THE K-COMPLETION OF E_{6}

DONALD M. DAVIS

Abstract. We compute the 2-primary v_{1}-periodic homotopy groups of the K-completion of the exceptional Lie group E_{6}. This is done by computing the Bendersky-Thompson spectral sequence of E_{6}. We conjecture that the natural map from E_{6} to its K-completion induces an isomorphism in v_{1}-periodic homotopy, and discuss issues related to this conjecture.

1. Introduction

The p-primary v_{1}-homotopy groups of a space X, denoted v_{1}^{−1}π_{∗}(X; p) and defined in [11], are a localization of the portion of the homotopy groups of X detected by p-local K-theory. In [10], the author completed the determination of the oddprimary v_{1}-periodic homotopy groups of all compact simple Lie groups. The groups v_{1}^{−1}π_{∗}(X; 2) have been determined for X = SU(n) ([3]), Sp(n) ([5]), G_{2} ([12]), and F_{4} ([4]). Joint work of the author and Bendersky is very close to completing the computation of v_{1}^{−1}π_{∗}(SO(n); 2). That will leave E_{6}, E_{7}, and E_{8} to be determined, which would complete a program suggested to the author by Mimura in 1989. In this paper, we determine v_{1}^{−1}π_{∗}(Ê_{6}; 2), where X̂ denotes the K-completion of X, as defined in [6]. We conjecture that the natural map E_{6} → Ê_{6} induces an isomorphism in v_{1}^{−1}π_{∗}(−; 2).

In [6], Bendersky and Thompson defined the K-completion X̂ of 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 → X̂ induces an isomorphism of p-primary v_{1}-periodic homotopy groups. They also constructed a spectral sequence (BTSS) which, for many spaces X, converges to v_{1}^{−1}π_{∗}(X̂; p). It was shown in [7] and [4] that, localized at any prime, if X is K_{∗}-strongly spherically

Date: August 29, 2001.

1991 Mathematics Subject Classification. 55T15,55Q52,57T20.

Key words and phrases. K-theory, homotopy groups, exceptional Lie groups.

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resolved (KSSR), then X satisfies the CTP. This condition means that X can be built by fibrations from spheres S^{2n_{i}+1} such that K_{∗}X is built as a K_{∗}K-coalgebra as an extension of the K_{∗}S^{2n_{i}+1}.

In [4], it was proved that F_{4} satisfies the CTP at 2, and a general result ([4, 5.8,5.11,5.15] was proved which implies that if E_{6}/F_{4} satisfies the CTP, then so does E_{6}. Many standard functors of algebraic topology would lead one to expect that there is a fibration

S^{9} → E_{6}/F_{4} → S^{17},  (1.1)

which would imply that E_{6}/F_{4} is KSSR and then that E_{6} 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

ΩS^{9} → Ω(E_{6}/F_{4}) → ΩS^{17}

would also imply the CTP for E_{6}. It is not known whether such a fibration exists.

It was proved in [7] that, localized at an odd prime p, if X is K-algebraically spherically resolved and K-durable, then it satisfies the CTP. The first condition (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 v_{1}-periodic homotopy groups. Both E_{6}/F_{4} and E_{6} 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 satisfied.

Our main result is the following determination of v_{1}^{−1}π_{∗}(Ê_{6}; 2). As discussed above, the expectation is that this equals v_{1}^{−1}π_{∗}(E_{6}; 2).

Theorem 1.2. Let e = min(12, ν(ℓ − 18) + 5) and f = min(12, 2ν(ℓ − 3) + 8). Then

v_{1}^{−1}π_{8ℓ+d}(Ê_{6}; 2) ≈ \begin{cases}Z/2^{f} & d = −3 \\ Z/2^{f} ⊕ Z/2 & d = −2 \\ Z/2^{e} ⊕ Z/2 ⊕ Z/2 & d = −1, 0 \\ Z/2^{5} ⊕ Z/2 ⊕ Z/2 & d = 1 \\ Z/2^{5} ⊕ Z/2 & d = 2 \\ Z/2^{3} & d = 3, 4 \end{cases}

Here and throughout, ν(−) 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, positively

K-COMPLETION OF E_{6} 3

sloping lines the action of the Hopf map η (or the element h_{1} in the spectral sequence), and negatively sloping lines differentials in the spectral sequence, which implies that the elements which they connect do not survive to give nonzero homotopy classes. The dotted lines that look like h_{1} means that h_{1} is usually present, but perhaps not always.

Diagram 1.3.

[figure: s = 1; s = 4; 8ℓ − 2; 8ℓ; 8ℓ + 2; 8ℓ + 4; 2^{f}; 2^{6}; 2^{e}; 2^{4}; 2^{4}; 2^{6}; 2^{f}; 2^{e}]

The reader should observe that this chart, and the homotopy groups which it depicts, has a form very similar to the charts for v_{1}^{−1}π_{∗}(G_{2}) and v_{1}^{−1}π_{∗}(F_{4}) depicted in [4, 4.9]. The only difference is the orders of the groups on the 1- and 2-lines, and the lack of some exotic extensions.

The determination of the E_{2}-term of the spectral sequence is rather straightforward, given the general results of [4] and calculations of the Adams operations in K^{∗}(E_{6}) performed in [10]. This is performed in Section 2. The d_{3}-differentials are determined by showing that there are maps of spaces which relate the classes in question to classes in spaces where d_{3} is known. This is performed in Sections 3 and 4. In Section 5, we expand our discussion of whether E_{6} satisfies the CTP.

The author would like to thank Martin Bendersky and Fred Cohen for helpful comments on this work.

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2. The E_{2}-term

In this section we compute the E_{2}-term of the BTSS converging to v_{1}^{−1}π_{∗}(Ê_{6}). We are always using the v_{1}-periodic BTSS localized at 2, and all v_{1}-periodic homotopy groups are 2-primary.

In [4, 1.1], it was proved that for spaces whose K-homology and K-cohomology form nice exterior algebras,

E_{2}^{s,t}(X) ≈ \begin{cases}Ext_{A}^{s}(QK^{1}(X; Z_{2}^{∧})/ im(ψ^{2}), QK^{1}(S^{t}; Z_{2}^{∧})) & if t is odd \\ 0 & if t is even. \end{cases}

(2.1)

Here Q(−) denote the indecomposable quotient, and A is the category of stable 2-adic Adams modules.

It was proved in [4, 5.5] that simply-connected mod p finite H-spaces whose rational homology is associative satisfy the required hypothesis, and hence (2.1) applies to E_{6}. We remark again that although the spectral sequence is denoted as E_{2}(X), it is only known to converge to v_{1}^{−1}π_{∗}(X̂), and indeed even this is not known to be always true. It was proved in [7, 1.4] that, localized at any prime, if X is KASR, then the BTSS of X converges to v_{1}^{−1}π_{∗}(X̂).

As input for the BTSS, we need the Adams module K^{∗}(X). We use Z/2-graded K-theory. First is the general result of Hodgkin ([14]) which states that, for a simplyconnected compact Lie group G, K^{∗}(G) is an exterior algebra on generators in K^{1}(G) obtained from the fundamental representations of G. For E_{6}, there are six such generators. Proposition 2.2 follows from [10, 3.9], which gives ψ^{k} for all integers k. It can also be deduced from [18] after a lot of manipulation.

Proposition 2.2. There is a basis for QK^{1}(E_{6}) on which the matrices of ψ^{−1}, ψ^{2}, and ψ^{3} are given by

(ψ^{−1}) = (\begin{matrix}−1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & −1 \end{matrix}) , (ψ^{2}) = (\begin{matrix}2 & 0 & 0 & 0 & 0 & 0 \\ 7 & 16 & 0 & 0 & 0 & 0 \\ −33 & 0 & 32 & 0 & 0 & 0 \\ 7 & 0 & −8 & 128 & 0 & 0 \\ −4 & −1 & 4 & 64 & 256 & 0 \\ 1 & 0 & −1 & −24 & 0 & 2048 \end{matrix}) ,

K-COMPLETION OF E_{6} 5

(ψ^{3}) = (\begin{matrix}3 & 0 & 0 & 0 & 0 & 0 \\ 39 & 81 & 0 & 0 & 0 & 0 \\ −264 & 0 & 3^{5} & 0 & 0 & 0 \\ 147 & 0 & −162 & 3^{7} & 0 & 0 \\ −87 & −27 & 81 & 3^{7} & 3^{8} & 0 \\ 82 & 0 & −81 & −3^{7} & 0 & 3^{11} \end{matrix})

Here each column expresses ψ^{k} of a basis element in terms of the basis elements.

By [4, 3.10],

Ext_{A}^{1,2n+1}(M)^{#} ≈ M/ im(ψ^{−1} − (−1)^{n}, ψ^{3} − 3^{n}).  (2.3)

Thus the Pontryjagin dual of E_{2}^{1,2n+1}(E_{6}) is the abelian group presented by the 18×6 matrix

(\begin{matrix}(ψ^{2})^{T} \\ (ψ^{−1})^{T} − (−1)^{n}I \\ (ψ^{3})^{T} − 3^{n}I \end{matrix}) ,

where (ψ^{k})^{T} refers to the transpose of the matrices of 2.2. Using Maple, we simplify the relations, removing generators one-at-a-time. For example, we begin by pivoting on the 1 in position (1,6), and then removing the top row, which expresses the sixth generator in terms of the others, and the last column, since that generator is no longer required. It is convenient to do separate reductions for the two parities of n.

If n = 2k is even, it is most efficient to express entries in terms of R := 3^{2k} − 3^{72}. Thus 3^{n} is replaced by R + 3^{72} at the outset. Of course, it is only in hindsight that one can think to do this. We have ν(R) = ν(k − 36) + 3. We can eliminate all but the first column, and five nontrivial relations remain on this single generator. Omitting odd coefficients, these are

2^{12} + 2^{6}R
2^{25} + 2^{14}R
2^{12} + 2^{1}R
2^{16} + 2^{1}R + R^{2}
2^{13} + 2^{5}R

If ν(R) < 11, then the relation with the smallest 2-exponent is the third (or fourth), giving 2^{ν(R)+1} as the single relation, the order of the cyclic group. If ν(R) ≥ 11, then the smallest relation is the first, giving 2^{12} as the order of the cyclic group. These

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combine to give min(12, ν(k − 36) + 4) as the exponent of this cyclic 2-group, and this gives the 1-line groups in Diagram 1.3 when t − s = 8ℓ or 8ℓ + 4.

Similarly, if n = 2k + 1 is odd, it turns out to be most efficient to write the entries of the matrix in terms of Q := 3^{2k+1} − 3^{11}. Then ν(Q) = ν(k − 5) + 3. The matrix again reduces to one column, with relations (omitting odd coefficients)

2^{14} + 2^{6}Q

2^{18} + 2^{14}Q
2^{12} + 2^{8}Q + Q^{2}
2^{12} + 2^{6}Q + Q^{2}
2^{7}Q + 2^{3}Q^{2}

One easily checks that this gives min(12, 2ν(k − 5) + 6) as the minimal 2-exponent, and this yields the 1-line groups in t − s = 8ℓ ± 2. We summarize with

Proposition 2.4. The nonzero groups on the 1-line of the BTSS of E_{6} are given by

E_{2}^{1,2n+1}(E_{6}) ≈ \begin{cases}Z/2^{min(12,ν(k−36)+4)} & n = 2k \\ Z/2^{min(12,2ν(k−5)+6)} & n = 2k + 1. \end{cases}

Next, we deal with the 2-line groups. By [4, 3.10], if M is a finite stable 2-adic Adams module and θ_{n} = ψ^{3} − 3^{n}, then there is a short exact sequence 0 → coker(θ_{n}|Q_{n}(M)) → Ext_{A}^{2,2n+1}(M)^{#} → ker(θ_{n}| coker(ψ^{−1} − (−1)^{n}))|_{M} → 0,

(2.5)

where the functor Q_{n} is defined by

Q_{n} = \frac{ker(ψ^{−1} − (−1)^{n})}{im(ψ^{−1} + (−1)^{n})}.  (2.6)

We evaluate the right-hand part of (2.5). A mild surprise occurred here in that the group is not always cyclic, as had been the case in most other examples.

Proposition 2.7. For M = QK^{1}(E_{6})/ im(ψ^{2}),

ker(θ_{n}| coker(ψ^{−1}−(−1)^{n})) ≈ \begin{cases}Z/2 ⊕ Z/2^{min(11,ν(k−36)+3)} & n = 2k \\ Z/2^{min(12,2ν(k−5)+6)} & n = 2k + 1. \end{cases}

K-COMPLETION OF E_{6} 7

Proof. Since the kernel and cokernel of an endomorphism of a finite abelian group have the same order, the group has the same order as the corresponding 1-line group given in Proposition 2.4.

We begin with the case n = 2k + 1. We argue similarly to [4, 4.4]. Let \overline{M} := M/ im(ψ^{−1} + 1), \overline{M}_{2} = ker(·2|\overline{M}), and K = ker((ψ^{3} − 3^{n})|\overline{M}). We will show that dim(\overline{M}_{2} ∩ K) = 1, which implies that K is cyclic. Using 2.2, the abelian group \overline{M} can be presented by the matrix

(\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 2 & 7 & −33 & 7 & −4 & 1 \\ 0 & 16 & 0 & 0 & −1 & 0 \\ 0 & 0 & 32 & −8 & 4 & −1 \\ 0 & 0 & 0 & 128 & 64 & −24 \\ 0 & 0 & 0 & 0 & 0 & 2048 \end{matrix})

If {v_{1}, …, , v_{6}} denotes the basis on which this matrix presents relations, then a basis for \overline{M}_{2} is given by ψ^{2}(v_{4})/2 and ψ^{2}(v_{6})/2, the last two rows divided by 2. Using 2.2, we have in \overline{M}

(ψ^{3} − 1)(ψ^{2}(v_{6})/2) = ψ^{2}(ψ^{3} − 1)(v_{6})/2 = ψ^{2}(\frac{3^{11}−1}{2}v_{6}) ≡ 0 and similarly

(ψ^{3} − 1)(ψ^{2}(v_{4})/2) = ψ^{2}(\frac{3^{7}−1}{2}v_{4} + \frac{3^{7}}{2} v_{5} − \frac{3^{7}}{2} v_{6}) ≡ ψ^{2}(v_{6})/2 ≢ 0. Thus \overline{M}_{2} ∩ K = ⟨\frac{ψ^{2}v_{6}}{2} ⟩.

The case n = 2k is slightly more delicate. Let M̃ = M/ im(ψ^{−1}−1); its presentation matrix can be reduced to

(\begin{matrix}−2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 16 & 0 & −1 & −1 & 0 & 0 \\ 32 & 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & −1 \\ 2^{12} & 0 & 0 & 0 & 0 & 0 \end{matrix})  (2.8)

Thus M̃ is spanned by v_{1} and v_{3} of order 2^{12} and 2, respectively. Using 2.2 and the relations in M̃, we have

(ψ^{3} − 3^{2k})v_{3} ≡ (3^{5} − 3^{2k})v_{3} − 162(16v_{1} − v_{3}) + 81 · 32v_{1} ≡ 0,

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since 2v_{3} = 0. Thus K has a Z/2-summand generated by v_{3}. Similarly (ψ^{3}−3^{2k})v_{1} = (3−3^{2k})v_{1}+39·2v_{1}+147(16v_{1}−v_{3})−87·32v_{1} ≡ v_{3}−(3^{2k}+351)v_{1}. Thus v_{1} is not in K, but 2v_{1} has a chance. Since 351 = 2^{11}u − 3^{72} with u odd,

(ψ^{3} − 3^{2k})(2v_{1}) = 2(3^{72} − 3^{2k} − 2^{11}u)v_{1} = 2^{3+ν(2k−72)}u′v_{1}, with u′ odd, where 2^{12}v_{1} = 0. Thus 2^{max(0,12−(3+ν(2k−72)))}2v_{1} generates the other summand, which will have 2-exponent min(12 − 1, 3 + ν(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 E_{6} are given by

E_{2}^{2,2n+1}(E_{6}) ≈ Z/2 ⊕ \begin{cases}Z/2^{min(12,ν(k−36)+4)} & n = 2k \\ Z/2^{min(12,2ν(k−5)+6)} & n = 2k + 1. \end{cases}

Finally, we determine the eta-towers. This refers to elements in filtration > 2, all of which occur in families related by h_{1}, also known as η. ([4, 3.6]) With Q_{n} as defined in (2.6), we have, from [4, 3.10], for s > 2, a short exact sequence 0 → coker(θ_{n}|Q_{s+n}(M)) → Ext_{A}^{s,2n+1}(M)^{#} → ker(θ_{n}|Q_{s+n−1}(M)) → 0.

(2.10)

We establish

Proposition 2.11. For M = QK^{1}(E_{6})/ im(ψ^{2}), Q_{n}(M) ≈ Z/2, generated by v_{3} if n is odd, and by 2^{17}v_{1} if n is even.

Proof. If n is odd, Q_{n}(M) equals ker(ψ^{−1} + 1 : M̃ → M), where M̃ is presented by (2.8). Then M̃ ≈ Z/2 ⊕ Z/2^{12}, with generators v_{3} and v_{1}. From 2.2, we see that v_{3} ∈ ker(ψ^{−1} +1); however, (ψ^{−1} +1)(2^{11}v_{1}) = 2^{11}v_{2}, which is nonzero in M. One way to see that 2^{11}v_{2} ≠ 0 is to use pivoting to obtain the following alternate presentation

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for M, with columns still v_{1}, …, v_{6}.

(\begin{matrix}16 & −8 & −40 & 0 & 0 & 1 \\ 0 & 16 & 0 & 0 & −1 & 0 \\ 2 & 7 & −1 & −1 & 0 & 0 \\ 2^{7}5 & 2^{6}27 & −2^{6}17 & 0 & 0 & 0 \\ 0 & 2^{12} & 0 & 0 & 0 & 0 \\ 2^{18} & 0 & 0 & 0 & 0 & 0 \end{matrix})  (2.12)

If n is even, we want ker(ψ^{−1} − 1 : \overline{M} → M), where \overline{M} is presented by (2.12) with the second and fifth columns omitted. We check ψ^{−1} − 1 on the elements of order 2. We have 2^{17}v_{1} ↦ −2^{18}v_{1} + 2^{17}v_{2} = 0, while

2^{6}5v_{1} − 2^{5}17v_{3} ↦ −2^{7}5v_{1} + 2^{6}5v_{2} + 2^{6}17v_{3} = 2^{6}(5 + 27)v_{2} = 2^{11}v_{2}, which has order 2 in M.

If X is a topological space, let Q_{n}(X) = Q_{n}(QK^{1}(X)/ im(ψ^{2})).

Corollary 2.13. For s > 2, E_{2}^{s,2n+1}(E_{6})^{#} ≈ Z/2 ⊕ Z/2, with generators 2^{17}v_{1} and v_{3}.

Proof. We use (2.10) and Proposition 2.11, and 2.2 to show that θ_{n}(v_{3}) = 0 in Q_{od}(E_{6}) and θ_{n}(2^{17}v_{1}) = 0 in Q_{ev}(E_{6}). Indeed, θ_{n}(v_{3}) = 2Av_{3} − 162v_{4} + 81v_{5} − 81v_{6}, which is 0 in a group presented by (2.8), while θ_{n}(2^{17}v_{1}) = 0 in a group presented by (2.12) since there 2^{18}v_{1} = 0 and 2^{17}v_{i} = 0 for 2 ≤ i ≤ 6.

This, with 2.4 and 2.9, completes the determination of the E_{2}-term, which is as suggested by Diagram 1.3. The h_{1}-extensions from the 1-line will be discussed in Section 4.

3. d_{3}-differentials

In this section, we determine the d_{3}-differentials on the eta towers in the BTSS of E_{6}. An eta tower consists of elements in E_{2}^{s,2s+i} for s ≥ s_{0} connected by h_{1}. Here s_{0} = 1, 2, or 3. We denote by η_{i}(X) the Z_{2}-vector space of eta towers passing through E_{2}^{s,2s+i}(X) for s > 2. Note that d_{3} is a homomorphism from η_{i}(X) to η_{i−4}(X). With E_{2}^{s,t} depicted as usual in position (x, y) = (t − s, s), then η_{i}(X) is a tower of elements whose position satisfies x − y = i. Since η^{4} = 0 in homotopy, d_{3}-differentials must

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annihilate all eta towers, except for a few elements at the bottom of the target tower. By (2.10), if QK^{1}(X) consists of elements whose dimensions are all of the same parity, then so does η_{∗}(X), and η_{i}(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 d_{3} : η_{i}(X) → η_{i−4}(X) is nonzero.

We make heavy use of the fibration

F_{4} → E_{6} → EIV,  (3.1)

where EIV is the group quotient E_{6}/F_{4}. This fibration was studied quite thoroughly in [18]. It was observed there that

H^{∗}(EIV; Z) ≈ Λ(x_{9}, x_{17}),

and the Serre spectral sequence of (3.1) collapses. From this, the spectral sequence

H^{∗}(EIV; K^{∗}(F_{4})) ⇒ K^{∗}(E_{6})

implies that there is a short exact sequence

0 → QK^{1}(EIV) → QK^{1}(E_{6}) → QK^{1}(F_{4}) → 0,  (3.2)

with each of the three algebras K^{∗}(X) being exterior. The following key proposition implies that the Adams operations in K(EIV) are as they would be if (1.1) existed.

Proposition 3.3. There is a basis {y_{1}, y_{2}} for QK^{1}(EIV) on which for all k

ψ^{k}(y_{1}) = k^{4}y_{1} + u\frac{k^{4}−k^{8}}{16} y_{2},  (3.4)

with u a unit in Z_{(2)}, and ψ^{k}(y_{2}) = k^{8}y_{2}.

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 QK^{1}(EIV) = ker(QK^{1}(E_{6}) → QK^{1}(F_{4})). This must be the subspace of QK^{1}(E_{6}) spanned by v_{2} and v_{5} (in the basis used in Section 2). Perhaps the easiest way to see this is to use (ψ^{−1}) of 2.2, which shows that ψ^{−1} = 1 on this subspace, while it equals −1 on QK^{1}(F_{4}).

From 2.2, we find that 240v_{2} +v_{5} is an eigenvector for ψ^{2} with eigenvalue 16. Thus, considering the rational splitting of E_{6} as a product of spheres as in [10], we deduce that ψ^{k}(240v_{2} + v_{5}) = k^{4}(240v_{2} + v_{5}) and ψ^{k}(v_{5}) = k^{8}v_{5} for all integers k. Thus ψ^{k}v_{2} = k^{4}v_{2} + \frac{k^{4}−k^{8}}{240} v_{5}, from which follows the result with y_{1} = v_{2} and y_{2} = v_{5}.

K-COMPLETION OF E_{6} 11

In [4, 4.2], it was shown that there is a basis {x_{1}, x_{2}, x_{3}, x_{4}} of QK^{1}(F_{4}) on which ψ^{−1} = −1 and the matrix of ψ^{2} is

(\begin{matrix}2 & 0 & 0 & 0 \\ 3 & 32 & 0 & 0 \\ 1 & −8 & 128 & 0 \\ 0 & −1 & −24 & 2048 \end{matrix})

We easily verify the following result.

Proposition 3.5. In terms of the bases introduced, the exact sequence (3.2) is given by y_{1} ↦ v_{2}, y_{2} ↦ v_{5}, v_{1} ↦ −x_{1} − x_{2}, v_{3} ↦ x_{2}, v_{4} ↦ x_{3}, and v_{6} ↦ x_{4}.

Let \overline{K}(X) := QK^{1}(X)/ im(ψ^{2}). Since ψ^{2} acts injectively in our modules, the Snake Lemma applied to (3.2) yields a short exact sequence

0 → \overline{K}(EIV) → \overline{K}(E_{6}) → \overline{K}(F_{4}) → 0  (3.6)

and hence by (2.1) a long exact sequence 0 → E_{2}^{1,t}F_{4} → E_{2}^{1,t}E_{6} → E_{2}^{1,t}EIV → E_{2}^{2,t}F_{4} → E_{2}^{2,t}E_{6} → E_{2}^{2,t}EIV → E_{2}^{3,t}F_{4} → . (3.7)

In [4, 4.6], it was shown that

E_{2}^{s,t}(F_{4}) ≈ \begin{cases}Z/2 & s = 1, t ≡ 1(4) \\ Z/2^{f} & s = 1, t = 4k + 3 \\ Z/2^{f} ⊕ Z/2 & s = 2, t = 4k + 3 \\ Z/2 ⊕ Z/2 & s = 2, t ≡ 1(4) \\ Z/2 ⊕ Z/2 & s > 2, t odd \\ 0 & t even \end{cases}

with f = min(12, 2ν(k −5)+6). Using 3.3 and the method of [4, 4.11-4.12], we obtain

E_{2}^{s,t}(EIV) ≈ \begin{cases}Z/2 & s = 1, t ≡ 3(4) \\ Z/2^{e} & s = 1, t = 4k + 1 \\ Z/2^{e} ⊕ Z/2 & s = 2, t = 4k + 1 \\ Z/2 ⊕ Z/2 & s = 2, t ≡ 3(4) \\ Z/2 ⊕ Z/2 & s > 2, t odd \\ 0 & t even \end{cases}

with e = min(12, ν(k − 36 − 2^{8}) + 3). By just substituting the known E_{2}-groups into (3.7), we can deduce quite a bit about the morphisms; in particular, the rank

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of every morphism beginning with E_{2}^{2,t}EIV → E_{2}^{3,t}F_{4} is 1. This follows by counting the alternating sum of the exponents of the orders of the groups. Thus of the two eta-towers in E_{2}^{s,t}(E_{6}) for any odd value of t − 2s, one comes from F_{4} and the other maps nontrivially to EIV.

We need to know more specifically which classes map across in (3.7). For this, we use the following commutative diagram of exact sequences, where Q_{n}(X) is the functor of (2.6) applied to M = \overline{K}X.

⟶^{θ} Q_{s+n}(EIV) ⟶^{α_{1}} Ext_{A}^{s,2n+1}(\overline{K}EIV)^{#} ⟶^{β_{1}} Q_{s+n−1}(EIV) ⟶^{θ}
f_{1}↓  ^{f_{2}}↓  ^{f_{3}}↓
⟶^{θ} Q_{s+n}(E_{6}) ⟶^{α_{2}} Ext_{A}^{s,2n+1}(\overline{K}E_{6})^{#} ⟶^{β_{2}} Q_{s+n−1}(E_{6}) ⟶^{θ}
g_{1}↓  ^{g_{2}}↓  ^{g_{3}}↓
⟶^{θ} Q_{s+n}(F_{4}) ⟶^{α_{3}} Ext_{A}^{s,2n+1}(\overline{K}F_{4})^{#} ⟶^{β_{3}} Q_{s+n−1}(F_{4}) ⟶^{θ}

(3.8)

Proposition 3.9. In (3.8) with s + n even, g_{3}(v_{3}) = x_{2} pulls back nontrivially to g_{2}. If s + n is odd, then g_{3}(2^{17}v_{1}) = 2^{17}x_{1} pulls back nontrivially to g_{2}. If s + n is even (resp. odd), there is an element z ∈ Ext_{A}^{s,2n+1}(\overline{K}EIV)^{#} such that β_{1}(z) = 2^{7}y_{2} (resp. y_{1}) and f_{2}(z) = α_{2}(2^{17}v_{1}) (resp. α_{2}(v_{3})).

Proof. The exact sequence for F_{4} is given in [4, 4.5]. Actually, it is more convenient here to use the presentation

(\begin{matrix}0 & −32 & 8 & 1 \\ 2 & 3 & 1 & 0 \\ 2^{7}5 & 2^{6}27 & 0 & 0 \\ 2^{18} & 0 & 0 & 0 \end{matrix})  (3.10)

for \overline{K}(F_{4}) 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 Q_{ev}(F_{4}), and θ sends the first element to the second. This is a computation best done in Maple, using the rows of (3.10) to reduce (ψ^{3} − 1)(2^{6}5x_{1} + 2^{6}27x_{2}), obtained from [4, 4.2], to 2^{17}x_{1}. As in [4, 4.5], Q_{od}(F_{4}) = ⟨x_{1}, x_{2} ∼ x_{3}⟩ with θ sending the first to the second. Now the g_{2}-part of the proposition in straightforward, using Proposition 3.5.

We have Q_{od}(EIV) ≈ Z/2 with generator 2^{7}y_{2}, and Q_{ev}(EIV) ≈ Z/2 with generator y_{1}. One readily checks, using 3.5 and 2.11, that f_{1} = 0 and f_{3} = 0 in (3.8). Thus

K-COMPLETION OF E_{6} 13

the morphism f_{2} must be as claimed because of our previous observation that it has rank 1.

We can deduce d_{3} on half of the eta towers in E_{6} from its behavior in F_{4}. By Corollary 2.13, each η_{od}(E_{6}) has basis {2^{17}v_{1}, v_{3}}.

Proposition 3.11. The differential d_{3} : η_{8ℓ+1}(E_{6}) → η_{8ℓ−3}(E_{6}) sends the v_{3}-tower to the v_{3}-tower. The differential d_{3} : η_{8ℓ+7}(E_{6}) → η_{8ℓ+3}(E_{6}) sends the 2^{17}v_{1}-tower to the 2^{17}v_{1}-tower.

Proof. It was shown in [4, 4.13] that in the BTSS of F_{4}, d_{3} : η_{8ℓ+1}(F_{4}) → η_{8ℓ−3}(F_{4}) is the “identity map,” while d_{3} : η_{8ℓ+3}(F_{4}) → η_{8ℓ−1}(F_{4}) sends the x_{1}-tower to the x_{1}-tower, and d_{3} : η_{8ℓ+7}(F_{4}) → η_{8ℓ+3}(F_{4}) sends the 2^{17}x_{1}-tower to the 2^{17}x_{1}-tower. We dualize (3.8) and use 3.9 to see that the v_{3}-tower is in the image of η_{4∗+1}(F_{4}) → η_{4∗+1}(E_{6}), and that η_{4∗+3}(F_{4}) → η_{4∗+3}(E_{6}) sends the 2^{17}x_{1}-tower to the 2^{17}v_{1}-tower. Naturality of d_{3} implies the result.

By using the inclusion map S^{9} → EIV, we deduce the third family of d_{3}-differentials.

Proposition 3.12. The differential d_{3} : η_{8ℓ+3}(E_{6}) → η_{8ℓ−1}(E_{6}) sends the v_{3}-tower to the v_{3}-tower.

Proof. Using the dual of f_{2} in Proposition 3.9, it suffices to show that d_{3} : η_{8ℓ+3}(EIV) → η_{8ℓ−1}(EIV) sends the y_{1}-tower to the y_{1}-tower. The map j : S^{9} → EIV induces

QK^{1}EIV ⟶^{j^{∗}} QK^{1}S^{9}

satisfying j^{∗}(y_{1}) = g and j^{∗}(y_{2}) = 0. The generator g ∈ QK^{1}S^{9} gives rise to the stable eta towers in BTSS(S^{9}), and by [1, p.58] or [3, p.488] d_{3} : η_{8ℓ+3}(S^{9}) → η_{8ℓ−1}(S^{9}) sends the stable eta tower to the stable eta tower. Our conclusion follows by naturality using the map j.

The final d_{3} on eta towers requires a more elaborate argument. The conclusion is that d_{3} is as it would be if the fibration (1.1) existed. Our desired result is

Proposition 3.13. The differential d_{3} : η_{8ℓ+1}(E_{6}) → η_{8ℓ−3}(E_{6}) sends the 2^{17}v_{1}-tower to the 2^{17}v_{1}-tower.

Proof. By Proposition 3.9, it suffices to show η_{8ℓ+1}(EIV) → η_{8ℓ−3}(EIV) sends the 2^{7}y_{2}-tower to the 2^{7}y_{2}-tower. This is a consequence of the following two propositions.

Proposition 3.14. There is an isomorphism of BTSSs

E_{r}^{s,t}(ΩEIV) ≈ E_{r}^{s,t+1}(EIV)

for r ≥ 2.

Proposition 3.15. The differential d_{3} : η_{8ℓ}(ΩEIV) → η_{8ℓ−4}(ΩEIV) sends the 2^{7}y_{2}- tower to the 2^{7}y_{2}-tower.

Proof of Proposition 3.14. We need the following result.

Proposition 3.16. The algebra K_{∗}(ΩEIV) is a polynomial algebra on two classes in K_{0}(−).

Proof. We first note that H_{∗}(ΩEIV) is a polynomial algebra on classes in H_{8}(−) and H_{16}(−). To see this, we first use the computations of the rings H_{∗}(ΩF_{4}; R) and H_{∗}(ΩE_{6}; R) for R = Z/2 in [15, 2.3] and for R = Z/3 in [13, Thm.1], which imply that H_{∗}(ΩEIV; R) is polynomial on 8- and 16-dimensional generators. With coefficients Q or Z/p for p > 3, this is also true because the localizations of F_{4} and E_{6} are products of spheres whose dimensions are well known. The result with integral coefficients follows by standard methods.

Now the Atiyah-Hirzebruch spectral sequence H_{∗}(ΩEIV; K_{∗}) ⇒ K_{∗}(ΩEIV) implies the result for K_{∗}(−).

Returning to the proof of 3.14, we adapt an argument first used in [2, 6.1] to show that for the BP -based unstable Novikov spectral sequence (UNSS) E_{2}^{s,t}(ΩS^{2n+1}) ≈ E_{2}^{s,t+1}(S^{2n+1}). It was shown in [5, 3.3] that the same argument works if S^{2n+1} is replaced by a space Y for which BP_{∗}(ΩY) is the polynomial algebra on classes {y_{2i}}, and BP_{∗}Y is isomorphic to an exterior algebra on the primitive classes σy_{2i}. It was noted in [6, 4.12] that these arguments can be adapted to the K-based BTSS. This, with Proposition 3.16, yields the proposition when r = 2.

K-COMPLETION OF E_{6} 15

To prove the result for all r, we note that the isomorphism of E_{2}-terms is induced by a map of towers. To see this, we use the following natural map of augmented cosimplicial spaces, where K(X) = Ω^{∞}(K ∧ Σ^{∞}X).

[figure: ΩX; ΩKX; ΩK(KX); ΩK^{3}X; ⋯; ΩX; KΩX; K(K(ΩX)); K^{3}ΩX; ⋯]

Applying π_{∗}(−) and taking homology of the alternating sum to the first row yields E_{2}^{∗,∗}(ΩX), and doing this to the second yields E_{2}^{∗,∗−1}(X). The induced morphism in homology is the E_{2} isomorphism observed above. But these cosimplicial spaces give rise, by the Tot construction, to the towers that define the entire spectral sequence, and so the morphism induces a morphism of spectral sequences.

Proof of Proposition 3.15. Let F denote the fiber of the inclusion map S^{9} → EIV. The Serre spectral sequence of this fibration shows that H^{∗}F, like H^{∗}(ΩS^{17}), is a divided polynomial algebra on a 16-dimensional class. Thus H_{∗}F = ⟨x_{16i}⟩ with

ψ(x_{16i}) = ∑ \binom{i}{j}x_{16j} ⊗ x_{16(i−j)}.

It follows that the Z/2-graded K_{∗}F has the same coalgebra structure, although the grading is lost. Thus there is an abstract isomorphism of coalgebras K_{∗}F ≈ K_{∗}(ΩS^{17}).

This implies that the fibration

ΩS^{9} → ΩEIV → F

induces a relatively injective extension sequence

0 → K_{∗}(ΩS^{9}) → K_{∗}(ΩEIV) ⟶^{p_{∗}} K_{∗}(F) → 0,

and hence, by [2, 4.3], a long exact sequence of BTSS, commuting with d_{3}

→ E_{2}^{s}(ΩS^{9}) → E_{2}^{s}(ΩEIV) ⟶^{p_{∗}} E_{2}^{s}(F) → E_{2}^{s+1}(ΩS^{9}) → .

We remark here that the notion of relatively injective extension sequence was defined in [4, 5.14]. It is the notion which was intended in [2, 4.3].

The Hurewicz Theorem gives a map f : S^{16} → F, and we have the map p : ΩEIV → F from the above fibration. Both of these maps induce morphisms of

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BTSS, commuting with d_{3}. The image under p_{∗} of the eta towers 2^{7}y_{2} equal the image under f_{∗} of eta towers in S^{16} which map, in the EHP sequence, to the unstable eta towers in η_{4∗+1}(S^{17}). By [1, p.58] or [3, p.488], the d_{3}-differential on the unstable eta towers in S^{17} is nonzero from η_{8ℓ+1}(S^{17}) to η_{8ℓ−3}(S^{17}). Hence it is nonzero from η_{8ℓ}(S^{16}) to η_{8ℓ−4}(S^{16}), hence also in F, and thence in ΩEIV.

4. Fine Tuning

In this section we determine the d_{3}-differentials from the 1-line, prove Proposition 2.9, and show that there are no exotic extensions (·2) in the BTSS.

In general, the h_{1}-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 d_{3} on the 1-line is to determine d_{3} on the eta towers and then use the h_{1}-action from the 1-line to the eta towers to deduce d_{3} on the 1-line. A different method was used for Sp(n) in [5] and illustrated in [5, 4.3] the subtle things that can happen. In that case, the relevant 2-line group contained some classes supporting d_{3}-differentials and some which did not, and we argued indirectly to see whether d_{3}x was nonzero, which was equivalent to determining whether h_{1}x contained any classes of the first type.

In our situation here, the 1-line classes in E_{6} map to or from classes in EIV or F_{4} on which h_{1} and d_{3} are known, from which we deduce the following result, which will be proved along with Proposition 2.9.

Proposition 4.1. The h_{1}-extensions and d_{3}-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_{A}^{∗,2n+1})^{#} and (Ext_{GInv}^{∗,2n+1})^{#} induced by (3.6). Here GInv denotes the category of profinite abelian groups with involution. We will also use Inv, the category of abelian groups with involution.

The analysis is much more delicate when n = 2k is even, and so we focus on this case. Some details of the proof (but not final conclusions) are slightly different when ν(k − 36) = 8 due to the ν(k − 36 − 2^{8}) which occurs in E_{2}(EIV), so we assume ν(k − 36) ≠ 8 to simplify the exposition.

K-COMPLETION OF E_{6} 17

The exact sequences (2.5) and (2.10) are obtained in [4, §3] from a long exact sequence

← Ext_{A}^{s}(M, N)^{#} ← Ext_{GInv}^{s}(M, N)^{#} ⟵^{ψ_{M}^{3} −ψ_{N}^{3}} Ext_{GInv}^{s}(M, N)^{#} ← Ext_{A}^{s+1}(M, N)^{#} ←

and isomorphisms

Ext_{GInv}^{s+1,2n+1}(M)^{#} = Ext_{GInv}^{s+1}(M, QK^{1}S^{2n+1})^{#}
≈ Ext_{Inv}^{s}(Z_{(2)}^{((−1)^{n})}, M^{#})^{#} ≈ \begin{cases}Q_{s+n+1}(M) & s > 0 \\ M/(ψ^{−1} − (−1)^{n}) & s = 0. \end{cases}

Then

0 ← Ext_{GInv}^{1,4k+1}(\overline{K}F_{4})^{#} ← Ext_{GInv}^{1,4k+1}(\overline{K}E_{6})^{#} ← Ext_{GInv}^{1,4k+1}(\overline{K}EIV)^{#}
← Ext_{GInv}^{2,4k+1}(\overline{K}F_{4})^{#} ← Ext_{GInv}^{2,4k+1}(\overline{K}E_{6})^{#} ← Ext_{GInv}^{2,4k+1}(\overline{K}EIV)^{#} ← Ext_{GInv}^{3,4k+1}(\overline{K}F_{4})^{#}

is

0 ← \overline{K}F_{4}/(ψ^{−1} − 1) ← \overline{K}E_{6}/(ψ^{−1} − 1) ← \overline{K}EIV/(ψ^{−1} − 1)
← Q_{ev}(\overline{K}F_{4}) ← Q_{ev}(\overline{K}E_{6}) ← Q_{ev}(\overline{K}EIV) ← Q_{od}(\overline{K}F_{4}).

Using results and notation from Sections 2 and 3, this sequence becomes, with x_{c} = 2^{6}5x_{1} + 2^{5}27x_{2}, 0 ← ⟨x_{2}⟩ ← ⟨v_{3}, v_{1}⟩ ← ⟨y_{1}⟩ ← ⟨x_{c}, 2^{17}x_{1}⟩ ← ⟨2^{17}v_{1}⟩ ← ⟨y_{1}⟩ ← ⟨x_{1}, x_{2}⟩, with all elements having order 2 except v_{1} and y_{1}, which have order 2^{12}. Using 3.5, the explicit morphisms, yielding an exact sequence, are

\begin{aligned}x_{2} & ↩ & v_{3} & & 0 & ↩ & 2^{17}x_{1} & ↩ & 2^{17}v_{1} \\ x_{1} + x_{2} & ↩ & v_{1} & & 2^{11}y_{1} & ↩ & x_{c} & & 0 & ↩ & y_{1} & ↩ & x_{1} \\ & & 2v_{1} & ↩ & y_{1} \end{aligned} (4.2)

With ν = min(12, ν(k − 36) + 3), the nonzero occurrences of ψ^{3} − 3^{2k} on the groups in this sequence are x_{1} ↦ x_{2}, v_{1} ↦ 2^{ν}v_{1} + v_{3}, y_{1} ↦ 2^{ν}y_{1}, x_{c} ↦ 2^{17}x_{1}, and x_{1} ↦ x_{2}. 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 E_{2}^{s}(X) for Ext_{A}^{s,4k+1}(\overline{K}X), and denote by x(2^{e}) an element x of order 2^{e}. Elements not followed by parentheses have order

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2.

x_{1}  ←  v_{1}(2^{ν+1})  ⟵^{·2} y_{1}(2^{ν})
↓  ↓  ↓
E_{2}^{1}(F_{4})^{#} ←  E_{2}^{1}(E_{6})^{#}  ← E_{2}^{1}(EIV)^{#} ←
x_{c}  2^{17}v_{1}  y_{1}  ←  x_{1}
↓  ↓  ↓  ↓
E_{2}^{2}(F_{4})^{#} ←  E_{2}^{2}(E_{6})^{#}  ← E_{2}^{2}(EIV)^{#} ← E_{2}^{3}(F_{4})^{#}
↓  ↓  ↓
x_{2}  ← v_{3}, 2^{12−ν}v_{1}(2^{ν}) ⟵^{2} 2^{12−ν}y_{1}(2^{ν})

In order for the E_{2}^{s}(−)^{#} sequence to be exact, 2^{12−ν}v_{1} must hit x_{c} in E_{2}^{2}(F_{4})^{#}, and 2^{11}y_{1} must hit 2^{17}v_{1} in E_{2}^{2}(E_{6})^{#}. The latter implies the nontrivial extension in E_{2}^{2}(E_{6})^{#}, that it contains a Z/2^{ν+1} with 2^{17}v_{1} the element of order 2 in this summand. This concludes the proof of 2.9.

The above analysis showed E_{2}^{1,4k+1}(E_{6})^{#} ← E_{2}^{1,4k+1}(EIV)^{#} injective; hence

E_{2}^{1,4k+1}(E_{6}) → E_{2}^{1,4k+1}(EIV)

is surjective. In 3.9 and the proof of 3.12, we showed that the stable eta tower in E_{2}^{∗,4k+2∗−1}(E_{6}) maps to the one in EIV which comes from S^{9}. If ν(k) ≤ 1, E_{2}^{1,4k+1}(S^{9}) → E_{2}^{1,4k+1}(EIV) is an isomorphism. So the h_{1}-extension and d_{3}-differential from E_{2}^{1,8ℓ+5}(S^{9}) implies the same in E_{6}. A nonzero h_{1}-extension from E_{2}^{1,8ℓ+1}(E_{6}) is implied when ℓ is odd. It is not so important because d_{3} must be 0 on E_{2}^{1,8ℓ+1}(E_{6}) since h_{1}^{2} times the Z/2 in (8ℓ+1, 2) is in the image of d_{3}. It is likely that an adaptation of [4, 3.7] to modules which do not satisfy ψ^{−1} = −1 would allow the determination of h_{1} on E_{2}^{1,8ℓ+1}(E_{6}).

Since E_{2}^{1,4k+3}(F_{4}) → E_{2}^{1,4k+3}(E_{6}) is an isomorphism, the h_{1}-extensions and d_{3}- differentials from E_{2}^{1,4k+3}(E_{6}) follow from those in F_{4}. In [4, 4.14], it was shown that h_{1} on E_{2}^{1,4k+3}(F_{4}) hits the stable eta tower iff ν(k − 5) ≠ 3 and hits the unstable eta tower iff ν(k − 5) = 3. This implies the nonzero h_{1} and d_{3} from E_{2}^{1,8ℓ+3}(E_{6}) pictured in Diagram 1.3, and that h_{1} is nonzero from E_{2}^{1,8ℓ−1}(E_{6}) unless ℓ ≡ 7 mod 8, in which case it is 0. The latter is because the unstable eta tower in F_{4} in this dimension maps to 0 in E_{6}. That h_{1}x is sometimes 0 here is mildly interesting because in [4, 3.8] it was shown that if ψ^{−1} = −1 then h_{1} acts injectively on E_{2}^{1}(X)/2.

K-COMPLETION OF E_{6} 19

The analysis of exact sequences earlier in this proof implied that E_{2}^{2,4k+1}(E_{6}) → E_{2}^{2,4k+1}(EIV) sends the Z/2^{e} summand surjectively, and that the generator is dual to 2^{17}v_{1}, which is the name of the unstable eta tower whose d_{3}-differential is described in 3.13. The h_{1}-extension and d_{3}-differential on it follow immediately.

The exact sequence in Ext_{A}^{1,4k+3}(−) induced by (3.6) easily implies that E_{2}^{2,4k+3}(F_{4}) → E_{2}^{2,4k+3}(E_{6}) sends the Z/2^{f} bijectively. The h_{1}-extension (for all k) and d_{3}-differential when k is odd follow from those in F_{4} established in [4].

Finally, we have the following result about extensions in the spectral sequence.

Proposition 4.3. There are no nontrivial extensions (·2) from E_{2}^{s,t}(E_{6}) to E_{2}^{s+2,t+2}(E_{6}).

Proof. For s = 1 or 2, the group E_{2}^{s,8ℓ+3}(E_{6}) ≈ Z/2^{6} is mapped isomorphically onto from E_{2}^{s,8ℓ+3}(F_{4}). In [4, 4.14], it is shown that when s = 1 (resp. s = 2) there is a nonzero extension into the element of E_{2}^{s+2,8ℓ+5}(F_{4}) which is part of the stable (resp. unstable) eta tower. In Proposition 3.9, it is proved that these eta towers map trivially to E_{2}^{s+2,8ℓ+5}(E_{6}), hence this extension is 0 in E_{6}.

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 E_{2}^{s,8ℓ+1}(E_{6}) is in the image from E_{2}^{s,8ℓ+1}(F_{4}), which does not support an extension in [4, 4.9].

5. The CTP for E_{6}

In this brief section, we mention several issues related to the CTP for E_{6}. In other words, how might we prove that the groups v_{1}^{−1}π_{∗}(Ê_{6}), which we have computed, are in fact isomorphic to v_{1}^{−1}π_{∗}(E_{6}), which is what we really want?

There are no known examples of spaces which do not satisfy the CTP. On the other hand, the only spaces which are known to satisfy it are certain spaces which can, 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 E_{6} satisfies the CTP. However, the proof of the result of [7] relies on much delicate machinery developed at the odd primes by Bousfield, especially in [8]. One crucial difference between the odd primes and the

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prime 2 is that Ext_{A}^{s}(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_{1}^{−1}π_{∗}(X) is the possibility that there could be elements in v_{1}^{−1}π_{∗}(X) not seen by the spectral sequence because they correspond to a family of elements in π_{n_{i}}(X) for n_{i} → ∞, related to one another by ·v_{1}^{e} and having increasing filtrations. The algebraic ·v_{1} operation in the UNSS or BTSS preserves filtration, but the homotopy-theoretic operation can increase filtration. For S^{2n+1}, the E_{∞}-term of the v_{1}-periodic BTSS yields exactly the v_{1}-periodic homotopy classes, which we know because they were calculated by another method in [16] (p = 2) and [17] (p odd). Spaces built from odd spheres, or perhaps their loop spaces, by fibrations can then guarantee nonexistence of undetected v_{1}-periodic families by exactness properties. The proof in [7] that at the odd primes X KASR and K-durable implies the CTP for X ultimately boils down to building the localization X_{K} from various S_{K}^{2n+1}.

The isomorphism E_{2}^{s,t}(S^{2n+1}) ≈ E_{2}^{s,t−1}(ΩS^{2n+1}) implies that if a space is nicely fibered by ΩS^{2n+1}’s, then it satisfies the CTP. For spaces fibered by Ω^{2}S^{2n+1}, it is not so clear.

If it could be proved that F, the fiber of S^{9} → EIV considered in the proof of 3.15, has the same homotopy type as ΩS^{17}, then the CTP for E_{6} could be deduced. In [9, 2.1], a map Ω^{2}S^{17} ⟶^{f} ΩS^{9} was constructed. If it could be shown that the composite

Ω^{2}S^{17} ⟶^{f} ΩS^{9} ⟶^{Ωi} ΩEIV

is null-homotopic, then one could deduce existence of a fibration

Ω^{2}S^{9} → Ω^{2}EI → Ω^{2}S^{17}.

It is likely that such a fibration would imply the CTP for E_{6}, although a detailed argument has not been produced.

References

[1] M. Bendersky, The v_{1}-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.

[3] M. Bendersky and D. M. Davis, 2-primary v_{1}-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, v_{1}-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 Conference Proceedings.

[8] A. K. Bousfield, The K-theory localization and v_{1}-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 v_{1}-periodic homotopy groups, London Math Society Lecture Notes 176 (1992) 293-327.

[12] , Three contributions to the homotopy theory of the exceptional Lie groups G_{2} and F_{4}, 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 (1997) 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 v_{1}-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 simplyconnected exceptional Lie group E_{6}, Osaka Jour Math 28 (1991) 663-681.

Lehigh University, Bethlehem, PA 18015

E-mail address: dmd1@lehigh.edu