Pontryagin square

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.