On the action of β_{1} in the stable homotopy of spheres at the prime 3

Katsumi Shimomura

Abstract. Let S^{0} denote the sphere spectrum localized away from 3. The element β_{1} is the generator of the homotopy group π_{10}(S^{0}). Toda showed that β_{1}^{5} ≠ 0 and β_{1}^{6} = 0. In this paper, we generalize his result and show that β_{1}^{4}β_{9t+1} ≠ 0 and β_{1}^{5}β_{9t+1} = 0 for β_{9t+1} ∈ π_{144t+10}(S^{0}) with t ≥ 0. In particular, β_{1}^{4}β_{10} ≠ 0 and β_{1}^{5}β_{10} = 0, where the existence of β_{10} was shown by Oka. This is proved by determining subgroups of π_{∗}(L_{2}S^{0}). Here L_{2} denotes the Bousfield localization functor with respect to v_{2}^{−1}BP.

1. Introduction

Let p be a prime number and S^{0} the sphere spectrum localized away from p. We write E_{r}^{∗}(X) as the E_{r}-term of the Adams-Novikov spectral sequence converging to π_{∗}(X) for a spectrum X localized away from p. Miller, Ravenel and Wilson [1] introduced β-elements β_{s/j,i+1} in E_{2}^{2}(S^{0}) for (s, j, i+1) ∈ B^{+}, where

B^{+} = {(s, j, i + 1) ∈ Z^{3} | s = mp^{n}, n ≥ 0, p |̸ m ≥ 1, j ≥ 1, i ≥ 0, subject to
i) j ≤ p^{n} if m = 1, ii) p^{i}|j ≤ a_{n−i}, and iii) a_{n−i−1} < j if p^{i+1}|j}

for integers a_{k} defined by a_{0} = 1 and a_{k} = p^{k} + p^{k−1} − 1. Here we use the abbreviation β_{s/j,1} = β_{s/j} and β_{s/1,1} = β_{s}.

Let V (1) denote the Toda-Smith spectrum, which is a cofiber of the Adams map α : Σ^{2p−2}V (0) → V (0), where V (0) is the mod p Moore spectrum. Since there exists a map β : Σ^{2p^{2}−2}V (1) → V (1) which induces v_{2} on BP -homology at a prime p > 3 by [9], we have homotopy elements β_{t} ∈ π_{2t(p^{2}−1)−2p}(S^{0}) with t > 0. On the other hand, there is no such self map at the prime 3. However there are homotopy elements β_{i} for i = 1, 2, 3, 5, 6, 10 in this case due to Toda and Oka (cf. [2]). Besides, assuming the existence of the self map B : Σ^{144}V (1) → V (1) that induces v_{2}^{9} on BP -homology, we see that there exists a family {β_{9t+i} | i = 0, 1, 2, 5, 6, t ≥ 0} in π_{∗}(S^{0}). The existence of B seems to be shown by Pemmaraju in his thesis. Furthermore, the existence of β_{6/3} ∈ π_{82}(S^{0}) is shown by Ravenel [4].

2000 Mathematics Subject Classification. 55Q45, 55Q51.

Key words and phrases. Homotopy groups of spheres, L_{2}-localization, β-elements, Adams-Novikov spectral sequence.

In this paper, we obtain the following relations among β_{9t+1}, β_{2} and β_{6/3}:

Theorem A. Let t, i, j and k be non-negative integers. Then in the homotopy groups π_{∗}(S^{0}) of sphere spectrum localized away from 3,

β_{9t+1}β_{1}^{i} ≠ 0 ∈ π_{∗}(S^{0}) if and only if i < 5,
β_{9t+1}β_{2}β_{1}^{j} ≠ 0 ∈ π_{∗}(S^{0}) if and only if j < 2, and
β_{9t+1}β_{6/3}β_{1}^{k} ≠ 0 ∈ π_{∗}(S^{0}) if and only if k < 4.

As is seen in [3, p.624], we have a relation

uvβ_{s}β_{t} = stβ_{u}β_{v} for s + t = u + v

in the E_{2}-term E_{2}^{4}(S^{0}). This implies the following:

Corollary B. In the homotopy groups π_{∗}(S^{0}) localized away from 3,

∏_{i=1}^{k} β_{9t_{i}+1} ≠ 0 if and only if k < 6,and
(∏_{i=1}^{k} β_{9t_{i}+1})β_{9t+2} ≠ 0 if and only if k < 2,

for integers t, t_{i} ≥ 0. In particular, β_{9t+1}^{k} ≠ 0 if and only if k < 6.

Remark. If the self-map B does not exist, the above theorems are valid only for the homotopy elements such as β_{1} and β_{10}.

We prove Theorem A by determining subgroups of π_{∗}(L_{2}S^{0}), where L_{2}: S_{(3)} → S_{(3)} denotes the Bousfield localization functor on the category S_{(3)} of spectra localized away from 3 with respect to the Johnson-Wilson spectrum E(2). In π_{∗}(L_{2}S^{0}), we have generalized β-elements β_{s/j,i+1} ∈ E_{2}^{2}(L_{2}S^{0}) for (s, j, i + 1) ∈B, where

B = {(s, j, i + 1) ∈ Z^{3} | s = mp^{n}, n ≥ 0, 3 |̸ m ∈ Z, j ≥ 1, i ≥ 0,
such that 3^{i}|j ≤ a_{n−i} and either 3^{i+1} |̸ j or a_{n−i−1} < j}.

Consider the Z/3[β_{1}]-modules

Ĝ = ∑_{t∈Z}(B_{5}{β_{9t+1}}⨁B_{4}{β_{9t+1}β_{6/3}}

⨁B_{3}\{\overline{β_{9t+7}α_{1}}\}

⨁B_{2}{β_{9t+1}α_{1}, [β_{9t+2}β′_{1}], [β_{9t+5}β′_{1}]}),

Ĝ^{∗} = ∑_{t∈Z}(B_{5}{g_{16(9t+7)+15}}⨁B_{4}{g_{16(9t+3)+7}}

⨁B_{2}{g_{144t}, g_{16(9t+5)+2}, g_{16(9t+8)+2}}

⨁∑_{n≥1}(B_{3}{g_{16(3^{n+2}t+9u+3)} | u ∈ Z − I(n)}

⨁B_{2}{g_{16(3^{n+2}t+9u+3)} | u ∈ I(n)})).

Here B_{k} = Z/3[β_{1}]/(β_{1}^{k}),

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

\overline{x} denotes a homotopy element detected by x in the E_{2}-term, [x] is an element of π_{∗}(L_{2}S^{0}) such that i_{∗}([x]) = x ∈ π_{∗}(L_{2}V (0)) for the inclusion i: S^{0} → V (0) = S^{0} ⋃_{3} e^{1}, and g_{i} ∈ π_{i}(L_{2}S^{0}) is the generator. Then the direct sum Ĝ⨁Ĝ^{∗} is generated by

S = \{β_{9t+1}, β_{9t+1}β_{6/3}, \overline{β_{9t+7}α_{1}}, β_{9t+1}α_{1}, [β_{9t+2}β′_{1}], [β_{9t+5}β′_{1}],
g_{16(9t+7)+15}, g_{16(9t+3)+7}, g_{144t}, g_{16(9t+5)+2},
g_{16(9t+8)+2}, g_{16(9t+3)} | t ∈ Z}

as a Z/3[β_{1}]-module. Our key lemma is the following:

Theorem C. The homotopy groups π_{∗}(L_{2}S^{0}) contain the subgroups Ĝ⨁Ĝ^{∗}.

Consider the localization map ι: S^{0} → L_{2}S^{0}. Then we immediately see the following:

Corollary D. For any element x ∈ π_{∗}(S^{0}) such that ι_{∗}(x) ∈ S, we have xβ_{1} ≠ 0 ∈ π_{∗}(S^{0}).

In [7], we showed that the β-elements β_{s/j,i+1} for (s, j, i + 1) ∈B^{c} do not exist in π_{∗}(L_{2}S^{0}), where

\begin{matrix}B^{c} = \{(9t + 4, 1, 1), (9t + 7, 1, 1), (9t + 8, 1, 1), (9t + 3, 3, 1), (9s, 3, 2), (3^{i}s, 3^{i}, 1) \\ | t ∈ Z, s ∈ Z − 3Z, i > 1\}. \end{matrix}

Moreover β-elements β_{s/j,i+1} for (s, j, i + 1) ∈B^{c} do not exist in π_{∗}(S^{0}) if t ≥ 1, 3 |̸ s ≥ 1 and i > 1. We further showed in [7] the existence of βelements β_{9t}, β_{9t+1} and β_{9t+5} in π_{∗}(L_{2}S^{0}) for t ∈ Z. Here we extend the existence theorem of β-elements in π_{∗}(L_{2}S^{0}):

Theorem E. β_{s/j,i+1} for (s, j, i + 1) ∈ B − B^{c} survives to a homotopy element of π_{∗}(L_{2}S^{0}).

Note that β_{s/j,i+1} are homotopy elements for all (s, j, i + 1) ∈ B at a prime > 5 ([8]).

The following sections 2 to 6 are devoted to proving Theorem C and to giving subgroups of π_{∗}(M^{2}) for an L_{2}-local spectrum M^{2} with E(2)_{∗}(M^{2}) = E(2)_{∗}/(3^{∞}, v_{1}^{∞}). Theorems A and E are actually corollaries of Theorem C, and proved in §7.

2. Basic properties of H^{∗}M_{0}^{2}

Let E(2) be the Johnson-Wilson spectrum with coefficient ring E(2)_{∗} = Z_{(3)}[v_{1}, v_{2}^{±1}]. Then E(2)_{∗}E(2) is a Hopf algebroid over E(2)_{∗} with E(2)_{∗}E(2) = E(2)_{∗}[t_{1}, t_{2},…]/(η_{R}(v_{i}) : i > 2). For an E(2)_{∗}E(2)-comodule M, Ext_{E(2)_{∗}E(2)}^{∗} (E(2)_{∗}, M) is the cohomology of the cobar complex Ω^{∗}M = Ω_{E(2)_{∗}E(2)}^{∗}M, and we will denote it by H^{∗}M.

The chromatic comodules N_{j}^{i} and M_{j}^{i} are defined inductively by N_{0}^{0} = E(2)_{∗}, N_{1}^{0} = E(2)_{∗}/(3), N_{2}^{0} = E(2)_{∗}/(3, v_{1}), M_{j}^{i} = v_{i+j}^{−1}N_{j}^{i} and the short exact sequence 0 → N_{j}^{i} → M_{j}^{i} → N_{j}^{i+1} → 0 for i + j + 1 ≤ 2 [1]. Note that N_{j}^{i} = M_{j}^{i} if i+j = 2. These have E(2)_{∗}E(2)-comodule structure induced from the right unit η_{R}: E(2)_{∗} → E(2)_{∗}E(2). Consider the long exact sequences associated to these short ones 0 → N_{0}^{0} → M_{0}^{0} → N_{0}^{1} → 0 and 0 → N_{0}^{1} → M_{0}^{1} →^{f} M_{0}^{2} → 0, whose connecting homomorphisms are δ′: H^{s}N_{0}^{1} → H^{s+1}E(2)_{∗} and δ: H^{s}M_{0}^{2} → H^{s}N_{0}^{1}. Then we see that δ′δ: H^{s}M_{0}^{2} → H^{s+2}E(2)_{∗} is an epimorphism if s ≥ 1, and an isomorphism if s > 1, since H^{s}M_{0}^{0} = 0 for s ≥ 1 and H^{s}M_{0}^{1} = 0 for s > 1 by [1]. In particular, Lemma 2.1. H^{s}E(2)_{∗} for s ≥ 3 consists of torsion elements.

We have a short exact sequence 0 → M_{1}^{1} →^{i} M_{0}^{2} →^{3} M_{0}^{2} → 0 (i(x) = x/3) which induces a long exact sequence

⋯ ⟶ H^{s−1}M_{0}^{2} ⟶^{δ} H^{s}M_{1}^{1} ⟶^{i_{∗}} H^{s}M_{0}^{2} ⟶^{3} H^{s}M_{0}^{2} ⟶^{δ} · · · .

An easy diagram chasing shows the following:

Lemma 2.2. ([1, Remark 3.11]) Consider the following commutative diagram of modules with horizontal exact sequences and a 3 torsion module B^{s}:

B^{s−1}  H^{s}M_{1}^{1}  B^{s}  B^{s}  H^{s+1}M_{1}^{1} H^{t−1}M_{0}^{2}  H^{t}M_{1}^{1}  H^{t}M_{0}^{2}  H^{t}M_{0}^{2}  H^{t+1}M_{1}^{1}^{δ} w_{_{f^{s−1}} u}  ^{i_{∗}} w_{_{g^{s}} u}  ^{3}  w_{_{f^{s}} u}  ^{δ} w_{_{f^{s}} u}  _{_{g^{s+1}} u}_{^{δ} w}  _{^{i_{∗}} w}  _{^{3} w}  _{^{δ} w}

If g^{s} and g^{s+1} are isomorphisms, then f^{s} is an epimorphism. Moreover, if f^{s−1} is an epimorphism, then f^{s} is an isomorphism.

Let b_{10} denote the element of H^{2}E(2)_{∗} represented by the cocycle −t_{1} ⊗ t_{1}^{2} − t_{1}^{2} ⊗ t_{1}. Then b_{10} acts on H^{∗}M for any comodule M. In [7], we show the following:

Proposition 2.3. The multiplication by b_{10} yields an isomorphism H^{s}M_{1}^{1} → H^{s+2}M_{1}^{1} if s > 3 and an epimorphism if s = 3.

This together with Lemma 2.2 implies

Corollary 2.4. The multiplication by b_{10} yields an isomorphism H^{s}M_{0}^{2} → H^{s+2}M_{0}^{2} if s > 3 and an epimorphism if s = 3.

Corollary 2.5. H^{∗}M_{0}^{2} ≅ (H^{4}M_{0}^{2}⨁H^{5}M_{0}^{2})⨂Z/3[b_{10}] for ∗ > 3.

3. Some formulae in Ω^{∗}E(2)_{∗}

The Adams-Novikov E_{2}-term for computing π_{∗}(L_{2}X) is the cohomology H^{∗}E(2)_{∗}(X) of the cobar complex Ω^{∗}E(2)_{∗}(X), and in particular, H^{∗}E(2)_{∗} = H^{∗}N_{0}^{0} is the E_{2}-term for π_{∗}(L_{2}S^{0}).

Take X to be the Toda-Smith spectrum V (1). Then E(2)_{∗}(V (1)) = K(2)_{∗} = Z/3[v_{2}^{±1}] and H^{∗}K(2)_{∗} = K(2)_{∗}[b_{10}]⨂F ⨂Λ(ζ_{2}). Here F denotes the module Z/3{1, h_{10}, h_{11}, b_{11}, ξ, ψ_{0}, ψ_{1}, b_{11}ξ} satisfying the following relations (cf. [6, Prop. 5.9]):

(3.1)  \begin{matrix}v_{2}^{2}h_{10}b_{10} = h_{11}b_{11}, v_{2}h_{11}b_{10} = −h_{10}b_{11} \\ b_{11}ξ = v_{2}h_{10}ψ_{1} = v_{2}h_{11}ψ_{0}, b_{10}ξ = −h_{10}ψ_{0} = v_{2}^{−1}h_{11}ψ_{1}, \\ v_{2}^{3}b_{10}^{2} = −b_{11}^{2}, b_{10}ψ_{1} = −v_{2}^{−1}b_{11}ψ_{0}, b_{10}ψ_{0} = v_{2}^{−2}b_{11}ψ_{1} \end{matrix}

If X is the mod 3 Moore spectrum V (0), then the E_{2}-term for V (0) is H^{∗}E(2)_{∗}/(3), which is determined in [7]. In particular, we see the following:

Lemma 3.2.

\begin{aligned}H^{2,0}E(2)_{∗}/(3) & = & 0 \\ H^{3,0}E(2)_{∗}/(3) & = & \{v_{2}^{−1}ψ_{0}, v_{2}^{−1}h_{10}b_{10}\} \\ H^{4,0}E(2)_{∗}/(3) & = & \{v_{2}^{−3}b_{10}b_{11}, v_{2}^{−1}ψ_{0}ζ_{2}, v_{2}^{−1}h_{10}b_{10}ζ_{2}\} \\ H^{5,0}E(2)_{∗}/(3) & = & \{v_{2}^{−3}b_{10}b_{11}ζ_{2}\}. \end{aligned}

Consider the long exact sequence

⋯ ⟶ H^{∗−1}E(2)_{∗}/(3) ⟶^{δ} H^{∗}E(2)_{∗} ⟶^{3} H^{∗}E(2)_{∗} ⟶^{j_{∗}} H^{∗}E(2)_{∗}/(3) ⟶^{δ} ⋯

associated to the short exact sequence 0 → E(2)_{∗} →^{3} E(2)_{∗} →^{j} E(2)_{∗}/(3) → 0.

Lemma 3.3. The map d_{1} = j_{∗}δ: H^{∗}E(2)_{∗}/(3) → H^{∗+1}E(2)_{∗}/(3) sends v_{2}^{−1}h_{10}b_{10} (resp. v_{2}^{−1}h_{10}b_{10}ζ_{2}) to v_{2}^{−3}b_{10}b_{11} (resp. v_{2}^{−3}b_{10}b_{11}ζ_{2}).

Proof. Note that v_{2}^{−1}h_{10}b_{10} is represented by a cochain whose leading term is v_{2}^{−3}t_{1}^{3} ⊗ b_{11}. Since d(t_{1}^{3^{i+1}}) = 3b_{1i} by definition, we compute

(3.4)  d(v_{2}^{−3}t_{1}^{3} ⊗ b_{11}) = 3(v_{2}^{−3}b_{10} ⊗ b_{11} + ⋯),

which shows δ(v_{2}^{−1}h_{10}b_{10}) = v_{2}^{−3}b_{10}b_{11}+· · ·. For v_{2}^{−1}h_{10}b_{10}ζ_{2}, the result follows immediately from (3.4) and Proposition 4.2 in the next section. q.e.d.

Let x denote a cochain that represents ξ. Then it is shown in [7, Lemma 4.4] that d(x) ≡ v_{1}^{2}f_{0} mod (3) for f_{0} that represents −v_{2}^{−1}ψ_{0} mod (3, v_{1}) (In [7], x is denoted by X(0)). So we have a cochain A such that

(3.5)  d(x) ≡ v_{1}^{2}f_{0} mod (3) and d(f_{0}) = 3A

in the cobar complex Ω^{∗}E(2)_{∗}. Then A is a cocycle of H^{4,0}E(2)_{∗}. Furthermore, we have

Lemma 3.6. d(f_{0}) = ±3f_{0} ⊗ z mod (9) in the cobar complex Ω^{4}E(2)_{∗}. Here z denotes a cocycle that represents the generator ζ_{2}.

Proof. The projection E(2)_{∗} → E(2)_{∗}/(3) sends A in (3.5) to a cocycle, which is also denoted by A. By virtue of Lemmas 3.2 and 3.3, we put

[A] = k_{1}v_{2}^{−1}h_{10}b_{10}ζ_{2} + k_{2}v_{2}^{−1}ψ_{0}ζ_{2},

where [A] denotes a cohomology class represented by A ∈ Ω^{∗}E(2)_{∗}/(3). In fact, f_{0} may be replaced by f_{0} + kv_{2}^{−3}t_{1}^{3} ⊗ b_{11} for some k ∈ Z/3 if necessary. Since d_{1}([A]) = 0 and d_{1}(v_{2}^{−1}ψ_{0}) = 3[A] by definition,

0 = k_{1}v_{2}^{−3}b_{10}b_{11}ζ_{2} + k_{2}(k_{1}v_{2}^{−1}h_{10}b_{10}ζ_{2} + k_{2}v_{2}^{−1}ψ_{0}ζ_{2})ζ_{2},

by Lemma 3.3. Noticing that ζ_{2}^{2} = 0 and v_{2}^{−3}b_{10}b_{11}ζ_{2} ≠ 0, we see that k_{1} = 0. On the other hand, if A represents 0, then we have an element v_{2}^{−1}ψ_{0} in H^{3,0}E(2)_{∗}. Since H^{2,0}E(2)_{∗}/(3) = 0 by Lemma 3.2, v_{2}^{−1}ψ_{0} generates a Z_{(3)}- free submodule in H^{3,0}E(2)_{∗}, which contradicts Lemma 2.1. Therefore A represents a non-zero element. This means that k_{2} = ±1. q.e.d.

Lemma 3.7. Put ṽ_{2}t_{1} = v_{2}t_{1} + v_{1}τ and ṽ_{2}^{2}t_{1}^{3} = v_{2}^{2}t_{1}^{3} + v_{1}v_{2}t_{1}^{6}. Then d(ṽ_{2}t_{1}) = v_{1}^{2}b_{10} and d(ṽ_{2}^{2}t_{1}^{3}) = −v_{1}^{2}b_{11}. Furthermore, there exists a cochain u ∈ Ω^{2}E(2)_{∗} such that

d(u) ≡ ṽ_{2}t_{1} ⊗ b_{11} + ṽ_{2}^{2}t_{1}^{3} ⊗ b_{10} mod (3, v_{1}^{2}).

Proof. The first statement is checked by a routine computation.

Turn to the second. We obtain an element u′ ∈ Ω^{2}E(2)_{∗} such that d(u′) ≡ v_{2}t_{1} ⊗ b_{11} + b_{10} ⊗ v_{2}^{2}t_{1}^{3} mod (3, v_{1}) from the relation v_{2}h_{11}b_{10} = −h_{10}b_{11} in H^{3}E(2)_{∗}/(3, v_{1}) of (3.1). Put d(u′) ≡ ṽ_{2}t_{1} ⊗ b_{11} + b_{10} ⊗ ṽ_{2}^{2}t_{1}^{3} + v_{1}w mod (3, v_{1}^{3}) for some cochain w. Sending this by d, we have 0 ≡ v_{1}^{2}b_{10} ⊗ b_{11} − b_{10} ⊗ v_{1}^{2}b_{11} + v_{1}d(w) mod (3, v_{1}^{3}). Then w ∈ H^{3,52}E(2)_{∗}/(3, v_{1}^{2}), which is 0 since H^{3,52}E(2)_{∗}/(3, v_{1}) = {v_{2}b_{11}ζ_{2}} by [6, Th. 5.8] and d(v_{2}b_{11}ζ_{2}) ≢ 0 mod (3, v_{1}^{2}) by [7, Lemma 3.3]. Therefore, we see that there is a cochain \overline{w} such that d(\overline{w}) ≡ w, and put u′′ = u′ − v_{1}\overline{w} to obtain d(u′′) ≡ ṽ_{2}t_{1} ⊗ b_{11} + b_{10} ⊗ ṽ_{2}^{2}t_{1}^{3} mod (3, v_{1}^{3}). There is also a cochain a such that d(a) ≡ ṽ_{2}^{2}t_{1}^{3} ⊗ b_{10} − b_{10} ⊗ ṽ_{2}^{2}t_{1}^{3} mod (3, v_{1}^{2}), and so we have the lemma by putting u = u′′ + a. q.e.d.

Lemma 3.8. There exists a cochain w such that

d(w) ≡ ṽ_{2}t_{1} ⊗ f_{0} − x′ ⊗ b_{10} mod (3, v_{1}^{3}).

Here x′ denotes a cocycle that represents ξ mod (3, v_{1}) and t_{1} ⊗ x′ is homologous to t_{1} ⊗ x mod (3, v_{1}^{3}).

Proof. This is shown in the same way as the above lemma. By the equation h_{10}ψ_{0} = −ξb_{10} in (3.1), we have a cochain w′ such that d(w′) ≡ v_{2}t_{1}⊗f_{0}−b_{10}⊗x mod (3, v_{1}). Put d(w′) ≡ ṽ_{2}t_{1}⊗f_{0}−b_{10}⊗x+v_{1}a for a cochain a, and send this by d. Then we see that a is a cocycle of Ω^{4,16}E(2)_{∗}/(3, v_{1}^{2}). Since we see that H^{4,16}E(2)_{∗}/(3, v_{1}^{2}) = {h_{10}b_{10}ζ_{2}} by [7], a = kt_{1} ⊗ z ⊗ b_{10} for some k ∈ Z/3. Furthermore, b_{10} ⊗ x is homologous to x ⊗ b_{10}, which yields a cochain w such that d(w) ≡ ṽ_{2}t_{1} ⊗ f_{0} − (x − kv_{1}t_{1} ⊗ z) ⊗ b_{10}. Now put x′ = x − kv_{1}t_{1} ⊗ z, and we have the lemma. q.e.d.

Lemma 3.9. In the cobar complex Ω^{∗}E(2)_{∗}, there exists a cochain y such that

d(y) ≡ t_{1} ⊗ x − v_{1}v_{2}^{−1}f_{1} − v_{1}z ⊗ x − kv_{1}v_{2}^{−2}t_{1} ⊗ b_{11} mod (3, v_{1}^{2})

for some k ∈ Z/3. Here f_{1} denotes a cocycle that represents ψ_{1}.

Proof. It is shown in [7, Lemma 6.4] that there exists a cochain Y_{0} such that d(Y_{0}) ≡ t_{1} ⊗ X + v_{1}v_{2}^{−3}τ^{3} ⊗ X + v_{1}^{2}v_{2}^{−1}t_{1}^{3} ⊗ X mod (3, v_{1}^{3}). It is also shown that x ≡ X + v_{1}v_{2}^{−1}Y_{1} + kv_{1}v_{2}^{−2}b_{11} for some k ∈ Z/3 in [7, Proof of Lemma 4.4.]. Take now y to be Y_{0}, and we obtain

d(y) ≡ t_{1} ⊗ (x − v_{1}v_{2}^{−1}Y_{1} − kv_{1}v_{2}^{−2}b_{11}) − v_{1}z ⊗ X + v_{1}v_{2}^{−1}t_{2} ⊗ X

mod (3, v_{1}^{2}). Since f_{1} = t_{1} ⊗ Y_{1} − t_{2} ⊗ X by the proof of [7, Lemma 4.4], we have the result. q.e.d.

4. The E_{2}-terms H^{s}M_{0}^{2} for s > 3.

Let E(2, 1)_{∗} denote Z/3[v_{1}, v_{2}^{±3}]. In [7], H^{∗}M_{1}^{1} is given as the direct sum of three E(2, 1)_{∗}-modules A_{i}:

H^{∗}M_{1}^{1} = A_{0}⨁A_{1}⨁A_{2}.

In order to describe the modules A_{i}, we use the following notation:

k(1)_{∗} = Z/3[v_{1}]
K(1)_{∗} = Z/3[v_{1}^{±1}]
P E = Z/3[b_{10}]⨂Λ(ζ_{2})
E(2, n)_{∗} = Z/3[v_{1}, v_{2}^{±3^{n}}]
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}}
F_{n} = E(2, 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}}}.

Then the modules A_{i} are given as follows:

A_{0} = (K(1)_{∗}/k(1)_{∗})⨂Λ(h_{10}, ζ_{2})
A_{1} = ∑_{n≥0} F_{n}⨂Λ(ζ_{2})
A_{2} = (F_{(h)}⨁F_{(t)}⨁F_{(h)}^{∗}⨁F_{(t)}^{∗})⨂P E,

Consider the exact sequence H^{1,0}M_{1}^{1} →^{δ} H^{2,0}E(2)_{∗}/(3, v_{1}^{3^{i}}) → H^{2,−4·3^{i}}M_{1}^{1}

associated to the short exact sequence 0 → E(2)_{∗}/(3, v_{1}^{3^{i}}) →^{1/v{3^{i} \atop 1}} M_{1}^{1} →^{v{3^{i} \atop 1}} M_{1}^{1} → 0. Then the structure of H^{∗}M_{1}^{1} shows immediately the following:

Lemma 4.1. For each i > 0, each element of H^{2,0}E(2)_{∗}/(3, v_{1}^{3^{i}}) is divisible by v_{1}^{3^{i}−3^{i−1}}.

Now we obtain the next proposition corresponding to [5, Lemma 2.6]. In fact, the proof of [5, Lemma 2.6] also works at the prime 3 by the above lemma.

Proposition 4.2. For each integer i > 0, there exists a cocycle z_{i} of Ω^{1,0}E(2)_{∗}/(3^{i+1}, v_{1}^{3^{i}}) such that z_{i} = z ∈ Ω^{1}K(2)_{∗}.

By virtue of this proposition, we abuse the notation z for a cocycle that represents ζ_{2} as we did in the previous papers for a prime > 3.

Consider the connecting homomorphism δ: H^{s}M_{0}^{2} → H^{s+1}M_{1}^{1} associated to the short exact sequence 0 → M_{1}^{1} →^{i} M_{0}^{2} →^{3} M_{0}^{2} → 0.

Lemma 4.3. δ(v_{2}^{2}ξ/3v_{1}^{3}) = v_{2}ψ_{0}/v_{1}.

Proof. In [7, Lemma 4.4], it is shown that there exists a cochain X(2) such that d(X(2)) ≡ v_{1}^{4}z^{3}⊗X^{3}−v_{1}^{4}v_{2}^{−3}f_{1}^{3} mod (3, v_{1}^{5}). Since H^{3,20}M_{2}^{0} = 0, the congruence holds mod (3, v_{1}^{6}) if we replace X(2) by a suitable cochain x′′. Put now d(x′′) ≡ v_{1}^{4}z^{3}⊗X^{3}−v_{1}^{4}v_{2}^{−3}f_{1}^{3}+3A mod (9, v_{1}^{6}). Then 0 ≡ 3v_{1}^{3}t_{1}⊗z^{3}⊗X^{3}− 3v_{1}^{3}t_{1} ⊗ v_{2}^{−3}f_{1}^{3} + 3d(A) mod (9, v_{1}^{4}). This implies that A ∈ H^{3,40}E(2)_{∗}/(3, v_{1}^{3}), which is Z/3{v_{1}^{2}v_{2}ψ_{0}} by [7]. Note that 3v_{1}^{3}t_{1} ⊗ z^{3} ⊗ X^{3} and v_{1}^{3}t_{1} ⊗ v_{2}^{−3}f_{1}^{3} are homologous to zero and v_{1}^{3}b_{11} ⊗ X mod (9, v_{1}^{4}), respectively, by Lemma 3.9 and (3.1), and that d(−v_{1}^{2}v_{2}(v_{2}f_{0})) ≡ v_{1}^{3}v_{2}t_{1}^{3} ⊗ f_{0} + v_{1}^{3}b_{11} ⊗ X mod (3, v_{1}^{4}) by [7, Lemma 4.3] whose right hand side is homologous to −v_{1}^{3}b_{11} ⊗ X by (3.1). Thus d(A) = d(−v_{1}^{2}v_{2}^{2}f_{0} + ⋯), and we see that A is homologous to −v_{1}^{2}v_{2}^{2}f_{0}, and that d(x′′) ≡ −3v_{1}^{2}v_{2}^{2}f_{0} mod (9, v_{1}^{3}). Since x′′ and f_{0} represent v_{2}^{2}ξ and −v_{2}^{−1}ψ_{0}, respectively, we have the lemma. q.e.d.

Put

G^{s} = (i_{∗}(F_{(h)}⨁F_{(h)}^{∗})⨂E(2, 1)_{∗}[b_{10}]⨂Λ(ζ_{2}))^{s} ⊂ H^{s}M_{0}^{2}

for i_{∗}: H^{∗}M_{1}^{1} → H^{∗}M_{0}^{2} given by i_{∗}(x) = x/3.

Lemma 4.4. For the connecting homomorphism δ: H^{s}M_{0}^{2} → H^{s+1}M_{1}^{1}, δ(x) for a generator x of G^{s} is obtained by the following equations:

δ(v_{2}/3v_{1}) = −v_{2}h_{10}/v_{1}^{2},
δ(v_{2}h_{10}/3v_{1}) = v_{2}^{−1}b_{11}/v_{1} + v_{2}h_{10}ζ_{2}/v_{1},
δ(v_{2}^{2}h_{11}/3v_{1}) = v_{2}^{2}b_{10}/v_{1} + v_{2}^{2}h_{11}ζ_{2}/v_{1},
δ(v_{2}b_{11}/3v_{1}) = v_{2}^{2}h_{11}b_{10}/v_{1}^{2};
δ(v_{2}(v_{2}^{−1}ψ_{0})/3v_{1}) = ξb_{10}/v_{1}^{2} ± v_{2}(v_{2}^{−1}ψ_{0})ζ_{2}/v_{1},
δ(ξ/3v_{1}) = v_{2}^{−1}ψ_{1}/v_{1} + (1 ± 1)ξζ_{2}/v_{1} + kv_{2}^{−1}h_{11}b_{10}/v_{1}
δ(b_{11}ξ/3v_{1}) = v_{2}^{2}(v_{2}^{−1}ψ_{0})b_{10}/v_{1} + (1 ± 1)b_{11}ξζ_{2}/v_{1} + kv_{2}h_{10}b_{10}^{2}/v_{1},
δ(v_{2}ψ_{1}/3v_{1}) = b_{11}ξ/v_{1}^{2} ± v_{2}ψ_{1}ζ_{2}/v_{1}.

Proof. Note that v_{2}h_{10} is represented by a cochain ṽ_{2}t_{1} = η_{R}(v_{2})t_{1}−v_{1}t_{2} of Lemma 3.7. In the cobar complex Ω^{∗}E(2)_{∗}/(9, v_{1}^{3}), we compute

(4.5)  \begin{aligned}d(v_{1}^{2}v_{2}) & = & 6v_{1}t_{1}η_{R}(v_{2}) + 3v_{1}^{2}t_{2} = 6v_{1}ṽ_{2}t_{1}; \\ d(v_{1}^{2}v_{2}t_{1}) & = & 6v_{1}t_{1} ⊗ v_{2}t_{1} + 3v_{1}^{2}t_{2} ⊗ t_{1} \\ & = & \underline{6v_{1}v_{2}t_{1} ⊗ t_{1}}_{1} + \underline{6v_{1}^{2}τ ⊗ t_{1}}_{2} \\ d(3v_{1}v_{2}t_{1}^{2}) & = & \underline{3v_{1}^{2}t_{1}^{3} ⊗ t_{1}^{2}}_{3} − \underline{6v_{1}v_{2}t_{1} ⊗ t_{1}}_{1} \\ d(3v_{1}^{2}t_{1}τ) & = & −3v_{1}^{2}(\underline{t_{1}^{4} ⊗ t_{1}}_{2} + \underline{t_{1}^{3} ⊗ t_{1}^{2}}_{3} + t_{1} ⊗ τ + \underline{τ ⊗ t_{1}}_{2}) \\ d(−3v_{1}^{2}v_{2}^{−2}t_{3}) & = & 3v_{1}^{2}v_{2}^{−2}(t_{1} ⊗ t_{2}^{3} + \underline{t_{2} ⊗ t_{1}^{9}}_{2} + v_{2}b_{11}); \\ d(v_{1}^{2}v_{2}^{2}t_{1}^{3}) & = & 6v_{1}t_{1} ⊗ v_{2}^{2}t_{1}^{3} + 6v_{1}^{2}v_{2}t_{2} ⊗ t_{1}^{3} + 3v_{1}^{2}v_{2}^{2}b_{10} \\ & = & \underline{6v_{1}v_{2}^{2}t_{1} ⊗ t_{1}^{3}}_{4} + \underline{12v_{1}^{2}v_{2}t_{1}^{4} ⊗ t_{1}^{3}}_{7} \\ & & +\underline{6v_{1}^{2}v_{2}t_{2} ⊗ t_{1}^{3}}_{7} + \underline{3v_{1}^{2}v_{2}^{2}b_{10}}_{6} \\ d(6v_{1}v_{2}^{2}t_{2}) & = & \underline{12v_{1}^{2}v_{2}t_{1}^{3} ⊗ t_{2}}_{5} − \underline{6v_{1}v_{2}^{2}t_{1} ⊗ t_{1}^{3}}_{4} − \underline{6v_{1}^{2}v_{2}^{2}b_{10}}_{6} \\ d(3v_{1}^{2}v_{2}^{−7}t_{3}^{3}) & = & −3v_{1}^{2}v_{2}^{−7}(t_{1}^{3} ⊗ t_{2}^{9} + t_{2}^{3} ⊗ t_{1}^{27} + v_{2}^{3}b_{11}^{3}) \\ & = & −3v_{1}^{2}v_{2}^{−7}(\underline{v_{2}^{8}t_{1}^{3} ⊗ t_{2}}_{5} + \underline{v_{2}^{−6}t_{2}^{3} ⊗ t_{1}^{3}}_{7} + \underline{v_{2}^{9}b_{10})}_{6}. \end{aligned}

The underlined terms with the same number sum up to zero except for the terms numbered 6 and 7. The terms numbered 6 and 7 sum up to 3v_{1}^{2}v_{2}^{2}b_{10} and 6v_{1}^{2}v_{2}^{2}ζ_{2} ⊗ t_{1}^{3}, respectively. These imply the first three equations. In fact, δ([a/3v_{1}]) = [(i_{∗})^{−1}d(v_{1}^{2}a)/9v_{1}^{3}], where [a] denotes a homology class represented by a. Since δ(b_{11}a) = b_{11}δ(a), the first equation gives δ(v_{2}b_{11}/3v_{1}) = −v_{2}h_{10}b_{11}/v_{1}^{2}, which equals v_{2}^{2}h_{11}b_{10}/v_{1}^{2} by Lemma 3.7. Thus we have the fourth equation.

By Lemma 3.6 and the first equation in (4.5), we compute

(4.6)  d(v_{1}^{2}v_{2}f_{0}) ≡ 6v_{1}ṽ_{2}t_{1} ⊗ f_{0} ± 3v_{1}^{2}v_{2}f_{0} ⊗ z mod (9, v_{1}^{3}).

Now by Lemma 3.8, we have the fifth equation. Multiplying h_{10} to the fifth equation yields δ(h_{10}ψ_{0}/3v_{1}) = h_{10}ξb_{10}/v_{1}^{2} + h_{10}ψ_{0}ζ_{2}/v_{1}. Lemma 3.9 says that h_{10}ξ = v_{1}v_{2}^{−1}ψ_{1} + v_{1}ξζ_{2} + kv_{1}v_{2}^{−2}h_{10}b_{11} for some k ∈ Z/3. Since h_{10}ψ_{0} = −ξb_{10}, we have the sixth equation. The seventh follows immediately from the product of b_{11} and the sixth equation. The multiplication of b_{10} and fifth equation gives the last one by the relations (3.1). Here note that b_{10}^{2} = −v_{2}^{−3}b_{11}^{2} holds in H^{4}E(2)_{∗}/(3, v_{1}^{2}). q.e.d.

Proposition 4.7. H^{s}M_{0}^{2} = G^{s} for s = 4, 5. In other words, H^{4}M_{0}^{2} and H^{5}M_{0}^{2} are Z/3[v_{2}^{±3}]-modules generated by

G^{4} : \begin{matrix}v_{2}b_{10}^{2}/3v_{1}, v_{2}h_{10}b_{10}ζ_{2}/3v_{1}, v_{2}^{2}h_{11}b_{10}ζ_{2}/3v_{1}, v_{2}b_{11}b_{10}/3v_{1}; \\ ψ_{0}ζ_{2}/3v_{1}, ξb_{10}/3v_{1}, b_{11}ξ/3v_{1}, v_{2}ψ_{1}ζ_{2}/3v_{1}, \end{matrix}

and

G^{5} : \begin{matrix}v_{2}b_{10}^{2}ζ_{1}/3v_{1}, v_{2}h_{10}b_{10}^{2}/3v_{1}, v_{2}^{2}h_{11}b_{10}^{2}/3v_{1}, v_{2}b_{11}b_{10}ζ_{2}/3v_{1}; \\ ψ_{0}b_{10}/3v_{1}, ξb_{10}ζ_{2}/3v_{1}, b_{11}ξζ_{2}/3v_{1}, v_{2}ψ_{1}b_{10}/3v_{1}, \end{matrix}

respectively.

Proof. Put B^{s} = G^{s}. Then there is a canonical map f^{s}: B^{s} → H^{s}M_{0}^{2} sitting in the commutative diagram

H^{s−1}M_{0}^{2}  H^{s}M_{1}^{1}  B^{s}  B^{s}  H^{s+1}M_{1}^{1} H^{s−1}M_{0}^{2}  H^{s}M_{1}^{1}  H^{s}M_{0}^{2}  H^{s}M_{0}^{2}  H^{s+1}M_{1}^{1}.^{δ} w  ^{i_{∗}} w  ^{3}  w_{_{f^{s}} u}  ^{δ} w_{_{f^{s}} u}_{^{δ} w}  _{^{i_{∗}} w}  _{^{3} w}  _{^{δ} w}

Lemma 4.4 implies that the δ-images of the generators of B^{s} are linearly independent. Therefore we see that the above sequence is exact, and Lemma 2.2 shows that f^{s} is an isomorphism. q.e.d.

5. On the E_{2}-terms H^{s}M_{0}^{2} for s ≤ 3 We write down the submodules A_{2}^{s} ⊂ H^{s}M_{1}^{1}:

A_{2}^{0} = Z/3{v_{2}/v_{1}, v_{2}^{−1}/v_{1}}
A_{2}^{1} = Z/3{v_{2}h_{10}/v_{1}^{2}, v_{2}h_{10}/v_{1}, v_{2}^{2}h_{11}/v_{1}^{2}, v_{2}^{2}h_{11}/v_{1},

v_{2}ζ_{2}/v_{1}, v_{2}^{−1}ζ_{2}/v_{1}}

A_{2}^{2} = Z/3{v_{2}b_{11}/v_{1}, v_{2}^{−1}b_{11}/v_{1}, v_{2}h_{10}ζ_{2}/v_{1}^{2}, v_{2}h_{10}ζ_{2}/v_{1},

v_{2}^{2}h_{11}ζ_{2}/v_{1}^{2}, v_{2}^{2}h_{11}ζ_{2}/v_{1}, v_{2}b_{10}/v_{1}, v_{2}^{−1}b_{10}/v_{1}, ξ/v_{1}^{2}, ξ/v_{1}}

A_{2}^{3} = Z/3{v_{2}b_{11}ζ_{2}/v_{1}, v_{2}^{−1}b_{11}ζ_{2}/v_{1}, v_{2}h_{10}b_{10}/v_{1}^{2}, v_{2}h_{10}b_{10}/v_{1},

v_{2}^{2}h_{11}b_{10}/v_{1}^{2}, v_{2}^{2}h_{11}b_{10}/v_{1}, v_{2}b_{10}ζ_{2}/v_{1}, v_{2}^{−1}b_{10}ζ_{2}/v_{1}, ψ_{0}/v_{1}, v_{2}ψ_{0}/v_{1}, v_{2}ψ_{1}/v_{1}, v_{2}^{−1}ψ_{1}/v_{1}, ξζ_{2}/v_{1}^{2}, ξζ_{2}/v_{1}}.

Now consider the map d_{1} = δi_{∗}: H^{s}M_{1}^{1} → H^{s+1}M_{1}^{1}. Then [1, Prop. 6.9] shows

(5.1)  d_{1}(v_{2}^{3}/v_{1}^{3}) = v_{2}^{2}h_{11}/v_{1}^{2}.

Here we compute:

Lemma 5.2. The Bockstein differential d_{1} = δi_{∗} acts up to sign as follows:

\begin{aligned}d_{1}(v_{2}/v_{1}) & = & v_{2}h_{10}/v_{1}^{2}, \\ d_{1}(v_{2}^{−1}/v_{1}) & = & v_{2}^{−1}ζ_{2}/v_{1}; \\ d_{1}(v_{2}h_{10}/v_{1}) & = & v_{2}^{−1}b_{11}/v_{1} + v_{2}h_{10}ζ_{2}/v_{1}, \\ d_{1}(v_{2}^{2}h_{11}/v_{1}) & = & v_{2}^{−1}b_{10}/v_{1} + v_{2}^{−1}h_{11}ζ_{2}/v_{1}, \\ d_{1}(v_{2}ζ_{2}/v_{1}) & = & v_{2}h_{10}ζ_{2}/v_{1}^{2}; \\ d_{1}(v_{2}b_{11}/v_{1}) & = & v_{2}^{2}h_{11}b_{10}/v_{1}^{2}, \\ d_{1}(v_{2}h_{10}ζ_{2}/v_{1}) & = & v_{2}^{−1}b_{11}ζ_{2}/v_{1}^{2}, \\ d_{1}(v_{2}^{2}h_{11}ζ_{2}/v_{1}) & = & v_{2}^{2}b_{10}ζ_{2}/v_{1}, \\ d_{1}(v_{2}b_{10}/v_{1}) & = & v_{2}h_{10}b_{10}/v_{1}^{2}, \\ d_{1}(ξ/v_{1}^{2}) & = & ξζ_{2}/v_{1}^{2}, \\ d_{1}(ξ/v_{1}) & = & v_{2}^{−1}ψ_{1}/v_{1} + ξζ_{2}/v_{1}; \\ d_{1}(v_{2}b_{11}ζ_{2}/v_{1}) & = & v_{2}^{2}h_{11}b_{10}ζ_{2}/v_{1}^{2}, \\ d_{1}(v_{2}h_{10}b_{10}/v_{1}) & = & v_{2}^{−1}b_{11}b_{10}/v_{1}^{2} + v_{2}h_{10}b_{10}ζ_{2}/v_{1}, \\ d_{1}(v_{2}^{2}h_{11}b_{10}/v_{1}) & = & v_{2}^{−1}b_{10}^{2}/v_{1} + v_{2}^{−1}h_{11}b_{10}ζ_{2}/v_{1}, \\ d_{1}(v_{2}b_{10}ζ_{2}/v_{1}) & = & v_{2}h_{10}b_{10}ζ_{2}/v_{1}^{2}; \\ d_{1}(ξζ_{2}/v_{1}) & = & v_{2}^{−1}ψ_{1}ζ_{2}/v_{1}, \\ d_{1}(ψ_{0}/v_{1}) & = & ξb_{10}/v_{1}^{2} + ψ_{0}ζ_{2}/v_{1}, \\ d_{1}(v_{2}ψ_{1}/v_{1}) & = & b_{11}ξ/v_{1}^{2}. \end{aligned}

The other elements of A_{2} missing in the left hand sides are in the image of d_{1}.

Proof. Lemma 4.3 and (5.1) show that v_{2}^{−2}ψ_{0}/v_{1} and v_{2}^{2}h_{11}/v_{1}^{2} are in the image of d_{1}. The other parts follow from Lemma 4.4, except for d_{1} on v_{2}^{−1}/v_{1} and ξ/v_{1}^{2}.

For the exceptional cases, consider the diagram

H^{s−1}M_{0}^{2}  H^{s}M_{1}^{1}  H^{s}M_{0}^{2}  H^{s}M_{0}^{2}  H^{s+1}M_{1}^{1} H^{s+1}M_{0}^{2}  H^{s+2}M_{1}^{1}  H^{s+2}M_{0}^{2}  H^{s+2}M_{0}^{2}  H^{s+3}M_{1}^{1}.^{δ} w_{_{b_{10}} u}  ^{i_{∗}} w_{_{b_{10}} u}  ^{3} w_{_{b_{10}} u}  ^{δ} w_{_{b_{10}} u}  _{_{b_{10}} u}_{^{δ} w}  _{^{i_{∗}} w}  _{^{3} w}  _{^{δ} w}

If we have a relation δ(α/3) = βb_{10} + αζ_{2} in Lemma 4.4, then we see that β/3 = −αζ_{2}/3 in H^{∗}M_{0}^{2}, since i_{∗}(x) = x/3. Therefore, we compute

b_{10}δ(β/3) = δ(βb_{10}/3) = −δ(αζ_{2}/3) = −δ(α/3)ζ_{2} = −βb_{10}ζ_{2},

and so we obtain

δ(β/3) = −βζ_{2}

up to Ker b_{10}. Note that b_{10} acts monomorphically on A_{2}. Now take β to be the exceptional cases, and we have all d_{1}. q.e.d.

Hence, we have

Proposition 5.3. H^{s}M_{0}^{2} contains E(2, 1)_{∗}/(3, v_{1})-module as follows:

H^{0}M_{0}^{2} ⊃ E(2, 1)_{∗}{v_{2}^{±1}/3v_{1}}
H^{1}M_{0}^{2} ⊃ E(2, 1)_{∗}{v_{2}h_{10}/3v_{1}, v_{2}^{2}h_{11}/3v_{1}, v_{2}ζ_{2}/3v_{1}}
H^{2}M_{0}^{2} ⊃ E(2, 1)_{∗}{v_{2}b_{11}/3v_{1}, v_{2}h_{10}ζ_{2}/3v_{1}, v_{2}^{2}h_{11}ζ_{2}/3v_{1}, v_{2}b_{10}/3v_{1}
ξ/3v_{1}^{2}, ξ/3v_{1}}
H^{3}M_{0}^{2} ⊃ E(2, 1)_{∗}{v_{2}b_{11}ζ_{2}/3v_{1}, v_{2}h_{10}b_{10}/3v_{1}, v_{2}^{2}h_{11}b_{10}/3v_{1}, v_{2}b_{10}ζ_{2}/3v_{1}
ξζ_{2}/3v_{1}, ψ_{0}/3v_{1}, v_{2}ψ_{1}/3v_{1}}.

6. The Adams-Novikov differentials

Now consider spectra defined by cofiber sequences:

(6.1) S^{0} → p^{−1}S^{0} → N^{1}, N^{1} → L_{1}N^{1} → N^{2}, V (0) → v_{1}^{−1}V (0) → W,

and M^{2} = L_{2}N^{2}. The Adams-Novikov differentials on π_{∗}(L_{2}W) is determined in [7]. Let ι: L_{2}W → M^{2} denote the canonical map that induces i: M_{1}^{1} → M_{0}^{2}. Suppose that d_{r}(x) = y in the E_{r}-term for L_{2}W. Then d_{r}(x/3) = d_{r}(i_{∗}x) = i_{∗}y = y/3. In this way, we determine the differentials except for d_{9}(v_{2}^{−1}/3v_{1}) and d_{5}(v_{2}^{3t}ξ/3v_{1}^{2}).

Lemma 6.2. The Adams-Novikov differential d_{r} is given (up to sign) by

d_{r}(v_{2}/3v_{1}) = 0, d_{5}(v_{2}^{4}/3v_{1}) = v_{2}^{2}h_{11}b_{10}^{2}/3v_{1}, d_{5}(v_{2}^{7}/3v_{1}) = v_{2}^{5}h_{11}b_{10}^{2}/3v_{1}, d_{r}(v_{2}^{2}/3v_{1}) = 0, d_{r}(v_{2}^{5}/3v_{1}) = 0, d_{9}(v_{2}^{−1}/3v_{1}) = v_{2}^{−5}b_{11}b_{10}^{3}ζ_{2}/3v_{1}, d_{r}(v_{2}h_{10}/3v_{1}) = 0, d_{9}(v_{2}^{4}h_{10}/3v_{1}) = v_{2}b_{10}^{5}/3v_{1}, d_{r}(v_{2}^{7}h_{10}/3v_{1}) = 0, d_{r}(v_{2}^{2}h_{11}/3v_{1}) = 0, d_{r}(v_{2}^{5}h_{11}/3v_{1}) = 0, d_{9}(v_{2}^{8}h_{11}/3v_{1}) = v_{2}^{4}b_{11}b_{10}^{4}/3v_{1},

d_{5}(v_{2}b_{11}/3v_{1}) = v_{2}h_{10}b_{10}^{3}/3v_{1}, d_{r}(v_{2}^{4}b_{11}/3v_{1}) = 0, d_{5}(v_{2}^{7}b_{11}/3v_{1}) = v_{2}^{7}h_{10}b_{10}^{3}/3v_{1}, d_{5}(ξ/3v_{1}^{2}) = v_{2}^{−3}b_{11}ξb_{10}ζ_{2}/3v_{1}, d_{r}(v_{2}^{3}ξ/3v_{1}^{2}) = 0, d_{5}(v_{2}^{6}ξ/3v_{1}^{2}) = v_{2}^{3}b_{11}ξb_{10}ζ_{2}/3v_{1},

d_{r}(ξ/3v_{1}) = 0, d_{r}(v_{2}^{3}ξ/3v_{1}) = 0, d_{9}(v_{2}^{6}ξ/3v_{1}) = v_{2}^{3}ψ_{0}b_{10}^{4}/3v_{1}, d_{5}(ψ_{0}/3v_{1}) = v_{2}^{−3}b_{11}ξb_{10}^{2}/3v_{1}, d_{r}(v_{2}^{3}ψ_{0}) = 0, d_{5}(v_{2}^{6}ψ_{0}/3v_{1}) = v_{2}^{3}b_{11}ξb_{10}^{2}/3v_{1}, d_{5}(v_{2}ψ_{1}/3v_{1}) = ξb_{10}^{3}/3v_{1}, d_{5}(v_{2}^{4}ψ_{1}/3v_{1}) = v_{2}^{3}ξb_{10}^{3}/3v_{1}, d_{r}(v_{2}^{7}ψ_{1}/3v_{1}) = 0. d_{9}(b_{11}ξ/3v_{1}) = v_{2}^{−2}ψ_{1}b_{10}^{5}/3v_{1}, d_{r}(v_{2}^{3}b_{11}ξ/3v_{1}) = 0, d_{r}(v_{2}^{6}b_{11}ξ/3v_{1}) = 0.

Proof. Here we show the exceptional cases. Lemma 4.4 shows

v_{2}^{−1}b_{10}/3v_{1} = −v_{2}^{−1}h_{11}ζ_{2}/3v_{1} and v_{2}^{3t}ξb_{10}/3v_{1}^{2} = ±v_{2}^{3t}ψ_{0}ζ_{2}/3v_{1}.

Now we compute

b_{10}d_{9}(v_{2}^{−1}/3v_{1}) = d_{9}(v_{2}^{−1}b_{10}/3v_{1}) = −d_{9}(v_{2}^{−1}h_{11}ζ_{2}/3v_{1}) = v_{2}^{−5}b_{11}b_{10}^{4}ζ_{2}/3v_{1},

and we have d_{9}(v_{2}^{−1}/3v_{1}) = v_{2}^{−5}b_{11}b_{10}^{3}/3v_{1} as desired. In the same way, we have the other case. q.e.d.

Now we display the chart of the Adams-Novikov spectral sequence:

20 0  12v_{2}  v_{2}  v_{2}  v_{2} v_{2}  v_{2}  v_{2}  v_{2}  v_{2} [figure]  [figure]  [figure]  [figure]  [figure] v_{2}  v_{2}  v_{2}  v_{2}  v_{2} [figure]  [figure]  [figure]  [figure]  [figure] 28v_{2} 15h_{0}  h_{0}  h_{0}  h_{0} h_{0}  h_{0}  h_{0}  h_{0}  h_{0} h_{0}  h_{0}  h_{0}  h_{0}  h_{0} [figure]  [figure]  [figure]  [figure]  [figure] 39h_{1}  h_{1} h_{1}  h_{1}  h_{1}  h_{1}  h_{1} [figure]  [figure]  [figure]  [figure]  [figure] h_{1}  h_{1}  h_{1}  h_{1}  h_{1} 46 b_{1} [figure] b_{1}  b_{1}  b_{1}  b_{1}  b_{1} [figure]  [figure]  [figure]  [figure]  [figure] b_{1}  b_{1}  b_{1}  b_{1}  b_{1}  ξ ξ  ξ  ξ  ξ  ξ ξ  ξ  ξ  ξ  ξ [figure]  [figure]  [figure]  [figure]  [figure] ξ  ξ  ξ ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0} [figure]  [figure]  [figure]  [figure] ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0} [figure]  [figure]  [figure]  [figure]  [figure] ψ_{0}  ψ_{0}  ψ_{0}  33ψ_{1}  ψ_{1} [figure]  [figure] ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1} ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1} [figure]  [figure]  [figure]  [figure]  [figure] 36bξ  bξ [figure]  [figure] bξ  bξ  bξ  bξ  bξ bξ  bξ  bξ  bξ  bξ

20 0  60v_{2}  v_{2}  v_{2}  v_{2} [figure]  [figure]  [figure]  [figure] v_{2}  v_{2}  v_{2}  v_{2}  v_{2} v_{2}  v_{2}  v_{2}  v_{2}  v_{2} [figure]  [figure]  [figure]  [figure]  [figure] 76v_{2} 63h_{0}  h_{0}  h_{0}  h_{0} [figure]  [figure]  [figure]  [figure] h_{0}  h_{0}  h_{0}  h_{0}  h_{0} h_{0}  h_{0}  h_{0}  h_{0}  h_{0} 87h_{1}  h_{1} h_{1}  h_{1}  h_{1}  h_{1}  h_{1} h_{1}  h_{1}  h_{1}  h_{1}  h_{1} [figure]  [figure]  [figure]  [figure]  [figure] 94 b_{1} b_{1}  b_{1}  b_{1}  b_{1}  b_{1} [figure]  [figure]  [figure]  [figure]  [figure] b_{1}  b_{1}  b_{1}  b_{1}  b_{1} [figure]  [figure]  [figure]  [figure]  [figure]  ξ [figure]^{ζ_{2}} ξ  ξ  ξ  ξ  ξ ξ  ξ  ξ  ξ  ξ ξ  ξ  ξ [figure]  [figure]  [figure] ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0} ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0} [figure]  [figure]  [figure]  [figure]  [figure] ψ_{0}  ψ_{0}  ψ_{0} [figure]  [figure]  [figure]  81ψ_{1}  ψ_{1} [figure]  [figure] ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1} [figure]  [figure]  [figure]  [figure]  [figure] ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1} 84bξ  bξ bξ  bξ  bξ  bξ  bξ [figure]  [figure]  [figure]  [figure]  [figure] bξ  bξ  bξ  bξ  bξ^{20 0  108 v_{2}  v_{2}  v_{2}  v_{2} [figure]  [figure]  [figure]  [figure] v_{2}  v_{2}  v_{2}  v_{2}  v_{2} [figure]  [figure]  [figure]  [figure]  [figure] v_{2}  v_{2}  v_{2}  v_{2}  v_{2} 124 v_{2} [figure] d_{9} : ζ_{2} 111 h_{0}  h_{0}  h_{0}  h_{0} h_{0}  h_{0}  h_{0}  h_{0}  h_{0} [figure]  [figure]  [figure]  [figure]  [figure] h_{0}  h_{0}  h_{0}  h_{0}  h_{0} 135 h_{1}  h_{1} [figure]  [figure] h_{1}  h_{1}  h_{1}  h_{1}  h_{1} h_{1}  h_{1}  h_{1}  h_{1}  h_{1} 142 b_{1} [figure] b_{1}  b_{1}  b_{1}  b_{1}  b_{1} b_{1}  b_{1}  b_{1}  b_{1}  b_{1} [figure]  [figure]  [figure]  [figure]  [figure]  ξ [figure]^{ξ : ζ_{2}} ξ  ξ  ξ  ξ  ξ [figure]  [figure]  [figure]  [figure]  [figure] ξ  ξ  ξ  ξ  ξ ξ  ξ  ξ ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0} [figure]  [figure]  [figure]  [figure] ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0}  ψ_{0} ψ_{0}  ψ_{0}  ψ_{0} [figure]  [figure]  [figure]  129 ψ_{1}  ψ_{1} ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1} [figure]  [figure]  [figure]  [figure]  [figure] ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1}  ψ_{1} [figure]  [figure]  [figure]  [figure]  [figure] 132 bξ  bξ bξ  bξ  bξ  bξ  bξ bξ  bξ  bξ  bξ  bξ [figure]  [figure]  [figure]  [figure]  [figure]}

17

Some of survivors are killed by other differentials derived from [7]: (6.3)

\begin{aligned}d_{5}(v_{2}^{9t+3}/3v_{1}^{3}) & = & v_{2}^{9t+1}h_{10}b_{10}^{2}/3v_{1}, \\ d_{5}(v_{2}^{9t−1}h_{11}/9v_{1}^{2}) & = & v_{2}^{9t−2}h_{10}b_{10}^{2}ζ_{2}/3v_{1}, \\ d_{5}(v_{2}^{3^{n+2}t+3^{n+1}}h_{10}/3v_{1}^{2·3^{n+1}+1}) & = & ±v_{2}^{3^{n+2}t+3(3^{n}−1)/2}ξb_{10}^{2}/3v_{1} (n ≥ 0), \\ d_{5}(v_{2}^{3^{n+2}t+8·3^{n}}h_{10}/3v_{1}^{10·3^{n}+1}) & = & −v_{2}^{3^{n+2}t+5·3^{n}+3(3^{n−1}−1)/2}ξb_{10}^{2}/3v_{1} (n > 1). \end{aligned}

This shows that,

Theorem 6.4. The E_{∞}-term of π_{∗}(M^{2}) contains the module G̃⨁G̃^{∗}⨁G̃Z⨁G̃Z^{∗}. Here E(2, 1)_{∗}-modules are given as follows:

G̃ = B_{5}(2, 2)_{∗}{v_{2}/3v_{1}}⨁B_{4}(2, 2)_{∗}{v_{2}^{4}b_{11}/3v_{1}}
⨁B_{3}(2, 2)_{∗}{v_{2}^{7}h_{10}/3v_{1}}
⨁B_{2}(2, 2)_{∗}{v_{2}h_{10}/3v_{1}, v_{2}^{2}h_{11}/3v_{1}, v_{2}^{5}h_{11}/3v_{1}},
G̃^{∗} = B_{5}(2, 2)_{∗}{v_{2}^{7}ψ_{1}/3v_{1}}⨁B_{4}(2, 2)_{∗}{v_{2}^{3}ψ_{0}/3v_{1}}
⨁B_{2}(2, 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)}),
G̃Z = B_{5}(2, 2)_{∗}{v_{2}ζ_{2}/3v_{1}}
⨁B_{3}(2, 2)_{∗}{v_{2}^{4}b_{11}ζ_{2}/3v_{1}}

⨁B_{2}(2, 2)_{∗}{v_{2}h_{10}ζ_{2}/3v_{1}, v_{2}^{2}h_{11}ζ_{2}/3v_{1}, v_{2}^{5}h_{11}ζ_{2}/3v_{1}, v_{2}^{7}h_{10}ζ_{2}/3v_{1}},

G̃Z^{∗} = B_{5}(2, 2)_{∗}{v_{2}^{7}ψ_{1}ζ_{2}/3v_{1}}⨁B_{4}(2, 2)_{∗}{v_{2}^{3}ψ_{0}ζ_{2}/3v_{1}}
⨁B_{2}(2, 2)_{∗}{ξζ_{2}/3v_{1}}
⨁B_{1}(2, 2)_{∗}{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 B_{k}(2, n)_{∗} = (Z/3)[v_{2}^{±3^{n}}, b_{10}]/(b_{10}^{k}) and I(n) are given in the introduction.

Proof. Suppose that d_{r}(x) = y ≠ 0 in the Adams-Novikov spectral sequence for π_{∗}(M^{2}). Then y is in the image of i_{∗}: H^{∗}M_{1}^{1} → H^{∗}M_{0}^{2}, since y has filtration ≥ 5. Lemma 5.2 shows that δ(y) ≠ 0 for the connecting homomorphism δ: H^{∗}M_{0}^{2} → H^{∗}M_{1}^{1}, and so we have δ(x) ≠ 0 and d_{r}(δ(x)) = δ(y) in the Adams-Novikov spectral sequence for π_{∗}(L_{2}W). Observing the differentials given in [7] with Lemma 4.4, we see that there is no more new differentials, and obtain the theorem. q.e.d.

7. Application to β-elements

In [1], H^{0}M_{0}^{2} is determined and we see that

v_{2}^{s}/3^{i+1}v_{1}^{j} ∈ H^{0}M_{0}^{2} if and only if (s, j, i + 1) ∈ B.

Consider the universal Greek letter map η = δ′δ: H^{0}M_{0}^{2} → H^{2}E(2)_{∗}, where δ: H^{0}M_{0}^{2} → H^{1}N_{0}^{1} and δ′: H^{1}N_{0}^{1} → H^{2}E(2)_{∗} are the connecting homomorphisms associated to the short exact sequences 0 → N_{0}^{1} → M_{0}^{1} → M_{0}^{2} → 0 and 0 → E(2)_{∗} → M_{0}^{0} → N_{0}^{1} → 0, respectively. Then the β-elements are defined by

β_{s/j,i+1} = η(v_{2}^{s}/3^{i+1}v_{1}^{j}).

We obtain the following immediately.

Lemma 7.1. Mod (3, v_{1}), β_{1} ≡ b_{10}, β_{2} ≡ v_{2}h_{11}ζ_{2} and β_{6/3} ≡ v_{2}^{3}b_{11} in the E_{2}-term E_{2}(S^{0}).

Furthermore, note that β′_{1} = h_{11} ∈ E_{2}(V (0)) and α_{1} = h_{10} ∈ E_{2}(S^{0}). The generators of G̃ then yields the following elements:

η(v_{2}^{9t+1}/3v_{1}) = β_{9t+1}, η(v_{2}^{9t+4}b_{11}/3v_{1}) = β_{9t+1}β_{6/3},
η(v_{2}^{9t+7}h_{10}/3v_{1}) = β_{9t+7}α_{1}, η(v_{2}^{9t+1}h_{10}/3v_{1}) = β_{9t+1}α_{1},
η(v_{2}^{9t+2}h_{11}/3v_{1}) = [β_{9t+2}β′_{1}], η(v_{2}^{9t+5}h_{11}/3v_{1}) = [β_{9t+5}β′_{1}].

Now we prove the theorems in the introduction.

Proof of Theorem C. Consider the long exact sequences

⋯ → π_{∗}(L_{0}S^{0}) → π_{∗}(L_{2}N^{1}) → π_{∗+1}(L_{2}S^{0}) → ⋯

and

⋯ → π_{∗}(L_{1}N^{1}) → π_{∗}(M^{2}) → π_{∗+1}(L_{2}N^{1}) → ⋯

associated to the cofiber sequences of (6.1). Note that π_{∗}(L_{0}S^{0}) = Q and

π_{∗}(L_{1}N^{1}) = Q/Z_{(3)}⨂Λ(y)⨁A

shown in [1], where A is the Z_{(3)}-module generated by v_{1}^{sp^{i}}/3^{i+1} for i ≥ 0 and 3 |̸ s ∈ Z. Therefore, the module G̃⨁G̃^{∗}⨁G̃Z⨁G̃Z^{∗} given in Theorem 6.4 is isomorphically sent to π_{∗}(L_{2}S^{0}). Theorem C now follows. q.e.d.

Proof of Theorem A. Consider the localization map ι: S^{0} → L_{2}S^{0}. Since the induced map ι_{∗}π_{∗}(S^{0}) → π_{∗}(L_{2}S^{0}) sends a β-element to the corresponding β-element, the non-triviality of products of β-elements in π_{∗}(S^{0}) is deduced from the one in π_{∗}(L_{2}S^{0}). The necessity follows immediately from Theorem C except for β_{2}. For β_{2}, note that β_{9t+1}β_{2} = [β_{9t+2}β′_{1}]ζ_{2} ∈ η(G̃Z) for the universal Greek letter map η. Thus the necessity for β_{2} is shown.

In Lemma 6.2, we have d_{9}(v_{2}^{9t+4}h_{10}/3v_{1}) = v_{2}^{9t+1}b_{10}^{5}/3v_{1} and d_{9}(v_{2}^{9t+8}h_{11} /3v_{1}) = v_{2}^{9t+4}b_{11}b_{10}^{4}/3v_{1}, which yield

(7.2)  d_{9}(β_{9t+4}α_{1}) = β_{1}^{5}β_{9t+1} and d_{9}(β_{9t+8}h_{11}) = β_{9t+1}β_{6/3}β_{1}^{4}

in the E_{9}-term E_{9}^{∗}(L_{2}S^{0}) as the image of the universal Greek letter map. In the same manner, the equation d_{5}(v_{2}^{9t+4}ζ_{2}/3v_{1}) = v_{2}^{9t+2}h_{11}ζ_{2}b_{10}^{2}/3v_{1} in Lemma 6.2 yields

(7.3)  d_{5}(β_{9t+4}ζ_{2}) = β_{9t+1}β_{2}b_{10}^{2}

in the E_{9}-term E_{5}^{∗}(L_{2}S^{0}). If t ≥ 0, then the equations (7.2) and (7.3) also hold in the Adams-Novikov spectral sequence for π_{∗}(S^{0}), since the elements appeared in (7.2) and (7.3) are also defined in E_{2}(S^{0}). q.e.d.

Proof of Theorem E. In the proof of Theorem 6.4, we read off that the elements on the 0-th line hit nothing except for the β-elements given by B^{c}. Therefore, we obtain Theorem E. q.e.d.

References

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[3] S. Oka and K. Shimomura, On products of the β-elements in the stable homotopy groups of spheres, Hiroshima Math. J. 12 (1982), 611–626.

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

[5] K. Shimomura, Non-triviality of some products of β-elements in the stable homotopy of spheres, Hiroshima Math. J. 17 (1987), 349–353.

[6] K. Shimomura, The homotopy groups of the L_{2}-localized Toda-Smith spectrum at the prime 3, Trans. Amer. Math. Soc., 349 (1997), 1821–1850.

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