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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:

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|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

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

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|Steenrod square]]).

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:

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

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 [[#References|[2]]], [[#References|[3]]]). In final form this generalization is as follows (see [[#References|[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====
<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====
For a definition of  $  \Gamma ( \pi ) $
see [[Ring with divided powers|Ring with divided powers]].