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One of the generalizations of the one-dimensional cohomology group; a concept which is, in a certain sense, dual to that of [[Homotopy group|homotopy group]].
 
One of the generalizations of the one-dimensional cohomology group; a concept which is, in a certain sense, dual to that of [[Homotopy group|homotopy group]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231801.png" /> be the set of homotopy classes of continuous mappings from a pointed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231802.png" /> to the pointed sphere. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231803.png" /> does not always have a natural group structure. (This is the case only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231804.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231805.png" /> is then a group.) The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231806.png" /> is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231807.png" />.
+
Let $  \pi  ^ {n} ( X) = [ X , S  ^ {n} ] $
 +
be the set of homotopy classes of continuous mappings from a pointed space $  X $
 +
to the pointed sphere. The set $  \pi  ^ {n} ( X) $
 +
does not always have a natural group structure. (This is the case only for $  n = 1 , 3 $,  
 +
since $  S  ^ {n} $
 +
is then a group.) The group $  \pi  ^ {1} ( X) $
 +
is the same as $  H  ^ {1} ( X , \mathbf Z ) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231808.png" /> is a [[CW-complex|CW-complex]] of dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c0231809.png" />, then a group structure can be defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318010.png" /> in the following way. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318011.png" /> one considers the mapping
+
If $  X $
 +
is a [[CW-complex|CW-complex]] of dimension at most $  2 n - 2 $,  
 +
then a group structure can be defined on $  \pi  ^ {n} ( X) $
 +
in the following way. For $  [ \alpha ] , [ \beta ] \in \pi  ^ {n} ( X) $
 +
one considers the mapping
  
<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/c/c023/c023180/c02318012.png" /></td> </tr></table>
+
$$
 +
( \alpha \times \beta ) \circ \Delta : \
 +
X  \rightarrow  S  ^ {n} \times S  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318013.png" /> is the diagonal mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318014.png" /> are representatives of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318015.png" />. In view of the restriction on the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318016.png" /> there is a unique homotopy class of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318017.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318018.png" /> is a bouquet of pointed spheres) the composite of which with the natural inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318019.png" /> is the same as the homotopy class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318020.png" />. The homotopy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318023.png" /> is the folding mapping, is set equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318024.png" />. With respect to this operation the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318025.png" /> is an Abelian group; therefore, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318026.png" /> is often regarded as a functor defined only on the category of CW-complexes of dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318027.png" />, with values in the category of Abelian groups. For CW-complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318028.png" /> of dimension less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318030.png" />. Thus, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318031.png" /> is of interest in dimensions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318032.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318033.png" />, that is, in the so-called stable dimensions.
+
where $  \Delta : X \rightarrow X \times X $
 +
is the diagonal mapping and $  \alpha , \beta : X \rightarrow S  ^ {n} $
 +
are representatives of the classes $  [ \alpha ] , [ \beta ] $.  
 +
In view of the restriction on the dimension of $  X $
 +
there is a unique homotopy class of mappings $  f : X \rightarrow S  ^ {n} \lor S  ^ {n} $(
 +
here $  S  ^ {n} \lor S  ^ {n} $
 +
is a bouquet of pointed spheres) the composite of which with the natural inclusion $  S  ^ {n} \lor S  ^ {n} \subset  S  ^ {n} \times S  ^ {n} $
 +
is the same as the homotopy class of $  ( \alpha \times \beta ) \circ \Delta $.  
 +
The homotopy class $  [ \theta \circ f  ] \in \pi  ^ {n} ( X) $
 +
of $  \theta \circ f : X \rightarrow S  ^ {n} $,  
 +
where $  \theta : S  ^ {n} \lor S  ^ {n} \rightarrow S  ^ {n} $
 +
is the folding mapping, is set equal to $  [ \alpha ] + [ \beta ] \in \pi  ^ {n} ( X) $.  
 +
With respect to this operation the set $  \pi  ^ {n} ( X) $
 +
is an Abelian group; therefore, the functor $  \pi  ^ {n} $
 +
is often regarded as a functor defined only on the category of CW-complexes of dimension at most $  2 n - 2 $,  
 +
with values in the category of Abelian groups. For CW-complexes $  X $
 +
of dimension less than $  n $,  
 +
$  \pi  ^ {n} ( X) = 0 $.  
 +
Thus, the functor $  \pi  ^ {n} $
 +
is of interest in dimensions from $  n $
 +
to $  2 n - 2 $,  
 +
that is, in the so-called stable dimensions.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318036.png" /> is the [[Suspension|suspension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318037.png" />. This isomorphism is given by the suspension functor: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318039.png" /> is an arbitrary finite-dimensional CW-complex, then for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318040.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318041.png" /> has a group structure (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318042.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318043.png" />). The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318044.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318045.png" /> is called the stable cohomotopy group of the CW-complex. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318046.png" /> are defined for all integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318047.png" /> (and not merely positive integers). If one chooses for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318048.png" /> two points (one of which is distinguished), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318051.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318052.png" /> are the stable homotopy groups of spheres for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318053.png" />.
+
If $  \mathop{\rm dim}  X \leq  2 n - 2 $,  
 +
then $  \pi  ^ {n} ( X) \approx \pi  ^ {n+} 1 ( S X ) $,  
 +
where $  S X $
 +
is the [[Suspension|suspension]] of $  X $.  
 +
This isomorphism is given by the suspension functor: $  [ X , S  ^ {n} ] \rightarrow [ S X , S S  ^ {n} ] = [ S X , S  ^ {n+} 1 ] $.  
 +
If $  X $
 +
is an arbitrary finite-dimensional CW-complex, then for sufficiently large $  N $
 +
the set $  \pi  ^ {n+} N ( S  ^ {N} X ) $
 +
has a group structure (for $  N \geq  \mathop{\rm dim}  X - 2 n + 2 $
 +
one has $  \mathop{\rm dim}  ( S  ^ {n} X ) = N +  \mathop{\rm dim}  X \leq  2 ( n + N ) - 2 $).  
 +
The group $  \pi _ {S}  ^ {n} ( X) = \pi  ^ {n+} N ( S  ^ {N} X ) $
 +
with $  N \geq  \mathop{\rm dim}  X - 2 n + 2 $
 +
is called the stable cohomotopy group of the CW-complex. The groups $  \pi _ {S}  ^ {n} ( X) $
 +
are defined for all integer $  n $(
 +
and not merely positive integers). If one chooses for $  X $
 +
two points (one of which is distinguished), then $  \pi _ {S}  ^ {n} ( X) = 0 $
 +
for $  n \geq  0 $,  
 +
$  \pi _ {S}  ^ {0} ( x) = \mathbf Z $,  
 +
and $  \pi _ {S}  ^ {n} ( X) = \pi _ {N-} n ( S  ^ {N} ) $
 +
are the stable homotopy groups of spheres for $  n < 0 $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318054.png" /> is a pair of CW-complexes of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318055.png" />, then when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318056.png" />, the relative cohomotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318057.png" /> is defined. One has the following exact sequence of Abelian groups:
+
If $  ( X ;  A ) $
 +
is a pair of CW-complexes of dimension $  m $,  
 +
then when $  m \leq  2 n - 2 $,  
 +
the relative cohomotopy group $  \pi  ^ {n} ( X , A ) = \pi  ^ {n} ( X / A ) $
 +
is defined. One has the following exact sequence of Abelian groups:
  
<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/c/c023/c023180/c02318058.png" /></td> </tr></table>
+
$$
 +
\pi  ^ {i} ( X)  \rightarrow \
 +
\pi  ^ {i} ( A)  \rightarrow \
 +
\pi  ^ {i+} 1 ( X , A )
 +
\rightarrow  \pi  ^ {i+} 1 ( X) \rightarrow
 +
$$
  
<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/c/c023/c023180/c02318059.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
\pi  ^ {i+} 1 ( A)  \rightarrow  \pi  ^ {i+} 2 ( X , A )  \rightarrow \dots ,
 +
$$
  
extending indefinitely to the right; however, from some term onwards all groups are trivial: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318060.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318061.png" />. This sequence extends to the left only as far as those values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318062.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318063.png" />. In this sequence the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318065.png" /> are induced by the natural mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318067.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318068.png" /> is constructed as follows. For a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318069.png" /> and a representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318070.png" /> of it, one chooses an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318072.png" /> defined on the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318073.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318074.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318075.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318076.png" />, the homotopy class of which (an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318077.png" />) is put in correspondence with the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318078.png" />.
+
extending indefinitely to the right; however, from some term onwards all groups are trivial: $  \pi  ^ {i} ( X , A ) = \pi  ^ {i} ( X) = \pi  ^ {i} ( A) = 0 $
 +
when $  i > m $.  
 +
This sequence extends to the left only as far as those values of $  i $
 +
for which $  m \leq  2 i - 2 $.  
 +
In this sequence the homomorphisms $  \pi  ^ {i} ( X) \rightarrow \pi  ^ {i} ( A) $
 +
and $  \pi  ^ {i} ( X / A ) \rightarrow \pi  ^ {i} ( X) $
 +
are induced by the natural mappings $  A \subset  X $
 +
and $  X \rightarrow X / A $.  
 +
The homomorphism $  \pi  ^ {i} ( A) \rightarrow \pi  ^ {i+} 1 ( X / A ) $
 +
is constructed as follows. For a class $  [ f ] \in \pi  ^ {i} ( A) = [ A , S  ^ {i} ] $
 +
and a representative $  f : A \rightarrow S  ^ {i} $
 +
of it, one chooses an extension $  F : X \rightarrow D  ^ {i+} 1 $
 +
of $  f $
 +
defined on the subspace $  A \subset  X $
 +
with values in $  S  ^ {i} \subset  D  ^ {i+} 1 $.  
 +
The mapping $  F $
 +
induces a mapping $  X / A \rightarrow D  ^ {i+} 1 / S  ^ {i} = S  ^ {i+} 1 $,  
 +
the homotopy class of which (an element of $  \pi  ^ {i+} 1 ( X , A ) $)  
 +
is put in correspondence with the class $  [ f ] \in \pi  ^ {i} ( A) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318079.png" /> is a pair of pointed CW-complexes of finite dimension, then there is the exact sequence of stable cohomotopy groups
+
If $  ( X , A ) $
 +
is a pair of pointed CW-complexes of finite dimension, then there is the exact sequence of stable cohomotopy groups
  
<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/c/c023/c023180/c02318080.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  \pi _ {S}  ^ {i}
 +
( X)  \rightarrow  \pi _ {S}  ^ {i}
 +
( A)  \rightarrow  \pi _ {S}  ^ {i+} 1
 +
( X , A )  \rightarrow  \pi _ {S}  ^ {i+} 1 ( X)  \rightarrow \dots ,
 +
$$
  
extending indefinitely in both directions. This circumstance enables one to convert the stable cohomotopy groups into a generalized cohomology theory. For an arbitrary (non-pointed) finite-dimensional CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318081.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318082.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318083.png" /> is the pointed CW-complex obtained as the disjoint union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318084.png" /> with a distinguished point. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023180/c02318085.png" />, defined on the category of finite-dimensional CW-complexes, provides a generalized cohomology theory by setting
+
extending indefinitely in both directions. This circumstance enables one to convert the stable cohomotopy groups into a generalized cohomology theory. For an arbitrary (non-pointed) finite-dimensional CW-complex $  X $,
 +
let  $  \pi _ {S}  ^ {i} ( X) = \pi _ {S}  ^ {i} ( X \cup x _ {0} , x _ {0} ) $,  
 +
where $  ( X \cup x _ {0} , x _ {0} ) $
 +
is the pointed CW-complex obtained as the disjoint union of $  X $
 +
with a distinguished point. The functor $  \pi _ {S}  ^ {*} $,  
 +
defined on the category of finite-dimensional CW-complexes, provides a generalized cohomology theory by setting
  
<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/c/c023/c023180/c02318086.png" /></td> </tr></table>
+
$$
 +
\pi _ {S}  ^ {*} ( X , A )  = \
 +
{\pi _ {S}  ^ {*} } tilde
 +
( X / A )  = \
 +
\mathop{\rm Ker} [ \pi _ {S}  ^ {*} ( X / A )
 +
\rightarrow \pi _ {S}  ^ {*} (  \mathop{\rm pt} ) ] .
 +
$$
  
 
The value at a point of this theory is the same as the stable homotopy groups of spheres.
 
The value at a point of this theory is the same as the stable homotopy groups of spheres.
Line 33: Line 145:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


One of the generalizations of the one-dimensional cohomology group; a concept which is, in a certain sense, dual to that of homotopy group.

Let $ \pi ^ {n} ( X) = [ X , S ^ {n} ] $ be the set of homotopy classes of continuous mappings from a pointed space $ X $ to the pointed sphere. The set $ \pi ^ {n} ( X) $ does not always have a natural group structure. (This is the case only for $ n = 1 , 3 $, since $ S ^ {n} $ is then a group.) The group $ \pi ^ {1} ( X) $ is the same as $ H ^ {1} ( X , \mathbf Z ) $.

If $ X $ is a CW-complex of dimension at most $ 2 n - 2 $, then a group structure can be defined on $ \pi ^ {n} ( X) $ in the following way. For $ [ \alpha ] , [ \beta ] \in \pi ^ {n} ( X) $ one considers the mapping

$$ ( \alpha \times \beta ) \circ \Delta : \ X \rightarrow S ^ {n} \times S ^ {n} , $$

where $ \Delta : X \rightarrow X \times X $ is the diagonal mapping and $ \alpha , \beta : X \rightarrow S ^ {n} $ are representatives of the classes $ [ \alpha ] , [ \beta ] $. In view of the restriction on the dimension of $ X $ there is a unique homotopy class of mappings $ f : X \rightarrow S ^ {n} \lor S ^ {n} $( here $ S ^ {n} \lor S ^ {n} $ is a bouquet of pointed spheres) the composite of which with the natural inclusion $ S ^ {n} \lor S ^ {n} \subset S ^ {n} \times S ^ {n} $ is the same as the homotopy class of $ ( \alpha \times \beta ) \circ \Delta $. The homotopy class $ [ \theta \circ f ] \in \pi ^ {n} ( X) $ of $ \theta \circ f : X \rightarrow S ^ {n} $, where $ \theta : S ^ {n} \lor S ^ {n} \rightarrow S ^ {n} $ is the folding mapping, is set equal to $ [ \alpha ] + [ \beta ] \in \pi ^ {n} ( X) $. With respect to this operation the set $ \pi ^ {n} ( X) $ is an Abelian group; therefore, the functor $ \pi ^ {n} $ is often regarded as a functor defined only on the category of CW-complexes of dimension at most $ 2 n - 2 $, with values in the category of Abelian groups. For CW-complexes $ X $ of dimension less than $ n $, $ \pi ^ {n} ( X) = 0 $. Thus, the functor $ \pi ^ {n} $ is of interest in dimensions from $ n $ to $ 2 n - 2 $, that is, in the so-called stable dimensions.

If $ \mathop{\rm dim} X \leq 2 n - 2 $, then $ \pi ^ {n} ( X) \approx \pi ^ {n+} 1 ( S X ) $, where $ S X $ is the suspension of $ X $. This isomorphism is given by the suspension functor: $ [ X , S ^ {n} ] \rightarrow [ S X , S S ^ {n} ] = [ S X , S ^ {n+} 1 ] $. If $ X $ is an arbitrary finite-dimensional CW-complex, then for sufficiently large $ N $ the set $ \pi ^ {n+} N ( S ^ {N} X ) $ has a group structure (for $ N \geq \mathop{\rm dim} X - 2 n + 2 $ one has $ \mathop{\rm dim} ( S ^ {n} X ) = N + \mathop{\rm dim} X \leq 2 ( n + N ) - 2 $). The group $ \pi _ {S} ^ {n} ( X) = \pi ^ {n+} N ( S ^ {N} X ) $ with $ N \geq \mathop{\rm dim} X - 2 n + 2 $ is called the stable cohomotopy group of the CW-complex. The groups $ \pi _ {S} ^ {n} ( X) $ are defined for all integer $ n $( and not merely positive integers). If one chooses for $ X $ two points (one of which is distinguished), then $ \pi _ {S} ^ {n} ( X) = 0 $ for $ n \geq 0 $, $ \pi _ {S} ^ {0} ( x) = \mathbf Z $, and $ \pi _ {S} ^ {n} ( X) = \pi _ {N-} n ( S ^ {N} ) $ are the stable homotopy groups of spheres for $ n < 0 $.

If $ ( X ; A ) $ is a pair of CW-complexes of dimension $ m $, then when $ m \leq 2 n - 2 $, the relative cohomotopy group $ \pi ^ {n} ( X , A ) = \pi ^ {n} ( X / A ) $ is defined. One has the following exact sequence of Abelian groups:

$$ \pi ^ {i} ( X) \rightarrow \ \pi ^ {i} ( A) \rightarrow \ \pi ^ {i+} 1 ( X , A ) \rightarrow \pi ^ {i+} 1 ( X) \rightarrow $$

$$ \rightarrow \ \pi ^ {i+} 1 ( A) \rightarrow \pi ^ {i+} 2 ( X , A ) \rightarrow \dots , $$

extending indefinitely to the right; however, from some term onwards all groups are trivial: $ \pi ^ {i} ( X , A ) = \pi ^ {i} ( X) = \pi ^ {i} ( A) = 0 $ when $ i > m $. This sequence extends to the left only as far as those values of $ i $ for which $ m \leq 2 i - 2 $. In this sequence the homomorphisms $ \pi ^ {i} ( X) \rightarrow \pi ^ {i} ( A) $ and $ \pi ^ {i} ( X / A ) \rightarrow \pi ^ {i} ( X) $ are induced by the natural mappings $ A \subset X $ and $ X \rightarrow X / A $. The homomorphism $ \pi ^ {i} ( A) \rightarrow \pi ^ {i+} 1 ( X / A ) $ is constructed as follows. For a class $ [ f ] \in \pi ^ {i} ( A) = [ A , S ^ {i} ] $ and a representative $ f : A \rightarrow S ^ {i} $ of it, one chooses an extension $ F : X \rightarrow D ^ {i+} 1 $ of $ f $ defined on the subspace $ A \subset X $ with values in $ S ^ {i} \subset D ^ {i+} 1 $. The mapping $ F $ induces a mapping $ X / A \rightarrow D ^ {i+} 1 / S ^ {i} = S ^ {i+} 1 $, the homotopy class of which (an element of $ \pi ^ {i+} 1 ( X , A ) $) is put in correspondence with the class $ [ f ] \in \pi ^ {i} ( A) $.

If $ ( X , A ) $ is a pair of pointed CW-complexes of finite dimension, then there is the exact sequence of stable cohomotopy groups

$$ \dots \rightarrow \pi _ {S} ^ {i} ( X) \rightarrow \pi _ {S} ^ {i} ( A) \rightarrow \pi _ {S} ^ {i+} 1 ( X , A ) \rightarrow \pi _ {S} ^ {i+} 1 ( X) \rightarrow \dots , $$

extending indefinitely in both directions. This circumstance enables one to convert the stable cohomotopy groups into a generalized cohomology theory. For an arbitrary (non-pointed) finite-dimensional CW-complex $ X $, let $ \pi _ {S} ^ {i} ( X) = \pi _ {S} ^ {i} ( X \cup x _ {0} , x _ {0} ) $, where $ ( X \cup x _ {0} , x _ {0} ) $ is the pointed CW-complex obtained as the disjoint union of $ X $ with a distinguished point. The functor $ \pi _ {S} ^ {*} $, defined on the category of finite-dimensional CW-complexes, provides a generalized cohomology theory by setting

$$ \pi _ {S} ^ {*} ( X , A ) = \ {\pi _ {S} ^ {*} } tilde ( X / A ) = \ \mathop{\rm Ker} [ \pi _ {S} ^ {*} ( X / A ) \rightarrow \pi _ {S} ^ {*} ( \mathop{\rm pt} ) ] . $$

The value at a point of this theory is the same as the stable homotopy groups of spheres.

As for homotopy groups, the cohomotopy groups cannot be explicitly calculated even in the simplest cases, and this severely restricts the possibility of practical application of the above functors.

References

[1] S.-T. Hu, "Homotopy theory" , Acad. Press (1959)
[2] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)

Comments

References

[a1] G.W. Whitehead, "Elements of homotopy theory" , Springer (1978)
How to Cite This Entry:
Cohomotopy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomotopy_group&oldid=46395
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article