Cohomology mod 3 of the classifying space of the exceptional Lie group $E_6$, I : structure of Cotor

0.2 MB·English

by  Mamoru Mimura

COHOMOLOGY MOD 3 OF THE CLASSIFYING SPACE OF THE
EXCEPTIONAL LIE GROUP E , I : STRUCTURE OF COTOR
6
MAMORU MIMURA, YURIKO SAMBE, AND MICHISHIGE TEZUKA
Abstract. We study the structure of the E2-term of the Rothenberg-Steenrod spectral
sequenceconvergingtotheclassifyingspaceofthecompact,connected,simplyconnected,
exceptional Lie group of rank 6.
2
2010 Mathematics Subject Classification. 55R35,55R40, 55T20
1
0
Key words and phrases, exceptional Lie group, spectral sequence, Cotor
2

n
a
J
1. Introduction
6
2
Let E6 be the compact, connected, simply connected, exceptional Lie group of rank 6
]
T and BE its classifying space. Then we have the Rothenberg-Steenrod spectral sequence
6
A {E ,d } ([RS]) such that
r r
.
h
at E2 = CotorH∗(E6;Z3)(Z3,Z3) and E∞ = grH∗(BE6;Z3).
m
[
The E -term was calculated by Mimura and Sambe [MS], where some details of the
2

2
calculations are somewhat omitted. This is one of the reasons why we give here a detailed
v
1 account of them. After the calculations of the E -term, the collapsing of the spectral
2
1
sequence was proved by Kono and Mimura [KM]; thus one can obtain the module struc-
8
5 ture of H∗(BE ;Z ). (See also Toda [T], in which the collapsing is shown by a different
. 6 3
2 method.) Further in order to determine the ring structure of H∗(BE ;Z ), we will need
1 6 3
1 more information (see [MST1]), that is, we need to add supplement to the results of [MS].
1
: This is the main reason for the present work.
v
i The paper is organized as follows. In Section 2 we state our results. In Section 3 we
X
prove Theorem 2.1 and in Section 4 we prove Theorems 2.3.A and 2.3.B. In Section 5, we
r
a
compare our calculation to the May spectral sequence.
Thus the present work is essentially a supplement, and hence a more detailed version, of
[KM] and [MS]. The results of the present work are announced in [MST2].
In the series of our work, we will determine H∗(BE ;Z ). Continuation follows.
6 3
Throughout the paper, the coefficient field is always Z , which is hereafter omitted.
3
Partially supported by the Grant-in-Aid for Scientific Research (C) 21540104, Japan Society for the
Promotion of Science.
1
2 M.MIMURA,Y.SAMBE, ANDM.TEZUKA
2. Results
Let us start with
Notation For a commutative ring R, we denote by R{x ,...,x } a free R-module
1 n
generated by elements x , ..., x . For a graded algebra S, we denote by S+ the ideal
1 n
generated by the elements with positive degrees. We also denote by T(x ,...,x ) a tensor
1 n
algebra generated by elements x , ..., x , and by ∆(x) an algebra additively isomorphic
1 n
to Z ⊕Z {x}.
3 3
The following is essentially the restatement of Theorem 5.20 of [MS].
Theorem 2.1. Cotor (Z ,Z ) is additively isomorphic to the following module:
H∗(E6) 3 3
(C ⊕D)⊗Z [x ,x ,x ],
3 36 48 54
where C = Z [a ,a ,a ]{1,y ,y2 ,y ,y2 ,y y ,y ,y ,y }
3 4 8 10 20 20 22 22 20 22 58 60 76
⊕Z [a ,a ]{y ,y2 ,y y ,y y ,y }
3 8 10 26 26 20 26 22 26 64
and D = Z [x ]+{1,a ,a ,a ,y ,y ,a y ,y }
3 26 4 8 10 20 22 10 20 26
⊕Z [x ]{a ,y ,y ,y ,y a ,y a ,y a ,y y }.
3 26 9 21 25 27 21 8 21 10 25 10 21 26
By an easy calculation one obtains
Corollary 2.2. The Poincar´e polynomials PS of them are given respectively by
g(t) 1
PS(C ⊗Z [x ,x ,x ]) = · ,
3 36 48 54
(1−t4)(1−t8)(1−t10) (1−t36)(1−t48)(1−t54)
h(t) 1
PS(D⊗Z [x ,x ,x ]) = · ,
3 36 48 54
1−t26 (1−t36)(1−t48)(1−t54)
where g(t) = 1+t20 +t22 +t26 −t30 +t40 +t42 +t44 +t46 +t48 −t50 −t56 +t58
+t60 +t64 −t68 +t76
and h(t) = t9 +t21 +t25 +t26 +t27 +t29 +t30 +t31 +t34 +t35 +t36 +t46 +t47
+t48 +t52 +t56.
The following two theorems will play an important role when determining the ring struc-
ture of H∗(BE ) (see [MST1]).
6
Theorem 2.3.A The extension
0 → D ⊗Z [x ,x ,x ] → Cotor (Z ,Z ) → C ⊗Z [x ,x ,x ] → 0
3 36 48 54 H∗(E6) 3 3 3 36 48 54
is split as algebras.
Theorem 2.3.B We have an algebra isomorphism
Cotor (Z ,Z ) ∼= Z [a ,x ,y ,y ,y ,a ,a ,a ,x ,
H∗(E6) 3 3 3 9 26 21 25 27 4 8 10 36
x ,x ,y ,y ,y ,y ,y ,y ,y ]/I,
48 54 20 22 26 58 60 64 76
where I is the ideal generated by the following
COHOMOLOGY MOD 3 OF THE CLASSIFYING SPACE OF E6 3
i) a y = −a y +a y ,
4 26 8 22 10 20
a y = −a y −a3y2 −a y ,
4 64 8 60 8 22 10 58
y3 = −a3x −a2a2a2 y +a a a y2 +a a2 y y +a3 x ,
22 4 54 4 8 10 22 4 8 10 22 4 10 20 22 10 36
y2 y = −a y +a2a x ,
20 22 4 58 8 10 36
y y2 = −a y +a a2 x ,
20 22 4 60 8 10 36
y y = −a y −a a2y +a3a2 x +a a x y ,
58 22 4 76 4 8 60 8 10 36 8 10 36 26
y y = a2x y +a a2a2 y y +a a a y +a a2 y
60 22 4 54 20 4 8 10 20 22 4 8 10 60 4 10 58
+a2a3 x −a2 x y ,
8 10 36 10 36 26
y y = −a x y2 −a2a2x y +a2a2 x y +a a4a2 y y
76 22 4 54 20 4 8 54 20 4 10 48 22 4 8 10 20 22
+a a2a2 y +a a a y −a4a3 x +a x y2 ,
4 8 10 58 4 8 10 76 8 10 36 10 36 26
y y2 = −a a2x −a4a2 y +a2a y y +a y ,
22 26 4 8 54 8 10 22 8 10 22 26 10 64
y y y = a y −a y ,
20 22 26 8 60 10 58
y2 y = a2a x +a a3a2 y −a2a y2 −a a2 y y −a y ,
22 26 4 8 54 4 8 10 22 8 10 22 8 10 20 22 10 60
y y = a2a3x −a a x y +a a5a2 y −a4a y2 +a3a2 y y
64 22 4 8 54 4 8 54 20 4 8 10 22 8 10 22 8 10 20 22
−a a2 y +a y ,
8 10 58 10 76
y3 = a3x −a2a4y −a a2y2 +a3x ,
20 4 48 4 8 20 4 8 20 8 36
y2 y = −a y +a2a x ,
20 22 4 58 8 10 36
y y = −a2x y −a a2y +a a4y y +a2x y +a4a x ,
58 20 4 48 22 4 8 58 4 8 20 22 8 36 26 8 10 36
y y = −a y −a a2y +a3a2 x −a a x y ,
60 20 4 76 4 8 60 8 10 36 8 10 36 26
y y = a x y2 +a a4y +a x y2 −a5a2 x ,
76 20 4 48 22 4 8 60 8 36 26 8 10 36
y y2 = a a2 x +a y −a3y y −a4a2 y −a2a y y ,
20 26 4 10 48 8 64 8 22 26 8 10 20 8 10 20 26
y2 y = a2a x −a a4a y +a y −a2a y2 ,
20 26 4 10 48 4 8 10 20 8 58 8 10 20
y y = a a x y +a y −a3y −a4a y y +a2a y ,
64 20 4 10 48 22 8 76 8 60 8 10 20 22 8 10 58
y2 = −a2x y −a2a2x y2 −a2a6y +a a7a2 x +a a a2 x x +a2x y ,
58 4 48 60 4 8 48 22 4 8 60 4 8 10 36 4 8 10 36 48 8 36 64
y y = −a4x x +a3a4x y +a3a2a2 x y +a2a2x y2 −a2a6a2 y y
58 60 4 48 54 4 8 54 20 4 8 10 48 22 4 8 54 20 4 8 10 20 22
−a2a4a2 y −a2a3a y +a2a a x y2 +a2a2 x y y −a a3x x
4 8 10 58 4 8 10 76 4 8 10 48 22 4 10 48 20 22 4 8 36 54
+a a6a3 x +a a3 x x −a5a2 x y −a2a2 x y y −a a x y ,
4 8 10 36 4 10 36 48 8 10 36 22 8 10 36 20 26 8 10 36 64
y y = a3x x y +a3a4a2 x y −a2a4x y2 −a2a8a2 y y +a2a7a y
58 76 4 48 54 20 4 8 10 48 22 4 8 54 20 4 8 10 20 22 4 8 10 60
−a2a6a2 y +a2a5a y +a2a2a2 x y y +a2a a x y
4 8 10 58 4 8 10 76 4 8 10 48 20 22 4 8 10 48 60
+a2a2 x y +a a2a3 x x +a3x x y −a6a3 x y
4 10 48 58 4 8 10 36 48 8 36 54 20 8 10 36 20
+a4a2 x y y −a a2 x x y −a3 x x y ,
8 10 36 20 26 8 10 36 48 22 10 36 48 20
y y = −a a x y +a y +a3y +a4a y y −a2a y ,
58 26 4 10 48 22 8 76 8 60 8 10 20 22 8 10 58
y y = a3a x x −a2a5x y +a a3x y2 −a a6a y −a a4a y
58 64 4 8 48 54 4 8 54 20 4 8 54 20 4 8 10 60 4 8 10 76
−a a a2 x y y +a a x y +a4x x +a7a3 x −a5a2 x y
4 8 10 48 20 22 4 10 48 60 8 36 54 8 10 36 8 10 36 26
−a3a x y2 +a a3 x x ,
8 10 36 26 8 10 36 48
y2 = a2x y +a3a a3 x y −a2a5a3 y y −a2a3a3 y −a2a2a2 y
60 4 54 58 4 8 10 48 22 4 8 10 20 22 4 8 10 58 4 8 10 76
−a2a a x y2 +a2a2 x y2 +a a5a4 x −a a2a x x −a4a3 x y
4 8 10 54 20 4 10 48 22 4 8 10 36 4 8 10 36 54 8 10 36 22
+a3a4 x y +a2a2 x y y +a a3 x y y +a2 x y ,
8 10 36 20 8 10 36 22 26 8 10 36 20 26 10 36 64
y y = a3x x y +a4a a x x −a3a5a x y +a3a3a3 x y −a2a4x y y
60 76 4 48 54 22 4 8 10 48 54 4 8 10 54 20 4 8 10 48 22 4 8 54 20 22
−a2a7a3 y y +a2a6a2 y −a2a5a3 y +a2a4a2 y −a2a a3 x y y
4 8 10 20 22 4 8 10 60 4 8 10 58 4 8 10 76 4 8 10 48 20 22
+a2a2 x y +a a4a x x +a a a4 x x +a3x x y
4 10 48 60 4 8 10 36 54 4 8 10 36 48 8 36 54 22
+a4a2 x y y +a2a x x y −a2a2 x y −a3 x x y ,
8 10 36 22 26 8 10 36 54 20 8 10 36 64 10 36 48 22
y y = −a a x y −a3a2 y y +a2a y −a a2 y −a y ,
60 26 4 8 54 20 8 10 20 22 8 10 60 8 10 58 10 76
4 M.MIMURA,Y.SAMBE, ANDM.TEZUKA
y y = a4a x x +a3a4a x y +a3a2a3 x y −a2a x y
60 64 4 10 48 54 4 8 10 54 20 4 8 10 48 22 4 8 54 58
−a2a3x y y +a2a6a3 y y +a2a5a2 y +a2a4a3 y +a2a a2 x y2
4 8 54 20 22 4 8 10 20 22 4 8 10 60 4 8 10 58 4 8 10 48 22
−a2a3 x y y +a a6a4 x −a a3a x x
4 10 48 20 22 4 8 10 36 4 8 10 36 54
−a a4 x x +a5a3 x y −a4a4 x y ,
4 10 36 48 8 10 36 22 8 10 36 20
y3 = a3x −a4a2 y −a2a y2 +a3 x ,
26 8 54 8 10 26 8 10 26 10 48
y y = −a a4x +a2x y −a6a2 y +a4a y y +a2a y +a2 x y ,
64 26 4 8 54 8 54 20 8 10 22 8 10 22 26 8 10 64 10 48 22
y2 = −a2a7a x +a2a a x x −a a6x y −a a9a3 y +a a5a x y
64 4 8 10 54 4 8 10 48 54 4 8 54 22 4 8 10 22 4 8 10 54 20
+a a3a3 x y +a2x y −a4x y y −a7a3 y y +a5a3 y
4 8 10 48 22 8 54 58 8 54 20 22 8 10 20 22 8 10 58
−a4a2 y −a2a2 x y2 −a a3 x y y −a2 x y ,
8 10 76 8 10 48 22 8 10 48 20 22 10 48 60
y2 = −a2x x y y −a4a3a x x −a3a2x x y +a3a7a x y
76 4 48 54 20 22 4 8 10 48 54 4 8 48 54 22 4 8 10 54 20
+a3a5a3 x y −a3a a x x y −a2a4x y +a2a6x y y
4 8 10 48 22 4 8 10 48 54 20 4 8 54 58 4 8 54 20 22
−a2a9a3 y y −a2a8a2 y −a2a7a3 y +a2a5a x y2
4 8 10 20 22 4 8 10 60 4 8 10 58 4 8 10 54 20
−a2a4a2 x y2 −a2a a3 x y +a2a2 x y −a a9a4 x
4 8 10 48 22 4 8 10 48 58 4 10 48 76 4 8 10 36
−a a3a4 x x +a2x x y y −a5x x y +a8a3 x y
4 8 10 36 48 8 36 54 20 26 8 36 54 22 8 10 36 22
−a7a4 x y −a5a3 x y y +a4a2 x y −a2a3 x x y
8 10 36 20 8 10 36 20 26 8 10 36 64 8 10 36 48 22
+a2 x x y y ,
10 36 48 22 26
y y = a a3x y −a a a2 x y +a x y2 −a5a2 y y
76 26 4 8 54 20 4 8 10 48 22 8 54 20 8 10 20 22
+a4a y −a3a2 y −a2a y +a x y2 ,
8 10 60 8 10 58 8 10 76 10 48 22
y y = −a3a2a x x −a2a x x y −a2a6a x y −a2a x x y
64 76 4 8 10 48 54 4 8 48 54 22 4 8 10 54 20 4 10 48 54 20
−a a3x y −a a5x y y +a a8a3 y y +a a7a2 y
4 8 54 58 4 8 54 20 22 4 8 10 20 22 4 8 10 60
+a a6a3 y −a a5a2 y −a a4a x y2 −a a3a2 x y2
4 8 10 58 4 8 10 76 4 8 10 54 20 4 8 10 48 22
+a a a2 x y −a a3 x y +a3x x y +a8a4 x
4 8 10 48 60 4 10 48 58 8 36 54 26 8 10 36
−a6a3 x y +a5a x x +a5a2 y y +a4a2 x y2
8 10 36 26 8 10 36 54 8 10 58 22 8 10 36 26
+a2a4 x x +a3 x x y .
8 10 36 48 10 36 48 26
ii) a2, y2 , y2 , y2 ,
9 21 25 27
a y +x a , a y +x a , a y +x a ,
9 21 26 4 9 25 26 8 9 27 26 10
y y +x y , y y −x y , y y −x y ,
21 25 26 20 21 27 26 22 25 27 26 26
iii) a ∂2Q, y ∂2Q, y ∂2Q, y ∂2Q, x ∂2Q,
9 21 25 27 26
a a , a a , a a ,
9 4 9 8 9 10
y a , y a , y a ,
21 4 25 8 27 10
y a +a y = y a −a y ,
21 8 9 20 25 4 9 20
y a −a y = y a +a y ,
21 10 9 22 27 4 9 22
y a −a y = y a +a y ,
25 10 9 26 27 8 9 26
y y , y y , y y , y y , y y , y y ,
21 20 25 20 21 22 27 22 25 26 27 26
y y −y y = y y +y y = −y y −y y .
27 20 25 22 25 22 21 26 27 20 21 26
Here the definition of ∂ is given in (3.3), and the table of ∂2-image is given in Lemma
4.2.
COHOMOLOGY MOD 3 OF THE CLASSIFYING SPACE OF E6 5
3. Proof of Theorem 2.1
First let us recall some results from [MS]. We consider the differential algebra (V,d)
defined as follows:
V = T(a ,a ,a ,b ,b ,b ,a ,c )/J,
4 8 10 12 16 18 9 17
where J is the ideal generated by
a b −b a +c a for j = 12,16,18
9 j j 9 17 j−8
and
αβ −(−1)|β||α|βα for all the pairs (α,β) of the generators
except (a ,b ) with j = 12,16,18 and (a ,c );
9 j 9 17
the differential d is given over generators by
(3.1) da = 0 for i = 4,8,9,10, dc = a2, db = −a a for j = 12,16,18.
i 17 9 j 9 j−8
Then the differential algebra (V,d) is seen to be a resolution of Z over H∗(E ). (For a
3 6
proof, see Theorem 5.7 and Corollary 5.8 in [MS].) So by the definition of Cotor we have
Theorem ([MS]) H∗(V,d) = Cotor (Z ,Z ).
H∗(E6) 3 3
We will use a spectral sequence to calculate H∗(V,d). To that end we need to introduce
a filtration in V. To begin with, we fix an additive basis of V. The following lemma is
straightforward from the definition of the ideal J.
Lemma 3.1. We have an additive isomorphism
V ∼= T(a ,c )⊗Z [a ,a ,a ,b ,b ,b ].
9 17 3 4 8 10 12 16 18
Hence as a basis of V we can choose the following monomials:
(3.2) aℓ1cm1 ...aℓpcmpai1ai2ai3bj1bj2bj3.
9 17 9 17 4 8 10 12 16 18
We define a weight by
w(a ) = 2 for i = 4,8,10;
i
w(b ) = 0 for j = 12,16,18;
j
w(a ) = 1, w(c ) = 2, w(0) = ∞.
9 17
For a monomial of the form (3.2), we define the weight by
w(α ...α ) = w(α )+···+w(α ).
1 n 1 n
For an element x = λ x , where x is a monomial given in (3.2) and λ ∈ Z , we define
i i i i 3
P
that w(x) = infw(x ). Then we introduce a filtration in V by setting
i
i
FpV = {x ∈ V | w(x) ≥ p} for p ∈ N.
6 M.MIMURA,Y.SAMBE, ANDM.TEZUKA
It is easily seen that {FpV} is a family of differential graded algebras giving a decreasing
filtration. Hence from this filtration we can construct a spectral sequence {E ,d } such
r r
that E = V and E = H∗(V,d), namely, converging to Cotor (Z ,Z ).
0 ∞ H∗(E6) 3 3
It is convenient for later use to introduce here an auxiliary derivation ∂ on a subalgebra
Z [a ,a ,a ,b ,b ,b ] of V defined as in (5.9) of [MS]:
3 4 8 10 12 16 18
(3.3) ∂a = 0 for i = 4,8,10; ∂b = −a for j = 12,16,18.
i j j−8
Since the only non trivial differential on the generators of E is d (c ) = a2 by (3.1), we
0 0 17 9
have that
E = Z [x ]⊗Z [a ,a ,a ]⊗Λ(a )⊗Z [b ,b ,b ],
1 3 26 3 4 8 10 9 3 12 16 18
where x = a c −c a . Since d = d = 0 for degree reasons, we have
26 9 17 17 9 1 2
E = E = E .
1 2 3
We remark here that E = E is a commutative algebra in the graded sense and also
3 1
that d is a derivative differential, i.e.,
3
d (xy) = d (x)y ±xd (y), x,y ∈ E
3 3 3 3
such that
d (b ) = −a a , d (b ) = −a a , d (b ) = −a a ,
3 12 4 9 3 16 8 9 3 18 10 9
which are obtained by the differential formula (3.1).
In the next steps, we will use the following
Convention Let (M,d) be a differential module such that M = M ⊕M ⊕N and
1 2 1
that d(M ) ⊂ N and d| = 0. Then we denote Ker d∩M by Ker d and N /d(M )
1 1 M2⊕N1 1 1 1
by Coker d.
Under our convention, we have that
H(M) = Ker d⊕Coker d⊕M .
2
Moreover, for simplicity we will describe the differential d only on M . If not specifically
1
denoted, we understand that d(a) = 0 for a ∈ M ⊕N .
2 1
To calculate the E -term, we introduce another spectral sequence starting with the E -
4 3
term. That is, we define a weight w by
e
(3.4) w(b ) = 0 for i = 12,16,18, w(x ) = w(a ) = w(a ) = 0, w(a ) = 1, w(a ) = 3.
i 26 9 4 8 10
e e e e e e
By similarly introducing a filtration using the weight w as before, we obtain a new spectral
sequence {E ,d } such that E = E and E = E . e
r r 0 3 ∞ 4
At the Ee-teerm we see thaet the relationed (b ) = −a a gives d (b ) = −a a , from
0 3 12 4 9 0 12 4 9
which we oebtain that e
E = Z [a ,a ,a ,x ]{1,a b2 }⊕Z [a ,a ,x ]{a ,a b } ⊗Z [b ,b ]⊗Z [b3 ].
1 (cid:16) 3 4 8 10 26 9 12 3 8 10 26 9 9 12 (cid:17) 3 16 18 3 12
e
COHOMOLOGY MOD 3 OF THE CLASSIFYING SPACE OF E6 7
Since the relation d (b ) = −a a gives d (b ) = −a a , we have that
3 16 8 9 1 16 9 8
e
E = Z [a ,a ,a ,x ]⊕Z [a ,a ,a ,x ]{a b ,a b2 }
2 (cid:16) 3 4 8 10 26 3 4 8 10 26 4 16 4 16
e
⊕Z [a ,x ]{a ,a b }⊕Z [a ,a ,x ]{a b2 }
3 10 26 9 9 16 3 8 10 26 9 16
⊕Z [a ,a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 }
3 4 8 10 26 9 12 9 12 16 9 12 16
⊕Z [a ,a ,x ]{a b ,a b b ,a b b2 } ⊗Z [b ]⊗Z [b3 ,b3 ].
3 8 10 26 9 12 9 12 16 9 12 16 (cid:17) 3 18 3 12 16
It follows from the relation d(a b2 + a b b ) = −a2a b + a a2c giving d (a b2 +
4 16 8 12 16 8 9 12 4 8 17 3 4 16
a b b ) = −a2a b in E that d (a b2 ) = −a2a b , which implies that
8 12 16 8 9 12 3 2 4 16 8 9 12
e
E = Z [a ,a ,a ,x ]{1,a b ,a2b2 }⊕Z [a ,x ]{a ,a b }
3 (cid:16) 3 4 8 10 26 4 16 4 16 3 10 26 9 9 16
e
⊕Z [a ,a ,x ]{a b2 }⊕Z [a ,a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 }
3 8 10 26 9 16 3 4 8 10 26 9 12 9 12 16 9 12 16
⊕Z [a ,x ]⊗∆(a ){a b }⊕Z [a ,a ,x ]{a b b ,a b b2 }
3 10 26 8 9 12 3 8 10 26 9 12 16 9 12 16 (cid:17)
⊗Z [b3 ,b3 ]⊗Z [b ].
3 12 16 3 18
Using the expression Z [b ] = Z {1,b ,b2 }⊗Z [b3 ], the E -term may be rewritten as
3 18 3 18 18 3 18 3
e
E = Z [a ,a ,a ,x ]{1,b ,b2 ,a b ,a b b ,a b b2 ,a2b2 ,a2b2 b ,a2b2 b2 }
3 (cid:16) 3 4 8 10 26 18 18 4 16 4 16 18 4 16 18 4 16 4 16 18 4 16 18
e ⊕Z [a ,x ]{a ,a b ,a b2 ,a b ,a b b ,a b b2 }
3 10 26 9 9 18 9 18 9 16 9 16 18 9 16 18
⊕Z [a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 } (4)
3 8 10 26 9 16 9 16 18 9 16 18
⊕Z [a ,a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 ,a b2 b ,a b2 b b ,a b2 b b2 ,
3 4 8 10 26 9 12 9 12 18 9 12 18 9 12 16 9 12 16 18 9 12 16 18
a b2 b2 ,a b2 b2 b ,a b2 b2 b2 } (5)
9 12 16 9 12 16 18 9 12 16 18
⊕Z [a ,x ]⊗∆(a ){a b ,a b b ,a b b2 }
3 10 26 8 9 12 9 12 18 9 12 18
⊕Z [a ,a ,x ]{a b b ,a b b b ,a b b b2 ,
3 8 10 26 9 12 16 9 12 16 18 9 12 16 18
a b b2 ,a b b2 b ,a b b2 b2 } (7)
9 12 16 9 12 16 18 9 12 16 18 (cid:17)
⊗Z [b3 ,b3 ,b3 ],
3 12 16 18
where (4), (5) and (7) will appear in Lemma 3.6. The only non trivial differential
d : Z [a ,a ,a ,x ]{b ,b2 } −→ Z [a ,x ]{a ,a b }
3 3 4 8 10 26 18 18 3 10 26 9 9 18
e
is given by d (b ) = −a a which is obtained from d(b ) = −a a , and hence d (b2 ) =
3 18 9 10 18 9 10 3 18
a a b . It ies immediate to see that e
9 10 18
Ker d = Z [a ,a ,a ,x ]{a b ,a b2 }⊕Z [a ,a ,x ]{a b ,a b2 }, 1′
3 3 4 8 10 26 4 18 4 18 3 8 10 26 8 18 8 18 ¯
Coker de = Z [x ]{a ,a b }. 2′
3 3 26 9 9 18
¯
e
8 M.MIMURA,Y.SAMBE, ANDM.TEZUKA
Thus we have that
E = Z [a ,a ,a ,x ]{a b ,a b2 }⊕Z [a ,a ,x ]{a b ,a b2 }
4 (cid:16) 3 4 8 10 26 4 18 4 18 3 8 10 26 8 18 8 18
e ⊕Z [a ,a ,a ,x ]{1,a b ,a b b ,a b b2 ,a2b2 ,a2b2 b ,a2b2 b2 }
3 4 8 10 26 4 16 4 16 18 4 16 18 4 16 4 16 18 4 16 18
⊕Z [x ]{a ,a b }
3 26 9 9 18
⊕Z [a ,x ]{a b2 ,a b ,a b b ,a b b2 }
3 10 26 9 18 9 16 9 16 18 9 16 18
⊕Z [a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 }
3 8 10 26 9 16 9 16 18 9 16 18
⊕Z [a ,a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 ,a b2 b ,a b2 b b ,a b2 b b2 ,
3 4 8 10 26 9 12 9 12 18 9 12 18 9 12 16 9 12 16 18 9 12 16 18
a b2 b2 ,a b2 b2 b ,a b2 b2 b2 }
9 12 16 9 12 16 18 9 12 16 18
⊕Z [a ,x ]⊗∆(a ){a b ,a b b ,a b b2 }
3 10 26 8 9 12 9 12 18 9 12 18
⊕Z [a ,a ,x ]{a b b ,a b b b ,a b b b2 ,
3 8 10 26 9 12 16 9 12 16 18 9 12 16 18
a b b2 ,a b b2 b ,a b b2 b2 }
9 12 16 9 12 16 18 9 12 16 18 (cid:17)
⊗Z [b3 ,b3 ,b3 ].
3 12 16 18
The relation d (a b b −a b b −a b b ) = −a a a b in E implies that the only
3 4 16 18 8 12 18 10 12 16 8 9 10 12 3
non trivial differential
d : Z [a ,a ,a ,x ]{a b b ,a b b2 } → Z [a ,x ]⊗∆(a ){a b ,a b b }
4 3 4 8 10 26 4 16 18 4 16 18 3 10 26 8 9 12 9 12 18
e
is given by d (a b b ) = −a a a b , and hence d (a b b2 ) = a a a b b . So we have
4 4 16 18 8 9 10 12 4 4 16 18 8 9 10 12 18
that e e
Ker d = Z [a ,a ,a ,x ]{a2b b ,a2b b2 }
4 3 4 8 10 26 4 16 18 4 16 18 3′
e ⊕Z [a ,a ,x ]{a a b b ,a a b b2 }, ¯
3 8 10 26 4 8 16 18 4 8 16 18
Coker d = Z [a ,x ]{a b ,a b b }⊕Z [x ]{a a b ,a a b b }. 4′
4 3 10 26 9 12 9 12 18 3 26 9 8 12 9 8 12 18
¯
e
Thus we have that
E = Z [a ,a ,a ,x ]{a b ,a b2 }⊕Z [a ,a ,x ]{a b ,a b2 }
5 (cid:16) 3 4 8 10 26 4 18 4 18 3 8 10 26 8 18 8 18
e ⊕Z [a ,a ,a ,x ]{1,a b ,a2b2 ,a2b2 b ,a2b2 b2 }
3 4 8 10 26 4 16 4 16 4 16 18 4 16 18
⊕Z [a ,a ,a ,x ]{a2b b ,a2b b2 }⊕Z [a ,a ,x ]{a a b b ,a a b b2 }
3 4 8 10 26 4 16 18 4 16 18 3 8 10 26 4 8 16 18 4 8 16 18
⊕Z [x ]{a ,a b }⊕Z [a ,x ]{a b2 ,a b ,a b b ,a b b2 }
3 26 9 9 18 3 10 26 9 18 9 16 9 16 18 9 16 18
⊕Z [a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 }
3 8 10 26 9 16 9 16 18 9 16 18
⊕Z [a ,a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 ,a b2 b ,a b2 b b ,a b2 b b2 ,
3 4 8 10 26 9 12 9 12 18 9 12 18 9 12 16 9 12 16 18 9 12 16 18
a b2 b2 ,a b2 b2 b ,a b2 b2 b2 }
9 12 16 9 12 16 18 9 12 16 18
⊕Z [a ,x ]{a b ,a b b }⊕Z [x ]{a a b ,a a b b }
3 10 26 9 12 9 12 18 3 26 9 8 12 9 8 12 18
⊕Z [a ,x ]⊗∆(a ){a b b2 }
3 10 26 8 9 12 18
⊕Z [a ,a ,x ]{a b b ,a b b b ,a b b b2 ,
3 8 10 26 9 12 16 9 12 16 18 9 12 16 18
a b b2 ,a b b2 b ,a b b2 b2 }
9 12 16 9 12 16 18 9 12 16 18 (cid:17)
⊗Z [b3 ,b3 ,b3 ].
3 12 16 18
COHOMOLOGY MOD 3 OF THE CLASSIFYING SPACE OF E6 9
The relation d (a b2 + a b b ) = −a2 a b in E implies that the only non trivial
3 8 18 10 16 18 10 9 16 3
differential
d : Z [a ,a ,x ]{a b2 } → Z [a ,x ]{a b }
5 3 8 10 26 8 18 3 10 26 9 16
e
is given by d (a b2 ) = −a2 a b . Hence we obtain that
5 8 18 10 9 16
e
Ker d = Z [a ,a ,x ]{a2b2 }, 5′
5 3 8 10 26 8 18 ¯
Coker de = ∆(a )⊗Z [x ]{a b }. 6′
5 10 3 26 9 16
¯
e
Thus we have that
E = Z [a ,a ,a ,x ]{a b ,a b2 }⊕Z [a ,a ,x ]{a b ,a2b2 }
6 (cid:16) 3 4 8 10 26 4 18 4 18 3 8 10 26 8 18 8 18
e ⊕Z [a ,a ,a ,x ]{1,a b ,a2b2 ,a2b2 b ,a2b2 b2 ,a2b b ,a2b b2 }
3 4 8 10 26 4 16 4 16 4 16 18 4 16 18 4 16 18 4 16 18
⊕Z [a ,a ,x ]{a a b b ,a a b b2 }
3 8 10 26 4 8 16 18 4 8 16 18
⊕Z [x ]{a ,a b ,a a b ,a a b b }
3 26 9 9 18 9 8 12 9 8 12 18
⊕Z [a ,x ]{a b2 ,a b b ,a b b2 ,a b ,a b b }
3 10 26 9 18 9 16 18 9 16 18 9 12 9 12 18
⊕Z [x ]⊗∆(a ){a b }
3 26 10 9 16
⊕Z [a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 }
3 8 10 26 9 16 9 16 18 9 16 18
⊕Z [a ,a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 ,a b2 b ,a b2 b b ,a b2 b b2 ,
3 4 8 10 26 9 12 9 12 18 9 12 18 9 12 16 9 12 16 18 9 12 16 18
a b2 b2 ,a b2 b2 b ,a b2 b2 b2 }
9 12 16 9 12 16 18 9 12 16 18
⊕Z [a ,x ]⊗∆(a ){a b b2 }
3 10 26 8 9 12 18
⊕Z [a ,a ,x ]{a b b ,a b b b ,a b b b2 ,
3 8 10 26 9 12 16 9 12 16 18 9 12 16 18
a b b2 ,a b b2 b ,a b b2 b2 }
9 12 16 9 12 16 18 9 12 16 18 (cid:17)
⊗Z [b3 ,b3 ,b3 ].
3 12 16 18
Similarly the relation d (a b2 ) = a a2 b in E implies that the only non trivial differ-
3 4 18 9 10 12 3
ential
d : Z [a ,a ,a ,x ]{a b2 } → Z [a ,x ]{a b }
6 3 4 8 10 26 4 18 3 10 26 9 12
e
is given by d (a b2 ) = a a2 b , from which we have that
6 4 18 9 10 12
e
Ker d = Z [a ,a ,a ,x ]{a2b2 }⊕Z [a ,a ,x ]{a a b2 }, 7′
6 3 4 8 10 26 4 18 3 8 10 26 4 8 18 ¯
Coker de = Z [x ]⊗∆(a ){a b }. 8′
6 3 26 10 9 12
¯
e
10 M.MIMURA,Y.SAMBE, ANDM.TEZUKA
Thus we have that
E = Z [a ,a ,a ,x ]{a b ,a2b2 }⊕Z [a ,a ,x ]{a b ,a b2 ,a a b2 }
7 (cid:16) 3 4 8 10 26 4 18 4 18 3 8 10 26 8 18 8 18 4 8 18
e ⊕Z [a ,a ,a ,x ]{1,a b ,a2b2 ,a2b2 b ,a2b2 b2 ,a2b b ,a2b b2 }
3 4 8 10 26 4 16 4 16 4 16 18 4 16 18 4 16 18 4 16 18
⊕Z [a ,a ,x ]{a a b b ,a a b b2 }
3 8 10 26 4 8 16 18 4 8 16 18
⊕Z [x ]{a ,a b ,a a b ,a a b b }
3 26 9 9 18 9 8 12 9 8 12 18
⊕Z [a ,x ]{a b2 ,a b b ,a b b2 ,a b b }
3 10 26 9 18 9 16 18 9 16 18 9 12 18
⊕Z [x ]⊗∆(a ){a b ,a b }
3 26 10 9 12 9 16
⊕Z [a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 }
3 8 10 26 9 16 9 16 18 9 16 18
⊕Z [a ,a ,a ,x ]{a b2 ,a b2 b ,a b2 b2 ,a b2 b ,a b2 b b ,a b2 b b2 ,
3 4 8 10 26 9 12 9 12 18 9 12 18 9 12 16 9 12 16 18 9 12 16 18
a b2 b2 ,a b2 b2 b ,a b2 b2 b2 }
9 12 16 9 12 16 18 9 12 16 18
⊕Z [a ,x ]⊗∆(a ){a b b2 }
3 10 26 8 9 12 18
⊕Z [a ,a ,x ]{a b b ,a b b b ,a b b b2 ,
3 8 10 26 9 12 16 9 12 16 18 9 12 16 18
a b b2 ,a b b2 b ,a b b2 b2 }
9 12 16 9 12 16 18 9 12 16 18 (cid:17)
⊗Z [b3 ,b3 ,b3 ].
3 12 16 18
Then for degree reasons we see that d = 0 for r ≥ 7 so that E = E , which gives the
r 7 ∞
E -term of the original spectral sequencee. e e
4
Now we can summarize our calculations from (4) to (7) and from 1′ to 8′ as follows;
¯ ¯
for the kernel parts, we have
Z [a ,a ,a ,x ]{1,a b ,a2b2 ,a b ,a2b2 ,a2b b ,a2b2 b ,a2b b2 ,a2b2 b2 }
3 4 8 10 26 4 16 4 16 4 18 4 18 4 16 18 4 16 18 4 16 18 4 16 18
⊕Z [a ,a ,x ]{a b ,a2b2 ,a a b b ,a a b2 ,a a b b2 }; (1)
3 8 10 26 8 16 8 16 4 8 16 18 4 8 18 4 8 16 18
for the cokernel parts, we have
Z [x ]{a ,a b ,a b ,a b ,a a b ,a a b ,a a b ,a a b b } (2)
3 26 9 9 12 9 16 9 18 9 8 12 9 10 12 9 10 16 9 8 12 18
⊕Z [a ,x ]{a b b }.
3 10 26 9 12 18
Hence we can write
(3.5) E = E = (1)⊕(2)⊕Z [a ,x ]{a b b }⊕Z [a ,x ]{a b b ,a b b2 ,a b2 }
7 ∞ 3 10 26 9 12 18 3 10 26 9 16 18 9 16 18 9 18
e e
⊕(4)⊕(5)⊕Z [a ,x ]⊗∆(a ){a b b2 }⊕(7).
3 10 26 8 9 12 18
Then one can verify that the following elements are chosen to be generators of the filtered
algebra E :
0
y = −a b , y = a b , y = a b ,
20 4 16 22 4 18 26 8 18
y = −a2b2 b , y = −a2b b2 , y = −a2b b2 , y = −a2b2 b2 ,
58 4 16 18 60 4 16 18 64 8 12 18 76 4 16 18
y = a b , y = a b , y = a b .
21 9 12 25 9 16 27 9 18
Numbering the term Z [a ,x ]{a b b ,a b b ,a b b2 ,a b2 } by (3) and the term
3 10 26 9 12 18 9 16 18 9 16 18 9 18
Z [a ,x ]⊗∆(a ){a b b2 } by (6), we have proved the following
3 10 26 8 9 12 18