THE INTEGRAL COHOMOLOGY RING OF E8=T 1.E7 MASAKI NAKAGAWA* December 27, 2006 Abstract.We determine the integral cohomology ring of the ho- mogeneous space E8=T1.E7by the Borel presentation and a method due to Toda. Then using the Gysin exact sequence associated with the circle bundle S1 ! E8=E7! E8=T1.E7, we also determine the integral cohomology ring of E8=E7. 1.introduction Let G be a compact connected Lie group and H a centralizer of a toral subgroup. Then the homogeneous space G=H is called a gener- alized flag manifold and plays an important role in the modern math- ematics such as algebraic topology, differential geometry and algebraic geometry. In fact, G=H admits a complex structure, K"ahler structure and symplectic structure. In algebraic topology, it is a classical prob- lem to determine the integral cohomology ring H*(G=H; Z) of G=H. Since the Chow ring of G=H, A(G=H) is canonically isomorphic to H*(G=H; Z) ([10]), the determination of H*(G=H; Z) is of fundamen- tal importance. There are two descriptions of the cohomology ring H*(G=H; Z); One is the Borel presentation which uses the invariants of Weyl groups and the spectral sequence technique ([2]). The other is the Schubert pre- sentation which uses the so called Schubert calculus ([1]). Using the Borel presentation of rational cohomology and the results on mod p cohomology H*(G; Z=pZ) for all primes p, Toda initiated the research for computing the integral cohomology ring of G=H in [17]. Along the lines of his idea, the integral cohomology rings of various flag mani- folds have been computed explicitly([18], [19], [12], [11], [15]). On the other hand, recently H. Duan developed extensively a multiplicative formula of Schubert classes which is a generalization of the Littlewood- Richerdson rule of Grassmann manifold ([8]). Furthermore, H. Duan __________ 2000 Mathematics Subject Classification. Primary 57T15; Secondary 55T10. Key words and phrases. cohomology, Lie groups, flag manifolds. * Partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society of the Promotion of Science. 1 and X. Zhao also computed the integral cohomology rings of the above flag manifolds independently of Toda's method ([9]). Until recently none of these two methods have been successful in computing the in- tegral cohomology rings of homogeneous spaces of the exceptional Lie group E8. The exceptional Lie group E8 contains a closed connected subgroup T 1.E7 whose local type is T 1x E7, where T 1is a certain one dimensional torus (see [12], x2). It is obtained as a centralizer of a certain one dimensional torus (see 2.1). Hence the homogeneous space E8=T 1.E7 is a generalized flag manifold. In this paper, using the above method due to Borel and Toda, we compute the integral cohomology ring of E8=T 1.E7 explicitly. The motivation of the current work is to study the cohomology of the irreducible symmetric space EIX = E8=S3.E7 (see [12], x1), as well as the integral cohomology ring of the full flag manifold E8=T , where T is a maximal torus in E8, and the Chow ring of the corresponding complex algebraic group ([13], [14]). In fact, we will use Theorem 3.8 to compute H*(E8=T ; Z), which is the only remaining case among G=T 's for G simple, in our forthcoming paper([16]). Moreover, the homogeneous space E8=T 1.E7 is a generating variety of E8 in the sense of Bott ([5]) and its integral cohomology ring is needed to compute the Pontrjagin ring H*( E8; Z), where E8 denotes the based loop space of E8. The paper is organized as follows: In x2, we compute the rings of invariants of the Weyl groups of E8 and T 1.E7. In x3, using these results and the Borel presentation of rational cohomology ring, we compute the rational cohomology ring of E8=T 1.E7. Then, by investigating the integral cohomology ring of E8=T in low degrees, we determine the integral cohomology ring of E8=T 1.E7 explicitly. Furthermore, using the Gysin exact sequence associated with the circle bundle S1 -! E8=E7 -! E8=T 1.E7, we also determine the integral cohomology ring of E8=E7. Throughout this paper, oei(x1, . .,.xn) denotes the i-th elementary symmetric function in variables x1, . .,.xn. The author is grateful to Professor M. Mimura for many useful dis- cussions and advice during the preparation of the manuscript. 2. the rational invariant subalgebras of the Weyl groups 2.1. Notations. Let T be a maximal torus of E8. According to [7], the Dynkin diagram of E8 is as follows: 2 ff1 ff3 ff4 ff5 ff6 ff7 ff8 e______e______e______e______e_____e______e_ | | | | e|ff2 where ffi's are the simple roots. As usual we may regard each root as an element of H1(T ; Z) ~=H2(BT ; Z). Let C8 be the centralizer of a one dimensional torus determined by ffi= 0 (i 6= 8). Then as shown in ([12], x2), C8 = T 1.E7 and T 1\ E7 ~=Z=2Z. The Weyl groups W (.) of E8, C8 are respectively given as follows: W (E8) = , W (C8) = , where sidenotes the simple reflection corresponding to the simple root ffi. Let {!i}1 i 8be the fundamental weights corresponding to the sys- tem of the simple roots {ffi}1 i 8. We also regard each weight as an elememt of H2(BT ; Z) and then {!i}1 i 8forms a basis of H2(BT ; Z). The action of si's on {!i}1 i 8is given as follows: 8 >< X8 2(ffi|ffj) !i- _______!j if k = i, si(!k) = (ffj|ffj) >: j=1 !k if k 6= i. Next we define the elements t8= !8, t7= s8(t8) = !7- !8, t6= s7(t7) = !6- !7, t5= s6(t6) = !5- !6, t4= s5(t5) = !4- !5, t3= s4(t4) = !2+ !3- !4, t2= s3(t3) = !1+ !2- !3, t1= s1(t2) = -!1+ !2, ci= oei(t1, . .,.t8), 1 t= !2 = _c1. 3 3 Then t and ti's span H2(BT ; Z), since each !i is an integral linear combination of t and ti's and we have the following isomorphism: H*(BT ; Z) = Z[t1, . .,.t8, t]=(c1- 3t). Furthermore, the action of si's on {ti}1 i 8and t is given by TABLE 1, where blanks indicate the trivial action. __________________________________________|||||||||| |___|s1_|____s2_______|s3_|s4s|5|s6s|7|s8_||||||||||| |_t1t|2|__t_-_t2-_t3___|_|__|__|___|__|__|_|||||||||| |_t2t|1|__t_-_t1-_t3__t|3|__|__|___|__|__|_|||||||||| |_t3_|_|__t_-_t1-_t2__t|2|t4_|_|___|__|__|_|||||||||| |_t4_|_|_____________|___|t3_|t5_|_|__|__|_|||||||||| |_t5_|_|_____________|___|__|t4_|t6_|_|__|_|||||||||| |_t6_|_|_____________|___|__|__|t5_|t7_|_|_|||||||||| |_t7_|_|_____________|___|__|__|___|t6_|t8_||||||||||| |_t8_|_|_____________|___|__|__|___|__|t7_||||||||||| |_t_|__|2t_-_t1-_t2-_t3_||__|__|___|__|__|_ Table 1 Putting u= t8, 3 o= t - _u, 2 1 oi= ti- _u (1 i 7), 2 we have H*(BT ; Q) = Q[u, o, o1, . .o.7]=(~c1- 3o) for ~c1= o1+ . .+.o7. The action of si's on {oi}1 i 7and o is given by TABLE 2, where blanks also indicate the trivial action. Since E7\ T = T 0is a maximal torus of E7, we have a commutative diagram of natural maps: E7=T 0--~-!C8=T --i-! E8=T ? ? ? (2.1) ?y ?y ?y'0 g = BT 0 ---! BT ---! BT. Since E8 is 2-connected, '*0: H2(BT ; Z) -~! H2(E8=T ; Z). Under this isomorphism, we denote the '*0-images of ti(1 i 8), t by the same letters. Thus we have the generators ti(1 i 8), t 2 H2(E8=T ; Z) 4 ____________________________________________||||||||| |___|s1_|_______s2_________|s3_|s4s|5|s6_|s7_|||||||||| |_o1o|2|____o_-_o2-_o3_____|__|__|___|__|__|_||||||||| |_o2o|1|____o_-_o1-_o3_____|o3_|_|___|__|__|_||||||||| |_o3_|_|____o_-_o1-_o2_____|o2_|o4_|_|__|__|_||||||||| |_o4_|_|___________________|__|o3_|o5_|_|__|_||||||||| |_o5_|_|___________________|__|__|o4_|o6_|_|_||||||||| |_o6_|_|___________________|__|__|___|o5_|o7_|||||||||| |_o7_|_|___________________|__|__|___|__|o6_|||||||||| |_o__|_|-o_+_o4+_o5+_o6+_o7_|_|__|___|__|__|_||||||||| |_u_|__|___________________|__|__|___|__|__|_ Table 2 with a relation c1 = 3t. We donote the generators ti(1 i 7), x in [19] by t0i(1 i 7), t0. Then, by a similar arguments to that in ([19], x1), we have (2.2) g*(ti) = t0i(1 i 7), g*(t8) = 0, g*(t) = t0. 2.2. Invariant subalgebra of W (C8). We recall the rational invari- ant forms for E7 given in [19]. We put x0i= 2t0i- t0(1 i 7), x08= t0. Then the set S0= { (x0i+ x0j) (1 i < j 8)} H2(BT 0; Q) is invariant under the action of W (E7). Thus we have W (E7)-invariant forms X (2.3) I0n= yn 2 H2n(BT 0; Q)W(E7). y2S0 Then direct computation using the same method as in ([19], x2) yields the following results: (2.4) I02 = -25. 3(c02- 4t02), I06 28. 32(c032+ 8c06) mod (t0, a60), I08 212. 5(2c042- 3c03c05) mod (t0, a80), I010 212. 32. 5 . 7(c052- 4c03c07) mod (t0, a100), I012 215. 32. 5(-54c062+ 18c05c07- c03c04c05) mod (t0, a120), I014 216. 3 . 7 . 11 . 29(2c072+ 2c03c04c07- c03c05c06) mod (t0, a140), I018 221. 5 . 1229(-126c05c06c07- 5c03c04c05c06) mod (t0, a180), 5 where c0i= oei(t01, . .,.t07) and ai0denotes the ideal of H*(BT 0; Q) gener- ated by I0jfor j < i with j 2 {2, 6, 8, 10, 12, 14, 18}. We also recall the following result: Proposition 2.1 ([19], Lemma 2.1). The rational invariant subalgebra of the Weyl group W (E7) is given as follows: H*(BT 0; Q)W(E7)= Q[I02, I06, I08, I010, I012, I014, I018]. TABLE 2 shows that the action of W (C8) on o, o1, . .,.o7 is the same as that of W (E7) on t0, t01, . .,.t07. Therefore we have Lemma 2.2. If we represent I0n= _n(t0, t01, . .,.t07) 2 H2n(BT 0; Q)W(E7), the rational invariant subalgebra of the Weyl group W (C8) is given as follows: H*(BT ; Q)W(C8)= Q[u, J2, J6, J8, J10, J12, J14, J18], where Jn = _n(o, o1, . .,.o7) 2 H2n(BT ; Q)W(C8). 2.3. Invariant subalgebra of W (E8). We put 2 2 ,i= 2ti- _t (1 i 8), ,9 = -_ t. 3 3 Then we see from TABLE 1 that the set S = { (,i- ,j) (1 i < j 9), (,i+ ,j+ ,k) (1 i < j < k 9)} is invariant under the action of W (E8). In fact, S is an orbit of 2!8 under the action of W (E8). Thus we have W (E8)-invariant forms X In = yn 2 H2n(BT ; Q)W(E8). y2S Let us compute In's in the following way; We put sn = ,1n+ . .+.,9n, dn = oen(,1, . .,.,9). Then sn's and dn's are related to each other by the Newton formula: n-1X (2.5) sn = (-1)i-1sn-idi+ (-1)n-1ndn (dn = 0 forn > 9). i=1 Note that s0 = 9, s1 = d1 = ,1+ . .+.,9 = 0. 6 Then X In X X X X __= e,i-,j+ e-,i+,j e,i+,j+,k+ e-,i-,j-,k n n! i. Then A is a subalgebra of H*(BT ; Q) containing H*(BT ; Q)W(C8). More explicitly, we have (2.10) A = Q[u, c1, c2, . .,.c7]. In fact, we can show (2.10) as follows; Putting "ci= oei(t1, . .,.t7), we have cn = "cn+ u"cn-1 (1 n 8), since X8 Y8 7Y X7 cn = (1 + ti) = (1 + u) (1 + ti) = (1 + u) "cn. n=0 i=1 i=1 n=0 Conversely, one can express "cn= cn - ucn-1+ u2cn-2- . .+.(-1)nun (1 n 7). In particular, the following relation holds: (2.11) c8 = uc7- u2c6+ u3c5- u4c4+ u5c3- u6c2+ u7c1- u8. 9 Therefore, by TABLE 1, we have A = H*(BT ; Q) = Q[t1, t2, . .,.t7, u] = Q[u, "c1, "c2, . .,."c7] = Q[u, c1, c2, . .,.c8]=(c8- uc7+ u2c6- . .+.u8) = Q[u, c1, c2, . .,.c7], which has shown (2.10). Denote by ai A ( resp.bi H*(BT ; Q)W(C8)), the ideal of A (resp. of H*(BT ; Q)W(C8)) generated by Ij's for j < i where j 2 {2, 8, 12, 14, 18, 20, 24, 30}. The remainder of this section is devoted to proving the next lemma: Lemma 2.5. In H*(BT ; Q)W(C8) = Q[u, J2, J6, J8, J10, J12, J14, J18], we have (i)I2 = 5J2+ 120u5, I8 = 22. 3J8+ (decomp.), 23. 3 . 5 I12= -22. 7J12+ (decomp.), I14= _______J14+ (decomp.), 29 24. 3 . 52. 7 . 13 I18= -_____________ J18+ (decomp.), 1229 where (decomp.) means decomposable elements. (ii)I20 227. 32. 52. 11 . 17 . 41"I20 mod b20, I24 232. 33. 5 . 7 . 11 . 19 . 199"I24 mod b20, I30 235. 34. 55. 7 . 11 . 13 . 61"I30 mod b20. Suppose Lemma 2.5 for the moment. Then we have the following: Lemma 2.6. The rational invariant subalgebra of the Weyl group W (E8) is given as follows: H*(BT ; Q)W(E8)= Q[I2, I8, I12, I14, I18, I20, I24, I30]. Proof.By Lemma 2.5, Ii is not a polynomial of Ij's for j < i where i = 2, 8, 12, 14, 18, 20, 24, 30. Since H*(BT ; Q)W(E8)~=H*(BE8; Q) = Q[y4, y16, y24, y28, y36, y40, y48, y60] with deg(yi) = i, we have the required result. 10 1 3 Proof of Lemma 2.5 (i).Since oi= ti- _u (1 i 7) and o = t - _u, 2 2 we have oi ti(1 i 7), o t mod (u). Therefore, putting ~ci= oei(o1, . .,.o7), we obtain cn ~cn(1 n 7), c8 0 mod (u). Then, in view of (2.4) and Lemma 2.2, we have (2.12) J2 = -25. 3(~c2- 4o2) -25. 3c2 mod (t, u), J6 28. 32(~c23+ 8~c6) mod (o, ~a6) 28. 32(c23+ 8c6) mod (t, u, ~a6), J8 212. 5(2~c24- 3~c3~c5) mod (o, ~a8) 212. 5(2c24- 3c3c5) mod (t, u, ~a8), J10 212. 32. 5 . 7(~c25- 4~c3~c7) mod (o, ~a10) 212. 32. 5 . 7(c25- 4c3c7) mod (t, u, ~a10), J12 215. 32. 5(-54~c26+ 18~c5~c7- ~c3~c4~c5) mod (o, ~a12) 215. 32. 5(-54c26+ 18c5c7- c3c4c5) mod (t, u, ~a12), J14 216. 3 . 7 . 11 . 29(2~c27+ 2~c3~c4~c7- ~c3~c5~c6) mod (o, ~a14) 216. 3 . 7 . 11 . 29(2c27+ 2c3c4c7- c3c5c6) mod (t, u, ~a14), J18 221. 5 . 1229(-126~c5~c6~c7- 5~c3~c4~c5~c6) mod (o, ~a18) 221. 5 . 1229(-126c5c6c7- 5c3c4c5c6) mod (t, u, ~a18), where ~aidenotes the ideal of H*(BT ; Q)W(C8)generated by Jj's for j < i with j 2 {2, 6, 8, 10, 12, 14, 18}. Now we prove the last formula of (i): For degree reasons, in H*(BT ; Q)W(C8) = Q[u, J2, J6, J8, J10, J12, J14, J18], we can put (2.13) I18= ff18J18+ (decomp.) 11 for some ff182 Q. On the other hand, by using (2.10) and (2.12), we have A=(t, u, ~a18)A==(t, u, J2, J6, J8, J10, J12, J14) = Q[u, c1, c2, c3, c4, c5, c6, c7] 0 1 , 2 2 3_ 2 B t, u, c2, c3+ 8c6, c4- 2c3c5, c5- 4c3c7,C @ 1 1 1 A c26- _c5c7+ __c3c4c5, c27+ c3c4c7- _c3c5c6 3 54 2 = Q[c3, c4, c5, c6, c7] 0 1 , 2 2 3_ 2 B c3+ 8c6, c4- 2c3c5, c5- 4c3c7, C @ 1 1 1 A . c26- _c5c7+ __c3c4c5, c27+ c3c4c7- _c3c5c6 3 54 2 We consider (2.13) in the ring A=(t, u, ~a18). Then, by Lemma 2.3 (v) and (2.12), we have I18 -225. 3 . 53. 7 . 13(-126c5c6c7- 5c3c4c5c6), J18 221. 5 . 1229(-126c5c6c7- 5c3c4c5c6). 24. 3 . 52. 7 . 13 Therefore ff18= -_____________ . Similar tedious computation gives 1229 the other formulas. Before proceeding the proof of Lemma 2.5 (ii), we need the following lemma: Lemma 2.7. Explicit forms of W (C8)-invariants J6, J10are given as follows: J6= 28. 3{24~c6+ 3~c23- 4~c2~c4- 2~c32+ (-12~c5- 6~c2~c3)o + (31~c22+ 16~c4)o2 + 12~c3o3 - 136~c2o4 + 188o6}, J10= 212. 3{105~c25- 420~c3~c7+ 90~c2~c3~c5- 60~c2~c24+ 300~c22~c6+ 15~c22~c23 - 20~c32~c4- 2~c52+ (-30~c32~c3- 330~c22~c5+ 480~c2~c7+ 90~c2~c3~c4- 210~c4~* *c5)o + (270~c22~c4- 210~c2~c23- 150~c3~c5- 2220~c2~c6+ 75~c42+ 345~c24)o2 + (480c~22~c3- 570c~3~c4+ 2070~c2~c5- 660~c7)o3 + (-1050~c2~c4+ 4080~c6- 950~c32+ 705~c23)o4 + (-2250~c2~c3- 3420~c5)o5 + (1580~c4+ 5165~c22)o6 + 2820~c3o7 - 12360~c2o8 + 10868o10}, where ~ci= oei(o1, . .,.o7). 12 Proof.Since Jn has the same expression as I0nreplacing t0, c0iwith o, ~ci (Lemma 2.2), we have to compute the W (E7)-invariant forms I06, I010 explicitly. But this can be done from the data in ([19], x2). We can rewrite J6, J10in terms of t, u, ci(2 i 7); Since u = t8 1 and oi= ti- _u (1 i 7), we have 2 ` ' 7 ` ' 7 ` ' 7 ` ' 1 X 1 Y 1 Y 1 1 + _u ~cn= 1 + _u (1 + oi) = 1 + _u 1 - _u + ti 2 n=0 2 i=1 2 i=1 2 Y8 ` ' 8 ` ' 8-i 1 X 1 = 1 - _u + ti= 1 - _u ci i=1 2 i=0 2 and hence n ` '` 'n-i 1 X 8 - i 1 n-i ~cn+ _u~cn-1= -_ ciu (1 n 7). 2 i=0 n - i 2 From this, we obtain (2.14) 9 ~c1= 3t - _u, 2 37 2 ~c2= c2- 12tu + __u , 4 7 87 2 93 3 ~c3= c3- _c2u + __tu - __u , 2 4 8 11 2 3 163 4 ~c4= c4- 3c3u + __c2u - 24tu + ___u , 2 16 5 2 21 3 297 4 219 5 ~c5= c5- _c4u + 4c3u - __c2u + ___tu - ___u , 2 4 16 32 11 2 13 3 57 4 45 5 247 6 ~c6= c6- 2c5u + __c4u - __c3u + __c2u - __tu + ___u , 4 4 16 4 64 3 7 2 15 3 31 4 63 5 381 6 255 7 ~c7= c7- _c6u + _c5u - __c4u + __c3u - __c2u + ___tu - ___u . 2 4 8 16 32 64 128 On the other hand, by Lemma 2.3 (ii) and (2.11), we have I8 214. 3 . 5(2c24- 3c3c5) mod (t, c8, a8), c8 = uc7- u2c6+ u3c5- u4c4+ u5c3- u6c2+ u7c1- u8 uc7- u2c6+ u3c5- u4c4+ u5c3- u8 mod (t, a8), 13 and hence 3 c24 _c3c5 mod (t, c8, a12), 2 u8 uc7- u2c6+ u3c5- u4c4+ u5c3 mod (t, c8, a8). Therefore, by (2.9) and Lemma 2.7, we obtain (2.15) 1 273 6 v = _____J6- ___u 46080 640 2 1 2 1 1 2 1 3 _c6+ __c3- _c5u + _c4u - _c3u mod (t, a8), 5 20 2 3 2 1 55 4 666919 10 w = ________J10- __u v - ______u 15482880 24` 645120' ` ' 1 2 1 1 1 1 2 1 3 __c5- _c3c7+ _c3c6- _c4c5 u - _c3c5u + -c7+ _c3c4 u 12 3 2 6 6 3 1 2 4 1 6 1 7 -_ c3u + _c4u + _c3u mod (t, c8, a12). 2 3 2 Under these preparations, we will prove Lemma 2.5 (ii). Proof of Lemma 2.5 (ii).First note that H*(BT ; Q)W(C8)= Q[u, J2, J6, J8, J10, J12, J14, J18] = Q[u, I2, v, I8, w, I12, I14, I18] by (i). Since I202 H*(BT ; Q)W(E8) H*(BT ; Q)W(C8), we can put (*) I20 227. 33. 52. 11 . 17 . 41(~1u20+ ~2u14v + ~3u8v2 + ~4u2v3 + ~5u10w + ~6u4vw + ~7w2) mod b20 for some ~i2 Q. In order to determine the coefficients ~i, we need the following lemma, which is directly verified by making use of (2.10) and Lemma 2.3. Lemma 2.8. A=(t, c8, a20)= A=(t, I2, c8, I8, I12, I14, I18) = Q[u, c3, c4, c5, c6, c7]=J, 14 where J is the ideal generated by uc7- u2c6+ u3c5- u4c4+ u5c3- u8, 3 c24- _c3c5, 2 5 5 1 2 1 4 c26- _c5c7+ __c3c4c5- _c3c6+ __c3, 3 54 6 24 1 1 1 2 c27- _c3c5c6+ _c3c4c7+ _c4c5, 2 3 6 29 3 2 3 476 c63- 7c43c6+ __c3c4c5+ 182c3c5c7+ 75c3c5- ___c3c4c5c6- 24c5c6c7. 9 3 In particular, A=(t, c8, a20) has a basis {uicj3ck4cl5cm6cn7(0 i 7, 0 j 5, 0 l, 0 k, m, n 1)} as a Q-vector space. Now we consider the relation (*) in the ring A=(t, c8, a20). By Lemma 2.3 (vi), we have ` ' 1 4 1 2 1 2 2 1 3 1 3 I20 227.33.52.11.17.41 ___c5- __c3c5c7- __c3c4c5- __c3c4c7+ __c3c5c6. 144 18 54 27 18 On the other hand, using (2.15) and Lemma 2.8, we can rewrite each monomial in the right hand side of (*). For example, we have 1 4 1 2 1 2 2 1 3 1 3 w2 ___c5- __c3c5c7- __c3c4c5+ __c3c5c6- __c3c4c7 144 18 54 18 27 1 7 2 1 7 2 1 6 3 1 6 5 6 - __u c3c7+ __u c3c5+ __u c3c5+ _u c3c5c6- _u c3c4c7 12 12 24 3 9 + . ... Then using the second half of Lemma 2.8, the coefficients in (*) are obtained as follows: 10 ~1 = 3, ~2 = 15, ~3 = 20, ~4 = __, ~5 = 4, ~6 = 10, ~7 = 1. 3 Thus we have obtained ` 10 2 3 10 I20 227. 33. 52. 11 . 17 . 41 3u20+ 15u14v + 20u8v2 + __u v + 4u w 3 j + 10u4vw + w2 227. 32. 52. 11 . 17 . 41(9u20+ 45u14v + 60u8v2 + 10u2v3 + 12u10w + 30u4vw + 3w2) 227. 32. 52. 11 . 17 . 41"I20 mod b20. 15 Putting 24 18 12 2 6 3 (**) I24 232. 33. 52. 7 . 11 . 19 . 199 ~1u + ~2u v + ~3u v + ~4u v + ~5v4 + ~6u14w + ~7u8vw + ~8u2v2w + ~9u4w2) mod b20 for some ~i2 Q, we will proceed quite similarly. By Lemma 2.3 (vii), we have ` 31 5 1 4 337 3 3 I24 232. 33. 52. 7 . 11 . 19 . 199 ____c3c4c5+ ___c3c5c7+ _____c3c5 8640 480 25920 ' 71 3 31 3 31 3 22 2 1 4 -____c3c4c5c6+ ___c3c5c6c7+ ___c3c5c6- ___c3c4c5c7- ___c4c5. 4320 240 480 135 120 On the other hand, in A=(t, c8, a20), we have, for example, 31 5 1 4 337 3 3 71 3 31 3 v4 ____c3c4c5+ ___c3c5c7+ _____c3c5- ____c3c4c5c6+ ___c3c5c6c7 8640 480 25920 4320 240 31 3 22 2 1 4 11 7 4 7 2 + ___c3c5c6- ___c3c4c5c7- ___c4c5+ __u c3c5- u c3c5c6 480 135 120 16 11 7 2 9 6 4 619 6 3 - __u c3c4c5+ ___u c3c6- ___u c3c4c5+ . ... 18 160 480 Then using the second half of Lemma 2.8, the coefficients in (**) are obtained as follows: 11 21 ~1 = __, ~2 = 12, ~3 = 21, ~4 = 12, ~5 = 1, ~6 = __, ~7 = 12, 5 5 9 ~8 = 6, ~9 = _. 5 Thus we have obtained ` 11 24 18 12 2 6 3 I24 232. 33. 52. 7 . 11 . 19 . 199 __u + 12u v + 21u v + 12u v 5 ' 21 14 8 2 2 9 4 2 +v4 + __u w + 12u vw + 6u v w + _u w 5 5 232. 33. 5 . 7 . 11 . 19 . 199(11u24+ 60u18v + 105u12v2 + 60u6v3 + 5v4 + 21u14w + 60u8vw + 30u2v2w + 9u4w2) 232. 33. 5 . 7 . 11 . 19 . 199"I24 mod b20. Finally, we can also put (* * *) I30 238. 34. 55. 7 . 11 . 13 . 61( 1u30+ 2u24v + 3u18v2 + 4u12v3 + 5u6v4 + 6v5 + 7u20w + 8u14vw + 9u8v2w + 10u2v3w + 11u10w2 + 12u14vw2 + 13w3) mod b20 16 for some i2 Q. Then, by Lemma 2.3 (viii), we have ` 599 5 47 5 3 1519 4 I30 238. 34. 55. 7 . 11 . 13 . 61 -_____c3c4c5c6+ _____c3c5+ _____c3c5c6c7 51840 34560 25920 6293 3 2 32537 3 3 189919 2 4 2012 2 16693 4 + ____c3c4c5c7- _____c3c5c6+ ______c3c4c5+ ____c3c4c5c6c7- _____c3c5c7 7290 25920' 466560 1215 25920 223 4 1 6 -____c4c5c6- ____c5 . 6480 1728 On the other hand, in A=(t, c8, a20), we have, for example, 31 5 47 5 3 1 4 1993 3 3 91 3 2 v5 _____c3c4c5c6- _____c3c5+ ___c3c5c6c7+ ____c3c5c6- ___c3c4c5c7 11520 69120 640 5760 360 1279 2 4 49 2 7 4 299 4 - _____c3c4c5- ___c3c4c5c6c7+ ____c4c5c6+ ____c3c5c7+ . .,. 11520 135 1440 1920 1 5 5 4 365 3 3 1043 3 2 1079 2 4 w3 ___c3c4c5c6- __c3c5c6c7+ ___c3c5c6- ____c3c4c5c7- ____c3c4c5 162 81 648 2916 5832 226 2 2 4 431 4 1 6 - ___c3c4c5c6c7+ __c4c5c6+ ____c3c5c7+ ____c5+ . ... 243 81 1296 1728 Then using the second half of Lemma 2.8, the coefficients in (* * *) are obtained as follows: 9 35 3 1 = -_ , 2 = -3, 3 = 0, 4 = -5, 5 = -__ , 6 = -2, 7 = -_ 8 2 2 9 3 8 = _, 9 = 15, 10= -5, 11= -_ , 12= 3, 13= -1. 2 2 Thus we have obtained ` 9 30 24 12 3 35 6 4 5 I30 238. 34. 52. 7 . 11 . 13 . 61 -_ u - 3u v - 5u v - __u v - 2v 8 2 ' 3 20 9 14 8 2 2 3 3 10 2 4 2 3 -_ u w + _u vw + 15u v w - 5u v w - _u w + 3u vw - w 2 2 2 235. 34. 52. 7 . 11 . 13 . 61(-9u30- 24u24v - 40u12v3 - 140u6v4 - 16v5 - 12u20w + 36u14vw + 120u8v2w - 40u2v3w - 12u10w2 + 24u4vw2 - 8w3) 235. 34. 52. 7 . 11 . 13 . 61"I30 mod b20. Consequently, we have established Lemma 2.5. 3. Cohomology of E8=T 1.E7 3.1. Rational cohomology ring of E8=T 1.E7. With the above re- sults, we will compute the rational cohomology ring of E8=C8. First of all we review the classical results of Borel ([2]); Let G be a compact 17 connected Lie group, H a closed connected subgroup of G of maximal rank and T a commnon maximal torus. Consider the fibration ae G=H -'! BH -! BG. Since H*(BG; Q) is a polynomial ring generated by elements of even degrees and H*(G=H; Q) has vanishing odd dimensional part (Hirsch formula [2]), the rational cohomology spectral sequence for this fibra- tion collapses. In particular, we have the following description of the rational cohomology ring of G=H: '*~ H*(G=H; Q) - H*(BH; Q)=(ae*H+ (BG; Q)) ~=H*(BT ; Q)W(H)=(H+ (BT ; Q)W(G)), where H+ = i>0Hi and ( ) means the ideal generated by the ele- ments in parenthesis. We apply this result to the fibration: ae E8=C8 -'!BC8 -! BE8. Then using Lemmas 2.2, 2.6, 2.5 and (2.9), we have H*(E8=C8; Q) ~=H*(BT ; Q)W(C8)=(H+ (BT ; Q)W(E8)) ~=Q[u, J2, J6, J8, J10, J12, J14, J18]=(I2, I8, I12, I14, I18, I20,* * I24, I30) ~=Q[u, I2, v, I8, w, I12, I14, I18]=(I2, I8, I12, I14, I18, I20, I2* *4, I30) ~=Q[u, v, w]=("I20, "I24, "I30). Thus we have obtained Lemma 3.1. The rational cohomology ring of E8=T 1.E7 is given as follows: H*(E8=T 1.E7; Q) = Q[u, v, w]=("I20, "I24, "I30), where degu = 2, degv = 12, degw = 20 and "I20, "I24and "I30are given by (2.8). 3.2. Integral cohomology ring of E8=T in low degrees. Consider the fibration p E7=T 0~=C8=T -i! E8=T -! E8=C8. Since H*(E8=C8; Z) and H*(E7=T 0; Z) have no torsion and vanishing odd dimensional part by Bott [4], the Serre spectral sequence for the above fibration collapses and the following sequence p* * i* * * 0 Z ! H*(E8=C8; Z) -! H (E8=T ; Z) -! H (C8=T ; Z) ~=H (E7=T ; Z) ! Z 18 is co-exact; that is, p* is injective, i* is surjective and Keri* = (p*H+ (E8=C8; Z)), the ideal generatedpby*H+ (E8=C8; Z). Therefore we will obtain some information about the generators of H*(E8=C8; Z) by considering Keri*. In order to investigate Keri*, we will determine H*(E8=T ; Z) up to degrees 36. First we need a simple lemma. Lemma 3.2. For the elements t and ci= oei(t1, . .,.t8) 2 H*(BT ; Z), we have (i) Sq2(c2) c3+ tc2, Sq4(c3) c5+ tc4+ c2c3, Sq8(c5+ tc4) tc8+ c2c7+ c3c6+ c4c5+ tc24+ t2c7+ t3c6+ t2c2c5 + t2c3c4, Sq14(c8+ c24+ t2c6+ t4c4+ t8) (c8+ t2c6+ t4c4+ t8)(c7+ tc6) mod 2. (ii)P1(c2- t2) c4+ c22+ t4, P3(c4- t4) c25+ 2c4c6+ 2c3c7+ 2c2c8+ c23c4+ c2c24+ c22c6+ 2c2c3c5 + 2t10 mod 3. (iii)P1(c2+ t2) c6+ 2c23+ 4c2c4+ 2c32+ 2tc5+ tc2c3+ 4t2c4+ 4t2c22 + 3t3c3+ t4c2+ 2t6 mod 5. Proof.(i) follows immediately from the Wu formula: Sq2i-2(ci) i-1X c2i-1-jcj and c1 = 3t t mod 2. j=0 (ii) Put pi= ti1+ . .+.ti8 (power sum). Then pi's and ci's are related to each other by the Newton formula: n-1X pn = (-1)i-1pn-ici+ (-1)n-1ncn. i=1 In particular, considering with mod 3 coefficients, we have p1 c1 0, p2 c2, 19 p4 2c4+ 2c22, p10 2c25+ c4c6+ c3c7+ c2c8+ 2c23c4+ 2c2c24+ 2c22c6+ c32c4+ c2c3c5 + c52. On the other hand, we have _ ! X X X X P1(p2) P1 t2i P1(t2i) 2tiP1(ti) 2ti. t3i i i i i X 2t4i 2p4, i _ ! X X X P3(p4) P3 t4i P3(t4i) (2P3(t2i)t2i+ 2P2(t2i)P1(t2i)) i i i X X (2t6i. 2t4i) t10i p10. i i Using these facts, we have easily the required results. (iii) Similar computation yields the required results. Lemma 3.3. The integral cohomology ring of E8=T for degrees 36 is given as follows: H*(E8=T ; Z)= Z[t1, . .,.t8, t, fl3, fl4, fl5, fl6, fl9, fl10, fl15] =(ae1, ae2, ae3, ae4, ae5, ae6, ae8, ae9, ae10, ae12, ae14, ae1* *5, ae18), where t1, . .,.t8, t 2 H2 are as in x2, fli 2 H2i(i = 3, 4, 5, 6, 9, 10, 15) and ae1= c1- 3t, ae2= c2- 4t2, ae3= c3- 2fl3, ae4= c4+ 2t4- 3fl4, ae5= c5- 3tfl4+ 2t2fl3- 2fl5, ae6= c6- 2fl32- tfl5+ t2fl4- 2t6- 5fl6, ae8= -3c8+ 3fl42- 2fl3fl5+ t(2c7- 6fl3fl4) + t2(2fl32- 5fl6) + 3t3fl5 + 4t4fl4- 6t5fl3+ t8, ae9= 2c6fl3+ tc8+ t2c7- 3t3c6- 2fl9, ae10= fl52- 2c7fl3- t2c8+ 3t3c7- 3fl10, 20 ae12= 15fl62+ 2fl3fl4fl5- 2c7fl5+ 2fl34+ 10fl32fl6- 3c8fl4- 2fl43 + t(c8fl3- 2fl32fl5+ 4c7fl4+ 6fl3fl42) + t2(3fl10- 25fl4fl6- c7fl3- 16fl32fl* *4) + t3(25fl3fl6- 3fl4fl5+ 10fl33) + t4(3c8+ 3fl3fl5+ 5fl42) + t5(-3c7- 5fl3fl4) + 4t6fl32- 7t8fl4+ 4t9fl3, ae14= c27- 3c8fl6+ 6fl4fl10- 4c8fl32+ 6c7fl3fl4- 6fl32fl42- 12fl42fl6- 2fl3fl5f* *l6 + t(24fl3fl4fl6- 8c7fl32- 8c7fl6+ 4c8fl5- 6fl3fl10+ 12fl33fl4) + t2(-2fl3fl4fl5+ 6fl43+ 2fl32fl6+ 20fl62- 4fl34- c7fl5) + t3(-12fl3fl42+ 8c8fl3- 5c7fl4+ 3fl5fl6) + t4(3fl10- 26fl4fl6+ 6c7fl3- 4fl3* *2fl4) + t5(24fl3fl6+ 3fl4fl5+ 12fl33) + t6(-6c8+ 2fl42) - 4t7c7+ t8(6fl6- 6fl32) - 6t10fl4+ 12t11fl3- 2t14, ae15= (c8- t2c6+ 2t3fl5+ 3t4fl4- t8)(c7- 3tc6) - 2(fl32+ c6)(fl9- c6fl3) - 2fl1* *5, ae18= -fl29- 27c8fl10- 18fl24fl10+ 4fl33fl9+ 10fl3fl6fl9+ 6fl3fl5fl10- 6fl3fl4f* *l5fl6 - 18c7fl3fl24+ 9c8fl4fl6+ 3c8fl23fl4+ 18fl23fl34+ 36fl34fl6+ 6c27fl4+ 6c7fl2* *3fl5 - 6fl33fl4fl5+ 6c7fl5fl6- 4fl63- 30fl36- 26fl43fl6- 55fl23fl26- 27c7c8fl3 + t(2fl3fl5fl9- 72c7fl4fl6+ 24c8fl4fl5+ 12c7fl23fl4+ 12c7fl10+ c8fl9+ 6c27fl3 - 8c8fl3fl6+ 36fl3fl4fl10- 108fl3fl24fl6- 4fl23fl5fl6- 5c8fl33+ 2fl43fl5- 54* *fl33fl24) + t2(88fl43fl4- 7c28+ c7fl9- 45fl23fl10- 2fl3fl4fl9- 10c8fl3fl5+ 28c7fl3fl6-* * 9c7fl4fl5 + 225fl4fl26- 18fl44- 27c8fl24+ 283fl23fl4fl6- 39fl6fl10+ 12fl3fl24fl5- 19c7* *fl33) + t3(-9fl5fl10- 150fl3fl26- 165fl33fl6- 6fl23fl9- 10c7fl3fl5+ 63c7fl24+ 43c7* *c8 + 9fl4fl5fl6- fl23fl4fl5+ 54fl3fl34- 15fl6fl9+ 46c8fl3fl4- 42fl53) + t4(-9c8fl6+ 6fl4fl10- 16c27- 103fl23fl24- 3fl5fl9+ 36fl3fl5fl6- 135fl24fl6 - 119c7fl3fl4+ 9fl33fl5- 19c8fl23) + t5(39c7fl6- 18fl24fl5+ 117c7fl23+ 195fl3fl4fl6- 12c8fl5+ 3fl4fl9+ 36fl3fl1* *0+ 87fl33fl4) + t6(33fl34+ 4fl3fl9+ 18c7fl5+ 3fl3fl4fl5- 32fl43- 62fl23fl6) + t7(-87fl3fl24- 16fl23fl5- 6c7fl4- 45fl5fl6+ 4c8fl3) + t8(-39fl10+ 115fl23fl4- 77c7fl3+ 81fl4fl6) + t9(-6fl9+ 18fl4fl5- 30fl33- 57fl3fl6) + t10(-9c8- 27fl24+ 9fl3fl5) + t11(33fl3fl4+ 48c7) - 34t12fl23- 18t13fl5+ 9t14fl4+ 12t15fl3- 6t18. 21 Proof.According to Toda ([17], Proposition 3.2), one can give the gen- eral description of H*(E8=T ; Z) as follows: H*(E8=T ; Z) =Z[t1, . .,.t8, t, fl3, fl4, fl5, fl6, fl9, fl10, fl15] =(ae1, ae2, ae3, ae4, ae5, ae6, ae8, ae9, ae10, ae12, ae14, ae15, * *ae18, ae20, ae24, ae30), where t1, . .,.t8, t 2 H2 as above and ae1= c1- 3t, aei= ffii- 2fli(i = 3, 5, 9, 15), aei= ffii- 3fli(i = 4, 10), ae6= ffi6- 5fl6. Here ffii(i = 3, 4, 5, 6, 9, 10, 15) is an arbitrary element satisfying ffi3 Sq2(ae2), ffi5 Sq4(ffi3), ffi9 Sq8(ffi5), ffi15 Sq14(ae8) mod 2, ffi4 P1(ae2), ffi10 P3(ffi4) mod 3, ffi6 P1(ae2) mod 5. Other relation aej (j = 2, 8, 12, 14, 18, 20, 24, 30) is determined by the maximality of the integer nj in (3.1) nj. aej '*0(Ij) mod (aei; i < j). Now let us determine the generators and the relations explicitly; (1) In view of Lemma 2.3 (i) and (3.1), we can take ae2 = c2- 4t2. (2) By Lemma 3.2 (i), we have ffi3 Sq2(ae2) Sq2(c2) c3+ tc2 c3 mod (2, ae1, ae2) and we can take ffi3 = c3 so that ae3 = c3- 2fl3. (3) By Lemma 3.2 (ii), we have ffi4 P1(ae2) P1(c2- 4t2) c4+ c22+ t4 c4+ 2t4 mod (3, ae1, ae2) and we can take ffi4 = c4+ 2t4 so that ae4 = c4+ 2t4- 3fl4. (4) By Lemma 3.2 (i), we have ffi5 Sq4(ffi3) Sq4(c3) c5+ tc4+ c2c3 c5+ tc4 mod (2, ae1, ae2, ae3, ae4) c5- 3tfl4+ 2t2fl3 mod (2, ae1, ae2, ae3, ae4) and we can take ffi5 = c5- 3tfl4+ 2t2fl3 so that ae5 = c5- 3tfl4+ 2t2fl3- 2fl5. 22 (5) By Lemma 3.2 (iii), we have ffi6 P1(ae2) P1(c2- 4t2) c6+ 2c23+ 4c2c4+ 2c32+ 2tc5+ tc2c3 + 4t2c4+ 4t2c22+ 3t3c3+ t4c2+ 2t6 mod (5, ae1) c6+ 3fl23+ 4tfl5+ t2fl4+ 3t6 mod (5, ae1, ae2, ae3, ae4, ae5) c6- 2fl23- tfl5+ t2fl4- 2t6 mod (5, ae1, ae2, ae3, ae4, ae5) and we can take ffi6 = c6- 2fl23- tfl5+ t2fl4- 2t6 so that ae6 = c6- 2fl23- tfl5+ t2fl4- 2t6- 5fl6. (6) By Lemma 2.3 (ii), we have I8 214. 3 . 5{-18c8- 3c3c5+ 2c24+ t(12c7- 3c3c4) + t2(-6c6+ 3c23) + 12t3c5+ 2t4c4- 12t5c3+ 14t8} mod (I2) 215. 32. 5{-3c8+ 3fl24- 2fl3fl5+ t(2c7- 6fl3fl4) + t2(2fl23- 5fl6) + 3t3fl5+ 4t4fl4- 6t5fl3+ t8} mod (aei; i < 8). Hence, by (3.1), we have 215. 32. 5 ae8 '*0(I8) mod (aei; i < 8) and it follows the form of ae8. (7) By Lemma 3.2 (i) , we have ffi9 Sq8(ffi5) Sq8(c5+ tc4) tc8+ c2c7+ c3c6+ c4c5+ tc24+ t2c7 + t3c6+ t2c2c5+ t2c3c4 mod (2, ae1) tc8+ t2c7+ t3c6 mod (2, aei; i < 9) tc8+ t2c7- 3t3c6+ 2c6fl3 mod (2, aei; i < 9) and we can take ffi9 = tc8+ t2c7- 3t3c6+ 2c6fl3 so that ae9 = tc8+ t2c7- 3t3c6+ 2c6fl3- 2fl9. (8) By Lemma 3.2 (ii), we have ffi10 P3(ffi4) P3(c4+ 2t4) c25+ 2c4c6+ 2c3c7+ 2c2c8+ c23c4 + c2c24+ c22c6+ 2c2c3c5+ 2t10 mod (3, ae1) fl25+ c7fl3+ 2t2c8 mod (3, aei; i < 10) fl52- 2c7fl3+ 3t3c7- t2c8 mod (3, aei; i < 10) and we can take ffi10= fl25- 2c7fl3+ 3t3c7- t2c8 so that ae10= fl25- 2c7fl3+ 3t3c7- t2c8. 23 (9) By (2.6), (2.5) and (2.7), we obtain ae 3 2 1 2 3 1 2 1 4 I12 218. 34. 5 . 7 _c6 - c5c7- c4c8+ _c3c4c5- __c4- __c3c6+ __c3 5 6 27 10 40 ` ' 1 7 3 1 2 1 2 + t _c3c8+ _c4c7- _c5c6- _c3c5+ _c3c4 2 3 5 5 6 ` ' 2 2 5 2 1 2 + t2 _c5- _c3c7- _c4c6- _c3c4 5 2 3 6 ` ' ` ' ` ' 19 2 1 3 4 1 2 19 5 14 1 + t3 __c3c6- _c4c5- _c3 + t -_ c4+ __c3c5 + t __c7+ _c3c4 10 3 5 9 30 3 2 ` ' oe 56 23 2 2 7 5 8 22 9 154 12 +t6 -__ c6+ __c3 - __t c5- _t c4- __t c3+ ___t mod (I2) 15 30 15 9 15 135 218. 34. 5 . 7{15fl62+ 2fl3fl4fl5- 2c7fl5+ 2fl34+ 10fl32fl6- 3c8fl4- 2fl43 + t(c8fl3- 2fl32fl5+ 4c7fl4+ 6fl3fl42) + t2(3fl10- 25fl4fl6- c7fl3- 16fl32fl4) + t3(25fl3fl6- 3fl4fl5+ 10fl33) + t4(3c8+ 3fl3fl5+ 5fl42) + t5(-3c7- 5fl3fl4) + 4t6fl32- 7t8fl4+ 4t9fl3} mod (aei; i < 12). Hence, by (3.1), we have 218. 34. 5 . 7 ae12 '*0(I12) mod (aei; i < 12) and it follows the form of ae12. Quite similarly, we have 220. 32. 52. 7 . 11 ae14'*0(I14) mod (aei; i < 14), 226. 33. 52. 7 . 13 ae18'*0(I18) mod (aei; i < 18) and it follows the forms of ae14and ae18. (10) Finally, we will determine the relation ae15. Since ae8= -3c8+ 3fl42- 2fl3fl5+ t(2c7- 6fl3fl4) + t2(2fl32- 5fl6) + 3t3fl5 + 4t4fl4- 6t5fl3+ t8 c8+ c24+ t2c6+ t4c4+ t8 mod (2, ae1, ae2, ae3, ae4, ae5, ae6), we have, by Lemma 3.2 (i), ffi15 Sq14(ae8) Sq14(c8+ c24+ t2c6+ t4c4+ t8) (c8+ t2c6+ t4c4+ t8)(c7+ tc6) mod (2, ae1) (c8- t2c6+ 2t3fl5+ 3t4fl4- t8)(c7- tc6) - 2(c6+ fl23)(fl9- c6fl3) mod (2, aei; i < 15) 24 and we can take ffi15= (c8- t2c6+ 2t3fl5+ 3t4fl4- t8)(c7- tc6) - 2(c6+ fl23)(fl9- c6fl3) so that ae15= (c8-t2c6+2t3fl5+3t4fl4-t8)(c7-tc6)-2(c6+fl23)(fl9-c6fl3)-2fl15. Consequently, we have established the lemma. In order to determine Keri*, we need the result on H*(E7=T 0; Z) in ([15], Theorem 5.9), which can be restated as follows: Theorem 3.4. The integral cohomology ring of E7=T 0is given as follows: H*(E7=T 0; Z)= Z[t01, . .,.t07, t0, fl03, fl04, fl05, fl09] =(ae01, ae02, ae03, ae04, ae05, ae06, ae08, ae09, ae010, ae01* *2, ae014, ae018), where t01, . .,.t07, t02 H2 are as in x2, fl0i2 H2i(i = 3, 4, 5, 9) and ae01= c01- 3t0, ae02= c02- 4t02, ae03= c03- 2fl03, ae04= c04+ 2t04- 3fl04, ae05= c05- 3t0fl04+ 2t02fl03- 2fl05, ae06= fl032+ 2c06- 2t0fl5- 3t02fl04+ t06, ae08= 3fl042- 2fl03fl05+ t0(2c07- 6fl03fl04) - 9t02c06+ 12t03fl05+ 15t04fl04- * *6t05fl03- t08, ae09= 2c06fl03+ t02c07- 3t03c06- 2fl09, ae010= fl052- 2c07fl03+ 3t03c07, ae012= 3c026- 2fl043- 2c07fl05+ 2fl03fl04fl05+ t0(4c07fl04- 2c06fl05+ 6fl03fl04* *2) + t02(-3c07fl03+ 3c06fl04) + t03(-12fl04fl05+ 5c06fl03) + t04(-2fl03fl05- 15* *fl042) - 10t06c06+ 12t07fl05+ 19t08fl04- 6t09fl03- 2t012, ae014= c072+ 6c07fl03fl04- 2c06fl03fl05- t02c07fl05+ t03(-9c07fl04+ 3c06fl05) -* * 6t04c07fl03 + 9t07c07, ae018= -fl092+ 2c06c07fl05+ 6c07fl03fl042- 2c072fl04- 2c06fl03fl04fl05+ 2c06fl0* *3fl09 + t0(-6c072fl03+ 24c06c07fl04) + t02(-25c07fl04fl05+ c07fl09- 18c06c07fl03) + t03(-45c07fl042+ 20c07fl03fl05+ 3c06fl04fl05- 3c06fl09) 25 + t04(11c027+ 2c06fl03fl05+ 48c07fl03fl04) + 51t05c06c07- 53t06c07fl05 + t07(-69c07fl04- 3c06fl05) + 16t08c07fl03+ 15t011c07. Remark 3.5. Using the result in [15], we expressed the relations ae012, ae014 and ae018in terms of the generators t01, . .,.t07, t0, fl03, fl04, fl05, fl09. Corollary 3.6. For the induced homomorphism i* : H*(E8=T ; Z) -! H*(C8=T ; Z) ~=H*(E7=T 0; Z), we obtain that Keri* = (u, "fl6, fl10, fl15), where "fl6= 2fl6+ fl23- t2fl4+ t6. Proof.By (2.2), we have (3.2) i*(ti) = t0i(1 i 7), i*(t8) = 0, i*(t) = t0 and therefore (3.3) i*(cn) = c0n(1 n 7), i*(c8) = 0. Then it is verified directly that i*(fli)= fl0i(i = 3, 4, 5, 9), i*(fl6)= c06- t0fl50- t02fl40, (3.4) i*(fl10)= 0, i*(fl15)= 0. Now we put I = (u, "fl6, fl10, fl15), the ideal of H*(E8=T ; Z) generated by the elements in the parenthesis. Using (3.2), (3.4) and Theorem 3.4, we see that I is contained in Keri*. Hence there is an induced map H*(E8=T ; Z)=I -! H*(C8=T ; Z) ~=H*(E7=T 0; Z). Then, by Lemma 3.3, we have H*(E8=T ; Z)=I =Z[t1, . .,.t8, t, fl3, fl4, fl5, fl6, fl9, fl10, fl15] =(u, ae1, ae2, ae3, ae4, ae5, ae6, "fl6, ae8, ae9, ae10, fl10, a* *e12, ae14, ae15, fl15, ae18) = Z[t1, . .,.t7, t, fl3, fl4, fl5, fl9] =(ae1, ae2, ae3, ae4, ae5, "fl6, ae8, ae9, ae10, ae12, ae14, ae1* *8) for degrees 36. Then it follows from Lemma 3.3, Theorem 3.4, (3.2), (3.3) and (3.4) that i*(aei) ae0i(i = 1, 2, 3, 4, 5, 8, 9, 10, 12, 14, 18), i*("fl6) ae06. 26 Therefore this map induces an isomorphism and the assertion follows. 3.3. Generators of H*(E8=T 1. E7; Z). From Corollary 3.6, we see that H*(E8=C8; Z) is generated by some four elements "u2 H2, "v2 H12, w"2 H20 and "x2 H30 such that (3.5) ("u, "v, "w, "x) = (u, "fl6, fl10, fl15) as ideals. So our next task is to describe these generators in the ring H*(E8=T ; Z). Hereafter we identify H*(E8=C8; Z) with the subalgebra Im p* of H*(E8=T ; Z). Firstly, by Lemma 2.7 and (2.14), we have (3.6) "J6= ___1____J6 210. 32. 5 ` ' ` ' 2 1 2 1 1 1 1 2 _c6+ __c3+ c5 -_ t - _u + c4 _tu + _u 5 20 5 2 2 3 ` ' 1 3 1 2 1 3 1 6 5 1 4 2 3 3 2 4 5 +c3 -_ t - _t u - _u + _t + t u - _t u + t u + t u - tu 5 2 2 5 3 273 6 +___ u mod (I2); 640 (3.7) "J10= _____1_____J10 214. 33. 5 . 7 ` ' ` ' 1 2 1 1 1 1 1 2 23 2 __c5- _c3c7+ _c3c6u + c4c5 -_ t - _u + c3c5 _t - __u 12 3 2 6 6 6 84 ` ' ` ' 1 2 1 1 2 2 1 4 23 2 2 37 4 +c24 __t + _tu + __u + c3 __t + __t u - __u 12 6 14 12 84 96 ` ' 1 3 1 2 23 2 1 3 +c3c4 -_ t - _t u - __tu + _u ` 6 6 84 '3 ` ' 1 2 17 2 5 3 3 3 3 2 2 1 3 37 4 +c7 t3+ _t u - __tu + __u + c6 -_ t u - __t u + _tu - __u 6 42 14 2 14 2 84 ` ' 1 5 1 4 23 3 2 1 2 3 1 4 71 5 +c5 -_ t + _t u + __t u - _t u + __tu + ___u 3 6 21 3 24 336 ` ' 1 6 1 5 17 4 2 2 3 3 1 2 4 5 5 131 6 +c4 _t + _t u + __t u - _t u - _t u + __tu - ___u 3 6 42 3 3 16 504 ` ' 1 7 1 6 23 5 2 1 4 3 37 3 4 73 2 5 3 6 239 7 +c3 -_ t + _t u - __t u - _t u + __t u + __t u - _tu + ___u 3 6 21 3 24 48 2 336 27 1 10 1 9 7 8 2 2 7 3 7 6 4 35 5 5 233 4 6 7 3 7 +_ t - _t u + _t u + _t u - _t u - __t u + ___t u + __t u 3 3 6 3 8 8 72 24 129 2 8 215 9 208601 10 -___ t u + ___tu - ______u mod (I2). 56 168 645120 On the other hand, by Lemma 3.3, we have c3 = 2fl3, c4 = 3fl4- 2t4, (3.8) c5 = 2fl5+ 3tfl4- 2t2fl3, c6 = 5fl6+ 2fl32+ tfl5- t2fl4+ 2t6 in H*(E8=T ; Z) ,! H*(E8=T ; Q). Substituting (3.8) into (3.6), we have J"6 = ___1____J6 210. 32. 5 (3.9) = 2fl6+ fl32- ufl5+ fl4(-t2+ u2) - u3fl3+ t6- t4u2 273 6 +t3u3+ t2u4- tu5+ ___u . 640 Similarly, substituting (3.8) into (3.7) and using the relations ae8, ae9 and ae10, we have (3.10) "J10= _____1_____J10 214. 33. 5 . 7 = fl10+ ufl9- u3c7- ufl4fl5+ 2u2fl42- 2u2fl3fl5+ fl3fl4(-6tu2+ 2u3) ` ' ` ' 7 4 2 2 3 55 4 +fl32 2t2u2+ 2tu3+ __u + fl6 -5t u + 5tu + __u 24 12 ` ' 55 5 +fl5 t4u + 3t3u2+ t2u3- __u 24 ` ' 103 2 4 5 79 6 +fl4 6t4u2- 3t3u3- ___t u - tu + __u 24 24 ` ' 31 7 +fl3 -6t5u2- 2t4u3+ 4t3u4+ 6t2u5- 4tu6- __u 24 55 6 4 5 5 7 4 6 79 3 7 31 2 8 55 9 +4t7u3+ __t u - 6t u - __t u + __t u + __t u - __tu 24 24 24 24 24 666919 10 +______u . 645120 Now let us determine our generators "u, "v, "wand "x; Obviously, we can take u = t8 as our generator "u. 28 Next, since H*(E8=C8; Q) is generated by u, J6 and J10(see 3.1), we can put v"= ffJ"6+ fiu6 for some ff, fi 2 Q. On the other hand, by (3.5), we can express "v= "fl6+ f for some element f 2 (u) \ H12(E8=T ; Z). Then using (3.9), we have 2fl6+ fl32- t2fl4+ t6+ f = ff(2fl6+ fl32- t2fl4+ t6) ` ' 273 6 + ff(-ufl5+ u2fl4- u3fl3- t4u2+ t3u3+ t2u4- tu4) + ___ff + fiu , 640 273 and we can take ff = 1, fi = -___ . Thus we see that 640 (3.11) 1 273 6 v = ________J6- ___u 210. 32. 5 640 = 2fl6+ fl32- ufl5+ fl4(-t2+ u2) - u3fl3+ t6- t4u2+ t3u3+ t2u4- tu5 can be chosen as our generator "v. Similarly, we can put w"= ~J"10+ ~u4v + u10 for some ~, ~, 2 Q. On the other hand, by (3.5), we can express "w= fl10+ g for some element g 2 (u, "fl6) \ H20(E8=T ; Z). Then using (3.10), we can take ~ = 1 and hence fl10+ g= fl10+ ufl9- u3c7- ufl4fl5+ 2u2fl42- 2u2fl3fl5+ fl3fl4(-6tu2+ 2u3) ae ` ' oe 7 4 + fl322t2u2+ 2tu3+ ~ + __ u 24 ae ` ' oe 55 4 + fl6 -5t2u2+ 5tu3+ 2~ + __ u 12 ae ` ' oe 55 5 + fl5 t4u + 3t3u2+ t2u3+ -~ - __ u 24 ae ` ' ` ' oe 103 2 4 5 79 6 + fl4 6t4u2- 3t3u3+ -~ - ___ t u - tu + ~ + __ u 24 24 ae ` ' oe 31 7 + fl3 -6t5u2- 2t4u3+ 4t3u4+ 6t2u5- 4tu6+ -~ - __ u 24 29 ` ' ` ' ` ' 55 6 4 5 5 7 4 6 79 3 7 + 4t7u3+ ~ + __ t u - 6t u + -~ - __ t u + ~ + __ t u 24 24 24 ` ' ` ' ` ' 31 2 8 55 9 666919 10 + ~ + __ t u + -~ - __ tu + + ______ u , 24 24 645120 55 666919 and we can take ~ = -__ , = -______. Thus we see that 24 645120 (3.12) 1 55 4 666919 10 w = ___________J10- __u v - ______u 214. 33. 5 . 7 24 645120 = fl10+ ufl9- u3c7- ufl4fl5+ 2u2fl42- 2u2fl3fl5+ fl3fl4(-6tu2+ 2u3) +fl32(2t2u2+ 2tu3- 2u4) + fl6(-5t2u2+ 5tu3) + fl5(t4u + 3t3u2+ t2u3) +fl4(6t4u2- 3t3u3- 2t2u4- tu5+ u6) +fl3(-6t5u2- 2t4u3+ 4t3u4+ 6t2u5- 4tu6+ u7) +4t7u3- 6t5u5+ 2t4u6+ t3u7- t2u8 can be chosen as our generator "w. Finally, we have to find an element x of degree 30 such that x fl15 mod (u, v, w) in H*(E8=T ; Z). Consider the element 1 15 30 x = _u inH (E8=T ; Q). 2 In fact, it can be shown that x is an integral cohomology class and is contained in Imp*. Furthermore, we can check directly that x fl15 mod (u, v, w) (see appendix). Hence the element x can be chosen as our generator "x. Remark 3.7. We can describe the element x explicitly in the ring H*(E8=T ; Z). But it is too lengthy to write down here, so we will give the explicit form of x in the appendix. 3.4. Integral cohomology ring of E8=T 1.E7. Using the element x, we can rewrite "I30: "I30= -9u30- 24u24v - 12u20w + 36u14vw - 40u12v3 - 12u10w2 + 120u8v2w - 140u6v4 + 24u4vw2 - 40u2v3w - 16v5 - 8w3 = -36x2- 48u9vx - 24u5wx + 36u14vw - 40u12v3 - 12u10w2 + 120u8v2w - 140u6v4 + 24u4vw2 - 40u2v3w - 16v5 - 8w3 = 4(-9x2- 12u9vx - 6u5wx + 9u14vw - 10u12v3 - 3u10w2 + 30u8v2w - 35u6v4 + 6u4vw2 - 10u2v3w - 4v5 - 2w3). 30 Therefore, in view of Lemma 3.1, we obtain the following main result: Theorem 3.8. The integral cohomology ring of E8=T 1.E7 is given as follows: H*(E8=T 1.E7; Z) = Z[u, v, w, x]=(r15, r20, r24, r30), where degu = 2, degv = 12, degw = 20, degx = 30 and r15= u15- 2x, r20= 9u20+ 45u14v + 12u12w + 60u8v2 + 30u4vw + 10u2v3 + 3w2, r24= 11u24+ 60u18v + 21u14w + 105u12v2 + 60u8vw + 60u6v3 + 9u4w2 + 30u2v2w + 5v4, r30= -9x2- 12u9vx - 6u5wx + 9u14vw - 10u12v3 - 3u10w2 + 30u8v2w - 35u6v4 + 6u4vw2 - 10u2v3w - 4v5 - 2w3. In order to determine the integral cohomology ring of E8=E7, we consider the Gysin exact sequence associated with the following circle bundle (3.13) S1 -! E8=E7 -ss!E8=T 1.E7, where ss is the natural projection. In this case, it reduces to the fol- lowing short exact sequeces: (3.14) 0 -! Hodd(E8=E7; Z) -! H*(E8=T 1.E7; Z) -xu!H*(E 1 ss*even 8=T .E7; Z) -! H (E8=E7; Z) -! 0, where Heven= i 0H2iand Hodd= i 0H2i+1. From the exactness of (3.14), it follow that Heven(E8=E7; Z) is iso- morphic to H*(E8=T 1.E7; Z)=(u). Define the elements zi(i = 12, 20, 30) of H*(E8=E7; Z) as follows: z12= ss*(v), z20= ss*(w), z30= ss*(x). Then, by Theorem 3.8, we obtain Heven(E8=E7; Z)= Z[z12, z20, z30]=(2z30, 3z202, 5z124, 4z125+ 2z203+ 9z302). = Z[z12, z20, z30]=(2z30, 3z202, 5z124, z125+ z203+ z302). By Poincar'e duality there exist elements zi 2 Hi(E8=E7; Z) (i = 59, 71, 79, 83, 91, 95, 103, 115) such that z312z20z59= z212z20z71= z312z79= z12z20z83= z212z91= z20z95= z12z103= z115. 31 Then it is not hard to show that z71= z12z59, z79= z20z59, z83= z212z59, z91= z12z20z59, z103= z212z20z59, z115= z312z20z59. Summing up the results, we obtain the following: Corollary 3.9. The structure of H*(E8=E7; Z) is given by the following table: ____________________________________|k| | |_nontrivial_H_(E8=E7;_Z)b|asis_elements_||0|| |________H__=_Z________|____1______|_|12|| |________H__=_Z________|____z12_____||20|| |________H__=_Z________|____z20_____||24|2| |________H__=_Z________|___z12______||30|| |_______H__=_Z2_______|_____z30_____||32|| |________H__=_Z________|___z12z20___||36|3| |________H__=_Z________|___z12______||40|2| |_______H__=_Z3_______|____z20______||42|| |_______H__=_Z2_______|____z12z30___||44|2| |________H__=_Z________|__z12z20____||48|4| |_______H__=_Z5_______|____z12______||50|| |_______H__=_Z2_______|____z20z30___||52|2| |_______H__=_Z3_______|___z12z20____||54|2| |_______H__=_Z2_______|___z12z30____||56|3| |________H__=_Z________|__z12z20____||59|| |________H__=_Z________|____z59_____||62|| |_______H__=_Z2_______|__z12z20z30__||64|22| |_______H__=_Z3_______|____z12z20___||66|3| |_______H__=_Z2_______|____z12z30___||68|4| |_______H__=_Z5_______|____z12z20___||71|| |________H__=_Z________|___z12z59___||74|2| |_______H__=_Z2_______|__z12z20z30__||76|32| |_______H__=_Z3_______|____z12z20___||79|| |________H__=_Z________|___z20z59___||83|2| |________H__=_Z________|__z12z59____| 32 ____________________________________|k| | |_nontrivial_H_(E8=E7;_Z)b|asis_elements_||86|3| |_______H__=_Z2_______|__z12z20z30__||91|| |________H__=_Z________|_z12z20z59__||95|3| |________H__=_Z________|__z12z59____||103|2| |_______H___=_Z_______|__z12z20z59__||115|3| |_______H___=_Z_______|__z12z20z59__ | 4.appendix In 3.3, we defined the element x as a rational cohomology class given by 1 15 30 x = _u in H (E8=T ; Q). 2 We need to show that x is in fact an integral cohomology class. By Lemma 3.3, the following relations hold in H*(E8=T Z): c1 = 3t, c2 = 4t2, c3 = 2fl3, (4.1) c 4 4 = 3fl4- 2t , c5 = 2fl5+ 3tfl4- 2t2fl3, c6 = 5fl6+ 2fl23+ tfl5- t2fl4+ 2t6. Note that, by (2.11) and (4.1), the following relation holds: (4.2) c8 = uc7- 5u2fl6- 2u2fl23+ (-tu2+ 2u3)fl5+ (t2u2+ 3tu3- 3u4)fl4 +(-2t2u3+ 2u5)fl3- 2t6u2+ 2t4u4- 4t2u6+ 3tu7- u8. Using (4.2), we can rewrite the higher relations ae8, ae9, ae10, ae12, ae14and ae15. For example, ae8= -3c8+ 3fl24- 2fl3fl5+ t(2c7- 6fl3fl4) + t2(2fl23- 5fl6) + 3t3fl5 +4t4fl4- 6t5fl3+ t8 = 3fl24- 2fl3fl5+ (2t - 3u)c7- 6tfl3fl4+ (-5t2+ 15u6)fl6 +(2t2+ 6u2)fl23+ (3t3+ 3tu2- 6u3)fl5+ (4t4- 3t2u2- 9tu3+ 9u4)fl4 +(-6t5+ 6t2u3- 6u5)fl3+ t8+ 6t6u2- 6t4u4+ 12t2u6- 9tu7+ 3u8. 33 Using the relations aei(i = 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15), we rewrite 1 15 the element x = _u as follows: 2 1 15 x = _u 2 1 15 3 4 2 3 = _ u - ae15+ uae14- u ae12+ (t u - t u )ae10 2 +(fl6+ tfl5+ ufl5+ u2fl4+ u3fl3+ t6+ t4u2+ t3u3+ t2u4+ tu5+ u6)ae9 3 2 4 -39u7ae8+ (-tufl3+ 2t u - 5tu )ae10 = fl15- 20fl3fl62+ 3fl32fl9- 23fl33fl6- 6fl35+ 4fl6fl9 + 3ufl4fl10- ufl5fl9- 3ufl32fl42+ 3uc7fl3fl4- 6ufl42fl6+ (-3t + 2u)fl33fl5 + (-4t + 4u)fl3fl5fl6 + (-t2- u2)fl4fl9+ (t2+ tu - u2)c7fl32+ (9t2+ 12tu + 5u2)fl3fl4fl6 + (5t2+ 6tu + 2u2)fl33fl4+ (3t2+ 4tu + u2)c7fl6 + (-6t3- 2t2u - 6tu2+ 5u3)fl34- u3fl3fl9+ (3t2u + u3)fl43+ (2t2u + 3tu2)c7fl5 + (-45t3+ 10t2u - 40tu2)fl62+ (t3- 2t2u + tu2- u3)fl3fl4fl5 + (-33t3+ t2u - 31tu2+ 13u3)fl32fl6 + (-2t4- 4t3u - 3tu3+ 3u4)c7fl4+ (-9t4- 6t3u - 18t2u2+ 5tu3- 3u4)fl5fl6 + (-3t4- 3t3u - 7t2u2+ 5tu3- 4u4)fl32fl5+ (-t4- 6t3u - t2u2- 3tu3)fl3fl42 + (-3t4u - 6t3u2+ 3t2u3+ 15tu4)fl10+ (-3t4u + t3u2+ 5t2u3+ 10tu4- u5)c7fl3 + (15t5- 2t4u + 3t3u2+ 14t2u3- 16tu4+ 3u5)fl32fl4 + (39t5- 13t4u + 8t3u2+ 35t2u3- 31tu4- 3u5)fl4fl6 + (t6- t4u2- t3u3- t2u4- tu5- u6)fl9 + (-13t6+ 12t5u + 5t4u2- 56t3u3+ 8t2u4+ 21tu5+ 2u6)fl3fl6 + (6t6+ 3t5u + 2t4u2+ 7t3u3+ t2u4- 8tu5+ 3u6)fl4fl5 + (-8t6+ 6t5u + 2t4u2- 22t3u3+ 6t2u4+ 8tu5- 2u6)fl33 + (-6t7+ t6u - 7t4u3+ 5t3u4+ 3t2u5+ 3tu6- 63u7)fl42 + (-t7+ 2t6u + t5u2- 11t4u3+ 6t3u4+ 5t2u5+ 6tu6+ 39u7)fl3fl5 + (2t8+ 6t7u + 3t6u2- 4t5u3- 15t4u4+ 6t3u5+ 3t2u6- 40tu7+ 59u8)c7 + (3t8+ t6u2+ 11t5u3+ 14t4u4- 20t3t5- 4t2u6+ 118tu7+ 3u8)fl3fl4 + (-48t9+ 3t8u - 41t7u2+ 18t6u3+ 16t5u4- 13t4u5- 67t3u6+ 125t2u7 - 15tu8- 291u9)fl6 34 + (-18t9- 3t8u - 16t7u2+ 10t6u3- 4t5u4- 8t4u5- 16t3u6- 23t2u7- 10tu8 - 115u9)fl32 + (-6t10- 3t9u - 9t8u2+ 5t7u3- 5t6u4- 14t4u6- 52t3u7+ 6t2u8- 60tu9 + 117u10)fl5 + (18t11- 3t10u + 5t9u2+ 11t8u3- 28t7u4+ 8t6u5+ 20t5u6- 64t4u7- 15t3u8 + 54t2u9+ 178tu10- 177u11)fl4 + (-2t12+ 6t11u + 2t10u2- 20t9u3+ 11t8u4+ 22t7u5- 8t6u6+ 83t5u7 + 15t4u8+ 5t3u9- 116t2u10+ tu11+ 117u12)fl3 - 12t15- t14u - 10t13u2+ 6t12u3+ 7t11u4- 13t10u5- 31t9u6+ 9t8u7- t7u8 - 118t6u9- 18t5u10+ 131t4u11- 6t3u12- 233t2u13+ 175tu14- 58u15, which has shown that x is an integral cohomology class. 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