By Hans-Joachim Baues

The algebra of basic cohomology operations computed via the well known Steenrod algebra is without doubt one of the strongest instruments of algebraic topology. This e-book computes the algebra of secondary cohomology operations which enriches the constitution of the Steenrod algebra in a brand new and unforeseen method.

The ebook solves a long-standing challenge at the algebra of secondary cohomology operations by way of constructing a brand new algebraic concept of such operations. the implications have powerful influence at the Adams spectral series and for this reason at the computation of homotopy teams of spheres.

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An Amodule X satisﬁes B(n) · X n = 0 for all n if and only if X is unstable. 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.

The theory of Eilenberg-MacLane spaces 23 Let Hp ⊂ Kp0 be the full subcategory generated by objects H(Fx1 ⊕· · ·⊕Fxr ) with n1 , . . , nr ≥ 1 and r ≥ 0. Then the cohomology functor yields an isomorphism of categories where Hop p is the opposite category of Hp , H ∗ : Kp = Hop p . 2) holds. Compare also [BJ4]. We have the following commutative diagram of functors corresponding to the well-known equation ˜ n (X) = [X, K(n)] H for n ≥ 0. 3) model(Kp ) qV [X,−] qqq Top∗0 / qqq qqq vv vv vvv H ∗ vvv v8 Kp0 Here [X, −] is the model which carries an object A in Kp to the set [X, A].

1) V = Fx1 ⊕ · · · ⊕ Fxn Let Vec be the category of F-vector spaces and F-linear maps. The zero-vector space is denoted by V = 0. Let S(V ) = F[x1 , . . 1) and let Λ(V ) = Λ(x1 , . . 1). 2) µ : An ⊗ Am −→ An+m denoted by µ(x, y) = x · y and a unit 1 ∈ A0 with 1 · x = x · 1 = x. The algebra A is connected if A0 = F and A is augmented if an algebra map A → F is given. In particular each connected algebra is augmented. Of course A0 is a subalgebra of A and all An , n ≥ 1, are A0 -bimodules. Moreover A is commutative if x · y = (−1)|x||y|y·x.