Editing Homotopy groups of spheres (section)

=== 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}}