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Concept
Steenrod algebra
Summary
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod cohomology.
For a given prime number , the Steenrod algebra is the graded Hopf algebra over the field of order , consisting of all stable cohomology operations for mod cohomology. It is generated by the Steenrod squares introduced by for , and by the Steenrod reduced th powers introduced in and the Bockstein homomorphism for .
The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.
Cohomology operation
A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring , the cup product squaring operation yields a family of cohomology operations:
Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below.
These operations do not commute with suspension—that is, they are unstable. (This is because if is a suspension of a space , the cup product on the cohomology of is trivial.) Steenrod constructed stable operations
for all greater than zero. The notation and their name, the Steenrod squares, comes from the fact that restricted to classes of degree is the cup square. There are analogous operations for odd primary coefficients, usually denoted and called the reduced -th power operations:
The generate a connected graded algebra over , where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case , the mod Steenrod algebra is generated by the and the Bockstein operation associated to the short exact sequence
In the case , the Bockstein element is and the reduced -th power is .
We can summarize the properties of the Steenrod operations as generators in the cohomology ring of Eilenberg–Maclane spectra
since there is an isomorphism
giving a direct sum decomposition of all possible cohomology operations with coefficients in .
Official source
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