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''stationary duality, Spanier duality''
 
''stationary duality, Spanier duality''
  
A [[Duality|duality]] in homotopy theory which exists (in the absence of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830002.png" />-homotopy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830003.png" />-cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830005.png" />-category, is a [[Category|category]] whose objects are topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830006.png" />, while its morphisms are classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830007.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830009.png" />-homotopic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300010.png" /> from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300011.png" />-fold [[Suspension|suspension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300012.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300015.png" /> being considered as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300016.png" />-homotopic if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300017.png" /> such that the suspensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300019.png" /> are homotopic in the ordinary sense. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300020.png" /> of such classes, which are known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300022.png" />-mappings, constitutes an Abelian group with respect to the so-called track addition [[#References|[1]]], [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300023.png" /> is the limit of the direct spectrum of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300024.png" /> of ordinary homotopy classes with suspension mappings as projections; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300025.png" /> is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300026.png" /> in which the corresponding elements are represented by one and the same mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300028.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300030.png" />-dual polyhedron of the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300031.png" /> in a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300032.png" /> is an arbitrary polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300034.png" /> which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300035.png" />-deformation retract of the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300036.png" />, i.e. the morphism corresponding to the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300037.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300038.png" />-equivalence. The polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300039.png" /> exists for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300041.png" /> may be considered as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300042.png" />.
+
A [[Duality|duality]] in homotopy theory which exists (in the absence of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the $  S $-
 +
homotopy and $  S $-
 +
cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or $  S $-
 +
category, is a [[Category|category]] whose objects are topological spaces $  X $,  
 +
while its morphisms are classes $  \{ f  \} $
 +
of $  S $-
 +
homotopic mappings $  f $
 +
from a $  p $-
 +
fold [[Suspension|suspension]] $  S  ^ {p} X _ {1} $
 +
into $  S  ^ {p} X _ {2} $,  
 +
$  f $
 +
and $  g: S  ^ {q} X _ {1} \rightarrow S  ^ {q} X _ {2} $
 +
being considered as $  S $-
 +
homotopic if there exists an $  r \geq  \max ( p, q) $
 +
such that the suspensions $  S  ^ {r-} p f $
 +
and $  S  ^ {r-} q g $
 +
are homotopic in the ordinary sense. The set $  \{ X _ {1} , X _ {2} \} $
 +
of such classes, which are known as $  S $-
 +
mappings, constitutes an Abelian group with respect to the so-called track addition [[#References|[1]]], [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]. The group $  \{ X _ {1} , X _ {2} \} $
 +
is the limit of the direct spectrum of the sets $  [ S  ^ {k} X _ {1} , S  ^ {k} X _ {2} ] $
 +
of ordinary homotopy classes with suspension mappings as projections; if $  k $
 +
is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism $  S: \{ X _ {1} , X _ {2} \} \rightarrow \{ SX _ {1} , SX _ {2} \} $
 +
in which the corresponding elements are represented by one and the same mapping $  S  ^ {p} X _ {1} \rightarrow S  ^ {p} X _ {2} $,  
 +
$  p \geq  1 $.  
 +
The $  n $-
 +
dual polyhedron of the polyhedron $  X $
 +
in a sphere $  S  ^ {n} $
 +
is an arbitrary polyhedron $  D _ {n} X $
 +
in $  S  ^ {n} $
 +
which is an $  S $-
 +
deformation retract of the complement $  S  ^ {n} \setminus  X $,  
 +
i.e. the morphism corresponding to the imbedding $  D _ {n} X \subset  S  ^ {n} \setminus  X $
 +
is an $  S $-
 +
equivalence. The polyhedron $  D _ {n} X $
 +
exists for all $  X $,  
 +
and $  X $
 +
may be considered as $  D _ {n}  ^ {2} X $.
  
For any polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300043.png" /> and any polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300045.png" /> which are dual to them, there exists a unique mapping
+
For any polyhedra $  X _ {1} , X _ {2} $
 +
and any polyhedra $  D _ {n} X _ {1} $
 +
and $  D _ {n} X _ {2} $
 +
which are dual to them, there exists a unique 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/s/s083/s083000/s08300046.png" /></td> </tr></table>
+
$$
 +
D _ {n} : \{ X _ {1} , X _ {2} \}  \rightarrow \
 +
\{ D _ {n} X _ {2} , D _ {n} X _ {1} \}
 +
$$
  
 
satisfying the following conditions:
 
satisfying the following conditions:
  
a) It is an involutory contravariant functorial isomorphism, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300047.png" /> is a homomorphism such that if
+
a) It is an involutory contravariant functorial isomorphism, i.e. $  D _ {n} $
 +
is a homomorphism such that if
  
<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/s083/s083000/s08300048.png" /></td> </tr></table>
+
$$
 +
i : X _ {1}  \subset  X _ {2} \  \textrm{ and } \  i  ^  \prime  : D _ {n} X _ {2}  \subset  D _ {n} X _ {1} ,
 +
$$
  
 
then
 
then
  
<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/s083/s083000/s08300049.png" /></td> </tr></table>
+
$$
 +
D _ {n} \{ i \}  = \{ i  ^  \prime  \} ;
 +
$$
  
 
if
 
if
  
<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/s083/s083000/s08300050.png" /></td> </tr></table>
+
$$
 +
\{ f _ {1} \}  \in  \{ X _ {1} , X _ {2} \} \  \textrm{ and } \ \
 +
\{ f _ {2} \}  \in  \{ X _ {2} , X _ {3} \} ,
 +
$$
  
 
then
 
then
  
<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/s083/s083000/s08300051.png" /></td> </tr></table>
+
$$
 +
D _ {n} ( \{ f _ {2} \} \cdot \{ f _ {1} \} )  = \
 +
D _ {n} \{ f _ {1} \} \cdot D _ {n} \{ f _ {2} \} ;
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300052.png" /> is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300053.png" /> or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300055.png" />.
+
if $  \theta $
 +
is an element of $  \{ X _ {1} , X _ {2} \} $
 +
or of $  \{ D _ {n} X _ {2} , D _ {n} X _ {1} \} $,  
 +
then $  D _ {n} D _ {n} \theta = \theta $.
  
 
b) The following relations are valid:
 
b) The following relations are valid:
  
<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/s083/s083000/s08300056.png" /></td> </tr></table>
+
$$
 +
SD _ {n}  = D _ {n+} 1 \  \textrm{ and } \  D _ {n+} 1 S  = D _ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300058.png" /> are considered as polyhedra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300059.png" />-dual to polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300060.png" /> and, correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300062.png" /> this means that it does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300063.png" /> and is stationary with respect to suspension.
+
where $  SD _ {n} X _ {i} $
 +
and $  D _ {n} X _ {i} $
 +
are considered as polyhedra, $  ( n + 1 ) $-
 +
dual to polyhedra $  X _ {i} $
 +
and, correspondingly, $  SX _ {i} $,
 +
$  i = 1, 2; $
 +
this means that it does not depend on $  n $
 +
and is stationary with respect to suspension.
  
 
c) It satisfies the equation
 
c) It satisfies the equation
  
<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/s083/s083000/s08300064.png" /></td> </tr></table>
+
$$
 +
D _ {a}  ^ {n} \theta _ {*}  = ( D _ {n} \theta )  ^ {*} D _ {a}  ^ {n} ,
 +
$$
  
 
where
 
where
  
<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/s083/s083000/s08300065.png" /></td> </tr></table>
+
$$
 +
\theta _ {*} : H _ {p} ( X _ {1} )  \rightarrow  H _ {p} ( X _ {2} )
 +
$$
  
 
and
 
and
  
<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/s083/s083000/s08300066.png" /></td> </tr></table>
+
$$
 +
( D _ {n} \theta )  ^ {*} : H  ^ {n-} p- 1
 +
( D _ {n} X _ {1} )  \rightarrow  H  ^ {n-} p- 1 ( D _ {n} X _ {2} )
 +
$$
  
are homomorphisms of the above homology and cohomology groups, induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300067.png" />-mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300069.png" />, and
+
are homomorphisms of the above homology and cohomology groups, induced by $  S $-
 +
mappings $  \theta \in \{ X _ {1} , X _ {2} \} $
 +
and $  D _ {n} \theta $,  
 +
and
  
<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/s083/s083000/s08300070.png" /></td> </tr></table>
+
$$
 +
D _ {a} : H _ {p} ( X _ {i} )  \rightarrow  H  ^ {n-} p- 1
 +
( D _ {n} X _ {i} ) ,\  i= 1 , 2 ,
 +
$$
  
is an isomorphism obtained from the isomorphism of [[Alexander duality|Alexander duality]] by replacing the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300071.png" /> by its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300072.png" />-deformation retract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300073.png" />.
+
is an isomorphism obtained from the isomorphism of [[Alexander duality|Alexander duality]] by replacing the set $  S  ^ {n} \setminus  X _ {i} $
 +
by its $  S $-
 +
deformation retract $  D _ {n} X _ {i} $.
  
The construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300074.png" /> is based on the representation of a given mapping as the composition of an imbedding and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300075.png" />-deformation retract.
+
The construction of $  D _ {n} $
 +
is based on the representation of a given mapping as the composition of an imbedding and an $  S $-
 +
deformation retract.
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300077.png" />-homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300078.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300079.png" /> is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300080.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300082.png" />-cohomotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300084.png" /> is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300085.png" />. As in ordinary homotopy theory, one defines the homomorphisms
+
The $  S $-
 +
homotopy group $  \Sigma _ {p} ( X) $
 +
of a space $  X $
 +
is the group $  \{ S  ^ {p} , X \} $,  
 +
and the $  S $-
 +
cohomotopy group $  \Sigma  ^ {p} ( X) $
 +
of $  X $
 +
is the group $  \{ X, S  ^ {p} \} $.  
 +
As in ordinary homotopy theory, one defines the homomorphisms
  
<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/s083/s083000/s08300086.png" /></td> </tr></table>
+
$$
 +
\phi _ {p} : \Sigma _ {p} ( X)  \rightarrow  H _ {p} ( X) ,
 +
$$
  
<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/s083/s083000/s08300087.png" /></td> </tr></table>
+
$$
 +
\phi  ^ {p} : \Sigma  ^ {p} ( X)  \rightarrow  H  ^ {p} ( X) .
 +
$$
  
Regarding the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300089.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300090.png" />-dual leads to the isomorphisms
+
Regarding the spheres $  S  ^ {p} $
 +
and $  S  ^ {n-} p- 1 $
 +
as $  n $-
 +
dual leads to the isomorphisms
  
<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/s083/s083000/s08300091.png" /></td> </tr></table>
+
$$
 +
D _ {n} : \Sigma _ {p} ( X)  \rightarrow  \Sigma  ^ {n-} p- 1 ( D _ {n} X)
 +
$$
  
 
and to the commutative diagram
 
and to the commutative diagram
  
<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/s083/s083000/s08300092.png" /></td> </tr></table>
+
$$
  
Thus, the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300093.png" /> connects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300094.png" />-homotopy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300095.png" />-cohomotopy groups, just as the isomorphism of Alexander duality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300096.png" /> connects the homology and cohomology groups. Any duality in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300097.png" />-category entails a duality of ordinary homotopy classes if the conditions imposed on the space entail the existence of a one-to-one correspondence between the set of the above classes and the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300098.png" />-homotopy classes.
+
Thus, the isomorphism $  D _ {n} $
 +
connects $  S $-
 +
homotopy and $  S $-
 +
cohomotopy groups, just as the isomorphism of Alexander duality $  D _ {a}  ^ {n} $
 +
connects the homology and cohomology groups. Any duality in the $  S $-
 +
category entails a duality of ordinary homotopy classes if the conditions imposed on the space entail the existence of a one-to-one correspondence between the set of the above classes and the set of $  S $-
 +
homotopy classes.
  
Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300099.png" /> converts one of these theorems into the other, which means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000100.png" />-homotopy groups are replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000101.png" />-cohomotopy groups, homology groups by cohomology groups, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000102.png" /> by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000103.png" />, the smallest dimension with a non-trivial homology group by the largest dimension with a non-trivial cohomology group, and vice versa. In ordinary homotopy theory the definition of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000104.png" />-cohomotopy group requires that the dimension of the space does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000105.png" /> (or, more generally, that the space be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000106.png" />-coconnected, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000107.png" />), which impairs the perfectly general nature of duality.
+
Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. $  D _ {n} $
 +
converts one of these theorems into the other, which means that $  S $-
 +
homotopy groups are replaced by $  S $-
 +
cohomotopy groups, homology groups by cohomology groups, the mapping $  \phi _ {p} $
 +
by the mapping $  \phi  ^ {n-} p- 1 $,  
 +
the smallest dimension with a non-trivial homology group by the largest dimension with a non-trivial cohomology group, and vice versa. In ordinary homotopy theory the definition of an $  n $-
 +
cohomotopy group requires that the dimension of the space does not exceed $  2n - 2 $(
 +
or, more generally, that the space be $  ( 2n - 1) $-
 +
coconnected, $  n > 1 $),  
 +
which impairs the perfectly general nature of duality.
  
There are several trends of generalization of the theory: e.g. studies are made of spaces with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000108.png" />-homotopy type of polyhedra, the relative case, a theory with supports, etc. [[#References|[3]]], [[#References|[5]]], , [[#References|[7]]]. The theory was one of the starting points in the development of stationary homotopy theory [[#References|[8]]].
+
There are several trends of generalization of the theory: e.g. studies are made of spaces with the $  S $-
 +
homotopy type of polyhedra, the relative case, a theory with supports, etc. [[#References|[3]]], [[#References|[5]]], , [[#References|[7]]]. The theory was one of the starting points in the development of stationary homotopy theory [[#References|[8]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.H. Spanier, "Duality and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000109.png" />-theory" ''Bull. Amer. Math. Soc.'' , '''62''' (1956) pp. 194–203 {{MR|0085506}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" ''Mathematika'' , '''2''' : 3 (1955) pp. 56–80 {{MR|0074823}} {{ZBL|0064.17202}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" ''Ann. of Math.'' , '''67''' : 2 (1958) pp. 203–238 {{MR|0105105}} {{ZBL|0092.15701}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.G. Barratt, "Track groups 1; 2" ''Proc. London Math. Soc.'' , '''5''' (1955) pp. 71–106; 285–329</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "The theory of carriers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000110.png" />-theory" , ''Algebraic geometry and Topology (A Symp. in honor of S. Lefschetz)'' , Princeton Univ. Press (1957) pp. 330–360 {{MR|0084772}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" ''C.R. Acad. Sci. Paris'' , '''246''' : 17 (1958) pp. 2444–2447 {{MR|0100261}} {{ZBL|0092.39901}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" ''C.R. Acad. Sci. Paris'' , '''246''' : 18 (1958) pp. 2555–2558 {{MR|0100262}} {{ZBL|0092.40001}} </TD></TR><TR><TD valign="top">[6c]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" ''C.R. Acad. Sci. Paris'' , '''246''' : 21 (1958) pp. 2991–2993 {{MR|0100263}} {{ZBL|0092.40101}} </TD></TR><TR><TD valign="top">[6d]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" ''C.R. Acad. Sci. Paris'' , '''247''' : 6 (1958) pp. 620–623 {{MR|0100264}} {{ZBL|0092.40102}} </TD></TR><TR><TD valign="top">[6e]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Décomposition homologique d'un polyhèdre simplement connexe" ''C.R. Acad. Sci. Paris'' , '''248''' : 14 (1959) pp. 2054–2056</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) {{MR|0309097}} {{ZBL|0217.48601}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.H. Spanier, "Duality and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000109.png" />-theory" ''Bull. Amer. Math. Soc.'' , '''62''' (1956) pp. 194–203 {{MR|0085506}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" ''Mathematika'' , '''2''' : 3 (1955) pp. 56–80 {{MR|0074823}} {{ZBL|0064.17202}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" ''Ann. of Math.'' , '''67''' : 2 (1958) pp. 203–238 {{MR|0105105}} {{ZBL|0092.15701}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.G. Barratt, "Track groups 1; 2" ''Proc. London Math. Soc.'' , '''5''' (1955) pp. 71–106; 285–329</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "The theory of carriers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000110.png" />-theory" , ''Algebraic geometry and Topology (A Symp. in honor of S. Lefschetz)'' , Princeton Univ. Press (1957) pp. 330–360 {{MR|0084772}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" ''C.R. Acad. Sci. Paris'' , '''246''' : 17 (1958) pp. 2444–2447 {{MR|0100261}} {{ZBL|0092.39901}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" ''C.R. Acad. Sci. Paris'' , '''246''' : 18 (1958) pp. 2555–2558 {{MR|0100262}} {{ZBL|0092.40001}} </TD></TR><TR><TD valign="top">[6c]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" ''C.R. Acad. Sci. Paris'' , '''246''' : 21 (1958) pp. 2991–2993 {{MR|0100263}} {{ZBL|0092.40101}} </TD></TR><TR><TD valign="top">[6d]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" ''C.R. Acad. Sci. Paris'' , '''247''' : 6 (1958) pp. 620–623 {{MR|0100264}} {{ZBL|0092.40102}} </TD></TR><TR><TD valign="top">[6e]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Décomposition homologique d'un polyhèdre simplement connexe" ''C.R. Acad. Sci. Paris'' , '''248''' : 14 (1959) pp. 2054–2056</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) {{MR|0309097}} {{ZBL|0217.48601}} </TD></TR></table>

Revision as of 08:12, 6 June 2020


stationary duality, Spanier duality

A duality in homotopy theory which exists (in the absence of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the $ S $- homotopy and $ S $- cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or $ S $- category, is a category whose objects are topological spaces $ X $, while its morphisms are classes $ \{ f \} $ of $ S $- homotopic mappings $ f $ from a $ p $- fold suspension $ S ^ {p} X _ {1} $ into $ S ^ {p} X _ {2} $, $ f $ and $ g: S ^ {q} X _ {1} \rightarrow S ^ {q} X _ {2} $ being considered as $ S $- homotopic if there exists an $ r \geq \max ( p, q) $ such that the suspensions $ S ^ {r-} p f $ and $ S ^ {r-} q g $ are homotopic in the ordinary sense. The set $ \{ X _ {1} , X _ {2} \} $ of such classes, which are known as $ S $- mappings, constitutes an Abelian group with respect to the so-called track addition [1], [2], [4], [5]. The group $ \{ X _ {1} , X _ {2} \} $ is the limit of the direct spectrum of the sets $ [ S ^ {k} X _ {1} , S ^ {k} X _ {2} ] $ of ordinary homotopy classes with suspension mappings as projections; if $ k $ is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism $ S: \{ X _ {1} , X _ {2} \} \rightarrow \{ SX _ {1} , SX _ {2} \} $ in which the corresponding elements are represented by one and the same mapping $ S ^ {p} X _ {1} \rightarrow S ^ {p} X _ {2} $, $ p \geq 1 $. The $ n $- dual polyhedron of the polyhedron $ X $ in a sphere $ S ^ {n} $ is an arbitrary polyhedron $ D _ {n} X $ in $ S ^ {n} $ which is an $ S $- deformation retract of the complement $ S ^ {n} \setminus X $, i.e. the morphism corresponding to the imbedding $ D _ {n} X \subset S ^ {n} \setminus X $ is an $ S $- equivalence. The polyhedron $ D _ {n} X $ exists for all $ X $, and $ X $ may be considered as $ D _ {n} ^ {2} X $.

For any polyhedra $ X _ {1} , X _ {2} $ and any polyhedra $ D _ {n} X _ {1} $ and $ D _ {n} X _ {2} $ which are dual to them, there exists a unique mapping

$$ D _ {n} : \{ X _ {1} , X _ {2} \} \rightarrow \ \{ D _ {n} X _ {2} , D _ {n} X _ {1} \} $$

satisfying the following conditions:

a) It is an involutory contravariant functorial isomorphism, i.e. $ D _ {n} $ is a homomorphism such that if

$$ i : X _ {1} \subset X _ {2} \ \textrm{ and } \ i ^ \prime : D _ {n} X _ {2} \subset D _ {n} X _ {1} , $$

then

$$ D _ {n} \{ i \} = \{ i ^ \prime \} ; $$

if

$$ \{ f _ {1} \} \in \{ X _ {1} , X _ {2} \} \ \textrm{ and } \ \ \{ f _ {2} \} \in \{ X _ {2} , X _ {3} \} , $$

then

$$ D _ {n} ( \{ f _ {2} \} \cdot \{ f _ {1} \} ) = \ D _ {n} \{ f _ {1} \} \cdot D _ {n} \{ f _ {2} \} ; $$

if $ \theta $ is an element of $ \{ X _ {1} , X _ {2} \} $ or of $ \{ D _ {n} X _ {2} , D _ {n} X _ {1} \} $, then $ D _ {n} D _ {n} \theta = \theta $.

b) The following relations are valid:

$$ SD _ {n} = D _ {n+} 1 \ \textrm{ and } \ D _ {n+} 1 S = D _ {n} , $$

where $ SD _ {n} X _ {i} $ and $ D _ {n} X _ {i} $ are considered as polyhedra, $ ( n + 1 ) $- dual to polyhedra $ X _ {i} $ and, correspondingly, $ SX _ {i} $, $ i = 1, 2; $ this means that it does not depend on $ n $ and is stationary with respect to suspension.

c) It satisfies the equation

$$ D _ {a} ^ {n} \theta _ {*} = ( D _ {n} \theta ) ^ {*} D _ {a} ^ {n} , $$

where

$$ \theta _ {*} : H _ {p} ( X _ {1} ) \rightarrow H _ {p} ( X _ {2} ) $$

and

$$ ( D _ {n} \theta ) ^ {*} : H ^ {n-} p- 1 ( D _ {n} X _ {1} ) \rightarrow H ^ {n-} p- 1 ( D _ {n} X _ {2} ) $$

are homomorphisms of the above homology and cohomology groups, induced by $ S $- mappings $ \theta \in \{ X _ {1} , X _ {2} \} $ and $ D _ {n} \theta $, and

$$ D _ {a} : H _ {p} ( X _ {i} ) \rightarrow H ^ {n-} p- 1 ( D _ {n} X _ {i} ) ,\ i= 1 , 2 , $$

is an isomorphism obtained from the isomorphism of Alexander duality by replacing the set $ S ^ {n} \setminus X _ {i} $ by its $ S $- deformation retract $ D _ {n} X _ {i} $.

The construction of $ D _ {n} $ is based on the representation of a given mapping as the composition of an imbedding and an $ S $- deformation retract.

The $ S $- homotopy group $ \Sigma _ {p} ( X) $ of a space $ X $ is the group $ \{ S ^ {p} , X \} $, and the $ S $- cohomotopy group $ \Sigma ^ {p} ( X) $ of $ X $ is the group $ \{ X, S ^ {p} \} $. As in ordinary homotopy theory, one defines the homomorphisms

$$ \phi _ {p} : \Sigma _ {p} ( X) \rightarrow H _ {p} ( X) , $$

$$ \phi ^ {p} : \Sigma ^ {p} ( X) \rightarrow H ^ {p} ( X) . $$

Regarding the spheres $ S ^ {p} $ and $ S ^ {n-} p- 1 $ as $ n $- dual leads to the isomorphisms

$$ D _ {n} : \Sigma _ {p} ( X) \rightarrow \Sigma ^ {n-} p- 1 ( D _ {n} X) $$

and to the commutative diagram

$$

Thus, the isomorphism $ D _ {n} $ connects $ S $- homotopy and $ S $- cohomotopy groups, just as the isomorphism of Alexander duality $ D _ {a} ^ {n} $ connects the homology and cohomology groups. Any duality in the $ S $- category entails a duality of ordinary homotopy classes if the conditions imposed on the space entail the existence of a one-to-one correspondence between the set of the above classes and the set of $ S $- homotopy classes.

Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. $ D _ {n} $ converts one of these theorems into the other, which means that $ S $- homotopy groups are replaced by $ S $- cohomotopy groups, homology groups by cohomology groups, the mapping $ \phi _ {p} $ by the mapping $ \phi ^ {n-} p- 1 $, the smallest dimension with a non-trivial homology group by the largest dimension with a non-trivial cohomology group, and vice versa. In ordinary homotopy theory the definition of an $ n $- cohomotopy group requires that the dimension of the space does not exceed $ 2n - 2 $( or, more generally, that the space be $ ( 2n - 1) $- coconnected, $ n > 1 $), which impairs the perfectly general nature of duality.

There are several trends of generalization of the theory: e.g. studies are made of spaces with the $ S $- homotopy type of polyhedra, the relative case, a theory with supports, etc. [3], [5], , [7]. The theory was one of the starting points in the development of stationary homotopy theory [8].

References

[1] E.H. Spanier, "Duality and -theory" Bull. Amer. Math. Soc. , 62 (1956) pp. 194–203 MR0085506
[2] E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" Mathematika , 2 : 3 (1955) pp. 56–80 MR0074823 Zbl 0064.17202
[3] E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" Ann. of Math. , 67 : 2 (1958) pp. 203–238 MR0105105 Zbl 0092.15701
[4] M.G. Barratt, "Track groups 1; 2" Proc. London Math. Soc. , 5 (1955) pp. 71–106; 285–329
[5] E.H. Spanier, J.H.C. Whitehead, "The theory of carriers and -theory" , Algebraic geometry and Topology (A Symp. in honor of S. Lefschetz) , Princeton Univ. Press (1957) pp. 330–360 MR0084772
[6a] B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" C.R. Acad. Sci. Paris , 246 : 17 (1958) pp. 2444–2447 MR0100261 Zbl 0092.39901
[6b] B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" C.R. Acad. Sci. Paris , 246 : 18 (1958) pp. 2555–2558 MR0100262 Zbl 0092.40001
[6c] B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" C.R. Acad. Sci. Paris , 246 : 21 (1958) pp. 2991–2993 MR0100263 Zbl 0092.40101
[6d] B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" C.R. Acad. Sci. Paris , 247 : 6 (1958) pp. 620–623 MR0100264 Zbl 0092.40102
[6e] B. Eckmann, P.J. Hilton, "Décomposition homologique d'un polyhèdre simplement connexe" C.R. Acad. Sci. Paris , 248 : 14 (1959) pp. 2054–2056
[7] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303
[8] G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) MR0309097 Zbl 0217.48601
How to Cite This Entry:
S-duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=S-duality&oldid=48599
This article was adapted from an original article by G.S. Chogoshvili (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article