The homotopy groups of L_{2}V (1\frac{1}{2}) and an invertible spectrum at the prime three

By Ippei Ichigi and Katsumi Shimomura

Abstract. In [5], an E(2)-invertible spectrum X at the prime 3 is constructed from the Toda spectrum V (1 \frac{1}{2} ) with a map f: X → Σ^{−21}L_{2}V (1 \frac{1}{2} ) inducing the projection f_{∗}: E(2)_{∗} → E(2)_{∗}/(3). It satisfies X^{3} = X ∧ X ∧ X = L_{2}S^{0}, though X ≠ L_{2}S^{0}. The inclusion i: S^{0} → V (0) for the mod 3 Moore spectrum V (0) yields another map i ∧ 1_{X}: X → V (0) ∧ X that induces the projection E(2)_{∗} → E(2)_{∗}/(3). In this paper, we show the difference between Σ^{−21}L_{2}V (1 \frac{1}{2} ) and V (0) ∧ X by the structures of the homotopy groups of them. Our starting point is the equivalence v_{2}^{3} : V (1) ∧ X ≃ L_{2}V (1) for the Smith-Toda spectrum V (1) shown in [4]. We determine the homotopy groups π_{∗}(V (0) ∧ X^{k}) of the k-fold smash product X^{k} from the equivalence v_{2}^{3} by the v_{1}-Bockstein spectral sequence, and then the homotopy groups π_{∗}(X^{k}) by the mod 3 Bockstein spectral sequence. We also show that the cofiber of f is X^{2} = X ∧X. Last, we obtain the homotopy groups π_{∗}(L_{2}V (1 \frac{1}{2} )) from these results by examining the cofiber sequence associated to f.

1. Introduction

Consider the stable homotopy category S_{p} localized at the prime p, and let E(n) ∈ S_{p} denote the n-th Johnson-Wilson spectrum characterized by π_{∗}(E(n)) = E(n)_{∗} = ℤ_{(p)}[v_{1}, …, v_{n}, v_{n}^{−1}] ⊂ v_{n}^{−1}BP_{∗} = v_{n}^{−1}ℤ_{(p)}[v_{1}, v_{2}, …]. The Bousfield localization L_{n}: S_{p} → S_{p} with respect to E(n) is then characterized by the property that a map f: L_{n}W → L_{n}W′ is an equivalence if and only if the map induces an isomorphism E(n)_{∗}(f): E(n)_{∗}(W) ≅ E(n)_{∗}(W′). In the category L_{n} = L_{n}S_{p}, a spectrum U is called invertible if there is a spectrum W such that U ∧ W = L_{n}S^{0}. A suspension of the sphere spectrum L_{n}S^{0} is an example of invertible spectra, and it is shown in [3] that there is no other invertible spectrum in L_{n} if n^{2} + n < 2p − 2. It is shown in [3] and [5] that a spectrum U is invertible if and only if E(n)_{∗}(U) = E(n)_{∗} as an E(n)_{∗}E(n)-comodule. In [3], it is further shown that there is an invertible spectrum other than a suspension of the sphere spectrum if n = 1 and p = 2. The homotopy groups of it are told from a result of [6].

From now on, we work in the category S_{3}. In [5], it is constructed for n = 2 and p = 3 another example X of invertible spectra that is not a suspension of the sphere spectrum, so that there is a map f: X → Σ^{−21}L_{2}V (1\frac{1}{2}) that induces the projection E(2)_{∗} = E(2)_{∗}(X) → E(2)(Σ^{−21}V (1\frac{1}{2})) = E(2)_{∗}/(3). Here V (1\frac{1}{2}) denotes the Toda spectrum obtained as a cofiber of the map [βi_{1}]: Σ^{16}V (0) → V (1) of the Smith-Toda spectra given in [12] that induces v_{2} : BP_{∗}/(3) → BP_{∗}/(3, v_{1}) on the BP_{∗}-homologies. In this paper, X^{k} denotes the k-fold smash products of X. Note that X in [5] is our X^{2}. The result seems to indicate that V (0) ∧ X ≃ Σ^{−21}L_{2}V (1\frac{1}{2}), but it is not true, since V (0) is a ring spectrum, while V (1\frac{1}{2}) is not a V (0)-module spectrum [12]. In this paper, we observe the difference from the viewpoint of the homotopy groups. We make computations of homotopy groups by the E(2)-based Adams spectral sequence E_{r}^{∗}(W) converging to π_{∗}(L_{2}W) for a spectrum W. Since there is an isomorphism E_{2}^{∗}(V (0) ∧ X^{k}) = E_{2}^{∗}(V (0)), the difference appears in the differentials on them. Our computation of the differentials is based on the equivalence v_{2}^{3} : Σ^{48}V (1) ∧ X ≃ L_{2}V (1) given in [4]. Since the Bockstein spectral sequences behave well on the E_{1}-terms of the chromatic spectral sequences, we consider the E(2)_{∗}E(2)-comodules M_{2}^{0} = E(2)_{∗}/(3, v_{1}), M_{1}^{1} = E(2)_{∗}/(3, v_{1}^{∞}) and M_{0}^{2} = E(2)_{∗}/(3^{∞}, v_{1}^{∞}). Let M^{2}W and M^{1}W′ for spectra W and W′ with HQ_{∗}(W′) = 0 be spectra defined by the cofiber sequences

L_{2}W →^{η_{0}} L_{0}W → N^{1}W, N^{1}W →^{η_{1}} L_{1}N^{1}W → M^{2}W and L_{2}W′ →^{η_{1}} L_{1}W′ → M^{1}W′,

where η_{n} denotes the L_{n}-localization. Then the E_{2}-terms of the spectra that we concern are given as follows:

E_{2}^{∗}(V (1) ∧ X^{k}) = H^{∗}M_{2}^{0}, E_{2}^{∗}(M^{1}V (0) ∧ X^{k}) = H^{∗}M_{1}^{1} = E_{2}^{∗}(Σ^{21}M^{1}V (1\frac{1}{2})) and E_{2}^{∗}(M^{2}X^{k}) = H^{∗}M_{0}^{2}, where H^{∗}M for a comodule M denotes Ext_{E(2)_{∗}E(2)}^{∗}(E(2)_{∗}, M). These modules fit in the long exact sequences

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

Key words and phrases. Invertible spectrum, Smith-Toda spectrum, Homotopy groups.

The second author is partially supported by Grant-in-Aid for Scientific Research (No. 14540083), Ministry of Education, Culture, Sports, Science and Technology, Japan.

\begin{matrix}0 → H^{0}M_{2}^{0} → H^{0}M_{1}^{1} →^{v_{1}} H^{0}M_{1}^{1} →^{δ} H^{1}M_{2}^{0} → ⋯ → H^{s}M_{2}^{0} → H^{s}M_{1}^{1} →^{v_{1}} H^{s}M_{1}^{1} →^{δ} ⋯ and \\ 0 → H^{0}M_{1}^{1} →^{φ} H^{0}M_{0}^{2} →^{3} H^{0}M_{0}^{2} →^{δ} H^{1}M_{1}^{1} → ⋯ → H^{s}M_{1}^{1} →^{φ} H^{s}M_{0}^{2} →^{3} H^{s}M_{0}^{2} →^{δ} ⋯, \end{matrix}

which define the v_{1}- and mod 3 Bockstein spectral sequences, respectively. In [9], the module H^{∗}M_{1}^{1} is shown to be the direct sum of the three modules A_{i}’s for i = 0, 1, 2 such that the Adams differentials on A_{0} and A_{1} are trivial. Furthermore, in [10], A_{2} is decomposed into two summands (A_{2})_{(h)} and (A_{2})_{(t)} so that φ((A_{2})_{(h)}) = \overline{(A_{2})}_{(h)} ⊂ H^{∗}M_{0}^{2} is isomorphic to (A_{2})_{(h)}. Furthermore, it is shown that δ(\overline{(A_{2})}_{(h)}) = (A_{2})_{(t)}. The equivalence v_{2}^{3} : Σ^{48}V (1) ∧ X ≃ L_{2}V (1) and the results of [8] show that the Adams differentials act on (A_{2})_{(h)} and (A_{2})_{(t)}. By this, we obtain the homotopy groups π_{∗}(L_{2}V (0)) in [9], in which a differential is missing, and here we correct it in the last section. After modifying modules A_{i} slightly, we obtain

Theorem 1.1. The homotopy groups π_{∗}(M^{1}V (0) ∧ X^{k}) are the direct sum of A_{0}, A_{1}, { \atop v_{2}^{9−3k}(A_{2})}_{(h)} and

{ \atop v_{2}^{9−3k}(A_{2})}_{(t)}.

See (4.10) for the generators of these modules. Here we write M̃ as a resulted submodule of the E_{∞}-term obtained from a differential submodule M of the E_{2}-term. The result for k = 0 gives a correction of [9, Th. 2.8] (see §7).

Let \overline{A_{i}} for i = 0, 1, 2 denote the module fitting in the exact sequence A_{i} →^{φ} \overline{A_{i}} →^{3} \overline{A_{i}} →^{δ} A_{i}, which is determined in [11].

Theorem 1.2. The homotopy groups π_{∗}(M^{2}X^{k}) are the direct sum of \overline{A_{0}}, \overline{A_{1}} and { \atop v_{2}^{9−3k}\overline{A_{2}}}.

The structures of \overline{A_{0}}, \overline{A_{1}} and \overline{A_{2}} are slightly difference from [11]. See (5.9) for the definitions of these modules in this paper. Note that the description of the modules in [11] had many misprints.

We show that there is a cofiber sequence X^{2} → X →^{f} Σ^{−21}L_{2}V (1\frac{1}{2}) → ΣX^{2} in the next section, and we

have the induced cofiber sequence M^{2}X^{2} → M^{2}X →^{f} Σ^{−20}L_{2}M^{1}V (1\frac{1}{2}) → ΣM^{2}X^{2}. It induces the long exact

sequence H^{∗−1}M_{0}^{2} →^{δ} H^{∗}M_{1}^{1} →^{φ_{∗}} H^{∗}M_{0}^{2} →^{3} H^{∗}M_{0}^{2} of the E_{2}-terms. Since the submodules behave δ(\overline{(A_{2})}_{(h)}) = (A_{2})_{(t)} and φ_{∗}((A_{2})_{(h)}) = \overline{(A_{2})}_{(h)} under the maps δ and φ_{∗} in the exact sequence, we read off the homotopy groups π_{∗}(M^{1}V (1\frac{1}{2})) from the above results.

Theorem 1.3. The homotopy groups π_{∗}(Σ^{−21}M^{1}V (1\frac{1}{2})) are the direct sum of A_{0}, A_{1}, { \atop v_{2}^{3}(A_{2})}_{(h)} and { \atop v_{2}^{6}(A_{2})}_{(t)}.

The modules { \atop v_{2}^{3}(A_{2})}_{(h)} and { \atop v_{2}^{6}(A_{2})}_{(t)} are those given in (4.10). Thus the structure of the homotopy groups π_{∗}(M^{1}V (1\frac{1}{2})) is a mixture of the homotopy groups π_{∗}(M^{1}V (0) ∧ X) and π_{∗}(M^{1}V (0) ∧ X^{2}).

Let K(2) denote the second Morava K-theory and L_{K(2)} : S_{3} → S_{3} the Bousfield localization functor with respect to K(2). Then L_{K(2)}V (0) ∧ X^{k} = lim_{j} L_{2}M(3, v_{1}^{j}) ∧ X^{k} and L_{K(2)}X^{k} = lim_{i,j} L_{2}M(3^{i}, v_{1}^{j}) ∧ X^{k} for the generalized Smith-Toda spectrum M(3^{i}, v_{1}^{j}) such that BP_{∗}(M(3^{i}, v_{1}^{j})) = BP_{∗}/(3^{i}, v_{1}^{j}). Furthermore,

L_{K(2)}X^{2} → L_{K(2)}X →^{f} Σ^{−21}L_{K(2)}V (1\frac{1}{2}) is a cofiber sequence. Since M^{1}V (0) ∧ X^{k} = colim_{j}L_{2}M(3, v_{1}^{j}) ∧ X^{k}

and M^{2}X^{k} = colim_{i,j}L_{2}M(3^{i}, v_{1}^{j}) ∧ X^{k}, these results show immediately the structure of the homotopy groups π_{∗}(L_{K(2)}V (0) ∧ X^{k}), π_{∗}(L_{K(2)}V (1\frac{1}{2})) and π_{∗}(L_{K(2)}X^{k}).

Corollary 1.4. The homotopy groups π_{∗}(L_{K(2)}V (0) ∧ X^{k}), π_{∗}(L_{K(2)}V (1\frac{1}{2})) and π_{∗}(L_{K(2)}X^{k}) are as follows:

π_{∗}(L_{K(2)}V (0) ∧ X^{k}) = Â_{0}⨁Â_{1}⨁v_{2}^{9−3k}{ \atop (A_{2})_{(h)}}⨁v_{2}^{9−3k}{ \atop (A_{2})_{(t)}},
π_{∗}(L_{K(2)}V (1\frac{1}{2})) = Â_{0}⨁Â_{1}⨁v_{2}^{3}{ \atop (A_{2})_{(h)}}⨁v_{2}^{6}{ \atop (A_{2})_{(t)}} and
π_{∗}(L_{K(2)}X^{k}) = ˆ \overline{A_{0}}⨁ˆ\overline{A_{1}}⨁v_{2}^{9−3k}\begin{matrix} \\ \overline{(A_{2})} & (h) \end{matrix}.

The homotopy groups of L_{2}V (1 \frac{1}{2} ) and an invertible spectrum at the prime three 3

Here Â_{0} = k(1)_{∗}^{∧}⨂Λ(h_{10}, ζ_{2}) for k(1)_{∗}^{∧} = lim ℤ/3[v_{1}]/(v_{1}^{k}), the modules Â_{1}, v_{2}^{9−3k}{ \atop (A_{2})_{(h)}} and v_{2}^{9−3k}{ \atop (A_{2})_{(t)}}

are isomorphic to A_{1}, { \atop v_{2}^{9−3k}(A_{2})_{(h)}} and { \atop v_{2}^{9−3k}(A_{2})_{(t)}}, respectively, ˆ \overline{A_{0}} = (ℤ_{p}⨁k(1)_{∗}^{∧}h_{10}) ⨂Λ(ζ_{2}), and the

modules ˆ \overline{A_{1}} and v_{2}^{9−3k}\begin{matrix} \\ \overline{(A_{2})} & (h) \end{matrix} are isomorphic to \overline{A_{1}} and { \atop v_{2}^{9−3k}\overline{(A_{2})}_{(h)}}, respectively.

Furthermore, these theorems imply the structure of the homotopy groups π_{∗}(V (0) ∧ X^{k}), π_{∗}(X^{k}) and π_{∗}(L_{2}V (1\frac{1}{2})) from the chromatic spectral sequences [6], since the homotopy groups π_{∗}(L_{1}V (0) ∧ X^{k}) and π_{∗}(L_{1}V (1\frac{1}{2})) are all isomorphic, the homotopy groups π_{∗}(L_{1}N^{1} ∧ X^{k}) are isomorphic to π_{∗}(L_{1}N^{1}S^{0}), and π_{∗}(L_{0}X^{k}) = π_{∗}(SQ).

Corollary 1.5. The homotopy groups π_{∗}(L_{2}V (0) ∧ X^{k}), π_{∗}(L_{2}V (1\frac{1}{2})) and π_{∗}(L_{2}X^{k}) are as follows:

π_{∗}(L_{2}V (0) ∧ X^{k}) = A′_{0}⨁Â_{1}⨁v_{2}^{9−3k}{ \atop (A_{2})_{(h)}}⨁v_{2}^{9−3k}{ \atop (A_{2})_{(t)}},
π_{∗}(L_{2}V (1\frac{1}{2})) = A′_{0}⨁Â_{1}⨁v_{2}^{3}{ \atop (A_{2})_{(h)}}⨁v_{2}^{6}{ \atop (A_{2})_{(t)}} and
π_{∗}(L_{2}X^{k}) = \overline{A_{0}}′⨁ˆ\overline{A_{1}}⨁v_{2}^{9−3k}\begin{matrix} \\ \overline{(A_{2})} & (h) \end{matrix}.

Here A′_{0} = k(1)_{∗}⨂Λ(h_{10})⨁Σ^{−1}(K(1)_{∗}/k(1)_{∗})ζ_{2}⨂Λ(h_{10}) for k(1)_{∗} = ℤ/3[v_{1}] and K(1) = v_{1}^{−1}k(1)_{∗}, and \overline{A_{0}}′ = ℤ_{(3)}⨁A_{+}⨁Σ^{−1}(A_{−}⨁Q/ℤ_{(3)}⟨y⟩)ζ_{2} for the modules A_{+} and A_{−} and the element y given in [11].

This paper is organized as follows: In the next section, we show the cofiber sequence X →^{f} Σ^{−21}V (1\frac{1}{2}) → ΣX^{2} assuming Theorem 1.2. In section three, we study the b_{10} action on the chromatic E_{1}-term H^{∗}M_{1}^{1}, which is the key lemma to determine the Adams differentials of E_{r}^{∗}(M^{1}V (0) ∧ X^{k}). We compute differentials on A_{2} for M^{1}V (0) ∧ X^{k}, and then determine all the differentials on E_{r}^{∗}(M^{1}V (0) ∧ X^{k}) in section four. In section five, we determine the homotopy groups π_{∗}(M^{2}X^{k}). By use of the results in sections two and five, we prove Theorem 1.3 in section six. In the last section, we give a correction of [9], [10] and [11], as well as some misprints in [11].

2. The Toda spectrum V (1\frac{1}{2}) and the invertible spectrum X

Let V (i) denote the i-th Smith-Toda spectrum defined by the cofiber sequences

(2.1)  S^{0} ⟶^{3} S^{0} ⟶^{i} V (0) ⟶^{j} ΣS^{0} and Σ^{4}V (0) ⟶^{α} V (0) ⟶^{i_{1}} V (1) ⟶^{j_{1}} Σ^{5}V (0),

where α denotes the Adams map that induces v_{1} on the BP_{∗}-homology. Toda showed the existence of the map B: Σ^{16}V (0) → V (1), which cannot be extended to β: Σ^{16}V (1) → V (1), that induces v_{2} on the BP_{∗}-homology (cf. [12]). Then the spectrum V (1\frac{1}{2}) is the cofiber of B, and we have the commutative diagram

(2.2)  0  ΣE(2)_{∗}(V (1_{\frac{1}{2}}))  Σ^{−16}E(2)_{∗}(V (0))  E(2)_{∗}(V (1))  0 0  Σ^{−20}E(2)_{∗}/(3)  Σ^{−16}E(2)_{∗}/(3)  Σ^{−16}K(2)_{∗}  0 w  w  ^{B_{∗}} w  w w  _{^{v_{1}} w} u_{≅}  w u_{v_{2} ≅}  w

of E(2)_{∗}E(2)-comodules, where E(2) denotes the Johnson-Wilson spectrum such that E(2)_{∗} = ℤ_{(3)}[v_{1}, v_{2}^{±1}], and K(2)_{∗} = E(2)_{∗}/(3, v_{1}). Therefore, we have an isomorphism E(2)_{∗}(V (1\frac{1}{2})) = Σ^{−21}E(2)_{∗}/(3). Here we set Y = Σ^{21}V (1\frac{1}{2}), and obtain

E(2)_{∗}(Y) = E(2)_{∗}/(3)

as an E(2)_{∗}E(2)-comodule. Consider the E(2)-based Adams spectral sequence E_{r}^{∗}(W) converging to π_{∗}(L_{2}W). Then E_{2}^{∗}(Y) = E_{2}^{∗}(V (0)), since the E_{2}-terms reflect only the comodule structures. Using the cofiber sequence, we have Lemma 2.3. The element v_{1}v_{2}^{−3}b_{11}b_{10} ∈ E_{2}^{4,4}(Y) is a permanent cycle, and d_{9}(v_{1}v_{2}^{−1}ψ_{0}) = v_{2}^{−5}b_{11}ξb_{10}^{4} up to sign in E_{9}^{12,12}(Y).

Proof. Consider the exact sequence

π_{5}(V (0)) ⟶ π_{21}(V (1)) ⟶^{j_{Y} _{∗}} π_{0}(Y) ⟶ π_{4}(V (0)) ⟶^{B_{∗}} π_{20}(V (1))

associated to the cofiber sequence

(2.4)  V (1) →^{j_{Y}} Y →^{v_{Y}} Σ^{17}V (0) →^{B} ΣV (1).

Using Toda’s computation [12], we have

π_{21}(V (1)) = ℤ/3{i_{1}j_{1}Biβ_{1}}, π_{4}(V (0)) = ℤ/3{αi} and π_{5}(V (0)) = 0,

and B_{∗}(αi) = i_{0}β_{1} ≠ 0. Therefore, we have π_{0}(Y) = ℤ/3{j_{Y} i_{1}j_{1}Biβ_{1}}. Here, the element j_{Y} i_{1}j_{1}Biβ_{1} represents δi_{1}_{∗}δ′(v_{2})b_{10} in the E_{2}-term, where δ and δ′ are the connecting homomorphisms associated to the cofiber sequences (2.4) and (2.1). Since we compute δi_{1}_{∗}δ′(v_{2})b_{10} = δ(h_{11}b_{10}) = v_{1}v_{2}^{−3}b_{11}b_{10}, we obtain the first part of the lemma.

For the second part, consider the exact sequence of E_{2}-terms

E_{2}^{s}(V (0)) ⟶^{v_{2}_{∗}} E_{2}^{s}(V (1)) ⟶^{δ} E_{2}^{s+1}(Y)

associated to (2.4). Then we read off from [9, Lemma 3.4] that δ(v_{2}ξ) = v_{1}v_{2}^{−1}ψ_{0} and δ(v_{2}^{−2}ψ_{0}b_{10}^{4}) = v_{2}^{−5}b_{11}ξb_{10}^{4}. We also see that d_{9}(v_{2}ξ) = ±v_{2}^{−2}ψ_{0}b_{10}^{4} in E_{9}^{11}(V (1)) from [8, Prop. 10.5] (in which k = 1 by [2]). The δ-image of it is the desired equality. □

In [5], Kamiya and the second author constructed an invertible spectrum X with the map f: X → Y that induces the projection E(2)_{∗} → E(2)_{∗}/(3). The invertible spectrum X is characterized by the Adams differential

d_{5}(1) = −v_{2}^{−2}h_{11}b_{10}^{2} ∈ E_{5}^{5,4}(X).

Note that this X is X^{2} in [5].

We obtain the following lemma as a corollary of Theorem 1.2:

Lemma 2.5. If r > 9, then the Adams differential d_{r} acts trivially on E_{r}^{∗}(X).

Note that the map i ∧ 1_{X}: X → V (0) ∧ X for i in (2.1) also induces the projection E(2)_{∗} → E(2)_{∗}/(3). Furthermore, it is shown that there is a possibility of the existence of another invertible spectrum U such that U^{3} = U ∧ U ∧ U = L_{2}S^{0} other than L_{2}S^{0}, X and X^{2} = X ∧ X, characterized by the differential d_{5}(1) = v_{2}^{−1}ξb_{10}ζ_{2} ∈ E_{5}^{5,4}(U).

Theorem 2.6. The fiber of f is X^{2}. That is, X^{2} → X →^{f} Y → ΣX^{2} is a cofiber sequence.

Proof. Since f induces an epimorphism fitting in the short exact sequence

(2.7)  0 ⟶ E(2)_{∗} ⟶^{3} E(2)_{∗} ⟶^{f_{∗}} E(2)_{∗}/(3) ⟶ 0,

we see that the E(2)_{∗}-homology of the fiber F_{f} of f is E(2)_{∗}, which shows that F_{f} is invertible by [5, Th.1.1]. Therefore, F_{f} is one of L_{2}S^{0}, X, X ∧ X, U and U^{2} = U ∧ U by [5].

F_{f} is not X. Indeed, if F_{f} = X, then [X, X]_{0} = π_{0}(L_{2}S^{0}) = ℤ_{(3)}. Then 3: E(2)_{∗} → E(2)_{∗} is induced from 3 ∈ [X, X]_{0}, and Y = V (0) ∧ X, which is a V (0)-module spectrum. This contradicts to the fact that Y is not a V (0)-module spectrum [12, Th. 6.4]. (Note that the obstruction survives after the localization by L_{2}.)

Suppose that F_{f} = L_{2}S^{0}. Then we have the map g: Y → L_{2}S^{1} in the cofiber sequence L_{2}S^{0} → X →^{f} Y →^{g} L_{2}S^{1}, which induces the connecting homomorphism δ: E_{2}^{∗}(Y) = E_{2}^{∗}(V (0)) → E_{2}^{∗+1}(S^{0}) associated to (2.7). We compute that δ(v_{1}v_{2}^{−3}b_{11}b_{10}) = v_{2}^{−2}h_{11}b_{10}^{2}, and we see that d_{9}(v_{2}^{−2}h_{11}b_{10}^{2}) = v_{2}^{3}b_{11}b_{10}^{6} by [8]. It follows that v_{1}v_{2}^{−3}b_{11}b_{10} is not a permanent cycle, which contradicts to Lemma 2.3.

Last suppose that F_{f} = U or U^{2}. We write g_{U^{k}} as the generator of E(2)_{∗}-module E(2)_{∗}(U^{k}) ≅ E(2)_{∗}. Then U^{k} for k = 1, 2 is characterized by the Adams differential d_{5}(g_{U^{k}}) = (−1)^{k+1}v_{2}^{−1}ξb_{10}ζ_{2}g_{U^{k}} ∈ E_{5}^{5,4}(U^{k}). Since δ(v_{1}v_{2}^{−1}ψ_{0}ζ_{2}) = v_{2}^{−1}ξb_{10}ζ_{2}g_{U^{k}}, we obtain d_{13}(3g_{X}) = ±v_{2}^{−5}b_{11}ξb_{10}^{4}ζ_{2}g_{X} ≠ 0 by Lemma 2.3. This contradicts to Lemma 2.5. □

3. The action of b_{10}

We first study the multiplication by b_{10} in H^{∗}M_{1}^{1}, whose structure is given in [9] as follows:

(3.1)  H^{∗}M_{1}^{1} = A_{0}⨁A_{1}⨁A_{2}.

The homotopy groups of L_{2}V (1 \frac{1}{2} ) and an invertible spectrum at the prime three 5

Here,

A_{0} = (K(1)_{∗}/k(1)_{∗})⨂Λ(h_{10}, ζ_{2}),
A_{1} = ∑_{n≥0} F_{n}⨂Λ(ζ_{2}) and
A_{2} = (F_{(h)}⨁F_{(t)}⨁F_{(h)}^{∗}⨁F_{(t)}^{∗})⨂ℤ/3[b_{10}]⨂Λ(ζ_{2})

for the modules:

k(1)_{∗} = ℤ/3[v_{1}],
K(1)_{∗} = ℤ/3[v_{1}^{±1}],
F_{(h)} = ℤ/3[v_{2}^{±3}]{v_{2}/v_{1}, v_{2}h_{10}/v_{1}, v_{2}^{2}h_{11}/v_{1}, v_{2}b_{11}/v_{1}},
F_{(t)} = ℤ/3[v_{2}^{±3}]{v_{2}^{−1}/v_{1}, v_{2}h_{10}/v_{1}^{2}, v_{2}^{2}h_{11}/v_{1}^{2}, v_{2}^{−1}b_{11}/v_{1}},
F_{(h)}^{∗} = ℤ/3[v_{2}^{±3}]{ξ/v_{1}, ψ_{0}/v_{1}, v_{2}ψ_{1}/v_{1}, b_{11}ξ/v_{1}},
F_{(t)}^{∗} = ℤ/3[v_{2}^{±3}]{ξ/v_{1}^{2}, v_{2}^{−2}ψ_{0}/v_{1}, v_{2}^{−1}ψ_{1}/v_{1}, b_{11}ξ/v_{1}^{2}} and
F_{n} = ℤ/3[v_{1}, v_{2}^{±3^{n+2}}]{v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−1}, v_{2}^{3^{n+1}}h_{10}/v_{1}^{6×3^{n}+1},
v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}+1}, v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}}}.

The degrees of these elements are:

\begin{matrix}x & ||x|| & x & ||x|| \\ v_{2}^{±1}/v_{1} & 12, 60, 108, (−20, 28, 76) & v_{2}h_{10}/v_{1}^{2} & 11, 59, 107, (15, 63, 111) \\ v_{2}^{2}h_{11}/v_{1}^{2} & 35, 83, 131, (39, 87, 135) & v_{2}^{±1}b_{11}/v_{1} & 46, 94, 142, (14, 62, 110) \\ ξ/v_{1}^{2} & −2, 46, 94, (2, 50, 98) & v_{2}^{−1±1}ψ_{0}/v_{1} & 9, 57, 105, (−23, 25, 73) \\ v_{2}^{±1}ψ_{1}/v_{1} & 33, 81, 129, (1, 49, 97) & b_{11}ξ/v_{1}^{2} & 32, 80, 128, (36, 84, 132) \\ v_{2}^{±3}/v_{1}^{3} & 36, −60 & v_{2}^{3}h_{10}/v_{1}^{7} & 23 \\ v_{2}^{8}h_{10}/v_{1}^{11} & 87 & v_{2}^{5±3}ξ/v_{1}^{4} & 118, 22 \\ v_{2}^{±9}/v_{1}^{11} & 100 & v_{2}^{9}h_{10}/v_{1}^{19} & 71 \\ v_{2}^{24}h_{10}/v_{1}^{31} & 119 & v_{2}^{3(5±3)+1}ξ/v_{1}^{12} & 70 \\ v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−1} & 4 & v_{2}^{3^{n+1}}h_{10}/v_{1}^{6×3^{n}+1} & 71 \\ v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}+1} & 71 & v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}} & 70 \end{matrix}

In this table, the numbers in the column of ||x|| are considered mod 144.

Lemma 3.2. The action of b_{10} on A_{1} is as follows:

v_{2}^{3(3t±1)}b_{10}/v_{1}^{3} = ±v_{2}^{9t±3−2}b_{11}/v_{1},
v_{2}^{3^{n}(3t±1)}b_{10}/v_{1}^{4×3^{n−1}−1} = ±v_{2}^{3^{n−1}(9t−1±3)−1}b_{11}/v_{1} + λ_{1}v_{2}^{3^{n−1}(9t−1±3)+1}h_{10}ζ_{2}/v_{1} (n > 1),
v_{2}^{3^{n}(3t+1)}h_{10}b_{10}/v_{1}^{2×3^{n}+1} = (−1)^{n+1}v_{2}^{3^{n+1}t+(3^{n}−1)/2}ψ_{1}/v_{1} (n > 0),
v_{2}^{9t+8}h_{10}b_{10}/v_{1}^{11} = v_{2}^{9t+5}ψ_{1}/v_{1} + λ_{2}v_{2}^{9t+6}ξζ_{2}/v_{1},
v_{2}^{3^{n}(9t+8)}h_{10}b_{10}/v_{1}^{10×3^{n}+1} = −v_{2}^{3^{n+2}t+5×3^{n}+(3^{n}−1)/2}ψ_{1}/v_{1} (n > 0),
v_{2}^{3^{n}(9t+5±3)+(3^{n}−1)/2}ξb_{10}/v_{1}^{4×3^{n}} = ±v_{2}^{3^{n+1}(3t+1±1)+3(3^{n}−1)/2}b_{11}ξ/v_{1}^{2} (n ≥ 0).

Here the double signs are the same order, and λ_{i} for i = 1, 2 is an integer of ℤ/3.

Proof. Suppose that δ(x) = y for x ∈ A_{1} ∩ H^{s−2,t−12}M_{1}^{1}, and δ(w) = yb_{10} for w ∈ A_{2} ∩ H^{s,t}M_{1}^{1}. Then δ(xb_{10} − w) = 0, and xb_{10} = w if H^{s,t−4}M_{1}^{1} = 0. By [9, Prop. 3.4], we have the table:

\begin{matrix}x & y & w & (s, t) & H^{s,t−4}M_{1}^{1} \\ v_{2}^{3(3t±1)}/v_{1}^{3} & ±v_{2}^{9t±3−1}h_{10} & ±v_{2}^{9t±3−2}b_{11}/v_{1} & (2, ±48) & 0 \\ v_{2}^{3^{n}(3t±1)}/v_{1}^{4×3^{n−1}−1} & ∓v_{2}^{3^{n−1}(9t−1±3)}h_{10} & ±v_{2}^{3^{n−1}(9t−1±3)−1}b_{11}/v_{1} & (2, 112), (2, 16) & \begin{matrix}(2, 60 ± 48) \\ v_{2}^{4±3}h_{10}ζ_{2}/v_{1}^{2} \end{matrix} \\ v_{2}^{3^{n}(3t+1)}h_{10}/v_{1}^{2×3^{n}+1} & (−1)^{n+1}v_{2}^{3^{n+1}t+(3^{n}−1)/2}ξ & (−1)^{n+1}v_{2}^{3^{n+1}t+(3^{n}−1)/2}ψ_{1}/v_{1} & (3, 36), (3, 84) & 0 \\ v_{2}^{9t+8}h_{10}/v_{1}^{11} & −v_{2}^{9t+5}ξ & v_{2}^{9t+5}ψ_{1}/v_{1} & (3, 100) & \begin{matrix}(3, 96) \\ v_{2}^{5}h_{11}b_{10}/v_{1}^{2} \\ v_{2}^{4}b_{11}ζ_{2}/v_{1} \\ v_{2}^{6}ξζ_{2}/v_{1}^{2} \end{matrix} \\ v_{2}^{3^{n}(9t+8)}h_{10}/v_{1}^{10×3^{n}+1} & −v_{2}^{3^{n+2}t+5×3^{n}+(3^{n}−1)/2}ξ & −v_{2}^{3^{n+2}t+5×3^{n}+(3^{n}−1)/2}ψ_{1}/v_{1} & (3, 132), (3, 84) & 0 \\ v_{2}^{a(t,±)}ξ/v_{1}^{4×3^{n}} & ±v_{2}^{a(t,±)−3^{n}−1}ψ_{1} & ±v_{2}^{a(t,±)−3^{n}−1}b_{11}ξ/v_{1}^{2} & \begin{matrix}(4, 132), (4, 36) \\ (4, 84) \end{matrix} & 0 \end{matrix}

Here a(t, ±) = 3^{n}(9t + 5 ± 3) + (3^{n} − 1)/2. If the image of v_{1} contains an element of the form ub_{10}/v_{1}, then we replace x by x − λu/v_{1} for an integer λ so that xb_{10} = w. □

We decompose the module F_{n} into the direct sum of F′_{n}, (F′′_{n})_{(h)} and (F′′_{n})_{(t)} defined by

F′_{n} = ℤ/3[v_{1}, v_{2}^{±3^{n+2}}]{v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−2}, v_{2}^{3^{n+1}}h_{10}/v_{1}^{6×3^{n}}, v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}}, v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}−2}},
(F′′_{0})_{(h)} = Z/3[v_{2}^{±9}]{v_{2}^{±3}/v_{1}^{3}, v_{2}^{3}h_{10}/v_{1}^{7}, v_{2}^{5±3}ξ/v_{1}^{3}},
(F′′_{0})_{(t)} = Z/3[v_{2}^{±9}]{v_{2}^{8}h_{10}/v_{1}^{11}, v_{2}^{5±3}ξ/v_{1}^{4}},
(F′′_{n})_{(h)} = ℤ/3[v_{2}^{±3^{n+2}}]{v_{2}^{3^{n+1}}h_{10}/v_{1}^{6×3^{n}+1}, v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}+1}, v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}−1}} for n > 0,
(F′′_{n})_{(t)} = ℤ/3[v_{2}^{±3^{n+2}}]{v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−1}, v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}}} for n > 0.

We define A′_{1}, (A′_{2})_{(h)} and (A′_{2})_{(t)} by

(3.3)  \begin{aligned}A′_{1} & = & ∑_{n≥0} F′_{n}, \\ (A′_{2})_{(x)} & = & (F_{(x)}⨁F_{(x)}^{∗})⨂ℤ/3[b_{10}]⨁ ∑_{n≥0}(F′′_{n})_{(x)} for x = h, t. \end{aligned}

Note that A_{1}⨁A_{2} = (A′_{1}⨁(A′_{2})_{(h)}⨁(A′_{2})_{(t)}) ⨂Λ(ζ_{2}), and b_{10}(∑_{n≥0} F′′_{n})_{(x)} ⊂ F_{(x)}⨁F_{(x)}^{∗} for x = h, t by Lemma 3.2.

4. The homotopy groups of M^{1}V (0) ∧ X^{k}

For a finite spectrum W of type 1, we define M^{1}W as the cofiber of the localization map L_{2}W → L_{1}W. Then, the E_{2}-terms E_{2}^{∗}(M^{1}V (0) ∧ X^{k}) and E_{2}^{∗}(M^{1}Y) are all isomorphic to H^{∗}M_{1}^{1} (see (3.1)). Here E_{r}^{∗}(W) denotes the E_{r}-term of the E(2)-based Adams spectral sequence converging to the homotopy groups π_{∗}(L_{2}W). In this section, we determine the homotopy groups π_{∗}(M^{1}V (0) ∧ X^{k}) as well as π_{∗}(M^{1}V (0)). The result of [9] has an error, and we correct it here.

As in [8], we consider the homotopy element β_{6/3} ∈ π_{82}(L_{2}S^{0}), which is detected by the element v_{2}^{3}b_{11} mod (3, v_{1}^{2}) of the E_{2}-term E_{2}^{2,84}(L_{2}S^{0}) (cf. [8, Lemma 2.4]). By the relation b_{11}^{2} = −v_{2}^{3}b_{10}^{2} in the E_{2}-term E_{2}^{∗}(L_{2}V (1)) ([8, Prop. 5.9]), we see that the action of β_{6/3} induces an action B_{11} on A_{2} ⊂ H^{∗}M_{1}^{1}.

Lemma 4.1. There is an action B_{11} on A_{2} such that B_{11}^{2} = −v_{2}^{9}, which induces an isomorphism on E_{r}-terms of the Adams spectral sequences converging to π_{∗}(M^{1}V (0) ∧ X^{i}) (i = 0, 1, 2) and π_{∗}(M^{1}Y). In particular,

\begin{aligned}B_{11}(v_{2}^{3t±1}/v_{1}) & = & v_{2}^{3t+3±1}b_{11}/v_{1}, \\ B_{11}(v_{2}^{3t+1}h_{10}/v_{1}^{2}) & = & −v_{2}^{3t+5}h_{11}/v_{1}^{2}, \\ B_{11}(v_{2}^{3t}ξ/v_{1}^{2}) & = & v_{2}^{3t+3}b_{11}ξ/v_{1}^{2}, \\ B_{11}(v_{2}^{3t−1±1}ψ_{0}/v_{1}) & = & −v_{2}^{3t+3±1}ψ_{1}/v_{1} and \end{aligned} \begin{aligned}B_{11}(v_{2}^{3t±1}b_{11}/v_{1}) & = & −v_{2}^{3t+6±1}/v_{1}, \\ B_{11}(v_{2}^{3t+2}h_{11}/v_{1}^{2}) & = & v_{2}^{3t+7}h_{10}/v_{1}^{2}, \\ B_{11}(v_{2}^{3t}b_{11}ξ/v_{1}^{2}) & = & −v_{2}^{3t+6}ξ/v_{1}^{2}, \\ B_{11}(v_{2}^{3t±1}ψ_{1}/v_{1}) & = & v_{2}^{3t+5±1}ψ_{0}/v_{1}. \end{aligned}

Proof. We define the action B_{11}(x) = y if v_{2}^{3}b_{11}(x) = y with y is not divisible of b_{10} on A_{2}, or if v_{2}^{3}b_{11}(x) = yb_{10}. Then the relations are certified by [8, Prop. 5.9]. Since β_{1} ∈ π_{10}(S^{0}) and β_{6/3} ∈ π_{82}(S^{0}) define the actions (β_{1})_{∗} = b_{10} and (β_{6/3})_{∗} = v_{2}^{3}b_{11} and E_{2}-term is free on b_{10}, we see that B_{11} defines the action on the spectral sequence. □

By use of the action, we write F_{(h)}⨂ℤ/3[b_{10}] = (F̈_{(h)}^{+}⨁F̈_{(h)}^{−}) ⨂Λ(B_{11}), F_{(t)}⨂ℤ/3[b_{10}] = (F̈_{(t)}^{+}⨁F̈_{(t)}^{−}) ⨂Λ(B_{11}),

F_{(h)}^{∗}⨂ℤ/3[b_{10}] = ((F̈_{(h)}^{+})^{∗}⨁(F̈_{(h)}^{−})^{∗}) ⨂Λ(B_{11}) and F_{(t)}^{∗}⨂ℤ/3[b_{10}] = ((F̈_{(t)}^{+})^{∗}⨁(F̈_{(t)}^{−})^{∗}) ⨂Λ(B_{11}), where

(4.2)  \begin{aligned}F̈_{(h)}^{+} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{v_{2}/v_{1}\}, \\ F̈_{(h)}^{−} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{v_{2}h_{10}/v_{1}\}, \\ (F̈_{(h)}^{−})^{∗} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{ξ/v_{1}\}, \\ (F̈_{(h)}^{+})^{∗} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{v_{2}ψ_{1}/v_{1}\} and \end{aligned} \begin{aligned}F̈_{(t)}^{−} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{v_{2}^{−1}/v_{1}\}, \\ F̈_{(t)}^{+} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{v_{2}h_{10}/v_{1}^{2}\}, \\ (F̈_{(t)}^{+})^{∗} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{ξ/v_{1}^{2}\}, \\ (F̈_{(t)}^{−})^{∗} & = & ℤ/3[v_{2}^{±3}, b_{10}]\{v_{2}^{−1}ψ_{1}/v_{1}\}. \end{aligned}

In [4], we showed the isomorphism v_{2}^{3} : Σ^{48}L_{2}V (1) ≃ V (1) ∧ X^{2}, where X is the invertible spectrum such that d_{5}(1) = −v_{2}^{−2}h_{11}b_{10}^{2} ∈ E_{5}^{5}(X). Then we obtain a homotopy element.

Lemma 4.3. Let X^{k} denote the k-fold smash product of the invertible spectrum X, and E_{r}^{∗}(X^{k}), the E_{r}-term of the E(2)-based Adams spectral sequence converging to π_{∗}(X^{k}). Then for k = 1, 2, β_{10−3k} ∈ E_{2}^{2,156−48k}(X^{k}) is a permanent cycle.

Proof. It is shown in [4] that there is an isomorphism v_{2}^{3} : Σ^{48}V (1) → V (1) ∧ X^{2}, which also defines an

isomorphism v_{2}^{6} : Σ^{96}V (1) →^{v_{2}^{3}} Σ^{48}V (1)∧X^{2} →^{v_{2}^{3}∧1} V (1)∧X^{4} = V (1)∧X. Then the composites S^{106} →^{i} Σ^{106}V (0) →^{B}

Σ^{90}V (1) →^{v_{2}^{6}} Σ^{−6}V (1) ∧ X →^{j_{0}∧1} X and S^{58} →^{i} Σ^{58}V (0) →^{B} Σ^{42}V (1) →^{v_{2}^{3}} Σ^{−6}V (1) ∧ X^{2} →^{j_{0}∧1} X^{2} are represented by

β_{7} and β_{4}, respectively. Here j_{0} = jj_{1} : V (1) → S^{6} denotes the projection to the top cell. □

Note that β_{3k+1} is represented by v_{2}^{3k}b_{10} mod (3, v_{1}^{2}). Indeed, for k < 3,

β_{3k+1} = δδ′(v_{2}^{3k+1})
= δ(v_{2}^{3k}t_{1}^{3} + v_{1}^{2}(kv_{2}^{3k−2}t_{1}^{9} − v_{2}^{3k}t_{1}) + v_{1}^{3}x) for some x
= δ(v_{2}^{3k}t_{1}^{3} + (k − 1)v_{1}^{2}v_{2}^{3k}t_{1} + v_{1}^{3}x′) for some x′
≡ v_{2}^{3k}b_{10} + kv_{1}v_{2}^{3k−1}t_{1}^{3} ⊗ t_{1}^{3} + kv_{1}^{2}v_{2}^{3k−2}t_{1}^{6} ⊗ t_{1}^{3} + 2(k − 1)v_{1}v_{2}^{3k}t_{1} ⊗ t_{1} mod (3, v_{1}^{3})
≡ v_{2}^{3k}b_{10} − kv_{1}^{2}v_{2}b_{11} mod (3, v_{1}^{3}).

Here δ and δ′ denote the connecting homomorphisms associated to the short exact sequences 0 → E(2)_{∗} → E(2)_{∗} → E(2)_{∗}/(3) → 0 and 0 → E(2)_{∗}/(3) → E(2)_{∗}/(3) → E(2)_{∗}/(3, v_{1}) → 0, respectively, and η_{R}(v_{3}) ≡ v_{3}+v_{2}t_{1}^{9}+v_{1}t_{2}^{3}−v_{1}^{9}t_{2}+v_{1}^{2}V −t_{1}η_{R}(v_{2}^{3}) mod (3), where V is the element defined by 3v_{1}V ≡ v_{2}^{3}+v_{1}^{3}t_{1}^{9}−v_{1}^{9}t_{1}^{3}−η_{R}(v_{2}^{3}) mod (9).

The submodule (A_{2}, d_{5}^{k}) with the Adams differential is the differential submodule of the E(2)-based Adams E_{2}-term (E_{2}^{∗}(M^{1}V (0) ∧ X^{k}), d_{5}^{k}) for k = 1, 2, 3. We also see that (H^{∗}(A_{2}, d_{5}^{k}), d_{9}^{k}) is also a differential module. We write d_{r}^{0} as d_{r}^{3}, since X^{3} = L_{2}S^{0} = X^{0}.

Lemma 4.4. The multiplication by v_{2}^{9−3k} induces an isomorphism H^{∗}(H^{∗}(A_{2}, d_{5}^{0}), d_{9}^{0}) → H^{∗}(H^{∗}(A_{2}, d_{5}^{k}), d_{9}^{k}) for the modules H^{∗}(H^{∗}(A_{2}, d_{5}^{k}), d_{9}^{k}) with k = 1, 2. Furthermore, d_{r}^{k} = 0 on E_{r}-term for r > 9.

Proof. From the structure of A_{2}, we see that v_{2}^{9−3k} is an isomorphism. The identity map X^{3} ∧ X^{k} = X^{k} induces the pairing E_{r}^{∗}(X^{3})⨂E_{r}^{∗}(X^{k}) → E_{r}^{∗}(X^{k}), that makes the Adams differential a derivation. We compute

(4.5)  d_{5}^{k}(v_{2}^{9−3k}xb_{10}) = d_{5}^{k}(xβ_{10−3k}) = d_{5}^{0}(x)β_{10−3k} = v_{2}^{9−3k}d_{5}^{0}(x)b_{10}

by Lemma 4.3. It follows that d_{5}^{k}(v_{2}^{9−3k}x) = v_{2}^{9−3k}d_{5}^{0}(x), and so v_{2}^{9−3k} is a map of differential modules with differential d_{5}. Therefore, v_{2}^{9−3k} induces an isomorphism H^{∗}(A_{2}, d_{5}^{0}) ≅ H^{∗}(A_{2}, d_{5}^{k}). In a similar computation as (4.5) with Lemma 4.3, we see that v_{2}^{9−3k} is a map of differential modules with d_{9}, and obtain an isomorphism H^{∗}(H^{∗}(A_{2}, d_{5}^{0}), d_{9}^{0}) ≅ H^{∗}(H^{∗}(A_{2}, d_{5}^{k}), d_{9}^{k}). Since H^{∗}(H^{∗}(A_{2}, d_{5}^{0}), d_{9}^{0}) has a horizontal vanishing line at the level 12 [9], we see that d_{r}^{k} = 0 for r > 9. □

Recall in [9, Prop.s 8.5 and 8.9] the action of the Adams differentials on A_{2} in the E_{2}-term of the E(2)-based Adams spectral sequence for computing π_{∗}(M^{1}V (0)). Since B_{11} commutes with the Adams differentials, the differentials on A_{2} are read off from the following:

(4.6) \begin{aligned}d_{5}(v_{2}^{3t+1}/v_{1}) & = & −tv_{2}^{3t−1}h_{11}b_{10}^{2}/v_{1} & (h), \\ d_{9}(v_{2}^{9t−1}/v_{1}) & = & v_{2}^{9t−4}h_{11}b_{10}^{4}/v_{1}^{2} & (t), \\ d_{5}(v_{2}^{3t+1}h_{10}/v_{1}^{2}) & = & tv_{2}^{3t−1}b_{10}^{3}/v_{1} & (t), \\ d_{9}(v_{2}^{9t+4}h_{10}/v_{1}) & = & v_{2}^{9t+1}b_{10}^{5}/v_{1} & (h), \end{aligned} \begin{aligned}d_{5}(v_{2}^{3t+1}ψ_{1}/v_{1}) & = & −(t + 1)v_{2}^{3t}ξb_{10}^{3}/v_{1} & (h), \\ d_{9}(v_{2}^{9t+5}ψ_{1}/v_{1}) & = & v_{2}^{9t+3}ξb_{10}^{5}/v_{1}^{2} & (t), \\ d_{5}(v_{2}^{3t}ξ/v_{1}^{2}) & = & −(t − 1)v_{2}^{3t−2}ψ_{0}b_{10}^{2}/v_{1} & (t), \\ d_{9}(v_{2}^{9t+6}ξ/v_{1}) & = & v_{2}^{9t+3}ψ_{0}b_{10}^{4}/v_{1} & (h). \end{aligned}

The elements of F_{(h)}⨁F_{(t)}⨁F_{(h)}^{∗}⨁F_{(t)}^{∗} that do not support a non-trivial differential in the table are all permanent cycles. Here in the table, the names (h) and (t) after equations means that the equations are in a module with suffix (h) and (t), respectively, This shows that d_{5} assigns the module with the plus sign in (4.2) to the module with the minus sign. The structure of H^{∗}(H^{∗}(A_{2}, d_{5}^{0}), d_{9}^{0}) is read off from (4.6) as follows:

Lemma 4.7. (1) H^{∗}(A_{2}, d_{5}^{0}) is isomorphic to the tensor product of Λ(ζ_{2}) and the direct sum of

Ḟ_{(h)}^{+}, Ḟ_{(t)}^{+}, (Ḟ_{(h)}^{+})^{∗}, (Ḟ_{(t)}^{+})^{∗}, F_{(h)}^{−}, F_{(t)}^{−}, (F_{(h)}^{−})^{∗} and (F_{(t)}^{−})^{∗}.

(2) H^{∗}(H^{∗}(A_{2}, d_{5}^{0}), d_{9}^{0}) is isomorphic to the tensor product of Λ(ζ_{2}) and the direct sum of

F_{(h)}^{+}, F_{(t)}^{+}, (F_{(h)}^{+})^{∗}, (F_{(t)}^{+})^{∗}, F_{(h)}^{−}, F_{(t)}^{−}, (F_{(h)}^{−})^{∗} and (F_{(t)}^{−})^{∗}.

Here,

Ḟ_{(h)}^{+} = ℤ/3[v_{2}^{±9}, b_{10}]{v_{2}/v_{1}, v_{2}^{4}h_{10}/v_{1}}⨂Λ(B_{11}),
Ḟ_{(t)}^{+} = ℤ/3[v_{2}^{±9}, b_{10}]{v_{2}^{−1}/v_{1}, v_{2}h_{10}/v_{1}^{2}}⨂Λ(B_{11}),
(Ḟ_{(h)}^{+})^{∗} = ℤ/3[v_{2}^{±9}, b_{10}]{v_{2}^{6}ξ/v_{1}, v_{2}^{7}ψ_{1}/v_{1}}⨂Λ(B_{11}),
(Ḟ_{(t)}^{+})^{∗} = ℤ/3[v_{2}^{±9}, b_{10}]{v_{2}^{3}ξ/v_{1}^{2}, v_{2}^{5}ψ_{1}/v_{1}}⨂Λ(B_{11}),
F_{(h)}^{−} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{2}h_{11}/v_{1}, v_{2}^{5}h_{11}/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{3}){v_{2}h_{10}/v_{1}, v_{2}^{7}h_{10}/v_{1}},
F_{(t)}^{−} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{2}/v_{1}, v_{2}^{5}/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{3}){v_{2}^{−1}b_{11}/v_{1}, v_{2}^{5}b_{11}/v_{1}},
(F_{(h)}^{−})^{∗} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{3}b_{11}ξ/v_{1}, v_{2}^{6}b_{11}ξ/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{3}){ξ/v_{1}, v_{2}^{3}ξ/v_{1}},
(F_{(t)}^{−})^{∗} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{4}ψ_{0}/v_{1}, v_{2}^{7}ψ_{0}/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{3}){v_{2}^{2}ψ_{1}/v_{1}, v_{2}^{8}ψ_{1}/v_{1}},
F_{(h)}^{+} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{5}){v_{2}/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{4}b_{11}/v_{1}},
F_{(t)}^{+} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{5}){v_{2}h_{10}/v_{1}^{2}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{5}h_{11}/v_{1}^{2}},
(F_{(h)}^{+})^{∗} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{5}){v_{2}^{7}ψ_{1}/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{3}ψ_{0}/v_{1}} and
(F_{(t)}^{+})^{∗} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{5}){v_{2}^{3}ξ/v_{1}^{2}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{6}b_{11}ξ/v_{1}^{2}}.

This together with Lemma 4.4 shows the following:

Corollary 4.8. H^{∗}(H^{∗}(A_{2}, d_{5}^{k}), d_{9}^{k}) for k = 1, 2 is isomorphic to v_{2}^{9−3k}H^{∗}(H^{∗}(A_{2}, d_{5}^{0}), d_{9}^{0}).

For each generator x/v_{1}^{a} of (F′′_{n})_{(h)}⨁(F′′_{n})_{(t)}, we see that there is an element y/v_{1}^{b} of A_{2} such that xb_{10}/v_{1}^{a} = y/v_{1}^{b} by Lemma 3.2. By Lemma 4.4, we read off an equation d_{r}(v_{2}^{3k−9}y/v_{1}^{b}) = zb_{10}/v_{1}^{c} from the table (4.6). Then we see that d_{r}(x/v_{1}^{a}) = v_{2}^{9−3k}z/v_{1}^{c} in E_{r}^{∗}(M^{1}V (0) ∧ X^{k}). In this way, we obtain the differentials on (F′′_{n})_{(h)}⨁(F′′_{n})_{(t)} in the following table, in which d_{r}(x) = y in E_{r}^{∗}(M^{1}V (0) ∧ X^{k}) if y is in the column k:

x  r  0  r  1  r  2
v_{2}^{9t+3}/v_{1}^{3}  5  v_{2}^{9t+1}h_{10}b_{10}^{2}/v_{1}  ∞  0  5  v_{2}^{9t+1}h_{10}b_{10}^{2}/v_{1}
v_{2}^{9t−3}/v_{1}^{3}  ∞  0  5 v_{2}^{9t−5}h_{10}b_{10}^{2}/v_{1}  5  v_{2}^{9t−5}h_{10}b_{10}^{2}/v_{1}
v_{2}^{27t±9}/v_{1}^{11}  ∞  0  ∞  0  9  v_{2}^{27t±9−5}h_{10}b_{10}^{4}/v_{1}^{2}
v_{2}^{3^{n+1}t±3^{n}}/v_{1}^{4×3^{n−1}−1}  ∞  0  9 v_{2}^{b(n,t,±)}h_{10}b_{10}^{4}/v_{1}^{2} ∞  0

Here n > 2,

v_{2}^{9t+3}h_{10}/v_{1}^{7}  5  v_{2}^{9t}ξb_{10}^{2}/v_{1}  5  v_{2}^{9t}ξb_{10}^{2}/v_{1}  ∞  0
v_{2}^{3^{n+1}t+3^{n}}h_{10}/v_{1}^{2×3^{n}+1}  5  v_{2}^{3^{n+1}t−1}ξ_{n}b_{10}^{2}/v_{1}  ∞  0  5  v_{2}^{3^{n+1}t−1}ξ_{n}b_{10}^{2}/v_{1}

Here n > 1,

v_{2}^{9t+8}h_{10}/v_{1}^{11}  9  v_{2}^{9t+3}ξb_{10}^{4}/v_{1}^{2}  ∞  0  ∞  0
v_{2}^{27t+24}h_{10}/v_{1}^{31}  ∞  0  5 v_{2}^{27t+15}ξb_{10}^{2}/v_{1}  5  v_{2}^{27t+15}ξb_{10}^{2}/v_{1}
v_{2}^{3^{n+2}t+8×3^{n}}h_{10}/v_{1}^{10×3^{n}+1} 5 v_{2}^{3^{n+2}t+5×3^{n}−1}ξ_{n}b_{10}^{2}/v_{1} ∞  0  5 v_{2}^{3^{n+2}t+5×3^{n}−1}ξ_{n}b_{10}^{2}/v_{1}

Here n > 1,

v_{2}^{9t+2}ξ/v_{1}^{4}  5  v_{2}^{9t−1}ψ_{1}b_{10}^{2}/v_{1}  5  v_{2}^{9t−1}ψ_{1}b_{10}^{2}/v_{1}  ∞  0
v_{2}^{9t+8}ξ/v_{1}^{4}  ∞  0  5  v_{2}^{9t+5}ψ_{1}b_{10}^{2}/v_{1}  5  v_{2}^{9t+5}ψ_{1}b_{10}^{2}/v_{1}
v_{2}^{3^{n+2}t+3^{n}(5±3)}ξ_{n}/v_{1}^{4×3^{n}}  5  v_{2}^{c(n,t,±)}ψ_{1}b_{10}^{2}/v_{1}  ∞  0  5  v_{2}^{c(n,t,±)}ψ_{1}b_{10}^{2}/v_{1}

Here n > 0,

v_{2}^{9t+2}ξ/v_{1}^{3}  9  v_{2}^{9t−2}ψ_{1}b_{10}^{4}/v_{1}  ∞  0  ∞  0
v_{2}^{9t+8}ξ/v_{1}^{3}  ∞  0  9  v_{2}^{9t+4}ψ_{1}b_{10}^{4}/v_{1}  ∞  0
v_{2}^{3^{n+2}t+3^{n}(5±3)}ξ_{n}/v_{1}^{4×3^{n}−1} ∞  0  ∞  0  9  v_{2}^{c(n,t,±)−1}ψ_{1}b_{10}^{4}/v_{1}

Here n > 0,

Here ξ_{n} = v_{2}^{(3^{n}−1)/2}ξ^{1}, b(n, t, ±) = 3^{n+1}t±3^{n} −3^{n−1} −2 and c(n, t, ±) = 3^{n+2}t+3^{n+1}(1±1)+3(3^{n} −1)/2−1. Remark. The differential d_{9}(v_{2}^{9t+2}ξ/v_{1}^{3}) = v_{2}^{9t−2}ψ_{1}b_{10}^{4}/v_{1} in the last third line is missing in [9].

We obtain the homology groups H^{∗}(H^{∗}(A′_{2}, d_{5}^{k}), d_{9}^{k}) from this table with Lemma 4.7 and Corollary 4.8. Proposition 4.9. Let A′_{2} be the direct sum of the submodules (A′_{2})_{(h)} and (A′_{2})_{(t)} in (3.3). Then the module H^{∗}(H^{∗}(A′_{2}, d_{5}^{k}), d_{9}^{k}) for k = 0, 1, 2 is isomorphic to the direct sum of the modules (F′′_{n})_{(h)}^{(k)}⨁(F′′_{n})_{(t)}^{(k)} for n ≥ 0,

\begin{matrix}v_{2}^{9−3k}F_{(h)}^{+}, v_{2}^{9−3k}F_{(t)}^{+}^{(k)}, v_{2}^{9−3k}(F_{(h)}^{+})^{∗}^{(′)}, v_{2}^{9−3k}(F_{(t)}^{+})^{∗}^{(k)}, v_{2}^{9−3k}F_{(h)}^{−}^{(k)}, \\ v_{2}^{9−3k}F_{(t)}^{−}, v_{2}^{9−3k}(F_{(h)}^{−})^{∗}^{(k)} and v_{2}^{9−3k}(F_{(t)}^{−})^{∗}^{(′)}. \end{matrix}

Here,

\begin{aligned}(F′′_{0})_{(h)}^{(0)} & = & ℤ/3[v_{2}^{±9}]\{v_{2}^{−3}/v_{1}^{3}, v_{2}^{8}ξ/v_{1}^{3}\}, \\ (F′′_{1})_{(h)}^{(0)} & = & ℤ/3[v_{2}^{±27}]\{v_{2}^{24}h_{10}/v_{1}^{31}, v_{2}^{16±9}ξ/v_{1}^{11}\}, \\ (F′′_{n})_{(h)}^{(0)} & = & ℤ/3[v_{2}^{±3^{n+2}}]\{v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}−1}\}, \end{aligned} \begin{aligned}(F′′_{0})_{(t)}^{(0)} & = & ℤ/3[v_{2}^{±9}]\{v_{2}^{8}ξ/v_{1}^{4}\}, \\ (F′′_{1})_{(t)}^{(0)} & = & ℤ/3[v_{2}^{±27}]\{v_{2}^{±9}/v_{1}^{11}\}, \\ (F′′_{n})_{(t)}^{(0)} & = & ℤ/3[v_{2}^{±3^{n+2}}]\{v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−1}\}; \end{aligned}

\begin{aligned}(F′′_{0})_{(h)}^{(1)} & = & ℤ/3[v_{2}^{±9}]\{v_{2}^{3}/v_{1}^{3}, v_{2}^{2}ξ/v_{1}^{3}\}, \\ (F′′_{1})_{(h)}^{(1)} & = & ℤ/3[v_{2}^{±27}]\{v_{2}^{9}h_{10}/v_{1}^{19}, v_{2}^{16±9}ξ/v_{1}^{11}\}, \\ (F′′_{n})_{(h)}^{(1)} & = & ℤ/3[v_{2}^{±3^{n+2}}]\{v_{2}^{3^{n+1}}h_{10}/v_{1}^{2×3^{n+1}+1}, v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}+1}, v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}−1}\}, \\ (F′′_{0})_{(t)}^{(1)} & = & ℤ/3[v_{2}^{±9}]\{v_{2}^{8}h_{10}/v_{1}^{11}\}, \\ (F′′_{1})_{(t)}^{(1)} & = & ℤ/3[v_{2}^{±27}]\{v_{2}^{±9}/v_{1}^{11}, v_{2}^{16±9}ξ/v_{1}^{12}\}, \\ (F′′_{n})_{(t)}^{(1)} & = & ℤ/3[v_{2}^{±3^{n+2}}]\{v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}}\}; \end{aligned}

1In [11], ξ_{n} = v_{2}^{−3^{n}+(3^{n}−1)/2}ξ.

\begin{aligned}(F′′_{0})_{(h)}^{(2)} & = & ℤ/3[v_{2}^{±9}]\{v_{2}^{3}h_{10}/v_{1}^{7}, v_{2}^{8}ξ/v_{1}^{3}, v_{2}^{2}ξ/v_{1}^{3}\}, \\ (F′′_{1})_{(h)}^{(2)} & = & 0, \\ (F′′_{n})_{(h)}^{(2)} & = & 0, \end{aligned} \begin{aligned}(F′′_{0})_{(t)}^{(2)} & = & ℤ/3[v_{2}^{±9}]\{v_{2}^{8}h_{10}/v_{1}^{11}, v_{2}^{2}ξ/v_{1}^{4}, \}, \\ (F′′_{1})_{(t)}^{(2)} & = & 0, \\ (F′′_{n})_{(t)}^{(2)} & = & ℤ/3[v_{2}^{±3^{n+2}}]\{v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−1}\}; \end{aligned}

F_{(t)}^{+}^{(0)} = F_{(t)}^{+},
F_{(t)}^{+}^{(1)} = ℤ/3[b_{10}]/(b_{10}^{5}){v_{2}^{s}h_{10}/v_{1}^{2} | s ∈ 9ℤ − J_{1}}⨁ℤ/3[b_{10}]/(b_{10}^{4}){v_{2}^{s}h_{10}/v_{1}^{2} | s ∈ J_{1}}
⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{5}h_{11}/v_{1}^{2}},
F_{(t)}^{+}^{(2)} = ℤ/3[v_{2}^{±27}, b_{10}]/(b_{10}^{5}){v_{2}^{19}h_{10}/v_{1}^{2}}⨁ℤ/3[v_{2}^{±27}, b_{10}]/(b_{10}^{4}){v_{2}^{19±9}h_{10}/v_{1}^{2}}
⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{5}h_{11}/v_{1}^{2}},
(F_{(h)}^{+})^{∗}^{(′)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{7}ψ_{1}/v_{1}, v_{2}^{3}ψ_{0}/v_{1}},
(F_{(t)}^{+})^{∗}^{(0)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{4}){v_{2}^{3}ξ/v_{1}^{2}, v_{2}^{6}b_{11}ξ/v_{1}^{2}},
(F_{(t)}^{+})^{∗}^{(k)} = (F_{(t)}^{+})^{∗}  for k = 1, 2,
F_{(h)}^{−}^{(0)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{2}h_{11}/v_{1}, v_{2}^{5}h_{11}/v_{1}, v_{2}h_{10}/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{3}){v_{2}^{7}h_{10}/v_{1}},
F_{(h)}^{−}^{(1)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{2}h_{11}/v_{1}, v_{2}^{5}h_{11}/v_{1}, v_{2}^{7}h_{10}/v_{1}}⨁ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{3}){v_{2}h_{10}/v_{1}},
F_{(h)}^{−}^{(2)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{2}h_{11}/v_{1}, v_{2}^{5}h_{11}/v_{1}, v_{2}^{7}h_{10}/v_{1}, v_{2}h_{10}/v_{1}},
(F_{(h)}^{−})^{∗}^{(0)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{3}b_{11}ξ/v_{1}, v_{2}^{6}b_{11}ξ/v_{1}, ξ/v_{1}}
⨁ℤ/3[b_{10}]/(b_{10}^{3}){v_{2}^{9t+3}ξ/v_{1} | t ∉ J}⨁ℤ/3[b_{10}]/(b_{10}^{2}){v_{2}^{9t+3}ξ/v_{1} | t ∈ J},
(F_{(h)}^{−})^{∗}^{(1)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{3}b_{11}ξ/v_{1}, v_{2}^{6}b_{11}ξ/v_{1}, v_{2}^{3}ξ/v_{1}}⨁ℤ/3[v_{2}^{±27}, b_{10}]/(b_{10}^{2}){v_{2}^{9}ξ/v_{1}}
⨁ℤ/3[v_{2}^{±27}, b_{10}]/(b_{10}^{3}){v_{2}^{9±9}ξ/v_{1}},
(F_{(h)}^{−})^{∗}^{(2)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{3}b_{11}ξ/v_{1}, v_{2}^{6}b_{11}ξ/v_{1}}
⨁ℤ/3[v_{2}^{±27}, b_{10}]/(b_{10}^{2}){v_{2}^{12}ξ/v_{1}}⨁ℤ/3[v_{2}^{±27}, b_{10}]/(b_{10}^{3}){v_{2}^{12±9}ξ/v_{1}}
⨁ℤ/3[b_{10}]/(b_{10}^{3}){v_{2}^{9t}ξ/v_{1} | t ∉ J}⨁ℤ/3[b_{10}]/(b_{10}^{2}){v_{2}^{9t}ξ/v_{1} | t ∈ J} and
(F_{(t)}^{−})^{∗}^{(′)} = ℤ/3[v_{2}^{±9}, b_{10}]/(b_{10}^{2}){v_{2}^{4}ψ_{0}/v_{1}, v_{2}^{7}ψ_{0}/v_{1}, v_{2}^{2}ψ_{1}/v_{1}, v_{2}^{8}ψ_{1}/v_{1}},

in which

J_{1} = {s | s = 3^{n+1}t ± 3^{n} − 3^{n−1} − 8 for n > 2 and t ∈ ℤ} and
J = {m ∈ ℤ | m = 3^{n}t + (3^{n−1} − 1)/2 or m = 3^{n+1}t + 5 × 3^{n−1} + (3^{n−1} − 1)/2 for n > 0}.

Note that each integer m has the form

3^{n+1}t ± 3^{n} + (3^{n} − 1)/2 for some t ∈ ℤ and n ≥ 0.

Thus, we see that {c(n, t, ±) | n > 0, t ∈ ℤ} = {m | m = 9t + 2, t ∈ ℤ}. Indeed, integers 3^{n+1}t + 3^{n}(1 ± 1) + (3^{n} − 1)/2 for n > 0 ran through 3u + 1.

For each element x of A′_{1}, there is an element x̂ ∈ ∑_{n≥0} F′′_{n} and integer j > 0 such that v_{1}^{j}x̂ = x. Therefore, we read off from the above table that d_{r}(x) = 0 if j > 1, and d_{5}(x) = 0 if j = 1. Suppose that j = 1 and d_{9}(x) = y. Then we see that δ(y) ≠ 0 by the above table. Indeed, if δ(y) = 0, then d_{9}(x̂) = y/v_{1}, and such y is killed by d_{5}. Since δ(x) = 0 for the connecting homomorphism δ: H^{∗}M_{1}^{1} → H^{∗}M_{2}^{0}, we see that there is an element z ∈ H^{∗}M_{2}^{0} such that d_{5}(z) = δ(y) and d_{5}(x̂) = z/v_{1}, and that there is no such element z. Therefore, the Adams differentials on A′_{1} are trivial, and we obtain the E_{∞}-term as E_{∞}^{∗}(M^{1}V (0)∧X^{k}) = A_{0}⨁ (A′_{1}⨁H^{∗}(H^{∗}(A′_{2}, d_{5}^{k}), d_{9}^{k})) ⨂Λ(ζ_{2}). Moreover, the spectral sequence has no extension problem, since π_{∗}(M^{1}V (0)∧X^{k}) is ℤ/3-module, and we obtain the homotopy groups.

Proof of Theorem 1.1. The theorem follows by setting

A_{0} = A_{0}, A_{1} = A′_{1}⨂Λ(ζ_{2}), and { \atop v_{2}^{9−3k}(A_{2})_{(h)}}⨁{ \atop v_{2}^{9−3k}(A_{2})_{(t)}} = H^{∗}(H^{∗}(A′_{2}, d_{5}^{k}), d_{9}^{k})⨂Λ(ζ_{2}).

Here the modules A_{0} and A′_{1} on the right hand sides of the equations are those given in (3.1) and (3.3).

Indeed, (4.10)

\begin{aligned}{ \atop v_{2}^{9−3k}(A_{2})_{(h)}} & = & (∑_{n≥0}(F′′_{n})_{(h)}^{(k)}⨁v_{2}^{9−3k}F_{(h)}^{+}⨁v_{2}^{9−3k}(F_{(h)}^{+})^{∗}^{(′)}⨁v_{2}^{9−3k}F_{(h)}^{−}^{(k)}⨁v_{2}^{9−3k}(F_{(h)}^{−})^{∗}^{(k)}) ⨂Λ(ζ_{2}) and \\ { \atop v_{2}^{9−3k}(A_{2})_{(t)}} & = & (∑_{n≥0}(F′′_{n})_{(t)}^{(k)}⨁v_{2}^{9−3k}F_{(t)}^{+}^{(k)}⨁v_{2}^{9−3k}(F_{(t)}^{+})^{∗}^{(k)}⨁v_{2}^{9−3k}F_{(t)}^{−}⨁v_{2}^{9−3k}(F_{(t)}^{−})^{∗}^{(′)}) ⨂Λ(ζ_{2}). \end{aligned}

□

5. The homotopy groups π_{∗}(M^{2}X^{k})

For a spectrum W, M^{2}W is defined by the cofiber sequences

L_{2}W ⟶^{η_{0}} L_{0}W ⟶ N^{1}W and N^{1}W ⟶^{η_{1}} L_{1}N^{1}W ⟶ M^{2}W.

Here η_{i} denotes the L_{i}-localization. Consider the cofiber sequence X^{k} →^{3} X^{k} → V (0) ∧ X^{k}. Then we have the following commutative diagrams:

X_{k}  X_{k}  V (0) ∧ X^{k} L_{0}X^{k}  L_{0}X^{k}  ∗ N^{1}X^{k}  N^{1}X^{k}  ΣV (0) ∧ X^{k},^{3}  w_{_{η_{0}} u}  w_{_{η_{0}} u}  u_{^{3} w} u  w u  u_{^{3} w}  w  N^{1}X^{k}  N^{1}X^{k}  ΣV (0) ∧ X^{k} L_{1}N^{1}X^{k}  L_{1}N^{1}X^{k}  ΣL_{1}V (0) ∧ X^{k} M^{2}X^{k}  M^{2}X^{k}  ΣM^{1}V (0) ∧ X^{k}.^{3} w_{_{η_{1}} u}  w_{_{η_{1}} u}  u_{^{3} w} u  w u  u_{^{3} w}  w

These yield a cofiber sequence

(5.1)  M^{2}X^{k} ⟶^{3} M^{2}X^{k} ⟶^{ι} ΣM^{1}V (0) ∧ X^{k},

which induces the short exact sequence

(5.2)  0 ⟶ M_{1}^{1} ⟶^{φ} M_{0}^{2} ⟶^{3} M_{0}^{2} ⟶ 0,

where φ is given by φ(x) = x/3. In order to determine the E_{2}-term for π_{∗}(M^{2}X^{k}), we consider the long exact sequence

(5.3)  \begin{matrix}0 ⟶ H^{0}M_{1}^{1} ⟶^{φ_{∗}} H^{0}M_{0}^{2} ⟶^{3} H^{0}M_{0}^{2} ⟶^{δ} H^{1}M_{1}^{1} ⟶^{φ_{∗}} H^{1}M_{0}^{2} ⟶^{3} H^{1}M_{0}^{2} ⟶^{δ} ⋯ \\ ⟶^{δ} H^{s}M_{1}^{1} ⟶^{φ_{∗}} H^{s}M_{0}^{2} ⟶^{3} H^{s}M_{0}^{2} ⟶^{δ} H^{s+1}M_{1}^{1} ⟶^{φ_{∗}} ⋯ \end{matrix}

induced from the short exact sequence (5.2).

Proof of Theorem 1.2. The E_{2}-term H^{∗}M_{0}^{2} is decomposed into the submodules \overline{A_{i}} for i = 0, 1, 2 such

that A_{i}^{s} →^{φ_{∗}} \overline{A_{i}}^{s} →^{3} \overline{A_{i}}^{s} →^{δ} A_{i}^{s+1}, where M^{s} = M ∩ H^{s}N if M ⊂ H^{∗}N. In particular, 3: \overline{A_{2}}^{s} → \overline{A_{2}}^{s} is

a zero map and φ_{∗} is an isomorphism between (A_{2})_{(h)} and \overline{A_{2}}, except for some elements of (A_{2})_{(t)}. Here, (A_{2})_{(h)} = (F_{(h)}⨁F_{(h)}^{∗})⨂ℤ/3[b_{10}]⨂Λ(ζ_{2}). Indeed, it follows from [10, Lemmas 4.4 and 5.2] that the map φ_{∗} acts trivially on (A_{2})_{(t)} except for

(5.4)  \begin{matrix}φ_{∗}(v_{2}^{−1}/v_{1}) = v_{2}^{−1}/3v_{1}, φ_{∗}(v_{2}^{2}h_{11}/v_{1}^{2}) = v_{2}^{2}h_{11}/3v_{1}^{2}, \\ φ_{∗}(ξ/v_{1}^{2}) = ξ/3v_{1}^{2}, φ_{∗}(v_{2}^{−2}ψ_{0}/v_{1}) = v_{2}^{−2}ψ_{0}/3v_{1}, \end{matrix}

(5.5)  \begin{matrix}φ_{∗}(v_{2}^{−1}b_{10}/v_{1}) = v_{2}^{−1}h_{11}ζ_{2}/3v_{1}, φ_{∗}(v_{2}^{2}h_{11}b_{10}/v_{1}^{2}) = v_{2}b_{11}ζ_{2}/3v_{1}, \\ φ_{∗}(ξb_{10}/v_{1}^{2}) = ψ_{0}ζ_{2}/3v_{1}, φ_{∗}(v_{2}^{−2}ψ_{0}b_{10}/v_{1}) = v_{2}^{−3}ξb_{11}ζ_{2}/3v_{1}, \\ φ_{∗}(v_{2}^{−1}b_{11}/v_{1}) = v_{2}h_{10}ζ_{2}/3v_{1}, φ_{∗}(ξb_{11}/v_{1}^{2}) = v_{2}ψ_{1}ζ_{2}/3v_{1} and φ_{∗}(v_{2}^{−1}ψ_{1}/v_{1}) = ξζ_{2}/3v_{1}, \end{matrix}

where the equations are all up to sign. Some equations in [10, Lemmas 4.4 and 5.2] should be replaced by δ(v_{2}b_{11}/3v_{1}) = v_{2}^{2}h_{11}b_{10}/v_{1}^{2} + v_{2}b_{11}ζ_{2}/v_{1} and d_{1}(v_{2}ψ_{1}/v_{1}) = b_{11}ξ/v_{1}^{2} + v_{2}ψ_{1}ζ_{2}/v_{1}. The φ_{∗}-images of the Adams differentials on the elements of (5.4) are

(5.6)  \begin{aligned}d_{9}(v_{2}^{9t+8−3k}/3v_{1}) & = & v_{2}^{9t+4−3k}b_{11}b_{10}^{3}ζ_{2}/3v_{1}, \\ d_{5}(v_{2}^{9t+2−3k}h_{11}/3v_{1}^{2}) & = & v_{2}^{9t+1−3k}h_{10}b_{10}^{2}ζ_{2}/3v_{1}, \\ d_{5}(v_{2}^{9t+8−3k}h_{11}/3v_{1}^{2}) & = & v_{2}^{9t+7−3k}h_{10}b_{10}^{2}ζ_{2}/3v_{1}, \\ d_{5}(v_{2}^{9t−3k}ξ/3v_{1}^{2}) & = & v_{2}^{9t−3−3k}ξb_{11}b_{10}ζ_{2}/3v_{1}, \\ d_{5}(v_{2}^{9t+6−3k}ξ/3v_{1}^{2}) & = & v_{2}^{9t+3−3k}ξb_{11}b_{10}ζ_{2}/3v_{1} and \\ d_{9}(v_{2}^{9t+1−3k}ψ_{0}/3v_{1}) & = & v_{2}^{9t−2−3k}ψ_{1}b_{10}^{4}ζ_{2}/3v_{1} \end{aligned}

up to sign in E_{r}^{∗}(X^{k}), by use of the second, the fifth, the fifth, the fourth, the fourth and sixth ones of (5.5). We write d_{5}(χ′_{k,i}) = χ_{k,i} as the i-th equation of (5.6).

Furthermore, each element x in (F′′_{n})_{(h)} is also sent to the non-zero element x/3 of H^{∗}M_{0}^{2} under φ_{∗}. Indeed, each x ∈ (F′′_{n})_{(h)} supports a non-trivial differential d_{r}(x) = y ≠ 0 ∈ (A_{2})_{(h)} in E_{r}^{∗}(M^{1}V (0) ∧ X^{k}) for some k = 0, 1, 2, and φ_{∗}(y) ≠ 0. If φ_{∗}(x) = x/3 = 0, then there is an element z/3 such that δ(z/3) = x, and z/3 supports a non-trivial differential. Therefore, in our case, it follows that r = 9 and d_{5}(z/3) = w ≠ 0 ∈ \overline{A_{2}} for some element w ∈ \overline{A_{2}}. Since δ(w) = d_{5}(δ(z/3)) = d_{5}(x) = 0, w is divisible by 3, and so w = 0, which is a contradiction. Therefore, x/3 ≠ 0. Thus, φ_{∗} induces an isomorphism on (A′_{2})_{(h)}. It follows that the Adams differentials on φ_{∗}((A′_{2})_{(h)}) = \overline{(A′_{2})}_{(h)} behave in the same way as those on (A′_{2})_{(h)}, and so there is a

map φ̃: { \atop v_{2}^{9−3k}(A_{2})_{(h)}} → E_{∞}^{∗}(X^{k}), whose kernel consists of elements x such that φ_{∗}(x) is hit by an element of \overline{(F′′_{n})_{(t)}} ⊂ \overline{A_{1}} or appearing in (5.6).

Suppose that d_{r}(x) = y in E_{5}^{∗}(X^{k}) with x ∉ φ_{∗}(H^{∗}M_{1}^{1}). Then δ(y) ∈ (A_{2})_{(t)} ⊂ E_{2}^{∗}(M^{1}V (0) ∧ X^{k}), since filt y ≥ 5. If δ(x) = 0, then there is x̂ ∈ H^{∗}M_{0}^{2} such that 3x̂ = x and d_{s}(x̂) ≠ 0 with s < r. So we assume that δ(x) ≠ 0. Then d_{r}(δ(x)) = δ(y). It follows that δ(x) ∈ (A′_{2})_{(t)}. Here, the differentials of the Bockstein spectral sequence associated to (5.3) on (F′′_{n})_{(h)}⨁(F′′_{n})_{(t)} are read off from the action of the connecting homomorphism given in [6, Prop. 6.9] and [11, Prop.s 5.4, 5.6] by: (5.7)

\begin{aligned}δ(v_{2}^{3^{n+2}t±3^{n+1}}/3v_{1}^{4×3^{n}−1}) & = & v_{2}^{3^{n+2}t±3^{n+1}}h_{10}/v_{1}^{4×3^{n}}, \\ δ(v_{2}^{3^{n+2}t+3^{n+1}}h_{10}/3v_{1}^{2×3^{n+1}+1}) & = & ±(−1)^{n}v_{2}^{3^{n+2}t+2×3^{n}+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}} (n ≥ 0), \\ δ(v_{2}^{9t+8}h_{10}/3v_{1}^{11}) & = & v_{2}^{9t+8}h_{10}ζ_{2}/v_{1}^{11}, \\ δ(v_{2}^{3^{n+2}t+8×3^{n}}h_{10}/3v_{1}^{10×3^{n}+1}) & = & (−1)^{n−1}v_{2}^{3^{n+2}t+3^{n+1}+8×3^{n−1}+(3^{n−1}−1)/2}ξ/v_{1}^{4×3^{n−1}} (n > 0), \\ δ(v_{2}^{3^{n+2}t+8×3^{n}+(3^{n}−1)/2}ξ/3^{n+1}v_{1}^{4×3^{n}}) & = & v_{2}^{3^{n+2}t+8×3^{n}}(v_{2}^{(3^{n}−1)/2}ξζ_{2} + v_{2}^{3^{n}−1}ψ_{1})/v_{1}^{4×3^{n}} t ≢ 1 mod 3, \\ δ(v_{2}^{3^{n+2}t+3^{n}(5±3)+(3^{n}−1)/2}ξ/3v_{1}^{4×3^{n}−1}) & = & v_{2}^{3^{n+2}t+3^{n}(5±3)}(v_{2}^{(3^{n}−1)/2}ξζ_{2} + v_{2}^{3^{n}−1}ψ_{1})/v_{1}^{4×3^{n}−1}. \end{aligned}

Thus, we see that x = v_{2}^{3^{n+2}t+8×3^{n}+(3^{n}−1)/2}ξ/3^{n+1}v_{1}^{4×3^{n}} for n > 0 and k = 0, 2, since it is not in the image of φ_{∗}, and obtain

(5.8) d_{5}(v_{2}^{3^{n+2}t+8×3^{n}+(3^{n}−1)/2}ξ/3^{n+1}v_{1}^{4×3^{n}}) = v_{2}^{c(n,t,+)+1}ξb_{10}^{2}ζ_{2}/3v_{1} = v_{2}^{3^{n+2}t+2×3^{n+1}+3(3^{n}−1)/2}ξb_{10}^{2}ζ_{2}/3v_{1},

which we write as χ(t, n). Therefore, for the element 3x, we see that ι_{∗}(3x) = v_{2}^{3^{n+2}t+2×3^{n+1}+3(3^{n}−1)/2}ξb_{10}^{2}ζ_{2}/v_{1}, since v_{2}^{3^{n+2}t+2×3^{n+1}+3(3^{n}−1)/2}ξb_{10}^{2}ζ_{2}/v_{1} is a permanent cycle of E_{r}^{∗}(M^{1}V (0) ∧ X^{k}). It follows that there arises

no other differentials. Now put \overline{A_{0}}, \overline{A_{1}} and { \atop v_{2}^{9−3k}\overline{A_{2}}} as follows to obtain the theorem.

(5.9)  \begin{aligned}\overline{A_{0}} & = & (\overline{A_{−}}⨁ℚ/ℤ)⨂Λ(ζ_{2}), \\ \overline{A_{1}} & = & (X̃_{1}⨁X̃_{2}⨁H̃⨁H̃I⨁X̃_{1}ζ_{2}⨁X̃_{2}^{∗}⨁H̃ζ_{2})/S, \\ ˜\overline{A_{2}} & = & φ_{∗}( ˜(A_{2})_{(h)})/C_{2}^{(0)}⨁B_{2}^{(0)}, \\ { \atop v_{2}^{6}\overline{A_{2}}} & = & φ_{∗}({ \atop v_{2}^{6}(A_{2})}_{(h)})/C_{2}^{(1)}⨁B_{2}^{(1)} \\ & & ⨁ ∑_{n>0} ℤ/3^{n+1}[v_{2}^{±3^{n+2}}]\{v_{2}^{8×3^{n}+(3^{n}−1)/2}ξ/3^{n+1}v_{1}^{4×3^{n}}\} and \\ { \atop v_{2}^{3}\overline{A_{2}}} & = & φ_{∗}({ \atop v_{2}^{3}(A_{2})}_{(h)})/C_{2}^{(2)}⨁B_{2}^{(2)}. \end{aligned}

Here, the modules are defined above or in [11] except for

S = ∑_{n>0} (φ_{∗}((F′′_{n})_{(h)})⨁ℤ/3^{n+1}[v_{2}^{±3^{n+2}}]{v_{2}^{8×3^{n}+(3^{n}−1)/2}ξ/3^{n+1}v_{1}^{4×3^{n}}}) ,
B_{2}^{(k)} = ℤ/3[v_{2}^{±3}]{v_{2}^{−1}/3v_{1}, v_{2}^{2}h_{11}/3v_{1}^{2}, ξ/3v_{1}^{2}, v_{2}^{−2}ψ_{0}/3v_{1}}/(χ′_{k,i} : 1 ≤ i ≤ 5),
C_{2}^{(k)} = ℤ/3[v_{2}^{±9}]{χ_{k,i} | 1 ≤ i ≤ 5}⨁ ∑_{n≥0} ℤ/3^{n+1}{χ(3t, n), χ(3t + 2, n) | t ∈ ℤ} for k = 0, 2, and
C_{2}^{(1)} = ℤ/3[v_{2}^{±9}]{χ_{1,i} | 1 ≤ i ≤ 5}.
□

Remark. The differential (5.8) is also missing in [11].

6. The homotopy groups π_{∗}(M^{1}V (1\frac{1}{2}))

In this section, we determine the homotopy groups π_{∗}(M^{1}Y) by observing the cofiber sequence X → Y → X^{2} given in section 2. By the definition of M^{i} for i = 1, 2, the cofiber sequence induces another cofiber sequence

Σ^{−1}M^{2}X ⟶ M^{1}Y ⟶ M^{2}X^{2} ⟶ M^{2}X,

which induces the long exact sequence (5.3) of the E_{2}-terms, since E_{2}^{∗}(M^{1}Y) = H^{∗}M_{1}^{1}. As in [11], δ (resp. φ_{∗}) assigns F_{(h)}⨁(F_{(h)})^{∗} to F_{(t)}⨁(F_{(t)})^{∗} (resp. F_{(h)}⨁(F_{(h)})^{∗}). Therefore, the differential pattern follows immediately.

Lemma 6.1. On A_{2}, the Adams differentials for computing π_{∗}(M^{1}Y) act as follows:

(6.2)

\begin{aligned}v_{2}^{3}× & d_{5}(v_{2}^{3t+1}/v_{1}) & = & −tv_{2}^{3t−1}h_{11}b_{10}^{2}/v_{1} & (h), \\ v_{2}^{6}× & d_{9}(v_{2}^{9t−1}/v_{1}) & = & v_{2}^{9t−4}h_{11}b_{10}^{4}/v_{1}^{2} & (t), \\ v_{2}^{6}× & d_{5}(v_{2}^{3t+1}h_{10}/v_{1}^{2}) & = & tv_{2}^{3t−1}b_{10}^{3}/v_{1} & (t), \\ v_{2}^{3}× & d_{9}(v_{2}^{9t+4}h_{10}/v_{1}) & = & v_{2}^{9t+1}b_{10}^{5}/v_{1} & (h), \end{aligned} \begin{aligned}v_{2}^{3}× & d_{5}(v_{2}^{3t+1}ψ_{1}/v_{1}) & = & −(t + 1)v_{2}^{3t}ξb_{10}^{3}/v_{1} & (h), \\ v_{2}^{6}× & d_{9}(v_{2}^{9t+5}ψ_{1}/v_{1}) & = & v_{2}^{9t+3}ξb_{10}^{5}/v_{1}^{2} & (t), \\ v_{2}^{6}× & d_{5}(v_{2}^{3t}ξ/v_{1}^{2}) & = & −(t − 1)v_{2}^{3t−2}ψ_{0}b_{10}^{2}/v_{1} & (t), \\ v_{2}^{3}× & d_{9}(v_{2}^{9t+6}ξ/v_{1}) & = & v_{2}^{9t+3}ψ_{0}b_{10}^{4}/v_{1} & (h). \end{aligned}

Remark. This result implies that there is no self map that represents v_{1} on Y. Indeed, v_{1}(v_{2}^{7}h_{10}/v_{1}^{2}) = v_{2}^{7}h_{10}/v_{1} in the E_{2}-term, and so v_{2}^{7}h_{10}/v_{1} is a permanent cycle if v_{1} exists. This is a contradiction.

Lemma 6.3. In the Adams spectral sequence (E_{r}^{∗}(M^{1}Y), d_{r}^{Y}), we have the following:

(1) H^{∗}(A_{2}, d_{5}^{Y}) is isomorphic to the tensor product of Λ(ζ_{2}) and the direct sum of

v_{2}^{3} (Ḟ_{(h)}^{+}⨁F_{(h)}^{−}⨁(Ḟ_{(h)}^{+})^{∗}⨁(F_{(h)}^{−})^{∗}) and v_{2}^{6} (Ḟ_{(t)}^{+}⨁F_{(t)}^{−}⨁(Ḟ_{(t)}^{+})^{∗}⨁(F_{(t)}^{−})^{∗}) .

(2) H^{∗}(H^{∗}(A_{2}, d_{5}^{Y}), d_{9}^{Y}) is isomorphic to the tensor product of Λ(ζ_{2}) and the direct sum of

v_{2}^{3} (F_{(h)}^{+}⨁F_{(h)}^{−}⨁(F_{(h)}^{+})^{∗}⨁(F_{(h)}^{−})^{∗}) and v_{2}^{6} (F_{(t)}^{+}⨁F_{(t)}^{−}⨁(F_{(t)}^{+})^{∗}⨁(F_{(t)}^{−})^{∗}) .

Proof of Theorem 1.3. Since the Adams differentials send the modules with (h) (resp. (t)) to those with (h) (resp. (t)), Lemmas 6.3 and the table above Proposition 4.9 show that H^{∗}(H^{∗}(A′_{2}, d_{5}^{Y}), d_{9}^{Y}) is isomorphic

to { \atop v_{2}^{3}(A_{2})_{(h)}}⨁{ \atop v_{2}^{6}(A_{2})_{(t)}} for the modules in (4.10). Therefore, we obtain the Theorem 1.3. □

Remark. Theorem 1.3 is also obtained directly by use of the cofiber sequence Σ^{16}V (0) →^{B} V (1) → V (1\frac{1}{2}).

7. Appendix: Corrections of [9], [10] and [11]

In [9], a differential is missing (see §4 above). Therefore, F̃_{0} and F̃^{∗} in [9, (2.7), Th. 2.8] is replaced by

F̃_{0} = E(2, 2)_{∗}{v_{2}^{3}/v_{1}^{2}, v_{2}^{−3}/v_{1}^{3}, v_{2}^{3}h_{10}/v_{1}^{6}, v_{2}^{8}h_{10}/v_{1}^{10}, v_{2}^{8}ξ/v_{1}^{4}},
F̃^{∗} = B_{4}(2, 2)_{∗}{v_{2}^{7}ψ_{1}/v_{1}, v_{2}^{3}ξ/v_{1}^{2}, v_{2}^{3}ψ_{0}/v_{1}, v_{2}^{6}b_{11}ξ/v_{1}^{2}}
⨁B_{2}{v_{2}^{−1}ψ_{1}/v_{1}, ξ/v_{1}, v_{2}^{2}ψ_{1}/v_{1}, v_{2}^{4}ψ_{0}/v_{1}, v_{2}^{3}b_{11}ξ/v_{1}, v_{2}^{7}ψ_{0}/v_{1}, v_{2}^{6}b_{11}ξ/v_{1}}
⨁ ∑_{n≥1} (B_{3}(2, n + 2)_{∗}{v_{2}^{9u+3}ξ/v_{1} | u ∈ ℤ − I(n)}⨁B_{2}(2, n + 2)_{∗}{v_{2}^{9u+3}ξ/v_{1} | u ∈ I(n)}) .

The missing differential is on the modules with subscript (h), and there is another missing differential (5.8). Therefore, we should replace G̃^{∗} and { \atop GZ^{∗}} in [10] and [11] by

G̃^{∗} = B_{4}(2, 2)_{∗}{v_{2}^{7}ψ_{1}/3v_{1}, v_{2}^{3}ψ_{0}/3v_{1}}⨁B_{2}{ξ/3v_{1}, v_{2}^{3}b_{11}ξ/3v_{1}, v_{2}^{6}b_{11}ξ/3v_{1}},

⨁ ∑_{n≥1} (B_{3}(2, n + 2)_{∗}{v_{2}^{9u+3}ξ/3v_{1} | u ∈ ℤ − I(n)}⨁B_{2}(2, n + 2)_{∗}{v_{2}^{9u+3}ξ/3v_{1} | u ∈ I(n)}) ,

{ \atop GZ^{∗}} = B_{4}(2, 2)_{∗}\{v_{2}^{7}ψ_{1}ζ_{2}/3v_{1}, v_{2}^{3}ψ_{0}ζ_{2}/3v_{1}\}⨁B_{2}(2, 2)\{ξζ_{2}/3v_{1}\}

⨁B_{1}{v_{2}^{3}b_{11}ξζ_{2}/3v_{1}, v_{2}^{6}b_{11}ξζ_{2}/3v_{1}}

⨁ ∑_{n≥1} (B_{3}(2, n + 2)_{∗}{v_{2}^{9u+3}ξζ_{2}/3v_{1} | u ∈ ℤ − I′(n)}⨁B_{2}(2, n + 2)_{∗}{v_{2}^{9u+3}ξζ_{2}/3v_{1} | u ∈ I′(n)}) ,

where

I′(n) = {x ∈ ℤ | x = (3^{n−1} − 1)/2, x = 2 × 3^{n−1} + (3^{n−1} − 1)/2 or x = 5 × 3^{n−2} + (3^{n−2} − 1)/2}.

There are many misprints in [11], though the proofs of statements are correct. Here, we correct the propositions that we need to show our theorem. Propositions 3.1, 5.3 and 5.4 of [11] should be replaced by

Proposition 3.1. The connecting homomorphisms δ: \overline{A_{1}}^{s} → \overline{A_{1}}^{s+1} map X̃_{1}, X̃_{2}, H̃, H̃I, X̃_{1}ζ_{2}, X̃_{2}^{∗} and H̃ζ_{2} to H^{∗}, X_{2}ζ_{2}, X_{1}^{∗}, HIζ_{2}, H^{∗}ζ_{2}, X_{2}^{∗}ζ_{2} and X_{1}^{∗}ζ_{2}, respectively. Furthermore, the images of generators under δ are linearly independent.

To help to understand this, we write down a chart of differential pattern:

H → X_{1}^{∗}
X_{1} → H^{∗}  X_{2}^{∗} → X_{2}^{∗}ζ_{2}
HI → HIζ_{2}
X_{2} → X_{2}ζ_{2}  Hζ_{2} → X_{1}^{∗}ζ_{2}
X_{1}ζ_{2} → H^{∗}ζ_{2}

Proposition 5.3. For n ≥ 0 and s with 3 | (s − 1) or 9 | (s + 1),

δ(v_{2}^{3^{n}s}h_{10}/3^{n+1}v_{1}) = v_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}(+v_{2}^{s−2}b_{11}/v_{1} if n = 0).

Proposition 5.4. Let s, n, j, k be integers with 3 |̸ s, k > 0 and 0 ≤ i ≤ n. Then the Bockstein differential on v_{2}^{3^{n}s}h_{10}/v_{1}^{3^{i}k+1} is given as follows :

1. If 3^{i}k ≤ 2 × 3^{n−i}, then

δ(v_{2}^{3^{n}s}h_{10}/3^{i+1}v_{1}^{3^{i}k+1}) = −kv_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}^{3^{i}k+1} − (−1)^{n−i}sv_{2}^{3^{n}s}ξ_{n−i−1}/v_{1}^{2^{i}k−2×3^{n−i−1}} + · · · .
2. If s = 9t − 1, 3^{i}k ≤ 8 × 3^{n} + 2 × 3^{n−i+1} and i > 0, then
δ(v_{2}^{3^{n}s}h_{10}/3^{i}v_{1}^{3^{i}k+1}) = (−1)^{n−i}v_{2}^{3^{n+1}(3t−1)}ξ_{n−i}/v_{1}^{3^{i}k−8×3^{n}−2×3^{n−i}} + · · · .
3. If s = 9t − 1, 3^{i+1} |̸ 3^{i}k ≤ 8 × 3^{n} + 2 × 3^{n−i} and i < n, then
δ(v_{2}^{3^{n}s}h_{10}/3^{i+1}v_{1}^{3^{i}k+1}) = −kv_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}^{3^{i}k+1} + · · · .
4. If s = 9t − 1, 3^{n}k ≤ 8 × 3^{n} + 2 and 3 |̸ (k + 1), then
δ(v_{2}^{3^{n}s}h_{10}/3^{n+1}v_{1}^{3^{n}k+1}) = −(k + 1)v_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}^{3^{n}k+1} + · · · .

5. If s = 9t − 1, 3^{n}k ≤ 8 × 3^{n} and 3 | (k + 1) (i.e. k = 2, 5, 8), then

δ(v_{2}^{3^{n}s}h_{10}/3^{n+2}v_{1}^{2×3^{n}+1}) = −v_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}^{2×3^{n}+1} + ⋯,
δ(v_{2}^{3^{n}s}h_{10}/3^{n+2}v_{1}^{8×3^{n}+1}) = v_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}^{8×3^{n}+1} + ⋯ and
δ(v_{2}^{3^{n}s}h_{10}/3^{n+3}v_{1}^{5×3^{n}+1}) = −v_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}^{5×3^{n}+1} + · · · .

Here ⋯ denotes an element killed by a lower power of v_{1} than is shown.

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Ippei Ichigi

Department of Mathematics

Faculty of Science Kochi University Kochi, 780-8520, Japan Email:95sm004@math.kochi-u.ac.jp

Katsumi Shimomura

Department of Mathematics

Faculty of Science Kochi University Kochi, 780-8520, Japan Email:katsumi@math.kochi-u.ac.jp