# 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)

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