Steenrod square

A stable 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 and 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 [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 [5], [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). This underlines the fundamental role played by Steenrod squares in algebraic topology and its applications. For example, bordism groups are calculated using Steenrod squares.

Steenrod squares were introduced by N. Steenrod [4].

References

[1]	N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)
[2]	D.B. Fuks, A.T. Fomenko, V.L. Gutenmakher, "Homotopic topology" , Moscow (1969) (In Russian)
[3]	M.K. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[4]	N.E. Steenrod, "Products of cocylces and extensions of mappings" Ann. of Math. , 48 (1947) pp. 290–320
[5]	D. Epstein, "Steenrod operations in homological algebra" Invent. Math. , 1 : 2 (1966) pp. 152–208
[6]	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
[7]	Matematika , 5 : 2 (1961) pp. 3–11; 11–30; 30–49; 50–102

References

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