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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>
+
$$
 +
 
 +
\begin{array}{ccc}
 +
{\Sigma _ { p }  { ( X) } }  & \stackrel{ \phi _ p  }{\rightarrow}  &{H _ { p }  { ( X) } }  \\
 +
{ { {D _ { n }  } } \downarrow }  &{}  &{\downarrow { {D _ { a }  ^ { n } } } }  \\
 +
{\Sigma ^ {  { {n-p} -1}  } { ( D _ { n }  ^ { X } ) } }  & \stackrel{\phi ^{n-p-1}}{\rightarrow}  &{H ^ {  { {n-p} -1}  } { ( D _ { n }  X) } }  \\
 +
\end{array}
 +
 
 +
$$
  
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>

Latest revision as of 13:45, 8 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

$$ \begin{array}{ccc} {\Sigma _ { p } { ( X) } } & \stackrel{ \phi _ p }{\rightarrow} &{H _ { p } { ( X) } } \\ { { {D _ { n } } } \downarrow } &{} &{\downarrow { {D _ { a } ^ { n } } } } \\ {\Sigma ^ { { {n-p} -1} } { ( D _ { n } ^ { X } ) } } & \stackrel{\phi ^{n-p-1}}{\rightarrow} &{H ^ { { {n-p} -1} } { ( D _ { n } X) } } \\ \end{array} $$

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=23971
This article was adapted from an original article by G.S. Chogoshvili (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article