# Steenrod reduced power

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 .

#### References

[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 |

#### Comments

For more references see Steenrod algebra.

**How to Cite This Entry:**

Steenrod reduced power.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Steenrod_reduced_power&oldid=15254