THE TRIPLE LOOP SPACE APPROACH TO THE TELESCOPE CONJECTURE MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK April 18, 2000 Contents 1. The telescope conjecture and Bousfield localization 3 1.1. Telescopes 3 1.2. Bousfield localization and Bousfield classes 5 1.3. The telescope conjecture 7 1.4. Some other open questions 8 2. Some variants of the Adams spectral sequence 9 2.1. The classical Adams spectral sequence 10 2.2. The Adams-Novikov spectral sequence 11 2.3. The localized Adams spectral sequence 14 2.4. The Thomified Eilenberg-Moore spectral sequence 18 2.5. Hopf algebras and localized Ext groups 22 3. The spectra y(n) and Y (n) 25 3.1. The EHP sequence and some Thom spectra 25 3.2. The homotopy of Lny(n) and Y (n) 29 3.3. The triple loop spacen 33 4. Properties of 3S1+2p 35 4.1. The Snaith splitting 35 4.2. Ordinary homology 36 4.3. Morava K-theory n 40 4.4. The computation of Y (n)*(3S1+2p ) via the Eilenberg- Moore spectral sequence 44 5. Toward a proof of the differentials conjecture 49 5.1. The E2-term of the localized Thomified Eilenberg-Moore spectral sequence 49 5.2. Short differentials 54 5.3. Excluding spurious differentials 60 1 2 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK References 63 The purpose of this paper is to describe an unsuccessful attempt to prove that the telescope conjecture (see 1.13 below for the precise statement) is false for n 2 and each prime p. At the time the it was originally formulated over 20 years ago (see [Rav84 ]), the telescope conjecture appeared to be the simplest and most plausible statement about the question at hand, namely the relation between two different localization functors. We hope the present paper will demonstrate that this is no longer the case. We will set up a spectral sequence converging to the homotopy of one of the two localizations (the geometrically de- fined telescope) of a certain spectrum, and it will be apparent that only a bizarre pattern of differentials would lead to the known homotopy of the localization defined in terms of BP -theory, the answer predicted by the telescope conjecture. While we cannot exclude such a pattern, it is certainly not favored by Occam's razor. No use will be made here made of the parametrized Adams spectral sequence of [Rav92b ]; we will say more about that approach in a fu- ture paper. Instead we will rely on some constructions related to the EHP sequence which are described in x3, where we define the spectra y(n) and Y (n), and a variant of the Eilenberg-Moore spectral sequence (which we call the Thomified Eilenberg-Moore spectral sequence) de- scribed in x2.4. x1 is an expository introduction to the telescope conjecture. We define telescopes and recall the nilpotence (1.1), periodicity (1.4) and thick subcategory (1.12) theorems of Devinatz, Hopkins and Smith ([DHS88 ] and [HS98 ]). We also recall the definitions of Bousfield lo- calization and related concepts and the Bousfield localization theorem (1.8). We then state four equivalent formulations of the telescope con- jecture in 1.13. In x2 we introduce the various spectral sequences that we will use. These include the classical Adams (x2.1) and Adams-Novikov (x2.2) spectral sequences. We also need the localized Adams spectral sequence of Miller [Mil81 ] (x2.3), for which we prove a convergence theorem 2.13. This is the spectral sequence we will use to compute the homotopy of our telescope Y (n) and see that it may well differ from the answer pre- dicted by the telescope conjecture. In x2.4 we introduce the Thomified Eilenberg-Moore spectral sequence and its localized form. In certain cases (2.26 and 2.27) we identify its E2-term as Ext over a Massey- Peterson algebra. All of these spectral sequences require the use of THE TELESCOPE CONJECTURE 3 Ext groups over various Hopf algebras, and we review the relevant ho- mological algebra in x2.5. This includes two localizations ((2.34) and (2.35)) of the Cartan-Eilenberg spectral sequence which are new as far as we know. In x3 we use the EHP sequence to construct the spectrum y(n) and its telescope Y (n). We describe the computation of ss*(Lny(n)) using the Adams-Novikov spectral sequence, and then state our main compu- tational conjecture, 3.16, which says that the localized Adams spectral sequence gives a different answer for ss*(Y (n)) when n > 1. This would disprove the telescope conjecture, which implies that Lny(n) = Y (n). Our construction of y(n) gives us a map n f 3S1+2p -! y(n); with which we originally hoped to prove Conjecture 3.16 and is the rea-n son for the title of this paper. In x4 we recall some properties 3S1+2p , including the Snaith splitting (4.2) and its ordinary homology as a mod- ule over the Steenrod algebra (Lemma 4.7). In x4.3 we recall Tamaki's unpublished computation of its Morava K-theory using his formula- tion [Tam94 ] of the Eilenberg-Moore spectral sequence, and in x4.4 we show that similar methods can be used to compute its Y (n)*-theory. These are not needed for our main results and are included due to their independent interest. In x5 we describe our program to prove Conjecture 3.16 and thereby disprove the telescope conjecture for n > 1. Our method is to construct a map (derived from the map f above) to the localized Adams spectral sequence for Y (n)* from a localized ThomifiednEilenberg-Moore spec- tral sequence converging to Y (n)*(3S1+2p ). This map turns out to be onto in each Er, so differentials in the latter spectral sequence are determined by those in the former, which are described in Conjecture 5.15. The source spectral sequence has far more structure than the target, and we had hoped to use this to prove 5.15. There are three such structures, each of which figures in the program, namely: n (i)3S1+2p is an H-space, so the spectral sequence is one of Hopf algebras. (ii)It has a Snaith splitting which must be respected by differentials. (iii)The pth Hopf map induces an endomorphism of our spectral se- quence, which is identified in Lemma 5.16. Previously we had thought that this structure could be used to con- struct certain permanent cycles "yi;jmapping to bn+i;jin the localized Adams spectral sequence that would force the latter to collapse from 4 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK a certain stage. Unfortunately, this is not the case. For more details, see the comments after Conjecture 5.12. It is pleasure to acknowledge helpful conversations with Fred Co- hen, Bill Dwyer, Emmanuel Dror Farjoun, Mike Hopkins, Nick Kuhn, Haynes Miller, Hal Sadofsky, Brooke Shipley, and Dai Tamaki. 1. The telescope conjecture and Bousfield localization 1.1. Telescopes. The telescope conjecture is a statement about the stable homotopy groups of finite complexes. There is not a single non- trivial example for which such groups are completely known. There are many partial results, especially about the stable homotopy groups of spheres. Unstably the situation is only slightly better. We have complete knowledge of ss*(X) for a finite complex X only in the cases where X is known to be an Eilenberg-Mac Lane space, such as when X is a surface of positive genus. Experience has shown that one can get interesting information about ss*(X) in the stable case in the following way. Suppose one has a stable map of the form f dX -! X for which all iterates are essential; this can only happen if d 0. Such a map is said to be periodic. We say that f is nilpotent if some iterate of it is null. In any case we can define the telescope f-1 X to be the direct limit of the system f -d f -2d f X -! X -! X -! . . .: This will be contractible if f is nilpotent. In the (rare) cases when f is periodic, the computation of ss*(f-1 X) is far more tractable than that of ss*(X). The map f induces an endomorphism of ss*(X), which we will denote abusively by f, making ss*(X) a module over the ring Z [f]. Since homotopy commutes with direct limits, we have ss*(f-1 X) = ss*(X) Z [f]Z[f; f-1 ]: The telescope conjecture is a statement about this group. Before stating it we will describe some motivating examples. We assume that all spaces and spectra in sight are localized at a prime p. o For any spectrum X let f be the degree p map. It induces multi- plication by p in homotopy and homology and induces an isomor- phism in rational homology. If H*(X; Q ) is nontrivial, i.e., if the integer homology of X is not all torsion, then all iterates of the degree p map are essential. THE TELESCOPE CONJECTURE 5 In this case the telescope p-1X is the rationalization XQ of X with ss*(XQ ) = ss*(X) Q = H*(X; Q ); the rational homotopy of X. It is a rational vector space. o Let V (0) be the mod p Moore spectrum. For each prime p Adams [Ada66 ] constructed a map ae 8 ifp = 2 dV (0) -ff!V (0) where d = 2p - 2 if p is odd. This map induces an isomorphism in classical K-theory and all iterates of it are nontrivial. ss*(ff-1V (0)) has been computed ex- plicitly by Mahowald [Mah81 ] for p = 2 and Miller [Mil81 ] for odd primes. It is finitely presented as a module over Z [ff; ff-1]. The image of ss*(V (0)) in ss*(ff-1V (0)) is known, and this gives us a lot of information about the former. By analogy with the previous example, one might expect ss*(ff-1V (0)) to be K*(V (0)), but the situation here is not so simple. The an- swer is however predictable by K-theoretic or BP-theoretic meth- ods; we will say more about this later. o For odd p let V (1) denote the cofiber of the Adams map ff. It is a CW-complex with one cell each in dimensions 0, 1, 2p - 1 and 2p. Smith [Smi71 ] and Toda [Tod71 ] have shown that for p 5 there is a periodic map 2-2 fi 2p V (1) -! V (1): In this case the homotopy of the telescope is not known. The results of Devinatz-Hopkins-Smith ([DHS88 ] and [HS98 ]) allow us to study telescopes in a very systematic way. They indicate that BP-theory and Morava K-theory are very useful here. First we have the nilpotence theorem characterizing nilpotent maps. Theorem 1.1 (Nilpotence theorem). For a finite p-local spectrum X, a map f dX -! X is nilpotent if and only if the induced map on BP*(X) is nilpotent. Equivalently, it is nilpotent if and only if the induced map on K(n)*(X) is nilpotent for each n. For the study of periodic maps two definitions are useful. Definition 1.2. A p-local finite complex X has type n if n is the small- est integer for which K(n)*(X) is nontrivial. 6 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK Definition 1.3. A map f dX -! X is a vn-map if K(n)*(f) is an isomorphism and K(m)*(f) = 0 for m 6= n. (The spectrum X here need not be finite.) A finite complex of type n does not admit a vm -map for m > n; this follows from the algebraic properties of the target category of the BP-homology functor. For m < n, the trivial map is a vm -map. The cofiber of a vn-map on a type n complex is necessarily a complex of type n + 1. In the three examples above we have a such a map for n = 0, 1 and 2 respectively. Now we can state the periodicity theorem of [HS98 ]. Theorem 1.4 (Periodicity theorem). Every type n finite complex ad- mits a vn-map. Given two such maps f and g there are positive integers i and j such that fi = gj. Corollary 1.5. For a type n p-local finite complex X, any vn-map f : dX ! X yields the same telescope f-1 X, which we will denote by v-1nX or Xb. 1.2. Bousfield localization and Bousfield classes. Definition 1.6. Given a homology theory h*, a spectrum X is h*-local for each spectrum W with h*(W ) = 0, [W; X] = 0. An h*-localization X ! LhX is an h*-equivalence from X to an h*-local spectrum. We denote the fiber of this map by ChX. If h* is represented by a spectrum E we will write LE and CE for Lh and Ch. The case E = v-1nBP is of special interest, and we denote the corresponding functors by Ln and Cn. The following properties of localization are formal consequences of these definitions. Proposition 1.7. If LhX exists it is unique and the functor Lh is idempotent. The map X ! LhX is terminal among all h*-equivalences from X and initial among all maps from X to h*-local spectra. ChX is h*-acyclic and the map ChX ! X is terminal among all maps from h*-acyclics to X. The homotopy inverse limit of h*-local spectra is h*-local, although the functor Lh (if it exists) need not commute with homotopy inverse or direct limits. The homotopy direct limit of local spectra need not be local. The definitive theorem in this subject is due to Bousfield [Bou79 ]. THE TELESCOPE CONJECTURE 7 Theorem 1.8 (Bousfield localization theorem). The localization LhX exists for all spectra X and all homology theories h*. Roughly speaking, one constructs ChX by taking the direct limit of all h*-acyclic spectra mapping to X. (This is not precisely correct because of set theoretic problems; there are too many such maps to form a direct limit. Bousfield found a way around this difficulty.) A variant on this procedure is to consider the homotopy direct limit of all finite h*-acyclic spectra mapping to X, which we denote by CfhX. (Here f stands for finite, and there are no set theoretic problems.) We denote the cofiber of CfhX ! X by LfhX. Definition 1.9. A localization functor Lh is finite if Lh = Lfh, i.e., if ChX is always a homotopy direct limit of finite h*-acyclic spectra mapping to X. Proposition 1.10. If the functor Lh is finite then (i) it commutes with homotopy direct limits, (ii)the homotopy direct limit of h*-local spectra is local, (iii)LhX = X ^ LhS0 for all X, and (iv) Lh is the same as Bousfield localization with respect to the homol- ogy theory represented by LhS0. It can be shown [Rav84 , Prop. 1.27] that the four properties listed in 1.10 are equivalent. We say that a localization functor is smashing if it has them. Thus 1.10 says that every finite localization functor is smashing. Bousfield conjectured [Bou79 , 3.4] the converse, that every smashing localization functor is finite. The functor Ln is known to be smashing [Rav92a , Theorem 7.5.6], but if the telescope conjecture fails, it is not finite for n 2. Definition 1.11. Two spectra E and F are Bousfield equivalent if E*(X) = 0 iff F*(X) = 0, or equivalently if LE = LF . The corre- sponding equivalence class is denotes by , the Bousfield class of E. We say that if E*(X) = 0 implies F*(X) = 0. Dror Farjoun [Far96 ] uses the notation X Y (Y can be built from X by cofibrations) in an unstable context to mean > . The following consequence of 1.1 is very useful, e.g. it was used to prove 1.4. A subcategory of the stable homotopy category of finite complexes is thick if it is closed under cofibrations and retracts. One example is the subcategory of h*-local finite spectra for a given h. The following result of [DHS88 ] classifies all thick subcategories. 8 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK Theorem 1.12 (Thick subcategory theorem). Any nontrivial thick sub- category of the stable homotopy category of p-local finite complexes is the category C n of K(n - 1)*-acyclic spectra for some n 0. Note that C0 is the entire category, C0 C1 C2 : :;: and the intersection of all these is the trivial subcategory consisting of a point. This dry sounding theorem is a useful tool. Suppose one wants to prove that all p-local finite spectra of type n satisfy a certain prop- erty, say that they are all demented. (This example is due to John Harper.) If one can show that the subcategory of demented spectra is thick, then all that remains is to show that a single one of type n is demented. If one is demented they all are demented. Conversely, if we can find a single type n spectrum that is not demented, then none of them are. 1.3. The telescope conjecture. Now we will discuss several equiva- lent formulations of the telescope conjecture. Telescope conjecture 1.13. Choose a prime p and an integer n 0. Let X be a p-local finite complex of type n (1.2) and let Xb be the associated telescope (1.5). Then (i) bX = LnX. (ii) = . (iii)The Adams-Novikov spectral sequence for Xb converges to ss*(Xb). (iv) The functors Ln and Lfnare the same if Ln-1 = Lfn-1. We will sketch the proof that the four statements above are equiva- lent. The set of K(n - 1)*-acyclic finite p-local spectra satisfying (i) is thick. The same is true for (ii) and for the statement that (1.14) = : Thus if we can find a type n X with this property it will follow that (i) and (ii) are equivalent. One can show that (1.14) holds if the Adams- Novikov E2-term for X has a horizontal vanishing line; this means that LnX can be built out of K(n) with a finite number of cofibrations. Such an X can be constructed using the methods described in [Rav92a , x8.3]. For the third statement, the Adams-Novikov spectral sequence for LnX (which is BP*-equivalent to bX) was shown in [Rav87 ] to converge to its homotopy, so it also converges to that of Xb iff (i) holds. THE TELESCOPE CONJECTURE 9 For the fourth statement, since the functors Ln and Lfnare both smashing, they commute with homotopy direct limits. This means that if they agree on finite complexes, they agree on all spectra. For K(n - 1)*-acyclic X it is known that LfnX = Xb (see [Rav93b ], Miller [Mil92 ] or Mahowald-Sadofsky [MS95 ]) so (i) says the two functors agree on such X. For finite p-local X of smaller type, the methods of [Rav93b , x2] show that Cfn-1(the fiber of X ! Lfn-1X) is a homotopy direct limit of type n finite complexes, so LnX = LfnX. Any attempt to prove 1.13 is likely to rely on 1.12. It is easy to show that the set of K(n - 1)*-acyclic finite spectra satisfying 1.13(i) is thick. Thus one can prove or disprove the telescope conjecture if we can compare ss*(LnX) with ss*(Xb) for a single type n spectrum X. The telescope conjecture for n = 1 follows from the computations of Mahowald [Mah81 ] and Miller [Mil81 ] of ss*([V (0)) which showed that agrees with the previously known value of ss*(L1V (0)). Alternately we can disprove the telescope conjecture by finding a spectrum Y (which need not be finite) for which LfnY 6= LnY . The groups ss*(LnX) (or ss*(LnY )) and ss*(Xb) (or ss*(LfnY )) can be computed with variants of the Adams spectral sequence. These meth- ods will be discussed in the next section. The spectrum we will use, y(n), is a certain Thom spectrum which will be constructed in x3. We will use the Adams-Novikov spectral sequence to show (Corollary 3.12) that ss*(Lny(n)) is finitely gener- ated over a certain ring R(n)* defined below in (3.13); this is relatively easy. A far more difficult calculation (Conjecture 3.16) using the local- ized Adams spectral sequence (described in x2.3) comes quite close to showing that ss*(Lfny(n)) is not finitely generated over R(n)* for n > 1, which would disprove the telescope conjecture. 1.4. Some other open questions. The spectrum y(n) of x3 has a telescope Y (n) associated with it. Conjecture 3.9 below says that 1.13(ii) holds with K(n) replaced by Y (n). Computing ss*(Y (n)) is the main object of this paper. Each Y (n) is a module over a spectrum T1 (4.3) and we suspect (4.4) that it has the same Bousfield class as the sphere spectrum. The functors Ln could be called chromatic localizations. There are natural transformations from Ln+1 to Ln, so for each spectrum X we have an inverse system L0X - L1X - L2X - . . .; and we can ask if the natural map from X to the homotopy inverse limit is an equivalence. This is the chromatic convergence question. The 10 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK chromatic convergence theorem of Hopkins and the author [Rav92a , 7.5.7] says that this is the case for p-local finite spectrum X. The telescopic convergence question concerns the inverse limit of the LfnX, its telescopic localizations. We know that there are maps X -! LfnX -! LnX and that holim LnX ' X, so X is a retract of holim LfnX. It suffices to answer this question for the case X = S0, since LfnX = X ^ LfnS0 (Lfnis smashing) and smashing with a finite complex preserves inverse limits. 2. Some variants of the Adams spectral sequence The Adams spectral sequence for ss*(X) is derived from the following Adams diagram. X0 u______X1 u______X2 u______. . . | | | | | | (2.1) g0| g1| g2| | | | |u |u |u K0 K1 K2 Here Xs+1 is the fiber of gs. We get an exact couple of homotopy groups and a spectral sequence with Es;t1= sst-s(Ks) and dr : Es;tr! Es+r;t+r-1r: This spectral sequence converges to ss*(X) if the homotopy inverse limit lim Xs is contractible and certain lim1 groups vanish. When X is connective, it is a first quadrant spectral sequence. For more background, see [Rav86 ]. Now suppose we have a generalized homology theory represented by a ring spectrum E. Then the canonical E-based Adams resolution for X is the diagram (2.1) with Ks = E ^ Xs. More generally an E-based Adams resolution for X is such a diagram where Ks is such that the map gs ^ E is the inclusion of a retract. Under certain hypotheses on E the resulting E2-term is independent of the choice of resolution and can be identified as an Ext group. The classical Adams spectral sequence is the case where E = H=p, the mod p Eilenberg-Mac Lane spectrum, and the Adams-Novikov spectral sequence is the case where E = BP , the Brown-Peterson spectrum. We will have occasion to use a noncanonical Adams resolution below for a case where E = H=p. Then the condition is that H*(gs) be monomorphic for each s. THE TELESCOPE CONJECTURE 11 2.1. The classical Adams spectral sequence. Here we have Es;t2= Ext A*(Z =(p); H*(X)); where A* is the dual Steenrod algebra, H*(X) is the mod p homology of X, and Ext is taken in the category of A*-comodules. This group is the same as ExtA (H*(X); Z=(p)), where H*(X) is regarded as a module over the Steenrod algebra A. This group is not easy to compute in most cases. There is not a single nontrivial example where X is finite and this group is completely known, although there are good algorithms for computing it in low dimensions. We recall the structure of A*. When working over a field k we will use the notation P (x) and E(x) to denote polynomial and exterior algebras over k on x. As an algebra we have 8 >> P (1; 2; : :): with |i| = 2i- 1 >> < forp = 2 A* = P (1; 2; : :): E(o0; o1; : :):with |i| = 2pi- 2 >> i >> and |oi| = 2p - 1 : forp > 2: For odd primes we will denote the polynomial and exterior factors by P* and Q* respectively. For p = 2, P* and Q* will denote P (2i) and E(i) respectively. The coproduct is given by X j (i) = pi-j j where 0 = 1: 0ji X j and (oi) = oi 1 + pi-j oj: 0ji In x2.5 we will review some facts about Ext groups over Hopf algebras such as A*, which we will refer to here when needed. In x3 we will construct a spectrum y(n) with ae P (1; : :;:n) for p = 2 H*(y(n)) = P ( 1; : :;:n) E(o0; : :;:on-1) for p > 2: Let ae A*=(1; : :;:n) forp = 2 (2.2) B(n)* = A *=(1; : :;:n; o0; : :;:on-1)forp > 2 Then we have H*(y(n)) = A*2B(n)*Z=(p), and we can use the change- of-rings isomorphism (2.30) to prove Proposition 2.3. With notation as above ExtA*(Z =(p); H*(y(n))) = Ext B(n)*(Z =(p); Z=(p)): 12 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK For future reference we record some information about this Ext group. For a fixed value of n let ae P (n+1; n+2; : :): for p > 2 (2.4) P*0 = P (2 2 ae n+1; n+2; : :): for p = 2 E(on; on+1; : :): for p > 2 (2.5) Q0* = E( n+1; n+2; : :): for p = 2 Then we have a Hopf algebra extension (2.31) (2.6) P*0-! B(n)* -! Q0* and a Cartan-Eilenberg spectral sequence (2.32) converging to the group of 2.3 with E2 = ExtP0*(Z =(p); ExtQ0*(Z =(p); Z=(p))) = ExtP0*(Z =(p); V 0): where (2.7) V 0= P (vn; vn+1; : :):: The elements vn+k for 0 k n are permanent cycles. In x2.3 we will consider the effect of inverting vn. 2.2. The Adams-Novikov spectral sequence. Here we have Es;t2= Ext BP*(BP)(BP*; BP*(X)): Here we are taking Ext in the category of comodules over the Hopf alge- broid BP*(BP ). The difficulty of computing this group is comparable to the classical case. The structure of BP*(BP ) is as follows. As algebras we have BP*(BP ) = BP*[t1; t2; : :]: with |ti| = 2pi- 2: It is not a Hopf algebra (i.e., a cogroup object in the category of alge- bras), but a Hopf algebroid, which is a cogroupoid object in the cat- egory of algebras. (For more discussion of this definition see [Rav86 , A1.1] or [Rav92a , B.3].) This means that in addition to a coproduct map there is a right unit map jR : BP* ! BP*(BP ). The for- mulas for these maps involve the formal group law and are somewhat complicated. We will give approximations for them now. Let I = (p; v1; v2; : :): BP*: THE TELESCOPE CONJECTURE 13 Then we have X j (ti) tj tpi-j mod I where t0 = 1 Xj j and jR (vi) vjtpi-j mod I2 where v0 = p: j There is an analog of (2.30) for Hopf algebroids stated as A1.3.12 in [Rav86 ]. We have (2.8) BP*(y(n)) = BP*=In[t1; : :;:tn]: The analog of 2.3 is the following. Corollary 2.9. Ext BP*(BP)(BP*; BP*(y(n))) = Ext BP*(BP)=(t1;:::;tn)(BP*; BP*=In): When X is a finite complex of type n, the Adams-Novikov E2-term for Xb is surprisingly easy to compute. In many cases we can get a complete description of it, much unlike the situation for X itself. It was this computability that originally motivated the second author's interest in this problem. For such X we know that BP*(Xb) = BP*(LnX) = v-1nBP*(X), and BP*(X) is always annihilated by some power of the ideal In = (p; v1; : :;:vn-1) BP*: More generally if X is a connective spectrum in which each element of BP*(X) is annihilated by some power of In, we have BP*(LnX) = v-1nBP*(X). The results of [Rav87 ] and the smash product theorem [Rav92a , 7.5.6] imply that the Adams-Novikov spectral sequence for LnX converges to ss*(X). Now assume for simplicity that BP*(X) is annihilated by In itself; this condition is satisfied in all of the examples we shall study here. This means that v-1nBP*(X) is a comodule over v-1nBP*(BP )=In, which turns out to be much more manageable than BP*(BP ) itself. There is a change-of-rings isomorphism (originally conceived by Morava and first proved in [MR77 ]) that enables us to replace v-1nBP*(BP )=In with a smaller Hopf algebra (n), which we now describe. Let ae Q forn = 0 K(n)* = Z=(p)[v -1 n; vn ]forn > 0; (this is the coefficient ring for Morava K-theory) and define a BP*- module structure on it by sending vm to zero for m 6= n . K(n)* for 14 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK n > 0 is a graded field in the sense that every graded module over it is free. Then let (n) = K(n)* BP* BP*(BP ) BP* K(n)*; where the tensor product on the right is with respect to the BP*-module structure on BP*(BP ) induced by the right unit map jR . Using more precise information about jR , we get the following explicit description of (n) as an algebra. n pi (n) = K(n)*[t1; t2; : :]:=(vntpi - vn ti): It is a Hopf algebra with coproduct inherited from that on BP*(BP ). For a BP*(BP )-comodule M, K(n)* BP* M is a comodule over (n). Now we can state the change-of-rings theorem of [MR77 ]. Theorem 2.10. Let M be a BP*(BP )-comodule that is annihilated by the ideal In. Then there is a natural isomorphism ExtBP*(BP)(BP*; v-1nM) = Ext (n)(K(n)*; K(n)* BP* M): The Ext group on the right is explicitly computable in many inter- esting cases. It is related to the continuous mod p cohomology of the strict automorphism group of the height n formal group law. This connection was first perceived by Morava and is explained in [Rav86 , Chapter 6]. The methods given there lead to the following analog of 2.9. Corollary 2.11. With BP*(y(n)) as in (2.8), Ext BP*(BP)(BP*; v-1nBP*(y(n))) n p pn pn = Ext (n)=(t1;:::;tn)(K(n)*; K(n)*[vntp1 - vnt1; : :;:vntn - vn tn]) = P (vn+1; : :;:v2n) Ext(n)=(t1;:::;tn)(K(n)*; K(n)*) = K(n)*[vn+1; : :;:v2n] E(hn+i;j:1 i n; 0 j n - 1); j(pn+i-1) where hn+i;j2 Ext 1;2p corresponds to the primitive element j tpn+i2 (n)=(t1; : :;:tn): 2.3. The localized Adams spectral sequence. The classical Adams spectral sequence is useless for studying the telescope Xb because its homology is trivial. We need to replace it with the localized Adams spectral sequence; the first published account of it is due to Miller [Mil81 ]. It is derived from the Adams spectral sequence in the follow- ing way. The telescope Xb is obtained from X by iterating a vn-map f : X ! -dX. Suppose there is a lifting f": X ! -dXs0 THE TELESCOPE CONJECTURE 15 (where Xs0 is as in (2.1)) for some s0 0. This will induce maps "f: Xs ! -dXs+s0 for s 0. This enables us to define Xbs to be the homotopy direct limit of f" "f f" Xs __________w-dXs+s0 ______w-2dXs+2s0 __________.w. . Let Xs = X for s < 0. Thus we get the following diagram, similar to that of (2.1). . . .u_____Xb-1_u______Xb0 u______Xb1_u_______. . . | | | | | (2.12) g-1| g0| g1| | | | |u |u |u Kb-1 Kb0 Kb1; where the spectra bKsare defined after the fact as the obvious cofibers. This leads to a full plane spectral sequence (the localized Adams spec- tral sequence) with Es;t1= sst-s(Kbs) and dr : Es;tr! Es+r;t+r-1r as before. This spectral sequence converges to the homotopy of the ho- motopy direct limit ss*(lim! bX-s) if the homotopy inverse limit lim Xbs is contractible. Theorem 2.13 (Convergence of the localized Adams spectral sequence). For a spectrum X equipped with maps f and "fas above, in the localized Adams spectral sequence for ss*(Xb) we have o The homotopy direct limit lim! Xb-s is the telescope Xb. o The homotopy inverse limit lim Xbs is contractible if the original (unlocalized) Adams spectral sequence has a vanishing line of slope s0=d at Er for some finite r, i.e., if there are constants c and r such that Es;tr= 0 for s > c + (t - s)(s0=d): (In this case we say that f has a parallel lifting f".) 16 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK Proof. For the assertion about the homotopy direct limit, note that bXs = lim -diXs+is !i 0 so lim bXs = lim lim-diXs+is0 !s !s !i = lim lim-diXs+is0 !i !s = lim -diX !i = Xb: Next we will prove the assertion about the vanishing line. Let Es;tr(X) denote the Er-term of the Adams spectral sequence for X, and Es;tr(Xb) that of the localized Adams spectral sequence. Then f" induces homomorphisms f" s+s ;t-d s+s ;t+d Es;tr(X) -! Er 0 ( X) = Er 0 (X) and we have Es;tr(Xb) = limEs+ks0;t+kdr(X); ! k so the vanishing line of the localized Adams spectral sequence follows from that of the unlocalized Adams spectral sequence. Next we will show that lim (X^i) is contractible. Recall that X^i = lim-kdXi+ks ! k 0 so ssm (X^i) = limssm+kd (Xi+ks0): ! k Now the vanishing line implies that the map g : Xs ! Xs-r+1 sat- isfies ssm (g) = 0 for m < (sd + c)=s0. To see this, note that a perma- nent cycle of filtration s corresponds to a coset (modulo the image of ss*(Xs+1)) in ss*(Xs). It is dead in the Er-term if and only if its image in ss*(Xs-r+1) is trivial. It follows that for each k > 0 we have a diagram g Xs ______________wXs-r+1 | | | | | | |u g |u -dkXs+s0k ______-dkXs+s0k-r+1w THE TELESCOPE CONJECTURE 17 in which both maps g vanish on ssm for m < (sd + c)=s0. Hence the map Xbs -^g!Xbs-r+1 has the same property. It follows that if we fix m and s, the homomorphism (2.14) ssm (X^i) -! ssm (X^s) is trivial for sufficiently large i, and the image of limss*(X^i) -! ss*(X^s) is trivial for each s, so lim ss*(X^i) = 0: To complete the proof that lim (X^i) is contractible, we need to show that lim1ss*(X^i) = 0: However, (2.14) implies that the inverse system of homotopy groups is Mittag-Leffler, so lim1 vanishes. According to Boardman [Boa81 , x10], the convergence of a whole plane spectral sequence such as ours requires, in addition to the con- tractibility just proved, the vanishing of a certain obstruction group that he calls W . (It measures the failure of certain direct and inverse limits to commute.) However, his Lemma 10.3 says that our vanishing __ line implies that W = 0. |__| Here are some informative examples. o If we start with the Adams-Novikov spectral sequence, then the map f cannot be lifted since BP*(f) is nontrivial. Thus we have s0 = 0 and the lifting condition requires that X has a horizontal vanishing line in its Adams-Novikov spectral sequence. This is not known (or suspected) to occur for any nontrivial finite X, so we do not get a convergence theorem about the localized Adams-Novikov spectral sequence, which is merely the standard Adams-Novikov spectral sequence applied to Xb. o If we start with the classical Adams spectral sequence, an un- published theorem of Hopkins-Smith says that a type n X (with n > 0) always has a vanishing line of slope 1=|vn| = 1=(2pn - 2). Thus we have convergence if f has a lifting with s0 = d=|vn|. This does happen in the few cases where Toda's complex V (n - 1) 18 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK exists. Then V (n - 1) is a type n complex with a vn-map with d = |vn| and s0 = 1. o In favorable cases (such as Toda's examples and y(n)) the E2-term of the localized Adams spectral sequence can be identified as an Ext groups which can be computed explicitly. We will discuss the last example in more detail. For simplicity we assume until further notice that p is odd. Recall from x2.1 that Ext A*(Z =(p); H*(y(n))) = Ext B(n)*(Z =(p); Z=(p)) and that the latter can be computed using Cartan-Eilenberg spectral sequence (2.32) for the extension (2.6) with E2 = Ext P0*(Z =(p); V 0): The effect of localization is to invert vn as in (2.35). The comodule structure on V 0is given by X n+k (v2n+i) = 1 v2n+i + pn+i-k vn+k 0k 0 recursively by ! X k (2.15) w2n+i = v-1n v2n+i - vn+k wp2n+i-k ; 0> E(hn+i;j: i > 0; 0 j < n) >> < P (bn+i;j: i > 0; 0 j < n) (2.20) = v-1nP (vn; : :;:v2n) for p odd >> >> P (hn+i;j: i > 0; 0 j < n) : for p = 2: Since the elements vn+i for 0 i n are permanent cycles, the Cartan-Eilenberg spectral sequence collapses. There are no multiplica- tive extensions since E1 has no zero divisors. Hence the above is a description of the E2-term of the localized Adams spectral sequence for Y (n). 2.4. The Thomified Eilenberg-Moore spectral sequence. We will use a Thomified form of the Eilenberg-Moore spectral sequence which is introduced in [MRS ]. Let (2.21) X -!i E -h! B 20 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK be a fiber sequence with simply connected base space B, and suppose that we also have a p-local stable spherical fibration over E which is oriented with respect to mod p homology. Let Y , and K be the Thomifications of X and E. In [MRS ] we construct a diagram Y _______Y0 u______Y1 u______Y2 u______. . . | | | (2.22) g0| g1| g2 | | | | |u |u |u K0 K1 K2 where Ys+1 is the fiber of gs and ___ (s) H*(Ks) = -sH*(K) H *(B ): This is similar to the Adams diagram of (2.1), but H*(gs) need not be a monomorphism in general. As before the associated exact couple of homotopy groups leads to a spectral sequence, which we will call this the Thomified Eilenberg-Moore spectral sequence. To identify the E2-term n certain cases, note that H*(K) is simul- taneously a comodule over A* and (via the Thom isomorphism and the map h*) H*(B), which is itself a comodule over A*. Following Massey-Peterson [MP67 ], we combine these two structures by defining the semitensor product coalgebra (2.23) R* = H*(B) A* THE TELESCOPE CONJECTURE 21 in which the coproduct is the composite H*(B) A* | | BA | |u H*(B) H*(B) A* A* | H*(B) BA*A* | | |u (2.24) H*(B) A* H*(B) A* A* | H*(B)A*TA* | | |u H*(B) A* A* H*(B) A* | H*(B)mAH*(B)A* | | |u (H*(B) A*) (H*(B) A*); where A and B are the coproducts on A* and H*(B), T is the switching map, B : H*(B) ! A* H*(B) is the comodule structure map, and mA is the multiplication in A*. Massey-Peterson gave this definition in cohomological terms. They denoted the semitensor algebra R by H*(B) A, which is additively isomorphic to H*(B) A with multiplication given by (x1 a1)(x2 a2) = x1a01(x2) a001a2; where xi 2 H*(B), ai 2 A, and a01a001denotes the coproduct expansion of a1 given by the Cartan formula. Our definition is the homological reformulation of theirs. Note that given a map f : V ! B and a subspace U V , H* (V =U) = H*(V; U) is an R-module since it is an H*(V )-module via relative cup products, even if the map f does not extend to the quotient V =U. In our case we have maps Gs ! B for all s 0 given by (e; b1; : :;:bs) 7! he: These are compatible with all of the maps ht, so H*(Ys) and H*(Ks) are R*-comodules, and the maps between them respect this structure. We will see in the next theorem that under suitable hypotheses, the E2-term of the Thomified Eilenberg-Moore spectral sequence is ExtR* (Z =(p); H*(K)) when B is an H-space. When B is an H-space we have a Hopf algebra extension (2.31) A* -! R* -! H*(B): 22 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK This gives us a Cartan-Eilenberg spectral sequence (2.32) converging to this Ext group with (2.25) E2 = Ext A*(Z =(p); ExtH*(B)(Z =(p); H*(K))): Note that the inner Ext group above is the same as ExtH*(B)(Z =(p); H*(E)), the E2-term of the classical Eilenberg-Moore spectral sequence con- verging to H*(X). If the latter collapses from E2(which it does in the examples we will study), then the Ext group of (2.25) can be thought of as Ext A*(Z =(p); H*(Y )); where H*(Y ) is equipped with the Eilenberg-Moore bigrading. This is the usual Adams E2-term for Y when H*(Y ) is concentrated in Eilenberg-Moore degree 0, but not in general. Theorem 2.26. (i)Suppose that H*(K) is a free A-module and B is simply connected. Then the Thomified Eilenberg-Moore spec- tral sequence associated with the homotopy of (2.22) converges to ss*(Y ) with E2 = Ext R*(Z =(p); H*(K)); where R* is the Massey-Peterson coalgebra of (2.23). (ii)If in addition the map i : X ! E induces a monomorphism in mod p homology, then the Thomified Eilenberg-Moore spectral sequence coincides with the classical Adams spectral sequence for Y . This is proved in [MRS ]. Now we give a corollary that indicates that the hypotheses are not as restrictive as they may appear. Corollary 2.27. Given a fibration X -! E -! B with X p-adically complete, a p-local spherical fibration over E, and B simply connected, there is a spectral sequence converging to ss*(Y ) (where Y is the Thomification of X) with E2 = Ext H*(B)A* (Z =(p); H*(K)); where K as usual is the Thomification of E. Proof. We can apply 2.26 to the product of the given fibration with pt.! 2S3 ! 2S3, where 2S3 is equipped with the p-local spherical fibration of Lemma 3.3 below. Then the Thomified total space is K ^ H=p, so its cohomology is a free A-module. Thus the E2-term is Ext H*(B^H=p)A* (Z =(p); H*(K ^ H=p)) = Ext H*(B)A* (Z =(p); H*(K)): THE TELESCOPE CONJECTURE 23 __ |__| 2.5. Hopf algebras and localized Ext groups. In this subsection we will collect some results about Ext groups over Hopf algebras and their localizations. We refer the reader to [Rav86 , A1.3] for details of the unlocalized theory. Given a connected graded cocommutative Hopf algebra over a field k (always Z=(p) in this paper) and a left -comodule M, there is a cobar complex C (M) whose cohomology is Ext (k; M); see [Rav86 , A1.2.11] where it is denoted by C (k; M). Additively we have Cs(M) = s M: The coboundary on C0(M) = M is given by (2.28) d(m) = (m) - 1 m where : M ! M is the comodule structure map. When M is a comodule algebra, C (M) is a differential graded algebra. The product is somewhat complicated and is given in [Rav86 , A1.2.15]. For future reference we record the formula for C1(M) C1(M) -[! C2(M); namely (2.29) (fl1 m1) [ (fl2 m2) = fl1 m01fl2 m001m2; where m01 m001denotes the comodule expansion of m1. Given a Hopf algebra map f : ! and a left -comodule M, there is a spectral sequence converging to Ext (k; M) with (2.29) Ei;j2= Ext (k; Ext (k; M)) and dr : Es;tr! Es+r;t-r+1r: It is derived from the double complex C (C ( M)) by filtering by the first degree. More explicitly we have Ci(Cj ( M)) = i j M The jth row is C (j M), which is acyclic since the comodule j M is free over . This means that filtering by the second degree and computing the cohomology of each row first gives us C (M) in the 0th column. This shows that the total complex is chain homotopy equivalent to C (M) and its cohomology is Ext (k; M). On the other hand, the ith column is i C ( M) 24 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK so its cohomology is i Ext (k; M) giving Ei;j1= Ci(Ext j(k; M)) and Ei;j2= Ext i(k; Extj (k; M)) as claimed. There are two interesting cases of this spectral sequence, occurring when f is surjective and when it is injective. When it is surjective the inner Ext group is 2 M concentrated in degree 0 since is a free -comodule. Hence the spectral sequence collapses and we have (2.30) Ext (k; M) = Ext (k; 2 M): This is the change-of-rings isomorphism due originally to Milnor-Moore [MM65 ]. The other interesting case of the spectral sequence occurs when we have an extension of Hopf algebras f g (2.31) -! -! ; this means that = both as -modules and as -comodules. Applying (2.30) to the surjection g gives Ext (k; M) = Ext (k; 2 M) = Ext (k; M) so the E2-term of the spectral sequence associated with f is (2.32) Ei;j2= Ext i(k; Extj(k; M)): This is the Cartan-Eilenberg spectral sequence of [CE56 , page 349]. Now we will discuss localized Ext groups. Suppose a Hopf algebra has an odd dimensional (this is not needed if k has characteristic 2) primitive element t. Then there is a corresponding element v 2 Ext1 (k; k) which we would like to invert. The class v is represented by a short exact sequence 0 -! k -! L -! |t|k -! 0 of -comodules. Now suppose we have an injective -resolution (such as the one associated with the cobar complex or the double complex THE TELESCOPE CONJECTURE 25 above) of a left -comodule M, d0 d1 0 _______wM ______wI0 _______I1w______w. . . and let Js = kerds = cokerds-1. Then for each s 0 we have a diagram 0 __________wJs ________Lw Js ______w|t|Js _________w0 ||||||||||||| | ||||||||||||| || | ||||||||||||| ||| || : |||||||||||||||||||||||||||| ||||||||||||| || | ||||||||||||| || | ||||| | |u |u 0 __________wJs __________wIs _________Js+1w _________w0 Using this we get a diagram M _________-|t|J1w ______w-2|t|J2 ________.w. . | | | | | | | | | | | | | | | | | | | |u |u |u I0 _________-|t|I1w ______w-2|t|I2 ________w. .;. (where the maps in the bottom row exist because their targets are in- jective and the vertical maps are inclusions) and hence a direct limit of injective resolutions, of the corresponding cochain complexes obtained by cotensoring over with k, and of Ext groups. We denote the direct limit of cobar complexes by v-1 C (M) (the localized cobar complex ) and its cohomology by v-1 Ext (M), the localized Ext group. Now suppose we have a map f : ! as before with an odd dimen- sional primitive t 2 corresponding to v 2 Ext 1(k; M). We can replace the double complex C (C (M)) by v-1 C (C (M)). The equiv- alence between C (C ( M)) and C (M) is preserved by inverting v in this way, so we get a spectral sequence converging to v-1 Ext (k; M). The ith column of the double complex is v-1 Ci(C ( M)), and we get Ei;j2= v-1 Exti(k; Extj (k; M)): When f is onto, the inner Ext group collapses as before and we get a localized change-of-rings isomorphism (2.33) v-1 Ext (k; M) = v-1 Ext (k; 2 M); 26 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK and when f is the injection in a Hopf algebra extension as in (2.31) we get the first form of the localized Cartan-Eilenberg spectral sequence (2.34) v-1 Ext (k; Ext (k; M)) =) v-1 Ext (k; M): We can also consider the case where the odd dimensional primitive t is in but not in . Then we replace the double complex C (C ( M)) by C (v-1 C ( M)). Then again we have acyclic rows and taking their cohomology gives v-1 C (M) in the 0th column. Thus our spectral sequence converges again to v-1 Ext (k; M) with Ei;j2= Ext i(k; v-1 Extj(k; M)): In the case of an extension we use (2.33) to identify the inner Ext group, and we get the second form of the localized Cartan-Eilenberg spectral sequence (2.35) Ext (k; v-1 Ext (k; M)) =) v-1 Ext (k; M): 3. The spectra y(n) and Y (n) We will now construct the spectrum y(n) whose homology and E2- terms were discussed previously, along with the associated telescope Y (n). 3.1. The EHP sequence and some Thom spectra. Recall that S3 is homotopy equivalent to a CW-complex with a single cell in every even dimension. Let Jm S2 (the mth James product of S2) denote its 2m-skeleton. James [Jam55 ] showed that there is a splitting _ S3 ' S2i+1: i>0 These lead to the James-Hopf maps Hi: S3 ! S2i+1 which are surjective in homology. We will denote Hp simply by H. When i is a power of a prime p, the p-local fiber of this map is a skeleton, i.e., there is a p-local fiber sequence n+1 (3.1) Jpn-1S2 -! S3 -! S2p : Definition 3.2. y(n) is the Thom spectrum of the p-local spherical fi- bration over Jpn-1S2 induced from the one over 2S3 given by Lemma 3.3 below. THE TELESCOPE CONJECTURE 27 y(n) is an A1 ring spectrum, since it is the Thom spectrum of a bundle induced by a loop map. It may be that in the cases where Toda's complex V (n - 1) exists and p is odd, that y(n) ' V (n - 1) ^ T (n) (but probably not as A1 ring spectra), where T (n) is the spectrum of [Rav86 , x6.5] with BP*(T (n)) = BP*[t1; t2; : :;:tn]: It is a p-local summand of the Thom spectrum of the canonical complex bundle over SU(pn). The following is proved in [MRS ]. Lemma 3.3. For each prime p there is a p-local spherical fibration over 2S3 whose Thom spectrum is the mod p Eilenberg-Mac Lane spec- trum H=p. For the rest of this section we assume the p is odd to avoid notational complications. We have H*(y(n)) = E(o0; o1; . .o.n-1) P (1; . .n.) as comodules over A*. Lemma 3.4. y(n) is a split ring spectrum, i.e., y(n) ^ y(n) is a wedge of suspensions of y(n) with one summand for each basis element of H*(y(n)). In particular y(n)*(y(n)) = ss*(y(n)) H*(y(n)): Proof. Consider the Atiyah-Hirzebruch spectral sequence for y(n)*(y(n)) with E2 = H*(y(n); ss*(y(n))): It suffices to show that each multiplicative generator of H*(y(n)) is a permanent cycle. These generators all have dimensions no more than |vn|, and below that dimension y(n) is equivalent to H=p. It follows that there are no differentials in the Atiyah-Hirzebruch spectral sequence_in_ that range. |__| The classical Adams E2-term for y(n) was described in Corollary 2.3. In low dimensions there is no room for any differentials, and we have Lemma 3.5. Below dimension 2p2n+1 - 2pn-1 - 2, the Adams spectral sequence for ss*(y(n)) collapses from E2 (for formal reasons), with E2 = P (vn; . .;.v2n) E(hn+i;j: i > 0; j 0) P (bn+i;j: i > 0; j 0); 28 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK where n+i-1 vn+i 2 E1;2p2 n+i+j-2pj hn+i;j 2 E1;2p2 n+i+j+1-2pj+1 bn+i;j 2 E2;2p2 : Proof. From the Hopf algebra extension H*(y(n)) -! A* -! B(n)* we see that ExtB (Z =(p); Z=(p)) (our E2-term) is a comodule over H*(y(n)), regarded as a subalgebra of A*. From 3.4 we see that this subalgebra of A* is part of the coalgebra of co-operations in y(n)*-theory. This means that the corresponding quotient of A acts on the Adams spectral sequence. Routine calculations give fi(vn+i) = hn+i;0; j P p(hn+i;j) = hn+i-1;j+1 j+1 and P p (bn+i;j) = bn+i-1;j+1: Hence if we can show that vn+i for i n and bn+i;0for i n - 1 are permanent cycles, then the same will be true of all generators in our range of dimensions. We will show this by proving that there are no elements (besides hn+i;0) in dimension |vn+i| - 1 or |bb+i;0| - 1 for these i. This can be done by organizing the information in a suitable way. define the weight ||x|| of an element x by ||vn+i|| = pi; ||hn+i;j||= pi+j; ||bn+i;j||= pi+j+1 and ||xy|| = ||x|| + ||y||: The generator x having the lowest dimension for its weight is 8 < vn if||x|| = 1; hn+1;0 if||x|| = p and : b j+2 n+1;j if||x|| = p for j 0: and the one with the highest weight is always vn+i. Next observe that for i n, (pi+ 1)|bn+1;j| > |vn+i+j+2| and (pi- 1)|vn+k | < |bn+1;i+k-2|: THE TELESCOPE CONJECTURE 29 This means that in our range the target of a differential on a generator x must have the same weight as x. The first possible exceptions to this occur just outside our range, namely it is possible that (3.6) d1(v2n+1) = vnhn+1;n j and d2pj(h2n+1-j;j) = vnbpn+1;n-1-j for0 j n - 1: We will see below (5.18) that these differentials actually occur, the first being apparent from the structure of B(n)*. Now consider the quantity (x) = |x| - 2pn||x||; which satisfies (xy) = (x) + (y). Then we have (vn+i) = -2; (hn+i;j) = -1 - 2pj and (bn+i;j-1) = -2 - 2pj: From this we can see that for any monomial x of weight pi, (vn+i) exceeds 1 + (x) except when x = hn+i;0, and (bn+i-1;0) exceeds it except when x is one of the three generators with a higher value of , namely vn+i, hn+i;0and hn+i-1;1. We know that dr(vn+1) must have weight pi and that (dr(vn+i)) = (vn+i)-1, so there is no possible nontrivial value of dr(vn+1). Similarly_ there can be no differential on bn+i-1;0. |__| The first positive dimensional element in ss*(y(n)) is vn 2 ss2pn-2(y(n)). We can use the multiplication on y(n) to extend vn to a self-map. The telescope Y (n) is the homotopy colimit of (3.7) y(n) -vn!-|vn|y(n) -vn!-2|vn|y(n) -vn!. . . Theorem 3.8. The telescope Y (n) defined above is Lfny(n). Proof. We will adapt the methods used by Hopkins-Smith [HS98 ] to prove the periodicity theorem, as explained in [Rav92a , Chapter 6]. Let X by a finite complex of type n with a vn-map f such that K(n)*(f) is multiplication by vkn; see [Rav92a , 6.1.1]. Let R = DX ^ X, which is a finite ring spectrum. We will compute in ss*(R ^ y(n)), which is a noncommutative Z =(p)-algebra. Let F 2 ss*(R ^ y(n)) denote the image of f under map R ! R ^ y(n), and let G be the image of g = vkn under the map y(n) ! R ^ y(n). 30 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK Now R ^ y(n) has an Adams vanishing line of slope 1=|vn| since y(n) does. The map F -G represents an element above the line of this slope through the origin, so it is nilpotent. (In the proof of the periodicity theorem, the nilpotence theorem was needed at this point. We do not need an analog of it here because we have the vanishing line.) The methods of [Rav92a , 6.1.2] can be applied here to show that for some i pi i > 0, F p and G commute. Now replace F and G by their commuting powers. F - G is still nilpotent for the same reason as before, and for j 0 we have j pj pj 0 = (F - G)p = F - G : j pj Thus F p = G . Replacing the original f and g by suitable powers we get a commutative diagram (ignoring suspensions) f^y(n) X ^ y(n) ______w X ^ y(n) | | | | X^g | |X^g | | |u f^y(n) |u X ^ y(n) ______wX ^ y(n): It follows that Xb ^ y(n) = X ^ Y (n) = bX^ Y (n): Thus the map y(n) ! Y (n) is an Xb*-equivalence. The result will follow if we can show that Y (n) is Xb*-local. We have Y (n) ^ Cf = 0, Cf being the cofiber of f and therefore a finite complex of type n + 1. Given this, is follows from the thick subcategory theorem that Y (n) __ annihilates all finite K(n)*-acyclic complexes, so it is Xb*-local. |__| Conjecture 3.9. Y (n) has the same Bousfield class as the telescope associated with a finite complex of type n. This could be regarded as a new formulation of the telescope con- jecture, with Y (n) taking the place of K(n). See also Conjecture 4.4 below. A stronger conjecture is the following. Conjecture 3.10. Y (n) has the same homotopy type as an infinite wedge of finite type n telescopes. 3.2. The homotopy of Lny(n) and Y (n). We have BP*(Lny(n)) = v-1nBP*(y(n)) = v-1nBP*=In[t1; : :;:tn]: and we know that the Adams-Novikov spectral sequence converges to ss*(Lny(n)). Its E2-term was given above in 2.11, namely E2 = K(n)*[vn+1; : :;:v2n] E(hn+i;j:1 i n; 0 j n - 1): THE TELESCOPE CONJECTURE 31 It follows from 3.5 that each vn+i is a permanent cycle, as is hn+i;j for i + j n. This accounts for just over half of the n2 exterior generators. Perhaps other exterior generators are permanent cycles for dimensional reasons, but we will see below that similar elements in the localized Adams spectral sequence are not. Question 3.11. Does the Adams-Novikov spectral sequence for Lny(n) collapse? It does for sparseness reasons when 2p > n2. In any case we have the following result. Corollary 3.12. ss*(Lny(n)) is finitely presented as a module over the ring (3.13) R(n)* = K(n)*[vn+1; : :;:v2n]: We had hoped to show this is not true of ss*(Y (n)) for n > 1, showing that Y (n) (which is Lfny(n)) differs from Lny(n), thereby disproving the telescope conjecture. We can compute ss*(Y (n)) with the localized Adams spectral se- quence. Its E2-term was identified in (2.20) as E2 = R(n)* E(hn+i;j: i > 0; 0 j n - 1) = P (bn+i;j: i > 0; 0 j n - 1): As remarked above, the hn+i;jfor i + j n and the vn+i are permanent cycles. Conjecture 3.14. For i > 0 and 0 j n - 1, the element h2n+i-j;j survives to E2pj and supports a nontrivial differential j d2pj(h2n+i-j;j) = vnbpn+i;n-1-j: Each bn+i;jfor i > 0 and 0 j n - 2 survives to E1+2pn-1. Note that if in addition each bn+i;jwere a permanent cycle, then we would have n-1-j (3.15) E1 = R(n)* E(hn+i;j:i + j n) P (bn+i;j)=(bpn+i;j): For n > 1, this E1 and hence ss*(Y (n)) would be infinitely generated as a module over R(n)*, which is incompatible with the telescope con- jecture. However we cannot prove that each bn+i;jis a permanent cycle for n > 1, and it seems unlikely to be true. Hence we expect E1 to be more complicated than indicated by (3.15). We have 32 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK Conjecture 3.16 (Differentials conjecture). In the localized Adams spec- tral sequence for Y (n) the elements hn+i;0and hn+i;1survive to E2 and E2p respectively, and there are differentials d2(h2n+i;0+ d2n+i;0) = vnbn+i;n-1 and d2p(h2n+i-1;1+ d2n+i-1;1) = vnbpn+i;n-2 for decomposables d2n+i-j;j. The elements bn+i;jfor j < n - 1 survive to E2p+1, so E2p+1 = R(n)* E(hn+i;0: 1 i n) E(hn+i;1: 0 i n - 1) E(hn+i;j: i > 0; 2 j n - 1) P (bn+i;n-2: i > 0)=(bpn+i;n-2) P (bn+i;j: i > 0; 0 j n - 3): This will be discussed in x5. The strategy is to lift the computa- tion back to a localized ThomifiednEilenberg-Moore spectral sequence converging to Y (n)*(3S1+2p ) in a manner to be described in x3.3, specifically using the map of (3.18) below. Curiously, its E2-term is essentially the one above tensored with itself. The corresponding state- ment about differentials there is Conjecture 5.15. The advantage of this lifting is that the localized Thomified Eilenberg- Moore spectral sequence has far more structure than the localized Adams spectral sequencenabove, due in large part to the structure of the space 3S1+2p . Its properties are developed in x4. It is an H- space (which makes the spectral sequence one of Hopf algebras) with a Snaith splitting described in x4.1. The pth Hopf map induces an endomorphism of the spectral sequence that is described in Lemma 5.16. For n = 1 Conjecture 3.16 gives the following. E2 = R(1)* E(h2;0; h3;0; : :): P (b2;0; b3;0; : :): with differentials d2(hi+1;0) = v1bi;0 fori 2; which leaves E3 = E1 = R(1)* E(h2;0): Thus for n = 1, the localized Adams spectral sequence and the Adams- Novikov spectral sequence give the same answer. Miller [Mil81 ] proved the telescope conjecture for n = 1 and p odd by doing a similar calcu- lation with y(1) replaced by V (0). For n = 2 we have E2 = R(2)* E(h3;0; h3;1; h4;0; h4;1; : :): P (b3;0; b3;1; b4;0; b4;1; : :):: THE TELESCOPE CONJECTURE 33 The first differential, d2(hi+2;0) = v2bi;1 fori 3 gives E3 = R(2)* E(h3;0; h4;0) E(h3;1; h4;1; : :): P (b3;0; b4;0; : :):: A pattern of higher differentials consistent with the telescope conjecture would be i+1 d1+pi+1(hi+2;1) = vp2 bi;0 fori 3: Notice that these get arbitrarily long for large i, and they are preempted by the differentials of 3.16, d2p(hi+1;1) = v2bpi;0: The splitting of Lemma 3.4 has implications for the spectral se- quences we are studying. For any space or spectrum X, y(n)*(X) is a left comodule over y(n)*(y(n)) = H*(y(n)) y(n)*: The same goes for Er of a spectral sequence converging to y(n)*(X), in which case y(n)* may be filtered in some way. Similarly Y (n)*(X) is a left comodule over Y (n)*(Y (n)) = H*(y(n)) Y (n)*: Lemma 3.17. The comodule structure of the localized E2-term of (2.20) is given by X n+k (vn+i) = pi-k vn+k ; 0ki X __pj (hn+i;j) = k hn+i-k;j+k 0kn-1-j X __pj+1 and (bn+i;j) = k bn+i-k;j+k 0kn-1-j In the localized Adams spectral sequence for Y (n), if x is any of the hn+i;jor bn+i;jand d2(x) is nontrivial, it cannot be divisible by vn+k for any k > 0. Proof. The coalgebra structure in H*(y(n)) can be read off by injecting it into the dual Steenrod algebra. The values of (vn+i) and (hn+i;j) __pj can be read off from the coproducts on on+i and n+iin A*, and (bn+i;j) is the transpotent of the latter. The divisibility of d2(x) by vn+k would contradict this comodule __ structure. |__| 34 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK 3.3. The triplenloop space. Now we will explain why the triple loop space 3S1+2p is relevant to the proof of the main theorem. Consider the following diagram in which each row is a fiber sequence. (3.18) n n n Jpn-1S2 x 3S1+2p ______w2S3 ______w2S1+2p x 2S1+2p ||||||||| | | ||||||||||||| | | | |||||||||||||||| | | |||||||||||||||| | |u ||||||||| | |u i Jpn-1S2 ____________w2S3 ____________w2S1+2pn Here the top row is the Cartesiannproduct of the bottom row with the path fibration on 2S1+2p . The right vertical map is loop space multiplication, while the left one is the product of the identity map on the first factor with the inclusion of the fiber of i on the second factor. We will look at the Thomified Eilenberg-Moore spectral sequence for each row where the spherical fibration over 2S3 is the one given by 3.3. Then the bottom row satisfies the hypotheses of Theorem 2.26(ii), so we get the Adams spectral sequence for y(n). For the top row, the E2-term is described by the following special- ization of 2.26. Theorem 3.19. Consider the Thomified Eilenberg-Moore spectral se- quence associated with E = 2S3, equipped with the spherical fibration given by 3.3. Suppose the defining fibration has the form i _______________h X| |_____________wE Bw ||||||||||||| | ||||||||| | ||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||||||| ||||||||||||||||||||| i1xi2 h1x* X1 x B2 ______w2S3 x pt. _______wB1 x B2 where h is an H-map and H*(i1) is monomorphic, and Y = Y1^ B2+. Then H*(Y1) is a subalgebra of A* = H*(K), and we let = A* H*(Y1)Z =(p) Then the E2-term of the Thomified Eilenberg-Moore spectral sequence is Ext H*(B2) (Z =(p); Z=(p)); where H*(B2) is a semitensor product coalgebra with coproduct as in (2.23). In the next section we will see that the top row of (3.18) satisfies the hypotheses of 3.19. In this case the Hopf algebra is B(n)* of (2.2). THE TELESCOPE CONJECTURE 35 Proof. We have a Hopf algebra extension H*(B2) A* -! H*(B1) H*(B2) A* -! H*(B1) and hence a Cartan-Eilenberg spectral sequence converging to the E2 of the Thomified Eilenberg-Moore spectral sequence with E2 = Ext H*(B2)A*(Z =(p); ExtH*(B1)(Z =(p); H*(K))): In our case H*(K) is a free comodule over H*(B1), so the prespectral sequence collapses to (3.20) Ext H*(B2)A* (Z =(p); H*(Y1)): Using the Hopf algebra extension H*(B2) -! H*(B2) A* -! H*(Y1) we can equate (3.20) with Ext H*(B2) (Z =(p); Z=(p)): __ as claimed. |__| Theorem 3.21. The Thomified Eilenberg-Moore spectral sequence for the top row of (3.18) can be localized in the same way that the one for the bottom row can. Proof. The spectral sequencenin question is based on the diagram (2.22) with Y0 = y(n) ^ 3S1+2p+. This diagram has suitable multiplicative properties. In order to getna localized resolution as in (2.12), we need to lift the map vn ^ 3S1+2p+ to Y1. This lifting exists if and only if n g0(vn ^ 3S1+2p+) is null, which it is since K0 = H=p and H*(vn) = 0. Thus the Thomified Eilenberg-Moore spectral sequence for the top row of (3.18) can be localized compatibly with our localization of the Adams spectral sequence associated with the bottom row. Convergence of the localization of the top row (which is not actually needed for our purposes) can be proved using the argument of Theorem 2.13 provided we have a suitable vanishing line. Our E2-term is a subquotient of ExtB(n)*(Z =(p); ExtH*(2S1+2pn)(Z =(p); Z=(p))): n The connectivities of B(n)* and 2S1+2p imply that both factors have __ a vanishing line of slope 1=|vn| as required. |__| 36 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK n 4. Properties of 3S1+2p 4.1. The Snaith splitting. For each n > 0 we have a fibration of spaces (3.1) n Jpn-1S2 ! S3 ! S1+2p which leads to a stable map n f (4.1) 3S1+2p+ -! y(n): n We know that 3S1+2p+ has a Snaith splitting [Sna74 ] n _ |v |i (4.2) 3S1+2p+ ' n Ti: i0 Here Ti is a certain finite complex (independent of n) with bottom cell in dimension 0 and top cell in dimension 2i - 2ff(i), where ff(i) denotes the sum of the digits in the p-adic expansion of i. In particular T1 = S0. Moreover there are pairings Ti^ Tj ! Ti+j: Thus we get a ring spectrum (4.3) T1 = limTi: ! Using the map vn of (3.7) and the map f of (4.1), for each i 0 we get a diagram i|vn|Ti ______wi|vn|Ti+1 ________w. . .________i|vn|T1w | | | f| f| b|f | | | |u v |u v |u y(n) _______-|vn|y(n)wn________w.n. ._________wY (n) The map f on the left is vinon the bottom cell of its source. The map bfon the right makes Y (n) a module spectrum over T1 . If 3.9 is true, then the following seems likely. Conjecture 4.4. The Bousfield class of T1 is that of the p-local sphere spectrum. T1 is also the Thom spectrum of a bundle over 30S3 obtained as follows. From the EHP sequence we get a fiber sequence 2S2p-1 -! 30S3 -! 3S2p+1: THE TELESCOPE CONJECTURE 37 We get our bundle from one over 3S2p+1 obtained by extending the map S2p-2 ! BU corresponding to a generator of ss2p-2(BU). Equivalently, our bundle is the one obtained from the map 30S3 = 30SU(2) -! 30SU = BU: There is also a Hopf map n H 3 1+2pn+1 (4.5) 3S1+2p -! S which is surjective in ordinary homology. It induces a map from the pith Snaith summand of the source to the ith one of the target, (4.6) Tpi- H! 2(p-1)iTi; which has degree one on the top celln(in dimension 2pi - 2ff(i)). We will use this map to study 3S1+2p and T1 below. Recall (4.3) that the spectrum T1 is the homotopy direct limit of the Snaith summands of a certain triple loop space. The analogous spectrum obtained from the Snaith splitting of the double loop space of an odd dimensional sphere is H=p, but T1 is far more interesting. It turns out that K(n)*(T1 ) bears a remarkable resemblance to the supposed value of ss*(Y (n)). (Compare Conjecture 3.16 and Theorem 4.17 below.) 4.2. Ordinary homology. H*(3S2dp+1) has long been known [CLM76 ] as a module over the Steenrod algebra A, and is as follows. Lemma 4.7. 8 < P (ui : i 0) P (xi;j: i > 0; j 0) for p = 2 H*(3S2dp+1) = P (ui : i 0) E(xi;j: i > 0; j 0) : P (y i;j: i > 0; j 0) for p > 2 where |ui| = 2(pi+1d - 1), |xi;j| = 2pj(pi+1d - 1) - 1 and |yi;j| = 2pj+1(pi+1d - 1) - 2. For all primes this group can be identified with Ext H*(2S1+2pn)(Z =(p); Z=(p)); i.e., the Eilenberg-Moorenspectral sequence in mod p homology for the path fibration on 2S1+2p collapses. 38 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK For p = 2 the action of the Steenrod algebra A is given by 8 k < xi;02 if k = 0 Sq2*(ui) = ui-1 if k = 1 : x2 8 i-k+1;k-2 otherwise 2 if k = 0 and j > 0 k < xi;j-1 Sq 2*(xi;j)= xi-1;j+1 if k = j + 1 : 0 otherwise. For p odd we have fi*(ui) = xi;0 ae p k -ui-1 if k = 0 P*p(ui) = 0 otherwise fi*(xi;j)= yi;j-1 forj > 0 ae k xi-1;j+1 if k = j P*p(xi;j) = 0 otherwise fi*(yi;j)= 0 8 p k < -yi;j-1 if k = 0 and j > 0 P*p(yi;j) = yi-1;j+1 if k = j + 1 : 0 otherwise. We will also need to know the action of the Milnor primitives Qk, which can be read off from Lemma 4.7. Up to sign we have ae xi-k;k ifk < i Qk(ui) = 0 otherwise 8 k >< ypi;j-k-1 for0 k < j (4.8) Qk(xi;j) = 0 fork = j >: pj yi+j-k;k-j-1 forj < k < i + j where yi;j= x2i;jwhen p = 2. Proof of Lemma 4.7. We will prove this for p odd, leaving the case p = 2 (which is easier) as an exercise for the reader. We will relate the description of the Lemma to the one given by Cohen in [CLM76 ]. There he speaks of Dyer-Lashof operations with upper indices Qs : Hq ! Hq+2(p-1)s. Within this proof Qs will denote a reindexed Dyer- Lashof operation rather than the Milnor operation. We define Qs : Hq ! Hpq+(p-1)s, when q and s have the same parity, by Qs = Q(s+q)=2 THE TELESCOPE CONJECTURE 39 with ae 0 fors < 0 Qs(x) = xp fors = 0: With this in mind, Cohen's result says that H*(3S2pd+1) = P (Qi2(u0) : i 0) E(Qj1fiQi2(u0) : i > 0; j 0) P (fiQj+11fiQi2(u0) : i > 0; j 0); where u0 2 H2pd-2 is the fundamental class. We define ui = Qi2(u0); xi;j = Qj1fiQi2(u0) and yi;j = fiQj+11fiQi2(u0): These elements have the indicated dimensions. It remains to show that the action of the Steenrod algebra is as stated. The action of Steenrod operations on Dyer-Lashof operations is given by the Nishida relations. For operations on a q-dimensional class, these are X (p - 1) s+q_- r (4.9) P*rQs = (-1)r+i 2 Qs-2r+2piP*i i r - pi and (4.10) X (p - 1) s+q_- r - 1 P*rfiQs = (-1)r+i 2 fiQs-2r+2piP*i i r - pi X (p - 1) s+q_- r - 1 + (-1)r+i 2 Qs+1-2r+2piP*ifi: i r - pi - 1 In particular we have q P*1Q2 = __Q0; 2 so P*1(ui) = P*1Q2(ui-1) = -Q0(ui-1) = -upi-1: 40 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK For k > 0 (4.9) gives k pk-1 P*pQ2 = Q2P* ; k pkk so P*p(ui) = P* Q2(ui-k) = Qk2P*1(ui-k) = -Qk2(upi-k-1) = 0 by the Cartan formula. We have fi(ui) = xi;0and fi(xi;j+1) = yi;jby definition, and it follows that fi(xi;0) = 0 and fi(yi;j) = 0. The Nishida relations also give ae k 0 k = 0 P*pQ1 = pk-1 Q1P* k > 0; and for s = 1 or 2 ae k -Qs-1fi k = 0 P*pfiQs = pk-1 fiQsP* k > 0: It follows that P*1(xi;0)= P*1fiQi2(u0) = -Q1fiQi-12(u0) = -xi-1;1; and for k > 0 k pk i P*p(xi;0) = P* fiQ2(u0) ae i-k fiQk2P*1Q2 (u0) k < i = i pk-i fiQ2P* (u0) k i = 0: For j > 0, P*1(xi;j) = P*1Qj1fiQi2(u0) = 0; THE TELESCOPE CONJECTURE 41 and when k > 0 k pkj P*p(xi;j) = P* Q1(xi;0) ae 0 k < j = j pk-j Q1P* (xi;0) k j 8 < 0 k < j = Qj1(xi-1;1) k = j : 0 k > j ae xi-1;j+1 k = j = 0 k 6= j as claimed. Finally we have for j > 0, P*1(yi;j)= P*1fiQj+11(xi;0) = -Q0fiQj1(xi;0) = -Q0(yi;j-1) = -ypi;j-1; and for all j when k > 0 k pk j+1 P*p(yi;j) = P* fiQ1 (xi;0) k-1j = fiQ1P*p Q1(xi;0) k-1 = fiQ1P*p (xi;j) ae 0 k - 1 6= j = fiQ 1(xi-1;j+1) k - 1 = j ae 0 k 6= j + 1 = y i-1;j+1 k = j + 1 __ as claimed. |__| 4.3. Morava K-theory. In this subsection we will study the Eilenberg- Moore spectral sequence for K(n)*(3S2dp+1) for d > 0. First we need to know K(n)*(2S2dp+1), which was computed by Yamaguchi [Yam88 ]. We will assume for simplicity that p is odd. We could find K(n)*(2S2dp+1) with either the Atiyah-Hirzebruch spectral sequence or the Eilenberg-Moore spectral sequence starting with K(n)*(S2pd+1). (Since S2pd+1 splits after a single suspension, it is easy to work out its Morava K-theory.) It turns out that the two spectral sequences are the same up to reindexing, and we will describe the former. We have (4.11) H*(2S2dp+1) = E(ei : i 0) P (fi : i 0) 42 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK with |ei| = 2dpi- 1 and |fi| = 2dpi+1 - 2. (When p = 2, fi = e2i.) In terms of the Dyer-Lashof operations Qi we have ei = Qi1(e0) and fi = fiQi+11(e0) The coaction of the dual Steenrod algebra A* is given by P __ pn+k ei 7! 1 ei+ 0k n are {xm-j;j: 1 j n - 1} [ ` (4.18) {ypm-`-k-1;k:0 k n - 2; 0 ` + k m - 2} [ m-k {upk :0 k n}; for 1 m n we have to add the element xm;0. The dimensions of these elements are 8 >< |xm-j;j| = 2dpm - 2pj - 1; ` m `+k+1 ` (4.19) |ypm-`-k-1;k|= 2dp - 2p - 2p >: pm-k and |uk | = 2dpm - 2pm-k For m n, these dimensions are all within |vn| of each other, so there can be no higher differentials in these Snaith degrees. In particular the elements uk for 0 k n, xi;jfor i + j n and yi;jfor i + j < n are all permanent cycles. For m = n + 1, (4.18) reads {xn+1-j;j:1 j n - 1} [ k {ypn-j-k;j:0 j n - 2; 0 j + k n - 1} [ n+1-k {upk : 0 k n}: THE TELESCOPE CONJECTURE 45 The dimensions of these are compatible with the desired differentials j (4.20) d2pj-1(xn+1-j;j) = vnyp1;n-1-j for1 j n - 1; which can be inferred from (4.8). The only remaining primitives in this Snaith degree are even dimensional, all of the odd dimensional ones having been accounted for. Now consider the primitives in Snaith degree pn+i for i > 1. We claim that the only differentials that occur here are j (4.21) d2pj-1(xi+n-j;j) = vnypi;n-j-1 for j n - 1: Here we make use of the Hopf map H. Its (i - 1)th iterate sends the source and target of (4.21) to those of (4.20). Thus (4.21) holds provided that there is no earlier differential on xi+n-j;j. Once these differentials have been taken into account, there are no odd dimensional primitives left in Snaith degrees above pn, so the spectral sequence collapses from E2pn-1. To prove (4.21) note that the elements of (4.19) with ` + k < n + 1 have dimensions too high to be a target of a differential on xn+i-j;j, the ones with ` + k = n + 1 are the proposed targets, and the one with ` + k > n + 1 are powers of elements of lower Snaith degree that n+i-k have already been killed. The elements upk have either too high a dimension (if i > k) or too high a filtration (if i k) to be the target of an earlier differential on xn+i-j;j. It follows that E1 = E2pn-1 = K(n)*[u0; . .u.n] E(xi;j: i + j n) n-j-1 P (yi;j: j n - 2)=(ypi;j ) This spectral sequence converges to K(n)*(3S2dp+1). To show there are no multiplicative extensions, we need to show that n-1-j ypi;j = 0. If it is nonzero, it must be a K(n)*-linear combination of n+i-k the elements upk for 0 k n, but the latter do not have the right__ dimensions modulo |vn|, except possibly for p = 2. |__| n 4.4. The computation of Y (n)*(3S1+2p ) via the Eilenberg-Moore spectral sequence. In this subsection we will prove the following. Theorem 4.22. For each n > 0 there is an additive isomorphism Y (n)*(3S2dp+1) = Y (n)* K(n)*K(n)*(3S2dp+1): 46 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK The isomorphism here need not be multiplicative. We will say more about this below after the proof. Before proving his result we need the following. Lemma 4.23. There is an additive isomorphism Y (n)*(2S2dp+1) = Y (n)* K(n)*K(n)*(2S2dp+1): Proof. Consider first the Atiyah-Hirzebruch spectral sequence for y(n)*(2S2dp+1* *). We have the differentials of (4.13), which leaves n E2pn = y(n)* E(e0; e1; . .e.n) P (fi)=(vnfpi): It follows that in the Atiyah-Hirzebruch spectral sequence for Y (n)*(2S2dp+1) we have n E2pn = Y (n)* E(e0; e1; . .e.n) P (fi)=(fpi): Now we can argue as in the proof of 4.14 that there can be no higher __ differentials for dimensional reasons. |__| Proof of Theorem 4.22. To prove the theorem we will use Tamaki's spectral sequence and the computation is essentially the same as that of x4.3. It is not necessary to know ss*(Y (n)) explicitly. In order to use Tamaki's spectral sequence we will use the computation of Y (n)*(2S2dp+1) of 4.23. The group Y (n)*(2S2dp+1) is a free Y (n)*- module. This means that the E2-term of the Tamaki spectral sequence can be identified as a Cotor group as in the Morava K-theory case. In order to use the Tamaki spectral sequence we need to know the coalgebra structure. As in the computation for Morava K-theory, it is convenient to ignore the nontriviality of the Verschiebung and proceed as if the Hopf algebra were primitively generated. The resulting Cotor group can be regarded as the E1-term of the Tamaki spectral sequence. With this in mind, we can define ui, xi+1;j, and yi+1;jas in (4.16). As before we have E1 = Y (n)* P (ui : 0 i n) E(xi;0: i n) P (yi;j: j n - 1) E(xi;j: 0 j n - 1): This Eilenberg-Moore spectral sequence maps to the one for Morava K-theory where we have the differentials of (4.21), namely j d2pj-1(xi+n-j;j) = vnypi;n-j-1 for j n - 1: Similar differentials will occur in the spectral sequence at hand if they are not preempted by earlier ones. In Snaith degrees < pn+1 we can compare with the Eilenberg-Moore spectral sequence for y(n)*(3S2pd+1) and conclude that there are no differentials for dimensional reasons, so THE TELESCOPE CONJECTURE 47 again the elements ui for i n, xi;jfor i + j n and yi;jfor i + j < n are all permanent cycles. In Snaith degree pn+1, the primitives are {xn+1-j;j: 0 j n - 1} [ k {ypn-j-k;j: 0 j n - 1; 0 j + k n - 1} [ n+1-k {upk : 0 k n} As in the proof of 4.17,we can use the Hopf map H to rule out many dif- ferentials. If some xn+1-j;jsupports a nontrivial differential, its target must be in the kernel of H since H(xn+1-j;j) = xn-j;j is a perma- j nent cycle. Thus the target must be a multiple of either yp1;n-jfor n+1 0 j n - 1 or up0 . Hence the expected differentials j d2pj-1(xn+1-j;j) = vnyp1;n-1-j follow by induction on j. In larger degrees we can rule out other differentials on xn+i;jas in the proof of 4.17. In the proof of 4.17 we knew that each yi;jwith j n - 2 was a permanent cycle because the remaining primitives in the same Snaith degree were also even dimensional. However, this argument is not good enough here because we do not know (and we will see that it is not true) that Y (n)* is even dimensional. We have to consider the possibility that i+j+1-k (4.24) dpi+j+1-k-2(yi;j) = ffupk for some ff 2 Y (n)* and n k 2n. We can use the Hopf map to exclude such a differential in the fol- lowing way. The Snaith summands of 3S2dp+1 are independent of d. If d is divisible bypm then our spectral sequence is in the image of the mth iterate of the Hopf map H. Thus we get a diagram yi;j_________ffupi+j+1-kwdr u k u | || m | ||| (4.25) H | || | || | yi+m;j ______wff0upi+j+m+1-k0dr0 k0 where r + 2 = pi+j+1-k 0 and r0+ 2 = pi+j+m+1-k 48 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK with n ; k; k0 2n. Since Hm induces a map of spectral sequences, we have r r0, so i + j + 1 - k i + j + m + 1 - k0 k0 k + m This is incompatible with the upper bound on k0 since we can do this for any value of m, so there can be no nontrivial differential of the form_ (4.24). |__| We will now explain why the isomorphism of 4.22 need not be mul- tiplicative. The argument given in the proof of 4.17 to show that there are no multiplicative extensions does not carry over to the computation n pn-1-j of Y (n)*(3S1+2p ). Indeed yi;j could be a nontrivial Y (n)*-linear n+i-k combination of the elements upk for 0 k n, e.g., n+i v-1nbi+j+1;n-2-jup0 when i + j n. Here bi;jdenotes the image of yi-n;junder the map of (3.18). The use of the Hopf map in the last paragraph of the proof above is similar to its use by the first author in [Mah77 ], and it deserves further comment. In that paper there was a 2-local stable splitting _ 2S9 ' 7iBi i>0 where the stable summand Bi is known now to be the Brown-Gitler spectrum B([i=2]) with bottom cell in dimension 0 and top cell in dimension i - ff(i). There one wanted to show that a certain element j xj 2 Ext 1;1+2A*(Z =(2); H*(B2j)) was a permanent cycle. The Hopf map induces j+1 H -2j -2 B2j+1- ! B2j which sends xj+1 to xj. Now suppose there is a nontrivial differential j dr(xj) = zj 2 Ext 1+r;r+2A*(Z =(2); H*(B2j)): It was shown that no such zj is in the image of the Hopf map H, so the differential cannot occur. This amounts to saying that there is an element j x 2 ss0(lim -2 B2j) THE TELESCOPE CONJECTURE 49 which projects to an element representing xj. Using properties of Brown-Gitler spectra one can produce maps j S0 ! RP--21! lim-2 B2j where the first map is Spanier-Whitehead dual to the transfer map t : RP11 ! S0, and the composite is the desired x. Dually one can look for an element j x* 2 ss0(lim 2 DB2j) ! which is given by the composite j 1 t 0 lim 2 DB2j -! RP1 -! S : ! j * 1 In [Car83 ] Carlsson showed that H*(lim! 2 DB2j) has H (RP1 ) as a direct summand as an A-module, and that both are unstable injectives. Using results of Goerss-Lannes [GL87 ] or Lannes-Schwartz [LS89 ], one j 1 can deduce that the spectrum lim! 2 DB2j has RP1 as a retract. One can ask analogous questions about the triple loop summands Ti. The Hopf map induces -2piTpi- H! -2iTi for each i. This map sends yi+1;jto yi;j. Our proof shows that for each 0 j n - 2 there is an element k yj 2 limY (n)-2pj+1-2(-2p Tpk) k which projects to yi;jfor each i. Note that lim Y (n)*(-2p Tpk) need k not be the same as Y (n)*(lim -2p Tpk) since homology does not com- mute with inverse limits, but it is the former group which interests us here. The limit problem disappears when we dualize, since homology and cohomology do commute with direct limits. We have an element j+1+2 2pk y*j2 Y (n)2p (lim DTpk): ! and similarly for Morava K-theory. An analog of Carlsson's theorem would be the following. k Conjecture 4.26. The spectrum lim! 2p DTpk has a nontrivial re- tract of the suspension spectrum of the Eilenberg-Mac Lane space K(Z =(p); 2) as a retract. Kuhn [Kuh ] has recently proved the corresponding statement in co- homology for p = 2. 50 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK A proof of 4.26 (or the construction of a suitable map from the direct limit to K(Z =(p); 2)) might lead to an independent construction of the elements y*jas follows. K(n)*(K(Z =(p); 2)) is known [RW80 ] and it is likely that Y (n)*(K(Z =(p); 2)) has a similar description. The former has n - 1 algebra generators which might map to the desired y*j. For more details, see [Rav98 ] 5. Toward a proof of the differentials conjecture 5.1. The E2-term of the localized Thomified Eilenberg-Moore spectral sequence. Now we are ready to describe our program to prove our conjecture about differentials, 3.16. We will use the map of (3.18) from the localizednThomified Eilenberg-Moore spectral sequence for Y (n)*(3S1+2p ) to the localized Adams spectral sequence for Y (n). From x3.3 we know that the E2-term of the former is v-1nExtH*(2S1+2pn)B(n)* (Z =(p); Z=(p)); n where B(n)* is as in (2.2) and H*(2S1+2p ) is given in (4.11). Lemma 5.1. (i)The E2-term of the localizednThomified Eilenberg- Moore spectral sequence for Y (n)*(3S1+2p ) is -1 Ext P0 Z =(p); vn ExtQH*(2S1+2pn) (Z =(p); Z=(p)) = R(n)* E(hn+i;j) P (bn+i;j) P (u0; : :;:un) E("xi;j) P ("yi;j): where the indices i and j satisfy i > 0 and 0 j n - 1, and R(n)* = K(n)*[vv+1; : :;:v2n]: The elements "xi;jand "yi;jare related to the homology classes xi;j n and yi;jin H*(3S1+2p ) and will be defined below in (5.9) and (5.10). (ii)Under the map of (3.18), vn+k 7! vn+k ; hn+i;j 7! hn+i;j; bn+i;j 7! bn+i;j; uk 7! vn+k ; "xi;j7! hn+i;j; and "yi;j7! bn+i;j: THE TELESCOPE CONJECTURE 51 (iii)The H*(y(n))-comodule structure (as in Lemma 3.17) on these generators is given by X n+k (vn+i) = pi-k vn+k ; 0ki X __pj (hn+i;j) = k hn+i-k;j+k; 0kn-1-j X __pj+1 (bn+i;j) = k bn+i-k;j+k; 0kn-1-j X __ (ui) = 1 ui+ ok "xi-k;k; 0ki X __pj ("xi;j)= k "xi-k;j+k 0kn-1-j X __pj+1 and ("yi;j)= k "yi-k;j+k: 0kn-1-j n Proof. (i) We begin by describing the coproduct in H*(2S1+2p ) B(n)* using (2.24) and (4.12). Elements in 1 B(n)* have theirnusual coproduct, while coproducts for the generators of H*(2S1+2p ) 1 are given by ei 1 7! ei 1 1 1 + 1 1 ei 1 X __ n+k + 1 on+k fpi-n-k 1 0k 2 and F = P (e2 2 0; e1; : :):forp = 2: The coalgebra structure of F P*0and the structure of U V 0as a left comodule over it for odd primes are given by 8 P k >> 7! p ; >> i 0ki Pi-k k __ < fi 7! fi 1 + pk (5.3) P 0k> vi 7! p vk >> 0ki i-k __k : and u P p i 7! 1 ui+ 0k 0 by a similar formula, ! X k (5.5) zn+i = v-1n un+i - vn+k zpn+i-k : 0 0 and 0 j n - 1, "xi;jcor- responds (roughly speaking; see (5.9) below for the precise definition) j pn to thenelement -fpi (which is primitive in the quotient F=(fi ) P*0=(pn+i)), and "yi;jis its transpotent. In particular this Ext group is vn-torsion free. As in (2.20), localizing inverts vn, and we get the stated value of E2 for the Cartan-Eilenberg spectral sequence for (5.2). To show that the spectral sequence collapses from E2, it suffices to show that the elements uk and vn+k for 0 k n are permanent cycles. This follows from the fact that ek and on+k are primitive for these k. Next we need to define the elements "xi;jand "yi;j. The localized double complex associated with (5.2) is CFP0* (v-1nCEQ0* (Z =(p))): The algebra E Q0*is primitively generated, so we can replace its local- ized cobar complex by its localized Ext group U v-1nV 0. The coaction of F P*0on it was described above in (5.3), and the corresponding local Ext group was given in (5.8). In the cobar complex we have by (5.7) and (2.28), n X __pn pk d(zn+i) = -fpi 1 + k zn+i-k; n0v2n+itp ; __v(t)= v-1 (t); P the functional inverse of v(t); i u(t) = i0 uitp ; P n+i ^u(t) = P i>0un+itp ; (5.11) n+i w(t) = P i>0w2n+itp ; n+i z(t) = P i>0zn+itpn ; i (t) = i0 pitp ; __ P __pn i (t) = i0 i tp ; P pn n+i f(t) = i0 fi tp where f0 = 1; __ P __pn n+i and f(t) = i0 fi tp : THE TELESCOPE CONJECTURE 55 In what follows we will drop the variable t and denote functional com- position by the symbol O. In this way (5.3) can be rewritten as (v) = (1 v) O ( 1) __ __ so (__v) = ( 1) O (1 v); __ (^u) = 1 ^u+ (1 v) O ((f - 1) 1); __ and (f - 1) = ( 1) O (1 (f - 1)) + (f - 1) 1 __ __ __ so O f = 1 + - f since in any connected Hopf algebra, (x) = x0x00implies that x0x00= 0. Similarly (2.15) and (5.5) can be rewritten as __ (^u) = 1 ^u+ (1 v) O ((f - 1) 1); ^u = v O z so z = __vO ^u; and ^v = v O w so w = __vO ^v: Thus we have (z) = (__v) O (^u) __ __ __ = ( 1) O (1 v) O (1 ^u+ (1 v) O ((f - 1) 1)) __ __ __ __ = ( 1) O (1 vO ^u) + ( 1) O ((f - 1) 1) __ __ __ = ( 1) O (1 z) + O (f - 1) 1 __ = ( 1) O (1 z) + (1 - f) 1; __ which is a reformulation of (5.7). |__| 5.2. Short differentials. It follows from 5.1(iii) that a differential on "xi;jforces a similar one on hn+i;j, and if "yi;jis a permanent cycle so is bn+i;j. As in the computations of x4.3 and x4.4 we have more control over such differentials because they must respect the Snaith splitting and the Hopf algebra structure. Thus Conjecture 3.14 is a consequence of the following. Conjecture 5.12. For i > 0 and 0 j n - 1, the element "xn+i-j;j survives to E2pj and supports a nontrivial differential j d2pj("xn+i-j;j) = vn"ypi;n-1-j: Each "yi;jfor i > 0 and 0 j n - 2 survives to E1+2pn-1. 56 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK Note that if in addition each "yi;jwere a permanent cycle, then we would have E1 = R(n)* P (u0; : :;:un) E("xi;j; hn+i;j:i + j n) n-1-j pn-1-j (5.13) P ("yi;j; bn+i;j)=("ypi;j ; bn+i;j ): One might think that the above is a consequence of Theorems 4.17 and 4.22, but this is not the case. The differentials of 4.17 suggest but do not actually imply those of 5.12, because the two spectral sequences are based on different filtrations. The method used in the proof of 4.22 to show that the yi;jare permanent cycles does not imply that the "yi;j are. That method can be used to show that there is an element congruent to "yi;jmodulo decomposables, namely Hn+1 ("yi+n+1;j) (the image of "yi+n+1;junder the (n + 1)th iterate of the Hopf map H), which is a permanent cycle. We can use Lemma 5.16 below to identify it as X i+j+k+1-n (5.14) Hn+1 ("yi+n+1;j) = "yi;j- fii+k;j(v-1nun-k )p 0kn for certain coefficients fii+k;jdefined in 5.16. The image of this element under the map of (3.18) is X i+j+k+1-n bn+i;j- fii+k;j(v-1nv2n-k)p = 0 by 5.16 below. 0kn Previously we had thought this image was congruent to bn+i;jmodulo decomposables, which would imply the collapsing of the localized Adams spectral sequence for Y (n), but unfortunately this is not the case. Thus the survival of the element of (5.14) is of no help in determining the structure of Y (n)*. On the other hand, the low dimensional compu- tation showing that "xi;jsurvives does imply the survival of hn+i;jfor i + j n. We have Conjecture 5.15 (Second differentials conjecture). In the localized Adams spectral sequence for Y (n) for n > 1 the elements hn+i;0and hn+i;1sur- vive to E2 and E2p respectively, and there are differentials d2("xn+i;0+ dn+i;0)= vn"yi;n-1 and d2p("xn+i-1;1+ dn+i-1;1)= vn"ypi;n-2 for decomposables dn+i-j;j(not to be confused with, but mapping to the decomposables d2n+i-j;jof Theorem 3.16). The elements "yi;jfor THE TELESCOPE CONJECTURE 57 j < n - 1 survive to E2p+1, so E2p+1 = R(n)* P (u0; : :;:un) E("xi;0; hn+i;0: 1 i n) E("xi;1; hn+i;1: 0 i n - 1) E("xi;j; hn+i;j: i > 0; 2 j n - 1) P ("yi;n-2; bn+i;n-2: i > 0)=("ypi;n-2; bpn+i;n-2) P ("yi;j; bn+i;j: i > 0; 0 j n - 3): We will make use of the Hopf map as before. Consider the following diagram in which both rows are fiber sequences. n 2 3 n n Jpn-1S2 x 3S1+2p _______w S x pt. ________2S1+2pw x 2S1+2p ||||||||| | | | |||||||||||||||| | 2 m | |||||||||||||||| |2 1+2pn m Jpn-1S xH | ||||||||||||| | | S xH | |||||||||||||||| | |u ||||||||| | |u m+n ______ 2 3 ______ n m+n Jpn-1S2 x 3S1+2p w S x pt. w2S1+2p x 2S1+2p This gives us a map from the Thomified Eilenberg-Moore spectral se- n 3 1+2pn+k quence for y(n)*(3S1+2p ) to the one for y(n)*( S ). Now the second triple loop space has the same Snaith summands as the first, so the two spectral sequences are isomorphic. Thus the Hopf map H in- duces an endomorphism of our spectral sequence which is multiplicative and linear over R(n)* E(hn+i;j) P (bn+i;j). Lemma 5.16. The Hopf map described above sends uk+1 7! uk; i+j-n "xi+1;j7! "xi;j- ji;j(v-1nun)p i+j+1-n and "yi+1;j7! "yi;j- fii;j(v-1nun)p : where the coefficients ji;jand fii;jvanish for i n and are defined recursively by X i+j+k-n hn+i;j = ji+k;j(v-1nv2n-k)p 0kn jn+i;jmod (vn+1; : :;:v2n) X i+j+k+1-n and bn+i;j = fii+k;j(v-1nv2n-k)p 0kn fin+i;jmod (vn+1; : :;:v2n): This along with Conjecture 3.16 implies that j (5.17) d2pj(jn+i-j;j) = vnfipi;n-1-j 58 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK for j = 0; 1. Proof. The value on uk+1 is immediate. To evaluate H("xi+1;j) we need first to compute H(zn+i+1) for zn+i+1 as defined in (5.5). Let ( 0 i j fori = 0 ri = -1 P pk vn vn+i - 0 0: We will show by induction on i that ae v-1nun for i = 0 H(zn+i+1) = -1 pi zn+i - (vn un) rifor i > 0: This is immediate for i = 0. For the inductive step with i > 0, write ! X k zn+i+1 = v-1n un+i+1 - vn+k zpn+i+1-k ; 00 so we have v O r = v - vn: It follows that r = __vO (v - vn) = 1 - __vO vn so r O v-1nO (v - ^v)= v-1nO (v - ^v) - __vO (v - ^v) = v-1nO (v - ^v) - 1 + __vO ^v = v-1nO (v - ^v) - 1 + w Now the expression v-1nO (v - ^v) is concentrated in dimensions below that of w2n+1. Thus for each i > 0 we have X i+k w2n+i = ri+k(v-1nv2n-k)p : 0kn Taking the coboundary gives X i+k hn+i;n= ji+k;n(v-1nv2n-k)p : 0kn so X i+j+k-n hn+i;j= ji+k;j(v-1nv2n-k)p : 0kn Taking the transpotent of the above gives the desired formula for __ H("yi+1;j). |__| The proof of 5.1(i) can be modified to give a similar description of the unlocalized E2-term in Snaith degrees less than pn+1. One can make an argument similar to that of 3.5 to show that there are no differentials in that range. Alternately, onencan look at the ordinary Adams spectral sequence for y(n)*(3S1+2p ). Using the skeletal filtration one gets a prespectral sequence converging to the Adams E2-term with n E2 = H*(3S1+2p ) ExtB(n)*(Z =(p); Z=(p)): Again there is no room for differentials in the range of the homology of the pnth Snaith summand. The first differentials occur in Snaith 60 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK degree pn+1. They are induced by the Milnor operation Qn given in (4.8), namely d2pn-1(un+1) = vnx1;n and (5.18) j d2pn-1(xn+1-j;j) = vnyp1;n-1-j for 0 j n - 1; where differentials are indexed by the skeletal filtration. These give the differentials of Conjecture 5.15 for i = 1, and (via the map of (3.18)) those of (3.6). This enables us to proceed by induction on Snaith degree, using the Hopf endomorphism described in Lemma 5.16. Assume induc- tively that the differentials on "xn+i-j-1;jand jn+i-j-1;jare as stated in 3.16 and (5.17). Differentials must commute with the Hopf map, so if "xn+i-j;jsurvives to E2pj we have H(d2pj("xn+i-j;j)) = d2pj(H("xn+i-j;j)) i-1 = d2pj("xn+i-j-1;j- jn+i-j-1;j(v-1nun)p )) j pj -1 pi-1 = vnypi-1;n-1-j- vnfii-1;n-1-j(vn un) j = H(vn"ypi;n-1-j): This means that if "xn+i-j;jsurvives to E2pj then j (5.19) d2pj("xn+i-j;j) = vn"ypi;n-1-j+ ci;j; where the error term ci;jmust be a Hopf algebra primitive of Snaith degree pn+i that is in the kernel of the Hopf endomorphism H. The other possibility is that "xn+i-j;jdoes not survive to E2pj but supports an earlier differential of the form (5.20) dr("xn+i-j;j) = ci;j for2 r < 2pj; where ci;jis as above. Similarly we can assume inductively that "yn+i-j;j-1 survives to E2p+1, so a differential on "yn+i-j;j-1must have the form (5.21) dr("yn+i-j;j-1) = c0i;j for2 r 2p; where c0i;jhas the same properties as ci;j. We will refer to such unwanted differentials as spurious and show they cannot occur by showing that there are no nontrivial elements ci;jand c0i;jas above. We will use the n structure of the triple loop space 3S1+2p . THE TELESCOPE CONJECTURE 61 5.3. Excluding spurious differentials. The fact that the error terms ci;jand c0i;jare primitives of Snaith degree pn+i in the kernel of the Hopf endomorphism means that they must have the form n+i X pn+i-k-2 (5.22) ci;j= fli;jup0 + ffi;j;k"y1;k 0kn-1 n+i X 0 pn+i-k-2 (5.23) and c0i;j= fl0i;jup0 + ffi;j;k"y1;k 0kn-1 where the coefficients fli;j,ffi;j;k, fl0i;jand ff0i;j;kare in the Er-term of t* *he localized Adams spectral sequence for Y (n) as follows. n+i; 3pn+i-2pj+r-2 fli;j2 E1+r-pr n+i-k-2; 2pn+i-1-2pj+r-1 ffi;j;k2 E1+r-2pr n+i; 3pn+i-2pj+r-2 fl0i;j2 E2+r-pr n+i-k-2; 2pn+i-1-2pj+r-1 ff0i;j;k2 E2+r-2pr Note that there is no hope of excluding these coefficients by simple sparseness arguments, because there are only finitely positive values of t for which the group Es;rs|vn|+tvanishes for small r. These filtrations of these coefficients are negative, but the spuri- ous differentials must lift back to the unlocalized Thomified Eilenberg- Moore spectral sequence, and there the coefficients must have nonneg- ative filtration. The element "xi;jor "yi;jneed not be in the image of the unlocalized Er-term, but some vn-multiple of each must be until we get to the stage where it supports a localized differential. The power of vn could increase with r if there is an unlocalized dr with a target in the vn-torsion. Thus in order to exclude spurious differentials, we will proceed as follows. (i)Find the smallest vn-multiple of "xi;jwhich is in the image of the unlocalized E2-term. (ii)Get an upper bound (depending on dimension) of the vn-torsion in the unlocalized E2-term. (iii)Show that the torsion created in E3 by the expected d2s does not exceed this upper bound. (iv) Use the torsion estimate to get information about smallest vn- multiple of "xi;jwhich is in the image of the unlocalized Er-term. This will lead to restrictions on the coefficients in (5.22) and (5.23) which will enable us to exclude spurious differentials. 62 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK We do not know how to control the torsion in E2p+1 created by the d2ps, and this difficulty prevents us from proving Conjectures 3.14 and 5.12 for j > 1. For step (i) above, let ae 0 fori 0 (5.24) e(i; j) = pi+j-pj_ p-1 fori > 1: Then we can combine (5.6) with (5.9) to conclude that for i > 1 __pj pi+j-1 (5.25) ve(i-1;j+1)n"xn+i;j (-1)ive(i-2;j+1)n+1n+1 un+1 mod (vn): i+j-1 This represents (-1)ive(i-2;j+1)n+1hn+1;jupn+1 , which is nontrivial in the appropriate Ext group. Similarly one can show that ve(i-1;j+2)n"yn+i;j has a nontrivial reduction modulo vn. For step (ii) above we have the following torsion estimate. Lemma 5.26. All vn-torsion in the E2 of the Thomified Eilenberg- Moore spectral sequence below dimension 2pn(pn+i + pn+1 - 2) is killed by ve(i;0)n. Proof. Recall from the proof of 5.1 that our E2-term is isomorphic (up to regrading) to Ext FP0* (Z =(p); U V 0): Consider the short exact sequence of comodules over F P*0, 0 ! U V 0! Z ! Z=(U V 0) ! 0; where Z is an in (5.4). We know by (5.8) that the Ext group for Z is torsion free, so the torsion in E2 all comes from the Ext group for the quotient comodule via the connecting homomorphism. It follows that the torsion in E2 is controlled by that in the quotient itself. The first element there not killed by ve(i;0)nis zn+1zn+i, which is in the indicated_ dimension. |__| For (iii), the localized differential d2("xn+i;0) = vn"yi;n-1pulls back to d2(ve(i-1;1)n"xn+i;0)=ve(i;0)n"yi;n-1 ( ve(i;0)n"yi;n-1 for i n + 1 = e(n+1;0) e(i-n-1;n+1) vn (vn yi;n-1)for i n + 2; THE TELESCOPE CONJECTURE 63 so in E3 the element ve(i-n-1;n+1)nyi;n-1 for i n + 2, which is not divisible by vn, has dimension 2pn(pn+1_-_1)(pi-_1)_ - 2; p - 1 and is killed by ve(n+1;0)n. This exponent does not exceed the one given by 5.26, so no higher torsion exists in E3. We now turn to step (iv). In (5.22) and (5.23) we can ignore the terms with k = n - 1 since we know that "y1;n-1is killed by d2. The remaining coefficients with the largest filtrations are ffi;j;n-2and ff0i;j;n-2 with Filt(ffi;j;n-2)= 1 + r - 2pi Filt(ff0i;j;n-2)= 2 + r - 2pi Thus in order to get a spurious value of d2(ve(i-1;1)n"xn+i;0) we would need the quantity 3 + e(i - 1; 1) - 2pi = (3 - 2p)e(i; 0) to be positive, but it never is. Thus d2("xn+i;0) is as claimed. For the differential on "xn+i-1;1, we need to estimate the smallest vn- multiple of it which is in the image of the unlocalized Er-term. If we assume the worst, namely that at each stage there is a differential with a target having the largest order of vn-torsion allowed by 5.26, namely e(i; 0). Then the filtration of this element is at most (5.27) 1 + e(i - 1; 1) + (r - 2)e(i; 0) = (r - 1)e(i; 0): It follows that for r < 2p the filtration of ve(i-1;1)+(r-2)e(i;0)nffi;1;n-2is * *at most 2p - 3. The product of any other coefficient with this power of vn would have negative filtration, so the other coefficients must vanish. Now we make use of the comodule structure of Lemma 5.1(iii). It implies that ffi;1;n-2cannot be divisible by vn+k for k > 0. This means it suffices to consider it modulo (vn+1; : :;:v2n). Its image there is a linear combination of elements of the form venx with e 0 and x in the subring generated by the surviving hn+i;jand bn+i;j. The filtration of x must be a nonnegative multiple of 2p - 2, so it must be 0. This means our spurious differential has the form i dr("xn+i-1;1) = ven"yp1;n-2: The exponent e is positive, which contradicts our assumption that r < 2p. 64 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK However we cannot exclude the case r = 2p, so in the localized Thomified Eilenberg-Moore spectral sequence for some i 3 we could have i d2p("xn+i-1;1)= vn"ypn+i-2;0+ ffn-2yp1;n-2 i-p p so d2p("xn+i-1;1- v-1nffn-2yp1;n-2"xn;1)=vn"yn+i-2;0: We still need to show that the elements "yn+i-j;j-1for 0 < j < n and "xn+i-j;jfor 1 < j < n survive to E2p+1. 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