A STABLE APPROACH TO AN UNSTABLE HOMOTOPY SPECTRAL SEQUENCE

MARTIN BENDERSKY AND DONALD M. DAVIS

Abstract. Recently, Bendersky and Thompson introduced a spectral sequence which, for many spaces X, converges to the v_{1}- periodic homotopy groups of X. It is proved that the E_{2}-term of this spectral sequence is often given by Ext in the category of stable p-adic Adams modules of QK^{1}(X; Z_{p}^{∧})/ im(ψ^{p}). We compute this spectral sequence when p = 2 and X is the exceptional Lie group F_{4}, yielding as a new result the 2-primary v_{1}-periodic homotopy groups of F_{4}. 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 simplyconnected spaces X; it converges, on a class of spaces which includes finite Hspaces and K_{∗}-strongly spherically resolved spaces, to the homotopy groups of the K-completion of X, denoted X̂. This convergence, and other convergence issues, will be discussed in Section 5.

We will work with the v_{1}-periodic version of this spectral sequence, localized 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 E_{2}-term of the BTSS, denoted E_{2}(X), can, for many spaces X, be computed directly from the indecomposables QK^{1}(X; Z_{p}^{∧}) and the Adams operations ψ^{k}. This should be contrasted with the method used in [10] to compute E_{2}, which involved delicate manipulations with the unstable cobar complex.

Let A denote the abelian category of stable p-adic Adams modules.([12, 2.6]) An object in A is a p-profinite abelian group with Adams operations ψ^{k} for k ∈ Z − pZ,

Date: August 23, 2001.

1991 Mathematics Subject Classification. 55T15,55Q52.

Key words and phrases. homotopy groups, Adams operations, exceptional Lie groups.

2 BENDERSKY AND DAVIS

satisfying certain axioms. Our main theorem applies to simply-connected spaces X for which there is a torsion-free K_{∗}K-subcomodule M ⊂ P K_{odd}(X; Z) such that K_{∗}(X; Z) ≈ Λ(M) as (Z-graded) K_{∗}(K)-coalgebras, while the Z/2-graded K^{∗}(X; Z_{p}^{∧}) is isomorphic to Λ̂(M_{1} ⊗ Z_{p^{∞}})^{#} with ψ^{p} is monic on QK^{1}(X; Z_{p}^{∧}). Here (−)^{#} denotes Pontrjagin duality, P (−) denotes the primitives in a coalgebra, and Λ an exterior algebra. We prove in Proposition 5.5 that simply-connected mod p finite H-spaces whose rational homology is associative and K_{∗}-strongly spherically resolved spaces (see 5.3) satisfy these conditions.

Theorem 1.1. If X is a space satisfying the above conditions, then the E_{2}-term of the BTSS satisfies

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

We prove Theorem 1.1 in Section 2. In Section 3, we develop a method of computing Ext_{A}(−, −), especially when p = 2, which case is much more delicate than the oddprimary. These results build upon earlier work of Bousfield ([12],[14],[15]) and of the authors ([7]).

In Section 4, we apply these results to compute the BTSS and v_{1}-periodic homotopy groups of the exceptional Lie group F_{4}. Determination of d_{3}-differentials requires comparison with d_{3} in other spaces related to F_{4} by fibrations. We obtain

Theorem 1.2. Let e = min(12, 2ν(i − 3) + 8). Then the 2-primary v_{1}-periodic homotopy groups of F_{4} are given by

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

Here, and throughout, ν(−) denotes the exponent of 2 in an integer. The 2ν(−) occurring in the answer is a surprise, compared to previous computations for other Lie groups.

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 3

Theorem 1.2 leaves SO(n), E_{6}, E_{7}, and E_{8} as the only compact simple Lie groups whose 2-primary v_{1}-periodic homotopy groups have not been computed. In [19], the second author completed the computation of all odd-primary v_{1}-periodic homotopy groups of all compact simple Lie groups. The authors expect to use Theorem 1.1 in a future paper to compute v_{1}^{−1}π_{∗}(SO(n); 2).([9])

The distinction between v_{1}^{−1}π_{∗}X, which we want, and v_{1}^{−1}π_{∗}X̂, which the BTSS often computes, is presented in the following definition.

Definition 1.3. A space X satisfies the Completion Telescope Property (CTP) if the natural map X → X̂ induces an isomorphism in v_{1}^{−1}π_{∗}(−).

It follows easily from [10, 4.12] and [11, 1.4] that S^{2n+1} and ΩS^{2n+1} satisfy the CTP. In Section 5, we prove

Theorem 1.4. The spaces S^{2n}, ΩS^{2n}, G_{2}, and F_{4} 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 not-yet-published work.

2. Proof of Theorem 1.1

We shall adopt the following notation and conventions. K^{∗}(−) will denote Z/2graded K-cohomology with Z_{p}^{∧} coefficients, while K_{∗}(−) denotes Z-graded K-homology with Z coefficients. This difference in gradings is primarily for convenience of exposition: K-homology needs Z-gradings for its unstable condition, while our use of K-cohomology is primarily in K^{1}(−). The Bott element v_{1} ∈ K_{2} ≈ K^{−2} gives isomorphisms

K_{i}(X) ⟶^{v_{1}} K_{i+2}(X), K^{i}(X) ⟶^{v_{1}} K^{i−2}(X)

for all integers i, allowing passage between Z/2-graded and Z-graded theories. Coactions K_{∗}X ⟶^{ψ} K_{∗}K ⊗ K_{∗}X and Adams operations ψ^{k} in K^{∗}(X) are passed along by ψ(v_{1}x) = v_{1}ψ(x) and ψ^{k}(v_{1}x) = kv_{1}ψ^{k}(x).

Note that if K_{i}(X) is torsion-free, then K^{i}(X) and K_{i}(X) ⊗ Z_{p^{∞}} are Pontrjagin dual to one another. We will denote (M ⊗ Z_{p^{∞}})^{#} by M^{#} for notational simplicity.

4 BENDERSKY AND DAVIS

Recall that a profinite abelian group or K^{∗}-module is a stable p-adic Adams module if it admits operations ψ^{k} for k ∈ Z − pZ satisfying the properties of [12, 2.6]. A p-adic Adams module admits operations ψ^{k} for all k ∈ Z as in [12, 2.8]. If M is a stable p-adic Adams module, then the free p-adic Adams module, F̃(M), generated by M is defined as follows.

Definition 2.1. ([12, 3.1]) As abelian groups or K^{∗}-modules,

F̃(M) = M × M × ⋯,

with Adams operations defined by

ψ^{k}(x_{1}, x_{2}, …) = \begin{cases}(ψ^{k}x_{1}, ψ^{k}x_{2}, …) & if k ≢ 0 mod p \\ (0, x_{1}, x_{2}, …) & if k = p. \end{cases}

The following result plays a key role in the proof of Theorem 1.1.

Proposition 2.2. There is an isomorphism of Z/2-graded p-adic Adams modules

QK^{∗}(SU) ≈ F̃(Γ),

where Γ 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 ∧ S^{1})⟨2⟩, the 1-connected cover of K ∧ S^{1} localized at Z/p. We have Ω^{∞}E = SU = U⟨2⟩. Theorem 3.3 of [12] asserts that

K^{∗}(Ω^{∞}E) ≈ Λ̂F̃K^{1}(E).  (2.3)

Since K^{1}(E) ≈ Γ 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) = K̃^{∗}(ΣCP^{∞}) = K^{∗}{ξ_{1}, ξ_{2}, … }, the free K^{∗}-module with basis ξ_{k} = ξ^{k} − 1, with ξ the canonical line bundle over CP^{∞}. Note that ξ_{k} has grading 1 in K^{∗}(SU). The Adams operations act by ψ^{r}ξ_{k} = ξ_{rk} for r ≥ 0. For a ≥ 0, define M_{a} to be the K^{∗}-submodule of K^{∗}{ξ_{1}, ξ_{2}, … } generated by {ξ_{p^{a}k}|(k, p) = 1}. Denote M_{0} by Γ.

We claim that Γ is a projective stable p-adic Adams module on one generator. First note that Γ admits Adams operations ψ^{k} for positive k prime to p. Since Γ ≈

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 5

K^{∗}(CP^{∞})/ im(ψ^{p}), it also admits the operation ψ^{−1}, since ψ^{−1}(im(ψ^{p})) ⊂ im(ψ^{p}). Next observe that Γ has one generator as stable Adams module since ξ_{k} = ψ^{k}(ξ_{1}) for positive k prime to p. Finally, to show that Γ is projective, let B ⟶^{ϕ} C be a surjection of stable Adams modules and Γ ⟶^{g} C a morphism of stable Adams modules. Define Γ ⟶^{\overline{g}} B by \overline{g}(ξ_{1}) = b_{0} for some b_{0} satisfying ϕ(b_{0}) = g(ξ_{1}) and \overline{g}(ξ_{k}) = ψ^{k}b_{0} for positive k prime to p. We must show that \overline{g} also respects the action of ψ^{−1}.

For this, note that the p-adic stable Adams modules Γ and B are inverse limits of finite Adams modules. By a property of stable p-adic Adams modules ([12, 2.6]), for each n, there exists m such that ψ^{k} ≡ ψ^{k+m} mod p^{n} in Γ and B for all integers k. Thus \overline{g} commutes with ψ^{−1} mod p^{n} for all positive integers n. Passing to the inverse limit shows that \overline{g} respects the action of ψ^{−1} .

The map Γ → M_{a} defined on generators by ξ_{k} ↦ ξ_{p^{a}k} is an isomorphism of stable p-adic Adams modules. Thus

QK^{∗}(SU) = M_{0} × M_{1} × ⋯ ≈ Γ × Γ × · · · .

The action of ψ^{p} on Γ × Γ × Γ × ⋯ is given on generators by

ψ^{p}(ξ_{k_{1}}, ξ_{k_{2}}, …) = (0, ξ_{k_{1}}, ξ_{k_{2}}, …).

Here we have used that ψ^{p}(ξ_{k}) in the ith factor corresponds to ψ^{p}(ξ_{kp^{i}}) = ξ_{kp^{i+1}}, which is ξ_{k} in the (i + 1)st factor. Hence QK^{∗}(SU) ≈ F̃(Γ), yielding the result.

We denote by M the category of free (Z-graded) K_{∗}-modules, and by S the homotopy category of topological spaces. We recall the definition ([10]) of the functor V from M to itself, and the V-resolution of certain M ∈ M.

M → V (M) → V (V M) → V (V^{2}M) → ⋯  (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 Ω^{∞}KM. Note that π_{∗}(KM) ≈ M, ∗ ≥ 0. For a space X with free K_{∗}-homology, KX is defined to be KK_{∗}(X) (equivalently KX = Ω^{∞}(K∧Σ^{∞}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 transformations δ : V → V^{2} and ϵ : V → I satisfying certain identities. (See [4, 5.2].)

6 BENDERSKY AND DAVIS

Note that if all basis elements of M have odd dimension, then the same is true of V (M). This follows since K_{r} = U if r is odd, and K_{∗}(U) is generated by odddimensional elements. The 0-part of Theorem 1.1 now follows from (2.7), (2.10), and (2.11).

The category V of unstable K_{∗}K-comodules consists of objects M ∈ M equipped with a K_{∗}-homomorphism η_{M} : M → V (M) with the usual commutative diagrams ([3, 2.15]). If X is as in 1.1, K_{∗}(X) is a Hopf algebra with M = P K_{∗}(X) = QK_{∗}(X) ∈ 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 η_{M} : 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_{∗}K-comodule. 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 odd-dimensional classes.

There are two maps in V from V (M) → V (V (M)), namely V (η_{M}) and η_{V (M)}. In general, there are n + 1 coface maps V (V^{n−1}(M)) → V (V^{n}(M)). There is also the map V (V (M)) → V (M), which is not in V. In general, there are n codegeneracy maps V (V^{n}(M)) → V (V^{n−1}(M)). These maps fit together to generate the V cosimplicial resolution C (we omit the codegeneracies):

[figure: M; V (M); V^{2}(M); ⋯]  (2.6)

The coboundary maps in the resolution (2.4) are the alternating sums of the coface maps in (2.6). Note that, whereas each V^{s}(M) satisfies V^{s}(M)_{t} ≈ V^{s}(M)_{t+2} for all t, the coboundary maps in (2.4) do not share this period-2 behavior, since they do not commute with the periodicity operator. That is, the groups can be considered as being Z/2-graded, but the morphisms cannot.

As usual, Ext_{V} is defined as the derived functors of Hom_{V}:

Ext_{V}^{s,t}(K_{∗}, M) = H_{s}(Hom_{V}(K_{∗}(S^{t}), C))
= H_{∗}(M_{t} → V (M)_{t} → V^{2}(M)_{t} → ⋯), (2.7)

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 7

where the coboundary maps in (2.7) are the alternating sums of the coface maps in

[figure: M; V (M); ⋯,]  (2.8)

which is obtained by applying the adjointness isomorphism

Hom_{V}(K_{∗}(S^{t}), V (N)) = N_{t}  (2.9)

to (2.6).

The connection with E_{2}(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 injectives in the non-abelian category G introduced in [4, §6]. We are using the fact that F V (M) = K_{∗}KM, which are the injectives in the category G. Applying Hom_{G}(K_{∗}(S^{t}), −) to F (C) also gives (2.8). Thus

Ext_{G}^{s}(K_{∗}S^{t}, K_{∗}X) ≈ Ext_{V}^{s}(K_{∗}S^{t}, P K_{∗}X)  (2.10)

if X is as in 1.1. From [10, 4.3], we have

E_{2}^{s,t}(X) = Ext_{G}^{s}(K_{∗}(S^{t}), K_{∗}X) for t − s > 0.  (2.11)

If N is a free K_{∗}-module with basis B and N_{ev} = 0, then V (N) = ⨁_{b∈B} P K_{∗}K_{b} (recall P K_{∗}K_{i} = QK_{∗}K_{i} if i is odd). Each K_{b} is a copy of U = KS^{|b|}, the 0space in the Ω-spectrum of K ∧ Σ^{∞}S^{|b|}. The Pontryagin dual V (N)^{#} is isomorphic to QK^{∗}(KN), which gives it the structure of p-adic Adams module. We restrict attention to the Z/2-graded module, and note that it is 0 in grading 0. Using 2.2, we obtain

V (N)_{1}^{#} ≈ ⨁_{b∈B} QK^{1}(S^{1} × SU_{(b)})
≈ ⨁_{b}(K^{1}(S^{1})_{b} × F̃(Γ)_{b})
≈ ⨁_{b} K^{1}(S^{1})_{b} × F̃(Γ ⊗ N_{1}^{#}),  (2.12)

where Γ ⊗ N_{1}^{#} has the extended stable Adams module structure, i.e. for k prime to p, ψ^{k}(γ ⊗ n) = ψ^{k}(γ) ⊗ n. Here Γ has grading 0, since it is tensored with classes of grading 1. We define Ṽ (N)_{1}^{#} to be F̃(Γ ⊗ N_{1}^{#}). We remind the reader that in this paragraph and throughout the remainder of this section L^{#} means (L ⊗ Z_{p^{∞}})^{#} for any abelian group L.

Now we deduce our main theorem.

8 BENDERSKY AND DAVIS

Proof of 1.1. Let X and M be as in Theorem 1.1 and its preamble. The Pontrjagin dual of the complex obtained by tensoring (2.4) with Z_{p^{∞}},

0 ← M^{#} ← V (M)^{#} ← V (V M)^{#} ← ⋯,  (2.13)

is acyclic. The maps in the complex (2.13) are in the category of p-adic Adams modules. This follows from dualizing (2.5). In particular, the following diagram of exact sequences commutes.

0 ←  M^{#}  ←  (V (M))^{#}  ←  (V (V M))^{#}  ← ⋯
↓^{ψ^{p}}  ↓^{ψ^{p}}  ↓^{ψ^{p}}
0 ←  M^{#}  ←  (V (M))^{#}  ←  (V (V M))^{#}  ← ⋯
↓  ↓  ↓
0 ← M^{#}/ im(ψ^{p}) ← (V (M))^{#}/ im(ψ^{p}) ← (V (V M))^{#}/ im(ψ^{p}) ← ⋯
↓  ↓  ↓
0  0  0

Since ψ^{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 resolution of M^{#}/ im(ψ^{p}). Now ψ^{p} is an isomorphism on the factors of K^{∗}(S^{1}), and by (2.12) and Definition 2.1 there is an isomorphism of stable p-adic Adams modules (Ṽ (V^{s}M))^{#}/ im(ψ^{p}) ≈ Γ⊗(V^{s}M)^{#}. So the bottom row is a resolution of M^{#}/ im(ψ^{p}) by A-projectives.

The boundary d_{s} : V^{s}(M) → V^{s+1}(M) in the resolution (2.4) satisfies d_{s} = V (d_{s−1}) − η_{V^{s}(M)}. We wish to show that the following diagram commutes, where

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 9

t is a positive odd integer.

Hom_{V}(K_{∗}S^{t}, V^{s}(M))  ⟶^{(d_{s})_{∗}}  Hom_{V}(K_{∗}S^{t}, V^{s+1}(M))
↓^{≈}  ↓^{≈}
Hom_{K_{∗}}(K_{∗}S^{t}, V^{s−1}(M))  Hom_{K_{∗}}(K_{∗}S^{t}, V^{s}(M))
↓^{≈}  ↓^{≈}
Hom_{AbGp}(V^{s−1}(M)_{1}^{#}, K^{1}S^{t})  Hom_{AbGp}(V^{s}(M)_{1}^{#}, K^{1}S^{t})
↓^{≈}  ↓^{≈}
Hom_{A}(V^{s}(M)_{1}^{#}/ im(ψ^{p}), K^{1}S^{t}) ⟶^{(d_{s}^{#})^{∗}} Hom_{A}(V^{s+1}(M)_{1}^{#}/ im(ψ^{p}), K^{1}S^{t})

(2.14)

The first of the vertical isomorphisms is due to (2.9). The second of the vertical isomorphisms is Pontrjagin duality. The third of the vertical isomorphisms is a consequence of V^{s}(M)^{#}/ im(ψ^{p}) ≈ Ṽ (V^{s−1}M)^{#}/ im(ψ^{p}) ≈ F̃(Γ⊗(V^{s−1}M)^{#})/ im(ψ^{p}) ≈ Γ⊗(V^{s−1}M)^{#} with Γ projective in A on one generator.

For the V (d_{s−1}) portion of d_{s}, commutativity of (2.14) is true because (d_{s−1})_{∗} and (d_{s−1}^{#})^{∗} can be placed as intermediate horizontal arrows, yielding three commutative squares. Commutativity of the η_{V^{s}(M)} portion of (2.14) is proved using consideration of the unstable cobar complex, which we now describe.

The way in which Ext_{V}(−) has been computed in papers such as [10], [4], and [3] is by viewing V^{s}M as the subset of

\overbrace{E_{∗}E ⊗_{E_{∗}} ⋯ ⊗_{E_{∗}} E_{∗}E}^{s} ⊗_{E_{∗}}E_{∗}M

satisfying an “unstable condition.” Here E is a spectrum such as K or BP, and M is an unstable E_{∗}E-comodule which is free as an E_{∗}-module. Under this identification,

10 BENDERSKY AND DAVIS

η_{V N} : V N → V^{2}N sends γ ⊗ n to ψ(γ) ⊗ n, and in the commutative diagram

Hom_{V}(K_{∗}S^{t}, V N) ⟶^{η_{V N}_{∗}} Hom_{V}(K_{∗}S^{t}, V^{2}N)
↓^{≈}  ↓^{≈}
Hom_{K_{∗}}(K_{∗}S^{t}, N) ⟶ Hom_{K_{∗}}(K_{∗}S^{t}, V N)
↓^{≈}  ↓^{≈}
N_{t}  ⟶^{ϕ}  (V N)_{t}

the corresponding morphism ϕ sends n to 1⊗n. Here we are thinking of N as V^{s−1}M. Similarly, with ϕ as above, there is a commutative diagram

Γ ⊗ N_{1}^{#}  ⟵^{Γ⊗ϕ^{#}}  Γ ⊗ (V N)_{1}^{#}
↑^{≈}  ↑^{≈}
F̃(Γ ⊗ N_{1}^{#})/ im(ψ^{p}) ⟵ F̃(Γ ⊗ (V N)_{1}^{#})/ im(ψ^{p})
↑^{≈}  ↑^{≈}
(V N)_{1}^{#}/ im(ψ^{p}) ⟵^{η_{V N}^{#}} (V^{2}N)_{1}^{#}/ im(ψ^{p})

With Hom_{A}(−, K^{1}S^{t}) applied to the second, these diagrams imply commutativity of the η_{V^{s}M} portion of (2.14) and hence of the diagram itself.

Thus the homology of the (Hom_{V}(K_{∗}S^{t}, V^{s}M), (d_{s})_{∗})-sequence is isomorphic to the homology of the (Hom_{A}(V^{s}M_{1}^{#}/ im(ψ^{p}), K^{1}S^{t}), (d_{s}^{#})^{∗})-sequence. These are the two groups which the theorem asserts to be isomorphic.

The following example might be instructive. Note that X = S^{2n+1} satisfies the conditions of 1.1. In this case, M^{#} = M_{n}, a free K^{∗}-module on a single generator with ψ^{k} = k^{n}. The short exact sequence

0 → M_{n} ⟶^{ψ^{p}} M_{n} → M_{n}/ im(ψ^{p}) → 0

induces an exact (Bockstein) sequence in Ext_{A}, which relates the unstable E_{2} for S^{2n+1} with the stable E_{2} for the sphere spectrum. Here Ext_{A}(M_{n}) is the E_{2}-term of a K-based spectral sequence, indexed so as to converge to the stable v_{1}-periodic homotopy groups of S^{2n+1}.

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 11

3. Computing Ext_{A}(−, −)

In this section, we develop a method of computing Ext_{A}^{s,t}(M) for a stable p-adic Adams module M. For simplicity of exposition, we focus mostly on modules in which ψ^{−1} = −1, which is all we need in this paper. The general case, described in Theorem 3.10, requires only minor modifications.

If t = 2n + 1, we let Ext_{A}^{s,t}(M) = Ext_{A}^{s}(M, S_{t}), where S_{t} = QK^{1}(S^{t}; Z_{p}^{∧}) is Z_{p}^{∧} with ψ^{k} = ·k^{n}. In this section, (−)^{#} denotes ordinary Pontrjagin duality.

Theorem 3.1. Let M be a finite stable p-adic Adams module with ψ^{−1} = −1. a. If p is odd and r denotes a generator of (Z/p^{2})^{×}, then

Ext_{A}^{s,2n+1}(M)^{#} ≈ \begin{cases}M/ im(ψ^{r} − r^{n}) & s = 1 \\ ker((ψ^{r} − r^{n})|M) & s = 2 \\ 0 & otherwise. \end{cases}

b. Let p = 2, M_{2} = ker(2|M), and θ = ψ^{3} − 1. If n is odd, there is an isomorphism

Ext_{A}^{1,2n+1}(M)^{#} ≈ coker((ψ^{3} − 3^{n})|M)

and a split short exact sequence

0 → coker(θ|M/2) → Ext_{A}^{2,2n+1}(M)^{#} → ker((ψ^{3} − 3^{n})|M) → 0. If n is even, then

Ext_{A}^{1,2n+1}(M)^{#} ≈ coker(θ|M/2).

If s + n is odd and s > 2, there is a split short exact sequence

0 → coker(θ|M/2) → Ext_{A}^{s,2n+1}(M)^{#} → ker(θ|M_{2}) → 0. If s + n is even and s > 1, there is a split short exact sequence

0 → coker(θ|M_{2}) → Ext_{A}^{s,2n+1}(M)^{#} → ker(θ|M/2) → 0.

The case s = 1 was proved in [7]. The odd-primary case is proved in [12, §8].

The proof of Theorem 3.1 when p = 2 will utilize the following elementary result. As in [15, §3], let Inv denote the category of abelian groups with involution ψ^{−1}. Let M^{(ϵ)} denote a 2-local abelian group M with ψ^{−1} = ϵ.

Proposition 3.2. If M is a 2-local abelian group with ψ^{−1} = −1, then

Ext_{Inv}^{s}(Z_{(2)}^{((−1)^{n})}, M) ≈ \begin{cases}M_{2} & if s + n even and s ≥ 0 \\ M/2 & if s + n odd and s > 0 \\ M & if s = 0 and n odd \end{cases}

Proof. Let P denote the object of Inv which is Z_{(2)} ⊕ Z_{(2)} with ψ^{−1} switching the summands. Note that P is projective. Let ϵ = (−1)^{n}. A projective resolution of Z_{(2)}^{(ϵ)} is given by

⋯ ⟶^{d_{2}} C_{2} ⟶^{d_{1}} C_{1} ⟶^{d_{0}} C_{0} → Z_{(2)}^{(ϵ)} → 0

with each C_{i} = P and d_{i} = 1 + (−1)^{i+1}ϵψ^{−1}. The complex

Hom_{Inv}(C_{0}, M) ⟶^{d_{0}^{∗}} Hom_{Inv}(C_{1}, M) ⟶^{d_{1}^{∗}} Hom_{Inv}(C_{2}, M) ⟶^{d_{2}^{∗}} ⋯

is isomorphic to

M ⟶^{1+ϵ} M ⟶^{1−ϵ} M ⟶^{1+ϵ} ⋯,

and the homology of this is as claimed in the proposition.

Proof of Theorem 3.1. Let GInv denote the category of 2-profinite abelian groups with involution. Similarly to [12, 8.3], we have

Proposition 3.3. If M and N are stable 2-adic Adams modules, there is a natural exact sequence

0 → Hom_{A}(M, N) → Hom_{GInv}(M, N) ⟶^{ψ_{M}^{3} −ψ_{N}^{3}} Hom_{GInv}(M, N)
→ Ext_{A}^{1}(M, N) → Ext_{GInv}^{1}(M, N) ⟶^{ψ_{M}^{3} −ψ_{N}^{3}} Ext_{GInv}^{1}(M, N)
→ Ext_{A}^{2}(M, N) → Ext_{GInv}^{2}(M, N) ⟶^{ψ_{M}^{3} −ψ_{N}^{3}} Ext_{GInv}^{2}(M, N)
→ Ext_{A}^{3}(M, N) → · · · .

The exact sequence of 3.3 is obtained from a short exact sequence

0 → U(M) ⟶^{Uψ^{3}−ψ^{3}} U(M) → M → 0,

where U : GInv → A is left adjoint to the forgetful functor. This U is a profinite version of the functor of [15, 6.6], and satisfies Ext_{A}^{s}(U(M), N) ≈ Ext_{GInv}^{s}(M, N). Its existence, adjointness, exactness, and the fundamental short exact sequence above

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 13

all follow by Pontrjagin duality from the analogous results for the functor \overline{U} in [15, pp.145-6].

Since GInv is dual to the torsion subcategory of Inv, we have

Ext_{GInv}^{s}(M, N) ≈ Ext_{Inv}^{s}(N^{#}, M^{#}).  (3.4)

If N = QK^{1}(S^{2n+1})^{∧}, then, since the Pontrjagin dual of Z_{2}^{∧} is (Q/Z)_{(2)}, we obtain

Ext_{GInv}^{s,2n+1}(M) ≈ Ext_{Inv}^{s−1}(Z_{(2)}^{((−1)^{n})}, M^{#})

by Proposition 3.9 and the Ext-sequence 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,

0 → Ext_{A}^{0,2n+1}(M) → 0 → 0 → Ext_{A}^{1,2n+1}(M) → M^{#} ⟶^{ψ^{3}−3^{n}} M^{#} → Ext_{A}^{2,2n+1}(M)
→ M_{2}^{#} ⟶^{ψ^{3}−1} M_{2}^{#} → Ext_{A}^{3,2n+1}(M) → (M/2)^{#} ⟶^{ψ^{3}−1} (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 h_{1}-action 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 Ext_{A} and an element

h_{1} ∈ Ext_{A}^{1}(QK^{1}(S^{2n+1})^{∧}, QK^{1}(S^{2n+3})^{∧})

satisfying 1. 2h_{1} = 0;

2. Yoneda product with h_{1} corresponds to the h_{1}-action 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(θ|M/2) ⟶ Ext_{A}^{s+1,2n+3}(M)^{#} ⟶ ker(θ|M_{2}) ⟶ 0
1↓  ^{h_{1}^{#}}↓  ^{1}↓
0 ⟶ coker(θ|M/2) ⟶ Ext_{A}^{s,2n+1}(M)^{#} ⟶ ker(θ|M_{2}) ⟶ 0

and a similar one when s + n is even.

14 BENDERSKY AND DAVIS

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 ·h_{1} is an isomorphism on Ext_{A}^{s,2n+1}(M), and so, since 2h_{1} = 0, Ext_{A}^{s,2n+1}(M) can have no elements of order 4.

Proposition 3.6 shows that the h_{1}-action is an isomorphism from Ext_{A}^{s,2n+1}(M) to Ext_{A}^{s+1,2n+3}(M) for s ≥ 2. When s = 1 and n is even, the diagram of 3.6(3) applies to describe the action of h_{1}, which is injective. When s = 1 and n is odd, the situation is a bit more delicate. The following result is proved in [9]. We forego giving the proof here because it is quite involved, and the result plays only a minor role in our applications in Section 4.

Proposition 3.7. Suppose ψ^{−1} = −1 in QK^{1}(X). Let N = (QK^{1}(X)/ im(ψ^{2}))^{#}, N_{2} = ker(2|N), and θ = ψ^{3} − 3^{n} with n odd. There is a commutative diagram of exact sequences

0 ⟶ E_{2}^{1,2n+1}(X) ⟶^{i} N ⟶^{θ^{#}} N
↓  ^{h_{1}·}↓  ^{ρ_{2}}↓  ^{ρ_{2}}↓
N_{2} ⟶^{θ^{#}} N_{2} ⟶^{j} E_{2}^{2,2n+3}(X) ⟶ N/2 ⟶^{θ^{#}} N/2

If ρ_{2}(i(x)) = 0, then h_{1}x = j(θ^{#}(x/2)), which is well-defined as an element of coker(θ^{#}|N_{2}).

Corollary 3.8. With the hypotheses of Proposition 3.7, h_{1}· is injective on E_{2}^{1,2n+1}(X)/2.

Proof. Suppose h_{1}x = 0. Then θ^{#}(i(x)) = 0 and i(x) = 2y for some y ∈ N. By 3.7, h_{1}x = j(θ^{#}(y)) considered as an element of coker(θ^{#}|N_{2}). Since h_{1}x is assumed to be 0, we deduce θ^{#}(y) = θ^{#}(z) with 2z = 0. Then y − z ∈ ker(θ^{#}). Hence there is an element y′ ∈ E_{2}^{1,2n+1}(X) satisfying i(y′) = y − z and 2y′ = x. Hence x = 0 ∈ E_{2}^{1,2n+1}(X)/2.

The following proposition was used earlier in this section.

Proposition 3.9. If M is a finite object of Inv, then Ext_{Inv}^{s}(Q^{(ϵ)}, M) = 0 for s ≥ 0.

Proof. The object M must be isomorphic to a sum of (Z/2^{n})^{(ϵ′)}’s plus copies of P/2^{n} = Z/2^{n} ⊕ Z/2^{n} with ψ^{−1} interchanging factors.

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 15

By [15, 3.10], Ext_{Inv}^{s}(Q^{(ϵ)}, M) = 0 for s > 1. For s = 0, we have

Hom_{Inv}(Q^{(ϵ)}, M) ⊂ Hom_{AbGp}(Q, M) = 0.

Let 0 → R → F → Q → 0 be a projective resolution in AbGp. Then 0 → R^{(ϵ)} → F^{(ϵ)} → Q^{(ϵ)} → 0 is a projective resolution in Inv by [15, 3.6]. We first consider the case ϵ′ = ϵ. Here Hom_{Inv}(F^{(ϵ)}, (Z/2^{n})^{(ϵ)}) = Hom_{AbGp}(F, Z/2^{n}), and so Ext_{Inv}^{1}(Q^{(ϵ)}, (Z/2^{n})^{(ϵ)}) = Ext_{AbGp}(Q, Z/2^{n}). Using the injective resolution in AbGp

0 → Z/2^{n} → Q/Z ⟶^{2^{n}} Q/Z → 0,

one readily verifies Ext_{AbGp}(Q, Z/2^{n}) = 0.

With ϵ′ = −ϵ, we have Hom_{Inv}(F^{(ϵ)}, (Z/2^{n})^{(−ϵ)}) = Hom_{AbGp}(F, Z/2). Arguing as above with n = 1, we obtain Ext_{Inv}^{1}(Q^{(ϵ)}, (Z/2^{n})^{(−ϵ)}) = 0. Finally, Ext_{Inv}^{1}(Q^{(ϵ)}, P/2^{n}) = 0 follows from Hom_{Inv}(F^{(ϵ)}, P/2^{n}) = 0 by a similar argument.

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 3.10. a. Let p = 2 and let M be a finite stable p-adic Adams module. Let θ_{n} = ψ^{3} − 3^{n}, and

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

Then • Ext_{A}^{1,2n+1}(M)^{#} ≈ coker(θ_{n}| coker(1 − (−1)^{n}ψ^{−1}));

• there is a short exact sequence 0 → coker(θ_{n}|Q_{n}) → Ext_{A}^{2,2n+1}(M)^{#} → ker(θ_{n}| coker(1−(−1)^{n}ψ^{−1})) → 0;

• for s > 2, there is a short exact sequence

0 → coker(θ_{n}|Q_{s+n}) → Ext_{A}^{s,2n+1}(M)^{#} → ker(θ_{n}|Q_{s+n−1}) → 0. b. If M is a 2-local abelian group with involution ψ^{−1}, then

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

We complete this section by proving Proposition 3.6.

16 BENDERSKY AND DAVIS

Proof of Proposition 3.6. We apply Proposition 3.3 and (3.4) to obtain an exact sequence

Hom_{Inv}((Q/Z)^{((−1)^{n+1})}, (Q/Z)^{((−1)^{n})}) → Ext_{A}^{1}(QK^{1}(S^{2n+1})^{∧}, QK^{1}(S^{2n+3})^{∧})

→ Ext_{Inv}^{1}((Q/Z)^{((−1)^{n+1})}, (Q/Z)^{((−1)^{n})}) ⟶^{2·odd} .

Using (3.5), and then arguing as in the proof of Proposition 3.9, we obtain Ext_{Inv}^{1}((Q/Z)^{((−1)^{n+1})}, (Q/Z)^{((−1)^{n})}) ≈ Ext_{Inv}^{0}(Z_{(2)}^{((−1)^{n+1})}, (Q/Z)^{((−1)^{n})}), and, similarly to Proposition 3.2, this is Z/2, generated by \frac{1}{2} ∈ Q/Z on the RHS. The nonzero element is called h_{1}. Since h_{1} ∈ Z/2, 2h_{1} = 0. Part 2 of the proposition follows since the isomorphism of Theorem 1.1 respects Yoneda products, and the two notions of h_{1} must agree since they are the only nonzero element in isomorphic groups.

For part 3, we first consider the Yoneda product in Ext_{Inv}

Ext_{Inv}^{s}(Z_{(2)}^{(ϵ)}, M) ⊗ Ext_{Inv}^{1}(Z_{(2)}^{(−ϵ)}, Z_{(2)}^{(ϵ)}) → Ext_{Inv}^{s+1}(Z_{(2)}^{(−ϵ)}, M). With P as in the proof of 3.2, composition with h_{1} is defined by the diagram

0 ← Z_{(2)}^{(−ϵ)} ← P ⟵^{1+ϵψ^{−1}} P ⟵^{1−ϵψ^{−1}} P ← ⋯ P
↙ ↓^{1} ↓^{1} ↓^{1}
Z_{(2)}^{(ϵ)} ← P ⟵^{1−ϵψ^{−1}} P ← ⋯ P
↓
M

Since the chain map of resolutions can be chosen to be the identity, the composition 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 Yoneda products.

4. The BTSS of F_{4}

In this section, we prove Theorem 1.2. There are three steps.

1. Use Theorems 1.1 and 3.1 to compute the E_{2}-term of the BTSS converging to v_{1}^{−1}π_{∗}(F̂_{4}).

2. Use the fibration

G_{2} → F_{4} → F_{4}/G_{2}  (4.1)

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 17

to determine the differentials and extensions in the spectral sequence.

3. Show that F_{4} → F̂_{4} induces an isomorphism in v_{1}^{−1}π_{∗}(−). This is done in Theorem 1.4.

From [19, 3.8] we have

Proposition 4.2. There is a basis {v_{1}, v_{2}, v_{3}, v_{4}} of QK^{1}(F_{4}) on which ψ^{−1} = −1 and the transposes of the matrices of ψ^{2} and ψ^{3} are given by

(ψ^{2})^{T} = (\begin{matrix}2 & 3 & 1 & 0 \\ 0 & 32 & −8 & −1 \\ 0 & 0 & 128 & −24 \\ 0 & 0 & 0 & 2048 \end{matrix}) and (ψ^{3})^{T} = (\begin{matrix}3 & 24 & 15 & −1 \\ 0 & 3^{5} & −162 & −81 \\ 0 & 0 & 3^{7} & −3^{7} \\ 0 & 0 & 0 & 3^{11} \end{matrix}) .

This can be shown to agree with the Chern character calculation of [26, 4.8]. By Theorems 1.1 and 3.1, E_{2}^{1,4k+3}(F_{4})^{#} is obtained from the following result.

Proposition 4.3. If (ψ^{2})^{T} and (ψ^{3})^{T} are as in Proposition 4.2, then the abelian group presented by the matrix

(\begin{matrix}(ψ^{2})^{T} \\ (ψ^{3} − 3^{2k+1})^{T} \end{matrix} )

is Z/2^{min(12,6+2ν(k−5))}.

Proof. Replace 3^{2k+1} by 3^{10}(R + 3) in the matrix. Then ν(R) = ν(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 to odd multiples, are

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

Then the exponent of 2 in the fourth relation is min(12, 2ν(R)), and all other relations are at least that 2-divisible.

18 BENDERSKY AND DAVIS

The order of ker((ψ^{3}−3^{2k+1})|QK^{1}(F_{4})/ im(ψ^{2})), which is a summand of E_{2}^{2,4k+3}(F_{4})^{#}, equals that of the cokernel, which was determined in the preceding proposition. For the group structure, we need

Proposition 4.4. The group ker((ψ^{3} − 3^{2k+1})|QK^{1}(F_{4})/ im(ψ^{2})) is cyclic.

Proof. Let M = QK^{1}(F_{4})/ im(ψ^{2}), M_{2} = ker(·2|M), and K = ker((ψ^{3} − 3^{2k+1})|M). The number of summands in K equals the dimension of M_{2} ∩ K. Note that ψ^{3} − 3^{2k+1} = ψ^{3} − 1 on M_{2}.

A basis for M_{2} is given by ψ^{2}(v_{3})/2 and ψ^{2}(v_{4})/2. We have

(ψ^{3} − 1)(ψ^{2}(v_{4})/2) = ψ^{2}(ψ^{3} − 1)(v_{4})/2 = ψ^{2}(\frac{3^{11}−1}{2}v_{4}) ≡ 0 ∈ M, and

(ψ^{3} − 1)(ψ^{2}(v_{3})/2) = ψ^{2}(\frac{3^{7}−1}{2}v_{3} − \frac{3^{7}}{2} v_{4}) ≡ ψ^{2}(v_{4}/2) ∈ M.

Thus M_{2} ∩ K = ⟨ψ^{2}(v_{4})/2⟩ is 1-dimensional.

For the elements of higher filtration in E_{2}(F_{4}), we need

Proposition 4.5. Let M = QK^{1}(F_{4})/ im(ψ^{2}). Then (ψ^{3} − 1)|(M/2) has kernel ≈ Z/2 with basis {v_{2} ∼ v_{3}} and cokernel ≈ Z/2 with basis {v_{1}}, while (ψ^{3} − 1)|M_{2} has kernel ≈ Z/2 with basis {ψ^{2}(v_{4})/2} and cokernel ≈ Z/2 with basis {ψ^{2}(v_{3})/2}.

Proof. We have M/2 ≈ Z/2⊕Z/2 with basis {v_{1}, v_{2} ∼ v_{3}}. Mod (2, im(ψ^{2})), we have (ψ^{3} − 1)v_{1} = v_{3} and (ψ^{3} − 1)v_{2} = 0. In the proof of Proposition 4.4, ψ^{3} − 1 on M_{2} was analyzed.

We obtain the following diagram of E_{2}(F_{4}) with e = 6 and f = min(12, 8+2ν(ℓ−3)).

Diagram 4.6.

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

Here, as usual with Adams spectral sequence types of diagrams, the horizontal grading is t − s, and classes in E_{∞}^{∗,∗+i}(X) provide an associated graded for π_{i}(X̂). Each dot represents Z/2, and an integer represents a cyclic summand of that order. The diagonal lines indicate multiplication by h_{1} in the BTSS (3.6), which corresponds to η in homotopy. We call these “η-towers.” The action of h_{1} on the 1-line can be computed explicitly using Proposition 3.7; see Proposition 4.15. For most of our purposes, we just need that it is nonzero on the 1-line (Corollary 3.8), and so we do not indicate explicitly which element it hits in Diagram 4.6.

Because η^{4} = 0 in π_{n+4}(S^{n}), there must be a pattern of d_{3}-differentials which annihilates all η-towers in large filtration. However, careful consideration is required to determine whether a particular η-tower supports a d_{3}-differential or is hit by one. This will affect whether or not a few elements at the bottom of the η-tower survive the spectral sequence.

In order to determine the d_{3}-differentials in F_{4}, we use the fibration (4.1). The groups v_{1}^{−1}π_{∗}(G_{2}; 2) were computed in [21] using homotopy theoretic methods. We now show how these groups can be seen in the BTSS.

20 BENDERSKY AND DAVIS

From [19, 3.7], QK^{1}(G_{2}) has a basis {g_{1}, g_{2}} on which (ψ^{2})^{T} and (ψ^{3})^{T} are given by

(ψ^{2})^{T} = (\begin{matrix}2 & −15 \\ 0 & 32 \end{matrix}) and (ψ^{3})^{T} = (\begin{matrix}3 & −120 \\ 0 & 3^{5} \end{matrix}) .

By methods similar to those employed above for F_{4}, we obtain

Proposition 4.7. Let M′ = QK^{1}(G_{2})/ im(ψ^{2}).

1. E_{2}^{1,4k+3}(G_{2})^{#} ≈ Z/2^{min(6,ν(k−2)+3)} ≈ ker((ψ^{3} − 3^{2k+1})|M′);

2. M′/2 ≈ Z/2, generated by g_{1}, with ψ^{3} − 1 = 0 on M′/2, and M′_{2} = Z/2, generated by 16g_{2}, with ψ^{3} − 1 = 0 on M′_{2}.

Thus by Theorems 1.1 and 3.1 E_{2}(G_{2}) has the form of Diagram 4.6, with e = min(6, ν(ℓ − 1) + 4) and f = 3.

The following result will be proved in Section 5, simultaneously with the proof that Theorem 1.4 holds for G_{2}. The proof utilizes the map G_{2} → S^{6} with fiber SU(3), the analysis of v_{1}^{−1}π_{∗}(G_{2}) in [21], and the fact that S^{6} satisfies the CTP.

Theorem 4.8. The differentials and extensions in the BTSS of G_{2} are as in Diagram 4.9, with e = min(6, ν(ℓ − 1) + 4) and f = 3.

Diagram 4.9.

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

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 21

Note that this diagram for ℓ − 1 would provide additional η-towers which are not displayed on the left side of the diagram.

We need also the BTSS and v_{1}-periodic homotopy groups of F_{4}/G_{2}. We use [21, 1.1], which states that there is a 2-local fibration

S^{15} → F_{4}/G_{2} → S^{23}.  (4.10)

Proposition 4.11. F_{4}/G_{2} satisfies the CTP.

Proof. In 5.3, we define the notion of a K_{∗}-strongly spherically resolved space, and in 5.5 we prove that such spaces satisfy the CTP. This notion is slightly stronger than previous notions of spherical resolvability, in that we require that each K_{∗}(X_{i}) is isomorphic to an exterior algebra as a K_{∗}K-coalgebra. Since the X_{i} will not usually have a product structure, it will not be automatic that the coproduct and coaction on classes that appear to be products of primitive elements satisfy the requisite formulas. However, for odd-sphere bundles over odd spheres, this is not a problem.

In the case at hand, X = F_{4}/G_{2}, using Z/2-graded K-homology, K_{1}(X) has basis consisting of two primitive classes, x_{1} coming from S^{15}, and x_{2} mapping to S^{23}, while K_{0}(X) has basis consisting of a single class x_{3} whose coproduct involves x_{1} ⊗ x_{2} + x_{2} ⊗ x_{1}, as can be seen by duality from the algebra K^{∗}(X). The K_{∗}K-coaction must satisfy ψ(x_{1}) = 1 ⊗ x_{1} by naturality, ψ(x_{3}) = 1 ⊗ x_{3} since (K_{∗}K)_{od} = 0, and ψ(x_{2}) = 1 ⊗ x_{2} + h ⊗ x_{1} for some h ∈ K_{∗}K. Since x_{1}^{2} = 0 in an exterior algebra, these formulas for ψ are consistent with an exterior algebra in which x_{3} = x_{1}x_{2}, i.e., ψ(x_{3}) = ψ(x_{1})ψ(x_{2}).

Since the attaching map in F_{4}/G_{2} is σ, by [1, 7.5,7.17], we have

Proposition 4.12. QK^{1}(F_{4}/G_{2}) has basis {w_{1}, w_{2}} with ψ^{k}w_{2} = k^{11}w_{2} and

ψ^{k}w_{1} = k^{7}w_{1} + \frac{uk^{7}(k^{4} − 1)}{16}w_{2},

with u odd.

Applying to this the methods of 4.3, 4.4, 4.5, and 4.7, we obtain that the BTSS-E_{2} for F_{4}/G_{2} has the form of Diagram 4.6 with e = 6 and f = min(12, 7 + ν(ℓ − 19)).

22 BENDERSKY AND DAVIS

Theorem 4.13. The differentials and extensions in the BTSS of F_{4}/G_{2} are as in Diagram 4.9 with e = 6 and f = min(12, 7 + ν(ℓ − 19)).

Proof. We need the fact ([6, p.488],[2, p.352]) that the BTSS of S^{8m−1} has the form of Diagram 4.9 with e = 3 and f = min(4m − 1, ν(ℓ − m) + 4). The fibration (4.10) induces a short exact sequence in QK^{1}(−), a long exact sequence in E_{2} of the BTSS, and a long exact sequence in v_{1}^{−1}π_{∗}(−). The d_{3}-differentials on the η-towers emanating from (t − s, s) = (8ℓ + 3, 2) in S^{15} and S^{23} force similar d_{3}-differentials in F_{4}/G_{2}, as do the d_{3}-differentials on the η-towers emanating from (8ℓ + 4, 1). Since E_{2}^{2,8ℓ+3}(F_{4}/G_{2}) → E_{2}^{2,8ℓ+3}(S^{23}) maps onto the Z/8, which supports a d_{3}-differential, d_{3} must be nonzero on Z/2^{6} ⊂ E_{2}^{2,8ℓ+3}(F_{4}/G_{2}) and on the η-tower arising from it.

For s = 1 and 2, E_{2}^{s,8ℓ+3}(S^{15}) → E_{2}^{s,8ℓ+3}(F_{4}/G_{2}) is a monomorphism Z/8 ↪ Z/2^{6}, and so the nontrivial extension (·2) from E_{2}^{s,8ℓ+3}(F_{4}/G_{2}) to E_{2}^{s+2,8ℓ+5}(F_{4}/G_{2}) will follow from that in S^{15} once we know that the η-tower into which E_{2}^{s,8ℓ+3}(S^{15}) extends maps across. When s = 1, this is clear, since the extension is into h_{1}^{2}E_{2}^{1,8ℓ+1}, and E_{2}^{1}(S^{15}) → E_{2}^{1}(F_{4}/G_{2}) must be injective.

The case s = 2 requires more care. For s ≥ 2, the two classes of E_{2}^{s,8ℓ−3+2s}(S^{15}) can be characterized as “stable” and “unstable.” From the point of view of Theorem 3.1, the one in ker(θ|M/2) is stable, while the one in coker(θ|M_{2}) is unstable. This is true because elements of M_{2}, being ψ^{2}x/2, depend on the dimension of the sphere, while elements in M/2 are generators of M and independent of the dimension of the sphere. The extension in BTSS(S^{15}) in t − s = 8ℓ + 1 is into the unstable class. This is true because the large summand in E_{2}^{2}(S^{2n+1}) is unstable.

Now consider the commutative diagram of exact sequences, with \overline{K}(X) = QK^{1}(X)/ im(ψ^{2}),

⟶^{θ} \overline{K}(F_{4}/G_{2})_{2} ⟶ Ext_{A}^{s,8ℓ−3+2s}(\overline{K}(F_{4}/G_{2}))^{#} ⟶ \overline{K}(F_{4}/G_{2})/2 ⟶^{θ}
↓  ↓  ↓
⟶^{θ} \overline{K}(S^{15})_{2} ⟶ Ext_{A}^{s,8ℓ−3+2s}(\overline{K}(S^{15}))^{#} ⟶ \overline{K}(S^{15})/2 ⟶^{θ}

With w_{i} as in Proposition 4.12, (ψ^{3} − 1)(ψ^{2}w_{1}/2) = ψ^{2}(\frac{3^{7}−1}{2}w_{1} + \frac{3^{7}(3^{4}−1)}{2·16}w_{2}) ≡ ψ^{2}(\frac{w_{2}}{2} ) mod im ψ^{2} and

(ψ^{3} − 1)(ψ^{2}w_{2}/2) = ψ^{2}(\frac{3^{11}−1}{2}w_{2}) ≡ 0 mod im ψ^{2}.

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 23

Thus coker(θ|\overline{K}(F_{4}/G_{2})_{2}) = ⟨\frac{ψ^{2}w_{1}}{2} ⟩, which maps nontrivially to coker(θ|\overline{K}(S^{15})_{2}). Dually, the unstable class in Ext_{A}^{s,8ℓ−3+2s}(\overline{K}(S^{15})) maps nontrivially to F_{4}/G_{2}, and hence the extension from E_{∞}^{2,8ℓ+3}(F_{4}/G_{2}) to E_{∞}^{4,8ℓ+5}(F_{4}/G_{2}) is nontrivial.

We must check that the element into which we have just seen the nontrivial extension in v_{1}^{−1}π_{8ℓ+1}(F_{4}/G_{2}) hits is not killed by a d_{3}-differential. It looks that way in Diagram 4.9, but we have not been precise here in depicting which element is hit by h_{1} from position (8ℓ + 2, 1), which affects which element is hit by d_{3} from (8ℓ + 2, 1). In Proposition 4.15, we will see that, using \{w_{2}, \frac{1}{2}ψ^{2}w_{1}\} as basis for the eta towers in E_{2}^{s,2n+1}(F_{4}/G_{2}) with s+n even, h_{1} from E_{2}^{1,8ℓ+3} hits the sum of the basis elements and hence so does d_{3}, since d_{3}h_{1} = h_{1}d_{3}. Thus the extension is into a nonzero homotopy class.

The result for v_{1}^{−1}π_{∗}(F_{4}/G_{2}) that can be read off from Theorem 4.13 differs slightly from [20, 8.10]. A mistake in [20] was discussed in [8]. The key lemma [20, 8.16] is false, and this caused the evaluation of a d_{6}-differential in [20, p.1045] to be incorrect.

Now we can prove the following result, from which Theorem 1.2 follows immediately, once we know 1.4 for F_{4}.

Theorem 4.14. The differentials and extensions in the BTSS of F_{4} are as in Diagram 4.9 with e = 6 and f = min(12, 8 + 2ν(ℓ − 3)).

Proof. The fibration (4.1) induces a short exact sequence in QK^{1}(−) and a long exact sequence in E_{2}. The d_{3}-differentials on the η-towers emanating from (t−s, s) = (8ℓ+3, 2) and (8ℓ+4, 1) are implied by their existence in G_{2} and F_{4}/G_{2}. The nonzero d_{3} from (8ℓ + 2, 1) follows by Corollary 3.8.

The Z/2^{6} summand in E_{2}^{2,8ℓ+3} maps isomorphically from F_{4} to F_{4}/G_{2}, as does the η-tower arising from it. Thus the d_{3} on this η-tower in F_{4}/G_{2} implies the same in F_{4}, and the nontrivial extension from E_{2}^{2,8ℓ+3}(F_{4}/G_{2}) implies the same in F_{4}.

Finally, E_{2}^{1,8ℓ+3}(G_{2}) → E_{2}^{1,8ℓ+3}(F_{4}) is injective, as is E_{2}^{1,8ℓ+1}(G_{2}) → E_{2}^{1,8ℓ+1}(F_{4}), and so the extension in G_{2} from E_{∞}^{1,8ℓ+3} to h_{1}^{2} · E_{2}^{1,8ℓ+1} implies the same in F_{4}.

Although it seems to be not necessary for our conclusions, except perhaps as it was used at the end of the proof of 4.13, we list the result of applying Proposition 3.7 to

24 BENDERSKY AND DAVIS

obtain the h_{1}-action on the 1-line of the spaces we are studying here. It is slightly easier to state and prove if we dualize the diagram of 3.7.

Proposition 4.15. Let X = G_{2}, F_{4}/G_{2}, or F_{4}, and M = QK^{1}(X)/ im(ψ^{2}). Then E_{2}^{2,4k+5}(X)^{#} has basis consisting of one nonzero element from coker(θ|M_{2}), which we call “unstable,” and one nonzero element from ker(θ|M/2), which we call “stable.” In the notation used earlier in this section, the unstable elements are, respectively, 16g_{2}, \frac{1}{2}ψ^{2}w_{1}, and \frac{1}{2}ψ^{2}v_{3}, while the stable elements are, respectively, g_{1}, w_{2}, and v_{2}. For G_{2}, h_{1}^{#} is nonzero on the stable class iff ν(k − 2) ≤ 3, and is nonzero on the unstable class iff ν(k − 2) ≥ 3. For F_{4}/G_{2}, h_{1}^{#} is nonzero on the stable class iff ν(k − 5) = 0 or 5, and is nonzero on the unstable class except perhaps when ν(k − 5) = 5. For F_{4}, h_{1} is nonzero on the stable class iff ν(k − 5) ≠ 3 and is nonzero on the unstable class iff ν(k − 5) = 3.

Proof. We illustrate with the proof for F_{4}. The others are similar and slightly easier.

The dual of the diagram of 3.7 is

M_{2} ⟶^{θ} M_{2} ⟶ E_{2}^{2,4k+5}(X)^{#} ⟶ M/2 ⟶^{θ} M/2
ι↓  ^{ι}↓  ^{h_{1}^{#}}↓  ↓
M ⟶^{θ} M ⟶ E_{2}^{1,4k+3}(X)^{#} ⟶ 0

An unstable element is the image in E_{2}^{2,4k+5}(X)^{#} of some \frac{1}{2}ψ^{2}x ∈ M_{2}. It will have a nonzero image under h_{1}^{#} iff adjoining the relation \frac{1}{2}ψ^{2}x to the presentation matrix of 4.3 decreases the order of the group. A stable element is the pullback to E_{2}^{#} of an element y ∈ M/2 for which θ(y) = 2z. By Proposition 3.7, this will have a nonzero image under h_{1}^{#} iff adjoining the relation z (= \frac{1}{2}θ(y)) decreases the order of the group.

If \frac{1}{2}ψ^{2}v_{3} is added to the presentation matrix for F_{4}, this has the effect of dividing the first of the five relations listed in the proof of 4.3 by 2, yielding 2^{13} + 2^{5}R. This lowers the order of the group only when ν(R) = 6.

Note that θ(v_{2}) is divisible by 2 in M because θ(v_{2}) − ψ^{2}(v_{2}) is divisible by 2 in QK^{1}(F_{4}). We add \frac{1}{2}(θ(v_{2}) − ψ^{2}(v_{2})) to the presentation matrix, which has the effect of dividing the third of the five relations by 2, yielding 2^{11} + 2^{7}R + \frac{1}{2}R^{2}. This lowers the order of the group unless ν(R) = 6.

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 25

5. The completion telescope property

In this section we first show that the BTSS converges to v_{1}^{−1}π_{∗}(X̂) for a class of spaces X which includes spheres and simply-connected finite H-spaces. Then we prove Theorem 1.4, which is the isomorphism v_{1}^{−1}π_{∗}(X) ≈ v_{1}^{−1}π_{∗}(X̂) for certain important spaces X.

We begin by recalling some of the results of [10]. For a space X, the K-completion of X, denoted X̂, is constructed as Tot of a cosimplicial space constructed from the K-theory spectrum. There is a natural transformation X → X̂. The Bousfield-Kan spectral sequence associated to the standard filtration of the K-theory Tot is the BTSS. We are using here the v_{1}-periodic BTSS, although the difference between this and the unlocalized BTSS is inconsequential for our purposes here, since they agree in sufficiently large dimensions.

Proposition 5.1. ([10, 2.3]) Suppose there is an N and r such that E_{r}^{s,t} = 0 if s > N. Then the BTSS converges to the v_{1}-periodic homotopy groups of X̂.

Proof. It is shown in [10, 2.3] that, under this hypothesis, the unlocalized BTSS converges to π_{∗}(X̂) . Because there is a horizontal vanishing line, the v_{1}-periodic BTSS converges to v_{1}^{−1}π_{∗}(X̂), since there can be no v_{1}-periodic family of classes or differentials of increasingly large filtration.

One family of spaces for which we can prove there is a horizontal vanishing line is the algebraically spherically resolved spaces, ([11]).

Definition 5.2. A space X is K_{∗}-algebraically spherically resolved (ASR) if: 1. K_{∗}(X) is a free K_{∗}-module.

2. There is a K_{∗}K-subcomodule M ⊂ K_{od}(X) such that K_{∗}(X) ≈ Λ(M) as K_{∗}K-comodules.

3. If Λ(M) is made into a coalgebra by making M primitive, then the isomorphism is as K_{∗}(K)-coalgebras.

4. One can choose a basis {m_{1}, m_{2}, … } for M so that each sequence 0 → K_{∗}{m_{1}, …, m_{n−1}} → K_{∗}{m_{1}, …, m_{n}} → K_{∗}{m_{n}} → 0 is a short exact sequence of K_{∗}(K)-comodules.

26 BENDERSKY AND DAVIS

A space is ASR if from the point of view of K-theory it appears as if it is built 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 K_{∗}-strongly spherically resolved (SSR) if there are spaces ∗ = X_{0}, X_{1}, …, X_{k} = X and fibrations

X_{i−1} → X_{i} → S^{n_{i}}  (5.4)

with n_{i} odd such that the K-homology groups of (5.4) form a split extension

Λ(x_{1}, …, x_{i−1}) → Λ(x_{1}, …, x_{i}) → Λ(x_{i})

as K_{∗}K-coalgebras.

Proposition 5.5. Suppose X is either K_{∗}-SSR or a simply-connected finite H-space with H_{∗}(X; Q) associative. Then X is K_{∗}-ASR and satisfies the hypotheses of Theorem 1.1.

Proof. The proof when X is SSR is completely straightforward. If X is a simplyconnected finite H-space 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}^{∧}) ≈ Λ̂(P K^{1}(X; Z_{p}^{∧})). In [19, proof of 3.5], an algorithm was described for finding a basis of QK^{1}(X) on which the matrix of the action of all Adams operations is triangular. We will sketch the algorithm in the next paragraph. Dualizing using [12, 2.3] and [14, §10], we obtain a basis of P K_{1}X on which the matrix of the K_{∗}K-coaction is triangular. This provides the required filtration by exterior K_{∗}K-subcoalgebras.

Sketch of algorithm: Since rationally X is a product of S^{2n_{i}+1}, there are elements x_{i} of QK^{1}(X) on which ψ^{k}(x_{i}) = k^{n_{i}}x_{i} for all k and such that rationally {x_{i}} is a basis. Let M be the integer matrix expressing the x_{i} in terms of some initial basis B of the free abelian group QK^{1}(X). If det(M) = ±1, then {x_{i}} is a basis of QK^{1}(X) and has diagonal matrix for all ψ^{k}. If not, we will modify {x_{i}} to remove divisors of det(M) and maintain the triangular property that, for all i and k, ψ^{k}(x′_{i}) involves only x′_{j} for j ≥ i. Suppose {x′_{i}} is such a set, with M′ the matrix expressing it in terms of B, and p is a prime divisor of det(M′). Then there is a nonzero vector v = x′_{i_{0}} + ∑ α_{j}x′_{j} with all j > i_{0} and α_{j} ∈ Z such that v is divisible by p. Replace x′_{i_{0}} by x′′_{i_{0}} = \frac{1}{p}v in the set \{x′_{i}\}. The determinant of the matrix M′′ expressing this new set in terms of B is det(M′)/p, and ψ^{k}(x′′_{i_{0}}) = k^{n_{i_{0}}}x′′_{i_{0}}+ a combination of x′_{j} with j > i_{0}, so the matrix of ψ^{k} stays triangular.

The following result was proved in [11].

Proposition 5.6. The v_{1}-periodic BTSS converges to v_{1}^{−1}π_{∗}(X̂) if X is K_{∗}-algebraically spherically resolved.

Proof. Briefly (for p = 2), by the same argument as in [2, 5.4], the conditions guarantee that E_{2}(X) is generated as an h_{1}-module by classes of filtration ≤ 2. In a forthcoming paper of Bousfield ([17]), it will be shown that the Yoneda product in E_{2} of the BTSS is associated to composition in homotopy. Since η^{4} = 0 in homotopy, E_{4} must have a horizontal vanishing line, and so Proposition 5.1 applies.

There are other important examples of spaces which satisfy 5.1. Although ΩS^{2n+1} is not ASR, it has E_{2} isomorphic to E_{2}(S^{2n+1}). Similarly, ΩS^{2n} is not ASR, but satisfies the condition of 5.1 because of 5.16.

We have established that the BTSS converges to v_{1}^{−1}π_{∗}(X̂) for many spaces. However our interest is in v_{1}^{−1}π_{∗}(X). Recall that X satisfies the CTP (1.3) if v_{1}^{−1}π_{∗}(X) → v_{1}^{−1}π_{∗}(X̂) is an isomorphism. The rest of the paper is devoted to proving 1.4, which states that S^{2n}, ΩS^{2n}, G_{2}, and F_{4} satisfy the CTP. The following lemma, which is a simple application of the Five Lemma, will be useful.

Lemma 5.7. Suppose X → Y → Z is a fibration for which X̂ → Ŷ → Ẑ is also a fibration. If the CTP is true for any two of the spaces, then the third space satisfies the CTP.

We now look for conditions that guarantee that a fibration satisfies 5.7. We observe that there is an easy solution to this problem at the odd primes. The K-homology of a finite simply-connected H-space X is an exterior algebra generated by P K_{od}(X) ≈ QK_{od}(X). In this case E_{2}(X) ≈ Ext_{V}(K_{∗}, QK_{∗}(X)), by 2.10 and 2.11. Furthermore at odd primes, by [11], E_{2}^{s,2n+1}(X) ≈ v_{1}^{−1}π_{2n+1−s}(X̂) for s ∈ {1, 2}, and E_{2}^{s}(X) = 0 if s > 2. Thus if the fibration induces a short exact sequence in QK_{∗}(−), its long exact sequence is E_{2} gives a long exact sequence in v_{1}-periodic homotopy of the Kcompletions, and hence the Five Lemma will imply that if two of the spaces satisfy the

28 BENDERSKY AND DAVIS

CTP, then so does the third. Because of differentials and extensions, this argument does not work for p = 2.

Associated to a fibration X → Y → Z is a K-homology cobar spectral sequence

E_{s}^{2} = Cotor_{s}^{K_{∗}(Z)}(K_{∗}, K_{∗}(Y)) ⟹ K_{∗}(X).  (5.8)

This spectral sequence, which generalizes that of Eilenberg-Moore ([23]) and is studied in [17], does not always converge to K_{∗}(X). However, Proposition 5.10, which is implied by [25, A1.2.8b], states that under favorable conditions it converges in a very strong way.

Definition 5.9. Suppose X → Y → Z is a fibration of connected spaces whose Khomologies are free over Z_{(2)}. We say the K-homology cobar spectral sequence strongly collapses if it collapses from E^{2} to the isomorphism

E_{s}^{2} ≈ \begin{cases}0 & s > 0 \\ K_{∗}X & s = 0. \end{cases}

Proposition 5.10. The K-homology cobar spectral sequence of a fibration which induces a relatively injective extension sequence (5.14) in K-homology strongly collapses.

Fibrations for which (5.8) strongly collapses are of some importance because of the following recent theorem of Bousfield ([17]).

Theorem 5.11. Suppose the fibration X → Y → Z induces a strongly collapsing K-homology cobar spectral sequence. Then the induced sequence X̂ → Ŷ → Ẑ is of the homotopy type of a fibration.

The notion of injective extension sequence was defined in [5, 3.1], following [13, 2.1]. However, there was a difference in the contexts. Bousfield was working over a field in [13], while [5] was working over BP_{∗}. This difference is pointed out on [5, p.378], where it is remarked that this “causes no change in the theory.” Actually it does cause a change. One requirement for C′ → C → C′′ to be an injective extension sequence is that C must be injective as a C′′-comodule. In the context of BP_{∗}-modules, this will not be the case in many important examples, such as one on [5, p.381]. Instead, the authors of [5] meant to be dealing with relatively injective modules and relatively

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 29

injective extension sequences. With this change of meaning, all the statements in [5] are valid.

As in [25, A1.2.7], we make the following definition.

Definition 5.12. If Γ is a coalgebra over a commutative ring A, then an extended Γ-comodule is one of the form Γ⊗_{A}N, where N is an A-module. A relatively injective Γ-comodule is a direct summand of an extended one.

The relationship that this has to the ordinary notion of injective is given in the following proposition.

Proposition 5.13. ([25, A1.2.8]) If M → N is a monomorphism of Γ-comodules which is split over A (in particular, if M and N are free A-modules), then any map from M to a relatively injective Γ-comodule S extends to N.

As pointed out in [25, A1.2.9,A1.2.10,A1.2.11], the usual methods of determining Ext using injective resolutions work for relative injectives as long as we restrict attention to free modules over the ground ring.

Now we make the following definition, totally analogous to [5, 3.1].

Definition 5.14. A relatively injective extension sequence (RIES) is a sequence of coalgebra maps C′ ⟶^{f} C ⟶^{g} C′′ such that

• g is an epimorphism of A-modules;

• f is the inclusion A□_{C′′}C → C;

• C is a relatively injective C′′-comodule.

The following result implies that the EHP fibration

S^{2n−1} → ΩS^{2n} → ΩS^{4n−1}  (5.15)

induces an RIES in K_{∗}(−) and hence its K-homology cobar spectral sequence strongly collapses.

Lemma 5.16. K_{∗}(ΩS^{2n}) ≈ K_{∗}(S^{2n−1}) ⊗ K_{∗}(ΩS^{4n−1}) as coalgebras.

Proof. The result is true with H_{∗} replacing K_{∗}. The Chern character shows that the result is true rationally. The E_{2}-term of the AHSS is isomorphic to

K_{∗} ⊗ H_{∗}(S^{2n−1}) ⊗ H_{∗}(ΩS^{4n−1}) ≈ Λ_{K_{∗}}(x) ⊗_{K_{∗}} P_{K_{∗}}[y],

30 BENDERSKY AND DAVIS

where Λ_{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 differential in the AHSS would violate the rational calculation. This proves the isomorphism in the lemma as K_{∗}-modules. We need to show y ∈ K_{4n−2}(ΩS^{4n−1}) is primitive. If not, the reduced coproduct has the form Δ(y) = x ⊗ x. But, this would imply that \overline{x} ∈ K^{1}(ΩS^{2n}), the dual of x, has nontrivial square, which is a contradiction.

Now the CTP for ΩS^{2n} is immediate from 5.11, 5.7, and the fact (noted near the end of Section 1) that ΩS^{2m−1} satisfies the CTP.

Exactly these same ingredients imply the CTP for S^{2n}, using the 2-primary fibration

S^{2n} → ΩS^{2n+1} → ΩS^{4n+1}.

This fibration induces an RIES in K_{∗}(−) by the argument used to deduce the similar statement for BP_{∗}(−) in [5, p.388].

The proof of 1.4 for G_{2} is more computational. Along with it, we prove Theorem 4.8.

Proof of 1.4 for G_{2} and 4.8. We consider the diagram induced by the map G_{2} → S^{6} (with fiber SU(3))

\begin{matrix}v_{1}^{−1}π_{∗}(G_{2}) & ⟶ & v_{1}^{−1}π_{∗}(S^{6}) \\ ↓ & & ↓ \\ v_{1}^{−1}π_{∗}(Ĝ_{2}) & ⟶ & v_{1}^{−1}π_{∗}(Ŝ^{6}) \end{matrix}  (5.17)

The analysis of v_{1}^{−1}π_{∗}(G_{2}) in [21], especially the diagram on page 667, can be viewed as a determination of the exact sequence

→ v_{1}^{−1}π_{∗}SU(3) → v_{1}^{−1}π_{∗}G_{2} → v_{1}^{−1}π_{∗}S^{6} ⟶^{∂} v_{1}^{−1}π_{∗−1}SU(3) → . (5.18)

That analysis is used implicitly in the following paragraphs.

A chart for v_{1}^{−1}π_{∗}(SU(3)), obtained from v_{1}^{−1}π_{∗}(S^{3}) (•) and v_{1}^{−1}π_{∗}(S^{5}) (∘), is the sum of Diagram 5.19 with an isomorphic chart, displaced (−1, −2) units. As usual, lines with negative slope represent boundary morphisms in the exact sequence of the fibration S^{3} → SU(3) → S^{5}.

Diagram 5.19.

[figure: 8ℓ − 4; 8ℓ + 2; ·; ·; ·; ·; ·; ·; ·; ·; ·]

Thus v_{1}^{−1}π_{∗}(SU(3)) appears as the upper part (∘) of Diagram 5.20, in which the lower part (•) is v_{1}^{−1}π_{∗}(S^{6}), from [22] or [21, p.667]. The long lines of negative slope represent ∂ in (5.18).

Diagram 5.20.

[figure: ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 8ℓ − 4; 8ℓ + 2,; ℓ even]

If ℓ is odd, the only change involves the differentials from 8ℓ + 2. In this case, remove the two indicated differentials from 8ℓ + 2. For exactly one of the two mod 4 congruences of odd ℓ, a differential from the bottom element in 8ℓ + 2 hits the top element in 8ℓ + 1. In [21, p.668,top], it was asserted that this differential is nonzero iff ℓ ≡ 3 mod 4. Although, as we shall see, this assertion is correct, there was a flaw

32 BENDERSKY AND DAVIS

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 F_{4}/G_{2} 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 ℓ. Thus v_{1}^{−1}π_{∗}(G_{2}) is as in Diagram 5.21, with the differential from 8ℓ + 2 being d_{2} if ℓ is even, and d_{3} for one of the mod 4 congruences of odd ℓ. Actually, the transition from 5.20 to 5.21 does not make some of the η-extensions clear, but they were established in [21].

Diagram 5.21.

[figure: 8ℓ − 3; 8ℓ + 2]

From Diagram 5.20, we see that the kernel of v_{1}^{−1}π_{∗}(G_{2}) → v_{1}^{−1}π_{∗}(S^{6}) consists of 0, the element of order 2 in 8ℓ + 2, and, for one of the mod 4 congruences of odd ℓ, the element of order 2 in 8ℓ + 1. Since S^{6} satisfies the CTP, (5.17) implies that the kernel of v_{1}^{−1}π_{∗}(G_{2}) → v_{1}^{−1}π_{∗}(Ĝ_{2}) consists of, at most, the elements described in the preceding sentence.

Recall that the BTSS of Diagram 4.6 converges to v_{1}^{−1}π_{∗}(Ĝ_{2}), and it must admit families of d_{3}-differentials annihilating all η-towers (except for a few elements at the bottom of some of the η-towers). The preceding paragraph has shown that the four parts of Diagram 5.21 involving η ≠ 0 map nontrivially to v_{1}^{−1}π_{∗}(Ĝ_{2}). Thus the

STABLE APPROACH TO UNSTABLE SPECTRAL SEQUENCE 33

pattern of differentials in the BTSS of G_{2} must be as in Diagram 4.9 to allow for these elements of v_{1}^{−1}π_{∗}(Ĝ_{2}).

The nonzero extension in dimension 8ℓ + 2 in Diagram 4.9 for G_{2} must occur because v_{1}^{−1}π_{8ℓ+1}(G_{2}) → v_{1}^{−1}π_{8ℓ+1}(Ĝ_{2}) must send the element x of highest filtration in Diagram 5.21 nontrivially (it corresponds to the top element in v_{1}^{−1}π_{8ℓ+1}(S^{6}) in Diagram 5.20), and since ηx is divisible by 2, the same must be true of its image in v_{1}^{−1}π_{8ℓ+1}(Ĝ_{2}). A similar argument implies the nontrivial extension in t − s = 8ℓ + 1 in Diagram 4.9.

Thus v_{1}^{−1}π_{8ℓ+2}(G_{2}) → v_{1}^{−1}π_{8ℓ+2}(Ĝ_{2}) is injective (since the element of order 2 maps across),

v_{1}^{−1}π_{8ℓ+2}(Ĝ_{2}) ≈ Z/2^{min(6,ν(ℓ−1)+4)},

while v_{1}^{−1}π_{8ℓ+2}(G_{2}) ≈ Z/2^{4} if ℓ is even, and is Z/2^{5} for one odd mod 4 congruence of ℓ, and Z/2^{6} for the other. This implies that the Z/2^{5} must occur for ℓ ≡ 3 mod 4, and v_{1}^{−1}π_{8ℓ+2}(G_{2}) → v_{1}^{−1}π_{8ℓ+2}(Ĝ_{2}) is bijective. Thus [21, 4.1,4.5] are valid, as are Theorems 4.8 and 1.4 for G_{2}.

Finally we deduce the CTP for F_{4} from the fact that (4.1) induces an RIES in K_{∗}(−), 5.10, 5.11, 5.7, and the fact that G_{2} and F_{4}/G_{2} both satisfy the CTP, the latter by 4.11. That

K_{∗}(G_{2}) → K_{∗}(F_{4}) → K_{∗}(F_{4}/G_{2})

is an RIES follows from Proposition 5.5 and the fact that there is a short exact sequence in P K_{1}(−).

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