CUBO A Mathematical Journal

Vol.10, N
o
¯ 03, (43–55). October 2008

The Modulo Two Homotopy Groups of the

L2-Localization of the Ravenel Spectrum

Ippei Ichigi and Katsumi Shimomura

Department of Mathematics, Faculty of Science,

Kochi University, Kochi, 780-8520, Japan

email: 95sm004@math.kochi-u.ac.jp

email: katsumi@math.kochi-u.ac.jp

ABSTRACT

The Ravenel spectra T (m) for non-negative integers m interpolate between the sphere

spectrum and the Brown-Peterson spectrum. Let L2 denote the Bousfield-Ravenel

localization functor with respect to v−1
2

BP . In this paper, we determine the homotopy

groups π∗(L2T (m) : Z/2) = [M2, L2T (m)]∗ for m > 1, where M2 denotes the modulo

two Moore spectrum.

RESUMEN

El espectro de Ravenel T (m) para enteros no negativos m interpola entre el espectro

esferico y el espectro de Brown-Peterson. Denotemos por L2 el funtor de localización

de Bousfield-Ravenel con respecto a v−1
2

BP . En este art́ıculo, determinamos el grupo

de homotopia π∗(L2T (m) : Z/2) = [M2, L2T (m)]∗ para m > 1, donde M2 denota el

espectro de Moor modulo dos.

Key words and phrases: homotopy groups, Bousfield-Ravenel localization, Ravenel spectrum.

Math. Subj. Class.: 55Q99, 55Q51, 20J06.



44 Ippei Ichigi and Katsumi Shimomura CUBO
10, 3 (2008)

1 Introduction

Let S(2) denote the stable homotopy category of 2-local spectra, and BP ∈ S(2) denote the Brown-

Peterson ring spectrum. Then, BP∗ = π∗(BP ) = Z(2)[v1, v2, . . . ] and BP∗(BP ) = π∗(BP ∧BP ) =

BP∗[t1, t2, . . . ], which form a Hopf algebroid. The Adams-Novikov spectral sequence for computing

the homotopy groups π∗(X) of a spectrum X has the E2-term E
∗

2
(X) = Ext∗BP∗(BP )(BP∗, BP∗(X)).

Let L2 : S(2) → S(2) be the Bousfield-Ravenel localization functor with respect to v
−1

2
BP . Then,

the E2-term E
∗

2
(L2S

0) for the sphere spectrum S0 is determined in [12], but the homotopy

groups π∗(L2S
0) stay undetermined. The Ravenel spectrum T (m) for m > 0 is a ring spec-

trum characterized by BP∗(T (m)) = BP∗[t1, t2, . . . tm] ⊂ BP∗(BP ) as a BP∗(BP )-comodule.

The spectrum T (m) interpolates between the sphere spectrum and the Brown-Peterson spec-

trum, and so the homotopy groups π∗(L2T (m)) seem accessible if m is sufficiently large. Indeed,

π∗(L2T (∞)) = π∗(L2BP ) is determined by Ravenel [8]. Let Mk denote the mod k Moore spectrum

defined by the cofiber sequence

S0 w2 S0 wi Mk wj S1. (1.1)
For m = 1, T (1) ∧ M2 is the Mahowald spectrum X〈1〉 and the homotopy groups of L2X〈1〉 are

determined in [11]. But even the homotopy groups of L2T (1) ∧ M4 are too complicated to be

determined completely (cf. [2], [3]). Consider a spectrum T (m)/(va
1
) defined as a cofiber of the

self-map va
1
: Σ2aT (m) → T (m) defined by the generator v1 ∈ π2(T (m)). We use the notation:

Vm(0) = T (m) ∧ M2 and Vm(1)a = T (m)/(v
a
1
) ∧ M2, (1.2)

and abbreviate Vm(1)1 to Vm(1). In this paper, we consider the case where m > 1, and deter-

mine π∗(L2Vm(1)) and π∗(L2Vm(0)). The Adams-Novikov E2-term E
∗

2
(L2Vm(1)) for m > 1 is

determined by Ravenel [10] as follows:

E∗
2
(L2Vm(1)) = Km(2)∗ ⊗ ∧(h1,0, h1,1, h2,0, h2,1) (1.3)

for generators hi,j ∈ E
1,2

m+i+j+1
−2

j+1

2
(L2Vm(1)) and Km(2)∗ = v

−1

2
Z/2[v2, v3, . . . , vm+2]. We show

that Vm(1) is a T (m)-module spectrum with M2-action, and then that all additive generators of

the E2-term are permanent cycles and the extension problem of the spectral sequence is trivial.

Theorem 1.4. π∗(L2Vm(1)) = Km(2)∗ ⊗ ∧(h1,0, h1,1, h2,0, h2,1) as a Z/2-module.

Let α : Σ8M2 → M2 denote the Adams map such that BP∗(α) = v
4

1
, and Ka

2
denote a cofiber

of αa. Then, we show that Vm(1)4a = T (m) ∧ K
a
2

in Lemma 2.4 and denote the telescope of

Vm(1)4
α
→ · · ·

α
→ Vm(1)4a

α
→ Vm(1)4a+4

α
→ · · · by Vm(1)∞. By the v1-Bockstein spectral sequence,

we determine the Adams-Novikov E2-term E
∗

2
(L2Vm(1)∞), whose structure is given in [4] without



CUBO
10, 3 (2008)

The Modulo Two Homotopy Groups ... 45

proof. Here we give a proof of it. Consider the integers en and an defined by

en =
8n − 1

7
and an =



















1 n = 0

3ek+1 − 1 n = 3k + 1

6ek+1 n = 3k + 2

12ek+1 n = 3k + 3.

(1.5)

We introduce modules

Em(2)∗ = v
−1

2
Z(2)[v1, v2, . . . , vm+2],

Q(k) = Em−1(2)∗/(2, v
ak
1

)[xk+1]〈xk/v
ak
1
〉,

where xn ∈ Em(2)∗ is an element defined in (4.1) such that xn ≡ v
2

n

m+2 modulo (2, v1), and

xn/v
an
1

∈ E0
2
(L2Vm(1)∞) by Proposition 4.3. We also introduce homology classes ζ and ζn of

E1
2
(Vm(0)), which correspond to elements vm+2h1,1 and v

2
l
ek

m+2ζl ∈ E
1

2
(L2Vm(1)) for n = 3k + l with

l ∈ {1, 2, 3}, respectively, where ζl corresponds to h1,0 if l = 1, and h2,l−2 if l = 2, 3.

Proposition 1.6. (cf. [4]) The E2-term of Adams-Novikov spectral sequence for computing

π∗(L2Vm(1)∞) is isomorphic to the direct sum of Q(0) ⊗ ∧(h1,0, h2,0, h2,1) and the tensor product

of ∧(ζ) and

Em−1(2)∗/(2, v
∞

1
) ⊕

⊕

k>0

Q(k) ⊗ ∧(ζk+1, ζk+2)

as a Z/2[v1]-module.

By noticing that xn ∈ E
0

2
(L2Vm(1)an ) survives to π∗(L2Vm(1)an ) in Lemma 5.1, we see that

all additive generators of Proposition 1.6 are permanent cycles.

Theorem 1.7. The homotopy groups π∗(L2Vm(1)∞) are isomorphic to the Adams-Novikov E2-

term given in Proposition 1.6.

Consider the cofiber sequence

Vm(0) wη v−11 Vm(0) wp Vm(1)∞ w ΣVm(0) (1.8)
for the localization map η. Here, we introduce algebras

km(1)∗ = Z/2[v1, v2, . . . , vm+1] and Km(1)∗ = v
−1

1
km(1)∗.

Ravenel showed the following

Proposition 1.9. (cf. [10]) The homotopy groups π∗(v
−1

1
Vm(0)) are isomorphic to Km(1)∗ ⊗

∧(h1,0).



46 Ippei Ichigi and Katsumi Shimomura CUBO
10, 3 (2008)

There is a relation between h1,0 and ζ, which is shown in section four:

Lemma 1.10. The induced homomorphism p∗ from p in (1.8) assigns h1,0/v
j
1
∈ E1

2
(v−1

1
Vm(0)) to

ζ/v
j−2
1

∈ E1
2
(L2Vm(1)∞).

Observing the correspondence in the Adams-Novikov E2-terms, we obtain

Corollary 1.11. The homotopy groups π∗(L2Vm(0)) are isomorphic to the direct sum of Σ
−1Q(0)⊗

∧(h1,0, h2,0, h2,1) and the tensor product of ∧(ζ) and

km(1)∗ ⊕ Σ
−1km(1)∗/(2, v

∞

1
, v∞

2
) ⊕

⊕

k>0

Σ−1Q(k) ⊗ ∧(ζk+1, ζk+2)

as a Z/2[v1]-module.

In the next section, we observe about an action of the Moore spectrum M2 on Vm(1)t and

a ring structure of Vm(1)4t, in order to study the Adams-Novikov differential and the extension

problem of the spectral sequence in the following sections. We prove Theorem 1.4 in section three.

Section four is devoted to show Proposition 1.6. We end by proving Theorem 1.7 in the last section.

2 The spectrum T (m) ∧ Ktk

We work in the stable homotopy category of spectra localized at the prime two. Let BP denote

the Brown-Peterson spectrum. Then, we have the Adams-Novikov spectral sequence

E
s,t
2

(X) = Ext
s,t

Γ
(A, BP∗(X)) =⇒ π∗(X).

Here (A, Γ) is the associated Hopf algebroid such that

(A, Γ) = (BP∗, BP∗(BP )) = (Z(2)[v1, v2, . . . ], BP∗[t1, t2, . . . ])

for the Hazewinkel generators vk ∈ BP2k+1−2 and the generators tk ∈ BP2k+1−2(BP ).

Let Mk and K
t
k

for k = 2, 4 and t > 0 denote spectra defined by the cofiber sequences

S0 w2 S0 wi Mk wj S1 and Σ8tMk wαt Mk witk Ktk wjtk Σ8t+1Mk.
Here α denotes the Adams map such that BP∗(α) = v

4

1
. Note that M4 and K

t
4

are ring spectra

(cf. [5]). The Ravenel spectrum T (m) is characterized by BP∗(T (m)) = A[t1, . . . , tm] ⊂ Γ as

Γ-comodules, and is a ring spectrum, whose multiplication and unit map we denote by µ and ι, re-

spectively. Throughout the paper, we fix a positive integer m. Let (A, Γm) = (A, Γ/(t1, t2, . . . , tm))

be the Hopf algebroid associated with (A, Γ), and consider a spectrum X such that BP∗(X) =

M ⊗A A[t1, . . . , tm] for a Γ-comodule M . Then, we have an isomorphism

E∗
2
(X) = Ext∗

Γm
(A, M ) (2.1)



CUBO
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The Modulo Two Homotopy Groups ... 47

by the change of rings theorem (cf. [10]). By observing the reduced cobar complex for the Ext

group, we have

Lemma 2.2. The E2-term has the vanishing line of the slope 1/(qm − 1) if M is (−1)-connected.

Hereafter, we put

qm = 2
m+2 − 2 (2.3)

which is the degree of u1 = vm+1 and s1 = tm+1. This shows π2(T (m)) = BP2 = Z(2){v1} if

m > 0. Let T (m)/(va
1
) for an integer a > 0 denote the cofiber of ṽa

1
: Σ8aT (m) → T (m), where

ṽ1 : Σ
8T (m) → T (m) is the composite

ṽ1 : Σ
8
T (m) = S8 ∧ T (m) T (m) ∧ T (m)wv1∧T (m) wµ T (m).

Lemma 2.4. For k = 2, 4 and a > 0, T (m)/(v4a
1

) ∧ Mk = T (m) ∧ K
a
k
. In particular, T (m) ∧ Ka

2
∧

M4 = T (m)/(v
4a
1

) ∧ M2 ∧ M4 = T (m) ∧ M2 ∧ K
a
4
.

Proof. Since π8(T (m)∧Mk) = BP8/(k) = Z/k{v
4

1
, v1v2} by Lemma 2.2, we see that v

4

1
∧Mk = ι∧

αi ∈ π8(T (m)∧Mk). Indeed, both of these elements are assigned to v
4

1
∈ BP8(T (m)∧Mi) under the

homomorphism induced from the unit map of BP . It extends to v4
1
∧Mk = ι∧α : Mk → T (m)∧Mk,

since [Mk, T (m) ∧ Mk]8 = π8(T (m) ∧ Mk). Indeed, π9(T (m) ∧ Mk) = BP9/(k) = 0. We further

extend it to a self-map A = ṽ4
1
∧ Mk = T (m) ∧ α : T (m) ∧ Mk → T (m) ∧ Mk by the ring structure

of T (m). Now the cofiber of Aa is T (m)/(v4a
1

) ∧ Mk = T (m) ∧ K
a
k
.

�

This lemma implies

Vm(1)4a = T (m) ∧ K
a
2

(2.5)

for the spectrum Vm(1)4a in (1.2).

Lemma 2.6. Let F denote one of the spectra Mk and K
a
k

for k = 2, 4 and a > 0. Then, there

is a pairing νF : F ∧ F → T (m) ∧ F such that νF ◦ (F ∧ iF ) = ι ∧ F : F → T (m) ∧ F for m > 0.

Here iF : S
0 → F denotes the inclusion to the bottom cell.

Proof. The pairing for F = M4 or K
a
4

is the composite (ι∧F ∧F )(T (m)∧µF ) for the multiplication

µF of the ring spectrum of F (see [5]).

For F = M2, we see that π0(T (m)∧M2) = BP0/(2) = Z/2 and π1(T (m)∧M2) = BP1/(2) = 0

by Lemma 2.2, and so [M2, T (m) ∧ M2]0 = Z/2.



48 Ippei Ichigi and Katsumi Shimomura CUBO
10, 3 (2008)

Note that M2 ∧ M4 = M2 ∨ ΣM2. Then, by Lemma 2.4,

T (m) ∧ M2 ∧ K
a
4

= T (m)/(v4a
1

) ∧ M2 ∧ M4 = T (m)/(v
4a
1

) ∧ (M2 ∨ ΣM2)
= T (m)/(v4a

1
) ∧ M2 ∨ ΣT (m)/(v

4a
1

) ∧ M2 = T (m) ∧ K
a
2
∨ ΣT (m) ∧ Ka

2
.

We also see that T (m) ∧ Ka
2
∧ Ka

4
= T (m)/(v4a

1
) ∧ Ka

2
∧ M4 = T (m)/(v

4a
1

) ∧ (Ka
2
∨ ΣKa

2
), and so

T (m) ∧ Ka
2
∧ Ka

4
∧ M2 = T (m) ∧ K

a
2
∧ Ka

2
∨ ΣT (m) ∧ Ka

2
∧ Ka

2
. Then,

T (m) ∧ M2∧K
a
4
∧Ka

4
∧M2 = T (m)∧K

a
2
∧Ka

4
∧M2 ∨ ΣT (m)∧K

a
2
∧Ka

4
∧M2

= T (m)∧Ka
2
∧Ka

2
∨ ΣT (m)∧Ka

2
∧Ka

2
∨ ΣT (m)∧Ka

2
∧Ka

2
∧M2.

Let µK : K
a
4
∧Ka

4
→ Ka

4
denote the multiplication of the ring spectrum Ka

4
, and ν̃ be the composite

T (m) ∧ M2 ∧ M2 T (m) ∧ T (m) ∧ M2wT (m)∧νM2 wµ∧M2 T (m) ∧ M2. Then the desired pairing is a
composite

Ka
2
∧ Ka

2
wι∧K∧K T (m) ∧ Ka

2
∧ Ka

2
winc∧Ka2 T (m) ∧ M2 ∧ Ka4 ∧ Ka4 ∧ M2 wswitch

T (m)∧M2∧M2∧K
a
4
∧Ka

4
wν̃ T (m) ∧ M2∧Ka4 ∧Ka4 wT (m)∧M2∧µK T (m) ∧ M2∧Ka4 wprj T (m)∧Ka2 .

�

Corollary 2.7. The spectra Vm(0) and Vm(1)4a for a > 0 are ring spectra.

We say that a spectrum X has M2-action, if there is a pairing ϕX : X ∧ M2 → X such that

ϕX (X ∧i) = idX . Here i : S
0 → M2 is the inclusion of (1.1) and idX : X → X denotes the identity

map.

Lemma 2.8. Vm(1)t has M2-action.

Proof. Since T (m) is an associative ring spectrum, T (m)/(vt
1
) is a T (m)-module spectrum. The

action ϕVm(1)t is defined by the composite Vm(1)t∧M2 = T (m)/(v
t
1
)∧M2∧M2 wT (m)/(vt1)∧νM2

T (m)/(vt
1
)∧T (m)∧M2 w T (m)/(vt1)∧M2 = Vm(1)t. �

Since Vm(1)t is a T (m)-module spectrum, it implies the following

Corollary 2.9. Vm(1)t is a Vm(0)-module spectrum.

3 The homotopy groups of L2Vm(1)

Note that if BP∗(X) is (2, v1)-nil, then BP∗(L2X) = v
−1

2
BP∗(X), since L2 is smashing (cf. [8],

[9]). Therefore, the Adams-Novikov E2-term E
∗

2
(L2Vm(1)t) is Ext

∗

Γ
(A, v−1

2
BP∗/(2, v

t
1
)[t1, . . . , tm]),

which is isomorphic to

E∗
2
(L2Vm(1)t) = Ext

∗

Γm
(A, v−1

2
BP∗/(2, v

t
1
))



CUBO
10, 3 (2008)

The Modulo Two Homotopy Groups ... 49

by (2.1). Consider a spectrum

Em(2) = v
−1

2
BP 〈m + 2〉

for the Johnson-Wilson spectrum BP 〈m + 2〉. Then we obtain a Hopf algebroid

(Em(2)∗, Σm(2)) = (v
−1

2
Z(2)[v1, v2, . . . , vm+2], Em(2)∗ ⊗A Γm ⊗A Em(2)∗).

Since

v
−1

2
BP∗/J w1⊗ηR Em(2)∗/J ⊗A Γm

for an invariant regular ideal J = (2b, va
1
) is a faithfully flat extension, we have an isomorphism

Ext∗
Γm

(A, BP∗/J) ∼= Ext
∗

Σm(2)
(Em(2)∗, Em(2)∗/J)

by a theorem of Hopkins’ (cf. [1, Th. 3.3]). Note that m + 2 is the smallest number n, for which

v
−1

2
BP∗/J w1⊗ηR v−12 BP 〈n〉∗/J ⊗A Γm is a faithfully flat extension. We use the abbreviation

H∗M = Ext∗
Σm(2)

(Em(2)∗, M ) (3.1)

for a Σm(2)-comodule M . We compute the Ext group H
∗M by the reduced cobar complex

˜Ω∗
Σm(2)

M (cf. [10]). Since the differentials of the cobar complex are defined by the right unit

ηR : Em(2)∗ → Σm(2) and the diagonal ∆ : Σm(2) → Σm(2) ⊗Em(2)∗ Σm(2), we write down here

some formulas on them based on the Hazewinkel and the Quillen formulas:

vn = 2ℓn −
∑n−1

k=1
ℓkv

2
k

n−k
∈ Q ⊗ A = Q[ℓ1, ℓ2, . . . ],

ηR(ℓn) =
∑n

k=0
ℓkt

2
k

n−k
∈ Q ⊗ Γ = Q ⊗ A[t1, t2, . . . ] and

∑

i+j=n
ℓi∆(t

2
i

j ) =
∑

i+j+k=n
ℓit

2
i

j ⊗ t
2

i+j

k
∈ Q ⊗ Γ ⊗A Γ.

(3.2)

Hereafter, we put v2 = 1 and use the following notation:

ui = vm+i and si = tm+i.

Since the structure maps preserve degrees, we may recover v2’s from its degrees. Then, we obtain

the following two lemmas immediately from (3.2) by a routine computation:

Lemma 3.3. The right unit ηR : A → Γm/(2) acts as follows:

ηR(vn) = vn for n ≤ m + 1,

ηR(u2) = u2 + v1s
2

1
+ v2

m+1

1
s1,

ηR(u3) ≡ u3 + s
4

1
+ s1 + v1r1 mod (2, v

2
m+2

1
),

ηR(u4) ≡ u4 + s
4

2
+ s2 + v3s

8

1
+ v2

m+1

3
s1 mod (2, v1)

for r1 = s
2

2
+ v1u2s

2

1
.

This yields the relations in Σm(2):

s4
1
+ s1 ≡ v1r1 mod (2, v

2
m+2

1
) and s4

2
+ s2 + v3s

8

1
+ v2

m+1

3
s1 ≡ 0 mod (2, v1). (3.4)



50 Ippei Ichigi and Katsumi Shimomura CUBO
10, 3 (2008)

Lemma 3.5. The diagonal ∆ behaves on the generators si as follows:

∆(s1) = s1 ⊗ 1 + 1 ⊗ s1,
∆(s2) = s2 ⊗ 1 + 1 ⊗ s2 + v1s1 ⊗ s1,
∆(s3) ≡ s3 ⊗ 1 + 1 ⊗ s3 + v2s

2

1
⊗ s2

1
mod (2, v

1
).

Lemma 3.6. Let z denote an element defined by r4
1
+r1 +v

2

3
s4
1
+v2

m+2

3
s2
1

= v1z. Then the cochains

r1, z ∈ ˜Ω
1

Σm(2)
Em(2)∗/(2) are cocycles. Besides, z ≡ u2s

2

1
modulo (v2

1
).

Proof. Since v1 ∈ ˜Ω
0

Σm(2)
Em(2)∗/(2) and s1 ∈ ˜Ω

1

Σm(2)
Em(2)∗/(2) are both cocycles, so is r1

by the relation v1r1 = s
4

1
+ s1 ∈ Σm(2) in (3.4). Furthermore, v3 ∈ ˜Ω

0

Σm(2)
Em(2)∗/(2) is

a cocycle. It follows similarly from its definition that z is a cocycle. By the definition of r1,

r4
1

+ r1 ≡ s
8

2
+ s2

2
+ v1u2s

2

1
≡ v1u2s

2

1
+ v2

3
s16
1

+ v2
m+2

3
s2
1

modulo (2, v2
1
) by (3.4). �

We now work as [6].

Lemma 3.7. ut
2
∈ E0

2
(Vm(1)) and u

t
2
h2,0 ∈ E

1

2
(Vm(1)) for each t > 0 are permanent cycles.

Proof. For t = 1, the lemma is seen by Lemma 2.2. Consider the cofiber sequence Σ2Vm(0)
v1
→

Vm(0)
i1
→ Vm(1)

j1
→ Σ3Vm(0). Put d(u

t
2
) = v1k

′

t ∈
˜Ω1

Σm(2)
Em(2)∗/(2) by virtue of Lemma 3.3,

and let kt ∈ E
1

2
(Vm(0)) be the homology class of the cocycle k

′

t. Then, k1 = h1,1, v1kt = 0

and kt+1 = 〈k1, v1, kt〉. Indeed, 〈k1, v1, kt〉 is the class of k
′

1
ηR(u

t
2
) + u2k

′

t = d(u
t+1
2

)/v1 = k
′

t+1.

Besides, δ(ut
2
) = kt for the connecting homomorphism associated to the cofiber sequence. Let

ξ1 ∈ πqm−1(Vm(0)) denote the homotopy element detected by k1. Then, v1ξ1 = ξ1v1 = 0.

Suppose now that ut
2
∈ E0

2
(Vm(1)) is a permanent cycle. Then, kt is a permanent cycle that

detects the element ξt = j1u
t
2

by the Geometric Boundary Theorem. Since v1ξt = 0, the Toda

bracket {ξ1, v1, ξk} is defined, which is detected by the Massey product 〈k1, v1, kt〉. Note here that

the Toda bracket is defined since Vm(0) is a ring spectrum. It follows that kt+1 is a permanent cycle

and detects a homotopy element, which we denote by ξt+1. Since the Massey product 〈v1, k1, v1〉 is

zero in the E2-term E
0,qm+4
2

(Vm(0)), we see that {v1, ξ1, v1} = 0 by Lemma 2.2. Now we compute

v1{ξ1, v1, ξk} = {v1, ξ1, v1}ξk = 0, and ξt+1 is pulled back to u
t+1
2

under the map j1.

Turn to ut
2
h2,0. In this case a similar argument works. For the connecting homomorphism

δ, δ(ut
2
h2,0) = 〈h

2

1,0, v1, kt〉, which detects a homotopy element {η
2

0
, v1, ξt}, where η0 denotes an

element detected by h1,0. Applying v1 shows {v1, η
2

0
, v1}ξt = 0. Indeed, {v1, η

2

0
, v1} is detected by

E
s,2qm+4+s
2

(Vm(0)) for s > 2. �

Lemma 3.8. The elements h1,0, h1,1 ∈ E
1

2
(Vm(0)) and h2,1 ∈ E

1

2
(L2Vm(0)) are permanent cycles.

Proof. h1,0, h1,1 are seen immediately by Lemma 2.2.



CUBO
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The Modulo Two Homotopy Groups ... 51

The cobar module ˜Ω
4,4qm+6

Γm
BP∗/(2) is generated by v

3

1
s
⊗4

1
and v2s

⊗4

1
by degree reason. The

first generator cobounds v2
1
s2 ⊗ s1 ⊗ s1, and we obtain E

4,4qm+6
2

(Vm(0)) = Z/2{v2h
4

1,0}. Put

d3(h2,1) = av2h
4

1,0 ∈ E
4,4qm+6
2

(Vm(0)) for a ∈ Z/2. Let w be an element fit in d(s3) = v2s
2

1
⊗

s2
1

+ v1w by virtue of Lemma 3.5. Then, d(w) = 0 in the cobar complex ˜Ω
3

Σm(2)
Em(2)∗/(2), and

we see that s⊗4
1

cobounds s2
3
⊗ s1 ⊗ s1 + v1w

2 ⊗ s2 + (r1 ⊗ s1 + s1 ⊗ r1 + v1r1 ⊗ r1) ⊗ s2 (in

which we set v2 = 1). It follows that d3(h2,1) = av2h
4

1,0 = 0 ∈ E
4

2
(L2Vm(0)) as desired. Indeed,

v2h
4

1,0 = v1gh
2

1,0 = 0, since v2h
2

1,0 = v1g for an element g and v1h
2

1,0 = 0 by d(s2) = v1s1 ⊗ s1.

�

Proof of Theorem 1.4. Every element x ∈ Es
2
(L2Vm(1)) is decomposed as x = x

′x′′ for x′ ∈

Z/2[u2]⊗∧(h2,0) and x
′′ ∈ Km−1(2)∗⊗∧(h1,0, h1,1, h2,1). Note that Km−1(2)∗⊗∧(h1,0, h1,1, h2,1) ⊂

E∗
2
(L2Vm(0)). Since x

′ (resp. x′′) is a permanent cycle of the Adams-Novikov spectral sequence

for computing π∗(L2Vm(1)) (resp. π∗(L2Vm(0))) by Lemma 3.7 (resp. 3.8), we obtain that the

element x is a permanent cycle from Corollary 2.9. We see that the extension problem is trivial

by Lemma 2.8. Indeed, Z/2 = π0(M2) acts on π∗(L2Vm(1)). �

4 The elements xn

We introduce the integer bn for n ≥ 0 by

bn =











an − 8 n ≡ 1 (3)

an − 3 n ≡ 2 (3)

0 n ≡ 0 (3),

and the elements xn ∈ Em(2)∗ defined by

xn = x
2

n−1 + v
bn
1

yn−1, where yn =



















0 n ≤ 0 or n ≡ 2 (3)

x0 n = 1

x2 + v
2

1
v4
3
x2

1
+ v4

1
v2

m+3

3
x1 n = 3

xn−2yn−3 n ≡ 0, 1 (3) and n ≥ 4.

(4.1)

We also consider cocycles zn ∈ Σm(2):

zn =











s2
n+1

1
n = 0, 1

r2
n−1

1
n = 2, 3

xn−3zn−3 n > 3.

(4.2)

Proposition 4.3. For the differential d : Ω0
Σm(2)

Em(2)∗/(2) → Ω
1

Σm(2)
Em(2)∗/(2) of the cobar

complex,

d(xn) = v
an
1

zn.



52 Ippei Ichigi and Katsumi Shimomura CUBO
10, 3 (2008)

Proof. For n = 0 and 1, it is immediate from Lemma 3.3, and the cases for n = 2 and 3 follow

from the computation d(x2) = d(u
4

2
+ v3

1
u2) = v

4

1
s8
1

+ v4
1
s2
1

= v6
1
r2
1

by (3.4). For n = 4,

d(x4) ≡ d(x
4

2
+ v18

1
x2 + v

20

1
v4
3
x2

1
+ v22

1
v2

m+3

3
x1)

≡ v24
1

r8
1

+ v24
1

r2
1

+ v24
1

v4
3
s8
1
+ v24

1
v2

m+3

3
s4
1

≡ v26
1

z2 ≡ v26
1

x1z1 mod (2, v
28

1
)

by the definition of z.

Suppose inductively that d(x3k+1) = v
a3k+1
1

x3k−2z3k−2 mod (2, v
a3k+1+2

1
) for k > 0.

d(x2
3k+1

) ≡ v
2a3k+1
1

x2
3k−2

z2
3k−2

mod (2, v
2a3k+1+4

1
)

d(v
a3k+2−3

1
y3k+1) ≡ d(v

a3k+2−3

1
x3k−1y3k−2)

≡ v
a3k+2−3

1
x3k−1(v1z

2

3k−2
+ v3

1
z3k−1) mod (2, v

a3k+2−3+a3k−1
1

)

and the sum shows d(x3k+2) ≡ v
a3k+2
1

x3k−1z3k−1 mod (2, v
a3k+2+2

1
). Similarly,

d(x4
3k+2

) ≡ v
4a3k+2
1

x4
3k−1

z4
3k−1

mod (2, v
4a3k+2+8

1
)

d(v
a3k+4−8

1
y3k+3) ≡ d(v

a3k+4−8

1
x3k+1y3k)

≡ v
a3k+4−8

1
x3k+1(v

6

1
z2
3k

+ v8
1
z3k+1) mod (2, v

a3k+4−8+a3k+1
1

)

and we have d(x3k+4) = v
a3k+4
1

x3k+1z3k+1 mod (2, v
a3k+4+2

1
), which completes the induction. �

Proof of Lemma 1.10. It suffices to show that h1,0/v
j
1
∈ E1

2
(L2Vm(1)∞) equals ζ/v

j−2
1

. The

element h1,0/v
j
1

is represented by s1/v
j
1
. We make a computation in the cobar complex

d(u2
2
/v

j+2
1

) = s4
1
/v

j
1

= s1/v
j
1

+ r1/v
j−1
1

d(v2
3
u2

2
/v

j+1
1

) = v2
3
s4
1
/v

j−1
1

d(v2
m+2

3
u2/v

j
1
) = v2

m+2

3
s2
1
/v

j−1
1

d(x2
2
/v

j+11
1

) = r4
1
/v

j−1
1

by Lemma 3.3 and Proposition 4.3. The sum yields the homologous relation s1/v
j
1
∼ z/v

j−2
1

by

Lemma 3.6, and so h1,0/v
j
1

= ζ/v
j−2
1

in E1
2
(L2Vm(1)∞). �

Proof of Proposition 1.6. We consider the v1-Bockstein spectral sequence given by the short

exact sequence 0 → Em(2)∗(Vm(1))
ϕ
→ Em(2)∗(Vm(1)∞)

v1
→ Em(2)∗(Vm(1)∞) → 0 for ϕ given

by ϕ(x) = x/v1. Let B
∗ denote the Z/2[v1]-module of the proposition. Then, it is easy to

see that Bs contains the image of ϕ∗ : E
s
2
(L2Vm(1)) → E

s
2
(L2Vm(1)∞) and that Proposition 4.3

defines a homomorphism f : Bs → Es
2
(L2Vm(1)∞). We also consider the composite ∂ = δ ◦

f : Bs → Es+1
2

(L2Vm(1)), where δ : E
s
2
(L2Vm(1)∞) → E

s+1
2

(L2Vm(1)) denotes the connecting

homomorphism associated to the short exact sequence. By [7, Remark 3.11], it suffices to show

the sequence

0 w Coker ∂ wϕ∗ B∗ wv1 B∗ w∂ Im ∂ w 0 (4.4)



CUBO
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The Modulo Two Homotopy Groups ... 53

is exact.

We decompose E∗
2
(L2Vm(1)) into the direct sum of MC = Km−1(2)∗[u

2

2
]{u2}⊗∧(h10, h20, h21),

MI = Km−1(2)∗[u
2

2
]{h11} ⊗ ∧(h10, h20, h21) and N ⊗ ∧(ζ) = Km−1(2)∗[u

2

2
] ⊗ ∧(h10, h20, h21, ζ).

We notice that for non-negative integers n and r with r < 8, there exist uniquely non-negative
integers t and q such that n = 8qt + req. By this fact, we decompose summands of N as follows:

Km−1(2)∗[u
2
2]

= Km−1(2)∗ ⊕
⊕

k≥1
xkKm−1(2)∗[xk+1]

A
,

Km−1(2)∗[u
2
2]h10

=
⊕

q≥0

(

(

x3q+2Km−1(2)∗[x3q+3]
a
⊕ x3q+3Km−1(2)∗[x3q+4]

b

)

ζ3q+4 ⊕ Km−1(2)∗[x3q+2]ζ3q+1
A

)

,

Km−1(2)∗[u
2
2]h20

=
⊕

q≥0

(

x3q+3Km−1(2)∗[x3q+4]ζ3q+5
c
⊕

(

x3q+1Km−1(2)∗[x3q+2]
d
⊕ Km−1(2)∗[x3q+3]

A

)

ζ3q+2

)

,

Km−1(2)∗[u
2
2]h21

=
⊕

q≥0

(

x3q+1Km−1(2)∗[x3q+2]
e
⊕ x3q+2Km−1(2)∗[x3q+3]

f
⊕ Km−1(2)∗[x3q+4]

A

)

ζ3q+3,

Km−1(2)∗[u
2
2]h10h20

=
⊕

q≥0

(

Km−1(2)∗[x3q+3]ζ3q+4ζ3q+2
a

⊕ x3q+3Km−1(2)∗[x3q+4]ζ3q+4ζ3q+5
B

⊕Km−1(2)∗[x3q+2]ζ3q+1ζ3q+2
d

)

,

Km−1(2)∗[u
2
2]h20h21

=
⊕

q≥0

(

Km−1(2)∗[x3q+4]ζ3q+3ζ3q+5
c

⊕

(

x3q+1Km−1(2)∗[x3q+2]
B
⊕Km−1(2)∗[x3q+3]

f

)

ζ3q+2ζ3q+3

)

,

Km−1(2)∗[u
2
2]h10h21

=
⊕

q≥0

(

(

Km−1(2)∗[x3q+3]x3q+2
B
⊕Km−1(2)∗[x3q+4]

b

)

ζ3q+4ζ3q+3⊕Km−1(2)∗[x3q+2]ζ3q+1ζ3q+3
e

)

,

Km−1(2)∗[u
2
2]h10h20h21

=
⊕

k≥1
Km−1(2)∗[xk+1]ζkζk+1ζk+2

B

.

Here, M X and M
′

X
for modules M and M ′ mean that M and M ′ are isomorphic under a

Bockstein differential dr for some r so that dr(M ) = M
′, which is seen by Proposition 4.3. Let

NC (resp. NI ) be the direct sum of single (resp. double) underlined submodules of N , and put
˜M = Q(0) ⊗ ∧(h1,0, h2,0, h2,1), ˜N =

⊕

k>0
Q(k) ⊗ ∧(ζk+1, ζk+2). Then we have the three exact

sequences

0 → MC
ϕ∗
→ ˜M

v1
→ ˜M → MI → 0, 0 → NC

ϕ∗
→ ˜N

v1
→ ˜N → NI → 0 and

0 → Km−1(2)∗ → Em−1(2)∗/(2, v
∞

1
) → Em−1(2)∗/(2, v

∞

1
) → 0,

the direct sum of which yields the sequence (4.4). �

5 The Adams-Novikov E∞-term for π∗(L2T (m) ∧ M2)

We first show that all elements of the Adams-Novikov E2-term for π∗(L2Vm(1)∞) are permanent

cycles. Take an element x/vt
1
∈ E0

2
(L2Vm(1)∞). Then x ∈ E

0

2
(L2Vm(1)t). Thus, if x = y

2/vt
1

for



54 Ippei Ichigi and Katsumi Shimomura CUBO
10, 3 (2008)

some y ∈ E0
2
(L2Vm(1)4t), then x is a permanent cycle. So it is sufficient to show that d3(xn) =

0 ∈ E3
2
(L2Vm(1)an ) for each n ≥ 0. We consider the integer

εn =

{

2 n 6≡ 0 (3)

0 n ≡ 0 (3)

so that Vm(1)an+εn is a ring spectrum by Corollary 2.7.

Lemma 5.1. d3(xn) = 0 ∈ E
3

2
(L2Vm(1)an ) for n ≥ 0.

Proof. For n = 0, it is shown in Lemma 3.7.

Suppose that d3(xn) = ξ ∈ E
3

2
(L2Vm(1)an ) for n > 0. Send this to E

3

2
(L2Vm(1)an−1 ), and we

see that ξ = d3(xn) = d3(x
2

n−1) ∈ E
3

2
(L2Vm(1)an−1 ). Then, the map v

εn−1
1

: E3
2
(L2Vm(1)an−1 ) →

E3
2
(L2Vm(1)an−1+εn−1 ) assigns v

εn−1
1

ξ to v
2εn−1
1

ξ = d3((v
εn−1
1

xn−1)
2), which is zero, since

v
εn−1
1

xn−1 ∈ E
0

2
(L2Vm(1)an−1+εn−1 ) and Vm(1)an−1+εn−1 is a ring spectrum. It follows that

ξ = v
an−1−εn−1
1

ξ′ for some ξ′ ∈ E3
2
(L2Vm(1)an−an−1+εn−1 ). Note that this works even if n = 1,

though Vm(1) is not a ring spectrum. Consider the commutative diagram

Vm(1) Vm(1) ∗ Vm(1)

Vm(1)an−a+1 Vm(1)an+1 Vm(1)a Vm(1)an−a+1

Vm(1)an−a Vm(1)an Vm(1)a Vm(1)an−a,

uvan−a1 uvan1 w u w u van−a1
u wv

a

1 u wiv wjv u pwva1 wi′v wj′v
(a = an−1 − εn−1) in which rows and columns are cofiber sequences. Let 〈x〉 ∈ π∗(X) denote

a homotopy element detected by x ∈ E∗
2
(X). Noticing that xn ∈ E

0

2
(L2Vm(1)an−1−εn−1 ) is

a permanent cycle, we see that jv∗(〈xn〉) = 〈v
an−an−1+εn−1
1

ζn〉 and j
′

v∗
(〈xn〉) = 〈ξ

′〉, and so

p∗(〈v
an−an−1+εn−1
1

ζn〉) = 〈ξ
′〉. Since 〈ζn〉 ∈ π∗(L2Vm(1)) by Theorem 1.4, we obtain 〈ξ

′〉 = 0,

and 〈xn〉 is in the image under the map i
′

v∗
. It follows that there is a permanent cycle x′n ∈

E0
2
(L2Vm(1)an ), whose leading term is xn, such that iv∗(〈x

′

n〉) = 〈xn〉 ∈ π∗(L2Vm(1)an−1−εn−1 ).

The lemma now follows by replacing xn by x
′

n. �

Received: February 2008. Revised: April 2008.



CUBO
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The Modulo Two Homotopy Groups ... 55

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