CHAPTER 1 Computing v1-periodic homotopy groups of spheres and some compact Lie groups Donald M. Davis Lehigh University Bethlehem, PA 18015 dmd1@lehigh.edu Contents 1.Introduction:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: * * 3 2.Definition of v1-periodic homotopy groups::::::::::::::::::::::::::::::::::::* *::: 5 3.The isomorphism v-11ss*(S2n+1) v-11sss*-2n-1(Bqn):::::::::::::::::::::::::::* *:: 7 4.J-homology :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: * *12 5.The v1-periodic homotopy groups of spectra ::::::::::::::::::::::::::::::::::* *::: 23 6.The v1-periodic UNSS for spheres:::::::::::::::::::::::::::::::::::::::::::::* *: 26 7.v1-periodic homotopy groups of SU(n) ::::::::::::::::::::::::::::::::::::::::* *:: 36 8.v1-periodic homotopy groups of some Lie groups ::::::::::::::::::::::::::::::* *::: 44 References:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: * * 55 HANDBOOK OF ALGEBRAIC TOPOLOGY Edited by I.M. James cO1995 Elsevier Science B.V. All rights reserved 1 2 Donald M. Davis Chapter 1 Section 1 Computing periodic homotopy groups 3 1.Introduction In this paper, we present an account of the principal methods which have been u* *sed to compute the v1-periodic homotopy groups of spheres and many compact simple Lie groups. The two main tools have been J-homology and the unstable Novikov spectral sequence (UNSS), and we shall strive to present all requisite backgrou* *nd on both of these. The v1-periodic homotopy groups of a space X, denoted v-11ss*(X; p), are a cer- tain localization of the actual p-local homotopy groups ss*(X)(p). We shall dro* *p the p from the notation except where it seems necessary. Very roughly, v-11ss*(X) i* *s a periodic version of the portion of ss*(X) detectable by real and complex K-theo* *ry and their operations. It is the first of a hierarchy of theories, v-1nss*(X), w* *hich should account for all of ss*(X). Each group v-11ssi(X) is a direct summand of * *some actual group ssi+L(X), at least if X has an H-space exponent and v-11ssi(X) is a finitely-generated abelian group, which is the case in all examples discussed h* *ere. The v1-periodic homotopy groups are important because for spaces such as spheres and compact simple Lie groups they give a significant portion of the ac* *tual homotopy groups, and yet are often completely calculable. The goal of this pape* *r is to explain how those calculations can be made. One might hope that these methods can be adapted to learning about vn-periodic homotopy groups for n > 1. One application of v1-periodic homotopy groups is to obtain lower bounds for t* *he exponents of spaces. The p-exponent of X is the largest e such that some homoto* *py group of X contains an element of order pe. We can frequently determine the lar* *gest p-torsion summand in v-11ss*(X). Such a summand must also exist in some ssi(X), although we cannot usually specify which ssi(X). Thus we obtain lower bounds for the p-exponents of spaces, which we conjecture to be sharp in many of the cases studied here. It is known to be sharp for S2n+1 if p is odd. See Corollaries 7.* *8 and 8.9 for estimates of the p-exponent of SU (n) when p is odd. Another application, which we will not discuss in this paper, is to James numb* *ers, which are an outgrowth of work on vector fields in the 1950's. It is proved in * *[23] that, for sufficiently large values of the parameters, the unstable James numbe* *rs equal the stable James numbers, and these equal a certain value which had been conjectured by a number of workers. In Section 2, we present the definition and basic properties of the v1-periodic homotopy groups. In Section 3, we describe the reduction of the calculation of the unstable groups v-11ss*(S2n+1) to the calculation of stable groups v-11sss** *(Bqn). Here Bqn is a space which will be defined in Section 3; when p = 2, it is the r* *eal projective space P 2n. Here we have begun the use, continued throughout the pap* *er, of q as 2p - 2. The original proof of this result, 3.1, appeared in [40] and [5* *6]; it involved delicate arguments involving the lambda algebra. We present a new proo* *f, due to Langsetmo and Thompson, which involves completely different techniques, primarily K-theoretic. In Section 4, we explain how to compute J*(Bqn), while in Section 5, we sketch the proof that if X is a spectrum, then v-11ss*(X) v-11J*(X). Combining the results of Sections 3, 4, and 5 yields a nice complete result for v-11ss*(S2n+1* *), which 4 Donald M. Davis Chapter 1 can be summarized as v-11ss2n+1+i(S2n+1) v-11sssi(Bqn) v-11Ji(Bqn) and, if p is odd, ae min(n;p(a)+1) Z=p if i = qa - 2 or qa - 1 0 if i 6 -1 or -2 mod q. We will use Z=n and Zn interchangeably, and let p(n) denote the exponent of p in n. The subscript p of will sometimes be omitted if it is clear from the con* *text. The final result for v-11ss*(S2n+1) when p = 2 is more complicated; see Theorem 4.2. In Section 6, we sketch the formation of the UNSS and its v1-localization, and* * for S2n+1 we compute the entire v1-localized UNSS and part of the unlocalized UNSS. In Section 7, we discuss the computation of the v1-periodic UNSS and v1-periodic homotopy groups in general for spherically resolved spaces and specifically for* * the special unitary groups SU(n). This is considerably easier at the odd primes tha* *n at the prime 2. The following key result of Bendersky ([4]) will be proved by obse* *rving that the homotopy-theoretic calculation and UNSS calculation agree for S2n+1. If p is odd, and X is built by fibrations from finitely many odd-dimensional sp* *heres, then v-11Es;t2(X) = 0 in the v1-periodic UNSS unless s = 1 or 2 and t is odd, in which case ae-1 1;i+1 v-11ssi(X; p) v1-E212;(X)i+2if i is even v1 E2 (X) if i is odd. In Section 7 we also review the computation of E12(SU (n)) in [6] and combine * *it with Theorem 1.1 to obtain the following result, which was the main result of [* *23]. Let p(m) denote the exponent of p in m, and define integers a(k; j) and ep(k; n) by X xk (ex - 1)j = a(k; j)__ ; kj k! and ep(k; n) = min{p(a(k; j)) : n j k}: If p is odd, then v-11ss2k(SU (n)) Z=pep(k;n), and v-11ss2k-1(SU (n)) is an ab* *elian group of order pep(k;n), although not always cyclic. In Section 8, we illustrate the two principal methods used in computing v1- periodic homotopy groups of the exceptional Lie groups, focusing on v-11ss*(G2;* * 5) Section 2 Computing periodic homotopy groups 5 for UNSS methods, and on v-11ss*(F4=G2; 2) for homotopy (J-homology) methods. We also discuss the recent thesis of Yang ([58]), which gives formulas more tra* *ctable than that of Definition 1.2 for the numbers ep(k; n) which appear in Theorem 1.* *3, provided n p2- p. 2.Definition of v1-periodic homotopy groups In this section, we present the definition and basic properties of the v1-perio* *dic homotopy groups. We work toward the definition of v-11ss*(X) by recalling the definition of v-11ss*(X; Z=pe). Let Mn(k) denote the Moore space Sn-1 [k en. The mod k homotopy group ssn(X; Z=k) is defined to be the set of homotopy classes [Mn(k); X]. With the prime p implicit, and q = 2(p - 1), let aee-1 s(e) = pmaxq(8; 2e-1if)piisfoddp = 2. (2.1) Let A : Mn+s(e)(pe) ! Mn(pe) denote a map, as introduced by Adams in [1], which induces an isomorphism in K-theory. Such a map exists provided n 2e+3. ([28, 2.11]) Then v-11ssi(X; Z=pe) is defined to be dirlimN[Mi+Ns(e)(pe); X], w* *here the maps A are used to define the direct system. The map is what Hopkins and Smith would call a v1-map, and they showed in [35] that any two vn-maps of a finite complex which admits such maps become homotopic after a finite number of iterations (of suspensions of the same map), and hence v-11ss*(X; Z=pe) does not depend upon the choice of the map A. Note that although v-11ss*(X; Z=pe) is a theory yielding information about the unstable homotopy groups of X, the maps A which define the direct system may be assumed to be stable maps, since the direct limit only cares about large values of i + Ns(e). Note also that the groups v-11ssi(X; Z=pe) are defined for all integers i and satisfy v-11ssi(X; Z* *=pe) v-11ssi+s(e)(X; Z=pe). There is a canonical map ae : Mn(pe+1) ! Mn(pe) which has degree p on the top cell, and degree 1 on the bottom cell. It satisfies the following compatibility* * with Adams maps. [34, p. 633] If A : Mn+s(e)(pe) ! Mn(pe) and A0: Mn+s(e+1)(pe+1) ! Mn(pe+1) are v1-maps, then there exists k so that the following diagram commutes. Mn+ks(e+1)(pe+1)ae-!Mn+ks(e+1)(pe)?? ?yA0k ?y 0 Akp Mn(pe+1) -ae! Mn(pe) Here p0= p unless p = 2 and e < 4, in which case p0= 1. 6 Donald M. Davis Chapter 1 Thus, after sufficient iteration of the Adams maps, there are morphisms ae* be- tween the direct systems used in defining v-11ss*(X; Z=pe) for varying e, and p* *assing to direct limits, we obtain a direct system * ae* v-11ss*(X; Z=pe) -ae!v-11ss*(X; Z=pe+1) -! . .:. (2.2) The following definition was given in [28], following less satisfactory definit* *ions in [30] and [23]. For any space X and any integer i, v-11ssi(X) = dirlimev-11ssi+1(X; Z=pe); using the direct system in (2.2). The reason for the use of the (i+1)st mod-pe periodic homotopy groups in defini* *ng the ith (integral) periodic groups is that the maps ae of Moore spaces have deg* *ree 1 on the bottom cells, but mod-pe homotopy groups are indexed by the dimension of the top cell. The mod-pe periodic homotopy groups have received more attention in the liter- ature, especially when e = 1. For spaces with H-space exponents, there is a clo* *se relationship between the integral periodic groups and the mod-pe groups, which * *we recall after giving the relevant definition. A space X has H-space exponent pe if for some positive integer L the pe-power map ffiLX ! ffiLX is null-homotopic. By [20] and [36], spheres and compact Lie groups have H-space exponents. (i)[28, 1.7] If X has H-space exponent pe, then there is a split short exact sequence 0 ! v-11ssi(X) ! v-11ssi(X; Z=pe) ! v-11ssi-1(X) ! 0: (ii)On the category of spaces with H-space exponents, there is a natural tran* *s- formation ss*(-)(p)! v-11ss*(-; p). (iii)If X has an H-space exponent, and v-11ssi(X) is a finitely generated abe* *lian group, then v-11ssi(X) is a direct summand of ssi+L(X) for some non-negative in- teger L. The proof of part ii utilizes the fibration e map *(Mn+1(pe); X) ! ffinX -p! ffinX; where map *(-; -) denotes the space of pointed maps. If the second map is null- homotopic, then the first map admits a section s. The natural transformation is induced by ffinX -s! map*(Mn+1(pe); X) ! dirlimkmap*(Mn+1+ks(e)(pe); X): Section 3 Computing periodic homotopy groups 7 Techniques of [28] imply naturality of this construction. To prove part iii, we* * note that the map s allows v-11ssi(X) to be written as dirlimkssi+ks(e)(X), which is* * a direct summand of one of the groups in the direct system, provided the direct l* *imit is finitely generated. If X is a spectrum, then v-11ss*(X) is defined in exactly the same way as for * *spaces, that is, as in Definition 2.2. If X is a space, then stable groups, v-11sss*(X)* *, can be defined either as v-11ss*(21 X), where 21 X denotes the suspension spectrum of X, or as v-11ss*(QX), where QX = ffi1 21 X is the associated infinite loop spac* *e. 3.The isomorphism v-11ss*(S2n+1) v-11sss*-2n-1(Bqn) In this section, we sketch a new proof, due to Thompson and Langsetmo ([39, 4.2* *]), of the following crucial result. There is a map ffi2n+1S2n+1! QBqn (3.1) which induces an isomorphism in v-11ss*(-). Here QX = ffi1 21 X, and Bqn is the qn-skeleton of the p-localization of the cl* *as- sifying space B 2p of the symmetric group 2p on p letters. Note that if p = 2, * *then Bqn is the real projective space RP 2n. The original proof, from [40] when p = 2 and [56] when p is odd, involved delicate arguments involving the lambda algebra and unstable Adams spectral sequences. We feel that the following argument, pri- marily K-theoretic, will speak to a broader cross-section of readers. The follo* *wing elementary result shows that it is enough to show that the map (3.1) induces an iso in v1-periodic mod p homotopy. If a map X ! Y induces an isomorphism in v-11ss*(-; Z=p), then it induces an isomorphism in v-11ss*(-; p). Proof. There are cofibrations of Moore spaces which induce natural exact se- quences * ! v-11ssn(X; Z=pe)ae-!v-11ssn(X; Z=pe+1) ! v-11ssn(X; Z=p)! v-11ssn-1(X; Z=pe) ! : Induction on e using the 5-lemma implies that there are isomorphisms v-11ss*(X; Z=pe) ! v-11ss*(Y ; Z=pe) for all e, compatible with the maps ae* which define the direct system (2.2). T* *he desired isomorphism of the direct limits is immediate. || The construction of the map (3.1) takes us far afield, and is not used elsewhe* *re in the computations. For completeness, we wish to say something about it, but we w* *ill 8 Donald M. Davis Chapter 1 be extremely sketchy. The map is due to Snaith. Work of many mathematicians, especially Peter May, is important in the construction. However, we shall just * *refer the reader to [37], where the proof of naturality of these maps is given, along* * with references to the earlier work. i. There are maps sn : ffi2n+1S2n+1! QBqn which are compatible with respect to inclusion maps as n increases, and such th* *at the adjoint map 21 ffi2n+1S2n+1!21 Bqn is the projection onto a summand in a decomposition of 21 ffi2n+1S2n+1 as a wed* *ge of spectra. g ii. There is a map QS2n+1 -! Q 22n+1B(n+1)q-1whose fiber, F, satisfies v-11ss*(ffi2n+1F; Z=p) v-11ss*(QBqn; Z=p): Sketch of proof. Let CN (k) denote the space of ordered k-tuples of disjoint l* *ittle cubes in IN . If X is a based space, let CN X denote the space of finite collec* *tions of disjoint little cubes of IN labeled with points of X. More formally, a CN X = CN (k) x2k Xk= ~; k1 where [(c1; : :;:ck); (x1; : :;:xk-1; *)] ~ [(c1; : :;:ck-1); (x1; : :;:xk-1)]: There are natural maps, due to May, CN X ! ffiN 2N X; (3.2) which are weak equivalences if X is connected. The space CN X is filtered by defining Fm (CN X) to be the subspace of m or fewer little labeled cubes. The successive quotients are defined by DN;mX = Fm (CN X)=Fm-1 (CN X) CN (m)+ ^2m X[m]; where X[m]denotes the m-fold smash product. Snaith proved that if X is path- connected, there is a weak equivalence of suspension spectra _ 21 CN X ' 21 DN;mX: (3.3) m1 Section 3 Computing periodic homotopy groups 9 Let N = 2n + 1 and X = S0. Stabilize the equivalence of (3.2), and project onto the summand of (3.3) with m = p to get 21 ffi2n+1S2n+1!21 C2n+1(p)= 2p : The identification of C2n+1(p)= 2p as the nq-skeleton of B 2p after localizatio* *n at p was obtained by Fred Cohen in [19, p. 246]. Note that (3.2) and (3.3) yield a map ffiN 2N X ! QDN;mX. The map g of ii) is obtained from the case N = 1 and X = S2n+1 as the composite QS2n+1! QD1;pS2n+1= Q(B 2+p^2p(S2n+1)[p]) ' Q 22n+1B(n+1)q-1: Here we have used results of [45] for the last equivalence. The compatible maps sn of 3.3(i) combine to yield a map s0: QS0 ! QB1 , and one can show that the right square commutes in the diagram of fibrations below. 2n+1g ffi2n+1F!?QS0?ffi-! QB(n+1)q-1? ?yt ?ys0 ?y= QBqn ! QB1 ! QB(n+1)q-1 The map t then follows, and will induce an isomorphism in v-11ss*(-; Z=p), as asserted in Theorem 3.3(ii), once we know that s0does. Kahn and Priddy showed that there is an infinite loop map : QB1 ! QS0 such that O s0 induces an isomorphism in ssj(-) for j > 0. It is easily verified using methods of the next two sections that the associated stable map B1 ! S0 induces an isomorphism in v-11ss*(-; Z=p). Hence so does s0. || Throughout this section, let K*(-) denote mod p K-homology, and Mk denote the mod p Moore space Mk(p). The following two theorems, whose proofs occupy most of the rest of this section, imply that j : S2n+1 ! F induces an iso in v-11ss*(-; Z=p). [16, 14.4] If k 2, and OE : X ! Y is a map of k-connected spaces such that ffi* *kOE is a K*-equivalence, then OE induces an isomorphism in v-11ss*(-; Z=p). [39] Let F be as in Theorem 3.3ii. The map S2n+1 ! QS2n+1 lifts to a map j : S2n+1! F such that ffi2j is a K*-equivalence. The map j of Theorem 3.5 can be chosen so that t O ffi2n+1j = sn, where t is t* *he map in the proof of Theorem 3.3ii which induces the isomorphism in v-11ss*(-; Z* *=p). Theorem 3.1 is now proved by applying Lemma 3.2 to sn. Bousfield localization is involved in the proof of Theorem 3.4 and several oth* *er topics later in the paper, and so we review the necessary material. If E is a s* *pec- trum, then a space (or spectrum) X is E*-local if every E*-equivalence Y ! W induces a bijection [W; X] ! [Y; X]. The E*-localization of X is an E*-local sp* *ace 10 Donald M. Davis Chapter 1 (or spectrum) XE together with an E*-equivalence X ! XE . The existence and uniqueness of these localizations were established in [18] and [17]. We will ne* *ed the following result of Bousfield, the proof of which is easily obtained using the * *ideas at the beginning of Section 5. [18, 4.8] A spectrum is K*-local if and only if its mod p homotopy groups are periodic under the action of the Adams map. We will omit the proof of Theorem 3.4, as it requires many peripheral ideas. Instead, we will sketch a proof of the following result, which, although weaker* * than 3.4, has the same flavor, and predated it. An alternate proof of Theorem 3.1 ca* *n be given by using Theorem 3.7 and strengthening Theorem 3.5 to show that K*(ffi3j) is bijective. As the calculations for ffi3 seem significantly more difficult th* *an for ffi2, we omit that approach. [57] Let p be an odd prime, and let X and Y be 3-connected spaces. Suppose that f : X ! Y is a map such that K*(ffikf) is an isomorphism for k = 0, 1, 2, and 3. Then f induces an isomorphism in v-11ss*(-; Z=p). Sketch of proof. If F ! E ! B is a principal fibration, there is a bar spec- tral sequence converging to K*(B) with E2s;t T orK*Fs;t(K*E; K*). (See [55] for* * a discussion of this spectral sequence.) Applied to the commutative diagram of principal fiber sequences ffi3Xp-!ffi3X?!map*(M3;?X)? ?yffi3f?yffi3f ?yf0 ffi3Yp-!ffi3Ym!ap*(M3; Y ); the spectral sequence and the hypothesis of the theorem imply that f0 induces an isomorphism in K*(-). Hence there is an equivalence of the K*-localizations 0 (map *(M3; X))K -fK!(map *(M3; Y ))K : Let V (X) denote the mapping telescope of * A* map *(M3; X) A-!map *(M3+q; X) -! . .:. Then V (X) is K*-local, since it is ffi1 of a periodic spectrum which is K*-lo* *cal by Theorem 3.6. This implies that there are maps i0 making the following diagram commute. 0 map *(M3;?X)! (map *(M3;?X))K -i!V?(X) ?yf0 ?yf0 ? K 0y V (f) map *(M3; Y!)(map *(M3; Y ))K -i!V (Y ) Section 3 Computing periodic homotopy groups 11 Since v-11ss*(X; Z=p) ss*(V (X); Z=p), the desired isomorphism is a consequence of the following construction of an inverse to V (f)* : ss*(V (X); Z=p) ! ss*(V (Y ); Z=p): An element ff 2 ssk(V (Y ); Z=p) can be represented by a map Mk ! map*(M3+qpj; Y ); or, adjointing, by a map Mk+qpj! map*(M3; Y ). Here some care is required to see that we can switch the Moore space factor on which the map A is performed. The element which corresponds to ff is the composite Mk+qpj ! map *(M3; Y ) ! (map *(M3; Y ))K (f0K)-1-!(map 3 i0 *(M ; X))K -! V (X): || The proof of Theorem 3.5 involves a good bit of delicate computation. The hard- est part is the determination of K*(ffi2F) as a Hopf algebra. In order to conve* *niently obtain the coalgebra structure of K*(ffi2F), we proceed in two steps. We first * *calcu- late the algebra K*(ffi3F), using the bar spectral sequence associated to the p* *rincipal fibration ffi4QS2n+1! ffi4Q 22n+1Bq(n+1)-1! ffi3F: This spectral sequence is calculated in [38], obtaining an algebra isomorphism K*(ffi3F) P [y1; y2] E[z]: (3.4) Here yi (resp. z) has bidegree (1; 1) (resp. (0; 1)) in the spectral sequence, * *and hence even (resp. odd) degree in K*(ffi3F). The calculation of this spectral se* *quence requires some preliminary computation regarding the algebra structure of K*(ffi* *2F), and this requires major input from [44]. Now we calculate the bar spectral sequence associated to the principal fibrati* *on ffi3F ! * ! ffi2F: This spectral sequence, with E2 T orP[y1;y2]E[z](K*; K*), collapses to yield an isomorphism of Hopf algebras K*(ffi2F) E[a1; a2] .[b]; (3.5) where . denotes the divided polynomial algebraPover Zp. The coproduct has a1, a* *2, and fl1(b) as the primitives, and (fli(b)) = flj(b) fli-j(b). 12 Donald M. Davis Chapter 1 Dualizing eq. (3.5) yields an isomorphism of algebras K*(ffi2F) E[ff1; ff2] P [fi]; with |ffi| odd and |fi| even. This matches nicely with the following result fro* *m [52, 3.8]. For any prime p, there is an isomorphism of algebras K*(ffi2S2n+1) E[u0; u1] P [w]; where E and P denote exterior and polynomial algebras over K*. We will use the Atiyah-Hirzebruch spectral sequence to show that the map 2j ffi2S2n+1 ffi-!ffi2F of Theorem 3.5 sends the generators of the isomorphic K*(-)-algebras across. Th* *is will imply the second half of Theorem 3.5. In [52], it is shown how the generators u0, w, and u1 of K*(ffi2S2n+1) arise i* *n the Atiyah-Hirzebruch spectral sequence whose E2-term is H*(ffi2S2n+1; K*). Indeed, they arise from the bottom three cohomology classes, of grading 2n - 1, 2pn - 2, and 2pn - 1, respectively. The map ffi2j induces an isomorphism in Hi(-; Zp) for i < 2pn-1+min(q; 2n-2), and so it maps onto the three generators of K*(ffi2S2n+* *1). 4.J-homology In this section, we show how to compute J*(Bqn). When combined with Theorems 3.1 and 5.1, this gives an explicit computation of v-11ss*(S2n+1), which we sta* *te at the end of this section as Theorem 4.2. This will be extremely important in our calculation of v-11ss*(Y ) for other spaces Y . We begin with the case p odd, w* *here the results are somewhat simpler to state. Historically it worked in the other * *order, with Mahowald's 2-primary results in [40] preceding Thompson's odd-primary work in [56]. Let p be an odd prime. We follow quite closely the exposition in [24] and [56,* * x3]. The spectrum bu(p)splits as a wedge of spectra 22i` satisfying H*(`; Zp) A==E, where A is the mod p Steenrod algebra, and E is the exterior subalgebra generat* *ed by Q0 = fi and Q1 = P 1fi - fiP 1. The spectrum ` is sometimes written BP <1>. Then `* = ss*(`) is calculated from the Adams spectral sequence (ASS) with Es;t2 Exts;tA(H*`; Zp) Exts;tE(Zp; Zp) Zp[a0; a1]; (4.1) where ai has bigrading (1; iq + 1). Here we have used the change-of-rings theor* *em in the middle step. There are no possible differentials in the spectral sequenc* *e, and since multiplication by a0 corresponds to multiplication by p in homotopy, we f* *ind Section 4 Computing periodic homotopy groups 13 that ss*(`) is a polynomial algebra over Z(p)on a class of grading q. Using the* * ring structure of `, one easily sees that there is a cofibration 2q ` ! ` ! HZ(p): Let k be a (p - 1)st root of unity mod p but not mod p2, and let k denote the Adams operation. The map k-1 : ` ! ` lifts to a map : ` !2q `. The connective J-spectrum, J, is defined to be the fiber of . The homotopy exact sequence of easily implies that ( Z (p) if i = 0 ssi(J) Z=pp(j)+1 if i = qj - 1 with j > 0 0 otherwise. The image of the classical J-homomorphism is mapped isomorphically onto these groups by the map S0 ! J; this is the reason for the name of the spectrum. We now proceed toward the calculation of J*(Bqn). We let B be the p-localizati* *on of the suspension spectrum of B 2p. Then, with coefficients always in Zp, the o* *nly nonzero groups Hi(B) occur when i 0 or -1 mod q, and i > 0. These groups are cyclic of order p with generator xisatisfying Q0xaq-1= xaqand Q1xaq-1= x(a+1)q. We will work with the skeleta Bqn and the quotients Bq(n+1)-1= B=Bqn; these are suspension spectra of the spaces which appeared in Theorem 3.3. There is a p-local map B ! S0 constructed by Kahn and Priddy. If R de- notes its cofiber, there is a filtration of the E-module H*R with subquotients 2qiE==E0 for i 0. Here E0 is the exterior subalgebra of E generated by Q0. Sin* *ce ExtE(E==E0) ExtE0(Zp) Zp[a0], we find that ExtE(H*R) has a "spike," con- sisting of the powers of a0, for each nonnegative value of t - s which is a mul* *tiple of q. Here we have begun a practice of omitting Zp from the second variable of ExtB(-; -) if B is a subalgebra of the Steenrod algebra. The action of ExtE(Zp) on ExtE(H*R) has a1 always acting nontrivially. We draw ASS pictures with coordinates (t - s; s), so that horizontal component refers to homotopy group. A chart for the ASS of R ^ ` is given in fig. 1. The short exact sequence 0 ! H*(2 B) ! H*(R) ! H*(S0) ! 0 induces an exact sequence ! Exts;tE(H*S0) -i*!Exts;tE(H*R) ! Exts;tE(H*(2 B)) ! Exts+1;tE(H*S0) ! : (4.2) These morphisms are ExtE(Zp)-module maps, and the action of a1 implies that i* 14 Donald M. Davis Chapter 1 Figure 1. ASS for `*(R) |6 |6 |6 | | | | | | | | | | | | |r |r |r | | | |r |r |r. . . | | | _____________________________________________|||rrr t - s =0 q 2q is injective. Thus there are elements xiq-12 Ext0;iq-1E(H*B) for i > 0 such that ae Exts;tE(H*B) = Zp0 ifott-hse=riqw-i1,sie>,0, 0 s < i with generators as0xiq-1. Hence ae (i+1)=q `i(B) Z=p if i -1 mod q, and i > 0 0 otherwise. There is an isomorphism of E-modules H*(Bq(n+1)-1) 2qnH*(B), and so `*(Bq(n+1)-1) `*(2qnB): The Ext calculation easily implies that the morphism `*(B) ! `*(Bq(n+1)-1) in- duced by the collapse map is surjective, and so the exact sequence of the cofib* *ration Bqn ! B ! Bq(n+1)-1implies that ae min((i+1)=q;n) `i(Bqn) Z=p if i -1 mod q, and i > 0 0 otherwise. The ASS chart for B4q is illustrated in fig. 2. The map S0 ! R implies that * : `qj(X) ! `(q-1)j(X) is multiplication by the same number for X = R as it was for X = S0. Thus it is multiplication by pp(j)+* *1. Now the map R !2 B implies that * : `qj-1(B) ! `(q-1)j-1(B) is multiplication by pp(j)+1. The maps Bqn ! B ! Bq(n+1)-1imply that the same is true in Bqn and Bq(n+1)-1. We obtain 8 min(n; (j)+1) > 0 Ji(Bqn) > Z=pmin(n-1p; (jif)i)= jq - 2, j > n (4.3) : Z=p p if i = jq - 2, 0 < j n 0 otherwise. Section 4 Computing periodic homotopy groups 15 Figure 2. ASS for `*(B4q) for t - s < 6q r | r r| | | r r| r| | | | r r| r| r| | | | r r| r| r| | | | ____________________________________________________________rrrr||| q - 1 2q - 1 3q - 1 4q - 1 5q - 1 6q - 1 This is illustrated in fig. 3, which is not quite an ASS chart. It is a combina* *tion of the charts for `*(Bqn) and `*-q+1(Bqn) and the homomorphism * between them, which is represented by lines of negative slope. The exact sequence 0 ! coker(*+1) ! J*(Bqn) ! ker(*) ! 0 says that elements which are not involved in these boundary morphisms com- prise J*(Bqn). There are several reasons for our having elevated the filtration* *s of `*-q+1(Bqn) by 1 in this chart. One is that it makes all the boundary morphisms* * go up, so that it looks like an ASS chart. Another is that (by [40]) there is a re* *solution of Bqn^J (which is not an Adams resolution) for which the homotopy exact couple is depicted by this chart. A third is that if J1 is defined to be the fiber of * *J ! HZ2, then the ASS chart for Bqn^ J1 will agree with this chart in filtration greater* * than 1. See [13, x6] for an elaboration on this. Figure 3. Beginning of chart for J*(B4q) r r |A | r r r| r| |@ | |AA| r r r|@r| r|Ar| |@| |@ | |A | r r r|r|@ r|@r| r|Ar| |A| |@| |@ | r r r|r| r|r|@ r|@r| @ | A| @ | ____________________________________________________________rrrr|||@A@ q - 1 2q - 1 3q - 1 4q - 1 5q - 1 6q - 1 If X is a space or spectrum, then v-11Ji(X) is defined analogously to Definiti* *on 2.2 to be dirlime;k[Mi+1+ks(e)(pe); X ^J]. Since J is a stable object, we can S-dua* *lize the Moore space, obtaining dirlime;kJi+1+ks(e)(X ^ M(pe)), where the Moore spectrum M(n) has cells of degree 0 and 1. The "+1" in this J-group is present due to the maps M(pe) ! M(pe+1) having degree 1 on the 1-cell. 16 Donald M. Davis Chapter 1 One can often compute v-11J*(X) directly from J*(X) without having to worry about the "^M(pe)". This can be done by extending the periodic behavior which occurs in positive filtration down into negative filtrations and negative stems* *. For example (cf. fig. 3), a chart for v-11J*(Bnq) has, for all integers a, adjacent* * towers of height n in aq - 2 and aq - 1 with d(a)+1-differential. If (a) + 1 n, then * *the differential is 0. This interpretation of v-11J*(-) can be justified using the * *following result. Let p be an odd prime, and let KU be the spectrum for periodic K-theory localiz* *ed at p. Let k be ak(p--11)st root of unity mod p but not mod p2, and let Ad denot* *e the fiber of KU - ! KU. Then v-11J*(-) Ad*(-). This follows from the fact that if v-11`*(-) is defined as dirlime;k`i+1+ks(e)(* *X ^ M(pe)), then v-11`*(-) KU*(-), which is a consequence of the fact that A ^ = v ^ 1M :2q M ! M ^ `, where A :2q M ! M and v : Sq ! `. When p = 2, the results are a bit messier to state and picture. If bsp denotes* * the 2-local connected ffi-spectrum whose (8k)th space is BSp[8k], then 24 bsp ' bo[* *4], the spectrum formed from bo by killing ssi(-) for i < 4. The map 3 - 1 : bo ! * *bo lifts to a map : bo !24 bsp, and J is defined to be the fiber of . Let A1 denote the subalgebra of the mod 2 Steenrod algebra A generated by Sq1 and Sq2. Then H*bo A==A1 and H*bsp A A1 N, where N = <1; Sq2; Sq3>. Hence, using the change of rings theorem, the E2-term of the ASS converging to ss*(bo) is ExtA1(Z2), while that for bsp is ExtA1(N). These are easily computed to begin as in fig. 4, with each chart acted on freely by an element in (t - s;* * s) = (8; 4). Positively sloping diagonal lines indicate the action of h1 2 Ext1;2A1(* *Z2). It corresponds to the Hopf map j in homotopy. Figure 4. Part of ASS for bo and bsp bo bsp | |6 |6 | | | | |r | |6 | |6 | | |r | |r | | | | |r |r |r |r r | | | |r r |r |rr | | | |r r |r |r | | _________________|r _______________________|r 0 4 0 4 Section 4 Computing periodic homotopy groups 17 There are no possible differentials in these ASS's, and so we obtain ( Z (2) if i 0 mod 4, i 0 ssi(bo) Z2 if i 1; 2 mod 8, i > 0 0 otherwise, in accordance with Bott periodicity. From Adams' work, we have * : ss4j(bo) ! ss4j(24 bsp) hitting all multiples of 22(j)+3, while * is 0 on the Z2's. This yields ssi(J) = 0 if i < 0, while for i* * 0 8 >>>Z(2) if i = 0 < Z=22(i)+1 if i 3 mod 4 ssi(J) > Z2 if i 0; 2 mod 8; i > 0 (4.4) >>:Z2 Z2 if i 1 mod 8; i > 1 0 if i 4; 5; 6 mod 8. From Adams' work and the confirmation of the Adams Conjecture, it is known that ss*(S0) ! ss*(J) sends the image of the classical J-homomorphism plus Adams' elements j and jj isomorphically onto ss*(J). A chart for ssi(J) with i 18 is given in fig. 5. Here the elements coming from bo are indicated by o's, while t* *hose from 24 bsp are indicated by O's. Figure 5. 2-primary ssi(J), i 18 |6 b| |BB |6 bBB | |b b|BB r | |6b b|@||6r@ b|||6rrBB |6 |6 | B| | b| b|A b|@|r@ b||rBb | |6 A |6 | | b|@r|@ b|A|rA b|@|r@ b|b | | A | pp | b|@r|@ b|A|rA r b| b| | | A | |r b|@r|@ b|A|rAr b| | | A | |r b|@r|@ b|A|r b | | |r b|@r|@ b| b | pp |r r b| b| | |r r b| | __________________________________________________|r 0 3 7 11 15 The first dotted j-extension can be deduced from the fact that * : H4(24 bsp) ! H4(bo) hits Sq4, together with the relation j3 = 4. This j-action is then pushed 18 Donald M. Davis Chapter 1 along by periodicity. Another argument which is frequently useful for deducing j-extensions such as this involves Toda brackets. The generator of ss8i+4(bo) is obtained from the element ff 2 ss8i+2(bo) as . Clearly ff pulls back * *to ss*(J), and if ffj were 0 here, then the bracket could also be formed in J. However, the boundary morphism on ss8i+4(bo) implies that this bracket cannot be formed, and so ffj must be nonzero in J. Moreover, it must be the element fi such that 2fi * *is ffi(). We will rename Bqn as P 2nwhen p = 2, with P denoting the suspension spectrum of RP 1. As in the odd primary case, there is a map : P ! S0 with nontriv- ial cohomology operations in its mapping cone R. This 2-primary map can be viewed more geometrically than its odd-primary analogue, as an amalgamation of composites P n! SO(n + 1) -J!ffinSn: With A0 denoting the exterior subalgebra of A generated by Sq1, H*R can be filtered as an A1-module with subquotients 24i A1==A0 for i 0, and so ExtA(H*(R ^ bo)) consists of h0-spikes rising from each position (t - s; s) = (* *4i; 0) for i 0. Here h0 is the element of Ext1;1A(Z2) or Ext1;1A1(Z2) corresponding t* *o Sq1 and to multiplication by 2 in homotopy. Also, we begin a practice of using with* *out comment the relation ExtA(H*(X ^ bo)) ExtA(H*X A==A1) ExtA1(H*X): Thus the nonzero groups boi(R) occur only when i 0 mod 4 and i 0, and these groups are Z(2). We also need ( Z (2) if i 0 mod 4, i 0 bspi(R) boi(R ^ (S0 [j e2[2e3)) Z2 if i 2 mod 4, i > 0 0 otherwise. The Z2's are obtained from the exact sequence in bo*(-) associated to the cofib* *ra- tion R ^ S2 ! R ^ (S2 [2e3) ! R ^ S3 -2!: Analogous to (4.2) is an exact sequence which allows us to compute ExtA1(H*P ) from ExtA1(H*S0) and ExtA1(H*R). This is most easily seen in the chart of fig. * *6, in which o's are from ExtA1(H*S0), and O's are from ExtA1(H*R). The groups are read off from this as 8 4j+3 > Z=2 i = 8j - 1, j > 0 : Z2 i = 8j + 1 or 8j + 2, j 0 0 otherwise. Section 4 Computing periodic homotopy groups 19 Figure 6. ExtA1(H*P), from ExtA1(H*S0) and ExtA1(H*R), t - s < 15 |6 b| |@r@||6 |6 b | b| b|r@@| b||6||6 b|@r||6@ ppb| |6 | | r b| b|@r|@ b|@r|@r b| @ |6 @ | @ | b|@r| b|@r| b|@r| b| @ | @ | b|@r| b|@r| b| b| @ | pp b|@r| r b| b| b| @ | _______________________________________b|r|@rb|b|b| @ | @ r| 3 7 11 Next on the agenda is bsp*(P ), which is computed from bsp*(S0) (in o) and bsp*(R) (in O) as in fig. 7. Figure 7. bsp*(P), from bsp*(S0) and bsp*(R), * 11 |6 ||6 | | |6||6 b | | | b|@r@| |6|| b | | | | b|r@@| b|@r@| |6 ||6 b | | |b | b|r|@@ b|r@@| b| @ | @ | pp |b@r| b|r|@ r b| b| @ | @ | |b@r| b|r|@r b| b| @ | @ | |_______________________________________br|@bbbb|r|@b|b| @ | @ r| 1 3 7 11 Note that in positive filtration bsp*(P ) looks like bo*(24 P ) pushed up by 1 filtration. The explanation for this is the short exact sequence of A1-modules 0 !25 Z2 ! A1==A0 ! N ! 0; (4.5) 20 Donald M. Davis Chapter 1 where N = <1; Sq2; Sq3>, as before. If this is tensored with any A0-free A1-mod* *ule M, such as P , then the exact ExtA1-sequence reduces to isomorphisms Exts-1;tA1(25 M) ! Exts;tA1(N M) when s > 1. When M = P , an iso is also obtained when s = 1. The isomorphism of A1-modules H*(P4n+1) H*(24n P ) allows one to imme- diately obtain bo*(P4n+1) and bsp*(P4n+1) from the above calculations. One way of determining bo*(P4n+3) and bsp*(P4n+3) is from the short exact sequence of A1-modules 0 ! H*(24n+4Z2) ! H*(P4n+3) ! H*(24n+3R) ! 0: This yields as bsp*(P4n+3) a chart which begins as in fig. 8, while bo*(P4n+7) bo*(24 P4n+3) is obtained from this chart by deleting all classes in filtration* * 0. Figure 8. bsp*(P4n+3), * 15 r | r r| || || r r| r| r r| r| | | r r| r| | | | _______________________________________rrrrr|r|r| 4n+ 3 7 11 15 Next we compute bo*(P 2m) and bsp*(P 2m) using the exact ExtA1-sequence cor- responding to the short exact sequence 0 ! H*(P2m+1) ! H*(P ) ! H*(P 2m) ! 0: For example, this yields the calculation of bo*(P 8n) indicated in fig. 9, wher* *e bo*(P ) is in o's, while bo*(P8n+1) is in O's. Next we form J*(P 2m) from bo*(P 2m) and (23 bsp)*(P 2m), with filtrations of * *the latter pushed up by 1, similarly to the odd primary case. The boundary morphism bo4i-1(P 2m) ! (24 bsp)4i-1(P 2m) is pictured by a differential in the chart, a* *nd, for the same reason as in the odd-primary case, its value is the same as in bo4i-1(* *S0) ! (24 bsp)4i-1(S0), namely a nonzero d(i)+1wherever possible. If m k, then the charts for J*(P 2m) and J*(P 2k) are isomorphic through dimension 2k - 1. This * *is illustrated in fig. 10 for k = 8. For * 2m - 1, the form of the chart for J*(P 2m) depends upon the mod 4 value of m. The last of the filtration-1 Z2's occurs in * = 2m or 2m + 2. The c* *hart near * = 2m is indicated in fig. 11. Note how the bsp-part is like the bo-part * *shifted Section 4 Computing periodic homotopy groups 21 Figure 9. bo*(P8n), from bo*(P) and bo*(P8n+1) | | | | | | | | r | | | | r | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | r| | | | | | | | r r| | | |r | | | | r r| | |r b |r | | | @ | r r| | b |r b@||r | | p pp | @ | @ | r r| r| | b b|@|r b@||r | | | | @ | @ | r r| r| r| | b b|@|r b@||r | | | |pp @ | @ | _______________________________________________________________________rr|r|r|* *||@|@rr 1 3 7 11 8n - 1 8n + 7 Figure 10. J*(P2m) in * 15, provided m 8 r|| r| r| || || r r| b r| r b r| b|r|A @ | A| r b b|@r| b|r|AA | @ | A| r r| b b|@r| b|r|AA | | @ | A| r r| b b r| b b|@r| b bbr||AA | | @ | A| ________________________________________rr|r|r|@r|A 1 3 7 11 15 22 Donald M. Davis Chapter 1 one unit left and two units down. Differentials dr with r > 1 are omitted from * *this chart; they occur on the towers in 8n - 1 and 8n + 7. Figure 11. J*(P2m) where it starts to ascend |r r| P 8n P 8n+2 P 8n+4 r br||rr P 8n+6brrrr| | r r b b|||r@ r b b|r|@ r r |r r ||||rbb||r@ r||b b|r|@ r ||r r ||r b ||r r r|| b ||r b|||r@ b r| b|r||@ 4n _ _ r b |r r b |r b||r rb r| b|||||rb||r@ b|r|| b|r|@ r b b||r@ b b@|r| b||r bb|r|@ b|||r b||r@ b|r| |@|pp r||b b|||r@ b b@||r| p|p| b b|r||@ b|||r @pp|| b|r| p p b r| b||r@ b@|r| p p b|r|@ p||p| p p r p|p| @b|r| |pp| |p|@p @||pp b||r |@|pp p p b|r|r@ p p b|r||@ p p p p p p b||r p p b|r| b|r|b@ b||r b|r|@ b|r| b|r|@ b|r||@rb| @b||r b|r| b|b b|||r r b| b|r|| r b| b|r|@r b| b|||r@r b| b| b||rr b| b|r|rb b| b b|r|b@ b b||r@br b| b||rb _____________|r_______________|| __________________________|||____________|r@ 8n -61 8n +63 8n +63 8n +611 8n +67 To obtain v-11J*(P 2m) from J*(P 2m), one removes the filtration-1 Z2's, and extends into negative filtration the periodic behavior which is present in the * *towers to the right of dimension 2m. The justification for this is similar to that in * *the odd- primary case, namely Proposition 4.1. For example, if i is any integer, v-11J*(* *P 8n+4) for 8i + 6 * 8i + 13 looks like the portion of fig. 11 for P 8n+4between 8n +* * 6 and 8n + 13, with a d(4i+4)-differential on the tower in 8i + 7. One might find it easier to compute v-11J*(P 2m) directly without bothering to first compute t* *he non-periodic J; however, one sometimes needs the non-periodic J-groups. We combine the results of this section with Theorem 3.1 and Theorem 5.1 to obtain the following extremely important result, Theorem 4.2. If p is odd, then ae min(n;p(a)+1) v-11ss2n+1+i(S2n+1; p) Z=p if i = qa - 2 or qa - 1 0 if i 6 -1 or -2 mod q. If n 1 or 2 mod 4, then 8 > Z2 Z2 min(3;n+1) if i 1; 4 mod 8 : Z2 Z=2 if i 2; 3 mod 8 Z=2min(n-1;2(j)+4)if i = 8j - 2 or 8j - 1. Section 5 Computing periodic homotopy groups 23 If n 0 or 3 mod 4, then 8 >>>Z2 Z2 Z2 if i 0; 1 mod 8 >> Z8 if i 3 mod 8 >>>0 if i 4; 5 mod 8 >:Z=2min(n;2(j)+4) if i = 8j - 2 Z2 Z=2min(n;2(j)+4)if i = 8j - 1. 5.The v1-periodic homotopy groups of spectra In this section, we sketch three proofs of the following central result. If X is a spectrum, then v-11ss*(X) v-11J*(X). This result was first stated, at least for mod p v1-periodic homotopy groups, i* *n [56]. Theorem 5.1 is a consequence of the following result, which is the special case where X is the mod p Moore spectrum M = S0 [p e1. Let v-11M denote the mapping telescope of M !2-s M !2-2sM ! . .;. where s = 8 if p = 2 and s = q if p is odd, and the maps are all suspensions of* * an Adams map A. Then the Hurewicz morphism ss*(v-11M) ! J*(v-11M) is an isomorphism. This theorem implies that for any spectrum X, the map X ^ v-11M ! X ^ v-11M ^ J is an equivalence, which, after dualizing the Moore spectra, implies that Theor* *em 5.1 is true with mod p coefficients. The general case of the theorem then follo* *ws from Lemma 3.2. As an aside, we note that these results are equivalent to the validity of Rave* *nel's Telescope Conjecture ([53]) when n = 1. This result, for which the analogue with n = 2 has been shown to be false, can be stated in the following way. The v1-telescope equals the K*-localization, i.e., v-11M = MK . Proof. Since the Adams maps induce isomorphisms in K*(-), the inclusion M ! v-11M is a K*-equivalence. Since v-11M ' v-11M ^ J ' M ^ v-11J, and, similarly to Proposition 4.1, there is a cofibration v-11J ! KO ! KO; 24 Donald M. Davis Chapter 1 it follows readily that v-11M is K*-local. || The remainder of this section is concerned with proofs of Theorem 5.2. Three distinct proofs have been given, although each is too complicated to present in detail here. We sketch each, relegating details to the original papers. The first proof, when p = 2, was given by Mahowald in [40], although he was offering sketches of this proof as early as 1970. The odd-primary analogue was * *given in [24]. A sketch of Mahowald's proof, involving bo-resolutions, follows. Using self-duality of M, it suffices to show that dirlimi[2k+8iM; S0] ! dirlimi[2k+8iM; J] (5.1) is an isomorphism. The target groups are easily determined by the methods of the preceding section to be given by two sequences of "lightning flashes" as in fig* *. 12. It is easily seen, for example from the upper edge of Adams spectral sequence, * *that these elements all come from actual stable homotopy classes, i.e., the morphism (5.1) is surjective. Figure 12. dirlimi[2k+8iM; J] r p r r p p | r r| r r r r | r r| r k -2 0 2 3 mod 8 Thefinjectivity of the morphism (5.1) will be proved by showing thatiif 2kf M -! S0 ! J is trivial, then for i sufficiently large, 2k+8iM -A! 2k M -! S0 is trivial._This will be done using bo-resolutions. Let bo denote the cofiber of the inclusion S0 ! bo. There is a tower of (co)fibrations __ __ __ S0 2-1 bo 2-2 bo^ bo # #_ __# __ bo 2-1 bo^ bo 2-2 bo^ bo^ bo The homotopy exact couple of this tower gives the bo-ASS for S0. It was proved in [25], following [41], that the E2-term of this spectral sequence vanishes ab* *ove a Section 5 Computing periodic homotopy groups 25 line_of slope 1/5. That is, Es;t2(S0) = 0 if s > 1_5(t - s) + 3. One can show t* *hat bo^ bo '24 bsp _ W , where W can be written explicitly, and the map __ 3 2-1 bo -ffi!2-1 bo^ bo !2 bsp may be used as the map whose fiber is J. Here ffi induces the lowest d1 in the bo-ASS, and the_second_map collapses W . Let Es = (2-1 bo)^s, the sth stage of the tower. By explicit calculation it can be shown that, if s > 1 or if s = 1 and the map is detected entirely in the W -* *part, a map X ! Es of Adams (HZ=2) filtration greater than 1 can be varied so that its projection to Es-1 is unchanged, while the new map lifts to Es+1. Originall* *y it was thought that this was true for maps of Adams filtration greater than 0, but* * a complication was noted inf[27]. Now suppose that 2k M -! S0 ! J is trivial. Then f lifts to a map into E1 whose projection into 23 bsp is trivial. The Adams map A can be written as the composite of two maps, each of HZ=2-Adams filtration greater than 1. Thus, by the result of the previous paragraph, f OAilifts to E2i+1. If i is chosen large* * enough that 2i + 1 > 1_5(k + 8i + 1) + 3, then this map 2k+8iM ! S0 will have bo-filtr* *ation so large that all such maps are trivial by the vanishing line result, completin* *g the proof. The first proof of Theorem 5.2 for p odd was given by Haynes Miller. A proof analogous to his for p = 2 has not been achieved; [32] was a step in that direc* *tion. Miller's work did not involve the spectrum J. Instead, in [46], he defined a lo* *calized ASS for M, and computed its E2-term. Then, in [47], using a clever comparison with the BP -based Novikov spectral sequence, he computed the differentials in * *the ASS, obtaining the following result. If p is odd, then ss*(v-11M) is free over Zp[v11] on two classes, namely [S0 ,! M ! v-11M] and [Sq-1 ff1-!M ! v-11M]. The methods of Section 4 show easily that these map isomorphically to v-11J*(M). We provide a little more detail about Miller's calculations. In [46] he obtain* *ed as an E2-term for the localized ASS v-11E[hi;0: i 1] P [bi;0: i 1]; (5.2) wherePhi;0corresponds to [i] and has bigrading (1; 2(pi-1)), while bi;0correspo* *nds to 1_ppj[ji|p-ji] and has bigrading (2; 2p(pi-1)). The first step in obtainin* *g this is to use a change-of-rings theorem to write the E2-term as v-11CotorA(1)*(Zp; * *Zp), where A(1)* is the quotient A*=(o0). This is then shown to be isomorphic to Zp[v11] CotorP(1)(Zp; Zp); where P (1) = Zp[1; 2; : :]:=(p1; p2; : :):, and this yields eq. (5.2). In [47], the differential d2(hi;0) = v1bi-1;0is established in the localized A* *SS. This leaves Zp[v11] E[h1;0] as E3 = E1 , and this is easily translated into Pr* *opo- 26 Donald M. Davis Chapter 1 sition 5.4. Miller first established this differential in an algebraic spectral* * sequence converging to the E2-term of the BP -based Novikov spectral sequence, and then showed that this implies the desired differential in the ASS by a comparison th* *eo- rem. Somewhat later, Crabb and Knapp ([21]) gave a proof of Theorem 5.1 for finite spectra X which was much less computational than those just discussed. Their proof utilized the solution of the Adams conjecture, and some refinements there* *of. They let Ad*(-) be the generalized cohomology theory corresponding to the fiber of k - 1 : KO ! KO. By Proposition 4.1, this is just our v-11J*. They prove the following result about stable cohomotopy, which by S-duality is equivalent * *to Theorem 5.1 for finite spectra. If X is a finite spectrum, then the Hurewicz morphism v-11ss*s(X; Z=pe) -h! Ad*(X; Z=pe) is bijective. Their main weapon is a result of May and Tornehave which says that if A*(-) is the connective theory associated to Ad*(-), and j is the morphism given by a solution of the Adams conjecture, then the composite A0(X) -j!ss0s(X) -h!A0(X) is bijective for a connected space X. This is used to show that, for k sufficie* *ntly large, there is a stable Adams map 2ks(e)M(pe) Ae-!M(pe) which is in the image under j from A-ks(e)(M(pe); Z=pe). This is then used to show that for any eleme* *nt x of ssns(X; Z=pe), for L sufficiently large, ALex is in the image of the morph* *ism j, and this easily implies injectivity in Theorem 5.5. Care is required throughout* * in distinguishing stable maps from actual maps. 6.The v1-periodic unstable Novikov spectral sequence for spheres In this section, we review the basic properties of the unstable Novikov spectral sequence (UNSS) based on the Brown-Peterson spectrum BP , and sketch the de- termination of the 1- and 2-lines of this spectral sequence when applied to S2n* *+1. Then we show how the v1-periodic UNSS is defined, and compute it completely for S2n+1. The spectrum BP associated to the prime p is a commutative ring spectrum satisfying BP* = ss*(BP ) = Z(p)[v1; v2; : :]:and BP*(BP ) = BP*[h1; h2; : :]:,* * with |vi| = |hi| = 2pi-2. The generators viare those of Hazewinkel, while hiis conju* *gate to Quillen's generator ti. We shall often abbreviate BP*BP as .. We will make frequent use of the right unit jR : BP* ! BP*BP . Section 6 Computing periodic homotopy groups 27 jR(v1) = v1- ph1, and Xp jR(v2) = v2- ph2+ (pp-1- 1)hp1v1+ (p + 1)vp1h1+ aivp+1-i1pihi1; i=2 where ai2 Z. In writing hp1v1 here, we have begun the practice of writing jR(v)h as hv. Thus hp1v1 6= v1hp1. Proposition 6.1 is easily derived from formulas relating vi to * *mi, and for jR(mi). See [14, 2.6], where the following formula for the comultiplica* *tion #: BP*BP ! BP*BP BP*BP is also computed. All tensor products in this and subsequent sections are over BP*. # (h1) = h1 1 + 1 h1, and p-1X # (h2) = h2 1 + 1 h2+ 1_ppihi1 hp-i1v1+ hp1 h1: i=1 Let BP n denote the nth space in the ffi-spectrum for BP . If X is a space, th* *en a space BP (X) is defined as limnffin(BP n ^ X). Define D1(X) to be the fiber of the unit map X ! BP (X), and inductively define Ds(X) to be the fiber of Ds-1(X) ! Ds-1(BP (X)). This gives rise to a tower of fibrations . .!.D2(X) ! D1(X) ! X: The homotopy exact couple of this tower is the UNSS of X; if X is simply connec* *ted, it converges to the localization at p of ss*(X). In general, computing this spectral sequence can be extremely difficult, but if BP*X is free as a BP*-module, and cofree as a coalgebra, then it becomes somewh* *at tractable. Indeed, in such a case Es;t2(X) ExtsU(At; P (BP*X)); (6.1) where At denotes a free BP*-module on a generator of degree t, P (-) denotes the primitives in a coalgebra, and U denotes the category of unstable .-comodules. * *We sketch a definition of the category U and the proof of eq. (6.1), referring the* * reader to [6, p. 744] or [8, x7] for more details. If M is a free BP*-module, then U(M) is defined to be the BP*-submodule of . M spanned by all elements of the form hI m satisfying the unstable condition 2(i1+ i2+ . .).< |m|; (6.2) where hI = hi11hi22...I.f M is not BP*-free, then U(M) is defined as coker(U(F1* *) ! U(F0)), where F0 and F1 are free BP*-modules with M = coker(F1 ! F0). We define Us(M) by iterating U(-). The category U consists of BP*-modules equipped 28 Donald M. Davis Chapter 1 with morphisms M -! U(M), U(M) -ffi!U2(M), and U(M) -ffl!M satisfying certain properties. The unstable condition (6.2) is analogous to the one for un* *stable right modules over the Steenrod algebra, but its proof relies on deep work of R* *avenel and Wilson in [54]. The category U is abelian. We abbreviate ExtsU(At; N) to Exts;tU(N). These groups_may be calculated_as the homology groups of the_unstable_cobar complex C*;*(N), defined by Cs;t(N) = Us(N)t, with boundary Cs -d!C s+1defined by d[fl1| . .|.fls]m=[1|fl1| . .|.fls]m X + (-1)j[fl1| . .|.fl0j|fl00j| . .|.fls]m X + (-1)s+1[fl1| . .|.fls|fl0]m00; P P where # (flj) = fl0j fl00jand (m) = fl0 m00. __ We will use a reduced complex C*;*(N), which is chain equivalent to C*;*(N). This is obtained from eU(N) = ker(U(N) -ffl!N) and its iterates eUsby Cs;t(N) = eUs(N)t. Finally, we illustrate how d(v) = jR(v) - v for v 2 BP* comes into pla* *y. P Suppose # (h) = h 1 + 1 h + h0 h00and (I) = 1 I, and let v; v02 BP*. Then X d([vh]v0I)=[1|vh]v0I - [vh|1]v0I - [v|h]v0I - [vh0|h00]v0I + [vh|v0]I X = [jR(v) - v|h]v0I - [vh0|h00]v0I - [vh|jR(v0) - v0]I We abbreviate C*;*(BP*(X)) to C*;*(X). The first result about the UNSS, both historically and pedagogically, is the f* *ol- lowing, which appeared in [8, 9.12]. We repeat their proof because it gives a g* *ood first example of working with the unstable cobar complex. Recall that q = 2(p -* * 1). Let p be an odd prime. If k > 0, then E1;2n+1+kq2(S2n+1) Z=pmin(n;p(k)+1): If t 6 2n + 1 mod q, or if t < 2n + 1, then Es;t2(S2n+1) = 0. Proof. Since |vi| and |hi| are divisible by q, all nonzero elements in BP*(S2n+* *1) have degree congruent to 2n + 1 mod q, and so the only possible nonzero elements in Es;t2(S2n+1) occur when t 2n + 1 mod q, and t 2n + 1. There is an injective chain map C*;*(S2n-1) ! C*;*+2(S2n+1) defined by A2n-17! A2n+1, corresponding to the double suspension homomor- phism of homotopy groups. Since the boundaries in C1(S2n-1) are sent bijectively to those in C1(S2n+1), the morphism E1;t2(S2n-1) ! E1;t+22(S2n+1) is injective. Section 6 Computing periodic homotopy groups 29 We quote a result, originally due to Novikov (but see [51, x5.3] for the proof* *), about the stable groups: if n is sufficiently large, then E1;2n+1+kq2(S2n+1) Z=pp(k)+1 with generator d(vk1)2n+1=pp(k)+1. We will prove Theorem 6.3 by showing that if n p(k) + 1, then d(vk1)=pn is defined on S2n+1, but not on S2n-1. We begin by observing d(vk1)=pn=((jR(v1))k - vk1)=pn = ((v1- ph1)k - vk1)=pn Xk = (-1)j kjpj-nvk-j1hj1: (6.3) j=1 k k Note that the coefficients jpj-n have nonnegative powers of p, since p( j )+j p(k) + 1 n for j 1. Now we work mod terms that are defined on S2n-1. This allows us to ignore terms in the sum (6.3) for j < n. For the other terms, we write pj-nhj1as (v1- jRv1)j-nhn1, and note that when this is expanded by the binomial theorem, all terms except vj-n1hn1may be ignored, since (jRv1)ihn12n-1= hn1vi12n-1 satisfies eq. (6.2) when i > 0. Thus the sum (6.3) reduces to Xk n-1X (-1)j kjvk-n1hn1= - (-1)j kjvk-n1hn1; j=n j=0 P k k k since j=0(-1)j j = 0. If j > 0 in the right-hand sum, then j is divisible by p, and then phn1can be written as v1hn-11- hn-11v1, so that the term is defined* * on S2n-1. Thus, mod S2n-1, (6.3) reduces to -vk-n1hn1. This class is not defined on S2n-1. || If p = 2, a similar argument establishes the following result. If p = 2, then, for u > 0, 8 ><0 u odd E1;2n+1+u2(S2n+1) > Z=2 2(u) = 1 : Z=4 u = 4 Z=2min(n;2(u)+1)u 0 mod 4, and u > 4. If u = 2k in the three nonzero cases, then the generators are, respectively, d(* *vk1)=2, d(v21)=4, and d(vk1+ 22(k)+1vk-31v2)=22(k)+2. When p is odd, the element -(d(vk1)=pj)2n+1 2 E1;2n+1+kq2(S2n+1) is denoted ffk=j. If j = 1, this will frequently be shortened to ffk. We note the followin* *g from the proof of Theorem 6.3. If n j, then ffk=j2n+1= vk-j1hj12n+1 mod terms defined on S2j-1. 30 Donald M. Davis Chapter 1 Next we cull from [5] information about unstable elements in E2;*2(S2n+1), whi* *ch form a subgroup which we shall denote by eE2;*2(S2n+1). By "unstable," we mean an element in the kernel of the iterated suspension. The main theorem of [5] is* * the following. Let p be an odd prime, and let t = p(a). Then 8 < Z=pn if n t + 1 eE2;qa+2n+12(S2n+1) Z=pt+1 if t + 1 n < a - t : Z=pa-n if a - t - 1 n < a. 2 2;qa+2n+1 The homomorphism eE2;qa+2n-12(S2n-1) -2! eE2 (S2n+1) is ( injective if n t + 1 .p if t + 1 < n < a - t surjectiveif a - t n < a. Let m = min(n; a - t - 1). Then pj times the generator of eE2;qa+2n+12(S2n+1) is h1 va-m+j-11hm-j12n+1 mod terms defined on S2(m-j)-1. This is illustrated in the chart below, where we list just leading terms, an el* *ement connected to one just below it by a vertical line is p times that element, and * *el- ements at the same horizontal level are related by the iterated double suspensi* *on Section 6 Computing periodic homotopy groups 31 homomorphism. We omit the subscript from h1 and v1, and the . __S3______S5_________________S2t+3______S2t+5___ hva-2h hva-2h . . . hva-2h | | hva-3h2 . . . hva-3h2 hva-3h2 | | ... .. .. .. . . . | | hva-t-2ht+1 hva-t-2ht+1 | hva-t-3ht+2 ... _________________________________________________________ _S2(a-t)-1__S2(a-t)+1______________S2a-1__ ... hv2tha-2t-1 | hv2t-1ha-2thv2t-1ha-2t | | ... .. .. .. . . . hvtha-t-1 hvtha-t-1 . . . hvtha-t-1 The proof of Theorem 6.6 requires results about Hopf invariants which we will address shortly. We begin by describing a plausibility argument for it using on* *ly ele- mentary ideas about the unstable cobar complex. We continue to omit the subscri* *pt of h1 and v1. (i)The lead term h va-n+j-1hn-j does not pull back to S2(n-j)-1because hn-j2(n-j)-1does not satisfy the unstable condition. (ii)By Proposition 6.1, ph = v - jR(v), a fact which we will begin to use frequently. It implies that p . h va-n+j-1hn-j = h va-n+jhn-j-1- h va-n+j-1hn-j-1v: The second term desuspends below S2(n-j)-1, and the first term is the next term up the unstable tower. (iii)We show that 22 applied to the element of order p on S2n+1 is a boundary when n t + 1. To do this, we give a more precise description of this element of order p as d(va-n-1hn+1)2n+1. This clearly double suspends to the boundary 32 Donald M. Davis Chapter 1 d(va-n-1hn+12n+3). Note how we had to wait until S2n+3 in order to put the in- side the d(-), since hn+12n+1does not satisfy the unstable condition. It remain* *s to show that the lead term is correct, which is the content of the following propo* *sition. If t = p(a), then d(va-n-1hn+1) h va+t-n-1hn-t mod terms defined on S2(n-t-1)+1. Proof. Replacing v by ph + jR(v) implies va-n-1hn+1 pa-n-1ha (mod S2n+1): Since boundaries on S2n+1 desuspend to S2(n-t)-1, we obtain d(va-n-1hn+1) pa-n-1d(ha) P a mod the indeterminacy stated in the proposition. Now d(ha) = j hj ha-j, and, since p( aj) t + 2 - j for j > 1, we find pa-n-1d(ha) spa-n-1+th ha-1 sh va+t-n-1hn-t mod terms defined on S2(n-t-1)+1. Here a = sptwith s not a multiple of p, and we have freely replaced ph by v - jR(v). || The main detail in the proof of Theorem 6.6 which is lacking in the plausibili* *ty argument above is an argument for why these are the only unstable elements on the 2-line. For this, we need the following result, which will also be useful i* *n other contexts. i. There is an unstable .-comodule W (n) and an exact sequence -P2!Es;t-1 2n-1 22 s;t+1 2n+1 H2 s-1;t-1 P2 s+1;t-1 2n-1 2 (S ) -! E2 (S ) -! ExtU (W (n)) -! E2 (S ): (6.4) ii. W (n) is a free module over BP*=p on classes x2pin-1for i > 0 with coaction X npi (x2pkn-1) = pk-ihk-i x2pin-1: i iii. Ext0U(W (n)) Zp[v1]x2pn-1. P iv. If z 2 ExtU(W (n)) is represented by flk x2pkn-1, then X P2(z) = d( flk pk-1hnk) 2n-1: Section 6 Computing periodic homotopy groups 33 v. Every element x 2 Es2(S2n+1)Pmay be represented, mod terms which desuspend to S2n-1, by a cyclePof the form flk pk-1hnk 2n+1, with flk 2 C*(A2pkn-1 Zp). Then H2(x) = flk x2pkn-1. Recall in (v.) that At is the free BP*-module on a generator of degree t, and t* *hat C*(-) denotes the reduced unstable cobar complex. We provide a bare outline of the proof, beginning with the construction from [* *9]. There is a nonabelian category G of unstable .-coalgebras, and a notion of ExtG such that if BP*X is a free BP*-module of finite type, then E2(X) (of the UNSS) is ExtG(BP*X). Letting PG(-) denote the primitives in G, one finds that if M is an object of G, then PG(M) is in the category U. By considering an appropriate double complex, one can construct a composite functor spectral sequence converg* *ing to ExtG(M) with Ep;q2 ExtpU(RqPG(M)): Here RqPG denotes the qth right derived functor of PG. If M satisfies RqPGM = 0 for q > 1, then the spectral sequence has only two nonzero columns, and reduces to an exact sequence ! ExtsU(PGM) ! ExtsG(M) ! Exts-1U(R1PGM) ! : (6.5) This will be the case when M = BP*(ffiS2n+1). One verifies that PGBP*(ffiS2n+1) A2n, and that R1PGBP*(ffiS2n+1) is the comodule W (n) described in Theorem 6.8, and the exact sequence (6.5) reduces to the sequence (6.4) in this case. T* *he descriptions of the morphisms in (iv.) and (v.) are obtained in [3] using an al* *ternate construction of the exact sequence. Part (iii.) is proved in [5, p. 535] by stu* *dying explicit cycles. Now we complete our observations on the proof of Theorem 6.6. If x is any nonzero unstable element on the 2-line, then there must be k and n so that 2 2;*+2 22kx 6= 0 2 ker(E2;*2(S2n-1) 2-!E2 (S2n+1)): Then by Theorem 6.8, there must be an element ve1x2pn-12 Ext0U(W (n)) such that P2(ve1x2pn-1) = d(ve1hn1)2n-1=22kx: But this is exactly the description of the unstable elements on the 2-line whic* *h was given in the third part of our plausibility argument for Theorem 6.6. When p = 2, the discussion above about the unstable elements on the 2-line goes through almost without change, as described in [10, pp. 482-484]. The result is* * that if n a - 2(a) - 2, then ae eE2;2n+1+2a2(S2n+1) Z=2 if a is odd Z=2min(2(a)+2;n)if a is even. 34 Donald M. Davis Chapter 1 The orders when a is even are 1 larger than in the odd primary case because v2 * *can be used to obtain 1 additional desuspension. Now we construct the v1-periodic UNSS, following [4] for the most part. In [7]* * a UNSS converging to map*(Y; X) was constructed. If Y = Mn(pe), the E2-term is the homology of C*(P (BP*(X))) Z=pe. The Adams map A induces * UNSS (map *(Mn(pe); X)) A-!UNSS (map *(Mn+s(e)(pe); X)); where s(e) is as in eq. (2.1). By [35], on E2 this is just multiplication by a * *power of v1 after iterating sufficiently. As in our definition of v-11ss*(-), we defi* *ne the v1-periodic UNSS of X by v-11E*;*r(X) = dirlime;kE*;*+1r(map *(Mks(e)(pe); X)): The direct system over e utilizes the maps ae : Mn(pe+1) ! Mn(pe) used in Secti* *on 1, and the shift of one dimension is done for the same reason as in our definit* *ion of v-11ss*(-). Similarly to Proposition 2.4, we have On the category of spaces with H-space exponents, there is a natural transforma* *tion from the UNSS to the v1-periodic UNSS. One of the main theorems of [4] is the following determination of the v1-perio* *dic UNSS of S2n+1. Let p be an odd prime. The v1-periodic UNSS of S2n+1 collapses from E2 and satisfies ae min(n; (a)+1) v-11Es;2n+1+u2(S2n+1) Z=p p if s = 1 or 2, and u = qa 0 otherwise. The morphism Es;t2(S2n+1) ! v-11Es;t2(S2n+1) is an isomorphism if s = 1 and t > 2n + 1, while for s = 2 it sends the unstable towers injectively, and bijec* *tively unless n a - p(a) - 1, where t = 2n + 1 + qa. Proof. It is readily verified that the elements on the 1- and 2-lines described* * in Theorems 6.3 and 6.6 form v1-periodic families. The main content of this theorem is that there is nothing else which is v1-periodic. In order to prove this, we use a v1-periodic version of the double suspension sequence (6.4). It is proved in [4] that the morphisms of (6.4) behave nicely w* *ith respect to v1-action, yielding an exact sequence -P2!v-1 s;t-1 2n-1 22 -1 s;t+1 2n+1 H2 -1 s-1;t-1 P2 1 E2 (S ) -! v1 E2 (S ) -! v1 Ext U (W (n)) -! : (6.6) By Theorem 6.8(ii.), there is a spectral sequence converging to v-11ExtU(W (n)) Section 6 Computing periodic homotopy groups 35 with M s;t Es;t1 v-11ExtU(A2pin-1 Zp): (6.7) i1 There is a short exact sequence given by the universal coefficient theorem 0 ! v-11Es;t2(Sn) Zp ! v-11Exts;tU(An Zp) ! T or(v-11Es-1;t2(Sn); Zp) ! 0: (6.8) We will use eqs. (6.6), (6.7), and (6.8) to show inductively that there are no unexpected elements in v-11E2(S2n+1). But first we show how the known elements fit into this framework. By (6.8), each summand in v-11E12(Sn) gives two Zp's, * *called stable, in v-11ExtU(An Zp), and similarly each summand in v-11E22(Sn) gives two summands in v-11ExtU(An Zp), called unstable. We claim that n v-11Exts;tU(W (n)) Zp if s = 0 or 1, and t 2n - 1 mod(q6.9) 0 otherwise. In [4, p. 57], the relationship between (6.9) and (6.7) is discussed: in the sp* *ectral sequence (6.7), stable classes from the (i+1)-summand hit unstable classes from* * the i-summand, yielding in v-11E1 only the stable classes from the 1-summand. These are the elements described in eq. (6.9). On the other hand, in the exact.sequen* *cep (6.6), let t = 2n + kq with e = p(k). If e < n - 1, then 22 is Z=pe -! Z=pe when s = 2, yielding the elements in v-11Exts;t-1U(W (n)) for s = 0 and 1, whil* *e if e n - 1, then for s = 1 and 2, 22 is Z=pn-1 ,! Z=pn, also yielding elements in v-11Exts;t-1U(W (n)) for s = 0 and 1. It seems useful for this proof to have one more bit of input, namely the resul* *t for the v1-periodic stable Novikov spectral sequence, which can be defined as a dir* *ect limit over e of stable Novikov spectral sequences of M(pe). [13, x2] There is a v1-periodic stable Novikov spectral sequence for S0, satisf* *ying v-11Es;tr(S0) = dirlimnv-11Es;t+2n+1r(S2n+1); and ae (t)+1 v-11Es;t2(S0) Z=pp if s = 1 and t 0 mod q 0 otherwise. We prove by induction on s that, for all n, v-11Es2(S2n+1), v-11Es+12(S2n+1), * *and v-11ExtsU(W (n)) contain only the elements described in Theorem 6.10 and eq. (6* *.9). This is easily seen to be true when s = 0, where we know all groups completely.* * As- sume it is true for all s < oe. Then v-11Eoe2(S2n+1) contains no unexpected ele* *ments 36 Donald M. Davis Chapter 1 because oe = (oe - 1) + 1. If v-11Eoe+12(S2n+1) contains an unexpected element,* * then some v-11Eoe+12(S2L+1) must contain one in ker(22) by Theorem 6.11. Such an ele- ment must be P2(x), where x is an unexpected element of v-11Extoe-1U(W (n)), bu* *t no such element exists by our induction hypothesis. Finally, v-11ExtoeU(W (n)) con* *tains no unexpected elements by (6.7) and (6.8) since, as just established, v-11Eoe2(* *S2m+1) and v-11Eoe+12(S2m+1) contain no unexpected elements for any m. || The UNSS and v1-periodic UNSS are considerably more complicated at the prime 2 than at the odd primes, but the v1-periodic UNSS of S2n+1 is still completely understood. We shall not discuss it in detail because most of our applications * *in this paper will be at the odd primes. The reader desiring more detail is referr* *ed to [4], which gives a chart with UNSS names of the elements. We reproduce in figs. 13 and 14 the charts from [10, p. 488] of the v1-periodic UNSS of S2n+1 at the prime 2. Here "3" means Z=23, and "" means Z=2 , where = min(2(8k + 8) + 1; n): Differentials emanating from a summand of order greater than 2 are nonzero only on a generator of the summand. Note how Z=8 in periodic homotopy is obtained as an extension by the Z2 in filtration 3 of the elements in the 1-line group w* *hich are divisible by 2 in a Z=8. Figure 14 applies to S2n+1 when n 1 or 2 mod 4, with n > 2. The reader is referred to [10, p.487] for the minor changes required when n 2. In fig. 14, t* *he dotted differential is present if and only if = n. In both charts, the left j-* *action on the Z=2 on line 1, which is usually indicated by positively sloping solid l* *ines, is indicated by the dotted line if n < (8k + 8) + 1. The argument establishing these charts appears in [4]. Note that v-11Es;*4(S2n* *+1) = 0 if s > 4, and hence no higher differentials are possible. 7.v1-periodic homotopy groups of SU (n) In this section we show how the v1-periodic UNSS determines the v1-periodic ho- motopy groups of spherically resolved spaces. This relationship is particular n* *ice when localized at an odd prime, and it is this case on which we focus most of o* *ur attention. At the end of the section, we discuss the changes required when p = * *2. After proving the general result for spherically resolved spaces, we specialize* * to SU(n), where the result is, in some sense, explicit. It is not clear that the v1-periodic UNSS of a space X must converge to the p-primary v1-periodic homotopy groups of X. There might be periodic homotopy classes which are not detected in the periodic UNSS because multiplication by v1 repeatedly increases BP filtration. It is also possible that a v1-periodic fami* *ly in E2 might support arbitrarily long differentials in the unlocalized spectral seq* *uence, in which case it would exist in all v-11Er, but would not represent an element of periodic homotopy. We now show that neither of these anomalies can occur Section 7 Computing periodic homotopy groups 37 Figure 13. v-11Es;2n+1+j2(S2n+1), n 0, 3 mod 4, p = 2 rr BrrB BrrBBB rrBB BBBB BB BB B s = 4 |rr BB BrrB B BrrBBB rrBB |BB| BB BB B ||B BB B B rr ||B BrrBBB| B BrrBBBB rrBB rr || B || BB B B B|| B || BB B B rr 3Br B || rrBBB rB prr B|| B ppp B|| B ppp s = 1 r 3 Br p j - s = 8k+ 1 3 5 7 Figure 14. v-11Es;2n+1+j2(S2n+1), n 1, 2 mod 4, n > 2, p = 2 r rr rrBB BBrrBB BB BBBB BB B B BB B B s = 4 r B rr BB rr|BB B BBrrBB BB B BB BB |pp| B B B B BB |pp| B r B rr| B B rrBBB||pB BBrrBBB BBrr B || B B BB|pp| B B || B B BB||pp B r 3 r B || rrB B rB p rrBp B|| B pp ppp B|| B ppppp s = 1 r 3 Br p j - s = 8k+ 1 3 5 7 38 Donald M. Davis Chapter 1 for a spherically resolved space, essentially because they cannot happen for an* * odd sphere, where the v1-periodic homotopy groups are known to agree with the v-11E* *2- term. A space X is spherically resolved if there are spaces X0; : :;:XL, with X0 = *, XL = X, and fibrations Xi-1! Xi! Sni (7.1) with ni odd, and algebra isomorphisms H*(Xi) H*(Xi-1) H*(Sni): The following result was stated as Theorem 1.1. It was proved in [4]. If p is odd, and X is spherically resolved, then v-11Es;t2(X) = 0 unless s = 1 * *or 2, and t is odd. The v1-periodic UNSS collapses to the isomorphisms ae-1 1;i+1 v-11ssi(X) v1-E212;(X)i+2if i is even v1 E2 (X) if i is odd. Proof. Each algebra BP *(Xi) is free, and so eq. (6.1) applies to give E2(Xi) ExtU(M(xn1; : :;:xni)), where M(-) denotes a free BP*-module on the indicated generators. There are short exact sequences in U 0 ! M(xn1; : :;:xni-1) ! M(xn1; : :;:xni) ! M(xni) ! 0; and hence long exact sequences ! Es;t2(Xi-1) ! Es;t2(Xi) ! Es;t2(Sni) ! Es+1;t2(Xi-1)(!7:.2) These exact sequences are compatible with the direct system of v1-maps whose li* *mit is the v1-periodic groups. Thus there is a v1-periodic version of (7.2), and he* *nce by Theorem 6.10 and induction on i, v-11Es;t2(X) = 0 unless s = 1 or 2, and t is o* *dd. Thus v-11E2(X) v-11E1 (X). If u > max{ni}, s = 1 or 2, and s + u is odd, there are natural edge morphisms ssu(X) ! Es;s+u2(X), and these are compatible with the direct system of v1-maps, giving morphisms v-11ssu(X) ! v-11Es;s+u2(X). These yield a commutative diagram of exact sequences Section 7 Computing periodic homotopy groups 39 0 ! v-11ss2k(Xi-1)!? v-11ss2k(Xi)? ! v-11ss2k(Sni)!? ?yOEi-1 ?yOEi ?y 0 !v-11E1;2k+12(Xi-1)! v-11E1;2k+12(Xi)v-1!1E1;2k+12(Sni)! ! v-11ss2k-1(Xi-1)!? v-11ss2k-1(Xi)? v!-11ss2k-1(Sni)!?0 ?yOE0i-1 ?yOE0 ? 0 i y ! v-11E2;2k+12(Xi-1)! v-11E2;2k+12(Xi)v-1!1E2;2k+12(Sni)! 0: (7.3) The zeros at the ends of the v-11E2-sequence follow from the previous paragraph. The zero morphism coming into v-11ss2k(Xi-1) follows from v-11E02(Sni) = 0 and OEi-1being an isomorphism, which is inductively known. The zero morphism coming out from v-11ss2k-1(Sni) follows since anything in the image must have filtrati* *on 2, but, by induction, v-11ss2k-2(Xi-1) is 0 above filtration 1. Comparison of Theorems 4.2 and 6.10 shows that the groups related by and by 0are isomorphic, and it is easy to see that and 0induce the isomorphisms. (* *See [23] for reasons.) Since X1 is a sphere, this comparison also shows that OE1 an* *d OE01 are isomorphisms, which starts the induction. Thus all OEiand OE0iare isomorphi* *sms by induction and the 5-lemma. || Theorem 7.2 is a nice result, but it still leaves the formidable task of calcu* *lating v-11E2(X). In [6], Bendersky proved the following seminal result, whose proof we will discuss throughout much of the remainder of this section. If k n, then in the UNSS E1;2k+12(SU (n)) Z=pep(k;n), where ep(k; n) is as defined in Definition 1.2, and p is any prime. This allows us to easily deduce Theorem 1.3, now demoted to corollary status, which we restate for the convenience of the reader. If p is odd, then v-11ss2k(SU (n)) Z=pep(k;n), and v-11ss2k-1(SU (n)) is an ab* *elian group of the same order. Proof of corollary. The first part of the corollary is a straightforward applic* *ation of Theorems 7.2 and 7.3, once we know that the groups in Theorem 7.3 are v1- periodic. This can be seen by observing how they arise, from exact sequences bu* *ilt from spheres, where the classes are all v1-periodic. In [23], a slightly differ* *ent proof of this part of the corollary was given, before the v1-periodic UNSS had been hatc* *hed. The exact sequence like the top row of (7.3) for the fibration SU (n - 1) ! SU(n) ! S2n-1 (7.4) implies, by induction on n, that |v-11ss2k-1(SU (n))| = |v-11ss2k(SU (n))|. Ind* *eed, the orders are equal for S2n-1 by Theorem 6.10, and so if they are equal for SU(n-1* *), 40 Donald M. Davis Chapter 1 then they will be equal for SU (n), since the alternating sum of the exponents * *of p in an exact sequence is 0. The fact that SU (2) = S3 starts the induction. || In [23], an example of a noncyclic group v-11ss2k-1(SU (n); 3) was given, and * *in [13] it was shown that v-11ss2k-1(SU (n); 2) will often have many summands (in addition to a regular pattern of Z2's). In order to prove Theorem 7.3, it is convenient to work with the UNSS based on MU, rather than BP . This allows us to work with the ordinary exponential series, rather than its p-typical analogue. The facts about MU that we need are summarized in the following result. i. MU*(MU) is a polynomial algebra over MU* with generators Bi of grading 2i for i > 0. There are elements fii2 MU2i(CP 1) which form a basis for MU*(CP 1) as an MU*-module. P ii. Let B denote the formal sum 1 + i>0Bi. The coaction MU*(CP 1) -! MU*MU MU* MU*(CP 1) satisfies X (fin) = (Bj)n-j fij: j Here (Bj)n-j denotes the component in grading 2(n - j) of the jth power of the formal sum B. iii. There is a ring homomorphism _e: MU*(MU) Q ! Q satisfying o_e(Bi) = 1=(i + 1). o_e(jR(a)) = 0 if a 2 MUi with i > 0. o_einduces an injection E1;2n+1+2k2(S2n+1) ! Q=Z. iv. The BP -based UNSS is the p-localization of the MU-based UNSS. Proof. Part (i.) is standard (e.g., [2]), while part (ii.) is [2, 11.4]. Part (* *iii.) is from [6]. There are elements mi 2 MU2i Q such that MU*MU Q is a polynomial algebra over Q on all miand jR(mi). One defines _eto be the ring homomorphism which sends mito 1=(i + 1) and jR(mi) to 0. The second property in (iii.) is cl* *ear, and the first follows by conjugating [2, 9.4] to obtain X mn = jR(mi)(Bi+1)n-i; and then applying _eto obtain _e(mn) = _e(Bn). One way to see the third property is to localize at p and pass to BP . Then mM* *Upi-1 passes_to mBPi, and so our _epasses to that of [6, 4.3]. It is shown on [6, p.7* *51] that eBP sends the p-local 1-line injectively, and this works for all p. || Now we can state a general theorem which incorporates most of the work in proving Theorem 7.3. This theorem was stated without proof in [11, 3.10], where Section 7 Computing periodic homotopy groups 41 it was applied to X = Sp(n), p = 2. We will outline the proof, which is a direct generalization of [6], later in this section. Suppose X is spherically resolved as in Definition 7.1 with n1 < n2 < . .,.and L possibly infinite. Then MU*(Xk) is an exterior algebra over MU* on classes y1; : :;:yk; with |yi| = ni. Let __1;t 1;t 1;t E (Xk) = ker(E2 (Xk) ! E2 (XL)): Let flk;j2 MU*(MU) be defined in terms of the coaction in MU*(Xi) by Xk (yk) = flk;j yj; (7.5) j=1 and let bk;j= _e(flk;j) 2 Q. Then the matrix B = (bk;j) is lower triangular wit* *h 1's on the diagonal. Let C = (ck;j) be the inverse of B, and let !k(m) = l:c:m:{den(ck;j) : m j k}: __1;nk __1;nk Then coker(E (Xm-1 ) ! E (Xk-1)) is cyclic of order !k(m). Now we specialize to SU (n), where we need the following result. In the UNSS (i)Es;t2(SU ) = 0 if s > 0. (ii)E1;2k+12(SU (n)) = 0 if n > k. (iii)E1;2k+12(SU (k)) Z=k!. (iv)If i < j k, the inclusion SU (i) ! SU(j) induces an injection in E1;2k+1* *2. (v)SU is spherically resolved as in Theorem 7.6 with Xk = SU (k + 1), nk = 2k + 1, and if the MU-coaction on SU is as in eq. (7.5), then X _ e(flk;j)xk = (- log(1 - x))j: (7.6) kj Proof. Part i is a nontrivial consequence of the fact that SU is an H-space with torsion-free homology and homotopy. See [6, 3.1]. The coalgebra MU*(SU (n)) is cofree with primitives isomorphic to MU*(2 CP n-1). Hence by (6.1) E2(SU (n)) ExtU(MU*(2 CP n-1)); and so the fibrations (7.4) induce exact sequences in E2. Part ii follows from * *part i and the exact sequence, and part iv is also immediate from the exact sequence. Let y2k+12 MU2k+1(SU (n)) be the generator corresponding to 2 fik 2 MU2k+1(2 CP n-1) 42 Donald M. Davis Chapter 1 for k < n. Part iii is proved on [6, p.748] by showing that the generator of E0;2k+12(SU ) is of the form k!y2k+1+ lower terms, so that E0;2k+12(SU (k + 1)) ! E0;2k+12(S2k+1) sends the generator of one Z to k! times the generator of the other Z. This imp* *lies part iii. By Proposition 7.5(ii.) and the relationship of MU*(SU (n)) with MU*(2 CP n-1) noted above, we find X (y2k+1) = (Bj)k-j y2j+1: P _ By Proposition 7.5(iii.), we have e(Bi)xi+1= - log(1-x). These facts yield pa* *rt v. || Now we can prove Theorem 7.3. If f(x) is a power series with constant termP0, let [f(x)] denote the infinite matrix whose entries ak;jsatisfy f(x)j = kak;j* *xk. One easily verifies that [g(x)][f(x)] = [f(g(x))]. Hence the inverse of the mat* *rix [- log(1 - x)] is [1 - e-x]. This observation, with Theorem 7.6 and Proposition* * 7.7, implies that there is a short exact sequence 0 ! E1;2k+12(SU (n)) ! E1;2k+12(SU (k)) ! Z=!k(n) ! 0 with middle group Z=k! and !k(n) = l:c:m:{den(coef(xk; (1 - e-x)j)) : n j k}: Thus E1;2k+12(SU (n)) is cyclic of order k!= l:c:m:{den(coef(xk; (ex - 1)j)) : n j k} k = gcd{coef(x_k!; (ex - 1)j) : n j k}: Looking at exponents of p yields Theorem 7.3 by Proposition 7.5(iv.). || It remains to prove Theorem 7.6, the notation of which we employ without com- ment. Define bk;j(m) for m k recursively by bk;j(k) = bk;jand bk;j(m) = bk;j(m + 1) - bk;m(m + 1)bm;j: (7.7) We begin by noting that if row reduction is performed on (B|I) so as to get at * *each step one more diagonal of 0's below the main diagonal of B, we find that the en* *tries ck;jof B-1 satisfy 8 < 0 if j > k ck;j= : 1 if j = k -bk;j(j + 1)if j < k. Section 7 Computing periodic homotopy groups 43 Then !k(m) = max(ord(bk;j(j + 1)) : m j k): (7.8) Here and throughout this proof,_ord(-) refers to_order in Q=Z. Fix k, and let o(m) = |coker(E 1;nk(Xm-1 ) ! E 1;nk(Xm ))|. We drop the subscript k from eqs. (7.7) and (7.8)._The fibration Xk-1 ! Xk ! Snk im- P plies that the generator g(k - 1) of E 1;nk(Xk-1) is d(yk) = j 2. ov-11E4(X) = v-11E1 (X), and v-11Es4(X) = 0 if s > 4. oIf the groups v-11E1;t2(X) are cyclic, then the v1-periodic UNSS converges to v-11ss*(X). 8.v1-periodic homotopy groups of some Lie groups In this section we focus on two examples. One uses UNSS methods to determine v-11ss*(G2; 5), while the other uses ASS methods to determine v-11ss*(F4=G2; 2). Here G2 and F4 are the two simplest exceptional Lie groups. The first example is just one of many discussed in [14]. We also discuss how these UNSS methods can * *be used to give tractable formulas for v-11ss*(SU (n); p) when p is odd and n p2-* * p. We close by summarizing the status of the program, initially proposed by Mimura, of computing the v1-periodic homotopy groups of all compact simple Lie groups. Our first theorem concerns the v1-periodic homotopy groups of certain sphere bundles over spheres, which appear frequently as direct factors of compact simp* *le Lie groups localized at p, according to the decompositions given in [49]. Let p be an odd prime, and let B1(p) denote an S3-bundle over S2p+1with attachi* *ng map ff1. Then the only nonzero v1-periodic homotopy groups of B1(p) are p-1)) v-11ss2p+qm-1(B1(p)) v-11ss2p+qm(B1(p)) Z=pmin(p+1;1+p(m-p : This is the case k = 1 of [14, 2.1]. The spaces B1(p) were called B(3; 2p+1) i* *n [14]. The following result follows immediately from the 5-local equivalence G2 ' B1(5* *). ( min(6;1+ (i-5009)) Z=5 5 if i 1 mod 8 v-11ssi(G2; 5) Z=5min(6;1+5(i-5010))if i 2 mod 8 0 otherwise Section 8 Computing periodic homotopy groups 45 The following result is the central part of the proof of Theorem 8.1. Indeed, * *this theorem, 8.3, along with Theorem 6.3, gives the order of each group E1;t2(B1(p)* *), and Theorem 8.5 shows the group is cyclic. Then Theorem 7.2 shows that this giv* *es v-11ss*(B1(p)) when * is even, and the proof of Corollary 7.4 shows that |v-11ss2k-1(B1(p))| = |v-11ss2k(B1(p))|: Finally, v-11ss*(B1(p)) is shown to be cyclic when * is odd in Theorem 8.6. In the exact sequence 0 ! E1;qm+2p+12(S3)i*-!E1;qm+2p+12(B1(p)) j*-!E1;qm+2p+1 2p+1 @ 2;qm+2p+1 3 2 (S ) -! E2 (S ); the morphism @ is a surjection to Z=p unless p(m) p - 1 and m=pp-1 1 mod p; in which case it is 0. We will use the double suspension Hopf invariant H2 discussed in Theorem 6.8. We denote by H0 the morphism H0: E22(S2n+1) H2-!Ext1U(W (n)) ! E12(M); obtained by following H2 by the stabilization. Here Es2(M) ExtsBP*BP(BP*; BP*=p) denotes the E2-term of the stable NSS for the mod p Moore spectrum M. We will eventually need the following facts about E2(M). (i)E2(M) is commutative. (ii)vk1h1 6= 0 2 E12(M). (iii)If x 2 E2(M), then v2x = xv2 = 0. (iv)v1hk(p-1)+11= vk(p-1)+11h1 in E2(M). e-1-1 (v)If s 6 0 mod p, then ffspe-1=e= -svsp1 h1 in E2(M). Proof. Part i is well-known, part ii is [51, p 157], and part iii follows from * *[48, 2.10]. Part iv follows from Proposition 6.1, part iii of thiselemma,-and1the fa* *ct that ph1 = 0 2 E2(M). To prove part v, we use 6.1 to expand vsp1 , obtaining e-1 spe-1 ffspe-1=e=1_pe(jR(vsp1 ) - (ph1+ jR(v1)) ) spe-1Xspe-1 e-1 = - pj-ehj1vsp1 -j: j=1 j 46 Donald M. Davis Chapter 1 All terms except j = 1 are divisible by p, and hence are 0. To insure that term* *s with j large are p times an admissible element, write pj-ehj1as p(v1-jR(v1))j-e-1he+* *1. || Now we begin the proof of Theorem 8.3. We begin with the case (m) < p - 1. In this case, @(gen) = ffm=(m)+1 ff13 -ffm=(m)+1 h13; (8.1) mod terms that desuspend to S1. Here we have used [6, 4.9] and Proposition 6.5. By Proposition 6.5, the assumption that (m) < p - 1 implies that ffm=(m)+1 is defined on S2p-1, and hence Theorem 6.8.v implies that H0(@(gen)) = -ffm=(m)+1 6= 0; where the last step uses parts v and ii of Lemma 8.4. Thus @ 6= 0 in this case,* * as claimed. Now we complete the proof of Theorem 8.3 by considering the case (m) p-1. We let s = m=p(m) and ae ffl = 01 ifi(m)f>(pm-)1= p - 1. We will establish the following string of equations in the next paragraph, and * *then we will further analyze whether these terms are 0 by studying their Hopf invari* *ant. The following string is valid mod terms which desuspend to S1. @(gen)= ffm=p ff13 = p-p(jR(vm1) - (ph1+ jR(v1))m ) ff13 (8.2) mX m = - pj-phj1 vm-j1ff13 j=1 j = -pm-p hm1 ff13- fflsh1 vm-11ff13 (8.3) = -vm-p1hp1 ff13- vm-p-11hp1 v1h13+ fflsh1 vm-11h13(8.4) = A + B + C; (8.5) where A, B, and C denote the three terms in the preceding line. Line (8.2) follows from Propositions 6.5 and 6.1. Line (8.3) has been obtained by observing that in the sum all terms desuspend to S1 except j = m and, if (m) = p - 1, j = 1. To see this, we observe that we need to have a p to make ff13 desuspend. This factor will be present unless j = 1 and (m) = p - 1. The requirement that j be 1_2times the degree of the symbols following hj1will only be a problem for large values of j. When j is large, write the term as m p(v - j (v ))j-p-1hp+1 vm-j ff : j 1 R 1 1 1 13 Section 8 Computing periodic homotopy groups 47 Since p times anything which is defined on S3 desuspends to S1, this desuspends* * to S1 provided p + 1 (p - 1)(m - j + 1) + 1, which simplifies to 1 (p - 1)(m - j* *), i.e., j < m. To obtain (8.4), we have rewritten the first term of eq. (8.3) as -p(v1- jR(v1))m-p-1hp+11 ff13; observed that when this is expanded, all terms except the first desuspend, and * *in that first term we write ph1 = v1- jR(v1). We note first that, by Proposition 6.2, A is d(h2) mod S1, and so H0(A) = 0. We can evaluate the Hopf invariant of B and C by Theorem 6.8.v; using Lemma 8.4, we obtain H0(B + C) = (-1 + ffls)vm-11h1: Hence H0(@(gen)) = 0 if and only if -1 + ffls 0 mod p. Since H0 is injective on E22(S3), this completes the proof of Theorem 8.3. || Now we settle the extension in the exact sequence of Theorem 8.3. The groups E1;qm+2p+12(B1(p)) in Theorem 8.3 are cyclic. Proof. We will show that whenever ker(@) 6= 0 in the exact sequence of 8.3, the* *re is an element z 2 E1;qm+2p+12(B1(p)) such that j*(z) = ffm 2p+1, the element of order p, and pz = i*(gen). Since @(ffm 2p+1) = 0 in these cases, there is w3 2 C1;qm+2p+1(S3) such that d(w3) = ffm ff13. Let z = ffm 2p+1- w3: Then z is a cycle, since d(z) = ffm ff13 - ffm ff13, and clearly j*(z) is as required. Since pffm = d(vm1), we have pz - d(vm12p+1)=d(vm1)2p+1- pw3- d(vm1)2p+1+ vm1ff13 = i*((vm1ff1- pw)3) We show (vm1ff1-pw)3 6= 0 2 E1;qm+2p+12(S3) by noting from [7, x7] that the Hopf invariant H2 : E12(S3) ! Ext0(W (1)) factors through the mod p reduction of the unstable cobar complex. Thus H2((vm1ff1- pw)3) = H2(vm1h13) = vm16= 0: The second "=" uses Theorem 6.8v, and the "6=" uses 6.8iii. || The following result completes the proof of Theorem 8.1 according to the outli* *ne given after Corollary 8.2. In Theorem 8.1, the group v-11ssqm+2p-1(B1(p)) is cyclic. 48 Donald M. Davis Chapter 1 Proof. We use the exact sequence in v-11ss*(-) of the fibration which defines B1(p). The cyclicity follows from that of v-11ssqm+2p-1(S2p+1) unless @ = 0 : v-11ssqm+2p(S2p+1) ! v-11ssqm+2p-1(S3): (8.6) If eq. (8.6) is satisfied, then Off1 6= 0 : v-11ssqm+2(S2p+1) ! v-11ssqm+2p-1(S2p+1) by [23, 6.2], and @ : v-11ssqm+2(S2p+1) ! v-11ssqm+1(S3) is an isomorphism of Z=p's by Theorem 8.3. Let G denote a generator of v-11ssqm+2(S2p+1), and let Y 2 v-11ssqm+2p-1(B1(p)) project to G O ff1. By [50, 2.1], pY = i*(<@G; ff1; p>) = @(G) O v1 6= 0: || It is shown in [49] that B1(p) is a direct factor of SU (n)(p)if p < n < 2p, a* *nd hence v-11ssi(SU (n); p) is given by Theorem 8.1 if p < n < 2p and i 1 or 2 mod q. This yields the following number theoretic result. If p is an odd prime, k 1 mod p - 1, and p < n < 2p, then the number ep(k; n) defined in 1.2 equals min(p; p(k - p - pp + pp-1)) + 1. The author has been unable to prove this result without the UNSS. In fact, the only tractable result for ep(k; n) which follows easily from Definition 1.2 see* *ms to be that if n p and k n - 1 mod p - 1, then ep(k; n) min(n - 1; p(k - n + 1) + 1* *), which is proved using the Little Fermat Theorem as in [22, p.792]. Using methods similar to those in our proof of Theorem 8.1, Yang ([58]) has proved the following tractable result for v-11ss*(SU (n); p) when p is odd and n p2 - p. Of course this can also be interpreted as a theorem about the numbers ep(k; n). We emphasize that the proof of Theorem 8.8 does not involve the use of Theorem 1.3. Suppose p is odd, k = c + (p - 1)d with 1 c < p, and c + (p - 1)b + 1 n c + (p - 1)(b + 1) with 0 b p - 1. Define j by 1 j p and d j mod p. Then v-11ss2k(SU (n); p) Section 8 Computing periodic homotopy groups 49 is cyclic of order pe, with 8 >>>min(c + (p - 1)j; b + (d - j) + 1) >>> if b < c and 1 j b >>> >>> j c j(p-1) > if c b and 1 j b >>> >>>min(c; b + 1 + (d)) >>> if b < c and b < j p >>> : b ifc b and b < j p One can easily read off from Theorem 8.8 the precise value of the numbers ep(n) which appeared in Corollary 7.8, yielding the following result for the p-expone* *nt of the space SU (n). If p is odd and n p2- p, then ae expp(SU (n)) ep(n) = nn - 1 ifoi(pt-h1)e+r2winseip.+ 1 for some i When p = 2, UNSS methods of computing v-11ss*(-; p) become more complicated because of the j-towers. Then ASS techniques become more useful, as they did for v-11ss*(G2; 2) in [29]. Here we show how to use ASS methods to determine v-11ss*(F4=G2; 2). It is hoped that the result of the calculations of v-11ss*(G* *2; 2) and v-11ss*(F4=G2; 2) might be combined to yield v-11ss*(F4; 2), but this involves * *one difficulty not yet resolved. Our main reason for including this example is to g* *ive a new illustration of this method. The proof of the following theorem will consume most of the remainder of this paper. If G denotes an abelian group, then mG denotes the direct sum of m copies of G. Let G(n) denote some group of order n. 8 >>>4Z2 i 0 mod 8 >>>Z8 Z8 Z2 i 1 mod 8 > 0 i 3 or 4 mod 8 >>>G(2min(15;(i-21)+4)) i 5 mod 8 >>> min(15;(i-22)+4) : Z2 Z2 Z=2 i 6 mod 8 5Z2 i 7 mod 8 There is a fibration S15 -i!F4=G2 -p!S23; (8.7) 50 Donald M. Davis Chapter 1 derived in [29, 1.1]. Here and throughout this proof all spaces and spectra are localized at 2. In [31], it was shown that for any spherically resolved space Y , there is a f* *inite torsion spectrum X satisfying v-11ss*(Y ) v-11J*(X). In our case, we have There is a spectrum X such that (i.) v-11J*(X) v-11ss*(F4=G2), and (ii.) there is a cofibration 215P 14! X !223+LP 22; (8.8) where L equals 0 or a large 2-power. We present in fig. 15 a chart which depicts an initial part of the ASS for v-1* *1J*(X) if X is as in Proposition 8.11 and L = 0. It depicts the direct sum of the spec* *tral sequences for v-11J*(215P 14) and v-11J*(223P 22), together with one differenti* *al, which will be established in Proposition 8.13. The o's are elements from P 14, * *while the O's are from P 22. Charts such as these for v-11J*(P n) were derived in Sec* *tion 4. To see that fig. 15 is also valid when L in Proposition 8.11 is a large 2-powe* *r, we use the following result. If L is a large 2-power, then the attaching map 222+LP 22! v-11215P 14 in Proposition 8.11 has filtration L=2 + 1. Using results of [40], this implies that a resolution of v-11X can be formed fr* *om v-11215P 14and OEL=2v-11223+L P 22, where OEj increases filtrations by j, and t* *his yields fig. 15. Proof of Lemma 8.12. Under S-duality, the generator corresponds to an element of v-11JL+7(P 14^ D(P 22)). This group is isomorphic to v-11JL+6(P 14^ P--223): (8.9) Using methods of [26], one can show that the relevant chart is as in fig. 16, w* *here the class indicated with a bigger o is the generator of (8.9), and has filtration L* *=2 + 1. Borel ([15]) showed that Sq8(x15) = x23 in H*(F4; Z2). This implies that the attaching map in F4=G2 and in X is the Hopf map oe, and hence corresponds to the generator of (8.9). Thus the attaching map has filtration L=2 + 1. || We can now establish the d2-differentials in fig. 15. There are d2-differentials as indicated in fig. 15. Proof. This follows from the oe attaching map just observed, together with the observation that the first of the pair of elements that are related by the diff* *erential in fig. 15 are v1-periodic versions of j23and joe15. || Section 8 Computing periodic homotopy groups 51 Figure 15. Initial chart for v-11ss*(F4=G2) r | 8k + 21 r |r | | | r |r | |r r | | A |r r A b | A |r |r A Ab b| | |0 |rd|r Ab r b| | | | |r b|r br r| | | | |r b||rrb r| | | b|rb||rr | b||b|rr | b||b|r d0= d(k+1)+2 b| b| b|db| d = d(k)+2 b| b| b b| b| b b| | b| b| b b| | b| b| b | | b| b | b| 8k + 26 Figure 16. The generator of v-11JL+6(P14^ P-2-23) r| | r| | r| r| |A | r| r|u |AA| r|Ar| r |AA| r|Ar|r |A | r|Ar|r | r| r | r| 52 Donald M. Davis Chapter 1 This d2-differential implies there is a nontrivial extension in v-11ss8k+26(F4* *=G2) as follows. v-11ss8k+26(F4=G2) Z=64. Proof. This follows from fig. 15 and a standard Toda bracket argument ([50, 2.1]), which in this situation says the following. Let A be the element support* *ing the higher of the two d2-differentials, and let D be the lowest o in 8k + 26. L* *et @ denote the boundary morphism in the exact sequence in v-11ss*(-) associated to the fibration (8.7). Then D lies in the Toda bracket <@(A); j; 2>, and so there* * exists an element E 2 v-11ss8k+26(F4=G2) such that p*(E) = A O j and i*(D) = 2E. || As indicated in fig. 15, there are d(k)+2-differentials between O-towers in 8k* * + 22 and 8k + 21, and there are d(k+1)+2-differentials between o-towers in 8k + 22 a* *nd 8k+21. This follows just from standard J*(-)-considerations. But there may also* * be differentials from the O-tower in 8k+22 to the o-tower in 8k+21. These differen* *tials from O to o are determined from the homomorphism v-11ss8k+22(S23) -@!v-11ss8k+21(S15); (8.10) which is evaluated in the following result. The image of the homomorphism (8.10) consists of all multiples of 8 if (k) 7, and is 0 if (k) > 7. The following result plays a central role in the proof of Proposition 8.15. Let (S15)K denote the K*-localization as constructed in [42]. There is a commu- tative diagram ffiS23? -@! S15? ?y` ?ye ffi1 (215P 14^-J)h!(S15)K in which @ is obtained from the fiber sequence (8.7), e is the localization, an* *d h induces an isomorphism in ssj(-) for j = 22 and j 28. Proof. The map h is constructed as in [29, pp. 669-670], using results of [42]* *. It induces an isomorphism in ssj(-) for many other small values of j, but we only * *care about values of j which are positive multiples of 22. The obstructions to its b* *eing an isomorphism for all small values of j are Z2-classes in filtration 1 in Jj(2* *15P 14) for j = 19, 23, and 27. The map ` is obtained by obstruction theory, since ffiS* *23has cells only in dimensions which are positive multiples of 22. || Now we prove Proposition 8.15. Let ` be as in Proposition 8.16. The morphism ss*(`) can be factored as ss*(ffiS23) ! sss*(ffiS23) ! J*(215P 14): Section 8 Computing periodic homotopy groups 53 There is a splitting M sss*(ffiS23) sss*(S22i): i>0 We will use the method of [43] to deduce that sss8k+21(S22) ! J8k+21(215P 14) sends the v1-periodic generator aek to 8 times the generator. Indeed, the stabl* *e map S22 !215P 14^ J which induces the morphism factors through 215P 8^ J, from which it projects nontrivially to 215P78^ J. We then use [43, 2.8] to deduce th* *at aek goes to the nonzero element of J8k+21(215 P78). This implies that its image in J8k+21(215 P 8) is the generator, and this maps to 8 times the generator of J8k+21(215P 14). The composite of v1-periodic summands of ss8k+21(ffiS23) ! sss8k+21(ffiS23) ! sss8k+21(S22) (8.11) is bijective if (k) 7, but is not surjective if (k) > 7. Thus when the composi* *te (8.11) is followed into J8k+21(215 P 14), the image of a v1-periodic generator * *is 8 times the generator if (k) 7 and 0 2 Z=16 if (k) > 7. Once we observe that, in the diagram of Proposition 8.16, h induces an isomorphism in ss8k+21(-) and e sends the v1-periodic summand isomorphically, we obtain the desired conclusion * *of Proposition 8.15. || The differentials implied by Proposition 8.15 have an interesting and unexpect* *ed implication about fig. 15. Since dr from the O-tower in 8k + 22 hits the top O * *in 8k + 21 and the o just above it with the same r, and since dr respects the acti* *on of h0, there must be an h0-extension between these classes in 8k + 21. If L = 0* * in (8.8), then this extension can only be accounted for by a failure of the map (8* *.8) to induce a split short exact sequence of A1-modules in cohomology. Indeed, we have If L = 0 in (8.8), then there is a splitting of A1-modules H*X H*(215P 30) H*(223P 6): This splitting is caused by having Sq2 6= 0 on the class in H*X corresponding to the top cell of H*(215P 14), i.e., Sq2 : H29X ! H31X is an isomorphism. This is the only way to account for the h0-extension in fig.* * 15. Proposition 8.17 implies that the h0-extensions are present in the chart for va* *lues of k ((k) > 7) where they cannot be deduced from differentials. It also implies 54 Donald M. Davis Chapter 1 that there is an h0-extension on the top O in 8k + 22. If L > 0 in (8.8), the s* *ame conclusion about the charts can be deduced from a more complicated analysis. It causes fig. 15 to take the form of fig. 17. Figure 17.|Final chart for v-11ss*(F4=G2) | | k odd r || k even r | | | 8k + 21 r |r | 8k + 21 r |r | | || | | | r |r | | r |r | r r | | | r r | | | A | | | | A | |r r A b | |r r A b | A | | A r |r A Ab b| | r |r A Ab b| | | | |J| |r|rA A b r b| | |r|r A b r b| |A|A | | |J|J | |rbDD|rAbr r| | |rb|rJ br r| |AA| | | |JJ| | |rb||rArb r| | |rb||rJrb r| |AA| | |J | b|b||rrAr | b|b||rrJr |AA | | b|b||rAr | b|b||rr |AA | | b|b||rAA || b|b||r b|b|AAA || b|b| b|b|AAA | b|b|d d = d(k)+2 | b|b|AAA b | b|b| b | b|b|AAA b b| | b|b| b b| | | | b|b|AbAA b| | | b|b| b b| | | b|b|Ab | | b|b| b | | | | b|b | | b|b | | b| 8k + 26 | b| 8k + 26 We can read off almost all of Theorem 8.10 from fig. 17. We must show that d6 * *is 0 on the O's near the bottom in 8k + 23 and 8k + 24. This is done by the argume* *nt used to prove Proposition 8.15. The classes involved are present in all spaces * *in the diagram in Proposition 8.16, but they are not mapped across by `*, since it fac* *tors through sss*(ffiS23). All that remains is the verification of the abelian group structure. Most of t* *he extensions are trivial due to the relation 2j = 0. The extension in 8k + 22 whe* *n k is odd was present before the exotic extension was deduced, and remains true. T* *he cyclicity of this 27 summand can also be deduced by consideration of the kernel* * of the homomorphism in the fibration which defines J, but that seems unnecessary. Note that no claim is made about the group structure in 8k + 21. || Section References Computing periodic homotopy groups 55 In 1989, Mimura suggested to the author that he try to calculate v-11ss*(G) for all compact simple Lie groups G. If p is odd, and G = Sp(n) or SO (n), then the result follows from Theorem 1.3 and [33]. With great effort, v-11ss*(Sp(n); 2) * *was calculated in [13]. The result involves a surpising pattern of differentials am* *ong Z2's from the various spheres which build Sp(n), resulting in [log2(4n=3)] copies of* * Z2 in certain v-11ssi(Sp(n)). Of the classical groups, only v-11ss*(SO (n); 2) rem* *ains. All torsion-free exceptional Lie groups were handled in [14], using the UNSS. In [2* *9] and [12], the torsion cases (G2; 2), (F4; 3), and (E6; 3) were handled. Remaini* *ng then are seven cases of (G; p) yet to be calculated. At least a few of these sh* *ould lend themselves to the methods of this paper. References [1]J. F. Adams, On the groups J(X), IV, Topology 5 (1966) 21-71. [2]_________, Stable homotopy and generalized homology, Univ of Chicago Press (* *1974). [3]M. Bendersky, The derived functors of the primitives for BP*(ffiS2n+1), Tran* *s Amer Math Soc 276 (1983) 599-619. [4]_________, The v1-periodic unstable Novikov spectral sequence, Topology 31 (* *1992) 47-64. [5]_________, Unstable towers in the odd primary homotopy groups of spheres, Tr* *ans Amer Math Soc 276 (1985) 529-542. [6]_________, Some calculations in the unstable Adams-Novikov spectral sequence* *, Publ RIMS 16 (1980) 739-766. [7]_________, The BP Hopf invariant, Amer Jour Math 108 (1986) 1037-1058. [8]M. Bendersky, E. B. Curtis, and H. R. Miller, The unstable Adams spectral se* *quence for generalized homology, Topology 17 (1978) 229-248. [9]M. Bendersky, E. B. Curtis, and D. C. Ravenel, The EHP sequence in BP-theory* *, Topology 21 (1982) 373-391. [10]M. Bendersky and D. M. Davis, 2-primary v1-periodic homotopy groups of SU(n* *), Amer Jour Math 114 (1991) 465-494. [11]________, The unstable Novikov spectral sequence for Sp(n), and the power s* *eries sinh-1(x), London Math Society Lecture Notes Series 176 (1992) 73-86. [12]________, 3-primary v1-periodic homotopy groups of F4 and E6, to appear in * *Trans Amer Math Soc. [13]M. Bendersky, D. M. Davis, and M. Mahowald, v1-periodic homotopy groups of * *Sp(n), to appear in Pacific Jour Math. [14]M. Bendersky, D. M. Davis, and M. Mimura, v1-periodic homotopy groups of ex* *ceptional Lie groups: torsion-free cases, Trans Amer Math Soc 333 (1992) 115-135. [15]A. Borel, Sur l'homologie et la cohomologie des groupes de Lie compacts con* *nexes, Amer Jour Math 76 (1954) 273-342. [16]A. K. Bousfield, Localization and periodicity in unstable homotopy theory, * *preprint. [17]________, The localization of spaces with respect to homology, Topology 14 * *(1978) 133-150. [18]________, The localization of spectra with respect to homology, Topology 18* * (1979) 257-281. [19]F. R. Cohen, T. L. Lada, and J. P. May, The homology of iterated loop space* *s, Springer-Verlag Lecture Notes in Math 533 (1976). [20]F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and* * exponents of the homotopy groups of spheres, Annals of Math 110 (1979) 549-565. [21]M. C. Crabb and K. Knapp, Adams periodicity in stable homotopy, Topology 24* * (1985) 475-486. [22]________, The Hurewicz map on stunted complex projective spaces, Amer Jour * *Math 110 (1988) 783-809. 56 Donald M. Davis Chapter 1 [23]D. M. Davis, v1-periodic homotopy groups of SU(n) at an odd prime, Proc Lon* *don Math Soc 43 (1991) 529-544. [24]________, Odd primary bo-resolutions and K-theory localization, Ill Jour Ma* *th 30 (1986) 79-100. [25]________, The bo-Adams spectral sequence: Some calculations and a proof of * *its vanishing line, Springer-Verlag Lecture Notes in Math 1286 (1987) 267-285. [26]________, Generalized homology and the generalized vector field problem, Qu* *ar Jour Math 25 (1974) 169-193. [27]D. M. Davis, S. Gitler, and M. Mahowald, Correction to The stable geometric* * dimension of vector bundles over real projective spaces, 280 (1983) 841-843. [28]D. M. Davis and M. Mahowald, Some remarks on v1-periodic homotopy groups, L* *ondon Math Soc Lecture Note Series 176 (1992) 55-72. [29]________, Three contributions to the homotopy theory of the exceptional Lie* * groups G2 and F4, Jour Math Soc Japan 43 (1991) 661-671. [30]________, v1-periodicity in the unstable Adams spectral sequence, Math Zeit* * 204 (1990) 319-339. [31]________, v1-localizations of torsion spectra and spherically resolved spac* *es, Topology 32 (1993) 543-550. [32]________, v1-periodic Ext over the Steenrod algebra, Trans Amer Math Soc 30* *9 (1988) 503-516. [33]B. Harris, On the homotopy groups of the classical groups, Annals of Math 7* *4 (1961) 407-413. [34]P. Hoffman, Relations in the stable homotopy rings of Moore spaces, Proc Lo* *ndon Math Society 18 (1968) 621-634. [35]M. J. Hopkins and J. Smith, to appear in Annals of Math. [36]I. M. James, On Lie groups and their homotopy groups, Proc Camb Phil Soc 55* * (1959) 244-247. [37]N. J. Kuhn, The geometry of the James-Hopf maps, Pac Jour Math 102 (1982) 3* *97-412. [38]L. Langsetmo, The K-theory localization of loops on an odd sphere and appli* *cations, Topology 32 (1993) 577-586. [39]L. Langsetmo and R. D. Thompson, Some applications of K(1)*(W(n)), preprint. [40]M. Mahowald, The image of J in the EHP sequence, Annals of Math 116 (1982) * *65-112. [41]________, bo-resolutions, Pac Jour Math 92 (1981) 365-383. [42]M. Mahowald and R. D. Thompson, The K-theory localization of an unstable sp* *here, Topol- ogy 31 (1992) 133-141. [43]________, Unstable compositions related to the image of J, Proc Amer Math S* *oc 102 (1988) 431-436. [44]J. McClure, The mod p K-theory of QX, Springer Verlag Lecture Notes in Math* * 1176 (1986) 291-383. [45]R. J. Milgram, Group representations and the Adams spectral sequence, Pac J* *our Math 41 (1972) 157-182. [46]H. R. Miller, A localization theorem in homological algebra, Math Proc Camb* * Phil Soc 84 (1978) 73-84. [47]________, On relations between Adams spectral sequences, with an applicatio* *n to the stable homotopy of a Moore space, Jour Pure Appl Alg 20 (1981) 287-312. [48]H. R. Miller and D. C. Ravenel, Morava stabilizer algebras and the localiza* *tion of Novikov's E2-term, Duke Math Jour 44 (1977) 433-447. [49]M. Mimura and H. Toda, Cohomology operations and the homotopy of compact Li* *e groups-I, Topology 9 (1970) 317-336. [50]________, Homotopy groups of SU(3), SU(4), and Sp(2), Jour Math Kyoto Univ * *3 (1964) 217-250. [51]D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Aca* *demic Press (1986). [52]________, The homology and Morava K-theory of ffi2SU(n), Forum Math 5 (1993* *) 1-21. Section References Computing periodic homotopy groups 57 [53]________, Localization with respect to certain periodic homology theories A* *mer Jour Math 106 (1984) 351-414. [54]D. C. Ravenel and W. S. Wilson, The Hopf ring for complex cobordism, Jour P* *ure and Appl Alg 9 (1977) 241-280. [55]________, The Morava K-theories of Eilenberg-MacLane spaces and the Conner-* *Floyd conjecture, Amer Jour Math 102 (1980) 691-748. [56]R. D. Thompson, The v1-periodic homotopy groups of an unstable sphere at od* *d primes, Trans Amer Math Soc 319 (1990) 535-560. [57]________, A relation between K-theory and unstable homotopy groups with an * *application to B 2p, Contemporary Math 146 (1993) 431-440. [58]H. Yang, Tractable formulas for v-11ss*(SU (n)) when n p2- p, Lehigh Univ.* * thesis, 1994.