Editing Homotopy groups of spheres (section)

==History==
In the late 19th century [[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 [[Henri Poincaré]] in his 1895 set of papers ''[[Analysis Situs (paper)|Analysis situs]]'' where the related concepts of [[homology (mathematics)|homology]] and the [[fundamental group]] were also introduced.{{sfn|O'Connor|Robertson|1996}}

Higher homotopy groups were first defined by [[Eduard Čech]] in 1932.{{sfn|Čech|1932|p=203}} (His first paper was withdrawn on the advice of [[Pavel Alexandrov|Pavel Sergeyevich Alexandrov]] and Heinz Hopf, on the grounds that the groups were commutative so could not be the right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.{{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 [[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 [[George W. Whitehead]] showed that there is a metastable range for the homotopy groups of spheres. [[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 [[José Adem]], [[Hiroshi Toda]], [[Frank Adams]], [[J. Peter May]], [[Mark Mahowald]], [[Daniel Isaksen]], [[Guozhen Wang]], and [[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|2023}}