A STABLE APPROACH TO AN UNSTABLE HOMOTOPY
SPECTRAL SEQUENCE
MARTIN BENDERSKY AND DONALD M. DAVIS
Abstract.Recently, Bendersky and Thompson introduced a spec
tral sequence which, for many spaces X, converges to the v1
periodic homotopy groups of X. It is proved that the E2term
of this spectral sequence is often given by Ext in the category of
stable padic Adams modules of QK1(X; Z^p)= im( p). We com
pute this spectral sequence when p = 2 and X is the exceptional
Lie group F4, yielding as a new result the 2primary v1periodic ho
motopy groups of F4. Some new general results about convergence
of this spectral sequence are also proved.
1.Statement of results
In [10], a spectral sequence, which we call the BTSS, was constructed for sim*
*ply
connected spaces X; it converges, on a class of spaces which includes finite H*
*spaces
and strongly spherically resolved spaces, to the homotopy groups of the Kcompl*
*etion
of X, denoted cX. This convergence, and other convergence issues, will be discu*
*ssed
in Section 5.
We will work with the v1periodic version of this spectral sequence, localize*
*d at
any prime p, although the main thrust of this paper is the case p = 2. Our main
result, Theorem 1.1, shows that the E2term of the BTSS, denoted E2(X), can, for
many spaces X, be computed directly from the indecomposables QK1(X; Z^p) and
the Adams operations k. This should be contrasted with the method used in [10]*
* to
compute E2, which involved delicate manipulations with the unstable cobar compl*
*ex.
Let A denote the abelian category of stable padic Adams modules.([12, 2.6]) *
*An
object in A is a pprofinite abelian group with Adams operations k for k 2 Z *
* pZ,
__________
Date: December 26, 2000.
1991 Mathematics Subject Classification. 55T15,55Q52.
Key words and phrases. homotopy groups, Adams operations, exceptional Lie
groups.
1
2 BENDERSKY AND DAVIS
satisfying certain axioms. Our main theorem applies to simplyconnected spaces
X for which there is a torsionfree K*Ksubcomodule M P Kodd(X; Z) such that
K*(X; Z) (M) as (Zgraded) K*(K)coalgebras, while the Z=2graded K*(X; Z^p)
is isomorphic to b(M1 Zp1)# with p is monic on QK1(X; Z^p). Here ()# denotes
Pontrjagin duality, P () denotes the primitives in a coalgebra, and an exteri*
*or
algebra. We prove in Proposition 5.5 that simplyconnected mod p finite Hspaces
whose rational homology is associative and strongly spherically resolved spaces*
* (see
5.3) satisfy these conditions.
Theorem 1.1. If X is a space satisfying the above conditions, then the E2term*
* of
the BTSS satisfies
8
>>Z=2e d = 3
>>> e
>>>Z=2 Z=2 d = 2
><
Z=2 Z=2 Z=2 d = 1; 0
v11ss8i+d(F4; 2) > 6
>>>Z=2 Z=2 d = 1
>>> 6
>>>Z=2 d = 2
:0 d = 3; 4:
Here, and throughout, () denotes the exponent of 2 in an integer. The 2()
occurring in the answer is a surprise, compared to previous computations for ot*
*her
Lie groups.
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 3
Theorem 1.2 leaves SO(n), E6, E7, and E8 as the only compact simple Lie groups
whose 2primary v1periodic homotopy groups have not been computed. In [19], the
second author completed the computation of all oddprimary v1periodic homotopy
groups of all compact simple Lie groups. The authors expect to use Theorem 1.1 *
*in
a future paper to compute v11ss*(SO(n); 2).
One point requiring care here is the distinction between v11ss*X and v11ss**
*cX. We
say that a space X satisfies the Completion Telescope Property (CTP) if X ! cX
induces an isomorphism in v11ss*(; 2). It follows easily from [10, 4.12] and *
*[11, 1.5]
that S2n+1and S2n+1satisfy the CTP. In Section 5, we prove
Theorem 1.3. The spaces S2n, S2n, G2, and F4 satisfy the CTP.
The authors wish to thank Johns Hopkins University JAMI program, where both
authors spent Spring Semester 2000 and much of this work was performed. They
would also like to thank Pete Bousfield for many helpful comments and allowing
them to use his notyetpublished work.
2. Proof of Theorem 1.1
We shall adopt the following notation and conventions. K*() will denote Z=2
graded Kcohomology with Z^pcoefficients, while K*() denotes Zgraded Khomolo*
*gy
with Z coefficients. This difference in gradings is primarily for convenience o*
*f expo
sition: Khomology needs Zgradings for its unstable condition, while our use *
*of
Kcohomology is primarily in K1(). The Bott element v1 2 K2 K2 gives iso
morphisms
Ki(X) v1!Ki+2(X); Ki(X) v1!Ki2(X)
for all integers i, allowing passage between Z=2graded and Zgraded theories. *
*Coac
k *
tions K*X ! K*K K*X and Adams operations in K (X) are passed along
by (v1x) = v1 (x) and k(v1x) = kv1 k(x).
Note that if Ki(X) is torsionfree, then Ki(X) and Ki(X) Zp1 are Pontrjagin
dual to one another. We will denote (M Zp1)# by M# for notational simplicity.
Recall that a profinite abelian group or K*module is a stable padic Adams m*
*odule
if it admits operations k for k 2 Z  pZ satisfying the properties of [12, 2.6*
*]. A
padic Adams module admits operations k for all k 2 Z as in [12, 2.8]. If M is*
* a
4 BENDERSKY AND DAVIS
stable padic Adams module, then the free padic Adams module, eF(M), generated
by M is defined as follows.
Definition 2.1.([12, 3.1]) As abelian groups or K*modules,
eF(M) = M x M x . .;.
with Adams operations defined by
8
<( kx1; kx2; : :):if k 6 0 mod p
k(x1; x2; : :):= :
(0; x1; x2; : :):if k =:p
The following result plays a key role in the proof of Theorem 1.1.
Proposition 2.2. There is an isomorphism of Z=2graded padic Adams modules
QK*(SU) eF();
where is a projective object of A on one generator of grading 1.
Proposition 2.2 is a special case of [12, 3.3]. To see this, we let E = (K ^ *
*S1)<2>,
the 1connected cover of K ^ S1 localized at Z=p. We have 1 E = SU = U<2>.
Theorem 3.3 of [12] asserts that
K*(1 E) bFeK1(E): (2.3)
Since K1(E) by [18, p.24], 2.2 follows from (2.3).
We present the following alternative proof.
Alternate proof of 2.2.We have QK*(SU) = fK*(CP 1) = K*{1; 2; : :}:, the free
K*module with basis k = k  1, with the canonical line bundle over CP 1. Note
that k has grading 1 in K*(SU). The Adams operations act by rk = rk for
r 0. For a 0, define Ma to be the K*submodule of K*{1; 2; : :}:generated by
{pak(k; p) = 1}. Denote M0 by .
We claim that is a projective stable padic Adams module on one generator.
First note that admits Adams operations k for positive k prime to p. Since
K*(CP 1)= im( p), it also admits the operation 1, since 1(im( p)) im( p).
Next observe that has one generator as stable Adams module since k = k(1) for
OE
positive k prime to p. Finally, to show that is projective, let B ! C be a su*
*rjection
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 5
of stable Adams modules and g! C a morphism of stable Adams modules. Define
_g _ _
! B by g(1) = b0 for some b0 satisfying OE(b0) = g(1) and g(k) = kb0 for
positive k prime to p. We must show that _galso respects the action of 1.
For this, note that the padic stable Adams modules and B are inverse limits*
* of
finite Adams modules. By a property of stable padic Adams modules ([12, 2.6]),*
* for
each n, there exists m such that k k+m mod pn in and B for all integers k.
Thus _gcommutes with 1 mod pn for all positive integers n. Passing to the inv*
*erse
limit shows that _grespects the action of 1 .
The map ! Ma defined on generators by k 7! pakis an isomorphism of stable
padic Adams modules. Thus
QK*(SU) = M0x M1x . . . x x . .:.
The action of p on x x x . .i.s given on generators by
p(k1; k2; : :):= (0; k1; k2; : :)::
Here we have used that p(k) in the ith factor corresponds to p(kpi) = kpi+1, *
*which
is k in the (i + 1)st factor. Hence QK*(SU) eF(), yielding the result. 
We denote by M the category of free (Zgraded) K*modules, and by S the homo
topy category of topological spaces. We recall the definition ([10]) of the fun*
*ctor V
from M to itself, and the Vresolution of certain M 2 M.
M ! V (M) ! V (V M) ! V (V 2M) ! . . . (2.4)
To define V (M), we first let KM be the spectrum realizing the homology theory
K*(; M). We then define KM to be 1 KM. Note that ss*(KM) M, * 0.
For a space X with free K*homology, KX is defined to be KK*(X) (equivalently
KX = 1 (K^1 X)). We use K to denote both the functor M ! S and the functor
S ! S. Then V (M) is defined to be the indecomposable quotient Q(K*KM). This V
is the functor of a cotriple on M, which means that there are natural transform*
*ations
ffi : V ! V 2and ffl : V ! I satisfying certain identities. (See [5, 5.2].)
Note that if all basis elements of M have odd dimension, then the same is true
of V (M). This follows since Kr = U if r is odd, and K*(U) is generated by odd
dimensional elements. The 0part of Theorem 1.1 now follows from (2.7), (2.10),*
* and
(2.11).
6 BENDERSKY AND DAVIS
The category V of unstable K*Kcomodules consists of objects M 2 M equipped
with a K*homomorphism jM : M ! V (M) with the usual commutative diagrams ([4,
2.15]). If X is as in 1.1, K*(X) is a Hopf algebra with M = P K*(X) = QK*(X) 2 *
*M.
The unit map h : X ! KX = KK*(X) induces the unstable coaction, K*(h) :
K*(X) ! K*(KX), which in turn induces a morphism
jM : M = P K*(X) ! P K*(KX) ! P K*(KQK*X) = P K*(KM) = V (M)
(2.5)
which gives M the structure of an unstable K*Kcomodule. The map KX !
KQK*X which induces the second homomorphism comes from KX = KK*X and the
natural morphism K*X ! QK*X. The last equality follows because P K*(KM) =
QK*(KM) if M is generated by odddimensional classes.
There are two maps in V from V (M) ! V (V (M)), namely V (jM ) and jV (M). In
general, there are n + 1 coface maps V (V n1(M)) ! V (V n(M)). There is also t*
*he
map V (V (M)) ! V (M), which is not in V. In general, there are n codegeneracy *
*maps
V (V n(M)) ! V (V n1(M)). These maps fit together to generate the V cosimplici*
*al
resolution C (we omit the codegeneracies):
 2 ____
M ___V (M)____V (M)__. . . (2.6)
The coboundary maps in the resolution (2.4) are the alternating sums of the cof*
*ace
maps in (2.6). Note that, whereas each V s(M) satisfies V s(M)t V s(M)t+2for all
t, the coboundary maps in (2.4) do not share this period2 behavior, since they*
* do
not commute with the periodicity operator. That is, the groups can be considere*
*d as
being Z=2graded, but the morphisms cannot.
As usual, ExtV is defined as the derived functors of Hom V:
Exts;tV(K*; M)= Hs(Hom V(K*(St); C))
= H*(Mt! V (M)t! V 2(M)t! . .).; (2.7)
where the coboundary maps in (2.7) are the alternating sums of the coface maps *
*in
 ____
M ____V (M)__. .;. (2.8)
which is obtained by applying the adjointness isomorphism
Hom V(K*(St); V (N)) = Nt (2.9)
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 7
to (2.6).
The connection with E2(X) is given by applying the free commutative algebra
functor F to the resolution (2.6) to obtain a resolution F (C) of K*(X) by inje*
*ctives in
the nonabelian category G introduced in [5, x6]. We are using the fact that F *
*V (M) =
K*KM, which are the injectives in the category G. Applying Hom G(K*(St); ) to
F (C) also gives (2.8). Thus
ExtsG(K*St; K*X) ExtsV(K*St; P K*X) (2.10)
if X is as in 1.1. From [10, 4.3], we have
Es;t2(X) = ExtsG(K*(St); K*X) fort  s > 0:(2.11)
L
If N is a free K*module with basis B and Nev= 0, then V (N) = b2BP K*Kb
(recall P K*Ki = QK*Ki if i is odd). Each Kb is a copy of U = KSb, the 0
space in the spectrum of K ^ 1 Sb. The Pontryagin dual V (N)# is isomorphic
to QK*(KN), which gives it the structure of padic Adams module. We restrict
attention to the Z=2graded module, and note that it is 0 in grading 0. Using 2*
*.2, we
obtain
M
V (N)#1 QK1(S1 x SU(b))
b2B
M i j
K1(S1)bx eF()b
Mb
K1(S1)bx eF( N#1); (2.12)
b
where N#1has the extended stable Adams module structure, i.e. for k prime to
p, k(fl n) = k(fl) n. Here has grading 0, since it is tensored with classe*
*s of
grading 1. We define eV(N)#1to be eF( N#1). We remind the reader that in this
paragraph and throughout the remainder of this section L# means (L Zp1)# for
any abelian group L.
Now we deduce our main theorem.
Proof of 1.1.Let X and M be as in Theorem 1.1 . The Pontrjagin dual of the comp*
*lex
obtained by tensoring (2.4) with Zp1,
0 M# V (M)# V (V M)# . .;. (2.13)
8 BENDERSKY AND DAVIS
is acyclic. The maps in the complex (2.13) are in the category of padic Adams
modules. This follows from dualizing (2.5). In particular, the following diagra*
*m of
exact sequences commutes.
0 M#? (V (M))#? (V (V?M))# . . .
??p ? p ? p
y ?y ?y
0 M#? (V (M))#? (V (V?M))# . . .
?? ? ?
y ?y ?y
0 M# = im(?p) (V (M))#=?im( p) (V (V M))#=?im( p) . . .
?? ? ?
y ?y ?y
0 0 0
Since p is injective, the vertical sequences of the above diagram are short *
*exact.
The induced long exact sequence in homology implies the bottom row is a reso
lution of M# = im( p). Now p is an isomorphism on the factors of K*(S1), and
by (2.12) and Definition 2.1 there is an isomorphism of stable padic Adams mod*
*ules
(Ve(V sM))#= im( p) (V sM)#. So the bottom row is a resolution of M# = im( p)
by Aprojectives.
The boundary ds : V s(M) ! V s+1(M) in the resolution (2.4) satisfies ds =
V (ds1)  jV s(M). We wish to show that the following diagram commutes, where
t is a positive odd integer.
(ds)* t s+1
Hom V(K*St; V s(M)) ! Hom V (K*S ; V (M))
?? ?
?y ??y
Hom K*(K*St; V s1(M)) Hom K*(K*St; V s(M))
?? ?
?y ??y
(2.14)
Hom AbGp(V s1(M)#1; K1St) Hom AbGp(V s(M)#1; K1St)
?? ?
?y ??y
(d#s)* s+1 # p 1 t
Hom A(V s(M)#1= im( p); K1St)!Hom A(V (M)1= im( ); K S )
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 9
The first of the vertical isomorphisms is due to (2.9). The second of the vert*
*ical
isomorphisms is Pontrjagin duality. The third of the vertical isomorphisms is a*
* con
sequence of
V s(M)#= im( p) eV(V s1M)#= im( p) eF((V s1M)#)= im( p) (V s1M)#
with projective in A on one generator.
For the V (ds1) portion of ds, commutativity of (2.14) is true because (ds1*
*)* and
(d#s1)* can be placed as intermediate horizontal arrows, yielding three commut*
*ative
squares. Commutativity of the jV s(M)portion of (2.14) is proved using consider*
*ation
of the unstable cobar complex, which we now describe.
The way in which ExtV() has been computed in papers such as [10], [5], and [*
*4]
is by viewing V sM as the subset of
z_______s_"_________
E*E E*. .E.*E*E E*E*M
satisfying an "unstable condition." Here E is a spectrum such as K or BP , and *
*M is
an unstable E*Ecomodule which is free as an E*module. Under this identificati*
*on,
jV N: V N ! V 2N sends fl n to (fl) n, and in the commutative diagram
jV N* t 2
Hom V(K*St; V N)! Hom V(K*S ; V N)
?? ?
?y ??y
Hom K*(K*St; N) ! Hom K*(K*St; V N)
?? ?
?y ??y
OE
Nt ! (V N)t
the corresponding morphism OE sends n to 1n. Here we are thinking of N as V s1*
*M.
Similarly, with OE as above, there is a commutative diagram
OE# #
N#1  (V N)1
x? x
?? ???
Fe( N#1)= im( p)  eF( (V N)#1)= im( p)
x? x
?? ???
j#V N 2 # p
(V N)#1= im( p)  (V N)1= im( )
10 BENDERSKY AND DAVIS
With Hom A(; K1St) applied to the second, these diagrams imply commutativity of
the jV sMportion of (2.14) and hence of the diagram itself.
Thus the homology of the (Hom V(K*St; V sM); (ds)*)sequence is isomorphic to*
* the
homology of the (Hom A(V sM#1= im( p); K1St); (d#s)*)sequence. These are the t*
*wo
groups which the theorem asserts to be isomorphic. 
The following example might be instructive. Note that X = S2n+1 satisfies the
conditions of 1.1. In this case, M# = Mn, a free K*module on a single generator
with k = kn. The short exact sequence
p p
0 ! Mn ! Mn ! Mn= im( ) ! 0
induces an exact (Bockstein) sequence in ExtA, which relates the unstable E2 for
S2n+1 with the stable E2 for the sphere spectrum. Here ExtA(Mn) is the E2term
of a Kbased spectral sequence, indexed so as to converge to the stable v1peri*
*odic
homotopy groups of S2n+1.
3.Computing ExtA(; )
In this section, we develop a method of computing Exts;tA(M) for a stable pa*
*dic
Adams module M. For simplicity of exposition, we focus mostly on modules in whi*
*ch
1 = 1, which is all we need in this paper. The general case, described in Th*
*eorem
3.8, requires only minor modifications.
If t = 2n + 1, we let Exts;tA(M) = ExtsA(M; St), where St= QK1(St; Z^p) is Z^*
*pwith
k = .kn. In this section, ()# denotes ordinary Pontrjagin duality.
Theorem 3.1. Let M be a finite stable padic Adams module with 1 = 1. a. If
p is odd and r denotes a generator of (Z=p2)x, then
8
>>ker(( r  rn)M) s = 2
>:
0 otherwise.
b. Let p = 2, M2 = ker(2M), and = 3  1. If n is odd, there is an isomorphism
Ext1;2n+1A(M)# coker(( 3  3n)M)
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 11
and a split short exact sequence
0 ! coker(M=2) ! Ext2;2n+1A(M)# ! ker(( 3  3n)M) ! 0:
If n is even, then
Ext1;2n+1A(M)# coker(M=2):
If s + n is odd and s > 2, there is a split short exact sequence
0 ! coker(M=2) ! Exts;2n+1A(M)# ! ker(M2) ! 0:
If s + n is even and s > 1, there is a split short exact sequence
0 ! coker(M2) ! Exts;2n+1A(M)# ! ker(M=2) ! 0:
The case s = 1 was proved in [8]. The oddprimary case is proved in [12, x8].
The proof of Theorem 3.1 when p = 2 will utilize the following elementary res*
*ult.
Let M(ffl)denote a 2local abelian group M with 1 = ffl.
Proposition 3.2. If M is a 2local abelian group with 1 = 1, then
8
>>M2 if s + n even and s 0
n) <
ExtsInv(Z((1)(2); M) >M=2 if s + n odd and s > 0
>:
M if s = 0 and n odd
Proof.Let P denote the object of Inv which is Z(2) Z(2)with 1 switching the
summands. Note that P is projective. Let ffl = (1)n. A projective resolution o*
*f Z(ffl)(2)
is given by
. . .d2!C2 d1!C1 d0!C0 ! Z(ffl)(2)! 0
with each Ci= P and di= 1 + (1)i+1ffl 1. The complex
d*0 d*1 d*2
Hom Inv(C0; M) ! Hom Inv(C1; M) ! Hom Inv(C2; M) ! . . .
is isomorphic to
M 1+ffl!M 1ffl!M 1+ffl!.;. .
and the homology of this is as claimed in the proposition. 
Proof of Theorem 3.1.Let GInv denote the category of 2profinite abelian groups
with involution. Similarly to [12, 8.3], we have
12 BENDERSKY AND DAVIS
Proposition 3.3. If M and N are stable 2adic Adams modules, there is a natural
exact sequence
3M 3N
0 ! Hom A(M; N) ! Hom GInv(M; N) ! Hom GInv(M; N)
3M 3N 1
! Ext1A(M; N) ! Ext1GInv(M; N) ! ExtGInv(M; N)
3M 3N 2
! Ext2A(M; N) ! Ext2GInv(M; N) ! ExtGInv(M; N)
! Ext3A(M; N) ! . .:.
The exact sequence of 3.3 is obtained from a short exact sequence
U 3 3
0 ! U(M) ! U(M) ! M ! 0;
where U : GInv ! A is left adjoint to the forgetful functor. This U is a profin*
*ite
version of the functor of [15, 6.6], and satisfies ExtsA(U(M); N) ExtsGInv(M; *
*N).
Since GInv is dual to the torsion subcategory of Inv, we have
ExtsGInv(M; N) ExtsInv(N# ; M# ): (3.4)
If N = QK1(S2n+1)^, then, since the Pontrjagin dual of Z^2is (Q=Z)(2), we obtain
n) #
Exts;2n+1GInv(M) Exts1Inv(Z((1)(2); M )
by Proposition 3.7 and the Extsequence induced from
0 ! Z(2)! Q(2)! Q=Z(2)! 0: (3.5)
If n is odd, the exact sequence of 3.3 becomes, using Proposition 3.2,
33n # 2;2n+1
0 ! Ext 0;2n+1A(M) ! 0 ! 0 ! Ext1;2n+1A(M) ! M# ! M ! ExtA (M)
31 # 3;2n+1 # 31 #
! M#2 ! M2 ! ExtA (M) ! (M=2) ! (M=2) ! . .;.
which yields the case n odd of Theorem 3.1 after dualization. The case n even *
*is
similar.
For the splitting of the short exact sequences of 3.1, we use an h1action on*
* the Ext
groups, as described in the following proposition, which will be proved at the *
*end of
this section.
Proposition 3.6. There is a Yoneda (composition) product in ExtA and an element
h1 2 Ext1A(QK1(S2n+1)^; QK1(S2n+3)^)
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 13
satisfying
1. 2h1 = 0;
2. Yoneda product with h1 corresponds to the h1action in the BTSS
under the isomorphism of Theorem 1.1;
3. Under the short exact sequences of 3.1 with s + n odd, there is a
commutative diagram
0 ! coker(M=2) ! Exts+1;2n+3A(M)#! ker(M2)  ! 0
?? ?? ??
1?y h#1?y 1?y
0 ! coker(M=2) ! Exts;2n+1A(M)#! ker(M2)  ! 0
and a similar one when s + n is even.
The splitting in Theorem 3.1 follows now, since the Five Lemma, applied to the
dual of the diagram of 3.6(3), implies that .h1 is an isomorphism on Exts;2n+1A*
*(M),
and so, since 2h1 = 0, Exts;2n+1A(M) can have no elements of order 4. 
The following proposition was used earlier in this section.
Proposition 3.7. If M is a finite object of Inv, then ExtsInv(Q(ffl); M) = 0 fo*
*r s 0.
0) *
* n
Proof.The object M must be isomorphic to a sum of (Z=2n)(ffl's plus copies of P*
*=2 =
Z=2n Z=2n with 1 interchanging factors.
By [15, 3.10], ExtsInv(Q(ffl); M) = 0 for s > 1. For s = 0, we have
Hom Inv(Q(ffl); M) Hom AbGp(Q; M) = 0:
Let 0 ! R ! F ! Q ! 0 be a projective resolution in AbGp. Then 0 !
R(ffl)! F (ffl)! Q(ffl)! 0 is a projective resolution in Inv by [15, 3.6]. We *
*first
consider the case ffl0= ffl. Here Hom Inv(F (ffl); (Z=2n)(ffl)) = Hom AbGp(F; Z*
*=2n), and so
Ext1Inv(Q(ffl); (Z=2n)(ffl)) = ExtAbGp(Q; Z=2n). Using the injective resolution*
* in AbGp
n
0 ! Z=2n ! Q=Z 2! Q=Z ! 0;
one readily verifies ExtAbGp(Q; Z=2n) = 0.
With ffl0= ffl, we have Hom Inv(F (ffl); (Z=2n)(ffl)) = Hom AbGp(F; Z=2). A*
*rguing as
above with n = 1, we obtain Ext1Inv(Q(ffl); (Z=2n)(ffl)) = 0. Finally, Ext1Inv*
*(Q(ffl); P=2n) =
0 follows from Hom Inv(F (ffl); P=2n) = 0 by a similar argument. 
14 BENDERSKY AND DAVIS
The generalization of 3.1 and 3.2 to an arbitrary M is given by the following*
* result,
whose proof is a straightforward generalization of methods used above.
Theorem 3.8. a. Let p = 2 and let M be a finite stable padic Adams module. Let
n = 3  3n, and
ker(1  (1)m 1)
Qm = ________________:
im (1 + (1)m 1)
Then
o Ext1;2n+1A(M)# coker(n coker(1  (1)n 1));
o there is a short exact sequence
0 ! coker(nQn) ! Ext2;2n+1A(M)# ! ker(n coker(1(1)n 1)) ! 0;
o for s > 2, there is a short exact sequence
0 ! coker(nQs+n) ! Exts;2n+1A(M)# ! ker(nQs+n1) ! 0:
b. If M is a 2local abelian group with involution 1, then
8
n) 0
ExtsInv(Z((1)(2); M) :
ker(1  (1)n 1) if s = 0.
We complete this section by proving Proposition 3.6.
Proof of Proposition 3.6.We apply Proposition 3.3 and (3.4) to obtain an exact *
*se
quence
n+1) ((1)n) 1 1 2n+1 ^ 1 2n+3 ^
Hom Inv((Q=Z)((1) ; (Q=Z) ) ! ExtA(QK (S ) ; QK (S ) )
n+1) ((1)n)2.odd
! Ext 1Inv((Q=Z)((1) ; (Q=Z) ) ! :
Using (3.5), and then arguing as in the proof of Proposition 3.7, we obtain
n+1) ((1)n) 0 ((1)n+1) ((1)n)
Ext1Inv((Q=Z)((1) ; (Q=Z) ) ExtInv(Z(2) ; (Q=Z) );
and, similarly to Proposition 3.2, this is Z=2, generated by 1_22 Q=Z on the RH*
*S.
The nonzero element is called h1. Since h1 2 Z=2, 2h1 = 0. Part 2 of the propos*
*ition
follows since the isomorphism of Theorem 1.1 respects Yoneda products, and the *
*two
notions of h1must agree since they are the only nonzero element in isomorphic g*
*roups.
For part 3, we first consider the Yoneda product in ExtInv
ExtsInv(Z(ffl)(2); M) Ext1Inv(Z(ffl)(2); Z(ffl)(2)) ! Exts+1Inv(Z(ffl)*
*(2); M):
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 15
With P as in the proof of 3.2, composition with h1 is defined by the diagram
1+ffl 1 1ffl 1
0 Z(ffl)(2)P  P?  P? . . .P?
? ? ?
. ?y1 ?y1 ?y1
1ffl 1
Z(ffl)(2) P  P . . .P
#
M
Since the chain map of resolutions can be chosen to be the identity, the compos*
*i
tion is the identity under the identifications given in Proposition 3.2. Part 3*
* of the
proposition follows since the morphisms of 3.3 and (3.4) are compatible with Yo*
*neda
products. 
4.The BTSS of F4
In this section, we prove Theorem 1.2. There are three steps.
1. Use Theorems 1.1 and 3.1 to compute the E2term of the BTSS
converging to v11ss*(cF4).
2. Use the fibration
G2 ! F4 ! F4=G2 (4.1)
to determine the differentials and extensions in the spectral se
quence.
3. Show that F4 ! cF4induces an isomorphism in v11ss*(). This is
done in Theorem 1.3.
From [19, 3.8] we have
Proposition 4.2. There is a basis {v1; v2; v3; v4} of QK1(F4) on which 1 = 1
and the transposes of the matrices of 2 and 3 are given by
0 1 0 1
2 3 1 0 3 24 15 1
B0 32 8 1 C B0 35 162 81 C
( 2)T = BB@00 128 24 CCAand( 3)T = BB@00 37 37CCA:
0 0 0 2048 0 0 0 311
This can be shown to agree with the Chern character calculation of [25, 4.8].
By Theorems 1.1 and 3.1, E1;4k+32(F4)# is obtained from the following result.
16 BENDERSKY AND DAVIS
Proposition 4.3. If ( 2)T and ( 3)T are as in Proposition 4.2, then the abelian
group presented by the matrix
2 T !
( )
( 3  32k+1)T
is Z=2min(12;6+2(k5)).
Proof.Replace 32k+1by 310(R + 3) in the matrix. Then (R) = (k  5) + 3. Pivot
the matrix on the entries in position (1,3), then (2,4), and then (5,2), which *
*will
have become odd. This leaves five relations on the first generator, which, up t*
*o odd
multiples, are
214+ 26R
218+ 214R
212+ 28R + R2
212+ 26R + R2
27R + 23R2
Then the exponent of 2 in the fourth relation is min(12; 2(R)), and all other r*
*elations
are at least that 2divisible. 
The order of ker(( 332k+1)QK1(F4)= im( 2)), which is a summand of E2;4k+32(*
*F4)#,
equals that of the cokernel, which was determined in the preceding proposition.*
* For
the group structure, we need
Proposition 4.4. The group ker(( 3  32k+1)QK1(F4)= im( 2)) is cyclic.
Proof.Let M = QK1(F4)= im( 2), M2 = ker(.2M), and K = ker(( 3  32k+1)M).
The number of summands in K equals the dimension of M2 \ K. Note that 3 
32k+1= 3  1 on M2.
A basis for M2 is given by 2(v3)=2 and 2(v4)=2. We have
111
( 3  1)( 2(v4)=2) = 2( 3  1)(v4)=2 = 2(3___2v4) 0 2 M;
and
71 37 2
( 3  1)( 2(v3)=2) = 2(3___2v3 __2v4) (v4=2) 2 M:
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 17
Thus M2\ K = < 2(v4)=2> is 1dimensional. 
For the elements of higher filtration in E2(F4), we need
Proposition 4.5. Let M = QK1(F4)= im( 2). Then ( 3  1)(M=2) has kernel
Z=2 with basis {v2 ~ v3} and cokernel Z=2 with basis {v1}, while ( 3  1)M2
has kernel Z=2 with basis { 2(v4)=2} and cokernel Z=2 with basis { 2(v3)=2}.
Proof.We have M=2 Z=2Z=2 with basis {v1; v2 ~ v3}. Mod (2; im( 2)), we have
( 3  1)v1 = v3 and ( 3  1)v2 = 0. In the proof of Proposition 4.4, 3  1 on *
*M2
was analyzed. 
We obtain the following diagram of E2(F4) with e = 6 and f = min(12; 8+2(`3)*
*).
Diagram 4.6.
        
        
        
        
___________________________________________________________*
*
        
   r r  r r  r r 
        
s = 4        
___________________________________________________________*
*
        
  r r  r r   r r   r r 
        
___________________________________________________________*
*
 f     e    
 r 2   r r  r 2  r r 
        
___________________________________________________________*
*
        
r   2f   r   2e   r 
        
s = 1        
___________________________________________________________
8`  2 8` 8` + 2 8` + 4
Here, as usual with Adams spectral sequence types of diagrams, the horizontal
grading is t  s, and classes in E*;*+i1(X) provide an associated graded for ss*
*i(cX).
Each dot represents Z=2, and an integer represents a cyclic summand of that ord*
*er.
The diagonal lines indicate multiplication by h1 in the BTSS (3.6), which corre*
*sponds
to j in homotopy. We call these "jtowers." The action of h1 on the 1line is d*
*elicate;
by omitting it from the diagram, we do not mean to say that this h1action is 0.
18 BENDERSKY AND DAVIS
Because j4 = 0 in ssn+4(Sn), there must be a pattern of d3differentials which
annihilates all jtowers in large filtration. However, careful consideration is*
* required
to determine whether a particular jtower supports a d3differential or is hit *
*by one.
This will affect whether or not a few elements at the bottom of the jtower sur*
*vive
the spectral sequence.
In order to determine the d3differentials in F4, we use the fibration (4.1).*
* The
groups v11ss*(G2; 2) were computed in [21] using homotopy theoretic methods. We
now show how these groups can be seen in the BTSS.
From [19, 3.7], QK1(G2) has a basis {g1; g2} on which ( 2)T and ( 3)T are giv*
*en
by ! !
2 15 3 T 3 120
( 2)T = 0 32 and ( ) = 0 35 :
By methods similar to those employed above for F4, we obtain
Proposition 4.7. Let M0= QK1(G2)= im( 2).
1. E1;4k+32(G2)# Z=2min(6;(k2)+3) ker(( 3  32k+1)M0);
2. M0=2 Z=2, generated by g1, with 3  1 = 0 on M0=2, and
M02= Z=2, generated by 16g2, with 3  1 = 0 on M02.
Thus by Theorems 1.1 and 3.1 E2(G2) has the form of Diagram 4.6, with e =
min(6; (`  1) + 4) and f = 3.
The following result will be proved in Section 5, simultaneously with the pro*
*of that
Theorem 1.3 holds for G2. The proof utilizes the map G2 ! S6 with fiber SU(3), *
*the
analysis of v11ss*(G2) in [21], and the fact that S6 satisfies the CTP.
Theorem 4.8. The differentials and extensions in the BTSS of G2 are as in Diag*
*ram
4.9, with e = min(6; (`  1) + 4) and f = 3.
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 19
Diagram 4.9.
     BB  B B 
       B B 
     Br r  B Br 
    r  BBB BB B BBB 
___________________________________________________________*
*BBBBBBBBBB
    B   BB B B BBB 
    B r tr BB rB B BBB 
       B   
s = 4   t  BBB   B BBB B BBB 
___________________________________________________________*
*BBBBBBBBBB
B  B B B
     B   BB BB  BB 
    t t B B  t B BB Br BBrB
  t   B B   BBrBB  B
___________________________________________________________*
*BBBBBB
     B B  B B B B
   t t  t BB  Br BBr rB
 2f     B2eB  B 
___________________________________________________________*
*BB
     B  B 
  2f  t  B2e  Br 
        
s = 1        
___________________________________________________________
8`  2 8` 8` + 2 8` + 4
Note that this diagram for `  1 would provide additional jtowers which are not
displayed on the left side of Diagram 4.9.
We need also the BTSS and v1periodic homotopy groups of F4=G2. We use [21,
1.1], which states that there is a 2local fibration
S15! F4=G2 ! S23: (4.10)
By [11], F4=G2, being strongly spherically resolved, satisfies the CTP. Since t*
*he at
taching map in F4=G2 is oe, by [1, 7.5,7.17], we have
Proposition 4.11. QK1(F4=G2) has basis {w1; w2} with kw2 = k11w2 and
uk7(k4  1)
kw1 = k7w1+ __________w2;
16
with u odd.
Applying to this the methods applied in 4.3, 4.4, 4.5, and 4.7, we obtain tha*
*t the
BTSSE2 for F4=G2 has the form of Diagram 4.6 with e = 6 and f = min(12; 7 +
(`  19)).
Theorem 4.12. The differentials and extensions in the BTSS of F4=G2 are as in
Diagram 4.9 with e = 6 and f = min(12; 7 + (`  19)).
20 BENDERSKY AND DAVIS
Proof.We need the fact ([7, p.488],[3, p.352]) that the BTSS of S8m1 has the f*
*orm
of Diagram 4.9 with e = 3 and f = min(4m  1; (`  m) + 4). The fibration
(4.10) induces a short exact sequence in QK1(), a long exact sequence in E2 of*
* the
BTSS, and a long exact sequence in v11ss*(). The d3differentials on the jto*
*wers
emanating from (t  s; s) = (8` + 3; 2) in S15and S23force similar d3different*
*ials in
F4=G2, as do the d3differentials on the jtowers emanating from (8` + 4; 1). S*
*ince
E2;8`+32(F4=G2) ! E2;8`+32(S23) maps onto the Z=8, which supports a d3differen*
*tial,
d3 must be nonzero on Z=26 E2;8`+32(F4=G2) and on the jtower arising from it.
For s = 1 and 2, Es;8`+32(S15) ! Es;8`+32(F4=G2) is a monomorphism Z=8 ,! Z=2*
*6,
and so the nontrivial extension (.2) from Es;8`+32(F4=G2) to Es+22(F4=G2) will *
*follow
from that in S15 once we know that the jtower into which Es;8`+32(S15) extends
maps across. When s = 1, this is clear, since the extension is into h21E1;8`+1*
*2, and
E12(S15) ! E12(F4=G2) must be injective.
The case s = 2 requires more care. For s 2, the two classes of Es;8`3+2s2(S*
*15)
can be characterized as "stable" and "unstable." From the point of view of Theo*
*rem
3.1, the one in ker(M=2) is stable, while the one in coker(M2) is unstable. T*
*his
is true because elements of M2, being 2x=2, depend on the dimension of the sph*
*ere,
while elements in M=2 are generators of M and independent of the dimension of t*
*he
sphere. The extension in BTSS(S15) in t  s = 8` + 1 is into the unstable class*
*. This
is true because the large summand in E22(S2n+1) is unstable.
Now consider the commutative diagram of exact sequences
! QK1(F s;8`3+2s1 # 1
4=G2)2? ! ExtA (QK?(F4=G2)) ! QK (F4=G2)=2? !
?? ? ?
y ?y ?y
! QK1(S15) s;8`3+2s1 15 # 1 15
2 ! ExtA (QK (S )) ! QK (S )=2  !
With wias in Proposition 4.11,
71 37(341) 2 w 2
( 3  1)( 2w1=2) = 2(3___2w1+ ______2.16w2) (_2_2) mod im
and
111 2
( 3  1)( 2w2=2) = 2(3___2w2) 0 mod im :
2w1 1 15
Thus coker(QK1(F4=G2)2) = <____2>, which maps nontrivially to coker(QK (S )2*
*).
Dually, the unstable class in Exts;8`3+2sA(QK1(S15)) maps nontrivially to F4=G*
*2, and
hence the extension from E2;8`+31(F4=G2) to E4;8`+51(F4=G2) is nontrivial. 
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 21
The result for v11ss*(F4=G2) that can be read off from Theorem 4.12 differs *
*slightly
from [20, 8.10]. A mistake in [20] was discussed in [9]. The key lemma [20, 8.1*
*6] is
false, and this caused the evaluation of a d6differential in [20, p.1045] to b*
*e incorrect.
Now we can prove the following result, from which Theorem 1.2 follows immedia*
*tely,
once we know 1.3 for F4.
Theorem 4.13. The differentials and extensions in the BTSS of F4 are as in Dia
gram 4.9 with e = 6 and f = min(12; 8 + 2(`  3)).
Proof.The fibration (4.1) induces a short exact sequence in QK1() and a long
exact sequence in E2. The d3differentials on the jtowers emanating from (ts;*
* s) =
(8` + 3; 2) and (8` + 4; 1) are implied by their existence in G2 and F4=G2. The*
* Z=26
summand in E2;8`+32maps isomorphically from F4 to F4=G2, as does the jtower
arising from it. Thus the d3 on this jtower in F4=G2 implies the same in F4, a*
*nd the
nontrivial extension from E2;8`+32(F4=G2) implies the same in F4.
Finally, E1;8`+32(G2) ! E1;8`+32(F4) is injective, as is E1;8`+12(G2) ! E1;8`*
*+12(F4),
and so the extension in G2 from E1;8`+31to h21. E1;8`+12implies the same in F4.*
* 
5. The completion telescope property
In this section we first show that the BTSS converges to v11ss*(cX) for a cl*
*ass of
spaces X which includes spheres and simplyconnected finite Hspaces. Then we p*
*rove
Theorem 1.3, which is the isomorphism v11ss*(X) v11ss*(cX) for certain impor*
*tant
spaces X.
We begin by recalling some of the results of [10]. For a space X, the Kcompl*
*etion
of X, denoted cX, is constructed as Tot of a cosimplicial space constructed fro*
*m the
Ktheory spectrum. There is a natural transformation X ! cX. The Bousfield
Kan spectral sequence associated to the standard filtration of the Ktheory Tot*
* is
the BTSS. We use a slightly weaker version of the spectral sequence obtained by
turning the BousfieldKan tower "upside down" ([10, x2]); this is associated to*
* cX.
We are using here the v1periodic BTSS, although the difference between this and
the unlocalized BTSS is inconsequential for our purposes here, since they agree*
* in
sufficiently large dimensions.
22 BENDERSKY AND DAVIS
Proposition 5.1. ([10, 2.3]) Suppose there is an N and r such that Es;tr= 0 if *
*s > N.
Then the BTSS converges to the v1periodic homotopy groups of cX.
Proof.It is shown in [10, 2.3] that, under this hypothesis, the unlocalized BTSS
converges to ss*(cX) . Because there is a horizontal vanishing line, the v1pe*
*riodic
BTSS converges to v11ss*(cX), since there can be no v1periodic family of clas*
*ses or
differentials of increasingly large filtration. 
One family of spaces for which we can prove there is a horizontal vanishing l*
*ine is
the algebraically spherically resolved spaces, ([11]).
Definition 5.2.A space X is algebraically spherically resolved (ASR) if:
1. K*(X) is a free K*module.
2. There is a K*Ksubcomodule M Kod(X) such that K*(X)
(M) as K*Kcomodules.
3. If (M) is made into a coalgebra by making M primitive, then
the isomorphism is as K*(K)coalgebras.
4. One can choose a basis {m1; m2; : :}:for M so that each sequence
0 ! K*{m1; : :;:mn1} ! K*{m1; : :;:mn} ! K*{mn} ! 0
is a short exact sequence of K*(K)comodules.
A space is ASR if from the point of view of Ktheory it appears as if it is b*
*uilt out
of a finite sequence of fibrations over odd spheres. The geometric analogue is *
*given
in the following definition.
Definition 5.3.A space X is strongly spherically resolved (SSR) if there are
spaces * = X0, X1; : :;:Xk = X and fibrations
Xi1! Xi! Sni (5.4)
with niodd such that the cohomology groups of (5.4) form the split extension
(x1; : :;:xi1) (x1; : :;:xi) (xi)
with xi = ni.
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 23
Proposition 5.5. Suppose X is either SSR or a simplyconnected finite Hspace w*
*ith
H*(X; Q) associative. Then X is ASR and satisfies the hypotheses of Theorem 1.1.
Proof.If X is SSR, then the AtiyahHirzebruch spectral sequence (AHSS) and anti
commutativity of the product imply that K*(X) is an exterior algebra on generat*
*ors
corresponding to the spheres. The collapsing of the AHSS can be seen since rati*
*onally
X is a product of spheres, because of the finiteness of the positive even stems.
Duality between Ki() and Ki() (see [2]) implies (1), (3), and (4) of the ASR
criteria. Criterion (2), that these generators can be chosen to be a K*Ksubcom*
*odule,
follows from the fact that rationally X is a product of spheres. The hypotheses*
* of 1.1
are similar, except for monicity of p which also follows by rationalizing.
If X is a simplyconnected finite Hspace with H*(X; Q) associative, then by *
*[16,
10.3,10.4] K*(X; Z(p)) is Z(p)free and K*(X; Z^p) b(P K1(X; Z^p)). The hypoth*
*eses
of 1.1 and (1)(3) of the ASR definition follow easily from this and the fact t*
*hat ratio
nally X is a product of spheres. The SES of 5.2(4), while perhaps not topologic*
*ally
realizable, follows by duality from the fact that K*(X) is an exterior algebra.*
* 
The following result was proved in [11].
Proposition 5.6. The v1periodic BTSS converges to v11ss*(cX) if X is algebrai*
*cally
spherically resolved.
Proof.Briefly (for p = 2), by the same argument as in [3, 5.4], the conditions *
*guar
antee that E2(X) is generated as an h1module by classes of filtration 2. In a
forthcoming paper of Bousfield ([17]), it will be shown that the Yoneda product*
* in
E2 of the BTSS is associated to composition in homotopy. Since j4 = 0 in homoto*
*py,
E4 must have a horizontal vanishing line, and so Proposition 5.1 applies. 
There are other important examples of spaces which satisfy 5.1. Although S2n+1
is not ASR, it has E2 isomorphic to E2(S2n+1). Similarly, S2n is not ASR, but
satisfies the condition of 5.1 because of 5.14.
We have established that the BTSS converges to v11ss*(cX) for many spaces. H*
*ow
ever our interest is in v11ss*(X). Spaces for which the BTSS actually converge*
*s to
v11ss*(X) are said to satisfy the completion telescope property. Precisely
24 BENDERSKY AND DAVIS
Definition 5.7.A space X satisfies the completion telescope property (CT P )
if the map X ! cXinduces an isomorphism in v11ss*().
The rest of the paper is devoted to proving 1.3, which states that S2n, S2n, *
*G2,
and F4 satisfy the CTP. The following lemma, which is a simple application of t*
*he
Five Lemma, will be useful.
Lemma 5.8. Suppose X ! Y ! Z is a fibration for which cX! bY! bZis also a
fibration. If the CTP is true for any two of the spaces, then the third space s*
*atisfies
the CTP.
We now look for conditions that guarantee that a fibration satisfies 5.8. We *
*observe
that there is an easy solution to this problem at the odd primes. The Khomolog*
*y of
a finite simplyconnected Hspace X is an exterior algebra generated by P Kod(X)
QKod(X). In this case E2(X) ExtV(K*; QK*(X)), by 2.10 and 2.11. Furthermore
at odd primes, by [11], Es;2n+12(X) v11ss2n+1s(cX) for s 2 {1; 2}, and Es2(X*
*) = 0
if s > 2. Thus if the fibration induces a short exact sequence in QK*(), its *
*long
exact sequence is E2 gives a long exact sequence in v1periodic homotopy of the*
* K
completions, and hence the Five Lemma will imply that if two of the spaces sati*
*sfy the
CTP, then so does the third. Because of differentials and extensions, this argu*
*ment
does not work for p = 2.
Associated to a fibration X ! Y ! Z is a Khomology cobar spectral sequence
E2s= CotorK*(Z)s(K*(Y ); K*) =) K*(X): (5.9)
This spectral sequence, which generalizes that of EilenbergMoore ([23]) and is*
* studied
in [17], does not always converge to K*(X). However, Proposition 5.11, which fo*
*llows
easily from [13], states that under favorable conditions it converges in a very*
* strong
way.
Definition 5.10.Suppose X ! Y ! Z is a fibration of connected spaces whose
Khomologies are free over Z(2). We say the Khomology cobar spectral sequence
strongly collapses if it collapses from E2 to the isomorphism
8
<0 s > 0
E2s :
K*X s = 0:
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 25
Proposition 5.11. The Khomology cobar spectral sequence of a fibration which i*
*n
duces an injective extension sequence in Khomology strongly collapses.
Fibrations for which (5.9) strongly collapses are of some importance because *
*of the
following recent theorem of Bousfield, [17].
Theorem 5.12. Suppose the fibration X ! Y ! Z induces a strongly collapsing
Khomology cobar spectral sequence. Then the induced sequence cX! bY! bZis of
the homotopy type of a fibration.
The following result implies that the EHP fibration
S2n1! S2n ! S4n1 (5.13)
induces an injective extension sequence and hence its Khomology cobar spectral
sequence strongly collapses.
Lemma 5.14. K*(S2n) K*(S2n1) K*(S4n1) as coalgebras.
Proof.The result is true with H* replacing K*. The Chern character shows that t*
*he
result is true rationally. The E2term of the AHSS is isomorphic to
K* H*(S2n1) H*(S4n1) K*(x) K* PK*[y];
where K*(x) is the exterior algebra over K* on a class x in degree 2n  1 and P*
*K*[y]
is a polynomial algebra over K* on a class in degree 4n  2. A nonzero differen*
*tial
in the AHSS would violate the rational calculation. This proves the isomorphism
in the lemma as K*modules. We need to show y 2 K4n2(S4n1) is primitive. If
not, the reduced coproduct has the form (y) = x x. But, this would imply that
__x2 K1(S2n), the dual of x, has nontrivial square, which is a contradiction. *
*
Now the CTP for S2n is immediate from 5.12, 5.8, and the fact (noted near the e*
*nd
of Section 1) that S2m1 satisfies the CTP.
Exactly these same ingredients imply the CTP for S2n, using the 2primary fib*
*ration
S2n ! S2n+1! S4n+1:
This fibration induces an injective extension sequence in K*() by the argument*
* used
to deduce the similar statement for BP*() in [6, p.388].
26 BENDERSKY AND DAVIS
The proof of 1.3 for G2 is more computational. Along with it, we prove Theorem
4.8.
Proof of 1.3 for G2 and 4.8.We consider the diagram induced by the map G2 ! S6
(with fiber SU(3))
v11ss*(G2)!v11ss*(S6)
?? ?
?y ??y (5.15)
v11ss*(Gc2)!v11ss*(cS6)
The analysis of v11ss*(G2) in [21], especially the diagram on page 667, can be*
* viewed
as a determination of the exact sequence
! v11ss*SU(3) ! v11ss*G2 ! v11ss*S6 @! v11ss*1SU(3) ! :
(5.16)
That analysis is used implicitly in the following paragraphs.
A chart for v11ss*(SU(3)), obtained from v11ss*(S3) (o) and v11ss*(S5) (O)*
*, is the
sum of Diagram 5.17 with an isomorphic chart, displaced (1; 2) units. As usua*
*l,
lines with negative slope represent boundary morphisms in the exact sequence of*
* the
fibration S3 ! SU(3) ! S5.
Diagram 5.17.r
A
r r bA b
@ A A .
r r@ bA bAb
@ A  .
r @ b Ab .
...
b.. .
8`  4 8` + 2
Thus v11ss*(SU(3)) appears as the upper part (O) of Diagram 5.18, in which t*
*he
lower part (o) is v11ss*(S6), from [22] or [21, p.667]. The long lines of nega*
*tive slope
represent @ in (5.16).
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 27
Diagram 5.18.bb bb
DD ....
bb bbD ...
DD DDD ...
bb D bb DD bb ....
DD D DDDD EE ...
bbbbD D bbDDDE ...
DDD DD D DDD DDE ...
bb DD D D DD DDrE ....
DDDD D D DD DDE...
bbDDD D D DD r DrrDE..
DDD DD D D DD DE 
DD DD D rDDDr DrEr
DD DD D DD E
DD DrDrD r DrrDrE
DD D D D A E
DD DrrrD Dr rArE
DD  A
DrD r rAr
D 
rD r r.

r r .
.
. r r .

8`  4rr 8` + 2;
 ` even
r
If ` is odd, the only change involves the differentials from 8` + 2. In this*
* case,
there is no d2differential from the bottom element. For exactly one of the two*
* mod
4 congruences of odd `, a differential from the bottom element in 8` + 2 hits t*
*he top
element in 8` + 1. In [21, p.668,top], it was asserted that this differential i*
*s nonzero
iff ` 3 mod 4. Although, as we shall see, this assertion is correct, there was*
* a flaw
in the argument. Diagram 4.12 of [21] does not commute, and [21, 4.6] is false.*
* This
is the same mistake that appeared for F4=G2 in [20] and was discussed near the *
*end
of Section 4 of this paper.
However, the Toda bracket argument of [21, p.668] correctly implies the claim*
* about
the differential being nonzero for half the odd values of `. Thus v11ss*(G2) i*
*s as in
Diagram 5.19, with the differential from 8` + 2 being d2 if ` is even, and d3 f*
*or one of
the mod 4 congruences of odd `. Actually, the transition from 5.18 to 5.19 does*
* not
make some of the jextensions clear, but they were established in [21].
28 BENDERSKY AND DAVIS
Diagram 5.19. r

r r

r rrr
 
r r rr
  
r r r r
  
  
r r r

r r

r r
r r

r r


rr
8`  3 8` + 2
From Diagram 5.18, we see that the kernel of v11ss*(G2) ! v11ss*(S6) consis*
*ts of
0, the element of order 2 in 8` + 2, and, for one of the mod 4 congruences of o*
*dd `,
the element of order 2 in 8` + 1. Since S6 satisfies the CTP, (5.15) implies th*
*at the
kernel of v11ss*(G2) ! v11ss*(Gc2) consists of, at most, the elements describ*
*ed in the
preceding sentence.
Recall that the BTSS of Diagram 4.6 converges to v11ss*(Gc2), and it must ad*
*mit
families of d3differentials annihilating all jtowers (except for a few elemen*
*ts at the
bottom of some of the jtowers). The preceding paragraph has shown that the four
parts of Diagram 5.19 involving j 6= 0 map nontrivially to v11ss*(Gc2). Thus *
*the
pattern of differentials in the BTSS of G2 must be as in Diagram 4.9 to allow f*
*or
these elements of v11ss*(Gc2).
The nonzero extension in dimension 8` + 2 in Diagram 4.9 for G2 must occur
because v11ss8`+1(G2) ! v11ss8`+1(Gc2) must send the element x of highest fil*
*tration
in Diagram 5.19 nontrivially (it corresponds to the top element in v11ss8`+1(S*
*6) in
Diagram 5.18), and since jx is divisible by 2, the same must be true of its ima*
*ge in
v11ss8`+1(Gc2). A similar argument implies the nontrivial extension in t  s =*
* 8` + 1
in Diagram 4.9.
STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 29
Thus v11ss8`+2(G2) ! v11ss8`+2(Gc2) is injective (since the element of orde*
*r 2 maps
across),
v11ss8`+2(Gc2) Z=2min(6;(`1)+4);
while v11ss8`+2(G2) Z=24 if ` is even, and is Z=25 for one odd mod 4 congruen*
*ce
of `, and Z=26 for the other. This implies that the Z=25 must occur for ` 3 mod
4, and v11ss8`+2(G2) ! v11ss8`+2(Gc2) is bijective. Thus [21, 4.1,4.5] are va*
*lid, as are
Theorems 4.8 and 1.3 for G2. 
Finally we deduce the CTP for F4 from the fact that (4.1) induces an injective
extension sequence in K*(), 5.11, 5.12, 5.8, and the fact that G2 and F4=G2 bo*
*th
satisfy the CTP, the latter since it is strongly spherically resolved by [21, 1*
*.1]. That
K*(G2) ! K*(F4) ! K*(F4=G2)
is an injective extension sequence follows from Proposition 5.5 and the fact th*
*at there
is a short exact sequence in P K1().
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Hunter College, CUNY, NY, NY 10021
Email address: mbenders@shiva.hunter.cuny.edu
Lehigh University, Bethlehem, PA 18015
Email address: dmd1@lehigh.edu