Steenrod Algebra - Axiomatic Characterization

Axiomatic Characterization

Steenrod & Epstein (1962) showed that the Steenrod squares Sqn:Hm→Hm+n are characterized by the following 5 axioms:

1. Naturality: Sqn is an additive homomorphism from Hm(X,Z/2Z) to Hm+n(X,Z/2Z), and is natural meaning that for any map f : X → Y, f*(Sqnx) = Sqnf*(x).
2. Sq0 is the identity homomorphism.
3. Sqn is the cup square on classes of degree n.
4. If n>dim(X) then Sqn(x) = 0
5. Cartan Formula:

In addition the Steenrod squares have the following properties:

• Sq1 is the Bockstein homomorphism of the exact sequence
• They satisfy the Adem relations, described below.
• They commute with the suspension homomorphism and the boundary operator.

Similarly the following axioms characterize the reduced p-th powers for p > 2.

1. Naturality: Pn is an additive homomorphism from Hm(X,Z/pZ) to Hm+2n(p−1)(X,Z/pZ), and is natural.
2. P0 is the identity homomorphism.
3. Pn is the cup p-th power on classes of degree 2n.
4. If 2n>dim(X) then Pn(x) = 0
5. Cartan Formula:

As before, the reduced p-th powers also satisfy Adem relations and commute with the suspension and boundary operators.