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May, 2019
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Homotopy Groups

When the domain X is a suspension of a CW complex with few cells, the based homotopy set [ X,Y ] * is called a homotopy group. Although homotopy groups are as fundamental as homology groups in algebraic topology, most elementary textbooks only treat the fundamental groups . It should be noted, however, homotopy groups ( homotopy sets ) are more fundamental than homology groups, since homology groups can be described in terms of homotopy groups by the Dold-Thom theorem .

One of the most famous textbooks on algebraic topology which contains enough materials on homotopy groups is the book [ Whi78 ] by G.W. Whitehead. And Gray’s book [ Gra75 ] . There are two good Japanese books, an old book by Komatsu, Nakaoka, and Sugawara and more recent book by Nishida.

Both Whitehead’s and Gray’s books are entitled “Homotopy Theory”. One of the central objects of study in homotopy theory has been homotopy groups, and especially homotopy groups of spheres .

Many people tried to compute homotopy groups of spheres. One of the first systematic attempts was Adams’ work [ Ada58 ] , but Adam’s method initially only worked for stable homotopy groups. Toda’s method [ Tod62 ] is more elementary and provide a powerful computational method.

One of the most popular method to generalize homotopy groups is to replace the spheres in the definition of homotopy groups by other spaces. And many more variations have been introduced.