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Difference between revisions of "Pontryagin square"

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m (fix tex)
m (fixing subscripts)
 
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defined for any pair of topological spaces  $  ( X , Y ) $
 
defined for any pair of topological spaces  $  ( X , Y ) $
 
and such that for any continuous mapping  $  f :  ( X , Y ) \rightarrow ( X  ^  \prime  , Y  ^  \prime  ) $
 
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 ^ { * } $(
+
the equality  $  f ^ { * } {\mathcal P} _ {2} = {\mathcal P} _ {2} f ^ { * } $ (naturality) holds.
naturality) holds.
 
  
 
Pontryagin squares have the following properties:
 
Pontryagin squares have the following properties:
Line 64: Line 63:
  
 
where  $  u \in C  ^ {2n} ( X ;  \mathbf Z ) $
 
where  $  u \in C  ^ {2n} ( X ;  \mathbf Z ) $
is a cocycle modulo  $  2  ^ {k} $(
+
is a cocycle modulo  $  2  ^ {k} $ (for the  $  \cup _ {i} $-products see [[Steenrod square|Steenrod square]]).
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  $  p $
 
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 ) $
 
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 $-
+
and is called the  $  p $-th Pontryagin power  $  {\mathcal P} _ {p} $.  
th Pontryagin power  $  {\mathcal P} _ {p} $.  
 
 
The operation  $  {\mathcal P} _ {p} $
 
The operation  $  {\mathcal P} _ {p} $
 
is uniquely defined by the following formulas:
 
is uniquely defined by the following formulas:
Line 77: Line 73:
 
$$  
 
$$  
 
{\mathcal P} _ {p} ( u + v )  =  {\mathcal P} _ {p} u + {\mathcal P} _ {p} v + i
 
{\mathcal P} _ {p} ( u + v )  =  {\mathcal P} _ {p} u + {\mathcal P} _ {p} v + i
\left ( \sum _ { i= } 1 ^ { p- } 1
+
\left ( \sum _ { i= 1} ^ { p-   1 }
 
\frac{1}{p}
 
\frac{1}{p}
 
  \left ( \begin{array}{c}
 
  \left ( \begin{array}{c}
Line 103: Line 99:
 
is zero. For  $  p > 2 $
 
is zero. For  $  p > 2 $
 
the equality  $  {\mathcal P} _ {p} ( u v ) = ( {\mathcal P} _ {p} u ) ( {\mathcal P} _ {p} v ) $
 
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 $-
+
holds, the multiplication may be taken both as outer ( $  \times $-multiplication) or inner ( $  \cup $-multiplication). For  $  p = 2 $
multiplication) or inner ( $  \cup $-
 
multiplication). For  $  p = 2 $
 
 
the corresponding equality is valid only up to summands of order 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 $(
+
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
see [[#References|[2]]], [[#References|[3]]]). In final form this generalization is as follows (see [[#References|[6]]]). The Pontryagin square is a ring homomorphism
 
  
 
$$  
 
$$  
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where  $  \Gamma $
 
where  $  \Gamma $
 
is a functor which associates a ring with divided powers to an Abelian group. For  $  \pi = \mathbf Z _ {p} $,  
 
is a functor which associates a ring with divided powers to an Abelian group. For  $  \pi = \mathbf Z _ {p} $,  
the  $  p $-
+
the  $  p $-th component of this homomorphism coincides with the  $  p $-th Pontryagin power  $  {\mathcal P} _ {p} $ (for  $  p= 2 $
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} $).
 
with the Pontryagin square  $  {\mathcal P} _ {2} $).
  

Latest revision as of 10:04, 11 July 2022


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=52483
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