On the action of fi1 in the stable homotopy of spheres
at the prime 3
Katsumi Shimomura
Abstract. The element fi1 is the generator of the stable ho
motopy group ss10(S0). Here S0 denotes the 3localized sphere
spectrum. Toda showed that fi516= 0 and fi61= 0. Here we gen
eralize it to fi41fi9t+16= 0 and fi51fi9t+1= 0 for fi9t+12 ss144t+10(S0)
with t 0. In particular, fi41fi10 6= 0 and fi51fi10 = 0 for fi10
shown to exist by Oka. This is proved by determining subgroups
of ss*(L2S0), where L2 denotes the Bousfield localization functor
with respect to v12BP .
1. Introduction
Let E*r(X) denote the Erterm of the AdamsNovikov spectral sequence
converging to ss*(X). Then Miller, Ravenel and Wilson [1] introduced fi
elements fis=j;i+1in E22(S0) for (s; j; i + 1) 2 B+ , where p denotes a prime
number, S0 is the plocal sphere, and
B + = {(s; j; i + 1) 2 Z 3 s = mpn; n 0; p6 m 1; j 1; i 0; subject to
i) j pn if m = 1, ii) pij ani, and iii) ani1 < j if pi+1j}
for integers ak defined by a0 = 1 and ak = pk + pk1  1. Here we use the
abbreviation fis=j;1= fis=jand fis=1;1= fis.
Let V (1) denote the TodaSmith spectrum, which is a cofiber of the
Adams map ff : 2p2V (0) ! V (0), where V (0) is the mod p Moore spec
22
trum. Since there exists a map fi : 2p V (1) ! V (1) which induces
v2 on BP homology at a prime > 3 by [9], we have homotopy elements
fit 2 ss2t(p21)2p(S0) with t > 0. On the other hand, there is no such self
map at the prime 3. But there are homotopy elements fiifor i = 1; 2; 3; 5; 6; 10
______________________________
1991 Mathematics Subject Classification. 55Q45, 55Q52.
Key words and phrases. Homotopy groups of spheres, L2localization, fieleme*
*nts,
AdamsNovikov spectral sequence.
1
in this case due to Toda and Oka (cf. [2]). Besides, if we assume the existence
of the self map B : 144V (1) ! V (1) that induces v92on BP homology, then
we see that there exists a family {fi9t+i i = 0; 1; 2; 5; 6; t 0} in ss*(S0).
The existence of B is claimed by Pemmaraju in his thesis. Furthermore, the
existence of fi6=32 ss82(S0) is shown by Ravenel [4]. Here we obtain a relation
among fi9t+1, fi2 and fi6=3as follows:
Theorem A. Let t, i, j and k be nonnegative integers. Then in the
homotopy groups ss*(S0) of sphere spectrum localized away from 3,
fi9t+1fii16= 0 2 ss*(S0) if and only if i < 5,
fi9t+1fi2fij16= 0 2 ss*(S0) if and only if j < 2, and
fi9t+1fi6=3fik16= 0 2 ss*(S0) if and only if k < 4.
As is seen in [3, p.624], we have a relation
uvfisfit = stfiufiv for s + t = u + v
in the E2term E42(S0). This implies
Corollary B. In the homotopy groups ss*(S0) localized away from 3,
Q k
Q i=1fi9tk+16= 0 if and only if k < 6,and
( ki=1fi9tk+1)fi9t+26= 0 if and only if k < 3,
for integers t; tk 0. In particular, fik9t+16= 0 if and only if k < 6.
Remark. If the self map B does not exist, the above theorems are valid
only for the homotopy elements such as fi1 and fi10.
We prove this by determining subgroups of ss*(L2S0), where L2 : S(3)!
S(3)denotes the Bousfield localization functor on the category S(3)of spectra
localized away from 3 with respect to the JohnsonWilson spectrum E(2).
In ss*(L2S0), we have generalized fielements fis=j;i+12 E22(L2S0) for (s; j; i*
* +
1) 2B, where
B = {(s; j; i + 1) 2 Z 3 s = mpn; n 0; 36 m 2 Z ; j 1; i 0;
such that 3ij ani and either 3i+16 j or ani1 <}j:
2
Consider the sets
Gb = X (B5{fi9t+1}L B4{fi9t+1fi6=3}
t2Z L _______
L B3{fi9t+7ff1}
X B2{fi9t+1ff1; [fi9t+2fi01]; [fi9t+5fi01]});
Gb* = (B5{g16(9t+7)+15}L B4{g16(9t+3)+7}
t2Z L
B2{g144t; g16(9t+5)+2; g16(9t+8)+2}
L X
(B3{g16(3n+2t+9u+3) u 2 Z  I(n)}
n1L
B2{g16(3n+2t+9u+3) u 2 I(n)})):
Here Bk = Z =3[fi1]=(fik1),
I(n) = {x 2 Z  x = (3n1  1)=2 or x = 5 . 3n2 + (3n2  1)=2};
__xdenotes a homotopy element detected by x in the E
2term, [x] is an element
of ss*(L2S0)Ssuch that i*([x]) = x 2 ss*(L2V (0)) for the inclusion i : S0 !
V (0) = S0 3e1, and gi 2 ssi(L2S0) is the generator. Then the subgroups
bGLGb* are generated by
_______
S = {fi9t+1; fi9t+1fi6=3; fi9t+7ff1; fi9t+1ff1; [fi9t+2fi01]; [fi9t+5f*
*i01];
g16(9t+7)+15; g16(9t+3)+7; g144t; g16(9t+5)+2;
g16(9t+8)+2; g16(9t+3) t 2 Z }
as a Z =3[fi1]module. In this paper, our key lemma is the following:
L *
Theorem C. The homotopy groups ss*(L2S0) contain subgroups Gb bG.
Consider the localization map : S0 ! L2S0. Then we see immediately
the following:
Corollary D. For any element x 2 ss*(S0) such that *(x) 2 S, we have
xfi1 6= 0 2 ss*(S0).
In [7], we showed that the fielements fis=j;i+1for (s; j; i + 1) 2Bc do not
exist in ss*(L2S0), where
B c= {(9t + 4; 1; 1); (9t + 7; 1; 1); (9t + 8; 1; 1); (9t + 3; 3; 1); (9s; 3; *
*2); (3is; 3i; 1)
 t 2 Z ; s 2 Z  3Z ; i > 1}:
3
Moreover fielements fis=j;i+1for (s; j; i + 1) 2Bc do not exist in ss*(S0) if
t 1, 36 s 1 and i > 1. We further showed in [7] the existence of fi
elements fi9t, fi9t+1 and fi9t+5 in ss*(L2S0) for t 2 Z . Here we show more
fielements in ss*(L2S0):
Theorem E. fis=j;i+1for (s; j; i + 1) 2 B  B c survives to a homotopy
element of ss*(L2S0).
Note that fis=j;i+1are homotopy elements for all (s; j; i + 1) 2 B at a
prime > 5 ([8]).
The following sections 2 to 6 are devoted to show Theorem C and we
give subgroups of ss*(M2) for an L2local spectrum M2 with E(2)*(M2) =
E(2)*=(31 ; v11). Theorems A and E are actually corollaries of Theorem C,
and proved in x7.
2. Basic properties of H*M20
Let E(2) be the JohnsonWilson spectrum with coefficient ring E(2)* =
Z(3)[v1; v12]. Then E(2)*E(2) is a Hopf algebroid over E(2)* with E(2)*E(2) =
E(2)*[t1; t2; . .].=(jR (vi) : i > 2). For an E(2)*E(2)comodule M, Ext*E(2)*E(*
*2)
(E(2)*; M) is the cohomology of the cobar complex *M = *E(2)*E(2)M, and
we will denote it by H*M.
Recall the chromatic comodules Nijand Mijdefined inductively by N00=
E(2)*, N01= E(2)*=(3), N02= E(2)*=(3; v1), Mij= v1i+jNijand the short
exact sequence 0 ! Nij! Mij! Ni+1j! 0 for i + j + 1 2 [1]. Note
that Nij= Mijif i + j = 2. These have E(2)*E(2)comodule structure
induced from the right unit jR : E(2)* ! E(2)*E(2). Consider the long
exact sequences associated to these short ones 0 ! N00! M00! N10! 0
and 0 ! N10! M10!f M20! 0, and denote ffi0 : HsN10! Hs+1E(2)* and
ffi : HsM20! HsN10for the connecting homomorphisms. Then we see that
ffi0ffi : HsM20! Hs+2E(2)* is an epimorphism if s 1, and an isomorphism
if s > 1, since HsM00= 0 for s 1 and HsM10= 0 for s > 1 by [1]. In
particular,
Lemma 2.1. HsE(2)* for s 3 consists of torsion elements.
4
We have a short exact sequence 0 ! M11!i M20!3 M20! 0 (i(x) = x=3)
which induces a long exact sequence
. ..! Hs1M20ffi!HsM11i*!HsM203! HsM20ffi!. .:.
An easy diagram chasing shows the following:
Lemma 2.2. ([1, Remark 3.11]) Consider the following commutative dia
gram
ffi _________i* __________3 ________ffi
Bs1 ________HsM11w wBs wBs Hs+1M11w
  
    
    
s1 s fs fs s+1
f  g    g 
    
    
u u u u u
ffi i* 3 ffi
Ht1M20 ______HtM11w _______wHtM20 _______HtM20w ______wHt+1M11
of modules with horizontal exact sequences and a 3 torsion module Bs. If gs
and gs+1 are isomorphisms, then fs is an epimorphism. Moreover, if fs1 is
an epimorphism, then fs is an isomorphism.
Let b10 denote the element of H2E(2)* represented by the cocycle t1
t21 t21 t1. Then b10 acts on H*M for any comodule M. In [7], we show the
following:
Proposition 2.3. The multiplication by b10yields an isomorphism HsM11!
Hs+2M11if s > 3 and an epimorphism if s = 3.
This together with Lemma 2.2 implies
Corollary 2.4. The multiplication by b10yields an isomorphism HsM20!
Hs+2M20if s > 3 and an epimorphism if s = 3.
L 5 2 N
Corollary 2.5. H*M20~=(H4M20 H M0) Z =3[b10] for * > 3.
5
3. Some formulae in *E(2)*
The AdamsNovikov E2term for computing ss*(L2X) is the cohomology
H*E(2)*(X) of the cobar complex *E(2)*(X), and in particular, H*E(2)* =
H*N00is the E2term for ss*(L2S0).
For X = V (1), the TodaSmithNspectrum,N E(2)*(V (1)) = K(2)* =
Z=3[v12] and H*K(2)* = K(2)*[b10] F (i2) for F = Z =3{1; h10; h11; b11;
; 0; 1; b11} (cf. [6]). Among these generators, we have relations (cf. [6,
Prop. 5.9]):
v22h10b10 = h11b11; v2h11b10 = h10b11
(3:1) b11 = v2h10 1 = v2h11 0; b10 = h10 0 = v12h11 1;
v32b210= b211; b10 1 = v12b11 0; b10 0 = v22b11 1
For X = V (0), the mod 3 Moore spectrum, the E2term H*E(2)*=(3) is
computed in [7], in particular, we see the following:
Lemma 3.2.
H2;0E(2)*=(3) = 0
H3;0E(2)*=(3) = {v12 0; v12h10b10}
H4;0E(2)*=(3) = {v32b10b11; v12 0i2; v12h10b10i2}
H5;0E(2)*=(3) = {v32b10b11i2}:
Consider the long exact sequence
j* * ffi
. ..! H*1E(2)*=(3) ffi!H*E(2)* 3! H*E(2)* ! H E(2)*=(3) ! . . .
associated to the short exact sequence 0 ! E(2)* !3 E(2)* !j E(2)*=(3) !
0.
Lemma 3.3. The map d1 = j*ffi : H*E(2)*=(3) ! H*+1E(2)*=(3) sends
v12h10b10 (resp. v12h10b10i2) to v32b10b11 (resp. v32b10b11i2).
Proof. Note that v12h10b10 is represented by a cochain whose leading
i+1
term is v32t31 b11. Since d(t31 ) = 3b1iby definition, we compute
(3:4) d(v32t31 b11) = 3(v32b10 b11+ . .).;
6
which shows ffi(v12h10b10) = v32b10b11 + . ... For v12h10b10i2, it follows i*
*m
mediately from (3.4) and Proposition 4.2 in the next section. q.e.d.
Let x denote a cochain that represents . Then it is shown in [7, Lemma
4.4] that d(x) v21f0 mod (3) for f0 that represents v12 0 mod (3; v1) (In
[7], x is denoted by X(0)). So we have a cochain A such that
(3:5) d(x) v21f0 mod ( 3) and d(f0) = 3A
in the cobar complex *E(2)*. Then A is a cocycle of H4;0E(2)*. Further
more, we have
Lemma 3.6. d(f0) = 3f0 z mod (9) in the cobar complex 4E(2)*.
Here z denotes a cocycle that represents the generator i2.
Proof. The projection E(2)* ! E(2)*=(3) sends A in (3.5) to a cocycle
which is also denoted by A. By virtue of Lemmas 3.2 and 3.3, we put
[A] = k1v12h10b10i2 + k2v12 0i2;
where [A] denotes a cohomology class represented by A 2 *E(2)*=(3). In
fact, f0 may be replaced by f0 + kv32t31 b11 for some k 2 Z =3 if necessary.
Since d1([A]) = 0 and d1(v12 0) = 3[A] by definition,
0 = k1v32b10b11i2 + k2(k1v12h10b10i2 + k2v12 0i2)i2;
by Lemma 3.3. Noticing that i22= 0 and v32b10b11i2 6= 0, we see that k1 = 0.
On the other hand, if A represents 0, then we have an element v12 0 in
H3;0E(2)*. Since H2;0E(2)*=(3) = 0 by Lemma 3.2, v12 0 generates a Z (3)
free submodule in H3;0E(2)*, which contradicts to Lemma 2.1. Therefore A
represents a nonzero element. This means that k2 = 1. q.e.d.
Lemma 3.7. Put vg2t1= v2t1 + v1o and vg22t31= v22t31+ v1v2t61. Then
d(vg2t1) = v21b10 and d(vg22t31) = v21b11. Furthermore, we have a cochain
u 2 2E(2)* such that
d(u) gv2t1 b11+ gv22t31 b10 mod ( 3; v21):
7
Proof. The first statement is checked by a routine computation.
For the second, we have an element u0 2 2E(2)* such that d(u0)
v2t1 b11+ b10 v22t31mod (3; v1), since we have a relation v2h11b10 = h10b11
in H3E(2)*=(3; v1) by (3.1). Put d(u0) gv2t1 b11+ b10 gv22t31+ v1w mod
(3; v31) for some cochain w. Send this by d, and we have 0 v21b10b11b10
v21b11 + v1d(w) mod (3; v31). Then w 2 H3;52E(2)*=(3; v21), which is 0 since
H3;52E(2)*=(3; v1) = {v2b11i2} by [6, Th. 5.8] and d(v2b11i2) 6 0 mod (3; v21)
by [7, Lemma 3.3]. Therefore, we see that there is a cochain __wsuch that
d(__w) w, and we have d(u00) gv2t1 b11+ b10 gv22t31mod (3; v31) if we put
u00= u0v1__w. There is also a cochain a such that d(a) gv22t31b10b10vg22t31
mod (3; v21), and so we have the lemma by putting u = u00+ a. q.e.d.
Lemma 3.8. There exists a cochain w such that
d(w) gv2t1 f0  x0 b10 mod ( 3; v31):
Here x0 denotes a cocycle that represents mod (3; v1) and t1 x0 is homol
ogous to t1 x mod (3; v31).
Proof. This is shown in the same way as the above lemma. By the
equation h10 0 = b10 in (3.1), we have a cochain w0 such that d(w0)
v2t1 f0  b10 x mod (3; v1). Put d(w0) gv2t1 f0  b10 x + v1a for a
cochain a. Send this by d, and we see that a is a cocycle of 4;16E(2)*=(3; v21).
Since we see that H4;16E(2)*=(3; v21) = {h10b10i2} by [7], a = kt1 z b10
for some k 2 Z =3. Furthermore, b10 x is homologous to x b10, and so we
have cochain w such that d(w) gv2t1 f0  (x  kv1t1 z) b10. Now put
x0= x  kv1t1 z, and we have the lemma. q.e.d.
Lemma 3.9. In the cobar complex *E(2)*, there exists a cochain y such
that
d(y) t1 x  v1v12f1  v1z x  kv1v22t1 b11 mod ( 3; v21)
for some k 2 Z =3. Here f1 denotes a cocycle that represents 1.
Proof. It is shown in [7, Lemma 6.4] that there exists a cochain Y0 such
that d(Y0) t1 X + v1v32o3 X + v21v12t31 X mod (3; v31). It is also
8
shown that x X + v1v12Y1 + kv1v22b11 for some k 2 Z =3 in [7, Proof of
Lemma 4.4.]. Take now y to be Y0. Then mod (3; v21),
d(y) t1 (x  v1v12Y1  kv1v22b11)  v1z X + v1v12t2 X:
Since f1 = t1Y1t2X by the proof of [7, Lemma 4.4], we have the result.
q.e.d.
4. The E2terms HsM20for s > 3.
In [7], H*M11is given as the direct sum of three E(2; 1)* = Z =3[v1; v32]
modules Ai: L L
H*M11= A0 A1 A2:
In order to describe the modules Ai, we use notations:
k(1)* = Z =3[v1]
K(1)* = Z =3[v11]
N
P E = Z =3[b10] (i2)
n
E(2; n)* = Z =3[v1; v32 ]
F(h) = Z =3[v32]{v2=v1; v2h10=v1; v22h11=v1; v2b11=v1}
F(t) = Z =3[v32]{v12=v1; v2h10=v21; v22h11=v21; v12b11=v1}
F(*h)= Z =3[v32]{=v1; 0=v1; v2 1=v1; b11=v1}
F(*t)= Z =3[v32]{=v21; v22 0=v1; v12 1=v1; b11=v21}
n+1 4.3n1 3n+1 6.3n+1
Fn = E(2; n + 2)*{v32 =v1 ; v2 h10=v1 ;
n 10.3n+1 3n(53)+(3n1)=2 4.3n
v8.32h10=v1 ; v2 =v1 }:
Then the modules Ai are given as follows:
N
A0 = (K(1)*=k(1)*) (h10; i2)
X N
A1 = Fn (i2)
n0
L L *L * N
A2 = (F(h) F(t) F(h) F(t)) P E;
i 2;4.3i1 v3i1
Consider the exact sequence H1;0M11!ffiH2;0E(2)*=(3; v31) ! H M1 !
i 1=v3i11v3i11
associated to the short exact sequence 0 ! E(2)*=(3; v31) ! M1 ! M1 !
0. Then the structure of H*M11shows immediately the following:
9
i
Lemma 4.1. For each i > 0, each element of H2;0E(2)*=(3; v31) is divisible
i3i1
by v31 .
In the same way as [5, Lemma 2.6] with the above lemma, we obtain
Proposition 4.2. For each integer i > 0, there exists a cocycle zi of
i 1
1;0E(2)*=(3i+1; v31) such that zi = z 2 K(2)*.
By virtue of this proposition, we abuse the notation z for a cocycle that
represents i2 as we did in the previous papers for a prime > 3.
Consider the connecting homomorphism ffi : HsM20! Hs+1M11associated
to the short exact sequence 0 ! M11!i M20!3 M20! 0.
Lemma 4.3. ffi(v22=3v31) = v2 0=v1.
Proof. In [7, Lemma 4.4], it is also shown that there exists a cochain
X(2) such that d(X(2)) v41z3X3v41v32f31mod (3; v51). Since H3;20M02=
0, the congruence holds mod (3; v61) if we replace X(2) by a suitable cochain
x00. Put now d(x00) v41z3 X3  v41v32f31+ 3A mod (9; v61). Then 0
3v31t1 z3 X3  3v31t1 v32f31+ 3d(A) mod (9; v41), and we see that A 2
H3;40E(2)*=(3; v31), which is seen to be {v21v2 0} by [7]. Note that 3v31t1
z3X3 and v31t1v32f31are homologous to zero and v31b11X mod (9; v41) by
Lemma 3.9 and (3.1), respectively, and that d(v21v2(v2f0)) v31v2t31 f0 +
v31b11X mod (3; v41) by [7, Lemma 4.3] which is homologous to v21b11X by
(3.1). Thus we see that A is homologous to v21v22f0, and d(x00) 3v21v22f0
mod (9; v31). Since x00and f0 represent v22 and v12 0, respectively, we have
the lemma. q.e.d.
Put
L * N N s s 2
Gs = (i*(F(h) F(h)) E(2; 1)*[b10] (i2)) H M0
for i* : H*M11! H*M20given by i*(x) = x=3.
Lemma 4.4. For the connecting homomorphism ffi : HsM20! Hs+1M11,
ffi(x) for a generator x of Gs is obtained by the following equations:
ffi(v2=3v1)= v2h10=v21;
10
ffi(v2h10=3v1)= v12b11=v1 + v2h10i2=v1;
ffi(v22h11=3v1)= v22b10=v1 + v22h11i2=v1;
ffi(v2b11=3v1)= v22h11b10=v21;
ffi(v2(v12 0)=3v1)= b10=v21 v2(v12 0)i2=v1;
ffi(=3v1) = v12 1=v1 + (1 1)i2=v1 + kv12h11b10=v1
ffi(b11=3v1)= v22(v12 0)b10=v1 + (1 1)b11i2=v1 + kv2h10b210=v1;
ffi(v2 1=3v1)= b11=v21 v2 1i2=v1:
Proof. Note that v2h10 is represented by a cochain gv2t1= jR (v2)t1v1t2
of Lemma 3.7. In the cobar complex *E(2)*=(9; v31), we compute
d(v21v2)= 6v1t1jR (v2) + 3v21t2 = 6v1vg2t1;
d(v21v2t1)= 6v1t1 v2t1 + 3v21t2 t1
= 6v1v2t1__t1_1+ 6v21o__t1_2
d(3v1v2t21)= 3v21t31_t21_3 6v1v2t1__t1_1
d(3v21t1o)= 3v21(t41_t1_2+ t31_t21_3+ t1 o + o__t1_2)
d(3v21v22t3)= 3v21v22(t1 t32+ t2 t91+ v2b11);
(4:5) 2 2 3 2 3 2 _______23 2 2
d(v1v2t1) = 6v1t1 v2t1 + 6v1v2t2 t1 + 3v1v2b10
= 6v1v22t1__t31_4+ 12v21v2t41_t31_7
+6v21v2t2__t31_7+ 3v21v22b10_6
d(6v1v22t2)= 12v21v2t31_t2_5 6v1v22t1__t31_4 6v21v22b10_6
d(3v21v72t33)= 3v21v72(t31 t92+ t32 t271+ v32b311)
= 3v21v72(v82t31_t2_5+ v62t32_t31_7+ v92b10)_6:
Underlined terms with the same number sum up to zero except for the terms
numbered 6 and 7. The terms numbered 6 and 7 sum up to 3v21v22b10 and
6v21v22i2 t31, respectively. These imply the first three equations. In fact,
ffi([a=3v1]) = [(i*)1d(v21a)=9v31], where [a] denotes a homology class repre
sented by a. Since ffi(b11a) = b11ffi(a), the first equation gives ffi(v2b11=3v*
*1) =
v2h10b11=v21, which equals to v22h11b10=v21by Lemma 3.7. Thus we have the
fourth equation.
By Lemma 3.6 and the first equation in (4.5), we compute
(4:6) d(v21v2f0) 6v1vg2t1 f0 3v21v2f0 z mod (9; v31):
Now by Lemma 3.8, we have the fifth equation. Multiply h10 to the fifth
equation, and we have ffi(h10 0=3v1) = h10b10=v21+ h10 0i2=v1: Lemma 3.9
11
says that h10 = v1v12 1 + v1i2 + kv1v22h10b11 for some k 2 Z =3. Since
h10 0 = b10, we have the sixth equation. The seventh follows immediately
from a product of b11 and the sixth equation. A multiplication of b10 and
fifth equation gives the last one by using relations in (3.1). Here note that
b210= v32b211holds in H4E(2)*=(3; v21). q.e.d.
Proposition 4.7. HsM20= Gs for s = 4; 5. In other words, H4M20and
H5M20are Z =3[v32]modules generated by
v2b210=3v1; v2h10b10i2=3v1; v22h11b10i2=3v1; v2b11b10=3v1;
G4 :
0i2=3v1; b10=3v1; b11=3v1; v2 1i2=3v1;
and
v2b210i1=3v1; v2h10b210=3v1; v22h11b210=3v1; v2b11b10i2=3v1;
G5 :
0b10=3v1; b10i2=3v1; b11i2=3v1; v2 1b10=3v1;
respectively.
Proof. Put Bs = Gs. Then there is a canonical map fs : Bs ! HsM20
sitting in the commutative diagram
ffi i* 3 ffi
Hs1M20 ______wHsM11 _________Bsw ___________Bsw ________wHs+1M11
  
    *
* 
 fs fs  *
* 
    *
* 
   u u  
ffi i* 3 ffi
Hs1M20 ______wHsM11 _______wHsM20 _______wHsM20 ______Hs+1M11:w
Lemma 4.4 implies that the ffiimages of the generators of Bs are linearly
independent. Therefore we see that the above sequence is exact, and Lemma
2.2 shows that fs is an isomorphism. q.e.d.
5. On the E2terms HsM20for s 3
The submodule As2 HsM11is:
A02 = Z =3{v2=v1; v12=v1}
A12 = Z =3{v2h10=v21; v2h10=v1; v22h11=v21; v22h11=v1;
12
v2i2=v1; v12i2=v1}
A22 = Z =3{v2b11=v1; v12b11=v1; v2h10i2=v21; v2h10i2=v1;
v22h11i2=v21; v22h11i2=v1; v2b10=v1; v12b10=v1;
=v21; =v1}
A32 = Z =3{v2b11i2=v1; v12b11i2=v1; v2h10b10=v21; v2h10b10=v1;
v22h11b10=v21; v22h11b10=v1; v2b10i2=v1; v12b10i2=v1;
0=v1; v2 0=v1; v2 1=v1; v12 1=v1;
i2=v21; i2=v1}:
Now consider the map d1 = ffii* : HsM11! Hs+1M11. Then [1, Prop. 6.9]
shows
(5:1) d1(v32=v31) = v22h11=v21:
Here we compute:
Lemma 5.2. The Bockstein differential d1 = ffii* acts up to sign as follows:
d1(v2=v1) = v2h10=v21;
d1(v12=v1) = v12i2=v1;
d1(v2h10=v1) = v12b11=v1 + v2h10i2=v1;
d1(v22h11=v1) = v12b10=v1 + v12h11i2=v1;
d1(v2i2=v1) = v2h10i2=v21;
d1(v2b11=v1) = v22h11b10=v21;
d1(v2h10i2=v1) = v12b11i2=v21;
d1(v22h11i2=v1) = v22b10i2=v1;
d1(v2b10=v1) = v2h10b10=v21;
d1(=v21) = i2=v21;
d1(=v1) = v12 1=v1 + i2=v1;
d1(v2b11i2=v1) = v22h11b10i2=v21;
d1(v2h10b10=v1) = v12b11b10=v21+ v2h10b10i2=v1;
d1(v22h11b10=v1) = v12b210=v1 + v12h11b10i2=v1;
d1(v2b10i2=v1) = v2h10b10i2=v21;
d1(i2=v1) = v12 1i2=v1;
d1( 0=v1) = b10=v21+ 0i2=v1;
d1(v2 1=v1) = b11=v21:
13
The other elements of A2 that do not appear in the left hand side are in the
image of d1.
Proof. Lemma 4.3 and (5.1) show that v22 0=v1 and v22h11=v21are in
the image of d1. The other parts follow from Lemma 4.4, except for d1 on
v12=v1 and =v21.
For the exceptional cases, consider the diagram
ffi i* 3 ffi
Hs1M20 _______wHsM11 ________wHsM20 ________wHsM20 _______wHs+1M11
    
b10 b10 b10 b10 b10
    
u u u u u
ffi i* 3 ffi
Hs+1M20 ______wHs+2M11 ______wHs+2M20 ______wHs+2M20 ______wHs+3M11:
If we have a relation ffi(ff=3) = fib10+ ffi2 in Lemma 4.4, then we see that
fi=3 = ffi2=3 in H*M20, since i*(x) = x=3. Therefore, we compute
b10ffi(fi=3) = ffi(fib10=3) = ffi(ffi2=3) = ffi(ff=3)i2 = fib10i2;
and so we obtain
ffi(fi=3) = fii2
up to Ker b10. Note that b10 acts monomorphically on A2. Now take fi to be
the exceptional cases, and we have all d1. q.e.d.
Hence, we have
Proposition 5.3. HsM20contains E(2; 1)*=(3; v1)module as follows:
H0M20 E(2; 1)*{v12=3v1}
H1M20 E(2; 1)*{v2h10=3v1; v22h11=3v1; v2i2=3v1}
H2M20 E(2; 1)*{v2b11=3v1; v2h10i2=3v1; v22h11i2=3v1; v2b10=3v1
=3v21; =3v1}
H3M20 E(2; 1)*{v2b11i2=3v1; v2h10b10=3v1; v22h11b10=3v1; v2b10i2=3v1
i2=3v1; 0=3v1; v2 1=3v1}:
14
6. The AdamsNovikov differentials
Now consider spectra defined by cofiber sequences:
(6:1) S0 ! p1S0 ! N1; N1 ! L1N1 ! N2; V (0) ! v11V (0) ! W;
and M2 = L2N2. The AdamsNovikov differentials on ss*(L2W ) is deter
mined in [7]. Let : L2W ! M2 denote the canonical map that induces
i : M11! M20. Suppose that dr(x) = y in the Erterm for L2W . Then
dr(x=3) = dr(i*x) = i*y = y=3. In this way, we have the following except for
d9(v12=3v1) and d5(v3t2=3v21):
Lemma 6.2. The AdamsNovikov differential dr is given (up to sign) by:
dr(v2=3v1) = 0; d5(v42=3v1) = v22h11b210=3v1; d5(v72=3v1) = v52h11b210=*
*3v1;
dr(v22=3v1) = 0; dr(v52=3v1) = 0; d9(v12=3v1) = v52b11b310i2=3v1;
dr(v2h10=3v1) = 0; d9(v42h10=3v1) = v2b510=3v1; dr(v72h10=3v1) = 0;
dr(v22h11=3v1) = 0; dr(v52h11=3v1) = 0; d9(v82h11=3v1) = v42b11b410=3*
*v1;
d5(v2b11=3v1) = v2h10b310=3v1; dr(v42b11=3v1) = 0; d5(v72b11=3v1) = v72h10*
*b310=3v1;
d5(=3v21) = v32b11b10i2=3v1; dr(v32=3v21) = 0; d5(v62=3v21) = v32b11b10i2*
*=3v1;
dr(=3v1) = 0; dr(v32=3v1) = 0; d9(v62=3v1) = v32 0b410=3v1;
d5( 0=3v1) = v32b11b210=3v1; dr(v32 0) = 0; d5(v62 0=3v1) = v32b11b210=*
*3v1;
d5(v2 1=3v1) = b310=3v1; d5(v42 1=3v1) = v32b310=3v1; dr(v72 1=3v1) = 0:
d9(b11=3v1) = v22 1b510=3v1; dr(v32b11=3v1) = 0; dr(v62b11=3v1) = 0:
Proof. Here we show the exceptional cases. Lemma 4.4 shows
v12b10=3v1 = v12h11i2=3v1 and v3t2b10=3v21= v3t2 0i2=3v1:
Now we compute
b10d9(v12=3v1) = d9(v12b10=3v1) = d9(v12h11i2=3v1) = v52b11b410i2=3v1;
and we have d9(v12=3v1) = v52b11b310=3v1 as desired. In the same way, we
have the other case. q.e.d.
Now we display the chart of the AdamsNovikov spectral sequence:
15


 DO p
 DDO p
 DO DD ppE
 DDO D ppph0E
 DOc DD E av2ppDdb
 r 0DDO D ph0E Dsp 1
 r DOc O DD EphE av2DsDdb pp
 c 0DDO E avDD d0b DO1 pp
 r DO O DD Eph s2D D qhE pp
20p D0DDOE Dav2D d0b 1DDO bb1D c1E pp
p O D ph0E sD1 qh1E Dr 0DDO p
p av2DD db DDO bb1D Ec O pp
p Ds 1 DO qh1E ODr DDO0 D ppph0
p D bb1DDOcE Dav2 DppOdb
pDO qh1E OrD0D D ph0 s 1Dp
pD bb1DDOcE D Dav2 DOdb O Dpp
pD qh1E OrD0D ph0 s 1DDO pp
pbbDO1cE D av2DDOdb O DD E ppbb1D
pOrDD0 ph0 s 1DDO D qh1E p
p D av2DDOdb DD E bb1D c pp
p ph0 s 1DDO D Eqh1 rD0 DO pp
p Dav2 DOdb DD E Dbb1 c D pp
pp s 1DDO E bbDDqh1Ec rD0 DO avD ph0dppbEE
p DD qhE D1r D phE D2sDO p
pE bbDD c1 0 avD d0bE D 1pp
pqh1E r1D0 ph0 s2D1 D ppqh1
pc av2 bDcpp
p ph0 qh1 b1 pp
___________________________________________________________________pppppppppp*
*pppppppppppppppppppppppppppppppppppppppppav2bv2
12 15 28 39 46
0  33 36

 DO p
 DO D p
 D DO D pp
 DO D E D ppph0
 D DOcE D av2ppDdb
 DO D E rD0D ph0 sp 1
 D E Dr DOcE D ph av2DDOsdb Opp
 D cE 0D avD d0b D DO1E pp
 Dr DO D ph s2 DO O DD Eqh pp
20p D0 Dav2 d0b DDDO1E Dbb1D c1 pp
p D ph0 s 1DDO O D qh1E Dr 0 p
p av2D db DDOE bb1DD c DDO pp
p s DDODO1 O D qh1E Dr 0 DO ppph0E
DOp O D E bb1DDc D av2DDppOdbE
D pDO D Eqh1 rD0 DO ph0E OsD1Dp
DpDE Dbb1D c D av2DDOdbE Dpp
pDEqh1 rD0 DO D ph0E OsD1D pp
Dpbb1c D av2 DOdbE D ppbb1D
prD0DO D ph0E ODs 1D qh1 p
p D av2 DOdbE D bb1DDOc pp
p D ph0E OsD1D qh1 r 0DDO pp
p av2 DOdbE D Dbb1 DOc DD E pp
pp sD1D bbD qh1c r 0DDO E avDDph0Eppdb
p D qh 1rDO DD phE D2s p
p bbD c1 D 0 E avDD d0b i1 pp
pqh1 r1 DDO0 D Eph0 s2D1 2 ppqh1
pc E Dav2 bcDppD
p D ph0E qh1 b1 pp
___________________________________________________________________pppppppppp*
*pppppppppppppppppppppppppppppppppppppppppav2Dbv2
60 63 76 87 94
0  81 84
16


 p
 DO DO p
 D D pp
 DO D DO D ppph0E
 D c D av2ppEdb
 DO D rD0 DO D ph0E ODsp 1
 D Dr c D phE av2sDODdbE pp
 D c 0 DO avD d0bE O D 1 pp
 Dr D phE s2DDO D qh pp
20p 0DDO av2D d0bE O D1 Dbb1 c1 pp
p ph0E sD1DDO D qh1 r 0DDO p
p av2D Edb O bb1D c DDOE pp
p ODs DDO1 D qh1 r DDO0DO O D ppph0E
DOp bb1DDOc O D E Dav2D ppdb
D p D qh1 r 0DDO D Eph0 sD1 p
Dp Dbb1 DOc O DD E Dav2D db pp
p qh1 r 0DDO D ph0E sD1 DO pp
DpbbDO1c O DD E av2D db D ppbb1
pr D0DO D ph0E Ds 1 DO D qh1E pO
pO DD E av2D db D bb1 DOcE pp
p D ph0E sD1 DO D qh1E OrD0D pp
p Dav2 db D bb1 DOcE D pp
pp sD1 DO bbD qh1cEE rD0D avD ph0dppb
p D qhE D1rDO D ph 2sDODOp
p bbD c1E D 0 avD d0b DD ppE1
pqh1E r1DDDO0 D ph0 s2 1 DD :ppqh1Ei
pcE Dav2 d9 : i2 E bDcppD 2
p D ph0 E qh1E b1 pp
___________________________________________________________________ppppppp*
*ppppppppppppppppppppppppppppppppppppppppppppav2Dbv2E
108 111 124 135 142
0  129 132
Furthermore, since the elements of the target are shown not to be killed
by the differentials in Lemma 6.2, we have other differentials derived from
[7]:
(6:3)
d5(v9t+32=3v31)= v9t+12h10b210=3v1;
d5(v9t12h11=9v21)= v9t22h10b210i2=3v1;
n+2t+3n+1 2.3n+1+1 3n+2t+3(3n1)=22
d5(v32 h10=3v1 ) = v2 b10=3v1 (n 0);
n+2t+8.3n 10.3n+1 3n+2t+5.3n+3(3n11)=22
d5(v32 h10=3v1 ) = v2 b10=3v1 (n > 1):
This shows that,
L *L L *
Theorem 6.4. The E1 term of ss*(M2) contains the module eG eG GgZ gGZ .
17
Here E(2; 1)*modules are given as follows:
eG = B5(2; 2)*{v2=3v1}L B4(2; 2)*{v42b11=3v1}
L 7
L B3(2; 2)*{v2h10=3v1}
B2(2; 2)*{v2h10=3v1; v22h11=3v1; v52h11=3v1};
eG* = B5(2; 2)*{v72 1=3v1}L B4(2; 2)*{v32 0=3v1}
L 3 6
L B2(2;P2)*{=3v1; v2b11=3v1; v2b11=3v1}
9u+3
n1 (B3(2;Ln + 2)*{v2 =3v1  u 2 Z  I(n)}
B2(2; n + 2)*{v9u+32=3v1  u 2 I(n)});
gGZ = B5(2; 2)*{v2i2=3v1}
L 4
L B3(2; 2)*{v2b11i2=3v1}
B2(2; 2)*{v2h10i2=3v1; v22h11i2=3v1; v52h11i2=3v1; v72h10i2=3v1};
GgZ * = B5(2; 2)*{v72 1i2=3v1}L B4(2; 2)*{v32 0i2=3v1}
L
L B2(2; 2)*{i2=3v1}
L B1(2;P2)*{v32b11i2=3v1; v62b11i2=3v1}
9u+3
n1 (B3(2;Ln + 2)*{v2 i2=3v1  u 2 Z  I(n)}
B2(2; n + 2)*{v9u+32i2=3v1  u 2 I(n)});
n k
for Bk(2; n)* = (Z =3)[v32 ; b10]=(b10) and I(n) given in the introduction.
Proof. Suppose that dr(x) = y 6= 0 in the AdamsNovikov spectral
sequence for ss*(M2). Then y is in the image of i* : H*M11! H*M20, since
y has filtration 5. Lemma 5.2 shows that ffi(y) 6= 0 for the connecting
homomorphism ffi : H*M20! H*M11, and so ffi(x) 6= 0 and we have dr(ffi(x)) =
ffi(y) in the AdamsNovikov spectral sequence for ss*(L2W ). Observing the
differentials given in [7] with Lemma 4.4, we see that there is no more new
differentials, and obtain the theorem. q.e.d.
7. Application to fielements
In [1], H0M20is determined and we see that
vs2=3i+1vj12 H0M20 if and only if(s; j; i + 1) 2 B :
Consider the universal Greek letter map j = ffi0ffi : H0M20! H2E(2)*, where
ffi : H0M20! H1N10and ffi0: H1N10! H2E(2)* are the connecting homomor
phisms associated to the short exact sequences 0 ! N10! M10! M20! 0
18
and 0 ! E(2)* ! M00! N10! 0, respectively. Then the fielements are
defined by
fis=j;i+1= j(vs2=3i+1vj1):
We obtain the following immediately.
Lemma 7.1. Mod (3; v1), fi1 b10, fi2 v2h11i2 and fi6=3 v32b11 in the
E2term E2(S0).
Furthermore, note that fi01= h11 2 E2(V (0)) and ff1 = h10 2 E2(S0).
The generators of Gethen yields the following elements:
j(v9t+12=3v1) = fi9t+1; j(v9t+42b11=3v1) = fi9t+1fi6=3;
j(v9t+72h10=3v1) = fi9t+7ff1; j(v9t+12h10=3v1) = fi9t+1ff1;
j(v9t+22h11=3v1) = [fi9t+2fi01]; j(v9t+52h11=3v1) = [fi9t+5fi01]:
Now we prove the theorems in the introduction.
Proof of Theorem C. Consider the long exact sequences
. .!.ss*(L0S0) ! ss*(L2N1) ! ss*+1(L2S0) ! . . .
and
. .!.ss*(L1N1) ! ss*(M2) ! ss*+1(L2N1) ! . . .
associated to the cofiber sequences of (6.1). Note that ss*(L0S0) = Q and
N L
ss*(L1N1) = Q =Z (3) (y) A
i i+1
shown in [1], where A is the Z (3)module generated by vsp1=3 for i 0
L *L L *
and 36 s 2 Z . Therefore, the module Ge eG GgZ gGZ given in Theorem
6.4 is isomorphically sent to ss*(L2S0). Theorem C now follows. q.e.d.
Proof of Theorem A. Consider the localization map : S0 ! L2S0.
Since the induced map * : ss*(S0) ! ss*(L2S0) sends a fielement to the same
named one, the nontriviality of products of fielements in ss*(S0) is deduced
from the one in ss*(L2S0). `if' part now follows immediately from Theorem
C except for fi2. For fi2, note that fi9t+1fi2 = [fi9t+2fi01]i2 2 j(GgZ ) for *
*the
universal Greek letter map j. Thus `if' part for fi2 is shown.
19
In Lemma 6.2, we have d9(v9t+42h10=3v1) = v9t+12b510=3v1 and d9(v9t+82h11
=3v1) = v9t+42b11b410=3v1, which yield
(7:2) d9(fi9t+4ff1) = fi51fi9t+1 and d9(fi9t+8h11) = fi9t+1fi6=3fi41
in the E9term E*9(L2S0) by considering the image of the universal Greek let
ter map. In the same manner, the equation d5(v9t+42i2=3v1) = v9t+22h11i2b210=3v1
in Lemma 6.2 yields
(7:3) d5(fi9t+4i2) = fi9t+1fi2b210
in the E9term E*5(L2S0). If t 0, then the equations (7.2) and (7.3) also
hold in the AdamsNovikov spectral sequence for ss*(S0), since the elements
appeared in (7.2) and (7.3) are also defined in E2(S0). q.e.d.
Proof of Theorem E. In the proof of Theorem 6.4, we read off that
the elements on the 0th line hit nothing except for the fielements given by
Bc. Therefore, we obtain Theorem E. q.e.d.
References
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Novikov spectral sequence, Ann. of Math. 106 (1977), 469516.
[2]S. Oka, Note on the fifamily in stable homotopy of spheres at the prime 3,
Mem. Fac. Sci. Kyushu Univ. 35 (1981), 367373.
[3]S. Oka and K. Shimomura, On products of the fielements in the stable homot*
*opy
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[4]D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Aca*
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[5]K. Shimomura, Nontriviality of some products of fielements in the stable *
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[7]K. Shimomura, The homotopy groups ss*(L2V (0)) at the prime 3, to appear in*
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Department of Mathematics,
Faculty of Science,
Kochi University,
21