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A [[Cohomology operation|cohomology operation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738101.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738102.png" />, i.e. a functorial mapping
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<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/p/p073/p073810/p0738103.png" /></td> </tr></table>
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defined for any pair of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738104.png" /> and such that for any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738105.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738106.png" /> (naturality) holds.
+
A [[Cohomology operation|cohomology operation]]  $  {\mathcal P} _ {2} $
 +
of type  $  ( \mathbf Z _ {2  ^ {k}  } , 2n ;  \mathbf Z _ {2  ^ {k+}  1 } , 4n ) $,
 +
i.e. a functorial mapping
 +
 
 +
$$
 +
{\mathcal P} _ {2} :  H  ^ {2n} ( X , Y ;  \mathbf Z _ {2  ^ {k}  } )  \rightarrow  H  ^ {4n}
 +
( X , Y ;  \mathbf Z _ {2  ^ {k+}  1 } ) ,
 +
$$
 +
 
 +
defined for any pair of topological spaces $  ( X , Y ) $
 +
and such that for any continuous mapping $  f : ( X , Y ) \rightarrow ( X  ^  \prime  , Y  ^  \prime  ) $
 +
the equality $  f ^ { * } {\mathcal P} _ {2} = {\mathcal P} _ {2} f ^ { * } $(
 +
naturality) holds.
  
 
Pontryagin squares have the following properties:
 
Pontryagin squares have the following properties:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738108.png" /> is the natural imbedding.
+
1) $  {\mathcal P} _ {2} ( u + v ) = {\mathcal P} _ {2} u + {\mathcal P} _ {2} v + i ( u v ) $,  
 +
where $  i : \mathbf Z _ {2  ^ {k}  } \rightarrow \mathbf Z _ {2  ^ {k+}  1 } $
 +
is the natural imbedding.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p0738109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381011.png" /> is the quotient homomorphism modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381012.png" />.
+
2) $  \rho {\mathcal P} _ {2} u = u  ^ {2} $
 +
and $  {\mathcal P} _ {2} \rho u = u  ^ {2} $,  
 +
where $  \rho : H  ^ {*} ( X , Y ;  \mathbf Z _ {2  ^ {k+}  1 } ) \rightarrow H  ^ {*} ( X , Y ;  \mathbf Z _ {2  ^ {k}  } ) $
 +
is the quotient homomorphism modulo $  2  ^ {k} $.
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381014.png" /> is the [[Suspension|suspension]] mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381015.png" /> is the [[Postnikov square|Postnikov square]] (in other words, the cohomology suspension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381016.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381017.png" />). If
+
3) $  {\mathcal P} _ {2} \Sigma = \Sigma {\mathcal P} $,  
 +
where $  \Sigma : H  ^ {2n-} 1 ( X ;  G ) \rightarrow H  ^ {2n} ( \Sigma X ;  G ) $
 +
is the [[Suspension|suspension]] mapping and $  {\mathcal P} $
 +
is the [[Postnikov square|Postnikov square]] (in other words, the cohomology suspension of $  {\mathcal P} _ {2} $
 +
is $  {\mathcal P} $).  
 +
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/p/p073/p073810/p07381018.png" /></td> </tr></table>
+
$$
 +
{\mathcal P} _ {2} : K ( \mathbf Z _ {2  ^ {k}  } , 2n )  \rightarrow  K ( \mathbf Z _ {2  ^ {k+}  1 } , 4n )
 +
$$
  
 
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/p/p073/p073810/p07381019.png" /></td> </tr></table>
+
$$
 +
{\mathcal P} : K ( \mathbf Z _ {2  ^ {k}  } , 2n - 1 )  \rightarrow  K ( \mathbf Z _ {2  ^ {k+}  1 } ,\
 +
4n - 1 )
 +
$$
  
are the representing mappings, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381020.png" />.
+
are the representing mappings, then $  \Omega {\mathcal P} _ {2} = {\mathcal P} $.
  
 
The properties 1), 2) uniquely characterize the Pontryagin square and thus can be taken as an axiomatic definition of it. Constructively the Pontryagin square is defined by the formula
 
The properties 1), 2) uniquely characterize the Pontryagin square and thus can be taken as an axiomatic definition of it. Constructively the Pontryagin square is defined by the formula
  
<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/p/p073/p073810/p07381021.png" /></td> </tr></table>
+
$$
 +
{\mathcal P} _ {2} \{ u \}  = \{ u \cup _ {0} u + u \cup _ {1} \delta u \}
 +
  \mathop{\rm mod}  2  ^ {k+} 1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381022.png" /> is a cocycle modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381023.png" /> (for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381024.png" />-products see [[Steenrod square|Steenrod square]]).
+
where $  u \in C  ^ {2n} ( X ;  \mathbf Z ) $
 +
is a cocycle modulo $  2  ^ {k} $(
 +
for the $  \cup _ {i} $-
 +
products see [[Steenrod square|Steenrod square]]).
  
There exists (see [[#References|[5]]], [[#References|[6]]]) a generalization of the Pontryagin square to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381025.png" /> is an arbitrary odd prime number. This generalization is a cohomology operation of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381026.png" /> and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381028.png" />-th Pontryagin power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381029.png" />. The operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381030.png" /> is uniquely defined by the following formulas:
+
There exists (see [[#References|[5]]], [[#References|[6]]]) a generalization of the Pontryagin square to the case when p $
 +
is an arbitrary odd prime number. This generalization is a cohomology operation of type $  ( \mathbf Z _ {p  ^ {k}  } , 2n ;  \mathbf Z _ {p ^ {k+}  1 } , 2pn ) $
 +
and is called the p $-
 +
th Pontryagin power $  {\mathcal P} _ {p} $.  
 +
The operation $  {\mathcal P} _ {p} $
 +
is uniquely defined by the following formulas:
  
<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/p/p073/p073810/p07381031.png" /></td> </tr></table>
+
$$
 +
{\mathcal P} _ {p} ( u + v )  = {\mathcal P} _ {p} u + {\mathcal P} _ {p} v + i
 +
\left ( \sum _ { i= } 1 ^ { p- }  1
 +
\frac{1}{p}
 +
\left ( \begin{array}{c}
 +
p \\
 +
i
 +
\end{array}
 +
\right )
 +
u  ^ {i} v  ^ {p-} 1 \right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381032.png" /> is the natural imbedding; and
+
where $  i : \mathbf Z _ {p ^ {k}  } \rightarrow \mathbf Z _ {p  ^ {k+}  1 } $
 +
is the natural imbedding; 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/p/p073/p073810/p07381033.png" /></td> </tr></table>
+
$$
 +
\rho {\mathcal P} _ {p} u = u  ^ {p} \  \textrm{ and } \  {\mathcal P} _ {p} \rho u = u  ^ {p} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381034.png" /> is the quotient homomorphism modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381035.png" /> generalizing the corresponding formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381036.png" />. The analogue of formula 3) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381037.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381038.png" />, which means that the cohomology suspension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381039.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381040.png" /> is zero. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381041.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381042.png" /> holds, the multiplication may be taken both as outer (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381043.png" />-multiplication) or inner (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381044.png" />-multiplication). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381045.png" /> the corresponding equality is valid only up to summands of order 2.
+
where $  \rho : H  ^ {*} ( X , Y ;  \mathbf Z _ {p  ^ {k+}  1 } ) \rightarrow H  ^ {*} ( X , Y ;  \mathbf Z _ {p ^ {k}  } ) $
 +
is the quotient homomorphism modulo p ^ {k} $
 +
generalizing the corresponding formulas for $  {\mathcal P} _ {2} $.  
 +
The analogue of formula 3) for $  {\mathcal P} _ {p} $
 +
has the form $  {\mathcal P} _ {p} \Sigma = 0 $,  
 +
which means that the cohomology suspension of $  {\mathcal P} _ {p} $
 +
for p > 2 $
 +
is zero. For p > 2 $
 +
the equality $  {\mathcal P} _ {p} ( u v ) = ( {\mathcal P} _ {p} u ) ( {\mathcal P} _ {p} v ) $
 +
holds, the multiplication may be taken both as outer ( $  \times $-
 +
multiplication) or inner ( $  \cup $-
 +
multiplication). For $  p = 2 $
 +
the corresponding equality is valid only up to summands of order 2.
  
In the most general way the Pontryagin square is defined for cohomology with coefficients in an arbitrary finitely-generated Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381046.png" /> (see [[#References|[2]]], [[#References|[3]]]). In final form this generalization is as follows (see [[#References|[6]]]). The Pontryagin square is a ring homomorphism
+
In the most general way the Pontryagin square is defined for cohomology with coefficients in an arbitrary finitely-generated Abelian group $  \pi $(
 +
see [[#References|[2]]], [[#References|[3]]]). In final form this generalization is as follows (see [[#References|[6]]]). The Pontryagin square is a ring homomorphism
  
<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/p/p073/p073810/p07381047.png" /></td> </tr></table>
+
$$
 +
{\mathcal P}  ^ {*} : \Gamma ( H  ^ {2n} ( X ; \pi ) )  \rightarrow  H  ^ {*}
 +
( X ; \Gamma ( \pi ) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381048.png" /> is a functor which associates a ring with divided powers to an Abelian group. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381049.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381050.png" />-th component of this homomorphism coincides with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381051.png" />-th Pontryagin power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381052.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381053.png" /> with the Pontryagin square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381054.png" />).
+
where $  \Gamma $
 +
is a functor which associates a ring with divided powers to an Abelian group. For $  \pi = \mathbf Z _ {p} $,  
 +
the p $-
 +
th component of this homomorphism coincides with the p $-
 +
th Pontryagin power $  {\mathcal P} _ {p} $(
 +
for $  p= 2 $
 +
with the Pontryagin square $  {\mathcal P} _ {2} $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Mappings of a 3-dimensional sphere into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381055.png" />-dimensional complex"  ''Dokl. Akad. Nauk SSSR'' , '''34'''  (1942)  pp. 35–37  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Boltyanskii,  "The homotopy theory of continuous mapping and vector fields" , Moscow  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,  "The classification of continuous mappings of a three-dimensional polyhedron into a simply connected polyhedron of arbitrary dimension"  ''Dokl. Akad. Nauk SSSR'' , '''64''' :  4  (1949)  pp. 461–462  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Browder,  E. Thomas,  "Axioms for the generalized Pontryagin cohomology operations"  ''Quart. J. Math.'' , '''13'''  (1962)  pp. 55–60</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Thomas,  "A generalization of the Pontrjagin square cohomology operation"  ''Proc. Nat. Acad. Sci. USA'' , '''42'''  (1956)  pp. 266–269</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Thomas,  "The generalized Pontryagin cohomology operations and rings with divided powers" , Amer. Math. Soc.  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Mappings of a 3-dimensional sphere into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381055.png" />-dimensional complex"  ''Dokl. Akad. Nauk SSSR'' , '''34'''  (1942)  pp. 35–37  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Boltyanskii,  "The homotopy theory of continuous mapping and vector fields" , Moscow  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,  "The classification of continuous mappings of a three-dimensional polyhedron into a simply connected polyhedron of arbitrary dimension"  ''Dokl. Akad. Nauk SSSR'' , '''64''' :  4  (1949)  pp. 461–462  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Browder,  E. Thomas,  "Axioms for the generalized Pontryagin cohomology operations"  ''Quart. J. Math.'' , '''13'''  (1962)  pp. 55–60</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Thomas,  "A generalization of the Pontrjagin square cohomology operation"  ''Proc. Nat. Acad. Sci. USA'' , '''42'''  (1956)  pp. 266–269</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Thomas,  "The generalized Pontryagin cohomology operations and rings with divided powers" , Amer. Math. Soc.  (1957)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For a definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073810/p07381056.png" /> see [[Ring with divided powers|Ring with divided powers]].
+
For a definition of $  \Gamma ( \pi ) $
 +
see [[Ring with divided powers|Ring with divided powers]].

Revision as of 08:07, 6 June 2020


A cohomology operation $ {\mathcal P} _ {2} $ of type $ ( \mathbf Z _ {2 ^ {k} } , 2n ; \mathbf Z _ {2 ^ {k+} 1 } , 4n ) $, i.e. a functorial mapping

$$ {\mathcal P} _ {2} : H ^ {2n} ( X , Y ; \mathbf Z _ {2 ^ {k} } ) \rightarrow H ^ {4n} ( X , Y ; \mathbf Z _ {2 ^ {k+} 1 } ) , $$

defined for any pair of topological spaces $ ( X , Y ) $ and such that for any continuous mapping $ f : ( X , Y ) \rightarrow ( X ^ \prime , Y ^ \prime ) $ the equality $ f ^ { * } {\mathcal P} _ {2} = {\mathcal P} _ {2} f ^ { * } $( naturality) holds.

Pontryagin squares have the following properties:

1) $ {\mathcal P} _ {2} ( u + v ) = {\mathcal P} _ {2} u + {\mathcal P} _ {2} v + i ( u v ) $, where $ i : \mathbf Z _ {2 ^ {k} } \rightarrow \mathbf Z _ {2 ^ {k+} 1 } $ is the natural imbedding.

2) $ \rho {\mathcal P} _ {2} u = u ^ {2} $ and $ {\mathcal P} _ {2} \rho u = u ^ {2} $, where $ \rho : H ^ {*} ( X , Y ; \mathbf Z _ {2 ^ {k+} 1 } ) \rightarrow H ^ {*} ( X , Y ; \mathbf Z _ {2 ^ {k} } ) $ is the quotient homomorphism modulo $ 2 ^ {k} $.

3) $ {\mathcal P} _ {2} \Sigma = \Sigma {\mathcal P} $, where $ \Sigma : H ^ {2n-} 1 ( X ; G ) \rightarrow H ^ {2n} ( \Sigma X ; G ) $ is the suspension mapping and $ {\mathcal P} $ is the Postnikov square (in other words, the cohomology suspension of $ {\mathcal P} _ {2} $ is $ {\mathcal P} $). If

$$ {\mathcal P} _ {2} : K ( \mathbf Z _ {2 ^ {k} } , 2n ) \rightarrow K ( \mathbf Z _ {2 ^ {k+} 1 } , 4n ) $$

and

$$ {\mathcal P} : K ( \mathbf Z _ {2 ^ {k} } , 2n - 1 ) \rightarrow K ( \mathbf Z _ {2 ^ {k+} 1 } ,\ 4n - 1 ) $$

are the representing mappings, then $ \Omega {\mathcal P} _ {2} = {\mathcal P} $.

The properties 1), 2) uniquely characterize the Pontryagin square and thus can be taken as an axiomatic definition of it. Constructively the Pontryagin square is defined by the formula

$$ {\mathcal P} _ {2} \{ u \} = \{ u \cup _ {0} u + u \cup _ {1} \delta u \} \mathop{\rm mod} 2 ^ {k+} 1 , $$

where $ u \in C ^ {2n} ( X ; \mathbf Z ) $ is a cocycle modulo $ 2 ^ {k} $( for the $ \cup _ {i} $- products see Steenrod square).

There exists (see [5], [6]) a generalization of the Pontryagin square to the case when $ p $ is an arbitrary odd prime number. This generalization is a cohomology operation of type $ ( \mathbf Z _ {p ^ {k} } , 2n ; \mathbf Z _ {p ^ {k+} 1 } , 2pn ) $ and is called the $ p $- th Pontryagin power $ {\mathcal P} _ {p} $. The operation $ {\mathcal P} _ {p} $ is uniquely defined by the following formulas:

$$ {\mathcal P} _ {p} ( u + v ) = {\mathcal P} _ {p} u + {\mathcal P} _ {p} v + i \left ( \sum _ { i= } 1 ^ { p- } 1 \frac{1}{p} \left ( \begin{array}{c} p \\ i \end{array} \right ) u ^ {i} v ^ {p-} 1 \right ) , $$

where $ i : \mathbf Z _ {p ^ {k} } \rightarrow \mathbf Z _ {p ^ {k+} 1 } $ is the natural imbedding; and

$$ \rho {\mathcal P} _ {p} u = u ^ {p} \ \textrm{ and } \ {\mathcal P} _ {p} \rho u = u ^ {p} , $$

where $ \rho : H ^ {*} ( X , Y ; \mathbf Z _ {p ^ {k+} 1 } ) \rightarrow H ^ {*} ( X , Y ; \mathbf Z _ {p ^ {k} } ) $ is the quotient homomorphism modulo $ p ^ {k} $ generalizing the corresponding formulas for $ {\mathcal P} _ {2} $. The analogue of formula 3) for $ {\mathcal P} _ {p} $ has the form $ {\mathcal P} _ {p} \Sigma = 0 $, which means that the cohomology suspension of $ {\mathcal P} _ {p} $ for $ p > 2 $ is zero. For $ p > 2 $ the equality $ {\mathcal P} _ {p} ( u v ) = ( {\mathcal P} _ {p} u ) ( {\mathcal P} _ {p} v ) $ holds, the multiplication may be taken both as outer ( $ \times $- multiplication) or inner ( $ \cup $- multiplication). For $ p = 2 $ the corresponding equality is valid only up to summands of order 2.

In the most general way the Pontryagin square is defined for cohomology with coefficients in an arbitrary finitely-generated Abelian group $ \pi $( see [2], [3]). In final form this generalization is as follows (see [6]). The Pontryagin square is a ring homomorphism

$$ {\mathcal P} ^ {*} : \Gamma ( H ^ {2n} ( X ; \pi ) ) \rightarrow H ^ {*} ( X ; \Gamma ( \pi ) ) , $$

where $ \Gamma $ is a functor which associates a ring with divided powers to an Abelian group. For $ \pi = \mathbf Z _ {p} $, the $ p $- th component of this homomorphism coincides with the $ p $- th Pontryagin power $ {\mathcal P} _ {p} $( for $ p= 2 $ with the Pontryagin square $ {\mathcal P} _ {2} $).

References

[1] L.S. Pontryagin, "Mappings of a 3-dimensional sphere into an -dimensional complex" Dokl. Akad. Nauk SSSR , 34 (1942) pp. 35–37 (In Russian)
[2] V.G. Boltyanskii, "The homotopy theory of continuous mapping and vector fields" , Moscow (1955) (In Russian)
[3] M.M. Postnikov, "The classification of continuous mappings of a three-dimensional polyhedron into a simply connected polyhedron of arbitrary dimension" Dokl. Akad. Nauk SSSR , 64 : 4 (1949) pp. 461–462 (In Russian)
[4] W. Browder, E. Thomas, "Axioms for the generalized Pontryagin cohomology operations" Quart. J. Math. , 13 (1962) pp. 55–60
[5] E. Thomas, "A generalization of the Pontrjagin square cohomology operation" Proc. Nat. Acad. Sci. USA , 42 (1956) pp. 266–269
[6] E. Thomas, "The generalized Pontryagin cohomology operations and rings with divided powers" , Amer. Math. Soc. (1957)

Comments

For a definition of $ \Gamma ( \pi ) $ see Ring with divided powers.

How to Cite This Entry:
Pontryagin square. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_square&oldid=48244
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article