FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS DONALD M. DAVIS 1.Introduction The p-primary v1-periodic homotopy groups of a space X, denoted v-11ss*(X; p)* * or just v-11ss*(X), were defined in [15]. They are a localization of the actual ho* *motopy groups, telling roughly the portion which is detected by K-theory and its opera* *tions. If X is a compact Lie group or spherically resolved space, each v-11ssi(X; p) i* *s a direct summand of some actual homotopy group of X. By use of a combination of homotopy-theoretic and unstable Novikov spectral s* *e- quence (UNSS) methods, the groups v-11ss*(X; p) were computed by the author and coworkers for the following compact simple Lie groups: o X a classical group and p odd ([12]); o X an exceptional Lie group with H*(X; Z) p-torsion-free ([8]); o (SU(n) or Sp(n), 2) ([6],[7]); o (G2; 2) ([16]), (F4 or E6, 3) ([5]), and (E7; 3) ([14]). In [10], Bousfield takes a new approach to v1-periodic homotopy groups. He sh* *ows that if X is a 1-connected finite H-space with H*(X; Q) associative, and p is a* *n odd prime, then v-11ss*(X; p) can be obtained explicitly from (K*(X; bZp); p; r),* * where r is a generator of the group of units (Z=p2)x. We will review his result in Th* *eorem 4.1. Let X be any compact simple Lie group, and p an odd prime. In this paper, we will show how to compute the second exterior power 2 of generators of the representation ring R(X), and use this to find a set of generators of K*(X; bZp* *) on which the Adams operations k behave in a nice way. From this, we use Bousfield* *'s theorem to determine v-11ss*(X; p). This approach is totally algebraic. There i* *s no __________ 1991 Mathematics Subject Classification. 55T15. Key words and phrases. v1-periodic homotopy groups, exceptional Lie groups, representation theory. 1 2 D. DAVIS homotopy theory (except that which went into proving Bousfield's theorem) and no UNSS. We have used this approach to check the results obtained earlier by homotopy theory and the UNSS for (X; p) if X = G2, F4, E6, or E7, and p 3, and X = E8 and p 7. All results are in agreement, except for one minor mistake in [5] in v-11ss*(F4; 3), which will be discussed in Section 8. Also in Section 8 we wil* *l show how this new approach resolved two minor matters for (E7; 3) which had been left unresolved in [14]. In this paper, we will focus our attention on the calculation of v-11ss*(E8; * *5) and v-11ss*(E8; 3), both of which are new. A main impediment toward finding v-11ss** *(E8; 5) had been uncertainty about a product decomposition which had been claimed by Harper in 1974 in [18, 4.4.1(b)]. In 1987, Kono questioned Harper's proof, and * *Harper agreed to Kono that his proof was flawed. Our methods show that indeed Harper's claim was incorrect; the asserted product decomposition does not exist. This wi* *ll be explained more fully in Proposition 3.6. Our main results are as follows, but we feel that the new methods introduced * *to obtain them are of much more interest than the results themselves. Let p(-) den* *ote the exponent of p in an integer. Theorem 1.1. Let r = min(5(m - k) + r; k). Then v-11ss2m(E8; 5) v-11ss2m-1(E8; 5) ( 0 if m is even Z=5max(4;4;4;4)if m: 3 mod 4 If m 1 mod 4, then v-11ss2m(E8; 5) Z=5e Z=5, and v-11ss2m-1(E8; 5) Z=5e+1, where e = max(2; 2; 2): Theorem 1.2. If i 3; 4 mod 4, then v-11ssi(E8; 3) = 0. For any integer k, v-11ss4k+1(E8; 3) v-11ss4k+2(E8; 3) 3e; FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 3 where 8 >>min(7 + 3(k - 9 - 313); 21)if k 0 mod 9 >> >>6 if k 1 mod 3 >> < min(9 + (k - 11); 24) if k 2 mod 9 e = 3 >>min(7 + 3(k - 6 - 2 . 37); 15)if k 3; 6 mod 9 >> >>min(10 + (k - 14); 30) if k 5 mod 9 >> 3 : min(9 + 3(k - 8); 18) if k 8 mod 9. This completes the computation of v-11ss*(X; p) for all compact simple Lie gr* *oups X and odd primes p, a project which was suggested to the author by Mimura in 19* *89. The situation when p = 2, which was part of Mimura's suggested project, is much more delicate. The author hopes to be able to adapt Bousfield's theorem to the * *prime 2, but that work is still in very preliminary stages. There are many computations in this project which would be intractable to do * *by hand. Specialized software LiE ([25]) is used to determine the second exterior * *power operations in R(E8). A nontrivial algorithm was required to get this informatio* *n into the form of an 8 x 8 matrix of integers, some of them 16 digits long, which can* * be interpreted as giving 2 on a canonical basis of P K1(E8), the primitive elemen* *ts. This portion of the work will be described in Section 2. The eigenvalues of this matrix are 2e for e 2 R = {1; 7; 11; 13; 17; 19; 23; * *29}, corre- Q sponding to the fact that rationally E8 is equivalent to e2RS2e+1. Using Mapl* *e, we find the associated eigenvectors. The determinant of the matrix of these eigenv* *ectors is D = 2613325107911413417319223229: This implies that localized at a prime p greater than 29, these eigenvectors sp* *an Q P K1(E8)(p), which then is isomorphic to P K1( R S2e+1)(p)as a module over all Adams operations k, and hence by Bousfield's theorem has Y v-11ss*(E8; p) v-11ss*( S2e+1; p): R If, for example, p = 29, we can find two of the eight eigenvectors, v and w, * *for which v0:= (v - w)=29 is integral. The set of vectors obtained from the eight eigenve* *ctors by replacing v by v0has its determinant equal to D=29, which is a unit in Z(29)* *, and so this set spans P K1(E8)(29). The eigenvectors v and w correspond to eigenval* *ues 4 D. DAVIS 21 and 229. Then k(v) = kv and k(w) = k29w for all integers k, and so we can determine k(v0), and from this use Bousfield's theorem to find v-11ss*(E8; 29)* *, which agrees with that deduced in [8] from the decomposition E8 '29B(3; 59) x S15x S23x S27x S35x S39x S47: The point is that the value of the determinant D, computed blindly from repre- sentation theory, is intimately related to the decomposition of E8 when localiz* *ed at each prime. Note also that the matrix analysis shows that the portion of K1(E8;* * Q) which must be modified to pass to K1(E8; Z(29)) is the portion related to S3 an* *d S59, consistent with the homotopy analysis. Because of the 510 factor in D, we must 10 times replace vectors by 1_5times a difference of vectors in order to find a set of vectors whose determinant is a * *unit in Z(5). On this set, an explicit formula for the Adams operations k can be give* *n. This portion of the work will be described in Section 3. We also show there how performing these basis changes for all relevant primes enables a determination * *of the Adams operations k in K*(X) (not localized at a prime) for all exceptional Lie groups X and all k. The Adams operation formulas are of the sort that allow us to draw inferences* * about attaching maps in the localized Lie groups. This new homotopy-theoretic informa* *tion has been derived here just from our representation-based calculations together * *with Adams' e-invariant work ([3]). This will be discussed in Sections 5 and 6. We feed this information into Bousfield's theorem, which, after a good deal of manipulation, yields the results for v-11ss*(E8; 5). The computations for v-11s* *s2m(E8) are given in Section 4, and those for v-11ss2m-1(E8) are given in Section 5, wh* *ich also includes some useful general results, such as periodicity of the number of summ* *ands, and the use of exact sequences. In Section 6, we perform a similar analysis to * *obtain the result in Theorem 1.2 for v-11ss*(E8; 3). Bendersky and Thompson ([9]) have recently constructed an unstable Bousfield- Kan spectral sequence based on K*K, where K represents periodic K-theory. It possesses some advantages over the BP -based UNSS used in papers such as [8] and [14], especially in that K*(E8)(p)is a free commutative algebra, whereas BP*(E8* *) is not. Using the result obtained in our Proposition 3.5 about the Adams operation* *s in K*(E8)(5), which effectively implies that there is an ff3 attaching map between* * cells FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 5 which would have been in separate factors if Harper's asserted splitting had be* *en cor- rect, together with their K*K-based spectral sequence, Bendersky and Thompson a* *re able to compute v-11ss*(E8; 5) in a manner which is arguably more insightful th* *an the computation here. Their method requires information about the homotopy theory of E8 (e.g. Steenrod operations), while ours requires information about the repres* *enta- tion theory. But so far homotopy theory has been unable to provide the complete picture (i.e., the ff3 attaching map), for which it had to rely on the represen* *tation- theoretic approach presented here. Moreover, our result for v-11ss*(E8; 3) stil* *l seems totally inaccessible to UNSS-type methods. The author would like to thank Martin Bendersky, Pete Bousfield, and Mamoru Mimura for helpful comments on this project. 2.Representation theory and 2 in K-theory In this section, we use representation theory to determine the Adams operation 2 in K*(E8), and present an algorithm by which this can be done for any compact simple Lie group. Let G be a simply-connected compact Lie group. Bousfield's approach to v-11ss* **(G; p) ([10]) requires as input certain Adams operations on the primitives P K1(G; bZp* *). Bousfield suggested to the author the relationship with exterior powers in the * *repre- sentation ring R(G) described in the next two paragraphs. Let I denote the augmentation ideal in R(G). Hodgkin's theorem ([19]) implies that there is an isomorphism I=I2 ! P K-1(G); (2.1) which may be viewed either as induced from the composition R(G) ! K0(BG; *) ! K-1(G) or from Hodgkin's function fi that views a representation ae : G ! U(n) * *as a homotopy class in [G; U] = K-1(G). Although Hodgkin doesn't write the isomor- phism (2.1), he describes R(G) in such a way that I=I2 is clearly the free abel* *ian group on the reduced basic representations eae1; : :;:eael, and shows that P K-* *1(G) is the free abelian group on fi(eae1); : :;:fi(eael). The simple description of fi* * makes it clear that (2.1) respects the exterior power operations n. 6 D. DAVIS Adams operations are related to exterior powers by the Newton formula n(a) - 1(a) n-1(a) + . .+.(-1)n-1n-1(a) 1(a) + (-1)nnn(a) = 0; which implies that n = (-1)n+1nn in I=I2. By [1, 5.3], n in K1(G) corresponds to n=n in K-1(G). Thus n in P K1(G) corresponds to (-1)n+1n in I=I2. Now let G be a compact simple Lie group of rank l (e.g., E8 of rank 8). The representation theory of G is equivalent to that of the associated Lie algebra * *G. Associated to G is a set of weights, a subset + of dominant weights, and a sub* *set {1; : :;:l} + such that (resp. +) is the free abelian group (resp. free abeli* *an monoid) generated by 1; : :;:l. (e.g., [20, p.67].) The set is given a partial* * order P P by mii m0iiif and only if mi m0ifor 1 i l. To each irreducible representation of G is associated a finite set of weights* * with multiplicities. It is a theorem that one of these weights is larger than all th* *e others, and it occurs with multiplicity 1. (e.g., [20, x21.1].) This highest weight i* *s domi- nant. The "highest weight" defines a function from the set of isomorphism class* *es of irreducible representations of G to +, and this function is bijective. It is a * *theorem (See, e.g., [19, 3.3]) that R(G) is a polynomial algebra generated by the irred* *ucible representations ae1; : :;:aelwhich have 1; : :;:las highest weights. If m = (m1; : :;:ml) is an l-tuple of nonnegative integers, let V (m) denote * *the unique irreducible representation with highest weight m11 + . .+.mll. We will need three types of information about representations. o The dimension of V (m) (as a complex vector space). P o The second exterior power 2(V (m)), expressed as cjV (kj) for nonnegative integers cj and l-tuples kj of nonnegative integers. P o The tensor product V (m) V (n), expressed as c0jV (k0j). There are algorithms for each of these, implemented conveniently in the softw* *are LiE ([25]). For dim(V (m)), Weyl's formula ([20, p.139]) is used. For 2(V (m)* *), a formula of [4] for symmetrized products is used. For V (m) V (n), Klimyk's for* *mula ([21]) is used. FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 7 Let eaei= aei- dim(aei). We need 2(eaei) as a linear combination of eae1; : :* *;:eaelin I=I2. P The software gives us 2(aei) as cjV (kj), and hence 2(eaei)= 2(aei) + 1(aei)1(- dim(aei)) + 2(- dim(aei)) X = cjV (kj) - dim(aei)aei+ -dim(aei)2:(2.2) We can iterate tensor product computations to write monomials aee11. .a.eellas * *linear combinations of V (k)'s, and then apply an easy row reduction to this result to* * write V (k)'s as polynomials in the aei's, or, after manipulating polynomials, in the* * eaei's. In I=I2, we ignore products of eaei's. Substituting the formulas for V (kj) as* * linear combinations of eaei's into (2.2) yields the desired expression of 2(eaei) as a* * linear combination of eaej's in I=I2. We illustrate how this works in the simple example of the exceptional Lie gro* *up G2, and then show how the computations can be expedited. The software tells us 2(ae1) = ae2 + ae1 and 2(ae2) = V (3; 0) + ae2. (Remember, ae2 = V (0; 1).) A* *lso, dim(ae1) = 7 and dim(ae2) = 14. Thus -7 2(eae1) = ae1+ae2-7ae1+ 2 = -6(eae1+7)+(eae2+14)+ 7.8_2= -6eae1+aee2: To find 2(eae2), we need to express V (3; 0) as a polynomial in ae1 and ae2. Th* *e software tells us ae1 ae1= V (2; 0) + ae1+ ae2+ 1 ae1 ae2= V (1; 1) + V (2; 0) + ae1 ae1 V (2; 0)= V (3; 0) + V (1; 1) + V (2; 0) + ae2+ ae1: This allows us to compute the second equation in 2(ae2) = V (3; 0) + ae2 = ae31- 2ae1ae2- ae21- ae1; from which we derive -14 2(eae2)= (ae31- 2ae1ae2- ae21- ae1) - 14ae2+ 2 = eae31+ 20eae21+ 104eae1- 2eae1eae2- 28eae2 104eae1- 28eae2modI2: (2.3) By our earlier remarks, this implies that P K1(G2) has basis {x1; x2} with 2(x* *1) = 6x1- x2 and 2(x2) = -104x1+ 28x2. 8 D. DAVIS This procedure can be expedited by just looking at linear terms. If f is an e* *lement of R(G), let L(f) denote the first-order terms when f is written as a polynomia* *l in eae1; : :;:eael. We have L(2(eae2)) = L(V (3; 0) + ae2) - 14eae2;(2.4) and from the tensor product equations above, a type of differentiation yields 2 dim(ae1)eae1=L(V (2; 0)) + eae1+ eae2 dim (ae2)eae1+ dim(ae1)eae2=L(V (1; 1)) + L(V (2; 0)) + eae1 dim(V (2; 0))eae1+ dim(ae1)L(V (2;=0))L(V (3; 0)) + L(V (1; 1)) +L(V (2; 0)) + eae2+ eae1: The first equation says L(V (2; 0)) = 13eae1-aee2, then the second says L(V (1;* * 1)) = 8eae2, and then, using dim(V (2; 0)) = 27, the third equation says L(V (3; 0)) = 104ea* *e1-15eae2, which when substituted into (2.4) yields (2.3). The LiE program that implements this expedited algorithm for E8 is listed and described in Section 7. The one subtlety is how to know which tensor products to compute. The dominant weights are ordered by height1, which is the sum of the coefficients when they are written as roots. For example, in G2 the weights ae1* * and ae2 correspond to roots 3ff1 + 2ff2 and 2ff1 + ff2, respectively, and so the he* *ight of V (m1; m2) is 5m1+3m2. An elementary result states that V (m)V (n) = V (m+n)+ P terms of height less than that of V (m + n). For every term V (l) with li> 1 * *which occurs in 2(aei), we choose a way of writing l = m + n, and differentiate the e* *quation V (m) V (n) = V (l) + terms of lower height to inductively obtain formulas for L(V (l)). It can happen that V (m) V (n) mi* *ght contain terms V (k) which did not appear in 2(aei). If so, we also find L(V (k)* *) by the same method, i.e., differentiating a formula for V (a) V (b) where a + b =* * k. When this algorithm is performed for E8, we obtain the matrix (2.5) for 2 on * *the basis {eae1; : :;:eae8} of I=I2. Thus, for example, L(2eae1) = -3628eae1- eae2+ eae3+ 3875eae8: __________ 1sometimes called "level" FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 9 0-3628 1829621 12625838007-1270362010619556 BB-1 -116621 -146298269 18170270443687 BB 1 -496 -5835130 582917207249 BB 0 1 3875 -468700376 BB 0 0 -177629 26815340999 BB 0 -247 12587859 -2027479372896 @ 0 150877 -392633383 78837408033778 3875 -454937510807381790-2860474034106800 2706011993074-401581533 0 0 1 -40039592220 6661497 0 0 CC -1242615998 185628 0 0 CC 1073250 -249 0 0 CC -68699627 27001 -1 0 CC (2.5) 5393300762 -2538745 248 0 CC -233942373952156457497 -30876 1 A 9317251205935-79993931702573495-247 We obtain the following important corollary of this computation, where fi is * *the isomorphism of (2.1). Corollary 2.6.With respect to the basis {fi(eae1); : :;:fi(eae8)} of P K1(E8), * *the matrix of - 2 is given by (2.5). 3. Nice form for k in P K1(E8)(5)and P K1(X) If p = 3 or 5, then 2 generates (Z=p2)x, and so Bousfield's theorem requires * *knowl- edge of 2 and p. A computation of p similar to that of the previous section * *could be made (provided enough computer time and space is available), and the results (Corollary 2.6 and its analogue) plugged into Bousfield's theorem to give resul* *ts for v-11ss*(E8; 5). However, the matrix (2.5) is so unwieldy that it would be very * *difficult to obtain a nice form for the resulting groups. For this reason, we find a new * *basis of P K1(E8)(5)on which the action of 2 has a nicer form. That is the purpose of t* *his section. Moreover, as we shall see, the new basis will be one on which every k* * can be determined at the same time as 2. We will also perform similar computations* * for Adams operations in K(X) (unlocalized) for all exceptional Lie groups X. To this end, we use Maple to find the eigenvalues and eigenvectors of a matri* *x M, which is defined to be the negative of the matrix (2.5). This is the matrix of * * 2 on P K1(E8). We are not surprised to find that the eigenvalues of M are 21, 27, 21* *1, 213, 10 D. DAVIS 217, 219, 223, and 229because of the rational equivalence E8 'Q S3 x S15x S23x S27x S35x S39x S47x S59 (3.1) and the fact that k acts as multiplication by kn on K1(S2n+1). A matrix whose columns are eigenvectors of M corresponding to the eigenvalues listed above in increasing order is 0 418105625 451155607289497-3133156733386433-2595116726135 BB 4168750 3797965233710 16166554278770 49280463350 BB 23125 20720212181 97338188051 212788925 BB 1 873857 5050967 7697 BB 377 326320702 2019676642 2670694 BB 119249 99339310201 777470688031 532675273 @ 27753998 16454843873197 398907853660267-470842549139 9022308750-26386060414578330-7953947450755023068079530157990 -7044348025-114112039 17026841 26910651 -3895749830 -1864130 -136850 -38570CC 21836035 56893 -7787 -1235CC 1471 1 1 1 CC 635402 206 446 -58 CC (3.2) 294773639 -86407 -48007 4409CC -4276040895712566861 3586541 -174307A 3836612952570-810585690-1913136906425670 The numbers in the columns are coefficients with respect to the basis of Coroll* *ary 2.6. The determinant of this matrix is a 68-digit integer which factors as 2613325107911413417319223229 (3.3) This determinant gives a lot of information. First, it says that localized at a* * prime greater than 29, the eigenvectors form a basis, consistent with the known resul* *t that E8 localized at such primes is equivalent to a product of spheres. Let {v1; v7; v11; v13; v17; v19; v23; v29} denote the columns of (3.2). It is very important to note that since these vec* *tors satisfy 2(vi) = 2ivi, they correspond to the sphere factors in (3.1), and henc* *e also satisfy k(vi) = kivi (3.4) for any positive integer k. FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 11 For primes p satisfying 11 p 29, it was shown in [24] that E8 is p-equivale* *nt to a certain product of spheres and sphere bundles over spheres with ff1 attaching* * maps. For such primes, the number of sphere bundles equals the exponent of p in (3.3)* *. We use p = 23 to illustrate how combinations of the eigenvectors in (3.2) correspo* *nd to these product decompositions, providing somewhat more detail than we did in our brief sketch for p = 29 in Section 1. We note that v01:= (v1 - v23)=23 and v07:= (v7 - 18v29)=23 are integral. If * *v1 and v7 are replaced by v01and v07in (3.2), then the determinant is divided by 2* *32, and so the new set of vectors is a basis for P K1(E8)(23). It follows from (3.4* *) that k(v01) = kv01+ 1_23(k-k23)v23and k(v07) = k7v07+ 18_23(k7-k29)v29. This agree* *s with the determination of k in sphere bundles B(3; 47) and B(15; 59) with attaching map* *s a1 given in [3]. Thus as Adams modules P K1(E8)(23) P K1(B(3; 47) x B(15; 59) x S23x S27x S35x S39)(23): By Bousfield's theorem, these two spaces will have isomorphic v1-periodic homot* *opy groups. Of course, we already knew that by the product decomposition of [24], but here we are getting it without relying on the [24] result. In [8], v1-peri* *odic homotopy groups of ff1 sphere bundles over spheres were determined by the UNSS, and v-11ss*(E8; p) deduced for p 11 using the product decomposition of [24]. * *In Proposition 5.5, we will determine the v1-periodic homotopy groups of these sph* *ere bundles by Bousfield's theorem, giving us a self-contained computation. This ca* *n be done for (E8; p) for any prime p 11. 12 D. DAVIS When p = 5, we use Maple to help us find 10 combinations of vectors that are divisible by 5. These are v01= (v1+ 2v13)=5 v013= (v13+ 3v17)=5 v017= (v17- v29)=5 v0013= (v013+ v017- v29)=5 v07= (v7- v11)=5 v011= (v11- 2v19)=5 v019= (v19- v23)=5 v007= (v07- v011+ 2v019)=5 v0011= (v011+ v019+ 2v23)=5 v0007= (v007- v019- 2v23)=5 The way that these turn out to be grouped, with 1, 13, 17, and 29 related in one group, and 7, 11, 19, and 23 related in the other group, is consistent with Wil* *kerson's product decomposition ([29]) of E8(5)as a product of two spaces whose rational * *types correspond to these two groupings. The matrix (v01; v0013; v017; v29; v0007; v0* *011; v019; v23) has determinant a unit in Z(5), and so its columns form a basis for P K1(E8)(5). We rename the classes (x1; x13; x17; x29; x7; x11; x19; x23) and compute k(xi) us* *ing (3.4). We obtain the following result. FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 13 Proposition 3.5. P K1(E8)(5)has basis {x1; x13; x17; x29; x7; x11; x19; x23} sa* *tisfying k(x1) = k1x1+ 10k(k12- 1)x13- 8k(k12- 1)x17+ 4_5k(k12- 1)x29 k(x13) = k13x13+ 4_5k13(k4 - 1)x17+ 2_25k13(1 + 2k4 - 3k16)x29 k(x17) = k17x17- 1_5k17(k12- 1)x29 k(x29) = k29x29 k(x7) = k7x7+ 2_5k7(1 - k4)x11+ 1_25k7(3 - 2k4 - k12)x19 +_1_125k7(32 + 16k4 - k12- 47k16)x23 k(x11) = k11x11+ 1_5k11(1 - k8)x19+ 1_25k11(9k12- k8 - 8)x23 k(x19) = k19x19+ 1_5k19(1 - k4)x23 k(x23) = k23x23 A formula k(xn) = knxn + u_5kn(k4m - 1)xn+4m + . .,.with u a unit in Z(5)and m 6 0 mod 5, is of the type that would be obtained if the space had cells of di* *mension 2n + 1 and 2n + 1 + 8m with attaching map ffm . The contributions of these cell* *s to v-11ss*(X) will be as if they had the ffm attaching map. The formula for k(x17* *) is, at the very least, strongly suggestive that in E8(5)the 35- and 59-cells are conne* *cted by ff3. This would contradict the product decomposition asserted in [18, 4.4.1b], * *which said that one Wilkerson factor could be further decomposed as X(3; 59) x X(27; * *35). One could make the argument more precise either by noting that it is impossible that a space whose Adams operations can be written as in Proposition 3.5 can be decomposed in this way (i.e., no change of basis can split the Adams operations* *), or by noting that v-11ss*(E8; 5), as computed in the next section, is incompatible* * with such a decomposition. Thus we obtain the following result. Proposition 3.6. In Wilkerson's decomposition ([29, 2.3]) of (E8)(5)as X0xX2, b* *oth factors are indecomposable. In particular, the product decomposition of X2 asse* *rted in [18, 4.4.1b] is not valid. By performing, for all relevant primes, changes of basis of the sort illustra* *ted above for E8 when p = 5 or 23, we obtain, for each exceptional Lie group X, bases for P K1(X) on which we can compute k. We obtain the following results. In all of them, let Bi= fi(eaei), where fi and eaeiare as in Section 2. 14 D. DAVIS Proposition 3.7. A basis for P K1(G2) is given by {y1; y5} with y1 = B1 and y5 = -4B1+ B2. For all integers k, k(y1) = ky1+ 1_30(k - k5)y5 k(y5) = k5y5 The nice feature of this result and the subsequent ones for the other excepti* *onal Lie groups is that they are a result about integral K-theory (i.e., not localized a* *t a prime) and the Adams operations have a nice form (triangular, among other features). S* *ince P K1(X) generates K*(X) for compact simple Lie groups X, multiplicativity of the Adams operations allows us to deduce the Adams operations on all of K*(X). Note that the classes y are subscripted by the exponent e such that k(y) = key+ oth* *er terms. The method of proof in each case is to o use LiE to obtain the matrix of 2 on {B1; : :;:Bl} similarly to (2.5), o use Maple to find eigenvectors of this matrix similarly to (3.2), and note that these vectors satisfy k(vi) = kivifor all k, o use Maple to repeatedly replace vectors v by (v - w)=p, where p is a prime which divides the determinant of the matrix of vectors and w is a linear combination of vectors which appear after v in the most recent set of vectors, until the determinant is 1, and o use Maple and (3.4) to compute k on the final set of vectors, since they are explicit combinations of the eigenvectors. In [26] and [27], Watanabe computed the Chern character on a certain set of g* *en- erators of K(G2), K(F4), and K(E6), and in [28] he explained the well-known way* * in which this would allow one to determine the Adams operations. Using LiE, his set of generators can be expressed as linear combinations of ours, and the results * *for k which could be read off from his results for ch can be related to ours. We chec* *ked this for G2 and F4 and found the results to be in agreement. However, it should* * be FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 15 pointed out that his generators do not have the nice feature of having a triang* *ular matrix for k. Also, in June 1998, Watanabe told the author that he felt that h* *is methods would not work for E7 and E8. An internal check on our results, which w* *as performed for each exceptional Lie group, is to transform the k formulas to th* *e basis of Bi's, and then set k = 2 and compare with the formulas given by LiE. Now we state the results for the other exceptional Lie groups. Proposition 3.8. A basis for P K1(F4) is given by {y1; y5; y7; y11} with y1 = -* *B4, y5 = B1- 3B4, y7 = -2B1+ B3- 15B4, and y11= -6B1+ B2- 11B3+ 102B4. For all integers k, k(y1) = ky11_1_10(k - k5)y5+ (-_1_70k + _1_120k5 + _1_168k7)y7 +(-_1_4620k + _1_6720k5 + _1__13440k7 - __1__147840k11)y11 k(y5) = k5y5+ 1_12(k5 - k7)y7+ (_1_672k5 - _1_960k7 - _1_2240k11)y11 k(y7) = k7y7+ 1_80(k7 - k11)y11 k(y11) = k11y11 Proposition 3.9. A basis for P K1(E6) is given by {y1; y4; y5; y7; y8; y11} with y1 = 2B1- B2+ 3B6 y4 = -B1+ B6 y5 = -2B1+ B2- 2B6 y7 = -B1- 3B2+ B5- 12B6 y8 = 11B1- B3+ B5- 11B6 y11 = 42B1- 21B2- 6B3+ B4- 6B5+ 42B6: 16 D. DAVIS For all integers k, k(y1) = ky1- 1_2(k - k4)y4+ 11_10(k - k5)y5+ (_1_70k - 11_120k5 + 13_168k7)y7 +(-_1_140k + _1_480k4 + 11_240k5 - 13_336k7 - _1_480k8)y8 +(_1_4620k - _11_6720k5 + _13_13440k7 + _67__147840k11)y11 k(y4) = k4y4+ _1_240(k4 - k8)y8 k(y5) = k5y5+ 1_12(k5 - k7)y7- 1_24(k5 - k7)y8+ (_1_672k5 - _1_960k7 - _1_22* *40k11)y11 k(y7) = k7y7- 1_2(k7 - k8)y8+ 1_80(k7 - k11)y11 k(y8) = k8y8 k(y11) = k11y11: Proposition 3.10. A basis for P K1(E7) is given by {y1; y5; y7; y9; y11; y13; y* *17} with y1 = 1873B1- 35B2+ 15B3- 2B5+ 287B6- 23056B7 y5 = -113B1- B2+ 2B3- B5+ 29B6- 547B7 y7 = 6B1- 6B2+ 7B6- 216B7 y9 = 2B1- B2+ B6- 30B7 y11 = -184B1- B2+ 3B3- B5+ 21B6- 292B7 y13 = 120B1- 5B2- 2B3+ B5- 22B6+ 328B7 y17 = 1672B1- 252B2- 34B3+ B4- 12B5+ 177B6- 1344B7: FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 17 For all integers k, k(y1) = ky1- 22544_5(k - k5)y5+ (29367_70k - 64814_15k5 + 3659_42k7)y7+ (350* *89_21k - 57769_45k5 -36590_63k7 + 8714_45k9)y9+ (-25409_2310k + 1409_210k5 + 3659_1680k7* * + 39031_18480k11)y11 +(-45223173_10010k + 85358629_18900k5 + 464693_181440k7 + 4357_5400k* *9 + 507403_221760k11- 24684827_5896800k13)y13 +(___1_1021020k + _2154361_154791000k5 + _3659_684288k7 + _4357_1296* *000k9 + _39031_11176704k11 -_24684827_707616000k13+ _8118218221_926269344000k17)y17 k(y5) = k5y5- 23_24(k5 - k7)y7+ (-_41_144k5 - 115_18k7 + 961_144k9)y9+ (_1_6* *72k5 + 23_960k7 -_57_2240k11)y11+ (60581_60480k5 + _2921_103680k7 + _961_34560k9 - 2* *47_8960k11- 106799_103680k13)y13 +(__1529_495331200k5 + _161_2737152k7 + _961_8294400k9 - _19__451584* *k11- _106799_12441600k13 +_1473021997_174356582400k17)y17 k(y7) = k7y7- 20_3(k7 - k9)y9+ 1_40(k7 - k11)y11+ (_127_4320k7 + 1_36k9 - 13* *_480k11 -_13_432k13)y13+ (__7__114048k7 + _1_8640k9 - _1__24192k11- _13_5184* *0k13+ _17__147840k17)y17 k(y9) = k9y9+ _1_240(k9 - k13)y13+ (__1_57600k9 - _1__28800k13+ _1__57600k17* *)y17 k(y11) = k11y11+ 13_12(k11- k13)y13+ (_5_3024k11- _13_1440k13+ _223_30240k17)* *y17 k(y13) = k13y13+ _1_120(k13- k17)y17 k(y17) = k17y17 Before stating the final of these results about Adams operations in the K-the* *ory of exceptional Lie groups, the case in which many of the numbers become ridiculous* *ly large, we point out two features. One is that the coefficient of each yj in k(* *yi) is actually an integer, yielding integrality results. The other is that at least t* *he second terms give information about attaching maps via primes occurring in denominator* *s. This follows from [3], where it was shown that in a sphere bundle over a sphere* * with attaching map fft with t 6 0 mod p the Adams operations will be as described in our Proposition 5.5. Thus, for example, the 5s in the denominators of the seco* *nd terms in the formulas for k(y1), k(y9), and k(y13) in Proposition 3.10 are, * *at the very least, strongly suggestive that there are ff1 attaching maps from 1 to 5, * *9 to 13, and 13 to 17 in E7, and a similar deduction can be made at the prime 3. The same conclusion can be made from somewhat simpler formulas of K(-)(p), but here we a* *re getting information about all primes at once. 18 D. DAVIS Proposition 3.11. A basis for P K1(E8) is given by {y1; y7; y11; y13; y17; y19;* * y23; y29} with y1 = -784157B1- 30713B2- 218B3+ 13B5- 919B6+ 950380B7- 153687494B8 y7 = 224745B1- 2221B2- 101B3+ 9B5- 950B6+ 69688B7- 3701825B8 y11 = 100088B1- 982B2- 45B3+ 4B5- 422B6+ 30898B7- 1636120B8 y13 = 100091B1- 982B2- 45B3+ 4B5- 422B6+ 30897B7- 1635950B8 y17 = 71682B1- 788B2- 32B3+ 3B5- 318B6+ 23441B7- 1244530B8 y19 = 26482B1- 223B2- 12B3+ B5- 105B6+ 7632B7- 403600B8 y23 = 28444B1- 195B2- 13B3+ B5- 104B6+ 7462B7- 392340B8 y29 = 2691065B1- 38570B2- 1235B3+ B4- 58B5+ 4409B6 -174307B7+ 6425670B8: FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 19 For all integers k, k(y1) = ky1- 1454165_42(k - k7)y7+ (182723259_154k - 650593421_336k7 + 39588* *1345_528k11)y11 +(-185634674_143k + 50198648299_24192k7 - 9105270935_12672k11- 91724* *41277_157248k13)y13 +(1116883611_6188k - 332043745267_1596672k7 - 4354694795_145152k11+ * *541174035343_9434880k13 +14942613523_135717120k17)y17+ (10741728047_149226k - 11326016865460* *3_1634592960k7 - 326918814701_38320128k11 +1201589807287_182891520k13- 1240236922409_1628605440k17+ 2656402375* *049_90348410880k19)y19 +(293377022893_2028117k - 2311774903640460613_13338278553600k7 - 603* *1690056666301_228248616960k11 +34318514761511237_627683696640k13+ 354124997881577_820817141760k17-* * 634880167636711_21683618611200k19 +31696200305279479_2454243253862400k23)y23+ (____1___38818159380k + * *_254146452881_7362729761587200k7 +_399916721902123_72128388948295680k11- _70600280509069_102437979291* *64800k13+ _21980584492333_53779939128115200k17 +_2656402375049_5724475313356800k19+ _31696200305279479_123693859994* *6649600k23- 205974908943845743817_8178638022847247155200k27)y27 k(y7) = k7y7- 2237_40(k7 - k11)y11+ (172603_2880k7 - 51451_960k11- 1825_288k* *13)y13 +(-1141699_190080k7 - 270677_120960k11+ 21535_3456k13+ 535673_266112* *k17)y17+ (-389433691_194594400k7 -_9236573_14515200k11+ 621595_870912k13- 44460859_3193344k17+ 897078* *5339_566092800k19)y19 +(-7948805340661_1587890304000k7 - 1874574277103_951035904000k11+ 13* *65640565_229920768k13 +12694914427_1609445376k17- 2144017696021_135862272000k19+ 794109865* *426391_88921857024000k23)y23 +(____873857_876515447808000k7 + __11299013179_27321359450112000k11-* * __561881_750461386752k13 +___787974983_105450861035520k17+ _8970785339_35867639808000k19 +_794109865426391_44816615940096000k23- 168990789500875519_940073335* *9594536960k29)y29 k(y11) = k11y11- 23_24(k11- k13)y13+ (-_121_3024k11- 1357_1440k13+ 29707_3024* *0k17)y17 +(-_4129_362880k11- 39169_362880k13- 2465681_362880k17+ 2508979_3628* *80k19)y19 +(-_837985819_23775897600k11- 86054063_95800320k13+ 704026193_182891* *520k17- 599645981_87091200k19 +129765607619_32691859200k23)y23+ (___5050967_683033986252800k11+ __* *177031__1563461222400k13 +___43698997_11983052390400k17+ _228089_2090188800k19+ _129765607619* *_16476697036800k23- _371040821209573_46446311065190400k29)y29 k(y13) = k13y13- 59_60(k13- k17)y17+ (-_1703_15120k13- 4897_720k17+ 5227_756k* *19)y19 +(-3741481_3991680k13+ 1398241_362880k17- 1249253_181440k19+ 3961099* *_997920k23)y23 +(___7697_65144217600k13+ __86789_23775897600k17+ _5227_47900160k19 +_3961099_502951680k23- _142072160653_17784371404800k29)y29 k(y17) = k17y17- 83_12(k17- k19)y19+ (23699_6048k17- 19837_2880k19+ 179587_60* *480k23)y23 +(__1471_396264960k17+ _83__760320k19+ _179587_30481920k23- _2617296* *1_4358914560k29)y29 k(y19) = k19y19- 239_240(k19- k23) + (__1_63360k19+ _239_120960k23- _265_1330* *56k29)y29 k(y23) = k23y23+ _1_504(k23- k29)y29 k(y29) = k29y29 20 D. DAVIS 4. Determination of v-11ss2m(E8; 5) Bousfield proved the following result in [10]. This is the result that has be* *en called "Bousfield's theorem" throughout this paper. Theorem 4.1. Let X be a 1-connected finite H-space with H*(X; Q) associative, p an odd prime, and r a generator of (Z=p2)x. Then v-11ss2m(X; p) = (coker(OEm ))# and v-11ss2m-1(X; p) = (ker(OEm ))#; where (-)# denotes the Pontryagin dual, and OEm = r - rm : P K1(X; bZp)= im( p) ! P K1(X; bZp)= im( p): For the finite abelian groups with which we deal, P K1(X; bZp)= im( p) is iso* *morphic to P K1(X; Z(p))= im( p), and the only effect of the Pontryagin dual is to reve* *rse the direction of arrows. The following result is immediate from Proposition 3.5 and Theorem 4.1. Proposition 4.2. v-11ss2m(E8; 5) Am Bm , where Am = Z(5)(x1; x13; x17; x29)=(r1; : :;:r8) and Bm = Z(5)(x7; x11; x19; x23)=(s1; : :;:s8); FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 21 where Z(5)(-) denotes the free Z(5)-module on the indicated generators, and the* * rela- tions are given as follows: r1 : 529x29; r2 : 517x17- 516(512- 1)x29; r3 : 513x13+ 4 . 512(54- 1)x17+ 2 . 511(1 + 2 . 54- 3 . 516)x29; r4 : 5x1+ 10 . 5(512- 1)x13- 8 . 5(512- 1)x17+ 4(512- 1)x29; r5 : (229- 2m )x29; r6 : (217- 2m )x17- 1_5. 217(212- 1)x29; r7 : (213- 2m )x13+ 12 . 213x17+ 3_25. 214(11 - 216)x29; r8 : (2 - 2m )x1+ 20(212- 1)x13- 16(212- 1)x17+ 8_5(212- 1)x29; s1 : 523x23; s2 : 519x19+ 518(1 - 54)x23; s3 : 511x11+ 510(1 - 58)x19+ 59(9 . 512- 58- 8)x23; s4 : 57x7+ 2 . 56(1 - 54)x11+ 55(3 - 2 . 54- 512)x19 +54(32 + 16 . 54- 512- 47 . 516)x23; s5 : (223- 2m )x23; s6 : (219- 2m )x19- 3 . 219x23; s7 : (211- 2m )x11- 51 . 211x19+ 183 . 214x23; s8 : (27- 2m )x7- 3 . 28x11- 165 . 27x19- 771 . 212x23: We shall analyze Am first. We will make frequent use of the following well-kn* *own fact proved in [2, 2.12]. Proposition 4.3. If r generates the group of units (Z=p2)x, then ( 0 if m 6 0 mod p - 1 p(rm - 1) = 1 + p(m) if m 0 mod p - 1: Relations r5, r6, r7, and r8 imply immediately that Am is 0 if m 6 1 mod 4. We will write m = 4k + 1 and divide r5, r6, r7, and r8 by the units 229, 217, 213,* * and 2, 22 D. DAVIS respectively. We use r6 to eliminate x29, and replace it by ((1 - 24k-16)=819)x* *17in the other relations, and then use the modified r7 to eliminate x17, replacing i* *t by a specific multiple of x13in all relations. Next we add 1_5(24k- 1)r4 to r8 to el* *iminate x1 from r8. Now the modified r4 is the only relation involving x1, and it is of th* *e form 5x1 + ff52x13. This implies that x1 + 5ffx13generates a Z=5 direct summand. The remaining relations on x13are as follows, obtained respectively from r1 and r5,* * r2, r3, and r8. t1 : 5min(28;(k-7))+(k-4)+(k-3)+3 t2 : 516(4095 + (512- 1)(24(k-4)- 1)) t3 : 513 6 . 819 + 3 . 2621(24k-16- 1) +1_5(24k-12- 1) 2(54- 1)819 - 1_5(24k-16- 1)(1 + 2 . 54- 3 . 516) t4 : 52 12 . 819 + 6 . 2621(24k-16- 1) - 1_5(24k-12- 1)(4 . 819 + 2_5(24k-16- * *1)) +1_5(24k- 1)(512- 1) 12 + 6.2621_819(24k-16- 1) -2_5(24k-12- 1)(2 + 1_5(24k-16- 1)=819) The relation t4 can be manipulated to a unit times 1_ 4k-16 4k-28 12 4k-12 12 4k 5(2 - 1)(2 - 1)(2 (2 - 1) - 5 (2 - 1)); (4.4) while t3 can be rewritten as - 216511(24k-16- 1)(24k-28- 1) - 213515(24k-12- 1)(24k-28- 1) + 3 . 528(24k-16* *- 1); (4.5) and t2 as 528(24k-16- 1) - 516212(24k-28- 1): (4.6) These four relations on x13result in a summand of order 5e, where o if k 0; 1 mod 5, then e = 2 from (4.4); o if k 3 mod 5, then e is the minimum of 2 + (k - 3) from (4.4) or 13 from (4.5); FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 23 o if k 4 mod 5, then e is the minimum of 2 + (k - 4) from (4.4) or 17 from (4.6); o if k 2 mod 5, then e is the minimum of 2 + (k - 7) from (4.4) or 29 from (4.5). Thus Am gives the groups claimed in Theorem 1.1 in the case m 1 mod 4. Now we use the relations s1; : :;:s8 to determine Bm . The analysis is simil* *ar to that just employed for Am , but we use a different technique near the end. Firs* *t we use s5; : :;:s8 to see that Bm = 0 if m 6 3 mod 4. Now we write m = 4k + 3, and divide the last four relations by their initial 2-power. We use s6 to eliminate* * x23, and then s7 to eliminate x19. When these expressions are substituted into s8, it be* *comes (1 - 24k-4)x7- ux11; where 5 4k-16 4k-8 4k-16 u = 6 + 165 + 257 . 2 (1 - 2 ) (1 - 2 )=(488 . 2 - 437) (4.7) is a unit. Thus x11 can be eliminated, and so Bm is cyclic with generator x7 a* *nd relations w1 : 5min(26;4+(k-5)+(k-4)+(k-2)+(k-1)); w2 : 520+(k-2)+(k-1)(15 + (1 - 54)(1 - 24k-16)); w3 : 510+(k-1) 52(488 . 24k-16- 437) 9 +(1 - 24k-8) 5 - 5 + 1_3(1 - 24k-16)(9 . 512- 58- 8) ; (1 - 24k-8) w4 : 54 53u + (1 - 24k-4) 2(52- 56) + _______________ 488 . 24k-16- 437 5 13 . 15 - 2 . 5 - 5 + 1_3(1 - 24k-16)(32 + 16 . 54- 512- 47 . 516) ; where u is as in (4.7). We easily read off from these that the order of Bm is 5min(7;4+(k-1))if k 1 or 3 mod 5. 24 D. DAVIS Now let P = 24k, and write w4 as a unit times 54 times the following expressi* *on. 16 16 8 53 6 . 212(488P - 437 . 216) + 24 165 . 2 + 257 . 32(2 - P ) (2 - P ) 16 5 * * 13 +(16 - P ) 50 . 28(1 - 54)(488P - 437 . 216) + (28- P ) 2 (15 - 2 . 5 - * *5 ) +1_3(216- P )(32 + 16 . 54- 512- 47 . 516) : We use Maple to write this expression as A + B . P + C . P 2+ D . P 3, where A,* * B, C, and D are certain explicit large integers. We note that for any positive int* *eger e, this cubic expression can be rewritten as a0+ a1(P - 2e) + a2(P - 2e)2+ a3(P - 2e)3; where a3 = D a2 = C + 3 . 2eD a1 = B + 2e+1C + 3 . 22eD a0 = A + 2eB + 22eC + 23eD: With e = 8, Maple computes these ai, from which we deduce that w4 can be written as 54(u157+ u253(24k- 28) + u35(24k- 28)2+ u4(24k- 28)3); where ui are units in Z(5). The relation given by this and w1 when k 2 mod 5 is the desired 5min(11;4+(k-2)), and w2 and w3 are easily seen to imply no addi* *tional restrictions. If k 4 mod 5, we use e = 16 in the above method (with the same values of A, * *B, C, and D) and obtain the following form of w4, where again uiare units in Z(5). 15 2 4k 16 2 4k 16 2 4k 16 3 54 u15 + u25 (2 - 2 ) + u35 (2 - 2 ) + u4(2 - 2 ) This and w1 give 5min(19;4+(k-4))as the relation. To see that w3 can give no sm* *aller relation, we can write it as 510((24k-16- 1)(u5(24k-16- 1) + 5u6) + 510u7); with uiunits. FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 25 If k 5 mod 5, we use e = 20 in the above method and find that w4 can be writ* *ten as 54(u1519+ u252(24k- 220) + u35(22k- 220)2+ u4(22k- 220)3): This and w1 give the desired 5min(24;4+(k-5)). To see that w2 cannot give a sma* *ller relation in this case, we write it as a unit times 520(24(24k-20- 1) + 54(24k-16- 1)): Finally, w3 may be rewritten, using the method employed above for w4, as 524u0+ 511u1(22k- 220) + 510u2(24k- 220)2; which implies that it cannot give a smaller relation here. This completes the calculation of Bm , showing that it yields exactly the val* *ues of v-11ss2m(E8; 5) claimed in Theorem 1.1 when m 3 mod 4. This completes the determination of v-11ss*(E8; 5) when * is even. 5.Determination of v-11ss2m-1(E8; 5) We could compute v-11ss2m-1(E8; 5) using Theorem 4.1 by computing the relevant kernel. These computations are probably even more difficult than those of Secti* *on 4, and so, uncharacteristically, we follow a program of avoiding calculations a* *s much as possible. A sequence of five propositions will yield the result for v-11ss2m* *-1(E8; 5) claimed in Theorem 1.1. The first proposition is elementary and well-known. Proposition 5.1. Let p be odd. If X is a simply-connected finite H-space with H*(X; Q) associative or X is a space built from a finite number of odd-dimensio* *nal spheres by fibrations, then v-11ss2m-1(X; p) and v-11ss2m(X; p) have the same o* *rder. Proof.For the first type of X, we use Theorem 4.1 and the observation that the * *kernel and cokernel of an endomorphism of a finite abelian group have the same order. * *For the second type of X, we use the argument employed for SU(n) in [12, p.530]. || Next we have the following recent observation of Bendersky. 26 D. DAVIS Proposition 5.2. If X is a space with an H-space exponent at p, the number of direct summands in v-11ssi(X; p) depends only on the residue mod q of i. Here we begin using q = 2(p - 1), and say that X has an H-space exponent at p if some pe-power map on some iterated loop space of X is null-homotopic. Compact Lie groups and spheres have H-space exponents at all primes. Proof of Proposition 5.2.There is a commutative diagram of isomorphisms ssi(X; Z=p) ssi(X) Z=p Tor(ssi-1(X); Z=p) # # ssi+kqpe(X; Z=p) ssi+kqpe(X) Z=p Tor(ssi-1+kqpe(X); Z=p) where the vertical arrows are as described in [12, x2]. These arrows are define* *d using the null-homotopy of the pe-power map. The one on the left can equivalently be defined using Adams maps of the mod p Moore space. The direct limit of these vertical morphisms defines the v1-periodic homotopy groups. The diagram commutes because the morphisms ss*X ! ss*+kqpeX commute with multiplication by p. Passing to the direct limit, we obtain an isomorphism v-11ssi(X; Z=p) v-11ssi(X) Z=p Tor(v-11ssi-1(X); Z=p): These will be finite Z(p)-modules, and if we denote by #(G) the number of direct summands in such a module, we obtain #(v-11ssi(X; Z=p)) = #(v-11ssi(X)) + #(v-11ssi-1(X)): e On the other hand, v-11ssi(X; Z=p) v-11ssi+q(X; Z=p) because the vp1-map whi* *ch defines the direct limit can be chosen to be the composite of maps qM(p) ! M(p). Thus #(v-11ssi(X))+#(v-11ssi-1(X)) has period q. This implies that #(v-11ssi(X)* *) has period q. To see this, we note that if = #(v-11ssqX) - #(v-11ss0X), it follows* * that #(v-11ssj+qX) - #(v-11ssjX) = (-1)j for all j, which implies that #(v-11ssj+Lq(X)) - #(v-11ssj(X)) = (-1)jL: If 6= 0, let j = 1 if > 0 and j = 0 if < 0. This implies the absurd statement that #(v-11ssj+Lq(X)) < 0 for L sufficiently large, and so we conclude = 0. * *|| FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 27 By these two propositions and the computation of Section 4, Theorem 1.1 will be proven once we show v-11ss2m-1(E8; 5) is cyclic for some m 1 mod 4 and for some m 3 mod 4. This could be done by an explicit calculation, but we prefer to approach it by exact sequences. The following result is immediate from the Sna* *ke Lemma. Proposition 5.3. Suppose 0 ! M1 ! M2 ! M3 ! 0 is a short exact sequence of p-adic Adams modules with p acting injectively on each. Let OEi= r - rm : Mi= im( p) ! Mi= im( p): Then there is an exact sequence 0 ! kerOE1 ! kerOE2 ! kerOE3 ! cokerOE1 ! cokerOE2 ! cokerOE3 ! 0: Theorem 4.1 and exactness of Pontryagin duality yield the following corollary. Corollary 5.4.Suppose maps X ! Y ! Z induce a short exact sequence of Adams modules 0 ! P K1(Z; bZp) ! P K1(Y ; bZp) ! P K1(X; bZp) ! 0; with p acting injectively on each. Then there is a long exact sequence ! v-11ssi+1Z ! v-11ssiX ! v-11ssiY ! v-11ssiZ ! v-11ssi-1X ! : Next we have the following basic calculation, which is the Bousfield approach* * to a sphere bundle over sphere with attaching map fft. This result is analogous to [* *8, 1.3], which was obtained using the UNSS. Proposition 5.5. If P K1(X; bZp) has generators x and y with ky = kn+t(p-1)y a* *nd kx = knx + u_pkn(kt(p-1)- 1)y, where u is a unit in Z(p), and t 6 0 mod p, the* *n the only nonzero groups v-11ss*(X; p) are v-11ss2n+qi(X; p) v-11ss2n+qi-1(X; p) Z=pe; 28 D. DAVIS where8 > 1 >: min(p(i - t - tpt(p-1)) + 1; t(p - 1)i+f2)i t mod p, and n = 1 Proof.We again make frequent use of Proposition 4.3. By Theorem 4.1, v-11ss2m(X* *; p) has generators x and y subject to relations pn+t(p-1)y pnx + upn-1(pt(p-1)- 1)y (rn+t(p-1)- rm )y (rn - rm )x + u_prn(rt(p-1)- 1)y The last two relations imply that the group is 0 unless m n mod p - 1. We let m = n + (p - 1)i. Since t 6 0 mod p, the fourth relation can be used to elimina* *te y. We obtain that the group is cyclic with generator x, and relations on x p(i)+1+min((i-t)+1;n+t(p-1)) (rt(p-1)- 1)pn-1- pn-1(pt(p-1)- 1)(1 - r(p-1)i): The second of these relations can be rewritten as t(p-1) (p-1)(i-t) t(p-1)(p-1)t pn-1 r (1 - r ) + p (r - 1) : Now inspection of the relations shows that if i - t 6 0 mod p, the first relati* *on becomes p(i)+2, and the second relation becomes pn. If i-t 0 mod p, then the f* *irst relation becomes pmin((i-t)+2;n+t(p-1)+1), while the second becomes pn+min((i-t* *);t(p-1)) unless (i - t) = t(p - 1). If i - t = ffpt(p-1)and we write rp-1= 1 + fip with * *fi 6 0 mod p, then the second relation becomes (i - ff)fipt(p-1)+1mod pt(p-1)+2. It is* * now routine to translate these relations when i - t 0 mod p into those claimed in * *the proposition. This establishes the results for the even-dimensional groups. The odd groups could be computed directly; however, we can avoid computation as follows. By Proposition 5.1, the odd groups have the asserted order, and by Proposition 5.2 it suffices to show that any one of them is cyclic. By Proposit* *ion 5.3, FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 29 there is an exact sequence 0 ! v-11ss2mS2n+1! v-11ss2mX ! v-11ss2mS2n+1+tqffi-!v-11ss2m-1S2n+1 ! v-11ss2m-1X ! v-11ss2m-1S2n+1+tq! 0; (5.6) where v-11ss*S2m+1 are the groups associated to the Adams module P K1(S2m+1; bZ* *p), which is cyclic with k = .km . It is easy to use Theorem 4.1 to compute that t* *he only nonzero groups v-11ss*S2m+1 are v-11ss2m+iqS2m+1 v-11ss2m+iq-1S2m+1 Z=pmin(m;(i)+1); (5.7) in agreement with well-known results. Choose m = n + (p - 1)(t + pn+t(p-1)). * *By (5.7) and our computation of v-11ssevX, the exact sequence (5.6) becomes 0 ! Z=p ! Z=pn+(p-1)t! Z=pn+(p-1)tffi-!Z=p ! v-11ss2n+qt+qpn+t(p-1)-1X ! Z=pn+(p-1)t! 0; which implies that ffi is surjective and v-11ss2n+qt+qpn+t(p-1)-1X Z=pn+(p-1)t* *. || Now, the determination of v-11ssodd(E8; 5) is completed by using the observat* *ion which precedes Proposition 5.3 and the following result. Proposition 5.8. If m = 4k+1 and k 0 or 1 mod 5, then v-11ss2m-1(E8; 5) Z=53. If m = 4k + 3 and k 3 mod 5, then v-11ss2m-1(E8; 5) Z=54. Proof.Let M291(resp. M237) denote the sub Adams module of P K1(E8)(5)generated by x1, x13, x17, and x29 (resp. x7, x11, x19, and x23). (See Proposition 3.5.* *) If i; j 2 {1; 13; 17; 29} with i j, let Mjidenote the subquotient of M291generate* *d by those xk in M291with i k j, and similarly for subquotients of M237. Let OEji= 2 - 2m : Mji= im( 5) ! Mji= im( 5) for a fixed value of m of the type specified in the proposition. Let Kji= ker(O* *Eji) and Cji= coker(OEji). We consider first the case m = 4k + 1 with k 0 or 1 mod 5. By (5.7) we have Kii Cii Z=5 for i 2 {1; 13; 17; 29}, and by Propositions 5.5 and 3.5, K1713 C1713 K2917 C2917 Z=52: 30 D. DAVIS By Theorem 1.1, C291 Z=5 Z=52, and, by a calculation similar to that of Am in Section 4 but much easier, we obtain C2913 Z=53: (5.9) We will sketch this calculation at the end of this section. The results about C* *- and K-groups just listed imply, by just diagram chasing, that K291 Z=53; (5.10) which implies the first part of the proposition by Theorem 4.1. Before we present the simple proof of (5.10), we wish to present the motivati* *on or underlying rationale for this argument and these results. It involves homotopy * *charts of the type used extensively in [14]. Diagram 5.11. 3 rJ] r 27 r|JJ r| 35 r||J r|| 59 r| Jr| 2m - 12m For the spheres S of dimensions 3, 27, 35, and 59 whose P K1-Adams modules bu* *ild this portion of P K1(E8)(5), and for the value of m being considered here, v-11ss2m(S) v-11ss2m-1(S) Z=5: These are indicated by the dots. The vertical lines from 59 to 35, and from 35 * *to 27, are nontrivial multiplication by 5 obtained from Proposition 5.5, which applies* * be- cause of the terms -1_5k17(k12-1)x29and 4_5k13(k4-1)x17in Proposition 3.5. (The* * first of these is the all-important term which implies that Harper's product decompos* *ition was false.) The computation in Section 4 of Am Z=5 Z=52 (for appropriate m) implies that there must be a differential from v-11ss2m(S59), but it wasn't cle* *ar whether it hit v-11ss2m-1(S27) or v-11ss2m-1(S3). (Actually, the UNSS or the 4_5k(k12-1* *)-term in 3.5 make it pretty clear that it hits v-11ss2m-1(S3).) Our computation that C29* *13 Z=53 implies that there was no differential from 59 to 27, and so it must go from 59* * to 3, as indicated in the diagram by the diagonal line. But this is just one way of thinking. The result (5.10) can be obtained by di* *agram- chasing as follows. The claims made above about certain K-groups and C-groups FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 31 imply that the relevant exact sequences of 5.3 yield short exact sequences in t* *he following commutative diagram, with all groups except the middle one given expl* *icitly in the second diagram. 0 0 0 ? ? ? ? ? ? y y y 0 ---! K2929---! K2917---! K1717---! 0 ? ? ? ?y ?y ?y 0 ---! K2929---! K2913---! K1713---! 0 ? ? ? ? y y K1313---! K1313 ? ? ? ? y y 0 0 Z=5 - --! Z=52 ---! Z=5 ? ? ? ?y ?y ?y Z=5 - --! K2913---! Z=52 ? ? ? ? y y Z=5 ---! Z=5 It is easy to verify that in such a diagram, we must have K2913 Z=53. Another exact sequence from 5.3 begins 0 ! K2913! K291! : Since |K291| = |C291| = 53, and K291contains a cyclic subgroup, K2913, of order* * 53, we must have K291 Z=53, as claimed in (5.10). The computation when m = 4k + 3 and k 3 mod 5 is easier. In this case, each of the four spheres yields a Z=5, and each pair of consecutive spheres has* * a nontrivial .5 extension because of 5.5 and the terms 2_5k7(1 - k4)x11, 1_5k11(1* * - k8)x19, and 1_5k19(1 - k4)x23in 3.5. Since |K237| = |C237| = 54, we can deduce cyclicit* *y of K237 32 D. DAVIS by chasing diagrams such as those used just above. The relevant homotopy chart * *is as below, with no differential. 15 r| r| 23 r|| r|| 39 r|| r|| 47 r| r| 2m - 12m We close this section by sketching the proof of (5.9). We are computing Am in Proposition 4.2, with x1, r4, and r8 removed, and m = 4k + 1 with k 0 or 1 mod 5. In the analysis in Section 4, we will have generator x13with relations t1, t* *2, and t3. The relation t1 says that 53x13= 0, and the other relations involve much la* *rger powers of 5. Thus the group is Z=53, as claimed. || 6. Calculation of v-11ss*(E8; 3) The computation of v-11ss*(E8; 3) is performed similarly to that of v-11ss*(E* *8; 5). One thing that makes the analysis more complicated is that all eight generators* * are related to one another by Adams operations, rather than being divided into two groups of four, as was the case for (E8; 5). It is more difficult to analyze an* * abelian group with 8 generators and 16 relations than to do it for two groups, each wit* *h 4 generators and 8 relations. We rely on Maple for every step of this computation. For i 2 {1; 7; 11; 13; 17; 19; 23; 29}, let vibe the columns of (3.2), satisf* *ying k(vi) = kivi. Then 32 times we replace vectors v by v0:= (v - w)=3, with w a linear com* *bi- nation of vectors in the set which follow v, and v0still integral, and obtain f* *inally a FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 33 basis {w1; w7; w11; w13; w17; w19; w23; w29} for P K1(E8; Z(3)) defined by w1 = v1=32+ 730v7=38+ 10v11=310+ 4v13=38+ 28v17=39- 109v19=38 +1364v23=39- 44078v29=310 w7 = (v7+ v11+ 7v13+ v17+ 92v19- 14v23- 994v29)=37 w11 = v11=39- v13=36- 2v17=38- 13v19=36+ 83v23=38- 11885v29=39 w13 = v13=35- v17=36+ 7v19=35+ 2v23=36+ 227v29=36 w17 = v17=35+ v19=33+ 4v23=35- 26v29=35 w19 = v19=32- 2v23=33- v29=33 w23 = (v23- v29)=32 w29 = v29: We chose to list here the basis in terms of eigenvectors virather than the clas* *ses Bi, as was done in Propositions 3.7 to 3.11, primarily for the sake of variety. By performing a matrix multiplication, we obtain the following formula for the Adams operations on the classes wi. This is totally analogous to Theorem 3.11, except that here we are localized at 3. 34 D. DAVIS Proposition 6.1. On the basis {wi} of P K1(E8; Z(3)) just defined, the Adams op* *er- ations are given, for any integer k, by k(w1) = kw1- 730_3(k - k7)w7+ (6560_3k - 2190k7 + 10_3k11)w11 +(918k - 24820_27k7 + 10_9k11+ 4_27k13)w13 +(1484_3k - 40150_81k7 + 50_81k11+ 4_81k13+ 28_81k17)w17 +(121_3k - 29200_729k7 + 10_243k11- 40_729k13- 28_243k17- 109_729k1* *9)w19 +(-998_3k + 727810_2187k7 - 1030_2187k11- 104_2187k13- 280_2187k17-* * 218_2187k19+ 1364_2187k23)w23 +(8479_9k - 6187480_6561k7 + 99320_59049k11- 316_6561k13+ _364_1968* *3k17- 109_6561k19 +_1364_19683k23- 44078_59049k29)w29 k(w7) = k7w7- 9(k7 - k11)w11+ (-34_9k7 + 3k11+ 7_9k13)w13+ (-55_27k7 + 5_3k* *11 +_7_27k13+ 1_9k17)w17+ (-_40_243k7 + 1_9k11- 70_243k13- 1_27k17+ 92* *_243k19)w19 +(997_729k7 - 103_81k11- 182_729k13- 10_243k17+ 184_729k19- 14_243k* *23)w23 +(-8476_2187k7 + 9932_2187k11- 553_2187k13+ _13_2187k17+ _92_2187k1* *9- _14_2187k23- 994_2187k29)w29 k(w11) = k11w11+ 1_3(k11- k13)w13+ (_5_27k11- 1_9k13- 2_27k17)w17 +(_1_81k11+ 10_81k13+ 2_81k17- 13_81k19)w19 +(-103_729k11+ 26_243k13+ 20_729k17- 26_243k19+ 83_729k23)w23 +(_9932_19683k11+ 79_729k13- _26_6561k17- 13_729k19+ _83_6561k23- 1* *1885_19683k29)w29 k(w13) = k13w13+ 1_3(k13- k17)w17+ (-10_27k13+ 1_9k17+ 7_27k19)w19 +(-26_81k13+ 10_81k17+ 14_81k19+ 2_81k23)w23 +(-_79_243k13- 13_729k17+ _7_243k19+ _2_729k23+ 227_729k29)w29 k(w17) = k17w17- 1_3(k17- k19)w19+ (-10_27k17+ 2_9k19+ 4_27k23)w23 +(_13_243k17+ 1_27k19+ _4_243k23- 26_243k29)w29 k(w19) = k19w19+ 2_3(k19- k23)w23+ (1_9k19- 2_27k23- 1_27k29)w29 k(w23) = k23w23+ 1_9(k23- k29)w29 k(w29) = k29w29: The second terms of the above expressions for k(wi) are, at the very least, * *strongly suggestive that E8 can be built by fibrations from Sn's according to the scheme FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 35 in Diagram 6.2. Here is Bousfield's functor ([10]) satisfying ss*(X) v-11ss*(* *X) (localized at 3). We do not wish to belabor this here, since it is not necessar* *y for our analysis. However, we note that it seems quite likely that this analysis could * *yield information about the attaching maps between cells of the 3-localizations of E8* * and E8, extending work in [23] and [17]. Our results here were previously inaccessi* *ble, since ffiis not detected by primary cohomology operations when i > 1. Diagram 6.2. ff5 ____|_______________|_ ____ ____ ____ ____ ____ | 3__|__|315f|f3| 23_|__|f27_|__|f35_|___|139_|__|f47_|__|f592|ff1ff2ff3 |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | We return now to the application of Proposition 6.1 to determining v-11ss*(E8* *; 3). By Theorem 4.1, v-11ss2m(E8; 3) has a presentation with 8 generators and 16 rel* *ations, corresponding to 3(wi) and ( 2 - 2m )(wi), where we use 6.1 for 2 and 3. By Proposition 4.3, which we shall use frequently, the 2 - 2m relations imply th* *at v-11ss2m(E8; 3) = 0 if m is even, and so we let m = 2k + 1. We divide the 2 - * *2m relations by the unit 2, and arrange the relations as rows of a 16 x 8 matrix, * *with the 3 relations listed first. We use Maple to manipulate the matrix. The entry in position (15,8) is a uni* *t, and so we pivot on it, and then delete the 15th row and the 8th column. This corresponds to writing the 8th generator as a combination of the other generato* *rs, then removing that generator and the relation which expressed it in terms of the others, while substituting this relation into all the others. In subsequent st* *eps, we pivot on and then eliminate positions (14,7), (13,6), (12,5), (11,4), (1,3), an* *d (8,2), ending up with a 9 x 1 matrix G. At each step, it is essential that the element* * on which we pivot is a unit. It was by no means clear at the outset that this coul* *d be done 7 times, for it has the important consequence that v-11ss2m(E8; 3) is cycl* *ic for each integer m. In manipulating the matrix, we let, for i 2 {0; 6; 10; 12; 16; 18; 22; 28}, P* *i= 22k-2i. Thus, for example, the elements in position (9,1) and (10,2) of the initial mat* *rix will be -P0 and -P6. After j steps of pivoting (j 7), all entries in the matrix wil* *l be quotients of polynomials in the Pi of degree j + 1. Since Pi 0 mod 3, mod 3 values of polynomials are determined by their constant term, which during the l* *ast 36 D. DAVIS few steps will be up to 30 digits long. Up to this point, we have only cared ab* *out mod 3 values, to find units on which we could pivot, but in the next step we ne* *ed more delicate information about exponents of 3. We will have v-11ss4k+2(E8; 3) 3e, where e is the smallest exponent of 3 of * *the nine entries of the matrix G. The denominator polynomials are units, and can be igno* *red. Maple does some factoring automatically. For example, the fourth entry of G has a factor 312P10P12, which we treat as 314+(k-5)+(k-6), since (P2j) = 1 + (k - j* *). Throughout this section, (-) = 3(-). Now we divide into cases depending upon the mod 9 value of k. We consider fir* *st the case k 2 11 mod 9. After performing the preliminary simplifications described in the preceding paragraph, we replace each occurrence of P2jby R + 222- 22j. T* *hus R is representing 22k- 222, which satisfies (R) = 1 + (k - 11). We find that the nine relations are, up to unit multiples of all terms, as follows. We emphasize* * that this assumes that k 2 mod 9, so that, for example, (k - 8) has been replaced b* *y 1, and (k - 10) by 0. We also point out that it seems to be infeasible to obtain t* *hese expressions by hand; their determination seems to require a computer. 325+ 38R + 36R2+ 34R3+ 32R4+ 3R5+ R6 32(322+ 37R + 38R2+ 33R3+ 33R4+ 32R5+ R6) 310(316+ 36R + 35R2+ 32R3+ 3R4+ R5) 315(312+ 34R + 32R2+ 3R3+ R4) 322(37+ 33R + 3R2+ R3) 327(33+ 3R + R2) 337+(k-11) 324+ 39R + 37R2+ 35R3+ 33R4+ 32R5+ 3R6+ R7 310+(k-11) Since (R) 3, we find that the term 38R in the first relation gives the smallest 3-power (39+(k-11)) if (k - 11) 15, while the term 324 in the second or eighth relation will be smallest if (k - 11) 16. This establishes the claim of Theore* *m 1.2 for v-11ss2m(E8; 3) when m = 2k + 1 and k 2 mod 9. The situation is similar when k 8 mod 9. If R = 22k- 216, then the first two relations are, up to unit multiples of all terms, 319+ 38R + 36R2+ 34R3+ 32R4+ 3R5+ R6 FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 37 and 318+ 39R + 38R2+ 35R3+ 35R4+ 34R5+ 32R6; while other relations involve larger exponents of 3. Note that these refer to p* *recisely the same relations as the first two of the nine listed above; however, R now re* *presents a different expression. Since (R) 3, the Ri-terms with i 2 are more highly 3- divisible than the R1-term, and so may be ignored. We find that the smallest ex* *ponent of 3 is min(9 + (k - 8); 18), with the 9 + (k - 8) coming from the first relati* *on and the 18 from the second. This yields the case k 8 mod 9 for v-11ss2m(-) in Theo* *rem 1.2. The case k 5 mod 9 follows similarly by noting that if R = 22k- 228, then* * the only significant terms are 332+ 39R + 36R2 in the first relation and 330+ 39R i* *n the second and eighth. If (R) = 3, there was a possibility of cancellation of the R* *1- and R2-terms in the first relation, but not in the second. The case k 1 mod 3 is easier. If R = 22k- 28, the first relation is 36 + 35R* * + 34R2+ 33R3+ 32R4+ 3R5+ R6, which gives a relation 36 if (R) 2, while the other relations are more highly 3-divisible. The case k 6 mod 9 introduces a second-order effect. If R = 22k- 212, the on* *ly significant terms are 314+ 36R from the first relation and 315+ 39R from the se* *cond. The smallest 3-exponent is 7 + (k - 6) if (k - 6) < 7, and is 14 if (k - 6) > 7. If (k - 6) = 7, the two terms in the first relation are both 314times a unit, a* *nd we must analyze the mod 3 values of these units to tell whether the sum of these t* *wo terms is 314times a unit or is divisible by 315. As the second relation is 315t* *imes a unit, we need only evaluate the first relation mod 315. Maple tells us that the unit coefficients of 314and 36R in the first relation* * are both 1 mod 3. Thus mod 315the first relation is k 6 314+ 36((1 + 3)k - (1 + 3)6) 314+ 36 3(k - 6) + 32 2 - 2 : If k = 37u, this becomes 314(1 + u), and so is 315if u 2 mod 3, and 314if u 1. This yields the min(7 + (k - 6 - 2 . 37); 15) in 1.2 when k 6 mod 9. The case k 0 mod 9 follows similarly once Maple tells us that if R = 22k-218, then the * *terms which can give smallest 3-power are (3a + 1)320+ (3b + 2)36R from the first rel* *ation and 321+ 39R from the second. 38 D. DAVIS The case k 3 mod 9 is easier. If R = 22k- 26, then the first relation begins 38+37R+34R2, and other relations involve larger 3-powers. Thus v-11ss2m(-) Z=38 in this case, which appears in 1.2 bundled along with the case k 6. This concludes the computation of v-11ss2m(E8; 3) in Theorem 1.2. The v-11ss2* *m-1(-) part of that theorem follows from Propositions 5.1 and 5.2 and the following re* *sult. Proposition 6.3. If m = 3, then v-11ss2m-1(E8; 3) Z=36. Proof.By Proposition 5.1 and the above computation that v-11ss6(E8; 3) Z=36, it suffices to show that v-11ss5(E8; 3) has an element x satisfying 35x 6= 0. For* * k = 2 and 3, let Mk denote the transpose of the matrix of k with respect to the basis {w1; w7; w11; w13; w17; w19; w23; w29} of Proposition 6.1. Thus Mk is the matri* *x whose rows have as entries the various coefficients in the equations of 6.1 with nume* *rical values obtained by substituting 2 or 3 for k. Let = M2- 8I, an invertible matr* *ix over Z(3). Let E = (1 1 1 1 1 1 1 1). Then EM3 is the sum of the rows of M3, and this combination of the wi's is in im( 3). Let A = EM3-1. Maple computes A = (-1_2; 548957_120; -81625061_2040; -3845766033_231880; -605269537661_75* *9778008; 1212671517798439_488157370140; 4278408406308902221_-248464589902573527757029703_ 200144521757400; 2238578548469645972700): The second and third denominators are divisible by 3, but the others are not. T* *hus 3A represents an element of P K1(E8; Z(3)), and 3A, which represents 2-23 appl* *ied to this vector, is in im( 3). Thus 3A 2 ker( 2 - 23 : P K1(E8; Z(3))= im( 3) ! P K1(E8; Z(3))= im( 3)). We will be done once we have shown that 36A 62 im( 3), for then 35(3A) is a nonzero element of the kernel, which, by Theorem 4.1, is isomorphic to v-11ss5(E8; 3). * *This is verified by Maple by successively subtracting multiples of the rows of M3 from * *36A to change the successive components of 36A to 0. For the first 7 steps, the req* *uired multiplier will be in Z(3), but when we get to the last step, our vector, which* * is equivalent to 36A mod im( 3), will be (0; 0; 0; 0; 0; 0; 0; 328u), with u a uni* *t in Z(3). Since the corresponding relation in im( 3) is 329, we conclude that 36A 62 im( * *3), as desired. || This concludes the proof of Theorem 1.2, with the only input being 2 in R(E8). We find it useful to interpret our result in terms of homotopy charts such as D* *iagram FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 39 5.11. We emphasize that this analysis is not part of our proof, but rather is a* *n attempt to understand how v-11ss*(E8; 3) is built from the v1-periodic homotopy groups * *of the eight spheres which build it. In [14], in which the UNSS was the primary tool, * *these charts were an integral part of the argument, but here we just use our computat* *ion to see how those charts must have been filled in. We are not saying that E8 is * *built from these eight spheres (It is not!), but rather that since P K1(E8; bZp) as a* *n Adams module is built from the related Adams modules for spheres (by our Theorem 3.5), v-11ss*(E8; 3) is built from that of the spheres, and we could probably conclud* *e that Bousfield's E8 is built from applied to the spheres. An important ingredient in the charts such as 5.11 and those of [14] is the c* *yclicity of the groups for the "sphere bundles" described in Proposition 5.5. In (E8; 3)* *, we also encounter "sphere bundles" with ffp as attaching map, a case which was not considered in 5.5. Here we have the following result, whose proof we merely ske* *tch. Proposition 6.4. If P K1(X; bZp) has generators x and y with ky = kn+p(p-1)y a* *nd kx = kn + u_pfflkn(kp(p-1)- 1)y, where u is a unit in Z(p)and ffl = 1 or 2, th* *en the only nonzero groups v-11ss*(X; p) are: 2 a. If ffl = 2, then v-11ss2n+qi(X; p) v-11ss2n+qi-1(X; p) Z=pe, where 8 >>min(n - 1; 2) if (i) = 0 >< min(n; (i) + 3) if (i - p) = 1 e = >>min(n + p(p - 1); (i - p) + 3) if (i - p) > 1 and n 4 >: min(4 + p(p - 1); (i - p - pp(p-1)+1) + ni-f1)(i - p) > 1 and 2 n 3 b. If ffl = 1 and n 2, then v-11ss2n+qi(X; p) v-11ss2n+qi-1(X; p) Z=pZ=pe, w* *here 8 >>1 if (i) = 0 >< min(n; (i) + 2) if (i - p) = 1 e = >>min(n + p(p - 1); (i - p) + 2) if (i - p) > 1 and n 3 >: min(3 + p(p - 1); (i - p - pp(p-1)+1)i+f1)(i - p) > 1 and n = 2 c. If ffl = 1 and n = 1, then v-11ss2n+qi(X; p) v-11ss2n+qi-1(X; p) Z=pmin(2+p(p-1);(i-p)+1): __________ 2If ffl = 2 and n = 1, then the coefficient of y in px is not in Z(3), and s* *o this case is excluded. 40 D. DAVIS Proof.The proof follows closely that of Proposition 5.5 with t = p. We let s = * *rp-1= 1 + fip, and note (si- 1) = 1 + (i). We consider first the case v-11ss2n+qi(X; * *p). First, let ffl = 2. As occurred in 5.5, the fourth relation allows one to eli* *minate y and deduce that the group is cyclic with relations (on generator x) p1+(i)pmin(n+p(p-1);(i-p)+1)andpn-2(pp(p-1)(si- 1) + sp- si): The only case in which it is not easily seen that the minimal exponent is as cl* *aimed occurs when n < 4 and i-p = u0pp(p-1)+1with u0 a unit in Z(p). In this case, th* *e first relation is pp(p-1)+4, and so we analyze the second relation mod pp(p-1)+4, obt* *aining i p i-p pn-2 pp(p-1)(1 + fip) - 1 - (1 + fip) (1 + fip) - 1 fipn+p(p-1)(1 - u0)(1 - fip=2); which yields the claim of the proposition. Now let ffl = 1 and n = 1. We use the second of the four relations similar to* * those in the proof of 5.5 to eliminate y, again obtaining that v-11ss2n+qi(X; p) is c* *yclic with relations on generator x pp(p-1)+2; p(si-p- 1); and pp(p-1)(si- 1) - sp(si-p- 1); whose minimal exponent of p is easily seen to be min(2 + p(p - 1); 1 + (i - p))* *, as claimed. Finally, let ffl = 1 and n > 1. All terms in all four relations are divisibl* *e by p. We split off a Z=p generated by 1=p times the last relation, and replace uy* * by (p(si- 1)=(sp- 1))x. The remaining summand has relations (on x) pn+p(p-1)+(i); p(i)+(i-p)+1; and pn-2(pp(p-1)(si- 1) + sp- si): The only case in which it is not easily seen that the smallest exponent is as c* *laimed occurs when i - p = u0pp(p-1)+1with u0 a unit in Z(p). Similarly to the case ff* *l = 2, the last relation becomes fip2+p(p-1)(1 - u0) mod pp(p-1)+3, which yields the c* *laim of the proposition. The result for v-11ss2n+qi-1(X; p) when ffl = 2 follows from the calculation * *of v2n+qi(X; p) just completed together with Propositions 5.1, 5.2, and 6.5. Proposition 6.5. If ffl = 2 and (i - p) p(p - 1) + n, then v-11ss2n+qi-1(X; p)* * is cyclic. FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 41 Proof.The proof is similar to that of Proposition 5.5. In the case considered h* *ere, |v-11ss2n+qi-1(X)| = pp(p-1)+nand, by (5.7), v-11ss2n+qi-1(S2n+pq+1) Z=pp(p-1)* *+n, and so the result follows from the exact sequence ! v-11ss2n+qi-1(S2n+1) ! v-11ss2n+qi-1(X) ! v-11ss2n+qi-1(S2n+pq+1) ! 0: || The result for v-11ss2n+qi-1(X) when ffl = 1 and n = 1 follows from Propositi* *ons 5.1 and 5.2 and the fact that if i 6 0 mod p, then v-11ss2n+qi(X) is cyclic, since * *it has order p. The proof for v-11ss2n+qi-1(X) when ffl = 1 and n 2 is more difficult, since* * it is not enough to verify it for one value of i. We focus on the case (i - p) = 1, (i) * *n - 2. Other cases are handled similarly. By Theorem 4.1 v-11ss2n+qi-1(X) ker(OE : G ! G); where G = OE(y)= rn(sp- si)y OE(x)= rn((1 - si)x + u_p(sp- 1)y): Using the second relation in G, we compute OE(pn-(i)-1x) = rnupn-(i)-2(sp- si)(1 + App(p-1))y; where A := (si- 1)=(sp- si) satisfies (A) = (i) - 1. Then z := pn-(i)-1x - upn-(i)-2(1 + App(p-1))y is in ker(OE), and p(i)+1z = -upn-1+p(p-1)u0y, where u0= (sp- 1)=(sp- si) is a * *unit. Thus this element p(i)+1z has order p in G, and so ker(OE) Z=p(i)+2 Z=p, with generators z and uu0pn+p(p-1)-2y + p(i)z. || We list in Diagrams 6.6 and 6.7 some representative charts indicating how v-1* *1ss4k+ffi(E8), ffi = 1; 2, is built from the various v-11ss4k+ffi(S2n+1). These charts are of * *the type of Diagram 5.11 and those of [14]. For each of the four cases considered in each d* *iagram, the left tower is building v-11ss4k+1(E8) and the right tower is building v-11s* *s4k+2(E8). 42 D. DAVIS The numbers on the left side of each diagram are the dimensions of the spheres.* * Dots represent Z=3, and a number e represents Z=pe. Vertical lines are extensions (m* *ul- tiplication by 3), as is the curved line from 23 to 3. Slanting arrows are boun* *dary morphisms in exact sequences. In the third and fourth case of both diagrams, t* *he boundary from the bottom generator is hitting a sum of two classes, which is re* *levant to the claim that the cokernel is cyclic. The extensions from 59 to 47, and from 15 to 3, are consequences of Propositi* *on 6.4 together with the terms 730_3(k -k7)w7 and 1_9(k23-k29)w29in Proposition 6.* *1. The other extensions are derived similarly from Proposition 5.5. The most unexpect* *ed part of the charts is the extension in Diagram 6.7 from 27 into the sum of 23 a* *nd 15 in dimension 4k + 2. This was seen to be necessary in order that v-11ss4k+2(E8)* * be cyclic in these cases. A check that this extension is really present was made b* *y hav- ing Maple compute v4k+2(X(15; 23; 27)), corresponding to the indicated subquoti* *ent Adams module from 6.1. This was done with k = 30, corresponding to the first ca* *se in 6.7, and Z=3 Z=36 was obtained, consistent with the unexpected extension. T* *his extension is probably also present in the cases of Diagram 6.6, but in those ca* *ses it would not affect the result. There is also an unexpected extension in 4k + 1 from 23 to p times the gener- ator in 15 when k 6 mod 9. This seemed necessary in order to get the cor- rect answer when (k - 6) 12, and was confirmed by a Maple computation that v-11ss4k+2(X(15; 23; 27)) Z=3 Z=3min(3+(k-6);13)if k 6 mod 9. The orders of * *the cyclic groups in the eight cases below are 6, (k - 11) + 9, 24, 24, 8, 8, (k - * *6) + 7, and 14, in agreement with Theorem 1.2. FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 43 Diagram 6.6. 3 r_HY r_ _rHY _r _rHY _r _rHY _r || H || || H || || H || || H || || H H || || H H || || H H || || H H || 15 |r|| Hr||| |||r H|||r |||r H|||r |||r H|||r |BBM | |BBM | |BBM | |BBM | | B | | B | | B | | B | 23 r||_B r||_ _2B _2 _2B _2 _2B _2 | B | | B | |BBMBBM | |ABAK | | | | | |B | |AAK | 27 r||B r|| r|B r| r|BBB r| r|BAA r| B B BBB JJ]BAA | B | || B || || BB || || BA || 35 r|| B r|| 2 B 2 2 BBBB 2 2J JBAAA2 | B | | B | | BBB | |@@IB@@I|JAA 39 r|| B r|| r|| B r|| r|| BBB r|| r||HYB@@r||JAA | B | | B | | BBB| |H H @B@|JAA | B | | B | | BBB| | HB@|JAA 47 r| Br| + 1 B+ 1 17 BB17 23HYHHYH23B@ | | | | | B | |H H | | B| | B| | B|BB | H HHB|H 59 r| r|B 2 2B 2 B2BB 2 H 2B = (k - 11) k 1 (3) (k - 11) = 16 (k - 11) 22 2 15 Diagram 6.7. 3 r_HY r_ _rHY r_ _rHY r_ _rHY r_ ||H@I || ||H@I || ||H@I || ||H@I || ||@ HH || ||@ HH || ||@ HH || f||@fHH_ || 15 4||A@KAH4||K 7||A@KAH7||KJ]@I@I2|| @ H2|| 2||@@I H2 || |BBML@L+A|A |BBML@L+A|AJ@@ |BBML@L+ | ||BBML@L+@ | |BL E|E |BLJ@ E|E |BL E|E ||BL@ EE| 23 r_BAA @ r_ _rBAAJ@@ r_ _rB @ r_ _rB_HY @ r_ | LAA E | LAA@@ E |ALK E |LH @ E | BLAA E | BLAAJ@E@ |ABL E |BLHH @E 27 2 BLAA 2 2 BLAAJ2 + 1BLA + 1 13QBLQkH13@I | BLAA | | BLAAJ | | BLA | |@BLQ@I@|JJ]AK | BL | | BL J| | BL | |A BLQ@ | 35 r| B AA r| |r B AA r| |r BA r| r|JB@AQ r| | L AA| | L AA| | L | | J L@@ | | BL A|A | BL A|A | BL | | AJBL@@| 39 2 B L2 2 B L2 2 B L2 2 A BJL2 | BL | | BL | | BL | | A BLJ| | B | | B | | B | | A B | 47 r| BLr| |r BLr| |r BLr| r| ABLr|J | | | | | | | | | B| | B| | B| | AB| 59 r| Br| |r Br| |r Br| r| Br|A = (k - 6) (k - 3) = 3 (k - 3) 6 (k - 6) 12 6 The charts when k 5,8 mod 9 are very similar to those when k 2 given in Diagram 6.6, while those when k 0 mod 9 are very similar to those when k 6 given in Diagram 6.7. Again we emphasize that these charts are not a part of our 44 D. DAVIS proof, but rather a way of interpreting our result in a way which is closer to * *past methods of calculating homotopy groups. 7. LiE program for computing 2 in R(E8) In this section we describe the program written in the specialized software L* *iE ([25]) to perform the calculation described in Section 2. We list the program * *and then describe what it is doing. setdefault E8 on + height mm=id(8) for r row mm do ext=alt_tensor(2,r); extt=ext; p2=0X null(8); x=1; while x==1 do x=0; for i=1 to length(extt) do u=expon(extt,i); if u[1]+u[2]+u[3]+u[4]+u[5]+u[6]+u[7]+u[8]>1 then j=1; while u[j]==0 do j=j+1 od; v=null(8); v[j]=u[j]; w=u-v; if w==null(8) then w[j]=1; v[j]=v[j]-1 fi; p1=tensor(v,w); n=length(p1); utop=expon(p1,n); if extt _ v==0 then x=1; p2=p2+1X v fi; if extt _ w==0 then x=1; p2=p2+1X w fi; for k=1 to n-1 do a=expon(p1,k); if extt _ a==0 then x=1; p2=p2+1X a fi od fi od; extt=extt+p2 od; pdim=0X null(8); pder1=1X[1,0,0,0,0,0,0,0]; pder2=1X[0,1,0,0,0,0,0,0]; pder3=1X[0,0,1,0,0,0,0,0]; pder4=1X[0,0,0,1,0,0,0,0]; pder5=1X[0,0,0,0,1,0,0,0]; pder6=1X[0,0,0,0,0,1,0,0]; pder7=1X[0,0,0,0,0,0,1,0]; FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 45 pder8=1X[0,0,0,0,0,0,0,1]; for i=1 to length(extt) do u=expon(extt,i); pdim=pdim+dim(u)X u; if u[1]+u[2]+u[3]+u[4]+u[5]+u[6]+u[7]+u[8]>1 then j=1; while u[j]==0 do j=j+1 od; v=null(8); v[j]=u[j]; w=u-v; if w==null(8) then w[j]=1; v[j]=v[j]-1 fi; p1=tensor(v,w); n=length(p1); c1=dim(v)*(pder1 _ w)+dim(w)*(pder1 _ v); c2=dim(v)*(pder2 _ w)+dim(w)*(pder2 _ v); c3=dim(v)*(pder3 _ w)+dim(w)*(pder3 _ v); c4=dim(v)*(pder4 _ w)+dim(w)*(pder4 _ v); c5=dim(v)*(pder5 _ w)+dim(w)*(pder5 _ v); c6=dim(v)*(pder6 _ w)+dim(w)*(pder6 _ v); c7=dim(v)*(pder7 _ w)+dim(w)*(pder7 _ v); c8=dim(v)*(pder8 _ w)+dim(w)*(pder8 _ v); for i=1 to n-1 do c1=c1-coef(p1,i)*(pder1 _ expon(p1,i)); c2=c2-coef(p1,i)*(pder2 _ expon(p1,i)); c3=c3-coef(p1,i)*(pder3 _ expon(p1,i)); c4=c4-coef(p1,i)*(pder4 _ expon(p1,i)); c5=c5-coef(p1,i)*(pder5 _ expon(p1,i)); c6=c6-coef(p1,i)*(pder6 _ expon(p1,i)); c7=c7-coef(p1,i)*(pder7 _ expon(p1,i)); c8=c8-coef(p1,i)*(pder8 _ expon(p1,i)) od; pder1=pder1+c1 X u; pder2=pder2+c2 X u; pder3=pder3+c3 X u; pder4=pder4+c4 X u; pder5=pder5+c5 X u; pder6=pder6+c6 X u; pder7=pder7+c7 X u; pder8=pder8+c8 X u fi od; der1=0; der2=0; der3=0; der4=0; der5=0; der6=0; der7=0; der8=0; for i=1 to length(ext) do 46 D. DAVIS der1=der1+coef(ext,i)*(pder1 _ expon(ext,i)); der2=der2+coef(ext,i)*(pder2 _ expon(ext,i)); der3=der3+coef(ext,i)*(pder3 _ expon(ext,i)); der4=der4+coef(ext,i)*(pder4 _ expon(ext,i)); der5=der5+coef(ext,i)*(pder5 _ expon(ext,i)); der6=der6+coef(ext,i)*(pder6 _ expon(ext,i)); der7=der7+coef(ext,i)*(pder7 _ expon(ext,i)); der8=der8+coef(ext,i)*(pder8 _ expon(ext,i)) od; print(der1);print(der2);print(der3);print(der4); print(der5);print(der6);print(der7);print(der8) od The first line says that the program is always working with the Lie algebra E* *8. The second line says that monomials are ordered by increasing height, a notion whic* *h was defined in Section 2. The first for loop, which extends throughout the program,* * lets r run over rows of an 8 x 8 identity matrix, with the ith iteration correspondi* *ng to computing 2(eaei). The variable ext is 2(aei) written as a polynomial whose exp* *onents are dominant weights, and coefficients are their multiplicity. The variable ext* *t will be a modified version of ext, expanded to include additional terms whose deriva* *tives must be computed, as described in Section 2. The variable x tells whether any new terms were adjoined to extt in the most recent iteration. Each exponent u in extt with sum of entries greater than 1 is decomposed as v+w. The polynomial p1 represents V (v) V (w). If v or w or any term of p1 does not appear in extt, t* *hen it is adjoined to extt, because we will need to know its derivatives. The purpo* *se of the 14-line while loop is just to adjoin these terms. The variable pdim is a polynomial whose coefficient of X u is dim(V (u)), whi* *le variables pderj are polynomials whose coefficient of X u is @j(V (u)), the coef* *ficient of eaejin L(V (u)). Here u ranges over all terms in extt; these will depend upo* *n which 2(eaei) we are computing. The long for loop computes these inductively. It st* *ill works with extt, which is 2(eaei) modified to include extra terms whose derivat* *ives are needed in the induction. The variable cj is @j(V (u)), i.e., the coefficien* *t of eaejin FROM REPRESENTATION THEORY TO HOMOTOPY GROUPS 47 V (u). It is determined by writing u = v + w, expanding X V (v) V (w) = V (u) + klV (tl) with height(tl)