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
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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)
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if ,
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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.
Steenrod reduced power. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_reduced_power&oldid=15254