On subgroups of π_{∗}(L_{2}T (1) ∧ M(2)) at the prime two

Ippei Ichigi, Katsumi Shimomura, and Xiangjun Wang

Abstract. Let L_{2} denote the Bousfield localization functor with respect to the v_{2}-localized Brown-Peterson spectrum v_{2}^{−1}BP on the stable homotopy category of spectra at the prime two, and T (1) denote the Ravenel spectrum. Then the Adams-Novikov spectral sequence is a tool to determine the homotopy groups π_{∗}(L_{2}T (1)). In [2], for s > 6, we determined the E_{∞}-term E_{∞}^{s} of the Adams-Novikov spectral sequence converging to π_{∗}(L_{2}T (1)). The s-th line E_{2}^{s} of the E_{2}-term is very complicated if 1 < s ≤ 6. Let M(k) denote the mod 2^{k} Moore spectrum. Then the complicated parts of the E_{2}-term for π_{∗}(L_{2}T (1) ∧ M(k)) also stay in the s-th lines for 1 < s ≤ 6. Here we determine the s-th lines of the Adams-Novikov E_{∞}-term for s = 0, 1 and s > 6 of the Adams-Novikov spectral sequence converging to π_{∗}(L_{2}T (1) ∧ M(2)). The result shows how disordered the structure of the E_{2}-term for π_{∗}(L_{2}T (1)) is.

1. Introduction

Let S_{(p)} denote the stable homotopy category of spectra localized away from a prime number p, and BP denote the Brown-Peterson spectrum characterized by the coefficient ring BP_{∗} = π_{∗}(BP) = Z_{(p)}[v_{1}, v_{2}, …]. Then we have the Bousfield localization functor L_{n}: S_{(p)} → S_{(p)} with respect to v_{n}^{−1}BP, and denote the image of it as L_{n}. The homotopy groups π_{∗}(L_{n}S^{0}) of the sphere spectrum S^{0} play an important role to understand L_{n}. The Adams-Novikov spectral sequence is a good tool to determine them. For n ≤ 2, the E_{2}-term for π_{∗}(L_{n}S^{0}) is determined in [4], [10], [8] and [9], and the homotopy groups of L_{n}S^{0} are also determined if n ≤ 2, except for the case where n = 2 and p = 2. Hereafter, we consider the exceptional case, and we set n = 2 and p = 2. Let T (1) denote the Ravenel spectrum characterized by the BP_{∗}-homology BP_{∗}(T (1)) = BP_{∗}[t_{1}] ⊂ BP_{∗}BP = BP_{∗}[t_{1}, t_{2}, …]. Then the homotopy groups π_{∗}(L_{2}T (1)) would help us to understand the homotopy groups π_{∗}(L_{2}S^{0}). Let \overline{M}(k) for k > 0 and \overline{V} (1) be cofibers of 2^{k}: T (1) → T (1) and v_{1} : Σ^{2}\overline{M}(1) → \overline{M}(1), respectively. Note that \overline{M}(k) = T (1) ∧ M(k) for the mod 2^{k} Moore spectrum M(k), and, in particular, \overline{M}(1) is the Mahowald spectrum X⟨1⟩ and v_{1} is the self map induced from the generator of π_{2}(X⟨1⟩)(cf. [3], [7]). Then the homotopy groups π_{∗}(L_{2}\overline{V} (1)) are determined in [3]. We consider the spectrum \overline{M}(1, ∞) defined by the cofiber sequence

(1.1)  \overline{M}(1) ^{η_{1}} wv_{1}^{−1}\overline{M}(1) w\overline{M}(1, ∞)

for the localization map η_{1} and obtain a cofiber sequence

(1.2)  \overline{V} (1) w\overline{M}(1, ∞) ^{v_{1}} w\overline{M}(1, ∞).

We determine π_{∗}(L_{2}\overline{M}(1, ∞)) by use of the cofiber sequence (1.2), and then π_{∗}(L_{2}\overline{M}(1)) by (1.1) in [7]. Our next target is to determine the homotopy groups π_{∗}(L_{2}T (1)) by using the mod 2 Bockstein spectral sequence. It is very hard to compute the Bockstein spectral sequence, but we get a partial results on π_{∗}(L_{2}T (1)) in [2]. Here we show how hard the other parts of the E_{2}-term are. (The paper [5] seems to require some more time to appear, because it is too hard to verify the complicated results.) To do so, we consider the mod 4 Moore spectrum M(2). Similarly, \overline{M}(2, ∞) denotes a cofiber of the localization map η_{1} : \overline{M}(2) → v_{1}^{−1}\overline{M}(2). The zeroth line of the E_{2}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{2}\overline{M}(2, ∞)) is not so complicated that we determine the structure in Theorem (2.10). Since we know the structure of E_{2}^{0}(L_{2}\overline{M}(1, ∞)), it seems easy to determine the structure of E_{2}^{0}(L_{2}\overline{M}(2, ∞)) by the exact sequence 0 → E_{2}^{0}(L_{2}\overline{M}(1, ∞)) →^{2} E_{2}^{0}(L_{2}\overline{M}(2, ∞)) → E_{2}^{0}(L_{2}\overline{M}(1, ∞)) →^{δ} E_{2}^{1}(L_{2}\overline{M}(1, ∞)), but it is unexpectedly hard to compute the connecting homomorphism. This is why we here employ the exact sequence

(1.3)  0 → E_{2}^{0}(L_{2}\overline{M}(2, 1)) → E_{2}^{0}(L_{2}\overline{M}(2, ∞)) → E_{2}^{0}(L_{2}\overline{M}(2, ∞)) →^{δ} E_{2}^{1}(L_{2}\overline{M}(2, 1))

to determine it. Here the homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)) are determined in Theorem (2.5). It seems easier to use this exact sequence even when we compute the first line E_{2}^{1}(L_{2}\overline{M}(2, ∞)) than to use that exact sequence. Since we determine the homotopy groups π_{∗}(v_{1}^{−1}\overline{M}(2)) in Proposition (4.1), the result on E_{2}^{0}(L_{2}\overline{M}(2, ∞)) gives rise to the zeroth and the first lines of the E_{2}-term for π_{∗}(L_{2}\overline{M}(2)). This displays how complicated the structure of the homotopy groups is. By use of the result [2], we also determine the s-th lines for s > 5, and we obtain subgroups of π_{∗}(L_{2}\overline{M}(2)). The second line is too complicated to determine here.

2000 Mathematics Subject Classification. Primary 55Q99; Secondary 55Q45, 55Q51.

Key words and phrases. Ravenel spectrum, Moore spectrum, Stable homotopy groups of spheres.

This paper is divided into six sections:

1. Introduction

2. Statement of results

3. The change of rings theorem and the relations in Σ(2)

4. The homotopy groups π_{∗}(v_{1}^{−1}T (1) ∧ M(2)) and π_{∗}(L_{2}\overline{M}(2, 1))

5. The elements x_{n}, g_{n}, R_{n} and X_{n}, and relations between them

6. The action of the connecting homomorphism

In the next section, we not only state our main results, but also prove some of main results. In section three, we introduce the Hopf algebroids which we consider in this paper and set up formulas on their right units η_{R}. In section four, we determine homotopy groups π_{∗}(v_{1}^{−1}\overline{M}(2)) and prove Theorems (2.5) and (2.6). We introduce some cochains in section five, and compute the behavior of the connecting homomorphism in the last section. We also prove Theorem (2.10) there.

2. Statement of results

We work in the stable homotopy category S_{(2)} of spectra localized away from the prime two. Let BP denote the Brown-Peterson spectrum with coefficient ring BP_{∗} = Z_{(2)}[v_{1}, v_{2}, …]. Then BP_{∗}BP = BP_{∗}[t_{1}, t_{2}, …] is a Hopf algebroid over BP_{∗}(cf. [6]). We compute homotopy groups of a spectrum X by the Adams-Novikov spectral sequence with E_{2}-term E_{2}^{s,t}(X) = Ext_{BP_{∗}BP}^{s,t} (BP_{∗}, BP_{∗}(X)). Let T (1) denote the Ravenel spectrum with BP_{∗}(T (1)) = BP_{∗}[t_{1}] ⊂ BP_{∗}BP. We have an element v_{1} + 2t_{1} ∈ Ext_{BP_{∗}BP}^{0,2} (BP_{∗}, BP_{∗}[t_{1}]) = E_{2}^{0,2}(T (1)), which survives to a homotopy element of π_{2}(T (1)). Since T (1) is a ring spectrum, the homotopy element defines a self map α^{j}: Σ^{2j}T (1) → T (1) for each j > 0, whose cofiber we denote by T (1)/v_{1}^{j}. For the mod 2^{i} Moore spectrum M(i) for i > 0, put \overline{M}(i, j) = M(i) ∧ (T (1)/v_{1}^{j}). Then, BP_{∗}(\overline{M}(i, j)) = BP_{∗}[t_{1}]/(2^{i}, (v_{1} + 2t_{1})^{j}) ⊂ BP_{∗}BP/(2^{i}, (v_{1} + 2t_{1})^{j}), and so we have the Adams-Novikov spectral sequence

E_{2}^{s,t}(L_{2}\overline{M}(i, j)) = Ext_{BP_{∗}BP}^{s,t} (BP_{∗}, v_{2}^{−1}BP_{∗}[t_{1}]/(2^{i}, (v_{1} + 2t_{1})^{j})) ⟹ π_{∗}(L_{2}\overline{M}(i, j)).

Note that \overline{M}(1, 1) is the \overline{V} (1) in the introduction. We first determine the structure of the homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)), which is obtained from the cofiber sequence

(2.1)  \overline{M}(1, 1) ⟶^{2} \overline{M}(2, 1) ⟶^{ĩ} \overline{M}(1, 1)

given by smashing T (1)/v_{1} with the cofiber sequence

(2.2)  M(1) ^{2} wM(2) wM(1)

of the Moore spectra. In order to state the result, we set up notation: In [3], the E_{2}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{2}\overline{M}(1, 1)) is determined as

E_{2}^{∗,∗}(L_{2}\overline{M}(1, 1)) = K(2)_{∗}[v_{3}, h_{20}] ⊗ Λ(h_{21}, h_{30}, h_{31}, ρ_{2}).

Here

(2.3)  K(2)_{∗} = Z/2[v_{2}, v_{2}^{−1}],

and h_{2i}, h_{3i} and ρ_{2} are the elements represented by the cocycles, whose leading terms are t_{2}^{2^{i}}, t_{3}^{2^{i}} and v_{2}^{−5}t_{4} + v_{2}^{−10}t_{4}^{2} in the cobar complex Ω_{BP_{∗}BP}^{1} v_{2}^{−1}BP_{∗}[t_{1}]/(2, v_{1}), respectively. By use of the generators \overline{h}_{21} and \overline{h}_{31} given in Lemma (4.3), we rewrite the E_{2}-term E_{2}^{∗}(L_{2}\overline{M}(1, 1)):

E_{2}^{∗}(L_{2}\overline{M}(1, 1)) = K_{∗}[v_{3}^{2}] ⊗ Z/2[h_{20}] ⊗ Λ(v_{2}, v_{3}, \overline{h}_{21}, h_{30}, \overline{h}_{31}, ρ_{2})

as a Z/2-module, where

(2.4)  K_{∗} = Z/2[v_{2}^{2}, v_{2}^{−2}].

Put

\begin{matrix}F^{0} = K_{∗}[v_{3}^{2}] ⊗ Λ(\overline{h}_{31}, ρ_{2}), F = K_{∗}[v_{3}^{2}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2}) and \\ C = K_{∗}[v_{3}^{2}] ⊗ Z/2[h_{20}] ⊗ Λ(v_{3}, \overline{h}_{21}, h_{30}, \overline{h}_{31}, ρ_{2}). \end{matrix}

We notice that F = F^{0} ⊗ Λ(\overline{h}_{21}), C = Z/2[h_{20}] ⊗ F ⊗ Λ(v_{3}, h_{30}) and E_{2}^{∗}(L_{2}\overline{M}(1, 1)) = C ⊗ Λ(v_{2}). Furthermore, the map 2 (resp. ĩ) in (2.1) induces a homomorphism 2: E_{2}^{∗}(L_{2}\overline{M}(1, 1)) → E_{2}^{∗}(L_{2}\overline{M}(2, 1)) (resp. π: E_{2}^{∗}(L_{2}\overline{M}(2, 1)) → E_{2}^{∗}(L_{2}\overline{M}(1, 1))), and 2M (resp. M) ⊂ E_{2}^{∗}(L_{2}\overline{M}(2, 1)) for a submodule M ⊂ E_{2}^{∗}(L_{2}\overline{M}(1, 1)) denotes the image of M under the homomorphism 2 (resp. π^{−1}).

Theorem (2.5). The E_{2}-term E_{2}^{∗}(L_{2}\overline{M}(2, 1)) is isomorphic to

2v_{2}C ⊕ h_{20}C ⊕ 2v_{3}F ⊕ h_{30}F ⊕ 2v_{3}h_{30}F^{0} ⊕ v_{3}h_{30}\overline{h}_{21}F^{0} ⊕ F̃.

Here F̃ = Z/4[v_{2}^{2}, v_{2}^{−2}, v_{3}^{2}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2}).

For describing the homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)), we introduce the modules

F^{0}′ = K_{∗}[v_{3}^{4}] ⊗ Λ(\overline{h}_{31}, ρ_{2}), F′ = K_{∗}[v_{3}^{4}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2}), F̃′ = Z/4[v_{2}^{2}, v_{2}^{−2}, v_{3}^{4}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2}), C′ = K_{∗}[v_{3}^{4}] ⊗ Z/2[h_{20}]/(h_{20}^{3}) ⊗ Λ(v_{3}, \overline{h}_{21}, h_{30}, \overline{h}_{31}, ρ_{2}) and C′′ = K_{∗}[v_{3}^{4}] ⊗ Z/2[h_{20}]/(h_{20}^{2}) ⊗ Λ(v_{3}, \overline{h}_{21}, h_{30}, \overline{h}_{31}, ρ_{2}).

Theorem (2.6). The Adams-Novikov E_{∞}-terms for the homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)) are isomorphic to the direct sum of the modules

2v_{2}C′, h_{20}C′′, 2v_{3}F, h_{30}F′, 2v_{3}h_{30}F^{0}, v_{3}h_{30}\overline{h}_{21}F^{0}′, F̃′ and 2v_{3}^{2}F′.

These theorems are proved in section four.

Next we consider the spectrum \overline{M}(2, ∞) = lim_{→} \overline{M}(2, n), and the exact sequence (1.3). Put

(2.7)  E_{∗} = Z/2[v_{1}, v_{2}^{2}, v_{2}^{−2}] and Ẽ_{∗} = Z/4[v_{1}, v_{2}^{2}, v_{2}^{−2}].

Then K_{∗} in (2.4) is E_{∗}/(v_{1}) and Ẽ_{∗}/(2) = E_{∗}. Let x_{k} = v_{3}^{4^{k}} + … denote a cochain in the cobar complex Ω_{BP_{∗}BP}^{∗} v_{2}^{−1}BP_{∗}[t_{1}], which is defined in (5.3), and a_{k} for each k ≥ 0 be the integer defined by

(2.8)  a_{k} = 4^{k} + 2e_{k} = e_{k+1} + e_{k}, for e_{k} = (4^{k} − 1)/3.

We further introduce modules:

M = 2v_{2}v_{3}K_{∗}[v_{3}^{2}] ⊕ 2v_{2}E_{∗}[v_{3}^{4}]⟨v_{3}^{2}/v_{1}^{3}⟩ ⊕ 2v_{2}E_{∗}[x_{n}^{2}]⟨x_{n}/v_{1}^{a_{n}+1}⟩ ⊕ 2v_{2}E_{∗}[x_{n+1}]⟨x_{n}^{2}/v_{1}^{2a_{n}}⟩,
L_{−1}^{3} = 2E_{∗}[v_{3}^{2}]⟨v_{3}/v_{1}⟩,
L_{k}^{0} = Ẽ_{∗}[x_{2k}^{4}]⟨x_{2k}^{2}/v_{1}^{c_{2k}}⟩ ⊕ 2E_{∗}[x_{2k}^{4}]{x_{2k}^{2}/v_{1}^{j} : c_{2k} < j ≤ 2c_{2k}},
L_{k}^{1} = Ẽ_{∗}[x_{2k+1}^{2}]⟨x_{2k+1}/v_{1}^{2c_{2k}}⟩ ⊕ 2E_{∗}[x_{2k+1}^{2}]{x_{2k+1}/v_{1}^{j} : 2c_{2k} < j ≤ c_{2k+1} + 2},
L_{k}^{2} = Ẽ_{∗}[x_{2k+1}^{4}]⟨x_{2k+1}^{2}/v_{1}^{c_{2k+1}+2}⟩ ⊕ 2E_{∗}[x_{2k+1}^{4}]{x_{2k+1}^{2}/v_{1}^{j} : c_{2k+1} + 2 < j ≤ 2c_{2k+1} + 6},
L_{k}^{3} = Ẽ_{∗}[x_{2k+2}^{2}]⟨x_{2k+2}/v_{1}^{2c_{2k+1}+6}⟩ ⊕ 2E_{∗}[x_{2k+2}^{2}]{x_{2k+2}/v_{1}^{j} : 2c_{2k+1} + 6 < j ≤ c_{2k+2}}.

Here, the integers c_{n} are defined by

(2.9)  c_{n} = 4^{n} + 3 × 4^{1+ε(n)}\overline{e}_{[ \frac{n}{2} ]} for n ≥ 0, ε(n) = \frac{1 − (−1)^{n}}{2} and \overline{e}_{k} = \frac{16^{k} − 1}{15},

and [x] denotes the greatest integer that does not exceed x.

Theorem (2.10). The zeroth line of the E_{2}-term for π_{∗}(L_{2}\overline{M}(2, ∞)) is given as follows:

E_{2}^{0}(L_{2}\overline{M}(2, ∞)) = v_{1}^{−1}Ẽ_{∗}/Ẽ_{∗} ⊕ 2v_{2}(v_{1}^{−1}E_{∗}/E_{∗}) ⊕ M ⊕ L_{−1}^{3} ⊕ ⨁_{k≥0} (L_{k}^{0} ⊕ L_{k}^{1} ⊕ L_{k}^{2} ⊕ L_{k}^{3}) .

We prove this in the last section.

This result shows that the structure of E_{2}^{0}(L_{2}\overline{M}(2, ∞)) is far complicated than that of E_{2}^{0}(L_{2}\overline{M}(1, ∞)). The difficulty to compute the E_{2}-term E_{2}^{1}(L_{2}\overline{M}(2, ∞)) appears in the cokernel of the connecting homomorphism δ: E_{2}^{0}(L_{2}\overline{M}(2, ∞)) → E_{2}^{1}(L_{2}\overline{M}(2, 1)), which involves ρ_{2} as in Proposition (6.2).

Let \overline{M}(∞, ∞) = lim_{→} \overline{M}(k, ∞). Then the Adams-Novikov E_{2}-term E_{2}^{s}(L_{2}\overline{M}(∞, ∞)) is isomorphic to (C̃_{0} ⊗ Λ(ρ_{2}))^{s} for s > 4 by [2], where

C̃_{0} = v_{2}v_{3}K_{∗}[v_{3}^{2}, h_{20}] ⊗ Λ(h_{21}, h_{30}, h_{31}).

This result yields the E_{2}-terms E_{2}^{s}(L_{2}\overline{M}(2, ∞)) for s > 5 from the long exact sequence

\begin{matrix}E_{2}^{s−1}(L_{2}\overline{M}(∞, ∞)) ^{4} wE_{2}^{s−1}(L_{2}\overline{M}(∞, ∞)) ^{δ′} wE_{2}^{s}(L_{2}\overline{M}(2, ∞)) \\ w^{1/4} E_{2}^{s}(L_{2}\overline{M}(∞, ∞)) ^{4} wE_{2}^{s}(L_{2}\overline{M}(∞, ∞)) \end{matrix}

induced from the cofiber sequence L_{2}\overline{M}(2, ∞) w^{1/4} L_{2}\overline{M}(∞, ∞) ^{4} wL_{2}\overline{M}(∞, ∞). Indeed, the homomorphism 4 is trivial for s > 5.

Theorem (2.11). For s > 5, the s-th line of the E_{2}-term for π_{∗}(L_{2}\overline{M}(2, ∞)) is given as follows:

E_{2}^{s}(L_{2}\overline{M}(2, ∞)) = (C̃_{0} ⊗ Λ(ρ_{2}))^{s} ⊕ δ′((C̃_{0} ⊗ Λ(ρ_{2}))^{s−1}).

Here the summand δ′((C̃_{0} ⊗ Λ(ρ_{2}))^{s−1}) denotes the δ′-image, which is isomorphic to (C̃_{0} ⊗ Λ(ρ_{2}))^{s−1}.

By definition, we have a cofiber sequence

(2.12)  \overline{M}(2) wv_{1}^{−1}\overline{M}(2) w\overline{M}(2, ∞),

which induces the long exact sequence

E_{2}^{s−1}(v_{1}^{−1}\overline{M}(2)) wE_{2}^{s−1}(L_{2}\overline{M}(2, ∞)) ^{δ} wE_{2}^{s}(L_{2}\overline{M}(2)) wE_{2}^{s}(v_{1}^{−1}\overline{M}(2)) wE_{2}^{s}(L_{2}\overline{M}(2, ∞)).

We notice that δ is an isomorphism if s > 2, since E_{2}^{s}(v_{1}^{−1}\overline{M}(2)) = 0 if s > 1 by Proposition (4.1). Now Theorems (2.10), (2.11) and Proposition (4.1) imply the following:

Theorem (2.13). The E_{2}-term for π_{∗}(L_{2}T (1) ∧ M(2)) is as follows:

E_{2}^{0}(L_{2}\overline{M}(2)) = Z/4[v_{1}, v_{2}^{2}] ⊕ 2v_{2}Z/2[v_{1}, v_{2}^{2}],
E_{2}^{1}(L_{2}\overline{M}(2)) = Z/4[v_{1}, v_{2}^{2}]/(v_{1}^{∞}, v_{2}^{∞}) ⊕ 2v_{2}Z/2[v_{1}, v_{2}^{2}]/(v_{1}^{∞}, v_{2}^{∞})
⊕δ (M ⊕ L_{−1}^{3} ⊕ ⨁_{k≥0} (L_{k}^{0} ⊕ L_{k}^{1} ⊕ L_{k}^{2} ⊕ L_{k}^{3})) ,
E_{2}^{s}(L_{2}\overline{M}(2)) = δ ((C̃_{0} ⊗ Λ(ρ_{2}))^{s−1}) ⊕ δδ′ ((C̃_{0} ⊗ Λ(ρ_{2}))^{s−2}) for s > 6.

Here δ(X) denotes the δ-image of X that is isomorphic to X.

Recall [2] the module

Ĉ_{0} = v_{2}v_{3}K_{∗}[v_{3}^{4}] ⊗ Λ(h_{20}, h_{21}, h_{30}, h_{31})⨁v_{2}v_{3}h_{20}^{2}K_{∗}[v_{3}^{4}] ⊗ Λ(h_{30}, h_{31}).

Then we showed in [2] that this is the submodule of the Adams-Novikov E_{∞}-term consisting of the survivors of the summand C_{0} of the E_{2}-term, and is also a submodule of π_{∗}(L_{2}\overline{M}(∞, ∞)). We also showed in [2] that the Adams-Novikov differential acts trivially on the other summands of the E_{2}-term.

Corollary (2.14). The Adams-Novikov E_{∞}-term for the homotopy groups π_{∗}(L_{2}\overline{M}(2)) contains the subgroups isomorphic to

Σ(M ⊕ L_{−1}^{3} ⊕ ⨁_{k≥0} (L_{k}^{0} ⊕ L_{k}^{1} ⊕ L_{k}^{2} ⊕ L_{k}^{3}) ) ⊕ ΣĈ_{0} ⊗ Λ(ρ_{2}) ⊕ Σ^{2}Ĉ_{0} ⊗ Λ(ρ_{2}).

Here Σ denotes a shift of dimension.

3. The change of rings theorem and the relations in Σ(2)

Let BP and BP ⟨n⟩ denote the Brown-Peterson ring spectrum and the unlocalized Johnson-Wilson spectrum, respectively. Then, BP gives rise to the Hopf algebroid (A, Γ) = (BP_{∗}, BP_{∗}BP) = (Z_{(2)}[v_{1}, v_{2}, …], BP_{∗}[t_{1}, t_{2}, …]) and BP ⟨n⟩_{∗} = Z_{(2)}[v_{1}, v_{2}, …, v_{n}] ⊂ BP_{∗}. Since BP ⟨3⟩ is a BP -module spectrum, v_{2} ∈ BP_{∗} yields the self map v_{2} : BP ⟨3⟩ → BP ⟨3⟩. Let F (2) denote the spectrum v_{2}^{−1}BP ⟨3⟩. Then, F (2)_{∗} = π_{∗}(F (2)) = Z_{(2)}[v_{1}, v_{2}, v_{2}^{−1}, v_{3}] and F (2)_{∗}(F (2)) = F (2)_{∗} ⊗_{A} Γ ⊗_{A} F (2)_{∗}. The Hopf algebroid structure of (A, Γ) defines the one on (B, Σ) = (F (2)_{∗}, F (2)_{∗}(F (2))). Consider the Hopf algebroid (E(2)_{∗}, E(2)_{∗}E(2)) associated to the localized Johnson-Wilson spectrum E(2) = v_{2}^{−1}BP ⟨2⟩. In [1], Hovey and Sadofsky showed the change of rings theorem: Ext_{Γ}^{s}(A, M) = Ext_{E(2)_{∗}E(2)}^{s}(E(2)_{∗}, E(2)_{∗} ⊗_{A} M) for a v_{2}-local Γ-comodule M. In the same manner as the “first proof” of it, the equivalence F (2) = ⋁_{k≥0} Σ^{k|v_{3}|}E(2) yields the isomorphism Ext_{Σ}^{s}(B, B ⊗_{A} M) = Ext_{E(2)_{∗}E(2)}^{s}(E(2)_{∗}, E(2)_{∗} ⊗_{A} M), and then, we have an isomorphism

Ext_{Γ}^{s}(A, M) = Ext_{Σ}^{s}(B, B ⊗_{A} M)

for a v_{2}-local Γ-comodule M. Indeed, F (2)- and E(2)-Adams resolutions of a spectrum X induce the same spectral sequence, and so the E_{2}-terms agree.

Consider Γ(2) = Γ/(t_{1}) = A[t_{2}, t_{3}, …]. Then the pair (A, Γ(2)) is a Hopf algebroid induced from (A, Γ) under the projection Γ → Γ(2). Since BP_{∗}(T (1)) = A[t_{1}], we have F (2)_{∗}(T (1)) = B[t_{1}], which is expressed as a cotensor product

(3.1)  B[t_{1}] = B□_{Σ(2)}Σ

for Σ(2) = B ⊗_{A} Γ(2) ⊗_{A} B. Here (B, Σ(2)) is a Hopf algebroid induced from (A, Γ(2)). Write H^{∗}M as Ext_{Σ(2)}^{∗}(B, M) for a Σ(2)-comodule M, and H^{∗}(M ⊗_{A} B) for a Γ-comodule M is isomorphic to

Ext_{Γ}^{s}(A, v_{2}^{−1}M ⊗_{A} A[t_{1}]) = Ext_{Σ}^{s}(B, M ⊗_{A} B[t_{1}]) = Ext_{Σ(2)}^{s}(B, M ⊗_{A} B)(= H^{∗}(M ⊗_{A} B)),

where the second equality follows from (3.1) by the change of rings theorem [6, A1.3.13]. In this paper, we employ the same method introduced in [4] to compute H^{∗}B/(4), which is isomorphic to the E_{2}-term Ext_{Γ}^{s}(A, v_{2}^{−1}A[t_{1}]/(4)). For this sake, we introduce the Σ(2)-comodules M_{2}^{0}(2) and M_{1}^{1}(2):

Definition (3.2). M_{2}^{0}(2) = B/(4, v_{1}) and M_{1}^{1}(2) is the cokernel of the inclusion B/(4) y w v_{1}^{−1}B/(4).

Note that the modules in the exact sequence (1.3) are:

(3.3)  E_{2}^{∗}(L_{2}\overline{M}(2, 1)) = H^{∗}M_{2}^{0}(2) and E_{2}^{∗}(L_{2}\overline{M}(2, ∞)) = H^{∗}M_{1}^{1}(2).

We study the module H^{∗}B/(4) by the long exact sequence

H^{s−1}v_{1}^{−1}B/(4) ⟶ H^{s−1}M_{1}^{1}(2) ⟶^{δ} H^{s}B/(4) ⟶ H^{s}v_{1}^{−1}B/(4) ⟶ H^{s}M_{1}^{1}(2)

associated to the short exact sequence 0 → B/(4) → v_{1}^{−1}B/(4) → M_{1}^{1}(2) → 0. We will determine H^{∗}v_{1}^{−1}B/(4) in Proposition (4.1). We compute H^{∗}M_{1}^{1}(2) by the v_{1}-Bockstein spectral sequence associated to the short exact

sequence 0 → M_{2}^{0}(2) ⟶^{1/v_{1}} M_{1}^{1}(2) ⟶^{v_{1}} M_{1}^{1}(2) → 0 of Σ(2)-comodules. For computing the Bockstein spectral sequence, we set up formulas on the structure map η_{R}: B → Σ(2) of the Hopf algebroid Σ(2) and some relations in Σ(2).

We begin with the behavior of the right unit η_{R}: A → Γ(2) = A[t_{2}, t_{3}, …]. Here, v_{i}’s are the Hazewinkel generators of A defined by

(3.4)  v_{i} = 2m_{i} − ∑_{k=1}^{i−1} m_{k}v_{i−k}^{2^{k}} ∈ 2^{−1}A = Q[m_{1}, m_{2}, …]

under the inclusion A y w 2^{−1}A. The unit map i: S^{0} → BP induces the right unit η_{R} = (i∧1)_{∗}: A = π_{∗}(BP) → Γ → Γ(2), and then its localization η_{R}: 2^{−1}A → 2^{−1}Γ(2), whose action is given by the Quillen formulas

(3.5)  η_{R}(m_{i}) = ∑_{k=0}^{i} m_{k}t_{i−k}^{2^{k}}.

With a routine computation with the formulas (3.4) and (3.5), we see that

Lemma (3.6). On the generators v_{i} for 0 < i ≤ 6, η_{R}: A → Γ(2) acts as follows:

η_{R}(v_{1}) = v_{1},
η_{R}(v_{2}) = v_{2} + 2t_{2},
η_{R}(v_{3}) = v_{3} + v_{1}t_{2}^{2} − v_{1}^{4}t_{2} + 2(t_{3} − v_{1}v_{2}t_{2} − v_{1}t_{2}^{2}),
η_{R}(v_{4}) ≡ v_{4} + v_{2}t_{2}^{4} + v_{2}^{4}t_{2} + v_{1}t_{3}^{2} + v_{1}^{2}v_{3}(t_{2}^{2} + v_{1}^{3}t_{2}) + v_{1}^{6}t_{2}^{3} + v_{1}^{8}t_{3} + v_{1}^{9}t_{2}^{2} mod (2),
η_{R}(v_{5}) ≡ v_{5} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2} + v_{2}t_{3}^{4} + v_{2}^{8}t_{3} + v_{1}t_{4}^{2} + v_{1}c(4) + v_{1}v_{2}^{8}t_{2}^{2} + v_{1}^{2}v_{2}v_{3}^{2}t_{2}^{4} mod (2, v_{1}^{3}) and
η_{R}(v_{6}) ≡ v_{6} + v_{4}t_{2}^{16} + t_{2}η_{R}(v_{4})^{4} + v_{3}t_{3}^{8} + v_{3}^{8}t_{3} + v_{2}t_{4}^{4} + v_{2}^{16}t_{4} + v_{2}c(4)^{2} + v_{2}^{17}t_{2}^{4} mod (2, v_{1}).

Here

2c(4) ≡ η_{R}(v_{4}^{2}) − (v_{4}^{2} + v_{2}^{2}t_{2}^{8} + v_{2}^{8}t_{2}^{2} + v_{1}^{2}t_{3}^{4}) mod (4, v_{1}^{4}).

Since η_{R}(v_{i}) = 0 in Σ(2) if i ≥ 4, we obtain relations in Σ(2) from Lemma (3.6):

(3.7)

v_{2}t_{2}^{4} ≡ v_{2}^{4}t_{2} + v_{1}t_{3}^{2} + v_{1}^{2}v_{3}t_{2}^{2} + v_{1}^{5}v_{3}t_{2} + v_{1}^{6}t_{2}^{3} + v_{1}^{8}t_{3} + v_{1}^{9}t_{2}^{2} mod (2),
v_{2}t_{3}^{4} = v_{2}^{8}t_{3} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2} + v_{1}t_{4}^{2} + v_{1}c(4) + v_{1}v_{2}^{8}t_{2}^{2} + v_{1}^{2}v_{2}v_{3}^{2}t_{2}^{4} mod (2, v_{1}^{3}) and
v_{2}t_{4}^{4} = v_{2}^{16}t_{4} + v_{3}t_{3}^{8} + v_{3}^{8}t_{3} + v_{2}c(4)^{2} + v_{2}^{17}t_{2}^{4} mod (2, v_{1}).

By the first relation of (3.7), we see that c(4) ≡ v_{2}^{5}t_{2}^{5} + v_{1}t_{3}^{2}(v_{2}t_{2}^{4} + v_{2}^{4}t_{2}) ≡ v_{2}^{5}t_{2}^{5} mod (2, v_{1}^{2}), and so v_{2}^{−10}c(4) + v_{2}^{−2}t_{2}^{2} ≡ v_{2}^{−5}t_{2}^{5} + v_{2}^{−2}t_{2}^{2} ≡ v_{1}v_{2}^{−6}t_{2}t_{3}^{2} mod (2, v_{1}^{2}). Then

(3.8)

v_{2}t_{3}^{4} ≡ v_{2}^{8}t_{3} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2} + v_{1}t_{4}^{2} + v_{1}^{2}(v_{2}^{4}t_{2}t_{3}^{2} + v_{2}v_{3}^{2}t_{2}^{4}) mod (2, v_{1}^{3})
≡ v_{2}^{8}t_{3} + v_{2}^{6}v_{3}t_{2}^{2} + v_{3}^{4}t_{2} + v_{1}t_{4}^{2} + v_{1}^{2}(v_{2}^{−2}v_{3}t_{3}^{4} + v_{2}^{4}t_{2}t_{3}^{2} + v_{2}v_{3}^{2}t_{2}^{4}) mod (2, v_{1}^{3})
≡ v_{2}^{8}t_{3} + v_{2}^{6}v_{3}t_{2}^{2} + v_{3}^{4}t_{2} + v_{1}t_{4}^{2}
+v_{1}^{2} (v_{3}(v_{2}^{5}t_{3} + v_{2}^{3}v_{3}t_{2}^{2} + v_{2}^{−3}v_{3}^{4}t_{2}) + v_{2}^{4}t_{2}t_{3}^{2} + v_{2}^{4}v_{3}^{2}t_{2}) mod (2, v_{1}^{3}) and
v_{2}t_{4}^{4} ≡ v_{2}^{16}t_{4} + v_{3}t_{3}^{8} + v_{3}^{8}t_{3} mod (2, v_{1})
≡ v_{2}^{16}t_{4} + v_{2}^{14}v_{3}t_{3}^{2} + v_{3}^{8}t_{3} + v_{2}^{13}v_{3}^{3}t_{2} + v_{2}^{−2}v_{3}^{9}t_{2}^{2} mod (2, v_{1}).

4. The homotopy groups π_{∗}(v_{1}^{−1}T (1) ∧ M(2)) and π_{∗}(L_{2}\overline{M}(2, 1))

Note that T (1) ∧ M(2) = \overline{M}(2). The Adams-Novikov E_{2}-term E_{2}^{∗}(v_{1}^{−1}\overline{M}(2)) is isomorphic to Ext_{Γ(2)}^{∗}(BP_{∗}, v_{1}^{−1}BP_{∗}/(4)) by the change of rings theorem [6, A1.3.13], since v_{1}^{−1}BP_{∗}[t_{1}]/(4) = v_{1}^{−1}BP_{∗}/(4)□_{Γ(2)}Γ.

Proposition (4.1). The Adams-Novikov E_{2}-term for π_{∗}(v_{1}^{−1}\overline{M}(2)) is isomorphic to

E_{2}^{∗}(v_{1}^{−1}\overline{M}(2)) = Z/4[v_{1}^{±1}, v_{2}^{2}] ⊗ Λ(v_{2}h_{20}) ⊕ 2v_{2}Z/2[v_{1}^{±1}, v_{2}^{2}] ⊕ h_{20}Z/2[v_{1}^{±1}, v_{2}^{2}].

Furthermore, this is isomorphic to the homotopy groups π_{∗}(v_{1}^{−1}\overline{M}(2)).

Proof. In [6, 6.5.6], it is shown that E_{2}^{∗}(v_{1}^{−1}\overline{M}(1)) = K(1)_{∗}[v_{2}] ⊗ Λ(h_{20}) for K(1)_{∗} = Z/2[v_{1}, v_{1}^{−1}]. Consider the long exact sequence

E_{2}^{∗}(v_{1}^{−1}\overline{M}(1)) ^{2} wE_{2}^{∗}(v_{1}^{−1}\overline{M}(2)) wE_{2}^{∗}(v_{1}^{−1}\overline{M}(1)) ^{δ} wE_{2}^{∗+1}(v_{1}^{−1}\overline{M}(1))

of the Adams-Novikov E_{2}-terms associated to the cofiber sequence (2.2). We make a computation in the cobar complex Ω_{Γ(2)}^{∗}v_{1}^{−1}BP_{∗}/(4) (cf. [6, A1.2.11]), and see that the action of the connecting homomorphism δ is obtained from the only relation δ(v_{2}) = h_{20}, which is verified by the equality η_{R}(v_{2}) = v_{2} + 2t_{2} in Γ(2) in Lemma (3.6). Now the E_{2}-term is obtained from the above exact sequence.

Since the E_{2}-term has the horizontal vanishing line s = 2, the spectral sequence collapses from the E_{2}-term, and the extension is trivial. q.e.d.

We turn to the homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)). The cofiber sequence (2.1) induces the long exact sequence

E_{2}^{∗}(L_{2}\overline{M}(1, 1)) ⟶^{2} E_{2}^{∗}(L_{2}\overline{M}(2, 1)) ⟶ E_{2}^{∗}(L_{2}\overline{M}(1, 1)) ⟶^{δ} E_{2}^{∗+1}(L_{2}\overline{M}(1, 1)).

Lemma (4.2). The connecting homomorphism δ acts as follows:

δ(v_{2}) = h_{20}, δ(v_{3}) = h_{30}, δ(h_{21}) = h_{20}^{2}, and δ(h_{31}) = h_{30}^{2} = (h_{21} + v_{2}h_{20})h_{31} + v_{2}v_{3}^{2}h_{20}h_{21}.

Here, we set v_{2}^{2} = 1.

Proof. This follows immediately from Lemma (3.6). The last equality is given in [3]. q.e.d.

We consider the elements

\overline{h}_{21} = h_{21} + v_{2}h_{20} and \overline{h}_{31} = h_{31} + v_{3}h_{30}.

Then Lemma (4.2) implies the following:

Lemma (4.3). The connecting homomorphism δ acts trivially on the elements

h_{20}, \overline{h}_{21}, h_{30} and \overline{h}_{31}.

Proof of Theorem (2.5). The connecting homomorphism δ acts as δ(v_{2}C) = h_{20}C, δ(F) = 0, δ(v_{3}F) = h_{30}F and δ(v_{3}h_{30}F^{0}) = v_{3}h_{30}\overline{h}_{21}F^{0}. The last correspondence follows from δ(v_{3}h_{30}) = \overline{h}_{21}(\overline{h}_{31} + v_{2}v_{3}^{2}h_{20} + v_{3}h_{30}) = v_{3}h_{30}\overline{h}_{21} + · · ·. Here, the correspondence is written as the leading term. q.e.d.

Proof of Theorem (2.6). The Adams-Novikov differential on E_{r}^{∗}(L_{2}\overline{M}(1, 1)) is shown in [3] to act as follows:

d_{3}(v_{3}^{4s+t}x) = \begin{cases}0 & t = 0, 1, \\ v_{2}^{2}v_{3}^{4s}xh_{20}^{3} & t = 2, \\ v_{2}^{2}v_{3}^{4s+1}xh_{20}^{3} & t = 3 \end{cases}

for x ∈ K(2)_{∗}[h_{20}] ⊗ Λ(\overline{h}_{21}, h_{30}, h_{31}, ρ_{2}). q.e.d.

5. The elements x_{n}, g_{n}, R_{n} and X_{n}, and relations between them

In order to define generators of the cohomology H^{∗}M_{1}^{1}(2) of the comodule M_{1}^{1}(2) in Definition (3.2), we introduce some elements in this section. First we redefine the elements x_{n}, which are used to give generators of H^{∗}M_{1}^{1} = E_{2}^{∗}(L_{2}\overline{M}(1, ∞)) in [7], and then observe the behavior of them under the differential d = η_{R} −η_{L}: B → Σ(2) of the cobar complex Ω_{Σ(2)}^{∗}B.

Lemma (5.1). Put x = v_{3}^{4} + v_{1}^{3}v_{2}^{6}v_{3} ∈ B. Then d(x) ≡ v_{1}^{6}v_{2}^{−2}g mod (2) in Σ(2). Here

g = t_{3}^{4} + v_{1}T_{2} + v_{1}^{8}v_{3}^{2}t_{2}^{2} + v_{1}^{10}(t_{2}^{6} + v_{2}^{2}t_{2}^{4}) + v_{1}^{14}t_{3}^{2} + v_{1}^{16}t_{2}^{4} ∈ Σ(2)

for T_{2} = v_{2}^{8}t_{2} + v_{1}v_{3}^{2}t_{2}^{4}.

Proof. This follows from the computation

d(v_{3}^{4}) ≡ v_{1}^{4}t_{2}^{8} + v_{1}^{16}t_{2}^{4} mod (2)
≡ v_{1}^{4}v_{2}^{−2}(v_{2}^{8}t_{2}^{2} + v_{1}^{2}t_{3}^{4} + v_{1}^{4}v_{3}^{2}(t_{2}^{4} + v_{1}^{6}t_{2}^{2}) + v_{1}^{12}t_{2}^{6} + v_{1}^{16}t_{3}^{2} + v_{1}^{18}t_{2}^{4}) + v_{1}^{16}t_{2}^{4} mod (2) by (3.7), and
d(v_{1}^{3}v_{2}^{6}v_{3}) ≡ v_{1}^{3}v_{2}^{6}(v_{1}t_{2}^{2} + v_{1}^{4}t_{2}) mod (2). q.e.d.

Hereafter, we put v_{2}^{2} = 1 for the sake of simplicity. In fact, we consider Z/4-module structure, and v_{2}^{2} is invariant mod (4) by Lemma (3.6). Therefore, v_{2}^{2} plays only a role adjusting the internal degrees, since every congruence is homogeneous.

Lemma (5.2). There is an element y of B such that

d(y) ≡ v_{1}^{10}g^{4} + v_{1}^{12}xg + v_{1}^{14}r^{4} + v_{1}^{16}r′ mod (2, v_{1}^{17}) ∈ Σ(2)

for r = t_{4}^{2} + t_{4} and r′ = v_{3}^{2}t_{3}^{8} + v_{3}^{8}t_{3}^{2}.

Proof. Put y′ = x^{2} + v_{1}^{9}v_{2}v_{3}^{2} + v_{1}^{11}v_{3}^{9} + v_{1}^{13}v_{3} + v_{1}^{13}v_{2}v_{3}^{10}, and we have d(y′) ≡ v_{1}^{11}T_{2} + v_{1}^{13}v_{3}t_{2}^{2} + v_{1}^{14}t_{4}^{4} + v_{1}^{16}(r′ + t_{2}^{2}t_{3}^{4} + v_{3}t_{2}) mod (2, v_{1}^{17}). Indeed, it is the sum of the following congruences mod (2, v_{1}^{17}):

d(x^{2}) ≡ v_{1}^{12}(t_{3}^{8} + v_{1}^{2}t_{2}^{2} + v_{1}^{4}v_{3}^{4}t_{2}^{8})
≡ v_{1}^{12}(t_{3}^{2} + v_{3}^{2}t_{2}^{4} + v_{3}^{8}t_{2}^{2} + v_{1}^{2}t_{4}^{4} + v_{1}^{4}v_{3}^{2}t_{3}^{8} + v_{1}^{4}t_{2}^{2}t_{3}^{4} + v_{1}^{2}t_{2}^{2}),
d(v_{1}^{9}v_{2}v_{3}^{2}) ≡ v_{1}^{11}v_{2}t_{2}^{4}
≡ v_{1}^{11}(t_{2} + v_{1}t_{3}^{2} + v_{1}^{2}v_{3}t_{2}^{2} + v_{1}^{5}v_{3}t_{2}),
d(v_{1}^{11}v_{3}^{9}) ≡ v_{1}^{12}v_{3}^{8}(t_{2}^{2} + v_{1}^{3}t_{2})
≡ v_{1}^{12}v_{3}^{8}(t_{2}^{2} + v_{1}^{3}(v_{2}t_{2}^{4} + v_{1}t_{3}^{2})),
d(v_{1}^{13}v_{3}) ≡ v_{1}^{14}t_{2}^{2} and
d(v_{1}^{13}v_{2}v_{3}^{10}) ≡ v_{1}^{15}v_{2}v_{3}^{8}t_{2}^{4},

in which we use relations in (3.7).

We now put y = y′ + v_{1}^{4}x + v_{1}^{8}v_{3}^{18} + v_{1}^{9}v_{3}^{5} + v_{1}^{12}v_{3}^{20}, and compute mod (2, v_{1}^{17}),

d(y′) ≡ v_{1}^{11}T_{2} + v_{1}^{13}v_{3}t_{2}^{2} + v_{1}^{14}t_{4}^{4} + v_{1}^{16}(r′ + t_{2}^{2}t_{3}^{4} + v_{3}t_{2}),
d(v_{1}^{4}x) ≡ v_{1}^{10}(t_{3}^{4} + v_{1}T_{2})
≡ v_{1}^{10}(t_{3}^{16} + v_{3}^{4}t_{2}^{8} + v_{3}^{16}t_{2}^{4} + v_{1}^{4}t_{4}^{8} + v_{1}T_{2})
≡ v_{1}^{10}(t_{3}^{16} + v_{3}^{4}(t_{2}^{2}1 + v_{1}^{2}t_{3}^{4} + v_{1}^{4}v_{3}^{2}t_{2}^{4}) + v_{3}^{16}t_{2}^{4} + v_{1}^{4}t_{4}^{8} + v_{1}T_{2}),
d(v_{1}^{8}v_{3}^{18}) ≡ v_{1}^{10}v_{3}^{16}t_{2}^{4} + v_{1}^{16}v_{3}^{16}t_{2}^{2},
d(v_{1}^{9}v_{3}^{5}) ≡ v_{1}^{10}v_{3}^{4}t_{2}^{2} + v_{1}^{13}v_{3}^{4}t_{2} + v_{1}^{13}v_{3}t_{2}^{8} + v_{1}^{14}t_{2}^{10} and
d(v_{1}^{12}v_{3}^{20}) ≡ v_{1}^{16}v_{3}^{16}t_{2}^{8} ≡ v_{1}^{16}v_{3}^{16}t_{2}^{2}

to obtain

d(y) ≡ v_{1}^{10}(t_{3}^{16} + v_{1}^{4}t_{2}^{4}) + v_{1}^{12}v_{3}^{4}(t_{3}^{4} + v_{1}t_{2} + v_{1}^{2}v_{3}^{2}t_{2}^{4})
+v_{1}^{14}(t_{4}^{8} + t_{4}^{4}) + v_{1}^{15}v_{3}(t_{3}^{4} + v_{1}t_{2}) + v_{1}^{16}r′
≡ v_{1}^{10}g^{4} + v_{1}^{12}xg + v_{1}^{14}r^{4} + v_{1}^{16}r′.

Here we use relations v_{1}^{13}v_{3}t_{2}^{2} + v_{1}^{13}v_{3}t_{2}^{8} = v_{1}^{15}v_{3}t_{3}^{4} and v_{1}^{16}t_{2}^{2}t_{3}^{4} + v_{1}^{14}t_{2}^{10} = v_{1}^{14}t_{2}^{4}, and notice that x = v_{3}^{4} + v_{1}^{3}v_{3} and g^{4} ≡ t_{3}^{16} + v_{1}^{4}t_{2}^{4} mod (2, v_{1}^{8}). q.e.d.

We now define elements x_{k} ∈ B for k ≥ 0 inductively by

(5.3)  x_{0} = v_{3}, x_{1} = x = v_{3}^{4} + v_{1}^{3}v_{3} and x_{k+1} = x_{k}^{4} + v_{1}^{a_{k+1}−12}x^{4e_{k−1}}y

for the integers a_{n} and e_{n} in (2.8), and consider elements g_{k} ∈ Σ(2) satisfying

(5.4)  g_{1} = g and d(x^{4e_{k−1}}y) ≡ v_{1}^{10}g_{k}^{4} + v_{1}^{12}g_{k+1} mod (2) for k > 0.

Note that g_{k} is a well-defined element if we consider it modulo (2).

Lemma (5.5). For k > 0,

d(x_{k}) ≡ v_{1}^{a_{k}}g_{k} mod (2).

Proof. This is verified inductively by definition. Indeed, d(x_{k+1}) ≡ d(x_{k}^{4}) + v_{1}^{a_{k+1}−12}d(x^{4e_{k−1}}y) ≡ v_{1}^{4a_{k}}g_{k}^{4} + v_{1}^{a_{k+1}−12}(v_{1}^{10}g_{k}^{4} + v_{1}^{12}g_{k+1}) mod (2), and a_{k+1} = 4a_{k} + 2. q.e.d.

Lemma (5.6). For k > 1,

g_{k} ≡ x^{e_{k−1}}g + v_{1}^{2}x^{4e_{k−2}}r^{4} mod (2, v_{1}^{3}).

Here r = t_{4}^{2} + t_{4} as above.

Proof. For k = 2, it follows from Lemma (5.2) and (5.4). Suppose that the lemma holds for k > 1. Then, Lemma (5.2) and the definition (5.4) show that

v_{1}^{12}g_{k+1} ≡ d(x^{4e_{k−1}}y)+v_{1}^{10}g_{k}^{4} ≡ x^{4e_{k−1}}(v_{1}^{10}g^{4}+v_{1}^{12}xg+v_{1}^{14}r^{4})+v_{1}^{10}x^{4e_{k−1}}g^{4} ≡ v_{1}^{12}x^{e_{k}}g+v_{1}^{14}x^{4e_{k−1}}r^{4} mod (2, v_{1}^{15}).

Since v_{1} acts monomorphically on the cobar complex, we obtain the lemma. q.e.d.

We introduce an element R_{n} ∈ Σ(2) satisfying

(5.7)  v_{1}^{e_{n}+1}R_{n} ≡ g_{n+1} + x_{n}g_{n} mod (2)

for n > 0. Note also that R_{n} is well-defined modulo (2).

Lemma (5.8). R_{1} ≡ r^{4} + v_{1}^{2}r′ mod (2, v_{1}^{3}) and R_{2} ≡ R_{1}^{4} mod (2, v_{1}^{6}). For k > 1, there is a cochain w_{k} such that

d(w_{k}) ≡ v_{1}^{e_{k+1}+13}(R_{k}^{4} + R_{k+1}) mod (2, v_{1}^{a_{k+1}−12}).

Proof. The congruences on R_{1} and R_{2} follow from (5.7), (5.3) and (5.4). Indeed, v_{1}^{14}R_{1} ≡ v_{1}^{12}g_{2} + v_{1}^{12}x_{1}g_{1} ≡ d(y) + v_{1}^{10}g^{4} + v_{1}^{12}xg mod (2) , which is congruent to v_{1}^{14}r^{4} + v_{1}^{16}r′ mod (2, v_{1}^{17}) by Lemma (5.2). Thus, the first congruence follows. For the second congruence, we compute

d(x^{4}y) ≡ v_{1}^{10}g_{2}^{4} + v_{1}^{12}g_{3} mod (2)
≡ v_{1}^{10}(x_{1}^{4}g_{1}^{4} + v_{1}^{8}R_{1}^{4}) + v_{1}^{12}(x_{2}g_{2} + v_{1}^{6}R_{2}) mod (2)
≡ x_{1}^{4}(v_{1}^{10}g_{1}^{4} + v_{1}^{12}g_{2}) + v_{1}^{18}(R_{1}^{4} + R_{2}) mod (2, v_{1}^{26})
≡ x_{1}^{4}d(y) + v_{1}^{18}(R_{1}^{4} + R_{2}) mod (2, v_{1}^{24}),

and obtain v_{1}^{18}(R_{1}^{4} + R_{2}) ≡ 0 mod (2, v_{1}^{24}), since d(x^{4}y) ≡ x_{1}^{4}d(y) mod (2, v_{1}^{24}).

By (5.4) and Lemma (5.5), we see that

d(x^{4e_{k}}y) ≡ v_{1}^{10}g_{k+1}^{4} + v_{1}^{12}g_{k+2} mod (2),
d(x_{k}^{4}x^{4e_{k−1}}y) ≡ v_{1}^{10}x_{k}^{4}g_{k}^{4} + v_{1}^{12}x_{k}^{4}g_{k+1} mod (2, v_{1}^{4a_{k}}).

Put w_{k} = x^{4e_{k}}y + x_{k}^{4}x^{4e_{k−1}}y. Then,

d(w_{k}) ≡ v_{1}^{10}(g_{k+1}^{4} + x_{k}^{4}g_{k}^{4}) + v_{1}^{12}(g_{k+2} + x_{k}^{4}g_{k+1}) mod (2, v_{1}^{4a_{k}})
≡ v_{1}^{10+4e_{k}+4}R_{k}^{4} + v_{1}^{12+e_{k+1}+1}R_{k+1} mod (2, v_{1}^{4a_{k}})

by (5.7). The last congruence now follows from the relations 10 + 4e_{k} + 4 = 12 + e_{k+1} + 1 and 4a_{k} = a_{k+1} − 2. q.e.d.

We have homologous relations between R_{n}’s:

Lemma (5.9). There are elements u_{n} and u′_{n} for n > 0 such that

d(u_{1}) ≡ v_{1}^{10}(R_{1}^{2} + R_{1} + x_{1}g_{1}^{2}) mod (2, v_{1}^{12}), and
d(u_{n}) ≡ v_{1}^{10×4^{n−1}}(R_{n}^{2} + R_{n} + v_{1}^{4e_{n−1}}x_{n}g_{n}^{2} + v_{1}^{a_{n−1}}x_{n}g_{n}) mod (2, v_{1}^{3×4^{n}}) for n > 1; and
d(u′_{n}) ≡ v_{1}^{5×4^{n}}(R_{n+1} + R_{n}^{2} + v_{1}^{2e_{n}}x_{n}^{2}g_{n+1} + v_{1}^{2a_{n−1}}x_{n}^{2}g_{n}^{2}) mod (2, v_{1}^{6×4^{n}}) for n > 0.

Proof. Put u_{1} = v_{1}^{4}v_{3}^{32}x + v_{3}^{32}y′ + v_{1}^{8}v_{3}^{14} + v_{1}^{11}v_{3}^{11} + v_{1}^{9}v_{3}^{37}. Then we compute

d(v_{1}^{4}v_{3}^{32}x) ≡ v_{1}^{10}v_{3}^{32}(t_{3}^{4} + v_{1}T_{2}) mod (2, v_{1}^{18}),
d(v_{3}^{32}y′) ≡ v_{3}^{32}(v_{1}^{11}T_{2} + v_{1}^{13}v_{3}t_{2}^{2} + v_{1}^{14}t_{4}^{4} + v_{1}^{16}(r′ + t_{2}^{2}t_{3}^{4} + v_{3}t_{2})) mod (2, v_{1}^{17}),
d(v_{1}^{8}v_{3}^{14}) ≡ v_{1}^{8}v_{3}^{8}(v_{3}^{4} + v_{1}^{4}t_{2}^{8})(v_{3}^{2} + v_{1}^{2}t_{2}^{4}) − v_{1}^{8}v_{3}^{14} mod (2, v_{1}^{14})
≡ v_{1}^{10}v_{3}^{12}t_{2}^{4} + v_{1}^{12}v_{3}^{10}t_{2}^{8} mod (2, v_{1}^{14})
≡ v_{1}^{10}v_{3}^{12}t_{2}^{4} + v_{1}^{12}v_{3}^{10}t_{2}^{2} mod (2, v_{1}^{14}) by (3.7),
d(v_{1}^{11}v_{3}^{11}) ≡ v_{1}^{11}v_{3}^{8}(v_{3}^{2} + v_{1}^{2}t_{2}^{4})(v_{3} + v_{1}t_{2}^{2}) − v_{1}^{11}v_{3}^{11} mod (2, v_{1}^{14})
≡ v_{1}^{12}v_{3}^{10}t_{2}^{2} + v_{1}^{13}v_{3}^{9}t_{2}^{4} mod (2, v_{1}^{14}) and
d(v_{1}^{9}v_{3}^{37}) ≡ v_{1}^{9}v_{3}^{32}(v_{3}^{4} + v_{1}^{4}t_{2}^{8})(v_{3} + v_{1}t_{2}^{2} + v_{1}^{4}t_{2}) − v_{1}^{9}v_{3}^{37} mod (2, v_{1}^{14})
≡ v_{1}^{10}v_{3}^{36}t_{2}^{2} + v_{1}^{13}(v_{3}^{36}t_{2} + v_{3}^{33}t_{2}^{8}) mod (2, v_{1}^{14})
≡ v_{1}^{10}v_{3}^{36}(t_{2}^{8} + v_{1}^{2}t_{3}^{4}) + v_{1}^{13}(v_{3}^{36}t_{2} + v_{3}^{33}t_{2}^{8}) mod (2, v_{1}^{14}) by (3.7)

and obtain

d(u_{1}) ≡ v_{1}^{10}(v_{3}^{32}t_{3}^{4} + v_{3}^{12}t_{2}^{4} + v_{3}^{36}t_{2}^{8}) + v_{1}^{12}v_{3}^{36}(t_{3}^{4} + v_{1}t_{2}) + v_{1}^{13}v_{2}v_{3}^{9}t_{2} mod (2, v_{1}^{14}).

Here, we have the relation 0 ≡ r^{8} + r^{4} + v_{3}^{4}t_{3}^{8} + v_{3}^{32}t_{3}^{4} + v_{3}^{12}t_{2}^{4} + v_{3}^{36}t_{2}^{8} mod (2, v_{1}^{4}). Indeed, we compute

r^{8} + r^{4} = t_{4}^{16} + t_{4}^{8} + t_{4}^{8} + t_{4}^{4} = t_{4}^{16} + t_{4}^{4}
= t_{4}^{4} + v_{3}^{4}t_{3}^{8} + v_{3}^{32}t_{3}^{4} + v_{3}^{12}t_{2}^{4} + v_{3}^{36}t_{2}^{8} + t_{4}^{4} mod (2, v_{1}^{4})

by the relation in (3.8). It follows that d(u_{1}) ≡ v_{1}^{10}(r^{8} + r^{4} + v_{3}^{4}t_{3}^{8}) mod (2, v_{1}^{12}). Notice that R_{1} ≡ r^{4} mod (2, v_{1}^{2}) by Lemma (5.8), v_{3}^{4} ≡ x_{1} mod (2, v_{1}^{3}) by (5.3) and g_{1} ≡ t_{3}^{4} mod (2, v_{1}) by (5.4) and Lemma (5.1), and we obtain

d(u_{1}) ≡ v_{1}^{10}(R_{1}^{2} + R_{1} + x_{1}g_{1}^{2}) mod (2, v_{1}^{12}).

We now turn to the case for n = 2. Square the above congruence, and we have

d(u_{1}^{2}) ≡ v_{1}^{20}(R_{1}^{4} + R_{1}^{2} + x_{1}^{2}g_{1}^{4}) mod (2, v_{1}^{24}).

By Lemmas (5.1) and (5.2),

d(v_{1}^{10}x_{1}^{2}y) ≡ v_{1}^{20}x_{1}^{2}(g_{1}^{4} + v_{1}^{2}g_{2}) + v_{1}^{22}yg_{1}^{2} mod (2, v_{1}^{24}),

where x = x_{1} and g = g_{1} by (5.3) and (5.4). Put u′′_{1} = u_{1}^{2} + v_{1}^{10}x_{1}^{2}y. Since y ≡ x_{1}^{2} mod (2, v_{1}^{2}) by the definition in Lemma (5.2), we obtain

d(u′′_{1}) ≡ v_{1}^{20}(R_{1}^{4} + R_{1}^{2} + v_{1}^{2}x_{1}^{2}g_{2} + v_{1}^{2}x_{1}^{2}g_{1}^{2}) mod (2, v_{1}^{24}),

whence u′_{1} = v_{1}^{2}w_{1} + u′′_{1} satisfies the lemma for n = 1 by Lemma (5.8). Suppose that

d(u′_{n}) ≡ v_{1}^{5×4^{n}}(R_{n+1} + R_{n}^{2} + v_{1}^{2e_{n}}x_{n}^{2}g_{n+1} + v_{1}^{2a_{n−1}}x_{n}^{2}g_{n}^{2}) mod (2, v_{1}^{6×4^{n}}).

Squaring this,

d(u′_{n}^{2}) ≡ v_{1}^{10×4^{n}}(R_{n+1}^{2} + R_{n}^{4} + v_{1}^{4e_{n}}x_{n}^{4}g_{n+1}^{2} + v_{1}^{4a_{n−1}}x_{n}^{4}g_{n}^{4}) mod (2, v_{1}^{3×4^{n+1}}).

The elements x_{n}^{4}, g_{n}^{4} and R_{n}^{4} are homologous to x_{n+1}, v_{1}^{2}g_{n+1} and R_{n+1} by (5.3), (5.4) and Lemma (5.8), respectively. We define u_{n+1} to be the sum of u′_{n}^{2} and the elements that give these homologous relations, and we obtain the congruence on d(u_{n+1}). Here we notice that a_{n} = 4a_{n−1} + 2.

Squaring the congruence on d(u_{n+1}), we have

d(u_{n+1}^{2}) ≡ v_{1}^{5×4^{n+1}}(R_{n+1}^{4} + R_{n+1}^{2} + v_{1}^{8e_{n}}x_{n+1}^{2}g_{n+1}^{4} + v_{1}^{2a_{n}}x_{n+1}^{2}g_{n+1}^{2}) mod (2, v_{1}^{6×4^{n+1}}).

We also have

d(v_{1}^{5×4^{n+1}+8e_{n}−10}x_{n+1}^{2}x^{4e_{n}}y) ≡ v_{1}^{5×4^{n+1}+8e_{n}}x_{n+1}^{2}(g_{n+1}^{4} + v_{1}^{2}g_{n+2}) mod (2, v_{1}^{6×4^{n+1}}).

Since 8e_{n} + 2 = 2e_{n+1}, we put u′_{n+1} = u_{n+1}^{2} + v_{1}^{5×4^{n+1}−e_{n+2}−13}w_{n+1} + v_{1}^{5×4^{n+1}+8e_{n}−10}x_{n+1}^{2}x^{4e_{n}}y, and obtain the congruence on d(u′_{n+1}) by Lemma (5.8). This completes the induction. q.e.d.

We also consider the elements x_{n,1} = 2x_{n} + v_{1}^{5×4^{n−1}}x_{n−1}^{2} and x′_{n,1} = 2x_{n}^{2} + v_{1}^{10×4^{n−1}}x_{n−1}^{4}.

Lemma (5.10). For n > 1,

d(x_{n,1}) ≡ 2v_{1}^{a_{n}+e_{n−1}+1}R_{n−1} (= 2v_{1}^{7×4^{n−1}}R_{n−1}) mod (4, v_{1}^{2×4^{n}+e_{n−1}−1}) = (4, v_{1}^{7×4^{n−1}+e_{n}−1}),
d(x′_{n,1}) ≡ 2v_{1}^{2a_{n}+2e_{n−1}+2}R_{n−1}^{2} (= 2v_{1}^{14×4^{n−1}}R_{n−1}^{2}) mod (4, v_{1}^{4^{n+1}+2e_{n−1}−2}) = (4, v_{1}^{14×4^{n−1}+2e_{n}−2}).

Proof. These follow from the computation:

d(2x_{n}) ≡ 2v_{1}^{a_{n}}g_{n} ≡ 2v_{1}^{a_{n}}x_{n−1}g_{n−1} + 2v_{1}^{a_{n}+e_{n−1}+1}R_{n−1} mod (4),
d(v_{1}^{5×4^{n−1}}x_{n−1}^{2}) ≡ 2v_{1}^{5×4^{n−1}+a_{n−1}}x_{n−1}g_{n−1} mod (4, v_{1}^{5×4^{n−1}+2a_{n−1}}) = (4, v_{1}^{2×4^{n}+e_{n−1}−1});
d(2x_{n}^{2}) ≡ 2v_{1}^{2a_{n}}g_{n}^{2} ≡ 2v_{1}^{2a_{n}}x_{n−1}^{2}g_{n−1}^{2} + 2v_{1}^{2a_{n}+2e_{n−1}+2}R_{n−1}^{2} mod (4),
d(v_{1}^{10×4^{n−1}}x_{n−1}^{4}) ≡ 2v_{1}^{10×4^{n−1}+2a_{n−1}}x_{n−1}^{2}g_{n−1}^{2} mod (4, v_{1}^{10×4^{n−1}+4a_{n−1}}) = (4, v_{1}^{4^{n+1}+2e_{n−1}−2}).

q.e.d.

Note that d(x_{n}^{2}) ≡ 2v_{1}^{a_{n}}x_{n}g_{n} mod (4, v_{1}^{2a_{n}}) and e_{n} + 1 + 2 × 4^{n−1} = 2a_{n−1} + 2. Then the above two lemmas imply the following:

Lemma (5.11). Put \overline{u}_{n} = 2v_{1}^{a_{n+1}+e_{n}+1−10×4^{n−1}}u_{n}+x_{n+1,1}+v_{1}^{e_{n+2}+a_{n−1}+1}x_{n}^{2} and \overline{u}′_{n} = 2v_{1}^{2a_{n+1}+2e_{n}+2−5×4^{n}}u′_{n}+ x′_{n+1,1} + v_{1}^{2e_{n+2}+2a_{n−1}+2}x_{n}^{4}. Then,

d(\overline{u}_{n}) ≡ 2v_{1}^{a_{n+1}+e_{n}+1}(R_{n}^{2} + v_{1}^{4e_{n−1}}x_{n}g_{n}^{2}) mod (4, v_{1}^{a_{n+1}+2a_{n−1}+2}) and
d(\overline{u}′_{n}) ≡ 2v_{1}^{2a_{n+1}+2e_{n}+2}(R_{n+1} + v_{1}^{2e_{n}}x_{n}^{2}g_{n+1}) mod (4, v_{1}^{2a_{n+1}+a_{n}+2})

for n > 0.

Lemma (5.12). There exist elements ũ_{n} and u′_{n} such that

d(ũ_{n}) ≡ 2v_{1}^{a_{n+1}+e_{n}+1}(R_{n}^{2} + v_{1}^{6×4^{n−2}}x_{n−1}^{6}R_{n−2}^{2}) mod (4, v_{1}^{a_{n+1}+2a_{n−1}+2}) for n > 2,
d(u′_{n}) ≡ 2v_{1}^{2a_{n+1}+2e_{n}+2}(R_{n+1} + v_{1}^{3×4^{n−1}}x_{n}^{3}R_{n−1}) mod (4, v_{1}^{2a_{n+1}+a_{n}+2}) for n ≥ 2.

10 Ippei Ichigi, Katsumi Shimomura, and Xiangjun Wang

Proof. For n > 2, put ũ_{n} = \overline{u}_{n} + v_{1}^{a_{n+1}+2e_{n}−2a_{n−2}}x_{n−1}^{6}x_{n−2}^{4}, and the first one follows from

d(\overline{u}_{n}) ≡ 2v_{1}^{a_{n+1}+e_{n}+1}(R_{n}^{2} + v_{1}^{4e_{n−1}}x_{n}g_{n}^{2})
≡ 2v_{1}^{a_{n+1}+e_{n}+1}(R_{n}^{2} + v_{1}^{4e_{n−1}}x_{n−1}^{6}(x_{n−2}^{2}g_{n−2}^{2} + v_{1}^{2e_{n−2}+2}R_{n−2}^{2})) mod (4, v_{1}^{a_{n+1}+2e_{n}+2e_{n−1}+2}),
d(v_{1}^{a_{n+1}+2e_{n}−2a_{n−2}}x_{n−1}^{6}x_{n−2}^{4})
≡ 2v_{1}^{a_{n+1}+2e_{n}}x_{n−1}^{6}x_{n−2}^{2}g_{n−2}^{2} mod (4, v_{1}^{a_{n+1}+2e_{n}+2a_{n−2}}).

Here, note that 2a_{n−1} = 2e_{n} + 2e_{n−1} = e_{n} + 2 × 4^{n−1} − 1.

In a similar way, the second congruence follows from

d(\overline{u}′_{n}) ≡ 2v_{1}^{2a_{n+1}+2e_{n}+2}(R_{n+1} + v_{1}^{2e_{n}}x_{n}^{2}g_{n+1})
≡ 2v_{1}^{2a_{n+1}+2e_{n}+2}(R_{n+1} + v_{1}^{2e_{n}}x_{n}^{3}(x_{n−1}g_{n−1} + v_{1}^{e_{n−1}+1}R_{n−1})) mod (4, v_{1}^{2a_{n+1}+5e_{n}+3}),
d(v_{1}^{2a_{n+1}+4e_{n}+2−a_{n−1}}x_{n}^{3}x_{n−1}^{2})
≡ 2v_{1}^{2a_{n+1}+4e_{n}+2}x_{n}^{3}x_{n−1}g_{n−1} mod (4, v_{1}^{2a_{n+1}+4e_{n}+2+a_{n−1}}).

We also notice that a_{n} − 1 = 5e_{n} = 2e_{n} + 4^{n} − 1. q.e.d.

In the same manner as x_{n,1} and x′_{n,1}, we consider

y_{n,1} = x_{n−1}^{4} + v_{1}^{2a_{n−1}−2a_{n−2}}x_{n−1}^{2}x_{n−2}^{4} and y′_{n,1} = x_{n}^{2} + v_{1}^{a_{n}−a_{n−1}}x_{n}x_{n−1}^{2}.

Lemma (5.13).

d(y_{n,1}) ≡ 2v_{1}^{2a_{n−1}+2e_{n−2}+2}x_{n−1}^{2}R_{n−2}^{2} (= 2v_{1}^{14×4^{n−2}}x_{n−1}^{2}R_{n−2}^{2}) mod (4, v_{1}^{2a_{n−1}+2a_{n−2}}) for n > 2,
d(y′_{n,1}) ≡ 2v_{1}^{a_{n}+e_{n−1}+1}x_{n}R_{n−1} (= 2v_{1}^{7×4^{n−1}}x_{n}R_{n−1}) mod (4, v_{1}^{a_{n}+a_{n−1}}) for n > 1.

Proof. These follow from

d(x_{n−1}^{4}) ≡ 2v_{1}^{2a_{n−1}}x_{n−1}^{2}g_{n−1}^{2} mod (4, v_{1}^{4a_{n−1}})
≡ 2v_{1}^{2a_{n−1}}x_{n−1}^{2}(x_{n−2}^{2}g_{n−2}^{2} + v_{1}^{2e_{n−2}+2}R_{n−2}^{2}) mod (4, v_{1}^{4a_{n−1}}),
d(v_{1}^{2a_{n−1}−2a_{n−2}}x_{n−1}^{2}x_{n−2}^{4}) ≡ 2v_{1}^{2a_{n−1}}x_{n−1}^{2}x_{n−2}^{2}g_{n−2}^{2} mod (4, v_{1}^{2a_{n−1}+2a_{n−2}}); and
d(x_{n}^{2}) ≡ 2v_{1}^{a_{n}}x_{n}g_{n} mod (4, v_{1}^{2a_{n}})
≡ 2v_{1}^{a_{n}}x_{n}(x_{n−1}g_{n−1} + v_{1}^{e_{n−1}+1}R_{n−1}) mod (4, v_{1}^{2a_{n}}),
d(v_{1}^{a_{n}−a_{n−1}}x_{n}x_{n−1}^{2}) ≡ 2v_{1}^{a_{n}}x_{n}x_{n−1}g_{n−1} mod (4, v_{1}^{a_{n}+a_{n−1}}). q.e.d.

Lemma (5.14). d(x_{n−1}^{2}) ≡ 2v_{1}^{a_{n−1}}g_{n} + 2v_{1}^{2×4^{n−1}}R_{n−1} mod (4, v_{1}^{2a_{n−1}}) for n ≥ 2 and d(x_{n−1}^{4}) ≡ 2v_{1}^{2a_{n−1}}g_{n}^{2} + 2v_{1}^{4^{n}}R_{n−1}^{2} mod (4, v_{1}^{4a_{n−1}}) for n ≥ 2. Proof. These follow from Lemma (5.5) and the definition (5.7). q.e.d.

Let b_{n,k} for n, k > 0 be integers defined by

(5.15)  b_{n,k} = 4^{n} + 3 × 4^{n−2k+3}\overline{e}_{k−1} − 2 × 4^{n−2k}

and consider the elements

(5.16)  \begin{aligned}X_{n} & = & x_{n,1} + ∑_{k=1}^{[ \frac{n}{2} ]−1} v_{1}^{b_{n,k}}x_{n−2k+2}^{3\overline{e}_{k−1}}u′_{n−2k}, \\ X′_{n} & = & x′_{n,1} + ∑_{k=1}^{[ \frac{n}{2} ]−1} v_{1}^{2b_{n,k}}x_{n−2k+2}^{6\overline{e}_{k−1}}ũ_{n−2k+1}, \\ Y_{n} & = & y_{n,1} + ∑_{k=1}^{[ \frac{n−1}{2} ]−1} v_{1}^{2b_{n−1,k}}x_{n−1}^{2}x_{n−2k+1}^{6\overline{e}_{k−1}}ũ_{n−2k}, \\ Y′_{n} & = & y′_{n,1} + ∑_{k=1}^{[ \frac{n}{2} ]−1} v_{1}^{b_{n,k}}x_{n}x_{n−2k+2}^{3\overline{e}_{k−1}}u′_{n−2k}, \end{aligned}

where integers [x] and \overline{e}_{n} are those given in (2.9).

Lemma (5.17). The elements in (5.16) satisfy X_{n} ≡ 2x_{n}, X′_{n} ≡ 2x_{n}^{2}, Y_{n} ≡ x_{n−1}^{4} and Y′_{n} ≡ x_{n}^{2} mod (4, v_{1}^{2}), and

d(X_{n}) ≡ 2v_{1}^{c_{n}}x_{2+ε(n)}^{3\overline{e}_{[ \frac{n}{2} ]−1}}R_{1+ε(n)} mod (4, v_{1}^{c_{n}+4^{1+ε(n)}}),
d(X′_{n}) ≡ 2v_{1}^{2c_{n}}x_{2+ε(n)}^{6\overline{e}_{[ \frac{n}{2} ]−1}}R_{1+ε(n)}^{2} mod (4, v_{1}^{2c_{n}+2×4^{1+ε(n)}}),
d(Y_{n}) ≡ 2v_{1}^{2c_{n−1}}x_{n−1}^{2}x_{2+ε(n−1)}^{6\overline{e}_{[ \frac{n−1}{2} ]−1}}R_{1+ε(n−1)}^{2} mod (4, v_{1}^{2c_{n−1}+2×4^{1+ε(n−1)}}) and
d(Y′_{n}) ≡ 2v_{1}^{c_{n}}x_{n}x_{2+ε(n)}^{3\overline{e}_{[ \frac{n}{2} ]−1}}R_{1+ε(n)} mod (4, v_{1}^{c_{n}+4^{1+ε(n)}})

for integers [x], c_{n}, ε(n) and \overline{e}_{n} in (2.9).

Proof. Note that the integers b_{n,k} in (5.15) satisfy

b_{n,1} = a_{n} + e_{n−1} − 2a_{n−1} − 2e_{n−2} − 1 and
b_{n,k+1} = b_{n,k} + 2a_{n−2k+1} + 2e_{n−2k} + 3 × 4^{n−2k−1} − 2a_{n−2k−1} − 2e_{n−2k−2};
2b_{n,1} = 2a_{n} + 2e_{n−1} − a_{n} − e_{n−1} + 1 and
2b_{n,k+1} = 2b_{n,k} + a_{n−2k+2} + e_{n−2k+1} + 6 × 4^{n−2k−1} − a_{n−2k} − e_{n−2k−1}.

Indeed, a_{n} + e_{n−1} = 7 × 4^{n−1} − 1.

The differentials on the elements in (5.16) then follow immediately from Lemmas (5.10), (5.12) and (5.13) as follows:

d(X_{n}) ≡ 2v_{1}^{a_{n}+e_{n−1}+1}R_{n−1}

+ ∑_{k=1}^{[ \frac{n}{2} ]−1} v_{1}^{b_{n,k}}x_{n−2k+2}^{3\overline{e}_{k−1}}(2v_{1}^{2a_{n−2k+1}+2e_{n−2k}+2}(R_{n−2k+1} + v_{1}^{3×4^{n−2k−1}}x_{n−2k}^{3}R_{n−2k−1}))

≡ 2v_{1}^{b_{n,[ \frac{n}{2} ]−1}+2a_{3+ε(n)}+2e_{2+ε(n)}+2+3×4^{1+ε(n)}}x_{2+ε(n)}^{3\overline{e}_{[ \frac{n}{2} ]−1}}R_{1+ε(n)} mod (4, v_{1}^{c_{n}+4^{1+ε(n)}}),
d(X′_{n}) ≡ 2v_{1}^{2a_{n}+2e_{n−1}+2}R_{n−1}^{2}

+ ∑_{k=1}^{[ \frac{n}{2} ]−1} v_{1}^{2b_{n,k}}x_{n−2k+2}^{6\overline{e}_{k−1}}(2v_{1}^{a_{n−2k+2}+e_{n−2k+1}+1}(R_{n−2k+1}^{2} + v_{1}^{6×4^{n−2k−1}}x_{n−2k}^{6}R_{n−2k−1}^{2}))

≡ 2v_{1}^{2b_{n,[ \frac{n}{2} ]−1}+a_{4+ε(n)}+e_{3+ε(n)}+1+6×4^{1+ε(n)}}x_{2+ε(n)}^{6\overline{e}_{[ \frac{n}{2} ]−1}}R_{1+ε(n)}^{2} mod (4, v_{1}^{2c_{n}+2×4^{1+ε(n)}}),
d(Y_{n}) ≡ 2v_{1}^{2a_{n−1}+2e_{n−2}+2}x_{n−1}^{2}R_{n−2}^{2}

+ ∑_{k=1}^{[ \frac{n−1}{2} ]−1} v_{1}^{2b_{n−1,k}}x_{n−1}^{2}x_{n−2k+1}^{6\overline{e}_{k−1}}(2v_{1}^{a_{n−2k+1}+e_{n−2k}+1}(R_{n−2k}^{2} + v_{1}^{6×4^{n−2k−2}}x_{n−2k−1}^{6}R_{n−2k−2}^{2}))

≡ 2v_{1}^{2b_{n−1,[ \frac{n−1}{2} ]−1}+a_{3+ε(n−1)}+e_{2+ε(n−1)}+1+6×4^{ε(n−1)}}x_{n−1}^{2}x_{2+ε(n−1)}^{6\overline{e}_{[ \frac{n−1}{2} ]−1}}R_{1+ε(n−1)}^{2} mod (4, v_{1}^{2c_{n−1}+2×4^{1+ε(n−1)}}),
d(Y′_{n}) ≡ 2v_{1}^{a_{n}+e_{n−1}+1}x_{n}R_{n−1}

+ ∑_{k=1}^{[ \frac{n}{2} ]−1} v_{1}^{b_{n,k}}x_{n}x_{n−2k+2}^{3\overline{e}_{k−1}}(2v_{1}^{2a_{n−2k+1}+2e_{n−2k}+2}(R_{n−2k+1} + v_{1}^{3×4^{n−2k−1}}x_{n−2k}^{3}R_{n−2k−1}))

≡ 2v_{1}^{b_{n,[ \frac{n}{2} ]−1}+2a_{3+ε(n)}+2e_{2+ε(n)}+2+3×4^{1+ε(n)}}x_{n}x_{2+ε(n)}^{3\overline{e}_{[ \frac{n}{2} ]−1}}R_{1+ε(n)} mod (4, v_{1}^{c_{n}+4^{1+ε(n)}}).

Note that 2[\frac{n}{2}] = n − ε(n) by definition (2.9), and we obtain the lemma. q.e.d.

Lemma (5.18). There exists an element χ such that d(χ) ≡ 2v_{1}^{5}g_{1}^{2} + 2v_{1}^{7}R_{1} + 2v_{1}^{7}v_{2}v_{3}^{3}\overline{t}_{21} + 2v_{1}^{7}v_{2}v_{3}^{6}t_{2} mod (4, v_{1}^{8}) for r = t_{4} + t_{4}^{2} in Lemma (5.2) and \overline{t}_{21} = t_{2}^{2} + v_{2}t_{2}.

Proof. Mod (4, v_{1}^{8})

2v_{1}^{5}g_{1}^{2} ≡ 2v_{1}^{5}(t_{3}^{2} + v_{3}^{2}t_{2}^{4} + v_{3}^{8}t_{2}^{2} + v_{1}^{2}t_{4}^{4} + v_{1}^{2}t_{2}^{2}),
d(2v_{1}^{2}v_{2}v_{3}^{2}) ≡ 2v_{1}^{4}v_{2}t_{2}^{4},
d(v_{1}^{4}v_{2}) ≡ 2v_{1}^{4}t_{2},
d(v_{1}^{3}v_{3}^{4}) ≡ 2v_{1}^{5}v_{3}^{2}t_{2}^{4} + v_{1}^{7}t_{2}^{8},
d(2v_{1}^{4}v_{3}^{9}) ≡ 2v_{1}^{5}v_{3}^{8}t_{2}^{2},
d(2v_{2}x_{1}) ≡ 2v_{1}^{6}v_{2}(t_{3}^{4} + v_{1}t_{2})
≡ 2v_{1}^{6}(t_{3} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2} + v_{1}t_{4}^{2} + v_{1}v_{2}t_{2}) by (3.7),
d(2v_{1}^{5}v_{3}^{2}) ≡ 2v_{1}^{7}t_{2}^{4},
d(3v_{1}^{6}v_{3}) ≡ 2v_{1}^{6}t_{3} + 3v_{1}^{7}t_{2}^{2},
d(v_{1}^{6}v_{2}v_{3}^{4}) ≡ 2v_{1}^{6}v_{3}^{4}t_{2}.

The sum of these congruences with (3.7) shows the existence of an element χ′′ such that 2v_{1}^{5}g_{1}^{2} + d(χ′′) ≡ 2v_{1}^{7}r^{2} mod (4, v_{1}^{8}). We further compute mod (4, v_{1}^{4}),

2v_{1}^{3}t_{4}^{2} ≡ 2v_{1}^{3}(t_{4}^{8} + v_{3}^{2}t_{3}^{16} + v_{3}^{16}t_{3}^{2})
≡ 2v_{1}^{3}(t_{4}^{8} + v_{3}^{2}t_{3}^{4} + v_{3}^{6}t_{2}^{32} + v_{3}^{18}t_{2}^{4} + v_{3}^{16}t_{3}^{2}) by (3.7),
d(2v_{1}^{2}v_{3}^{7}) ≡ 2v_{1}^{3}v_{3}^{6}t_{2}^{2},
d(v_{1}v_{3}^{20}) ≡ 2v_{1}^{3}v_{3}^{18}t_{2}^{4},
d(2v_{2}v_{3}^{18}) ≡ 2v_{1}^{2}v_{2}v_{3}^{16}t_{2}^{4},
d(v_{1}^{2}v_{2}v_{3}^{16}) ≡ 2v_{1}^{2}v_{3}^{16}t_{2}.

These with (3.7) also imply the existence of an element χ′ such that

2v_{1}^{5}g_{1}^{2} + d(χ′) ≡ 2v_{1}^{7}r^{4} + 2v_{1}^{7}v_{3}^{2}t_{3}^{4} mod (4, v_{1}^{8}).

The congruences

2v_{3}^{2}t_{3}^{4} ≡ 2v_{2}v_{3}^{2}(t_{3} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2}) and
d(v_{2}v_{3}^{3}) ≡ 2v_{2}v_{3}^{2}t_{3} + 2v_{3}^{3}t_{2}

show that 2v_{3}^{2}t_{3}^{4} is homologous to 2v_{2}v_{3}^{3}\overline{t}_{21} + 2v_{2}v_{3}^{6}t_{2}. Since r^{4} = R_{1} by Lemma (5.8), we obtain the lemma. q.e.d.

Lemma (5.19). There exists an element \overline{χ} such that d(\overline{χ}) ≡ 2v_{1}^{3}x_{1}g_{1}^{2} + 2v_{1}^{3}v_{3}^{6}\overline{t}_{21} mod (4, v_{1}^{4}) for \overline{t}_{21} in Lemma (5.18).

Proof. Note that t_{2}^{4} ≡ v_{2}t_{2} + v_{1}v_{2}t_{3}^{2} mod (2, v_{1}^{2}). Then the lemma follows as the sum of the congruences:

2v_{1}^{3}x_{1}g_{1}^{2} ≡ 2v_{1}^{3}x_{1}(t_{3}^{2} + v_{3}^{2}t_{2}^{4} + v_{3}^{8}t_{2}^{2}),
d(2v_{2}x_{1}v_{3}^{2}) ≡ 2v_{1}^{2}v_{2}x_{1}t_{2}^{4},
d(v_{1}^{2}v_{2}x_{1}) ≡ 2v_{1}^{2}x_{1}t_{2},
d(2v_{1}^{2}x_{1}v_{3}^{9}) ≡ 2v_{1}^{3}x_{1}v_{3}^{8}t_{2}^{2},
d(2v_{1}^{2}x_{1}v_{3}^{3}) ≡ 2v_{1}^{3}x_{1}v_{3}^{2}t_{2}^{2}.

q.e.d.

6. The action of the connecting homomorphism

In this section, we determine E_{2}^{∗}(L_{2}\overline{M}(2, ∞)) = H^{∗}M_{1}^{1}(2) by observing the long exact sequence (1.3) (see also (3.3)) by the method using [4, Remark 3.11]:

Lemma (6.1). Suppose that a submodule D^{s} of H^{s}M_{1}^{1}(2) fits in the exact sequence

H^{s−1}M_{1}^{1}(2) ^{δ} wH^{s}M_{2}^{0}(2) w^{1/v_{1}}D^{s} ^{v_{1}} wD^{s} ^{δ} wH^{s+1}M_{2}^{0}(2).

Then, H^{s}M_{1}^{1}(2) = D^{s}.

We read off the zeroth and the first lines of the E_{2}-term E_{2}^{∗}(L_{2}\overline{M}(2, 1)) = H^{∗}M_{2}^{0}(2) from Theorem (2.5) and (3.3):

H^{0}M_{2}^{0}(2) = 2v_{2}K_{∗}[v_{3}^{2}] ⊗ Λ(v_{3}) ⊕ 2v_{3}K_{∗}[v_{3}^{2}] ⊕ Z/4[v_{2}^{±2}, v_{3}^{2}] and
H^{1}M_{2}^{0}(2) = 2v_{2}K_{∗}[v_{3}^{2}] ⊗ Λ(v_{3})\{h_{20}, \overline{h}_{21}, h_{30}, \overline{h}_{31}, ρ_{2}\} ⊕ h_{20}K_{∗}[v_{3}^{2}] ⊗ Λ(v_{3})
⊕2v_{3}K_{∗}[v_{3}^{2}]\{\overline{h}_{21}, \overline{h}_{31}, ρ_{2}\} ⊕ h_{30}K_{∗}[v_{3}^{2}]
⊕2v_{3}h_{30}K_{∗}[v_{3}^{2}] ⊕ Z/4[v_{2}^{±2}, v_{3}^{2}]\{\overline{h}_{21}, \overline{h}_{31}, ρ_{2}\}.

Proposition (6.2). For the generators of H^{0}M_{1}^{1}(2), we see the behavior of the connecting homomorphism:

δ(2v_{2}v_{3}^{2t+1}/v_{1}) = 2v_{2}v_{3}^{2t}\overline{h}_{21},
δ(2v_{2}v_{3}^{4t+2}/v_{1}^{3} + v_{2}v_{3}^{4t}/v_{1}) = 2v_{3}^{4t+1}h_{30} + 2v_{3}^{4t}\overline{h}_{31},
δ(2v_{2}x_{n}^{2t+1}/v_{1}^{a_{n}+1}) = −x_{n}^{2t}v_{3}^{4e_{n−1}} \overline{h}_{21} + 2x_{n}^{2t}v_{3}^{4e_{n−1}}ρ_{2},
δ(2v_{2}x_{n}^{4t+2}/v_{1}^{2a_{n}}) = 2v_{2}x_{n}^{4t}v_{3}^{8e_{n−1}} \overline{h}_{31} + 2v_{2}x_{n}^{4t}v_{3}^{8e_{n−1}+1}h_{30};
δ(2v_{3}^{2t+1}/v_{1}) = 2v_{2}v_{3}^{2t}h_{20} + 2v_{3}^{2t}\overline{h}_{21};
δ(v_{3}^{4t+2}/v_{1}) = 2v_{2}v_{3}^{4t+1}h_{20} + 2v_{3}^{4t+1}\overline{h}_{21},
δ(2v_{3}^{4t+2}/v_{1}^{2}) = 2v_{3}^{4t}\overline{h}_{21},
δ(v_{3}^{8t+4}/v_{1}^{2}) = 2v_{3}^{8t+2}\overline{h}_{21},
δ(2x_{1}^{2t+1}/v_{1}^{6}) = 2v_{2}v_{3}^{8t+1}\overline{h}_{21},
δ(x_{1}^{4t+2}/v_{1}^{6}) = 2v_{2}v_{3}^{16t+5}\overline{h}_{21},
δ(2x_{1}^{4t+2}/v_{1}^{14}) = 2x_{1}^{4t}(ρ_{2} + v_{2}v_{3}^{3}\overline{h}_{21} + v_{2}v_{3}^{6}h_{20}),
δ(x_{2}^{2t+1}/v_{1}^{14}) = 2x_{1}^{8t+2}(ρ_{2} + v_{2}v_{3}^{3}\overline{h}_{21} + v_{2}v_{3}^{6}h_{20}),
δ(2x_{2k}^{2t+1}/v_{1}^{c_{2k}}) = 2v_{2}x_{2}^{2×4^{2k−2}t+3\overline{e}_{k−1}}v_{3}^{2}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}) k ≥ 1,
δ(x_{2k}^{4t+2}/v_{1}^{c_{2k}}) = 2v_{2}x_{2}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}v_{3}^{2}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}) k ≥ 1,
δ(2x_{2k}^{4t+2}/v_{1}^{2c_{2k}}) = 2v_{3}^{4^{2k+1}t+6\overline{e}_{k}} \overline{h}_{21} k ≥ 1,
δ(x_{2k+1}^{2t+1}/v_{1}^{2c_{2k}}) = 2v_{3}^{4^{2k}(8t+2)+6\overline{e}_{k}} \overline{h}_{21} k ≥ 1,
δ(2x_{2k+1}^{2t+1}/v_{1}^{c_{2k+1}+2}) = 2v_{2}x_{1}^{2×4^{2k}t+3\overline{e}_{k}}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}) k ≥ 1,
δ(x_{2k+1}^{4t+2}/v_{1}^{c_{2k+1}+2}) = 2v_{2}x_{1}^{4^{2k}(4t+1)+3\overline{e}_{k}}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}) k ≥ 1,
δ(2x_{2k+1}^{4t+2}/v_{1}^{2c_{2k+1}+6}) = 2x_{1}^{4^{2k+1}t+6\overline{e}_{k}}(ρ_{2} + v_{2}v_{3}^{3}\overline{h}_{21} + v_{2}v_{3}^{6}h_{20}) k ≥ 1,
δ(x_{2k}^{2t+1}/v_{1}^{2c_{2k−1}+6}) = 2x_{1}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}(ρ_{2} + v_{2}v_{3}^{3}\overline{h}_{21} + v_{2}v_{3}^{6}h_{20}) k > 1.

Proof. Throughout this proof, we use the relations in (3.7) freely. The first and the second equalities follow from

d(2v_{2}v_{3}^{2t+1}/v_{1}^{2} + v_{2}v_{3}^{2t}/v_{1}) = 2v_{2}v_{3}^{2t}t_{2}^{2}/v_{1} + 2v_{3}^{2t}t_{2}/v_{1} and
d(2v_{2}v_{3}^{4t+2}/v_{1}^{4} + v_{2}v_{3}^{4t}/v_{1}^{2}) = 2v_{2}v_{3}^{4t}t_{2}^{4}/v_{1}^{2} + 2v_{3}^{4t}t_{2}/v_{1}^{2}
= 2v_{3}^{4t}t_{3}^{2}/v_{1}.

Turn to the third equality. Suppose first that n = 1. Since

d(2v_{2}x_{1}^{2t+1}/v_{1}^{8}) = 2v_{2}x_{1}^{2t}g_{1}/v_{1}^{2} = 2v_{2}x_{1}^{2t}(t_{3}^{4} + v_{1}t_{2})/v_{1}^{2}
= 2x_{1}^{2t}(t_{3} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2} + v_{1}t_{4}^{2})/v_{1}^{2} + 2v_{2}x_{1}^{2t}t_{2}/v_{1},
d(x_{1}^{2t}v_{3}/v_{1}^{2}) = 2x_{1}^{2t}t_{3}/v_{1}^{2} + x_{1}^{2t}t_{2}^{2}/v_{1} + 2x_{1}^{2t}(v_{2}t_{2} + t_{2}^{2})/v_{1},
d(3x_{1}^{2t}v_{3}^{2}/v_{1}^{3}) = 2x_{1}^{2t}v_{3}t_{2}^{2}/v_{1}^{2} + 3x_{1}^{2t}t_{2}^{4}/v_{1},
d(v_{2}x_{1}^{2t}v_{3}^{4}/v_{1}^{2}) = 2x_{1}^{2t}v_{3}^{4}t_{2}/v_{1}^{2},

we obtain d(2v_{2}x_{1}^{2t+1}/v_{1}^{8} + ⋯) = 2x_{1}^{2t}t_{4}^{2}/v_{1} + x_{1}^{2t}t_{2}^{2}/v_{1} + 2x_{1}^{2t}(v_{2}t_{2} + t_{2}^{2})/v_{1} + x_{1}^{2t}t_{2}^{4}/v_{1}. Put \overline{t}′_{21} = t_{2}^{2} + t_{2}^{4} + 2t_{4}^{2} + 2v_{2}t_{2} + 2v_{2}t_{2}^{5} and recall \overline{t}_{21} in Lemma (5.18). Then, 3\overline{t}_{21} − \overline{t}′_{21} = 2t_{4}^{2} + 2v_{2}t_{4} mod (4, v_{1}), since η_{R}(v_{2}v_{4}) ≡ v_{2}v_{4} +v_{2}^{2}t_{2}^{4} −v_{2}^{5}t_{2} +2v_{2}t_{4} +2v_{4}t_{2} +2v_{2}t_{2}^{5} +2v_{2}^{4}t_{2}^{2} mod (4, v_{1}) in Γ(2). Thus, we see that \overline{t}′_{21} represents 3\overline{h}_{21} +2ρ_{2}. For n > 1, the equality follows similarly from the computation

d(2v_{2}x_{n}^{2t+1}/v_{1}^{a_{n}+2}) = 2v_{2}x_{n}^{2t}g_{n}/v_{1}^{2} = 2v_{2}x_{n}^{2t}x_{1}^{e_{n−1}}(t_{3}^{4} + v_{1}t_{2})/v_{1}^{2}
= 2x_{n}^{2t}x_{1}^{e_{n−1}}(t_{3} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2} + v_{1}t_{4}^{2})/v_{1}^{2} + 2v_{2}x_{n}^{2t}x_{1}^{e_{n−1}}t_{2}/v_{1},
d(x_{n}^{2t}x_{1}^{e_{n−1}}v_{3}/v_{1}^{2}) = 2x_{n}^{2t}x_{1}^{e_{n−1}}t_{3}/v_{1}^{2} + x_{n}^{2t}x_{1}^{e_{n−1}}t_{2}^{2}/v_{1}^{2} + 2x_{n}^{2t}x_{1}^{e_{n−1}}(v_{2}t_{2} + t_{2}^{2})/v_{1},
d(x_{n}^{2t}x_{2}^{e_{n−2}}v_{3}^{6}/v_{1}^{3}) = 2x_{n}^{2t}x_{1}^{e_{n−1}}v_{3}t_{2}^{2}/v_{1}^{2} + x_{n}^{2t}x_{1}^{e_{n−1}}t_{2}^{4}/v_{1} + 2x_{n}^{2t}x_{1}^{e_{n−1}}t_{2}^{4}/v_{1},
d(v_{2}x_{n}^{2t}x_{1}^{e_{n−1}}v_{3}^{4}/v_{1}^{2}) = 2x_{n}^{2t}x_{1}^{e_{n−1}}v_{3}^{4}t_{2}/v_{1}^{2}.

The fourth equality is verified by

d(2v_{2}x_{n}^{4t+2}/v_{1}^{2a_{n}+1}) = 2v_{2}x_{n}^{4t}g_{n}^{2}/v_{1} = 2v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}g_{1}^{2}/v_{1} = 2v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}t_{3}^{8}/v_{1}
= 2v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}(t_{3}^{2} + v_{3}^{2}t_{2}^{4} + v_{3}^{8}t_{2}^{2})/v_{1},
d(v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{2}/v_{1}) = 2x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{2}t_{2}/v_{1},
d(2v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{9}/v_{1}^{2}) = 2v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{8}t_{2}^{2}/v_{1}.

The fifth and the sixth ones follow from d(2v_{3}^{2t+1}/v_{1}^{2}) = 2v_{3}^{2t}t_{2}^{2}/v_{1} and d(v_{3}^{4t+2}/v_{1}^{2}) = 2v_{3}^{4t+1}t_{2}^{2}/v_{1}, respectively. The seventh and the eighth are checked as

d(2v_{3}^{4t+2}/v_{1}^{3}) = 2v_{3}^{4t}t_{2}^{4}/v_{1} = 2v_{3}^{4t}(\overline{t}_{21} + t_{2}^{2})/v_{1},
d(2v_{3}^{4t+1}/v_{1}^{2}) = 2v_{3}^{4t}t_{2}^{2}/v_{1}; and
d(v_{3}^{8t+4}/v_{1}^{3}) = 2v_{3}^{8t+2}t_{2}^{4}/v_{1},
d(2v_{3}^{8t+3}/v_{1}^{2}) = 2v_{3}^{8t+2}t_{2}^{2}/v_{1}.

The ninth and the tenth ones follow from the computation:

d(2x_{1}^{2t+1}/v_{1}^{7}) = 2x_{1}^{2t}g_{1}/v_{1} = 2x_{1}^{2t}(v_{2}t_{3} + v_{2}v_{3}t_{2}^{2} + v_{2}v_{3}^{4}t_{2})/v_{1},
d(v_{2}x_{1}^{2t}v_{3}/v_{1}) = 2x_{1}^{2t}v_{3}t_{2}/v_{1} + 2v_{2}x_{1}^{2t}t_{3}/v_{1},
d(2x_{1}^{2t+1}v_{3}^{2}/v_{1}^{3}) = 2x_{1}^{2t+1}t_{2}^{4}/v_{1}; and
d(x_{1}^{4t+2}/v_{1}^{7}) = 2x_{1}^{4t+1}g_{1}/v_{1} = 2x_{1}^{4t+1}(v_{2}t_{3} + v_{2}v_{3}t_{2}^{2} + v_{2}v_{3}^{4}t_{2})/v_{1},
d(v_{2}x_{1}^{4t+1}v_{3}/v_{1}) = 2x_{1}^{4t+1}v_{3}t_{2}/v_{1} + 2v_{2}x_{1}^{4t+1}t_{3}/v_{1},
d(2x_{1}^{4t+2}v_{3}^{2}/v_{1}^{3}) = 2x_{1}^{4t+2}t_{2}^{4}/v_{1},

in which 2v_{2}x_{1}^{2t}v_{3}t_{2}^{2}/v_{1}+2x_{1}^{2t}v_{3}t_{2}/v_{1} = 2v_{2}x_{1}^{2t}v_{3}\overline{t}_{21}/v_{1} and 2v_{2}x_{1}^{4t+1}v_{3}t_{2}^{2}/v_{1}+2x_{1}^{4t+1}v_{3}t_{2}/v_{1} = 2v_{2}x_{1}^{4t+1}v_{3}\overline{t}_{21}/v_{1}. By Lemma (5.18),

d(2x_{1}^{4t+2}/v_{1}^{15}) = 2x_{1}^{4t}g_{1}^{2}/v_{1}^{3},
d(x_{1}^{4t}χ/v_{1}^{8}) = 2x_{1}^{4t}g_{1}^{2}/v_{1}^{3} + 2x_{1}^{4t}(R_{1} + v_{2}v_{3}^{3}\overline{t}_{21} + v_{2}v_{3}^{6}t_{2})/v_{1}.

The element ρ_{2} is represented by the cocycle r^{4}, and r^{2} = r^{4} + s for s = t_{3}^{4} = v_{2}(t_{3} + v_{3}t_{2}^{2} + v_{3}^{4}t_{2}). Note that 2v_{2}t_{3} = 2v_{3}t_{2} up to homology. Then σ = [s] = v_{2}v_{3}\overline{h}_{21} + v_{2}v_{3}^{4}h_{20}. It follows that

δ(2x_{1}^{4t+2}/v_{1}^{14}) = 2x_{1}^{4t}ρ_{2} + 2v_{2}x_{1}^{4t}(v_{3}^{3}\overline{h}_{21} + v_{3}^{6}h_{20}).

In the same manner,

d(x_{2}^{2t}x_{1}^{4}/v_{1}^{15}) = 2x_{2}^{2t}x_{1}^{2}g_{1}^{2}/v_{1}^{3},
d(x_{2}^{2t}x_{1}^{2}χ/v_{1}^{8}) = 2x_{2}^{2t}x_{1}^{2}g_{1}^{2}/v_{1}^{3} + 2x_{2}^{2t}x_{1}^{2}r^{2}/v_{1},

and we obtain

δ(x_{2}^{2t+1}/v_{1}^{14}) = 2x_{1}^{8t+2}ρ_{2} + 2v_{2}x_{1}^{8t+2}(v_{3}^{3}\overline{h}_{21} + v_{3}^{6}h_{20}).

For n = 2k ≥ 2, by Lemma (5.17), we compute

d(x_{2k}^{2t}X_{2k}/v_{1}^{c_{2k}+1}) = 2x_{2k}^{2t}x_{2}^{3\overline{e}_{k−1}}R_{1}/v_{1},
d(2x_{2k}^{2t}x_{2}^{3\overline{e}_{k−1}}x_{1}^{2}/v_{1}^{15} + …) = 2x_{2k}^{2t}x_{2}^{3\overline{e}_{k−1}}(R_{1} + v_{2}v_{3}^{3}\overline{t}_{21} + v_{2}v_{3}^{6}t_{2})/v_{1}.

It follows that

δ(2x_{2k}^{2t+1}/v_{1}^{c_{2k}}) = 2v_{2}x_{2}^{2×4^{2k−2}t+3\overline{e}_{k−1}}v_{3}^{2}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}).

In the same way, for even n = 2k ≥ 2,

d(x_{n}^{4t}Y′_{n}/v_{1}^{c_{n}+1}) = 2x_{2k}^{4t+1}x_{2}^{3\overline{e}_{k−1}}R_{1}/v_{1}

with d(2x_{2k}^{4t+1}x_{2}^{3\overline{e}_{k−1}}x_{1}^{2}/v_{1}^{15} + …) = 2x_{2k}^{4t+1}x_{2}^{3\overline{e}_{k−1}}(R_{1} + v_{2}v_{3}^{3}\overline{t}_{21} + v_{2}v_{3}^{6}t_{2})/v_{1}, and we obtain

δ(x_{2k}^{4t+2}/v_{1}^{c_{2k}}) = 2v_{2}x_{2}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}v_{3}^{2}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}).

For n = 2k ≥ 2, by Lemmas (5.17), (5.9) and (5.19),

d(x_{n}^{4t}X′_{2k}/v_{1}^{2c_{2k}+1}) = 2x_{2}^{4^{2k−1}t+6\overline{e}_{k−1}}R_{1}^{2}/v_{1},
d(x_{2}^{4^{2k−1}t+6\overline{e}_{k−1}}X_{2}/v_{1}^{c_{2}+1}) = 2x_{2}^{4^{2k−1}t+6\overline{e}_{k−1}}R_{1}/v_{1},
d(2x_{2}^{4^{2k−1}t+6\overline{e}_{k−1}}u_{1}/v_{1}^{11}) = 2x_{2}^{4^{2k−1}t+6\overline{e}_{k−1}}(R_{1}^{2} + R_{1} + x_{1}g_{1}^{2})/v_{1},
d(x_{2}^{4^{2k−1}t+6\overline{e}_{k−1}} \overline{χ}/v_{1}^{4}) = 2x_{2}^{4^{2k−1}t+6\overline{e}_{k−1}}(x_{1}g_{1}^{2} + v_{3}^{6}\overline{t}_{21})/v_{1},

On subgroups of π_{∗}(L_{2}T (1) ∧ M(2)) at the prime two 15

and we have

δ(2x_{2k}^{4t+2}/v_{1}^{2c_{2k}}) = 2v_{3}^{4^{2k+1}t+6\overline{e}_{k}} \overline{h}_{21}.

For odd n = 2k + 1 ≥ 3, by Lemmas (5.17), (5.9) and (5.19),

d(x_{n}^{2t}Y_{n}/v_{1}^{2c_{n−1}+1}) = 2x_{n−1}^{8t+2}x_{2}^{6\overline{e}_{k−1}}R_{1}^{2}/v_{1} = 2x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}R_{1}^{2}/v_{1},
d(x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}X_{2}/v_{1}^{c_{2}+1}) = 2x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}R_{1}/v_{1},
d(2x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}u_{1}/v_{1}^{11}) = 2x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}(R_{1}^{2} + R_{1} + x_{1}g_{1}^{2})/v_{1},
d(x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}} \overline{χ}/v_{1}^{4}) = 2x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}(x_{1}g_{1}^{2} + v_{3}^{6}\overline{t}_{21})/v_{1}.

It follows that

δ(x_{2k+1}^{2t+1}/v_{1}^{2c_{2k}}) = 2v_{3}^{4^{2k}(8t+2)+6\overline{e}_{k}} \overline{h}_{21}.

Lemmas (5.17), (5.10) and (5.9) show the equalities

d(x_{2k+1}^{2t}X_{2k+1}/v_{1}^{c_{2k+1}+3}) = 2x_{3}^{2×4^{2k−2}t+3\overline{e}_{k−1}}R_{2}/v_{1}^{3},
d(x_{3}^{2×4^{2k−2}t+3\overline{e}_{k−1}}x′_{2,1}/v_{1}^{14×4+3}) = 2x_{3}^{2×4^{2k−2}t+3\overline{e}_{k−1}}R_{1}^{2}/v_{1}^{3},
d(2x_{3}^{2×4^{2k−2}t+3\overline{e}_{k−1}}u′_{1}/v_{1}^{23}) = 2x_{3}^{2×4^{2k−2}t+3\overline{e}_{k−1}}(R_{2} + R_{1}^{2} + v_{1}^{2}x_{1}^{2}g_{2})/v_{1}^{3}
= 2x_{3}^{2×4^{2k−2}t+3\overline{e}_{k−1}}(R_{2} + R_{1}^{2} + v_{1}^{2}v_{2}x_{1}^{3}(t_{3} + v_{3}t_{2}^{2} + v_{3}^{4}t_{2}))/v_{1}^{3},
d(v_{2}v_{3}x_{1}^{2×4^{2k}t+3\overline{e}_{k}}/v_{1}) = 2x_{1}^{2×4^{2k}t+3\overline{e}_{k}}(v_{3}t_{2} + v_{2}t_{3})/v_{1},

which give rise to

δ(2x_{2k+1}^{2t+1}/v_{1}^{c_{2k+1}+2}) = 2v_{2}x_{1}^{2×4^{2k}t+3\overline{e}_{k}}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}).

Consider the odd case where n = 2k + 1.

d(x_{n}^{4t}Y′_{n}/v_{1}^{c_{n}+3}) = 2x_{n}^{4t+1}x_{3}^{3\overline{e}_{k−1}}R_{2}/v_{1}^{3} = 2x_{3}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}R_{2}/v_{1}^{3},
d(x_{3}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}x′_{2,1}/v_{1}^{14×4+3}) = 2x_{3}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}R_{1}^{2}/v_{1}^{3},
d(2x_{3}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}u′_{1}/v_{1}^{23}) = 2x_{3}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}(R_{2} + R_{1}^{2} + v_{1}^{2}x_{1}^{2}g_{2})/v_{1}^{3}
= 2x_{3}^{4^{2k−2}(4t+1)+3\overline{e}_{k−1}}(R_{2} + R_{1}^{2} + v_{1}^{2}v_{2}x_{1}^{3}(t_{3} + v_{3}t_{2}^{2} + v_{3}^{4}t_{2}))/v_{1}^{3},
d(v_{2}v_{3}x_{1}^{4^{2k}(4t+1)+3\overline{e}_{k}}/v_{1}) = 2x_{1}^{4^{2k}(4t+1)+3\overline{e}_{k}}(v_{3}t_{2} + v_{2}t_{3})/v_{1},

and so

δ(x_{2k+1}^{4t+2}/v_{1}^{c_{2k+1}+2}) = 2v_{2}x_{1}^{4^{2k}(4t+1)+3\overline{e}_{k}}(v_{3}\overline{h}_{21} + v_{3}^{4}h_{20}).

For n = 2k + 1 ≥ 3, by Lemmas (5.17), (5.10), (5.9) and (5.18),

d(x_{n}^{4t}X′_{2k+1}/v_{1}^{2c_{2k+1}+7}) = 2x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}R_{2}^{2}/v_{1}^{7},
d(x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}x_{3,1}/v_{1}^{7×4^{2}+7}) = 2x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}R_{2}/v_{1}^{7},
d(2x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}u_{2}/v_{1}^{43}) = 2x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}(R_{2}^{2} + R_{2} + v_{1}^{4}x_{2}g_{2}^{2})/v_{1}^{7}
= 2x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}(R_{2}^{2} + R_{2} + v_{1}^{4}x_{1}^{6}g_{1}^{2})/v_{1}^{7},
d(x_{1}^{4^{2k+1}t+6\overline{e}_{k}}χ/v_{1}^{8}) = 2x_{1}^{4^{2k+1}t+6\overline{e}_{k}}(g_{1}^{2} + v_{1}^{2}R_{1} + v_{1}^{2}v_{2}v_{3}^{3}\overline{t}_{21} + v_{1}^{2}v_{2}v_{3}^{6}t_{2})/v_{1}^{3}.

It follows that

δ(2x_{2k+1}^{4t+2}/v_{1}^{2c_{2k+1}+6}) = 2x_{1}^{4^{2k+1}t+6\overline{e}_{k}}(ρ_{2} + v_{2}v_{3}^{3}\overline{h}_{21} + v_{2}v_{3}^{6}h_{20}).

Last, for n = 2k ≥ 4, by Lemmas (5.17), (5.10), (5.9) and (5.18),

d(x_{n}^{2t}Y_{n}/v_{1}^{2c_{n−1}+7}) = 2x_{n−1}^{8t+2}x_{3}^{6\overline{e}_{k−2}}R_{2}^{2}/v_{1}^{7} = 2x_{3}^{4^{2k−4}(8t+2)+6\overline{e}_{k−2}}R_{2}^{2}/v_{1}^{7},
d(x_{3}^{4^{2k−4}(8t+2)+6\overline{e}_{k−2}}x_{3,1}/v_{1}^{7×4^{2}+7}) = 2x_{3}^{4^{2k−4}(8t+2)+6\overline{e}_{k−2}}R_{2}/v_{1}^{7},
d(2x_{3}^{4^{2k−4}(8t+2)+6\overline{e}_{k−2}}u_{2}/v_{1}^{43}) = 2x_{3}^{4^{2k−4}(8t+2)+6\overline{e}_{k−2}}(R_{2}^{2} + R_{2} + v_{1}^{4}x_{2}g_{2}^{2})/v_{1}^{7}
= 2x_{3}^{4^{2k−4}(8t+2)+6\overline{e}_{k−2}}(R_{2}^{2} + R_{2} + v_{1}^{4}x_{1}^{6}g_{1}^{2})/v_{1}^{7},
d(x_{1}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}χ/v_{1}^{8}) = 2x_{1}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}(g_{1}^{2} + v_{1}^{2}R_{1} + v_{1}^{2}v_{2}v_{3}^{3}\overline{t}_{21} + v_{1}^{2}v_{2}v_{3}^{6}t_{2})/v_{1}^{3},

and we obtain δ(x_{2k}^{2t+1}/v_{1}^{2c_{n−1}+6}) = 2x_{1}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}(ρ_{2} + v_{2}v_{3}^{3}\overline{h}_{21} + v_{2}v_{3}^{6}h_{20}). q.e.d.

Proof of Theorem (2.10). Set D^{0} to be the right hand side of the formula for E_{2}^{0}(L_{2}\overline{M}(2, ∞)), which is H^{0}M_{1}^{1}(2). Then Proposition (6.2) shows that D^{0} satisfies the hypothesis of Lemma (6.1). q.e.d.

Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan

  E-mail address: 95sm004@math.kochi-u.ac.jp

Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan

  E-mail address: katsumi@math.kochi-u.ac.jp

Nankai University, Department of Mathematics, Tianjin 300071, P. R. China

  E-mail address: xiwang@nankai.edu.cn

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