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'' $  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|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 [[#References|[2]]]), so that  $  \pi _ {k}  ^ {s} ( X) = \pi _ {2k+2} ( E  ^ {k+2} X) $.

In calculating stable homotopy groups, the Adams [[Spectral sequence|spectral sequence]] is used (see [[#References|[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|Spheres, homotopy groups of the]]), for an infinite-dimensional real projective space and for various other spaces.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.B. Fuks,   A.T. Fomenko,   V.L. Gutenmakher,   "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Whitehead,   "Recent advances in homotopy theory" , Amer. Math. Soc.  (1970)</TD></TR></table>