Editing Homotopy groups of spheres (section)

{{Short description|How spheres of various dimensions can wrap around each other}}
{{More footnotes|date=September 2022}}
[[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]], the '''homotopy groups of spheres''' describe how spheres of various [[Dimension#In mathematics|dimension]]s can wrap around each other. They are examples of [[topological invariant]]s, which reflect, in [[abstract algebra|algebraic]] terms, the structure of spheres viewed as [[topological space]]s, forgetting about their precise geometry. Unlike [[homology group]]s, which are also topological invariants, the [[homotopy group]]s are surprisingly complex and difficult to compute.

[[Image:Hopf Fibration.png|right|thumb|The [[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]] ({{math|''S''<sup>1</sup>}}) and the ordinary [[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]]es of mappings are summarized.  An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an [[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>}}. In the smooth case, it follows directly from [[Sard's theorem|Sard's Theorem]]. Therefore the homotopy group is the [[trivial group]].
* When {{math|''i'' {{=}} ''n''}}, every map from {{math|''S''<sup>''n''</sup>}} to itself has a [[degree of a map|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]]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 90.{{sfn|Isaksen|Wang|Xu|2023}} 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]]s, a technique first applied to homotopy groups of spheres by [[Jean-Pierre Serre]]. Several important patterns have been established, yet much remains unknown and unexplained.