;0 >> >>Z=pmin((k)+1;n+k(p-1))if s = 1, t - s = 2n + qk, k <;0 >> >> min((k)+1;n) >>Z=p if s = 0, t - s = 2n + qk + 1, k >;0 >> min((k)+1;n+k(p-1)) >>Z=p if s = 0, t - s = 2n + qk + 1, k <;0 >: 0 otherwise: There are also evident analogues of parts (ii)-(iv) of Theorem 3.1 with coefficients in Q=Z(p). Note that the groups we are studying have lower homological degree than their integral counterparts, which is the reason that Q=Z(p)coefficients are convenient for the calculation we are about to make. From the short exact sequence of coefficient groups 0 ! Z=p -i!Q=Z(p)! Q=Z(p)! 0 we get a "Bockstein" long exact sequence . . .! Es;t2(S2n+1; Z=p) -i*!Es;t2(S2n+1; Q=Z(p)) ! Es;t2(S2n+1; Q=Z(p)) ! Es+1;t2(S2n+1; Z=p) ! : : : Now consider the -1 and -2 stems (with Q=Z(p)coefficients). There are two possibilities. Either the generator of E2;2n+12(S2n+1; Z=p) is in the kernel of the Bockstein i*, or it is not. In the former case, there would be a finite cyclic group in the -1 stem, which represents some portion of the stable Q=Z(p)desuspended to the 2n + 1 sphere. In the latter case the entire stable Q=Z(p)desuspends to the three sphere, and there is an additional Q=Z(p)in the -2 stem. It is this latter case which occurs, as stated in Theorem 7.1. Note that in either case there is one unstable Q=Z(p)in the -1 stem accounting for the fact that the element in E0;2n+12(S2n+1; Z=p) is the reduction of an integral class, namely the generator of the 0 stem. THE B-K SPECTRAL SEQUENCE FOR PERIODIC THEORIES 29 Since the statement of the Theorem is equivalent to the stable Q=Z(p) desuspending to S3, it suffices to do the calculation on the three sphere (algebraically we actually calculate on a two dimensional class in light of the fact that V (A[2n]) = U(A[2n+1])). The generator of E2;2n+12(S2n+1; Z=p) is represented by v-21h1 h1. Thus we need to show that v-21h1__h1_ 3 6= 0 p in the homology of the cobar complex for S3 with Q=Z(p)coefficients. The p-typical formal group law complicates the calculation. (An alternative to the following proof would be to give a direct proof that the stable Q=Z(p)desuspends to the three sphere. We have not been able to do so. It is left as an exercise to desuspend the class of order v-2h1 h1 p2 to S3.) In order to show that _________ 3 is not zero it is easier p to work with the K-theory spectrum. We recall the results of [2] and [1] and explain how to convert their notation to BP notation. Write u 2 ss2(K) for the Bott generator. u and 1 . u 2 K2(K) for its image under the left and right unit maps. ([2] and [1] denote the left action by u and the right action by v. We also identify the Bott class in homotopy with its K-theory Hurewicz image.) Let K_2i denote the 2i-th space in the -spectrum for K-theory. The Bott map gives a homotopy equivalence K_2i-! BU. The image of the map, (7.2) K*(CP 1) -! K*(BU) induced by the inclusion, generates K*(BU) as a ring (the product is induced by the Whitney sum). The K-theory of CP 1 is described in [2, (1.3)] . Definition 7.3. A polynomial f(!) 2 Q[!] is said to be numerical if f(n) is an integer for every integer, n. Let A denote the set of all numerical polynomials. A is a subring of Q[!] which contains Z[!]. Let ! !(! - 1) . .(.! - n + 1) (7.4) = _______________________: n n! ! ! ! Then 0 ; 1 ; 2 . . .is a Z basis for A. It is shown in [2] that A is isomorphic to K0(CP 1) as rings where the ring structure on K0(CP 1) is induced by the tensor produce of complex line bundles. K0(K) is the direct limit of iterated Bott maps, B* : K0(BU) - ! K0(BU) . 30 MARTIN BENDERSKY AND ROBERT D. THOMPSON The Bott map annihilates decomposables, and the following diagram commutes up to decomposables ([2]): A ____________-A ____________A- ____________.-. . | | | | | | | | | | | | |? B |? B |? B K0(BU) ______-K0(BU)* ______K0(BU)-* _________.-.*.________K0(K)- where the maps A -! A are multiplication by ! . Hence K0(K) A[!-1] . Since the multiplication on K0(K) is induced by the tensor product of bundles, the isomorphism is as rings. There is the relation ! ! ! ! = k + (k + 1) k k k + 1 which describes the Bott map. In K0(K) ! is identified with u-1 . u . The coproduct K0(CP 1) -! K0(K) K0(CP 1) is induced by (7.5) ! -! ! !: There is another description of the generators. We view BU as the 0- space in the spectrum for K theory. Recall that K*(CP 1) K*[[x]] where |x|= 2. We define fin 2 K2n(CP 1) by ae i ff 1 if i = n x ; fin= 0 if i 6= n x 2 K2(CP 1) corresponds to a map x : CP 1 ! K2 BU whichhinduces (7.2).i Define feinto be (x)*(fin) . Then K*(BU) w K* efi1; efi2;..T.h.e two sets of generators are related by efin= un n . We have homomorphisms (7.6) Kq(BU) Kq(K_2m) -oem!Kq-2m(K) Denote the class, efin, in Kq(K_2m) by fin;m. Then K*(K) is generated over Z[u; u-1; 1 . u-1] by oe0(fin;o) with oem (fin;m) = oe0(fin;o) . u-m . In Kn(K) we use bsnto denote oe2(fin+1;2) . This agrees the names for the generators defined in [28] (who does not use the s superscript). There is the Hopf Ring description of Kq(K_2m) (which is equivalent to the usual description of the terms in the unstable cobar complex) in [24], [15]. They denote by bi the classes fii;22 K2i(K_2) . We also have THE B-K SPECTRAL SEQUENCE FOR PERIODIC THEORIES 31 classes [ff] 2 K0(K-|ff|) defined as follows. If ff 2 K-r then ff 2 ss0(K_r). [ff] is the K-theory Hurewicz image of ff. The tensor product of virtual bundles induces the ring structure in K*(K) which induces a O-product K*(Km ) K*(Kn) ! K*(Km+n ) K*(Km ) also has a *-product inducted by the Whitney sum. As usual, Q denotes the indecomposables functor with respect to the *-product. Then [24], [15] show that QK*(K2m) is generated by the bi's and the [u]'s using the O-product. Since K*(Km ) is generated as a Z[u; u-1; 1 . u-1]-module by {bi} it follows that O-products of bi's must be a sum of bi's over Z[u; u-1; 1 . u-1]. In the unstable cobar complex, we identify QK*(K2m) with its image in K*(K) . In particular if ffi = oem (fl); fl 2 QK*(K2m) we will write ffi 2m for fl . For example bn = fin . u-1 2 and bsn= fin+1 . u-1. The Hopf ring class, [ff], suspends to 1 . ff and is written 1 ff-|u|(which has dimension 0, as it should on K-|u|). Notice that the shift in degree in (7.6)is realized by deleting the 2m in the unstable cobar notation. The bn's and [ff] enjoy simple coaction formulas. Let b = i0 bsithen (7.7) (b) = j0 bj bj; ([ff])= 1 [ff] It is an amusing exercise to show that (7.5)is compatible with (7.7). We will calculate on the 2-dimensional class. The first few terms of the K-theory unstable cobar complex for S2n are: ! ! (7.8) K*(S2n) ! K*(BU) ! K*(K(BU)) . . . ! Recall that K(BU) is the space defined in Section 2 for E = K and is the space whose homotopy groups are K*(BU). There is a homotopy equivalence Y K(BU) BU(fin)x Y where the copies of BU are indexed by the K* generators, fin and Y is a product of copies of BU indexed by the *-decomposable generators. On the two sphere the differential K*(S2) ! K*(BU) is given by d(un 2) = (1 . un - un) 2 = un(!n - 1) 2. The description of the generators in terms of numerical polynomials gives us formulas for the right action of un in terms of the b's. 32 MARTIN BENDERSKY AND ROBERT D. THOMPSON Example 7.9. Let p = 3. Then 1 . u2 = u2 + 6(efi3+ uefi2) An instructive way to check the validity of the right action formula such as this is to substitute ! = 0; 1; 2 . ...In our example we assert that ! ! -1 u2(!2 - 1) = 1 . u2 - u2 = 6(efi3+ uefi2) = 6u2( + )! 3 2 These are readily seen to be equal using the fact that ! = 0 for! = 0; 1; . .k.- 1 k k and k = 1. We now consider (7.8)tensored with Q=Z(p). An element f 2 Q[!-1] represents something in the rational unstable cobar complex on S2n if and only if f(!)!n is a polynomial in Q[!], and such a polynomial f(!)!n 2 Q[!] is zero if and only if f(!)!n is a numerical polynomial. More generally we may write an element in filtration k of the unstable cobar complex as f(!1; !2; . .!.k) . f is zero if and only if f(!1; !2; . .!.k)(!1!2. .!.k)n is integral. We now consider filtration 2. For integers, a and b we have the map a;b: Q[!1; ! 2] ! Q defined by sending !s1!t2to asbt . To clarify the notation, !1 denotes ! 1 and !2 denotes 1 !. a;bmaps A A to Z(p). So we have a well defined map (!1!2)n a;b a;b: C2 Q=Z(p)----! A A Q=Z(p)--! Q=Z(p) We now restrict to n = 1 (i.e. S2) and internal degree 0 and write C for C2 Q=Z(p). We say a polynomial, f(!1; !2) is symmetric if a;b= b;afor all integers a; b . v-2h1 h1 2 Theorem 7.10. _________ is a non-zero homology class on S . p Proof. h1 = -d(v)=p = -d(up-1)=p = up-1(1 - !p-1)=p so v-2h1 h1 p-1 p-1 p-1 3 f = ___________= ! (1 - ! ) (1 - ! )=p p THE B-K SPECTRAL SEQUENCE FOR PERIODIC THEORIES 33 which is not symmetric (p;p-1(f) = 0, p-1;p(f) 6= 0 mod Q=Z(p)). Now consider any f(!) 2 Q[!] in filtration 1 with internal degree 0 (i.e. there are no coefficients). We have d(f(!)) = 1 f(!) - f(! !) + f(!) 1 which is symmetric (since there are no coefficients to pass through the v-2h1 h1 tensor). So _________ cannot be in the image of the differential and p __ this proves the theorem. |__| v-2h1 h1 Remark 7.11. On S4 we have the differential d(v-2h21=p) = _________. p It is interesting to note how the above proof for S2 does not work on S4. We want to show that a;b= b;aon the 4 sphere. To see this notice that !p+1(1 - !p-1) !2(1 - !p-1)=p3 = !2(1 - !p-1) !2(1 - !p-1)=p3 - !2(1 - !p-1)2 !2(1 - !p-1)=p3 and the second term on the right, !2(1 - !p-1)2 !2(1 - !p-1)=p3, is numerical (i.e. a;b(!2(1 - !p-1)2 !2(1 - !p-1)=p3) 2 Z(p) for all integers (a; b)) and is therefore 0 with Q=Z(p)coefficients. The first term on the right, !2(1 - !p-1) !2(1 - !p-1)=p3, is clearly sym- metrical. Remark 7.12. The method of numerical polynomials gives an alter- native proof of the stable result Theorem 4.2 (b) of [21]. In terms of Q[!], their class y is ln(!), the formal natural log series. So dy = 0 follows from our formula for d(f(!)). 8. The double suspension spectral sequence The double suspension sequence in homotopy is the long exact se- quence of homotopy groups . .!.ssi(S2n-1) ! ssi+2(S2n+1) ! ssi-1(W (n)) ! : : : where W (n) is defined to be the homotopy fiber of the double suspen- sion map S2n-1 ! 2S2n+1. The double suspension sequence appears at the level of the E2-term of the classical unstable Adams spectral sequence: . .!.Es;t2(S2n-1) ! Es;t2(S2n+1) ! Es;t2((W (n))) ! Es+1;t2(S2n-1) ! : ::: 34 MARTIN BENDERSKY AND ROBERT D. THOMPSON This is obtained by filtering the -algebra by the odd spheres and defin- ing E*2((W (n))) for various n to be the homology of the subquotients of the filtration (with a suitable choice of indexing). See [18] and [14] for details. Note that E*2((W (n))) is not actually the E2-term of the unstable Adams spectral sequence for the space W (n). The double suspension sequence appears at the E2-level of the BP - based BKSS, i.e. the unstable Novikov spectral sequence: (8.1) 2 s 2n+1 H2 s-1 P2 s+1 2n-1 . .!.Es2(S2n-1) -oe!E2(S ) -! ExtA(U)(W (n)) -! E2 (S ) ! : : : where o W (n) BP*=p{x2pn-1; x2p2n-1; :::} with -coaction given by X i (x2pin-1) = pk-ihnpk-i x2pin-1; o If z = fl x2pn-12 Ext(W (n)), then P2(z) = d(fl phn) 2n-1; P P (In general if z = fli x2pin-1then P2(z) = d( fli pk-1hnk):) o If x 2 Es2(S2n+1) is represented in the unstable cobar complex by fl hn 2n+1 modulo terms which desuspend, and fl is a double suspension with respect to the dimension of hn 2n+1 (i.e. fl is defined on the 2n(p - 1) + 2n - 1 sphere) then H2(x) = fl x2pn-1 P (In general x can bePrepresented by a cocyle of the form flk pk-1hnkand H2(x) = flk x2pkn-1.) See [4] and [5] for details. The double suspension sequence for BP comes about as a special case of the composite functor spectral sequence 4.8 applied to the unstable -comodule BP*(S2n+1). In [4] it is shown that the machinery of [8] generalizes to the case of coalgebras over the graded ground ring BP*. Since BP*(BPn ) is cofree as a coalgebra, the G-derived functors of P are the same as the derived functors of P in the category of BP*- coalgebras. The calculations of [4] then show that the higher derived functors of BP*(S2n+1) vanish (specifically Ri = 0 for i > 1) and the CFSS collapses to two rows. After various identifications, this yields 8.1. THE B-K SPECTRAL SEQUENCE FOR PERIODIC THEORIES 35 In this section we make a few remarks concerning the fact that this is not what happens for E(1). In particular the above CFSS does not reduce to two rows: instead of a double suspension long exact sequence there is a double suspension spectral sequence. First we recall the explicit double complex construction given in [5] of the CFSS for E*(S2n+1). Let E satisfy 4.4. From Proposition 4.6 we have the functor V , defined as the indecomposables in G. From [24] we know that G(M(2n)) is isomorphic to the polynomial algebra E*[V (M(2n))]. For simplicity, in what follows denote M(2n) by M. Like U, there are maps V 2! V and 1 ! V making V into a triple. Hence there is a cosimplicial resolution d0-! -d0! -s0 0 s0 d1 (8.2) M d-! V (M) - V 2(M) -! . . . -d1! -s1 d2-! Apply the polynomial algebra functor to this resolution to obtain a resolution -d0! -d0! -s0 0 s0 d1 (8.3) E*(S2n+1) d-! G(M) - G(V M) -! . . . -d1! -s1 -d2! Now apply the primitive element functor to the un-augmented cosim- plicial object from 8.3 to give a cosimplicial object, U(2n), -d0! d0-! 1 (8.4) U(M) d1 U(V (M)) -d! . . . -! -d2! and the homology of the resulting chain complex is, by definition, RiGP E*(S2n+1). The cosimplicial object 8.4 fits into a bi-cosimplicial object defined by Di;j= UiV j(M); i > 0; j 0 36 MARTIN BENDERSKY AND ROBERT D. THOMPSON which we represent by the following diagram. .. . . .. o ! o ! " " " " " " ! ! U(U(M)) ! U(U(V (M)) ! . . . ! " " " " ! ! (U(M)) ! (UV (M)) ! . . . ! The associated bi-complex generates the composite functor spectral sequence. The complex associated to U(k)U(2n) is the k-th row. If we take vertical homology first, the result is concentrated in filtration zero and is the chain complex associated to ! ! 2 M ! V (M) ! V (M) . . . ! Taking homology horizontally, we get Ext A(V )(M(2n)) which is the same as Ext A(U)(M(2n + 1)) = E2(S2n+1) by 4.7 and 4.9. By taking homologies in the opposite order we obtain a spectral sequence ExtA(U)(RiGP (E*(S2n+1))) =) E2(S2n+1): The term ExtA(U)(R0GP E*(S2n+1)) is just Ext A(U)(M(2n)) = ExtA(U)(M(2n - 1)) = E2(S2n-1): We have R1GP E*(S2n+1) = W (n) by definition. In the case of E = BP , that is it. The G-derived functors of BP*(S2n+1) are computed in [4] and shown to vanish for i > 1. Hence the above spectral sequence degenerates to a long exact sequence. This cannot happen for K-theory. For example, W (n) is a mod-p vector space and if there were a double suspension long exact sequence of E2- terms, this would imply that E2(S2n+1) is bounded by pn as an abelian group (W (1) is essentially the three sphere). This contradicts the exis- tence of unbounded torsion in E2(S2n+1), specifically the three groups isomorphic to Q=Z. For this reason, the divisible groups can be thought of as obstructions to the vanishing of higher G-derived functors of P in the category M(G). They can also be thought of as corresponding in some sense to THE B-K SPECTRAL SEQUENCE FOR PERIODIC THEORIES 37 the Eilenberg-Mac Lane spaces that measure the failure of localization to commute with fiber sequences. Several things appear to be going on here. For one, E(1)*(E(1)2n ) is not cofree as a coalgebra, so the G-derived functors of P are not necessarily the same as the derived functors of P in the category of coalgebras over E(1)*. From [15] one sees that the spaces in the spectrum are limits of Wilson spaces [27]. But the category of coal- gebras over a non-connective ground ring is subtle, and even E(1)* of Wilson spaces are not cofree, e.g. E(1)*(CP 1). (Notice that E*(CP 1) is not a free algebra for E = E(n), K and even BP .) Secondly, even if there were some relationship between the G-derived functors of P , and the derived functors of P in the category of coalge- bras over a non-connective ground ring, the methods of [8], generalized to BP in [4], rely on irreducibility of the coalgebras under study, a condition which fails to hold for the K-theory of non-connected spaces. In particular the spaces E(n)m are non-connected. It would be useful and interesting to have a better understanding of M(G), along with some techniques for calculating these derived functors. References 1. J. F. Adams, A. S. Harris, and R. M. Switzer, Hopf algebras of cooperations* * for real and complex K-theory, Proc. London Math. Soc. (3) 23 (1971), 385-408. 2. Andrew Baker, Francis Clarke, Nigel Ray, and Lionel Schwartz, On the Kummer congruences and the stable homotopy of BU, Trans. Amer. Math. Soc. 316 (1989), 385-432. 3. M. Bendersky, E. B. Curtis, and H. R. Miller, The unstable Adams spectral sequence for generalized homology, Topology 17 (1978), 229-248. 4. M. Bendersky, E. B. Curtis, and D. Ravenel, EHP sequences in BP theory, Topology 21 (1982), no. 4, 373-391. 5. Martin Bendersky, The derived functors of the primitives for BP*(S2n+1), Trans. Amer. Math. Soc. 276 (1983), no. 2, 599-619. 6. ______, Unstable towers in the odd primary homotopy groups of spheres, Tran* *s. Amer. Math. Soc. 287 (1985), no. 2, 529-542. 7. ______, The v1-periodic unstable Novikov spectral sequence, Topology 31 (1992), no. 1, 47-64. 8. A. K. Bousfield, Nice homology coalgebras, Trans. Amer. Math. Soc. 148 (197* *0), 473-489. 9. ______, The localization of spaces with respect to homology, Topology 14 (1975), 133-150. 10. ______, The localization of spectra with respect to homology, Topology 18 (1979), 257-281. 11. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizatio* *ns, Lecture Notes in Mathematics, vol. 304, Springer-Verlag, 1972. 12. ______, The homotopy spectral sequence of a space with coefficients in a ri* *ng, Topology 11 (1972), 79-106. 38 MARTIN BENDERSKY AND ROBERT D. THOMPSON 13. A.D. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May, Modern foundations for stable homotopy theory, Handbook of Algebraic Topology (I.M. James, ed.), North-Holland, Amsterdam, 1995, pp. 213-253. 14. J. R. Harper and H. R. Miller, On the double suspension homomorphism at odd primes, Trans. Amer. Math. Soc. 273 (1982), no. 1, 319-331. 15. M. J. Hopkins and J. R. Hunton, On the structure of spaces representing a Landweber exact cohomology theory, Topology (19xx), xxx-xxx. 16. I. Kriz, Towers of E1 ring spectra with an application to BP , preprint. 17. P. S. Landweber, Homological properties of comodules over MU *MU and BP*BP , Amer. J. Math. 98 (1976), 591-610. 18. Mark Mahowald, On the double suspension homomorphism, Trans. Amer. Math. Soc. 214 (1975), 169-178. 19. Mark Mahowald and Robert Thompson, The K-theory localization of an un- stable sphere, Topology 31 (1992), no. 1, 133-141. 20. J.P. May, N. Ray, F. Quinn, and J. Tornehave, E1 ring spaces and E1 ring spectra, Lecture Notes in Math., vol. 577, Springer-Verlag, 1977. 21. H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. 106 (1977), 469-516. 22. Haynes R. Miller and Douglas C. Ravenel, Morava stabilizer algebras and the localization of Novikov's E2-term, Duke Math. J. 44 (1977), no. 2, 433-447. 23. G. Mislin, Localizations with respect to K-theory, J. Pure Appl. Algebra 10 (1977), 201-213. 24. D. C. Ravenel and W. S. Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1977), 241-280. 25. Douglas C. Ravenel, Localization with respect to certain periodic homology * *the- ories, Amer. J. Math. 106 (1984), 351-414. 26. W. S. Wilson, The -spectrum for Brown-Peterson cohomology, part I, Com- ment. Math. Helv. 48 (1973), 45-55. 27. ______, The -spectrum for Brown-Peterson cohomology, part II, Amer. J. Math. 97 (1975), 101-123. 28. W. Stephen Wilson, Brown-peterson homology, an introduction and sampler, Regional Conference Series in Math, vol. 48, AMS, Providence RI, 1980. Hunter College and the Graduate Center, CUNY E-mail address: mbenders@shiva.hunter.cuny.edu URL: http://math.hunter.cuny.edu/"benders Hunter College and the Graduate Center, CUNY E-mail address: thompson@math.hunter.cuny.edu URL: http://math.hunter.cuny.edu/"thompson