Steenrod square
In algebraic topology, a Steenrod algebra was defined by Cartan (1955) to be the algebra of stable cohomology operations for mod p cohomology.
For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p cohomology. It is generated by the Steenrod squares introduced by Steenrod (1947) for p=2, and by the Steenrod reduced pth powers introduced in Steenrod (1953) and the Bockstein homomorphism for p>2.
The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.
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 Y is a suspension of a space X, the cup product on the cohomology of Y is trivial.) Norman Steenrod constructed stable operations
for all i greater than zero. The notation Sq and their name, the Steenrod squares, comes from the fact that Sqn restricted to classes of degree n is the cup square. There are analogous operations for odd primary coefficients, usually denoted Pi and called the reduced p-th power operations:
The Sqi generate a connected graded algebra over Z/2, where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case p > 2, the mod p Steenrod algebra is generated by the Pi and the Bockstein operation β associated to the short exact sequence
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