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 AdamsNovikov spectral sequence 11
2.3. The localized Adams spectral sequence 14
2.4. The Thomified EilenbergMoore 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 Ktheory 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 E2term of the localized Thomified EilenbergMoore
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 EilenbergMoore spectral sequence
(which we call the Thomified EilenbergMoore 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 AdamsNovikov (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
EilenbergMoore spectral sequence and its localized form. In certain
cases (2.26 and 2.27) we identify its E2term 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 CartanEilenberg 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 AdamsNovikov 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 rean
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 Ktheory using his formula
tion [Tam94 ] of the EilenbergMoore 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 ThomifiednEilenbergMoore 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 Hspace, 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 EilenbergMac 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 f1 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*(f1 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*(f1 X) = ss*(X) Z [f]Z[f; f1 ]:
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 p1X 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 Ktheory and all
iterates of it are nontrivial. ss*(ff1V (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; ff1]. The
image of ss*(V (0)) in ss*(ff1V (0)) is known, and this gives us a
lot of information about the former.
By analogy with the previous example, one might expect ss*(ff1V (0))
to be K*(V (0)), but the situation here is not so simple. The an
swer is however predictable by Ktheoretic or BPtheoretic 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 CWcomplex 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
22 fi
2p V (1) ! V (1):
In this case the homotopy of the telescope is not known.
The results of DevinatzHopkinsSmith ([DHS88 ] and [HS98 ]) allow
us to study telescopes in a very systematic way. They indicate that
BPtheory and Morava Ktheory are very useful here. First we have
the nilpotence theorem characterizing nilpotent maps.
Theorem 1.1 (Nilpotence theorem). For a finite plocal 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 plocal 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 vnmap 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
BPhomology functor. For m < n, the trivial map is a vm map. The
cofiber of a vnmap 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 vnmap. 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 plocal finite complex X, any vnmap
f : dX ! X yields the same telescope f1 X, which we will denote by
v1nX 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 = v1nBP 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 plocal 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 plocal 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 plocal finite complex of type n (1.2) and let Xb be the
associated telescope (1.5). Then
(i) bX = LnX.
(ii) = .
(iii)The AdamsNovikov spectral sequence for Xb converges to ss*(Xb).
(iv) The functors Ln and Lfnare the same if Ln1 = Lfn1.
We will sketch the proof that the four statements above are equiva
lent.
The set of K(n  1)*acyclic finite plocal 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 E2term 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 AdamsNovikov 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 MahowaldSadofsky [MS95 ]) so (i) says the two functors
agree on such X. For finite plocal X of smaller type, the methods of
[Rav93b , x2] show that Cfn1(the fiber of X ! Lfn1X) 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 AdamsNovikov 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 plocal 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= ssts(Ks) and dr : Es;tr! Es+r;t+r1r:
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 Ebased Adams resolution for
X is the diagram (2.1) with Ks = E ^ Xs. More generally an Ebased
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 E2term 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 EilenbergMac Lane
spectrum, and the AdamsNovikov spectral sequence is the case where
E = BP , the BrownPeterson 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) = pij j where 0 = 1:
0ji
X j
and (oi) = oi 1 + pij 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; : :;:on1) for p > 2:
Let
ae
A*=(1; : :;:n) forp = 2
(2.2) B(n)* = A
*=(1; : :;:n; o0; : :;:on1)forp > 2
Then we have H*(y(n)) = A*2B(n)*Z=(p), and we can use the change
ofrings 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 CartanEilenberg 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 AdamsNovikov 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 tpij mod I where t0 = 1
Xj j
and jR (vi) vjtpij 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 AdamsNovikov E2term
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) =
v1nBP*(X), and BP*(X) is always annihilated by some power of the
ideal
In = (p; v1; : :;:vn1) 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) =
v1nBP*(X). The results of [Rav87 ] and the smash product theorem
[Rav92a , 7.5.6] imply that the AdamsNovikov 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 v1nBP*(X) is a comodule over v1nBP*(BP )=In, which
turns out to be much more manageable than BP*(BP ) itself. There
is a changeofrings isomorphism (originally conceived by Morava and
first proved in [MR77 ]) that enables us to replace v1nBP*(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 Ktheory) 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 changeofrings 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*; v1nM) = 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*; v1nBP*(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+i1)
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 vnmap
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 __________wdXs+s0 ______w2dXs+2s0 __________.w. .
Let Xs = X for s < 0. Thus we get the following diagram, similar
to that of (2.1).
. . .u_____Xb1_u______Xb0 u______Xb1_u_______. . .
 
  
(2.12) g1 g0 g1
  
u u u
Kb1 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= ssts(Kbs) and dr : Es;tr! Es+r;t+r1r
as before. This spectral sequence converges to the homotopy of the ho
motopy direct limit ss*(lim! bXs) 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! Xbs 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 limdiXs+is0
!s !s !i
= lim limdiXs+is0
!i !s
= lim diX
!i
= Xb:
Next we will prove the assertion about the vanishing line. Let
Es;tr(X) denote the Erterm 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 ;td 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 = limkdXi+ks
! k 0
so
ssm (X^i) = limssm+kd (Xi+ks0):
! k
Now the vanishing line implies that the map g : Xs ! Xsr+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 Erterm if and only if its image
in ss*(Xsr+1) is trivial.
It follows that for each k > 0 we have a diagram
g
Xs ______________wXsr+1
 
 
 
u g u
dkXs+s0k ______dkXs+s0kr+1w
THE TELESCOPE CONJECTURE 17
in which both maps g vanish on ssm for m < (sd + c)=s0. Hence the
map
Xbs ^g!Xbsr+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
MittagLeffler, 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 AdamsNovikov 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 AdamsNovikov 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 AdamsNovikov
spectral sequence, which is merely the standard AdamsNovikov
spectral sequence applied to Xb.
o If we start with the classical Adams spectral sequence, an un
published theorem of HopkinsSmith 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 vnmap with
d = vn and s0 = 1.
o In favorable cases (such as Toda's examples and y(n)) the E2term
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 CartanEilenberg 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+ik vn+k
0k 0 recursively by
!
X k
(2.15) w2n+i = v1n v2n+i  vn+k wp2n+ik ;
0> E(hn+i;j: i > 0; 0 j < n)
>>
< P (bn+i;j: i > 0; 0 j < n)
(2.20) = v1nP (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
CartanEilenberg spectral sequence collapses. There are no multiplica
tive extensions since E1 has no zero divisors. Hence the above is a
description of the E2term of the localized Adams spectral sequence
for Y (n).
2.4. The Thomified EilenbergMoore spectral sequence. We
will use a Thomified form of the EilenbergMoore 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 plocal 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 EilenbergMoore spectral sequence.
To identify the E2term 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
MasseyPeterson [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*.
MasseyPeterson 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 Rmodule 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 E2term of the Thomified EilenbergMoore spectral sequence is
ExtR* (Z =(p); H*(K)) when B is an Hspace. When B is an Hspace we
have a Hopf algebra extension (2.31)
A* ! R* ! H*(B):
22 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK
This gives us a CartanEilenberg 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 E2term of the classical EilenbergMoore 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 EilenbergMoore bigrading. This
is the usual Adams E2term for Y when H*(Y ) is concentrated in
EilenbergMoore degree 0, but not in general.
Theorem 2.26. (i)Suppose that H*(K) is a free Amodule and B
is simply connected. Then the Thomified EilenbergMoore 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 MasseyPeterson coalgebra of (2.23).
(ii)If in addition the map i : X ! E induces a monomorphism in mod
p homology, then the Thomified EilenbergMoore 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 padically complete, a plocal 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 plocal spherical
fibration of Lemma 3.3 below. Then the Thomified total space is K ^
H=p, so its cohomology is a free Amodule. Thus the E2term 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;tr+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 changeofrings isomorphism due originally to MilnorMoore
[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 E2term of the spectral sequence associated with f is
(2.32) Ei;j2= Ext i(k; Extj(k; M)):
This is the CartanEilenberg 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 ! tk ! 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 = cokerds1. Then for each s 0 we have a diagram
0 __________wJs ________Lw Js ______wtJs _________w0
 
  
  
: 
  
  
  u u
0 __________wJs __________wIs _________Js+1w _________w0
Using this we get a diagram
M _________tJ1w ______w2tJ2 ________.w. .

  
  
  
  
  
  
u u u
I0 _________tI1w ______w2tI2 ________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 v1 C (M) (the localized cobar complex )
and its cohomology by v1 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 v1 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 v1 Ext (k; M).
The ith column of the double complex is v1 Ci(C ( M)), and we
get
Ei;j2= v1 Exti(k; Extj (k; M)):
When f is onto, the inner Ext group collapses as before and we get a
localized changeofrings isomorphism
(2.33) v1 Ext (k; M) = v1 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 CartanEilenberg spectral sequence
(2.34) v1 Ext (k; Ext (k; M)) =) v1 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 (v1 C ( M)). Then again we have acyclic rows and
taking their cohomology gives v1 C (M) in the 0th column. Thus our
spectral sequence
converges again to v1 Ext (k; M) with
Ei;j2= Ext i(k; v1 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 CartanEilenberg
spectral sequence
(2.35) Ext (k; v1 Ext (k; M)) =) v1 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 CWcomplex with a single cell in every
even dimension. Let Jm S2 (the mth James product of S2) denote its
2mskeleton. James [Jam55 ] showed that there is a splitting
_
S3 ' S2i+1:
i>0
These lead to the JamesHopf 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 plocal fiber of this map is a skeleton, i.e., there
is a plocal fiber sequence
n+1
(3.1) Jpn1S2 ! S3 ! S2p :
Definition 3.2. y(n) is the Thom spectrum of the plocal spherical fi
bration over Jpn1S2 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 plocal 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 plocal spherical fibration
over 2S3 whose Thom spectrum is the mod p EilenbergMac 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.n1) 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 AtiyahHirzebruch 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 AtiyahHirzebruch spectral sequence_in_
that range. __
The classical Adams E2term 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  2pn1  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+i1
vn+i 2 E1;2p2
n+i+j2pj
hn+i;j 2 E1;2p2
n+i+j+12pj+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 E2term) 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 cooperations 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+i1;j+1
j+1
and P p (bn+i;j) = bn+i1;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 ifx = 1;
hn+1;0 ifx = p and
: b j+2
n+1;j ifx = 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+k2:
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+1j;j) = vnbpn+1;n1j 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  2pnx;
which satisfies (xy) = (x) + (y). Then we have
(vn+i) = 2;
(hn+i;j) = 1  2pj
and (bn+i;j1) = 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+i1;0) exceeds it
except when x is one of the three generators with a higher value of ,
namely vn+i, hn+i;0and hn+i1;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+i1;0. __
The first positive dimensional element in ss*(y(n)) is vn 2 ss2pn2(y(n)).
We can use the multiplication on y(n) to extend vn to a selfmap. The
telescope Y (n) is the homotopy colimit of
(3.7) y(n) vn!vny(n) vn!2vny(n) vn!. . .
Theorem 3.8. The telescope Y (n) defined above is Lfny(n).
Proof. We will adapt the methods used by HopkinsSmith [HS98 ] to
prove the periodicity theorem, as explained in [Rav92a , Chapter 6]. Let
X by a finite complex of type n with a vnmap 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)) = v1nBP*(y(n)) = v1nBP*=In[t1; : :;:tn]:
and we know that the AdamsNovikov spectral sequence converges to
ss*(Lny(n)). Its E2term 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 AdamsNovikov 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 E2term 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+ij;j
survives to E2pj and supports a nontrivial differential
j
d2pj(h2n+ij;j) = vnbpn+i;n1j:
Each bn+i;jfor i > 0 and 0 j n  2 survives to E1+2pn1.
Note that if in addition each bn+i;jwere a permanent cycle, then we
would have
n1j
(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;n1
and d2p(h2n+i1;1+ d2n+i1;1) = vnbpn+i;n2
for decomposables d2n+ij;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;n2: i > 0)=(bpn+i;n2)
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 ThomifiednEilenbergMoore 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 E2term 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 E2term of (2.20)
is given by
X n+k
(vn+i) = pik vn+k ;
0ki
X __pj
(hn+i;j) = k hn+ik;j+k
0kn1j
X __pj+1
and (bn+i;j) = k bn+ik;j+k
0kn1j
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
Jpn1S2 x 3S1+2p ______w2S3 ______w2S1+2p x 2S1+2p
 
   
  
  
u   u
i
Jpn1S2 ____________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 EilenbergMoore 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 E2term is described by the following special
ization of 2.26.
Theorem 3.19. Consider the Thomified EilenbergMoore 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 Hmap 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 E2term of the Thomified EilenbergMoore 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 CartanEilenberg spectral sequence converging to the E2
of the Thomified EilenbergMoore 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 EilenbergMoore 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 EilenbergMoore 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 E2term 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
Jpn1S2 ! 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 padic 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
ivnTi ______wivnTi+1 ________w. . .________ivnT1w
  
f f bf
  
u v u v u
y(n) _______vny(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 plocal 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
2S2p1 ! 30S3 ! 3S2p+1:
THE TELESCOPE CONJECTURE 37
We get our bundle from one over 3S2p+1 obtained by extending the
map S2p2 ! BU corresponding to a generator of ss2p2(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(p1)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 EilenbergMoorenspectral 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) = ui1 if k = 1
: x2
8 ik+1;k2 otherwise
2 if k = 0 and j > 0
k < xi;j1
Sq 2*(xi;j)= xi1;j+1 if k = j + 1
: 0 otherwise.
For p odd we have
fi*(ui) = xi;0
ae p
k ui1 if k = 0
P*p(ui) = 0 otherwise
fi*(xi;j)= yi;j1 forj > 0
ae
k xi1;j+1 if k = j
P*p(xi;j) = 0 otherwise
fi*(yi;j)= 0
8 p
k < yi;j1 if k = 0 and j > 0
P*p(yi;j) = yi1;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
xik;k ifk < i
Qk(ui) = 0 otherwise
8 k
>< ypi;jk1 for0 k < j
(4.8) Qk(xi;j) = 0 fork = j
>: pj
yi+jk;kj1 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 DyerLashof operations with upper indices Qs :
Hq ! Hq+2(p1)s. Within this proof Qs will denote a reindexed Dyer
Lashof operation rather than the Milnor operation. We define Qs :
Hq ! Hpq+(p1)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 H2pd2 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 DyerLashof operations is given
by the Nishida relations. For operations on a qdimensional class, these
are
X (p  1) s+q_ r
(4.9) P*rQs = (1)r+i 2 Qs2r+2piP*i
i r  pi
and
(4.10)
X (p  1) s+q_ r  1
P*rfiQs = (1)r+i 2 fiQs2r+2piP*i
i r  pi
X (p  1) s+q_ r  1
+ (1)r+i 2 Qs+12r+2piP*ifi:
i r  pi  1
In particular we have
q
P*1Q2 = __Q0;
2
so P*1(ui) = P*1Q2(ui1)
= Q0(ui1)
= upi1:
40 MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK
For k > 0 (4.9) gives
k pk1
P*pQ2 = Q2P* ;
k pkk
so P*p(ui) = P* Q2(uik)
= Qk2P*1(uik)
= Qk2(upik1)
= 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 = pk1
Q1P* k > 0;
and for s = 1 or 2
ae
k Qs1fi k = 0
P*pfiQs = pk1
fiQsP* k > 0:
It follows that
P*1(xi;0)= P*1fiQi2(u0)
= Q1fiQi12(u0)
= xi1;1;
and for k > 0
k pk i
P*p(xi;0) = P* fiQ2(u0)
ae ik
fiQk2P*1Q2 (u0) k < i
= i pki
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 pkj
Q1P* (xi;0) k j
8
< 0 k < j
= Qj1(xi1;1) k = j
: 0 k > j
ae
xi1;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;j1)
= ypi;j1;
and for all j when k > 0
k pk j+1
P*p(yi;j) = P* fiQ1 (xi;0)
k1j
= fiQ1P*p Q1(xi;0)
k1
= fiQ1P*p (xi;j)
ae
0 k  1 6= j
= fiQ
1(xi1;j+1) k  1 = j
ae
0 k 6= j + 1
= y
i1;j+1 k = j + 1
__
as claimed. __
4.3. Morava Ktheory. 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 AtiyahHirzebruch spectral sequence
or the EilenbergMoore spectral sequence starting with K(n)*(S2pd+1).
(Since S2pd+1 splits after a single suspension, it is easy to work out
its Morava Ktheory.) 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 DyerLashof 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
{xmj;j: 1 j n  1} [
`
(4.18) {ypm`k1;k:0 k n  2; 0 ` + k m  2} [
mk
{upk :0 k n};
for 1 m n we have to add the element xm;0.
The dimensions of these elements are
8
>< xmj;j = 2dpm  2pj  1;
` m `+k+1 `
(4.19) ypm`k1;k= 2dp  2p  2p
>: pmk
and uk  = 2dpm  2pmk
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+1j;j:1 j n  1} [
k
{ypnjk;j:0 j n  2; 0 j + k n  1} [
n+1k
{upk : 0 k n}:
THE TELESCOPE CONJECTURE 45
The dimensions of these are compatible with the desired differentials
j
(4.20) d2pj1(xn+1j;j) = vnyp1;n1j 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) d2pj1(xi+nj;j) = vnypi;nj1 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+nj;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
E2pn1.
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+ij;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+ik
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+ij;j.
It follows that
E1 = E2pn1
= K(n)*[u0; . .u.n] E(xi;j: i + j n)
nj1
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
n1j
ypi;j = 0. If it is nonzero, it must be a K(n)*linear combination of
n+ik
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 EilenbergMoore
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 AtiyahHirzebruch 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 AtiyahHirzebruch 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 E2term of the Tamaki spectral sequence
can be identified as a Cotor group as in the Morava Ktheory case.
In order to use the Tamaki spectral sequence we need to know the
coalgebra structure. As in the computation for Morava Ktheory, 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 E1term 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 EilenbergMoore spectral sequence maps to the one for Morava
Ktheory where we have the differentials of (4.21), namely
j
d2pj1(xi+nj;j) = vnypi;nj1 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 EilenbergMoore 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+1j;j: 0 j n  1} [
k
{ypnjk;j: 0 j n  1; 0 j + k n  1} [
n+1k
{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+1j;jsupports a nontrivial differential, its target
must be in the kernel of H since H(xn+1j;j) = xnj;j is a perma
j
nent cycle. Thus the target must be a multiple of either yp1;njfor
n+1
0 j n  1 or up0 . Hence the expected differentials
j
d2pj1(xn+1j;j) = vnyp1;n1j
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+1k
(4.24) dpi+j+1k2(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+1kwdr
u k u
 
m  
(4.25) H  
 

yi+m;j ______wff0upi+j+m+1k0dr0
k0
where
r + 2 = pi+j+1k
0
and r0+ 2 = pi+j+m+1k
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 pn1j
of Y (n)*(3S1+2p ). Indeed yi;j could be a nontrivial Y (n)*linear
n+ik
combination of the elements upk for 0 k n, e.g.,
n+i
v1nbi+j+1;n2jup0
when i + j n. Here bi;jdenotes the image of yin;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 2local stable splitting
_
2S9 ' 7iBi
i>0
where the stable summand Bi is known now to be the BrownGitler
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
BrownGitler spectra one can produce maps
j
S0 ! RP21! lim2 B2j
where the first map is SpanierWhitehead 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 Amodule, and that both are unstable injectives.
Using results of GoerssLannes [GL87 ] or LannesSchwartz [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+12(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 Ktheory.
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 EilenbergMac 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 E2term of the localized Thomified EilenbergMoore
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 EilenbergMoore spectral sequence
for Y (n)*(3S1+2p ) to the localized Adams spectral sequence for Y (n).
From x3.3 we know that the E2term of the former is
v1nExtH*(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 E2term 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) = pik vn+k ;
0ki
X __pj
(hn+i;j) = k hn+ik;j+k;
0kn1j
X __pj+1
(bn+i;j) = k bn+ik;j+k;
0kn1j
X __
(ui) = 1 ui+ ok "xik;k;
0ki
X __pj
("xi;j)= k "xik;j+k
0kn1j
X __pj+1
and ("yi;j)= k "yik;j+k:
0kn1j
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 fpink 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 Pik k __
< fi 7! fi 1 + pk
(5.3) P 0k> vi 7! p vk
>> 0ki ik __k
: and u P p
i 7! 1 ui+ 0k 0 by
a similar formula,
!
X k
(5.5) zn+i = v1n un+i  vn+k zpn+ik :
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
vntorsion free.
As in (2.20), localizing inverts vn, and we get the stated value of
E2 for the CartanEilenberg 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* (v1nCEQ0* (Z =(p))):
The algebra E Q0*is primitively generated, so we can replace its local
ized cobar complex by its localized Ext group U v1nV 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+ik;
n0v2n+itp ;
__v(t)= v1 (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+ij;j
survives to E2pj and supports a nontrivial differential
j
d2pj("xn+ij;j) = vn"ypi;n1j:
Each "yi;jfor i > 0 and 0 j n  2 survives to E1+2pn1.
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)
n1j pn1j
(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+1n
(5.14) Hn+1 ("yi+n+1;j) = "yi;j fii+k;j(v1nunk )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+1n
bn+i;j fii+k;j(v1nv2nk)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;n1
and d2p("xn+i1;1+ dn+i1;1)= vn"ypi;n2
for decomposables dn+ij;j(not to be confused with, but mapping to
the decomposables d2n+ij;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;n2; bn+i;n2: i > 0)=("ypi;n2; bpn+i;n2)
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
Jpn1S2 x 3S1+2p _______w S x pt. ________2S1+2pw x 2S1+2p
  
  
2 m   2 1+2pn m
Jpn1S xH     S xH
  
u   u
m+n ______ 2 3 ______ n m+n
Jpn1S2 x 3S1+2p w S x pt. w2S1+2p x 2S1+2p
This gives us a map from the Thomified EilenbergMoore 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+jn
"xi+1;j7! "xi;j ji;j(v1nun)p
i+j+1n
and "yi+1;j7! "yi;j fii;j(v1nun)p :
where the coefficients ji;jand fii;jvanish for i n and are defined
recursively by
X i+j+kn
hn+i;j = ji+k;j(v1nv2nk)p
0kn
jn+i;jmod (vn+1; : :;:v2n)
X i+j+k+1n
and bn+i;j = fii+k;j(v1nv2nk)p
0kn
fin+i;jmod (vn+1; : :;:v2n):
This along with Conjecture 3.16 implies that
j
(5.17) d2pj(jn+ij;j) = vnfipi;n1j
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
v1nun 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 = v1n un+i+1  vn+k zpn+i+1k ;
00
so we have
v O r = v  vn:
It follows that
r = __vO (v  vn)
= 1  __vO vn
so r O v1nO (v  ^v)= v1nO (v  ^v)  __vO (v  ^v)
= v1nO (v  ^v)  1 + __vO ^v
= v1nO (v  ^v)  1 + w
Now the expression v1nO (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(v1nv2nk)p :
0kn
Taking the coboundary gives
X i+k
hn+i;n= ji+k;n(v1nv2nk)p :
0kn
so
X i+j+kn
hn+i;j= ji+k;j(v1nv2nk)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 E2term 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 E2term 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
d2pn1(un+1) = vnx1;n and
(5.18) j
d2pn1(xn+1j;j) = vnyp1;n1j 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+ij1;jand jn+ij1;jare as stated in
3.16 and (5.17). Differentials must commute with the Hopf map, so if
"xn+ij;jsurvives to E2pj we have
H(d2pj("xn+ij;j)) = d2pj(H("xn+ij;j))
i1
= d2pj("xn+ij1;j jn+ij1;j(v1nun)p ))
j pj 1 pi1
= vnypi1;n1j vnfii1;n1j(vn un)
j
= H(vn"ypi;n1j):
This means that if "xn+ij;jsurvives to E2pj then
j
(5.19) d2pj("xn+ij;j) = vn"ypi;n1j+ 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+ij;jdoes not survive to E2pj but
supports an earlier differential of the form
(5.20) dr("xn+ij;j) = ci;j for2 r < 2pj;
where ci;jis as above. Similarly we can assume inductively that "yn+ij;j1
survives to E2p+1, so a differential on "yn+ij;j1must have the form
(5.21) dr("yn+ij;j1) = 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+ik2
(5.22) ci;j= fli;jup0 + ffi;j;k"y1;k
0kn1
n+i X 0 pn+ik2
(5.23) and c0i;j= fl0i;jup0 + ffi;j;k"y1;k
0kn1
where the coefficients fli;j,ffi;j;k, fl0i;jand ff0i;j;kare in the Erterm of t*
*he
localized Adams spectral sequence for Y (n) as follows.
n+i; 3pn+i2pj+r2
fli;j2 E1+rpr
n+ik2; 2pn+i12pj+r1
ffi;j;k2 E1+r2pr
n+i; 3pn+i2pj+r2
fl0i;j2 E2+rpr
n+ik2; 2pn+i12pj+r1
ff0i;j;k2 E2+r2pr
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;rsvn+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 Erterm, but some vnmultiple 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
vntorsion.
Thus in order to exclude spurious differentials, we will proceed as
follows.
(i)Find the smallest vnmultiple of "xi;jwhich is in the image of the
unlocalized E2term.
(ii)Get an upper bound (depending on dimension) of the vntorsion
in the unlocalized E2term.
(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 Erterm.
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+jpj_
p1 fori > 1:
Then we can combine (5.6) with (5.9) to conclude that for i > 1
__pj pi+j1
(5.25) ve(i1;j+1)n"xn+i;j (1)ive(i2;j+1)n+1n+1 un+1 mod (vn):
i+j1
This represents (1)ive(i2;j+1)n+1hn+1;jupn+1 , which is nontrivial in the
appropriate Ext group. Similarly one can show that ve(i1;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 vntorsion 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 E2term 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;n1pulls back to
d2(ve(i1;1)n"xn+i;0)=ve(i;0)n"yi;n1
(
ve(i;0)n"yi;n1 for i n + 1
= e(n+1;0) e(in1;n+1)
vn (vn yi;n1)for i n + 2;
THE TELESCOPE CONJECTURE 63
so in E3 the element ve(in1;n+1)nyi;n1 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;n1is killed by d2. The
remaining coefficients with the largest filtrations are ffi;j;n2and ff0i;j;n2
with
Filt(ffi;j;n2)= 1 + r  2pi
Filt(ff0i;j;n2)= 2 + r  2pi
Thus in order to get a spurious value of d2(ve(i1;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+i1;1, we need to estimate the smallest vn
multiple of it which is in the image of the unlocalized Erterm. If we
assume the worst, namely that at each stage there is a differential with
a target having the largest order of vntorsion 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(i1;1)+(r2)e(i;0)nffi;1;n2is *
*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;n2cannot 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+i1;1) = ven"yp1;n2:
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 EilenbergMoore spectral sequence for some i 3 we could
have
i
d2p("xn+i1;1)= vn"ypn+i2;0+ ffn2yp1;n2
ip p
so d2p("xn+i1;1 v1nffn2yp1;n2"xn;1)=vn"yn+i2;0:
We still need to show that the elements "yn+ij;j1for 0 < j < n and
"xn+ij;jfor 1 < j < n survive to E2p+1. An argument similar to the
one above shows that each survives to E2p. At that stage we know that
"yp1;n2gets killed, so we can ignore the coefficients ffi;j;n2and ff0i;j;n2,
concentrating instead on ffi;j;n3and ff0i;j;n3. Multiplying them by the
worst power of vn still gives an element with negative filtration, so we
can exclude spurious d2ps on these elements.
This completes the program to prove Conjectures 3.16 and 5.15.
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