A stable cohomology operation , , of the type , which raises the dimension by . This means that for every integer and every pair of topological spaces , a homomorphism
is defined such that , where is the coboundary homomorphism (stability) and for any continuous mapping (naturality). The Steenrod squares possess the following properties:
2) , where is the Bockstein homomorphism associated with the short exact sequence of coefficient groups ;
3) if , then ;
4) if , then ;
5) (Cartan's formula) ;
6) (Adem relation) if , then
where are binomial coefficients modulo 2.
In Cartan's formula, multiplication can be considered as outer (-multiplication) as well as interior (cup-multiplication). This is equivalent to the statement that the mapping , defined by the formula
The properties 1), 3) and 4) define the operations uniquely and can therefore be taken as defining axioms. The constructive definition of is based on the simplicial structure in chain groups and on the existence of a diagonal mapping . Let be the minimal acyclic free chain -complex i.e. the chain complex for which
where is the generator of . The existence of an equivariant chain mapping
for any simplex , is proved by the method of acyclic carriers or by an explicit construction (see ). The symbol here signifies the smallest subcomplex of the chain complex containing the element . Let . Any two cochains , are put in correspondence by the formula , for any simplex , with the cochain , which is called their cup--product. For the coboundary of this chain, the formula
holds, from which it follows that the formula correctly defines a homomorphism
which does not depend on the choice of the mapping .
The operations 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 are preserved then (for example, generally speaking, ) and there is yet (1991) no single general theory for the generalized operations (see , ).
Many cohomology operations which act on cohomology groups with coefficients in the groups and 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 .
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Steenrod square. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_square&oldid=48827