Steenrod reduced power

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A stable cohomology operation , , of the type , where is a fixed odd prime number, which is the analogue modulo of the Steenrod square, and which is a homomorphism

defined for every pair of topological spaces and any integer . The Steenrod reduced powers possess the following properties (apart from naturality and stability , where is the coboundary homomorphism):

1) ;

2) if , then ;

3) if , then ;

4) (Cartan's formula) ;

5) (Adem's relation)

if ,

if , where is the Bockstein homomorphism associated with the short exact sequence of coefficient groups , while are the binomial coefficients reduced modulo .

These properties are analogous to the corresponding properties of Steenrod squares, whereby the operation corresponds to the operation . Just as for Steenrod squares, the multiplication in 4) can be considered to be both exterior (-multiplication) and interior (-multiplication). Steenrod reduced powers commute with suspension and transgression.

The properties 1)–3) uniquely characterize , and can be constructed in the same way as Steenrod squares using the minimal acyclic free chain -complex .


[1] N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)
[2] Matematika , 5 : 2 (1961) pp. 3–11; 11–30; 30–49; 50–102


For more references see Steenrod algebra.

How to Cite This Entry:
Steenrod reduced power. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article