Stable homotopy group

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-stable homotopy group , of a topological space

The inductive limit of the sequence


where is the suspension over the topological space . The suspension homomorphism relates the class of the spheroid to the class of the spheroid , where is obtained by factorization from the mapping . The sequence (*) stabilizes at the -rd term (see [2]), so that .

In calculating stable homotopy groups, the Adams spectral sequence is used (see [1]). Up till now (1991), no stable homotopy group is known at all for any incontractible space. However, partial calculations are known for the homotopy groups of the spheres (cf. Spheres, homotopy groups of the), for an infinite-dimensional real projective space and for various other spaces.


[1] D.B. Fuks, A.T. Fomenko, V.L. Gutenmakher, "Homotopic topology" , Moscow (1969) (In Russian)
[2] J. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970)
How to Cite This Entry:
Stable homotopy group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.B. Fuks (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article