Stable homotopy group

$ k $-stable homotopy group $ \pi _ {k} ^ {s\star} ( X) $, of a topological space $ X $

The inductive limit of the sequence

$$ \tag{* } \pi _ {k} ( X) \rightarrow ^ { E } \pi _ {k+1} ( EX) \rightarrow ^ { E } \pi _ {k+2} ( E ^ {2} X) \rightarrow ^ { E } \dots , $$

where $ EY $ is the suspension over the topological space $ Y $. The suspension homomorphism $ E: \pi _ {m} ( Y) \rightarrow \pi _ {m+1} ( EY) $ relates the class of the spheroid $ f: S ^ {m} \rightarrow Y $ to the class of the spheroid $ Ef: ES ^ {m} = S ^ {m+1} \rightarrow EY $, where $ Ef $ is obtained by factorization from the mapping $ f \times \mathop{\rm Id} _ {( 0,1] } $. The sequence (*) stabilizes at the $ ( k+ 3) $-rd term (see [2]), so that $ \pi _ {k} ^ {s} ( X) = \pi _ {2k+2} ( E ^ {k+2} X) $.

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.

References

[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)