Download The Algebra of Secondary Cohomology Operations by Hans-Joachim Baues PDF

Show description

3)(5) and where j is the inclusion with j(y) = 1 ⊗ y. The operator Di is the composite Di = pri ◦ γx ◦ j, or equivalently we have for x = x ⊗ 1 ∈ Eβ (Fx) ⊗ H the equation: (2) γx (1 ⊗ y) = ωi (x) ⊗ Di (y). i 14 Chapter 1. Primary Cohomology Operations We set Di = 0 for i > pq and for i < 0. Moreover we say that the extended power algebra H is unitary if (3) and (5) hold. Di (y) = 0 for i > (p − 1)q and D(p−1)q (y) = ϑq · y (3) for y ∈ H q . )q with m = (p − 1)/2. We [SE]. 3 page 112 |y| is even and j ∈ {2m(p − 1), 2m(p − 1) − 1; m ≥ 0}, |y| is odd and j ∈ {(2m + 1)(p − 1), (2m + 1)(p − 1) − 1; m ≥ 0}.

We have the suspension functor Σ : U −→ U (3) defined by setting (ΣM )n = M n−1 . Let Σ : M n−1 → (ΣM )n be the map of degree 1 given by the identity of M n−1 . Then the A-action on ΣM is defined by θ(Σm) = (−1)|θ| Σ(θm) for m ∈ M , θ ∈ A. We obtain the A-module F (n) = Σn (A/B(n)) (4) which is the free unstable module on one generator [n] in degree n. Here [n] = Σn {1} ∈ F (n) is defined by the unit 1 ∈ A. A basis of A/B(n) is given by admissible monomials of excess ≤ n. Free objects in U are direct sums of modules F (n), n ≥ 0.

We also use the free module functor (6) R : ∆Set −→ ∆Mod which carries X to RX where (RX)n is the free R-module generated by Xn . We have the natural map [−] : X → φRX which carries x ∈ X to the corresponding generator [x] ∈ RX. Moreover for A ∈ ∆Mod and f : X → φA in ∆Set we have the unique map f¯ : RX → A in ∆Mod for which the composite (φf¯)[−] coincides with f . , M [n]i = 0 for i = n and M [n]n = M . Given a chain complex (C, d) in Chain we define the cochain complex C ∗ (M ) = Hom(C, M ) with (8) C n (M ) = HomMod (Cn , M ) and differential ∂ = Hom(d, 1M ).