The homotopy groups π_{∗}(L_{2}S^{0}) at the prime 3

Katsumi Shimomura and Xiangjun Wang

The homotopy groups π_{∗}(L_{2}S^{0}) of the L_{2}-localized sphere are determined by studying the Bockstein spectral sequence. The results indicate also the homotopy groups π_{∗}(L_{K(2)}S^{0}) and we see that the fiber of the localization map L_{2}S_{3}^{0} → L_{K(2)}S^{0} is homotopic to Σ^{−2}L_{1}S_{3}^{0}.

1. Introduction and statement of results

For each prime number p, there is the Bousfield localization functor L_{n} : S_{(p)} → S_{(p)} with respect to v_{n}^{−1}BP, where S_{(p)} denotes the stable homotopy category localized away from the prime p, BP the Brown-Peterson spectrum at p, and v_{n} the n-th generator of the coefficient algebra BP_{∗}. Consider the Morava K-theories K(n) and the Johnson-Wilson spectra E(n), where K(n)_{∗} = Z/p[v_{n}^{±1}] and E(n)_{∗} = Z_{(p)}[v_{1}, v_{2},…, v_{n}, v_{n}^{−1}]. Then L_{n} is also the localization with respect to K(0) ∨ K(1) ∨… ∨ K(n) or E(n).

Hopkins and Ravenel show the homotopy equivalence S_{(p)}^{0} ≃ holim_{⟵}_{n} L_{n}S^{0},

and so π_{∗}(L_{n}S^{0}) is an approximation of the homotopy groups of spheres. Actually, π_{∗}(L_{0}S^{0}) = Q at any prime and π_{∗}(L_{1}S^{0}) = Z_{(3)}⨁A⨁Q/Z_{(3)}⟨y⟩ at a prime > 2(cf. [3], [4]), where A denotes the module generated by the generalized α-elements (see below) and Q/Z_{(3)}⟨y⟩ ⊂ π_{−2}(L_{1}S^{0}) for the virtual generator y. The homotopy groups π_{∗}(L_{2}S^{0}) of E(2)_{∗}-localized spheres are determined at a prime > 3 in [9], which satisfy Hopkins’ chromatic splitting conjecture [2]. In this paper, we determine π_{∗}(L_{2}S^{0}) at the prime 3.

Theorem A. The homotopy groups π_{∗}(L_{2}S^{0}) at the prime 3 are direct sum of three modules G_{i}’s, which are described as follows :

G_{0} = Z_{(3)}⨁A_{+}⨁Σ^{−1}(A_{−}⨁Q/Z_{(3)}⟨y⟩)ζ_{2}
G_{1} = B⨁C⨁CI⨁B^{∗}⨁(B_{1}⨁C)ζ_{2}
G_{2} = Ĝ⨁Ĝ^{∗}⨁ĜZ⨁ĜZ^{∗}

Here the modules on the right hand sides are as follows:

A = ∑_{i≥0} Z/3^{i+1}⟨α_{3^{i}s/i+1} | 3 |̸ s ∈ Z⟩
A_{+} = Z_{(3)}{α_{3^{i}s/i+1} | α_{3^{i}s/i+1} ∈ A, s > 0}
A_{−} = Z_{(3)}{α_{3^{i}s/i+1} | α_{3^{i}s/i+1} ∈ A, s < 0}

for G_{0},

B = Z_{(3)}{β_{3^{n}s/3^{i}m,i+1} | n ≥ 0, 3 |̸ s ∈ Z, i ≥ 0, 1 ≤ m < 4 · 3^{n−2i−1}
and 3 |̸ m or 4 · 3^{n−2i−2} ≤ m}
B_{1} = Z_{(3)}{β_{3^{n}s/3^{i}m,i+1} | β_{3^{n}s/j,i+1} ∈ B, 3 |̸ (s + 1), 3|m,
or i ≠ k + 1 if s = 3^{k+2}t − 1 with k ≥ 0 and 3 |̸ t}
C = C_{1}⨁C_{2}
C_{1} = Z_{(3)}{α̃_{1}β_{3^{n}(3t+1)/3^{i}m+1,i} | 0 < i ≤ n, and 2 · 3^{n−i} < 3^{i}m ≤ 2 · 3^{n−i+1}}
C_{2} = Z_{(3)}{α̃_{1}β_{3^{n}(9t−1)/3^{i}m+1,i} | 0 < i ≤ n, and

2 · 3^{n−i−1} < 3^{i}m − 8 · 3^{n} ≤ 2 · 3^{n−i}}

CI = CI_{0}⨁CI_{1}⨁CI_{2}⨁CI_{3}
CI_{0} = Z_{(3)}{c_{3^{n}s} | n ≥ 0, s = 3t + 1 or s = 9t − 1 (t ∈ Z)}
CI_{1} = Z_{(3)}{α̃_{1}β_{3^{n}(3t+1)/3^{i}m+1,i+1} | 0 ≤ i ≤ n, 2 · 3^{n−i−1} < 3^{i}m ≤ 2 · 3^{n−i},
3 |̸ m}
CI_{2} = Z_{(3)}{α̃_{1}β_{3^{n}(9t−1)/3^{i}m+1,i+1} | 3 |̸ m ≤ 8 · 3^{n−i}, 0 ≤ i < n,
k + 1 > i if 3^{k}|t}
CI_{3} = Z_{(3)}\{α̃_{1}β_{3^{n}(9t−1)/3^{n}m+1,i} | i = \{ \begin{matrix}n + 1 & if m = 1, 3, 4, 6, 7, \\ n + 2 & if m = 2, 8, \\ n + 3 & if m = 5 \end{matrix} ,
k + 1 > n if 3^{k}|t}
B^{∗} = Z_{(3)}{β(n)_{3^{n+l}s/3^{i}m,i+1}^{∗} | 3 |̸ s ∈ Z, i, n ≥ 0, 0 < 3^{i}m ≤ 4 · 3^{n},

l > i and l > i + 1 if 3|(s + 1)} for G_{1} and

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}]})⨁B_{1}{[β_{9t−1/2}β′_{1}]},

Ĝ^{∗} = ∑_{t∈Z}(B_{5}{χ_{9t+7}^{1}}⨁B_{4}{χ_{9t+3}^{0}}

⨁B_{2}{β(0)_{9t+1}^{∗}, β_{6/3}β(0)_{9t+1}^{∗}, β_{6/3}β(0)_{9t+4}^{∗}}

⨁∑_{n≥1}(B_{3}{β(0)_{3^{n+2}t+9u+3}^{∗} | u ∈ Z − I(n)}

⨁B_{2}{β(0)_{3^{n+2}t+9u+3}^{∗} | u ∈ I(n)})),
ĜZ = ∑_{t∈Z}(B_{5}{ζβ_{9t+1}}⨁B_{3}{ζβ_{9t+1}β_{6/3}}

⨁B_{2}\{\overline{ζβ_{9t+7}α_{1}}, ζβ_{9t+1}α_{1}, ζ[β_{9t+2}β′_{1}], ζ[β_{9t+5}β′_{1}]\}),

ĜZ^{∗} = ∑_{t∈Z}(B_{5}{ζ_{2}χ_{9t+7}^{1}}⨁B_{4}{ζ_{2}χ_{9t+3}^{0}}⨁B_{2}{ζ_{2}β(0)_{9t+1}^{∗}}

⨁B_{1}{ζ_{2}β_{6/3}β(0)_{9t+1}^{∗}, ζ_{2}β_{6/3}β(0)_{9t+4}^{∗}}

⨁∑_{n≥1}(B_{3}{ζ_{2}β(0)_{3^{n+2}t+9u+3}^{∗} | u ∈ Z − I(n)}

⨁B_{2}{ζ_{2}β(0)_{3^{n+2}t+9u+3}^{∗} | u ∈ I(n)})).

for G_{2}. 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}. The generators are defined in section 4 and degrees of them are:

|α_{a/b}| = 4a − 1, |β′_{1}| = 11, |β_{a/b,c}| = 16a − 4b − 2, |c_{a}| = 16a − 7,

|α̃_{1}β_{a/b,c}| = 16a − 4b + 1, |β(a)_{b/c,d}^{∗}| = 16b − 8 · 3^{a} − 4c − 4,

|χ_{a}^{0}| = 16a + 7, |χ_{a}^{1}| = 16a + 15, |ζ_{2}| = −1,

and orders of them are:

\begin{matrix}o(α_{a/b}) = 3^{b}, o(β_{a/b,c}) = 3^{c}, o(c_{3^{n}s}) = 3^{n+1} if 3 |̸ s, \\ o(α̃_{1}β_{a/b,c}) = 3^{c}, o(β(a)_{b/c,d}^{∗}) = 3^{d}, o(χ_{a}^{0}) = 3, o(χ_{a}^{1}) = 3. \end{matrix}

Furthermore, we abbreviate as follows:

\begin{matrix}α_{a} = α_{a/1}, β_{a/b} = β_{a/b,1}, β_{a} = β_{a/1}, α̃_{1}β_{a/b} = α̃_{1}β_{a/b,1}, \\ α̃_{1}β_{a} = α̃_{1}β_{a/1} β(a)_{b/c}^{∗} = β(a)_{b/c,1}^{∗}, and β(a)_{b}^{∗} = β(a)_{b/1}^{∗}. \end{matrix}

Our computation of the differentials of the Bockstein spectral sequence π_{∗}(L_{2}V (0)) ⇒ π_{∗}(L_{2}S^{0}) also works for the Bockstein spectral sequence associated to the cofiber sequence L_{K(2)}S^{0} → L_{K(2)}S^{0} → L_{K(2)}V (0), and we obtain the homotopy groups π_{∗}(L_{K(2)}S^{0}) from the result on π_{∗}(L_{K(2)}V (0)) given in [6].

Theorem B. The homotopy groups π_{∗}(L_{K(2)}S^{0}) are the direct sum of (Z_{3}⨁A_{+})⨂Λ(ζ_{2}), G_{1} and G_{2}. Therefore, the homotopy fiber F (L_{1}S^{0}, L_{2}S_{3}^{0}) of the localization map L_{2}S_{3}^{0} → L_{K(2)}S^{0} of the 3-adic completed sphere S_{3}^{0} is homotopic to Σ^{−2}L_{1}S_{3}^{0}.

The second half of the theorem is seen in the same manner as the one at a prime > 3 [2]. This shows that there is the only one summand in F (L_{1}S^{0}, L_{2}S_{3}^{0}) while Hopkins’ chromatic splitting conjecture says that it has three summands.

Theorem A is proved by using the Adams-Novikov spectral sequence. Let N^{2} denote the spectrum such that BP_{∗}(N^{2}) = BP_{∗}/(3^{∞}, v_{1}^{∞}). We denote the Adams-Novikov E_{2}-term for computing π_{∗}(L_{2}N^{2}) by H^{∗}M_{0}^{2}. In the next section we show that the E_{2}-term H^{∗}M_{0}^{2} is the direct sum of the three modules \overline{A_{i}} for i = 0, 1, 2, and give the structures of \overline{A_{0}} and \overline{A_{2}}. Section 3 is devoted to determine the structure of the module \overline{A_{1}} using some results proven in the last section. The differentials of the Adams-Novikov spectral sequence are studied in [7], and we deduce the homotopy groups π_{∗}(L_{2}N^{2}) from these results, and then the chromatic spectral sequence shows Theorem A in section 4.

2. Notations and the structure of H^{∗}M_{0}^{2}

Consider the Hopf algebroid

(E(2)_{∗}, E(2)_{∗}(E(2))) = (Z_{(3)}[v_{1}, v_{2}^{±1}], E(2)_{∗}[t_{1}, t_{2},…]⨂_{BP_{∗}}E(2)_{∗})

associated to the Johnson-Wilson spectrum E(2), where BP denotes the Brown-Peterson spectrum with BP_{∗} = Z_{(3)}[v_{1}, v_{2},…] and BP_{∗} acts on E(2)_{∗} by sending v_{i} to v_{i} if i ≤ 2, and to zero if i > 2. In this paper, H^{∗}M for an E(2)_{∗}(E(2))-comodule M denotes Ext_{E(2)_{∗}(E(2))}^{∗}(E(2)_{∗}, M), which is given as a cohomology of the cobar complex Ω^{∗}M of E(2)_{∗}(E(2))-comodules (cf. §5).

Then we have the Adams-Novikov spectral sequence

E_{2}^{∗} = H^{∗}E(2)_{∗} ⟹ π_{∗}(L_{2}S^{0})

converging to the homotopy groups of E(2)_{∗}-localized sphere spectrum.

We now recall the definition of the chromatic comodules N_{j}^{i} and M_{j}^{i}. They are defined inductively by setting N_{j}^{0} = E(2)_{∗}/I_{j} for I_{0} = 0, I_{1} = (3) and I_{2} = (3, v_{1}), M_{j}^{i} = v_{i+j}^{−1}N_{j}^{i}, and N_{j}^{i+1} = M_{j}^{i}/N_{j}^{i}. Note that M_{j}^{i} = N_{j}^{i} if i + j = 2 and = 0 if i + j > 2. Then we see that the Adams-Novikov E_{2}- term H^{∗}E(2)_{∗} is obtained from H^{∗}M_{0}^{i} for i ≤ 2 and the long exact sequences H^{s}N_{0}^{i} → H^{s}M_{0}^{i} → H^{s}N_{0}^{i+1} → H^{s+1}N_{0}^{i}. Since the modules H^{∗}M_{0}^{i} for i < 2 are determined in [3], we here determine H^{∗}M_{0}^{2}. For this sake, we consider the comodule M_{1}^{1} = Z/3{x/v_{1}^{j} | j > 0, x ∈ K(2)_{∗}}, where K(2)_{∗} = Z/3[v_{2}^{±1}], and the short exact sequence 0 ⟶ M_{1}^{1} ⟶^{1/3} M_{0}^{2} ⟶^{3} M_{0}^{2} ⟶ 0. Here note that M_{0}^{2} is described as M_{0}^{2} = Z_{(3)}{x/3^{i}v_{1}^{j} | i, j > 0, x ∈ K(2)_{∗}}. Then we will get H^{∗}M_{0}^{2} from H^{∗}M_{1}^{1} which is determined in [6], by using the lemma given in [3, Remark 3.11].

In order to describe the module H^{∗}M_{1}^{1}, we set up some notations: H^{∗}M_{2}^{0} = H^{∗}K(2)_{∗} is determined (cf. [5]) to be F ⨂K(2)_{∗}[b_{10}]⨂Λ(ζ_{2}), where F is the Z/3-vector space spanned by 1, h_{10}, h_{11}, b_{11}, ξ, ψ_{0}, ψ_{1}, and b_{11}ξ. These generators ζ_{2}, h_{1i} and b_{1i}, are cohomology classes represented by cocycles v_{2}^{−1}(t_{2} − t_{1}^{4}) + v_{2}^{−3}t_{2}^{3}, t_{1}^{3^{i}} and −t_{1}^{3^{i}} ⊗ t_{1}^{2·3^{i}} − t_{1}^{2·3^{i}} ⊗ t_{1}^{3^{i}} of the cobar complex Ω^{∗}K(2)_{∗}, respectively. Besides, ξ and ψ_{i} are the generators of H^{2,8}K(2)_{∗} and H^{3,8(i+2)}K(2)_{∗}.

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

F_{(h)} = Z/3[v_{2}^{±3}]{v_{2}/v_{1}, v_{2}h_{10}/v_{1}, v_{2}^{−1}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}^{−1}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}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}^{2·3^{n+1}+1},
v_{2}^{8·3^{n}}h_{10}/v_{1}^{8·3^{n}+1}, v_{2}^{3^{n+1}(2±1)}ξ_{n}/v_{1}^{4·3^{n}}},
F′_{0} = E(2, 2)_{∗}{v_{2}^{±3}/v_{1}^{2}, v_{2}^{3}h_{10}/v_{1}^{7}, v_{2}^{8}h_{10}/v_{1}^{9}, v_{2}^{3(2±1)}ξ_{0}/v_{1}^{4}},

where ξ_{n} = v_{2}^{−3^{n}+(3^{n}−1)/2}ξ. Note that F_{0} = F′_{0}⨁F′′_{0} for F′′_{0} = Z/3[v_{2}^{±9}]{v_{2}^{±3}/v_{1}^{3}}. Then, in [7], H^{∗}M_{1}^{1} is shown to be the direct sum of F′′_{0} and the three modules A_{i}, where

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

Once we know the behavior of connecting homomorphism δ : H^{∗}M_{1}^{1} → H^{∗+1}M_{0}^{2}, we will obtain H^{∗}M_{0}^{2} by [3, Remark 3.11]. In [3] and [7], it is shown that δ(A_{i}) ⊂ A_{i}. Let A_{i}^{s} denote the submodule A_{i} ⋂ H^{s}M_{1}^{1}. We now define the submodule \overline{A_{i}}^{s} to fit in the commutative diagram of exact sequences

(2.1)  A_{i}^{s}  \overline{A_{i}}^{s}  \overline{A_{i}}^{s}  A_{i}^{s+1} H^{s}M_{1}^{1}  H^{s}M_{0}^{2}  H^{s}M_{0}^{2}  H^{s+1}M_{1}^{1}.^{1/3}  _{w}_{z u}  ^{3}  _{w}_{z u}  ^{δ}  _{w}_{y u}  _{y u}_{^{1/3} w}  _{^{3} w}  _{^{δ} w}

We deduce the following lemma from [3, Remark3.11]:

Lemma 2.2. If we have the commutative diagram of exact sequences

\overline{A_{i}}^{s−1}  A_{i}^{s}  B^{s}  B^{s}  A_{i}^{s+1} A_{i}^{s}  _{\overline{A_{i}}^{s}}  _{\overline{A_{i}}^{s}}  A_{i}^{s+1}, w  ^{1/3} w  ^{3} w_{_{f} u}  ^{δ} w_{u_{f}}_{^{1/3} w}  _{^{3} w}  _{^{δ} w}

then f is an isomorphism.

Note that [9, Prop. 7.2] is also valid at the prime 3, and the elements y′_{3t}, V and G_{1} there are v_{2}^{3t+2}h_{11}/9v_{1}^{2}, −v_{2}^{2}h_{11} and v_{2}^{−1}b_{10} in our notation, respectively. Therefore, we have

Proposition 2.3. For each integer t,

δ(v_{2}^{3t+2}h_{11}/9v_{1}^{2}) = v_{2}^{3t+2}(−b_{10} + h_{11}ζ_{2})/v_{1}^{2} + ⋯

Since δ(v_{2}^{3s}/3v_{1}^{3}) = sv_{2}^{3s−1}h_{11}/v_{1}^{2} by [3, Prop. 6.9] and δ(v_{2}^{9t−1}h_{11}/9v_{1}^{2}) = v_{2}^{9t−1}(−b_{10} + h_{11}ζ_{2})/v_{1}^{2} + ⋯ by Proposition 2.3, we see the following:

Proposition 2.4. H^{s}M_{0}^{2} is isomorphic to ∑_{i=0}^{2} \overline{A_{i}}^{s} if s ≠ 1, and H^{1}M_{0}^{2} ≅ ∑_{i=0}^{2} \overline{A_{i}}^{s}⨁ ∑_{t∈Z} Z/9\{v_{2}^{9t−1}h_{11}/9v_{1}^{2}\}.

In the same manner as [3, Theorem 4.2], we obtain Proposition 2.5. The module \overline{A_{0}} is given as follows :

\overline{A_{0}} = (\overline{A_{−}}⨁Q/Z_{(3)})⨂Λ(ζ_{2}),

where

\overline{A_{−}} = \{1/3^{i+1}v_{1}^{3^{i}m} | i ≥ 0, m > 0\}.

The module \overline{A_{2}} is determined in [7] as follows: Proposition 2.6. The module \overline{A_{2}} is the direct sum of Z/3[v_{2}^{±3}]\{v_{2}^{−1}/3v_{1}, ξ/3v_{1}^{2}\}

and

(\overline{F_{(h)}}⨁\overline{F_{(h)}^{∗}})⨂P E.

Here \overline{F_{(h)}} and \overline{F_{(h)}^{∗}} are the images of F_{(h)} and F_{(h)}^{∗} under the map H^{∗}M_{1}^{1} →^{φ} H^{∗}M_{0}^{2} given by φ(x) = x/3.

3. Determination of \overline{A_{1}} We devide A_{1} into 14 pieces:

A_{1} = ((X_{1}⨁X_{2})⨁(H⨁HI⨁H^{∗})⨁(X_{1}^{∗}⨁X_{2}^{∗}))⨂Λ(ζ_{2}).

Here,

X_{1} = X_{1,1}⨁X_{1,2}
X_{1,1} = Z/3{v_{2}^{3^{n}(3t+1)}/v_{1}^{j} | n ≥ 0, t ∈ Z, 0 < j < 4 · 3^{n−1},
such that j > 4 · 3^{n−i−1} − 1 if 3^{i}|j}

X_{1,2} = Z/3{v_{2}^{3^{n}(3^{k+2}u−1)}/v_{1}^{3^{k+1}m} | n ≥ 0, 3 |̸ u ∈ Z, 0 < 3^{k+1}m < 4 · 3^{n−1},
such that j > 4 · 3^{n−i−1} − 1 if 3^{i−k−1}|m}
X_{2} = X_{2,1}⨁X_{2,2}
X_{2,1} = Z/3{v_{2}^{3^{n}(3t−1)}/v_{1}^{3^{i}m} | n ≥ 0, 3 |̸ m, 1 ≤ m ≤ 4 · 3^{n−2i−1}}
X_{2,2} = Z/3{v_{2}^{3^{n}(3^{k+2}u−1)}/v_{1}^{3^{k+1}m} | 2 ≤ 2k ≤ n, 0 < m < 4 · 3^{n−2k−1}, 3 |̸ m},
H = H_{1}⨁H_{2}
H_{1} = Z/3{v_{2}^{3^{n}(3t+1)}h_{10}/v_{1}^{j+1} | 0 < i < n, j ≤ 2 · 3^{n}, 2 · 3^{n−i} ≤ j if 3^{i}|j}
H_{2} = Z/3{v_{2}^{3^{n}(9t−1)}h_{10}/v_{1}^{j} | n ≥ 0, 8 · 3^{n} < j ≤ 10 · 3^{n} + 1}
HI = HI_{1}⨁HI_{2}
HI_{1} = Z/3{v_{2}^{3^{n}(3t+1)}h_{10}/v_{1}^{j+1} | 0 ≤ i ≤ n, 4 · 3^{n−i−1} ≤ j ≤ 2 · 3^{n−i}
if 3^{i}|j and 3^{i+1} |̸ j}
HI_{2} = Z/3{v_{2}^{3^{n}(9t−1)}h_{10}/v_{1}^{j+1} | n ≥ 0, 0 ≤ j ≤ 8 · 3^{n}, 3^{k+1} |̸ (j + 4 · 3^{n})
if t = 3^{k}u with 3 |̸ u}
H^{∗} = H_{1}^{∗}⨁(H_{2,1}^{∗} + H_{2,2}^{∗})
H_{1}^{∗} = Z/3{v_{2}^{3^{n}(3t+1)}h_{10}/v_{1}^{j+1} | n ≥ 0, j > 0 and j < 4 · 3^{n−i−1} if 3^{i+1} |̸ j}
H_{2,1}^{∗} = Z/3{v_{2}^{3^{n}(9t−1)}h_{10}/v_{1}^{j+1} | n ≥ 0, j > 0 and j ≤ 4 · 3^{n−i−1} if 3^{i+1} |̸ j}
H_{2,2}^{∗} = Z/3{v_{2}^{3^{n}(9t−1)}h_{10}/v_{1}^{j+1} | n ≥ 0, j ≥ 0 and 3^{k+1}|(j + 4 · 3^{n}) ≤ 4 · 3^{n+1}
if t = 3^{k}u with 3 |̸ u}
X_{1}^{∗} = X_{1,1}^{∗}⨁X_{1,2}^{∗}
X_{1,1}^{∗} = Z/3{v_{2}^{3^{n+i}(3t+1)}ξ_{n}/v_{1}^{3^{i}k} | i > 0, 0 < 3^{i}k ≤ 4 · 3^{n}}
X_{1,2}^{∗} = Z/3{v_{2}^{3^{n+i+1}(3t−1)}ξ_{n}/v_{1}^{3^{i}k} | i > 0, 0 < 3^{i}k ≤ 4 · 3^{n}}
X_{2}^{∗} = Z/3{v_{2}^{3^{n+l}s}ξ_{n}/v_{1}^{3^{i}k} | 3 |̸ s, i, n ≥ 0, 0 < 3^{i}k ≤ 4 · 3^{n},
l > i and l > i + 1 if 3|(s + 1)}

We also consider the following submodules of H^{∗}M_{0}^{2}:

X̃ = Z_{(3)}{v_{2}^{3^{n}s}/3^{i+1}v_{1}^{j} | n ≥ 0, 3 |̸ s ∈ Z, i ≥ 0, j > 0,
with 3^{i}|j < 4 · 3^{n−i−1} and either 3^{i+1} |̸ j or 4 · 3^{n−i−2} < j}
= X̃_{1}⨁X̃_{2}

X̃_{1} = Z_{(3)}{v_{2}^{3^{n}s}/3^{i+1}v_{1}^{3^{i}m} | v_{2}^{3^{n}s}/3^{i+1}v_{1}^{3^{i}m} ∈ X̃, 3 |̸ (s + 1), 3|m,

or i ≠ k + 1 if s = 3^{k+2}t − 1 with k ≥ 0 and 3 |̸ t}

X̃_{2} = Z_{(3)}{v_{2}^{3^{n}s}/3^{i+1}v_{1}^{3^{i}m} | v_{2}^{3^{n}s}/3^{i+1}v_{1}^{3^{i}m} ∈ X̃, 3|(s + 1), 3 |̸ m

and i = k + 1 if s = 3^{k+2}t − 1 with k ≥ 0 and 3 |̸ t}

H̃ = H̃_{1}⨁H̃_{2}
H̃_{1} = Z_{(3)}{v_{2}^{3^{n}(3t+1)}h_{10}/3^{i}v_{1}^{3^{i}m+1} | 0 < i ≤ n, and

2 · 3^{n−i} < 3^{i}m ≤ 2 · 3^{n−i+1}}

H̃_{2} = Z_{(3)}{v_{2}^{3^{n}(9t−1)}h_{10}/3^{i}v_{1}^{3^{i}m+1} | 0 < i ≤ n, and

2 · 3^{n−i} < 3^{i}m − 8 · 3^{n} ≤ 2 · 3^{n−i+1}}

H̃I = H̃I_{0}⨁H̃I_{1}⨁H̃I_{2}⨁H̃I_{3}
H̃I_{0} = Z_{(3)}{v_{2}^{3^{n}s}h_{10}/3^{n+1}v_{1} | n ≥ 0, s = 3t + 1 or s = 9t − 1 (t ∈ Z)}
H̃I_{1} = Z_{(3)}{v_{2}^{3^{n}(3t+1)}h_{10}/3^{i+1}v_{1}^{3^{i}m+1} | 0 ≤ i ≤ n,

2 · 3^{n−i−1} < 3^{i}m ≤ 2 · 3^{n−i}, 3 |̸ m}

H̃I_{2} = Z_{(3)}{v_{2}^{3^{n}(9t−1)}h_{10}/3^{i+1}v_{1}^{3^{i}m+1} | 3 |̸ m < 8 · 3^{n−i},
0 ≤ i < n, k + 1 > i if 3^{k}|t}
H̃I_{3} = Z_{(3)}\{v_{2}^{3^{n}(9t−1)}h_{10}/3^{i}v_{1}^{3^{n}m+1} | i = \{ \begin{matrix}n + 1 & if m = 1, 3, 4, 6, 7, \\ n + 2 & if m = 2, 8, \\ n + 3 & if m = 5, \end{matrix}
k + 1 > n if 3^{k}|t}
X̃_{2}^{∗} = Z_{(3)}{v_{2}^{3^{n+l}s}ξ_{n}/3^{i+1}v_{1}^{3^{i}m} | 3 |̸ s ∈ Z, i, n ≥ 0, 0 < 3^{i}m ≤ 4 · 3^{n},
l > i and l > i + 1 if 3|(s + 1)}

The propositions of section 5 below show the behavior of the connecting homomorphism δ : \overline{A_{1}}^{s} → A_{1}^{s+1} as follows:

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

We now use Lemma 2.2 to obtain our main theorem:

Theorem 3.2. \overline{A_{1}}^{s} is isomorphic to X̃_{1}⨁X̃_{2} if s = 0, H̃⨁H̃I⨁X̃_{1}ζ_{2} if s = 1, X̃_{2}^{∗}⨁H̃ζ_{2} if s = 2, and 0 otherwise.

4. The homotopy groups π_{∗}(L_{2}S^{0})

Let E_{r}(X) denote the E_{r}-term of the Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(X). We start with a general result on the spectral sequence.

Lemma 4.1. Let X →^{f} Y →^{g} Z →^{h} ΣX be the cofiber sequence with BP_{∗}(h) = 0. Then we have the induced maps E_{r}^{s}(X) →^{f∗} E_{r}^{s}(Y) →^{g∗} E_{r}^{s}(Z) →^{δ} E_{r}^{s+1}(X). Suppose that g_{∗}(\overline{y}) = \overline{z} for non-zero elements y ∈ E_{r}^{s}(Y) and z ∈ E_{r}^{s+r}(Z). Here \overline{a} denotes a homotopy element that is detected by an element a of the E_{r}-term. Then if y = f_{∗}(x) for some x ∈ E_{r}^{s}(X), then d_{r}(x) = δ(z).

Let N^{1} and N^{2} denote the cofibers of the localization maps S^{0} → L_{0}S^{0} and N^{1} → L_{1}N^{1}, respectively. Then we have the Adams-Novikov spectral sequence E_{2}(L_{2}N^{2}) = H^{∗}M_{0}^{2} ⇒ π_{∗}(L_{2}N^{2}). The differentials of the spectral sequence are determined in [7], and we have the following

Proposition 4.2. The E_{∞}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{2}N^{2}) is the derect sum of the three modules \overline{A_{0}}, \overline{A_{1}} and Ã_{2}. Here \overline{A_{0}} and \overline{A_{1}} are determined in the previous sections, and

Ã_{2} = G̃⨁G̃^{∗}⨁G̃Z⨁G̃Z^{∗},

where these four modules are determined in [7]:

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}}⨁B_{1}(2, 2){v_{2}^{−1}h_{11}/3v_{1}^{2}},

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)}),

for B_{k}(2, n)_{∗} = (Z/3)[v_{2}^{±3^{n}}, b_{10}]/(b_{10}^{k}) and I(n) given in the introduction.

Lemma 4.3. There is no extension problem in the spectral sequence for π_{∗}(L_{2}N^{2}).

Proof. Let M(i, ∞) be a cofiber of the localization map M(i) → L_{1}M(i) of the mod 3^{i} Moore spectrum M(i). Then we have the cofiber sequence M(i, ∞) →^{φ} N^{2} →^{3^{i}} N^{2}. In [7], we show that if d_{r}(x) = y in E_{r}^{∗}(L_{2}N^{2}) for a non zero element y, then the order of x is three except for d_{5}(v_{2}^{9t−1}h_{11}/9v_{1}^{2}) = v_{2}^{9t−2}h_{10}b_{10}^{2}ζ_{2}/3v_{1}^{2}. This exceptional differential does not induce the extension. Thus we take i = 1, and M(1, ∞) is W of [6]. Then we see by [7] that the differentials on E_{r}^{∗}(L_{2}N^{2}) are obtained by sending ones on E_{r}^{∗}(L_{2}W) by the map φ_{∗} : E_{r}^{∗}(L_{2}W) → E_{r}^{∗}(L_{2}N^{2}), and so y cannot be a image of the connecting homomorphism. Now we apply Lemma 4.1 to show the lemma. q.e.d.

Corollary 4.4. The homotopy groups π_{∗}(L_{2}N^{2}) are the direct sum of the three modules \overline{A_{0}}, \overline{A_{1}} and Ã_{2}.

Proof of Theorem A. Consider the exact sequences ⋯ → π_{∗}(L_{2}S^{0}) →

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

⋯ associated to the cofiber sequence S^{0} → L_{0}S^{0} → N^{1} and N^{1} → L_{1}N^{1} → N^{2}. These also induce the connecting homomorphisms δ : E_{2}^{s}(L_{2}N^{1}) → E_{2}^{s+1}(L_{2}S^{0}) and δ′ : E_{2}^{s}(L_{2}N^{2}) → E_{2}^{s+1}(L_{2}N^{1}) of E_{2}-terms. Now define elements of the E_{2}-term E_{2}^{∗}(L_{2}S^{0}) by

α_{a/b} = δ(v_{1}^{a}/3^{b}), β′_{1} = h_{11} − v_{1}^{2}h_{10},
β_{a/b,c} = δδ′(v_{2}^{a}/3^{c}v_{1}^{b}),
c_{3^{n}s} = δδ′(v_{2}^{3^{n}s}h_{10}/3^{n+1}v_{1}) for 3 |̸ s,
α̃_{1}β_{a/b,c} = δδ′(v_{2}^{a}h_{10}/3^{c}v_{1}^{b}),
β(a)_{b/c,d}^{∗} = δδ′(v_{2}^{b}ξ_{a}/3^{d}v_{1}^{c}),
χ_{a}^{0} = δδ′(v_{2}^{a}ψ_{0}/3v_{1}), and
χ_{a}^{1} = δδ′(v_{2}^{a}ψ_{1}/3v_{1}).

Then \overline{A_{1}} and Ã_{2} are isomorphic to G_{1} and G_{2}, respectively. Since π_{∗}(L_{0}S^{0}) = Q and π_{∗}(L_{1}N^{1}) = Q/Z_{(3)}⨂Λ(y)⨁Al, an easy diagram chasing with Collorary 4.4 enables us to obtain G_{0} from \overline{A_{0}}, and shows Theorem A. Here Al is the Z_{(3)}-module generated by v_{1}^{3^{i}s}/3^{i+1} for i ≥ 0 and 3 |̸ s ∈ Z. q.e.d.

5. Computations in the cobar complex

In this section we work on the cobar complex (cf. [3]) based on the Hopf algebroid (E(2)_{∗}, E(2)_{∗}(E(2))) in order to study the connecting homomorphism δ : \overline{A_{1}}^{s} → A_{1}^{s+1}. The structure maps η_{R} : E(2)_{∗} → E(2)_{∗}(E(2)) and Δ : E(2)_{∗}(E(2)) → E(2)_{∗}(E(2)) behave as follows:

η_{R}(v_{1}) = v_{1} + 3t_{1}
η_{R}(v_{2}) = v_{2} + v_{1}t_{1}^{3} − t_{1}η_{R}(v_{1})^{3} − 3v_{1}t_{1}(v_{1}^{2} + 3v_{1}t_{1} + 3t_{1}^{2})
Δ(t_{1}) = t_{1} ⊗ 1 + 1 ⊗ t_{1}
Δ(t_{2}) = t_{2} ⊗ 1 + t_{1} ⊗ t_{1}^{2} + v_{1}b_{0}
Δ(t_{3}) ≡ t_{3} ⊗ 1 + t_{2} ⊗ t_{1}^{9} + t_{1} ⊗ t_{2}^{3} + 1 ⊗ t_{3} + v_{2}b_{1} − v_{1}b_{20} mod (9, v_{1}^{2}),

where 3b_{i} = t_{1}^{3^{i+1}} ⊗ 1 + 1 ⊗ t_{1}^{3^{i+1}} − (t_{1} ⊗ 1 + 1 ⊗ t_{1})^{3^{i+1}} and 3b_{20} = (t_{2}^{3} ⊗ 1 + t_{1}^{3} ⊗ t_{1}^{9} + 1 ⊗ t_{2}^{3}) − (t_{2} ⊗ 1 + t_{1} ⊗ t_{1}^{3} + 1 ⊗ t_{2})^{3}. Furthermore, we have the relations in E(2)_{∗}(E(2)) by setting η_{R}(v_{i}) = 0 in BP_{∗}(BP) for i > 2 such as v_{2}t_{i−2}^{9} ≡ v_{2}^{3^{i}}t_{i−2} mod (3, v_{1}) and v_{2}t_{1}^{9} ≡ v_{2}^{3}t_{1} − v_{1}t_{2}^{3} mod (3, v_{1}^{2}). For an E(2)_{∗}(E(2))- comodule M with structure map induced from η_{R}, the cobar complex is a family of E(2)_{∗}-modules Ω^{s}M = M⨂_{E(2)_{∗}}E(2)_{∗}(E(2))⨂_{E(2)_{∗}} ⋯ ⨂_{E(2)_{∗}}E(2)_{∗}(E(2)) (s factors) with differential d : Ω^{s}M → Ω^{s+1}M defined by d(m ⊗ x) = η_{R}(m)⊗x+ ∑_{i=1}^{s}(−1)^{i}m⊗Δ_{i}(x)−(−1)^{s}m⊗x⊗1 for m ∈ M and x ∈ Ω^{s}M, where Δ_{i}(x_{1} ⊗… ⊗ x_{n}) = x_{1} ⊗… ⊗ Δ(x_{i}) ⊗… ⊗ x_{n}.

Lemma 5.1. In the cobar complex Ω^{∗}E(2)_{∗}/(3, v_{1}^{3}), put t_{31} = v_{2}^{−6}t_{3}^{3} − t_{1}^{3}t_{2}^{3}, and we have

d(t_{31}) = t_{1}^{6} ⊗ t_{1}^{9} − t_{1}^{3} ⊗ t_{2}^{3} − v_{2}^{3}t_{1}^{3} ⊗ z^{3} − v_{2}^{−3}b_{11}^{3}.

Here z = v_{2}^{−1}(t_{2} − t_{1}^{4}) + v_{2}^{−3}t_{2}^{3}.

Proof. This follows immediately from the computation:

d(v_{2}^{−6}t_{3}^{3}) = −v_{2}^{−6}t_{1}^{3} ⊗ t_{2}^{9} − v_{2}^{−6}t_{2}^{3} ⊗ t_{1}^{27} − v_{2}^{−3}b_{11}^{3},
d(−t_{1}^{3}t_{2}^{3}) = t_{1}^{6} ⊗ t_{1}^{9} + t_{1}^{3} ⊗ t_{1}^{12} + t_{1}^{3} ⊗ t_{2}^{3} + t_{2}^{3} ⊗ t_{1}^{3}
= t_{1}^{6} ⊗ t_{1}^{9} − v_{2}^{3}t_{1}^{3} ⊗ z^{3} + v_{2}^{−6}t_{1}^{3} ⊗ t_{2}^{9} − t_{1}^{3} ⊗ t_{2}^{3} + t_{2}^{3} ⊗ t_{1}^{3}.

q.e.d.

Lemma 5.2. There exists an element Y_{n} such that

d(Y_{n}) = −v_{1}^{4·3^{n−1}−1}σ_{n+1} ⊗ V^{3^{n}} − v_{1}^{7·3^{n−1}}v_{2}^{2·3^{n}}x^{3^{n}} mod (3, v_{1}^{8}),

for each n > 0. Here σ_{n} = t_{1}+v_{1}z^{3^{n}}, V is 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) and x is a cocycle whose leading term is −v_{2}^{10}t_{3}^{3}⊗t_{1}^{3}−v_{2}^{−3}t_{1}⊗t_{3}.

Proof. First note that V ≡ −v_{2}^{2}t_{1}^{3} − v_{1}v_{2}t_{1}^{6} mod (3, v_{1}^{2}). Recall the element Y of Ω^{1}E(2)_{∗} ([1, Th. 4.8]) such that Y ≡ σ_{2}η_{R}(v_{2}^{3})−v_{1}^{2}V +v_{1}^{3}v_{2}^{−2}t_{1}^{18}+ v_{1}^{4}v_{2}^{−27}(t_{1}^{9}t_{2}^{27} − t_{3}^{9}) mod (3, v_{1}^{5}) and

d(Y) ≡ v_{1}^{7}x^{3} mod (3, v_{1}^{8}).

Here x^{3} denotes a cocycle whose leading term is −v_{2}^{30}t_{3}^{9}⊗t_{1}^{9}−v_{2}^{−9}t_{1}^{3}⊗t_{3}^{3}, which represents v_{2}ξ ∈ H^{2,8}M_{2}^{0}. By use of Y, we define elements Y_{i} inductively by

Y_{1} = η_{R}(v_{2}^{6})Y + v_{1}^{5}v_{2}^{5}t_{2}^{3} − v_{1}^{6}v_{2}^{4}t_{31} + v_{1}^{5}v_{2}^{8}z^{3}
Y_{n} = Y_{n−1}^{3} + v_{1}^{4·3^{n−1}−4}v_{2}V^{3^{n}}

Since d(η_{R}(v)t) = d(t)Δη_{R}(v) − t ⊗ d(v) for v ∈ E(2)_{∗} and t ∈ E(2)_{∗}E(2), we compute mod (3, v_{1}^{8}),

d(η_{R}(v_{2}^{6})Y) = v_{1}^{7}v_{2}^{6}x^{3} − Y ⊗ (−v_{1}^{3}v_{2}^{3}t_{1}^{9} + v_{1}^{6}t_{1}^{18})
= v_{1}^{7}v_{2}^{6}x^{3} − v_{1}^{3}Y ⊗ v_{2}^{−3}V^{3} + v_{1}^{6}Y ⊗ t_{1}^{18}
= v_{1}^{7}v_{2}^{6}x^{3} + v_{1}^{6}σ_{2}η_{R}(v_{2}^{3}) ⊗ t_{1}^{18}

−v_{1}^{3}(σ_{2}η_{R}(v_{2}^{3}) − v_{1}^{2}V + v_{1}^{3}v_{2}^{−2}t_{1}^{18} + v_{1}^{4}v_{2}^{−27}(t_{1}^{9}t_{2}^{27} − t_{3}^{9})) ⊗ v_{2}^{−3}V^{3}

= v_{1}^{7}v_{2}^{6}x^{3} + v_{1}^{6}(\underline{v_{2}t_{1}^{9}}_{4} + v_{1}t_{2}^{3} + v_{1}z^{9}η_{R}(v_{2}^{3})) ⊗ t_{1}^{18}
−v_{1}^{3}σ_{2} ⊗ V^{3} − v_{1}^{5}(\underline{−v_{2}^{2}t_{1}^{3}}_{1} − \underline{v_{1}v_{2}t_{1}^{6}}_{2} + v_{1}^{2}v_{2}^{2}t_{1}) ⊗ v_{2}^{3}t_{1}^{9}
+\underline{v_{1}^{6}v_{2}^{−2}t_{1}^{18} ⊗ v_{2}^{3}t_{1}^{9}}_{4} + v_{1}^{7}v_{2}^{−27}(t_{1}^{9}t_{2}^{27} − t_{3}^{9})) ⊗ v_{2}^{3}t_{1}^{9}
d(v_{1}^{5}v_{2}^{5}t_{2}^{3}) = −\underline{v_{1}^{6}v_{2}^{4}t_{1}^{3} ⊗ t_{2}^{3}}_{3} − \underline{v_{1}^{5}v_{2}^{5}t_{1}^{3} ⊗ t_{1}^{9}}_{1}
d(−v_{1}^{6}v_{2}^{4}t_{31}) = −v_{1}^{7}v_{2}^{3}t_{1}^{3} ⊗ t_{31} − v_{1}^{6}v_{2}^{4}(\underline{t_{1}^{6} ⊗ t_{1}^{9}}_{2} − \underline{t_{1}^{3} ⊗ t_{2}^{3}}_{3} − \underline{v_{2}^{3}t_{1}^{3} ⊗ z^{3}}_{5} − \underline{v_{2}^{−3}b_{11}^{3}}_{4})
d(v_{1}^{5}v_{2}^{8}z^{3}) = −\underline{v_{1}^{6}v_{2}^{7}t_{1}^{3} ⊗ z^{3}}_{5} + v_{1}^{7}v_{2}^{6}t_{1}^{6} ⊗ z^{3},

in which each sum of the underlined terms with the same number amount to 0. So we redefine the cocycle −x^{3} by the cocycle that appears in the sum of above congruences to satisfy

d(Y_{1}) ≡ −v_{1}^{3}σ_{2} ⊗ V^{3} − v_{1}^{7}v_{2}^{6}x^{3}.

Here x^{3} has the same leading term −v_{2}^{30}t_{3}^{9} ⊗t_{1}^{9} −v_{2}^{−9}t_{1}^{3} ⊗t_{3}^{3} as the above cocyle x^{3}.

Turn now to the case n. We assume the case for n − 1. Then

d(Y_{n−1}^{3}) ≡ −v_{1}^{4·3^{n−1}−3}σ_{n}^{3} ⊗ V^{3^{n}} − v_{1}^{7·3^{n−1}}v_{2}^{2·3^{n}}x^{3^{n}}
d(v_{1}^{4·3^{n−1}−4}v_{2}V^{3^{n}}) = v_{1}^{4·3^{n−1}−3}(t_{1}^{3} − v_{1}^{2}t_{1}) ⊗ V^{3^{n}}

Note that σ_{n}^{3} − (t_{1}^{3} − v_{1}^{2}t_{1}) = v_{1}^{2}σ_{n+1}, and we have the case for n. q.e.d.

The following is also shown in [8, Prop. 4.4] which also holds at the prime 3. Here the elements y_{3^{n}s}, t_{1} ⊗ ζ, and g_{0} are our v_{2}^{3^{n}s}h_{10}, h_{10}ζ_{2} and v_{2}^{−2}b_{11}, respectively.

Proposition 5.3. For n ≥ 0 and s ∈ I,

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}^{3^{n}s−2}b_{11}/v_{1}.

We have the similar results to [9, Prop.s 7.5, 7.6, 7.8]: Proposition 5.4. Let s, n, i, 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}^{3^{i}k−2·3^{n−i−1}} + · · · .

2. If s = 9t − 1 and 3^{i}k ≤ 8 · 3^{n} + 2 · 3^{n−i+1}, 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} 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} 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} + ⋯
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 it shown.

Proof. Let z̃ denote the element given in [7] such that z̃ ≡ v_{2}^{−1}(t_{2} −t_{1}^{4})+ v_{2}^{−3}t_{2}^{3} mod (3, v_{1}) and d(z̃) ≡ 0 mod (3^{i}, v_{1}^{3^{i−1}k}) for any i, k > 0, and denote σ = t_{1} + v_{1}z̃. We also consider a cocycle

y_{j,l} = ∑_{k>0} ( \begin{matrix}k + j − 2 \\ k − 1 \end{matrix} ) \frac{−(−t_{1})^{k}}{3^{l−k+1}kv_{1}^{j+k−1}}

of Ω^{1}M_{0}^{2} ([3]). Put σ_{a,b} = y_{a,b} + z̃/3^{b}v_{1}^{a−1}, and we see that 3^{b−1}σ_{a,b} = σ/3v_{1}^{a} and

d(σ_{3^{i}k+1,i+2}) = kt_{1} ⊗ z̃/3v_{1}^{3^{i}k+1}.

Note that v_{2}^{3^{n}s}h_{10}/v_{1}^{3^{i}k+1} is represented by a cocycle c(3^{n}s/3^{i}k+1) = η_{R}(v_{2}^{3^{n}s})σ/v_{1}^{3^{i}k+1}+ w/v_{1}^{3^{i}k−3^{n}} for some w ∈ E(2)_{∗}(E(2)).

For i = 0 and k < 3^{n} − 1, we define c(3^{n}s/k + 1, l) = η_{R}(v_{2}^{3^{n}s})σ_{k+1,l} for an integer l > 0, and replace the generator v_{2}^{3^{n}s}h_{10}/v_{1}^{k+1} by the element represented by the cocycle c(3^{n}s/k +1). Since d(v_{2}^{3}) ≡ 3v_{1}V mod (9, v_{1}^{3}) by definition, we see that d(v_{2}^{3^{n}s}) ≡ −3^{i+1}sv_{1}^{3^{n−i−1}}v_{2}^{3^{n−i}(3^{i}s−1)}V^{3^{n−i−1}} mod (3^{i+2}, v_{1}^{3^{n−i}}). We compute in Ω^{2}M_{0}^{2}:

d(c(3^{n}s/k + 1, 2)) = d(η_{R}(v_{2}^{3^{n}s})σ_{k+1,2})
= kt_{1} ⊗ η_{R}(v_{2}^{3^{n}s})z̃/3v_{1}^{k+1} − σ_{k+1,2} ⊗ d(v_{2}^{3^{n}s})
= kv_{2}^{3^{n}s}t_{1} ⊗ z̃/3v_{1}^{k+1} + sv_{2}^{3^{n}(s−1)}σ ⊗ V^{3^{n−1}}/3v_{1}^{k+1−3^{n−1}}

whose second term is homologous to −v_{2}^{3^{n}s−3^{n−1}}x^{3^{n−1}}/3v_{1}^{k−2·3^{n−1}} by Lemma 5.2. Since ξ_{n} is represented by (−1)^{n}v_{2}^{−3^{n}}x^{3^{n}}, this represents (−1)^{n}v_{2}^{3^{n}s}ξ_{n−1}/3v_{1}^{k−2·3^{n−1}} as desired. If k ≥ 3^{n}, then the case i = 0 follows from the formula v_{1}^{3n+3}δ(v_{2}^{3^{n}s}h_{10}/3v_{1}^{3^{i}k+1}) = δ(v_{2}^{3^{n}s}h_{10}/3v_{1}^{3^{i}k−3^{n}−2}).

Suppose the case for i. Then δ(v_{2}^{3^{n}s}h_{10}/3^{i+1}v_{1}^{3^{i+1}k+1}) = 0 if 3^{i+1}k ≤ 2 · 3^{n−i−1}. Since we compute

\begin{matrix}d(η_{R}(v_{2}^{3^{n}s})σ_{3^{i+1}k+1,i+3}) = kv_{2}^{3^{n}s}t_{1} ⊗ z/3v_{1}^{3^{i+1}k+1} \\ −v_{2}^{3^{n−i−1}(3^{i+1}s−1)}σ ⊗ V^{3^{n−i−2}}/3v_{1}^{3^{i+1}k+1−3^{n−i−2}} \end{matrix}

in Ω^{2}M_{0}^{2}, which shows the case for i + 1, and we obtain inductively the first part by Lemma 5.2. Thus if we denote a cocycle that represents v_{2}^{a}h_{10}/3^{c}v_{1}^{b} by c(a/b, c), then

(5.5)  \begin{matrix}d(c(3^{n}s/3^{i}k + 1, i + 2)) = kv_{2}^{3^{n}s}t_{1} ⊗ z/3v_{1}^{3^{i}k+1} \\ −(−1)^{n−i}v_{2}^{3^{n}s}x(n − i − 1)/3v_{1}^{3^{i}k−2·3^{n−i−1}}, \end{matrix}

where x(n) = (−1)^{n}v_{2}^{−3^{n}}x^{3^{n}} and so σ ⊗ V^{3^{n}} = (−1)^{n+1}v_{1}^{3^{n}+1}v_{2}^{3^{n+1}}x(n) up to homology.

Consider the case s = 9t − 1. The proof of [9, Lemma 7.7] works also at the prime 3 and we obtain:

(5.6) The element v_{2}^{3^{n}(9t−1)}h_{10}/3v_{1}^{3^{i}k+1} of H^{1}M_{0}^{2} is represented by a cochain c(3^{n}s/3^{i}k + 1, 1) = d(x_{n+2}^{t})/9tv_{1}^{4·3^{n}+3^{i}k} − c(3^{n+1}(3t − 1)/3^{i}k − 8 · 3^{n} + 1, 2).

In [3], they introduce the elements x_{i} ∈ E(2)_{∗} such that x_{i} ≡ v_{2}^{3^{i}} mod (3, v_{1}) and give the formulas on d(x_{i}). With a detail computaiton, we see that these elements satisfy d(x_{i}) ≡ v_{1}^{a_{i}}v_{2}^{2·3^{i−1}}σ_{n−1} mod (3, v_{1}^{2·3^{n}−1}) for i ≥ 2. We then compute with (5.5)

d(d(x_{n+2}^{t})/3^{i+2}tv_{1}^{4·3^{n}+3^{i}k}) = −kt_{1} ⊗ d(x_{n+2}^{t})/3tv_{1}^{4·3^{n}+3^{i}k+1}

= −kt_{1} ⊗ v_{2}^{3^{n+1}(3t−1)}σ/3v_{1}^{3^{i}k+1−8·3^{n}} + ⋯,

d(−kv_{2}^{3^{n+1}(3t−1)}t_{1}^{2}/3v_{1}^{3^{i}k+1−8·3^{n}}) = −kv_{2}^{3^{n+1}(3t−1)}t_{1} ⊗ t_{1}/3v_{1}^{3^{i}k+1−8·3^{n}}, d(c(3^{n+1}(3t − 1)/3^{i}k + 1 − 8 · 3^{n}, i + 2))

= kv_{2}^{3^{n+1}(3t−1)}t_{1} ⊗ z/3v_{1}^{3^{i}k+1−8·3^{n}} + (−1)^{n−i}v_{2}^{3^{n+1}(3t−1)}x(n − i)/3v_{1}^{3^{i}k−8·3^{n}−2·3^{n−i}}.

These amount to

d(c(3^{n}(9t − 1)/3^{i}k + 1, i + 1)) = (−1)^{n−i}v_{2}^{3^{n+1}(3t−1)}x(n − i)/3v_{1}^{3^{i}k−8·3^{n}−2·3^{n−i}}.

We also see the case n = i = 1 in the same manner, and we have the part 2. The parts 3 and 4 follow immediately from (5.6) and computation

d(d(x_{n+2}^{t})/3^{n+3}tv_{1}^{(4+k)3^{n}}) = (4 + k)t_{1} ⊗ d(x_{n+2}^{t})/9tv_{1}^{(4+k)3^{n}+1}.

In the same way, we obtain the part 5 by computing d(d(x_{n+2}^{t})/3^{n+4}tv_{1}^{(4+k)3^{n}}) for k = 2, 8 and d(d(x_{n+2}^{t})/3^{n+5}tv_{1}^{3^{n+2}}) for k = 5. q.e.d.

These imply that \overline{A_{1}}^{1} = H̃⨁H̃I⨁X̃_{1}ζ_{2}, and Propositions 5.3 and 5.4 show that the cokernel of δ : \overline{A_{1}}^{1} → A_{1}^{2} is isomorphic to X_{2}^{∗}.

Proposition 5.7. For an element v_{2}^{3^{n+l}s}ξ_{n}/v_{1}^{3^{i}k} of X_{2}^{∗}, the connecting homomorphism δ : \overline{A_{1}}^{2} → A_{1}^{3} acts as follows:

δ(v_{2}^{3^{n+l}s}ξ_{n}/3^{i+1}v_{1}^{3^{i}k}) = ±k(v_{2}^{3^{n+l}s}ξ_{n}ζ_{2}/v_{1}^{3^{i}k} + v_{2}^{3^{n+l}s−1}ψ_{1}/v_{1}^{3^{i}k} + ⋯).

Proof. Let c ∈ Ω^{2}M_{1}^{1} denote a cocycle that represents v_{2}^{3^{n+l}s}ξ_{n}/v_{1}^{3^{j}m} which is in the image of δ : \overline{A}_{1}^{1} → A_{1}^{2} with 3^{i}k ≤ 3^{j}m ≤ 4 · 3^{n} and j > i.

Since the cocycle c/3 ∈ Ω^{2}M_{0}^{2} is bounded, we have a cochain u ∈ Ω^{1}M_{0}^{2} such that d(u) = c/3. Then v_{2}^{3^{n+l}s}ξ_{n}/v_{1}^{3^{i}k} is represented by c′ = v_{1}^{3^{j}m−3^{i}k}c and so c′/3^{i+2} = (v_{1}^{3^{i}m−3^{i}k}/3^{i+1})d(u). Therefore, we compute in the cobar complex Ω^{3}M_{0}^{2},

d(c′/3^{i+2}) = d(1/3^{i+1}v_{1}^{3^{i}k−3^{j}m})d(u)
= −kt_{1} ⊗ c/3v_{1}^{3^{i}k−3^{j}m+1},

which represents ±k(v_{2}^{3^{n+l}s}ξ_{n}ζ_{2}/3v_{1}^{3^{i}k} + v_{2}^{3^{n+l}s−1}ψ_{1}/3v_{1}^{3^{i}k} + ⋯) by [7, Lemma 3.9] as desired. q.e.d.

References

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

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