Homotopy Groups of Spheres - History

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 (O'Connor & Robertson 2001). A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers Analysis situs where the related concepts of homology and the fundamental group were also introduced (O'Connor & Robertson 1996).

Higher homotopy groups were first defined by Eduard Čech in 1932 (Čech 1932, p. 203). (His first paper was withdrawn on the advice of Pavel Sergeyevich Alexandrov and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the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 and also for the Hurewicz theorem which can be used to calculate some of the groups (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 suspension theorem, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results (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 π_n(Sn) and π_4n−1(S2n). Others who worked in this area included José Ádem, Hiroshi Toda, Frank Adams and J. Peter May. The stable homotopy groups π_n+k(Sn) are known for k up to 64, and, as of 2007, unknown for larger k (Hatcher 2002, Stable homotopy groups, pp. 385–393).