The Adams-Novikov E_{2}-term for π_{∗}(L_{2}S^{0}) at the prime 2

Katsumi Shimomura and Xiangjun Wang

Abstract. In this paper we determine the E_{2}-term of the Adams-Novikov spectral sequence for computing the homotopy groups π_{∗}(L_{2}S^{0}) of L_{2}-localized sphere spectrum at the prime 2. The structure of the E_{2}-term indicates that the Q/Z_{(2)} factors of the homotopy groups π_{∗}(L_{2}S^{0}) appear at dimensions −4 and −5 as in the case for p > 3.

1. Introduction

Let E(2) be the Johnson-Wilson spectrum at a prime p, and L_{2} the Bousfield localization functor with respect to E(2). Then the homotopy groups π_{∗}(L_{2}S^{0}) are determined at a prime p > 2 (cf. [8], [7]). In this paper, we determine the E_{2}-term of the Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(L_{2}S^{0}) at the prime 2 (see Theorem 17.4). As a corollary, we find the Q/Z_{(2)} summands in π_{∗}(L_{2}S^{0}) which locate at dimensions −4 and −5. This is the same result as the case p > 3.

Hereafter we set p = 2. We compute the E_{2}-term of the Adams-Novikov spectral sequence converging to them by the Bockstein spectral sequence and the chromatic spectral sequence as in the previous papers. The chromatic modules M_{0}^{i} and N_{0}^{i} are defined as follows: N_{0}^{0} = E(2)_{∗} = Z_{(2)}[v_{1}, v_{2}^{±1}], M_{0}^{0} = 2^{−1}E(2)_{∗} and N_{0}^{1} is defined by the short exact sequence

(1.1)  0  w N_{0}^{0} y  w M_{0}^{0}  w N_{0}^{1}  w 0.

Furthermore, M_{0}^{1} = v_{1}^{−1}N_{0}^{1} and M_{0}^{2} is defined by the short exact sequence

(1.2)  0  w N_{0}^{1} y  w M_{0}^{1}  w M_{0}^{2}  w 0.

Consider the functor H^{∗}− =Ext_{E(2)_{∗}(E(2))}^{∗}(E(2)_{∗}, −) on the category of E(2)_{∗}(E(2))-comodules. Then these exact sequences give rise to the chromatic spectral sequence converging to the Adams-Novikov E_{2}- term H^{∗}E(2)_{∗} with E_{1}-term H^{∗}M_{0}^{i}. Since H^{∗}M_{0}^{0} and H^{∗}M_{0}^{1} are determined in [1], it suffices to determine H^{∗}M_{0}^{2}. The 0-th line of the E_{1}-term H^{0}M_{0}^{2} of the chromatic spectral sequence is determined in [3]. We compute the mod 2 Bockstein spectral sequence E_{1} = H^{∗}M_{1}^{1} ⇒ H^{∗}M_{0}^{2}, where M_{1}^{1} = E(2)_{∗}/(2, v_{1}^{∞}). We determined H^{∗}M_{1}^{1} in [5] by the v_{1}-Bockstein spectral sequence E_{1} = H^{∗}M_{2}^{0} ⇒ H^{∗}M_{1}^{1}, where M_{2}^{0} = K(2)_{∗} = E(2)_{∗}/(2, v_{1}) and the structure of H^{∗}M_{2}^{0} is given by Ravenel (cf. [2]). In this paper we determine H^{∗}M_{0}^{2} as in Theorem 17.2, and obtain the E_{2}-term for π_{∗}(L_{2}S^{0}) in Theorem 17.4. As a corollary, we obtain Q/Z_{(2)}-factors in the homotopy groups π_{∗}(L_{2}S^{0}) (Corollary 17.5).

This paper is divided into seventeen sections:

1. Introduction

2. Cocycles of the cobar complex Ω^{∗}A

3. The Miller-Ravenel-Wilson elements x_{i}

4. H^{∗}M_{0}^{1} revisited

5. H^{∗}M_{1}^{1} revisited

6. The cocycles R and B 7. The connecting homomorphism on J

8. The connecting homomorphism on K

9. The connecting homomorphism on E

10. The connecting homomorphism on F

11. The connecting homomorphism on D

12. The connecting homomorphism on L

13. The connecting homomorphism on M

14. The connecting homomorphism on N

15. The connecting homomorphism on P

16. The connecting homomorphism on Q

17. The Adams-Novikov E_{2}-terms

The sections 2 and 3 are devoted to make some preliminary computation in the cobar complexes for studying the mod 2 Bockstein spectral sequence H^{∗}M_{1}^{1} ⇒ H^{∗}M_{0}^{2} associated to the short exact sequence

(1.3)  0  w M_{1}^{1} ^{1/2} w M_{0}^{2} ^{2} w M_{0}^{2}  w 0.

In section 4, a Q/Z_{(2)} summand in H^{1}M_{0}^{2} is obtained from H^{1}M_{0}^{1}. We rewrite the E_{1}-term H^{∗}M_{1}^{1} = (D⨁E⨁F ⨁J⨁K⨁L⨁M⨁N ⨁P ⨁Q)⨂Λ(ρ_{2}) in section 5. In the next section, we study some cochains arisen from the structure of H^{∗}M_{1}^{1} including cocycles R on E(2)_{∗}/(2^{n+1}, v_{1}^{2^{n}}) and B on E(2)_{∗}/(2^{n+1}, v_{1}^{2^{n+1}}) for n ≥ 0, which represents the generator ρ_{2} and β, respectively. In sections 7 to 16, we study the behavior of the connecting homomorphism δ: H^{∗}M_{0}^{2} → H^{∗+1}M_{1}^{1} on each summand of H^{∗}M_{1}^{1} in order to see the differential d of the Bockstein spectral sequence. Note that each element used to state the results is expressed by its leading term as we did in the previous papers. In the last section, we compute the chromatic spectral sequence, or observe the exact sequence associated to (1.1) and (1.2), to obtain the module H^{∗}N_{0}^{1} and the desired Adams-Novikov E_{2}-term H^{∗}N_{0}^{0}. As a corollary, we obtain the Q/Z_{(2)} summands in the homotopy groups.

2. Cocycles of the cobar complex Ω^{∗}A

Let (A, Γ) denote the Hopf algebroid (E(2)_{∗}, E(2)_{∗}E(2)) = (Z_{(2)}[v_{1}, v_{2}^{±1}], E(2)_{∗}[t_{1}, t_{2},…]⨂_{BP_{∗}}E(2)_{∗}) associated to the Johnson-Wilson spectrum E(2). Then the cobar complex Ω^{∗}M for a right comodule M with structure map ψ: M → M⨂_{A}Γ is a differential graded module Ω^{s}M = M⨂_{A}Γ^{⊗s} with the differential d: Ω^{s}M → Ω^{s+1}M given by d(m) = ψ(m) − m, d(x) = 1 ⊗ x − Δ(x) + x ⊗ 1 and d(m ⊗ x ⊗ y) = d(m) ⊗ x ⊗ y + m ⊗ d(x) ⊗ y − m ⊗ x ⊗ d(y) for m ∈ M, x ∈ Ω^{1}A and y ∈ Ω^{s}A. Here, Δ: Γ → Γ⨂_{A}Γ denotes the diagonal map of Γ. Note that A is a Γ-comodule with the structure map η_{R}. In this paper, we consider comodules M induced from A and denote the structure map ψ by η_{R}. We denote the cohomology of the cobar complex by H^{∗}M. Then H^{∗}A is the E_{2}-term of the Adams-Novikov spectral sequence based on E(2) converging to π_{∗}(L_{2}S^{0}). In this section, we introduce some cocycles z̃, B̃ and G̃ of Ω^{∗}A that represent v_{1}v_{2}ζ, v_{1}^{4}β and v_{1}^{4}g in H^{∗}A/(2).

Here we write down some of the formulae on the structure maps of the Hopf algebroid Γ.

(2.1)  \begin{aligned}η_{R}(v_{1}) & = & v_{1} + 2t_{1}, \\ η_{R}(v_{2}) & = & v_{2} − 5v_{1}t_{1}^{2} − 3v_{1}^{2}t_{1} + 2t_{2} − 4t_{1}^{3}; \\ Δ(t_{1}) & = & t_{1} ⊗ 1 + 1 ⊗ t_{1}, \\ Δ(t_{2}) & = & t_{2} ⊗ 1 + t_{1} ⊗ t_{1}^{2} + 1 ⊗ t_{2} − v_{1}t_{1} ⊗ t_{1}. \end{aligned}

Furthermore, the equation η_{R}(v_{3}) = 0 in Γ implies the relation

(2.2)  v_{2}t_{1}^{4} + t_{1}η_{R}(v_{2}^{2}) + v_{1}t_{2}^{2} + v_{1}^{2}v_{2}t_{1}^{2} + v_{1}^{3}v_{2}t_{1} + v_{1}^{4}(t_{2} + t_{1}^{3}) ≡ 0 mod (2)

in Γ (cf. [5, (6.10)]). Recall that the generator ζ (which is denoted by ζ_{1} in [5] and ζ_{2} in [1]) is represented by the cochain z = v_{2}^{−1}(t_{2} + t_{1}^{3}) + v_{2}^{−2}t_{2}^{2} ∈ Ω^{1}A. Lemma 2.3. There is a cocycle z̃ of Ω^{1}A such that z̃ ≡ v_{1}v_{2}z mod (2, v_{1}^{2}).

We write ζ̃ as the homology class of z̃. Proof of Lemma 2.3. In Γ, put

(2.4) z̃ = η_{R}(v_{2})t_{1} + t_{1}^{4} + v_{1}t_{2} − v_{1}^{2}t_{1}^{2} − v_{1}^{3}t_{1}.

Then we see that z̃ ≡ v_{1}v_{2}z mod (2, v_{1}^{2}) by (2.2). We verify that z̃ is a cocycle by computation

d(t_{1}η_{R}(v_{2})) = −t_{1} ⊗ (−5v_{1}t_{1}^{2} − 3v_{1}^{2}t_{1} + 2t_{2} − 4t_{1}^{3})
= \underline{5v_{1}t_{1} ⊗ t_{1}^{2}}_{1} + \underline{10t_{1}^{2} ⊗ t_{1}^{2}}_{2} + \underline{3v_{1}^{2}t_{1} ⊗ t_{1}}_{3}
+\underline{12v_{1}t_{1}^{2} ⊗ t_{1}}_{4} + \underline{12t_{1}^{3} ⊗ t_{1}}_{5} − \underline{2t_{1} ⊗ t_{2}}_{6} + \underline{4t_{1} ⊗ t_{1}^{3}}_{7}
d(t_{1}^{4}) = −\underline{4t_{1} ⊗ t_{1}^{3}}_{7} − \underline{6t_{1}^{2} ⊗ t_{1}^{2}}_{2} − \underline{4t_{1}^{3} ⊗ t_{1}}_{5}
d(v_{1}t_{2}) = \underline{2t_{1} ⊗ t_{2}}_{6} − \underline{v_{1}t_{1} ⊗ t_{1}^{2}}_{1} + \underline{v_{1}^{2}t_{1} ⊗ t_{1}}_{3}
d(−v_{1}^{2}t_{1}^{2}) = −\underline{4v_{1}t_{1} ⊗ t_{1}^{2}}_{1} − \underline{4t_{1}^{2} ⊗ t_{1}^{2}}_{2} + \underline{2v_{1}^{2}t_{1} ⊗ t_{1}}_{3}
d(−v_{1}^{3}t_{1}) = −\underline{6v_{1}^{2}t_{1} ⊗ t_{1}}_{3} − \underline{12v_{1}t_{1}^{2} ⊗ t_{1}}_{4} − \underline{8t_{1}^{3} ⊗ t_{1}}_{5}.

q.e.d.

Lemma 2.5. There are cochains B̃_{i} ∈ Ω^{3}A for each i ≥ 0, and G̃ ∈ Ω^{4}A such that

a) B̃_{i} ≡ v_{1}^{4}B mod (2, v_{1}^{5}) and d(B̃_{i}) ≡ 0 mod (2^{i+1}, v_{1}^{2^{i}}).

b) G̃ ≡ v_{1}^{4}G mod (2, v_{1}^{5}) and d(G̃) = 0. Here B and G denote representatives of β ∈ H^{3}K(2)_{∗} and g ∈ H^{4}K(2)_{∗}, respectively. Proof. In [5, p.147], it is shown the existence of a cochain W such that d(W) ≡ z̃ ⊗ z^{4} ⊗ z^{4} + v_{1}^{4}B mod (2, v_{1}^{5}). The lemma is valid for B̃_{i} defined by the equation

d(W) = z̃ ⊗ ZZ_{i} + B̃_{i},

where ZZ_{i} is given by d(z^{2^{i+1}}) ≡ 2ZZ_{i} mod (v_{1}^{2^{i}}).

In the same way, we define G̃ by the equation d(U) = t_{1} ⊗ t_{1} ⊗ t_{1} ⊗ t_{1} + G̃, where U is a cochain satisfying d(U) ≡ t_{1} ⊗ t_{1} ⊗ t_{1} ⊗ t_{1} + v_{1}^{4}G mod (2, v_{1}^{5}) (cf. [5, p.152]). q.e.d.

For the later use, consider a cochain r ∈ Ω^{1}v_{1}^{−1}A given by

(2.6)  r = z̃ − t_{1}^{4} + 3v_{1}^{3}t_{1} + 2v_{1}(t_{1}^{3} − t_{2}) − 2t_{1}^{2}η_{R}(v_{1}^{−1}v_{2}) + 4v_{1}^{2}t_{1}^{2}.

Then we compute

(2.7) \begin{aligned}r & ≡ & z̃ − t_{1}^{4} + 3v_{1}^{3}t_{1} − 2v_{1}t_{2} + 2v_{1}t_{1}^{3} + 4v_{1}^{2}t_{1}^{2} \\ & & −2t_{1}^{2}(v_{1}^{−1} − 2v_{1}^{−2}t_{1})(v_{2} − v_{1}t_{1}^{2} + v_{1}^{2}t_{1} + 2t_{2}) mod (8) \\ & ≡ & z̃ − 3t_{1}^{4} + 3v_{1}^{3}t_{1} − 2v_{1}t_{2} + 4v_{1}^{2}t_{1}^{2} − 2v_{1}^{−1}v_{2}t_{1}^{2} + 4v_{1}^{−1}t_{1}^{2}t_{2} + 4v_{1}^{−1}t_{1}^{5} + 4v_{1}^{−2}v_{2}t_{1}^{3} mod (8) \end{aligned}

and we see that v_{1}^{2}r ∈ Ω^{1}A/(8). Lemma 2.8. In the cobar complex Ω^{1}A,

d(v_{2}^{2}) ≡ v_{1}^{2}(3r − z̃) − v_{1}^{3}(3v_{1}t_{1}^{2} + v_{1}^{2}t_{1}) + 4v_{2}^{2}Z mod (8).

for Z = z + v_{1}v_{2}^{−2}(v_{2}t_{1}^{2} + t_{1}^{5}) + v_{1}^{3}v_{2}^{−2}t_{2}. Proof. By (9.2), we have

3t_{1}^{4} ≡ z̃ − r + 3v_{1}^{3}t_{1} − 2v_{1}^{−1}v_{2}t_{1}^{2} − 2v_{1}t_{2} + 4v_{1}^{−2}v_{2}t_{1}^{3} + 4v_{1}^{−1}t_{1}^{2}t_{2} + 4v_{1}^{−1}t_{1}^{5} + 4v_{1}^{2}t_{1}^{2} mod (8), and so

t_{1}^{4} ≡ 3(z̃ − r) + v_{1}^{3}t_{1} + 2v_{1}^{−1}v_{2}t_{1}^{2} + 2v_{1}t_{2} + 4v_{1}^{−2}v_{2}t_{1}^{3} + 4v_{1}^{−1}t_{1}^{2}t_{2} + 4v_{1}^{−1}t_{1}^{5} + 4v_{1}^{2}t_{1}^{2} mod (8).

We then compute mod (8):

d(v_{2}^{2}) ≡ v_{1}^{2}t_{1}^{4} + \underline{v_{1}^{4}t_{1}^{2}} + 4t_{2}^{2} − 2v_{1}v_{2}t_{1}^{2} + \underline{2v_{1}^{2}v_{2}t_{1}} + 4v_{2}t_{2} − \underline{2v_{1}^{3}t_{1}^{3}} + 4v_{1}t_{1}^{2}t_{2} + \underline{4v_{1}^{2}t_{1}t_{2}}
≡ v_{1}^{2}t_{1}^{4} − v_{1}^{4}t_{1}^{2} + 4t_{2}^{2} − 2v_{1}v_{2}t_{1}^{2} + 2v_{1}^{2}η_{R}(v_{2})t_{1} + 4v_{2}t_{2} + 4v_{1}t_{1}^{2}t_{2}
≡ \underline{v_{1}^{2}t_{1}^{4}}_{1} − \underline{v_{1}^{4}t_{1}^{2}}_{2} + 4t_{2}^{2} − 2v_{1}v_{2}t_{1}^{2} + 2v_{1}^{2}(z̃ − \underline{t_{1}^{4}}_{1} − v_{1}t_{2} + \underline{v_{1}^{2}t_{1}^{2}}_{2} + v_{1}^{3}t_{1}) + 4v_{2}t_{2} + 4v_{1}t_{1}^{2}t_{2}
≡ −v_{1}^{2}t_{1}^{4} + v_{1}^{4}t_{1}^{2} + 4t_{2}^{2} − 2v_{1}v_{2}t_{1}^{2} + 2v_{1}^{2}(z̃ − v_{1}t_{2} + v_{1}^{3}t_{1}) + 4v_{2}t_{2} + 4v_{1}t_{1}^{2}t_{2}
≡ −v_{1}^{2}(3(\underline{z̃}_{1} − r) + \underline{v_{1}^{3}t_{1}}_{2} + \underline{2v_{1}^{−1}v_{2}t_{1}^{2}}_{3} + \underline{2v_{1}t_{2}}_{4} + 4v_{1}^{−2}v_{2}t_{1}^{3} + \underline{4v_{1}^{−1}t_{1}^{2}t_{2}}_{5} + 4v_{1}^{−1}t_{1}^{5} + \underline{4v_{1}^{2}t_{1}^{2}}_{6})
+\underline{v_{1}^{4}t_{1}^{2}}_{6} + 4t_{2}^{2} − \underline{2v_{1}v_{2}t_{1}^{2}}_{3} + 2v_{1}^{2}(\underline{z̃}_{1} − \underline{v_{1}t_{2}}_{4} + \underline{v_{1}^{3}t_{1}}_{2}) + 4v_{2}t_{2} + \underline{4v_{1}t_{1}^{2}t_{2}}_{5}
≡ v_{1}^{2}(3r − z̃) − v_{1}^{5}t_{1} + 4v_{1}v_{2}t_{1}^{2} + 4v_{1}^{3}t_{2} + 4v_{2}t_{1}^{3} + 4v_{1}t_{1}^{5} − 3v_{1}^{4}t_{1}^{2} + 4t_{2}^{2} + 4v_{2}t_{2}
≡ v_{1}^{2}(3r − z̃) − v_{1}^{3}(3v_{1}t_{1}^{2} + v_{1}^{2}t_{1}) + 4(v_{2}^{2}z + v_{1}(v_{2}t_{1}^{2} + t_{1}^{5}) + v_{1}^{3}t_{2})

q.e.d.

Lemma 2.9. In the cobar complex Ω^{1}v_{1}^{−1}A, we have

d(v_{1}v_{2}) = 2r + 4v_{1}t_{2} − 11v_{1}^{2}t_{1}^{2} − 7v_{1}^{3}t_{1} − 8v_{1}t_{1}^{3} + 4t_{1}^{2}(v_{1}^{−1} − 2v_{1}^{−2}t_{1} + 4v_{1}^{−2}t_{1}^{2}η_{R}(v_{1}^{−1}))η_{R}(v_{2}) Proof. Note that

(2.10)  η_{R}(v_{1}^{−1}) = v_{1}^{−1} − 2v_{1}^{−2}t_{1} + 4v_{1}^{−2}t_{1}^{2}η_{R}(v_{1}^{−1}).

The lemma follows from the equations:

r = η_{R}(v_{2})t_{1} − v_{1}t_{2} + 3v_{1}^{2}t_{1}^{2} + 2v_{1}^{3}t_{1} + 2v_{1}t_{1}^{3} − 2t_{1}^{2}(v_{1}^{−1} − 2v_{1}^{−2}t_{1} + 4v_{1}^{−2}t_{1}^{2}η_{R}(v_{1}^{−1}))η_{R}(v_{2}) and
d(v_{1}v_{2}) = 2t_{1}η_{R}(v_{2}) + v_{1}(−5v_{1}t_{1}^{2} − 3v_{1}^{2}t_{1} + 2t_{2} − 4t_{1}^{3})

by (2.4), (2.6) and (2.10). q.e.d. Lemma 2.11. In the cobar complex Ω^{1}A, we have

d(v_{1}^{15}v_{2}) ≡ −2v_{1}^{14}r + 16v_{1}^{12}t_{1}^{3}η_{R}(v_{2}) mod (32, v_{1}^{16}).

Proof. Since d(v_{1}^{15}v_{2}) ≡ v_{1}^{14}d(v_{1}v_{2})+d(v_{1}^{14})η_{R}(v_{1}v_{2}), we obtain d(v_{1}^{15}v_{2}) ≡ 2v_{1}^{14}r−4v_{1}^{14}r+16v_{1}^{12}t_{1}^{3}η_{R}(v_{2}) by Lemma 2.9 and the definition of r. q.e.d.

Lemma 2.12. In the cobar complex Ω^{1}A, there is a cochain zz such that

d(zz) ≡ −2v_{1}^{6}z̃ ⊗ z̃ + 8v_{1}^{6}X mod (16, v_{1}^{8})

for some X. Proof. By [5], there are cochains u and u′ such that d(u) ≡ t_{1} ⊗ z^{2} mod (2, v_{1}) and d(v_{1}u′) ≡ t_{1} ⊗ z̃ mod (2, v_{1}^{2}). Put zz = v_{1}^{6}z^{2} + 4v_{1}^{7}v_{2}^{2}u + 4v_{1}^{6}u^{′2}. By Lemma 2.3, we compute

d(v_{1}^{6}z^{2}) ≡ −4v_{1}^{4}(\underline{v_{1}t_{1}}_{1} + \underline{t_{1}^{2}}_{2}) ⊗ z^{2} − 2v_{1}^{6}z̃ ⊗ z̃
d(4v_{1}^{7}v_{2}^{2}u) ≡ 8v_{1}^{6}t_{1} ⊗ v_{2}^{2}u + 4v_{1}^{7}v_{2}^{2}(\underline{t_{1} ⊗ z^{2}}_{1} + 2X_{1})
d(4v_{1}^{4}(v_{1}u′)^{2}) ≡ \underline{4v_{1}^{4}t_{1}^{2} ⊗ z^{2}}_{2} + 8v_{1}^{6}X_{2}

mod (16, v_{1}^{8}) for some X_{1} and X_{2}. q.e.d.

3. The Miller-Ravenel-Wilson elements x_{i} By the definitions of z̃ and r, we have

(3.1)  r ≡ t_{1}^{4} + z̃ + v_{1}^{3}t_{1} ≡ v_{2}t_{1} + v_{1}(t_{2} + t_{1}^{3}) mod (2),

which represents the generator ρ_{1} of H^{1}M_{1}^{0} and will be used in this section. Define the elements y by

y = v_{1}^{2} − 4v_{1}^{−1}v_{2}

which is x_{1,1} of [1]. Lemma 3.2. In v_{1}^{−1}Γ, we have

d(y) = 8v_{1}^{−2}r ≡ 8v_{1}^{−2}(t_{1}^{4} + z̃ + v_{1}^{3}t_{1}) mod (16).

Proof. Noticing the equation (2.10), we compute

d(v_{1}^{2}) = 4v_{1}t_{1} + 4t_{1}^{2}
d(−4v_{1}^{−1}v_{2}) = −4(v_{1}^{−1} − 2v_{1}^{−2}t_{1} + 4v_{1}^{−2}t_{1}^{2}η_{R}(v_{1}^{−1}))(v_{2} − 5v_{1}t_{1}^{2} − 3v_{1}^{2}t_{1} + 2t_{2} − 4t_{1}^{3})
= 20t_{1}^{2} + 12v_{1}t_{1} − 8v_{1}^{−1}t_{2} + 16v_{1}^{−1}t_{1}^{3} + 8v_{1}^{−2}t_{1}η_{R}(v_{2}) − 16v_{1}^{−2}t_{1}^{2}η_{R}(v_{1}^{−1}v_{2}).

Therefore, using the relation η_{R}(v_{2})t_{1} = z̃ − t_{1}^{4} − v_{1}t_{2} + v_{1}^{2}t_{1}^{2} + v_{1}^{3}t_{1} by (2.4), the sum is

d(y) = 32t_{1}^{2} + 24v_{1}t_{1} − 16v_{1}^{−1}t_{2} + 16v_{1}^{−1}t_{1}^{3} − 16v_{1}^{−2}t_{1}^{2}η_{R}(v_{1}^{−1}v_{2}) + 8v_{1}^{−2}(z̃ − t_{1}^{4}).

and we see that the right hand side is 8v_{1}^{−2}r by (2.6). q.e.d.

Define the elements x_{i} for i ≥ 1 by

x_{1} = v_{2}^{2} + v_{1}^{3}v_{2},
x_{2} = v_{2}^{4} + v_{1}^{3}v_{2}^{3} + v_{1}^{6}v_{2}^{2},
x_{2n+1} = x_{2n}^{2} + v_{1}^{3·2^{2n+1}−12}x_{2} + v_{1}^{3·2^{2n+1}−3}v_{2} (n > 0),
x_{2n+2} = x_{2n+1}^{2} + v_{1}^{3·2^{2n+2}−12}x_{2} (n > 0).

Then we compute Lemma 3.3. In Γ, we have

d(x_{1}) ≡ v_{1}^{2}(r + z̃) mod (2).

Furthermore,

d(x_{n}) ≡ v_{1}^{2^{n}}z^{2^{n−1}} + v_{1}^{3·2^{n}−4}z̃ + \frac{1 + (−1)^{n}}{2} v_{1}^{3·2^{n}−1}t_{1} mod (2)

for n > 1.

Proof. By (2.1) and Lemma 2.8, we have

d(v_{2}^{2}) ≡ v_{1}^{2}(r + z̃) + v_{1}^{3}(v_{1}t_{1}^{2} + v_{1}^{2}t_{1}) mod (2) and
d(v_{1}^{3}v_{2}) ≡ v_{1}^{3}(v_{1}t_{1}^{2} + v_{1}^{2}t_{1}) mod (2).

The sum gives us the first one.

We show the others by induction. By (2.1), (2.2) and (3.1),

v_{2}d(v_{2}^{2}) + v_{1}^{2}t_{1}η_{R}(v_{2}^{2}) ≡ v_{1}^{2}v_{2}t_{1}^{4} + v_{1}^{2}t_{1}η_{R}(v_{2}^{2}) + v_{1}^{4}v_{2}t_{1}^{2}
≡ v_{1}^{3}t_{2}^{2} + v_{1}^{5}v_{2}t_{1} + v_{1}^{6}(t_{2} + t_{1}^{3}) ≡ v_{1}^{3}t_{2}^{2} + v_{1}^{5}r mod (2)

We also see that t_{1}^{2}η_{R}(v_{2}^{2}) ≡ r^{2} + v_{1}^{2}t_{2}^{2} + v_{1}^{4}t_{1}^{4} mod (2) by (2.1) and (3.1). Since d(v_{2}^{3}) = v_{2}d(v_{2}^{2}) + d(v_{2})η_{R}(v_{2}^{2}), we have

d(v_{2}^{3}) ≡ v_{1}^{3}t_{2}^{2} + v_{1}^{5}r + v_{1}r^{2} + v_{1}^{3}t_{2}^{2} + v_{1}^{5}t_{1}^{4} mod (2)
≡ v_{1}r^{2} + v_{1}^{5}z̃ + v_{1}^{8}t_{1} mod (2) by (3.1).

Then the first step d(x_{2}) ≡ v_{1}^{4}z^{2} + v_{1}^{8}z̃ + v_{1}^{11}t_{1} mod (2) is obtained as the sum of

d(x_{1}^{2}) ≡ v_{1}^{4}(r^{2} + z^{2}) mod (2) and
d(v_{1}^{3}v_{2}^{3}) ≡ v_{1}^{3}(v_{1}r^{2} + v_{1}^{5}z̃ + v_{1}^{8}t_{1}) mod (2).

Inductively suppose that

d(x_{2n}) ≡ v_{1}^{2^{2n}}z^{2^{2n−1}} + v_{1}^{3·2^{2n}−4}z̃ + v_{1}^{3·2^{2n}−1}t_{1} mod (2).

Then

d(x_{2n}^{2}) ≡ v_{1}^{2^{2n+1}}z^{2^{2n}} + \underline{v_{1}^{3·2^{2n+1}−8}z^{2}}_{1} + \underline{v_{1}^{3·2^{2n+1}−2}t_{1}^{2}}_{2} mod (2)
d(v_{1}^{3·2^{2n+1}−12}x_{2}) ≡ \underline{v_{1}^{3·2^{2n+1}−8}z^{2}}_{1} + v_{1}^{3·2^{2n+1}−4}z̃ + \underline{v_{1}^{3·2^{2n+1}−1}t_{1}}_{3} mod (2)
d(v_{1}^{3·2^{2n+1}−3}v_{2}) ≡ \underline{v_{1}^{3·2^{2n+1}−2}t_{1}^{2}}_{2} + \underline{v_{1}^{3·2^{2n+1}−1}t_{1}}_{3} mod (2)

and we obtain

d(x_{2n+1}) ≡ v_{1}^{2^{2n+1}}z^{2^{2n}} + v_{1}^{3·2^{2n+1}−4}z̃ mod (2).

Squaring this shows

d(x_{2n+1}^{2}) ≡ v_{1}^{2^{2n+2}}z^{2^{2n+1}} + \underline{v_{1}^{3·2^{2n+2}−8}z^{2}}_{a} mod (2).

Add

d(v_{1}^{3·2^{2n+2}−12}x_{2}) ≡ \underline{v_{1}^{3·2^{2n+1}−8}z^{2}}_{a} + v_{1}^{3·2^{2n+2}−4}z̃ + v_{1}^{3·2^{2n+2}−1}t_{1} mod (2),

and we obtain the case 2n + 2, and the induction completes. q.e.d.

Remark. In the following, v_{2}^{2^{n}} denotes sometimes x_{n}.

4. H^{∗}M_{0}^{1} revisited In [1, Th.4.16], it is shown that

(4.1)  H^{∗}M_{0}^{1} = ( ∑_{i>0,p|/s} Z/2^{i+2}{v_{1}^{2^{i}s}/2^{i+2}}) ⨁ (v_{1}Z/2[v_{1}^{±2}, h_{10}]⨂Λ(ρ_{1})) ⨁ (Q/Z_{(2)}⨂Λ(ρ_{1})) .

In this section we study the image of ρ_{1}/2^{i} (i > 0) under the map H^{1}M_{0}^{1} → H^{1}M_{0}^{2}.

Lemma 4.2. Put

r_{n} = z̃/v_{1}^{4} + ∑_{i>2} \binom{2^{n−1}}{i}2^{2i−n−2}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{−2i} + ∑_{i=1}^{2^{n}−4} \binom{2^{n} − 4}{i}2^{2i−1}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{−2i−4}η_{R}(v_{1}v_{2}).

Then r_{n} is a cochain of Ω^{1}v_{1}^{−1}A such that d(r_{n}) ≡ 0 mod (2^{n+1}). Proof. Consider y_{n} = v_{1}^{2^{n}} + 2^{n+1}v_{1}^{2^{n}−3}v_{2}. By (2.1), we compute

d(v_{1}^{2^{n}}) = (v_{1}^{2} + 4(v_{1}t_{1} + t_{1}^{2}))^{2^{n−1}}
= 2^{n+1}v_{1}^{2^{n}−2}(v_{1}t_{1} + t_{1}^{2}) + 2^{n+2}v_{1}^{2^{n}−4}(v_{1}t_{1} + t_{1}^{2})^{2} + ∑_{i>2} \binom{2^{n−1}}{i}2^{2i}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{2^{n}−2i}
d(2^{n+1}v_{1}^{2^{n}−3}v_{2}) = 2^{n+1}v_{1}^{2^{n}−4}(v_{1} + 2t_{1})(v_{2} − 5v_{1}t_{1}^{2} − 3v_{1}^{2}t_{1} + 2t_{2} − 4t_{1}^{3})

+2^{n+1} ∑_{i=1}^{2^{n}−4} \binom{2^{n} − 4}{i}2^{2i}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{2^{n}−2i−4}η_{R}(v_{1}v_{2})

= 2^{n+1}v_{1}^{2^{n}−4}(−5v_{1}^{2}t_{1}^{2} − 3v_{1}^{3}t_{1} + 2v_{1}t_{2} − 4v_{1}t_{1}^{3} + 2η_{R}(v_{2})t_{1})

+2^{n+1} ∑_{i=1}^{2^{n}−4} \binom{2^{n} − 4}{i}2^{2i}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{2^{n}−2i−4}η_{R}(v_{1}v_{2})

Then the sum is

d(y_{n}) = 2^{n+2}v_{1}^{2^{n}−4}(v_{1}t_{2} + η_{R}(v_{2})t_{1} − v_{1}^{3}t_{1} + v_{1}^{2}t_{1}^{2} + t_{1}^{4}) − 2^{n+3}v_{1}^{2^{n}−2}t_{1}^{2}

+ ∑_{i>2} \binom{2^{n−1}}{i}2^{2i}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{2^{n}−2i} + 2^{n+1} ∑_{i=1}^{2^{n}−4} \binom{2^{n} − 4}{i}2^{2i}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{2^{n}−2i−4}η_{R}(v_{1}v_{2})

= 2^{n+2}v_{1}^{2^{n}−4}z̃ + ∑_{i>2} \binom{2^{n−1}}{i}2^{2i}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{2^{n}−2i}

+2^{n+1} ∑_{i=1}^{2^{n}−4} \binom{2^{n} − 4}{i}2^{2i}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{2^{n}−2i−4}η_{R}(v_{1}v_{2})

This shows that the cochain r_{n} satisfies

d(y^{2^{n}}) = 2^{n+2}v_{1}^{2^{n}}r_{n}.

Then 0 ≡ 2^{2n+3}v_{1}^{2^{n}−2}(v_{1}t_{1} + t_{1}^{2}) ⊗ r_{n} + 2^{n+2}v_{1}^{2^{n}}d(r_{n}) mod (2^{2n+4}), and so we have

d(r_{n}) ≡ 2^{n+1}v_{1}^{−2}(v_{1}t_{1} + t_{1}^{2}) ⊗ r_{n} mod (2^{n+2}). q.e.d.

Proposition 4.3. The generator ρ_{1}/2^{n+1} of H^{1,0}M_{0}^{1} ≅ Q/Z_{(2)} is represented by r_{n}/2^{n+1}. Furthermore, the map H^{1}M_{0}^{1} → H^{1}M_{0}^{2} sends ρ_{1}/2^{n+2} to ζ̃/2^{n+2}v_{1}^{4}.

Proof. Since z̃/2v_{1}^{4} is homologous to r/2v_{1}^{4} by Lemma 3.3, we see that ρ_{1}/2 represented by r_{n}/2v_{1}^{4}, and we see the proposition by Lemma 4.2. q.e.d.

5. H^{∗}M_{1}^{1} revisited

Ravenel showed the structure of H^{∗}M_{2}^{0} = H^{∗}K(2)_{∗} (cf. [2]):

(5.1)  \begin{matrix}H^{∗}M_{2}^{0} = (K(2)_{∗}[h_{10}, h_{11}]/(h_{10}h_{11}, v_{2}h_{10}^{3} − h_{11}^{3})⨂Λ(β)⨂Z/2[g] \\ ⨁ζK(2)_{∗}[h_{0}]⨂Λ(ζ)) ⨂Λ(ρ_{2}). \end{matrix}

From this, H^{∗}M_{1}^{1} is determined in [5]:

(5.2)  H^{∗}M_{1}^{1} = (∑_{i=0}^{6} MB_{i}) ⨂Λ(ρ_{2}),

where

MB_{0} = K(1)_{∗}/k(1)_{∗}{h_{10}^{i}, h_{10}^{i}β, ζh_{0}^{i}, ζ^{2}h_{0}^{i}, ζ̃h_{0}^{i}, ζζ̃h_{0}^{i} | i = 0, 1, 2, 3}⨂Z[g]
MB_{1} = (v_{2}/v_{1})Z/2[v_{2}^{±2}]⨂Z/2[h_{11}]/(h_{11}^{3})⨂Λ(β)⨂Z/2[g]
MB_{2} = (v_{2}^{2}/v_{1}^{2})Z/2[v_{1}, v_{2}^{±4}]⨂Λ(h_{10}, v_{2}h_{10}, β)⨂Z/2[g]
MB_{3} = ∑_{n>0}(v_{2}^{2^{n}} ζ̃/v_{1}^{3·2^{n−1}+1})Z/2[v_{1}, v_{2}^{±2^{n+1}}]⨂Z/2[h_{0}]
MB_{4} = ∑_{n>1}(v_{2}^{2^{n}} ζ̃^{(n−1)}/v_{1}^{2^{n+1}})Z/2[v_{1}, v_{2}^{±2^{n+1}}]⨂Z/2[h_{0}]
MB_{5} = ∑_{n>1}(v_{2}^{2^{n}}/v_{1}^{3·2^{n−1}+3})Z/2[v_{1}, v_{2}^{±2^{n+1}}]⨂Z/2{v_{1}^{3}, v_{1}^{2}h_{10}, v_{1}h_{10}^{2}, h_{10}^{3}}⨂Z/2[g]
MB_{6} = ∑_{n>1}(v_{2}^{2^{n}} ζ̃^{(n−1)}/v_{1}^{2^{n+1}+4})Z/2[v_{1}, v_{2}^{±2^{n+1}}]⨂Z/2{ζ̃, v_{1}ζ̃h_{0}, v_{1}^{2}ζ̃h_{0}^{2}, v_{1}^{3}ζ̃h_{0}^{3}}⨂Z/2[g].

Here ζ̃^{(n−1)}/v_{1}^{j} denote the element represented by z^{2^{n−1}}/v_{1}^{j} ∈ Ω^{1}M_{1}^{1}, and so ζ̃^{(n−1)}/v_{1}^{2^{n−1}} = v_{2}^{2^{n−1}}ζ^{2^{n−1}} = 0.

Remark. The names of the generators are different from those of [5]. The generators ζh_{0}^{i}/v_{1}^{j}, ζ̃h_{0}^{i}/v_{1}^{j} and ζ̃^{(n)}h_{0}^{i}/v_{1}^{j} correspond to ζ_{a}g^{b}/v_{1}^{j}, v_{2}ζ_{a}g^{b}/v_{1}^{j−1}, and v_{2}^{2^{n}}ζ_{a}g^{b}/v_{1}^{j−2^{n}}, respectively, where a and b are integers such that a = 1, 2, 3, 4 and i = 4b + a − 1.

Note that the direct sum MB_{1}⨁MB_{2} is the tensor product of Z/2[v_{2}^{±4}, g] and the module displayed as follows:

[figure: 24; 0; 3; v_{2}/v_{1}; 4; 8; v_{2}^{2}/v_{1}^{2}; 10; 16; v_{2}^{3}/v_{1}]

in which the lines of the slope 1 (resp. 1/3, 0) denote the multiplication by h_{10} (resp. h_{11}, v_{1}). Now we consider the submodules of K(1)_{∗}/k(1)_{∗}:

A_{∞} = Z/2{1/v_{1}^{2j} | j > 0}
A_{n} = Z/2{1/v_{1}^{2j} | 0 < 2j ≤ 3 · 2^{n−1}}
A_{n}^{+} = Z/2{1/v_{1}^{2j} | 0 < 2j ≤ 3 · 2^{n−1} + 2}
A_{n}^{−} = Z/2{1/v_{1}^{2j} | 0 < 2j ≤ 3 · 2^{n−1} − 4}

for n ≥ 2. Then MB_{0} is divided into the following seven summands:

LC_{0} = v_{1}A_{∞}{1, h_{10}, h_{10}^{2}, h_{10}^{3}/v_{1}^{4})⨂Z/2[g]⨂Λ(β)⨁ζ̃A_{∞}{v_{1}, h_{0}}⨂Z/2[h_{0}^{2}]⨂Λ(ζ),
LI_{0} = A_{∞}{h_{10}, h_{10}^{2}, h_{10}^{3}, g)⨂Z/2[g]⨂Λ(β)⨁ζ̃A_{∞}{v_{1}h_{0}, h_{0}^{2}}⨂Z/2[h_{0}^{2}]⨂Λ(ζ),
JC = ζK(1)_{∗}/k(1)_{∗}⨂Z/2[h_{0}],
JI = ζ^{2}K(1)_{∗}/k(1)_{∗}⨂Z/2[h_{0}],
Q = Z/2{ζ̃/v_{1}^{4}}⨂Λ(ζ)
K_{0} = Z/2{h_{10}^{3}/v_{1}^{3}}⨂Λ(v_{1}^{2}, β)⨂Z/2[g]⨁Z/2{ζ̃/v_{1}^{2}}⨂Λ(ζ), and
P = A_{∞}⨂Λ(β)⨁(ζ̃/v_{1}^{4})A_{∞}⨂Λ(ζ).

The module MB_{1}⨁MB_{2} is the direct sum of four modules

K_{1} = (K_{1}^{0}⨁K_{1}^{2})⨂Z/2[g],
D_{1} = ∑_{n≥2,2 |̸s} v_{2}^{2^{n}s}K_{1}^{0}
F_{1} = ∑_{n≥2,2 |̸s} v_{2}^{2^{n}s}(gK_{1}^{0}⨁K_{1}^{2})⨂Z/2[g]
M = M^{0}⨂Z/2[v_{2}^{±4}, g].

Here

K_{1}^{0} = Z/2{v_{2}/v_{1}}⨂Λ(β)
K_{1}^{2} = Z/2{v_{2}^{−1}h_{11}^{2}/v_{1}}⨂Λ(β) and
M^{0} = (Z/2{v_{2}h_{11}/v_{1}, v_{2}^{3}/v_{1}}⨂Λ(h_{11})⨁Z/2{v_{2}^{2}/v_{1}^{2}}⨂Λ(v_{1}, h_{10}, v_{2}h_{10})) ⨂Λ(β)

The module M^{0} is displayed as follows:

[figure: 24; 0; 3; 8; v_{2}^{2}/v_{1}^{2}; 10; 16; v_{2}^{3}/v_{1}]

MB_{3} is the direct sum of

LC_{1} = Z/2{v_{2}^{2s}ζ̃/v_{1}^{3}, v_{2}^{2s}ζ̃h_{0}/v_{1}^{2}, v_{2}^{2s}ζ̃h_{0}/v_{1}^{4} | 2|/s ∈ Z}⨂Z/2[h_{0}^{2}],
LI_{1} = Z/2{v_{2}^{2s}ζ̃h_{0}/v_{1}^{3}, v_{2}^{2s}ζ̃h_{0}^{2}/v_{1}^{2}, v_{2}^{2s}ζ̃h_{0}^{2}/v_{1}^{4} | 2|/s ∈ Z}⨂Z/2[h_{0}^{2}],
LC_{2} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}ζ̃A_{n}{1/v_{1}, h_{0}}⨂Z/2[h_{0}^{2}],

LI_{2} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}ζ̃A_{n}{h_{0}/v_{1}, h_{0}^{2}}⨂Z/2[h_{0}^{2}],
D_{2} = ∑_{n≥2,2|/s} Z/2{v_{2}^{2^{n}s}ζ̃/v_{1}^{4}}⨂Λ(v_{1}^{2})
E_{1} = ∑_{2|/s} Z/2{v_{2}^{2s}ζ̃/v_{1}^{4}}⨂Λ(v_{1}^{2})
N_{1} = ∑_{n≥3,2|/s} v_{2}^{2^{n}s}(ζ̃/v_{1}^{4})A_{n}^{−} and
E_{2} = ∑_{2|/s} Z/2{v_{2}^{4s}ζ̃/v_{1}^{6}}⨂Λ(v_{1}^{2}).

MB_{4} is divided into

LC_{3} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζA_{n}{v_{1}, h_{0}}⨂Z/2[h_{0}^{2}],
LI_{3} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζA_{n}{v_{1}h_{0}, h_{0}^{2}}⨂Z/2[h_{0}^{2}],
N_{2} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζA_{n}.

MB_{5} is the direct sum of

LC_{4} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}A_{n}{v_{1}, h_{0}^{3}}⨂Z/2[g],
LC_{5} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}A_{n}^{+}{v_{1}h_{10}, v_{1}h_{10}^{2}}⨂Z/2[g],
LI_{4} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}A_{n}{h_{10}, g}⨂Z/2[g],
LI_{5} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}A_{n}^{+}{h_{10}^{2}, h_{10}^{3}}⨂Z/2[g],
F_{2} = ∑_{n≥2,2|/s} Z/2{v_{2}^{2^{n}s}h_{10}^{3}/v_{1}^{3}}⨂Λ(v_{1}^{2})⨂Z/2[g],
N_{3} = ∑_{n≥3,2|/s} v_{2}^{2^{n}s}A_{n} and
E_{3} = ∑_{2|/s} v_{2}^{4s}A_{2}.

MB_{6} is the direct sum of

LC_{6} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζζ̃A_{n}{h_{0}^{2}/v_{1}, h_{0}^{3}}⨂Z/2[g],
LC_{7} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζζ̃A_{n}^{+}{1/v_{1}, h_{0}}⨂Z/2[g],
LI_{6} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζζ̃A_{n}{h_{0}^{3}/v_{1}, g}⨂Z/2[g],
LI_{7} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζζ̃A_{n}^{+}{h_{0}/v_{1}, h_{0}^{2}}⨂Z/2[g],

F_{3} = ∑_{n≥2,2|/s} Z/2{v_{2}^{2^{n}s+2^{n−1}}ζζ̃g/v_{1}^{3·2^{n−1}+4}}⨂Λ(v_{1}^{2})⨂Z/2[g]
D_{3} = ∑_{n≥2,4|(s−1)} Z/2{v_{2}^{2^{n}s+2^{n−1}}ζζ̃/v_{1}^{3·2^{n−1}+4}}⨂Λ(v_{1}^{2})
D_{4} = ∑_{n≥3,2|/s} Z/2{v_{2}^{2^{n}s+2^{n−1}}ζζ̃/v_{1}^{4}}⨂Λ(v_{1}^{2})
E_{4} = ∑_{2|/s} Z/2{v_{2}^{4s+2}ζζ̃/v_{1}^{4}}⨂Λ(v_{1}^{2})
N_{4} = ∑_{n≥2,2|/s} v_{2}^{2^{n}s+2^{n−1}}ζ(ζ̃/v_{1}^{4})A_{n}^{−}⨁ ∑_{n≥2,4|(s+1)} Z/2{v_{2}^{2^{n}s+2^{n−1}}ζζ̃/v_{1}^{3·2^{n−1}+4}}⨂Λ(v_{1}^{2}).

Now put

\begin{matrix}D = ∑_{i=1}^{4} D_{i}, E = ∑_{i=1}^{4} E_{i}, F = ∑_{i=1}^{3} F_{i}, J = JC⨁JI, K = K_{0}⨁K_{1}, \\ L = ∑_{i=0}^{7}(LC_{i}⨁LI_{i}), and N = ∑_{i=1}^{4} N_{i}. \end{matrix}

Then

H^{∗}M_{1}^{1} = (D⨁E⨁F ⨁J⨁K⨁L⨁M⨁N⨁P ⨁Q) ⨂Λ(ρ_{2}).

6. The cocycles R and B

Consider the E(2)_{∗}-module M(i, j) = E(2)_{∗}/(2^{i}, v_{1}^{j}), which is also an E(2)_{∗}E(2)-comodule if 2^{i−1}|j. Then we have the exact sequence

(6.1)  H^{s−1,0}M_{1}^{1} ⟶^{δ} H^{s,0}M(1, j) ⟶^{1/v_{1}^{j}} H^{s,−2j}M_{1}^{1} ⟶^{v_{1}^{j}} H^{s,0}M_{1}^{1}.

Lemma 6.2. Every element of H^{s,0}M(1, 2^{n}) for s = 2, 3, 4 and n > 4 is divisible by v_{1}^{2^{n−2}} except for ζh_{0}, ζ^{2} and ζρ_{2} if s = 2, β, ζh_{0}^{2}, ζ^{2}h_{0}, v_{2}^{−1}h_{10}^{3}, ζh_{0}ρ_{2} and ζ^{2}ρ_{2} if s = 3, and g, ζh_{0}^{3}, ζ^{2}h_{0}^{2}, βρ_{2}, ζh_{0}^{2}ρ_{2}, ζ^{2}h_{0}ρ_{2} and v_{2}^{−1}h_{10}^{3}ρ_{2} if s = 4.

Proof. By (5.2), we see that

H^{1,0}M_{1}^{1} = Z/2{h_{10}/v_{1}, v_{2}ζ/v_{1}^{3}}
H^{2,0}M_{1}^{1} = Z/2{h_{10}^{2}/v_{1}^{2}, ζ̃h_{0}/v_{1}^{4}, ζζ̃/v_{1}^{4}, v_{2}^{−1}h_{11}^{2}/v_{1}, h_{10}ρ_{2}/v_{1}, v_{2}ζρ_{2}/v_{1}^{3}}
H^{3,0}M_{1}^{1} = Z/2{h_{10}^{3}/v_{1}^{3}, ζ̃h_{0}^{2}/v_{1}^{4}, ζζ̃h_{0}/v_{1}^{4}, h_{10}^{2}ρ_{2}/v_{1}^{2}, ζ̃h_{0}ρ_{2}/v_{1}^{4}, ζζ̃ρ_{2}/v_{1}^{4}, v_{2}^{−1}h_{11}^{2}ρ_{2}/v_{1}}.

Then Im δ = 0 if s = 2, = Z/2{v_{2}^{−1}h_{10}^{3}} if s = 3, and = Z/2{v_{2}^{−1}h_{10}^{3}ρ_{2}} if s = 4, which gives the exceptional elements.

If x/v_{1}^{j} ∈ H^{s,−2^{n+1}}M_{1}^{1} is in the image of 1/v_{1}^{2^{n}}, then v_{1}^{2^{n}−j}x ∈ H^{s,0}M(1, 2^{n}). So if the element v_{1}^{2^{n}−j}x of H^{s,0}M(1, 2^{n}) is divisible by v_{1}^{2^{n−2}}, then 2^{n} − j ≥ 2^{n−2}. We will find elements x/v_{1}^{j} with j > 2^{n} − 2^{n−2}. There is no such elements in MB_{1}⨁MB_{2}, since j ≤ 2. For a generator v_{2}^{2^{m}(2t+1)}ξ_{i}/v_{1}^{j} of MB_{i}, 2^{n} − 2^{n−2} < j ≤ 3 · 2^{m−1} + 4 and 6 · 2^{m}(2t + 1) + |ξ_{i}| − 2j = −2^{n+1}. Here |ξ_{i}| = 8 if i = 3, = 6 · 2^{m−1} if i = 4, = 2s if i = 5 and s = 2, 3, = 0 if i = 5 and s = 4, and = 6 · 2^{m−1} + 8 if i = 6. Then we see that there is no solution if n ≥ 4. In MB_{0}, we have

h_{10}^{2}/v_{1}^{2^{n}+2}, ζh_{0}/v_{1}^{2^{n}}, ζ^{2}/v_{1}^{2^{n}}, ζ̃h_{0}/v_{1}^{2^{n}+4}, ζζ̃/v_{1}^{2^{n}+4}, h_{10}ρ_{2}/v_{1}^{2^{n}+1}, ζρ_{2}/v_{1}^{2^{n}}, and ζ̃ρ_{2}/v_{1}^{2^{n}+4}

in H^{2,−2^{n+1}}M_{1}^{1},

\begin{matrix}h_{10}^{3}/v_{1}^{2^{n}+3}, β/v_{1}^{2^{n}}, ζh_{0}^{2}/v_{1}^{2^{n}}, ζ^{2}h_{0}/v_{1}^{2^{n}}, ζ̃h_{0}^{2}/v_{1}^{2^{n}+4}, ζζ̃h_{0}/v_{1}^{2^{n}+4} \\ h_{10}^{2}ρ_{2}/v_{1}^{2^{n}+2}, ζh_{0}ρ_{2}/v_{1}^{2^{n}}, ζ^{2}ρ_{2}/v_{1}^{2^{n}}, ζ̃h_{0}ρ_{2}/v_{1}^{2^{n}+4}, and ζζ̃ρ_{2}/v_{1}^{2^{n}+4}, \end{matrix}

in H^{3,−2^{n+1}}M_{1}^{1}, and

\begin{matrix}g/v_{1}^{2^{n}}, h_{10}β/v_{1}^{2^{n}+1}, ζh_{0}^{3}/v_{1}^{2^{n}}, ζ^{2}h_{0}^{2}/v_{1}^{2^{n}}, ζ̃h_{0}^{3}/v_{1}^{2^{n}+4}, ζζ̃h_{0}^{2}/v_{1}^{2^{n}+4}, \\ h_{10}^{3}ρ_{2}/v_{1}^{2^{n}+3}, βρ_{2}/v_{1}^{2^{n}}, ζh_{0}^{2}ρ_{2}/v_{1}^{2^{n}}, ζ^{2}h_{0}ρ_{2}/v_{1}^{2^{n}}, ζ̃h_{0}^{2}ρ_{2}/v_{1}^{2^{n}+4}, and ζζ̃h_{0}ρ_{2}/v_{1}^{2^{n}+4}, \end{matrix}

in H^{4,−2^{n+1}}M_{1}^{1}. Since the generators of the form x/v_{1}^{2^{n}} are pulled back to x of H^{∗}M(1, 2), we obtain the other exceptional elements. q.e.d.

Lemma 6.3. Let z_{i} denote a representative of ζh_{0}^{i−1} of H^{i,0}M(1, 2^{n}). Then

d(z_{1}) ≡ 2z_{1} ⊗ z_{1} mod (4, v_{1}^{2^{n}})
d(z_{2}) ≡ 2z_{1} ⊗ z_{2} + 2z_{3} mod (4, v_{1}^{2^{n}})
d(z_{3}) ≡ 2z_{1} ⊗ z_{3} mod (4, v_{1}^{2^{n}})
d(z_{4}) ≡ 2z_{1} ⊗ z_{4} + 2z_{1} ⊗ G mod (4, v_{1}^{2^{n}}).

Proof. Since z_{1} = z^{2^{n}} is primitive mod (2, v_{1}^{2^{n}}) and z_{1} is homologous to z_{1}^{2}, the first one is obvious if we replace z_{1} by z_{1}^{2}.

Put d(z_{2}) ≡ 2c_{n} mod (4, v_{1}^{2^{n+2}}). Then

(6.4)  c_{n} ≡ k_{1}B + k_{2}z_{3} + k_{3}z_{1} ⊗ z_{2} + k_{4}v_{2}^{−1}t_{1} ⊗ t_{1} ⊗ t_{1} + k_{5}z_{2} ⊗ R + k_{6}z_{1} ⊗ z_{1} ⊗ R + d(c′_{n})

mod (2, v_{1}^{2^{n}}) for some c′_{n} by Lemma 6.2, where R represents ρ_{2}. Recall [5, Lemma 6.8] that there are elements u_{i} such that d(u_{1}) ≡ z_{1} ⊗ t_{1} + v_{1}z_{2} and d(u_{2}) ≡ t_{1} ⊗ z_{2} + v_{1}z_{3} mod (2, v_{1}^{2^{n}}). Define elements w to fit in d(u_{1}) ≡ z_{1} ⊗ t_{1} + v_{1}z_{2} + 2w mod (4, v_{1}^{2^{n}}). Then

\begin{aligned}0 & ≡ & 2z_{1} ⊗ z_{1} ⊗ t_{1} + 2t_{1} ⊗ z_{2} + v_{1}d(z_{2}) + 2d(w) \\ & ≡ & 2d(z_{1} ⊗ u_{1}) + 2v_{1}z_{1} ⊗ z_{2} + 2d(u_{2}) + 2v_{1}z_{3} + 2v_{1}c_{n} + 2d(w) mod (4, v_{1}^{2^{n}}). \end{aligned}

Comparing with (6.4) shows the second.

In the same manner, we see the others. q.e.d.

Proposition 6.5. There is a cocycle R_{n} ∈ Ω^{1}M(n + 1, 2^{n}), which represents ρ_{2} of H^{1,0}M(1, 1) = H^{1,0}K(2)_{∗}.

Proof. Suppose first n > 3. Raising a representative of ρ_{2} of H^{1,0}K(2)_{∗} to 2^{3n} power yields a representative of ρ_{2} ∈ H^{1,0}M(1, 2^{3n}). Let R denote a representative of ρ_{2}. Then R^{2} also represents ρ_{2}, and we compute d(R^{2}) ≡ 2R ⊗ R mod (4, v_{1}^{3n}). The relation ρ_{2}^{2} = 0 of H^{∗}K(2)_{∗} shows the existence of an element S such that d(S) ≡ R ⊗ R mod (2, v_{1}^{3n}). Then R_{1} = R^{2} + 2S satisfies the relation d(R_{1}) ≡ 0 mod (4, v_{1}^{3n}).

Suppose inductively that there exists a cocycle R_{k} such that d(R_{k}) ≡ 0 mod (2^{k+1}, v_{1}^{2^{3n+2−2k}}) for k ≥ 1. Then d(R_{k}) ≡ 2^{k+1}x mod (2^{k+2}, v_{1}^{2^{3n+2−2k}}) for some x, and we see that x represents an element of H^{2,0}M(1, 2^{3n+2−2k}). Lemma 6.2 shows that x ≡ k_{1}z_{2} + k_{2}z_{1} ⊗ z_{1} + k_{3}z_{1}R_{k} + 2y + d(w) mod (4, v_{1}^{2^{3n−2k}}) for some cochains y and w. Since d(x) ≡ 0 mod (4, 2^{3n+2−2k}), we see that

0 = k_{1}(z_{1} ⊗ z_{2} + z_{3}) + k_{3}z_{1} ⊗ z_{1} ⊗ R_{k} + d(y)

mod (2, v_{1}^{2^{3n−2k}}) by Lemma 6.3, which implies a relation 0 = k_{1}(ζ^{2}h_{0} + ζh_{0}^{2}) + k_{3}ζ^{2}ρ_{2} of H^{3,0}K(2)_{∗} and we have k_{1} = 0 = k_{3}. Therefore, d(R_{k}) ≡ 2^{k+1}k_{2}z_{1} ⊗ z_{1} + d(2^{k+1}w) mod (2^{k+2}, v_{1}^{2^{3n−2k}}). Put R_{k+1} = R_{k} + 2^{k}k_{2}z_{1} + 2^{k+1}w and we have d(R_{k+1}) ≡ 0 mod (2^{k+2}, v_{1}^{2^{3n−2k}}), which completes the induction.

Now take R_{n} for a representative of ρ̃_{2} and we have the result for n > 3.

If n ≤ 3, just consider the projection M(5, 2^{4}) → M(n + 1, 2^{n}) → M(1, 1). q.e.d..

We write R as R_{n} as long as no confusion arises.

Corollary 6.6. In the cobar complex for M(n + 1, j) with 2^{n} | j, there is a cochain S such that

d(S) = R ⊗ R.

Proof. Since R is homologous to R^{2} in the cobar complex Ω^{1}M(1, j) for any j, we have cochains U_{j} and S_{j} such that

d(U_{j}) ≡ R^{2} − R − 2S_{j} mod (v_{1}^{j}).

Now send this by the differential d, and we have

0 ≡ 2R ⊗ R − 2d(S_{j}) mod (v_{1}^{j})

as desired. q.e.d.

Lemma 6.7. There is a cochain H such that H ≡ v_{1}^{3}v_{2}^{−1}h_{11}^{2} mod (2, v_{1}^{4}) and

d(H) ≡ 4v_{1}t_{1} ⊗ t_{1} ⊗ t_{1} mod (8, v_{1}^{4}).

Proof. Put H = v_{1}^{2}v_{2}^{−1}d(v_{2}t_{1}^{2}) − 2v_{1}^{3}v_{2}^{−2}t_{1}^{2}t_{2} ⊗ t_{1}^{2} − 4v_{1}v_{2}^{−1}D. Here D denotes a cochain such that d(D) ≡ t_{1}^{4} ⊗ t_{1} ⊗ t_{1} + t_{1}^{2} ⊗ t_{1}^{2} ⊗ t_{1}^{2} mod (2) given in [5, p.149]. Then, we obtain

d(H) ≡ 4v_{1}t_{1} ⊗ t_{1} ⊗ t_{1} mod (8, 4v_{1}^{2}, v_{1}^{4}).

from computation

d(v_{1}^{2}v_{2}^{−1}d(v_{2}t_{1}^{2})) ≡ v_{1}^{2}d(v_{2}^{−1}) ⊗ d(v_{2}t_{1}^{2}) + (4v_{1}t_{1} + 4t_{1}^{2}) ⊗ v_{2}^{−1}d(v_{2}t_{1}^{2})
≡ v_{1}^{2}v_{2}^{−2}(−v_{1}t_{1}^{2} + 2t_{2}) ⊗ (−v_{1}t_{1}^{2} + 2t_{2}) ⊗ t_{1}^{2} − 2v_{1}^{2}v_{2}^{−2}(−v_{1}t_{1}^{2} + 2t_{2}) ⊗ v_{2}t_{1} ⊗ t_{1}
+4v_{1}v_{2}^{−1}t_{1}^{2} ⊗ t_{1}^{2} ⊗ t_{1}^{2}
≡ 2v_{1}^{3}v_{2}^{−2}t_{1}^{3} ⊗ t_{1}^{2} ⊗ t_{1}^{2} − 2v_{1}^{3}v_{2}^{−2}t_{1}^{2} ⊗ t_{2} ⊗ t_{1}^{2} − 2v_{1}^{3}v_{2}^{−2}t_{2} ⊗ t_{1}^{2} ⊗ t_{1}^{2} + 2v_{1}^{3}v_{2}^{−2}t_{1}^{2} ⊗ v_{2}t_{1} ⊗ t_{1}
+4v_{1}v_{2}^{−1}t_{1}^{2} ⊗ t_{1}^{2} ⊗ t_{1}^{2}
d(−2v_{1}^{3}v_{2}^{−2}t_{1}^{2}t_{2} ⊗ t_{1}^{2}) ≡ 2v_{1}^{3}v_{2}^{−2}(t_{1}^{3} ⊗ t_{1}^{2} + t_{1} ⊗ t_{1}^{4} + t_{1}^{2} ⊗ t_{2} + t_{2} ⊗ t_{1}^{2}) ⊗ t_{1}^{2}
d(−4v_{1}v_{2}^{−1}D) ≡ −4v_{1}v_{2}^{−1}(t_{1}^{4} ⊗ t_{1} ⊗ t_{1} + t_{1}^{2} ⊗ t_{1}^{2} ⊗ t_{1}^{2}).

mod (8, 4v_{1}^{2}, v_{1}^{4}). Here note that v_{2}t_{1} ≡ t_{1}^{4} mod (2, v_{1}) in Γ by (2.2).

Thus we put d(H) ≡ 4v_{1}t_{1} ⊗ t_{1} ⊗ t_{1} + 4v_{1}^{2}C_{1} mod (8, v_{1}^{4}) for some C_{1}. Since d(C_{1}) ≡ 0 mod (2, v_{1}^{2}), C_{1} represents an element of H^{3,4}M(1, 2) = Z/2{h_{10}^{2}ρ_{2}, v_{1}v_{2}^{−1}h_{11}^{2}ρ_{2}}. Since d(v_{1}t_{1} ⊗ R) = 2t_{1} ⊗ t_{1} ⊗ R, we may assume that C_{1} represents a multiple of v_{1}v_{2}^{−1}h_{11}^{2}ρ_{2}. Checking the above computation carefully shows that there appears no R and we see that v_{1}^{2}C_{1} is homologous to zero. After suitable replacement, we may take C_{1} = 0. q.e.d.

Lemma 6.8. There is a cochain C of Ω^{3,0}M(4, 2^{n}) for n > 4 such that d(C) ≡ 8G mod (16, v_{1}^{2^{n}}) and C represents the element v_{2}^{−1}h_{10}^{3} ∈ H^{3,0}M(1, 2^{n}). Furthermore, v_{1}C is bounded in Ω^{3,0}M(3, 2^{n}).

Proof. We have d(H) ≡ v_{1}^{4}v_{2}^{−1}t_{1} ⊗ t_{1} ⊗ t_{1} + ⋯ mod (2). Therefore, we see that

(6.9)  d(H) ≡ 4v_{1}t_{1} ⊗ t_{1} ⊗ t_{1} + y^{2}C + 8U + 8C_{2} mod (16, 2^{n})

for some C_{2}, C = v_{2}^{−1}t_{1} ⊗ t_{1} ⊗ t_{1} + … and U in the proof of Lemma 2.5. Note that d(C) ≡ 0 mod (8, v_{1}^{2^{n}}) and d(C_{2}) ≡ 0 mod (2, v_{1}^{4}). Then we may put d(C_{2}) ≡ v_{1}^{4}C_{3} mod (2) for some C_{3}. The cochain C_{3} does not represents g since g/v_{1} is a generator of H^{4}M_{1}^{1}. Since we see that d(C) ≡ 8G + 8C_{3}, we replace the representative G by G + C_{3} to obtain the lemma.

q.e.d.

Corollary 6.10. There exists a cocycle G of Ω^{4,0}M(i+1, 2^{i}s) for any i, s that represents the polynomial generator g.

Lemma 6.11. There is a cochain C′ of Ω^{3,0}M(4, 8) for n > 4 such that d(C′) ≡ 8v_{1}^{7}v_{2}G mod (16, v_{1}^{8}) and C′ represents the element v_{1}^{7}h_{10}^{3} ∈ H^{3,0}M(1, 8).

Proof. By Lemmas 2.11 and 6.8, we compute

d(v_{1}^{7}v_{2}C) ≡ −2v_{1}^{6}r ⊗ C + 8v_{1}^{7}v_{2}G mod (16, v_{1}^{8})
≡ −2v_{1}^{2}r ⊗ (d(H) − 4v_{1}t_{1} ⊗ t_{1} ⊗ t_{1}) + 8v_{1}^{7}v_{2}G mod (16, v_{1}^{8}) by (6.9)
d(2v_{1}^{2}r ⊗ H) ≡ 2v_{1}^{2}r ⊗ d(H), since d(v_{1}^{2}r) ≡ 0 mod (8) by Lemma 3.2.

Since we see that v_{1}^{3}r ⊗ t_{1} ⊗ t_{1} ⊗ t_{1} is homologous to v_{1}^{6}ζ̃h_{0} mod (2, v_{1}^{8}), we have desired cocycle.

Lemma 6.12. In the cobar complex Ω^{2}M(n + 2, 2^{n+1}), there exists a cochain z_{n,s} for n > 3 and odd s such that z_{n,s} ≡ v_{1}^{2^{n+1}−4}v_{2}^{2^{n}s}z̃ mod (2, v_{1}^{2^{n+1}}) and

d(z_{n,s}) = 2^{n+1}v_{1}^{2^{n+1}−4}z̃ ⊗ v_{2}^{2^{n}s}Z + 2^{n+1}v_{1}^{2^{n+1}−2}X′,

for some X′.

Proof. Consider v_{1}^{2^{n+1}}η_{R}(v_{2}^{2^{n}s})r_{n+1} for r_{n+1} given in Lemma 4.2. We notice that d(v_{2}^{2^{n}s}) ≡ 0 mod (2^{n+5+ν(i)−2i}, v_{1}^{2i}) for i = 2^{ν(i)}i′ > 4 with i′ odd, since d(v_{2}^{2^{n}s}) ≡ 0 mod (2^{n−k}, v_{1}^{2^{k}}) and (2^{n−k}, v_{1}^{2^{k}}) ⊂ (2^{n−k}, v_{1}^{2i}) for k = 2i − ν(i) − 5 if i > 4. Then we compute

d(v_{1}^{2^{n+1}}η_{R}(v_{2}^{2^{n}s})r_{n+1}) = v_{1}^{2^{n+1}}r_{n+1} ⊗ d(v_{2}^{2^{n}s})
= v_{1}^{2^{n+1}} (z̃/v_{1}^{4} + ∑_{i>2} \binom{2^{n−1}}{i}2^{2i−n−2}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{−2i}

+ ∑_{i=1}^{2^{n}−4} \binom{2^{n} − 4}{i}2^{2i−1}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{−2i−4}η_{R}(v_{1}v_{2})) ⊗ d(v_{2}^{2^{n}s})

= v_{1}^{2^{n+1}}(v_{1}^{−4}z̃ + 8(v_{1}t_{1} + t_{1}^{2})^{3}v_{1}^{−6} + 8(v_{1}t_{1} + t_{1}^{2})^{4}v_{1}^{−8} + (v_{1}t_{1} + t_{1}^{2})v_{1}^{−6}η_{R}(v_{1}v_{2})) ⊗ d(v_{2}^{2^{n}s})
= v_{1}^{2^{n+1}−4}z̃ ⊗ d(v_{2}^{2^{n}s}) + 2^{n+1}v_{1}^{2^{n+1}}(v_{1}^{−1}t_{1}^{5} + v_{1}^{−2}t_{1}^{6} + v_{1}^{−4}t_{1}^{8}) ⊗ v_{2}^{2^{n}s−4}t_{1}^{8}
= v_{1}^{2^{n+1}−4}z̃ ⊗ (2^{n−1}v_{2}^{2^{n}s−2}(v_{1}^{2}(3r − z̃) + 4v_{2}^{2}Z) − 2^{n+1}v_{1}^{2}v_{2}^{2^{n}s−2}(3r − z̃)Z)

+2^{n+1}v_{1}^{2^{n+1}}(v_{1}^{−1}t_{1}^{5} + v_{1}^{−2}t_{1}^{6} + v_{1}^{−4}t_{1}^{8}) ⊗ v_{2}^{2^{n}s−4}t_{1}^{8}

= 2^{n−1}v_{1}^{2^{n+1}−4}z̃ ⊗ v_{2}^{2^{n}s−2}(v_{1}^{2}(3r − z̃) + 4v_{2}^{2}Z) − 2^{n+1}v_{1}^{2^{n+1}−2}z̃ ⊗ v_{2}^{2^{n}s−2}(3r − z̃)Z)

+2^{n+1}v_{1}^{2^{n+1}}(v_{1}^{−1}t_{1}^{5} + v_{1}^{−2}t_{1}^{6} + v_{1}^{−4}t_{1}^{8}) ⊗ v_{2}^{2^{n}s−4}t_{1}^{8}

We also compute

d(2^{n}v_{1}^{2^{n+1}−4}η_{R}(v_{2}^{2^{n}s−4})t_{1}^{16}) = 2^{n+1}v_{1}^{2^{n+1}−4}t_{1}^{8} ⊗ v_{2}^{2^{n}s−4}t_{1}^{8} + 2^{n+1}v_{1}^{2^{n+1}−2}t_{1}^{16} ⊗ v_{2}^{2^{n}s−6}t_{1}^{4}.

Next consider Y = −3z̃η_{R}(v_{1}^{15}v_{2}) + v_{1}^{8}zz for the cochain zz of Lemma 2.12, and we compute

d(Y) = 2v_{1}^{14}z̃ ⊗ (3r − z̃) + 8v_{1}^{14}X,

for some X by Lemmas 2.11 and 2.12, which equals

d(Y) = 2v_{1}^{12}z̃ ⊗ v_{1}^{2}(3r − z̃) + 8v_{1}^{12}(v_{1}t_{1} + t_{1}^{2})z̃ ⊗ (3r − z̃) + 8v_{1}^{14}X.

Put zz_{n,s} = v_{1}^{2^{n+1}}η_{R}(v_{2}^{2^{n}s})r_{n+1} + 2^{n}v_{1}^{2^{n+1}−4}η_{R}(v_{2}^{2^{n}s−4})t_{1}^{16} − 2^{n−2}v_{1}^{2^{n+1}−16}v_{2}^{2^{n}s−2}Y and we obtain the lemma. q.e.d.

Lemma 6.13. Suppose n > 4. Then there is a cocycle z̃z_{n} of Ω^{2}M(n + 1, 2^{n}) such that z̃z_{n} ≡ v_{1}^{2^{n}−4}z̃⊗ z^{4} + v_{1}^{2^{n}−2}Y_{1} + 2v_{1}^{2^{n}−4}W + 2Y_{2} mod (4, v_{1}^{2^{n}}) for the cochain W in the proof of Lemma 2.5 and cocycles Y_{i} (i = 1, 2) such that Y_{1} is a linear combination of h_{10}^{2}, 2v_{1}h_{10}ρ_{2} and h_{11}ρ_{2}.

Proof. Lemma 6.12 implies that c = d(zz_{2n,1})/2^{2n+1} yields a cocycle of Ω^{2}M(n+1, 2^{2n+1}) whose leading

term is v_{1}^{2^{2n+1}−4}z̃ ⊗ v_{2}^{2^{2n}}Z. Consider the exact sequence M(n + 1, 2^{n}) ⟶^{v{2^{2n+1}−2^{n} \atop 1}} M(n + 1, 2^{2n+1}) ⟶ M(n + 1, 2^{2n+1} − 2^{n}). Then by the definition of zz_{2n,1} we see that c is pull back to c′ of M(2^{n+1}, 2^{n}) whose leading term is v_{1}^{2^{n}−4}z̃ ⊗ v_{2}^{2^{2n}}Z. In fact, in Ω^{2}M(3n + 3, 2^{2n+1} − 2^{n}), we compute

d(v_{1}^{2^{2n+1}}η_{R}(v_{2}^{2^{2n}})r_{2n+1})

= v_{1}^{2^{2n+1}} ( ∑_{i>2^{n−1}} \binom{2^{2n−1}}{i}2^{2i−2n−2}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{−2i} + ∑_{i=2^{n−1}−1}^{2^{2n}−4} \binom{2^{2n−4}}{i}2^{2i−1}(v_{1}t_{1} + t_{1}^{2})^{i}v_{1}^{−2i−4}η_{R}(v_{1}v_{2})) .
= 0

if n > 5, since 2^{n} − 3 > 3n + 3. Then there is an isomorphism v_{2}^{2^{2n}}: M(n + 1, 2^{n}) → M(n + 1, 2^{n}) of Γ-comodules and we obtain a cocycle z̃z_{n}′ such that

z̃z_{n}′ ≡ v_{1}^{2^{n}−4}z̃ ⊗ Z + v_{1}^{2^{n}−2}X′′ mod (2, v_{1}^{2^{n}})

for some X′′. Since z̃ ⊗ Z is homologous to z̃ ⊗ z^{2^{5}} mod (2, v_{1}^{2}), we replace it by z̃z_{n} such that

z̃z_{n} ≡ v_{1}^{2^{n}−4}z̃ ⊗ z^{2^{5}} + v_{1}^{2^{n}−2}Y_{1} + 2v_{1}^{2^{n}−4}W + 2Y_{2} mod (8, v_{1}^{2^{n}})

for some Y_{1} and Y_{2}. Then we have v_{1}^{2^{n}−2}d(Y_{1}) ≡ 2d(Y_{2}) mod (4, v_{1}^{2^{n}}), and Y_{1} represents an element of H^{2,4}M(1, 2) = Z/2{h_{10}^{2}, v_{1}h_{10}ρ_{2}, h_{11}ρ_{2}, v_{1}v_{2}^{−1}h_{11}^{2}}. Therefore, we see that d(Y_{2}) ≡ kt_{1} ⊗ t_{1} ⊗ R for some k ∈ Z/2. Since h_{10}^{2}ρ_{2}/v_{1} is not zero in H^{1}M_{1}^{1}, we see that k = 0. Thus Y_{1} represents k_{1}h_{10}^{2} + 2k_{2}v_{1}h_{10}ρ_{2} + k_{3}h_{11}ρ_{2} + k_{4}v_{1}v_{2}^{−1}h_{11}^{2} of H^{2,4}M(2, 2) for some k_{i} ∈ Z/2. Then we have d(Y_{2}) ≡ 2k_{4}v_{1}^{2^{n}−1}t_{1} ⊗t_{1} ⊗t_{1} by Lemma 6.7. If k_{4} ≢ 0, then v_{1}^{2^{n}−1}t_{1} ⊗t_{1} ⊗t_{1} represents an element of H^{3}M(3, 2^{n}). Consider the composite

H^{3}M(3, 2^{n})^{1/8v_{1}^{2^{n}}}w H^{3}M_{0}^{2} ^{δ} w H^{4}M_{1}^{1},

where δ is the connecting homomorphism associated to the short exact sequence (1.3), which sends v_{1}^{2^{n}−1}t_{1} ⊗ t_{1} ⊗ t_{1} to v_{2}g/v_{1} by Lemma 6.11. On the other hand, h_{10}^{3}/8v_{1} = [t_{1} ⊗ t_{1} ⊗ t_{1}/8v_{1}] = [d(Y_{1})/16v_{1}^{2^{n}}] = 0, which contradict that v_{2}g/v_{1} is a generator. Therefore, k_{4} = 0. q.e.d.

Proposition 6.14. There is an element β ∈ H^{3,0}M(n, 2^{n}) for each n > 4 which goes to β ∈ H^{3,0}M(1, 2^{n}) by the projection.

Proof. Consider a commutative diagram with exact sequences

H^{3,0}M(n, 2^{n})  H^{3,0}M(n + 1, 2^{n})  H^{3,0}M(1, 2^{n}) H^{2,2^{n+1}}M(n + 1, 2^{n}) H^{3,0}M(1, 2^{n})  H^{3,0}M(2, 2^{n})  H^{3,0}M(1, 2^{n}) H^{2,2^{n+1}}M(2, 2^{n})^{2}  w_{_{pr} u}  _{u_{pr}}  ^{pr} w ^{’ ’ ’ ’’)}_{δ}_{_{pr} u}  _{^{2}  w}  _{^{pr} w} ^{’ ’ ’ ’’)}_{δ}

where δ denotes the connecting homomorphism associated to the short exact sequence 0 → M(i, 2^{n}) →^{v{2^{n} \atop 1}} M(i, 2^{2n}) → M(i, 2^{n}) → 0. We denote an element represented by z̃z_{n} by ζ̃ζ ∈ H^{2,2^{n+1}}M(n + 1, 2^{n}). By Lemma 6.13, we see that pr(ζ̃ζ) = v_{1}^{2^{n}−4}ζ̃ζ +v_{1}^{2^{n}−2}ξ_{1} +2ξ_{2} for some ξ_{i} such that ξ_{1} is a linear combination of h_{10}^{2}, 2v_{1}h_{10}ρ_{2} and h_{11}ρ_{2}. Since δ sends ξ_{1} to zero, δ(pr(ζ̃ζ)) = 2β + 2δ(ξ_{2}) and so prδ(pr(ζ̃ζ)) = 0. Therefore, we see that pr(˜2β) = 0 for ˜2β = δ(ζ̃ζ) ∈ H^{3,0}M(n + 1, 2^{n}) such that pr(˜2β) = 2β + 2δ(ξ_{2}). Thus we obtain β ∈ H^{3,0}M(n, 2^{n}) such that 2β = ˜2β as desired. q.e.d.

Lemma 6.15. Let B denote a cochain that represents β of the above lemma. Then there is a cochain W such that d(W) ≡ z̃ ⊗ ZZ_{5} + v_{1}^{4}B + 4v_{1}W_{1} + 2v_{1}^{2}W_{2} mod (8, v_{1}^{8}) for some cochains W_{i}, where W_{1} represents a linear combination of h_{10}^{3}, v_{2}ζ^{2}h_{0} and v_{2}ζh_{0}ρ_{2}.

Proof. As in the proof of Lemma 2.5, we have a cochain W such that

(6.16)  d(W) = z̃ ⊗ ZZ_{4} + v_{1}^{4}B + 2X_{1} + 2v_{1}^{2}X_{2}

for some cochains X_{1} and X_{2}. Then we have 0 ≡ 2d(X_{1}) + 2v_{1}^{2}d(X_{2}) mod (8, v_{1}^{2^{n}}) under the differential d, which shows that X_{1} + v_{1}^{2}X_{2} represents an element of H^{3,8}M(2, 4). We read off that H^{3,8}M(1, 4) is the Z/2-module generated by h_{11}^{2}ρ_{2}, v_{1}h_{10}^{3}, ζ̃h_{0}^{2}, ζ̃ζh_{0}, ζ̃h_{0}ρ_{2}, ζ̃ζρ_{2}, v_{1}^{2}h_{10}^{2}ρ_{2}, and v_{1}^{3}v_{2}^{−1}h_{11}^{2}ρ_{2} from the structure (5.1) of H^{∗}K(2)_{∗}. (Here v_{1}v_{2}β is taken away since δ(v_{1}v_{2}β) = v_{1}^{2}h_{11}β.) Therefore, we see that 2X_{1} represents a linear combination of v_{1}h_{10}^{3}, 4ζ̃ζh_{0}, 4ζ̃h_{0}ρ_{2} and 2ζ̃ζρ_{2}, and 2v_{1}^{2}X_{2} is a multiple of v_{1}^{3}v_{2}^{−1}h_{11}^{2}ρ_{2}. If we consider the equation (6.16) mod (8, v_{1}^{8}), then we see that 2X_{1} represents a linear combination of 4v_{1}h_{10}^{3} 4ζ̃ζh_{0} and 4ζ̃h_{0}ρ_{2}, since δ sends those generators to zero except for δ(2ζ̃ζρ_{2}) = v_{1}^{4}βρ_{2}. Here δ denotes the connecting homomorphis associated to the short exact sequence 0 → M(1, 8) →^{4} M(3, 8) → M(2, 8) → 0. Now put v_{1}W_{1} = X_{1} and W_{2} = X_{2}, and we see the lemma. q.e.d.

7. The connecting homomorphism on J

Proposition 7.1. For the connecting homomorphism δ : H^{∗}M_{0}^{2} → H^{∗+1}M_{1}^{1}, δ(ζh_{0}^{i}/2v_{1}^{j}) = ζ^{2}h_{0}^{i}/v_{1}^{j} + (i + j)ζh_{0}^{i+1} for each i ≥ 0 and j > 0.

Proof. Put t_{1}^{⊗i} = t_{1} ⊗ ⋯ ⊗ t_{1} ∈ Ω^{i}A. Then z ⊗ t_{1}^{⊗i}/v_{1}^{i+j} represents ζh_{0}^{i}/v_{1}^{j} of H^{i+1}M_{1}^{1}. Take an integer n to be 2^{n−1} > i + j, and we compute d(z^{2^{n}} ⊗ t_{1}^{⊗i}/4v_{1}^{i+j}) ≡ z^{2^{n−1}} ⊗ z^{2^{n−1}} ⊗ t_{1}^{⊗i}/2v_{1}^{i+j} + (i + j)z^{2^{n}} ⊗ t_{1}^{⊗(i+1)}/2v_{1}^{i+j+1} mod (4). q.e.d.

Put J̃ = Z/2{ζ/2v_{1}^{j} | j > 0}⨂Z/2[h_{0}], and the above lemma implies Lemma 7.2. We have the exact sequence

0  w JC ^{1/2} w J̃ ^{2} w J̃ ^{δ} w JI  w 0.

8. The connecting homomorphism on K

Consider the submodules KC and KI:

KC = Z/2{h_{10}^{3}/v_{1}, v_{2}^{−1}h_{11}^{2}/v_{1}}⨂Λ(β)⨂Z/2[g]⨁Z/2{v_{2}/v_{1}, ζζ̃/v_{1}^{2}}
KI = Z/2{v_{2}g/v_{1}, h_{10}^{3}/v_{1}^{3}}⨂Λ(β)⨂Z/2[g]⨁Z/2{ζ̃/v_{1}^{2}, v_{2}β/v_{1}}.

Proposition 8.1. For the generators of KC, we have

\begin{aligned}1. & δ(h_{10}^{3}/8v_{1}) & = & v_{2}g/v_{1} \\ 2. & δ(v_{2}^{−1}h_{11}^{2}/4v_{1}) & = & h_{10}^{3}/v_{1}^{3} \\ 3. & δ(h_{10}^{3}β/8v_{1}) & = & v_{2}βg/v_{1} + kh_{10}^{3}βρ_{2}/v_{1} for some k ∈ Z/2 \\ 4. & δ(v_{2}^{−1}h_{11}^{2}β/4v_{1}) & = & h_{10}^{3}β/v_{1}^{3} \\ 5. & δ(v_{2}/2v_{1}) & = & v_{2}ζ/v_{1} = ζ̃/v_{1}^{2} \\ 6. & δ(ζζ̃/8v_{1}^{2}) & = & v_{2}β/v_{1} \end{aligned}

Proof. We prove them one by one.

1. Lemma 6.11 shows this.

2. This follows from Lemma 6.7.

3. By Proposition 6.14, δ(ξβ) = δ(ξ)β for any ξ ∈ H^{∗}M_{0}^{2}. Therefore, this follows from the first one.

4. This also follows from Proposition 6.14 and the second one.

5. This is shown in [3].

6. Note that r_{3} ≡ v_{1}^{−4}z̃ mod (8) and d(r_{3}) ≡ 0 mod (16) by Lemma 4.2. Then we compute

d(yr_{3} ⊗ z^{2^{6}}/16) = r ⊗ z̃ ⊗ z^{2^{6}}/2v_{1}^{6} + yz̃ ⊗ ZZ_{5}/8.

for y of Lemma 3.2 and ZZ_{5} of Lemma 2.5. We see that the first term is bounded by computaiton:

d(x_{1}z̃ ⊗ z^{2^{5}}/2v_{1}^{8}) = (r + \underline{z̃}_{1}) ⊗ z̃ ⊗ z^{2^{5}}/2v_{1}^{6}
d(z^{2} ⊗ z^{2^{5}}/4v_{1}^{6}) = \underline{z̃ ⊗ z̃ ⊗ z^{2^{5}}/2v_{1}^{6}}_{1} + \underline{z^{2} ⊗ z^{2^{4}} ⊗ z^{2^{4}}/2v_{1}^{6}}_{2}
d(W^{2}/2v_{1}^{6}) = \underline{z^{2} ⊗ z^{2^{4}} ⊗ z^{2^{4}}/2v_{1}^{6}}_{2}.

On the other hand, by Lemma 6.15, we see that

d(yW/8v_{1}^{4}) = yz̃ ⊗ ZZ_{5}/8v_{1}^{4} + v_{2}B/2v_{1} + W_{1}/2v_{1}

as desired. q.e.d.

Note that Corollary 6.10 implies the following Lemma 8.2. If δ(ξ/2^{i}v_{1}^{j}) = χ/v_{1}^{k} for i ≤ 3, then δ(ξg^{l}/2^{i}v_{1}^{j}) = χg^{l}/v_{1}^{k} for l ≥ 0.

Put

K̃ = (Z/8{ζζ̃/8v_{1}^{2}, h_{10}^{3}/8v_{1}, h_{10}^{3}β/8v_{1}}⨁Z/4{v_{2}^{−1}h_{11}^{2}/4v_{1}, v_{2}^{−1}h_{11}^{2}β/4v_{1}}⨁Z/2{v_{2}/2v_{1}}) ⨂Z[g]. Then the above lemmas imply immediately the following Lemma 8.3. We have the exact sequence

0  w KC ^{1/2} w K̃ ^{2} w K̃  w KI  w 0.

9. The connecting homomorphism on E

Consider the submodules EC and EI:

EC = Z/2{v_{2}^{4s}/v_{1}^{2j}, v_{2}^{8t+6}ζ̃/v_{1}^{2j′} | s, t ∈ Z, 2|/s, j = 1, 2, 3, j′ = 1, 2}
EI = Z/2{v_{2}^{4s}ζ̃/v_{1}^{6}, v_{1}^{8t+2}ζ̃/v_{1}^{2j′}, v_{2}^{4s+2}ζζ̃/v_{1}^{2j′} | s, t ∈ Z, 2|/s, j′ = 1, 2}.

Then E = EI⨁EC. Proposition 9.1. For the generators of EC, we have

\begin{aligned}1. & δ(v_{2}^{4s}/8v_{1}^{2}) & = & v_{2}^{4s}ζ̃/v_{1}^{6} \\ 2. & δ(v_{2}^{4s}/2v_{1}^{2j}) & = & v_{2}^{4s−2}ζ̃/v_{1}^{2j−2}  (j = 2, 3) \\ 3. & δ(v_{2}^{8t+6}ζ̃/4v_{1}^{2j}) & = & v_{2}^{8t+6}ζζ̃/v_{1}^{2j}  (j = 1, 2) \end{aligned}

Proof. The first two equations are shown in [3, Lemma 3.17].

For the third, we compute by Lemma 2.8,

(9.2)  \begin{aligned}d(v_{2}^{8t+6}z̃/8v_{1}^{2j}) & ≡ & d(v_{1}^{4−2j}v_{2}^{8t+6}z̃/8v_{1}^{4}) \\ & ≡ & j(t_{1}^{2} + v_{1}t_{1}) ⊗ v_{2}^{8t+6}z̃/2v_{1}^{2j+2} − v_{2}^{8t+4}(r + z̃) ⊗ z̃/8v_{1}^{2j−2} \\ & & +v_{2}^{8t+6}Z ⊗ z̃/2v_{1}^{2j}. \end{aligned}

If j = 1, then the first term is v_{2}^{8t+5}t_{1}^{4} ⊗ v_{2}z/2v_{1}^{2}, since

d(v_{2}^{8t+7}z̃/2v_{1}^{5}) ≡ (v_{1}t_{1}^{2} + v_{1}^{2}t_{1}) ⊗ v_{2}^{8t+6}z̃/2v_{1}^{5} + v_{2}^{8t+5}t_{1}^{4} ⊗ z̃/2v_{1}^{3}.

Suppose that j = 2. Then d(v_{2}^{8t+4}(v_{1}^{15}v_{2})z̃/16v_{1}^{16}) = v_{2}^{8t+4}Z ⊗v_{1}v_{2}z̃/2v_{1}^{2} +v_{2}^{8t+4}r ⊗z̃/8v_{1}^{2} by Lemma 2.11 and d(v_{2}^{8t+4}z^{2}/16y) = r ⊗ v_{2}^{8t+4}z^{2}/2v_{1}^{6} + (v_{1}^{−1}v_{2})(r + z̃) ⊗ z^{2}/4v_{1}^{2} + v_{2}^{8t+4}Z ⊗ z^{2}/2v_{1}^{2} − v_{2}^{8t+4}z̃ ⊗ z̃/8v_{1}^{2} + v_{2}^{8t+5}z̃ ⊗ z̃/2v_{1}^{5}. Notice that z^{2}/4v_{1}^{2} = 0. These imply that the second term of (9.2) is homologous to an element of the form x/2v_{1}^{3} with x ∈ Γ⨂_{A}Γ. q.e.d.

Now put

Ẽ = Z/8{v_{2}^{4s}/8v_{1}^{2}, v_{2}^{4s}/2v_{1}^{2j}, v_{2}^{8t+6}ζ̃/4v_{1}^{2j′} | s, t ∈ Z, 2|/s, j = 2, 3, j′ = 1, 2},

and the above lemma implies Lemma 9.3. We have the exact sequence

0  w EC ^{1/2} w Ẽ ^{2} w Ẽ ^{δ} w EI  w 0.

10. The connecting homomorphism on F

Consider the submodules F C, F I and F̃:

F C = ∑_{n≥2,2|/s} v_{2}^{2^{n}s} (Z/2{v_{2}^{−1}h_{11}^{2}/v_{1}}⨂Λ(β)⨁Z/2{h_{10}^{3}/v_{1}, ζ̃^{(n−1)}ζ̃g/v_{1}^{2^{n+1}+2}}) ⨂Z/2[g]
F I = ∑_{n≥2,2|/s} v_{2}^{2^{n}s} (Z/2{v_{2}g/v_{1}}⨂Λ(β)⨁Z/2{h_{10}^{3}/v_{1}^{3}, ζ̃^{(n−1)}ζ̃g/v_{1}^{2^{n+1}+4}}) ⨂Z/2[g]
F̃ = ∑_{n≥2,2|/s} v_{2}^{2^{n}s} (Z{v_{2}^{−1}h_{11}^{2}/4v_{1}, v_{2}^{−1}h_{11}^{2}β/8v_{1}, h_{10}^{3}/8v_{1}, ζ̃^{(n−1)}ζ̃g/4v_{1}^{2^{n+1}+2}}) ⨂Z/2[g]

Lemma 10.1. We have the exact sequence

0  w F C ^{1/2} w F̃ ^{2} w F̃ ^{δ} w F I  w 0.

This follows from Lemma 8.2 and

Proposition 10.2. The connecting homomorphism δ acts on the generators of F̃ as follows :

\begin{aligned}1. & δ(v_{2}^{2^{n}s−1}h_{11}^{2}/4v_{1}) & = & v_{2}^{2^{n}s}h_{10}^{3}/v_{1}^{3} \\ 2. & δ(v_{2}^{2^{n}s−1}h_{11}^{2}β/8v_{1}) & = & v_{2}^{2^{n}s}ζ̃^{(n−1)}ζ̃g/v_{1}^{2^{n+1}+4} \\ 3. & δ(v_{2}^{2^{n}s}h_{10}^{3}/8v_{1}) & = & v_{2}^{2^{n}s+1}g/v_{1} \\ 4. & δ(v_{2}^{2^{n}s}ζ̃^{(n−1)}ζ̃g/4v_{1}^{2^{n+1}+2}) & = & v_{2}^{2^{n}s+1}βg/v_{1} \end{aligned}

Proof. We prove them one by one.

1. This follows from the second one of Proposition 8.1.

2. Suppose that n > 4. Then d(v_{2}^{2^{n}}) ≡ 0 mod (16, y^{2}). For the cochain H of Lemma 6.7 and the

cocycle B of Proposition 6.14, we have

d(v_{2}^{2^{n}s}B ⊗ H/16y^{2}) = v_{2}^{2^{n}s}B ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{3} + v_{2}^{2^{n}s}B ⊗ H_{1}/2v_{1}^{4} + v_{2}^{2^{n}s+1}B ⊗ H_{2}/2v_{1}^{3}. We also have

d(v_{2}^{2^{n}s}W ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{7}) = v_{2}^{2^{n}s}(z̃ ⊗ ZZ_{4} + v_{1}^{4}B + 2v_{1}^{2}W_{2}) ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{7}

for W of Lemma 6.15. Since ZZ_{4} ≡ z^{8} ⊗ z^{8} mod (2, v_{1}^{8}),

(10.3)  \begin{matrix}d(v_{2}^{2^{n}s}z^{2^{n−1}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{2^{n+1}+7}) \\ = & v_{1}^{2^{n}}v_{2}^{2^{n}(s−1)}z^{2^{n−1}} ⊗ z^{2^{n−1}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{2^{n+1}+7} \\ +2t_{1} ⊗ v_{2}^{2^{n}s}z^{2^{n−1}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{2^{n+1}+8} \\ +2v_{1}^{2^{n−1}}v_{2}^{2^{n−1}(2s−1)}z^{2^{n−2}} ⊗ z^{2^{n−1}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{2^{n+1}+7} \\ +2v_{2}^{2^{n}s}z^{2^{n−2}} ⊗ z^{2^{n−2}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{2^{n+1}+7} \\ = & v_{2}^{2^{n}s}z^{2^{n−1}} ⊗ z^{2^{n−1}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{7} \\ +t_{1} ⊗ v_{2}^{2^{n}s}z^{2^{n−1}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/2v_{1}^{2^{n+1}+8} \\ +v_{1}^{2^{n−1}}v_{2}^{2^{n−1}(2s−1)}z^{2^{n−2}} ⊗ z^{2^{n−1}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/2v_{1}^{2^{n+1}+7} \\ +v_{2}^{2^{n}s}z^{2^{n−2}} ⊗ z^{2^{n−2}} ⊗ z̃ ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/2v_{1}^{2^{n+1}+7}. \end{matrix}

This shows that v_{2}^{2^{n}s}B ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{3} is homologous to v_{2}^{2^{n}s}z^{2^{n−1}} ⊗ z̃⊗ t_{1} ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/2v_{1}^{2^{n+1}+8}, as desired.

For n = 4,

d(v_{2}^{16s}B ⊗ H/16y^{2})

= v_{2}^{16s−2}(r + z̃) ⊗ B ⊗ H/2v_{1}^{2} + v_{2}^{16s}B ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{3} + v_{2}^{16s}B ⊗ H_{1}/2v_{1}^{4} + v_{2}^{16s+1}B ⊗ H_{2}/2v_{1}^{3}

and we obtain the same result. For n = 3,

d(v_{2}^{8s}B ⊗ H/16y^{2})
= v_{2}^{8s−2}(r + z̃) ⊗ B ⊗ H/4v_{1}^{2} + v_{2}^{8s−1}d(v_{2}) ⊗ B ⊗ H/2v_{1}^{4}
+v_{2}^{8s}B ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{3} + v_{2}^{8s}B ⊗ H_{1}/2v_{1}^{4} + v_{2}^{8s+1}B ⊗ H_{2}/2v_{1}^{3}

and H ≡ 0 mod (4, v_{1}^{2}), which implies the same result as the above one. Similarly, for n = 2,

d(v_{2}^{4s}B ⊗ H/16y^{2})
= v_{2}^{4s−2}(r + z̃) ⊗ B ⊗ H/8v_{1}^{2} + v_{2}^{4s−1}d(v_{2}) ⊗ B ⊗ H/4v_{1}^{4}
+v_{2}^{4s}B ⊗ t_{1} ⊗ t_{1} ⊗ t_{1}/4v_{1}^{3} + v_{2}^{4s}B ⊗ H_{1}/2v_{1}^{4} + v_{2}^{4s+1}B ⊗ H_{2}/2v_{1}^{3}

and the congruence H ≡ 0 mod (4, v_{1}^{2}) also shows the result.

3. This follows from the first one of Proposition 8.1.

4. The cocycle x/2 that represents v_{2}^{2^{n}s+2^{n−1}}ζζ̃g/2v_{1}^{2^{n+1}+2} is homologous to v_{2}^{2^{n}s}t_{1} ⊗ t_{1} ⊗ t_{1} ⊗ B/4v_{1} by (10.3). The equation 3 above indicates the existence of a cochain c such that d(c) = v_{2}^{2^{n}s+1}g/2v_{1} and the leading term of c is v_{2}^{2^{n}s}t_{1} ⊗ t_{1} ⊗ t_{1}/16v_{1}. Then d(x/8) = d(c ⊗ B) = v_{2}^{2^{n}s+1}g ⊗ B/2v_{1} as desired. q.e.d.

11. The connecting homomorphism on D

Consider the submodules DC and DI:

DC = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}Z/2{v_{2}/v_{1}, ζ̃/v_{1}^{4}, v_{2}^{2^{n−1}}ζζ̃/v_{1}^{2}}⨁ ∑_{n≥2,4|(s−1)} v_{2}^{2^{n}s}Z/2{ζ̃^{(n−1)}ζ̃/v_{1}^{2^{n+1}+2}}
DI = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}Z/2{v_{2}β/v_{1}, ζ̃/v_{1}^{2}, v_{2}^{2^{n−1}}ζζ̃/v_{1}^{4}}⨁ ∑_{n≥2,4|(s−1)} v_{2}^{2^{n}s}Z/2{ζ̃^{(n−1)}ζ̃/v_{1}^{2^{n+1}+4}}

Proposition 11.1. The connecting homomorphism δ behaves as follows:

1. δ(v_{2}^{2^{n}s}ζ̃^{(n−1)}ζ̃/4v_{1}^{2^{n+1}+2}) = v_{2}^{2^{n}s+1}β/v_{1} s = 4t + 1
2.  δ(v_{2}^{2^{n−1}(4t+3)}ζζ̃/16v_{1}^{2}) = v_{2}^{2^{n−1}(4t+3)+1}β/v_{1} for n ≥ 3
3.  δ(v_{2}^{2^{n}s+1}/2v_{1}) = v_{2}^{2^{n}s}ζ̃/v_{1}^{2}
4.  δ(v_{2}^{2^{n}s}ζ̃/2^{n+2}v_{1}^{4}) = v_{2}^{2^{n}s+2^{n−1}}z^{2^{n−1}} ζ̃/v_{1}^{3·2^{n−1}+4} s ≡ 1 (4)
δ(v_{2}^{2^{n}s}ζ̃/2^{n+1}v_{1}^{4}) = v_{2}^{2^{n}s}ζζ̃/v_{1}^{4} s ≡ −1 (4)

Proof.

1. First we note that v_{2}^{2^{n}(4t+1)}z^{2^{n−1}} ⊗ z̃/2v_{1}^{2^{n+1}+2} is homologous to v_{2}^{2^{n+2}t+2^{n}}z^{2^{n}} ⊗ z̃/4v_{1}^{2} by

d(v_{2}^{2^{n+1}(2t+1)}z̃/4v_{1}^{3·2^{n}+2}) = v_{2}^{2^{n}(4t+1)}z^{2^{n−1}} ⊗ z̃/2v_{1}^{2^{n+1}+2} + v_{2}^{2^{n+2}t}z^{2^{n}} ⊗ z̃/4v_{1}^{2^{n}+2}.

We compute

d(v_{2}^{2^{n}(4t+1)}z^{2^{n−1}} ⊗ z̃/8v_{1}^{2^{n+1}+2}) = d(v_{2}^{2^{n+2}t+2^{n}}z^{2^{n}} ⊗ z̃/16v_{1}^{2})
= d(v_{2}^{2^{n+2}t+2^{n}}) ⊗ z^{2^{n}} ⊗ z̃/16v_{1}^{2} + v_{2}^{2^{n+2}t+2^{n}}d(z^{2^{n}} ⊗ z̃/16v_{1}^{2})

If n > 3, d(v_{2}^{2^{n}}) ≡ 0 mod (16, v_{1}^{2}). If n = 3, then d(v_{2}^{8}) ≡ 8v_{1}v_{2}^{7}t_{1}^{2} mod (16, v_{1}^{2}). Therefore, this follows from Proposition 8.1.6 if n ≥ 3. For the case n = 2, it also follows from Proposition 8.1.6, since z̃ ≡ 2(v_{2}^{−1}t_{3} + t_{1}t_{2}) mod (4, v_{1}).

2. This is similar to the above one.

3. This is immediate from Proposition 8.1.5.

4. Lemma 6.12 shows

(11.2)  d(z_{n,s}/2^{n+2}v_{1}^{2^{n+1}}) = z̃ ⊗ v_{2}^{2^{n}s}Z/2v_{1}^{4} + X′/2v_{1}^{2}

which implies the case s ≡ −1 mod p.

Since H^{2,3·2^{n+1}}M_{1}^{1} = Z/2{v_{2}^{2^{n}−1}h_{11}^{2}/v_{1}, v_{2}^{2^{n}+2^{n−1}}ζρ_{2}/v_{1}^{3·2^{n−1}}, v_{2}^{2^{n}+2^{n−1}}ζζ̃/v_{1}^{3·2^{n−1}+4}}, the right hand side of (11.2) for s = 1 is bounded, and so we have an element U/2 such that r_{n+1}η_{R}(v_{2}^{2^{n}})/2^{n+2} − v_{2}^{2^{n}−2}Y/16v_{1}^{16} + U/2 is a cocycle. Then by (10.3), we see the case where s = 1. In other words, there is a cochain X = r_{n+2}η_{R}(v_{2}^{2^{n}})/2^{n+3} − v_{2}^{2^{n}−2}Y/32v_{1}^{16} + U/4 + U′/2 for some U′ and d(X) = v_{2}^{2^{n}+2^{n−1}}z ⊗ z̃/v_{1}^{3·2^{n−1}+4} + · · ·. Since we compute that X ⊗ d(v_{2}^{2^{n+2}}) = 0, we have d(v_{2}^{2^{n+2}}X) = v_{2}^{2^{n+2}}d(X), and obtain the case s ≡ 1 mod (4). q.e.d.

Put

\begin{matrix}D̃ = ∑_{n≥2,2|/s} v_{2}^{2^{n}s}Z\{v_{2}/2v_{1}, v_{2}^{2^{n−1}}ζζ̃/16v_{1}^{2}\}⨁ ∑_{n≥2,4|(s−1)} v_{2}^{2^{n}s}Z\{ζ̃/2^{n+2}v_{1}^{4}, ζ̃^{(n−1)}ζ̃/4v_{1}^{2^{n+1}+2}\} \\ ⨁ ∑_{n≥2,4|(s+1)} v_{2}^{2^{n}s}Z\{ζ̃/2^{n+1}v_{1}^{4}\}, \end{matrix}

and we see the following

Lemma 11.3. We have the exact sequence

0  w DC ^{1/2} w D̃ ^{2} w D̃ ^{δ} w DI  w 0.

12. The connecting homomorphism on L

Put L̃ = 1/2(LC) for the map 1/2: H^{∗}M_{1}^{1} ⟶ H^{∗}M_{0}^{2}. We compute the δ-image of each element of L̃. We write M̃ as 1/2(M) for a comodule M. Then L̃ = ∑_{i=0}^{8} L̃C_{i}.

Proposition 12.1. For each element of L̃, we have

δ(ξ) = ξh_{0} + · · · .

Proof. Suppose that ξ is represented by an element x/2v_{1}^{2j+1}. Then we compute d(x/4v_{1}^{2j+1}) = t_{1}/2v_{1}^{2j+2} ⊗ x + x′/2v_{1}^{2j+1} for d(x) = 2x′, since d(v_{1}) = 2t_{1}. The part ⋯ is an element represented by x′/v_{1}^{2j+1}. q.e.d.

Corollary 12.2. The connecting homomorphism induces an isomorphism δ: L̃ → LI.

Proof. Since LI_{i} = h_{0}LC_{i} if i ≠ 1 and = ζ̃LC_{1} if i = 1, the above lemma shows an isomorphism L̃C_{i} → LI_{i} for each i. q.e.d.

Lemma 12.3. We have the exact sequence

0  w LC ^{1/2} w L̃ ^{2} w L̃ ^{δ} w LI  w 0.

Proof. Since each generator of LI is not trivial, the isomorphism δ of Corollary 12.2 implies that each generator of L̃ is not trivial, which shows that the homomorphism 1/2 is a monomorphism. Since 1/2 is an epimorphism by definition, we see that 1/2 is an isomorphism. q.e.d.

13. The connecting homomorphism on M

We begin with factoring M into the direct sum of MC and MI.

Consider the submodules MC and MI:

MC = Z/2{v_{2}^{2}/v_{1}^{2}, v_{2}^{2}/v_{1}, v_{2}^{2}h_{10}/v_{1}, v_{2}^{3}/v_{1}, v_{2}^{3}h_{10}/v_{1}, v_{2}^{3}h_{11}/v_{1}}⨂Λ(β)⨂Z/2[v_{2}^{±4}, g]
MI = Z/2{v_{2}^{2}h_{11}/v_{1}^{2}, v_{2}^{2}h_{10}/v_{1}^{2}, v_{2}h_{11}^{2}/v_{1}, v_{2}^{3}h_{10}/v_{1}^{2}, v_{2}^{3}h_{10}^{2}/v_{1}^{2}, v_{2}^{3}h_{10}^{2}/v_{1}}⨂Λ(β)⨂Z/2[v_{2}^{±4}, g]

Then we have

Proposition 13.1. For the generators of MC, we have

δ(v_{2}^{2}/2v_{1}^{2}) = v_{2}h_{11}/v_{1}
δ(v_{2}^{2}/2v_{1}) = v_{2}^{2}h_{10}/v_{1}^{2}
δ(v_{2}^{2}h_{10}/2v_{1}) = v_{2}^{2}h_{10}^{2}/v_{1}^{2}
δ(v_{2}^{3}/2v_{1}) = v_{2}^{3}h_{10}/v_{1}^{2}
δ(v_{2}^{3}h_{10}/2v_{1}) = v_{2}^{3}h_{10}^{2}/v_{1}^{2}
δ(v_{2}^{3}h_{11}/2v_{1}) = v_{2}^{3}h_{10}^{2}/v_{1}.

Proof. The first, the second and the fourth equations are shown in [3, Lemma 3.17]. The third and the fifth are obtained immediately from the second and the fourth ones, since h_{10} is represented by t_{1} which is primitive. The last one is also follows from the fourth. In fact, v_{2}^{3}h_{10}/v_{1}^{2} is represented by v_{2}r/v_{2} and h_{11} is represented by t_{1}^{2} + v_{1}t_{1}, and v_{2}r ⊗ (t_{1}^{2} + v_{1}t_{1})/v_{2} is homologous to v_{2}^{2}t_{1} ⊗ t_{1}^{4}/v_{1}. q.e.d.

This is displayed as follows:

[figure: 24; 0; 3; 8; v_{2}^{2}/v_{1}^{2}; 10; 16; v_{2}^{3}/v_{1}]

Now put

M̃ = Z/2{v_{2}^{2}/2v_{1}^{2}, v_{2}^{2}/2v_{1}, v_{2}^{2}h_{10}/2v_{1}, v_{2}^{3}/2v_{1}, v_{2}^{3}h_{10}/2v_{1}, v_{2}^{3}h_{11}/2v_{1}}⨂Λ(β)⨂Z/2[v_{2}^{±4}, g], and we see the following lemma by using Proposition 6.14. Lemma 13.2. We have the exact sequence

0  w MC ^{1/2} w M̃ ^{2} w M̃ ^{δ} w MI  w 0.

14. The connecting homomorphism on N

We introduce subsets of the set of triple integers:

T = {(n, i, k) ∈ Z^{3} | n ≥ 2, 0 ≤ i ≤ n − 1, 1 ≤ k ≤ i + 1}
T′ = {(n, i, k) ∈ Z^{3} | n ≥ 2, 0 ≤ i ≤ n − 1, 1 ≤ k ≤ i + 1, (i, k) ≠ (n − 1, 1), (n − 1, n − 1)},
T_{0} = {(n, i, k) ∈ Z^{3} | n ≥ 3, 2 ≤ i ≤ n − 1, 1 ≤ k ≤ i + 1}
T′_{0} = {(n, i, k) ∈ Z^{3} | n ≥ 2, (i, k) = (0, 1), (1, 1), (1, 2)}
T_{0}^{+} = {(n, i, k) ∈ T_{0} | n > i + k + 1}
T_{0}^{−} = {(n, i, k) ∈ T_{0} | n ≤ i + k + 1}

Then,

T = T_{0}⨁T′_{0} = T_{0}^{+}⨁T_{0}^{−}⨁T′_{0}.

Here T ⋂({n} × Z^{2}) for each n is described as follows:

[figure: T_{0}^{−}; T_{0}^{+}; T′_{0}; 1; 2; 0; n − 1; i; n; k]

Consider the modules

A(n, i, k) = Z/2{v_{2}^{2^{n}s}/v_{1}^{2^{k}m} | 2|/sm, 3 · 2^{i−1} < 2^{k}m ≤ 3 · 2^{i}},
Â(n, n − 1, 1) = Z/2{v_{2}^{2^{n}s}/v_{1}^{2m} | 2|/sm, 3 · 2^{n−2} < 2m ≤ 3 · 2^{n−1} − 4},
Z(n, i, k) = Z/2{v_{2}^{2^{i}(2^{n−i}s−1)}ζ̃^{(i−1)}/v_{1}^{2^{k}m} | 2|/sm, 2^{i−1} < 2^{k}m ≤ 2^{i+1}} for i > k,
Ẑ(n, n − 1, 1) = Z/2{v_{2}^{2^{n−1}(2s−1)}ζ̃^{(n−2)}/v_{1}^{2m} | 2|/sm, 2^{n−2} < 2m ≤ 2^{n} − 4}
Z(n, i, i) = Z/2{v_{2}^{2^{n}s−2^{i}} ζ̃^{(i−1)}/v_{1}^{2^{i+1}} | 2 |̸ s},
Z(n, i, i + 1) = Z/2{v_{2}^{2^{n}s−2^{i}} ζ̃^{(i−1)}/v_{1}^{2^{i}} | 2 |̸ s}.

Then N_{i}’s are rewritten as follows:

N_{3} = ∑_{(n,i,k)∈T} A(n, i, k)
NI_{1} = ∑_{(n,i,k)∈T −T_{0}^{−}}(ζ̃/v_{1}^{4})A(n, i, k)
NC_{1} = ∑_{(n,i,k)∈T′_{0}^{−}}(ζ̃/v_{1}^{4})A(n, i, k)
NI_{2} = ∑_{(n,i,k)∈T_{0}^{−}} Z(n, i, k)
NC_{2} = ∑_{(n,i,k)∈T_{0}^{+}} Z(n, i, k)
N_{4} = ∑_{(n,i,k)∈T_{0}}(ζ̃/v_{1}^{4})Z(n, i, k),

in which N_{i} = NI_{i}⨁NC_{i} for i = 1, 2. Now N is divided into two direct summands:

NI = NI_{1}⨁NI_{2}⨁N_{4}
NC = N_{3}⨁NC_{1}⨁NC_{2}.

We introduce modules to define Ñ.

Ã(n, i, k) = \{ \begin{matrix}Z/2\{v_{2}^{2^{n}s}/2^{n−i}v_{1}^{2^{k}m} | 2|/sm, 3 · 2^{i−1} < 2^{k}m ≤ 3 · 2^{i}\}, & n − i < k + 1 \\ Z/2\{v_{2}^{2^{n}s}/2^{k+2}v_{1}^{2^{k}m} | 2|/sm, 3 · 2^{i−1} < 2^{k}m ≤ 3 · 2^{i}\}, & n − i > k \end{matrix}
˜Â(n, n − 1, 1) = Z/2{v_{2}^{2^{n}s}/4v_{1}^{2m} | 2|/sm, 3 · 2^{n−2} < 2m ≤ 3 · 2^{n−1}}
Z̃(n, i, k) = Z/2{v_{2}^{2^{n}s−2^{i}} ζ̃^{(i−1)}/2^{k+2}v_{1}^{2^{k}m} | 2|/sm, 2^{i−1} < 2^{k}m ≤ 2^{i+1}}
Z̃(n, i, i) = Z/2{v_{2}^{2^{n}s−2^{i}} ζ̃^{(i−1)}/2^{i+2}v_{1}^{2^{i+1}} | 2 |̸ s},
Z̃(n, i, i + 1) = Z/2{v_{2}^{2^{n}s−2^{i}} ζ̃^{(i−1)}/2^{i+3}v_{1}^{2^{i}} | 2 |̸ s}.

The module Ñ is then

Ñ = ∑_{(n,i,k)∈T} Ã(n, i, k)⨁ ∑_{(n,i,k)∈T′_{0}^{−}}(ζ̃/v_{1}^{4})Ã(n, i, k)⨁ ∑_{(n,i,k)∈T_{0}^{+}} Z̃(n, i, k).

We read off the following from [3, Lemma 3.17]. Proposition 14.1. The connecting homomorphism δ behaves as follows:

δ(Ã(n, i, k)) = \{ \begin{matrix}(ζ̃/v_{1}^{4})A(n, i, k) & n > i + k + 1 \\ Z(n, i, k) & n ≤ i + k + 1 \end{matrix}
δ((ζ̃/v_{1}^{4})Ã(n, i, k)) = (ζ̃/v_{1}^{4})Z(n, i, k) for n ≤ i + k + 1
δ(Z̃(n, i, k)) = (ζ̃/v_{1}^{4})Z(n, i, k) for n > i + k + 1 .

Proof. The first equation is shown in [3, Lemma 3.17], and the second one is verified by Proposition 4.3.

For the third, note that

d(x_{2}^{2^{n}s}) ≡ 2^{n−i}sv_{1}^{2^{i}}v_{2}^{2^{n}s−2^{i}} ζ̃^{2^{i−1}} + ⋯ mod (v_{1}^{3·2^{i}})

obtained from Lemma 3.3, where ⋯ denotes an element divisible by 2^{n−i+1}. If we write 2^{n−i}sx(n, i, k) as the write hand side, then d(x(n, i, k)) ≡ 0 mod (v_{1}^{3·2^{i}}). Then if k < i, we compute

d(v_{2}^{2^{n}s−2^{i}} ζ̃^{(i−1)}/2^{k+3}v_{1}^{2^{k}m}) = d(x(n, i, k)/2^{k+3}v_{1}^{2^{k}m+2^{i}})
= r ⊗ x(n, i, k)/2v_{1}^{2^{k}m+2^{i}+4}

which homologous to z̃ ⊗ x(n, i, k)/2v_{1}^{2^{k}m+2^{i}+4}. In the same way we see the cases for k = i and i + 1. In fact, the exponent 2^{k}m + 2^{i} of v_{1} of the denominator is divisible by 2^{i+1} (resp. 2^{i}) if k = i (resp. k = i + 1).

q.e.d.

Lemma 14.2. We have the exact sequence

0  w NC ^{1/2} w Ñ ^{2} w Ñ ^{δ} w NI  w 0.

This is displayed as follows:

[figure: T0^{−}; T_{0}^{+}; T0^{′}; _{ζ}; v_{2}^{2^{n−1}} ζ; v_{2}^{2^{n−1}} ζ_{ζ}]

15. The connecting homomorphism on P

Put P C = A_{∞}⨁(ζζ̃/v_{1}^{4})A_{∞} and P I = (ζ̃/v_{1}^{4})A_{∞}⨁βA_{∞}.

Proposition 15.1. The connecting homomorphism δ acts as follows :

δ(1/2^{n+2}v_{1}^{2^{n}}) = ζ̃/v_{1}^{2^{n}+4}
δ(ζζ̃/2v_{1}^{2j+4}) = β/v_{1}^{2j}.

Proof. By Lemma 3.2, we see that d(1/2^{n+3}y^{2^{n−1}}) = r/2v_{1}^{2^{n}+4}. Lemma 3.3 shows that r/2v_{1}^{2^{n}+4} is homologous to z̃/2v_{1}^{2^{n}+4}, which implies the first equation.

Take n > 2j + 4. Then, we obtain the second by the computation

d(z̃ ⊗ z^{2^{n+1}}/4v_{1}^{2j+4}) = z̃ ⊗ z^{2^{n}} ⊗ z^{2^{n}}/2v_{1}^{2j+4}
d(W/2v_{1}^{2j+4}) = (z̃ ⊗ z^{2^{n}} ⊗ z^{2^{n}} + v_{1}^{4}B)/2v_{1}^{2j+4},

where W is the one of Lemma 6.14. q.e.d.

Put now

P̃ = ∑_{2 |̸s,n>0} Z/2^{n+2}{1/2^{n+2}v_{1}^{2^{n}s}}⨁ ∑_{j>0} Z/2{ζζ̃/2v_{1}^{2j+4}},

and we have Lemma 15.2. We have the exact sequence

0  w P C ^{1/2} w P̃ ^{2} w P̃ ^{δ} w P I  w 0.

16. The connecting homomorphism on Q By Lemma 6.13, we have a cocyle z̃z_{n−1}/2^{n}v_{1}^{2^{n−1}} ∈ Ω^{2,0}M_{0}^{2}, which represents ζ̃ζ/2^{n}v_{1}^{4} ∈ H^{2,0}M_{0}^{2}.

Proposition 16.1. The connecting homomorphism δ : H^{∗}M_{0}^{2} → H^{∗+1}M_{1}^{1} acts trivially on ζ̃/2^{n}v_{1}^{4} and ζ̃ζ/2^{n}v_{1}^{4} for each n. Proof. By the definition of δ, δ(ζ̃/2^{n}v_{1}^{4}) = 0 follows from Proposition 4.3. The other half follows from Lemma 6.13. q.e.d.

Now put Q̃ = Q/Z_{(2)}{ζ̃/v_{1}^{4}}⨂Λ(ζ), and we have Lemma 16.2. There is an exact sequence

0 w Q ^{1/2} w Q̃ ^{2} w Q̃ ^{δ} w 0

17. The Adams-Novikov E_{2}-terms We begin with restating the lemma given in [1, Remark 3.11].

Lemma 17.1. Suppose that H^{∗}M_{1}^{1} is a direct sum of submodules M_{i} which is also a direct sum of two modules MI_{i} and MC_{i}. If we have a submodule M̃_{i} of H^{∗}M_{0}^{2} in an exact sequence

0 w MC_{i} ^{1/2} w M̃_{i} ^{2} w M̃_{i} w MI_{i} w 0 for each i, where 1/2(x) = x/2, then H^{∗}M_{0}^{2} is the direct sum of M̃_{i}.

Now take M_{i} to be X and ρ_{2}X for X = D, E, F, J, K, L, M, N, P, and Q. Put a module

EM = D̃⨁Ẽ⨁F̃⨁J̃⨁K̃⨁L̃⨁M̃⨁Ñ⨁P̃⨁Q̃.

Then we have the following theorem from Lemmas 11.3, 9.3, 10.1, 7.2, 8.3, 12.3, 13.2, 14.2, 15.2, 16.2 and Proposition 6.5. Theorem 17.2. The module H^{∗}M_{0}^{2} is the tensor product of the exterior algebra Λ(ρ_{2}) and the module EM.

Consider the exact sequences

⋯ w H^{∗−1}M_{0}^{2} ^{δ_{1}} w H^{∗}N_{0}^{1} ^{i_{1}} w H^{∗}M_{0}^{1} ^{j_{1}} w H^{∗}M_{0}^{2} w⋯,

associated to the short exact sequence (1.2). Consider the submodules Ã and C̃ of H^{∗}M_{0}^{1}:

Ã = ∑_{i,2 |̸s} Z{v_{1}^{2^{i}s}/2^{i+2}}, C̃ = v_{1}Z/2[v_{1}^{±2}, h_{10}]⨂Λ(ρ_{1}).

Note that H^{∗}M_{0}^{1} = Ã⨁C̃⨁Q/Z_{(2)}⨂Λ(ρ_{1}) by (4.1). Furthermore, divide C̃ into the direct sum of the six submodules C̃_{i} given by

\begin{matrix}C̃_{1} = v_{1}Z/2[v_{1}^{2}]\{1, h_{10}, h_{10}^{2}, h_{10}^{3}\}⨂Z[g], C̃_{2} = v_{1}^{3}ρ_{1}Z/2[v_{1}^{2}, h_{0}^{2}], C̃_{3} = v_{1}^{3}ρ_{1}h_{10}Z/2[v_{1}^{2}, h_{0}^{2}], \\ C̃_{4} = v_{1}A_{∞}\{1, h_{10}, h_{10}^{2}, h_{10}^{3}\}⨂Z[g], C̃_{5} = v_{1}^{3}ρ_{1}A_{∞}[h_{0}^{2}] and C̃_{6} = v_{1}^{3}ρ_{1}h_{10}A_{∞}[h_{0}^{2}] \end{matrix}

Then the module Ã yields submodule Ã^{+} = ∑_{i,2 |̸s>0} Z{v_{1}^{2^{i}s}/2^{i+2}} of H^{0}N_{0}^{1} and kills the first summand of P̃. The direct sum C̃^{+} = ∑_{i=1}^{3} C̃_{i} is pulled back to H^{∗}N_{0}^{1} and ∑_{i=4}^{6} C̃_{i} kills the submodule v_{1}A_{∞}{1, h_{10}, h_{10}^{2}, h_{10}^{3}/v_{1}^{4}}⨂Z[g]⨁ζ̃A_{∞}{v_{1}, h_{0}}⨂Z/2[h_{0}^{2}] of LC_{0}. Furthermore, h_{10}^{3}/2v_{1} in K̃ is killed and h_{10}^{3}/2v_{1}^{3} yields Z/8 summand of H^{3}N_{0}^{1}. In fact, h_{10}^{3}/2v_{1}^{3} = 2δ(v_{2}^{−1}h_{11}^{2}/4v_{1}) for v_{2}^{−1}h_{11}^{2}/4v_{1} ∈ K̃. The submodule Q/Z_{(2)}⨂Λ(ρ_{1}) of H^{∗}M_{0}^{1} also yields Q/Z_{(2)} and kills the first summand of Q̃. We also put

\overline{EM} = D̃⨁Ẽ⨁F̃⨁J̃⨁M̃⨁Ñ.

Therefore, we have Proposition 17.3. The module H^{∗}N_{0}^{1} is the direct sum of Q/Z_{(2)}, Ã^{+}, C̃^{+}, ρ_{2}δ_{1}(EM), δ_{1}(\overline{EM}), K̃′, L̃′, P̃′ and Q̃′. Here

K̃′ = (Z/16{δ_{1}(ζζ̃/16v_{1}^{2})}⨁Z/8{δ_{1}(h_{10}^{3}β/8v_{1}), δ_{1}(v_{2}^{−1}h_{11}^{2}/4v_{1})}
⨁Z/4{δ_{1}(h_{10}^{3}/8v_{1}), δ_{1}(v_{2}^{−1}h_{11}^{2}β/4v_{1})}⨁Z/2{δ_{1}(v_{2}/2v_{1})}) ⨂Z[g],
L̃′ = L̃C′_{0}⨁ ∑_{i=1}^{8} δ_{1}(L̃C_{i}),
L̃C′_{0} = ∑_{j>0}(Z/2{δ_{1}(β/2v_{1}^{2j+1}), δ_{1}(h_{10}β/2v_{1}^{2j+1}), δ_{1}(h_{10}^{2}β/2v_{1}^{2j+1}), δ_{1}(h_{10}^{3}β/2v_{1}^{2j+5})}⨂Z/2[g]
⨁Z/2{ζ̃ζ/2v_{1}^{2j+1}, ζ̃ζh_{0}/2v_{1}^{2j+1}})⨂Z/2[h_{0}^{2}],
P̃′ = ∑_{j>0} Z/2{δ_{1}(ζζ̃/2v_{1}^{2j+4})}, and
Q̃′ = Q/Z_{(2)}{δ_{1}(ζζ̃/v_{1}^{4})}.

Consider the exact sequence

⋯  w H^{∗}N_{0}^{1} ^{δ_{0}} w H^{∗+1}N_{0}^{0}  w H^{∗+1}M_{0}^{0}  w· · · .

Here H^{∗}N_{0}^{0} = H^{∗}E(2)_{∗} is the Adams-Novikov E_{2}-term converging to the homotopy groups π_{∗}(L_{2}S^{0}). Since H^{s}M_{0}^{0} = Q concentrated at dimension zero, the above sequence splits into the exact sequence

0 → H^{0}N_{0}^{0} → Q → H^{0}N_{0}^{1} →^{δ} H^{1}N_{0}^{0} → 0 and the isomorphism δ: H^{s}N_{0}^{1} = H^{s+1}N_{0}^{0} for s > 0. The first

sequence yields Z_{(2)} and kills Q/Z_{(2)}, and we have the E_{2}-term.

Theorem 17.4. The E_{2}-term of the Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(L_{2}S^{0}) is isomorphic to the direct sum of Z_{(2)}, δ_{0}(Ã^{+}), δ_{0}(C̃^{+}), ρ_{2}δ_{0}δ_{1}(EM), δ_{0}δ_{1}(\overline{EM}), δ_{0}(K̃′), δ_{0}(L̃′), δ_{0}(P̃′) and δ_{0}(Q̃′).

Note that the α-elements are in Ã^{+} and the β-elements are in δ_{0}δ_{1}(Ñ). Furthermore, Q/Z_{(2)} summands are δ_{0}δ_{1}(ρ_{2}Q̃)⨁δ_{0}(Q̃′).

Corollary 17.5. The homotopy groups π_{∗}(L_{2}S^{0}) contain two Q/Z_{(2)} summands in dimension −4 and one in dimension −5. Furthermore, the group π_{i}(L_{2}S^{0}) is finite except for i = 0, −4, and −5.

Proof. The Q/Z_{(2)} summands of the E_{2}-term are permanent cycle by the naturality of the Adams-Novikov differentials. The other parts are told by Serre’s theorem. q.e.d.

References

[1] H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in Adams-Novikov spectral sequence, Ann. of Math. 106 (1977), 469–516.

[2] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, New York, 1986.

[3] K. Shimomura, Novikov’s Ext^{2} at the prime 2, Hiroshima Math. J. 11 (1981), 499–513.

[4] K. Shimomura, On the Adams-Novikov spectral sequence and products of β-elements, Hiroshima Math. J. 16 (1986), 209–224.

[5] K. Shimomura, The Adams-Novikov E_{2}-term for computing π_{∗}(L_{2}V (0)) at the prime 2, Topology and its Applications 96 (1999), 133–152.

[6] K. Shimomura and H. Tamura, Non-triviality of some compositions of β-elements in the stable homotopy of the Moore spaces, Hiroshima Math. J. 16 (1986), 121–133.

[7] K. Shimomura and X. Wang, The homotopy groups π_{∗}(L_{2}S^{0}) at the prime 3, preprint.

[8] K. Shimomura and A. Yabe, The homotopy groups π_{∗}(L_{2}S^{0}), Topology 34 (1995), 261–289.

Department of Mathematics,

Faculty of Science,

Kochi University,

Kochi, 780-8520

Japan Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520 Japan and Nankai University, Department of Mathematics, Tianjin 300071, P. R. China