Steenrod Algebra - Construction

Construction

Suppose that π is any degree n subgroup of the symmetric group on n points, u a cohomology class in Hq(X,B), A an abelian group acted on by π, and c a cohomology class in H_i(π,A). Steenrod (1953) showed how to construct a reduced power un/c in Hkq−i(X,(A⊗B⊗B⊗...⊗B)/π) as follows.

1. Taking the external product of u with itself n times gives an equivariant cocycle on Xn with coefficients in B⊗B⊗...⊗B.
2. Choose E to be a contractible space on which π acts freely and an equivariant map from E× X to Xn. Pulling back un by this map gives an equivariant cocyle on E× X and therefore a cocycle of E/π×X with coefficients in B⊗B⊗...⊗B.
3. Taking a slant product with c in H_i(E/π,A)gives a cocycle of X with coefficients in H₀(π,A⊗B⊗B⊗...⊗B)

The Steenrod squares and reduced powers are special cases of this construction where π is a cyclic group of prime order p=n acting as a cyclic permutation of n elements, and the groups A and B are cyclic of order p, so that H₀(π,A⊗B⊗B⊗...⊗B) is also cyclic of order p.