vn-ELEMENTS IN RING SPECTRA AND APPLICATIONS TO BORDISM THEORY MARK A. HOVEY October 1993 Introduction The work of Hopkins and Smith [HS ] has shown that the stable homotopy catego* *ry has layered periodic behavior. On the (p-local) sphere, the only non-nilpotent * *self- maps are multiplication by a power of p. But if we kill such a power to form t* *he Moore space M(pk), then we get a new family of non-nilpotent self maps, called * *the v1-self maps. Similarly, if we kill one of those, we get v2-self maps, and this* * behavior continues. One of the great advantages of the Brown-Peterson spectrum BP is that the pe- riodicities are not layered, but they all appear as homotopy classes vn 2 ss2(p* *n-1)BP. Another great advantage of BP is that it is comparatively simple algebraically* *. Its coefficient ring is polynomial, and it is possible to calculate in the Adams-No* *vikov spectral sequence based on the operations in BP -homology. In fact, most of t* *he spectra used by algebraic topologists are complex oriented, in that they admit * *maps from BP . But there is one crucial example that does not, namely, real K-theory KO. Hopkins and Miller [HMi ] have recently shown that KO is the tip of an iceb* *erg of non-complex oriented theories which have interesting torsion. It would be nice to have a bordism spectrum that did admit maps to the Hopkin* *s- Miller theories EOn. Recall that MO is the Thom spectrum arising from the k - 1-connected Postnikov cover BO of BO, and similarly for MU. Note that MO<4> = MSpin and MU<4> = MSU both admit orientations to KO. I hope that this also is the beginning of a general phenomenon, and that the MO and MU will admit orientations to EOn when k is sufficiently large. Such an orientation may have some analytic meaning. Witten interprets a (conj* *ec- tural) orientation from MO<8> to elliptic cohomology, which should be EO2, as t* *he index of an S1-equivariant Dirac operator on the free loop space of a manifold * *with MO<8> structure. However, it will be hard to get at the algebraic meaning of su* *ch an orientation because we know so little about the MO. This paper is an attempt to get some qualititative understanding of the MO and MU. We try to find analogues of vn in the homotopy of MO and MU. It turns out to be easier to study the existence and properties of such analogu* *es in 1 2 MARK A. HOVEY general ring spectra. This is the content of the first section. This section ow* *es much to the beautiful paper [HS ], and discusses many of the same ideas. The highlig* *ht of this section is a sufficient condition for the existence of a vn-element, defin* *ed below, in a ring spectrum. The idea here is to use the vn-elements we already know exi* *st in finite ring spectra. This condition takes a particularly nice form when the pri* *me p is odd and when n = 1. Then if any of the homotopy classes fft 2 ss2t(p-1)-1S0 map* * to 0 under the unit map to R, there is a v1-element in R. The second section is concerned with applications to MO and MU. Because fftis in the image of the J homomorphism, it is possible to determine when fftg* *oes to 0 in MO or MU. We therefore find v1-elements in MO and MU. Given a v1-element v in any ring spectrum R, we can get the Bousfield localization of R* * at the Morava K-theory K(1) by inverting v and completing at the prime p. On the other hand, it is also possible to calculate LK(1)MO in a different way by compari* *ng MO with MSpin. By comparing these two results, we can understand something about the homotopy of MO. We also show that, if the torsion in ss*MO is bounded, then the natural map MO*(X) MO*KO* ! KO*(X) is an isomorphism, extending the results of [HH ]. The author would like to thank Mike Hopkins. His ideas greatly simplified and improved this paper. The author would also like to thank Peter Landweber and Do* *ug Ravenel for sharing their unpublished notes on MO<8> [LS , PR ]. He would also * *like to thank David Johnson, Peter Landweber, Serge Ochanine, John Palmieri, and Hal Sadofsky for helpful discussions. All spectra are assumed to be p-local for some fixed prime p throughout this * *paper, unless explicitly stated otherwise. 1. Analogues of vn in general ring spectra 1.1. Notation. We fix notation. Given a ring spectrum R, we will use the standa* *rd 0 j notations for the multiplication map R ^ R -! R and j for the unit map S -! R. f Given a map X -! Y of spectra, and another spectrum E, we will write E*(f) for the induced map on E-homology. On the other hand, if f is a homotopy class in a ring spectrum R, and E is a ring spectrum, we will also use E*(f) to denote t* *he Hurewicz image of f in E*(R). The context should make clear which notation we mean. The reason for adopting such notation is the following lemma, which is well-k* *nown and implicit in many papers in the subject, but which we would like to make exp* *licit. Given v 2 sskR for a ring spectrum R, we can make a self-map ^v: kR -! R as the composition kR ' Sk ^ R v^1--!R ^ R -! R: vn-ELEMENTS IN RING SPECTRA 3 Lemma 1.1.1. Suppose E and R are ring spectra, and v 2 ssk(R). Then E*(^v) is multiplication by E*(v) in the ring E*(R). Thus, since homology commutes wi* *th direct limits, we have E*(v-1R) ~=E*(v)-1(E*R): Proof.Recall that ^vis the composite Sk ^ R v^1--!R ^ R -! R: Since the Hurewicz map is induced by the unit j S0 -! E; when we smash with E we get the composite 1^E*(v)^1 ^ E ^ Sk ^ R ------! E ^ E ^ R ^ R --! E ^ R: We compare this with multiplication by E*(v), which is the top row and last col* *umn of the following diagram. E*(v)^1^1 1^T^1 Sk ^ E ^ R ------! E ^ R ^ E ^ R ---! E ^ E ^ R ^ R ?? ?? ?? T^1?y f^1?y ^ ?y 1^E*(v)^1 ^ E ^ Sk ^ R ------! E ^ E ^ R ^ R ---! E ^ R Here T denotes the twist map, and f is the composite E ^ E ^ R T^1--!E ^ E ^ R 1^T--!E ^ R ^ E: Then the left square will commute by inspection, and the right square will comm* *ute if and only if E is commutative. However, even if E is not commutative, the who* *le j^v __ diagram will still commute, since E*(v) is the map Sk --! E ^ R: |_ | Note that in this lemma, if E = R, E*(v) is the right unit j^1 jR : S0 ^ R --! R ^ R applied to v. We will mostly be applying this lemma to the Morava K-theories K(n). We will include K(1) = HFp as a Morava K-theory unless otherwise stated. If R is a ring spectrum, note that K(n)*(j) is either injective or 0 since K(n)*(S0) is a grad* *ed field. We will denote K(n)*(j)(vkn) 2 K(n)*(R) by simply vkn, following Hopkin* *s- Smith [HS ]. 4 MARK A. HOVEY 1.2. vn-elements. We need to understand what we mean by an analog of vn. First, we recall the definition of Hopkins-Smith [HS ]. Definition 1.2.1.Given a finite ring spectrum R, and a homotopy class v 2 ss*(R* *), define v to be a vn-element if K(n)*(v) is a unit and K(i)*(v) is nilpotent if * *i 6= n. The advantage of this definition is that a vn-element induces a vn-self-map. * *How- ever, there are several problems with this definition applied to infinite ring * *spectra. First of all, in a general ring spectrum just knowing that K(n)*(v) is a unit w* *ill not be enough to conclude that K(n)*(vk) is a power of vn. The argument in [HS ] re* *lies on the finiteness of K(n)*(R) K(n)*Fp . But more importantly, vn 2 BP * is not a vn-element using this definition, because K(i)*(vn) is not nilpotent when i <* * n. These comments motivate the following definition. Definition 1.2.2.Given a ring spectrum R and v 2 ss*(R), define v to be a gen- eralized vn-element if K(n)*(vj) = vknfor some j; k and K(i)*(v) is nilpotent f* *or i > n. Hopkins and Smith show in [HS ] that if v is a vn-element then there are j; k* * such that K(n)*(vj) = vkn, so that v is a generalized vn-element. The converse is al* *so true. Theorem 1.2.3. If R is a finite ring spectrum, every generalized vn-element is* * in fact a vn-element. We need the following lemma. The proof given here is due to Hal Sadofsky. Lemma 1.2.4. Suppose X is a type n finite spectrum and f : kX -! X is a self- map with cofiber Y . Suppose k 6= 0. Then Y is either trivial, type n or type n* * + 1. Proof.Note that Y certainly has type at least n. Also, if k < 0, f is nilpotent* *. In that case K(n)*(f) can not be an isomorphism, so Y will have type n. So assume k > 0, and Y has some finite type greater than n + 1. Then K(n)*(f) and K(n + 1)*(f) are both isomorphisms. This implies that n > 0. Indeed, if n = 0, then f induces an isomorphism of the finite-dimensional vector space HQ*(X), which can't happen unless k = 0. Using the techniques of [HS ], we can find a vn-self map g of X and an intege* *r N such that K(n)*(fN ) = K(n)*(g). Recall the method is to use the ring spectrum R = DX ^X. The map f corresponds to a homotopy class v in R such that K(n)*(v) is a unit. Since K(n)*(R) K(n)*Fp is finite, there is an N such that K(n)*(vN )* * is a power of vn. Thus K(n)*(fN ) is also a power of vn. Replacing N by a possibly l* *arger value, we can find a vn-self map g which is multiplication by that same power o* *f vn. We can assume K(i)*(g) = 0 for i 6= n. Let Z be the cofiber of fN - g. Then K(n)*(Z) 6= 0, but K(n + 1)*(Z) = 0, whi* *ch is a contradiction. __|_ | vn-ELEMENTS IN RING SPECTRA 5 Note that this lemma is in general false when k = 0. Indeed, one can simply t* *ake the wedge of a type 0 spectrum X and a type 2 spectrum Z with the self-map which is the identity on X and trivial on Z. We can now prove Theorem 1.2.3. Proof of theorem.Suppose R has type m, and v 2 ssk(R) is a generalized vn-eleme* *nt. If n m, then K(i)*(R) = 0 for i < n, so v is a vn-element. So assume n > m. Note that the cofiber of ^vhas type n+1, which by the preceding lemma is impo* *ssible when k 6= 0. So we can assume k = 0. This also implies that K(m)*(v) must be a unit. First suppose that m > 0. From [HS ], there is an N such that K(m)*(vN ) * *= 1. By choosing a possibly larger value of N, we can assume K(i)*(vN ) = 0 for i > * *n: Let Y denote the cofiber of v^N- 1. Then K(m)*(Y ) is non-zero, while K(n + 1)** *(Y ) is zero, a contradiction. Now suppose that m = 0. Consider w, the image of v under the map of ring spectra R -! R ^ M(p2) induced by including the bottom cell. (We use M(p2) rath* *er than M(p) to avoid 2-primary problems). It is easy to see that w is also a gene* *ralized vn-element: see Theorem 1.3.1 below. So by the above, we must have n = 1, since R ^ M(p2) has type 1. But then ^wis a v1-self map of degree 0, which is impossi* *ble by [HS ]. __|_ | Because of this lemma, we will call our generalized vn-elements simply vn-ele* *ments from now on. We also point out the following fact, which is implicit in [HS ]. Lemma 1.2.5. Suppose R is a finite type n ring spectrum. Then R contains vn- elements. Proof.Take a vn-self map f of degee d of R such that K(n)*(f) = vknfor some k a* *nd K(i)*(f) = 0 for i 6= n. Let v denote the composite j d f Sd -! R -! R: Then v is a vn-element. Indeed, K(i)*(v) = K(i)*(f)(1). __|_ | 1.3. Properties of vn-elements. The properties of vn-elements that we need are summarized in the following theorem. f Theorem 1.3.1. (1) Suppose R -! S is a map of ring spectra and v 2 ss*R. If* * v is a vn-element, so is f(v). f (2) Suppose R -! S is a map of ring spectra such that K(i)*(f) is injective f* *or i n. Suppose v 2 ss*(R). If f(v) is a vn-element, so is v. (3) Suppose v; w 2 ss*(R) commute with each other. If v; w 2 ss*R are vn-elem* *ents, so is vw. 6 MARK A. HOVEY Proof.Note K(i)*(f)(K(i)*(v)k) = K(i)*(f(v))k: So if K(i)*(v) is nilpotent, so is K(i)*(f(v)). Conversely, if K(i)*(f) is inje* *ctive and K(i)*(f(v)) is nilpotent, so is K(i)*(v). Also, since f commutes with the unit* *, if K(n)*(vj) = vkn, then K(n)*(f(v)j) = K(n)*(f)(vkn) = vkn: Conversely, if K(n)*(f) is injective and K(n)*(f(v)j) = vkn, then K(n)*(f)(K(n)*(vj)) = K(n)*(f)(vkn) so K(n)*(vj) = vkn: This proves the first and second parts of the theorem. The * *third part just follows from the fact that the Hurewicz map is a ring homomorphism. * *__|_ | The main advantage of our definition of vn-elements comes from the second part of the above theorem. Corollary 1.3.2. Suppose R is a ring spectrum, v 2 ss*(R), and X is a finite ri* *ng spectrum of type m. Let f : R -! R ^ X be induced by the unit of X. If f(v) is a vn-element for some n m, then so is v. j Proof.The unit map S0 -! X is injective on K(i)-homology as long as K(i)*(X) 6=* * 0. Since X is type m, this will happen when i m. Since K(i)* is a field spectrum,* * so has a Kunneth isomorphism, K(i)*(f) = K(i)*(R) K(i)*K(i)*(j) is also injective when i m. The preceding proposition completes the proof. __* *|_ | 1.4. Finding vn-elements in ring spectra. We can now give our sufficient con- dition for the existence of a vn-element in a ring spectrum R. Theorem 1.4.1. Suppose we have a finite ring spectrum X, and let g : X -! Y j denote the cofiber of the unit S0 -! X. Suppose X is type m, and v 2 ssk(X) is* * a vn-element. Suppose R is a ring spectrum and suppose the composite g j^1 Sk -v!X -! Y --! R ^ Y is null. Then any lift of (j ^ 1)(v) to sskR is a vn-element. Proof.Since v is a vn-element and j ^ 1 is a map of ring spectra, (j ^ 1)(v) is* * also a vn-element. Now apply Corollary 1.3.2 to complete the proof. __|_ | Let us call vn-elements that arise from vn-elements in a finite ring spectrum* * in this way finitary vn-elements. Conjecture 1.4.2. Every vn-element is finitary. vn-ELEMENTS IN RING SPECTRA 7 We will see one advantage of finitary vn-elements in the next section. In pra* *ctice, it can be difficult to tell whether a vn-element is finitary. For example, show* *ing that v1 2 BP * is finitary at p = 2 requires finding a finite spectrum with a v1-sel* *f map of degree 2. This has been done by Davis and Mahowald [DM ], but I do not know if their example is a ring spectrum. The simplest case of Theorem 1.4.1 is when n = 1 and p is odd. Recall the ele* *ments fft2 ssqt-1S0, where q = 2p - 2. Corollary 1.4.3. Let p be an odd prime. Suppose R is a ring spectrum and j O ff* *t is null. Then there is a v1-element in ssqtR. Proof.Take the finite ring spectrum X in Theorem 1.4.1 to be the Moore spectrum M(p). The definition of fft is the composite of a v1 element in degree qt with* * the pinch map to the top cell. __|_ | We have a similar result at p = 2, which is complicated by the fact that M(2)* * is not a ring spectrum. Here we need the elements fft=22 ss8t-1S0. There is not comple* *te conformity on the name of this class: Ravenel calls it ff4t=2in [Rav1 ]. Corollary 1.4.4. Let p = 2. Suppose R is a ring spectrum such that j O fft=2is * *null. Then there is a v1-element in ss8tR. Proof.This is precisely the same as the previous corollary except we must use M* *(4). The relevant facts about M(4) can be found in [DM ]. __|_ | To apply Theorem 1.4.1 more generally, we need specific examples of type n fi* *nite ring spectra and vn-elements. These are provided by the generalized Moore spect* *ra M(pi0; vi11; : :;:vin-1n-1) which are discussed in [HS , Proposition 5.12] and * *[MS ]. These are type n spectra with the evident BP homology, which exist for a cofinal set * *of sequences. They can be chosen to be ring spectra, which will then have vn-eleme* *nts, for a smaller cofinal set of sequences by [Dev ]. Furthermore, Devinatz shows t* *hat we can assume these ring spectrum structures are coherent, in the sense that the n* *atural map M(pi0; : :;:vij-1j-1) -! M(pi0; : :;:vijj) is a map of ring spectra for all j n-1. Let us call sequences (i0; : :;:in-1) * *for which M(pi0; vi11; : :;:vin-1n-1) exists realizable, and sequences for which M(pi0; v* *i11; : :;:vin-1n-1) is a ring spectrum which is coherent in the above way multiplicative. Note that the M(pi0; vi11; : :;:vin-1n-1) have 2n cells, and the bottom 2n-1 * *cells and the top 2n-1 cells are both suspensions of M(pi0; : :;:vin-2n-2). Given a realizable sequence, one can get generalized Greek letter elements in* * the homotopy of the sphere by the Adams construction: include the bottom cell, iter* *ate the vn-self-map some number of times, and pinch off to the top cell. One of the outstanding problems in homotopy theory is to understand these elements: when 8 MARK A. HOVEY they exist, when they are non-trivial, and the extent to which they are all of * *the elements in the homotopy of the sphere. We can use the idea behind Theorem 1.4.1 to get some partial results about th* *ese Greek letter elements. Corollary 1.4.5. Suppose (i0; : :;:in-1) is multiplicative. Then the composite * *of in- cluding the bottom cell, iterating the vn-self-map, and pinching off to the top* * 2n-1 cells is non-trivial. Proof.Because the sequence is multiplicative, the map M(pi0; : :;:vin-2n-2) -! M(pi0; vi11; : :;:vin-1n-1) is a map of ring spectra. It is injective on K(i)-homology for i n, because th* *e maps used in making the M(pi0; vi11; : :;:vin-1n-1) induce the evident maps on BP -h* *omology. Hence, by Theorem 1.3.1(2), if the composite in the statement of the corollary * *were null, we would get a vn-element in the type n - 2 spectrum M(pi0; : :;:vin-2n-2* *). This is impossible. __|_ | 1.5. Generalized vn-elements and localization. We need to recall some facts relating to the failure of the telescope conjecture. These can be found in [Ho* *v ] or [MS ]. Recall that given a finite type n spectrum X, its telescope T el(X), de* *fined as the homotopy direct limit of any vn-self map, is well-defined. Furthermore,* * the Bousfield class of T el(X) is independant of the type n finite spectrum X. We d* *enote this Bousfield class by . Given a vn-element v in a finite ring spectr* *um R, we have LTel(n)(R) = v-1R: In particular, T el(n)*(v) is a unit, no matter which spectrum we use for T el(* *n). From this we get the following lemma. Lemma 1.5.1. Suppose R is a ring spectrum and v is a finitary vn-element. Then T el(n)*(v) is a unit. Proof.Since v is finitary, there is a finite type n ring spectrum X and a vn-el* *ement w such that (j ^ 1)(w) = f(v), where f : R -! R ^ X is the ring map induced by the inclusion of the bottom cell of X, and j ^ 1 : X -! R ^ X is induced by the unit of R. Since T el(n)*(w) is a unit, so is T el(n)*(f(v)). Another way to sa* *y this is that the self-map ^v: kR -! R becomes a homotopy equivalence upon smashing with T el(n) ^ X. We claim that = , so that in fact ^vbec* *omes a homotopy equivalence upon smashing with T el(n), as required. Indeed, since we are only worried about Bousfield classes, we can use w-1X for T el(n), and then* * note that w-1X ^ X = w-1X ^ w-1X which has the same Bousfield class as w-1X. __|_ | vn-ELEMENTS IN RING SPECTRA 9 Thus, if v is a finitary vn-element, the map R -! v-1R is a T el(n)-equivalen* *ce. Now v-1R is not T el(n)-local, but it is somewhat less local. Before describing* * this, we need a definition. Definition 1.5.2.Given a spectrum R, say that R satisfies the telescope conject* *ure if = : The usual telescope conjecture is that S0 satisfies the telescope conjecture,* * which we now know to be false for n = 2 [Rav2 ]. However, BP satisfies the telescope conjecture [Rav ]. Lemma 1.5.3. Suppose R is a ring spectrum and v is a vn-element. Then v-1R is T el(0) _ . ._.T el(n)-local. If R satisfies the telescope conjecture, then* * v-1R is T el(0) _ . ._.T el(n - 1) _ K(n) local. Proof.First note that K(i)*(v-1R) = 0 for i > n. Thus, if X is a ring spectrum * *of type n + 1, K(i)*(v-1R ^ X) = 0 for all i. Since v-1R ^ X is a ring spectrum, * *it must then be null, by the nilpotence theorem [HS ]. Then in fact v-1R ^ Y is nu* *ll for all finite spectra Y of type at least n + 1, by a thick subcategory argument. I* *n the terminology of [Hov ], the finite acyclics of v-1R are Cn+1. We showed in [Hov * *] that this means : Since v-1R is a ring spectrum, it is self-local, so also T el(0) _ . ._.T el(n)* * local. Now suppose R satisfies the telescope conjecture. Then = : __ |_ | We now must recall more results from [Hov ]. If X is a finite spectrum of typ* *e n, localization with respect to X is particularly simple. It is given by completio* *n with respect to p; v1; : :;:vn-1. That is, if Y is another spectrum, LX Y = lim-Y ^ M(pi0; vi11; : :;:vin-1n-1) = Yp;:::;vn-1: We show in [Hov ] that if Y is already T el(0) _ . ._.T el(n) local, then LTel(n)Y = Yp;:::;vn-1: Similarly, if Y is T el(0) _ . .T.el(n - 1) _ K(n) local, then LK(n)Y = Yp;:::;vn-1: We have then proved the following theorem. 10 MARK A. HOVEY Theorem 1.5.4. If R is a ring spectrum and v is a finitary vn-element, then LTel(n)R = (v-1R)p;:::;vn-1: If R satisfies the telescope conjecture and v is an arbitrary vn-element, then LK(n)R = (v-1R)p;:::;vn-1: Corollary 1.5.5. Suppose R is a ring spectrum. Then v 2 ss*R is a v1-element if and only if LK(1)R = (v-1R)p: Proof.Every spectrum satisfies the telescope conjecture when n = 1. __|_ | 2. The bordism spectra MO and MU 2.1. Introduction. Recall that the bordism spectrum MO is obtained by taking the Thom spectrum of the kth Postnikov cover BO of BO. They are ring spectr* *a, and the natural maps BO ! BO give maps of ring spectra MO ! MO: We have analogous statements for the complex analog MU. Proposition 2.1.1. The map induced by the unit S0 ! lim-MO is a homotopy equivalence. Similarly S0 ! lim-MU is a homotopy equivalence. Proof.We will just prove the real case as the complex case is similar. Note th* *at the inclusion of the basepoint into BO is an isomorphism on homotopy through dimension k - 1. It is therefore also an isomorphism on homology through dimens* *ion k - 1. If k 2, we can apply the Thom isomorphism to see that S0 ! MO is an isomorphism on homology through dimension k - 1. It is thus an isomorphism on homotopy through dimension k - 2 and an epimorphism on homotopy in dimension k - 1. __|_ | This is one of the motivations for studying the MO and the MU. But very little work has been done here. The spectra MSU = MU<4> and MSpin = MO<4> are reasonably well understood, though some open questions remain. References include [Pen , Bo, Ko ] for MSU and [ABP , GP ] for MSpin, as well as the stan* *dard reference for cobordism [St1]. The spectrum MO<8> has been studied at the prime 2 in [G , DM2 ]. Gorbunov and Mahowald have recently computed the first 50 or * *so vn-ELEMENTS IN RING SPECTRA 11 homotopy groups of MO<8> at p = 2 [GM ]. There are also some unpublished notes of Pengelley and Ravenel [PR ] concerning MO<8> at p = 3 (where, incidentally, * *it is much simpler than at p = 2) and at larger primes. Even the structure of the hom* *ology of MO and MU as modules over the Steenrod algebra is not known, though the homology groups and partial information about the Steenrod algebra structure are given by Stong [St], Singer [Sin], and Giambalvo [G2 ]. Giambalvo shows in [G2 * *] that the mod p cohomology of MO is free over the reduced power part of the Steenr* *od algebra when p k_2+ 1: The results of Singer [Sin] together with Giambalvo's p* *roof show that H*(MU; Fp) is also free over the reduced powers when p k_2+ 1. One would like to conclude from this that MO and MU split into a wedge of suspensions of BP when p k_2+ 1: But there may be torsion in the homology of MO when k > 8 and in the homology of MU when k > 6: It is true that MO<8> and MU<6> split into a wedge of suspensions of BP when p 5. However, neither splitting can not be multiplicative, and the ring structure even of MO<* *8>* at p = 5 is not known. More detailed information at p = 2 for MO<8> is given by Da* *vis in [D ]. Our primary motivation for studying the MO and the MU is that, if one had never heard of real K-theory, studying MSpin would lead one to it. Similarl* *y, we hope that studying MO and MU will lead us to higher analogs of real K- theory like the EOn constructed by Hopkins and Miller in [HMi ]. Those theories* * are both highly complete and nonconnective. We hope that there are uncompleted and connective versions of the EOn that will split off of appropriate MO, at lea* *st at p = 2. At odd primes, we expect to find analogues of the Pengelley spectrum BoP which splits off MSU, and which is a sort of amalgam of BP and ko. Note that there is an important difference between the behavior of the MO * *at odd primes and their behavior at p = 2. At odd primes, MSpin splits multiplicat* *ively into a wedge of suspensions of BP . Thus there is a ring spectrum map MO ! BP for k 4. At p = 2, on the other hand, Bahri and Mahowald [BM ] show that the Thom class gives a map A==A(r - 1) -! HF2 *(MO): Here A is the mod 2 Steenrod algebra and A(r - 1) is the Hopf subalgebra genera* *ted i b by the Sq2 for i r - 1: Also, for r = 4a + b with 0 b 3, OE(r) is 8a + 2 . This makes it seems likely that the jth Milnor primitive Qj acts freely on the * *mod 2 cohomology of MO. If so, since Qj is the first differential in the At* *iyah- Hirzebruch spectral sequence for K(j)*X, we would get that K(j)*MO = 0 for j r. We make the following conjecture about the MO. Conjecture 2.1.2. (1) (Ravenel) The Bousfield class of MU at all primes a* *nd of MO at odd primes is the same as that of BP . At p = 2, the Bousfield 12 MARK A. HOVEY class of MO is the same as that of BP : In particular, MO and MU satisfy the telescope conjecture. (2) There are vn-elements in MO* and MU* for all n. (3) At p = 2, MO admits an orientation to EOr-1, and the natural map MO*(X) MO*EOi*! EOi*(X) is an isomorphism. We expect MO to admit orientations to certain of the EOi at odd primes as well. The first case of this should be that MO<8> admits an orientation to EO2 * *at p = 3. However, it is unlikely that there could be a tensor product theorem he* *re, because MO<8> probably has BP summands in it at p = 3. The logical generalizat* *ion of this would be that MO<2p + 2> admits an orientation to EOp-1. We will see be* *low that this is impossible for p > 3. It is possible that MO<2p+2> admits an orien* *tation to EO2(p-1). 2.2. The image of J. The results of section 1.4 say that to find v1-elements in* * a ring spectrum R, we should determine which members of the ff family in ss*S0 map to 0 under the unit map j : S0 -! R. This problem is made much simpler by the f* *act that most of the ff family is in the image of the J homomorphism. This is descr* *ibed in [Rav1 ]. Let us recall, from for example [Ad ], the J homomorphism J : ss*(SO) -! ss*(* *S0): Consider a homotopy class x : Sk ! SO. The adjoint of x is a map "x: Sk+1 ! BSO: This defines an oriented vector bundle fl over Sk+1. The Thom spectrum M(fl) of* * fl is a 2-cell complex with a cell in dimension 0 and in dimension k + 1. The atta* *ching map of this complex is the J-homomorphism applied to x, as pointed out in [Ad ]* *. So we have a cofiber sequence Sk -Jx!S0 -! M(fl) -! Sk+1: We also have a map of Thom spectra M(fl) -! MSO and the composite S0 ! M(fl) ! MSO is the unit. Indeed, it is the Thom spectrum of the composite * ! Sk+1 ! BSO which is the unit of the H-space BSO. Notice that "xobviously lifts to y Sk+1 -! BO : Thus we get a lift of the Thom class M(fl) -! MO: vn-ELEMENTS IN RING SPECTRA 13 If we want to consider MU instead, we need to consider the complex J- homomorphism Jc, and start with a class x 2 sskU: In that case, we get a Thom class M(fl) -! MU: The results of [Bott] show that sskU ! sskSO is an isomorphism for k 3 (mod 8) but is multiplication by 2 for k 7 (mod 8). Thus Im Jc has index 2 in Im J in dimensions congruent to -1; 0; and 1 mod 8 and is all of Im J in dimensions con* *gruent to 3 modulo 8. Theorem 2.2.1. The composite Im J -! ss*S0 -! ss*MO is injective in dimensions k - 2 and 0 in dimensions k - 1. Similarly, the composite Im Jc -! ss*S0 -! ss*MU is injective in dimensions k - 2 and 0 in dimensions k - 1. Proof.The proof is exactly the same in the real and complex case, so we will ju* *st prove the real case. The unit map S0 ! MO is an k - 2-equivalence, as pointed out in Proposition 0.1, so the injectivity in those dimensions is clear. Given * *a class x 2 Im J in dimension n k - 1, we saw above that the unit map factors through the cofiber of x : Sn ! S0, so the theorem follows in the real case. __|_ | As an example, consider MU<8>. We have (HF2 )0MU<8> = (HF2 )8MU<8> = F2, and these are the only non-zero groups in dimensions 8. The Steenrod algebra a* *cts trivially here, because the Thom class pulls back to (HF2 )0MU and the map (HF2 )8MU -! (HF2 )8MU<8> is trivial. We use the standard notation for elements in the E2 term of the Ada* *ms spectral sequence at p = 2 [Rav1 ]. So in the Adams spectral sequence, the E2 t* *erm will have hi0h3 in filtration i + 1 and topological degree 7, for i 3. Each s* *uch element is a permanent cycle since it is a permanent cycle in the Adams spectral sequence for the sphere. No differential can hit h3 since it is in filtration * *1, so it survives to ss7MU<8>, despite the fact that it is in Im J. There is a tower beg* *inning in filtration 0 on a class x in topological degree 8. The preceeding theorem te* *lls us that d2(x) = h0h3. Thus we have a homotopy class v in filtration 4 which exists because h40h3 = 0. This class v is a v1-element. From Theorem 2.2.1 and the results of section 1, we get the following theorem. Recall that q = 2(p - 1) for odd primes p, and let q = 8 at p = 2. Also note th* *at fft=2 is in the image of the complex (in fact the quaternionic) J homomorphism at p =* * 2. 14 MARK A. HOVEY Theorem 2.2.2. There are v1-elements in ssqtMO and ssqtMU if and only if qt k. Furthermore, if k 4, any such element maps to a unit under the orientat* *ion to KO. Proof.The only thing we have not already shown is that a v1-element maps to a u* *nit in (the p-localization of) KO. To see this, note that the image must be a v1-el* *ement in KO, and any such is a unit. __|_ | Consider for a moment the manifold M corresponding to a v1-element v construc* *ted as above. Because v 2 ssqtMO can be lifted to ssqtMO simply by lifting the Thom class, the classifying map of the tangent bundle of M can be lifted to BO. So the tangent bundle is trivial on the qt - 1-skeleton. Such manifolds* * are called almost-parallelizable and have been studied, for example, by Milnor-Kerv* *aire in [MK ]. Note that M is not parallelizable, since we just showed above that v* * maps to a (p-local) unit in KO *, so the A^genus of M is nontrivial. The author's o* *riginal approach to this problem involved using these Milnor-Kervaire manifolds directl* *y, but the above approach (suggested by Hopkins) using the image of J gives more information. We still need the Milnor-Kervaire manifolds for a lemma later. We would like to extend this result to higher periodicities. We restate a pa* *rt of Conjecture 2.1.2. Conjecture 2.2.3. There are vn-elements in ss*MO and ss*MU for all n. In MO<8>, there is a v2-element in dimension 144 at p = 3, and in dimension 192 at p = 2. We give an outline of a possible proof of this conjecture. Consider a genera* *lized Moore spectrum M(pi0; vi11; : :;:vin-1n-1) for a multiplicative sequence, and l* *et Y denote the cofiber of the unit map, as in section 1.5. Given a vn-element v, let f : S* *k -! Y denote its image in Y , and let C denote its cofiber. The key fact that allows* * us to handle the case n = 1 is that C, or a suspension of it, is a Thom spectrum. * * In general, no suspension of C can be a Thom spectrum of an oriented bundle, becau* *se the Thom class would have to be in degree 0, and there will be a nontrivial (se* *condary) Bockstein on the Thom class in C. However, I make the following conjecture. Conjecture 2.2.4. A suspension of the Spanier-Whitehead dual of C is a Thom spectrum. If this conjecture were true, then it is easy to see that the base space, whi* *ch should probably be a suspension of the dual of Y , has to be highly connected, so we c* *ould repeat the argument above to find vn-elements in MO. The problem of finding the lowest dimension in which a vn-element can occur is extremely intricate. Note that there is an orientation from MO<8> to K(n) at odd primes, so any vn-element must map to an integral power of vn. Using the method of this paper, one would have to know which sequences are multiplicative. At p * *= 3, vn-ELEMENTS IN RING SPECTRA 15 M(p; v1) exists but is not a ring spectrum. M(p; v21) is a ring spectrum [Oka * *], but it is not known what power of v2 occurs. There is some reason to think it is v* *32, which is in dimension 48 [OT ]. But the computation of EO2 at p = 3, together with computations of the author in joint work with Ravenel, suggests that the f* *irst v2-element in ss*MO<8> is v92in dimension 144. The situation at p = 2 is even l* *ess well-understood. The results of Davis and Mahowald [DM ] suggest that M(4; v4* *1), which is a ring spectrum, should have a v82-self map. But their results were ba* *sed on an error in the calculations of ss*S0 which has recently been discovered by Hop* *kins and Mahowald. The corrected version suggests it is v322which occurs, in dimensi* *on 192. 2.3. K(1)-localizations. From the previous section, we know that LK(1)MO and LK(1)MU can be obtained by inverting a v1-element and completing at p. However, it is also possible to give a different description of the K(1)-locali* *zation. Lemma 2.3.1. For any prime p and any k 4, the natural map MO ! MSpin is a K(1)-equivalence. Similarly, the natural map MU -! MSU is a K(1)-equivalence for k 2. Proof.It suffices to show that the natural maps MO -! MO and MU -! MU are K(1)-equivalences. As the complex case is similar to the real case, we conc* *entrate on the real case. Consider the fibration K(ssk-1BO; k - 2) ! BO ! BO for k > 4. (When k = 4, MO = MSpin so there is nothing to prove.) Ravenel and Wilson have calculated the Morava K-theories of Eilenberg-MacLane spaces in [RW* * ]. Their paper is restricted to the odd prime case, but their results also hold fo* *r p = 2 [JW ]. In particular, K(1)*(K(Z ; j)) = K(1)* for j 3 and K(1)*(K(Z =2Z ; j) = K(1)* for j 2. Consider the (convergent) Atiyah-Hirzebruch Serre spectral sequence of this f* *ibra- tion: E2 = H*(BO ; K(1)*) ) K(1)*BO : 16 MARK A. HOVEY There is a natural map of this fibration to the trivial fibration over BO w* *hose fiber is a point. The resulting map of spectral sequences is an isomorphism on * *E2. Thus the map BO ! BO induces an isomorphism on K(1) homology. Now, the canonical vector bundles over BO and BO are both Spin bun- dles. Spin bundles are KO oriented, so in particular are K(1) oriented. The res* *ulting Thom isomorphism completes the proof. __|_ | The method above also leads to the following proposition. Proposition 2.3.2. Suppose p > 3 and k is arbitrary. Then there is no ring map MO -! EOp-1: Proof.There are two facts we need about EOp-1 from [HMi ]. Firstly, EOp-1 is K(p - 1)-local, and secondly, the image of the unit map on homotopy contains ff* *1 2 ssq-1EOp-1 among others. If we had an orientation MO -! EOp-1, it would fact* *or through LK(p-1)MO. The same argument as above shows that, if k p + 2, LK(p-1)MO ' LK(p-1)MO

: Thus we would have an orientation MO -! EOp-1 for some k p + 2: Thus ff1 2 ssq-1S0 must be mapped nontrivially to ssq-1MO: Since ff1 is in the * *image of J, by Therem 2.2.1, we must have q - 1 p, i.e. p 3. __|_ | Note that the above argument also shows that, if any MO admits an orientat* *ion to EO2(p-1), then MO<2p + 2> must admit such an orientation. This is no longer a contradiction, since MO<2p + 2> is the first MO to detect ff1 at p > 3. At odd primes both MSU and MSpin are wedges of suspensions of BP . Therefore LK(1)MO and LK(1)MU are wedges of suspensions of LK(1)BP , whose homo- topy groups are completely known. To understand LK(1)MSpin and LK(1)MSU at p = 2 we must do a little work. Recall that Anderson, Brown, and Peterson [ABP * * ] construct a 2-local splitting of MSpin _ _ _ MSpin '! 4n(J)ko_ 4n(J)-4ko<2>_ ?HF2 : 162J;n(J) even 162J;n(J) odd Here ko<2>denotes the 1-connected cover of ko, which is the connective cover of* * KO . J denotes a (possibly empty) partition, and n(J) denotes the integer of which J* * is a partition, i.e. the sum of the elements of J. Using this description, we can * *obtain the homotopy type of LK(1)MSpin. Lemma 2.3.3. The natural map ko -! KO vn-ELEMENTS IN RING SPECTRA 17 is a K(1)-equivalence. Thus LK(1)(ko) ' (KO )2: Proof.Denote the periodicity element in ss8ko by fi: Note that the fiber of xfi 8ko -! ko has a finite Postnikov tower. It is therefore K(j)-acyclic for all positive j. * *Hence xv is a K(1)-equivalence, and so ko -! KO is also. The K(1)-localization of any K-local spectrum is given by its p-comple* *tion, as is explained in [Hov , Section 2]. __|_ | Proposition 2.3.4. _ LK(1)MSpin ' ( KO J)2: 162J Proof.Note that, if n(J) is even, 4n(J)ko' ko<4n(J)>: Similarly, if n(J) is odd, 4n(J)-4ko<2>' ko<4n(J) - 2>: Indeed, xfin(J)=2 4n(J)ko-----! ko lifts to ko<4n(J)> and is easily seen to be a homotopy equivalence. The proof * *is similar when n(J) is odd. Hence, from the Anderson-Brown-Peterson splitting, _ MSpin -! KO J 162J is a K(1)-equivalence.W Here the subscript J just serves to distinguish the dif* *ferent copies of KO . Now 162JKOJ is a KO -module spectrum, and thus is K-local. I* *ts K(1)-localization is therefore obtained by completing at 2. __|_ | Note that this means that we understand LK(1)MSpin at p = 2 in terms of simpl* *er spectra, but at the moment we can not write LK(1)MSpin at odd primes in a simil* *ar way. Nor do we have a similar result for MSU at p = 2. This leads us to make the following conjecture. Conjecture 2.3.5. _ LK(n)BP ' ( E(n))p;v1;:::;vn-1: Similarly, at p = 2, _ _ LK(1)BoP ' ( KO _ K)2: 18 MARK A. HOVEY Recall that LK(n)BP us the completion of v-1nBP at the ideal (p; v1; : :;:vn* *-1): Baker and W"urgler [BW ] give a splitting of the Artinian completion of v-1nBP* * , but that is a much more drastic completion. Recall Ochanine's result [Och ] that MSU*(X) MSU* KO * ! KO *(X) is not an isomorphism. If this conjecture were true, it would indicate that one* * could not fix this by replacing MSU by BoP: There would be summands of K in (v-1BoP )2 that would indicate that such an isomorphism would be unlikely. We now have two descriptions of LK(1)MO and LK(1)MU. We can compare them to get some information about the homotopy of MO and MU. Let v denote any v1-element in MSpin* or MSU* of positive Adams filtration. All the v1-elements constructed above do have positive Adams filtration. First we need* * to know the homotopy of v-1MSpin and v-1MSU. Lemma 2.3.6. v-1MSpin* is a free Z(p)-module on countably many generators in dimensions 4k. It is, if p = 2, an F2 -vector space with countably infinite ba* *sis in dimensions 8k + 1 and 8k + 2. It is 0 in other dimensions. v-1MSU* is a free Z(p)-module on countably many generators in even degrees not congruent to 2 mod* *ulo 8. In dimensions congruent to 2 modulo 8 it is the direct sum of such a free mo* *dule with an F2-vector space of countably infinite dimension. In dimensions congruen* *t to 1 modulo 8 it is an F2-vector space of countably infinite dimension. Proof.We begin with MSpin. If p is odd, MSpin* is a polynomial ring with one generator in each dimension 4k for k > 0. v is not a multiple of p, so it is ea* *sy to see that v-1MSpin* has the required form. At p = 2, we must use the results of [ABP* * ]. Recall that there are maps ssJ : MSpin ! KO ; one for each partition J. The ideal I* of elements x in MSpin* such that ssJ(x)* * = 0 for all J is killed by v. Indeed, I* maps monomorphically under the Hurewicz ho* *mo- morphism to mod 2 homology, and v has trivial Hurewicz image. Thus v-1MSpin* = v-1(MSpin*=I*): Now MSpin*=I* is free in dimensions 4k on finitely many genera- tors, and an F2-vector space in dimensions 8k + 1 and 8k + 2. We will show that MSpin4k=I4k xv-!MSpin4k+|v|=I4k*+|v| is the inclusion of a summand, and the lemma will follow. To do this it suffices to show that the quotient is torsion-free, or, equival* *ently, that if x 2 MSpin*=I* is not divisible by 2k, neither is vx. Since x is not divisibl* *e by 2k, one of the ssJ(x) is not divisible by 2k either, because some of the ssJ give a* * 2-local splitting of MSpin*=I*. (See the next section or [ABP ] for more details.) R* *ecall that n(J) denotes the sum of the elements of J. Choose a J so that ssJ(x) is n* *ot divisible by 2k but all the ssK (x) with n(K) < n(J) are divisible by 2k. Ther* *e is vn-ELEMENTS IN RING SPECTRA 19 a formula to calculate ssJ(vx), involving ss0(v)ssJ(x) and terms with ssK (x) i* *n them with n(K) < n(J). All of the lower-order terms will be divisible by 2k, but, s* *ince ss0(v) is a power of the periodicity element by choice of v, ss0(v)ssJ(x) is no* *t. Thus vx cannot be divisible by 2k. To prove the theorem for MSU is the same for p odd. For p even we replace the results of Anderson-Brown-Peterson by those of Kochman [Ko ] and Botvinnik [Bo * *], which show that multiplication by vt1includes MSUj in MSUj+8tas a summand. __|* *_ | Now take a v1-element v in MO* with positive Adams filtration. We will den* *ote the image of v in MSpin* by v as well. Consider the cofiber sequence f -1 g -1 F -! v MO -! v MSpin: Because g becomes a homotopy equivalence upon p-completion, F is necessarily ra* *tio- nal, so a wedge of copies of HQ. Thus the image of f* in ss*(v-1MO) is divis* *ible. Since the map MO ! MSpin becomes the inclusion of a summand rationally, the same is true for v-1MO ! v-1MSpin: So the image of f* must be a direct sum of copies of Q=Z. In particular, f* can* *not be injective. Similar comments hold when MU and MSU replace MO and MSpin. We therefore get the following corollary. Corollary 2.3.7. v-1MO* is a free Z(p)-module in dimensions 4k. If p = 2, it* * is an F2-vector space on countably many generators in dimensions 8k + 1 and 8k + 2* *. It is a direct sum of copies of Q=Z in dimensions 4k -1. It is 0 in all other dime* *nsions. Similarly , v-1MU* is a free Z(p)-module in even dimensions not congruent to* * 2 modulo 8. In dimensions congruent to 2 modulo 8 it is the direct sum of such a * *module with an F2-vector space of countably infinite dimension. In dimensions congruen* *t to 1 modulo 8 it is the direct sum of such a vector space and copies of Q=Z. In al* *l other dimensions it is a direct sum of copies of Q=Z. This corollary does give some limited information about MO*. It says, for example, that multiplication by v acts nilpotently in dimensions 8k + 5 and 8k * *+ 6. It would be better if we knew that the divisible summands do not actually appea* *r. Conjecture 2.3.8. There are no divisible summands in v-1MO* or v-1MU*. Equivalently, the torsion in v-1MO* and v-1MU* is bounded. This conjecture would be true if the torsion in MO* and MU* were bounde* *d. I have recently proved this conjecture in some cases, in joint work with Ravene* *l. 20 MARK A. HOVEY 2.4. Tensor products. We show that Conjecture 2.3.8 implies that the natural map MO*(X) MO*KO * ! KO *(X) is an isomorphism, and that the natural map MU*(X) MU*KO * ! KO *(X) is an isomorphism after inverting 2. Recall that v is a v1-element of positive * *Adams filtration in MO* or MU*. We use the homotopy equivalences (v-1MO)p -! (v-1MSpin)p and (v-1MU)p -! (v-1MSU)p proved above, together with the facts that MSpin*(X) MSpin*KO * ! KO *(X) is an isomorphism [HH ] and MSU*(X) MSU* KO * ! KO *(X) is an isomorphism away from 2. Inverting v is fine, since v maps to a unit of K* *O *, but the completion is another story. The importance of Conjecture 2.3.8 is that* *, if it is true, passing to the completion does not lose any information. It is still s* *omewhat technically complicated to prove our theorem though, as readers of [HH ] and [H* *ov2 ] will recall. We begin with the following lemma. Lemma 2.4.1. v-1MU ! KO is surjective on homotopy groups, and therefore so is v-1MO ! KO : Proof.First note that the v1-elements found above have images in KO *which are unit multiples of vt1for all sufficiently large t. So upon inverting v we will* * hit all the powers of v1. This completes the proof for p = 3, but not for larger primes* *. In addition, the torsion classes in KO *are in the image of the unit map, so also* * in the image of MU*. So we need to show that a unit multiple of the generator in so* *me dimension congruent to 4 mod q is hit. For this we need the Milnor-Kervaire manifolds of [MK ]. These are almost- parallelizable manifolds M4n whose ^Agenus is given by the formula n-1B A^(M4n) = -an x numer((-1)____2n_): 4n Here an is 1 if n is even and 2 if n is odd, B2n is the 2nth Bernoulli number, * *and numer means the numerator of the fraction when expressed in lowest terms. Since M4n is almost-parallelizable, the classifying map of its tangent bundle factors throug* *h S4n, vn-ELEMENTS IN RING SPECTRA 21 so M4n defines a homotopy class in ss4nMO as long as 4n k. Furthermore, if n is odd, the map ss4nBU ! ss4nBO is an isomorphism, so we get homotopy classes in ss4nMU: We will show that A^(M4n) when 4n = qj + 4 is not divisible by p if p is odd * *and is not divisible by 4 when p = 2. This will show that the image of M4n in KO ** *is a (p-local) generator, and will complete the proof, since we can choose j to be* * odd and large enough to get a homotopy class in MU. To do this, we need to know about the divisibility of Bernoulli numbers. This subject has been much studied ever since Kummer showed that it was relevant to Fermat's Last Theorem. Many facts about Bernoulli numbers, with very simple proofs, can be found in [Joh ]. The simplest one is Von Staudt's theorem, found* * also in [Lang , p.49]. It is easy to use this to show that the numerator of B2n is * *always odd. This completes the proof at p = 2. For p > 3, we find in [Joh ] that, if p* * > 3 and p - 1 does not divide r, Br_ris a p-local integer, and that Br_ Br+p-1 _________ (mod p): r r + p - 1 In particular, _B(p-1)j+2_ B2 1 ___= ___6 0 (mod p): (p - 1)j + 2 2 12 Since the ^Agenus of M4k0is a unit multiple of the numerator of B2k_2k, it is n* *ot divisible by p either. This completes the proof. __|_ | Now, we know from [HH ] that v-1MSpin*(X) v-1MSpin*KO * ! KO *(X) is an isomorphism. In particular, from the preceeding lemma, v-1MSpin*(X) -! KO *(X) is surjective. If X is a finite torsion spectrum, then v-1MO*(X) = v-1MSpin*(X): Similarly, when p is odd, we know that v-1MSU*(X) v-1MSU* KO * ! KO *(X) is an isomorphism, so for X a finite torsion spectrum, v-1MU*(X) -! KO *(X) is surjective. From now on, let R denote either v-1MO or v-1MU. When R is v-1MU, we assume that p is odd. We apply the following proposition. 22 MARK A. HOVEY Proposition 2.4.2. Suppose we have a map E -! M of (p-local) spectra such that, for X a finite torsion spectrum, E*(X) -! M*(X) is surjective. Suppose as well * *that the torsion in E* is bounded, and that M* is a finitely generated Z(p)-module i* *n each dimension. Then E*(X) -! M*(X) is surjective for all finite X. Proof.We will show that there is a value of k such that E*(X) -! M*(X)=(pkM*(X)) is surjective. It is easy to see that, if X is finite, M*(X) must be finitely g* *enerated in each dimension. We then apply the following lemma in each dimension to deduce that E*(X) -! M*(X) is surjective. Lemma 2.4.3. Suppose f : G ! H is a homomorphism of (p-local) abelian groups and that H is finitely generated. Suppose the composite f n G ! H ! H=p H is surjective for some n. Then f is surjective. Proof of lemma.Consider the diagram of right exact sequences below. xpn n G ---! G - --! G=p G ?? ?? ?? f?y f?y ?y xpn n H ---! H - --! H=p H The proof of the snake lemma shows that the times pn map on the cokernal of f is surjective. Thus the cokernal of f is a finitely generated p-local, p-divisi* *ble group. So it must be 0. __|_ | We are assuming that E* has bounded torsion. It follows from [HH , Lemma 6] that for X finite, E*(X) also has bounded torsion. Choose k so large that pk k* *ills the torsion in E*(X). Consider the following short exact sequences, which come * *from the defining cofibration for M(pk), the mod pk Moore spectrum. E*(X)=(pkE*(X)) ---! E*(X ^ M(pk)) ---! Tor(E*-1(X)) ?? ? ? ?y ??y ??y M*(X)=(pkM*(X)) ---! M*(X ^ M(pk)) ---! Tor (M*-1(X); Z=pkZ) We have a similar diagram where k is replaced by 2k, and a map between the diagram for 2k and the diagram for k, which we will call r for reduction. The * *key vn-ELEMENTS IN RING SPECTRA 23 point is that the map r between the Tor terms is multiplication by pk, which is* * the zero map on the top rows. Now choose x 2 M*(X)=(pkM*(X)): Choose a lift y 2 M*(X)=(p2kM*(X)); so that r(y) = x. There is a class z 2 E*(X ^ M(p2k)) such that g(z) = i(y). Then g(r(z)) = r(g(z)) = ri(y) = ir(y) = i(x); but also j(r(z)) = r(j(z)) = 0, so there is a w 2 E*(X)=(pkE*(X)) such that i(w) = r(z). Thus f(w) = x, and we see that E*(X) ! M*(X)=(pkM*(X)) is surjective. __|_ | We have thus proved that, if Conjecture 2.3.8 is valid, v-1R*(X) ! KO *(X) is surjective for all finite X, and therefore that R*(X) R* KO * ! KO *(X) is surjective for all finite X. To prove injectivity, we follow the outline of [HH ]. The argument on page 19* *4 of that paper proves the following theorem. Theorem 2.4.4. Suppose E is a (p-local) ring spectrum and M an E-module spec- trum equipped with an E-module map f : E ! M. Suppose they satisfy the following conditions. (1) The natural map E ^ M(pk)*(X) E^M(pk)*M ^ M(pk)* ! M ^ M(pk)*(X) is an isomorphism for all sufficiently large k and for all finite X. (2) f* : E* ! M* is surjective. (3) E*(X) E* M* has no infinitely p-divisible elements for finite X. (4) E*(X) has no infinitely p-divisible elements for finite X. (5) E* has bounded torsion. Then the natural map E*(X) E* M* ! M*(X) is injective for all finite X. 24 MARK A. HOVEY We claim that, assuming Conjecture 2.3.8, these conditions are satisfied with* * p arbitrary and R = v-1MO or p odd and R = v-1MU and M = KO . Note that we have already proved the second condition. We have assumed the fifth con* *di- tion. Under this assumption, both the third and fourth condition follow from [H* *ov2 , Lemma 1]. (Take M = KO or M = R in that lemma.) We are left with verifying the first condition, that R ^ M(pk)*(X) R^M(pk)*KO ^ M(pk)* ! KO ^ M(pk)*(X) is an isomorphism. But this is the same as checking that MSpin ^ M(pk)*(X) MSpin^M(pk)*KO ^ M(pk)* ! KO ^ M(pk)*(X) is an isomorphism and MSU ^ M(pk)*(X) MSU^M(pk)* KO ^ M(pk)* ! KO ^ M(pk)*(X) is an isomorphism at odd primes. We did that for p = 2 in [HH ]. For odd primes* *, we will just work with MSpin. The proof is exactly the same for MSU. Note that (MSpin ^ M(pk))* ~=MSpin*=(pk): It is easy to check that, in general, if R is a ring and I is an ideal, and A a* *nd B are R=I-modules, then A R B ~=A R=IB: Thus MSpin ^ M(pk)*(X) MSpin^M(pk)*KO ^ M(pk)* ~=MSpin ^ M(pk)*(X) MSpin*KO * =(pk) ~=MSpin*(X ^ M(pk)) MSpin*KO * KO =(pk) KO * * ~=KO *(X ^ M(pk)) KO =(pk) ~ KO (X ^ M(pk)): KO * * = * We have therefore proved the following theorem. Theorem 2.4.5. If v-1MO* has bounded torsion, so in particular if MO* has bounded torsion, the natural map MO*(X) MO*KO * ! KO *(X) is an isomorphism for all X. Similarly, if v-1MU* has bounded torsion away from 2, the natural map MU*(X) MU*KO * ! KO *(X) is an isomorphism upon inverting 2. vn-ELEMENTS IN RING SPECTRA 25 Note that this statement can be interpreted both locally and globally. If al* *l the torsion in MO* is bounded, we get a global statement. If we only know the p-torsion in MO* is bounded, we still get that MO*(X) MO*KO * ! KO *(X) is an isomorphism after p-localization. In particular, MO<8>* has no p-torsion * *at all if p > 3, so we find that MO<8>*(X) MO<8>*KO * ! KO *(X) is an isomorphism upon inverting 6. In a sense, this is not very surprising, si* *nce the p-localization of MO<8> splits into a wedge of copies of BP when p > 3. But t* *his splitting is not multiplicative, and in fact the ring structure of MO<8>* is no* *t known even at such large primes. It is definitely not polynomial [PR ]. References [Ad]J. F. Adams, On the groups J(X)-IV, Topology, 5 (1966), 21-71. [ABP]D. W. Anderson, E. H. Brown, Jr., and F. P. Peterson, The structure of the* * Spin cobordism ring, Ann. of Math., 86 (1967), 271-298. [BM]A. Bahri and M. Mahowald, Stiefel-Whitney classes in H*BO, Proc. Am* *er. Math. Soc., 83 (1981), 653-655. [BW] A. Baker and U. W"urgler, Liftings of formal groups and the Artinian compl* *etion of v-1nBP, Math. Proc. Camb. Phil. Soc., 106 (1989), 511-530. [Bott]R. Bott, The stable homotopy of the classical groups, Ann. of Math., 70 (* *1959), 313-337. [Bo]B. Botvinnik, The structure of the ring MSU*, Math. USSR Sbornik, 69 (1991* *), 581-596. [Bous]A. K. Bousfield, The localization of spectra with respect to homology, To* *pology, 18 (1979), 257-281. [D]D. Davis, On the cohomology of MO<8>, Proceedings of the Adem symposium, C* *ontemporary Mathematics vol. 12, AMS (1982), 91-104. [DM]D. Davis and M. Mahowald, v1- and v2-periodicity in stable homotopy theory* *, Amer. J. Math., 103 (1981), 615-659. [DM2]D. Davis and M. Mahowald, A new spectrum related to 7-connected cobordism,* * Springer- Verlag Lecture Notes, 1370, 126-134. [Dev]E. Devinatz, Small ring spectra, Jour. Pure Appl. Alg., 81 (1992), 11-16. [G]V. Giambalvo, On <8> cobordism, Ill. J. Math. 15 (1971), 533-541, and 16 (* *1972), 704. [G2]V. Giambalvo, The mod p cohomology of BO<4k>, Proc. Amer. Math. Soc., 20 (* *1969), 593-597. [GP]V. Giambalvo and D. Pengelley, The homology of MSpin, Math. Proc. Camb. Ph* *il. Soc., 95 (1984), 427-436. [GM]V. Gorbunov and M. Mahowald, Some homotopy of the cobordism spectrum MO<8>, preprint, (1993). [HH]M. J. Hopkins and M. A. Hovey, Spin cobordism determines real K-theory, Ma* *th. Z., 210 (1992), 181-196. [HMi]M. J. Hopkins and H. R. Miller, Enriched multiplication on the cohomology * *theories En, to appear. [HS]M. J. Hopkins and J. Smith, Nilpotence and stable homotopy theory II, prep* *rint (1992). 26 MARK A. HOVEY [Hov]M. Hovey, Bousfield localization functors and Hopkins' chromatic splitting* * conjecture, to appear in the Proceedings of the Cech Centennial Conference on Homotopy Th* *eory. [Hov2]M. Hovey, Spin bordism and elliptic homology, to appear in Math. Z.. [JW]D. C. Johnson and W. S. Wilson, Projective dimension and Brown-Peterson ho* *mology, Topology, 12 (1973), 327-353. [Joh]W. Johnson, p-adic proofs of congruences for the Bernoulli numbers, J. Nu* *mber Theory, 7 (1975), 251-265. [Ko]S. Kochman, The ring structure of BoP*, in Algebraic Topology; Oaxtepec 19* *91, M. Tangora, ed., AMS, Contemporary Math. 146, 171-198. [LS]P. Landweber and R. Stong, Cobordism, complete intersections, and modular * *forms, unpub- lished notes (1987). [Lang]S. Lang, Elliptic functions, Springer-Verlag, New York, 1987. [Mah]M. Mahowald, bo-resolutions, Pac. J. Math., 192 (1981), 365-383. [MS]M. Mahowald and H. Sadofsky, vn telescopes and the Adams spectral sequence* *, preprint (1992). [MK]J. Milnor and M. Kervaire, Bernouilli numbers, homotopy groups, and a theo* *rem of Rohlin, Proceedings of the International Congress of Mathematicians, (1958), 454-4* *58. [Och]S. Ochanine, Modules de SU-bordisme. Applications, Bull. Soc. Math. Fr., 1* *15 (1987), 257- 289. [Oka]S. Oka, Ring spectra with few cells, Japan J. Math., 5 (1979), 81-100. [OT]S. Oka and H. Toda, 3-primary fi-family in stable homotopy, Hiroshima Math* *. J., 5 (1975), 447-460. [Pen]D. Pengelley, The homotopy type of MSU, Amer. J. Math., 104 (1982), 1101-* *1123. [PR]D. Pengelley and D. Ravenel, unpublished notes on MO<8>, (1986). [Rav]D. Ravenel, Localization with respect to certain periodic homology theorie* *s, Amer. J. Math., 106 (1984), 351-414. [Rav1]D. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Acad* *emic Press, New York, 1986. [Rav2]D. Ravenel, A counterexample to the telescope conjecture, preprint (1992). [RW]D. C. Ravenel and W. S. Wilson, The Morava K-theories of Eilenberg-MacLane* * spaces and the Conner-Floyd conjecture, Amer. J. Math., 102 (1980), 691-748. [Sin]W. Singer, Connective fiberings over BU and U, Topology, 7 (1968), 271-30* *3. [St]R. Stong, Determination of H*(BO(k; . .;.1); Z2) and H*(BU(k; . .;.1); Z2* *), Trans. Amer. Math. Soc., 107 (1963), 526-544. [St1]R. E. Stong, Notes on Cobordism Theory, Princeton University Press, New J* *ersey, 1968. University of Kentucky, Lexington, KY E-mail address: hovey@ms.uky.edu