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A stable [[Cohomology operation|cohomology operation]]  $  Sq  ^ {i} $, 
$  i \geq  0 $, 
of the type  $  ( \mathbf Z _ {2} , \mathbf Z _ {2} ) $, 
which raises the dimension by  $  i $. 
This means that for every integer  $  n $
and every pair of topological spaces  $  ( X, Y) $, 
a homomorphism

$$ 
Sq  ^ {i} :  H  ^ {n} ( X, Y;  \mathbf Z _ {2} )  \rightarrow  H  ^ {n+} i ( X, Y;  \mathbf Z _ {2} )
$$

is defined such that  $  \delta Sq  ^ {i} = Sq  ^ {i} \delta $, 
where  $  \delta $
is the coboundary homomorphism  $  \delta :  H  ^ {q} ( Y;  \mathbf Z _ {2} ) \rightarrow H  ^ {q+} 1 ( X, Y;  \mathbf Z _ {2} ) $(
stability) and  $  f ^ { \star } Sq  ^ {i} = Sq  ^ {i} f ^ { \star } $
for any continuous mapping  $  f:  ( X, Y) \rightarrow ( X  ^  \prime  , Y  ^  \prime  ) $(
naturality). The Steenrod squares  $  Sq  ^ {i} $
possess the following properties:

1)  $  Sq  ^ {0} =  \mathop{\rm id} $;

2)  $  Sq  ^ {1} = \beta $, 
where  $  \beta $
is the Bockstein homomorphism associated with the short exact sequence of coefficient groups  $  0 \rightarrow \mathbf Z _ {2} \rightarrow \mathbf Z _ {4} \rightarrow \mathbf Z _ {2} \rightarrow 0 $;

3) if  $  i =  \mathop{\rm dim}  x $, 
then  $  Sq  ^ {i} x = x  ^ {2} $;

4) if  $  i >  \mathop{\rm dim}  x $, 
then  $  Sq  ^ {i} x = 0 $;

5) (Cartan's formula)  $  Sq  ^ {i} ( xy) = \sum _ {j=} 0  ^ {i} ( Sq  ^ {i} x) \cdot ( Sq  ^ {i-} 1 y) $;

6) (Adem relation) if  $  a < 2b $, 
then

$$ 
Sq  ^ {a} Sq  ^ {b}  =  \sum _ { t= } 0 ^ { [ }  a/2] \left ( \begin{array}{c}
b- t- 1 \\
 a- 2t 
\end{array}
 \right ) _ {2} Sq  ^ {a+} b- t Sq  ^ {t} ,
$$

where  $  ( \cdot ) _ {2} $
are binomial coefficients modulo 2.

In Cartan's formula, multiplication can be considered as outer ( $  \times $-
multiplication) as well as interior (cup-multiplication). This is equivalent to the statement that the mapping  $  Sq:  H  ^  \star  ( X;  \mathbf Z _ {2} ) \rightarrow H  ^  \star  ( X;  \mathbf Z _ {2} ) $, 
defined by the formula

$$ 
Sqx  =  x + Sq  ^ {1} x + \dots + Sq  ^ {n-} 1 x + x  ^ {2} ,\ \ 
x \in H  ^ {n} ( X;  \mathbf Z _ {2} ),
$$

is a ring homomorphism. It follows from the stability condition that the Steenrod squares  $  Sq  ^ {i} $
commute with [[Suspension|suspension]] and [[Transgression|transgression]].

The properties 1), 3) and 4) define the operations  $  Sq  ^ {i} $
uniquely and can therefore be taken as defining axioms. The constructive definition of  $  Sq  ^ {i} $
is based on the simplicial structure in chain groups  $  C _  \star  ( X) $
and on the existence of a diagonal mapping  $  \Delta :  X \rightarrow X \times X $. 
Let  $  W $
be the minimal acyclic free chain  $  \mathbf Z _ {2} $-
complex i.e. the chain complex for which

$$ 
W _ {i}  =  \mathbf Z _ {2} [ e _ {i} , Te _ {i} ] ,\ \ 
de _ {i}  =  e _ {i-} 1 + (- 1)  ^ {i} Te _ {i-} 1 ,
$$

where  $  T $
is the generator of  $  \mathbf Z _ {2} $. 
The existence of an equivariant chain mapping

$$ 
\phi :  W \otimes C _  \star  ( X)  \rightarrow  C _  \star  ( X) \otimes C _  \star  ( X)
$$

such that

$$ 
\phi :  ( e _ {i} \otimes \sigma )  \in  C _  \star  ( \sigma \otimes \sigma )  \subset   C _  \star  ( X) \otimes C _  \star  ( X)  =  C _  \star  ( X \times X)
$$

for any simplex  $  \sigma \in C _  \star  ( X) $, 
is proved by the method of acyclic carriers or by an explicit construction (see [[#References|[4]]]). The symbol  $  C _  \star  ( \sigma \otimes \sigma ) $
here signifies the smallest subcomplex of the chain complex  $  C _  \star  ( X) \otimes C _  \star  ( X) $
containing the element  $  \sigma \otimes \sigma $. 
Let  $  i \geq  0 $. 
Any two cochains  $  u \in C  ^ {p} ( X) $, 
$  v \in C  ^ {q} ( X) $
are put in correspondence by the formula  $  ( u \cup _ {i} v)( \sigma ) = ( u \otimes v)( \phi ( e _ {i} \otimes \sigma )) $, 
for any simplex  $  \sigma \in C _ {p+} q- i ( X) $, 
with the cochain  $  u \cup _ {i} v \in C  ^ {p+} q- i ( X) $, 
which is called their cup- $  i $-
product. For the coboundary of this chain, the formula

$$ 
\delta \left ( u \cup _ { i } v \right )  = \ 
(- 1)  ^ {i} \delta u \cup _ { i } v + (- 1)  ^ {i+} p u \cup _ { i } \delta v +
$$

$$ 
+ 
(- 1)  ^ {i+} 1 u \cup _ { i- } 1 v + (- 1)  ^ {pq+} 1 v \cup _ { i- } 1 u
$$

holds, from which it follows that the formula  $  Sq  ^ {p-} i \{ u \} = \{ u \cup _ {i} u \} $
correctly defines a homomorphism

$$ 
Sq  ^ {p-} i :  H  ^ {p} ( X;  \mathbf Z _ {2} )  \rightarrow  H  ^ {2p-} i ( X;  \mathbf Z _ {2} )
$$

which does not depend on the choice of the mapping  $  \phi $.

The operations  $  Sq  ^ {i} $
are constructed in the same way in other simplicial structures with a diagonal mapping, for example, in cohomology groups of simplicial Abelian groups, of simplicial Lie algebras, etc. However, not all properties of the Steenrod squares  $  Sq  ^ {i} $
are preserved then (for example, generally speaking,  $  Sq  ^ {0} \neq  \mathop{\rm id} $) 
and there is yet (1991) no single general theory for the generalized operations  $  Sq  ^ {i} $(
see [[#References|[5]]], [[#References|[6]]]).

Many cohomology operations which act on cohomology groups with coefficients in the groups  $  \mathbf Z _ {2} $
and  $  \mathbf Z _ {p} $
can be expressed in terms of the Steenrod squares and their analogues (see [[Steenrod reduced power|Steenrod reduced power]]). This underlines the fundamental role played by Steenrod squares in algebraic topology and its applications. For example, [[Bordism|bordism]] groups are calculated using Steenrod squares.

Steenrod squares were introduced by N. Steenrod [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,   D.B.A. Epstein,   "Cohomology operations" , Princeton Univ. Press  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.B. Fuks,   A.T. Fomenko,   V.L. Gutenmakher,   "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.K. Tangora,   "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Steenrod,   "Products of cocylces and extensions of mappings"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 290–320</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Epstein,   "Steenrod operations in homological algebra"  ''Invent. Math.'' , '''1''' :  2  (1966)  pp. 152–208</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. May,   "A general algebraic approach to Steenrod operations" , ''The Steenrod Algebra and Its Applications'' , ''Lect. notes in math.'' , '''168''' , Springer  (1970)  pp. 153–231</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  ''Matematika'' , '''5''' :  2  (1961)  pp. 3–11; 11–30; 30–49; 50–102</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,   "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapt. 18</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Adams,   "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)  pp. Part III, Chapt. 12</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. V, Sect. 9</TD></TR></table>