COMPUTATION OF THE E_{3}-TERM OF THE ADAMS SPECTRAL SEQUENCE

HANS JOACHIM BAUES AND MAMUKA JIBLADZE

Abstract. An algorithm is described giving effective determination of the second differential in the Adams spectral sequence. The algorithm is based on the notion of secondary derived functor, and on the explicit algebraic model of the groupoid enriched category of stable maps and stable tracks between Eilenberg-Mac Lane spaces.

The algebra ℬ of secondary cohomology operations is a pair algebra with Σ-structure which as a Hopf algebra was explicitly computed in [1]. In particular the multiplication map A of ℬ was determined by an algorithm. In this paper we introduce algebraically the secondary Ext-groups Ext_{ℬ} and we prove that the E_{3}-term of the Adams spectral sequence (computing stable maps in {Y, X} _{p}^{∗}) is given by

E_{3}(Y, X) = Ext_{ℬ} (ℋ X, ℋ Y).

Here ℋ X is the secondary cohomology of the spectrum X which is the ℬ-module ^{Σ} if X is the sphere spectrum S^{0}. This leads to an algorithm for the computation of the group

E_{3}(S^{0}, S^{0}) = Ext_{ℬ} ( ^{Σ}, ^{Σ})

which is a new explicit approximation of stable homotopy groups of spheres improving the Adams approximation

E_{2}(S^{0}, S^{0}) = Ext_{𝒜} ( , ).

An implementation of our algorithm computed E_{3}(S^{0}, S^{0}) by now up to degree 40. In this range our results confirm the known results in the literature, see for example the book of Ravenel [6].

1. Modules over pair algebras

We here recall from [1] the notion of pair modules, pair algebras, and pair modules over a pair algebra B. The category B-Mod of pair modules over B is an additive track category in which we consider secondary resolutions as defined in [3]. Using such secondary resolutions we shall obtain the secondary derived functors Ext_{B} in section 3.

Let k be a commutative ring with unit and let Mod be the category of k-modules and k-linear maps. This is a symmetric monoidal category via the tensor product A⊗B over k of k-modules A, B. A pair of modules is a morphism

(1.1)  X = (X_{1} ⟶^{∂} X_{0})

in Mod. We write π_{0}(X) = ker∂ and π_{1}(X) = coker∂. A morphism f : X → Y of pairs is a commutative diagram

\xymatrix{X_{1} \ar[r]^{f_{1}} \ar[d]_{∂} & Y_{1} \ar[d]^{∂} \\ X_{0} \ar[r]^{f_{0}} & Y_{0}.}

Evidently pairs with these morphisms form a category���� (Mod) and one has functors

π_{0}, π_{1} :���� (Mod) → Mod.

A pair morphism is called a weak equivalence if it induces isomorphisms on π_{0} and π_{1}.

2000 Mathematics Subject Classification. Primary 55T15, 55U99, 18G99; Secondary 55S20, 18G10, 18G40.

Clearly a pair in Mod coincides with a chain complex concentrated in degrees 0 and 1. For two pairs X and Y the tensor product of the complexes corresponding to them is concentrated in degrees in 0, 1 and 2 and is given by

X_{1}⊗Y_{1} ⟶^{∂_{1}} X_{1}⊗Y_{0} ⊕ X_{0}⊗Y_{1} ⟶^{∂_{0}} X_{0}⊗Y_{0}

with ∂_{0} = (∂ ⊗ 1, 1 ⊗ ∂) and ∂_{1} = (−1 ⊗ ∂, ∂ ⊗ 1). Truncating X ⊗ Y we get the pair

X ¯⊗Y = ((X ¯⊗Y)_{1} = coker(∂_{1}) ⟶^{∂} X_{0} ⊗ Y_{0} = (X ¯⊗Y)_{0})

with ∂ induced by ∂_{0}.

(1.2) Remark. Note that the full embedding of the category of pairs into the category of chain complexes induced by the above identification has a left adjoint Tr given by truncation: for a chain complex

C = (... → C_{2} ⟶^{∂_{1}} C_{1} ⟶^{∂_{0}} C_{0} ⟶^{∂_{−1}} C_{−1} → ...) ,

one has

Tr(C) = (coker(∂_{1}) ⟶^{¯∂_{0}} C_{0}),

with ¯∂_{0} induced by ∂_{0}. Then clearly one has

X ¯⊗Y = Tr(X ⊗ Y).

Using the fact that Tr is a reflection onto a full subcategory, one easily checks that the category���� (Mod) together with the tensor product ¯⊗ and unit k = (0 → k) is a symmetric monoidal category, and Tr is a monoidal functor.

We define the tensor product A ⊗ B of two graded modules in the usual way, i. e. by

(A ⊗ B)^{n} = ⨁_{i+j=n} A^{i} ⊗ B^{j}.

A (graded) pair module is a graded object of���� (Mod), i. e. a sequence X^{n} = (∂ : X_{1}^{n} → X_{0}^{n}) of pairs in Mod. We identify such a pair module X with the underlying morphism ∂ of degree 0 between graded modules

X = (X_{1} ⟶^{∂} X_{0}).

Now the tensor product X ¯⊗Y of graded pair modules X, Y is defined by

(1.3)  (X ¯⊗Y)^{n} = ⨁_{i+j=n} X^{i} ¯⊗Y^{j}.

This defines a monoidal structure on the category of graded pair modules. Morphisms in this category are of degree 0.

For two morphisms f, g : X → Y between graded pair modules, a homotopy H : f ⇒ g is a morphism H : X_{0} → Y_{1} of degree 0 as in the diagram

(1.4)  [figure: X_{1}; f_{1}; g_{1}; ∂; Y_{1}; ∂; X_{0}; f_{0}; g_{0}; H; Y_{0},]

satisfying f_{0} − g_{0} = ∂H and f_{1} − g_{1} = H∂.

A pair algebra B is a monoid in the monoidal category of graded pair modules, with multiplication

μ : B ¯⊗B → B.

We assume that B is concentrated in nonnegative degrees, that is B^{n} = 0 for n < 0.

A left B-module is a graded pair module M together with a left action

μ : B ¯⊗M → M

of the monoid B on M.

More explicitly pair algebras and modules over them can be described as follows.

(1.5) Definition. A pair algebra B is a graded pair

∂ : B_{1} → B_{0}

in Mod with B_{1}^{n} = B_{0}^{n} = 0 for n < 0 such that B_{0} is a graded algebra in Mod, B_{1} is a graded B_{0}-B_{0}-bimodule, and ∂ is a bimodule homomorphism. Moreover for x, y ∈ B_{1} the equality

∂(x)y = x∂(y)

holds in B_{1}.

It is easy to see that there results an exact sequence of graded B_{0}-B_{0}-bimodules

0 → π_{1}B → B_{1} ⟶^{∂} B_{0} → π_{0}B → 0

where in fact π_{0}B is a k-algebra, π_{1}B is a π_{0}B-π_{0}B-bimodule, and B_{0} → π_{0}(B) is a homomorphism of algebras.

(1.6) Definition. A (left) module over a pair algebra B is a graded pair M = (∂ : M_{1} → M_{0}) in Mod such that M_{1} and M_{0} are left B_{0}-modules and ∂ is B_{0}-linear. Moreover a B_{0}-linear map

μ̄ : B_{1}⊗_{B_{0}}M_{0} → M_{1}

is given fitting in the commutative diagram

\xymatrix{B_{1} ⊗_{B_{0}} M_{1} \ar[r]^{1⊗∂} \ar[d]_{μ} & B_{1} ⊗_{B_{0}} M_{0} \ar[dl]^{μ̄} \ar[d]^{μ} \\ M_{1} \ar[r]_{∂} & M_{0},}

where μ(b ⊗ m) = ∂(b)m for b ∈ B_{1} and m ∈ M_{1} ∪ M_{0}.

For an indeterminate element x of degree n = |x| let B[x] denote the B-module with B[x]_{i} consisting of expressions bx with b ∈ B_{i}, i = 0, 1, with bx having degree |b| + n, and structure maps given by ∂(bx) = ∂(b)x, μ(b′⊗ bx) = (b′b)x and μ̄(b′⊗ bx) = (b′b)x.

A free B-module is a direct sum of several copies of modules of the form B[x], with x ∈ I for some set I of indeterminates of possibly different degrees. It will be denoted

B[I] = ⨁_{x∈I} B[x].

For a left B-module M one has the exact sequence of B_{0}-modules

0 → π_{1}M → M_{1} → M_{0} → π_{0}M → 0

where π_{0}M and π_{1}M are actually π_{0}B-modules.

Let B-Mod be the category of left modules over the pair algebra B. Morphisms f = (f_{0}, f_{1}) : M → N are pair morphisms which are B-equivariant, that is,f_{0} and f_{1} are B_{0}-equivariant and compatible with μ̄ above, i. e. the diagram

\xymatrix{B_{1} ⊗_{B_{0}} M_{0} \ar[r]^{μ̄} \ar[d]_{1⊗f_{0}} & M_{1} \ar[d]^{f_{1}} \\ B_{1} ⊗_{B_{0}} N_{0} \ar[r]^{μ̄} & N_{1}}

commutes.

For two such maps f, g : M → N a track H : f ⇒ g is a degree zero map

(1.7)  H : M_{0} → N_{1}

satisfying f_{0} − g_{0} = ∂H and f_{1} − g_{1} = H∂ such that H is B_{0}-equivariant. For tracks H : f ⇒ g, K : g ⇒ h their composition K H : f ⇒ h is K + H.

(1.8) Proposition. For a pair algebra B, the category B-Mod with the above track structure is a welldefined additive track category.

Proof. For a morphism f = (f_{0}, f_{1}) : M → N between B-modules, one has

Aut(f) = {H ∈ Hom_{B_{0}}(M_{0}, N_{1}) | ∂H = f_{0} − f_{0}, H∂ = f_{1} − f_{1}} Hom_{π_{0}B}(π_{0}M, π_{1}N).

Since this group is abelian, by [2] we know that B-Mod is a linear track extension of its homotopy category by the bifunctor D with D(M, N) = Hom_{π_{0}B}(π_{0}M, π_{1}N). It thus remains to show that the homotopy category is additive and the bifunctor D is biadditive.

By definition the set of morphisms [M, N] between objects M, N in the homotopy category is given by the exact sequence of abelian groups

Hom_{B_{0}}(M_{0}, N_{1}) → Hom_{B}(M, N) [M, N].

This makes evident the abelian group structure on the hom-sets [M, N]. Bilinearity of composition follows from consideration of the commutative diagram

\xymatrix{Hom_{B_{0}}(M_{0}, N_{1})⊗Hom_{B}(N, P) ⊕ Hom_{B}(M, N)⊗Hom_{B_{0}}(N_{0}, P_{1}) \ar[d] \ar[r]^{μ} & Hom_{B_{0}}(M_{0}, P_{1}) \ar[d] \\ Hom_{B}(M, N) ⊗ Hom_{B}(N, P) \ar[r] \ar@{->>}[d] & Hom_{B}(M, P) \ar@{->>}[d] \\ [M, N] ⊗ [N, P] \ar@{-->}[r] & [M, P]}

with exact columns, where μ(H⊗g + f ⊗K) = g_{1}H + Kf_{0}. It also shows that the functor B-Mod → B-Mod_{≃} is linear. Since this functor is the identity on objects, it follows that the homotopy category is additive.

Now note that both functors π_{0}, π_{1} factor to define functors on B-Mod_{≃}. Since these functors are evidently additive, it follows that D = Hom_{π_{0}B}(π_{0}, π_{1}) is a biadditive bifunctor.

(1.9) Lemma. If M is a free B-module, then the canonical map

[M, N] → Hom_{π_{0}B}(π_{0}M, π_{0}N)

is an isomorphism for any B-module N.

Proof. Let (g_{i})_{i∈I} be a free generating set for M. Given a π_{0}(B)-equivariant homomorphism f : π_{0}M → π_{0}N, define its lifting f̃to M by specifying f̃(g_{i}) = n_{i}, with n_{i} chosen arbitrarily from the class f ([g_{i}]) = [n_{i}].

To show monomorphicity, given f : M → N such that π_{0}f = 0, this means that imf_{0} ⊂ im∂, so we can choose H(g_{i}) ∈ N_{1} in such a way that ∂H(g_{i}) = f_{0}(g_{i}). This then extends uniquely to a B_{0}-module homomorphism H : M_{0} → N_{1} with ∂H = f_{0}; moreover any element of M_{1} is a linear combination of elements of the form b_{1}g_{i} with b_{1} ∈ B_{1}, and for these one has H∂(b_{1}g_{i}) = H(∂(b_{1})g_{i}) = ∂(b_{1})H(g_{i}). But f_{1}(b_{1}g_{i}) = b_{1}f_{0}(g_{i}) = b_{1}∂H(g_{i}) = ∂(b_{1})H(g_{i}) too, so H∂ = f_{1}. This shows that f is nullhomotopic.

2. Σ-structure

(2.1) Definition. The suspension ΣX of a graded object X = (X^{n})_{n∈} is given by degree shift, (ΣX)^{n} = X^{n−1}.

Let Σ : X → ΣX be the map of degree 1 given by the identity. If X is a left A-module over the graded algebra A then ΣX is a left A-module via

(2.2)  a · Σx = (−1)^{|a|}Σ(a · x)

for a ∈ A, x ∈ X. On the other hand if X is a right A-module then (Σx) · a = Σ(x · a) yields the right A-module structure on ΣX.

(2.3) Definition. A Σ-module is a graded pair module X = (∂ : X_{1} → X_{0}) together with an isomorphism

σ : π_{1}X Σπ_{0}X

of graded k-modules. We then call σ a Σ-structure of X. A Σ-map between Σ-modules is a map f between pair modules such that σ(π_{1}f) = Σ(π_{0}f)σ. If X is a pair algebra then a Σ-structure is an isomorphism of π_{0}X-π_{0}X-bimodules. If X is a left module over a pair algebra B then a Σ-structure of X is an isomorphism σ of left π_{0}B-modules. Let

(B-Mod)^{Σ} ⊂ B-Mod

be the track category of B-modules with Σ-structure and Σ-maps.

(2.4) Lemma. Suspension of a B-module M has by (2.2) the structure of a B-module and ΣM has a Σstructure if M has one.

Proof. Given σ : π_{1}M Σπ_{0}M one defines a Σ-structure on ΣM via

π_{1}ΣM = Σπ_{1}M ⟶^{Σσ} ΣΣπ_{0}M = Σπ_{0}ΣM.

Hence we get suspension functors between track categories

\xymatrix{B-Mod \ar[r]^{Σ} & B-Mod \\ (B-Mod)^{Σ} \ar[u] \ar[r]^{Σ} & (B-Mod)^{Σ}. \ar[u]}

(2.5) Lemma. The track category (B−Mod)^{Σ} is -additive in the sense of [3], with = Σ^{−1}, or as well -additive, with = Σ.

Proof. The statement of the lemma means that the bifunctor

D(M, N) = Aut(0_{M,N})

is either left- or right-representable, i. e. there is an endofunctor , respectively of (B-Mod)^{Σ} and a binatural isomorphism D(M, N) [ M, N], resp. D(M, N) [M, N].

Now by (1.7), a track in Aut(0_{M,N}) is a B_{0}-module homomorphism H : M_{0} → N_{1} with ∂H = H∂ = 0; hence

D(M, N) Hom_{π_{0}B}(π_{0}M, π_{1}N) Hom_{π_{0}B}(π_{0}Σ^{−1}M, π_{0}N) Hom_{π_{0}B}(π_{0}M, π_{0}ΣN).

(2.6) Lemma. If B is a pair algebra with Σ-structure then each free B-module has a Σ-structure.

Proof. This is clear from the description of free modules in 1.6.

3. The secondary differential over pair algebras

For a pair algebra B with a Σ-structure, for a Σ-module M over B, and a module N over B we now define the secondary differential

d_{(2)} : Ext_{π_{0}B}^{n}(π_{0}M, π_{0}N) → Ext_{π_{0}B}^{n+2}(π_{0}M, π_{1}N).

Here d_{(2)} = d_{(2)}(M, N) depends on the B-modules M and N and is natural in M and N with respect to maps in (B−Mod)^{Σ}. For the definition of d_{(2)} we consider secondary chain complexes and secondary resolutions. In [3] such a construction was performed in the generality of an arbitrary -additive track category. We will first present the construction of d_{(2)} for the track category of pair modules and then will indicate how this construction is a particular case of the more general situation discussed in [3].

(3.1) Definition. For a pair algebra B, a secondary chain complex M_{•} in B-Mod is given by a diagram of the form

[figure: ...; M_{n+2,1}; d_{n+1,1}; ∂_{n+2}; M_{n+1,1}; d_{n,1}; ∂_{n+1}; M_{n,1}; d_{n−1,1}; ∂_{n}; M_{n−1,1}; ∂_{n−1}; ...; ...; H_{n+1}; M_{n+2,0}; d_{n+1,0}; H_{n}; M_{n+1,0}; d_{n,0}; H_{n−1}; M_{n,0}; d_{n−1,0}; M_{n−1,0}; ...]

where each M_{n} = (∂_{n} : M_{n,1} → M_{n,0}) is a B-module, each d_{n} = (d_{n,0}, d_{n,1}) is a morphism in B-Mod, each H_{n} is B_{0}-linear and moreover the identities

d_{n,0}d_{n+1,0} = ∂_{n}H_{n}
d_{n,1}d_{n+1,1} = H_{n}∂_{n+2}

and

H_{n}d_{n+2,0} = d_{n,1}H_{n+1}

hold for all n ∈ . We thus see that in this case a secondary complex is the same as a graded version of a multicomplex (see e. g. [5]) with only two nonzero rows.

One then defines the total complex Tot(M_{•}) of the form

... ← M_{n−1,0} ⊕ M_{n−2,1} \underleftarrow{( \begin{matrix}d_{n−1,0} & −∂_{n−1} \\ H_{n−2} & −d_{n−2,1} \end{matrix} )} M_{n,0} ⊕ M_{n−1,1} \underleftarrow{( \begin{matrix}d_{n,0} & −∂_{n} \\ H_{n−1} & −d_{n−1,1} \end{matrix} )} M_{n+1,0} ⊕ M_{n,1} ← ...

Cycles and boundaries in this complex will be called secondary cycles, resp. secondary boundaries of M_{•}. Thus a secondary n-cycle in M_{•} is a pair (c, γ) with c ∈ M_{n,0}, γ ∈ M_{n−1,1} such that d_{n−1,0}c = ∂_{n−1}γ, H_{n−2}c = d_{n−2,1}γ and such a cycle is a boundary iff there exist b ∈ M_{n+1,0} and β ∈ M_{n,1} with c = d_{n,0}b + ∂_{n}β and γ = H_{n−1}b+d_{n−1,1}β. A secondary complex M_{•} is called exact if its total complex is, that is, if secondary cycles are secondary boundaries.

Let us now consider a secondary chain complex M_{•} in B-Mod. It is clear then that

π_{0}M_{•} :  ... → π_{0}M_{n+2} ⟶^{π_{0}d_{n+1}} π_{0}M_{n+1} ⟶^{π_{0}d_{n}} π_{0}M_{n} ⟶^{π_{0}d_{n−1}} π_{0}M_{n−1} → ...

is a chain complex of π_{0}B-modules. The next result corresponds to [3, lemma 3.5].

(3.2) Proposition. Let M_{•} be a secondary complex consisting of Σ-modules and Σ-maps between them. If π_{0}(M_{•}) is an exact complex then M_{•} is an exact secondary complex. Conversely, if π_{0}M_{•} is bounded below then secondary exactness of M_{•} implies exactness of π_{0}M_{•}.

Proof. The proof consists in translating the argument from the analogous general statement in [3] to our setting. Suppose first that π_{0}M_{•} is an exact complex, and consider a secondary cycle (c, γ) ∈ M_{n,0}⊕ M_{n−1,1}, i. e. one has d_{n−1,0}c = ∂_{n−1}γ and H_{n−2}c = d_{n−2,1}γ. Then in particular [c] ∈ π_{0}M_{n} is a cycle, so there exists [b] ∈ π_{0}M_{n+1} with [c] = π_{0}(d_{n})[b]. Take b ∈ [b], then c − d_{n,0}b = ∂_{n}β for some β ∈ M_{n+1,1}. Consider δ = γ−H_{n−1}b−d_{n−1,1}β. One has ∂_{n−1}δ = ∂_{n−1}γ−∂_{n−1}H_{n−1}b−∂_{n−1}d_{n−1,1}β = d_{n−1,0}c−d_{n−1,0}d_{n,0}b−d_{n−1,0}∂_{n}β = 0, so that δ is an element of π_{1}M_{n}. Moreover d_{n−2,1}δ = d_{n−2,1}γ − d_{n−2,1}H_{n−1}b − d_{n−2,1}d_{n−1,1}β = H_{n−2}c − H_{n−2}d_{n,0}b − H_{n−2}∂_{n}β = 0, i. e. δ is a cycle in π_{1}M_{•}. Since by assumption π_{0}M_{•} is exact, taking into account the Σ-structure π_{1}M_{•} is exact too, so that there exists ψ ∈ π_{1}M_{n} with δ = d_{n−1,1}ψ. Define β̃ = β + ψ. Then d_{n,0}b+∂_{n}β̃ = d_{n,0}b+∂_{n}β = c since ψ ∈ ker∂_{n}. Moreover d_{n−1,1}β̃ = d_{n−1,1}β+d_{n−1,1}ψ = d_{n−1,1}β+δ = γ−H_{n−1}b, which means that (c, γ) is the boundary of (b, β̃). Thus M_{•} is an exact secondary complex.

Conversely suppose M_{•} is exact, and π_{0}M_{•} bounded below. Given a cycle [c] ∈ π_{0}(M_{n}), represent it by a c ∈ M_{n,0}. Then π_{0}d_{n−1}[c] = 0 implies d_{n−1,0}c ∈ im∂_{n−1}, so there is a γ ∈ M_{n−1,1} such that d_{n−1,0}c = ∂_{n−1}γ. Consider ω = d_{n−2,1}γ − H_{n−2}c. One has ∂_{n−2}ω = ∂_{n−2}d_{n−2,1}γ − ∂_{n−2}H_{n−2}c = d_{n−2,0}∂_{n−1}γ − d_{n−2,0}d_{n−1,0}c = 0, i. e. ω is an element of π_{1}M_{n−2}. Moreover d_{n−3,1}ω = d_{n−3,1}d_{n−2,1}γ − d_{n−3,1}H_{n−2}c = H_{n−3}∂_{n−1}γ − H_{n−3}d_{n,0}c = 0, so ω is a n−2-dimensional cycle in π_{1}M_{•}. Using the Σ-structure, this then gives a n−3-dimensional cycle in π_{0}M_{•}. Now since π_{0}M_{•} is bounded below, we might assume by induction that it is exact in dimension n − 3, so that ω is a boundary. That is, there exists α ∈ π_{1}M_{n−1} with d_{n−2,1}α = ω. Define γ̃ = γ − α; then one has d_{n−2,1}γ̃ = d_{n−2,1}γ − d_{n−2,1}α = d_{n−2,1}γ − ω = H_{n−2}c. Moreover ∂_{n−1} γ̃ = ∂_{n−1}γ = d_{n−1,0}c since α ∈ ker(∂)n − 1. Thus (c, γ̃) is a secondary cycle, and by secondary exactness of M_{•} there exists a pair (b, β) with c = d_{n,0}b + ∂_{n}β. Then [c] = π_{0}(d_{n})[b], i. e. c is a boundary.

(3.3) Definition. Let B be a pair algebra with Σ-structure. A secondary resolution of a Σ-module M = (∂ : M_{1} → M_{0}) over B is an exact secondary complex F_{•} in (B−Mod)^{Σ} of the form

[figure: ...; ···; F_{31}; d_{21}; ∂_{3}; F_{21}; d_{11}; ∂_{2}; F_{11}; d_{01}; ∂_{1}; F_{01}; ϵ_{1}; ∂_{0}; M_{1}; ∂; 0; 0; ...; ···; ...; H_{2}; F_{30}; d_{20}; H_{1}; F_{20}; d_{10}; H_{0}; F_{10}; d_{00}; ϵ̂; F_{00}; ϵ_{0}; M_{0}; 0; 0; ...]

where each F_{n} = (∂_{n} : F_{n1} → F_{n0}) is a free B-module.

It follows from 3.2 that for any secondary resolution F_{•} of a B-module M with Σ-structure, π_{0}F_{•} will be a free resolution of the π_{0}B-module π_{0}M, so that in particular one has

Ext_{π_{0}B}^{n}(π_{0}M, U) H^{n}Hom(π_{0}F_{•}, U)

for all n and any π_{0}B-module U.

(3.4) Definition. Given a pair algebra B with Σ-structure, a Σ-module M over B, a module N over B and a secondary resolution F_{•} of M, we define the secondary differential

d_{(2)} : Ext_{π_{0}B}^{n}(π_{0}M, π_{0}N) → Ext_{π_{0}B}^{n+2}(π_{0}M, π_{1}N)

in the following way. Suppose given a class [c] ∈ Ext_{π_{0}B}^{n}(π_{0}M, π_{0}N). First represent it by some element in Hom_{π_{0}B}(π_{0}F_{n}, π_{0}N) which is a cocycle, i. e. its composite with π_{0}(d_{n}) is 0. By 1.9 we know that the natural maps

[F_{n}, N] → Hom_{π_{0}B}(π_{0}F, π_{0}N)

are isomorphisms, hence to any such element corresponds a homotopy class in [F_{n}, N] which is also a cocycle, i. e. value of [d_{n}, N] on it is zero. Take a representative map c : F_{n} → N from this homotopy class. Then cd_{n} is nullhomotopic, so we can find a B_{0}-equivariant map H : F_{n+1,0} → N_{1} such that in the diagram

[figure: F_{n+2,1}; d_{n+1,1}; ∂_{n+2}; F_{n+1,1}; d_{n,1}; ∂_{n+1}; F_{n,1}; c_{1}; ∂_{n}; N_{1}; ∂; F_{n+2,0}; d_{n+1,0}; H_{n}; F_{n+1,0}; d_{n,0}; H; F_{n,0}; c_{0}; N_{0}.]

one has c_{0}d_{n,0} = ∂H, c_{1}d_{n,1} = H∂_{n+1} and ∂c_{1} = c_{0}∂_{n}. Then taking Γ = c_{1}H_{n} − Hd_{n+1,0} one has ∂Γ = 0, Γ∂_{n+2} = 0, so Γ determines a map Γ̄ : coker∂_{n+2} → ker∂, i. e. from π_{0}F_{n+2} to π_{1}N. Moreover Γ̄π_{0}(d_{n+2}) = 0, so it is a cocycle in Hom(π_{0}(F_{•}), π_{1}N) and we define

d_{(2)}[c] = [Γ̄] ∈ Ext_{π_{0}B}^{n+2}(π_{0}M, π_{1}N).

(3.5) Definition. Let M and N be B-modules with Σ-structure. Then also all the B-modules Σ^{k}M, Σ^{k}N have Σ-structures and we get by 3.4 the secondary differential

[figure: Ext_{π_{0}B}^{n}(π_{0}M, π_{0}Σ^{k}N); d_{(2)}(M,Σ^{k}N); Ext_{π_{0}B}^{n+2}(π_{0}M, π_{1}Σ^{k}N); Ext_{π_{0}B}^{n}(π_{0}M, Σ^{k}π_{0}N); d; Ext_{π_{0}B}^{n+2}(π_{0}M, Σ^{k+1}π_{0}N).]

In case the composite

Ext_{π_{0}B}^{n−2}(π_{0}M, Σ^{k−1}π_{0}N) ⟶^{d} Ext_{π_{0}B}^{n}(π_{0}M, Σ^{k}π_{0}N) ⟶^{d} Ext_{π_{0}B}^{n+2}(π_{0}M, Σ^{k+1}π_{0}N)

vanishes we define the secondary Ext-groups to be the quotient groups

Ext_{B}^{n}(M, N)^{k} := kerd/ imd.

(3.6) Theorem. For a Σ-algebra B, a B-module M with Σ-structure and any B-module N, the secondary differential d_{(2)} in 3.4 coincides with the secondary differential

d_{(2)} : Ext_{a}^{n}(M, N) → Ext_{a}^{n+2}(M, N)

from [3, Section 4] as constructed for the -additive track category (B−Mod)^{Σ} in 2.5, relative to the subcategory b of free B-modules with a = b_{≃}.

Proof. We begin by recalling the appropriate notions from [3]. There secondary chain complexes A_{•} = (A_{n}, d_{n}, δ_{n})_{n∈} are defined in arbitrary additive track category B. They consist of objects A_{n}, morphisms d_{n} : A_{n+1} → A_{n} and tracks δ_{n} : d_{n}d_{n+1} ⇒ 0_{A_{n+2},A_{n}}, n ∈ , such that the equality of tracks

δ_{n}d_{n+2} = d_{n}δ_{n+1}

holds for all n. For an object X, an X-valued n-cycle in a secondary chain complex A_{•} is defined to be a pair (c, γ) consisting of a morphism c : X → A_{n} and a track γ : d_{n−1}c ⇒ 0_{X,A_{n−1}} such that the equality of tracks

δ_{n−2}c = d_{n−2}γ

is satisfied. Such a cycle is called a boundary if there exists a map b : X → A_{n+1} and a track β : c ⇒ d_{n}b such that the equality

γ = δ_{n−1}b d_{n−1}β

holds. A secondary chain complex is called X-exact if every X-valued cycle in it is a boundary. Similarly it is called b-exact, if it is X-exact for every object X in b, where b is a track subcategory of B. A secondary b-resolution of an object A is a b-exact secondary chain complex A_{•} with A_{n} = 0 for n < −1, A_{−1} = A, A_{n} ∈ b for n −1; the last differentials will be then denoted d_{−1} = ϵ : A_{0} → A, δ_{−1} = ϵ̂: ϵd_{0} → 0_{A_{1},A}and the pair (ϵ, ϵ̂) will be called augmentation of the resolution. It is clear that any secondary chain complex (A_{•}, d_{•}, δ_{•}) in B gives rise to a chain complex (A_{•}, [d_{•}]), in the ordinary sense, in the homotopy category B_{≃} of B. Moreover if B is Σ-additive, i. e. there exists a functor Σ and isomorphisms Aut(0_{X,Y}) [ΣX, Y], natural in X, Y, then b-exactness of (A_{•}, d_{•}, δ_{•}) implies b_{≃}-exactness of (A_{•}, [d_{•}]) in the sense that the chain complex of abelian groups [X, (A_{•}, [d_{•}])] will be exact for each X ∈ b. In [3], the notion of b_{≃}-relative derived functors has been developed using such b_{≃}-resolutions, which we also recall.

For an additive subcategory a = b_{≃} of the homotopy category B_{≃}, the a-relative left derived functors L_{n}^{a}F, n 0, of a functor F : B_{≃} → 𝒜 from B_{≃} to an abelian category 𝒜 are defined by

(L_{n}^{a}F)A = H_{n}(F(A_{•})),

where A_{•} is given by any a-resolution of A. Similarly, a-relative right derived functors of a contravariant functor F : B_{≃}^{op} → 𝒜 are given by

(R_{a}^{n}F)A = H^{n}(F(A_{•})).

In particular, for the contravariant functor F = [ , B] we get the a-relative Ext-groups

Ext_{a}^{n}(A, B) := (R_{a}^{n}[ , B])A = H^{n}([A_{•}, B])

for any a-exact resolution A_{•} of A. Similarly, for the contravariant functor Aut(0_{,B}) which assigns to an object A the group Aut(0_{A,B}) of all tracks α : 0_{A,B}⇒ 0_{A,B}from the zero map A → ∗ → B to itself, one gets the groups of a-derived automorphisms

Aut_{a}^{n}(A, B) := (R_{a}^{n}Aut(0_{,B}))(A).

It is proved in [3] that under mild conditions (existence of a subset of a such that every object of a is a direct summand of a direct sum of objects from that subset) every object has an a-resolution, and that the resulting groups do not depend on the choice of a resolution.

We next recall the construction of the secondary differential from [3]. This is the map of the form

d_{(2)} : Ext_{a}^{n}(A, B) → Aut_{a}^{n}(0_{A,B});

it is constructed from any secondary b-resolution (A_{•}, d_{•}, δ_{•}, ϵ, ϵ̂) of the object A. Given an element [c] ∈ Ext_{a}^{n}(A, B), one first represents it by an n-cocycle in [(A_{•}, [d_{•}]), B], i. e. by a homotopy class [c] ∈ [A_{n}, B] with [cd_{n}] = 0. One then chooses an actual representative c : A_{n} → B of it in B and a track γ : 0 ⇒ cd_{n}. It can be shown that the composite track Γ = cδ_{n} γd_{n+1} ∈ Aut(0_{A_{n+2},B}) satisfies Γd_{n+1} = 0, so it is an (n + 2)-cocycle in the cochain complex Aut(0_{(A_{•},[d_{•}]),B}) [(ΣA_{•}, [Σd_{•}]), B], so determines a cohomology class d(2)([c]) = [Γ] ∈ Ext_{a}^{n+2}(ΣA, B). It is proved in [3, 4.2] that the above construction does not indeed depend on choices.

Now turning to our situation, it is straightforward to verify that a secondary chain complex in the sense of [3] in the track category B-Mod is the same as the 2-complex in the sense of 3.1, and that the two notions of exactness coincide. In particular then the notions of resolution are also equivalent.

The track subcategory b of free modules is generated by coproducts from a single object, so b_{≃}- resolutions of any B-module exist. In fact it follows from [3, 2.13] that any B-module has a secondary b-resolution too.

Moreover there are natural isomorphisms

Aut(0_{M,N}) Hom_{π_{0}B}(π_{0}M, π_{1}N).

Indeed a track from the zero map to itself is a B_{0}-module homomorphism H : M_{0} → N_{1} with ∂H = 0, H∂ = 0, so H factors through M_{0} π_{0}M and over π_{1}N N_{1}.

Hence the proof is finished with the following lemma.

(3.7) Lemma. For any B-modules M, N there are isomorphisms

Ext_{a}^{n}(M, N) Ext_{π_{0}B}^{n}(π_{0}M, π_{0}N)

and

(R_{a}^{n}(Hom_{π_{0}B}(π_{0}( ), π_{1}N)))(M) Ext_{π_{0}B}^{n}(π_{0}M, π_{1}N).

Proof. By definition the groups Ext_{a}^{∗}(M, N), respectively (R_{a}^{n}(Hom_{B_{0}}(π_{0}( ), π_{1}N)))(M), are cohomology groups of the complex [F_{•}, N], resp. Hom_{π_{0}B}(π_{0}(F_{•}), π_{1}N), where F_{•} is some a-resolution of M. We can choose for _{•} some secondary b-resolution of M. Then π_{0}F_{•} is a free π_{0}B-resolution of π_{0}M, which makes evident the second isomorphism. For the first, just note in addition that by 1.9 [F_{•}, N] is isomorphic to Hom_{B_{0}}(π_{0}(F_{•}), π_{0}N).

4. The track category of spectra

In this section we introduce the notion of stable maps and stable tracks between spectra. This yields the track category of spectra. See also [1, section 2.5].

(4.1) Definition. A spectrum X is a sequence of maps

X_{i} ⟶^{r} ΩX_{i+1}, i ∈

in the category Top^{∗} of pointed spaces. This is an Ω-spectrum if r is a homotopy equivalence for all i.

A stable homotopy class f : X → Y between spectra is a sequence of homotopy classes f_{i} ∈ [X_{i}, Y_{i}] such that the squares

\xymatrix{X_{i} \ar[r]^{f_{i}} \ar[d]^{r} & Y_{i} \ar[d]^{r} \\ ΩX_{i+1} \ar[r]^{Ωf_{i+1}} & ΩY_{i+1}}

commute in Top^{∗}_{≃}. The category Spec consists of spectra and stable homotopy classes as morphisms. Its full subcategory Ω-Spec consisting of Ω-spectra is equivalent to the usual homotopy category of spectra considered as a Quillen model category.

A stable map f = (f_{i}, f̃_{i})_{i} : X → Y between spectra is a sequence of diagrams in the track category

Top^{∗} (i ∈ ) [figure: X_{i}; f_{i}; r; f̃_{i}; Y_{i}; r; ΩX_{i+1}; Ωf_{i+1}; ΩY_{i+1}.]

Obvious composition of such maps yields the category

Spec _{0} .

It is the underlying category of a track category Spec with tracks (H : f ⇒ g) ∈ Spec _{1} given by sequences

H_{i} : f_{i} ⇒ g_{i}

of tracks in Top^{∗} such that the diagrams

[figure: X_{i}; f_{i}; r; f̃_{i}; g_{i}; H_{i}; Y_{i}; r; ΩX_{i+1}; Ωf_{i+1}; Ωg_{i+1}; ΩH_{i+1}; ΩY_{i+1}]

paste to g̃_{i}. This yields a well-defined track category Spec . Moreover

Spec _{≃} Spec

is an isomorphism of categories. Let X, Y be the groupoid of morphisms X → Y in Spec _{0} and let X, Y _{1}^{0} be the set of pairs (f, H) where f : X → Y is a map and H : f ⇒ 0 is a track in Spec , i. e. a stable homotopy class of nullhomotopies for f.

For a spectrum X let Σ^{k}X be the shifted spectrum with (Σ^{k}X)_{n} = X_{n+k} and the commutative diagram

[figure: (Σ^{k}X)_{n}; r; Ω(Σ^{k}X)_{n+1}; X_{n+k}; r; Ω(X_{n+k+1})]

defining r for Σ^{k}X. A map f : Y → Σ^{k}X is also called a map f of degree k from Y to X.

5. The pair algebra ℬ and secondary cohomology of spectra as a ℬ-module

The secondary cohomology of a space was introduced in [1, section 6.3]. We here consider the corresponding notion of secondary cohomology of a spectrum.

Let be a prime field = /p and let Z denote the Eilenberg-Mac Lane spectrum with

Z^{n} = K( , n)

chosen as in [1]. Here Z^{n} is a topological -vector space and the homotopy equivalence Z^{n} → ΩZ^{n+1} is -linear. This shows that for a spectrum X the sets X, Σ^{k}Z _{0} and X, Σ^{k}Z _{1}^{0}, of stable maps and stable 0-tracks repectively, are -vector spaces.

We now recall the definition of the pair algebra ℬ = (∂ : ℬ_{1} → ℬ_{0}) of secondary cohomology operations from [1]. Let = /p^{2} and let

ℬ_{0} = T_{}(E_{𝒜} )

be the -tensor algebra generated by the subset

E_{𝒜} = \begin{cases}\{Sq^{1}, Sq^{2}, ...\} & for p = 2, \\ \{P^{1}, P^{2}, ...\}∪ \{β, βP^{1}, βP^{2}, ...\} & for odd p \end{cases}

of the mod p Steenrod algebra 𝒜. We define ℬ_{1} by the pullback diagram of graded abelian groups

(5.1)  [figure: Σ𝒜; ℬ_{1}; ∂; ; _{Z, Σ^{∗}Z {0 \atop 1}}; ∂; ℬ_{0}; s; _{Z, Σ^{∗}Z _{0}}; 𝒜 .]

in which the right hand column is an exact sequence. Here we choose for α ∈ E_{𝒜} a stable map s(α) : Z → Σ^{|α|}Z representing α and we define s to be the -linear map given on monomials a_{1} ⋯ a_{n} in the free monoid Mon(E_{𝒜} ) generated by E_{𝒜} by the composites

s(a_{1} ⋯ a_{n}) = s(a_{1}) ⋯ s(a_{n}).

It is proved in [1, 5.2.3] that s defines a pseudofunctor, that is, there is a well-defined track

Γ : s(a · b) ⇒ s(a) ∘ s(b)

for a, b ∈ ℬ_{0} such that for any a, b, c pasting of tracks in the diagram

[figure: Γ; s(a); s(b); s(a·b); Γ; s(c); s(b·c); Γ; s(a·b·c); Γ; s(a·b·c)]

yields the identity track. Now ℬ_{1} is a ℬ_{0}-ℬ_{0}-bimodule by defining

a(b, z) = (a · b, a • z)

with a • z given by pasting s(a)z and Γ. Similarly

(b, z)a = (b · a, z • a)

where z • a is obtained by pasting zs(a) and Γ. Then it is shown in [1] that ℬ = (∂ : ℬ_{1} → ℬ_{0}) is a well-defined pair algebra with π_{0}ℬ = 𝒜 and Σ-structure π_{1}ℬ = Σ𝒜.

For a spectrum X let

ℋ (X)_{0} = ℬ_{0} X, Σ^{∗}Z _{0}

be the free ℬ_{0}-module generated by the graded set X, Σ^{∗}Z _{0}. We define ℋ (X)_{1} by the pullback diagram

[figure: ΣH^{∗}X; ℋ (X)_{1}; ∂; ; _{X, Σ^{∗}Z {0 \atop 1}}; ∂; ℋ (X)_{0}; s; _{X, Σ^{∗}Z _{0}}; H^{∗}X]

where s is the -linear map which is the identity on generators and is defined on words a_{1} ⋯ a_{n} · u by the composite s(a_{1}) ⋯ s(a_{n})s(u) for a_{i} as above and u ∈ X, Σ^{∗}Z _{0}. Again s is a pseudofunctor and with actions • defined as above we see that the graded pair module

ℋ (X) = (ℋ (X)_{1} ⟶^{∂} ℋ (X)_{0})

is a ℬ-module. We call ℋ (X) the secondary cohomology of the spectrum X. Of course ℋ (X) has a Σ-structure in the sense of 2.3 above.

(5.2) Example. Let ^{Σ} be the ℬ-module given by the augmentation ℬ → ^{Σ} in [1]. Recall that ^{Σ} is the pair

Σ = ( ⊕ Σ ⟶^{∂} )

with ∂|_{}the inclusion nad ∂|_{Σ} = 0. Then the sphere spectrum S^{0} admits a weak equivalence of ℬ-modules

ℋ (S^{0}) ⟶^{∼} ^{Σ}.

Compare [1, 12.1.5].

6. The E_{3}-term of the Adams spectral sequence

We now are ready to formulate our main result describing the algebraic equivalent of the E_{3}-term of the Adams spectral sequence. Let X be a spectrum of finite type and Y a finite dimensional spectrum. Then for each prime p there is a spectral sequence E_{∗} = E_{∗}(Y, X) with

E_{∗} ⟹ [Y, Σ^{∗}X]_{p}
E_{2} = Ext_{𝒜} (H^{∗}X, H^{∗}Y).

(6.1) Theorem. The E_{3}-term E_{3} = E_{3}(Y, X) of the Adams spectral sequence is given by the secondary Ext group defined in 3.5

E_{3} = Ext_{ℬ} (ℋ ^{∗}X, ℋ ^{∗}Y).

(6.2) Corollary. If X and Y are both the sphere spectrum we get

E_{3}(S^{0}, S^{0}) = Ext_{ℬ} ( ^{Σ}, ^{Σ}).

Since the pair algebra ℬ is computed in [1] completely we see that E_{3}(S^{0}, S^{0}) is algebraically determined. This leads to the algorithm below computing E_{3}(S^{0}, S^{0}).

The proof of 6.1 is based on the following result in [1]. Consider the track categories

b ⊂ Spec
b′ ⊂ (ℬ−Mod)^{Σ}

where Spec is the track category of spectra in 4.1 and (ℬ−Mod)^{Σ} is the track category of ℬ-modules with Σ-structure in 2.3 with the pair algebra ℬ defined by (5.1). Let b be the full track subcategory of Spec consisting of finite products of shifted Eilenberg-Mac Lane spectra Σ^{k}Z^{∗}. Moreover let b′ be the full track subcategory of (ℬ−Mod)^{Σ} consisting of finitely generated free ℬ-modules. As in [3, 4.3] we obtain for spectra X, Y in 6.1 the track categories

{Y, X}b ⊂ Spec
b′{ℋ X, ℋ Y} ⊂ (ℬ−Mod)^{Σ}

with {Y, X}b obtained by adding to b the objects X, Y and all morphisms and tracks from X, Z , Y, Z for all objects Z in b. It is proved in [1, 5.5.6] that the following result holds which shows that we can apply [3, 5.1].

(6.3) Theorem [1]. There is a strict track equivalence

({Y, X}b)^{op} ⟶^{∼} b′{ℋ X, ℋ Y} .

Proof of 6.1. By the main result 7.3 in [3] we have a description of the differential d_{(2)} in the Adams spectral sequence by the following commutative diagram

\xymatrix{Ext_{a^{op}}^{n}(X, Y)^{m} \ar[r]^{d_{(2)}} \ar[d]^{} & Ext_{a^{op}}^{n+2}(X, Y)^{m+1} \ar[d]^{} \\ Ext_{𝒜}^{n} (H^{∗}X, H^{∗}Y)^{m} \ar[r]^{d_{(2)}} & Ext_{𝒜}^{n+2}(H^{∗}X, H^{∗}Y)^{m+1}}

where a = b_{≃}. On the other hand the differential d_{(2)} defining the secondary Ext-group Ext_{ℬ} (ℋ X, ℋ Y) is by 3.6 given by the commutative diagram

[figure: Ext_{a′}^{n}(ℋ X, ℋ Y)^{m}; Ext_{a′}^{n+2}(ℋ X, ℋ Y)^{m+1}; Ext_{𝒜}^{n} (H^{∗}X, H^{∗}Y)^{m}; Ext_{𝒜}^{n+2}(H^{∗}X, H^{∗}Y)^{m+1}]

where a′ = b′_{≃}. Now [3, 5.1] shows by 6.3 that the top rows of these diagrams coincide.

7. The multiplication map of ℬ

We recall notation = /4 , = _{2} = /2 from [1]. The quotient homomorphism will be denoted by π and the isomorphism 2 by i. Moreover we will need the set-theoretic section χ : ↪ of π given by χ(0) = 0, χ(1) = 1. In the pair algebra ℬ = (∂ : ℬ_{1} → ℬ_{0}) , recall that ℬ_{0} is the graded free associative -algebra on the generators Sq^{n} of degree n, for n 1; there is thus a surjective homomorphism of graded algebras π : ℬ_{0} 𝒜 onto the mod 2 Steenrod algebra. Its kernel is denoted by R, so that we have the short exact sequence

0 → R → ℬ_{0} → 𝒜 → 0.

It is well known that R is a graded two-sided ideal generated (as a two-sided ideal, i. e. as a ℬ_{0}-ℬ_{0}- bimodule) by 2ℬ_{0} and by the Adem elements

[a, b] := Sq^{a} Sq^{b}+ ∑_{k=max(0,a−b+1)}^{min(b−1,[\frac{a}{2}])} χ\binom{b − k − 1}{a − 2k}Sq^{a+b−k} Sq^{k},

for 0 < a < 2b. As shown in [1], one can generate R as a right ℬ_{0}-module by 2 ∈ R^{0} and the admissible relations Sq^{a_{k}} Sq^{a_{k−1}} ⋯ Sq^{a_{1}}[a_{0}, b] ∈ R^{a_{k}+...+a_{0}+b}, with a_{j+1} 2a_{j} for all k > j 0 and a_{0} < 2b.

As for the rest of the structure of ℬ, as an abelian group, ℬ_{1} is R ⊕ Σ𝒜, that is,

ℬ_{1}^{n} = R^{n} ⊕ 𝒜^{n−1},

and ∂ is the projection. Moreover the ℬ_{0}-bimodule structure of ℬ_{1} is given by

(r, a)b = (rb, aπ(b))

and

b(r, a) = (br, A(π(b), r) + π(b)a),

where

A : 𝒜 ⊗ R → 𝒜

is the multiplication map of degree -1 described in [1]. Algebraic properties characterizing the multiplication map A are achieved in [1, theorem 16.3.3]. In [1, section 16.6] an algorithm is obtained which computes the multiplication map A.

Particular important elements of ℬ_{1} get special notation; e. g. we have [2] := (2, 0) ∈ R^{0} ⊕ 0 = ℬ_{1}^{0} and Σ1 := (0, 1) ∈ R^{1} ⊕ 𝒜^{0} = ℬ_{1}^{1}. This pair algebra has an augmentation ϵ : ℬ → ^{Σ}, where ^{Σ} = ((i, 0) : ⊕ Σ → ) is the graded ℬ-module equal to i : 2 ⊂ in degree 0, to → 0 in degree 1 and zero in all other degrees. Components of ϵ are the augmentation ϵ_{0} : ℬ_{0} → and the homomorphism ϵ_{1} : R ⊕ Σ𝒜 → 2 ⊕ Σ given by (r, a) ↦ (ϵ_{0}(r), ϵ(a)).

8. The algorithm for the computation of d_{(2)} on Ext_{𝒜} ( , ) in terms of the multiplication maps

Suppose now given some projective resolution of the left 𝒜 -module . For definiteness, we will work with the minimal resolution

(8.1)  ← 𝒜 ⟨g_{0}^{0}⟩← 𝒜 ⟨g_{1}^{2^{n}} | n 0⟩ ← 𝒜 ⟨g_{2}^{2^{i}+2^{j}} | |i − j| 1⟩ ← ...,

where g_{m}^{d}, d m, is a generator of the m-th resolving module in degree d. Sometimes there are more than one generators with the same m and d, in which case the further ones will be denoted by′g_{m}^{d},′′g_{m}^{d}, · · · .

These generators and values of the differential on them can be computed effectively; for example, d(g_{1}^{2^{n}}) = Sq^{2^{n}} g_{0}^{0} and d(g_{m}^{m}) = Sq^{1} g_{m−1}^{m−1}; moreover e. g. an algorithm from [4] gives

d(g_{2}^{4}) = Sq^{3} g_{1}^{1} + Sq^{2} g_{1}^{2}
d(g_{2}^{5}) = Sq^{4} g_{1}^{1} + Sq^{2} Sq^{1} g_{1}^{2} + Sq^{1} g_{1}^{4}
d(g_{2}^{8}) = Sq^{6} g_{1}^{2} + (Sq^{4} + Sq^{3} Sq^{1})g_{1}^{4}
d(g_{2}^{9}) = Sq^{8} g_{1}^{1} + (Sq^{5} + Sq^{4} Sq^{1})g_{1}^{4} + Sq^{1} g_{1}^{8}
d(g_{2}^{10}) = (Sq^{8} + Sq^{5} Sq^{2} Sq^{1})g_{1}^{2} + (Sq^{5} Sq^{1} + Sq^{4} Sq^{2})g_{1}^{4} + Sq^{2} g_{1}^{8}
d(g_{2}^{16}) = (Sq^{12} + Sq^{9} Sq^{2} Sq^{1} + Sq^{8} Sq^{3} Sq^{1})g_{1}^{4} + (Sq^{8} + Sq^{7} Sq^{1} + Sq^{6} Sq^{2})g_{1}^{8}

⋯,

d(g_{3}^{6}) = Sq^{4} g_{2}^{2} + Sq^{2} g_{2}^{4} + Sq^{1} g_{2}^{5}
d(g_{3}^{10}) = Sq^{8} g_{2}^{2} + (Sq^{5} + Sq^{4} Sq^{1})g_{2}^{5} + Sq^{1} g_{2}^{9}
d(g_{3}^{11}) = (Sq^{7} + Sq^{4} Sq^{2} Sq^{1})g_{2}^{4} + Sq^{6} g_{2}^{5} + Sq^{2} Sq^{1} g_{2}^{8}
d(g_{3}^{12}) = Sq^{8} g_{2}^{4} + (Sq^{6} Sq^{1} + Sq^{5} Sq^{2})g_{2}^{5} + (Sq^{4} + Sq^{3} Sq^{1})g_{2}^{8} + Sq^{3} g_{2}^{9} + Sq^{2} g_{2}^{10}

⋯,

d(g_{4}^{11}) = Sq^{8} g_{3}^{3} + (Sq^{5} + Sq^{4} Sq^{1})g_{3}^{6} + Sq^{1} g_{3}^{10}
d(g_{4}^{13}) = Sq^{8} Sq^{2} g_{3}^{3} + (Sq^{7} + Sq^{4} Sq^{2} Sq^{1})g_{3}^{6} + Sq^{2} Sq^{1} g_{3}^{10} + Sq^{2} g_{3}^{11}

⋯,

d(g_{5}^{14}) = Sq^{10} g_{4}^{4} + Sq^{2} Sq^{1} g_{4}^{11}
d(g_{5}^{16}) = Sq^{12} g_{4}^{4} + Sq^{4} Sq^{1} g_{4}^{11} + Sq^{3} g_{4}^{13}

⋯,

d(g_{6}^{16}) = Sq^{11} g_{5}^{5} + Sq^{2} g_{5}^{14}

⋯,

etc.

By understanding the above formulæ literally (i. e. by applying χ degreewise to them), each such resolution gives rise to a sequence of ℬ-module homomorphisms

(8.2)  ^{Σ} ← ℬ ⟨g_{0}^{0}⟩← ℬ ⟨g_{1}^{2^{n}} | n 0⟩ ← ℬ ⟨g_{2}^{2^{i}+2^{j}} | |i − j| 1⟩ ← ...,

which is far from being exact — in fact even the composites of consecutive maps are not zero. In more detail, one has commutative diagrams

\xymatrix{2 \ar[d] & R^{0}g_{0}^{0} \ar[l]_{ϵ_{0}} \ar[d] & 0 \ar[l] \ar[d] & ... \ar[l] \\  & ℬ_{0}^{0}g_{0}^{0} \ar[l]_{ϵ_{0}} & 0 \ar[l] & ... \ar[l]}

in degree 0,

[figure: ; R^{1}g_{0}^{0} ⊕ 𝒜^{0}g_{0}^{0}; (0,ϵ); R^{0}g_{1}^{1}; ^{\binom{d}{0}}; 0; ...; 0; ℬ_{0}^{1}g_{0}^{0}; ℬ_{0}^{0}g_{1}^{1}; d; 0; ...]

in degree 1,

[figure: 0; R^{2}g_{0}^{0} ⊕ 𝒜^{1}g_{0}^{0}; (R^{1}g_{1}^{1} ⊕ R^{0}g_{1}^{2})⊕ 𝒜^{0}g_{1}^{1}; ^{( \begin{matrix}d & 0 \\ 0 & d \end{matrix})}; R^{0}g_{2}^{2}; ^{\binom{d}{0}}; 0; ...; 0; ℬ_{0}^{2}g_{0}^{0}; ℬ_{0}^{1}g_{1}^{1} ⊕ ℬ_{0}^{0}g_{1}^{2}; d; ℬ_{0}^{0}g_{2}^{2}; d; 0; ...]

in degree 2, ...

\xymatrix{0 \ar[d] & R^{n}g_{0}^{0} ⊕ 𝒜^{n−1}g_{0}^{0} \ar[l] \ar[d] & ⨁ _{2^{i} n} R^{n−2^{i}}g_{1}^{2^{i}} ⊕ ⨁ _{2^{i} n−1} 𝒜^{n−1−2^{i}} g_{1}^{2^{i}} \ar[l]_{^{( \begin{matrix}d & 0 \\ 0 & d \end{matrix})}} \ar[d] & ... \ar[l] \\ 0 & ℬ_{0}^{n}g_{0}^{0} \ar[l] & ⨁ _{2^{i} n} ℬ_{0}^{n−2^{i}}g_{1}^{2^{i}} \ar[l]_{d} & ... \ar[l]}

in degree n, etc.

Our task is then to complete these diagrams into an exact secondary complex via certain (degree preserving) maps

δ_{m} = \binom{δ_{m}^{R}}{δ_{m}^{𝒜}} : ℬ_{0} ⟨g_{m+2}^{n} | n⟩→ (R ⊕ Σ𝒜) ⟨g_{m}^{n} | n⟩.

Now for these maps to form a secondary complex, according to 3.1.1 one must have ∂δ = d_{0}d_{0}, δ∂ = d_{1}d_{1}, and d_{1}δ = δd_{0}. One sees easily that these equations together with the requirement that δ be left ℬ_{0}-module homomorphism are equivalent to

(8.3)  δ^{R} = dd,
(8.4)  δ^{𝒜} (bg)= π(b)δ^{𝒜} (g) + A(π(b), dd(g)),
(8.5)  dδ^{𝒜} = δ^{𝒜} d,

for b ∈ ℬ_{0}, g one of the g_{m}^{n}, and A(a, rg) := A(a, r)g for a ∈ 𝒜, r ∈ R. Hence δ is completely determined by the elements

δ_{m}^{𝒜} (g_{m+2}^{n}) ∈ ⨁_{k} 𝒜^{n−k−1} ⟨g_{m}^{k}⟩

which, to form a secondary complex, are only required to satisfy

dδ_{m}^{𝒜} (g_{m+2}^{n}) = δ_{m−1}^{𝒜}d(g_{m+2}^{n}),

where on the right δ_{m−1}^{𝒜} is extended to ℬ_{0} ⟨g_{m+1}^{∗}⟩via 8.4. Then furthermore secondary exactness must hold, which by 3.1 means that the (ordinary) complex

← ℬ_{0} ⟨g_{m−1}^{∗}⟩⊕ (R ⊕ Σ𝒜) ⟨g_{m−2}^{∗}⟩← ℬ_{0} ⟨g_{m}^{∗}⟩⊕ (R ⊕ Σ𝒜) ⟨g_{m−1}^{∗}⟩← ℬ_{0} ⟨g_{m+1}^{∗}⟩⊕ (R ⊕ Σ𝒜) ⟨g_{m}^{∗}⟩←

with differentials

( \begin{matrix}d_{m+1} & i_{m+1} & 0 \\ d_{m}d_{m+1} & d_{m} & 0 \\ δ_{m}^{𝒜} & 0 & d_{m} \end{matrix}) : ℬ_{0} ⟨g_{m+2}^{∗}⟩⊕ R ⟨g_{m+1}^{∗}⟩⊕ Σ𝒜 ⟨g_{m+1}^{∗}⟩→ ℬ_{0} ⟨g_{m+1}^{∗}⟩⊕ R ⟨g_{m}^{∗}⟩⊕ Σ𝒜 ⟨g_{m}^{∗}⟩

is exact. Then straightforward checking shows that one can eliminate R from this complex altogether, so that its exactness is equivalent to the exactness of a smaller complex

← ℬ_{0} ⟨g_{m−1}^{∗}⟩⊕ Σ𝒜 ⟨g_{m−2}^{∗}⟩← ℬ_{0} ⟨g_{m}^{∗}⟩⊕ Σ𝒜 ⟨g_{m−1}^{∗}⟩← ℬ_{0} ⟨g_{m+1}^{∗}⟩⊕ Σ𝒜 ⟨g_{m}^{∗}⟩←

with differentials

( \begin{matrix}d_{m+1} & 0 \\ δ_{m}^{𝒜} & d_{m} \end{matrix}): ℬ_{0} ⟨g_{m+2}^{∗}⟩⊕ Σ𝒜 ⟨g_{m+1}^{∗}⟩→ ℬ_{0} ⟨g_{m+1}^{∗}⟩⊕ Σ𝒜 ⟨g_{m}^{∗}⟩.

Note also that by 8.4 δ^{𝒜} factors through π to give

δ̄_{m}: 𝒜 ⟨g_{m+2}^{∗}⟩→ Σ𝒜 ⟨g_{m}^{∗}⟩.

It follows that secondary exactness of the resulting complex is equivalent to exactness of the mapping cone of this δ̄, i. e. to the requirement that δ̄is a quasiisomorphism. On the other hand, the complex (𝒜 ⟨g_{∗}^{∗}⟩, d_{∗}) is acyclic by construction, so any of its self-maps is a quasiisomorphism. We thus obtain (8.6) Theorem. Completions of the diagram 8.2 to an exact secondary complex are in one-to-one correspondence with maps δ_{m} : 𝒜 ⟨g_{m+2}^{∗}⟩→ Σ𝒜 ⟨g_{m}^{∗}⟩satisfying

(8.7)  dδg = δdg,

with δ(ag) for a ∈ 𝒜 defined by

δ(ag) = aδ(g) + A(a, ddg)

where A(a, rg) for r ∈ R is interpreted as A(a, r)g.

We can use this to construct the secondary resolution inductively. Just start by introducing values of δ on the generators as expressions with indeterminate coefficients; the equation (8.7) will impose linear conditions on these coefficients. These are then solved degree by degree. For example, in degree 2 one may have

δ(g_{2}^{2}) = η_{2}^{2}(Sq^{1}) Sq^{1} g_{0}^{0}

for some η_{2}^{2}(Sq^{1}) ∈ . Similarly in degree 3 one may have

δ(g_{3}^{3}) = η_{3}^{3}(Sq^{1}) Sq^{1} g_{1}^{1} + η_{3}^{3}(1)g_{1}^{2}.

Then one will get

dδ(g_{3}^{3}) = η_{3}^{3}(Sq^{1}) Sq^{1} d(g_{1}^{1}) + η_{3}^{3}(1)d(g_{1}^{2}) = η_{3}^{3}(Sq^{1}) Sq^{1} Sq^{1} g_{0}^{0} + η_{3}^{3}(1) Sq^{2} g_{0}^{0} = η_{3}^{3}(1) Sq^{2} g_{0}^{0}

and

δd(g_{3}^{3}) = δ(Sq^{1} g_{2}^{2})

= Sq^{1} δ(g_{2}^{2}) + A(Sq^{1}, dd(g_{2}^{2})) = η_{2}^{2}(Sq^{1}) Sq^{1} Sq^{1} g_{0}^{0} + A(Sq^{1}, d(Sq^{1} g_{1}^{1})) = A(Sq^{1}, Sq^{1} Sq^{1} g_{0}^{0}) = 0;

thus (8.7) forces η_{3}^{3}(1) = 0.

Similarly one puts δ(g_{m}^{d}) = ∑ _{m−2 d′ d−1} ∑ _{a} η_{m}^{d}(a)ag_{m−2}^{d′}, with a running over a basis in 𝒜^{d−1−d′}, and then substituting this in (8.7)gives linear equations on the numbers η_{m}^{d}(a). Solving these equations and choosing the remaining η’s arbitrarily then gives values of the differential δ in the secondary resolution.

Then finally to obtain the secondary differential

d_{(2)} : Ext_{𝒜}^{n} ( , )^{m} → Ext_{𝒜}^{n+2}( , )^{m+1}

from this δ, one just applies the functor Hom_{𝒜} ( , ) to the initial minimal resolution and calculates the map induced by δ on cohomology of the resulting cochain complex, i. e. on Ext_{𝒜}^{∗} ( , ). In fact since (8.1) is a minimal resolution, the value of Hom_{𝒜} ( , ) on it coincides with its own cohomology and is the -vector space of those linear maps 𝒜 ⟨g_{∗}^{∗}⟩ → which vanish on all elements of the form ag_{∗}^{∗} with a of positive degree.

Let us then identify Ext_{𝒜}^{∗} ( , ) with this space and choose a basis in it consisting of elements ĝ_{m}^{d}defined as the maps sending the generator g_{m}^{d} to 1 and all other generators to 0. One then has

(d_{(2)}(ĝ_{m}^{d}))(g_{m′}^{d′}) = ĝ_{m}^{d}δ(g_{m′}^{d′}).

The right hand side is nonzero precisely when g_{m}^{d} appears in δ(g_{m′}^{d′}) with coefficient 1, i. e. one has

d_{(2)}(ĝ_{m}^{d}) = ∑_{g_{m}^{d} appears in δ(g_{m+2}^{d+1})}ĝ_{m+2}^{d+1}.

For example, looking at the table of values of δ below we see that the first instance of a g_{m}^{d} appearing with coefficient 1 in a value of δ on a generator is

δ(g_{3}^{17}) = g_{1}^{16} + Sq^{12} g_{1}^{4} + Sq^{10} Sq^{4} g_{1}^{2} + (Sq^{9} Sq^{4} Sq^{2} + Sq^{10} Sq^{5} + Sq^{11} Sq^{4})g_{1}^{1}.

This means

d_{(2)}(ĝ_{1}^{16}) = ĝ_{3}^{17}

and moreover d_{(2)}(ĝ_{m}^{d}) = 0 for all g_{m}^{d}with d < 17 (one can check all cases for each given d since the number of generators g_{m}^{d} for each given d is finite).

Treating similarly the rest of the table below we find that the only nonzero values of d_{(2)} on generators of degree < 36 are as follows:

\begin{aligned}d_{(2)}(ĝ_{1}^{16}) & = ĝ_{3}^{17} \\ d_{(2)}(ĝ_{4}^{21}) & = ĝ_{6}^{22} \\ d_{(2)}(ĝ_{4}^{22}) & = ĝ_{6}^{23} \\ d_{(2)}(ĝ_{5}^{23}) & = ĝ_{7}^{24} \\ d_{(2)}(ĝ_{7}^{30}) & = ĝ_{9}^{31} \\ d_{(2)}(ĝ_{8}^{31}) & = ĝ_{10}^{32} \\ d_{(2)}(ĝ_{1}^{32}) & = ĝ_{3}^{33} \\ d_{(2)}(ĝ_{2}^{33}) & = ĝ_{4}^{34} \\ d_{(2)}(ĝ_{7}^{33}) & = ĝ_{9}^{34} \\ d_{(2)}(ĝ_{8}^{33}) & = ĝ_{10}^{34} \\ d_{(2)}(′ĝ_{3}^{34}) & = ĝ_{5}^{35} \\ d_{(2)}(ĝ_{8}^{34}) & = ĝ_{10}^{35}. \end{aligned}

Presently a computer calculation continues reaching degree 39 and showing that up to that degree there are the following further nonzero differentials:

\begin{aligned}d_{(2)}(ĝ_{7}^{36}) & = ĝ_{9}^{37} \\ d_{(2)}(ĝ_{8}^{37}) & = ĝ_{10}^{38}. \end{aligned}

These data can be summarized in the following picture, thus confirming calculations presented in Ravenel’s book [6].

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; ∘; ∘; •; •; •; •; •; •; •; •; •; •; ∘; •; •; •; •; •; •; ∘; •; •; •; •; •; ∘; ∘; •; •; •; •; •; •; ∘; •; ∘; •; •; •; ∘; •; •; •; •; •; •; •; •; •; •; ∘; •; •; •; •; ∘; •; •; •; •; •; •; •; ∘; ∘; •; •; •; •; •; •; •; •; •; ∘; ∘; ∘; ∘; •; •; •; •; •; •; ∘; ∘; ∘; •; ∘; •; •; •; •; •; ∘; •; •; ∘; ∘; ∘; •; •; •; •; ∘; •; ∘; ∘; ∘; •; •; •; ∘; •; •; •; d − m; m]

9. The table of values of the differential δ in the secondary resolution for ^{Σ}

The following table presents results of computer calculations of the differential δ. Note that it does not have invariant meaning since it depends on the choices involved in determination of the multiplication map A, of the resolution and of those indeterminate coefficients η_{m}^{d}(a) which remain undetermined after the conditions (8.7) are satisfied. The resulting secondary differential d_{(2)} however does not depend on these choices and is canonically determined.

\begin{aligned}δ(g_{2}^{2}) & = 0 \\ δ(g_{3}^{3}) & = 0 \\ δ(g_{2}^{4}) & = 0 \\ δ(g_{4}^{4}) & = 0 \\ δ(g_{2}^{5}) & = 0 \\ δ(g_{5}^{5}) & = 0 \\ δ(g_{3}^{6}) & = Sq^{4} g_{1}^{1} \end{aligned}

δ(g_{6}^{6}) = 0
δ(g_{7}^{7}) = 0
δ(g_{2}^{8}) = 0
δ(g_{8}^{8}) = 0
δ(g_{2}^{9}) = 0
δ(g_{9}^{9}) = 0
δ(g_{2}^{10}) = 0
δ(g_{3}^{10}) = (Sq^{4} Sq^{2} Sq^{1} + Sq^{7})g_{1}^{2}
+ Sq^{8} g_{1}^{1}
δ(g_{10}^{10}) = 0
δ(g_{3}^{11}) = (Sq^{7} Sq^{1} + Sq^{8})g_{1}^{2}
+ Sq^{6} Sq^{3} g_{1}^{1}
δ(g_{4}^{11}) = Sq^{5} g_{2}^{5}
+ Sq^{4} Sq^{2} g_{2}^{4}
δ(g_{11}^{11}) = 0
δ(g_{3}^{12}) = Sq^{7} Sq^{3} g_{1}^{1}
δ(g_{12}^{12}) = 0
δ(g_{4}^{13}) = Sq^{4} g_{2}^{8}
+(Sq^{7} + Sq^{5} Sq^{2})g_{2}^{5}
+(Sq^{8} + Sq^{6} Sq^{2})g_{2}^{4}
+(Sq^{7} Sq^{3} + Sq^{8} Sq^{2} + Sq^{10})g_{2}^{2}
δ(g_{13}^{13}) = 0
δ(g_{5}^{14}) = Sq^{4} Sq^{2} Sq^{1} g_{3}^{6}
+(Sq^{7} Sq^{3} + Sq^{8} Sq^{2})g_{3}^{3}
δ(g_{14}^{14}) = 0
δ(g_{2}^{16}) = 0
δ(g_{5}^{16}) = Sq^{3} g_{3}^{12}
+ Sq^{4} g_{3}^{11}
+ Sq^{5} g_{3}^{10}
+ Sq^{10} Sq^{2} g_{3}^{3}
δ(g_{6}^{16}) = 0
δ(g_{2}^{17}) = 0
δ(g_{3}^{17}) = g_{1}^{16}
+ Sq^{12} g_{1}^{4}
+ Sq^{10} Sq^{4} g_{1}^{2}
+(Sq^{9} Sq^{4} Sq^{2} + Sq^{10} Sq^{5} + Sq^{11} Sq^{4})g_{1}^{1}
δ(g_{6}^{17}) = (Sq^{5} + Sq^{4} Sq^{1})g_{4}^{11}
+(Sq^{12} + Sq^{10} Sq^{2})g_{4}^{4}
δ(g_{2}^{18}) = 0
δ(g_{3}^{18}) = (Sq^{11} Sq^{4} + Sq^{8} Sq^{4} Sq^{2} Sq^{1})g_{1}^{2}

+(Sq^{10} Sq^{4} Sq^{2} + Sq^{11} Sq^{5} + Sq^{12} Sq^{4} + Sq^{14} Sq^{2} + Sq^{16})g_{1}^{1}

δ(g_{4}^{18}) = (Sq^{6} Sq^{1} + Sq^{7})g_{2}^{10}

+(Sq^{6} Sq^{3} + Sq^{7} Sq^{2} + Sq^{9})g_{2}^{8} + Sq^{8} Sq^{4} g_{2}^{5} +(Sq^{10} Sq^{2} Sq^{1} + Sq^{13} + Sq^{11} Sq^{2} + Sq^{12} Sq^{1})g_{2}^{4} +(Sq^{9} Sq^{4} Sq^{2} + Sq^{15} + Sq^{12} Sq^{3} + Sq^{10} Sq^{5})g_{2}^{2}

δ(g_{7}^{18}) = Sq^{2} Sq^{1} g_{5}^{14}
δ(g_{4}^{19}) = Sq^{9} g_{2}^{9}

+(Sq^{10} + Sq^{8} Sq^{2})g_{2}^{8} + Sq^{11} Sq^{2} g_{2}^{5} +(Sq^{11} Sq^{2} Sq^{1} + Sq^{13} Sq^{1} + Sq^{8} Sq^{4} Sq^{2} + Sq^{10} Sq^{3} Sq^{1})g_{2}^{4} +(Sq^{14} Sq^{2} + Sq^{10} Sq^{4} Sq^{2} + Sq^{12} Sq^{4})g_{2}^{2}

δ(g_{5}^{19}) = Sq^{1} g_{3}^{17}

+ Sq^{4} Sq^{2} g_{3}^{12} + Sq^{4} Sq^{2} Sq^{1} g_{3}^{11} +(Sq^{6} Sq^{2} + Sq^{8})g_{3}^{10} +(Sq^{8} Sq^{4} + Sq^{11} Sq^{1})g_{3}^{6} +(Sq^{13} Sq^{2} + Sq^{10} Sq^{5} + Sq^{15} + Sq^{11} Sq^{4})g_{3}^{3}

δ(g_{2}^{20}) = 0
δ(g_{3}^{20}) = (Sq^{15} + Sq^{9} Sq^{4} Sq^{2})g_{1}^{4}

+(Sq^{12} Sq^{5} + Sq^{13} Sq^{4} + Sq^{16} Sq^{1})g_{1}^{2} +(Sq^{11} Sq^{5} Sq^{2} + Sq^{15} Sq^{3} + Sq^{18} + Sq^{12} Sq^{6})g_{1}^{1}

δ(g_{5}^{20}) = Sq^{4} Sq^{2} Sq^{1} g_{3}^{12}

+(Sq^{7} Sq^{1} + Sq^{8})g_{3}^{11} +(Sq^{10} Sq^{3} + Sq^{8} Sq^{4} Sq^{1} + Sq^{13} + Sq^{11} Sq^{2})g_{3}^{6} +(Sq^{13} Sq^{3} + Sq^{10} Sq^{4} Sq^{2} + Sq^{11} Sq^{5} + Sq^{12} Sq^{4})g_{3}^{3}

δ(′g_{5}^{20}) = Sq^{5} Sq^{2} g_{3}^{12}

+ Sq^{7} Sq^{2} g_{3}^{10} +(Sq^{12} Sq^{1} + Sq^{10} Sq^{3} + Sq^{8} Sq^{4} Sq^{1} + Sq^{10} Sq^{2} Sq^{1} + Sq^{11} Sq^{2})g_{3}^{6} +(Sq^{14} Sq^{2} + Sq^{13} Sq^{3} + Sq^{11} Sq^{5} + Sq^{16} + Sq^{12} Sq^{4})g_{3}^{3}

δ(g_{6}^{20}) = (Sq^{6} Sq^{2} + Sq^{8})g_{4}^{11}

+(Sq^{13} Sq^{2} + Sq^{15} + Sq^{11} Sq^{4})g_{4}^{4}

δ(g_{3}^{21}) = (Sq^{15} Sq^{2} Sq^{1} + Sq^{17} Sq^{1} + Sq^{12} Sq^{6})g_{1}^{2}

+(Sq^{13} Sq^{4} Sq^{2} + Sq^{15} Sq^{4} + Sq^{16} Sq^{3} + Sq^{17} Sq^{2} + Sq^{19})g_{1}^{1}

δ(g_{4}^{21}) = Sq^{3} g_{2}^{17}

+(Sq^{10} + Sq^{9} Sq^{1})g_{2}^{10} +(Sq^{9} Sq^{3} + Sq^{11} Sq^{1})g_{2}^{8} +(Sq^{15} + Sq^{13} Sq^{2} + Sq^{10} Sq^{5})g_{2}^{5} +(Sq^{13} Sq^{2} Sq^{1} + Sq^{12} Sq^{3} Sq^{1} + Sq^{12} Sq^{4} + Sq^{9} Sq^{4} Sq^{2} Sq^{1} + Sq^{10} Sq^{4} Sq^{2})g_{2}^{4} +(Sq^{16} Sq^{2} + Sq^{12} Sq^{6} + Sq^{15} Sq^{3})g_{2}^{2}

δ(g_{6}^{21}) = (Sq^{7} + Sq^{6} Sq^{1})g_{4}^{13}

+(Sq^{9} + Sq^{8} Sq^{1})g_{4}^{11} + Sq^{11} Sq^{5} g_{4}^{4}

δ(g_{3}^{22}) = Sq^{17} g_{1}^{4}

+(Sq^{16} Sq^{2} Sq^{1} + Sq^{13} Sq^{6} + Sq^{12} Sq^{4} Sq^{2} Sq^{1} + Sq^{12} Sq^{6} Sq^{1})g_{1}^{2} +(Sq^{13} Sq^{5} Sq^{2} + Sq^{17} Sq^{3} + Sq^{18} Sq^{2} + Sq^{14} Sq^{4} Sq^{2})g_{1}^{1}

δ(g_{4}^{22}) = Sq^{4} g_{2}^{17}

+ Sq^{11} g_{2}^{10} +(Sq^{12} + Sq^{9} Sq^{3})g_{2}^{9} +(Sq^{9} Sq^{4} + Sq^{13} + Sq^{8} Sq^{4} Sq^{1})g_{2}^{8} + Sq^{12} Sq^{4} g_{2}^{5} + Sq^{15} Sq^{2} g_{2}^{4} +(Sq^{13} Sq^{4} Sq^{2} + Sq^{19} + Sq^{13} Sq^{6} + Sq^{14} Sq^{5})g_{2}^{2}

δ(′g_{4}^{22}) = Sq^{2} Sq^{1} g_{2}^{18}

+(Sq^{8} Sq^{4} + Sq^{12})g_{2}^{9} +(Sq^{9} Sq^{4} + Sq^{13} + Sq^{12} Sq^{1})g_{2}^{8} +(Sq^{16} + Sq^{13} Sq^{3})g_{2}^{5} +(Sq^{15} Sq^{2} + Sq^{16} Sq^{1} + Sq^{13} Sq^{4} + Sq^{11} Sq^{4} Sq^{2})g_{2}^{4} +(Sq^{14} Sq^{5} + Sq^{19} + Sq^{17} Sq^{2})g_{2}^{2}

δ(g_{5}^{22}) = (Sq^{7} Sq^{2} + Sq^{6} Sq^{2} Sq^{1} + Sq^{6} Sq^{3})g_{3}^{12}

+ Sq^{10} g_{3}^{11} + (Sq^{9} Sq^{2} + Sq^{8} Sq^{3} + Sq^{11})g_{3}^{10} +(Sq^{14} Sq^{1} + Sq^{11} Sq^{3} Sq^{1} + Sq^{12} Sq^{3} + Sq^{13} Sq^{2})g_{3}^{6} + Sq^{13} Sq^{5} g_{3}^{3}

δ(g_{6}^{22}) = g_{4}^{21}

+(Sq^{6} Sq^{2} + Sq^{8} + Sq^{7} Sq^{1})g_{4}^{13} + Sq^{10} g_{4}^{11} +(Sq^{13} Sq^{4} + Sq^{15} Sq^{2} + Sq^{17})g_{4}^{4}

δ(g_{7}^{22}) = (Sq^{13} Sq^{3} + Sq^{14} Sq^{2} + Sq^{16})g_{5}^{5}
δ(g_{5}^{23}) = Sq^{4} g_{3}^{18}

+ Sq^{6} Sq^{3} Sq^{1} g_{3}^{12} +(Sq^{10} Sq^{1} + Sq^{11})g_{3}^{11} +(Sq^{8} Sq^{4} + Sq^{9} Sq^{3})g_{3}^{10} +(Sq^{13} Sq^{3} + Sq^{15} Sq^{1} + Sq^{9} Sq^{4} Sq^{2} Sq^{1} + Sq^{11} Sq^{5} + Sq^{14} Sq^{2} + Sq^{12} Sq^{4})g_{3}^{6} +(Sq^{16} Sq^{3} + Sq^{13} Sq^{6} + Sq^{15} Sq^{4})g_{3}^{3}

δ(g_{6}^{23}) = g_{4}^{22}

+ Sq^{9} g_{4}^{13} +(Sq^{10} Sq^{1} + Sq^{11} + Sq^{8} Sq^{3})g_{4}^{11} +(Sq^{16} Sq^{2} + Sq^{14} Sq^{4})g_{4}^{4}

δ(g_{7}^{23}) = (Sq^{6} + Sq^{4} Sq^{2})g_{5}^{16}

+ Sq^{7} Sq^{1} g_{5}^{14} + Sq^{15} Sq^{2} g_{5}^{5}

δ(g_{8}^{23}) = Sq^{5} g_{6}^{17}

+(Sq^{14} Sq^{2} + Sq^{13} Sq^{3})g_{6}^{6}

δ(g_{3}^{24}) = Sq^{11} Sq^{4} g_{1}^{8}

+(Sq^{19} + Sq^{17} Sq^{2})g_{1}^{4} +(Sq^{16} Sq^{4} Sq^{1} + Sq^{14} Sq^{4} Sq^{2} Sq^{1} + Sq^{17} Sq^{4} + Sq^{21})g_{1}^{2} +(Sq^{15} Sq^{7} + Sq^{14} Sq^{6} Sq^{2} + Sq^{18} Sq^{4} + Sq^{22} + Sq^{20} Sq^{2} + Sq^{15} Sq^{5} Sq^{2})g_{1}^{1}

δ(g_{4}^{24}) = Sq^{5} g_{2}^{18}

+(Sq^{12} Sq^{1} + Sq^{10} Sq^{2} Sq^{1} + Sq^{9} Sq^{4})g_{2}^{10} +(Sq^{12} Sq^{2} + Sq^{8} Sq^{4} Sq^{2} + Sq^{11} Sq^{3})g_{2}^{9} +(Sq^{13} Sq^{2} + Sq^{14} Sq^{1})g_{2}^{8} +(Sq^{12} Sq^{4} Sq^{2} + Sq^{16} Sq^{2} + Sq^{12} Sq^{6} + Sq^{14} Sq^{4} + Sq^{11} Sq^{5} Sq^{2} + Sq^{15} Sq^{3} + Sq^{13} Sq^{5})g_{2}^{5} +(Sq^{15} Sq^{4} + Sq^{19} + Sq^{14} Sq^{4} Sq^{1} + Sq^{13} Sq^{6} + Sq^{18} Sq^{1} + Sq^{13} Sq^{4} Sq^{2} + Sq^{15} Sq^{3} Sq^{1} + Sq^{17} Sq^{2} + Sq^{12} Sq^{6} Sq^{1})g_{2}^{4} +(Sq^{16} Sq^{5} + Sq^{14} Sq^{5} Sq^{2} + Sq^{18} Sq^{3} + Sq^{15} Sq^{6} + Sq^{13} Sq^{6} Sq^{2} + Sq^{15} Sq^{4} Sq^{2} + Sq^{21})g_{2}^{2}

δ(g_{7}^{24}) = g_{5}^{23}

+ Sq^{4} g_{5}^{19} +(Sq^{5} Sq^{2} + Sq^{7})g_{5}^{16} +(Sq^{9} + Sq^{8} Sq^{1})g_{5}^{14} +(Sq^{12} Sq^{6} + Sq^{16} Sq^{2} + Sq^{13} Sq^{5} + Sq^{14} Sq^{4})g_{5}^{5}

δ(g_{5}^{25}) = Sq^{4} g_{3}^{20}

+ Sq^{6} g_{3}^{18} + Sq^{7} g_{3}^{17} +(Sq^{8} Sq^{4} + Sq^{9} Sq^{2} Sq^{1} + Sq^{9} Sq^{3} + Sq^{10} Sq^{2})g_{3}^{12} +(Sq^{9} Sq^{4} + Sq^{10} Sq^{2} Sq^{1} + Sq^{12} Sq^{1})g_{3}^{11} +(Sq^{10} Sq^{5} Sq^{2} Sq^{1} + Sq^{12} Sq^{6} + Sq^{13} Sq^{5} + Sq^{18} + Sq^{13} Sq^{4} Sq^{1} + Sq^{16} Sq^{2} + Sq^{17} Sq^{1} + Sq^{14} Sq^{4} + Sq^{15} Sq^{3})g_{3}^{6} +(Sq^{18} Sq^{3} + Sq^{19} Sq^{2} + Sq^{16} Sq^{5})g_{3}^{3}

δ(g_{8}^{25}) = Sq^{7} g_{6}^{17}
δ(g_{4}^{26}) = (Sq^{4} Sq^{2} Sq^{1} + Sq^{6} Sq^{1})g_{2}^{18}

+(Sq^{8} + Sq^{6} Sq^{2})g_{2}^{17} +(Sq^{15} + Sq^{14} Sq^{1} + Sq^{8} Sq^{4} Sq^{2} Sq^{1})g_{2}^{10} +(Sq^{12} Sq^{4} + Sq^{14} Sq^{2})g_{2}^{9} +(Sq^{13} Sq^{4} + Sq^{14} Sq^{3} + Sq^{10} Sq^{5} Sq^{2} + Sq^{15} Sq^{2})g_{2}^{8} +(Sq^{14} Sq^{6} + Sq^{18} Sq^{2} + Sq^{15} Sq^{5} + Sq^{12} Sq^{6} Sq^{2})g_{2}^{5} +(Sq^{18} Sq^{2} Sq^{1} + Sq^{20} Sq^{1} + Sq^{17} Sq^{4} + Sq^{19} Sq^{2} + Sq^{13} Sq^{6} Sq^{2} + Sq^{14} Sq^{7} + Sq^{15} Sq^{6})g_{2}^{4} +(Sq^{17} Sq^{6} + Sq^{14} Sq^{7} Sq^{2} + Sq^{19} Sq^{4} + Sq^{16} Sq^{5} Sq^{2} + Sq^{14} Sq^{6} Sq^{3} + Sq^{15} Sq^{6} Sq^{2} + Sq^{20} Sq^{3} + Sq^{17} Sq^{4} Sq^{2})g_{2}^{2}

δ(g_{5}^{26}) = Sq^{5} g_{3}^{20}

+ Sq^{5} Sq^{2} g_{3}^{18} + Sq^{6} Sq^{2} g_{3}^{17} +(Sq^{10} Sq^{2} Sq^{1} + Sq^{10} Sq^{3} + Sq^{8} Sq^{4} Sq^{1})g_{3}^{12} +(Sq^{13} Sq^{2} + Sq^{11} Sq^{4} + Sq^{9} Sq^{4} Sq^{2} + Sq^{12} Sq^{3} + Sq^{10} Sq^{5})g_{3}^{10} +(Sq^{17} Sq^{2} + Sq^{11} Sq^{5} Sq^{2} Sq^{1} + Sq^{16} Sq^{2} Sq^{1} + Sq^{19} + Sq^{15} Sq^{4} + Sq^{14} Sq^{4} Sq^{1} + Sq^{12} Sq^{6} Sq^{1} + Sq^{12} Sq^{4} Sq^{2} Sq^{1} + Sq^{12} Sq^{5} Sq^{2})g_{3}^{6} +(Sq^{18} Sq^{4} + Sq^{14} Sq^{6} Sq^{2} + Sq^{16} Sq^{6} + Sq^{19} Sq^{3} + Sq^{17} Sq^{5} + Sq^{13} Sq^{6} Sq^{3})g_{3}^{3}

δ(g_{6}^{26}) = Sq^{3} ′g_{4}^{22}

+ Sq^{3} g_{4}^{22} + Sq^{4} g_{4}^{21} + Sq^{6} g_{4}^{19} +(Sq^{10} Sq^{2} + Sq^{9} Sq^{2} Sq^{1} + Sq^{12})g_{4}^{13} +(Sq^{13} Sq^{1} + Sq^{14} + Sq^{11} Sq^{3} + Sq^{12} Sq^{2})g_{4}^{11} +(Sq^{17} Sq^{4} + Sq^{15} Sq^{6} + Sq^{18} Sq^{3} + Sq^{21})g_{4}^{4}

δ(g_{9}^{26}) = (Sq^{6} Sq^{1} + Sq^{4} Sq^{2} Sq^{1})g_{7}^{18}

+(Sq^{15} Sq^{3} + Sq^{16} Sq^{2})g_{7}^{7}

δ(g_{4}^{27}) = Sq^{4} Sq^{2} g_{2}^{20}

+(Sq^{7} Sq^{2} + Sq^{9})g_{2}^{17} + Sq^{10} g_{2}^{16} +(Sq^{12} Sq^{4} + Sq^{11} Sq^{4} Sq^{1} + Sq^{13} Sq^{2} Sq^{1} + Sq^{16})g_{2}^{10} +(Sq^{17} + Sq^{10} Sq^{5} Sq^{2} + Sq^{13} Sq^{4} + Sq^{12} Sq^{5} + Sq^{15} Sq^{2})g_{2}^{9} +(Sq^{12} Sq^{4} Sq^{2} + Sq^{14} Sq^{4})g_{2}^{8} +(Sq^{15} Sq^{6} + Sq^{19} Sq^{2} + Sq^{12} Sq^{6} Sq^{3} + Sq^{13} Sq^{6} Sq^{2})g_{2}^{5} +(Sq^{17} Sq^{4} Sq^{1} + Sq^{12} Sq^{6} Sq^{3} Sq^{1} + Sq^{20} Sq^{2} + Sq^{15} Sq^{7} + Sq^{14} Sq^{7} Sq^{1})g_{2}^{4} +(Sq^{15} Sq^{6} Sq^{3} + Sq^{18} Sq^{6} + Sq^{16} Sq^{8} + Sq^{20} Sq^{4} + Sq^{18} Sq^{4} Sq^{2} + Sq^{15} Sq^{7} Sq^{2})g_{2}^{2}

δ(g_{5}^{28}) = (Sq^{6} Sq^{1} + Sq^{7})g_{3}^{20}

+ Sq^{9} g_{3}^{18} + Sq^{7} Sq^{3} g_{3}^{17} +(Sq^{12} Sq^{2} Sq^{1} + Sq^{9} Sq^{4} Sq^{2} + Sq^{8} Sq^{4} Sq^{2} Sq^{1} + Sq^{11} Sq^{3} Sq^{1} + Sq^{15} + Sq^{14} Sq^{1} + Sq^{12} Sq^{3})g_{3}^{12} + Sq^{12} Sq^{4} g_{3}^{11} +(Sq^{14} Sq^{3} + Sq^{11} Sq^{4} Sq^{2})g_{3}^{10} +(Sq^{14} Sq^{4} Sq^{2} Sq^{1} + Sq^{15} Sq^{6} + Sq^{21} + Sq^{13} Sq^{6} Sq^{2} + Sq^{19} Sq^{2} + Sq^{18} Sq^{3} + Sq^{15} Sq^{4} Sq^{2} + Sq^{17} Sq^{4} + Sq^{14} Sq^{5} Sq^{2} + Sq^{18} Sq^{2} Sq^{1} + Sq^{12} Sq^{6} Sq^{3} + Sq^{14} Sq^{7} + Sq^{12} Sq^{6} Sq^{2} Sq^{1})g_{3}^{6} +(Sq^{20} Sq^{4} + Sq^{24} + Sq^{18} Sq^{6} + Sq^{18} Sq^{4} Sq^{2} + Sq^{16} Sq^{8})g_{3}^{3}

δ(g_{9}^{28}) = Sq^{4} g_{7}^{23}

+(Sq^{20} + Sq^{18} Sq^{2})g_{7}^{7}

δ(g_{10}^{28}) = 0
δ(g_{5}^{29}) = Sq^{4} Sq^{2} g_{3}^{22}

+(Sq^{7} Sq^{1} + Sq^{6} Sq^{2})g_{3}^{20} + Sq^{10} g_{3}^{18} +(Sq^{11} + Sq^{9} Sq^{2})g_{3}^{17} +(Sq^{12} Sq^{4} + Sq^{9} Sq^{4} Sq^{2} Sq^{1} + Sq^{16} + Sq^{10} Sq^{4} Sq^{2} + Sq^{12} Sq^{3} Sq^{1} + Sq^{15} Sq^{1} + Sq^{14} Sq^{2})g_{3}^{12} +(Sq^{17} + Sq^{16} Sq^{1})g_{3}^{11} +(Sq^{11} Sq^{5} Sq^{2} + Sq^{12} Sq^{4} Sq^{2} + Sq^{13} Sq^{5} + Sq^{18})g_{3}^{10} +(Sq^{19} Sq^{2} Sq^{1} + Sq^{15} Sq^{6} Sq^{1} + Sq^{15} Sq^{4} Sq^{2} Sq^{1} + Sq^{14} Sq^{5} Sq^{2} Sq^{1} + Sq^{12} Sq^{6} Sq^{3} Sq^{1} + Sq^{15} Sq^{5} Sq^{2} + Sq^{17} Sq^{5} + Sq^{19} Sq^{3} + Sq^{22})g_{3}^{6} +(Sq^{16} Sq^{6} Sq^{3} + Sq^{20} Sq^{5} + Sq^{17} Sq^{6} Sq^{2} + Sq^{15} Sq^{7} Sq^{3} + Sq^{16} Sq^{7} Sq^{2} + Sq^{18} Sq^{5} Sq^{2} + Sq^{19} Sq^{6})g_{3}^{3}

δ(g_{6}^{29}) = (Sq^{12} Sq^{2} Sq^{1} + Sq^{11} Sq^{4} + Sq^{13} Sq^{2} + Sq^{14} Sq^{1} + Sq^{15} + Sq^{11} Sq^{3} Sq^{1})g_{4}^{13}

+(Sq^{12} Sq^{5} + Sq^{15} Sq^{2} + Sq^{13} Sq^{4} + Sq^{16} Sq^{1} + Sq^{17} + Sq^{11} Sq^{4} Sq^{2} + Sq^{14} Sq^{3})g_{4}^{11} +(Sq^{17} Sq^{5} Sq^{2} + Sq^{22} Sq^{2} + Sq^{15} Sq^{7} Sq^{2} + Sq^{18} Sq^{4} Sq^{2} + Sq^{20} Sq^{4})g_{4}^{4}

δ(g_{10}^{29}) = (Sq^{5} + Sq^{4} Sq^{1})g_{8}^{23}

+(Sq^{18} Sq^{2} + Sq^{20})g_{8}^{8}

δ(g_{7}^{30}) = Sq^{3} g_{5}^{26}

+ Sq^{4} Sq^{2} g_{5}^{23} + Sq^{7} g_{5}^{22} + Sq^{9} g_{5}^{20} + Sq^{9} ′g_{5}^{20} + Sq^{8} Sq^{2} g_{5}^{19} +(Sq^{10} Sq^{3} + Sq^{11} Sq^{2})g_{5}^{16} +(Sq^{15} + Sq^{12} Sq^{2} Sq^{1} + Sq^{10} Sq^{5} + Sq^{8} Sq^{4} Sq^{2} Sq^{1})g_{5}^{14} +(Sq^{21} Sq^{3} + Sq^{19} Sq^{5} + Sq^{18} Sq^{6} + Sq^{17} Sq^{7} + Sq^{22} Sq^{2} + Sq^{24})g_{5}^{5}

δ(g_{8}^{30}) = Sq^{2} Sq^{1} g_{6}^{26}

+ Sq^{6} g_{6}^{23} +(Sq^{6} Sq^{1} + Sq^{4} Sq^{2} Sq^{1})g_{6}^{22} +(Sq^{7} Sq^{2} + Sq^{8} Sq^{1} + Sq^{6} Sq^{3})g_{6}^{20} +(Sq^{10} Sq^{2} + Sq^{9} Sq^{3} + Sq^{8} Sq^{4})g_{6}^{17} +(Sq^{13} + Sq^{12} Sq^{1} + Sq^{10} Sq^{2} Sq^{1})g_{6}^{16} +(Sq^{21} Sq^{2} + Sq^{17} Sq^{6} + Sq^{18} Sq^{5} + Sq^{16} Sq^{7} + Sq^{17} Sq^{4} Sq^{2})g_{6}^{6}

δ(g_{11}^{30}) = Sq^{2} Sq^{1} g_{9}^{26}

δ(g_{8}^{31}) = Sq^{7} g_{6}^{23}

+(Sq^{5} Sq^{2} Sq^{1} + Sq^{7} Sq^{1})g_{6}^{22} +(Sq^{7} Sq^{3} + Sq^{10} + Sq^{8} Sq^{2})g_{6}^{20} +(Sq^{10} Sq^{3} + Sq^{13})g_{6}^{17} +(Sq^{8} Sq^{4} Sq^{2} + Sq^{13} Sq^{1} + Sq^{14} + Sq^{11} Sq^{2} Sq^{1})g_{6}^{16} +(Sq^{17} Sq^{5} Sq^{2} + Sq^{22} Sq^{2} + Sq^{21} Sq^{3} + Sq^{24} + Sq^{18} Sq^{6} + Sq^{17} Sq^{7})g_{6}^{6}

δ(g_{9}^{31}) = g_{7}^{30}

+ Sq^{4} Sq^{2} g_{7}^{24} +(Sq^{6} Sq^{1} + Sq^{4} Sq^{2} Sq^{1} + Sq^{7})g_{7}^{23} +(Sq^{12} + Sq^{8} Sq^{4} + Sq^{9} Sq^{2} Sq^{1})g_{7}^{18} +(Sq^{20} Sq^{3} + Sq^{18} Sq^{5})g_{7}^{7}

δ(g_{2}^{32}) = 0
δ(g_{6}^{32}) = Sq^{4} g_{4}^{27}

+ Sq^{5} g_{4}^{26} +(Sq^{8} Sq^{1} + Sq^{9} + Sq^{6} Sq^{3})′g_{4}^{22} + Sq^{7} Sq^{2} g_{4}^{22} +(Sq^{10} + Sq^{8} Sq^{2})g_{4}^{21} +(Sq^{12} + Sq^{8} Sq^{4})g_{4}^{19} + Sq^{13} g_{4}^{18} +(Sq^{16} Sq^{2} + Sq^{17} Sq^{1} + Sq^{13} Sq^{5} + Sq^{12} Sq^{4} Sq^{2} + Sq^{15} Sq^{2} Sq^{1} + Sq^{14} Sq^{3} Sq^{1})g_{4}^{13} +(Sq^{15} Sq^{4} Sq^{1} + Sq^{14} Sq^{6} + Sq^{16} Sq^{4} + Sq^{14} Sq^{4} Sq^{2} + Sq^{12} Sq^{6} Sq^{2} + Sq^{13} Sq^{6} Sq^{1})g_{4}^{11} +(Sq^{24} Sq^{3} + Sq^{17} Sq^{8} Sq^{2} + Sq^{20} Sq^{5} Sq^{2} + Sq^{19} Sq^{6} Sq^{2} + Sq^{18} Sq^{9} + Sq^{27})g_{4}^{4}

δ(g_{9}^{32}) = Sq^{7} g_{7}^{24}

+ Sq^{8} g_{7}^{23} +(Sq^{11} Sq^{2} + Sq^{12} Sq^{1} + Sq^{8} Sq^{4} Sq^{1} + Sq^{9} Sq^{4})g_{7}^{18} +(Sq^{18} Sq^{4} Sq^{2} + Sq^{21} Sq^{3} + Sq^{22} Sq^{2} + Sq^{24} + Sq^{20} Sq^{4})g_{7}^{7}

δ(′g_{9}^{32}) = (Sq^{7} + Sq^{5} Sq^{2})g_{7}^{24}

+ Sq^{8} g_{7}^{23} +(Sq^{8} Sq^{4} Sq^{1} + Sq^{10} Sq^{2} Sq^{1})g_{7}^{18} +(Sq^{18} Sq^{4} Sq^{2} + Sq^{19} Sq^{5} + Sq^{22} Sq^{2} + Sq^{20} Sq^{4})g_{7}^{7}

δ(g_{10}^{32}) = g_{8}^{31}

+ Sq^{6} g_{8}^{25} +(Sq^{7} Sq^{1} + Sq^{6} Sq^{2})g_{8}^{23} +(Sq^{21} Sq^{2} + Sq^{19} Sq^{4} + Sq^{23})g_{8}^{8}

δ(g_{2}^{33}) = 0
δ(g_{3}^{33}) = g_{1}^{32}

+ Sq^{24} g_{1}^{8} +(Sq^{28} + Sq^{25} Sq^{3})g_{1}^{4} +(Sq^{29} Sq^{1} + Sq^{30} + Sq^{23} Sq^{7} + Sq^{22} Sq^{7} Sq^{1} + Sq^{25} Sq^{5} + Sq^{23} Sq^{6} Sq^{1} + Sq^{23} Sq^{4} Sq^{2} Sq^{1})g_{1}^{2} +(Sq^{29} Sq^{2} + Sq^{20} Sq^{8} Sq^{3} + Sq^{22} Sq^{7} Sq^{2} + Sq^{27} Sq^{4} + Sq^{21} Sq^{7} Sq^{3} + Sq^{21} Sq^{8} Sq^{2} + Sq^{26} Sq^{5} + Sq^{28} Sq^{3} + Sq^{19} Sq^{9} Sq^{3} + Sq^{25} Sq^{6} + Sq^{19} Sq^{8} Sq^{4})g_{1}^{1}

δ(g_{7}^{33}) = Sq^{4} Sq^{2} g_{5}^{26}

+ Sq^{7} g_{5}^{25} +(Sq^{7} Sq^{2} + Sq^{8} Sq^{1} + Sq^{6} Sq^{3} + Sq^{6} Sq^{2} Sq^{1})g_{5}^{23} + Sq^{8} Sq^{2} g_{5}^{22} +(Sq^{8} Sq^{4} + Sq^{11} Sq^{1})g_{5}^{20} + Sq^{10} Sq^{2} ′g_{5}^{20} +(Sq^{13} Sq^{5} + Sq^{15} Sq^{2} Sq^{1})g_{5}^{14} +(Sq^{18} Sq^{6} Sq^{3} + Sq^{18} Sq^{7} Sq^{2} + Sq^{17} Sq^{8} Sq^{2} + Sq^{22} Sq^{5} + Sq^{23} Sq^{4} + Sq^{19} Sq^{6} Sq^{2} + Sq^{18} Sq^{9})g_{5}^{5}

δ(g_{8}^{33}) = Sq^{2} Sq^{1} g_{6}^{29}

+ Sq^{6} g_{6}^{26} +(Sq^{7} Sq^{2} + Sq^{9} + Sq^{6} Sq^{3})g_{6}^{23} +(Sq^{10} + Sq^{8} Sq^{2})g_{6}^{22} + Sq^{11} g_{6}^{21} +(Sq^{8} Sq^{4} + Sq^{10} Sq^{2})g_{6}^{20} +(Sq^{13} Sq^{2} + Sq^{11} Sq^{4} + Sq^{10} Sq^{5} + Sq^{9} Sq^{4} Sq^{2} + Sq^{15})g_{6}^{17} +(Sq^{15} Sq^{1} + Sq^{13} Sq^{2} Sq^{1} + Sq^{9} Sq^{4} Sq^{2} Sq^{1})g_{6}^{16} +(Sq^{20} Sq^{6} + Sq^{18} Sq^{8} + Sq^{19} Sq^{5} Sq^{2} + Sq^{21} Sq^{5} + Sq^{23} Sq^{3})g_{6}^{6}

δ(g_{10}^{33}) = (Sq^{7} + Sq^{6} Sq^{1})g_{8}^{25}

+(Sq^{9} + Sq^{8} Sq^{1})g_{8}^{23} + Sq^{19} Sq^{5} g_{8}^{8}

δ(g_{2}^{34}) = 0
δ(g_{3}^{34}) = (Sq^{21} Sq^{8} + Sq^{22} Sq^{5} Sq^{2})g_{1}^{4}

+(Sq^{19} Sq^{8} Sq^{4} + Sq^{23} Sq^{8} + Sq^{16} Sq^{8} Sq^{4} Sq^{2} Sq^{1} + Sq^{21} Sq^{10} + Sq^{24} Sq^{4} Sq^{2} Sq^{1} + Sq^{30} Sq^{1} + Sq^{21} Sq^{7} Sq^{2} Sq^{1} + Sq^{20} Sq^{10} Sq^{1} + Sq^{23} Sq^{7} Sq^{1})g_{1}^{2} +(Sq^{25} Sq^{7} + Sq^{22} Sq^{8} Sq^{2} + Sq^{25} Sq^{5} Sq^{2} + Sq^{30} Sq^{2} + Sq^{20} Sq^{9} Sq^{3} + Sq^{29} Sq^{3} + Sq^{23} Sq^{6} Sq^{3} + Sq^{23} Sq^{9} + Sq^{20} Sq^{8} Sq^{4} + Sq^{21} Sq^{8} Sq^{3} + Sq^{26} Sq^{4} Sq^{2} + Sq^{18} Sq^{8} Sq^{4} Sq^{2} + Sq^{20} Sq^{10} Sq^{2} + Sq^{24} Sq^{8} + Sq^{32})g_{1}^{1}

δ(′g_{3}^{34}) = (Sq^{26} Sq^{3} + Sq^{21} Sq^{8} + Sq^{27} Sq^{2} + Sq^{29})g_{1}^{4}

+(Sq^{23} Sq^{8} + Sq^{28} Sq^{2} Sq^{1} + Sq^{21} Sq^{7} Sq^{2} Sq^{1} + Sq^{22} Sq^{6} Sq^{2} Sq^{1} + Sq^{27} Sq^{4} + Sq^{24} Sq^{7} + Sq^{19} Sq^{8} Sq^{4} + Sq^{16} Sq^{8} Sq^{4} Sq^{2} Sq^{1})g_{1}^{2} +(Sq^{24} Sq^{6} Sq^{2} + Sq^{30} Sq^{2} + Sq^{27} Sq^{5} + Sq^{24} Sq^{8} + Sq^{21} Sq^{8} Sq^{3} + Sq^{25} Sq^{5} Sq^{2} + Sq^{23} Sq^{6} Sq^{3} + Sq^{18} Sq^{8} Sq^{4} Sq^{2} + Sq^{26} Sq^{4} Sq^{2} + Sq^{22} Sq^{8} Sq^{2} + Sq^{20} Sq^{8} Sq^{4} + Sq^{32} + Sq^{29} Sq^{3} + Sq^{23} Sq^{9})g_{1}^{1}

δ(g_{4}^{34}) = g_{2}^{33}

+ Sq^{13} g_{2}^{20} + Sq^{15} g_{2}^{18} +(Sq^{16} + Sq^{11} Sq^{5} + Sq^{10} Sq^{4} Sq^{2})g_{2}^{17} + Sq^{12} Sq^{5} g_{2}^{16} +(Sq^{23} + Sq^{17} Sq^{6} + Sq^{16} Sq^{4} Sq^{2} Sq^{1} + Sq^{15} Sq^{7} Sq^{1} + Sq^{16} Sq^{6} Sq^{1} + Sq^{18} Sq^{4} Sq^{1})g_{2}^{10} +(Sq^{21} Sq^{3} + Sq^{18} Sq^{6} + Sq^{16} Sq^{6} Sq^{2} + Sq^{15} Sq^{7} Sq^{2} + Sq^{17} Sq^{5} Sq^{2} + Sq^{15} Sq^{6} Sq^{3} + Sq^{19} Sq^{5})g_{2}^{9} +(Sq^{25} + Sq^{23} Sq^{2} + Sq^{21} Sq^{4} + Sq^{19} Sq^{4} Sq^{2} + Sq^{18} Sq^{5} Sq^{2} + Sq^{22} Sq^{3})g_{2}^{8} +(Sq^{26} Sq^{2} + Sq^{20} Sq^{6} Sq^{2} + Sq^{22} Sq^{4} Sq^{2} + Sq^{21} Sq^{5} Sq^{2} + Sq^{20} Sq^{8} + Sq^{19} Sq^{9} + Sq^{25} Sq^{3} + Sq^{24} Sq^{4} + Sq^{17} Sq^{8} Sq^{3} + Sq^{16} Sq^{8} Sq^{4} + Sq^{18} Sq^{8} Sq^{2})g_{2}^{5} +(Sq^{19} Sq^{6} Sq^{3} Sq^{1} + Sq^{20} Sq^{9} + Sq^{27} Sq^{2} + Sq^{20} Sq^{7} Sq^{2} + Sq^{23} Sq^{4} Sq^{2} + Sq^{28} Sq^{1} + Sq^{29} + Sq^{20} Sq^{8} Sq^{1} + Sq^{22} Sq^{7} + Sq^{22} Sq^{6} Sq^{1} + Sq^{24} Sq^{4} Sq^{1} + Sq^{22} Sq^{4} Sq^{2} Sq^{1} + Sq^{18} Sq^{8} Sq^{2} Sq^{1} + Sq^{25} Sq^{4} + Sq^{19} Sq^{7} Sq^{2} Sq^{1} + Sq^{21} Sq^{5} Sq^{2} Sq^{1} + Sq^{26} Sq^{2} Sq^{1})g_{2}^{4} +(Sq^{20} Sq^{9} Sq^{2} + Sq^{19} Sq^{8} Sq^{4} + Sq^{21} Sq^{7} Sq^{3} + Sq^{26} Sq^{5} + Sq^{21} Sq^{10} + Sq^{31} + Sq^{20} Sq^{8} Sq^{3} + Sq^{18} Sq^{9} Sq^{4} + Sq^{19} Sq^{9} Sq^{3} + Sq^{21} Sq^{8} Sq^{2} + Sq^{22} Sq^{7} Sq^{2})g_{2}^{2}

δ(g_{8}^{34}) = (Sq^{6} Sq^{1} + Sq^{4} Sq^{2} Sq^{1} + Sq^{7})g_{6}^{26}

+(Sq^{9} Sq^{2} + Sq^{10} Sq^{1} + Sq^{8} Sq^{2} Sq^{1})g_{6}^{22} + Sq^{12} g_{6}^{21} +(Sq^{11} Sq^{5} + Sq^{13} Sq^{3})g_{6}^{17} +(Sq^{13} Sq^{4} + Sq^{12} Sq^{4} Sq^{1} + Sq^{15} Sq^{2})g_{6}^{16} +(Sq^{18} Sq^{9} + Sq^{21} Sq^{6} + Sq^{22} Sq^{5} + Sq^{27} + Sq^{19} Sq^{6} Sq^{2} + Sq^{24} Sq^{3})g_{6}^{6}

δ(g_{9}^{34}) = g_{7}^{33}

+(Sq^{6} Sq^{3} + Sq^{9} + Sq^{6} Sq^{2} Sq^{1})g_{7}^{24} +(Sq^{14} Sq^{1} + Sq^{13} Sq^{2} + Sq^{15} + Sq^{11} Sq^{4} + Sq^{8} Sq^{4} Sq^{2} Sq^{1} + Sq^{11} Sq^{3} Sq^{1})g_{7}^{18} +(Sq^{19} Sq^{5} Sq^{2} + Sq^{20} Sq^{4} Sq^{2} + Sq^{24} Sq^{2} + Sq^{18} Sq^{8} + Sq^{21} Sq^{5} + Sq^{23} Sq^{3} + Sq^{26})g_{7}^{7}

δ(g_{10}^{34}) = g_{8}^{33}

+(Sq^{6} Sq^{2} + Sq^{8} + Sq^{5} Sq^{2} Sq^{1} + Sq^{7} Sq^{1})g_{8}^{25} + Sq^{8} Sq^{2} g_{8}^{23} +(Sq^{19} Sq^{6} + Sq^{21} Sq^{4} + Sq^{25} + Sq^{20} Sq^{5})g_{8}^{8}

δ(g_{11}^{34}) = Sq^{1} ′g_{9}^{32}

+(Sq^{21} Sq^{3} + Sq^{22} Sq^{2} + Sq^{24})g_{9}^{9}

δ(g_{4}^{35}) = Sq^{17} g_{2}^{17}

+(Sq^{18} + Sq^{16} Sq^{2})g_{2}^{16} +(Sq^{18} Sq^{6} + Sq^{16} Sq^{7} Sq^{1})g_{2}^{10} + Sq^{19} Sq^{6} g_{2}^{9} +(Sq^{18} Sq^{8} + Sq^{19} Sq^{5} Sq^{2} + Sq^{24} Sq^{2} + Sq^{18} Sq^{7} Sq^{1} + Sq^{16} Sq^{8} Sq^{2} + Sq^{26})g_{2}^{8} +(Sq^{25} Sq^{4} + Sq^{21} Sq^{6} Sq^{2} + Sq^{24} Sq^{5} + Sq^{26} Sq^{3} + Sq^{21} Sq^{8} + Sq^{23} Sq^{6} + Sq^{18} Sq^{9} Sq^{2} + Sq^{19} Sq^{7} Sq^{3} + Sq^{20} Sq^{7} Sq^{2})g_{2}^{5} +(Sq^{23} Sq^{7} + Sq^{20} Sq^{8} Sq^{2} + Sq^{20} Sq^{7} Sq^{2} Sq^{1} + Sq^{23} Sq^{4} Sq^{2} Sq^{1} + Sq^{27} Sq^{2} Sq^{1} + Sq^{16} Sq^{8} Sq^{4} Sq^{2} + Sq^{22} Sq^{6} Sq^{2} + Sq^{23} Sq^{6} Sq^{1} + Sq^{19} Sq^{8} Sq^{2} Sq^{1} + Sq^{28} Sq^{2} + Sq^{25} Sq^{5} + Sq^{21} Sq^{7} Sq^{2})g_{2}^{4} +(Sq^{30} Sq^{2} + Sq^{20} Sq^{9} Sq^{3} + Sq^{22} Sq^{8} Sq^{2} + Sq^{23} Sq^{6} Sq^{3} + Sq^{25} Sq^{5} Sq^{2} + Sq^{24} Sq^{8} + Sq^{26} Sq^{4} Sq^{2} + Sq^{24} Sq^{6} Sq^{2} + Sq^{21} Sq^{9} Sq^{2} + Sq^{23} Sq^{9} + Sq^{18} Sq^{8} Sq^{4} Sq^{2})g_{2}^{2}

δ(g_{5}^{35}) = ′g_{3}^{34}

+ Sq^{9} Sq^{4} g_{3}^{21} +(Sq^{8} Sq^{4} Sq^{2} + Sq^{10} Sq^{3} Sq^{1} + Sq^{11} Sq^{2} Sq^{1})g_{3}^{20} +(Sq^{14} Sq^{2} + Sq^{12} Sq^{4} + Sq^{13} Sq^{3})g_{3}^{18} +(Sq^{12} Sq^{5} + Sq^{17} + Sq^{14} Sq^{2} Sq^{1})g_{3}^{17} +(Sq^{16} Sq^{4} Sq^{2} + Sq^{14} Sq^{6} Sq^{2} + Sq^{12} Sq^{6} Sq^{3} Sq^{1} + Sq^{17} Sq^{5} + Sq^{14} Sq^{7} Sq^{1} + Sq^{21} Sq^{1} + Sq^{19} Sq^{3} + Sq^{15} Sq^{7})g_{3}^{12} +(Sq^{20} Sq^{2} Sq^{1} + Sq^{17} Sq^{6} + Sq^{15} Sq^{7} Sq^{1} + Sq^{18} Sq^{5} + Sq^{22} Sq^{1})g_{3}^{11} +(Sq^{24} + Sq^{20} Sq^{4} + Sq^{19} Sq^{5} + Sq^{17} Sq^{5} Sq^{2} + Sq^{18} Sq^{6})g_{3}^{10} +(Sq^{20} Sq^{8} + Sq^{23} Sq^{4} Sq^{1} + Sq^{24} Sq^{3} Sq^{1} + Sq^{24} Sq^{4} + Sq^{18} Sq^{6} Sq^{3} Sq^{1} + Sq^{21} Sq^{4} Sq^{2} Sq^{1} + Sq^{19} Sq^{8} Sq^{1} + Sq^{21} Sq^{7} + Sq^{25} Sq^{3} + Sq^{18} Sq^{7} Sq^{3} + Sq^{19} Sq^{9} + Sq^{27} Sq^{1} + Sq^{16} Sq^{8} Sq^{4} + Sq^{23} Sq^{5} + Sq^{17} Sq^{8} Sq^{3} + Sq^{21} Sq^{6} Sq^{1})g_{3}^{6} +(Sq^{31} + Sq^{26} Sq^{5} + Sq^{25} Sq^{6} + Sq^{19} Sq^{8} Sq^{4} + Sq^{21} Sq^{7} Sq^{3} + Sq^{19} Sq^{9} Sq^{3} + Sq^{29} Sq^{2} + Sq^{24} Sq^{5} Sq^{2} + Sq^{20} Sq^{8} Sq^{3} + Sq^{25} Sq^{4} Sq^{2} + Sq^{22} Sq^{6} Sq^{3} + Sq^{20} Sq^{9} Sq^{2} + Sq^{23} Sq^{8} + Sq^{27} Sq^{4} + Sq^{24} Sq^{7} + Sq^{28} Sq^{3})g_{3}^{3}

δ(g_{9}^{35}) = Sq^{4} g_{7}^{30}

+(Sq^{7} Sq^{3} + Sq^{6} Sq^{3} Sq^{1} + Sq^{10} + Sq^{8} Sq^{2})g_{7}^{24} +(Sq^{18} Sq^{9} + Sq^{19} Sq^{6} Sq^{2} + Sq^{23} Sq^{4} + Sq^{24} Sq^{3} + Sq^{21} Sq^{6} + Sq^{19} Sq^{8})g_{7}^{7} +(Sq^{11} + Sq^{8} Sq^{2} Sq^{1} + Sq^{10} Sq^{1})g_{7}^{23} + Sq^{12} g_{7}^{22} +(Sq^{13} Sq^{2} Sq^{1} + Sq^{12} Sq^{4} + Sq^{12} Sq^{3} Sq^{1} + Sq^{10} Sq^{4} Sq^{2} + Sq^{11} Sq^{4} Sq^{1} + Sq^{9} Sq^{4} Sq^{2} Sq^{1})g_{7}^{18}

δ(g_{10}^{35}) = g_{8}^{34}

+ Sq^{3} g_{8}^{31} +(Sq^{10} Sq^{1} + Sq^{11} + Sq^{9} Sq^{2})g_{8}^{23} +(Sq^{24} Sq^{2} + Sq^{19} Sq^{7})g_{8}^{8}

δ(g_{11}^{35}) = Sq^{2} ′g_{9}^{32}

+ Sq^{3} g_{9}^{31} + Sq^{6} g_{9}^{28} +(Sq^{8} + Sq^{7} Sq^{1})g_{9}^{26} + Sq^{25} g_{9}^{9}

δ(g_{12}^{35}) = Sq^{5} g_{10}^{29}

+(Sq^{22} Sq^{2} + Sq^{21} Sq^{3})g_{10}^{10}

References

[1] Hans-Joachim Baues, The algebra of secondary cohomology operations, 2004.

[2] Hans-Joachim Baues and Mamuka Jibladze, Classification of abelian track categories, K-Theory 25 (2002), 299–311.

[3] , Secondary derived functors and the Adams spectral sequence (2004) (in preparation).

[4] Robert R. Bruner, Calculation of large Ext modules, Computers in Geometry and Topology (Chicago, IL, 1986), Lecture Notes in Pure and Appl. Math., vol. 114, Dekker, New York, 1989, pp. 79–104.

[5] Jean-Pierre Meyer, Acyclic models for multicomplexes, Duke Math. J. 45 (1978), 67–85.

[6] Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, American Mathematical Society, University of Rochester - AMS, 2004, ISBN 0-8218-2967-X.

Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

  E-mail address: baues@mpim-bonn.mpg.de

Razmadze Mathematical Institute, M. Alexidze st. 1, Tbilisi 0193, Georgia

Current address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

  E-mail address: jib@rmi.acnet.ge