Subgroups of π_{∗}(L_{2}T (1)) at the prime two

Ippei Ichigi and Katsumi Shimomura

Abstract. Let T (1) be the Ravenel spectrum whose BP_{∗}-homology is BP_{∗}[t_{1}](⊂ BP_{∗}(BP)), and let L_{2} denote the Bousfield localization functor with respect to v_{2}^{−1}BP. In this paper, we show that the E_{4}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{2}T (1)) has horizontal vanishing line and is the E_{∞}-term. We also find subgroups of the homotopy groups π_{∗}(L_{2}T (1)).

1. Introduction

In this paper, everything is localized at the prime two. Let BP denote the Brown-Peterson ring spectrum at the prime two. Then the homotopy groups π_{∗}(BP) turn to the polynomial algebra BP_{∗} = Z_{(2)}[v_{1}, v_{2}, …] over the Hazewinkel generators v_{i} with |v_{i}| = 2^{i+1} − 2. The Ravenel spectrum T (1) is characterized by the Brown-Peterson homology as BP_{∗}(T (1)) = BP_{∗}[t_{1}] ⊂ BP_{∗}(BP) = BP_{∗}[t_{1}, t_{2}, …]. We consider the spectrum G = v_{2}^{−1}BP. Let L_{2} denote the Bousfield localization functor on the stable homotopy category of spectra with respect to G. One of the methods to determine the homotopy groups π_{∗}(L_{2}T (1)) is the Adams-Novikov spectral sequence E_{2}^{∗} = H^{∗}v_{2}^{−1}BP_{∗}[t_{1}] ⇒ π_{∗}(L_{2}T (1)), where H^{∗}− = Ext_{G_{∗}G}^{∗}(G_{∗}, −). We study the E_{2}-term by the chromatic spectral sequence ∑_{i=0}^{2} H^{∗}M_{0}^{i}[t_{1}] ⇒ H^{∗}v_{2}^{−1}BP_{∗}[t_{1}] and the mod 2 Bockstein spectral sequences H^{∗}M_{1}^{0}[t_{1}] ⇒ H^{∗}M_{0}^{1}[t_{1}] and H^{∗}M_{1}^{1}[t_{1}] ⇒ H^{∗}M_{0}^{2}[t_{1}]. Here, M_{0}^{0} = 2^{−1}BP_{∗}, M_{1}^{0} = v_{1}^{−1}BP_{∗}/(2), M_{0}^{1} = v_{1}^{−1}BP_{∗}/(2^{∞}), M_{1}^{1} = v_{2}^{−1}BP_{∗}/(2, v_{1}^{∞}) and M_{0}^{2} = v_{2}^{−1}BP_{∗}/(2^{∞}, v_{1}^{∞}). The modules H^{∗}M_{0}^{0}[t_{1}] and H^{∗}M_{1}^{0}[t_{1}] are given by Ravenel in [7]. In [5], Mahowald and the second author determined H^{∗}M_{2}^{0}[t_{1}] as the tensor product of the polynomial algebra K(2)_{∗}[v_{3}, h_{20}] and the exterior algebra Λ(h_{21}, h_{30}, h_{31}, ρ_{2}), where K(2)_{∗} = Z/2[v_{2}^{±1}]. In [8], the second author determined H^{∗}M_{1}^{1}[t_{1}] by the v_{1}-Bockstein spectral sequence H^{∗}M_{2}^{0}[t_{1}] ⇒ H^{∗}M_{1}^{1}[t_{1}] to be the tensor product of Λ(ρ_{2}) and the direct sum of modules A_{i}:

A_{0} = (v_{1}^{−1}K/K⨁ ∑_{n>1} x_{n}K/(v_{1}^{a_{n}})[x_{n+1}]⨂Λ(g_{n+1})) ⨂Λ(h̃_{20}),
A_{1} = v_{3}^{2}K/(v_{1}^{2})[x_{2}]⨂Λ(h_{30}, h_{31}) and
A_{2} = v_{3}K(2)_{∗}[v_{3}^{2}, h_{20}]⨂Λ(h_{21}, h_{30}, h_{31}).

Here K = Z/2[v_{1}, v_{2}^{±1}], a_{n} denotes the integer 2^{n} + \frac{2}{3}(2^{n} − 2^{ϵ(n)}) for ϵ(n) = (1 − (−1)^{n})/2, and the elements x_{n}, g_{n}, h_{i,j} and h̃_{20} denote the cohomology classes represented by the cocycles of the cobar

complex Ω_{G_{∗}G}^{∗}G_{∗}[t_{1}]/(2, v_{1}^{j}) for a suitable j > 0, whose leading terms are v_{3}^{2^{n}}, v_{3}^{4(2^{n−2}−2^{ε(n)})/3}t_{3}^{2^{ε(n)}}, t_{i}^{2^{j}} and v_{3}^{2}t_{2}, respectively. Consider the submodule

A_{21} = v_{3}K_{∗}^{2}[v_{3}^{2}]⨂Λ(h_{21}, h_{30}, h_{31}) ⊂ A_{2},

and put A_{2}^{0} = A_{2}/A_{21} as a module. We see that there is a submodule

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

of H^{∗}M_{0}^{2}[t_{1}], where K_{∗}^{2} = Z/2[v_{2}^{±2}] and x ∈ Ã_{2} is considered to be x/2v_{1} ∈ H^{∗}M_{0}^{2}[t_{1}]. Then we show that the map φ: H^{∗}M_{1}^{1}[t_{1}] → H^{∗}M_{0}^{2}[t_{1}] given by φ(x) = x/2 is restricted to φ: A_{2}^{0} → Ã_{2} and then the sequence 0 → (Ã_{2})^{s−1} →^{δ} (A_{2}^{0})^{s} →^{φ} (Ã_{2})^{s} → 0 for each s > 3 is exact, where (M)^{s} denotes the submodule of M consisting of elements of cohomology dimension s, and δ is the connecting homomorphism associated to the short exact sequence 0 → M_{1}^{1}[t_{1}] → M_{0}^{2}[t_{1}] → M_{0}^{2}[t_{1}] → 0. This shows our first result.

Theorem 1.1. H^{s}M_{0}^{2}[t_{1}] is isomorphic to (Ã_{2}⨂Λ(ρ_{2}))^{s} for s > 4.

Furthermore, we show that the mod 2 Bockstein spectral sequence splits (see Lemma 3.6). A summand of the spectral sequence is A_{2}^{0} ⇒ Ã_{2}. It seems very complicated to determine the other parts A_{1} = (A_{0}⨁A_{1}⨁A_{21})⨂Λ(ρ_{2}) ⇒ Ã_{1} (cf. [6], [2], [9]).

Let W be the spectrum such that BP_{∗}(L_{2}W) = M_{0}^{2}. Indeed, W is the cofiber of the localization map V → L_{1}V, where V is the cofiber of the localization map S^{0} → SQ. Then H^{∗}M_{0}^{2}[t_{1}] is isomorphic to the E_{2}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{2}W ∧ T (1)). We consider the submodule

Ã_{21} = v_{3}^{3}K_{∗}^{2}[v_{3}^{4}]⨂Λ(h_{30}, h_{31}) ⊂ H^{∗}M_{0}^{2}[t_{1}],

and see that Ã_{21}⨂Λ(ρ_{2}) ⊂ Ã_{1} (see Corollary 4.4). We write Ã_{1}^{0} = Ã_{1}/(Ã_{21}⨂Λ(ρ_{2})) as a module. We compute the differentials of the Adams-Novikov spectral sequence on Ã_{2} and Ã_{21}, and then show that the differentials on Ã_{1}^{0} are zero after a modification of Ã_{1}^{0} (see Corollary 4.8).

Theorem 1.2. The Adams-Novikov E_{∞}-term for the homotopy groups π_{∗}(L_{2}T (1) ∧ W) is isomorphic to the direct sum of Ã_{1}^{0} and Â_{2}⨂Λ(ρ_{2}), where

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

Note that we do not determine the structure of Ã_{1}^{0} of the theorem, though we know that the Adams-Novikov differentials are trivial on it.

By the definition of W, we have the composite η: W → ΣV → S^{2}, which induces the composite of connecting homomorphisms η_{∗}: H^{s}M_{0}^{2}[t_{1}] → H^{s+1}v_{2}^{−1}BP_{∗}/(2^{∞})[t_{1}] → H^{s+2}v_{2}^{−1}BP_{∗}[t_{1}] in the long exact sequences

\begin{matrix}H^{s}M_{0}^{1}[t_{1}] → H^{s}M_{0}^{2}[t_{1}] →^{δ} H^{s+1}v_{2}^{−1}BP_{∗}/(2^{∞})[t_{1}] → H^{s+1}M_{0}^{1}[t_{1}] and \\ H^{s}M_{0}^{0}[t_{1}] → H^{s}v_{2}^{−1}BP_{∗}/(2^{∞})[t_{1}] →^{δ} H^{s+1}v_{2}^{−1}BP_{∗}[t_{1}] → H^{s+1}M_{0}^{0}[t_{1}]. \end{matrix}

Since we see that both of H^{s}M_{0}^{0}[t_{1}] and H^{s}M_{0}^{1}[t_{1}] are zero for s > 0 (Theorem 2.5), we see that the connecting homomorphisms are isomorphisms for s > 0, and so is η_{∗}. In Proposition 4.7, we show that the E_{4}-term is the E_{∞}-term. Since η_{∗} is a map of spectral sequences, we have the results on π_{∗}(L_{2}T (1)).

Corollary 1.3. The Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(L_{2}T (1)) collapses from the E_{4}-term.

Corollary 1.4. The homotopy groups π_{∗}(L_{2}T (1)) contain the subgroups isomorphic to Â_{2}⨂Λ(ρ_{2}), which is the image of Â_{2}⨂Λ(ρ_{2}) under the map η_{∗}: π_{∗}(L_{2}T (1) ∧ W) → π_{∗}(L_{2}T (1)).

In the next section, we show that H^{s}M_{0}^{1}[t_{1}] is zero for s > 0 by determining it. In sections 3 and 4, we give proofs of Theorems 1.1 and 1.2, respectively. The authors would like to thank Professor Xiangjun Wang who pointed out mistakes in Lemmas 3.3 and 4.3 in a draft version of this paper.

2. H^{∗}M_{0}^{1}[t_{1}]

Let BP denote the Brown-Peterson spectrum at the prime two. Then BP_{∗} = Z_{(2)}[v_{1}, v_{2}, …] and BP_{∗}(BP) = BP_{∗}[t_{1}, t_{2}, …], and (BP_{∗}, BP_{∗}(BP)) is a Hopf algebroid. Hereafter, we write

H^{∗}M = Ext_{BP_{∗}(BP)}^{∗}(BP_{∗}, M)

for a BP_{∗}(BP)-comodule M. Consider the BP_{∗}(BP)-comodule M_{1}^{0} = v_{1}^{−1}BP_{∗}/(2). Then in [7, Th. 6.1.1 and Cor. 6.5.6], it is shown that

H^{∗}M_{1}^{0}[t_{1}] = K(1)_{∗}[v_{2}]⨂Λ(h_{20}).

Here H^{∗}M for a BP_{∗}(BP)-comodule M denotes Ext_{BP_{∗}(BP)}^{∗}(BP_{∗}, M), K(1)_{∗} = Z/2[v_{1}^{±1}] and h_{20} is the element represented by a cocycle of the cobar complex whose leading term is t_{2}. Consider the Hopf algebroid (A, Γ) = (BP_{∗}, BP_{∗}[t_{2}, t_{3} ⋯]), whose structure maps are induced from those of BP_{∗}(BP) under the projection BP_{∗}(BP) → Γ. We then have the change of rings theorem

H^{∗}M[t_{1}] = Ext_{Γ}^{∗}(A, M),

for a BP_{∗}(BP)-comodule M.

Lemma 2.1. In the Hopf algebroid (A, Γ),

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

Proof. This is based on the Hazewinkel’s and the Quillen’s formulas:

v_{n} = 2m_{n} − ∑_{i=1}^{n−1} m_{i}v_{n−i}^{2^{i}} ∈ Q⨂BP_{∗} = Q[m_{1}, m_{2}, …] and η_{R}(m_{n}) = ∑_{i=0}^{n} m_{i}t_{n−i}^{2^{i}} ∈ Q⨂BP_{∗}(BP).

We consider it in Q⨂BP_{∗}[t_{2}, t_{3}, …]. Then η_{R}(v_{1}) = 2η_{R}(m_{1}) = 2m_{1} = v_{1}, and η_{R}(v_{2}) = 2η_{R}(m_{2}) − m_{1}v_{1}^{2} = 2(m_{2} + t_{2}) − m_{1}v_{1}^{2} = v_{2} + 2t_{2}. For η_{R}(v_{3}), we compute

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

We define x_{1,n} ∈ v_{1}^{−1}A = v_{1}^{−1}BP_{∗} by

x_{1,0} = v_{2}, x_{1,1} = x_{1,0}^{2} + 2v_{1}^{3}v_{2} + 4v_{1}^{−1}v_{3}, and x_{1,n} = x_{1,n−1}^{2}.

Let d: v_{1}^{−1}A → v_{1}^{−1}A⨂_{A}Γ denote η_{R} − η_{L}. Then we have

Lemma 2.2. Let x_{1,i} be the elements defined above. Then we see that d(x_{1,n}) ≡ 2^{n+1}X_{n}t_{2} mod (2^{n+2}) for n ≥ 0, where X_{0} = 1 and X_{n} = x_{1,0}x_{1,1} ⋯ x_{1,n−1} for n > 0.

Proof. For n = 0, it follows from Lemma 2.1. For n = 1, we obtain the equation from the computations:

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

Here, the underlined terms with the same subscript cancel out.

Inductively, suppose that d(x_{1,n}) ≡ 2^{n+1}X_{n}t_{2} mod (2^{n+2}). Then

d(x_{1,n}^{2}) ≡ (x_{1,n} + 2^{n+1}X_{n}t_{2})^{2} − x_{1,n}^{2} mod (2^{n+3})
≡ 2^{n+2}x_{1,n}X_{n}t_{2} mod (2^{n+3}),

and obtain the congruence for n + 1. □

Lemma 2.3. H^{0}M_{0}^{1}[t_{1}] is the tensor product of Z_{(2)}[v_{1}, v_{1}^{−1}] and the direct sum of Q/Z_{(2)} and Z/(2^{n+1}) generated by x_{1,n}^{s}/2^{n+1} for each n ≥ 0 and odd s > 0.

Proof. Let B denote the module of the lemma. Then we have a sequence H^{∗}M_{1}^{0}[t_{1}] →^{φ} B →^{2} B fitting in the commutative diagram

0  H^{0}M_{1}^{0}[t_{1}] H^{0}M_{0}^{1}[t_{1}] H^{0}M_{0}^{1}[t_{1}] H^{1}M_{1}^{0}[t_{1}] 0  H^{0}M_{1}^{0}[t_{1}]  B  B  H^{1}M_{1}^{0}[t_{1}]. w  ^{φ} w  ^{2} w  ^{δ} w w  _{^{φ} w} u z_{i}  _{^{2}  w} u z_{i}  _{^{δ} w}

Here φ(x) = x/2. If the bottom sequence is exact, then the inclusion i is an isomorphism by [4, Remark 3.11]. To see the exactness, it suffices to show that Ker δ ⊂ Im 2, which is seen by δ(x_{1,n}^{s}/2^{n+1}) = v_{2}^{2^{n}(s−1)+2^{n}−1}h_{20} for odd s > 0. □

Corollary 2.4. The image of φ: H^{1}M_{1}^{0}[t_{1}] → H^{1}M_{0}^{1}[t_{1}] is zero.

Proof. Note that each integer s ≥ 0 is expressed uniquely as 2^{n+1}t + 2^{n} − 1 for some t, n ≥ 0. Therefore, each generator v_{2}^{s}h_{20} ∈ H^{1}M_{1}^{0}[t_{1}] for s ≥ 0 is the image of x_{1,n}^{2t+1}/2^{n+1} under δ. □

Theorem 2.5. H^{s}M_{0}^{1}[t_{1}] = 0 for s > 0.

3. Proof of Theorem 1.1

We will study H^{s}M_{0}^{2}[t_{1}] for s ≥ 0 by using the exact sequence

(3.1)  ⋯ ⟶ H^{s}M_{1}^{1}[t_{1}] ⟶^{φ} H^{s}M_{0}^{2}[t_{1}] ⟶^{2} H^{s}M_{0}^{2}[t_{1}] ⟶^{δ} H^{s+1}M_{1}^{1}[t_{1}] ⟶ ⋯

associated to the short exact sequence

(3.2)  0 ⟶ M_{1}^{1}[t_{1}] ⟶^{φ} M_{0}^{2}[t_{1}] ⟶^{2} M_{0}^{2}[t_{1}] ⟶ 0,

where φ(x) = x/2. Here, H^{∗}M = Ext_{BP_{∗}(BP)}^{∗}(BP_{∗}, M) as before. Consider the submodules

A_{2} = v_{3}K(2)_{∗}[v_{3}^{2}, h_{20}]⨂Λ(h_{21}, h_{30}, h_{31}) and A_{21} = v_{3}K_{∗}^{2}[v_{3}^{2}]⨂Λ(h_{21}, h_{30}, h_{31}),

of H^{∗}M_{1}^{1}[t_{1}], where K(2)_{∗} = Z/2[v_{2}^{±1}], K_{∗}^{2} = Z/2[v_{2}^{±2}] and an element x of the modules is considered to be an element x/v_{1} of H^{∗}M_{1}^{1}[t_{1}]. Put A_{2}^{0} = A_{2}/A_{21} as a module. Then, it is shown in [8, Th. 6.13] that

H^{s}M_{1}^{1}[t_{1}] = (A_{2}^{0}⨂Λ(ρ_{2}))^{s}

for s > 4 and H^{4}M_{1}^{1}[t_{1}] = (A_{2}^{0}⨂Λ(ρ_{2}))^{4}⨁v_{3}K_{∗}^{2}[v_{3}^{2}]{h_{21}h_{30}h_{31}ρ_{2}}, where (M)^{s} denotes the submodule of M consisting of elements of cohomology dimension s.

The exact sequence (3.1) defines the Bockstein spectral sequence H^{∗}M_{1}^{1}[t_{1}] ⇒ H^{∗}M_{0}^{2}[t_{1}]. The differential d_{1} is defined to be d_{1} = δφ: H^{s}M_{1}^{1}[t_{1}] → H^{s+1}M_{1}^{1}[t_{1}] for the maps δ and φ in (3.1). Then we have the following lemma.

Lemma 3.3. The differential d_{1} of the Bockstein spectral sequence acts on A_{2}^{0} as follows :

d_{1}(v_{2}^{2u+1}x) = v_{2}^{2u}xh_{20}

for an integer u and x ∈ A_{2}^{0} with v_{2} |̸ x.

Proof. Each cohomology class is represented as follows:

h_{20} = [t_{2}], h_{21} = [t_{2}^{2}], h_{30} = [t_{3}] and h_{31} = [t_{3}^{2}].

For the diagonal map Δ, Quillen’s formula Δ(t_{n}) = Ψ_{0}(n) + ∑_{k=1}^{n} m_{k}(Ψ_{k}(n) − Δ(t_{n−k})^{p^{k}}) together with Hazewinkel’s formula shows that Δ(t_{2}) = Ψ_{0}(2) = t_{2}⊗1+1⊗t_{2} and Δ(t_{3}) = Ψ_{0}(3)−v_{1}t_{2}⊗t_{2} ≡ t_{3}⊗1+1⊗t_{3} mod (4, v_{1}), where Ψ_{k}(l) = ∑_{i=0}^{l} t_{i}^{p^{k}} ⊗ t_{l−i}^{p^{k+i}} and t_{0} = 1. Thus this together with Lemma 2.1 shows that

(3.4)  d(v_{2}) ≡ 2t_{2}, d(v_{3}) ≡ 2t_{3}, d(t_{2}^{2}) ≡ 2t_{2} ⊗ t_{2} and d(t_{3}^{2}) ≡ 2t_{3} ⊗ t_{3}

mod (4, v_{1}) in Ω^{∗}v_{2}^{−1}BP_{∗}. By the definition of the differential of the cobar complex, the element d(v_{2}^{2u+1}x/4) of Ω^{∗}M_{0}^{2}[t_{1}] is computed

d(v_{2}^{2u+1}x/4) = d(v_{2}^{2u+1})x/4 + v_{2}^{2u+1}d(x/4) = v_{2}^{2u}t_{2}x/2 + v_{2}^{2u+1}d(x)/4 = v_{2}^{2u}xh_{20}/2 + v_{2}^{2u+1}y/2,

where y is an element of Ω^{∗}v_{2}^{−1}BP_{∗}/(4, v_{1}^{∞})[t_{1}] such that d(x) = 2y. We see that y ≢ ±v_{2}^{−1}xh_{20} mod (4, v_{1}) by (3.4). Note here that t_{3}⊗t_{3} represents the cohomology class h_{31}(v_{2}^{−1}h_{20}+v_{2}^{−2}h_{21})+v_{2}^{−3}v_{3}^{2}h_{20}h_{21} (see [5, p. 243, (1)]). □

The lemma indicates that h_{21} is redefined as

h_{21} = [t_{2}^{2} + v_{2}t_{2}]

and gives rise to the differential pattern on A_{2}^{0}:

0  1  2  3  4  ⋯
v_{2}v_{3}h_{20} ↦  v_{3}h_{20}^{2}  v_{2}v_{3}h_{20}^{3}  ↦  v_{3}h_{20}^{4}
v_{2}v_{3} ↦ v_{3}h_{20}  v_{2}v_{3}h_{20}^{2} ↦  v_{3}h_{20}^{3}  v_{2}v_{3}h_{20}^{4}  ↦
v_{2}v_{3}h_{21} ↦ v_{3}h_{20}h_{21}  v_{2}v_{3}h_{20}^{2}h_{21} ↦  v_{3}h_{20}^{3}h_{21}
v_{2}v_{3}h_{20}h_{21} ↦  v_{3}h_{20}^{2}h_{21}  v_{2}v_{3}h_{20}^{3}h_{21}  ↦
v_{2}v_{3}h_{20}h_{3i} ↦  v_{3}h_{20}^{2}h_{3i}  v_{2}v_{3}h_{20}^{3}h_{3i}  ↦
v_{2}v_{3}h_{3i} ↦ v_{3}h_{20}h_{3i}  v_{2}v_{3}h_{20}^{2}h_{3i} ↦  v_{3}h_{20}^{3}h_{3i}
v_{2}v_{3}h_{21}h_{3i} ↦ v_{3}h_{20}h_{21}h_{3i}  v_{2}v_{3}h_{20}^{2}h_{21}h_{3i} ↦
v_{2}v_{3}h_{20}h_{21}h_{3i} ↦  v_{3}h_{20}^{2}h_{21}h_{3i}
v_{2}v_{3}h_{20}h_{30}h_{31} ↦  v_{3}h_{20}^{2}h_{30}h_{31}
v_{2}v_{3}h_{30}h_{31} ↦ v_{3}h_{20}h_{30}h_{31}  v_{2}v_{3}h_{20}^{2}h_{30}h_{31} ↦
v_{2}v_{3}h_{21}h_{30}h_{31} ↦ v_{3}h_{20}h_{21}h_{30}h_{31}
v_{2}v_{3}h_{20}h_{21}h_{30}h_{31} ↦

in which x ↦ y denotes the d_{1}(x/v_{1}) = y/v_{1} for x/v_{1}, y/v_{1} ∈ A_{2}^{0}.

Observe the long exact sequence (3.1), and note that the module Ã_{2} given in the introduction is Im φ. Then the above differential pattern shows that δ is a monomorphism on Ã_{2}, since Ã_{2} is generated by the elements at the tails of the arrows.

Lemma 3.5. The module Ã_{2} given in the introduction fits in the short exact sequence

0 ⟶ (Ã_{2})^{s−1} ⟶^{δ} (A_{2}^{0})^{s} ⟶^{φ} (Ã_{2})^{s} ⟶ 0

for s > 3.

Proof of Theorem 1.1. Since H^{s}M_{1}^{1}[t_{1}] = (A_{2}^{0}⨂Λ(ρ_{2}))^{s} for s > 4, we have the commutative diagram

(Ã_{2}⨂Λ(ρ_{2}))^{s−1}  (A_{2}^{0}⨂Λ(ρ_{2}))^{s}  (Ã_{2}⨂Λ(ρ_{2}))^{s}  (Ã_{2}⨂Λ(ρ_{2}))^{s}  (A_{2}^{0}⨂Λ(ρ_{2}))^{s+1} H^{s−1}M_{0}^{2}[t_{1}]  H^{s}M_{1}^{1}[t_{1}]  H^{s}M_{0}^{2}[t_{1}]  H^{s}M_{0}^{2}[t_{1}]  H^{s+1}M_{1}^{1}[t_{1}]^{δ} w u  ^{φ} w  ^{2=0} w_{u_{g}}  ^{δ} w_{u_{g}}_{^{δ} w}  _{^{φ}  w}  _{^{2}  w}  _{^{δ} w}

of exact sequences by Lemma 3.5. If we show that the images of the left δ’s agree, then the map g is an isomorphism by [4, Remark 3.11]. We denote the maps δ and φ in the top sequence by δ′ and φ′. Then Im δ′ ⊂ Im δ. For any x ∉ Im δ′, φ′(x) = x/2 ≠ 0 and δ′(x/2) ≠ 0, which shows g(x/2) ≠ 0 since δ′ = δg. Therefore, φ(x) = g(φ′(x)) = g(x/2) ≠ 0, and x ∉ Im δ. □

Lemma 3.6. The Bockstein spectral sequence H^{∗}M_{1}^{1}[t_{1}] ⇒ H^{∗}M_{0}^{2}[t_{1}] splits into two spectral sequences A_{1} = (A_{0}⨁A_{1}⨁A_{21})⨂Λ(ρ_{2}) ⇒ Ã_{1} and A_{2}^{0}⨂Λ(ρ_{2}) ⇒ Ã_{2}⨂Λ(ρ_{2}). Here, the module Ã_{1} denotes a module fitting in the long exact sequence

0 → (A_{1})^{0} →^{φ} (Ã_{1})^{0} →^{2} (Ã_{1})^{0} →^{δ} (A_{1})^{1} →^{φ} ⋯ →^{δ} (A_{1})^{s} →^{φ} (Ã_{1})^{s} →^{2} (Ã_{1})^{s} →^{δ} (A_{1})^{s+1} →^{φ} · · · .

Proof. By Lemma 3.5, we have the subspectral sequence A_{2}^{0}⨂Λ(ρ_{2}) ⇒ Ã_{2}⨂Λ(ρ_{2}). Furthermore, Lemma 3.5 implies that all elements of A_{2}^{0}⨂Λ(ρ_{2}) do not survive to the E_{2}-term of the Bockstein spectral sequence. It follows that the differential d_{r} acts on A_{1}. Now Ã_{1} is generated by elements x̃_{r} such that 2^{r−1}x̃_{r} = x̃_{1} = φ(x) and δ(x̃_{r})’ are linearly independent. □

Remark. Ã_{1} is not determined here. Even the 0-dimensional part (Ã_{1})^{0} of it is very complicated (see. [6], [9]), though (Ã_{1})^{s} = 0 for s > 4.

4. Proof of Theorem 1.2

Recall [8] the spectrum C such that BP_{∗}(C) = BP_{∗}/(2, v_{1}^{∞})[t_{1}]. Then C fits in the cofiber sequence

C ⟶^{φ} W ∧ T (1) ⟶^{2} W ∧ T (1) ⟶ ΣC,

which induces the short exact sequence

0 ⟶ M_{1}^{1}[t_{1}] ⟶^{φ} M_{0}^{2}[t_{1}] ⟶^{2} M_{0}^{2}[t_{1}] ⟶ 0,

by applying BP_{∗}(L_{2}−). Let E_{r}^{s,t}(X) denote the E_{r}-term of the v_{2}^{−1}BP -based Adams spectral sequence converging to π_{t−s}(L_{2}X). Then the E_{2}-term is Ext_{v_{2}^{−1}BP_{∗}(v_{2}^{−1}BP)}^{∗,∗}(v_{2}^{−1}BP_{∗}, v_{2}^{−1}BP_{∗}(X)), which is iso-

morphic to H^{∗}v_{2}^{−1}BP_{∗}(X) by the change of rings theorem of Hovey and Sadofsky [1, Th. 3.1]. Indeed, we use the modified one [3, Th. 3.3]. In our case, we consider the spectral sequences E_{2}^{∗}(C) = H^{∗}M_{1}^{1}[t_{1}] ⇒ π_{∗}(L_{2}C) and E_{2}^{∗}(W ∧ T (1)) = H^{∗}M_{0}^{2}[t_{1}] ⇒ π_{∗}(L_{2}W ∧ T (1)).

For the sake of simplicity, we compute differentials by setting v_{2}^{2} = 1. In [8, Lemma 7.4], it is shown that for any v_{3}^{4t+3}x/v_{1} ∈ E_{2}^{s,u}(C) ∩ A_{2},

(4.1)  d_{3}(v_{3}^{4t+3}x/v_{1}) = v_{3}^{4t+1}xh_{20}^{3}/v_{1} ∈ E_{2}^{s+3,u+2}(C).

The other differentials on E_{r}^{∗}(C) are trivial except for the differentials

(4.2)

d_{3}(x_{n}^{s}h̃_{20}/v_{1}^{a_{n}}) = \begin{cases}v_{3}^{2^{n}(s−1)+4(2^{n−2}−1)/3+1}h_{20}^{2}h_{21}h_{30}/v_{1} & n is even \\ v_{2}v_{3}^{2^{n}(s−1)+8(2^{n−3}−1)/3+1}h_{20}^{2}h_{21}h_{31}/v_{1} & n is odd \end{cases}  and

d_{3}(x_{n}^{s}g_{n+1}h̃_{20}/v_{1}^{a_{n}}) = \begin{cases}v_{3}^{2^{n}s−3}h_{20}^{2}h_{21}h_{30}h_{31}/v_{1} & n is even \\ v_{2}v_{3}^{2^{n}s−3}h_{20}^{2}h_{21}h_{30}h_{31}/v_{1} & n is odd \end{cases}

for n ≥ 2 and odd s > 0, and a v_{2}-multiple of them ([8, Lemmas 7.6 and 7.8]). Here h̃_{20} is defined as the class represented by the cocycle t̃_{2} in the congruence d(v_{3}^{4}) ≡ 2v_{1}^{2}t̃_{2} mod (4), whose leading term is v_{2}^{3}v_{3}^{2}t_{2}.

Lemma 4.3. In the Adams-Novikov E_{3}^{∗}-term for π_{∗}(W ∧ T (1)),

d_{3}(v_{3}^{3}x/2v_{1}) = v_{2}v_{3}xh_{21}h_{20}^{2}/2v_{1} and d_{3}(v_{2}v_{3}^{3}y/2v_{1}) = v_{2}v_{3}yh_{20}^{3}/2v_{1}

for x ∈ K_{∗}^{2}[v_{3}^{4}]⨂Λ(h_{30}, h_{31}) and y ∈ K_{∗}^{2}[v_{3}^{4}, h_{20}]⨂Λ(h_{21}, h_{30}, h_{31}), and

d_{3}(x_{n}^{s}h̃_{20}/2v_{1}^{a_{n}}) = \begin{cases}v_{2}v_{3}^{2^{n}(s−1)+4(2^{n−2}−1)/3+1}h_{20}^{3}h_{30}/2v_{1} & n is even \\ v_{2}v_{3}^{2^{n}(s−1)+8(2^{n−3}−1)/3+1}h_{20}^{2}h_{21}h_{31}/2v_{1} & n is odd, \end{cases}

d_{3}(x_{n}^{s}g_{n+1}h̃_{20}/2v_{1}^{a_{n}}) = \begin{cases}v_{2}v_{3}^{2^{n}s−3}h_{20}^{3}h_{30}h_{31}/2v_{1} & n is even \\ v_{2}v_{3}^{2^{n}s−3}h_{20}^{2}h_{21}h_{30}h_{31}/2v_{1} & n is odd, \end{cases}

d_{3}(v_{2}x_{n}^{s}h̃_{20}/2v_{1}^{a_{n}}) = \begin{cases}v_{2}v_{3}^{2^{n}(s−1)+8(2^{n−3}−1)/3+1}h_{20}^{2}h_{21}h_{31}/2v_{1} & n is even \\ v_{2}v_{3}^{2^{n}(s−1)+4(2^{n−2}−1)/3+1}h_{20}^{3}h_{30}/2v_{1} & n is odd \end{cases}  and

d_{3}(v_{2}x_{n}^{s}g_{n+1}h̃_{20}/2v_{1}^{a_{n}}) = \begin{cases}v_{2}v_{3}^{2^{n}s−3}h_{20}^{2}h_{21}h_{30}h_{31}/2v_{1} & n is even \\ v_{2}v_{3}^{2^{n}s−3}h_{20}^{3}h_{30}h_{31}/2v_{1} & n is odd \end{cases}

for positive integers s and n with n > 1. Here the equations are all up to sign.

Proof. Note that v_{3}xh_{20}^{3}/2v_{1} = v_{2}v_{3}xh_{21}h_{20}^{2}/2v_{1} in E_{3}^{∗}(W ∧T (1)), since δ(v_{2}v_{3}xh_{20}^{2}/2v_{1}) = v_{3}xh_{20}^{3}/v_{1} + v_{2}v_{3}xh_{21}h_{20}^{2}/v_{1} by Lemma 3.3. In the same manner as this, we have the relations v_{3}^{2^{n}(s−1)+4(2^{n−2}−2^{ε(n)})/3+1}

h_{20}^{2}h_{21}h_{3ε(n)}/2v_{1} = v_{2}v_{3}^{2^{n}(s−1)+4(2^{n−2}−2^{ε(n)})/3+1}h_{20}^{3}h_{3ε(n)}/2v_{1} and v_{3}^{2^{n}s−3}h_{20}^{2}h_{21}h_{30}h_{31}/2v_{1} = v_{2}v_{3}^{2^{n}s−3} h_{20}^{3}h_{30}h_{31}/2v_{1}, since h_{21}^{2} = h_{20}^{2}. Then the differentials in (4.1) and (4.2) of the form d_{3}(x) = y (resp. d_{3}(x) = v_{2}y) yield differentials d_{3}(x/2) = v_{2}z/2 and d_{3}(v_{2}x/2) = v_{2}y/2 (resp. d_{3}(x/2) = v_{2}y/2 and d_{3}(v_{2}x/2) = v_{2}z/2) of E_{3}^{∗}(W ∧ T (1)), where z is an element such that δ(w) = y − z ∈ H^{∗}M_{1}^{1}[t_{1}] for an element w of H^{∗}M_{0}^{2}[t_{1}]. □

Corollary 4.4. The module Ã_{21} given in Introduction is a submodule of H^{∗}M_{0}^{2}[t_{1}]. In other words, the map sending an element x ∈ Ã_{21} to x/2v_{1} ∈ H^{∗}M_{0}^{2}[t_{1}] is a monomorphism.

Proof. It suffices to show that x/2v_{1} ≠ 0 ∈ H^{∗}M_{0}^{2}[t_{1}] for x ∈ Ã_{21}. The first equation of Lemma 4.3 shows d_{3}(x/2v_{1}) ≠ 0. □

Corollary 4.5. After a suitable modification of Ã_{1}^{0}, the v_{2}^{−1}BP based Adams differentials d_{3} originating

in Ã_{1}^{0} are all zero.

Proof. The only non-trivial differentials originating in Ã_{1}^{0} are given in Lemma 4.3, and their targets are all in the image of d_{3} originating in (Ã_{2}⨁Ã_{21})⨂Λ(ρ_{2}). □

Remark. This modification of Ã_{1}^{0} does not change the additive structure of Ã_{1}^{0} nor the E_{2}-term H^{∗}M_{0}^{2}[t_{1}]. In fact, each generator x ∈ Ã_{1}^{0} is just replaced by x + y for some y ∈ (Ã_{2}⨁Ã_{21})⨂Λ(ρ_{2}).

Theorem 4.6. The E_{4}-term of the v_{2}^{−1}BP based Adams spectral sequence contains Â_{2}⨂Λ(ρ_{2}), which is obtained from the subgroup Ã_{2}⨂Λ(ρ_{2}) of the E_{2}-term. Here, Â_{2} is the module given in Theorem 1.2.

Proof. The v_{2}^{−1}BP based Adams differential d_{3} makes (Ã_{2}, d_{3}) a differential module by Lemma 4.3, whose homology is

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

We decompose Â_{2}′ into the direct sum of the two modules

\begin{aligned}Â_{21}′ & = & v_{2}v_{3}K_{∗}^{2}[v_{3}^{4}]⨂Λ(h_{20}, h_{21}, h_{30}, h_{31})⨁v_{2}v_{3}h_{20}^{2}K_{∗}^{2}[v_{3}^{4}]⨂Λ(h_{30}, h_{31}) and \\ Â_{22}′ & = & v_{2}v_{3}h_{20}^{2}h_{21}K_{∗}^{2}[v_{3}^{4}]⨂Λ(h_{30}, h_{31}). \end{aligned}

The first differential in Lemma 4.3 gives the isomorphism d_{3} : Ã_{21} ≅ Â_{22}′, and we obtain the theorem by setting

Â_{2} = Â_{21}′. □

Proposition 4.7. The v_{2}^{−1}BP based Adams spectral sequence converging to π_{∗}(W ∧ T (1)) collapses from E_{4}-term. That is, E_{4}^{∗} = E_{∞}^{∗}.

Proof. Since (Ã_{1})^{s} = 0 for s > 4 and (Â_{2}⨂Λ(ρ_{2}))^{s} = 0 for s > 5, we see that E_{5}^{s} = 0 for s > 5. Therefore, the differentials d_{r} are all trivial for r > 5. Suppose that d_{5}(x/2^{l}) = y/2 for x/2^{l} ∈ Ã_{1}. Then y/2 ∈ Â_{21}′, and so δ(y/2) ≠ 0 ∈ E_{2}^{6}(C). Send the relation d_{5}(x/2^{l}) = y/2 by δ, and we see that d_{5}(δ(x/2^{l})) = δ(y/2) ∈ E_{5}^{6}(C). Since E_{5}^{6}(C) = 0 by [8, Corollary 7.9], there is an element z ≠ 0 ∈ E_{3}^{3} such that d_{3}(z) = δ(y/2). Then, φ_{∗}(z) must be hit by x/2^{l+1} under d_{3}. By Lemma 4.3, there is no such differential. □

From the proof of this together with Corollary 4.5, we obtain the following:

Corollary 4.8. The differentials d_{r} of the v_{2}^{−1}BP based Adams spectral sequence for π_{∗}(W ∧ T (1)) is

trivial on Ã_{1}^{0} ⊂ E_{2}^{∗}.

References

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8 Ippei Ichigi and Katsumi Shimomura

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