Editing Homotopy groups of spheres (section)

==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]] and [[abstract algebra]], with [[homotopy group]]s as a basic example.

=== {{mvar|n}}-sphere ===
An ordinary [[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 [[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]].{{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 (punctuation)|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]], 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 [[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]] is its continuity structure, formalized in terms of [[open set]]s or [[neighborhood (mathematics)|neighborhood]]s. A [[continuous map]] is a function between spaces that preserves continuity. A [[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]] of [[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]], {{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]]s) are grouped together into [[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]]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]].

A continuous map between two topological spaces induces a [[group homomorphism]] between the associated homotopy groups. In particular, if the map is a continuous [[bijection]] (a [[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}}