THE K-COMPLETION OF E6
DONALD M. DAVIS
Abstract.We compute the 2-primary v1-periodic homotopy groups
of the K-completion of the exceptional Lie group E6. This is done
by computing the Bendersky-Thompson spectral sequence of E6.
We conjecture that the natural map from E6 to its K-completion
induces an isomorphism in v1-periodic homotopy, and discuss is-
sues related to this conjecture.
1.Introduction
The p-primary v1-homotopy groups of a space X, denoted v-11ß*(X; p) and de-
fined in [11], are a localization of the portion of the homotopy groups of X de*
*tected
by p-local K-theory. In [10], the author completed the determination of the od*
*d-
primary v1-periodic homotopy groups of all compact simple Lie groups. The groups
v-11ß*(X; 2) have been determined for X = SU(n) ([3]), Sp(n) ([5]), G2 ([12]), *
*and
F4 ([4]). Joint work of the author and Bendersky is very close to completing t*
*he
computation of v-11ß*(SO(n); 2). That will leave E6, E7, and E8 to be determine*
*d,
which would complete a program suggested to the author by Mimura in 1989. In
this paper, we determine v-11ß*(Eb6; 2), where cXdenotes the K-completion of X,*
* as
defined in [6]. We conjecture that the natural map E6 ! bE6induces an isomorphi*
*sm
in v-11ß*(-; 2).
In [6], Bendersky and Thompson defined the K-completion cXof a space X to be
the homotopy limit of a certain tower of spaces under X. The space X is said to
satisfy the Completion Telescope Property (CTP) at the prime p if X ! cXinduces
an isomorphism of p-primary v1-periodic homotopy groups. They also constructed a
spectral sequence (BTSS) which, for many spaces X, converges to v-11ß*(cX; p). *
* It
was shown in [7] and [4] that, localized at any prime, if X is K*-strongly sphe*
*rically
__________
Date: August 29, 2001.
1991 Mathematics Subject Classification. 55T15,55Q52,57T20.
Key words and phrases. K-theory, homotopy groups, exceptional Lie groups.
1
2 DAVIS
resolved (KSSR), then X satisfies the CTP. This condition means that X can be b*
*uilt
by fibrations from spheres S2ni+1such that K*X is built as a K*K-coalgebra as an
extension of the K*S2ni+1.
In [4], it was proved that F4 satisfies the CTP at 2, and a general result ([*
*4,
5.8,5.11,5.15] was proved which implies that if E6=F4 satisfies the CTP, then s*
*o does
E6. Many standard functors of algebraic topology would lead one to expect that *
*there
is a fibration
S9 ! E6=F4 ! S17, (1.1)
which would imply that E6=F4 is KSSR and then that E6 satisfies the CTP at the
prime 2. However, it was proved by Cohen and Selick ([9]) that there can be no *
*such
fibration. A fibration
S9 ! (E6=F4) ! S17
would also imply the CTP for E6. It is not known whether such a fibration exist*
*s.
It was proved in [7] that, localized at an odd prime p, if X is K-algebraical*
*ly
spherically resolved and K-durable, then it satisfies the CTP. The first condit*
*ion
(KASR) means that K*(X) has the structure that it would if X were KSSR, and the
second that X and its K-localization have isomorphic v1-periodic homotopy group*
*s.
Both E6=F4 and E6 are KASR and K-durable at the prime 2. However, it is not
known whether, at the prime 2, this is enough to insure that the CTP is satisfi*
*ed.
Our main result is the following determination of v-11ß*(Eb6; 2). As discusse*
*d above,
the expectation is that this equals v-11ß*(E6; 2).
Theorem 1.2. Let e = min(12, (` - 18) + 5) and f = min(12, 2 (` - 3) + 8). Th*
*en
8
>>>Z=2f d = -3
>>> f
>>>Z=2 Z=2 d = -2
>< e
Z=2 Z=2 Z=2 d = -1, 0
v-11ß8`+d(Eb6; 2) > 5
>>>Z=2 Z=2 Z=2 d = 1
>>> 5
>>>Z=2 Z=2 d = 2
:Z=23 d = 3, 4
Here and throughout, (-) denotes the exponent of 2 in an integer. The picture *
*of
the BTSS which determines these groups is given in Diagram 1.3. This is a usual
sort of Adams spectral sequence type of chart, with dots representing Z=2's, po*
*sitively
K-COMPLETION OF E6 3
sloping lines the action of the Hopf map j (or the element h1 in the spectral s*
*equence),
and negatively sloping lines differentials in the spectral sequence, which impl*
*ies that
the elements which they connect do not survive to give nonzero homotopy classes.
The dotted lines that look like h1 means that h1 is usually present, but perhap*
*s not
always.
Diagram 1.3.
| | | | | B|B | B B| |
| | | | | | | B |B |
| | | | | B| r r | B |Br |
| | | | r | |BBB BB| B |BBB |
___________________________________________________________||||||||||*
*||||||||BBBBBBBBBB
| | | | B | |B B B| B| BBB |
| | | | B |r tr|B B |rB B| BBB |
| | | | | | B | | |
s = 4| | | t | B|BB | B |BBB |B BBB |
___________________________________________________________||||||||||*
*||||||||BBBBBBBBBBB
| | | | |BB | BB |BB |B BBB|
| | | | t t |B B | t B| BB |Br BB|rB
| | t | | |B B | BrB| BB | B|
___________________________________________________________||||||||||*
*||||||||BBBBBB
| | | | | B B | |B B B| 4 |B
| | | t 2e | |t B B| |Br BB|2 |rB
| 2f | |ppp | p|pp26B|B | |B |
___________________________________________________________||||||||||*
*||||||||pppBBppp
| | pp | | pp | |B | |B |
| |2fp | |2pe | |B26 | |B24 |
| | | | | | | | |
s = 1| | | | | | | | |
___________________________________________________________|||||||||
8` - 2 8` 8` + 2 8` + 4
The reader should observe that this chart, and the homotopy groups which it
depicts, has a form very similar to the charts for v-11ß*(G2) and v-11ß*(F4) de*
*picted
in [4, 4.9]. The only difference is the orders of the groups on the 1- and 2-li*
*nes, and
the lack of some exotic extensions.
The determination of the E2-term of the spectral sequence is rather straightf*
*orward,
given the general results of [4] and calculations of the Adams operations in K**
*(E6)
performed in [10]. This is performed in Section 2. The d3-differentials are det*
*ermined
by showing that there are maps of spaces which relate the classes in question t*
*o classes
in spaces where d3 is known. This is performed in Sections 3 and 4. In Section *
*5, we
expand our discussion of whether E6 satisfies the CTP.
The author would like to thank Martin Bendersky and Fred Cohen for helpful
comments on this work.
4 DAVIS
2. The E2-term
In this section we compute the E2-term of the BTSS converging to v-11ß*(Eb6).*
* We
are always using the v1-periodic BTSS localized at 2, and all v1-periodic homot*
*opy
groups are 2-primary.
In [4, 1.1], it was proved that for spaces whose K-homology and K-cohomology
form nice exterior algebras,
8
.
The case n = 2k is slightly more delicate. Let fM= M= im(_-1-1); its presenta*
*tion
matrix can be reduced0to 1
-2 1 0 0 0 0
BB0 0 2 0 0 0 C
BB CC
BB16 0 -1 -1 0 0 CC (2.8)
BB32 0 0 0 -1 0 CCC
@ 0 0 0 0 0 -1 A
212 0 0 0 0 0
Thus fM is spanned by v1 and v3 of order 212and 2, respectively. Using 2.2 and *
*the
relations in fM, we have
(_3 - 32k)v3 (35- 32k)v3- 162(16v1- v3) + 81 . 32v1 0,
8 DAVIS
since 2v3 = 0. Thus K has a Z=2-summand generated by v3. Similarly
(_3-32k)v1 = (3-32k)v1+39.2v1+147(16v1-v3)-87.32v1 v3-(32k+351)v1.
Thus v1 is not in K, but 2v1 has a chance. Since 351 = 211u - 372with u odd,
(_3 - 32k)(2v1) = 2(372- 32k- 211u)v1 = 23+ (2k-72)u0v1,
with u0 odd, where 212v1 = 0. Thus 2max(0,12-(3+ (2k-72)))2v1 generates the ot*
*her
summand, which will have 2-exponent min(12 - 1, 3 + (2k - 72) - 1), as claimed.
||
In Section 4, we will prove that a failure of (2.5) to split compensates for *
*the
unexpected splitting in Proposition 2.7, yielding
Proposition 2.9. The nonzero groups on the 2-line of the BTSS of E6 are given by
8
*
* 2, all of
which occur in families related by h1, also known as j. ([4, 3.6]) With Qn as d*
*efined
in (2.6), we have, from [4, 3.10], for s > 2, a short exact sequence
0 ! coker(`n|Qs+n(M)) ! Exts,2n+1A(M)# ! ker(`n|Qs+n-1(M)) ! 0.
(2.10)
We establish
Proposition 2.11. For M = QK1(E6)= im(_2), Qn(M) Z=2, generated by v3 if n
is odd, and by 217v1 if n is even.
Proof.If n is odd, Qn(M) equals ker(_-1 + 1 : fM ! M), where fM is presented by
(2.8). Then fM Z=2 Z=212, with generators v3 and v1. From 2.2, we see that
v3 2 ker(_-1+1); however, (_-1+1)(211v1) = 211v2, which is nonzero in M. One way
to see that 211v2 6= 0 is to use pivoting to obtain the following alternate pre*
*sentation
K-COMPLETION OF E6 9
for M, with columns still v1, . .,.v6.
0 1
16 -8 -40 0 0 1
BB 0 16 0 0 -1 0C
BB CC
BB 27 76 -16 -1 0 0CC (2.12)
BB2 5 2 27 -2 17 0 0 0CCC
@ 0 212 0 0 0 0A
218 0 0 0 0 0
___ ___
If n is even, we want ker(_-1 - 1 : M ! M), where M is presented by (2.12) w*
*ith
the second and fifth columns omitted. We check _-1 - 1 on the elements of order*
* 2.
We have 217v1 7! -218v1+ 217v2 = 0, while
265v1- 2517v3 7! -275v1+ 265v2+ 2617v3 = 26(5 + 27)v2 = 211v2,
which has order 2 in M. ||
If X is a topological space, let Qn(X) = Qn(QK1(X)= im(_2)).
Corollary 2.13. For s > 2, Es,2n+12(E6)# Z=2 Z=2, with generators 217v1 and
v3.
Proof.We use (2.10) and Proposition 2.11, and 2.2 to show that `n(v3) = 0 in Qo*
*d(E6)
and `n(217v1) = 0 in Qev(E6). Indeed, `n(v3) = 2Av3- 162v4+ 81v5- 81v6, which is
0 in a group presented by (2.8), while `n(217v1) = 0 in a group presented by (2*
*.12)
since there 218v1 = 0 and 217vi= 0 for 2 i 6. ||
This, with 2.4 and 2.9, completes the determination of the E2-term, which is *
*as
suggested by Diagram 1.3. The h1-extensions from the 1-line will be discussed *
*in
Section 4.
3.d3-differentials
In this section, we determine the d3-differentials on the eta towers in the B*
*TSS of
E6. An eta tower consists of elements in Es,2s+i2for s s0 connected by h1. He*
*re
s0 = 1, 2, or 3. We denote by ji(X) the Z2-vector space of eta towers passing t*
*hrough
Es,2s+i2(X) for s > 2. Note that d3 is a homomorphism from ji(X) to ji-4(X). Wi*
*th
Es,t2depicted as usual in position (x, y) = (t - s, s), then ji(X) is a tower o*
*f elements
whose position satisfies x - y = i. Since j4 = 0 in homotopy, d3-differentials *
*must
10 DAVIS
annihilate all eta towers, except for a few elements at the bottom of the targe*
*t tower.
By (2.10), if QK1(X) consists of elements whose dimensions are all of the same *
*parity,
then so does j*(X), and ji(X) depends only on i mod 4. What must be determined
for each family of eta towers is the mod 8 value of i for which d3 : ji(X) ! ji*
*-4(X)
is nonzero.
We make heavy use of the fibration
F4 ! E6 ! EIV, (3.1)
where EIV is the group quotient E6=F4. This fibration was studied quite thoroug*
*hly
in [18]. It was observed there that
H*(EIV ; Z) (x9, x17),
and the Serre spectral sequence of (3.1) collapses. From this, the spectral seq*
*uence
H*(EIV ; K*(F4)) ) K*(E6)
implies that there is a short exact sequence
0 ! QK1(EIV ) ! QK1(E6) ! QK1(F4) ! 0, (3.2)
with each of the three algebras K*(X) being exterior. The following key proposi*
*tion
implies that the Adams operations in K(EIV ) are as they would be if (1.1) exis*
*ted.
Proposition 3.3. There is a basis {y1, y2} for QK1(EIV ) on which for all k
4-k8
_k(y1) = k4y1+ uk___16y2, (3.4)
with u a unit in Z(2), and _k(y2) = k8y2.
Proof.This can be deduced from [18, Thm 2], which computes the Chern character
for EIV . We present a proof closer to our methods.
From (3.2), we deduce QK1(EIV ) = ker(QK1(E6) ! QK1(F4)). This must be the
subspace of QK1(E6) spanned by v2 and v5 (in the basis used in Section 2). Perh*
*aps
the easiest way to see this is to use (_-1) of 2.2, which shows that _-1 = 1 on*
* this
subspace, while it equals -1 on QK1(F4).
From 2.2, we find that 240v2+v5 is an eigenvector for _2 with eigenvalue 16. *
*Thus,
considering the rational splitting of E6 as a product of spheres as in [10], we*
* deduce
that _k(240v2 + v5) = k4(240v2 + v5) and _k(v5) = k8v5 for all integers k. Thus
4-k8
_kv2 = k4v2+ k___240v5, from which follows the result with y1 = v2 and y2 = v5.*
* ||
K-COMPLETION OF E6 11
In [4, 4.2], it was shown that there is a basis {x1, x2, x3, x4} of QK1(F4) o*
*n which
_-1 = -1 and the matrix of _2 is
0 1
2 0 0 0
BB3 32 0 0 C
B@1 -8 128 0 CCA
0 -1 -24 2048
We easily verify the following result.
Proposition 3.5. In terms of the bases introduced, the exact sequence (3.2) is *
*given
by y1 7! v2, y2 7! v5, v1 7! -x1- x2, v3 7! x2, v4 7! x3, and v6 7! x4.
__
Let K (X) := QK1(X)= im(_2). Since _2 acts injectively in our modules, the Sn*
*ake
Lemma applied to (3.2) yields a short exact sequence
__ __ __
0 ! K (EIV ) ! K (E6) ! K (F4) ! 0 (3.6)
and hence by (2.1) a long exact sequence
0 ! E1,t2F4 ! E1,t2E6 ! E1,t2EIV ! E2,t2F4 ! E2,t2E6 ! E2,t2EIV ! E3,t2F4 ! .
(3.7)
In [4, 4.6], it was shown that
8
>>>Z=2 s = 1, t 1(4)
>>> f
>>>Z=2 s = 1, t = 4k + 3
>< f
Z=2 Z=2 s = 2, t = 4k + 3
Es,t2(F4) >
>>>Z=2 Z=2 s = 2, t 1(4)
>>>
>>>Z=2 Z=2 s > 2, t odd
:0 t even
with f = min(12, 2 (k -5)+6). Using 3.3 and the method of [4, 4.11-4.12], we ob*
*tain
8
>>>Z=2 s = 1, t 3(4)
>>> e
>>>Z=2 s = 1, t = 4k + 1
>< e
Z=2 Z=2 s = 2, t = 4k + 1
Es,t2(EIV ) >
>>>Z=2 Z=2 s = 2, t 3(4)
>>>
>>>Z=2 Z=2 s > 2, t odd
:0 t even
with e = min(12, (k - 36 - 28) + 3). By just substituting the known E2-groups
into (3.7), we can deduce quite a bit about the morphisms; in particular, the r*
*ank
12 DAVIS
of every morphism beginning with E2,t2EIV ! E3,t2F4 is 1. This follows by count*
*ing
the alternating sum of the exponents of the orders of the groups. Thus of the t*
*wo
eta-towers in Es,t2(E6) for any odd value of t - 2s, one comes from F4 and the *
*other
maps nontrivially to EIV .
We need to know more specifically which classes map across in (3.7). For thi*
*s,
we use the following commutative diagram of exact sequences, where Qn(X) is the
__
functor of (2.6) applied to M = K X.
--`-! Q ff1 s,2n+1_ # fi1 `
s+n(EIV?) ---! ExtA (K?EIV ) ---! Qs+n-1(EIV?)---!
? ? ?
f1?y f2?y f3?y
--`-! Q ff2 s,2n+1_ # fi2 `
s+n(E6)? ---! ExtA ?(K E6) ---! Qs+n-1(E6)?---!
? ? ? (3.8)
g1?y g2?y g3?y
--`-! Q ff3 s,2n+1_ # fi3 `
s+n(F4) ---! ExtA (K F4) ---! Qs+n-1(F4) ---!
Proposition 3.9. In (3.8) with s + n even, g3(v3) = x2 pulls back nontrivially *
*to g2.
If s + n is odd, then g3(217v1) = 217x1 pulls back nontrivially to g2. If s + n*
* is even
__
(resp. odd), there is an element z 2 Exts,2n+1A(K EIV )# such that fi1(z) = 27y*
*2 (resp.
y1) and f2(z) = ff2(217v1) (resp. ff2(v3)).
Proof.The exact sequence for F4 is given in [4, 4.5]. Actually, it is more conv*
*enient
here to use the presentation
0 1
0 -32 8 1
BB2 3 1 0C
B@275 2627 0 0CCA (3.10)
218 0 0 0
__
for K (F4) instead of that of [4, 4.2]. This one can be obtained from that one*
* by
pivoting. Dividing the third and fourth rows of (3.10) by 2 gives a basis of Qe*
*v(F4),
and ` sends the first element to the second. This is a computation best done in*
* Maple,
using the rows of (3.10) to reduce (_3- 1)(265x1+ 2627x2), obtained from [4, 4.*
*2], to
217x1. As in [4, 4.5], Qod(F4) = with ` sending the first to the *
*second.
Now the g2-part of the proposition in straightforward, using Proposition 3.5.
We have Qod(EIV ) Z=2 with generator 27y2, and Qev(EIV ) Z=2 with gener-
ator y1. One readily checks, using 3.5 and 2.11, that f1 = 0 and f3 = 0 in (3.8*
*). Thus
K-COMPLETION OF E6 13
the morphism f2 must be as claimed because of our previous observation that it *
*has
rank 1. ||
We can deduce d3 on half of the eta towers in E6 from its behavior in F4. By
Corollary 2.13, each jod(E6) has basis {217v1, v3}.
Proposition 3.11. The differential d3 : j8`+1(E6) ! j8`-3(E6) sends the v3-towe*
*r to
the v3-tower. The differential d3 : j8`+7(E6) ! j8`+3(E6) sends the 217v1-tower*
* to the
217v1-tower.
Proof.It was shown in [4, 4.13] that in the BTSS of F4, d3 : j8`+1(F4) ! j8`-3(*
*F4)
is the "identity map," while d3 : j8`+3(F4) ! j8`-1(F4) sends the x1-tower to t*
*he
x1-tower, and d3 : j8`+7(F4) ! j8`+3(F4) sends the 217x1-tower to the 217x1-tow*
*er.
We dualize (3.8) and use 3.9 to see that the v3-tower is in the image of j4*+1(*
*F4) !
j4*+1(E6), and that j4*+3(F4) ! j4*+3(E6) sends the 217x1-tower to the 217v1-to*
*wer.
Naturality of d3 implies the result. ||
By using the inclusion map S9 ! EIV , we deduce the third family of d3-differ*
*entials.
Proposition 3.12. The differential d3 : j8`+3(E6) ! j8`-1(E6) sends the v3-towe*
*r to
the v3-tower.
Proof.Using the dual of f2in Proposition 3.9, it suffices to show that d3 : j8`*
*+3(EIV ) !
j8`-1(EIV ) sends the y1-tower to the y1-tower. The map j : S9 ! EIV induces
j* 1 9
QK1EIV - ! QK S
satisfying j*(y1) = g and j*(y2) = 0. The generator g 2 QK1S9 gives rise to the*
* stable
eta towers in BTSS(S9), and by [1, p.58] or [3, p.488] d3 : j8`+3(S9) ! j8`-1(S*
*9) sends
the stable eta tower to the stable eta tower. Our conclusion follows by natural*
*ity using
the map j. ||
The final d3 on eta towers requires a more elaborate argument. The conclusion*
* is
that d3 is as it would be if the fibration (1.1) existed. Our desired result is
Proposition 3.13. The differential d3 : j8`+1(E6) ! j8`-3(E6) sends the 217v1-t*
*ower
to the 217v1-tower.
14 DAVIS
Proof.By Proposition 3.9, it suffices to show j8`+1(EIV ) ! j8`-3(EIV ) sends t*
*he
27y2-tower to the 27y2-tower. This is a consequence of the following two propos*
*itions.
||
Proposition 3.14. There is an isomorphism of BTSSs
Es,tr( EIV ) Es,t+1r(EIV )
for r 2.
Proposition 3.15. The differential d3 : j8`( EIV ) ! j8`-4( EIV ) sends the 27y*
*2-
tower to the 27y2-tower.
Proof of Proposition 3.14.We need the following result.
Proposition 3.16. The algebra K*( EIV ) is a polynomial algebra on two classes
in K0(-).
Proof.We first note that H*( EIV ) is a polynomial algebra on classes in H8(-)
and H16(-). To see this, we first use the computations of the rings H*( F4; R) *
*and
H*( E6; R) for R = Z=2 in [15, 2.3] and for R = Z=3 in [13, Thm.1], which imply*
* that
H*( EIV ; R) is polynomial on 8- and 16-dimensional generators. With coefficien*
*ts Q
or Z=p for p > 3, this is also true because the localizations of F4 and E6 are *
*products
of spheres whose dimensions are well known. The result with integral coefficie*
*nts
follows by standard methods.
Now the Atiyah-Hirzebruch spectral sequence H*( EIV ; K*) ) K*( EIV ) im-
plies the result for K*(-). ||
Returning to the proof of 3.14, we adapt an argument first used in [2, 6.1] t*
*o show
that for the BP -based unstable Novikov spectral sequence (UNSS) Es,t2( S2n+1)
Es,t+12(S2n+1). It was shown in [5, 3.3] that the same argument works if S2n+1*
* is
replaced by a space Y for which BP*( Y ) is the polynomial algebra on classes {*
*y2i},
and BP*Y is isomorphic to an exterior algebra on the primitive classes oey2i. I*
*t was
noted in [6, 4.12] that these arguments can be adapted to the K-based BTSS. Thi*
*s,
with Proposition 3.16, yields the proposition when r = 2.
K-COMPLETION OF E6 15
To prove the result for all r, we note that the isomorphism of E2-terms is in*
*duced
by a map of towers. To see this, we use the following natural map of augmented
cosimplicial spaces, where K(X) = 1 (K ^ 1 X).
- ___-_-3 ___-_-
X ___-K X ___-_K(K( X))___-K X ___-_-. . .
| | | |
| | | |
|? |? - |? ___-_-3|? ___-_-
X ___- KX ___-_ K(KX) ___- K X ___-_-. . .
Applying ß*(-) and taking homology of the alternating sum to the first row yi*
*elds
E*,*2( X), and doing this to the second yields E*,*-12(X). The induced morphism*
* in
homology is the E2 isomorphism observed above. But these cosimplicial spaces gi*
*ve
rise, by the Tot construction, to the towers that define the entire spectral se*
*quence,
and so the morphism induces a morphism of spectral sequences. ||
Proof of Proposition 3.15.Let F denote the fiber of the inclusion map S9 ! EIV .
The Serre spectral sequence of this fibration shows that H*F , like H*( S17), i*
*s a
divided polynomial algebra on a 16-dimensional class. Thus H*F = with
X i ij
_(x16i) = j x16j x16(i-j).
It follows that the Z=2-graded K*F has the same coalgebra structure, although t*
*he
grading is lost. Thus there is an abstract isomorphism of coalgebras K*F K*( *
*S17).
This implies that the fibration
S9 ! EIV ! F
induces a relatively injective extension sequence
p*
0 ! K*( S9) ! K*( EIV ) -! K*(F ) ! 0,
and hence, by [2, 4.3], a long exact sequence of BTSS, commuting with d3
p* s s+1 9
! Es2( S9) ! Es2( EIV ) -! E2(F ) ! E2 ( S ) ! .
We remark here that the notion of relatively injective extension sequence was d*
*efined
in [4, 5.14]. It is the notion which was intended in [2, 4.3].
The Hurewicz Theorem gives a map f : S16 ! F , and we have the map p :
EIV ! F from the above fibration. Both of these maps induce morphisms of
16 DAVIS
BTSS, commuting with d3. The image under p* of the eta towers 27y2 equal the
image under f* of eta towers in S16which map, in the EHP sequence, to the unsta*
*ble
eta towers in j4*+1(S17). By [1, p.58] or [3, p.488], the d3-differential on th*
*e unstable
eta towers in S17 is nonzero from j8`+1(S17) to j8`-3(S17). Hence it is nonzero*
* from
j8`(S16) to j8`-4(S16), hence also in F , and thence in EIV . ||
4. Fine Tuning
In this section we determine the d3-differentials from the 1-line, prove Prop*
*osition
2.9, and show that there are no exotic extensions (.2) in the BTSS.
In general, the h1-action on the 1-line of the BTSS can be delicate. See, e.g*
*., [4,
4.15]. It is important, because one way to determine d3 on the 1-line is to det*
*ermine
d3 on the eta towers and then use the h1-action from the 1-line to the eta towe*
*rs to
deduce d3 on the 1-line. A different method was used for Sp(n) in [5] and illus*
*trated
in [5, 4.3] the subtle things that can happen. In that case, the relevant 2-lin*
*e group
contained some classes supporting d3-differentials and some which did not, and *
*we
argued indirectly to see whether d3x was nonzero, which was equivalent to deter*
*mining
whether h1x contained any classes of the first type.
In our situation here, the 1-line classes in E6 map to or from classes in EIV*
* or F4
on which h1 and d3 are known, from which we deduce the following result, which *
*will
be proved along with Proposition 2.9.
Proposition 4.1. The h1-extensions and d3-differentials from the 1- and 2-lines*
* are
as depicted in Diagram 1.3.
Proof of Propositions 2.9 and 4.1.We compare the exact sequences in (Ext*,2n+1A*
*)#
and (Ext*,2n+1GInv)# induced by (3.6). Here GInv denotes the category of profin*
*ite abelian
groups with involution. We will also use Inv, the category of abelian groups w*
*ith
involution.
The analysis is much more delicate when n = 2k is even, and so we focus on th*
*is
case. Some details of the proof (but not final conclusions) are slightly differ*
*ent when
(k - 36) = 8 due to the (k - 36 - 28) which occurs in E2(EIV ), so we assume
(k - 36) 6= 8 to simplify the exposition.
K-COMPLETION OF E6 17
The exact sequences (2.5) and (2.10) are obtained in [4, x3] from a long exact
sequence
_3M-_3N s # s+1 #
ExtsA(M, N)# ExtsGInv(M, N)# - ExtGInv(M, N) ExtA (M, N)
and isomorphisms
Exts+1,2n+1GInv(M)#=Exts+1GInv(M, QK1S2n+1)#
8
n) # # 0
ExtsInv(Z((-1)(2), M ) :
M=(_-1 - (-1)n) s = 0.
Then
__ # 1,4k+1_ # 1,4k+1_ #
0 Ext 1,4k+1GInv(K F4) ExtGInv(K E6) ExtGInv(K EIV )
__ # 2,4k+1_ # 2,4k+1_ # 3,4k+1_ #
Ext 2,4k+1GInv(K F4) ExtGInv(K E6) ExtGInv(K EIV ) ExtGInv(K F4)
is
__ -1 __ -1 __ -1
0 KF4=(_ - 1) K E6=(_ - 1) K EIV=(_ - 1)
__ __ __ __
Qev(K F4) Qev(K E6) Qev(K EIV ) Qod(K F4).
Using results and notation from Sections 2 and 3, this sequence becomes, with x*
*c =
265x1+ 2527x2,
0 <217v1> ,
with all elements having order 2 except v1 and y1, which have order 212. Using *
*3.5,
the explicit morphisms, yielding an exact sequence, are
x2 - v3 0 - 217x1 - 217v1
x1+ x2 - v1 211y1 - xc 0 - y1 - x1
2v1 - y1 (4.2)
With = min(12, (k - 36) + 3), the nonzero occurrences of _3 - 32kon the grou*
*ps
in this sequence are x1 7! x2, v1 7! 2 v1+ v3, y1 7! 2 y1, xc 7! 217x1, and x1 *
*7! x2.
The reader can verify that these commute with the morphisms of (4.2).
Now the short exact sequences of (2.5) and (2.10), and the isomorphism (2.3),
__
written vertically, become as follows. We write Es2(X) for Exts,4k+1A(K X), and*
* denote
by x(2e) an element x of order 2e. Elements not followed by parentheses have or*
*der
18 DAVIS
2.
x1 v1(2 +1) -.2 y1(2 )
# # #
E12(F4)# E12(E6)# E12(EIV )#
xc 217v1 y1 x1
# # # #
E22(F4)# E22(E6)# E22(EIV )# E32(F4)#
# # #
x2 v3, 212- v1(2-)2 212- y1(2 )
In order for the Es2(-)# sequence to be exact, 212- v1 must hit xc in E22(F4)#,
and 211y1 must hit 217v1 in E22(E6)#. The latter implies the nontrivial extens*
*ion
in E22(E6)#, that it contains a Z=2 +1 with 217v1 the element of order 2 in thi*
*s sum-
mand. This concludes the proof of 2.9.
The above analysis showed E1,4k+12(E6)# E1,4k+12(EIV )# injective; hence
E1,4k+12(E6) ! E1,4k+12(EIV )
is surjective. In 3.9 and the proof of 3.12, we showed that the stable eta tow*
*er
in E*,4k+2*-12(E6) maps to the one in EIV which comes from S9. If (k) 1,
E1,4k+12(S9) ! E1,4k+12(EIV ) is an isomorphism. So the h1-extension and d3-dif*
*ferential
from E1,8`+52(S9) implies the same in E6. A nonzero h1-extension from E1,8`+12(*
*E6) is
implied when ` is odd. It is not so important because d3 must be 0 on E1,8`+12(*
*E6)
since h21times the Z=2 in (8`+1, 2) is in the image of d3. It is likely that an*
* adaptation
of [4, 3.7] to modules which do not satisfy _-1 = -1 would allow the determinat*
*ion
of h1 on E1,8`+12(E6).
Since E1,4k+32(F4) ! E1,4k+32(E6) is an isomorphism, the h1-extensions and d3-
differentials from E1,4k+32(E6) follow from those in F4. In [4, 4.14], it was s*
*hown that
h1 on E1,4k+32(F4) hits the stable eta tower iff (k - 5) 6= 3 and hits the uns*
*table eta
tower iff (k - 5) = 3. This implies the nonzero h1 and d3 from E1,8`+32(E6) pi*
*ctured
in Diagram 1.3, and that h1 is nonzero from E1,8`-12(E6) unless ` 7 mod 8, in*
* which
case it is 0. The latter is because the unstable eta tower in F4 in this dimens*
*ion maps
to 0 in E6. That h1x is sometimes 0 here is mildly interesting because in [4, 3*
*.8] it
was shown that if _-1 = -1 then h1 acts injectively on E12(X)=2.
K-COMPLETION OF E6 19
The analysis of exact sequences earlier in this proof implied that E2,4k+12(E*
*6) !
E2,4k+12(EIV ) sends the Z=2e summand surjectively, and that the generator is d*
*ual to
217v1, which is the name of the unstable eta tower whose d3-differential is des*
*cribed
in 3.13. The h1-extension and d3-differential on it follow immediately.
The exact sequence in Ext1,4k+3A(-) induced by (3.6) easily implies that E2,4*
*k+32(F4) !
E2,4k+32(E6) sends the Z=2f bijectively. The h1-extension (for all k) and d3-di*
*fferential
when k is odd follow from those in F4 established in [4]. ||
Finally, we have the following result about extensions in the spectral sequen*
*ce.
Proposition 4.3. There are no nontrivial extensions (.2) from Es,t2(E6) to Es+2*
*,t+22(E6).
Proof.For s = 1 or 2, the group Es,8`+32(E6) Z=26 is mapped isomorphically on*
*to
from Es,8`+32(F4). In [4, 4.14], it is shown that when s = 1 (resp. s = 2) th*
*ere
is a nonzero extension into the element of Es+2,8`+52(F4) which is part of the *
*stable
(resp. unstable) eta tower. In Proposition 3.9, it is proved that these eta tow*
*ers map
trivially to Es+2,8`+52(E6), hence this extension is 0 in E6.
In the proof of Propositions 2.9 and 4.1, it was shown that for s = 1 and 2, *
*the
element of order 2 in Es,8`+12(E6) is in the image from Es,8`+12(F4), which doe*
*s not
support an extension in [4, 4.9]. ||
5. The CTP for E6
In this brief section, we mention several issues related to the CTP for E6. I*
*n other
words, how might we prove that the groups v-11ß*(Eb6), which we have computed, *
*are
in fact isomorphic to v-11ß*(E6), which is what we really want?
There are no known examples of spaces which do not satisfy the CTP. On the ot*
*her
hand, the only spaces which are known to satisfy it are certain spaces which ca*
*n, in
some sense, be built from spheres by fibrations.
Localized at an odd prime, the result of [7] which states that a space which *
*is
KASR and K-durable satisfies the CTP is very useful. If the 2-primary analogue
were known to be true, then we would know that E6 satisfies the CTP. However, t*
*he
proof of the result of [7] relies on much delicate machinery developed at the o*
*dd primes
by Bousfield, especially in [8]. One crucial difference between the odd primes *
*and the
20 DAVIS
prime 2 is that ExtsA(M, N) vanishes for s > 2 when p is odd. This is important*
* for
existence and uniqueness of realizations of certain morphisms of Adams modules.*
* An
adaptation to the prime 2 would be far from straightforward.
A main worry in considering convergence of spectral sequences related to v-11*
*ß*(X)
is the possibility that there could be elements in v-11ß*(X) not seen by the sp*
*ectral
sequence because they correspond to a family of elements in ßni(X) for ni ! 1,
related to one another by .ve1and having increasing filtrations. The algebraic*
* .v1
operation in the UNSS or BTSS preserves filtration, but the homotopy-theoretic
operation can increase filtration. For S2n+1, the E1 -term of the v1-periodic *
*BTSS
yields exactly the v1-periodic homotopy classes, which we know because they were
calculated by another method in [16] (p = 2) and [17] (p odd). Spaces built fro*
*m odd
spheres, or perhaps their loop spaces, by fibrations can then guarantee nonexis*
*tence
of undetected v1-periodic families by exactness properties. The proof in [7] th*
*at at
the odd primes X KASR and K-durable implies the CTP for X ultimately boils down
to building the localization XK from various S2n+1K.
The isomorphism Es,t2(S2n+1) Es,t-12( S2n+1) implies that if a space is nic*
*ely
fibered by S2n+1's, then it satisfies the CTP. For spaces fibered by 2S2n+1, *
*it is
not so clear.
If it could be proved that F , the fiber of S9 ! EIV considered in the proof *
*of 3.15,
has the same homotopy type as S17, then the CTP for E6 could be deduced. In [9,
f 9
2.1], a map 2S17- ! S was constructed. If it could be shown that the composi*
*te
f 9 i
2S17- ! S -! EIV
is null-homotopic, then one could deduce existence of a fibration
2S9 ! 2EI ! 2S17.
It is likely that such a fibration would imply the CTP for E6, although a detai*
*led
argument has not been produced.
References
[1]M. Bendersky, The v1-periodic unstable Novikov spectral sequence, Topology
31 (1992) 47-64.
[2]M. Bendersky, E.B. Curtis, and D.C. Ravenel, EHP sequences in BP theory,
Topology 21 (1982) 373-391.
K-COMPLETION OF E6 21
[3]M. Bendersky and D. M. Davis, 2-primary v1-periodic homotopy groups of
SU(n), Amer Jour Math 114 (1991) 529-544.
[4]________, A stable approach to an unstable homotopy spectral sequence, to
appear.
[5]M. Bendersky, D. M. Davis, and M. Mahowald, v1-periodic homotopy groups
of Sp(n), Pac Jour Math 170 (1995) 319-378.
[6]M. Bendersky and R. D. Thompson, The Bousfield-Kan spectral sequence for
periodic homology theories, to appear in Amer Jour Math.
[7]________, Some properties of the K-completion, to appear in JAMI Confer-
ence Proceedings.
[8]A. K. Bousfield, The K-theory localization and v1-periodic homotopy groups *
*of
H-spaces, Topology 38 (1999) 1239-1264.
[9]F. R. Cohen and P. S. Selick, Splittings of two function spaces, Quar Jour *
*Math
Oxford 41 (1990) 145-153.
[10]D. M. Davis, From representation theory to homotopy groups, to appear in
Mem Amer Math Soc.
[11]D. M. Davis and M. Mahowald, Some remarks on v1-periodic homotopy groups,
London Math Society Lecture Notes 176 (1992) 293-327.
[12]________, Three contributions to the homotopy theory of the exceptional Lie
groups G2 and F4, Jour Math Soc Japan 43 (1991) 661-671.
[13]H. Hamanaka and S. Hara, The mod 3 homology of the space of loops on the
exceptional Lie groups and the adjoint action, Jour Math Kyoto Univ 37 (199*
*7)
441-453.
[14]L. Hodgkin, On the K-theory of Lie groups, Topology 6 (1967) 1-36.
[15]A. Kono and K. Kozima, The mod 2 homology of the space of loops on the
exceptional Lie group, Proc Royal Soc Edin 112A (1989) 187-202.
[16]M. Mahowald, The image of J in the EHP sequence, Annals of Math 116
(1982) 65-112.
[17]R. D. Thompson, The v1-periodic homotopy groups of an unstable sphere at
odd primes, Trans Amer Math Soc 319 (1990) 535-559.
[18]T. Watanabe, The Chern character homomorphism of the compact simply-
connected exceptional Lie group E6, Osaka Jour Math 28 (1991) 663-681.
Lehigh University, Bethlehem, PA 18015
E-mail address: dmd1@lehigh.edu