<sub>14+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 6
| 30
| {{resize|92%|2520⋅6⋅2}}
| 6⋅2
| 12⋅2
| 24⋅4
| 240⋅24⋅4
| 16⋅4
| 16⋅2
| 16⋅2
| 48⋅4⋅2
|- style="background: #fee"
! style="background: #fff" | π<sub>15+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 30
| 30
| 30
| 30⋅2
| 60⋅6
| 120⋅2<sup>3</sup>
| 120⋅2<sup>5</sup>
| 240⋅2<sup>3</sup>
| 240⋅2<sup>2</sup>
| 240⋅2
| 240⋅2
|- style="background: #fdd"
! style="background: #eee" | π<sub>16+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 30
| 6⋅2
| 6<sup>2</sup>⋅2
| 2<sup>2</sup>
| 504⋅2<sup>2</sup>
| 2<sup>4</sup>
| 2<sup>7</sup>
| 2<sup>4</sup>
| 240⋅2
| 2
| 2
|- style="background: #fee"
! style="background: #fff" | π<sub>17+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 6⋅2
| 12⋅2<sup>2</sup>
| {{resize|78%|24⋅12⋅4⋅2<sup>2</sup>}}
| 4⋅2<sup>2</sup>
| 2<sup>4</sup>
| 2<sup>4</sup>
| 6⋅2<sup>4</sup>
| 2<sup>4</sup>
| 2<sup>3</sup>
| 2<sup>3</sup>
| 2<sup>4</sup>
|- style="background: #fdd"
! style="background: #eee" | π<sub>18+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 12⋅2<sup>2</sup>
| 12⋅2<sup>2</sup>
| {{resize|81%|120⋅12⋅2<sup>5</sup>}}
| 24⋅2<sup>2</sup>
| 24⋅6⋅2
| 24⋅2
| 504⋅24⋅2
| 24⋅2
| 24⋅2<sup>2</sup>
| 8⋅4⋅2
| 480⋅4<sup>2</sup>⋅2
|- style="background: #fee"
! style="background: #fff" | π<sub>19+''n''</sub>(''S''<sup>''n''</sup>)
| ⋅
| ⋅
| 12⋅2<sup>2</sup>
| 132⋅2
| 132⋅2<sup>5</sup>
| 264⋅2
| 1056⋅8
| 264⋅2
| 264⋅2
| 264⋅2
| 264⋅6
| 264⋅2<sup>3</sup>
| 264⋅2<sup>5</sup>
|}

<!-- Two blank lines between tables-->
{| class="toccontents" border="1" cellspacing="0" cellpadding="4" style="text-align: center"
|- style="background: #fff;"
! style="background: #eee" | ''S''<sup>''n''</sup> →
! style="width:4em" | ''S''<sup>13</sup>
! style="width:4em" | ''S''<sup>14</sup>
! style="width:4em" | ''S''<sup>15</sup>
! style="width:4em" | ''S''<sup>16</sup>
! style="width:4em" | ''S''<sup>17</sup>
! style="width:4em" | ''S''<sup>18</sup>
! style="width:4em" | ''S''<sup>19</sup>
! style="width:4em" | ''S''<sup>20</sup>
! style="width:4em" | ''S''<sup>≥21</sup>
|- style="background: #eef"
! style="background: #fff" | π<sub>12+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fee" | 2
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
| ⋅
|- style="background: #ddf"
! style="background: #eee" | π<sub>13+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fdd" | 6
| style="background: #fdd" | ∞⋅3
| 3
| 3
| 3
| 3
| 3
| 3
| 3
|- style="background: #eef"
! style="background: #fff" | π<sub>14+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fee" | 16⋅2
| style="background: #fee" | 8⋅2
| style="background: #fee" | 4⋅2
| 2<sup>2</sup>
| 2<sup>2</sup>
| 2<sup>2</sup>
| 2<sup>2</sup>
| 2<sup>2</sup>
| 2<sup>2</sup>
|- style="background: #ddf"
! style="background: #eee" | π<sub>15+''n''</sub>(''S''<sup>''n''</sup>)
| style="background: #fdd" | 480⋅2
| style="background: #fdd" | 480⋅2
| style="background: #fdd" | 480⋅2
| style="background: #fdd" | ∞⋅480⋅2
| 480⋅2
| 480⋅2
| 480⋅2
| 480⋅2
| 480⋅2
|- style="background: #fee"
! style="background: #fff" | π<sub>16+''n''</sub>(''S''<sup>''n''</sup>)
| 2
| 24⋅2
| 2<sup>3</sup>
| 2<sup>4</sup>
| 2<sup>3</sup>
| style="background: #eef" | 2<sup>2</sup>
| style="background: #eef" | 2<sup>2</sup>
| style="background: #eef" | 2<sup>2</sup>
| style="background: #eef" | 2<sup>2</sup>
|- style="background: #fdd"
! style="background: #eee" | π<sub>17+''n''</sub>(''S''<sup>''n''</sup>)
| 2<sup>4</sup>
| 2<sup>4</sup>
| 2<sup>5</sup>
| 2<sup>6</sup>
| 2<sup>5</sup>
| ∞⋅2<sup>4</sup>
| style="background: #ddf" | 2<sup>4</sup>
| style="background: #ddf" | 2<sup>4</sup>
| style="background: #ddf" | 2<sup>4</sup>
|- style="background: #fee"
! style="background: #fff" | π<sub>18+''n''</sub>(''S''<sup>''n''</sup>)
| 8<sup>2</sup>⋅2
| 8<sup>2</sup>⋅2
| 8<sup>2</sup>⋅2
| 24⋅8<sup>2</sup>⋅2
| 8<sup>2</sup>⋅2
| 8⋅4⋅2
| 8⋅2<sup>2</sup>
| style="background: #eef" | 8⋅2
| style="background: #eef" | 8⋅2
|- style="background: #fdd"
! style="background: #eee" | π<sub>19+''n''</sub>(''S''<sup>''n''</sup>)
| 264⋅2<sup>3</sup>
| 264⋅4⋅2
| 264⋅2<sup>2</sup>
| 264⋅2<sup>2</sup>
| 264⋅2<sup>2</sup>
| 264⋅2
| 264⋅2
| ∞⋅264⋅2
| style="background: #ddf" | 264⋅2
|}

==Table of stable homotopy groups==
The stable homotopy groups {{math|π{{su|lh=1|b=''k''|p=''S''}}}} are the products of cyclic groups of the infinite or prime power orders shown in the table. (For largely historical reasons, stable homotopy groups are usually given as products of cyclic groups of prime power order, while tables of unstable homotopy groups often give them as products of the smallest number of cyclic groups.) For {{math|''p'' > 5}}, the part of the {{mvar|p}}-component that is accounted for by the {{mvar|J}}-homomorphism is cyclic of order {{mvar|p}} if {{math|2(''p'' &minus; 1)}} divides {{math|''k'' + 1}} and 0 otherwise.<ref>{{harvnb|Fuks|2001}}. The 2-components can be found in {{harvtxt|Isaksen|Wang|Xu|2020}}, and the 3- and 5-components in {{harvtxt|Ravenel|2003}}.</ref> The mod 8 behavior of the table comes from Bott periodicity via the [[J-homomorphism]], whose image is underlined.

{| class="toccontents" border="1" cellpadding="4" style="text-align: center"
|-style="background: #eee"
!  ''n'' →
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! π<sub>0+''n''</sub><sup>''S''</sup>
| ∞
| <u>2</u>
| 2
| <u>8⋅3</u>
| ⋅
| ⋅
| 2
| <u>16⋅3⋅5</u>
|-
! π<sub>8+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅2
| <u>2</u>⋅2<sup>2</sup>
| 2⋅3
| <u>8⋅9⋅7</u>
| ⋅
| 3
| 2<sup>2</sup>
| <u>32</u>⋅2⋅<u>3⋅5</u>
|-
! π<sub>16+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅2
| <u>2</u>⋅2<sup>3</sup>
| 8⋅2
| <u>8</u>⋅2⋅<u>3⋅11</u>
| 8⋅3
| 2<sup>2</sup>
| 2⋅2
| <u>16</u>⋅8⋅2⋅<u>9</u>⋅3⋅<u>5⋅7⋅13</u>
|-
! π<sub>24+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅2
| <u>2</u>⋅2
| 2<sup>2</sup>⋅3
| <u>8⋅3</u>
| 2
| 3
| 2⋅3
| <u>64</u>⋅2<sup>2</sup>⋅<u>3⋅5⋅17</u>
|-
! π<sub>32+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅2<sup>3</sup>
| <u>2</u>⋅2<sup>4</sup>
| 4⋅2<sup>3</sup>
| <u>8</u>⋅2<sup>2</sup>⋅<u>27⋅7⋅19</u>
| 2⋅3
| 2<sup>2</sup>⋅3
| 4⋅2⋅3⋅5
| <u>16</u>⋅2<sup>5</sup>⋅3⋅<u>3⋅25⋅11</u>
|-
! π<sub>40+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅4⋅2<sup>4</sup>⋅3
| <u>2</u>⋅2<sup>4</sup>
| 8⋅2<sup>2</sup>⋅3
| <u>8⋅3⋅23</u>
| 8
| 16⋅2<sup>3</sup>⋅9⋅5
| 2<sup>4</sup>⋅3
| <u>32</u>⋅4⋅2<sup>3</sup>⋅<u>9</u>⋅3⋅<u>5⋅7⋅13</u>
|-
! π<sub>48+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅4⋅2<sup>3</sup>
| <u>2</u>⋅2⋅3
| 2<sup>3</sup>⋅3
| <u>8</u>⋅8⋅2⋅<u>3</u>
| 2<sup>3</sup>⋅3
| 2<sup>4</sup>
| 4⋅2
| <u>16</u>⋅3⋅<u>3⋅5⋅29</u>
|-
! π<sub>56+''n''</sub><sup>''S''</sup>
| <u>2</u>
| <u>2</u>⋅2<sup>2</sup>
| 2<sup>2</sup>
| <u>8</u>⋅2<sup>2</sup>⋅<u>9⋅7⋅11⋅31</u>
| 4
| ⋅
| 2<sup>4</sup>⋅3
| <u>128</u>⋅4⋅2<sup>2</sup>⋅<u>3⋅5⋅17</u>
|-
! π<sub>64+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅4⋅2<sup>5</sup>
| <u>2</u>⋅4⋅2<sup>8</sup>⋅3
| 8⋅2<sup>6</sup>
| <u>8</u>⋅4⋅2<sup>3</sup>⋅<u>3</u>
| 2<sup>3</sup>⋅3
| 2<sup>4</sup>
| 4<sup>2</sup>⋅2<sup>5</sup>
| <u>16</u>⋅8⋅4⋅2<sup>6</sup>⋅<u>27⋅5⋅7⋅13⋅19⋅37</u>
|-
! π<sub>72+''n''</sub><sup>''S''</sup>
| <u>2</u>⋅2<sup>7</sup>⋅3
| <u>2</u>⋅2<sup>6</sup>
| 4<sup>3</sup>⋅2⋅3
| <u>8</u>⋅2⋅9⋅<u>3</u>
| 4⋅2<sup>2</sup>⋅5
| 4⋅2<sup>5</sup>
| 4<sup>2</sup>⋅2<sup>3</sup>⋅3
| <u>32</u>⋅4⋅2<sup>6</sup>⋅<u>3⋅25⋅11⋅41</u>
|}

==References==
===Notes===
{{reflist}}

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{{refend}}

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==External links==
* {{citation
 |url= http://math.ucr.edu/home/baez/week102.html
 |title= This week's finds in mathematical physics 102
 |last= Baez
 |first= John
 |date=21 April 1997
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* {{citation
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 |title= Stable homotopy groups of spheres
 |first= Allen
 |last= Hatcher
 |access-date= 2007-10-20
}}
* {{citation
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 |title= A history of Topology
 |year= 1996
 |url= http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology_in_mathematics.html
 |access-date= 2007-11-14
 }} in [[MacTutor History of Mathematics archive]].
* {{citation
 |last1= O'Connor
 |first1= J. J.
 |last2= Robertson
 |first2= E. F.
 |title= Marie Ennemond Camille Jordan
 |year= 2001
 |url= http://www-history.mcs.st-andrews.ac.uk/Biographies/Jordan.html
 |access-date= 2007-11-14
 }} in MacTutor History of Mathematics archive.

[[Category:Homotopy theory]]
[[Category:Spheres]]

{{Sourceattribution|Homotopy groups of spheres}}