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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871202.png" />-stable homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871203.png" />, of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871204.png" />''
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'' $  k $-stable homotopy group $  \pi _ {k}  ^ {s\star} ( X) $, of a topological space $  X $''
  
 
The inductive limit of the sequence
 
The inductive limit of the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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\pi _ {k} ( X)  \rightarrow ^ { E }  \pi _ {k+1} ( EX)  \rightarrow ^ { E }  \pi _ {k+2} ( E  ^ {2} X) \rightarrow ^ { E }  \dots ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871206.png" /> is the [[Suspension|suspension]] over the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871207.png" />. The suspension homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871208.png" /> relates the class of the spheroid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s0871209.png" /> to the class of the spheroid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s08712010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s08712011.png" /> is obtained by factorization from the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s08712012.png" />. The sequence (*) stabilizes at the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s08712013.png" />-rd term (see [[#References|[2]]]), so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087120/s08712014.png" />.
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where $  EY $
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is the [[Suspension|suspension]] over the topological space $  Y $.  
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The suspension homomorphism $  E: \pi _ {m} ( Y) \rightarrow \pi _ {m+1} ( EY) $
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relates the class of the spheroid $  f: S  ^ {m} \rightarrow Y $
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to the class of the spheroid $  Ef:  ES  ^ {m} = S  ^ {m+1} \rightarrow EY $,  
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where $  Ef $
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is obtained by factorization from the mapping $  f \times  \mathop{\rm Id} _ {( 0,1] }  $.  
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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.
 
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.

Latest revision as of 16:00, 2 March 2022


$ 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)
How to Cite This Entry:
Stable homotopy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_homotopy_group&oldid=17968
This article was adapted from an original article by D.B. Fuks (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article