The homotopy groups of the L2-localized mod 3
Moore spectrum
Katsumi Shimomura*
(Received )
x1. Introduction
Let Sp denote the category of p-local spectra for each prime number p
and BP the Brown-Peterson spectrum at p. Then we have the Bousfield
localization functor Ln : Sp ! Sp with respect to v-1nBP for the generator
vn of BP* = Z (p)[vi : i > 0 ]. The category LnSp is easier to be understood
than Sp itself and reflects some properties of it. LnSp is, in a sense, gener-
ated by the Ln-localized sphere spectrum LnS0, because LnX = X ^ LnS0
for any spectrum X by the smash product theorem [10, Th. 7.5.6]. Besides,
we have the chromatic convergence theorem due to Hopkins and Ravenel
[10, Th. 7.5.7], which says that ho-limLnX = X for a finite spectrum X.
n
Therefore it is very important to compute the homotopy groups ss*(LnS0).
So far we know the homotopy groups ss*(LnS0) for n < 2 given in [8] and for
n = 2 and p > 3 in [12]. The next place to study is the case where n = 2
and p = 3. They are computed by using the Bockstein spectral sequences
ss*(LnV (k)) ) ss*(LnV (k - 1)), where V (n) denotes the Toda-Smith spec-
trum, and is known to exist if n < 4 and p > 2n (cf. [9]). (For LnV (k), we
have some other existence theorems in [13] and [14].) Note that V (-1) = S0
and V (0) is the mod p Moore spectrum. On the other hand, ss*(LnV (n - 1))
is computed by Ravenel (cf. [9]) in case of n < 4 and n < p-1, by Mahowald
[5] in case of n = p - 1 = 1, and by the author [11] and Henn and Mahowald
[4] in case of n = p - 1 = 2. In this paper we study the Bockstein spec-
tral sequence ss*(L2V (1)) ) ss*(L2V (0)) and determine ss*(L2V (0)) at the
______________________________
*Part of this work was done during the stay at Max-Plank-Institut f"ur Mathe*
*matik
1
prime number 3. Our main tool is the Adams-Novikov spectral sequence. In
[6] Miller, Ravenel and Wilson introduced the chromatic spectral sequence
converging to the E2-term of the Adams-Novikov spectral sequence for com-
puting the homotopy groups ss*(V (n)). We use here the modified chromatic
spectral sequence which converges to the E2-term of the Adams-Novikov
spectral sequence E*;*2(L2V (0)) ) ss*(L2V (0)) based on E(2) with E1-terms
H*M11and H*M01. Here E(2) denotes the Johnson-Wilson spectrum with co-
efficient E(2)* = Z (3)[v1; v12], M01= v-11E(2)*=(3) and M11= E(2)*=(3; v11)
are the E(2)*E(2)-comodules and H*M = Ext *E(2)*E(2)(E(2)*; M). H*M01
was determined by Ravenel [7], and so it suffices to determine H*M11for the
E2-term E*;*2(L2V (0)). In the first half of this paper, we actually determine
H*M11by using the Bockstein spectral sequence H*K(2)* ) H*M11, where
K(2)* = E(2)*=(3; v1) and H*K(2)* is determined by Ravenel [7]. The struc-
ture of H*M11is stated in Theorem 2.5, and obtain the E2-term E*;*2(L2V (0))
in Theorem 2.6. In [6] H0M11is determined, and we studied H1M11in [1].
Unfortunately Theorem 4.4 of [1] is incorrect, and so are Proposition 5.2 and
Theorem 1.1 consequently. Here we replace it by Lemma 4.2, which is proved
in x7, and obtain H1M11. In the second half of this paper, we determine the
Adams-Novikov differentials dr on E*;*r(L2W ) with E*;*2(L2W ) = H*M11,
and then the homotopy groups ss*(L2W ) which is described in Theorem 2.8.
Here W denotes a cofiber of the localization map V (0) ! holim-!V (0) for the
ff
Adams map ff : 4V (0) ! V (0). The homotopy groups ss*(L2V (0)), which
is our main result, are obtained in Theorem 2.11 as a corollary of Theorem
2.8. The results would have applications. Here we treat the fi-family of the
homotopy groups of ss*(L2S0) at the prime 3. We note that though the result
of [2] depends on a result of [1], it remains correct since the proof does not
require the incorrect part.
This paper is organized as follows: In the next section, we state our
results. Then we prove Theorem 2.5 in x3 assuming the behavior of the con-
necting homomorphisms ffis : HsM11! Hs+1K(2)* which will be studied in
the following sections. In x4, assuming the behavior of the differential of the
cobar complex *E(2)* which will be studied in xx6 and 7, we prove Proposi-
tion 3.4 which determines the differentials of the Bockstein spectral sequence
and is the key lemma to determine the E2-term E*;*2(L2V (0)). In order to
study the differential of the cobar complex *E(2)*, we need some relations
in E(2)*E(2), which is given in x5. In x8, we compute the differentials of the
2
Adams-Novikov spectral sequence, and prove Theorem 2.8. The last section
is devoted to applications for fi-elements.
x2. Statement of results
Throughout this paper everything is localized at the prime 3. Let V (0)
denotes the mod 3 Moore spectrum and W be the cofiber of the localization
map V (0) ! L1V (0). Since L1V (0) = holim-!V (0) for the Adams map ff :
ff
4V (0) ! V (0), we can define W as follows: Let V (1)j denote a cofiber
of ffj : 4jV (0) ! V (0). In particular, V (1)1 = V (1), the Toda-Smith
spectrum. Then we have canonical maps ssj : V (1)j ! V (1)j-1 and j :
V (1)j ! V (1)j+1. We define W = ho-lim!V (1)j. By definition, we have
j
vj-11 ssj
cofiber sequences V (1) ! V (1)j ! V (1)j-1, whose homotopy colimit yields
another one
(2:1) V (1) -i! W -v1!W:
Apply the Johnson-Wilson homology E(2)*(-) with coefficient E(2)* =
Z(3)[v1; v12] to the cofiber sequence we have a short exact sequence
(2:2) 0 -! K(2)* -i*!M11-v1!M11-! 0:
where K(2)* = (Z =3)[v12], M11= E(2)*=(3; v11) and i*(x) = x=v1. Note
that K(2)* is the coefficient ring of the second Morava K-theory K(2)*(-).
Apply the functor H*- = Ext*E(2)*E(2)(E(2)*; -) to the exact sequence (2.2),
and we obtain the Bockstein spectral sequence
E*;*2(L2V (1)) = H*K(2)* =) H*M11= E*;*2(L2W ):
Here E*;*2(X) denotes the E2-term of the Adams-Novikov spectral sequence
converging to the homotopy groups ss*(X).
In [3], Henn computed the E2-term H*K(2)* as follows:
Theorem 2.3. (cf. [11, Th. 5.8, Prop. 5.9].) The E2-term E*;*2(L2V (1)) =
H*K(2)* is isomorphic to the K(2)*[b10]-module
N N
F K(2)*[b10] (i2):
3
Here F = (Z =3){1; h10; h11; b11; ; 0; 1; b11}. Besides, we have relations :
h10h11 = 0; h10 = 0; h11 = 0;
v22h10b10 = h11b11; v2h11b10 = -h10b11;
b11 = v2h10 1 = v2h11 0; b10 = -h10 0 = v-12h11 1;
v32b210= -b211; b10 1 = -v-12b11 0; and b10 0 = v-22b11 1:
The bidegrees of these generators are as follows:
||v2|| = (0; 16); ||h10|| = (1; 4); ||h11|| = (1; 12); ||b10|| = (2; 12);
||b11|| = (2; 36); |||| = (2; 8); || 0|| = (3; 16); || 1|| = (3; 24):
In this paper, we first compute the Adams-Novikov E2-term E*;*2(L2W )
from Theorem 2.3 by using the Bockstein spectral sequence. In order to state
the E2-term, consider the algebra k(1)* = (Z =3)[v1] and k(1)*-modules
F = E(2; 1)*{v12=v1; v2h10=v21; v22h11=v21; v12b11=v1}
F * = E(2; 1)*{=v21; v212 0=v1; v12 1=v1; b11=v21}
(2:4) 3n+1 4.3n-1 3n+1 6.3n+1
Fn = E(2; n + 2)*{v2 =v1 ; v2 h10=v1 ;
n 10.3n+1 3n(53)+(3n-1)=2 4.3n
v8.32h10=v1 ; v2 =v1 }:
n
Here ||v1|| = (0; 4), E(2; n)* = (Z =3)[v1; v32 ], and k(1)* acts on an element
of the form x=vj1by the equation vi1(x=vj1) = x=vj-i1if j > i, and = 0
otherwise. We also use the notation K(1)* = v-11k(1)*. Then
Theorem 2.5. The E2-term E*;*2(L2W ) = H*M11of the Adams-Novikov
spectral sequence convergingNto ss*(L2W ) is isomorphic to the direct sum of
k(1)*-modules (K(1)*=k(1)*) (h10; i2),
X N L N N
Fn (i2); and (F F *) (Z =3)[b10] (i2):
n0
The short exact sequence associated to the cofiber sequence V (0) !
L1V (0) ! W yields the long exact sequence
H*E(2)*=(3) -! H*M01-! H*M11-@! H*+1E(2)*=(3);
in which M01=N(Z =3)[v11; v12] and the structures of H*M01is determined
to be K(1)* (h10) by Ravenel [7]. Let E*;*2(L2V (0)) denote the Adams-
Novikov E2-term converging to the homotopy groups ss*(L2V (0)). Observing
the exact sequence, we obtain the E2-term by Theorem 2.5:
4
Theorem 2.6. The E2-term E*;*2(L2V (0)) = H*E(2)*=(3) isNisomorphic
to the direct sum of the k(1)*-modules K(1)*=k(1)*{@(i2)} (h10),
N N X L L *N
k(1)* (h10) and (i2) @( Fn (F F ) (Z =3)[b10])
n0
In order to state the homotopy groups ss*(L2V (0)), we prepare more no-
tations:
eF0 = E(2; 2)*{v32=v21; v-32=v31; v32h10=v61; v82h10=v111;
v82=v41; v22=v31}
eF1 = E(2; 3)*{v92=v111; v92h10=v181; v242h10=v311; v1692=v111}
eFn = E(2; n + 2)*{v3n+12=v4.3n-11; v3n+12h10=v2.3n+11;
n 10.3n 3n(53)+(3n-1)=2 4.3n-1
v8.32h10=v1 ; v2 =v1 } (n 2)
Fe = B5(2; 2)*{v2=v1; v2h10=v21}
L 5 2 4
L B4(2; 2)*{v2h11=v1; v2b11=v1}
B3(2; 2)*{v22=v1; v52=v1; v72h10=v1}
(2:7) L B2(2; 2)*{v2h10=v1; v22h11=v1; v52h11=v1;
v52b11=v1; v-12b11=v1}
eF *= B5(2; 2)*{v72 1=v1}
L 3 2 3 6 2
L B4(2; 2)*{v2=v1; v2 0=v1; v2b11=v1}
L B3(2; 2)*{v-12 1=v1}
B2(2; 2)*{=v1; v22 1=v1; v42 0=v1; v32b11=v1;
L P v72 0=v1; v62b11=v1}
9u+3
n1L(B3(2; n + 2)*{v2 =v1 | u 2 Z - I(n)}
B2(2; n + 2)*{v9u+32=v1 | u 2 I(n)});
n k
where Bk(2; n)* = (Z =3)[v1; v32 ; b10]=(b10), and
I(n) = {x 2 Z | x = (3n-1 - 1)=2 or x = 5 . 3n-2 + (3n-2 - 1)=2}:
Studying the Adams-Novikov differentials d5 and d9 by results of [4] and
[11], we obtain the following
Theorem 2.8. The homotopy groups ss*(L2W ) are isomorphic to the
tensor product of theNexteriorPalgebra (i2) andLthe direct sum of k(1)*-
modules (K(1)*=k(1)*) (h10), n0 Fen and Fe eF.*
5
For describing more homotopy groups, we further introduce the k(1)*-
modules:
___ -4
F0_ = E(2; 2)*{v1v22h10; v2 h10; v1v22; v52; v62 1; v1 1}
F1 = E(2; 3)*{v9-32h10; v1v82; v162; v1v1292 1}
___ 3n+1-3n (3n+1-1)=2
Fn = E(2; n + 2)*{v2 h10; v1v2 ;
n+(3n-1)=2 3n+1(11)+3(3n-1)=2
__ v1v5.32 ;Lv1v2 1} (n 2)
F = B5(2;L2)*{h11; b10} B4(2; 2)*{v32b11; v32h11b11}
B3(2; 2)*{v2h11; v42h11; v1v62b10}
(2:9) L B2(2; 2)*{v1b10; v1b11; v1v32b11; v42h11b11; v-22h11b11}
___ L
F * = B5(2;L2)*{v72b10} B4(2; 2)*{v22 0; b11; v62 1b10}
L B3(2; 2)*{v-12b10}
B2(2; 2)*{v1v-12 0; v22b10; v22b11; v1v32 1b210;
L P v52b11; v1v62 1b10}
9u+2
n1 (B3(2;Ln + 2)*{v1v2 0 | u 2 Z - I(n)}
B2(2; n + 2)*{v1v9u+22 0 | u 2 I(n)});
___
where M is isomorphic to Mf for M = Fn; F; F *as k(1)*-modules while
there is one dimension shift. Furthermore, put k(1)^*= limk(1)*=(vj1). Since
j
holim-L2V (1)j = LK(2)V (0) by observing K(2)* homology, the above theorem
j
implies
Theorem 2.10. The homotopy groups ss*(LK(2)V (0)) are isomorphic to
the tensor productNof thePexterior_algebra_(i2)Land_the direct sum of k(1)*-
modules (k(1)^*) (h10), n0 Fn and F F *.
ObservingNthe cofibration L2V (0) ! L1V (0) ! L2W and ss*(L1V (0)) =
K(1)* (h10), we have
Theorem 2.11. TheNhomotopy groups ss*(L2VN(0)) arePisomorphic_toNthe
direct_sumLof_k(1)*N (h10), (K(1)*=k(1)*)@i2 (h10), n0 Fn (i2) and
(F F *) (i2).
Here recall the conjecture due to Ravenel on the fi-elements: fis 2 ss*(S0)
if and only if s 0; 1; 2; 3; 5; 6 mod 9. (See x9 for the definition of fi-elem*
*ents.)
6
`Only if' part is shown in [11], in which we also show that fis 2 ss*(L2S0) if
s 0; 1; 5 mod 9. On this conjecture, we have a supporting evidence:
Theorem 2.12. fis 2 ss*(L2S0) if and only if s 0; 1; 2; 3; 5; 6 mod 9.
In the E2-term, the fi-elements of the form fia=bare defined [6] for integers
a, b > 0 such that b 3(a) if (a) 1 and b < 4 . 3(a)-1 otherwise, where
the integer (a) denotes the maximal power of 3 that divides a. Then we
have homotopy fi-elements:
Theorem 2.13. In the Adams-Novikov spectral sequence E*2(L2S0) )
ss*(L2S0), we have
(a)The element fia with (a) = 0 is permanent if a 1; 2; 5 mod 9,
(b) The element fia=bwith (a) = 1 is permanent if 9 | (a - 3) and b < 3, or
if 9 | (a + 3) and b 3.
(c)Every element fia=bwith (a) 2 is permanent.
x3. Proof of Theorem 2.5
The proof of Theorem 2.5 is based on the following lemma due to [6,
Remark 3.11]. To state the lemma,Nwe set up notations: Let K denote a
Z=3-basis of the submodule F K(2)*[b10] of H*K(2)* given in Theorem
2.3, and __x=vj1denote an element of H*M11such that vj-11(__x=vj1) = x=v1 for
an element x 2 K and an integer j > 0. Consider the maps i* and ffis in the
long exact sequence associated to the short one (2.2)
(3:1) . . .-! HsK(2)* -i*!HsM11-v1!HsM11-ffis!Hs+1K(2)* -! . .:.
Note that i*(x) = x=v1. For each base x 2 K HsK(2)*, define an integer
j(x) by j(x) = j if ffis(__x=vj1) 6= 0, and j(x) = 1 otherwise. Define a k(1)*-
submodule B of H*M11by
B = k(1)*{__x=vj(x)1| x 2 K and i*(x) 6= 0 2 H*M11}:
7
N
Lemma 3.2. For the submodule B defined above, H*M11= B (i2) if B
satisfies the condition that the set {ffi(__x=vj(x)1)| x 2 K; j(x) < 1} is line*
*arly
independent.
Therefore, we will study the connecting homomorphism ffis : HsM11!
Hs+1K(2)* to find j(x) for each x 2 K. Note that if x 62 Im ffis, then
i*(x) 6= 0 in Hs+1M11. Thus a computation of ffis shows us all information
that we need. We will not distinguish x and __xin the sequel. The following
is our key lemma:
Lemma 3.3. The connecting homomorphism ffis : HsM11! Hs+1K(2)*
acts as follows :
1. On Fn (n 0),
n+1 4.3n-1 2.3n
ffi0(v32 =v1 )= v2 h10
n+1 6.3n+1 n (3n+1-1)=2
ffi1(v32 h10=v1 )= (-1) v2
n 10.3n+1 5.3n+(3n-1)=2
ffi1(v8.32h10=v1 )= -v2
n(53)+(3n-1)=2 4.3n 3n+1(11)+3(3n-1)=2
ffi2(v32 =v1 ) = v2 1 + . . .
Here + . .d.enots an element of K(2)*{i2}.
2. On F ,
ffi0(v2=v1)= h11
ffi1(v2h10=v21)= b10
ffi1(v22h11=v21)=b11
ffi2(v2b11=v1)= h11b11
3. On F *,
ffi2(=v21)= -v-12 0
ffi3(v2(v-12 0)=v1)= v-22b11
ffi3(v2 1=v1)= -v2b10
ffi4(b11=v21)= - 1b10
8
This gives rise to all the differentials of the Bockstein spectral sequence.
In fact, suppose that ffis(x=vj1) = y in the above lemma. Then for an element
a 2 HtE(2)*=(3; vj1), we havenffis+t(ax=vj1) = ay. Take a to be an element of
E(2; n)*=(v1) = (Z =3)[v32 ] or bt10i"2for t 0 and " = 0; 1. Then we have
ns t " j 3ns t "
ffis+2t+"(v32 xb10i2=v1) = v2 yb10i2;
for integers s and n such that 3n j. Therefore, we see that
Proposition 3.4. The connecting homomorphism ffis : HsM11! Hs+1K(2)*
acts as follows:
ffi0(vt2=v1)= tvt-12h11
nt 4.3n-1-1 3nt-3n-1
ffi0(v32 =v1 )= -tv2 h10(n > 0);
ffi1(v3t+12h10=v21)=v3t2b10
n(3t+1) 2.3n+1 3n+1t+(3n-1)=2
ffi1(v32 h10=v1 )= v2 (n > 0)
n(9t+8) 10.3n+1 3n+2t+5.3n+(3n-1)=2
ffi1(v32 h10=v1 )= -v2 (n 0)
ffi1(v3t+22h11=v21)=v3t2b11;
ffi2(v3t12b10=v1)= v3t-112h11b10
ffi2(v3t12b11=v1)= v3t-112h11b11 = v3t+112h10b10
ffi2(v3t2=v21)= v3t-12 0
n(9t+53)+(3n-1)=2 4.3n 3n+1(3t+11)+3(3n-1)=2
ffi2(v32 =v1 ) = v2 1 + . . .(n 0);
ffi2s+3(v3t+22h11bs+110=v21)=v3t2b11bs+110
ffi2s+3(v3t+12h10bs+110=v21)=v3t2bs+210
ffi2s+3(v3t12(v-12 0)bs10=v1)=v3t12(v-32b11bs10)
ffi2s+3(v3t12 1bs10=v1)=v3t12bs+110;
ffi2s+4(v3t12bs+210=v1)=v3t-112h11bs+210
ffi2s+4(v3t12b11bs+110=v1)=v3t+112h10bs+210
ffi2s+4(v3t2bs+110=v21)=v3t-12 0bs+110
ffi2s+4(v3t2b11bs10=v21)=-v3t2 1bs+110
for n; s 0 and t 2 Z , where + . .d.enots an element of K(2)*{i2}.
Corollary 3.5. The map i* : HsK(2)* ! HsM11sends each of the
9
following elements in K to a non-zero element:
ks 3t-1
h10; v32 h10; v2 h11;
for s; t 2 Z with s 1 (3) or s 8 (9).
v3t12b10; v3t12b11; vu2;
for t; u 2 Z with u 2 3Z or u = 3n(9t + 5 3) + (3n - 1)=2.
v3t-12h11bs+110; v3t+12h10bs+110; v3t-112 0bs10; v3t12 1bs10
for s; t 2 Z with s 0.
These elements in Corollary 3.5 form the set B, and Lemma 3.2 show
Theorem 2.5.
x4. Computation of the connecting homomorphism
In this section, we will prove Lemma 3.3 by assuming some results on the
cobar complex *E(2)* which will be shown in the next sections.
Let (E(2)*; E(2)*E(2)) denotes the Hopf algebroid associated to the Johnson-
WilsonNspectrum. For an E(2)*E(2)-comodule M with coaction : M !
M E(2)*E(2)*E(2), H*M = Ext *E(2)*E(2)(E(2)*;NM) is given as the coho-
mology of the cobar complex *M with sM = M E(2)*E(2)*E(2)s and
thePdifferential ds : sM ! s+1M defined by ds(x y) = (x) y +
s s s
i=1(-1) x y1 . . .(yi) . . .ys - (-1) x y 1 for x 2 M and
y = y1 . . .ys 2 E(2)*E(2)s . Consider the connecting homomorphism
ffis : HsM11! Hs+1K(2)* associated to (2.2). By definition, we see that
(4.1) If ds(x) vj1y mod (3; vj+11) in s+1E(2)*, then ffis([x]=vj1) = [y].
Now we state several lemmas:
Lemma 4.2. * There exists a cochain x(8 . 3n) 2 1E(2)* for each n 0
n
such that x(8 . 3n) v8.32t1 mod (3; v1) and
n+15.3n+(3n-1)=2 10.3n+2
d1(x(8 . 3n)) -v10.31 v2 X mod (3; v1 ):
______________________________
*This is the correction of the last congruence in [1, Prop. 5.2].
10
Here X denotes a cocycle that represents .
Lemma 4.3. There exist cochains X(2) and X(8) 2 2E(2)* such that
X(n) vn2X mod (3; v1) and
d2(X(2)) v41z3 X3 - v41v-32f13 mod ( 3; v51) and
d2(X(8)) -v41v32z3 X9 + v41v-62f19 mod ( 3; v51):
Here z and f1 represent i2 and 1, respectively.
Lemma 4.4. In the cobar complex 3E(2)* we have a cochain X0 such
that X0 X mod (3; v1) and
d2(X0) -v21v-12f0 mod ( 3; v31);
for a cocycle f0 representing 0.
Lemma 4.5. In the cobar complex 3E(2)*, we have cochains fi (i = 0; 1)
such that fi represents i in E*2(L2V (1)) and
d3(f0) v1v-22b11 X mod (3; v21);
d3(f1) 0 mod ( 3; v21):
Note that fi's of Lemmas 4.3 and 4.4 are the same as those of Lemma
4.5, which is seen by the proofs in x6.
Assuming these lemmas we will prove Lemma 3.3 by which we obtain
Proposition 3.4.
Proof of Lemma 3.3. In [6, Prop. 5.4], it is shown that d0(v2) v1t31
n 4.3n-1-12.3n-1 4.3n-1
mod (3; v31) and d0(v32) v1 v2 t1 mod (3; v1 ), which implies
i
the first equations in the parts 1 and 2 of Lemma 3.3. In fact, h1i = [t31].
Besides, we see that d2(v2b11) v1t31 b11 mod (3; v31), since d2(b11) 0 and
d0(v2) v1t31. Therefore the fourth one in the part 2 follows.
In [1, Prop. 5.2] and [1, Prop. 5.3], we show that
d1(x(1)) v21b10 mod ( 3; v31); d1(y(2)) v21b11 mod ( 3; v31);
n-1+1(3n-1)=2 6.3n-1+2
d1(x(3n)) -(-1)nv6.31 v2 X mod (3; v1 ) (n > 0);
11
where x(n) and y(n) are elements such that
x(n) vn2t1 and y(n) vn2t31 mod ( 3; v1):
These shows the second equation in the part 1 and the second and the third
ones in the part 2. The third one in the part 1 follows from Lemma 4.2.
n 3(3n-1)=2 3n (3n-1)=2
Since f31 v2 f1 and X v2 X mod (3; v1) up to homology
by Theorem 2.3, we see the fourth one of the part 1 from Lemma 4.3.
Now turn to the part 3. By Lemma 4.4 we obtain the first one. The forth
one also follows from it, since b10 1 = -v2b11 0 by Theorem 2.3. The second
one follows immediately from Lemma 4.5. For the third, since d0(v2) v1t31
and d3(f1) 0 mod (3; v21) by Lemma 4.5, we see that d3(v2f1) v1t31 f1
mod (3; v21), which is homologous to v1v2b10 X by a relation of Theorem
2.3. q.e.d.
x5. Some relations in E(2)*E(2)
Note that in E(2)*E(2) we have the following relations (cf. [1, (3.8),(3.9)]*
*):
t91= v-12t1jR (v32) - v1v-12t32- v21v-12V + v91v-12t2
v22t1 - v1v-12t32+ v21v2t31
(5:1) 3 -1 10 6 4 5 5
+v1(v2 t1 + t1) - v1v2t1 + v1t1 mod ( 3; v1)
n-1 -1 3 2
t9n v32 tn - v1v2 tn+1 mod (3; v1);
in which jR (v32) = v32+ v31t91- v91t31and
(5:2) V = -v22t31- v1v2t61+ v21v22t1 - v31v2t41+ v41t71- v51v2t21- v61t51:
Therefore we see the following relation in the cobar complex *E(2)*:
(5:3) c9 v2t2c mod ( 3; v1);
for a cochain c 2 *;4tE(2)*.
Note that d0 : E(2)* ! E(2)*E(2) is computed by d0(x) = jR (x) - x.
Since jR (v2) v2 + v1t31- v31t1 mod (3) by Landweber's formula, and so we
12
compute the followings mod (3; v81),
d0(v22) -v1v2t31+ v21t61+ v31v2t1 + v41t41+ v61t21
d0(v42) v1v32t31+ v41o3 + v51v22t31+ v61v2t61- v71v22t1
d0(v52) -v1v42t31+ v21v32t61- v31v22t91+ v41v2t32
(5:4) +v51(-v32t31+ t31t32+ t151) + v61(v22t61+ v32t21) + v71t1*
*31
d0(v72) v1v62t31+ v31v42t91- v41v32(t32+ t121) + v51v52t31
+v61v42t61+ v71t211
d0(v82) -v1v72t31+ v21v62t61 mod ( 3; v41);
where o = t41- t2.
Moreover, d1 : E(2)*E(2) ! E(2)*E(2)2 is given by d1(x) = 1 x -
(x) + x 1, and we have
d1(t1) = 0
(5:5) 3
d1(t2) = -t1 t1 - v1b10
for b10 = -t1 t21- t21 t1.
__
Lemma 5.6. There exist cochains T 0; T; and T in 1E(2)* such that
d1(T 0) -t91 z9 - b11- v-92g90 mod ( 3; v31);
d1(T ) -v2t91 z9 - v2b11- v-82g90+ v1v-272t31 (t93- t811t92) mod ( 3; v31);
__ 9 9 -26 09
d1(T ) -v2z t1 - v2b11- v2 g0
+v1v-272t31 (t93- t91t272) - v31v-272t1 (t93- t91t272) mod ( 3; *
*v61):
Here
z = v-12t2 + v-32t32- v-32t121; b11 = -t31 t61- t61 t31;
g0 = t1 t2 - t21 t31and g00= t32 t1 - t31 t101:
Proof. First consider the cochains bt3= t3- t91t2 and et3= t3- t1t32. Then
we see
d1(bt3) -v32t1 z - v2b11- v22g0 mod ( 3; v1) and
d1(et3) -v32z t1 - v2b11- g00 mod ( 3; v1);
13
by computing
d1(t3) -t1 t32- t2 t91- v2b11;
-t1 t32- v22t2 t1 - v2b11
d1(-t91t2) t101 t31+ t1 t121+ t91 t2 + t2 t91
-v22g0 + t1 t121- v22t1 t2 + t2 t91:
d1(-t1t32) t41 t91+ v31 t101+ t1 t32+ t32 t1
t121 t1 + t1 t32- g00- t32 t1
mod (3; v1).
Now put T 0= v-272bt93, and 9-th power of d1(bt3) yields d1(T 0): In fact, by
(5.3) we see that b911 v182b11 mod (3; v61).
We define T = v2T 0and compute d1(T ) d1(v2T 0) v1t31 T 0-
v-262(v272t91 z9 + v92b911+ v182g90) mod (3; v31)._ ___ ___
In the same manner we verify that the element T = v2T 0for T 0= v-272et93
satisfies the last congruence. q.e.d.
x6. Proofs of Lemmas 4.3, 4.4 and 4.5
Let x denote a cochain that represents in H2M02, and define X 2
2E(2)* by
X = v-42x9:
Lemma 6.1. The element X is a cocycle of *M02that represents in
H2M02, and satisfies the following in the cobar complex 3E(2)*:
d2X -v1v-12t31 X - v41v-42o3 X - v51v22t31 X mod (3; v61):
Proof. By definition with (5.4) we see that
0 d2(x9) d2(v42X)
v1v32t31 X + v41o3 X + v51v22t31 X
+v61v2t61 X + v42d2(X) mod ( 3; v71):
q.e.d.
As we noted in [1, (5.1)], we have an element w such that d1(w) = x3+v2x
in *K(2)*. Since x9 = v42x by (5.3), we obtain that d1(w3 - v32w) = 0, that
14
is w3 - v32w is a cocycle. Theorem 2.3 shows that w3 - v32w is bounded, but
nothing bounds it by degree reason. Therefore, w satisfies
(6:2) d1(w) = x3 + v2x and w3 = v32w in *M02:
By this, we have d1(w9) x27+ v92x9 v122X3 + v132X mod (3; v91), and so by
(5.4),
(6:3) d1(v-122w9) -v31v-152t91 w9 + v61v-182t181 w9 + X3 + v2X
mod (3; v91) since d0(v-122) v-182(d0(v22))3.
Since h10 = 0 in H3K(2)* by Theorem 2.3, we have cocycles y0 such
that d2(y0) = t1 x in 3K(2)*. Define y1 = -v-12(y30+ t31 w). Then
d2(y1) = -v-12(t31 x3 - t31 (x3 + v2x)) = t31 x.
Put Y0 = v-62y90and Y1 = v-102y91. Then we have
Lemma 6.4. Yi yi mod (3; v1) (i = 0; 1) and Y03 -v2Y1- v-182t271 w9
mod (3; v91). Besides,
d2(Y0) t1 X + v1v-32o3 X
+v21v-12t31 X mod (3; v31); and
d2(Y1) t31 X - v1v-12(t31 Y1 - t61 X)
+v31(v-32t91 Y1 - v-62t92 X) mod ( 3; v41):
Proof. The first one follows from (5.3). By definition, Y03= v-182y270
-v-92y91- v-182t271 w9 = -v2Y1 - v-182t271 w9 mod (3; v91).
A direct computation with (5.1) shows the following,
d2(v-62y90) v-62t91 x9 v-42(t1 - v1v-32t32+ v21v-12t31) v42X mod (3; v31);
and jR (v42) = v42+ d0(v42) is read off from (5.4).
For Y1, noticing that d0(v-102) v-182d0(v82) mod (3; v41), we compute with
(5.4),
d2(v-102y91) -v1v-112t31 y91+ v21v-122t61 y91+ v-102t271 x9
-v1v-112t31 v102Y1 + v21v-122t61 v102Y1
+v-42t31 v42X - v31v-132t92 v42X
t31 X - v1v-12(t31 Y1 - t61 X)
+v31v-32t91 Y1 - v31v-92t92 X mod (3; v41):
15
q.e.d.
Proof of Lemma 4.5. Define first f00= t21 X + t1 Y0 and f01=
t1 Y1 - t2 X. Then Lemma 6.4 shows that f0irepresents i =
for each i = 0; 1.
By Lemmas 5.6, 6.1 and 6.4 with (5.3) we compute
d3(f00) -v1(v-32t1 o3 X - v-12t21 t31 X)
v1(t1 z X - v-12g0 X)
d3(-v1v-32T X) v1v-32(v32t1 z + v2b11+ v22g0) X
d3(v1z Y0) -v1z t1 X
d3(-v1zt1 X) v1z t1 X + v1t1 z X
mod (3; v21). Then we have the first one by putting f0 = f00- v1v-32T X +
v1z Y0 - v1zt1 X.
Similarly, we compute
d3(f01) -t1 t31 X + v1v-12t1 (t31 Y1 - t61 X)
+t1 t31 X + v1b10 X - v1v-12t2 t31 X
d3(v1v-12t2 Y1) -v1v-12t1 t31 Y1 - v1v-12t2 t31 X
__3 -9 9 3 9 03
d3(v1v-92T X) v1v2 (-v2z t1 - v2b10- g0 ) X
d3(-v1z Y1) v1z t31 X
mod (3; v21). Notice that v1v-12(t2 t31- t1 t61) = v1v-92g030,_and we obtain
3
the second one by setting f1 = f01+ v1v-12t2 Y1 + v1v-92T X - v1z Y1.
q.e.d.
Proof of Lemma 4.3. Put X(2) = v2X3 - v1Y03- v31v-22Y13and note
that f1 t1 Y1 - t2 X mod (3; v1) for f1 in the proof of Lemma 4.5. We
then obtain d2(X(2)) from computation
d2(v2X3) v1t31 X3 - v31t1 X3 - v31v-22t91 X3
v1t31 X3 + v31(t1 + v1v-32t32) X3
d2(-v1Y03) -v1t31 X3 - v41v-92o9 X3
d2(-v31v-22Y13) -v41v-32t31 Y13- v31v-22t91 X3
-v41v-32t31 Y13- v31(t1 - v1v-32t32) X3
16
mod (3; v51) by Lemmas 6.1 and 6.4.
By defining X(8) = v42X9 - v1v-32Y19, we compute that
d(v42X9) v1v32t31 X9 + v41o3 X9
d(-v1v-32Y19) v41v-62t91 Y19- v1(v32t31- v31v-62t92) X9:
Since o3 + v-62t92 -v32z3 - v-62t92, we have the result. q.e.d.
__0 -1 __ -1
Proof of Lemma 4.4. Set X = X + v1v2 Y1 and f0 = v2 (t31 Y1 +
t61 X), and we obtain
__0 2 -1__ 3
d2(X ) v1v2 f0 mod (3; v1)
by the computation:
d2(X) -v1v-12t31 X
d2(v1v-12Y1) -v21v-22t31 Y1 + v1v-12t31 X - v21v-22(t31 Y1 - t61 X)
mod (3; v31). Indeed, these are seen by using Lemmas 6.1 and 6.4 and the
congruence d0(v-12) -v1v-22t31mod (3; v21) seen by (5.4). __
For a while, we argue in the E2-term E*;*2(L2V (1)). Notice that f0 repre-
sents an element in v-12, and 0 is an element of . By a
relation of the Massey products, we see that_h11 =
and = -b11. Therefore,_h11[f0] = -v-12b11 = -h11 0 by a
relation in Theorem 2.3, and so [f0] = - 0 mod Ker h11. Theorem 2.3 also
shows us that Ker h11 E3;162(L2V_(1)) is generated by h10b10. We have an
integer k_2_Z =3 such that [f0] = - 0 + kh10b10 and a cochain e0 such that
d2(e0) = f0 + f0 - kt1 b10 for f0 given in the proof of Lemma 4.5. Fur-
thermore the relation v22h10b10 = h11b11 certifies the existence of a cochain B
such that d1(B) =_v22t1 b10- t31 b11.
0 -1 2 -1 -3
Putting X0 = X + kv1v2 b11- v1(v2 e0 + kv2 B) leads us to the desired
congruence. q.e.d.
x7. Proof of Lemma 4.2
In this section, we correct [1, Th. 4.4], whose X should be replaced by
our x(8).
17
Lemma 7.1. There exists an element x(7) such that x(7) v72t1 mod
(3; v1), and
d1(x(7)) v21b311+ v71v52X mod (3; v81);
for a cocycle X that represents .
Proof._Put x(7)0= v52t91- v1v42t32+ v31v22t181-_v21T 03- v1v72z9 - v41v42t91*
*z9 +
v41v32T- v51v32t32- v61v-42T 3for T 0, T and T . Using (5.4), (5.5) and Lemma
5.6, we compute the followings mod (3; v81):
d1(v52t91) (-v1v42t31_1+ v21v32t61_(a1)- v31v22t91_2+ v41v2t32_c
+v51(-v32t31_6+ t31t32+ t151) + v61(v22t61_e+ v32t21)_d+ v7*
*1t131)) t91
d1(-v1v42t32) -v1(v1v32t31_(a1)+ v41o3 + v51v22t31_e+ v61v2t61) t32
+v1v42t31_t91_1+ v41v42b11_3
d1(v31v22t181) -v41v2t31_t181_c+ v51t61 t181+ v61v2t1__t181_d
+v71t41 t181+ v31v22t91_t91_2
d1(-v21T 03) v21t271_z9_b+ v21b311+ v21v-272g270_a
d1(-v1v72z9) -v21v62t31_z9_b- v41v42t91_z9_4+ v51v32t32 z9
+v51v32t121 z9 - v61v52t31_z9_7- v71v42t61 z9
d1(-v41v42t91z9) -v51v32t31 t91z9 + v41v42t91_z9_4+ v41v42z9__t91_5
__ 7 9 __ 4 3 9 9 -26 09
d1(v41v32T) v1t1 T + v1v2(-v2z t1 - v2b11- v2 g0
+v1v-272t31 (t93- t91t272) - v31v-272t1 (t93- t91t272))
v41v32(-v2z9__t91_5- v2b11_3- v-262g090_c
+v1v-272t31 (t93- t91t272))
d1(-v51v32t32) v51v32t31_t91_6
d1(-v61v-42T 3) v61v52t31_z9_7+ v61v52b10_d+ v61v22g30_e+ v71v-52t31 T 3:
The elements underlined with the same number are cancelled each other.
Since the sum of the elements underlined with (a1) is -v21v32g30, and g90
v102g0- v1(v92t41 t2+ v2t1 t33+ v72(t32 t2+ t1t32 t31)); the sum of the elements
with (a1) and a is:
v21v-272g270- v21v32g30 -v51(t121 t32+ v-242t31 t93+ v-62(t92 t32+ t31t92 t91)*
*):
18
Besides, the underlined parts with b, c and d are computed as follows:
v21t271 z9 - v21v62t31 z9 -v51v-32t92 z9
v41v2t32 t91- v41v2t31 t181- v41v-232g090 v71(v22t3 t1 - v42t2 t21); and
v61(v32t21 t91+ v2t1 t181) + v61v52b10 -v71v22(t21 t32- t1 t1t32)
using (5.1). Now we obtain
d1(x(7)0) v21b311+ v51Z + v71x0
for
Z = (t31t32+_t151)_At91- o3__t32_C+ t61 t181+ v32t32_z9_D
+v32t121_z9_D- v32t31_t91z9_B+ v-242t31 (t93_1- t91t272_B)
-(t121_t32_C+ v-242t31_t93_1+ v-62(t92_t32_C+ t31t92_t91_A)) - v-32t92_z*
*9_D
x0 = t131 t91+ t41 t181- v42t61 z9 + v-52t31 T 3
+v22t3 t1 - v42t2 t21- v22(t21 t32- t1 t1t32):
We introduce an element w = -z - v-12t2 = v-12t41+ v-12t2 - v-32t32. Notice
that z9 z3 mod (3; v31). Then the parts underlined with A, B, C and D
amount to v32t31 t91(w 1), v32t31 t91(1 w), v32w3 t32and v62w3 z3 mod
(3; v31), respectively, and so we have
Z v32t31 t91(w3 1 + 1 w3) + t61 t181- v62w3 w3:
Since we have d1(v2w) t1 t31mod (3; v1) by (5.5), we obtain
d1(v62w6) -(t31 t91(w3 1 + 1 w3) + t61 t181- v62w3 w3)
mod (3; v31). Therefore the cochain x(7) = x(7)0+ v51w6 satisfies the desired
congruence by putting x0= v52X. In fact, is represented by a cocycle whose
leading term is v-32t3 t1 + v-102t31 t33, and T is congruent to t3 mod (3; v1).
q.e.d.
Proof of Lemma 4.2. Put x(8) = V 3+ v41x(7) for V in (5.2). Since
d1(V ) v21b11 mod (3; v81) by [1, (3.7)], Lemma 7.1 implies the lemma for
n = 1. For large n, use (6.3) to obtain the lemma. q.e.d.
19
x8. The Adams-Novikov differentials on E*;*2(L2W )
In this section, we compute the Adams-Novikov differential dr : Es;tr(L2W ) !
Es+r;t+r-1r(L2W ) for r 2. Note that E*2(L2W ) is given in Theorem 2.5 and
that dr = 0 unless r 1 mod 4 by degree reason.
N
Proposition 8.1. For all r 2, dr = 0 on K(1)*=k(1)* (h10; i2).
Proof. Suppose that there are elements x 2 (h10; i2) and y 2 Es+rr(L2W )
with filt x = s for integers 0 s 2, r > 4 and j > 0, such that du(x=vj1) = 0
for u < r and
dr(x=vj1) = y 6= 0:
0
Then dr0(x=vj+11) = y0 6= 0 for some r0 r and y0 2 Es+rr0(L2W ). Since r is
finite, we may assume that du(x=vj+k1) = 0 for u < r and
dr(x=vj+k1) = y=vk16= 0 2 Es+rr(L2W )
from the beginning. Thus y generates a module isomorphic to K(1)*=k(1)*
in Es+rr(L2W ). On the other hand, Theorem 2.5 shows that Es+r2(L2W ) =
Hs+rM11does not contain such a module since s + r r > 4. This is a
contradiction. q.e.d.
In the following, an equation dr(x) = y means not only the indicated one
but also ds(x) = 0 for s < r.
L *
Lemma 8.2. Suppose that dr(x) = y on a element x of Fn or F F ,
then we get
dr(xbs10i"2) = ybs10i"2
for s 0, " = 0; 1 and xbs10i"22 E*;*2(L2W ).
Proof. Since b10 represents the homotopy element fi1, the relation
dr(x) = y implies dr(xbs10) = yb10.
The same proof that shows dr(x) = y in the spectral sequence for ss*(L2W )
works to show dr(xi2) = yi2 in it, since the proof depends on the result of
the differentials of the spectral sequence for ss*(L2V (1)) in which it is shown
in [11] that dr(x) = y if and only if dr(xi2) = yi2 q.e.d.
20
For the other differentials, we study the exact sequence
. .-.! Es;*2(L2V (1)) -i*!Es;*2(L2W ) -v1!Es;*2(L2W ) -ffis!Es+1;*2(L2V (1)) -!*
* . . .
associated to the cofiber sequence (2.1) in order to use the results on E*;*2(L*
*2V (1)):
(8.3) ([11, Prop.s 8.4, 9.13], [4]) The differential d5 of the sepectral sequen*
*ce
for ss*(L2V (1)) acts as follows :
d5(v3t+12) = tv3t-12h11b210; d5(v3t+12b11) = (1 - t)v3t+12h10b310;
d5(v3t+32 0) = tv3t2b11b210; d5(v3t+12 1) = (1 + t)v3t2b310;
d5(v3t-12) = (1 + t)v3t-32h11b210; d5(v3t-12b11) = tv3t-12h10b310;
d5(v3t+12 0) = (1 + t)v3t-22b11b210d5(v3t-12 1) = (1 - t)v3t-22b310;
d5(v3t2b10) = tv3t-22h11b310; d5(v3t2b11) = (1 - t)v3t2h10b310;
d5(v3t-12 0) = (1 - t)v3t-42b11b210;d5(v3t2 1b10) = (1 + t)v3t-12b310:
The differential d5 of the sepectral sequence for ss*(L2V (1)) acts as follows :
Here note that the undetermined integer k of [11] is shown to be 1 by [4]
and the results of (8.3) follow.
L *
Lemma 8.4. Let x be an element of Fn or F F . Then we see the
following :
(1) If x = i*(__x) and dr(__x) = __yin E*;*r(L2V (1)), then
dr(x) = i*(__y) in E*;*r(L2V (1)).
(2) If dr(ffis(x)) = ffis+r(y), then
dr(x) = y + . . .
Here . .d.enotes an element of J given by
N N
J = E(2; 1)*{v2h10=v1; v22h11=v1; =v1; b11=v1} (Z =3)[b10] (i2):
Furthermore the generators of J have the bidegrees :
||v3s+12h10=v1|| = (1; 16(3s + 1)); ||v3s+22h11=v1|| = (1; 16(3s + 2) + 8);
||v3s2=v1|| = (2; 48s + 4); ||v3s2b11=v1|| = (4; 48s + 40);
||b10|| = (2; 12) and ||i2|| = (1; 0):
21
Proof. Part 1) follows from the naturality of the differential dr.
The hypothesis dr(ffis(x)) = ffis+r(y) of 2) implies dr(x) y modLKer
ffis+r =LIm v1Nby naturality.NBesides, dr(x) 2 Es;*rfor s 5 and s5 Es;*r
G = (F F *) (Z =3)[b10] (i2). Therefore dr(x) y mod J = Im (v1 :
G ! G). The structure of J follows from Theorem 2.5. q.e.d.
L *N N
Proposition 8.5. The differentials on (FL F ) (Z =3)[b10] (i2) are
read off from the following results on F F * (by Lemma 8.2):
(a) d5(v3t+12=v1) = tv3t-12h11b210=v1
(a')d5(v3t-12=v1) = 0
(b) d5(v3t+12h10=v21) = tv3t-12b310=v1
(c)d5(v3t+22h11=v21) = (1 - t)v3t-12b11b210=v1 + kv3t+12h10b210i2=v1
for some k 2 Z =3
(d) d5(v3t+12b11=v1) = (1 - t)v3t+12h10b310=v1
(d')d5(v3t-12b11=v1) = 0
(a)*d5(v3t+32 0=v1) = tv3t2b11b210=v1
(a')*d5(v3t+12 0=v1) = 0
(b)* d5(v3t2=v21) = (1 - t)v3t-22 0b210=v1
(c)*d5(v3t2b11=v21) = (1 + t)v3t-12 1b310=v1
(d)* d5(v3t+12 1=v1) = (1 + t)v3t2b310=v1
(d')*d5(v3t-12 0=v1) = 0.
Proof. The first four equations of (8.3) give rise to (a), (d), (a)* and
(d)* by Lemma 8.4 (1).
Proposition 3.4 shows the following:
v3t-32h11b210= ffi4(v3t-22b210=v1); v3t-12h10b310= ffi6(v3t-22b11b210=v*
*1);
v3t-22b11b210= ffi8(v3t2 0b210=v1); v3t-22b310= ffi7(v3t-22 1b210=v1):
22
Since i*ffis(y) = 0, Lemma 8.4 (1) implies that the second four equations of
(8.3) yield (a'), (d'), (a')*, and (d')*.
Furthermore, Proposition 3.4 shows the following:
ffi1(v3t+12h10=v21) = v3t2b10; ffi1(v3t+22h11=v21) = v3t2b11;
ffi2(v3t2=v21) = v3t-12 0 ffi4(v3t2b11=v21) = -v3t2 1b10; and
v3t-22h11b310= -ffi6(v3t-12b310=v1); v3t2h10b310= -ffi6(v3t-12b11b210=v1*
*);
v3t-42b11b210= -ffi7(v3t-22 0b210=v1); v3t-12b310= ffi9(v3t-12 1b310=v1):
Therefore, we apply Lemma 8.4 (2) to show
d5(v3t+12h10=v21)= tv3t-12b310=v1 + . . .
d5(v3t+22h11=v21)= (1 - t)v3t-12b11b210=v1 + . . .
d5(v3t2=v21)= (1 - t)v3t-22 0b210=v1 + . . .
d5(v3t2b11=v21)= (1 + t)v3t-12 1b310=v1 + . . .
by using the last four equations of (8.3). Let (J)s;tdenotes the submodule of J
with bidegree (s; t). Then we see that (J)s;u= 0 if (s; u) = (6; 16(3t-1)+32),
(7; 16(3t - 2) + 36) or (9; 16(3t - 1) + 56), and = (Z =3){v3t+12h10b210i2=v1} *
*if
(s; u) = (6; 16(3t - 1) + 56). Therefore we obtain (b), (c), (b)* and (c)*.
q.e.d.
We here recall the folklore lemma which will be used later:
Lemma 8.6. Let X f! Y !g Z h! X be a cofiber sequence with
E(2)*(f) = 0. Then in the exact sequence E*2(L2X) f*!E*2(L2Y ) g*!E*2(L2Z) !ffi
E*+12(X), we have the followings :
(1) If we have a chart
y !g* z0
- d5
z ffi!x;
and x is a permanent cycle, then f*([x]) = [y], where [.] denotes a homotopy
class.
23
(2) If we have a chart
x0 !f* y0
- d5
y g*! z0
- d5
z !ffix;
then d9(x) = x0.
(3) If we have a chart
x0 -f*! y0
- d5
- d9 y -g*! z0
z - ffi! x
then d5(z) = z0.
v1 j
Corollary 8.7. Consider the cofiber sequence V (1) !i 4W ! W !
V (1). Then the induced map i* : ss*(L2V (1)) ! ss*-4(L2W ) acts as follows:
(a) i*(v9t+22h10) v9t+12h10b210=v21mod Ker fi1
(a') i*(v9t+22h10i2) v9t+12h10b210i2=v21mod Ker fi1
(b) i*(v3u+12) (1 + u)v3u2b210=v21+ . .m.od Ker fi1 (u 6 0 (3)).
(b') i*(v3u+12i2) (1 + u)v3u2b210i2=v21+ . .m.od Ker fi1 (u 6 0 (3)).
Proof. We obtain the following charts (up to signs)
v1 3
v2h10b310=v21-! v2h10b10=v1
- d5
v2b11=v1 -ffi!v22h10b10
and
v1 3u 3
(1 + u)v3u2b310=v21-! (1 + u)v2 b10=v1
- d5
v3u+12 1=v1 -ffi!v3u+12b10
24
from Propositions 3.4 and 8.5. Since v9t+22h10 and v3u+12 with u 6 0 mod
3 are permanent cycles by [11, Th. A], i*(v9t+22h10b10) = v9t+12h10b310=v21and
i*(v3u+12b10) = (1 + u)v3u2b310=v21by Lemma 8.6. Now divide it by b10 which
represents fi1 2 ss*(S0), we obtain (a) and (b).
In the same way, we obtain (a') and (b') by Lemma 8.2. q.e.d.
N
Proposition 8.8. The differentials on Fn (i2) are read off from the
following relation on Fn (by Lemma 8.2):
(
n+2t+3n+1 4.3n-1 v9t+12h10b210=v1 n = 0
(a) d5(v32 =v1 ) =
0 n > 0
n+2t-3n+1 4.3n-1
(a') d5(v32 =v1 ) = 0
n+2t+3n+1 2.3n+1+1 3n+2t+3(3n-1)=22
(b) d5(v32 h10=v1 ) = v2 b10=v1 (n 0)
(
n+2t+8.3n 10.3n+1 0 n = 0; 1
(c) d5(v32 h10=v1 ) = 3n+2t+5.3n+3(3n-1-1)=22
v2 b10=v1 n > 1
n(9t+53)+(3n-1)=2 n 3n+1(3t+11)+3(3n-1)=2-1
(d) d5(v32 =v4.31) = v2 1b210=v1
(n > 0)
(e) d5(v9t+82=v41) = 0
(e')d5(v9t+22=v41) = -v9t-12 1b210=v1
n b(n)-3n-1
Proof. For the cases (d) and (e'), d5(ffi2(vb(n)2=v4.31)) = d5(v2 1) =
n-2 b(n)-3n-2
vb(n)-32 b310= ffi7(v2 1b210=v1), since b(n) - 3n - 1 0 or 3
mod 9, where b(0) = 9t + 2 and b(n) = 3n(9t + 5 3) + (3n - 1)=2 for
n > 0. Therefore Lemma 8.4 (2) implies (d) and (e'), since (J)s;u= 0 for
(7; 16(b(n) - 3n - 2) + 44) and (7; 16(9t - 1) + 44).
Take x=va12 Fn. For the cases (a), (a'), (b), (c) and (e), d5(ffi*(x=va1)) =
0 by [11, Prop.s 8.4, 9.13]. Therefore, d5(x=va1) 2 J by Lemma 8.4 (2).
Comparing degrees, we have (a) for n > 0 and (a'), and (c) for n = 0.
Besides,
d5(v9t+32=v31)= k1v9t+12h10b210=v1 + k2v9t2b11i2=v1
d5(v9t-32=v31)= k3v9t-52h10b210=v1 + k4v9t-62b11i2=v1
25
n+2t+3n+1 2.3n+1+1 3n+2t+3(3n-1)=22
d5(v32 h10=v1 ) = k5v2 b10=v1 (n 0)
n+2t+8.3n 10.3n+1 3n+2t+5.3n+3(3n-1-1)=22
d5(v32 h10=v1 ) = k6v2 b10=v1 (n 1)
d5(v9t+82=v41)= k7v9t+62b210i2=v1 + k8v9t+52h11b310=v1
for some ki 2 Z =3 (1 i 8). Since v9t+52h11b310=v1 is hit by d5 of v9t+72b10=*
*v1
by Proposition 8.5, we take k8 = 0 by replacing v9t+82=v41by v9t+82=v41-
(k8)v9t+72b10=v1, that is,
d5(v9t+82=v41) = k7v9t+62b210i2=v1:
Now we determine the numbers ki for 1 i 7. Consider the following
charts (up to signs)
k2v-12 1b310i2=v1
- d5
v1 2
k1v2h10b210=v21+ k2b11i2=v21! k1v2h10b10=v1 + k2b11i2=v1
- d5
ffi02
v32=v31 ! v2h10
and
k3v-72b510=v1 + k4v-72 1b310i2=v1
- d5
v1 -5 2 -6
k3v-52h10b210=v21+ k4v-62b11i2=v21!k3v2 h10b10=v1 + k4v2 b11i2=v1
- d5
ffi0-4
v-32=v31 ! v2 h10
obtained from Propositions 3.4 and 8.5. Then k2 = 0 since v22h10 is a perma-
nent cycle by [11]. Besides Corollary 8.7 shows that k1 = 1. The numbers k3
and k4 are seen to be zero by Lemma 8.6, since v-42h10 is also a permanent
cycle by [11, Cor. 10.7]. By similar charts, the third and the fourth equations
imply
n+2t+(3n+1-1)=2 3n+2t+3(3n-1)=22 2
i*(v32 ) = k5v2 b10=v1 (n > 0)
n+2t+5.3n+(3n-1)=2 3n+2t+5.3n+3(3n-1-1)=22 2
i*(v32 ) = k6v2 b10=v1 (n 1)
n+2t+(3n+1-1)=2
up to sign using Propositions 3.4 and 8.5. In fact, for n > 0, v32
n+2t+5.3n+(3n-1)=2
and v32 are permanent cycles by [11, Cor. 10.7]. Now com-
pare with Corollary 8.7, and we obtain
26
(
0 n = 1
k5 = 1 if n > 0, and k6 =
1 n > 1:
If n = 0, then k5 = 1 by applying Lemma 8.6 (3) to the following chart (up
to sign):
v-22 0b410i*! v-22 0b410=v1
- d9 - d5
ffi1 2 2
v32h10=v71! v2 b10=v1:
Consider again the chart (up to sign)
k7v42 0b510i2=v1
- d5
v1 6 6
k7v62b310i2=v21! k7v2b10i2=v1
- d5
ffi2 6 7
v82=v41 ! v2 1 v2i2
obtained from Propositions 3.4 and 8.5. Since v62 1 v72i2 is a permanent
cycle by [11, Cor. 10.7], we obtain k8 = 0. q.e.d.
These propositions give us the following charts of E2-term with d5, in
which holizontal lines of length 4 denote multiplication by v1, lines of slope
1=3, multiplication by h10 and lines of slope 1=11, multiplication by h11. The
differential d5 is expressed by arrows of slope -5. Besides, the same pattern
of period (10; 2) denotes multiplicationNby b10. q
The following is the one on F (Z =3)[b10], where qqq_starting from
dimension 12 is multiples by b10 with
v-12b11=v1
v2h10=v21_____________v2h10=v1
i
ii i
v2=v1 i i i
The other one q q__q q is generated by
v22b10=v1
v22h11=v21_ v22h11=v1
v22=v1
27
| CO r rr DDDOO
|qqCO CrCOrr riiDDDOO r__rr_DD
|qCO r CriiDO D r rD
|qqrririDOCC DDDrO Drr__rD r
|q DDO r___rD r___ri
|qq r__rr_DD rD r_rr_ riDDODO
4 |qr rD r riiDO D r
|qq r rr__rii D r Drr_r_D
___________________________________________________||qqqqqqqqqqqqqqq*
*qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqrrr_iirrr_D
0| 12 28 50
| r__rr_ DDDOO
|qq r__rr_ riiDDDOO r__rr_DD
|q r riiDO D r rDDO
|qqrr_riiDO DDDrO Drr__rDDO DDDOr
|q DDO r___rD DDO Dr___ri
|qq r__rr_DD rDDDDOO Dr_rr_D Dri
4 |qr rDDO D r riiD r
|qq DDDOr rr__riiDD r rr_r_
___________________________________________________||qqqqqqqqqqqqqqq*
*qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqrrr_iirrr_DD
48| 60 76 98
| CO r__rr_
|qqCO Cr__rr_CO rii r__rr_
|qCO r Crii r r DO
|qqrr_riiCC r rr__rDO DDDOr
|q r___r DDO Dr___ri
|qq r__rr_ r DDDOO Dr_rr_D DriDDODO
4 |qr r DO D r riiDDO D r
|qq DDDOr rr__riiDD D r Drr_r_D
___________________________________________________||qqqqqqqqqqqqqqq*
*qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqrrr_iirrr_DDD
96| 108 124 146
N
The next one is the E2-term with d5 on F * (Z =3)[b10]. Each dot can be
read off from the degree of the generator.
|COr r DO
12|qqrrr_C r rr__rDO DDDrO
|q r___r DDO r___rD
|qq r__rr_ r DDDOO r__rr_DD rDDDODO
|q r DO D r rDDO D r
|qq DDDrO Drr__rD DDDOr rr__rDD
|q r___rD Dr___r
2|qqrr_ rD r Dr r___
___________________________________________________||qqqqqqqqqqqqqqq*
*qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
0| 2 25 46 50
DO
|COr DO DDDrO
12|qqrrr_DOC DDDOr rr__rDD r
|q DDO r___rD r___r
|qqDr__rr_D rD r__rr_ r DDODO
|q Dr r r DO D r
|qq r rr__r DDDOr rr__rDODD
|q r___r Dr___r D
2|qqrr_ r r Dr Dr___
___________________________________________________||qqqqqqqqqqqqqqq*
*qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
4|850 73 94 98
28
DO
| r DO DDDrO DO
12|qqrrr_DO DDDOr rr__rDODD DDDrO
|q DDO r___rD DDO r___rD
|qqDr__rr_D rDDDDOO r__rr_DD rD
|q Dr DO D r rD r
|qq DDDrO Drr__rD r rr__rDO
|q r___rD r___r D
2|qqrr_ rD r r Dr___
___________________________________________________||qqqqqqqqqqqqqqq*
*qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
9|698 121 142 146
For the part with i2, just shift the dimension. With these charts, the relation
fi61= 0 2 ss*(S0) implies the following
L *N *
* N
Proposition 8.9. On the elements of E*6(W ) originating (F F ) (Z =3)[b10*
*] (i2),
d9 : Es9(W ) ! Es+99(W ) is obtained by the followings :
(a) d9(v9t+22b11=v1) = v9t+12h10b510=v21
(b) d9(v9t+42h10=v1) = v9t+12b510=v1
(c)d9(v9t+82h11=v1) = v9t+42b11b410=v1
(d) d9(v9t+82b10=v1) = v9t+52h11b510=v21
(a)*d9(v9t+52 1=v1) = v9t+32b510=v21
(b)* d9(v9t+62=v1) = v9t+32 0b410=v1
(c)*d9(v9t2=v1) = v9t-22 1b510=v1
(d)* d9(v9t+12 0=v1) = v9t-32b11b410=v21
Proof. Since the proof of each equation is similar, we only prove (a)
here. The element named A in the following chart is v2b510=v21.
| Abr__rr_10 DDDOO
|qq A r__rr_ riiDDDOO r__rr_DD
|q r riiDO D r rDDO
|qqrr_riiDO DDDrO Drr__rDDO DDDOr
|q DDO r___rD DDO Dr___ri
|qq r__rr_DD rDDDDOO Dr_rr_D Dri
4 |qr rDDO D rB riiD r
|qq DDDOr rr__riiDD r rr_r_
___________________________________________________||qqqqqqqqqqqqqqq*
*qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqrrr_iirrr_DD
48| 60 76 98
29
__
Let E_n_denote the n-th term_of the Adams-Novikov resolution such that
holim-En = L2S0. Then Es2(E 12^ W ) ~=Es2(L2W ) if s < 12, and = 0 if s >
n
12. By observing the above chart, we see that v2h10=v21is a permanent_cycle in
the Adams-Novikov spectral sequence and yields an element of ss*(E12 ^ W ).
Since fi61= 0 2 ss*(S0) and ss*(S0) acts on any homotopy groups, the element
Ab10 = v2h10b610=v21must be killed by some element. Note here that fi1 is
represented by b10. The chart shows us the only candidate for_it is B in
the chart which is v22b11b10=v1. Since we_have_a map L2W ! E12 ^ W , the
relation d9(v22b11b10=v1) = Ab10in E*r(E12 ^W ) also holds in E*r(L2W ). Thus,
dividing by b10 gives (a). q.e.d.
The following completes the computation of the differentials.
N
Proposition 8.10. On the elements of E*6(W ) originating Fn (i2), d9
is given by the followings :
n+1(3t+1) 4.3n-1
(a) d9(v32 =v1 ) = 0
n+1(3t-1) 4.3n-1
(a')d9(v32 =v1 ) = 0
(c)d9(v9t+82h10=v111) = v9t+32b410=v21
(e)d9(v9t+82=v41) = 0
Proof. Consider the total degree
(
n+1(3t+1) 4.3n-1 100 mod 144 n = 1
|v32 =v1 |
8 4 mod 144 n 2;
n+1(3t-1) 4.3n-1 >< 84 mod 144
|v32 =v1 | > 100 mod 144
: 4 mod 144 :
Then the chart shows that nothing can be hit by d9 of these elements. Thus
(a) and (a') follows. In the same way, |v9t+82=v41| 118 mod 144, and nothing
can be the target. This is (e). For (c), we compute
ffi11(d9(v82h10=v111)) = d9(v52) = v22 0b410= ffi11(v32b410=v21):
q.e.d.
30
Theorem 8.11. The E10-termNE10(W ) isPisomorphicNto the directLsumNof
k(1)*-modules (K(1)*=k(1)*) (h10; i2), n0 Fen (i2) and (Fe Fe*) (i2)
for k(1)*-modules in (2.7).
Proof of Theorem 2.8. Since E*10(W ) has a holizontal vanishing line
by Theorem 8.11, we have E*;*10(W ) = E*;*1(W ). Furthermore, there arises
no extension problem in the spectral sequence, since ss*(L2W ) is a ss*(V (0))-
module and so (Z =3)-vector space. Therefore we obtain the homotopy groups
ss*(L2W ) = E*10(W ). q.e.d.
x9. fi-elements
The fi-elements in the E2-term for ss*(S0) are defined in [6]. Here we
modifies it in the E2-term H*E(2) for ss*(L2S0) as follows: Let 0 ! E(2)* !3
E(2)* ! E(2)*=(3) ! 0 and 0 ! E(2)*=(3) ! v-11E(2)*=(3) ! M11! 0 be
short exact sequences, and ffi : H*E(2)*=(3) ! H*+1E(2)* and ffi0: H*M11!
H*+1E(2)*=(3) the connecting homomorphisms associated to the short exact
sequences, respectively. Then for an element of the form va2=vb1in H0M11, we
define
fia=b= ffiffi0(va2=vb1) 2 H*E(2)*
and fia = fia=1, which is essential in the E2-term H*E(2)* for ss*(L2S0).
Consider the cofiber sequences defining the spectra V (0) and W : S0 !3
S0 !i V (0) !j S0 and V (0) ! L1V (0) ! W !ssV (0), respectively. If an
element va2=vb1is permanent cycle, then so is fia=b, by Geometric Boundary
theorem (cf. [9]).
Proof of Theorem 2.12. By Theorem 2.8, we see that vj2=v1 for
j 0; 1; 2; 3; 5; 6 mod 9 are permanent cycles. Thus `if' part is shown. `Only
if' part is shown in [11]. q.e.d.
Proof of Theorem 2.13. The element va2=vb1with 9|a is in eF1or eFnof
(2.7), and so it is permanent by Theorem 2.8. For the case 96 |a, the part (a)
follows from Theorem 2.12. v9t32=vb1comes from Fe0of (2.7), and we obtain
the part (b). q.e.d.
31
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32
Department of Mathematics,
Faculty of Science,
Kochi University,
Kochi, 780-8520,
Japan
33