The Steenrod Algebra and its Dual

Abstract

Having determined E_*(E) for E = MU, MSp, K and KO we turn now to a determination of E*(E)and E*(E) for E = H(ℤ₂). Although historically H*(H(ℤ₂); ℤ₂) and H*(H(ℤ₂); ℤ₂) were known as much as fifteen years before KO*(KO), for example, we shall see that the calculation of these Hopf algebras is at least as difficult as the calculation we just did for KO. In theory we could proceed as follows: for each n we have a fibration H(ℤ₂,n-1) → PH(ℤ₂, n) → H(ℤ₂, n), and the total space PH(ℤ₂,n)is contractible (the path space PY is contractible for any Y). Thus in the Serre spectral sequence for this fibration E^∞_pq= 0 if (p,q) ≠ (0,0). With a little luck this should enable us to determine H̃*(H (ℤ₂, n); ℤ₂) from a knowledge of H*(H(ℤ₂,n - 1); ℤ₂). The process could begin at n=1 since H̃*(H(ℤ₂,1); ℤ₂) ≅ H̃*(RP^∞; ℤ), which is a ℤ₂-vector space with basis {x_i:i⩾1}, x_i∈ H̃_i(RP^∞; ℤ₂). Then we would have H_q(H(ℤ₂); ℤ₂)≅ dirlim_n H̃_q+n(H(ℤ₂,n); ℤ₂). This is precisely how Cartan did determine this algebra (see [28]) using some heavy guns from homological algebra. We shall take a different approach, however; we shall construct some specific cohomology operations—the Steenrod squares Sq_i—and show that they generate the algebra A* = H*(H(ℤ₂); ℤ₂). It will then not be too difficult to determine the dual A* = H*(H(ℤ₂); ℤ₂). We shall also indicate the results for H(ℤ_p),p an odd prime.