Editing Homotopy group (section)

{{Short description|Algebraic construct classifying topological spaces}}
{{Use American English|date=January 2019}}
In [[mathematics]], '''homotopy groups''' are used in [[algebraic topology]] to classify [[topological space]]s. The first and simplest homotopy group is the [[fundamental group]], denoted <math>\pi_1(X),</math> which records information about [[Loop (topology)|loop]]s in a [[Mathematical space|space]].  Intuitively, homotopy groups record information about the basic shape, or ''[[Hole (topology)|holes]]'', of a topological space.

To define the ''n''th homotopy group, the base-point-preserving maps from an [[N-sphere|''n''-dimensional sphere]] (with [[base point]]) into a given space (with base point)  are collected into [[equivalence class]]es, called '''[[homotopy class]]es.''' Two mappings are '''homotopic''' if one can be continuously deformed into the other. These homotopy classes form a [[group (mathematics)|group]], called the''' ''n''th homotopy group''', <math>\pi_n(X),</math> of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never [[homeomorphic]], but topological spaces that {{em|are not}} homeomorphic {{em|can}} have the same homotopy groups.

The notion of homotopy of [[Path (topology)|path]]s was introduced by [[Camille Jordan]].<ref>{{Citation|title=Marie Ennemond Camille Jordan|url=http://www-history.mcs.st-and.ac.uk/~history/Biographies/Jordan.html}}</ref>