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 Brown-Peterson spectrum v_{2}^{−1}BP localized at v_{2} 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], we determined the E_{2}-term for π_{∗}(L_{2}T (1)) at s-line with s > 6, and studied a subgroup of π_{∗}(L_{2}T (1)). The s-line of the E_{2}-term seems very complicated if 1 < s ⩽ 6. Let M(k) denote the mod 2^{k} Moore spectrum. Then the complex parts of the E_{2}-term for π_{∗}(L_{2}M(k)) are s-lines with 0 < s ⩽ 5. Here we determine the Adams-Novikov E_{2}-term for π_{∗}(L_{2}T (1) ∧ M(2)) other than s-lines with 1 < s ⩽ 5, which shows how complex the structure of the E_{2}-term for π_{∗}(L_{2}T (1)) is.

1. Introduction

Let S_{(p)} denote the stable homotopy category at the 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 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, that is, we set n = 2 and p = 2. One of ways to understand the homotopy groups π_{∗}(L_{2}S^{0}) at the prime two is to understand the homotopy groups π_{∗}(L_{2}T (1)). Here T (1) denotes the Ravenel spectrum characterized by the BP_{∗}-homology BP_{∗}(T (1)) = BP_{∗}[t_{1}] ⊂ BP_{∗}(BP) = BP_{∗}[t_{1}, t_{2}, …]. Let \overline{M}(1) and \overline{V} (1) be cofibers of 2: T (1) → T (1) and v_{1} : Σ^{2}\overline{M}(1) → \overline{M}(1), respectively. Note that \overline{M}(1) is the Mahowald spectrum X⟨1⟩ (cf. [7]). Then the homotopy groups π_{∗}(L_{2}\overline{V} (1)) are determined in [3], with which the homotopy groups π_{∗}(L_{2}\overline{M}(1)) are determined in [7] by the v_{1}-Bockstein spectral sequence. Thus, our next target is to determine the homotopy groups π_{∗}(L_{2}T (1)) by using the mod 2 Bockstein spectral sequence. It seems very tough to compute the Bockstein spectral sequence, and we get a partial results on π_{∗}(L_{2}T (1)) in [2]. Here we show how tough the other parts of the E_{2}-term. (The paper [5] seems to require some more time to appear, because it is too complicated to verify the results.) To do so, we consider the mod 4 Moore spectrum M(2). The zero line is not so complicated and give the result here. We determine the first line of the E_{2}-term for π_{∗}(L_{2}T (1) ∧ M(2)) up to an extension of the Z-module structure, which shows how complex the structure of the homotopy groups is. In the same manner as [2], we also obtain the s-lines for s ⩾ 5, and we obtain subgroups of π_{∗}(L_{2}T (1) ∧ M(2)). The second line is too complicated to determine here.

This paper is divided into six sections:

1. Introduction

2. Statement of results

3. The action of η_{R} on BP_{∗}[t_{2}, t_{3}, …]

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

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

6. The action of the connecting homomorphism

In the next section, we state the our main results and give proofs of them except for the proof of the former part of Theorem 2.7. The other sections are devoted to show the exceptional part. In section three, we give formulas on the right unit η_{R}, and a change of rings theorem is given in section four. By the change of rings theorem, we work on a smaller Hopf algebroid. We introduce some cochains in section five, and compute the behavior of the connecting homomorphism in the last section.

2. Statement of results

We work in the stable homotopy category S_{(2)} of spectra localized away from two. Let BP denote the Brown-Peterson spectrum with coefficient ring BP_{∗} = Z_{(2)}[v_{1}, v_{2}, …]. We work in the category of L_{2} of v_{2}^{−1}BP - local spectra at the prime two. Let \overline{M}(i, j) be the generalized Moore spectrum such that BP_{∗}(\overline{M}(i, j)) =

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

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

2 Ippei Ichigi, Katsumi Shimomura, and Xiangjun Wang

BP_{∗}/(2^{i}, v_{1}^{j})[t_{1}] ⊂ BP_{∗}/(2^{i}, v_{1}^{j})[t_{1}, t_{2}, …] = BP_{∗}(BP/(2^{i}, v_{1}^{j})). Here BP/(2^{i}, v_{1}^{j}) denotes the spectrum such that π_{∗}(BP/(2^{i}, v_{1}^{j})) = BP_{∗}/(2^{i}, v_{1}^{j}). Then there is the Adams-Novikov spectral sequence

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

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

where h_{2i}, h_{3i} and ρ_{2} are the elements represented by the cocycles in the cobar complex Ω_{BP_{∗}(BP)}^{1}v_{2}^{−1}BP_{∗}[t_{1}]/(2, v_{1}), whose leading terms are t_{2}^{2^{i}}, t_{3}^{2^{i}} and v_{2}^{−5}t_{4} + v_{2}^{−10}t_{4}^{2}, respectively.

Now we consider the Adams-Novikov E_{2}-term for the homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)). Note that \overline{M}(2, 1) is obtained as the cofiber of the map Σ^{2}M(2) ∧ T (1) ⟶^{1∧v_{1}} M(2) ∧ T (1), where M(2) denotes the mod 4 Moore spectrum. By the Verdier’s axiom, we have the cofiber sequence

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

This induces the long exact sequence

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

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. We consider the elements

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

Lemma (2.1). The connecting homomorphism δ acts trivially on the elements

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

We rewrite the E_{2}-term E_{2}^{∗}(\overline{M}(1, 1)) as

E_{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}),

where K_{∗} = Z/2[v_{2}^{2}, v_{2}^{−2}]. Furthermore, put F = K_{∗}[v_{3}^{2}] ⊗ Λ(\overline{h}_{31}, ρ_{2}), E = K_{∗}[v_{3}^{2}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2}) and A = K_{∗}[v_{3}^{2}] ⊗ Z/2[h_{20}] ⊗ Λ(v_{3}, \overline{h}_{21}, h_{30}, \overline{h}_{31}, ρ_{2}). We notice that E = F ⊗ Λ(\overline{h}_{21}), A = Z/2[h_{20}] ⊗ E ⊗ Λ(v_{3}, h_{30}) and E_{2}^{∗}(\overline{M}(1, 1)) = A ⊗ Λ(v_{2}). Theorem (2.2). The E_{2}-term E_{2}^{∗}(\overline{M}(2, 1)) is isomorphic to

2v_{2}A ⊕ h_{20}A ⊕ 2v_{3}E ⊕ h_{30}E ⊕ 2v_{3}h_{30}F ⊕ v_{3}h_{30}\overline{h}_{21}F ⊕ Ẽ.

Here Ẽ = Z/4[v_{2}^{2}, v_{2}^{−2}, v_{3}^{2}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2}). Proof. The connecting homomorphism δ acts as δ(v_{2}A) = h_{20}A, δ(E) = 0, δ(v_{3}E) = h_{30}E and δ(v_{3}h_{30}F) = v_{3}h_{30}\overline{h}_{21}F. In the last correspondence, it 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} + · · ·. □

This plays an important role to determine the Adams-Novikov E_{2}-term for π_{∗}(L_{2}\overline{M}(2, ∞)). Here we jot down the structure of the homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)). In order to describe it, we introduce the modules

F′ = K_{∗}[v_{3}^{4}] ⊗ Λ(\overline{h}_{31}, ρ_{2}), E′ = K_{∗}[v_{3}^{4}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2}), Ẽ′ = Z/4[v_{2}^{2}, v_{2}^{−2}, v_{3}^{4}] ⊗ Λ(\overline{h}_{21}, \overline{h}_{31}, ρ_{2})

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

The Adams-Novikov differential on E_{r}^{∗}(\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}). Corollary (2.3). The homotopy groups π_{∗}(L_{2}\overline{M}(2, 1)) are isomorphic to the direct sum of the modules

2v_{2}A′, h_{20}A′′, 2v_{3}E, h_{30}E′, 2v_{3}h_{30}F, v_{3}h_{30}\overline{h}_{21}F′, Ẽ′ and 2v_{3}^{2}E′.

Next we consider the spectrum \overline{M}(k, ∞) = lim_{→} \overline{M}(k, n). Let x_{k}, t̃_{2} denote cochains in the cobar complex Ω_{BP_{∗}(BP)}^{∗}v_{2}^{−1}BP_{∗}[t_{1}], whose leading terms are v_{3}^{4^{k}} and v_{2}v_{3}^{2}t_{2}, respectively (see §5), and a_{k} for each k ⩾ 0 be the integer defined by

a_{n} = 4^{n} + 2e_{n} = e_{n+1} + e_{n},

where e_{n} = (4^{n} − 1)/3. We also consider the modules

\begin{matrix}E_{∗} = Z/2[v_{1}, v_{2}^{2}, v_{2}^{−2}], K_{∗} = E_{∗}/(v_{1}) = Z/2[v_{2}^{2}, v_{2}^{−2}], \\ C_{k} = E_{∗}/(v_{1}^{a_{k}})[x_{k}^{2}]⟨x_{k}/v_{1}^{a_{k}}⟩, C′_{k} = E_{∗}/(v_{1}^{2a_{k}})[x_{k+1}]⟨x_{k}^{2}/v_{1}^{2a_{k}}⟩ and C_{∞} = v_{1}^{−1}E_{∗}/E_{∗}. \end{matrix}

The module E_{2}^{∗}(\overline{M}(1, ∞)) is then given in [7], which is rewritten as

(2.4)  E_{2}^{∗}(\overline{M}(1, ∞)) = ((C_{∞} ⊕ A_{2}) ⊗ Λ(η_{20}) ⊕ A_{0} ⊕ A_{1}) ⊗ Λ(v_{2}, ρ_{2})

for the homology class η_{20} of t̃_{2}. Here A_{i}’s are submodules defined by

A_{0} = C_{0} ⊗ Z/2[h_{20}] ⊗ Λ(h_{21}, h_{30}, h_{31}) ≅ K_{∗}[v_{3}^{2}, h_{20}]⟨v_{3}/v_{1}⟩ ⊗ Λ(h_{21}, h_{30}, h_{31}),
A_{1} = C′_{0} ⊗ Λ(g_{1}, h_{31}) = E_{∗}/(v_{1}^{2})[x_{1}]⟨v_{3}^{2}/v_{1}^{2}⟩ ⊗ Λ(g_{1}, h_{31}),
A_{2} = ∑_{k⩾1} (C_{k} ⊗ Λ(g_{k}^{2}) ⊕ C′_{k} ⊗ Λ(g_{k+1})) ,

in which g_{n}’s are the element fitting in the congruences

d(x_{n}) ≡ v_{1}^{a_{n}}g_{n} mod (2) (≡ v_{1}^{a_{n}}v_{3}^{4e_{n−1}}h_{30} mod (2, v_{1}^{a_{n}+1}))

for n ⩾ 1 (cf. Lemma 5.5).

Consider the module

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

and H^{s}M_{0}^{2}[t_{1}] = E_{2}^{s}(\overline{M}(∞, ∞)) is isomorphic to (Ã_{2} ⊗ Λ(ρ_{2}))^{s} for s > 4 by [2]. The cofiber sequence

L_{2}\overline{M}(2, ∞)  L_{2}\overline{M}(∞, ∞) L_{2}\overline{M}(∞, ∞)^{1/4}w  ^{4} w  induces the long exact sequence
E_{2}^{∗}(\overline{M}(∞, ∞)) ^{4} wE_{2}^{∗}(\overline{M}(∞, ∞)) ^{δ′} wE_{2}^{∗+1}(\overline{M}(2, ∞)) w^{1/4} E_{2}^{∗+1}(\overline{M}(∞, ∞)) ^{4} wE_{2}^{∗+1}(\overline{M}(∞, ∞)).

We obtain the E_{2}-terms E_{2}^{s}(\overline{M}(2, ∞)) for a large s. For s = 0, it seems easy to compute E_{2}^{0}(\overline{M}(2, ∞)) from E_{2}^{0}(\overline{M}(1, ∞)) by the exact sequence 0 → E_{2}^{0}(\overline{M}(1, ∞)) →^{2} E_{2}^{0}(\overline{M}(2, ∞)) → E_{2}^{0}(\overline{M}(1, ∞)) →^{δ} E_{2}^{1}(\overline{M}(1, ∞)), but it is unexpectedly hard to compute, and here we compute it from \overline{M}(2, 1) by the exact sequence 0 → E_{2}^{0}(\overline{M}(2, 1)) → E_{2}^{0}(\overline{M}(2, ∞)) → E_{2}^{0}(\overline{M}(2, ∞)) →^{δ} E_{2}^{1}(\overline{M}(2, 1)). It seems easier to use this exact sequence even when we compute the first line E_{2}^{1}(\overline{M}(2, ∞)) than to use that exact sequence. In order to state the E_{2}-term, we introduce other 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_{k}^{0} = 2E_{∗}[x_{2k}^{2}]⟨x_{2k}/v_{1}^{c_{2k}}⟩ ⊕ E_{∗}[x_{2k}^{4}]⟨x_{2k}^{2}/v_{1}^{c_{2k}}⟩
L_{k}^{1} = 2E_{∗}[x_{2k}^{4}]⟨x_{2k}^{2}/v_{1}^{2c_{2k}}⟩ ⊕ E_{∗}[x_{2k+1}^{2}]⟨x_{2k+1}/v_{1}^{2c_{2k}}⟩
L_{k}^{2} = 2E_{∗}[x_{2k+1}^{2}]⟨x_{2k+1}/v_{1}^{c_{2k+1}+2}⟩ ⊕ E_{∗}[x_{2k+1}^{4}]⟨x_{2k+1}^{2}/v_{1}^{c_{2k+1}+2}⟩
L_{k}^{3} = 2E_{∗}[x_{2k+1}^{4}]⟨x_{2k+1}^{2}/v_{1}^{2c_{2k+1}+6}⟩ ⊕ E_{∗}[x_{2k+2}^{2}]⟨x_{2k+2}/v_{1}^{2c_{2k+1}+6}⟩

Here, E_{∗} = Z/2[v_{1}, v_{2}^{2}, v_{2}^{−2}] and K_{∗} = E_{∗}/(v_{1}) = Z/2[v_{2}^{2}, v_{2}^{−2}] as above, and integers c_{n} is defined by

c_{n} = 4^{n} + 3 × 4^{1+ε(n)}\overline{e}_{[ \frac{n}{2} ]} for n ⩾ 0,

where

ε(n) = \frac{1 − (−1)^{n}}{2} , \overline{e}_{k} = \frac{16^{k} − 1}{15}

and [x] denotes the largest integer that does not exceed x. We also consider the filtration of the E_{2}-term E_{2}^{0}(\overline{M}(2, ∞))

0 ⊂ F_{1} = 2E_{2}^{0}(\overline{M}(2, ∞)) ⊂ F_{2} = E_{2}^{0}(\overline{M}(2, ∞)),

and put GE_{2}^{0}(\overline{M}(2, ∞)) = F_{1} ⊕ F_{2}/F_{1}.

Theorem (2.5). The E_{2}-term for π_{∗}(L_{2}\overline{M}(2, ∞)) is given as follows:

GE_{2}^{0}(\overline{M}(2, ∞)) = v_{1}^{−1}E_{∗}/E_{∗} ⊗ Λ(2) ⊕ 2v_{2}(v_{1}^{−1}E_{∗}/E_{∗}) ⊕ M ⊕ ⨁_{k⩾0} (L_{k}^{0} ⊕ L_{k}^{1} ⊕ L_{k}^{2} ⊕ L_{k}^{3})
E_{2}^{s}(\overline{M}(2, ∞)) = (Ã_{2} ⊗ Λ(ρ_{2}))^{s} ⊕ δ′((Ã_{2} ⊗ Λ(ρ_{2}))^{s−1}) for s > 4.

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

This shows that the structure of E_{2}^{0}(\overline{M}(2, ∞)) is far complicated than that of E_{2}^{0}(\overline{M}(1, ∞)). The first obstruction to compute the E_{2}-term E_{2}^{1}(\overline{M}(2, ∞)) appears in the cokernel of the connecting homomorphism δ: E_{2}^{0}(\overline{M}(2, ∞)) → E_{2}^{1}(\overline{M}(2, 1)), which involves ρ_{2} as in Proposition 6.1. The another one is Z/4-module structure.

By definition, we have a cofiber sequence

(2.6)  T (1) ∧ M(2) wv_{1}^{−1}T (1) ∧ M(2) w\overline{M}(2, ∞).

The Adams-Novikov E_{2}-term for π_{∗}(v_{1}^{−1}T (1) ∧ M(2)) is isomorphic to

E_{2}^{∗}(v_{1}^{−1}T (1) ∧ 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}],

since E_{2}^{∗}(v_{1}^{−1}T (1) ∧ M(1)) = K(1)_{∗}[v_{2}] ⊗ Λ(h_{20}) and the long exact sequence

E_{2}^{∗}(v_{1}^{−1}T (1) ∧ M(1)) wE_{2}^{∗}(v_{1}^{−1}T (1) ∧ M(2)) wE_{2}^{∗}(v_{1}^{−1}T (1) ∧ M(1)) ^{δ} wE_{2}^{∗+1}(v_{1}^{−1}T (1) ∧ M(1)).

This shows that E_{2}^{s}(v_{1}^{−1}M(2) ∧ T (1)) = 0 if s > 1. Consider the long exact sequence

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

associated to the cofiber sequence (2.6), and we see that

E_{2}^{s}(T (1) ∧ M(2)) = E_{2}^{s−1}(\overline{M}(2, ∞)) if s > 2

and the sequence

0 wE_{2}^{0}(T (1) ∧ M(2)) wE_{2}^{0}(v_{1}^{−1}T (1) ∧ M(2)) wE_{2}^{0}(\overline{M}(2, ∞)) ^{δ} wE_{2}^{1}(T (1) ∧ M(2))
wE_{2}^{1}(v_{1}^{−1}T (1) ∧ M(2)) wE_{2}^{1}(\overline{M}(2, ∞)) ^{δ} wE_{2}^{2}(T (1) ∧ M(2)) w0

is exact.

Theorem (2.7). The E_{2}-term for π_{∗}(L_{2}T (1) ∧ M(2)) is as follows:

E_{2}^{0}(T (1) ∧ M(2)) = Z/4[v_{1}, v_{2}^{2}] ⊕ 2v_{2}Z/2[v_{1}^{±1}, v_{2}^{2}],
E_{2}^{1}(T (1) ∧ M(2)) = δ (M ⊕ ⨁_{k⩾0} (L_{k}^{0} ⊕ L_{k}^{1} ⊕ L_{k}^{2} ⊕ L_{k}^{3})) ,
E_{2}^{s}(T (1) ∧ M(2)) = δ ((Ã_{2} ⊗ Λ(ρ_{2}))^{s−1}) ⊕ δδ′ ((Ã_{2} ⊗ Λ(ρ_{2}))^{s−2}) for s > 5.

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

Recall [2] the module

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

Then this is the E_{∞}-term obtained from the summand A_{2} of the E_{2}-term, which is a submodule of π_{∗}(L_{2}\overline{M}(∞, ∞)), and there is no differential on the other summands of the E_{2}-term.

Corollary (2.8). The homotopy groups π_{∗}(L_{2}T (1) ∧ M(2)) contains the subgroups isomorphic to

M ⊕ ⨁_{k⩾0} (L_{k}^{0} ⊕ L_{k}^{1} ⊕ L_{k}^{2} ⊕ L_{k}^{3}) ⊕ Â_{2} ⊗ Λ(ρ_{2}) ⊕ Â_{2} ⊗ Λ(ρ_{2}).

3. The action of η_{R} on BP_{∗}[t_{2}, t_{3}, …]

Let BP denote the Brown-Peterson spectrum at the prime two. Then it associates the Hopf algebroid (A, Γ) = (BP_{∗}, BP_{∗}(BP)) = (Z_{(p)}[v_{1}, …], BP_{∗}[t_{1}, …]). Consider Γ(2) = BP_{∗}(BP)/(t_{1}) = BP_{∗}[t_{2}, t_{3}, …]. Then by the projection Γ → Γ(2), (A, Γ(2)) has the structure of the Hopf algebroid induced from (A, Γ). In this section, we determine the behavior of the right unit η_{R}: A → Γ(2) = A[t_{2}, …]. Here, v_{i}’s are the Hazewinkel generators of BP_{∗} defined by

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

under the inclusion BP_{∗} → 2^{−1}BP_{∗}. The unit map i: S^{0} → BP induces η_{R} = (i ∧ 1)_{∗}: BP_{∗} = π_{∗}(BP) → BP_{∗}(BP) → Γ(2), and η_{R} = (i∧1)_{∗}: 2^{−1}BP_{∗} = π_{∗}(2^{−1}BP) → BP_{∗}(2^{−1}BP) = 2^{−1}BP_{∗}(BP) = 2^{−1}BP_{∗}[t_{1}, …] → 2^{−1}Γ(2), whose action is given by the Quillen formulas

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

Lemma (3.3). On the generators v_{i} for 0 < i ⩽ 6, η_{R} 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} − t_{2}v_{3}^{4} + 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}).

Proof. Our computation is based on (3.1) and (3.2):

η_{R}(v_{1}) = η_{R}(2m_{1}) = 2(m_{1} + t_{1}) = 2m_{1} = v_{1},
η_{R}(v_{2}) = η_{R}(2m_{2} − m_{1}v_{1}^{2}) = 2(m_{2} + t_{2}) − m_{1}v_{1}^{2}
= v_{2} + 2t_{2} and
η_{R}(v_{3}) = η_{R}(2m_{3} − m_{2}v_{1}^{4} − m_{1}v_{2}^{2})
= 2(m_{3} + m_{1}t_{2}^{2} + t_{3}) − (m_{2} + t_{2})v_{1}^{4} − m_{1}(v_{2} + 2t_{2})^{2}
= 2m_{3} + v_{1}t_{2}^{2} + 2t_{3} − m_{2}v_{1}^{4} − v_{1}^{4}t_{2} − m_{1}v_{2}^{2} − 2v_{1}v_{2}t_{2} − 2v_{1}t_{2}^{2}
= 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}).

Since 4m_{1} = 2v_{1}, 4m_{2} = 2v_{2} + v_{1}^{3}, 4m_{3} = 2v_{3} + 2m_{2}v_{1}^{4} + v_{1}v_{2}^{2}, we have 4m_{1} ≡ 0 mod (2), 8m_{2} ≡ 0 mod (2) and 16m_{3} ≡ 0 mod (2). η_{R}(v_{4}) is given by the computation

η_{R}(v_{4}) = η_{R}(2m_{4} − m_{3}v_{1}^{8} − m_{2}v_{2}^{4} − m_{1}v_{3}^{2})
= 2(m_{4} + m_{2}t_{2}^{4} + m_{1}t_{3}^{2} + t_{4}) − (m_{3} + m_{1}t_{2}^{2} + t_{3})v_{1}^{8} − (m_{2} + t_{2})(v_{2} + 2t_{2})^{4}
−m_{1}(v_{3} + v_{1}t_{2}^{2} − v_{1}^{4}t_{2} + 2X)^{2}
≡ 2m_{4} + 2m_{2}t_{2}^{4} + v_{1}t_{3}^{2} + 2t_{4} − m_{3}v_{1}^{8} − m_{1}v_{1}^{8}t_{2}^{2} − v_{1}^{8}t_{3} − m_{2}v_{2}^{4} − v_{2}^{4}t_{2}
−m_{1}v_{3}^{2} − m_{1}v_{1}^{2}t_{2}^{4} − m_{1}v_{1}^{8}t_{2}^{2} − v_{1}(v_{3}(v_{1}t_{2}^{2} − v_{1}^{4}t_{2}) + v_{1}^{5}t_{2}^{3}) mod 2BP_{∗}
≡ v_{4} + v_{2}t_{2}^{4} + v_{1}t_{3}^{2} − v_{1}v_{1}^{8}t_{2}^{2} − v_{1}^{8}t_{3} − v_{2}^{4}t_{2} − v_{1}(v_{3}(v_{1}t_{2}^{2} − v_{1}^{4}t_{2}) + v_{1}^{5}t_{2}^{3}) mod 2BP_{∗}.

For η_{R}(v_{5}), we compute it mod (m_{1}^{3}). Then 4m_{1} ≡ 0 ≡ 4m_{2} mod 2BP_{∗}+(m_{1}^{3}) and 8m_{3} ≡ 0 mod 2BP_{∗}+(m_{1}^{3}).

η_{R}(v_{5}) = η_{R}(2m_{5} − m_{4}v_{1}^{16} − m_{3}v_{2}^{8} − m_{2}v_{3}^{4} − m_{1}v_{4}^{2})
= 2(m_{5} + m_{3}t_{2}^{8} + m_{2}t_{3}^{4} + m_{1}t_{4}^{2} + t_{5}) − (m_{4} + m_{2}t_{2}^{4} + m_{1}t_{3}^{2} + t_{4})v_{1}^{16}
−(m_{3} + m_{1}t_{2}^{2} + t_{3})(v_{2} + 2t_{2})^{8} − (m_{2} + t_{2})(v_{3} + v_{1}t_{2}^{2} − v_{1}^{4}t_{2} + 2X)^{4}
−m_{1}(v_{4} + v_{2}t_{2}^{4} + v_{1}t_{3}^{2} − v_{1}^{9}t_{2}^{2} − v_{1}^{8}t_{3} − v_{2}^{4}t_{2} − v_{1}(v_{3}(v_{1}t_{2}^{2} − v_{1}^{4}t_{2}) + v_{1}^{5}t_{2}^{3} + 2X′))^{2}
≡ 2m_{5} + 2m_{3}t_{2}^{8} + v_{2}t_{3}^{4} + v_{1}t_{4}^{2} + 2t_{5}
−(m_{3} + m_{1}t_{2}^{2} + t_{3})(v_{2}^{8} + 16Y) − (m_{2} + t_{2})(v_{3}^{4} + 2v_{1}^{2}v_{3}^{2}t_{2}^{4} + 4Y′)
−m_{1}(v_{4}^{2} + v_{2}^{2}t_{2}^{8} + v_{2}^{8}t_{2}^{2} + v_{1}^{2}t_{3}^{4} + 2c(4))
≡ 2m_{5} + 2m_{3}t_{2}^{8} + v_{2}t_{3}^{4} + v_{1}t_{4}^{2} + 2t_{5} − m_{3}v_{2}^{8} − m_{1}v_{2}^{8}t_{2}^{2} − v_{2}^{8}t_{3}
−m_{2}v_{3}^{4} − v_{1}^{2}v_{2}v_{3}^{2}t_{2}^{4} − t_{2}η_{R}(v_{3}^{4}) − m_{1}v_{4}^{2} − m_{1}v_{2}^{2}t_{2}^{8} − m_{1}v_{2}^{8}t_{2}^{2} − v_{1}c(4)
≡ v_{5} + v_{3}t_{2}^{8} + v_{2}t_{3}^{4} + v_{1}t_{4}^{2} + 2t_{5} − v_{1}v_{2}^{8}t_{2}^{2} − v_{2}^{8}t_{3} − v_{1}^{2}v_{2}v_{3}^{2}t_{2}^{4} − t_{2}η_{R}(v_{3}^{4}) − v_{1}c(4)

mod 2BP_{∗} + (m_{1}^{3}).

For η_{R}(v_{6}), we compute it mod (m_{1}). Then m_{1} ≡ 4m_{2} ≡ 4m_{3} ≡ 0 mod 2BP_{∗} + (m_{1}).

η_{R}(v_{6}) = η_{R}(2m_{6} − m_{5}v_{1}^{32} − m_{4}v_{2}^{16} − m_{3}v_{3}^{8} − m_{2}v_{4}^{4} − m_{1}v_{5}^{2})
≡ 2(m_{6} + m_{4}t_{2}^{16} + m_{3}t_{3}^{8} + m_{2}t_{4}^{4} + t_{6}) − (m_{4} + m_{2}t_{2}^{4} + t_{4})(v_{2} + 2t_{2})^{16}
−(m_{3} + t_{3})(v_{3} + 2X)^{8} − (m_{2} + t_{2})(v_{4} + v_{2}t_{2}^{4} − v_{2}^{4}t_{2} + 2X)^{4}
≡ 2m_{6} + 2m_{4}t_{2}^{16} + v_{3}t_{3}^{8} + v_{2}t_{4}^{4} − m_{4}v_{2}^{16} − m_{2}v_{2}^{16}t_{2}^{4} − v_{2}^{16}t_{4}
−m_{3}v_{3}^{8} − v_{3}^{8}t_{3} − m_{2}v_{4}^{4} − m_{2}(v_{2}^{4}t_{2}^{16} + v_{2}^{16}t_{2}^{4}) + v_{2}c(4)^{2} − t_{2}η_{R}(v_{4})^{4}
≡ v_{6} + v_{4}t_{2}^{16} + v_{3}t_{3}^{8} + v_{2}t_{4}^{4} − v_{2}^{17}t_{2}^{4} − v_{2}^{16}t_{4} − v_{3}^{8}t_{3} − v_{2}c(4)^{2} − t_{2}η_{R}(v_{4})^{4},

since

2c(4)^{2} ≡ η_{R}(v_{4}^{4}) − (v_{4}^{4} + v_{2}^{4}t_{2}^{16} + v_{2}^{16}t_{2}^{4}) mod (4, v_{1}^{4}). □

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

Let F (2)_{∗} denote the BP_{∗}-module Z_{(2)}[v_{1}, v_{2}, v_{2}^{−1}, v_{3}] with action given by sending v_{i} to 0 for i > 3, and F (2) the spectrum that represents the Landweber homology theory F (2)_{∗}(X) = F (2)_{∗} ⊗_{BP_{∗}} BP_{∗}(X). Then F (2)_{∗}(F (2)) = F (2)_{∗}⊗_{BP_{∗}} BP_{∗}(BP)⊗_{BP_{∗}} F (2)_{∗}, and the Hopf algebroid structure of (A, Γ) = (BP_{∗}, BP_{∗}(BP)) 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 second Johnson-Wilson spectrum E(2). In the same manner as the change of rings theorem Ext_{Γ}^{s}(A, M) = Ext_{E(2)_{∗}E(2)}^{s}(E(2)_{∗}, E(2)_{∗} ⊗_{A} M) for v_{2}-local A-comodule M given in [1], we obtain Ext_{Σ}^{s}(B, B ⊗_{A} M) = Ext_{E(2)_{∗}E(2)}^{s}(E(2)_{∗}, E(2)_{∗} ⊗_{A} M), and then

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

The Ravenel spectrum T (1) is characterized by BP_{∗}(T (1)) = BP_{∗}[t_{1}] = A[t_{1}]. It follows that F (2)_{∗}(T (1)) = F (2)_{∗}[t_{1}] = B[t_{1}], and

B[t_{1}] = B□_{Σ(2)}Σ

for Σ(2) = B ⊗_{A} Γ(2)⊗_{A} B = B ⊗_{A} A[t_{2}, t_{3}, …]⊗_{BP_{∗}} B. Now write H^{∗}M as Ext_{Σ(2)}^{∗}(B, M) for a Σ(2)-comodule M. Then, for a Γ-comodule M, we have

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).

In this paper, we compute H^{∗}B/(4) to determine the E_{2}-term Ext_{Γ}^{s}(A, v_{2}^{−1}BP_{∗}[t_{1}]/(4)). For this sake, we consider the chromatic spectral sequence associated to the short exact sequence 0 → B/(4) → v_{1}^{−1}B/(4) → M_{1}^{1}(2) → 0. We compute H^{∗}M_{1}^{1}(2) by using the v_{1}-Bockstein spectral sequence associated to the short exact

sequence 0 → B/(4, v_{1}) ⟶^{1/v_{1}} M_{1}^{1}(2) ⟶^{v_{1}} M_{1}^{1}(2) → 0.

From Lemma 3.3, we obtain the relations in Σ(2):

(4.1)

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 (4.1), 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

(4.2)

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}^{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}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})

by Lemma 3.3.

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

As we have defined in the previous section, M_{1}^{1}(2) denotes the cokernel of B/(4) → v_{1}^{−1}B/(4). In order to define generators of H^{∗}M_{1}^{1}(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} in [7].

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

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

On subgroups of π_{∗}(L_{2}T (1) ∧ M(2)) at the prime two 7

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}(\underline{v_{2}^{8}t_{2}^{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) and
d(v_{1}^{3}v_{2}^{6}v_{3}) ≡ v_{1}^{3}v_{2}^{6}(\underline{v_{1}t_{2}^{2}}_{2} + v_{1}^{4}t_{2}) mod (2). □

Hereafter, we put v_{2}^{2} = 1 for the sake of simplicity. Lemma (5.2). There is an element y 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}),

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}(\underline{t_{3}^{2}}_{1} + v_{3}^{2}t_{2}^{4} + \underline{v_{3}^{8}t_{2}^{2}}_{2} + v_{1}^{2}t_{4}^{4} + v_{1}^{4}(v_{3}^{2}t_{3}^{8} + t_{2}^{2}t_{3}^{4} + \underline{v_{3}^{4}t_{2}^{8}}_{a})
+\underline{v_{1}^{2}t_{2}^{2}}_{3} + \underline{v_{1}^{4}v_{3}^{4}t_{2}^{8}}_{a}),
d(v_{1}^{9}v_{2}v_{3}^{2}) ≡ v_{1}^{11}v_{2}t_{2}^{4}
≡ v_{1}^{11}(t_{2} + \underline{v_{1}t_{3}^{2}}_{1} + 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}(\underline{t_{2}^{2}}_{2} + v_{1}^{3}(\underline{v_{2}t_{2}^{4}}_{4} + v_{1}t_{3}^{2})),
d(v_{1}^{13}v_{3}) ≡ \underline{v_{1}^{14}t_{2}^{2}}_{3} and
d(v_{1}^{13}v_{2}v_{3}^{10}) ≡ \underline{v_{1}^{15}v_{2}v_{3}^{8}t_{2}^{4}}_{4}.

We further consider y = y′ + v_{1}^{4}x + v_{1}^{8}v_{3}^{18} + v_{1}^{9}v_{3}^{5}, and compute

d(y′) ≡ \underline{v_{1}^{11}T_{2}}_{2} + \underline{v_{1}^{13}v_{3}t_{2}^{2}}_{A} + v_{1}^{14}t_{4}^{4} + v_{1}^{16}(r′ + \underline{t_{2}^{2}t_{3}^{4}}_{B} + 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}(\underline{t_{2}^{2}}_{1} + v_{1}^{2}t_{3}^{4} + v_{1}^{4}v_{3}^{2}t_{2}^{4}) + \underline{v_{3}^{16}t_{2}^{4}}_{3} + v_{1}^{4}t_{4}^{8} + \underline{v_{1}T_{2}}_{2}),
d(v_{1}^{8}v_{3}^{18}) ≡ \underline{v_{1}^{10}v_{3}^{16}t_{2}^{4}}_{3} and
d(v_{1}^{9}v_{3}^{5}) ≡ \underline{v_{1}^{10}v_{3}^{4}t_{2}^{2}}_{1} + v_{1}^{13}v_{3}^{4}t_{2} + \underline{v_{1}^{13}v_{3}t_{2}^{8}}_{A} + \underline{v_{1}^{14}t_{2}^{10}}_{B} mod (2, v_{1}^{20})

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 the sums of the terms underlined with A and B are v_{1}^{15}v_{3}t_{3}^{4} and v_{1}^{14}t_{2}^{4}, respectively, 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}). □

Now define elements x_{k} 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.

We introduce an integer a_{n} = 4^{n} + 2e_{n} = e_{n+1} + e_{n} and an element g_{k} defined inductively by

(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).

Then we have the following relation by definition. Lemma (5.5). For k > 0,

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

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 = 1, it follows from Lemma 5.1 and (5.4). Suppose that the lemma holds for k. 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. □

We introduce an element R_{n} defined by

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

for n > 0.

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. 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 by (5.4), which equals v_{1}^{14}r^{4} + v_{1}^{16}r′ by Lemma 5.2. By the definitions (5.4), (5.7) and (5.3),

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 we 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}). The other part follows from the computation

v_{1}^{4e_{k}+4+10}R_{k}^{4} ≡ v_{1}^{10}g_{k+1}^{4} + v_{1}^{10}x_{k}^{4}g_{k}^{4} mod (2),
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}}).

Indeed, this shows that v_{1}^{4e_{k}+4+10}R_{k}^{4} is homologous to v_{1}^{12}g_{k+2} + v_{1}^{12}x_{k}^{4}g_{k+1} ≡ v_{1}^{12}(g_{k+2} + x_{k+1}g_{k+1}) = v_{1}^{12+e_{k+1}+1}R_{k+1} mod (2, v_{1}^{a_{k+1}−12}). Since 4e_{k} + 4 + 10 = 12 + e_{k+1} + 1, we see the lemma. □

The relations between R_{n}’s are given by

Lemma (5.9). a) 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} + \underline{v_{1}T_{2}}_{1}) mod (2, v_{1}^{18}),
d(v_{3}^{32}y′) ≡ v_{3}^{32}(\underline{v_{1}^{11}T_{2}}_{1} + \underline{v_{1}^{13}v_{3}t_{2}^{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} + \underline{v_{1}^{12}v_{3}^{10}t_{2}^{2}}_{3} mod (2, v_{1}^{14}),
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})
≡ \underline{v_{1}^{12}v_{3}^{10}t_{2}^{2}}_{3} + 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} + \underline{v_{3}^{33}t_{2}^{8}}_{2}) mod (2, v_{1}^{14})

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 (4.2). It follows that d(u_{1}) ≡ v_{1}^{10}(r^{8} + r^{4} + v_{3}^{4}t_{3}^{8}) mod (2, v_{1}^{12}). Since 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, 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 Lemma 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}. Since y ≡ x_{1}^{2} mod (2, v_{1}^{2}) by definition, 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}),

if we put u′_{1} = u_{1}^{2} + v_{1}^{10}x_{1}^{2}y, whence we have the lemma for n = 1 by 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 R_{n}^{4}, x_{n}^{4} and g_{n}^{4} are homologous to R_{n+1}, x_{n+1} and v_{1}^{2}g_{n+1}, respectively. We define u_{n+1} to be the sum of u′_{n}^{2} and the elments that give the 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 R_{n}^{4} is homologous to R_{n+1} by Lemma 5.8 and 8e_{n−1}+2 = 2e_{n}, we have u′_{n+1} = u_{n+1}^{2}+v_{1}^{5×4^{n+1}+8e_{n}−10}x_{n+1}^{2} ×x^{4e_{n}}y, and we obtain the congruence on d(u′_{n+1}). This completes the induction. □

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}).
□

Note that d(x_{n}^{2}) ≡ 2v_{1}^{a_{n}}x_{n}g_{n} mod (4, v_{1}^{2a_{n}}). Then the above two lemmas imply

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}+a_{n}}) 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}+2a_{n}})

for n > 0.

10 Ippei Ichigi, Katsumi Shimomura, and Xiangjun Wang

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.

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 − 1.

In a similar way, the second equation 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 5e_{n} = 2e_{n} + 4^{n} − 1 = a_{n} − 1. □

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}}),
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}}) and
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}}).
□

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 the definition (5.7) as follows:

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

Let b_{n,k} be integers defined by

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

On subgroups of π_{∗}(L_{2}T (1) ∧ M(2)) at the prime two 11

for k > 0. Here,

\overline{e}_{n} = (16^{n} − 1)/15 for n ⩾ 0.

Then consider the elements

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}.

Here [x] denotes the largest integer that does not exceed x. We introduce the integers c_{n}:

c_{n} = 4^{n} + 3 × 4^{1+ε(n)}\overline{e}_{[ \frac{n}{2} ]} for n ⩾ 0.

Here,

ε(n) = \frac{1 − (−1)^{n}}{2} = \begin{cases}0 & if n is even, \\ 1 & if n is odd. \end{cases}

Lemma (5.15). These elements 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)}}).

Proof. Note that the integers b_{n,k} 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.

12 Ippei Ichigi, Katsumi Shimomura, and Xiangjun Wang

The differentials of these elements 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), and we obtain the lemma. □ Lemma (5.16). There is 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}(\underline{t_{3}^{2}}_{1} + \underline{v_{3}^{2}t_{2}^{4}}_{2} + \underline{v_{3}^{8}t_{2}^{2}}_{3} + v_{1}^{2}t_{4}^{4} + \underline{v_{1}^{2}t_{2}^{2}}_{8}),
d(2v_{1}^{2}v_{2}v_{3}^{2}) ≡ \underline{2v_{1}^{4}v_{2}t_{2}^{4}}_{1},
d(v_{1}^{4}v_{2}) ≡ \underline{2v_{1}^{4}t_{2}}_{1},
d(v_{1}^{3}v_{3}^{4}) ≡ \underline{2v_{1}^{5}v_{3}^{2}t_{2}^{4}}_{2} + \underline{v_{1}^{7}t_{2}^{8}}_{5},
d(2v_{1}^{4}v_{3}^{9}) ≡ \underline{2v_{1}^{5}v_{3}^{8}t_{2}^{2}}_{3},
d(2v_{2}x_{1}) ≡ 2v_{1}^{6}v_{2}(t_{3}^{4} + v_{1}t_{2})
≡ 2v_{1}^{6}(\underline{t_{3}}_{4} + \underline{v_{3}t_{2}^{8}}_{1} + \underline{v_{3}^{4}t_{2}}_{6} + v_{1}t_{4}^{2} + \underline{v_{1}v_{2}t_{2}}_{7}),
d(2v_{1}^{5}v_{3}^{2}) ≡ \underline{2v_{1}^{7}t_{2}^{4}}_{7},
d(v_{1}^{6}v_{3}) ≡ \underline{2v_{1}^{6}t_{3}}_{4} + \underline{v_{1}^{7}t_{2}^{2}}_{5},
d(v_{1}^{6}v_{2}v_{3}^{4}) ≡ \underline{2v_{1}^{6}v_{3}^{4}t_{2}}_{6},
d(2v_{1}^{6}v_{3}) ≡ \underline{2v_{1}^{7}t_{2}^{2}}_{8}.

The sum is 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} + \underline{v_{3}^{6}t_{2}^{32}}_{1} + \underline{v_{3}^{18}t_{2}^{4}}_{2} + \underline{v_{3}^{16}t_{3}^{2}}_{3}),
d(2v_{1}^{2}v_{3}^{7}) ≡ \underline{2v_{1}^{3}v_{3}^{6}t_{2}^{2}}_{1},
d(v_{1}v_{3}^{20}) ≡ \underline{2v_{1}^{3}v_{3}^{18}t_{2}^{4}}_{2},
d(2v_{2}v_{3}^{18}) ≡ \underline{2v_{1}^{2}v_{2}v_{3}^{16}t_{2}^{4}}_{3},
d(v_{1}^{2}v_{2}v_{3}^{16}) ≡ \underline{2v_{1}^{2}v_{3}^{16}t_{2}}_{3}.

On subgroups of π_{∗}(L_{2}T (1) ∧ M(2)) at the prime two 13

This implies

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 computation

2v_{3}^{2}t_{3}^{4} ≡ 2v_{2}v_{3}^{2}(\underline{t_{3}}_{1} + v_{3}t_{2}^{8} + v_{3}^{4}t_{2}) and
d(v_{2}v_{3}^{3}) ≡ \underline{2v_{2}v_{3}^{2}t_{3}}_{1} + 2v_{3}^{3}t_{2}

shows 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. □ Lemma (5.17). There is 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} = t_{2}^{2} + v_{2}t_{2}. 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 from the computation:

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}.
□

6. The action of the connecting homomorphism We write H^{∗}W as E_{2}^{∗}(T (1) ∧ W) for a spectrum W, and X(2) denotes M(2, ∞), which is a cofiber of the

localization map M(2) → L_{1}M(2). Here, M(2) denotes the mod 4 Moore spectrum. Then we have the cofiber sequence M(2) → X(2) →^{v_{1}} X(2), which induces the long exact sequence

H^{∗}M(2) ⟶^{1/v_{1}} H^{∗}X(2) ⟶^{v_{1}} H^{∗}X(2) ⟶^{δ} H^{∗}M(2).

In this section, we determine H^{∗}X(2) by observing the long exact sequence in the method using [4, Remark 3.11]. We follow the method shown in [7]. First we determine H^{0}X(2). We have shown in the previous section

H^{0}M(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) = 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 (6.1). For the generators of H^{0}M(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. The first and the second ones 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}.

For the third equation, we consider the cases where n = 1 and n > 1. First, suppose that n = 1.

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}(\underline{t_{3}}_{1} + \underline{v_{3}t_{2}^{8}}_{2} + \underline{v_{3}^{4}t_{2}}_{3} + v_{1}t_{4}^{2})/v_{1}^{2} + \underline{2v_{2}x_{1}^{2t}t_{2}/v_{1}}_{4},
d(x_{1}^{2t}v_{3}/v_{1}^{2}) = \underline{2x_{1}^{2t}t_{3}/v_{1}^{2}}_{1} + x_{1}^{2t}t_{2}^{2}/v_{1},
d(x_{1}^{2t}v_{3}^{2}/v_{1}^{3}) = \underline{2x_{1}^{2t}v_{3}t_{2}^{2}/v_{1}^{2}}_{2} + x_{1}^{2t}t_{2}^{4}/v_{1},
d(v_{2}x_{1}^{2t}v_{3}^{4}/v_{1}^{2}) = \underline{2x_{1}^{2t}v_{3}^{4}t_{2}/v_{1}^{2}}_{3},
d(2x_{1}^{2t}v_{3}^{2}/v_{1}^{3}) = \underline{2x_{1}^{2t}t_{2}^{4}/v_{1}}_{4}.

Put \overline{t}′_{21} = t_{2}^{2} + t_{2}^{4} + 2t_{4}^{2} and \overline{t}_{21} = t_{2}^{2} + v_{2}t_{2}. Then, \overline{t}_{21} − \overline{t}′_{21} = 2t_{4}^{2} + 2v_{2}t_{4} mod (4, v_{1}) and we see that \overline{t}′_{21} represents \overline{h}_{21} + 2ρ_{2}. For n > 1, we compute

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

The fourth equation 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} + \underline{v_{3}^{2}t_{2}^{4}}_{1} + \underline{v_{3}^{8}t_{2}^{2}}_{2})/v_{1},
d(v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{2}/v_{1}) = \underline{2x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{2}t_{2}/v_{1}}_{1},
d(2v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{9}/v_{1}^{2}) = \underline{2v_{2}x_{n}^{4t}x_{1}^{2e_{n−1}}v_{3}^{8}t_{2}^{2}/v_{1}}_{2}.

The fifth and the sixth 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 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 tenth one are seen as follows:

d(2x_{1}^{2t+1}/v_{1}^{7}) = 2x_{1}^{2t}g_{1}/v_{1} = 2x_{1}^{2t}(\underline{v_{2}t_{3}}_{1} + \underline{v_{2}v_{3}t_{2}^{2}}_{2} + \underline{v_{2}v_{3}^{4}t_{2}}_{3})/v_{1},
d(v_{2}x_{1}^{2t}v_{3}/v_{1}) = \underline{2x_{1}^{2t}v_{3}t_{2}/v_{1}}_{2} + \underline{2v_{2}x_{1}^{2t}t_{3}/v_{1}}_{1},
d(2x_{1}^{2t+1}v_{3}^{2}/v_{1}^{3}) = \underline{2x_{1}^{2t+1}t_{2}^{4}/v_{1}}_{3},
d(x_{1}^{4t+2}/v_{1}^{7}) = 2x_{1}^{4t+1}g_{1}/v_{1} = 2x_{1}^{4t+1}(\underline{v_{2}t_{3}}_{4} + \underline{v_{2}v_{3}t_{2}^{2}}_{5} + \underline{v_{2}v_{3}^{4}t_{2}}_{6})/v_{1},
d(v_{2}x_{1}^{4t+1}v_{3}/v_{1}) = \underline{2x_{1}^{4t+1}v_{3}t_{2}/v_{1}}_{5} + \underline{2v_{2}x_{1}^{4t+1}t_{3}/v_{1}}_{4},
d(2x_{1}^{4t+2}v_{3}^{2}/v_{1}^{3}) = \underline{2x_{1}^{4t+2}t_{2}^{4}/v_{1}}_{6},

in which the terms with subscript 2 (resp. 5) amount to 2v_{2}x_{1}^{2t}v_{3}\overline{t}_{21}/v_{1} (resp. 2v_{2}x_{1}^{4t+1}v_{3}\overline{t}_{21}/v_{1}). By Lemma 5.16,

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 homologous. 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 have

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 5.15, we compute

d(x_{2k}^{2t}X_{2k}/v_{1}^{c_{2k}+1}) = \underline{2x_{2k}^{2t}x_{2}^{3\overline{e}_{k−1}}R_{1}/v_{1}}_{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}}(\underline{R_{1}}_{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 5.15, 5.9 and 5.17,

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

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}.

16 Ippei Ichigi, Katsumi Shimomura, and Xiangjun Wang

For odd n = 2k + 1 ⩾ 3, by Lemmas 5.15, 5.9 and 5.17,

d(x_{n}^{2t}Y_{n}/v_{1}^{2c_{n−1}+1}) = x_{n−1}^{8t+2}x_{2}^{6\overline{e}_{k−1}}R_{1}^{2}/v_{1} = \underline{x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}R_{1}^{2}/v_{1}}_{1},
d(x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}X_{2}/v_{1}^{c_{2}+1}) = \underline{2x_{2}^{4^{2k−2}(8t+2)+6\overline{e}_{k−1}}R_{1}/v_{1}}_{2},
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}}(\underline{R_{1}^{2}}_{1} + \underline{R_{1}}_{2} + \underline{x_{1}g_{1}^{2}}_{3})/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}}(\underline{x_{1}g_{1}^{2}}_{3} + 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 5.15, 5.10 and 5.9 show the equations

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

which give

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

and

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 5.15, 5.10, 5.9 and 5.16,

d(x_{n}^{4t}X′_{2k+1}/v_{1}^{2c_{2k+1}+7}) = \underline{2x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}R_{2}^{2}/v_{1}^{7}}_{1},
d(x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}x_{3,1}/v_{1}^{7×4^{2}+7}) = \underline{2x_{3}^{4^{2k−1}t+6\overline{e}_{k−1}}R_{2}/v_{1}^{7}}_{2},
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}}(\underline{R_{2}^{2}}_{1} + \underline{R_{2}}_{2} + \underline{v_{1}^{4}x_{1}^{6}g_{1}^{2}}_{3})/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}}(\underline{g_{1}^{2}}_{3} + 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 5.15, 5.10, 5.9 and 5.16,

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

Corollary (6.2). The zero line H^{0}(X(2)) is isomorphic to the module

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}}⟩
⊕⨁_{k⩾0} (2E_{∗}[x_{2k}^{2}]⟨x_{2k}/v_{1}^{c_{2k}}⟩ ⊕ E_{∗}[x_{2k}^{4}]⟨x_{2k}^{2}/v_{1}^{c_{2k}}⟩
⊕2E_{∗}[x_{2k}^{4}]⟨x_{2k}^{2}/v_{1}^{2c_{2k}}⟩ ⊕ E_{∗}[x_{2k+1}^{2}]⟨x_{2k+1}/v_{1}^{2c_{2k}}⟩
⊕2E_{∗}[x_{2k+1}^{2}]⟨x_{2k+1}/v_{1}^{c_{2k+1}+2}⟩ ⊕ E_{∗}[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}^{2c_{2k+1}+6}⟩ ⊕ E_{∗}[x_{2k+2}^{2}]⟨x_{2k+2}/v_{1}^{2c_{2k+1}+6}⟩) .

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: wxiangjun@eyou.com

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