The homotopy groups ss*(L2S0) at the prime 3
Katsumi Shimomura and Xiangjun Wang
The homotopy groups ss*(L2S0) of the L2-localized sphere are determined by
studying the Bockstein spectral sequence. The results indicate also the homotopy
groups ss*(LK(2)S0) and we see that the fiber of the localization map L2S03!
LK(2)S0 is homotopic to -2L1S03.
1. Introduction and statement of results
For each prime number p, there is the Bousfield localization functor Ln :
S(p)! S(p)with respect to v-1nBP , where S(p)denotes the stable homotopy
category localized away from the prime p, BP the Brown-Peterson spectrum
at p, and vn the n-th generator of the coefficient algebra BP*. Consider
the Morava K-theories K(n) and the Johnson-Wilson spectra E(n), where
K(n)* = Z =p[v1n] and E(n)* = Z (p)[v1; v2; . .;.vn; v-1n]. Then Ln is also the
localization with respect to K(0) _ K(1) _ . ._.K(n) or E(n).
Hopkins and Ravenel show the homotopy equivalence S0(p)' holim-LnS0,
n
and so ss*(LnS0) is an approximation of the homotopy groupsLofLspheres. Ac-
tually, ss*(L0S0) = Q at any prime and ss*(L1S0) = Z (3) A Q =Z (3) at
a prime > 2(cf. [3], [4]), where A denotes the module generated by the gen-
eralized ff-elements (see below) and Q =Z (3) ss-2(L1S0) for the virtual
generator y. The homotopy groups ss*(L2S0) of E(2)*-localized spheres are
determined at a prime > 3 in [9], which satisfy Hopkins' chromatic splitting
conjecture [2]. In this paper, we determine ss*(L2S0) at the prime 3.
Theorem A. The homotopy groups ss*(L2S0) at the prime 3 are direct
sum of three modules Gi's, which are described as follows :
L L -1 L
G0 = Z (3) A+ (A- Q =Z (3))i2
L L L *L L
G1 = B C CI B (B1 C)i2
L * L L *
G2 = Gb Gb dGZ GdZ
1
Here the modules on the right hand sides are as follows:
P i+1
A = i0 Z =3
A+ = Z (3){ff3is=i+1| ff3is=i+12 A; s > 0}
A- = Z (3){ff3is=i+1| ff3is=i+12 A; s < 0}
for G0,
B = Z(3){fi3ns=3im;i+1| n 0; 36 |s 2 Z ; i 0; 1 m < 4 . 3n-2i-1
and 36 |m or 4 . 3n-2i-2 m}
B1 = Z(3){fi3ns=3im;i+1| fi3ns=j;i+12 B; 36 |(s + 1); 3|m;
or i 6= k + 1 if s = 3k+2t - 1 with k 0 and 36}|t
L
C = C1 C2
C1 = Z(3){fgf1fi3n(3t+1)=3im+1;i| 0 < i n; and 2 . 3n-i < 3im 2 . 3n-i+1}
C2 = Z(3){fgf1fi3n(9t-1)=3im+1;i| 0 < i n; and
2 . 3n-i-1 < 3im - 8 . 3n 2 . 3n-i}
L L L
CI = CI0 CI1 CI2 CI3
CI0 = Z(3){c3ns | n 0; s = 3t + 1 or s = 9t - 1 (t 2 Z})
CI1 = Z(3){fgf1fi3n(3t+1)=3im+1;i+1| 0 i n; 2 . 3n-i-1 < 3im 2 . 3n-i;
36 |m}
CI2 = Z(3){fgf1fi3n(9t-1)=3im+1;i+1| 36 |m 8 . 3n-i; 0 i < n;
k + 1 > i if 3k|t}
8
> n + 2 if m = 2; 8; ;
: n + 3 if m = 5
k + 1 > n if 3k|t}
B* = Z(3){fi(n)*3n+ls=3im;i+1| 36 |s 2 Z , i; n 0, 0 < 3im 4 . 3n,
l > i and l > i + 1 if 3|(s +}1)
2
for G1 and
bG = X (B5{fi9t+1}L B4{fi9t+1fi6=3}L B3{_______fi9t+7ff1}
t2ZL L
X B2{fi9t+1ff1; [fi9t+2fi01]; [fi9t+5fi01]}) B1{[fi9t-1=2fi01*
*]};
bG* = (B5{O19t+7}L B4{O09t+3}
t2Z L
B2{fi(0)*9t+1; fi6=3fi(0)*9t+1; fi6=3fi(0)*9t+4}
L X *
(B3{fi(0)3n+2t+9u+3| u 2 Z - I(n)}
n1 L
X B2{fi(0)*3n+2t+9u+3| u 2 I(n)}));
dGZ = (B5{ifi9t+1}L B3{ifi9t+1fi6=3}
t2Z L _________
B2{ifi9t+7ff1; ifi9t+1ff1; i[fi9t+2fi01]; i[fi9t+5fi01]});
dGZ* = X (B5{i2O19t+7}L B4{i2O09t+3}L B2{i2fi(0)*9t+1}
t2Z L
B1{i2fi6=3fi(0)*9t+1; i2fi6=3fi(0)*9t+4}
L X *
(B3{i2fi(0)3n+2t+9u+3| u 2 Z - I(n)}
n1 L
B2{i2fi(0)*3n+2t+9u+3| u 2 I(n)})):
for G2. Here Bk = Z =3[fi1]=(fik1),
I(n) = {x 2 Z | x = (3n-1 - 1)=2 or x = 5 . 3n-2 + (3n-2 - 1)=2};
__xdenotes a homotopy element detected by x in the E
2-term, [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. The generators are defined in section 4 and degrees of them
are:
|ffa=b| = 4a - 1; |fi01| = 11; |fia=b;c| = 16a - 4b - 2; |ca| = 16a - *
*7;
|fgf1fia=b;c| = 16a - 4b + 1; |fi(a)*b=c;d| = 16b - 8 . 3a - 4c - 4;
|O0a| = 16a + 7; |O1a| = 16a + 15; |i2| = -1;
and orders of them are:
o(ffa=b) = 3b; o(fia=b;c) = 3c; o(c3ns) = 3n+1 if 36 |s;
o(fgf1fia=b;c) = 3c; o(fi(a)*b=c;d) = 3d; o(O0a) = 3; o(O1a) = 3:
Furthermore, we abbreviate as follows:
ffa = ffa=1; fia=b= fia=b;1; fia = fia=1; gff1fia=b= gff1fia=b;1;
fgf1fia= gff1fia=1fi(a)* * * *
b=c= fi(a)b=c;1; and fi(a)b = fi(a)b=1:
3
Our computation of the differentials of the Bockstein spectral sequence
ss*(L2V (0)) ) ss*(L2S0) also works for the Bockstein spectral sequence as-
sociated to the cofiber sequence LK(2)S0 ! LK(2)S0 ! LK(2)V (0), and we
obtain the homotopy groups ss*(LK(2)S0) from the result on ss*(LK(2)V (0))
given in [6].
TheoremLB.NThe homotopy groups ss*(LK(2)S0) are the direct sum of
(Z 3 A+ ) (i2), G1 and G2. Therefore, the homotopy fiber F (L1S0; L2S03)
of the localization map L2S03! LK(2)S0 of the 3-adic completed sphere S03is
homotopic to -2 L1S03.
The second half of the theorem is seen in the same manner as the one
at a prime > 3 [2]. This shows that there is the only one summand in
F (L1S0; L2S03) while Hopkins' chromatic splitting conjecture says that it has
three summands.
Theorem A is proved by using the Adams-Novikov spectral sequence. Let
N2 denote the spectrum such that BP*(N2) = BP*=(31 ; v11). We denote the
Adams-Novikov E2-term for computing ss*(L2N2) by H*M20. In the next sec-
tion_we show that the E2-term H*M20is the direct_sum_of_the_three modules
Ai for i = 0; 1; 2, and give the structures_of A0 and A2 . Section 3 is devoted
to determine the structure of the module A1 using some results proven in the
last section. The differentials of the Adams-Novikov spectral sequence are
studied in [7], and we deduce the homotopy groups ss*(L2N2) from these re-
sults, and then the chromatic spectral sequence shows Theorem A in section
4.
2. Notations and the structure of H*M20
Consider the Hopf algebroid
N
(E(2)*; E(2)*(E(2))) = (Z (3)[v1; v12]; E(2)*[t1; t2; . .]. BP*E(2)*)
associated to the Johnson-Wilson spectrum E(2), where BP denotes the
Brown-Peterson spectrum with BP* = Z (3)[v1; v2; . .].and BP* acts on E(2)*
by sending vi to vi if i 2, and to zero if i > 2. In this paper, H*M for an
E(2)*(E(2))-comodule M denotes Ext*E(2)*(E(2))(E(2)*; M), which is given as
a cohomology of the cobar complex *M of E(2)*(E(2))-comodules (cf. x5).
4
Then we have the Adams-Novikov spectral sequence
E*2= H*E(2)* =) ss*(L2S0)
converging to the homotopy groups of E(2)*-localized sphere spectrum.
We now recall the definition of the chromatic comodules Nijand Mij.
They are defined inductively by setting N0j= E(2)*=Ij for I0 = 0; I1 = (3)
and I2 = (3; v1), Mij= v-1i+jNij, and Ni+1j= Mij=Nij. Note that Mij= Nijif
i + j = 2 and = 0 if i + j > 2. Then we see that the Adams-Novikov E2-
term H*E(2)* is obtained from H*Mi0for i 2 and the long exact sequences
HsNi0! HsMi0! HsNi+10! Hs+1Ni0. Since the modules H*Mi0for i < 2
are determined in [3], we here determine H*M20. For this sake, we consider
the comodule M11= Z =3{x=vj1| j > 0; x 2 K(2)*}, where K(2)* = Z =3[v12],
1=3 2 3 2
and the short exact sequence 0 -! M11- ! M0 - ! M0 - ! 0. Here note
that M20is described as M20= Z (3){x=3ivj1| i; j > 0; x 2 K(2)*}. Then we
will get H*M20from H*M11which is determined in [6], by using the lemma
given in [3, Remark 3.11].
In order to describe the module H*M11,Nwe set up someNnotations: H*M02=
H*K(2)* is determined (cf. [5]) to be F K(2)*[b10] (i2), where F is
the Z =3-vector space spanned by 1; h10; h11; b11; ; 0; 1; and b11. These
generators i2, h1i and b1i, are cohomology classes represented by cocycles
i 3i 2.3i 2.3i 3i
v-12(t2 - t41) + v-32t32, t31 and -t1 t1 - t1 t1 of the cobar complex
*K(2)*, respectively. Besides, and i are the generators of H2;8K(2)*
and H3;8(i+2)K(2)*. N
Put k(1)*n= Z =3[v1], K(1)* = v-11k(1)*, P E = Z =3[b10] (i2), E(2; n)* =
Z=3[v1; v32 ] and
F(h) = Z =3[v32]{v2=v1; v2h10=v1; v-12h11=v1; v2b11=v1};
F(t) = Z =3[v32]{v-12=v1; v2h10=v21; v-12h11=v21; v-12b11=v1};
F(*h)= Z =3[v32]{=v1; 0=v1; v2 1=v1; b11=v1};
F(*t)= Z =3[v32]{=v21; v2 0=v1; v-12 1=v1; b11=v21};
n+1 4.3n-1 3n+1 2.3n+1+1
Fn = E(2; n + 2)*{v32 =v1 ; v2 h10=v1 ;
n 8.3n+1 3n+1(21) 4.3n
v8.32h10=v1 ; v2 n=v1 };
F00 = E(2; 2)*{v32=v21; v32h10=v71; v82h10=v91; v3(21)20=v41};
n+(3n-1)=2 L 9 3
where n = v-32 . Note that F0 = F00 F000for F000= Z =3[v2 ]{v2 =v31}.
Then, in [7], H*M11is shown to be the direct sum of F000and the three modules
5
Ai, where
N
A0 = (K(1)*=k(1)*) (h10; i2)
L X N
A1 = (F00 Fn) (i2)
L n>0 L L N
A2 = (F(h) F(t) F(*h) F(*t)) P E:
Once we know the behavior of connecting homomorphism ffi : H*M11 !
H*+1M20, we will obtain H*M20by [3, Remark 3.11]. In [3]Tand [7], it is
shown that ffi(Ai) Ai. Let_Asidenote the submodule Ai HsM11. We
s
now define the submodule Ai to fit in the commutative diagram of exact
sequences
1=3 ___s 3 ___s ffi s+1
Asi___________Aiw __________wAi __________wAi
z| z y y
(2:1) || || || ||
|u 1=3 |u |u |u
3 ffi
HsM11 _______wHsM20 _______wHsM20 ______Hs+1M11:w
We deduce the following lemma from [3, Remark3.11]:
Lemma 2.2. If we have the commutative diagram of exact sequences
___s-1______ s ________1=s3_______3 s _______fs+1fi
Ai wAi| | wB wB Aiw
||||||||||||||| | ||||||||
||||||||||||||| | ||||||||||||| |
||||||||||||||| | ||||||||||||| |
|||||||||||||||f |f ||||||||||||| |
||||||||||||||| | ||||||||||||| |
||||||||| | |u |u ||||||||||||| |
1=3 ___s 3 ___s ffi s+1
Asi_______wAi _______Aiw ______wAi ;
then f is an isomorphism.
Note that [9, Prop. 7.2] is also valid at the prime 3, and the elements
y03t, V and G1 there are v3t+22h11=9v21, -v22h11 and v-12b10 in our notation,
respectively. Therefore, we have
Proposition 2.3. For each integer t,
ffi(v3t+22h11=9v21) = v3t+22(-b10+ h11i2)=v21+ . . .
6
Since ffi(v3s2=3v31) = sv3s-12h11=v21by [3, Prop. 6.9] and ffi(v9t-12h11=9v2*
*1) =
v9t-12(-b10+ h11i2)=v21+ . .b.y Proposition 2.3, we see the following:
P 2 ___s 1 2
P Proposition 2.4. HsM20is isomorphic to i=0Ai if s 6= 1, and H M0 ~=
2 ___sLP 9t-1 2
i=0Ai t2ZZ =9{v2 h11=9v1}.
In the same manner as [3, Theorem 4.2], we obtain
___
Proposition 2.5. The module A0 is given as follows :
___ ___L N
A0 = (A- Q =Z (3)) (i2);
where ___ i
A- = {1=3i+1v31m | i 0; m > 0}:
___
The module A2 is determined in [7] as follows:
___ 3 -1
Proposition 2.6. The module A2 is the direct sum of Z =3[v2 ]{v2 =3v1; =3v21}
and ____L ____ N
(F(h) F(*h)) P E:
____ ____ '
Here F(h)and F(*h)are the images of F(h)and F(*h)under the map H*M11!
H*M20given by '(x) = x=3.
___
3. Determination of A1
We devide A1 into 14 pieces:
L L L L * L *L * N
A1 = ((X1 X2) (H HI H ) (X1 X2)) (i2):
Here,
L
X1 = X1;1 X1;2
n(3t+1) j n-1
X1;1 = Z =3{v32 =v1 | n 0; t 2 Z ; 0 < j < 4 . 3 ;
such that j > 4 . 3n-i-1 - 1 if 3i|j}
7
n(3k+2u-1) 3k+1m k+1 n-1
X1;2 = Z =3{v32 =v1 | n 0; 36 |u 2 Z ; 0 < 3 m < 4 . 3 ;
such that j > 4 . 3n-i-1 - 1 if 3i-k-1|m}
L
X2 = X2;1 X2;2
n(3t-1) 3im n-2i-1
X2;1 = Z =3{v32 =v1 | n 0; 36 |m; 1 m 4 . 3 }
n(3k+2u-1) 3k+1m n-2k-1
X2;2 = Z =3{v32 =v1 | 2 2k n; 0 < m < 4 . 3 ; 36 |m};
L
H = H1 H2
n(3t+1) j+1 n
H1 = Z =3{v32 h10=v1 | 0 < i < n; j 2 . 3 ; 2 . 3n-i j if 3i|j}
n(9t-1) j n n
H2 = Z =3{v32 h10=v1 | n 0; 8 . 3 < j 10 . 3 + 1}
L
HI = HI1 HI2
n(3t+1) j+1
HI1 = Z =3{v32 h10=v1 | 0 i n; 4 . 3n-i-1 j 2 . 3n-i
if 3i|j and 3i+16}|j
n(9t-1) j+1 n
HI2 = Z =3{v32 h10=v1 | n 0; 0 j 8 . 3 ; 3k+16 |(j + 4 . 3n)
if t = 3ku with 36 |u}
L * *
H* = H*1 (H2;1+ H2;2)
n(3t+1) j+1
H*1 = Z =3{v32 h10=v1 | n 0; j > 0 and j < 4 . 3n-i-1 if 3i+16}|j
n(9t-1) j+1
H*2;1= Z =3{v32 h10=v1 | n 0; j > 0 and j 4 . 3n-i-1 if 3i+16 |j}
n(9t-1) j+1
H*2;2= Z =3{v32 h10=v1 | n 0; j 0 and 3k+1|(j + 4 . 3n) 4 . 3n+1
if t = 3ku with 36 |u}
L *
X*1 = X*1;1X1;2
n+i(3t+1) 3ik i n
X*1;1= Z =3{v32 n=v1 | i > 0; 0 < 3 k 4 . 3 }
n+i+1(3t-1) 3ik i n
X*1;2= Z =3{v32 n=v1 | i > 0; 0 < 3 k 4 . 3 }
n+ls 3ik i n
X*2 = Z =3{v32 n=v1 | 36 |s; i; n 0; 0 < 3 k 4 . 3 ;
l > i and l > i + 1 if 3|(s +}1)
We also consider the following submodules of H*M20:
Xf = Z (3){v3ns2=3i+1vj1| n 0; 36 |s 2 Z ; i 0; j > 0;
with 3i|j < 4 . 3n-i-1 and either 3i+16 |j or 4 . 3n-i-2}< j
L
= gX1 gX2
8
gX1 = Z (3){v3ns2=3i+1v3im1| v3ns2=3i+1v3im12 fX; 36 |(s + 1); 3|m;
or i 6= k + 1 if s = 3k+2t - 1 with k 0 and 36}|t
gX2 = Z (3){v3ns2=3i+1v3im1| v3ns2=3i+1v3im12 fX; 3|(s + 1); 36 |m
and i = k + 1 if s = 3k+2t - 1 with k 0 and 36 |t}
Hf = gH1L gH2
gH1 = Z (3){v3n(3t+1)2h10=3iv3im+11| 0 < i n; and
2 . 3n-i < 3im 2 . 3n-i+1}
gH2 = Z (3){v3n(9t-1)2h10=3iv3im+11| 0 < i n; and
2 . 3n-i < 3im - 8 . 3n 2 . 3n-i+1}
HgI = gHI0L gHI1L gHI2L gHI3
gHI0 = Z (3){v3ns2h10=3n+1v1 | n 0; s = 3t + 1 or s = 9t - 1 (t 2 Z})
gHI1 = Z (3){v3n(3t+1)2h10=3i+1v3im+11| 0 i n;
2 . 3n-i-1 < 3im 2 . 3n-i; 36 |m}
gHI2 = Z (3){v3n(9t-1)2h10=3i+1v3im+11| 36 |m < 8 . 3n-i;
0 i < n; k + 1 > i if 3k|t}
8
>:
n + 3 if m = 5;
k + 1 > n if 3k|t}
gX2* = Z (3){v3n+ls2n=3i+1v3im1| 36 |s 2 Z , i; n 0, 0 < 3im 4 . 3n,
l > i and l > i + 1 if 3|(s +}1)
The propositions_of_section 5 below show the behavior of the connecting
s s+1
homomorphism ffi : A1 ! A1 as follows:
___s s+1
Proposition 3.1. The connecting homomorphism ffi : A1 ! A1 sends
gX1, gX2, fH, gHI, gX1i2 gX2*and gHIi2 to H*, X2i2, X*1, HIi2, X*2i2 and X*1i2,
respectively. Furthermore, the images of generaotrs under ffi are linearly in-
dependent.
We now use Lemma 2.2 to obtain our main theorem:
9
___s L L L
Theorem*3.2.LA1 is isomorphic to gX1 gX2if s = 0, fH gHI gX1i2 if
s = 1, gX2 fHi2 if s = 2, and 0 otherwise.
4. The homotopy groups ss*(L2S0)
Let Er(X) denote the Er-term of the Adams-Novikov spectral sequence
converging to the homotopy groups ss*(X). We start with a general result on
the spectral sequence.
Lemma 4.1. Let X f! Y !g Z h! X be the cofiber sequence with
BP*(h) = 0. Then we have the induced maps Esr(X) f*!Esr(Y ) g*!Esr(Z) !ffi
Es+1r(X). Suppose that g*(__y) = __zfor non-zero elements y 2 Esr(Y ) and
z 2 Es+rr(Z). Here __adenotes a homotopy element that is detected by an
element a of the Er-term. Then if y = f*(x) for some x 2 Esr(X), then
dr(x) = ffi(z).
Let N1 and N2 denote the cofibers of the localization maps S0 ! L0S0
and N1 ! L1N1, respectively. Then we have the Adams-Novikov spectral
sequence E2(L2N2) = H*M20) ss*(L2N2). The differentials of the spectral
sequence are determined in [7], and we have the following
Proposition 4.2. The E1 -term of the Adams-Novikov_spectral_sequence_ ___
for ss*(L2N2)_is the derect sum of the three modules A0, A1 and fA2. Here A0
and A1 are determined in the previous sections, and
fA2= eGL eG*LGgZ L gGZ*;
10
where these four modules are determined in [7]:
Ge = B5(2; 2)*{v2=3v1}L B4(2; 2)*{v42b11=3v1}L B3(2; 2)*{v72h10=3v1}
L 2 5 L -1 2
B2(2; 2)*{v2h10=3v1; v2h11=3v1; v2h11=3v1} B1(2; 2){v2 h11=3v1};
Ge* = 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.
Lemma 4.3. There is no extension problem in the spectral sequence for
ss*(L2N2).
Proof. Let M(i; 1) be a cofiber of the localization map M(i) ! L1M(i)
of the mod 3i Moore spectrum M(i). Then we have the cofiber sequence
i
M(i; 1) !' N2 !3 N2. In [7], we show that if dr(x) = y in E*r(L2N2) for a
non zero element y, then the order of x is three except for d5(v9t-12h11=9v21) =
v9t-22h10b210i2=3v21. This exceptional differential does not induce the extensi*
*on.
Thus we take i = 1, and M(1; 1) is W of [6]. Then we see by [7] that
the differentials on E*r(L2N2) are obtained by sending ones on E*r(L2W ) by
the map '* : E*r(L2W ) ! E*r(L2N2), and so y cannot be a image of the
connecting homomorphism. Now we apply Lemma 4.1 to show the lemma.
q.e.d.
Corollary 4.4. _The_homotopy_groups ss*(L2N2) are the direct sum of
the three modules A0 , A1 and fA2.
Proof of Theorem A. Consider the exact sequences . .!.ss*(L2S0) !
11
v1 2
ss*(L0S0) ! ss*(L2N1) ! . .a.nd . .!.ss*(L2N1) ! ss*(L1N1) ! ss*(L2N ) !
. .a.ssociated to the cofiber sequence S0 ! L0S0 ! N1 and N1 ! L1N1 !
N2. These also induce the connecting homomorphisms ffi : Es2(L2N1) !
Es+12(L2S0) and ffi0 : Es2(L2N2) ! Es+12(L2N1) of E2-terms. Now define
elements of the E2-term E*2(L2S0) by
ffa=b= ffi(va1=3b); fi01= h11- v21h10;
fia=b;c= ffiffi0(va2=3cvb1);
ns n+1
c3ns = ffiffi0(v32 h10=3 v1) for 36 |s,
fgf1fia=b;c= ffiffi0(va c b
2h10=3 v1);
fi(a)*b=c;d= ffiffi0(vb2a=3dvc1);
O0a = ffiffi0(va2 0=3v1); and
O1a = ffiffi0(va2 1=3v1):
___
Then A1 and fA2are isomorphicNtoLG1 and G2, respectively. Since ss*(L0S0) =
Q and ss*(L1N1) = Q =Z (3) (y) Al, an_easy_diagram chasing with Collo-
rary 4.4 enables us to obtain G0 from A0, and shows Theorem A. Here Al is
is i+1
the Z (3)-module generated by v31 =3 for i 0 and 36 |s 2 Z . q.e.d.
5. Computations in the cobar complex
In this section we work on the cobar complex (cf. [3]) based on the Hopf
algebroid (E(2)*;_E(2)*(E(2))) in order to study the connecting homomor-
s s+1
phism ffi : A1 ! A1 . The structure maps jR : E(2)* ! E(2)*(E(2)) and
: E(2)*(E(2)) ! E(2)*(E(2)) behave as follows:
jR (v1) = v1 + 3t1
jR (v2) = v2 + v1t31- t1jR (v1)3 - 3v1t1(v21+ 3v1t1 + 3t21)
(t1) = t1 1 + 1 t1
(t2) = t2 1 + t1 t21+ v1b0
(t3) t3 1 + t2 t91+ t1 t32+ 1 t3 + v2b1 - v1b20 mod ( 9; v21);
i+1 3i+1 3i+1 3 3
where 3bi = t31 1 + 1 t1 - (t1 1 + 1 t1) and 3b20 = (t2 1 + t1
t91+ 1 t32) - (t2 1 + t1 t31+ 1 t2)3. Furthermore, we have the relations
in E(2)*(E(2)) by setting jR (vi) = 0 in BP*(BP ) for i > 2 such as v2t9i-2
12
i 9 3 3 2
v32ti-2 mod (3; v1) and v2t1 v2t1 - v1t2 mod (3; v1). For an E(2)*(E(2))-
comodule M with structure map inducedNfrom jR , theNcobar complexNis a
family of E(2)*-modules sM = M E(2)*E(2)*(E(2)) E(2)*. . .E(2)*E(2)*(E(2))
(s factors) withPdifferential d : sM ! s+1M defined by d(m x) =
jR (m) x + si=1(-1)im i(x) - (-1)sm x 1 for m 2 M and x 2 sM,
where i(x1 . . .xn) = x1 . . .(xi) . . .xn.
Lemma 5.1. In the cobar complex *E(2)*=(3; v31), put t31 = v-62t33- t31t32,
and we have
d(t31) = t61 t91- t31 t32- v32t31 z3 - v-32b311:
Here z = v-12(t2 - t41) + v-32t32.
Proof. This follows immediately from the computation:
d(v-62t33)= -v-62t31 t92- v-62t32 t271- v-32b311;
d(-t31t32)= t61 t91+ t31 t121+ t31 t32+ t32 t31
= t61 t91- v32t31 z3 + v-62t31 t92- t31 t32+ t32 t31:
q.e.d.
Lemma 5.2. There exists an element Yn such that
n-1-1 3n 7.3n-12.3n3n 8
d(Yn) = -v4.31 oen+1 V - v1 v2 x mod ( 3; v1);
n 3 3 9 9 3
for each n > 0. Here oen = t1+v1z3 , V is defined by 3v1V v2+v1t1-v1t1-
jR (v32) mod (9) and x is a cocycle whose leading term is -v102t33t31-v-32t1t3.
Proof. First note that V -v22t31- v1v2t61mod (3; v21). Recall the
element Y of 1E(2)* ([1, Th. 4.8]) such that Y oe2jR (v32)-v21V +v31v-22t181+
v41v-272(t91t272- t93) mod (3; v51) and
d(Y ) v71x3 mod ( 3; v81):
Here x3 denotes a cocycle whose leading term is -v302t93t91-v-92t31t33, which
represents v2 2 H2;8M02. By use of Y , we define elements Yi inductively by
Y1 = jR (v62)Y + v51v52t32- v61v42t31+ v51v82z3
n-1-4 3n
Yn = Yn3-1+ v4.31 v2V
13
Since d(jR (v)t) = d(t)jR (v) - t d(v) for v 2 E(2)* and t 2 E(2)*E(2),
we compute mod (3; v81),
d(jR (v62)Y )= v71v62x3 - Y (-v31v32t91+ v61t181)
= v71v62x3 - v31Y v-32V 3+ v61Y t181
= v71v62x3 + v61oe2jR (v32) t181
-v31(oe2jR (v32) - v21V + v31v-22t181+ v41v-272(t91t272- t93)) *
* v-32V 3
= v71v62x3 + v61(v2t91_4+ v1t32+ v1z9jR (v32)) t181
-v31oe2 V 3- v51(-v22t31_1- v1v2t61_2+ v21v22t1) v32t91
+v61v-22t181_v32t91_4+ v71v-272(t91t272- t93)) v32t91
d(v51v52t32)= -v61v42t31_t32_3- v51v52t31_t91_1
d(-v61v42t31)= -v71v32t31 t31- v61v42(t61_t91_2- t31_t32_3- v32t31_z3_5- v-32b*
*311_4)
d(v51v82z3)= -v61v72t31_z3_5+ v71v62t61 z3;
in which each sum of the underlined terms with the same number amount to
0. So we redefine the cocycle -x3 by the cocycle that appears in the sum of
above congruences to satisfy
d(Y1) -v31oe2 V 3- v71v62x3:
Here x3 has the same leading term -v302t93t91-v-92t31t33as the above cocyle
x3.
Turn now to the case n. We assume the case for n - 1. Then
n-1-3 3 3n 7.3n-12.3n3n
d(Yn3-1) -v4.31 oen V - v1 v2 x
n-1-4 3n 4.3n-1-3 3 2 3n
d(v4.31 v2V ) = v1 (t1 - v1t1) V
Note that oe3n- (t31- v21t1) = v21oen+1, and we have the case for n. q.e.d.
The following is also shown in [8, Prop. 4.4] which also holds at the prime
ns -2
3. Here the elements y3ns, t1 i, and g0 are our v32 h10, h10i2 and v2 b11,
respectively.
Proposition 5.3. For n 0 and s 2 I,
ns n+1 3ns 3ns-2
ffi(v32 h10=3 v1) = v2 h10i2=v1 + v2 b11=v1:
14
We have the similar results to [9, Prop.s 7.5, 7.6, 7.8]:
Proposition 5.4. Let s; n; i; k be integers with 36 |s, k > 0 and 0 i
ns 3ik+1
n. Then the Bockstein differential on v32 h10=v1 is given as follows :
1. If 3ik 2 . 3n-i, then
ns i+1 3ik+1
ffi(v32 h10=3 v1 )
ns 3ik+1 n-i 3ns 3ik-2.3n-i-1
= -kv32 h10i2=v1 - (-1) sv2 n-i-1=v1 + . .:.
2. If s = 9t - 1 and 3ik 8 . 3n + 2 . 3n-i+1, then
ns i 3ik+1 n-i 3n+1(3t-1) 3ik-8.3n-2.3n-i
ffi(v32 h10=3 v1 ) = (-1) v2 n-i=v1 + . . .
3. If s = 9t - 1, 3i+16 |3ik 8 . 3n and i < n, then
ns i+1 3ik+1 3ns 3ik+1
ffi(v32 h10=3 v1 )= -kv2 h10i2=v1 + . . .
4. If s = 9t - 1, 3nk 8 . 3n and 36 |(k + 1), then
ns n+1 3nk+1 3ns 3nk+1
ffi(v32 h10=3 v1 )= -(k + 1)v2 h10i2=v1 + . . .
5. If s = 9t - 1, 3nk 8 . 3n and 3 | (k + 1) (i.e. k = 2; 5; 8), then
ns n+2 2.3n+1 3ns 2.3n+1
ffi(v32 h10=3 v1 )= -v2 h10i2=v1 + . . .
ns n+2 8.3n+1 3ns 8.3n+1
ffi(v32 h10=3 v1 )= v2 h10i2=v1 + . . .
ns n+3 5.3n+1 3ns 5.3n+1
ffi(v32 h10=3 v1 )= -v2 h10i2=v1 + . . .
Here . .d.enotes an element killed by a lower power of v1 than it shown.
Proof. Let ezdenote the element given in [7] such that ez v-12(t2-t41)+
i-1k
v-32t32mod (3; v1) and d(ez) 0 mod (3i; v31 ) for any i; k > 0, and denote
oe = t1 + v1ez. We also consider a cocycle
!
X k + j - 2 -(-t1)k
yj;l= ______________j+k-1
k>0 k - 1 3l-k+1kv1
15
of 1M20([3]). Put oea;b= ya;b+ ez=3bva-11, and we see that 3b-1oea;b= oe=3va1
and i
d(oe3ik+1;i+2) = kt1 ez=3v31k+1:
ns 3ik+1 n i 3ns *
* 3ik+1
Note that v32 h10=v1 is represented by a cocycle c(3 s=3 k+1) = jR (v2 )oe=*
*v1 +
ik-3n
w=v31 for some w 2 E(2)*(E(2)).
ns
For i = 0 and k < 3n - 1, we define c(3ns=k + 1; l) = jR (v32 )oek+1;lfor an
ns k+1
integer l > 0, and replace the generator v32 h10=v1 by the element repre-
sented by the cocycle c(3ns=k + 1). Since d(v32) 3v1V mod (9; v31) by defini-
ns i+1 3n-i-1 3n-i(3is-1)3n-i-1 i+2 3n-i
tion, we see that d(v32 ) -3 sv1 v2 V mod (3 ; v1 ).
We compute in 2M20:
ns
d(c(3ns=k + 1; 2)) = d(jR (v32 )oek+1;2)
ns k+1 3ns
= kt1 jR (v32 )ez=3v1 - oek+1;2 d(v2 )
ns k+1 3n(s-1) 3n-1 k+1-3n-1
= kv32 t1 ez=3v1 + sv2 oe V =3v1
ns-3n-1 3n-1 k-2.3n-1
whose second term is homologous to -v32 x =3v1 by Lemma
n 3n n 3ns k-2*
*.3n-1
5.2. Since n is represented by (-1)nv-32x , this represents (-1) v2 n-1=3v1
*
*ns 3ik+1
as desired. If k 3n, then the case i = 0 follows from the formula v3n+31ffi(v3*
*2 h10=3v1 ) =
ns 3ik-3n-2
ffi(v32 h10=3v1 ).
ns i+1 3i+1k+1 i+1
Suppose the case for i. Then ffi(v32 h10=3 v1 ) = 0 if 3 k
2 . 3n-i-1. Since we compute
ns 3ns 3i+1k+1
d(jR (v32 )oe3i+1k+1;i+3) = kv2 t1 z=3v1
n-i-1(3i+1s-1) n-i-2 3i+1k+1-3n-i-2
-v32 oe V 3 =3v1
in 2M20, which shows the case for i + 1, and we obtain inductively the first
part by Lemma 5.2. Thus if we denote a cocycle that represents va2h10=3cvb1
by c(a=b; c), then
ns 3ik+1
d(c(3ns=3ik + 1; i + 2)) = kv32 t1 z=3v1
(5:5) n 3ik-2.3n-i-1
-(-1)n-iv32sx(n - i - 1)=3v1 ;
n 3n 3n n+1 3n+1 3n+1
where x(n) = (-1)nv-32x and so oe V = (-1) v1 v2 x(n) up to
homology.
Consider the case s = 9t - 1. The proof of [9, Lemma 7.7] works also at
the prime 3 and we obtain:
16
n(9t-1) 3ik+1
(5.6) The element v32 h10=3v1 of H1M20is represented by a cochain
n+3ik n+1 i n
c(3ns=3ik + 1; 1) = d(xtn+2)=9tv4.31 - c(3 (3t - 1)=3 k - 8 . 3 + 1; 2).
i
In [3], they introduce the elements xi 2 E(2)* such that xi v32 mod
(3; v1) and give the formulas on d(xi). With a detail computaiton, we see
i-1 2.3n-1
that these elements satisfy d(xi) vai1v2.32 oen-1 mod (3; v1 ) for i 2.
We then compute with (5.5)
n+3ik t 4.3n+3ik+1
d(d(xtn+2)=3i+2tv4.31 ) = -kt1 d(xn+2)=3tv1
n+1(3t-1) 3ik+1-8.3n
= -kt1 v32 oe=3v1 + . .;.
n+1(3t-1)2 3ik+1-8.3n 3n+1(3t-1) 3ik+1-8.3n
d(-kv32 t1=3v1 ) = -kv2 t1 t1=3v1 ;
d(c(3n+1(3t - 1)=3ik + 1 - 8 . 3n; i + 2))
n+1(3t-1) 3ik+1-8.3n n-i 3n+1(3t-1) 3ik-8.3n-2.3n-i
= kv32 t1 z=3v1 + (-1) v2 x(n - i)=3v1 :
These amount to
n+1(3t-1) 3ik-8.3n-2.3n-i
d(c(3n(9t - 1)=3ik + 1; i + 1)) = (-1)n-iv32 x(n - i)=3v1 :
We also see the case n = i = 1 in the same manner, and we have the part 2.
The parts 3 and 4 follow immediately from (5.6) and computation
n t (4+k)3n+1
d(d(xtn+2)=3n+3tv(4+k)31) = (4 + k)t1 d(xn+2)=9tv1 :
n
In the same way, we obtain the part 5 by computing d(d(xtn+2)=3n+4tv(4+k)31)
n+2
for k = 2; 8 and d(d(xtn+2)=3n+5tv31 ) for k = 5. q.e.d.
___1 L L
These imply that A1 = Hf HgI_ gX1i2, and Propositions 5.3 and 5.4
1 2 *
show that the cokernel of ffi : A1 ! A1 is isomorphic to X2.
n+ls 3ik *
Proposition 5.7.__For_an element v32 n=v1 of X2, the connecting
2 3
homomorphism ffi : A1 ! A1 acts as follows:
n+ls i+1 3ik 3n+ls 3ik 3n+ls-1 3ik
ffi(v32 n=3 v1 ) = k(v2 ni2=v1 + v2 1=v1 + . .).:
n+ls 3jm
Proof. Let c 2 2M11denote a_cocycle_that represents v32 n=v1
1 2 i j n
which is in the image of ffi : A 1! A1 with 3 k 3 m 4 . 3 and j > i.
17
Since the cocycle c=3 2 2M20is bounded, we have a cochain u 2 1M20such
n+ls 3ik 0 3jm-3ik
that d(u) = c=3. Then v32 n=v1 is represented by c = v1 c and so
im-3ik i+1
c0=3i+2 = (v31 =3 )d(u). Therefore, we compute in the cobar complex
3M20,
ik-3jm
d(c0=3i+2) = d(1=3i+1v31 )d(u)
ik-3jm+1
= -kt1 c=3v31 ;
n+ls 3ik 3n+ls-1 3ik
which represents k(v32 ni2=3v1 + v2 1=3v1 + . .).by [7, Lemma
3.9] as desired. q.e.d.
References
[1]Y. Arita and K. Shimomura, The chromatic E1-term H1M11at the prime 3, Hi-
roshima Math. J. 26 (1996), 415-431.
[2]M. Hovey, Bousfield localization functors and Hopkins' chromatic splitting *
*conjec-
ture, Contemporary Math. 181, 225-250.
[3]H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in Adams-
Novikov spectral sequence, Ann. of Math. 106 (1977), 469-516.
[4]D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Aca*
*demic
Press, 1986.
[5]K. Shimomura, The homotopy groups of the L2-localized Toda-Smith spectrum at
the prime 3, Trans. A. M. S., 349 (1997), 1821-1850.
[6]K. Shimomura, The homotopy groups of the L2-localized mod 3 Moore spectrum,*
* to
appear in J. Math. Soc. Japan.
[7]K. Shimomura, On the action of fi1 in the stable homotopy of spheres at the*
* prime
3, preprint.
[8]K. Shimomura and A. Yabe, On the chromatic E1-term H*M20, Contemporary Math.
158 (1994), 217-228.
[9]K. Shimomura and A. Yabe, The homotopy groups ss*(L2S0), Topology, 34 (1995*
*),
261-289.
18
Department of Mathematics, Department of Mathematics,
Faculty of Science, Faculty of Science,
Kochi University, Kochi University,
Kochi, 780-8520 Kochi, 780-8520
Japan Japan
# and
# Department of Mathematics,
# Nankai University,
# Tianjin
# P. R. China
19