New York Journal of Mathematics

New York J. Math. 4 (1998) 185–221.

3-Primary v_{1}-Periodic Homotopy Groups of E_{7}

Donald M. Davis

Abstract. In this paper we compute the 3-primary v_{1}-periodic homotopy groups of the exceptional Lie group E_{7}. This represents the next stage in the author’s goal of calculating the v_{1}-periodic homotopy groups of all compact simple Lie groups (at least when localized at an odd prime). Most of the work goes into calculating the unstable Novikov spectral sequence of ΩE_{7}/Sp(2). Showing that this spectral sequence converges to the v_{1}-periodic homotopy groups in this case utilizes recent results of Bousfield and Bendersky-Thompson.

Contents

1. Introduction 185 2. Background in v_{1}-periodic homotopy and the UNSS 187 3. New results about extensions 192 4. Discussion of E_{7}/F_{4} 194 5. E_{2} of periodic UNSS of ΩE_{7}/Sp(2) 197 6. The final case 213 7. Periodic homotopy of E_{7} 216 References 220

1. Introduction

In this paper we compute the 3-primary v_{1}-periodic homotopy groups of the exceptional Lie group E_{7}.

The p-primary v_{1}-periodic homotopy groups of a space X, denoted v_{1}^{−1}π_{∗}(X; p) or just v_{1}^{−1}π_{∗}(X), were defined in [21]. They are a localization of the actual homotopy groups, telling roughly the portion which is detected by K-theory and its operations. If X is a compact Lie group, each v_{1}^{−1}π_{i}(X; p) is a direct summand of some actual homotopy group of X, and so summands of v_{1}-periodic homotopy groups of X give lower bounds for the p-exponent of X.

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York

ISSN 1076-9803/98

Received May 4, 1998.

Mathematics Subject Classification. 55T15.

Key words and phrases. v_{1}-periodic homotopy groups, exceptional Lie groups, unstable Novikov spectral sequence.

After the author computed v_{1}^{−1}π_{∗}(SU(n); p) for odd p in 1989, Mimura proposed the goal of calculating v_{1}^{−1}π_{∗}(X; p) for all compact simple Lie groups X. This has now been achieved in the following cases (X, p):

• X a classical group and p odd ([19]);

• X an exceptional Lie group with H_{∗}(X; Z) p-torsion-free ([13]);

• (SU(n) or Sp(n), 2) ([11],[12]);

• (G_{2}, 2) ([22]), (F_{4} or E_{6}, 3) ([10]), and (E_{7}, 3) (the current paper).

The only cases remaining then are (E_{8}, 2 or 3 or 5) and (SO(n) or F_{4} or E_{6} or E_{7}, 2). Several of these appear tractable.

Now we state our main theorem. We usually abbreviate v_{1}^{−1}π_{∗}(X; 3) as v_{∗}(X), and denote by ν(n) the exponent of 3 in the integer n.

Theorem 1.1. If j is even, then v_{2j}(E_{7}) = v_{2j−1}(E_{7}) = 0. If j is odd, then

v_{2j}(E_{7}) ≈ v_{2j−1}(E_{7}) ≈ \begin{cases}Z/3 ⊕ Z/3^{min(10,ν(j−9−2·3^{5})+4)} & if j ≡ 0 mod 3 \\ Z/3 ⊕ Z/3^{min(8,ν(j−43)+5)} & if j ≡ 1, 7 mod 9 \\ Z/3 ⊕ Z/3^{min(14,ν(j−13−4·3^{8})+5)} & if j ≡ 4 mod 9 \\ Z/9 ⊕ Z/3^{min(19,ν(j−17−2δ·3^{13})+4)} & if j ≡ 5, 8 mod 9, \end{cases}

where δ equals one of the numbers 2, 5, or 8. If j is odd and j ≡ 2 mod 9, then v_{2j−1}(E_{7}) ≈ Z/9 ⊕ Z/3^{min(13,ν(j−11)+4)}, while

v_{2j}(E_{7}) ≈ \begin{cases}Z/3^{3} ⊕ Z/3^{ν(j−11)+3} & if ν(j − 11) < 10 \\ Z/3^{3} ⊕ Z/3^{12} or Z/3^{4} ⊕ Z/3^{11} & if ν(j − 11) ≥ 10. \end{cases}

An immediate corollary of this work is a lower bound for the 3-exponent of E_{7}. Recall that the p-exponent of a space X, denoted exp_{p}(X), is the largest e such that π_{∗}(X) has an element of order p^{e}. We obtain

Corollary 1.2. The 3-exponent of E_{7} satisfies exp_{3}(E_{7}) ≥ 19.

Proof. If X is a compact Lie group, then v_{i}(X) ≈ dirlim_{k,m} π_{i+4k·3^{m}}(X). Hence an element of order 3^{19} in v_{2j}(E_{7}) when j ≡ 17 + 2δ · 3^{13} mod 2 · 3^{15} corresponds to an element of order 3^{19} in some π_{n}(E_{7}). □

By comparison, the result that we have obtained at other primes is ([13])

exp_{p}(E_{7}) \begin{cases}= 17 & if p > 17 \\ = 18 & if p = 17 \\ ≥ 17 & if p = 7, 11, or 13 \\ ≥ 18 & if p = 5. \end{cases}

This should be contrasted with the situation for spheres, where exp_{p}(S^{2n+1}) = n for all odd primes p and all positive integers n by [25] and [18].

Note that in Theorem 1.1, we determine the precise abelian group structure of all groups (with isolated exceptions), whereas in some earlier papers, such as [19], [13], and [10], we had been unable to determine the group structure of most groups v_{2j−1}(X). Because of the insights of [23], we are able not only to resolve the extension questions (group structure) in almost all cases occurring here, but also in those of [13] and [10]. These new results about group structure are presented in Section 3.

3-Primary Periodic Homotopy Groups of E_{7} 187

Most of the work is calculation of the v_{1}-periodic unstable Novikov spectral sequence (UNSS) of the space Y_{7} := ΩE_{7}/Sp(2). The main input is the detailed structure of H_{∗}(ΩE_{7}; Z/3) given in [26] and restated here in Proposition 5.2. The advantage of Y_{7} over ΩE_{7} is that BP_{∗}(Y_{7}) is a free commutative algebra, which makes its UNSS easier to calculate. Perhaps the most novel feature of the calculations here is the use of coassociativity to give detailed formulas for BP_{∗}-coaction. The terms which arise in this way play crucial roles in the calculations. The calculations of v_{∗}(F_{4}) in [10] are essential in the transition from v_{∗}(Y_{7}) to v_{∗}(ΩE_{7}).

Another delicate point is convergence of the v_{1}-periodic UNSS for Y_{7}. In Section 4, we use deep recent work of Bousfield and Bendersky-Thompson to prove that the v_{1}-periodic UNSS converges to v_{∗}(−) for E_{7}/F_{4}, which we will show implies similar convergence for Y_{7}.

The author would like to thank Mamoru Mimura, Pete Bousfield, and especially Martin Bendersky for useful suggestions.

Note added in proof (Oct. 5, 1998): Using a completely different method, based on [17, 9.2], the author has obtained an independent verification of the results for v_{2j}(E_{7}) given in Theorem 1.1, and shown that in the cases unresolved in Theorem 1.1, δ = 5 and, if j is odd and ν(j − 11) ≥ 10, then v_{2j}(E_{7}) ≈ Z/3^{3} ⊕ Z/3^{12}. This method relies very heavily on computer calculations, and will be discussed in a forthcoming paper.

2. Background in v_{1}-periodic homotopy and the UNSS

In this section, we review known results, and establish one new useful result about computing the UNSS. Although some of these results are also true when p = 2, it will simplify exposition to assume that p is an odd prime.

The v_{1}-periodic homotopy groups of any topological space X are defined by

v_{1}^{−1}π_{i}(X) = lim_{→}[M^{i+1+kqp^{e}}(p^{e}), X], (2.1)

where q = 2p − 2, a notation that will be used consistently throughout this paper, and M^{t}(n) denotes the Moore space S^{t−1} ∪_{n} e^{t}. Here the direct limit is taken over increasing values of e and k using Adams maps M^{t+qp^{e−1}}(p^{e}) → M^{t}(p^{e}) and canonical maps M^{t}(p^{e+1}) → M^{t}(p^{e}). This definition was given in [21], where their relationship with actual homotopy groups of many spaces was established.

A space X is said to have an H-space exponent at the prime p if, for some e and L, p^{e} : Ω^{L}X → Ω^{L}X is null homotopic. It was shown in [21, 1.9] that if X has an H-space exponent, then

v_{1}^{−1}π_{i}(X) ≈ lim_{→} π_{i+kqp^{e}}(X),

and hence v_{1}^{−1}π_{i}(X) is a direct summand of some group π_{i+kqp^{e}}(X). To make this final deduction, we need to know that the limit group is a finitely generated abelian group, but this will be the case.

Next we discuss the unstable cobar complex, which can be used to compute the UNSS for many spaces. We will modify and generalize previous treatments of this topic. Let BP be the Brown-Peterson spectrum corresponding to the prime p. Then

BP_{∗} = π_{∗}(BP) ≈ Z_{(p)}[v_{1}, v_{2}, …],

where v_{i} are the Hazewinkel generators of BP_{∗}. Let

Γ = BP_{∗}(BP) ≈ BP_{∗}[h_{1}, h_{2}, …],

where h_{i} are conjugates of Quillen’s generators t_{i}. We have |v_{i}| = |h_{i}| = 2(p^{i} − 1). Let η = η_{R} : BP_{∗} → BP_{∗}(BP) be the right unit. We write h_{i}v_{j} interchangeably with η(v_{j})h_{i}; this is the right action of BP_{∗} on Γ.

Let M be a Γ-comodule with coaction map ψ_{M} : M→Γ ⊗ M. Tensor products are always over BP_{∗}. The stable cobar complex SC^{∗}(M) is defined by

SC^{s}(M) = Γ ⊗ Γ ⊗ ⋯ ⊗ Γ ⊗ M,

with s copies of Γ, and differential d : SC^{s}(M) → SC^{s+1}(M) given by

(2.2) d(γ_{1} ⊗ ⋯ ⊗ γ_{s} ⊗ m) = 1 ⊗ γ_{1} ⊗ ⋯ γ_{s} ⊗ m

+ ∑_{j=1}^{s}(−1)^{j}γ_{1} ⊗ ⋯ ⊗ ψ(γ_{j}) ⊗ ⋯ ⊗ γ_{s} ⊗ m

+(−1)^{s+1}γ_{1} ⊗ ⋯ ⊗ γ_{s} ⊗ ψ_{M}(m). (2.3)

Our unstable cobar complex V C^{∗}(M) is a subcomplex of SC^{∗}(M), consisting of terms satisfying an unstable condition, introduced in the following definition.

Definition 2.4. [6, 3.3] If M is a nonnegatively graded free left BP_{∗}-module, then V (M) is defined to be the BP_{∗}-span of

{h^{I} ⊗ m : 2(i_{1} + i_{2} + ⋯) ≤ |m|} ⊂ Γ ⊗ M,

where I = (i_{1}, i_{2}, …) and h^{I} = h_{1}^{i_{1}}h_{2}^{i_{2}} · · ·.

This unstable condition will pervade our computations. Note that for odd dimensional classes, this agrees with the module U(M) which has been used most frequently in earlier work of the author and Bendersky. However, it also agrees with the V (M) construction employed in [12] on even-dimensional classes. The novelty here is that it will be applied to a module having classes of both parities.

Define V C^{0}(M) = M, and V C^{s}(M) = V (V C^{s−1}(M)). If M is a Γ-comodule, then the differential d of the stable cobar complex of M induces a differential on the subcomplex V C^{∗}(M). We will usually replace it by the chain-equivalent reduced complex obtained by replacing V (M) by ker(V (M)⟶^{ϵ}M) ([5, 2.16]). This has the effect of only looking at terms which have positive grading in each position. The homology groups of this unstable cobar complex are denoted by Ext_{V}^{s,t}(M). As observed in [6], these are the usual Ext groups in the abelian category V of Γ-comodules satisfying the unstable condition in Definition 2.4. Note there is a shift isomorphism

Ext_{V}^{s,t−1}(BP_{∗}S^{2n}) ≈ Ext_{V}^{s,t}(BP_{∗}S^{2n+1}), (2.5)

induced by a shift isomorphism of the unstable cobar complexes.

The following generalization of [8, §7] will be very useful to us. Its proof follows some suggestions of Martin Bendersky.

Theorem 2.6. If X is a simply-connected CW -space, there is a spectral sequence {E_{r}^{s,t}(X), d_{r}} which converges to the homotopy groups of X localized at p. If X is an H-space, and BP_{∗}(X) is a free commutative algebra, then

E_{2}^{s,t}(X) = Ext_{V}^{s,t}(Q(BP_{∗}X)),

where Q(BP_{∗}X) denotes the indecomposable quotient of BP_{∗}X.

This is the UNSS for the space X. We will write V C^{∗}(X) for the complex V C^{∗}(Q(BP_{∗}X)), whose homology is E_{2}(X). We denote by Fr the free commutative algebra functor. If N is a free BP_{∗}-module with basis B = B_{ev} ∪ B_{od}, then Fr(N) is the tensor product of a polynomial algebra over BP_{∗} on B_{ev} with an exterior algebra on B_{od}.

Proof. The spectral sequence was described in [8]. The determination of E_{2} when BP_{∗}(X) is a free commutative algebra is quite similar to that of [9, 6.1] and to the argument on [12, p.346]. Let M = Q(BP_{∗}X), a free BP_{∗}-module.

Let G denote the category of unstable Γ-coalgebras, and G(−) the associated functor considered in [8]. If N is a free BP_{∗}-module, then G(N) is defined to be BP_{∗}(BP (N)), where BP (N) is the 0th space of the Ω-spectrum representing the homology theory BP_{∗}(−) ⊗ N. If N has basis B, then

G(N) ≈ BP_{∗}(∏_{b∈B} BP_{|b|}) ≈ Fr(⟨h^{I}b : b ∈ B, 2|I| ≤ |b|⟩). (2.7)

Here h^{I} is as in Definition 2.4 with |I| = ∑ i_{j}, while |b| denotes the degree of the basis element b. Also BP_{n} denotes the nth space in the Ω-spectrum for BP. The first isomorphism in (2.7) is immediate from the definition of G given in [8, 6.3,6.7]. The second isomorphism follows from [33, p.51], which says that H_{∗}(BP_{n}) is a polynomial algebra if n is even, and an exterior algebra if n is odd, [31, 4.9], which says that the same thing is then true of BP_{∗}(BP_{n}), and [7, p.1040], which interprets conveniently the description of the indecomposables first given in [31]. Note that there is an isomorphism of BP_{∗}-modules

Q(G(N)) ≈ V (N). (2.8)

We claim that

BP^{∗}X⟶^{ξ}G(M) ⟶^{→} G(V (M)) ⟶_{→}^{→} G(V^{2}(M)) ⋯ (2.9)

is an augmented cosimplicial resolution in G. Here the augmentation ξ is the composite

BP_{∗}X⟶^{η_{X}}G(BP_{∗}X) ⟶^{G(ρ)} G(QBP_{∗}X),

where the second morphism applies G to the quotient morphism ρ. The cofaces are of two types:

• G(V^{q}M) ⟶^{η_{G(V ^{q} M)}}G(G(V^{q}M))⟶^{G(ρ)}G(V^{q+1}M), where ρ : G(−) → QG(−) =

V (−) is the quotient morphism.

• G(V^{i}(ψ_{V^{q−i}M})), 0 ≤ i ≤ q, where ψ_{N} : N → V (N) stabilizes to the Γ-

coaction. The degeneracies G(V^{q}M) → G(V^{q−1}M) just do the counit ϵ on one of the V - factors. It is clear that all of these morphisms are in G, and the cosimplicial identities are satisfied as usual. The argument of [12, 3.13] implies that the first type of coface map and the augmentation ξ are algebra morphisms. The second type of coface map is an algebra morphism since it is BP_{∗}(f) for an infinite loop map f, namely the map BP (N) ⟶^{BP (g)}BP (N′) induced by a BP_{∗}-morphism N⟶^{g}N′.

The exactness of the resulting augmented cochain complex

0 → BP_{∗}X⟶^{ξ}G(M) → G(V (M)) → G(V^{2}(M)) → ⋯ (2.10)

(obtained using the alternating sum of cofaces as boundaries) follows as in [9, p.387], but we provide details for completeness. (In comparing with [9], it is useful to note that V (N) ≈ σ^{−1}U(σN).) Since the coface operators are algebra homomorphisms, their alternating sum preserves the filtration of this augmented complex by powers of the augmentation ideal. Let E_{0} denote the quotients of the filtration. Then, using (2.8), we have

E_{0}(G(V^{q}(M))) ≈ Fr(Q(G(V^{q}(M)))) ≈ Fr(V^{q+1}(M)),

and E_{0}(BP_{∗}X) ≈ Fr(M). Thus E_{0}(2.10) is the free commutative algebra on the complex

0 → M → V (M) → V^{2}(M) → ⋯

with morphisms the alternating sum of ψ on each V and ψ_{M}, which is exact by [16, 7.8]. Since the free commutative algebra functor applied to an exact sequence yields an exact sequence, we deduce that (2.10) is exact, and hence yields a resolution in G of BP_{∗}X.

Hence Ext_{G}(BP_{∗}, BP_{∗}X) is equal to the cohomology of the complex obtained by applying Hom_{G}(BP_{∗}, −) to the portion of (2.10) after ξ. Since Hom_{G}(BP_{∗}, G(N)) ≈ N, we obtain that Ext_{G}(BP_{∗}, BP_{∗}X) is the homology of the complex

M → V (M) → V^{2}(M) → ⋯, (2.11)

with differentials as in (2.3). The claim of the theorem follows now from [8, 6.17], which states that E_{2}(X) ≈ Ext_{G}(BP_{∗}, BP_{∗}X), and the observation that (2.11) is just our unstable cobar complex, whose homology is Ext_{V}(M). □

The following definition will be extremely important.

Definition 2.12. The excess exc(γ) of an element γ of Γ̄^{s} is defined to be the smallest n such that γι_{2n+1} is an element of V C^{s}(S^{2n+1}).

This means that if γ = γ_{1} ⊗ ⋯ ⊗ γ_{s}, then for 1 ≤ i ≤ s,

γ_{i} ⊗ (γ_{i+1} ⋯ γ_{s}ι_{2n+1})

must satisfy 2.4. The following result, which was proved as [20, 4.2], gives a formula for the excess of certain monomials when s = 2.

Lemma 2.13. If a ≤ b and a ≤ d, then

exc(p^{a}h^{b}⊗v^{c}h^{d}v^{e}) = max(b−(p−1)(c+d), d)−min(a, |b−(p−1)c−pd|)−(p−1)e.

In [4], the v_{1}-periodic UNSS was defined and shown to satisfy the following very nice property.

Theorem 2.14. If p is odd and X is spherically resolved, the v_{1}-periodic UNSS of X satisfies

• v_{1}^{−1}E_{∞}^{s,t}(X) = v_{1}^{−1}E_{2}^{s,t}(X), and is 0 unless s = 1 or 2 and t is odd.

• v_{1}^{−1}E_{∞}^{s,t}(X) ≈ v_{1}^{−1}π_{t−s}(X) if s = 1 or 2 and t is odd.

• v_{1}^{−1}E_{2}^{s,t}(X) = dirlim E_{2}^{s,t+kqp^{e}}(X), where e is chosen sufficiently large, and the direct limit is taken over increasing values of k under multiplication by v_{1}^{p^{e}}.

Here we say that X is spherically resolved if it can be built from a finite number of odd-dimensional spheres by fibrations. In Section 4, we will show that Theorem 2.14 holds in a certain case in which we cannot prove that X is spherically resolved.

We will use the unstable cobar complex for the unlocalized UNSS, but, as we are dealing exclusively with v_{1}-periodic classes, we can, in effect, act as if it satisfies the first two properties of Theorem 2.14.

We will make frequent use of the following result for the spheres, which was proved in [20], following [8] and [3]. We introduce here terminology x ≡ y mod S^{2n−1} to mean that x − y desuspends to (or is defined on) S^{2n−1}. For elements of E_{2}^{s}(S^{2n+1}), we frequently abbreviate xι_{2n+1} as x.

Theorem 2.15. 1. The only nonzero groups v_{1}^{−1}E_{2}^{s,t}(S^{2n+1}) are

v_{1}^{−1}E_{2}^{s,2n+1+qm}(S^{2n+1}) ≈ Z/p^{e}

with s = 1 or 2 and e = min(n, ν(m) + 1).

2. The generator of E_{2}^{1,2n+1+qm}(S^{2n+1}) is α_{m/e} := d(v_{1}^{m})/p^{e} and satisfies

α_{m/e} ≡ −v_{1}^{m−e}h_{1}^{e} mod S^{2e−1}, (2.16)

and, if m = sp^{e−1} with s ≢ 0 mod p, and e > n, then

α_{m/e} ≡ −sv_{1}^{m−1}h_{1} mod p. (2.17)

3. If n ≤ ν(m) + 1 and 1 ≤ j ≤ n, then d(p^{m−ν(m)−1−j}h_{1}^{m})ι_{2n+1} has order p^{j} in E_{2}^{2,2n+1+qm}(S^{2n+1}). It equals v_{1}^{m−j−1}h_{1} ⊗ h_{1}^{j} mod S^{2j−1}.

4. If ν(m) + 1 ≤ n and 1 ≤ j ≤ ν(m) + 1, then d(p^{m−n−j}h_{1}^{m})ι_{2n+1} has order p^{j} in E_{2}^{2,2n+1+qm}(S^{2n+1}). It equals v_{1}^{m−n−j+ν(m)}h_{1} ⊗ h_{1}^{n+j−ν(m)−1} mod S^{2n+2j−2ν(m)−3}.

5. The homomorphism Σ^{2} : E_{2}^{2,2n−1+qm}(S^{2n−1}) → E_{2}^{2,2n+1+qm}(S^{2n+1}) is injective if n ≤ ν(m) + 1 and is multiplication by p otherwise.

Other more technical results proved in earlier works are as follows. Here we begin the practice, which will be continued throughout the paper, of often abbreviating h_{1} as h, and v_{1} as v. Also, we introduce the term “leading term” to refer to a monomial of largest excess in an element z of V C(X); all other monomials comprising z desuspend farther than does the leading term.

Proposition 2.18. 1. ([23, 2.9]) If a cycle of V C^{2}(S^{2n+1}) has order p^{f} in

E_{2}^{2,t}(S^{2n+1}) and leading term h ⊗ h^{j}ι_{2n+1}, then j + ν(|E_{2}^{2,t}(S^{2n+1})|) = f + n.

2. ([20, 4.6]) Let ν = ν(σ), and let

z = ϵv^{σ−e−1}h^{e} ⊗ h + L ∈ V C^{2,2n+1+qσ}(S^{2n+1})

be a cycle with ϵ ∈ Z_{(p)} and exc(L) < e − p + 1 ≤ n − ν. Then

z = d(uϵv^{σ−(e+ν−p+2)}h^{e+ν−p+2} + L′),

where u is a unit in Z_{(p)}, and exc(L′) < e + ν − p + 2. The same conclusion holds for z = ϵv^{σ−e+p−2}h ⊗ h^{e−p+1} + L.

We will need the following precise description of α_{2}.

Lemma 2.19. The element α_{2} which generates E_{2}^{1,2q+2n+1}(S^{2n+1}) is given by

α_{2} = −d(v_{1}^{2})/p = \frac{1}{p}(v^{2} − (v − ph)^{2}) = 2vh − ph^{2} = hv + vh.

We will make repeated use of the following result, especially part (1).

Lemma 2.20. Let p = 3. Then 1. η(v_{1}) = v_{1} − 3h_{1}

2. η(v_{2}) = v_{2} + 4v^{3}h − 18v^{2}h^{2} + 35vh^{3} − 24h^{4} − 3h_{2}

3. ψ(h_{1}) = h_{1} ⊗ 1 + 1 ⊗ h_{1}

4. ψ(h_{2}) = h_{2} ⊗ 1 + 1 ⊗ h_{2} + 4h^{3} ⊗ h + 6h^{2} ⊗ h^{2} + 3h ⊗ h^{3} − vh ⊗ h^{2} − vh^{2} ⊗ h

Proof. Parts (1) and (3) are standard, appearing in all referenced papers of the author and/or Bendersky. Part (2) is taken from Giambalvo’s tables ([24]). Part (4) is derived from [13, 2.6i], using part (1) of this lemma several times to replace a v on the right (which is interpreted as η(v)) by v − 3h. Note, however, that the sum in [13, 2.6i] should be preceded by a minus sign. □

The following result, proved in [23, 2.11,2.12,2.13], will be central to many of our calculations.

Lemma 2.21. 1. If n ≥ 1, then in E_{2}^{1}(S^{2n+1}), h_{1}^{p}v_{1} ≡ v_{1}^{p}h_{1} mod S^{1}. 2. h_{1}^{n+p−1} ⊗ h_{1} ≡ −v_{1}^{p−1}h_{1} ⊗ h_{1}^{n} mod S^{2n−1} if n > 1;

3. d(v_{1}^{ℓ}h_{1}^{n+1}) ≡ −(ℓ + n + 1)v_{1}^{ℓ}h_{1} ⊗ h_{1}^{n} mod S^{2n−1}.

3. New results about extensions

In this section, we show that v_{1}^{−1}π_{2j−1}(X) is cyclic when X is a sphere bundle over a sphere with attaching map α_{1} or α_{2}. This will be crucial to our proof of Theorem 1.1. We also determine the group structure of all groups v_{1}^{−1}π_{2j−1}(X) when X is an exceptional Lie group for which the orders |v_{1}^{−1}π_{∗}(X)| have been determined.

The first result of this section is the following, in which B_{k}(2n + 1, 2n + kq + 1) is an S^{2n+1}-bundle over S^{2n+kq+1} with attaching map α_{k}.

Theorem 3.1. Let n > 1, and k = 1 or 2. Then v_{2j−1}(B_{k}(2n + 1, 2n + kq + 1)) and v_{2j}(B_{k}(2n + 1, 2n + kq + 1)) are isomorphic cyclic p-groups with exponent

min(n, 2 + ν(j − n))  if j ≡ n mod p(p − 1)
min(n + k(p − 1), 2 + ν(j − n − k(p − 1))) if j ≡ n mod (p − 1)
and j ≢ n mod p(p − 1)
0  otherwise.

Proof. Let B = B_{k}(2n + 1, 2n + kq + 1). The determination of v_{2j}(B) when k = 1 was made in [13, 1.3(2)]. In [10, p.301], v_{2j}(B) was determined when k = 2, n = 4, and p = 3. The argument there adapts to the general case in a straightforward fashion.

The cyclicity of v_{2j−1}(B) when k = 1 is proved similarly to [23, p.613]. It is easy when |v_{2j}(S^{2n+q+1})| = p^{n+p−1}, for then ∂ : v_{2j}(S^{2n+q+1}) → v_{2j−1}(S^{2n+1}) is surjective, and so v_{2j−1}(B) ≈ v_{2j−1}(S^{2n+q+1}) is cyclic.

Now consider the case when |v_{2j}(S^{2n+q+1})| < p^{n+p−1}. By Theorem 2.15(4), the class d(p^{m−n−p}h^{m})ι_{2n+q+1} has order p in E_{2}^{2}(S^{2n+q+1}). Here m is an integer related to the stem of the class under consideration. Since ∂ annihilates this class, there is w ∈ V C^{2}(S^{2n+1}) such that z := d(p^{m−n−p}h^{m})ι_{2n+q+1} − w is a cycle in E_{2}^{2}(B). We wish to show that pz is the image of a generator of E_{2}^{2}(S^{2n+1}).

We use the formula

d(h^{t}ι) = d(h^{t})ι + h^{t}d(ι), (3.2)

which was explained as [23, 5.4]. This implies that

pz = d(p^{m−n−p+1}h^{m}ι_{2n+q+1}) − p^{m−n−p+1}h^{m} ⊗ hι_{2n+1} − pw.

The first term is a boundary. Using Lemma 2.20, the second term is, mod terms that desuspend below S^{2n+1}, −v_{1}^{m−n−p+1}h^{n+p−1} ⊗ hι_{2n+1}, and by Lemma 2.21(2), this is, mod lower terms, v_{1}^{m−n−p+1}h ⊗ h^{n}ι_{2n+1}, which is the leading term of a generator of E_{2}^{2}(S^{2n+1}), by Theorem 2.15 or Proposition 2.18. Also, as we shall show in the next paragraph, pw desuspends to S^{2n−1}. Since the double suspension from E_{2}^{2}(S^{2n−1}) to E_{2}^{2}(S^{2n+1}) is not surjective, this implies that pz is the image of a generator of E_{2}^{2}(S^{2n+1}).

One way to see that w can be chosen so that pw desuspends to S^{2n−1} is to note that pz = d(p^{m−n−p+1}h^{m})ι_{2n+q−1} − pw is a cycle in E_{2}^{2}(B(2n − 1, 2n + q − 1)); i.e., multiplying by p allows you to double desuspend the whole equation.

The argument when k = 2 is very similar. We will have w ∈ V C^{2}(S^{2n+1}) satisfying that z := d(p^{m−n−2p+1}h^{m})ι_{2n+2q+1} − w is a cycle in E_{2}^{2}(B), and, as in the previous paragraph, w can be chosen so that pw double desuspends. We obtain

pz = d(p^{m−n−2p+2}h^{m}ι_{2n+2q+1}) − p^{m−n−2p+2}h^{m} ⊗ α_{2}ι_{2n+1} − pw.

The first term is a boundary, the last term desuspends, while the middle term is, mod terms that desuspend, v_{1}^{m−n−2p+2}h^{n+2p−2} ⊗ (2vh − ph^{2})ι_{2n+1}. The term with ph^{2} desuspends, while the first term is, by 2.21(2), 2v_{1}^{m−n−p+1}h^{n+p−1} ⊗ hι_{2n+1}, which is the leading term of a generator of E_{2}^{2}(S^{2n+1}). The class {pz} generates E_{2}^{2}(S^{2n+1}) by Proposition 2.18(1). □

In [13] and [10], sphere bundles X of the type covered by Theorem 3.1 occurred as factors in product decompositions of exceptional Lie groups (localized at a prime p). In those papers, we merely asserted the order of the groups v_{1}^{−1}π_{2j−1}(X), but we can now declare that they are cyclic. There are a few other cases of factors Y of exceptional Lie groups for which only the order but not the group structure of v_{1}^{−1}π_{2j−1}(Y) was given in [13], but we can now complete the determination of the v_{1}-periodic homotopy of all torsion-free exceptional Lie groups by giving the group structure in these cases. The following result handles all of these, and those left unresolved in [10].

Proposition 3.3. 1. The 3-primary groups v_{2j−1}(B(11, 15)) and v_{2j−1}(E_{6}/F_{4}),

which occur in [10, 1.2], are cyclic.

2. The factors B(2n + 1, 2n + q + 1) which occur in G_{2} for p = 5, F_{4} and E_{6} for 5 ≤ p ≤ 11, E_{7} for 11 ≤ p ≤ 17 and for p = 5, and E_{8} for 11 ≤ p ≤ 29, as listed in [13, 1.1], have v_{2j−1}(B) cyclic of order given in [13, 1.3(2)].

3. The spaces B(11, 23, 35) and B(23, 35, 47, 59), which occur as factors in 7primary E_{7} and E_{8}, respectively, have v_{2j−1}(B) cyclic of order given in [13, 1.4].

4. The spaces B(3, 11, 19, 27, 35), B(3, 15, 27), and B(3, 15, 27, 39), which occur as factors of 5-primary E_{7}, 7-primary E_{7}, and 7-primary E_{8}, respectively, have v_{2j−1}(B) ≈ Z/p ⊕ Z/p^{e−1}, where e is the number given in [13, 1.4].

Proof. The first two parts are immediate from Theorem 3.1. The first space in part 3 is a factor of SU(18), and in the notation of [23, 1.5] it has N = 5 and i = 2. By [23, 1.9], its v_{2j−1}(−)-groups are cyclic. Similarly, the second space in part 3 is a quotient of a factor B = B(11, 23, 35, 47, 59) of SU(30). This factor B has N = 5 and i = 4 in the notation of [23, 1.5], and hence its groups v_{2j−1}(−) are cyclic by [23, 1.9]. Thus so are the groups of the desired space B(23, 35, 47, 59), since v_{2j−2}(S^{11}) = 0 for values of j under consideration.

The spaces in part 4 are also factors of SU(n) and hence are covered by [23, 1.9]. In the notation of [23, 1.5], these three spaces each have N = 1, while i = 4, 2, and 3, respectively, and m̂ > 0. Thus their v_{2j−1}(−) has a split Z/p by [23, 1.9]. □

4. Discussion of E_{7}/F_{4}

In this section we sketch a natural approach to Theorem 1.1. Although we will not follow it exactly, it is helpful in understanding the approach which we do employ. Also, the result here about the convergence of the v_{1}-periodic UNSS for E_{7}/F_{4} will play a key role in our later deduction of v_{1}^{−1}π_{∗}(E_{7}). Throughout the remainder of the paper, we will have p = 3.

The fibration

F_{4} → E_{7} → E_{7}/F_{4} (4.1)

induces a long exact sequence of v_{1}-periodic homotopy groups. The groups v_{∗}(F_{4}) were computed in [10], while v_{∗}(E_{7}/F_{4}) could be computed by the methods of this paper. Then we would need to determine the boundary homomorphism and extensions in the exact sequence associated to (4.1). This determination is complicated by the fact that the Bockstein β is nonzero in H^{∗}(F_{4}; Z/3), which causes BP_{∗}(F_{4}) to be not a free BP_{∗}-module, and therefore the UNSS of F_{4} cannot be calculated directly by known methods. (In [10], v_{∗}(F_{4}) was determined by a combination of topological and UNSS methods.) Moreover, applying Ω to the fibration does not help much, because BP_{∗}(ΩE_{7}) is not a free commutative algebra, and so we cannot apply Theorem 2.6 to compute its v_{1}-periodic UNSS. Hence UNSS methods cannot be used directly to analyze the exact sequence in v_{∗}(−) associated to (4.1).

Our proof could be expedited slightly if we were assured of the validity of the following conjecture, due to Mimura.

Conjecture 4.2. Localized at 3, E_{7}/F_{4} is spherically resolved by spheres of dimension 19, 27, and 35, and attaching maps α_{2}. That is, there is a fibration S^{19} → E_{7}/F_{4} → B_{2}(27, 35) and a fibration S^{27} → B_{2}(27, 35) → S^{35}, with attaching maps from 19 to 27 and from 27 to 35 both equal to the element α_{2} which generates π_{7}(S^{0})_{(3)} ≈ Z/3.

Although we cannot use this proposed topological description of E_{7}/F_{4}, we can say enough about this space to compute its v_{1}-periodic UNSS and prove that it converges to v_{∗}(E_{7}/F_{4}). However, the specific results of this computation will not be needed for the reasons cited earlier in this section, and the methods will be applied again in computing the v_{1}-periodic UNSS of the space Y_{7}, which will be our approach to v_{∗}(E_{7}), and so we shall wait until the next section to use them.

The following first steps toward proving Conjecture 4.2 will be useful to us later. They were pointed out by Mimura.

Proposition 4.3. (a) H^{∗}(E_{7}/F_{4}; Z) is an exterior algebra on classes of dimension 19, 27, and 35. (b) The 35-skeleton of E_{7}/F_{4} is S^{19} ∪_{±α_{2}} e^{27} ∪_{α_{2}} e^{35}, where α_{2} generates π_{7}(S^{0})_{(3)} ≈ Z/3.

The proof of this proposition requires the following result of Kono and Mimura.

Proposition 4.4. ([29]) There is an algebra isomorphism

H^{∗}(E_{7}; Z_{3}) ≈ Z_{3}[e_{8}]/(e_{8}^{3}) ⊗ Λ[e_{3}, e_{7}, e_{11}, e_{15}, e_{19}, e_{27}, e_{35}]

with only nonzero action of β or P^{p^{r}} on generators given by βe_{7} = e_{8}, βe_{15} = −e_{8}^{2}, P^{1}e_{3} = e_{7}, P^{1}e_{11} = e_{15}, P^{1}e_{15} = ±e_{19}, P^{3}e_{7} = e_{19}, P^{3}e_{15} = e_{27}.

Proof of Proposition 4.3. Part (a) was proved in [30, 9.4]. To prove part (b), let Φ denote the secondary cohomology operation associated with the relation P^{1}βP^{1}− βP^{2} − P^{2}β = 0. This secondary operation detects the map α_{2} and satisfies P^{3} = P^{1}Φ. (See [29, p.353].) In [29, 7.2], it is shown that Φ(ẽ_{27}) = ẽ_{35} in H^{∗}(Ẽ_{7}), where Ẽ_{7} denotes the fiber of E_{7} → K(Z, 3), from which it follows that Φ(e_{27}) = e_{35} in H^{∗}(E_{7}).

Since P^{1}e_{15} = ±e_{19} and P^{3}e_{15} = e_{27} in H^{∗}(E_{7}), we can use a dual relation P^{3} = ΦP^{1} to deduce that Φ(e_{19}) = ±e_{27}. The dual relation is deduced by applying the original relation in the S-dual, and then noting that P^{1}, P^{3}, and Φ are all selfdual. Here duality is given by the antiautomorphism of the Steenrod algebra, while Φ is self-dual since it is defined by a symmetric Adem relation involving self-dual terms. □

We close this section by proving the following result, which will be crucial for us, since we will use it later to deduce that the v_{1}-periodic UNSS of Y_{7} converges to v_{∗}(Y_{7}).

Theorem 4.5. The v_{1}-periodic UNSS of E_{7}/F_{4} converges to v_{∗}(E_{7}/F_{4}). Indeed, Theorem 2.14 holds if X = E_{7}/F_{4}.

Note that this result would be immediate from 2.14 if we knew that Conjecture 4.2 were true. Instead, we must call upon the following result, which was proved by Bendersky and Thompson at the request of the author. The statement and proof of this result rely heavily on the work of Bousfield ([17]), who defined a space to be K/p_{∗}-durable when its K/p_{∗}-localization map induces an isomorphism in v_{∗}(−).

Theorem 4.6. ([15]) Suppose X is a K/p_{∗}-durable space with K^{∗}(X; Ẑ_{p}) isomorphic as a Z/2-graded p-adic λ-ring to Λ̂(M), where M = M_{n} is a p-adic Adams module which admits a sequence of epimorphisms of p-adic Adams modules

M_{n} ⟶^{p_{n}} M_{n−1} ⟶^{p_{n−1}} ⋯ ⟶^{p_{2}} M_{1} = M(2m_{1} + 1)

with ker(p_{i}) = M(2m_{i} + 1) for 2 ≤ i ≤ n. Here M(2m + 1) ≈ K^{∗}(S^{2m+1}; Ẑ_{p}) as a p-adic Adams module, and Λ̂(M) denotes the exterior algebra on M. Then the (BP -based) v_{1}-periodic UNSS of X converges to v_{∗}(X).

Actually, what is proved in [15] is that if Xˆ denotes the K/p-completion, then X → Xˆ induces an isomorphism in v_{∗}(−). In [14], it is proved that the v_{1}-periodic UNSS converges to v_{∗}(Xˆ ), which then implies Theorem 4.6. Theorem 4.5 is an immediate consequence of Theorem 4.6 and the following two results.

Theorem 4.7. There is an isomorphism of Z/2-graded p-adic λ-rings

K^{∗}(E_{7}/F_{4}; Ẑ_{p}) ≈ Λ̂(M_{3}),

with short exact sequences of p-adic Adams modules

0 → M(35) → M_{3} → M_{2} → 0 and 0 → M(27) → M_{2} → M(19) → 0.

Theorem 4.8. E_{7}/F_{4} is K/3_{∗}-durable.

Proof of Theorem 4.7. We use Proposition 4.3 to give the E_{2}-term of the Atiyah-Hirzebruch spectral sequence converging to K^{∗}(E_{7}/F_{4}; Ẑ_{p}) as Λ[x_{19}, x_{27}, x_{35}] ⊗ KẐ_{p}^{∗}. The spectral sequence collapses to yield the claimed exterior algebra as K^{∗}(E_{7}/F_{4}; Ẑ_{p}). This collapsing can be deduced from Yagita’s result ([34]) that there is a 3-local isomorphism

BP^{∗}(E_{7}) ≈ BP^{∗}(F_{4}) ⊗ Λ[19, 27, 35],

or from Snaith’s result ([32]) that the spectral sequence

Tor_{R(G)}(Z, R(H)) ⇒ K^{∗}(G/H)

collapses.

The claim about the decomposition of M_{3} as a p-adic Adams module will follow once we show that the generators of the exterior algebra K^{1}(E_{7}/F_{4}) satisfy ψ^{k}(x_{35}) = k^{17}x_{35}, ψ^{k}(x_{27}) = k^{13}x_{27} + α_{1}x_{35}, and ψ^{k}(x_{19}) = k^{9}x_{19} + α_{2}x_{27} + α_{3}x_{35} for some integers α_{1}, α_{2}, and α_{3}. Note that K^{1}(E_{7}/F_{4}) is spanned by x_{19}, x_{27}, x_{35}, and x_{19}x_{27}x_{35}. We will show that the top cell of E_{7}/F_{4}, which corresponds to this product class, splits off stably, and so cannot be involved in Adams operations on the lower classes. Then the formula for the Adams operations follows from the inclusions S^{19} → E_{7}/F_{4}, S^{27} → (E_{7}/F_{4})/S^{19}, and S^{35} → (E_{7}/F_{4})/(E_{7}/F_{4})^{(27)}.

To prove the stable splitting, we argue similarly to [22, 1.1]. By [2, 3.3], the S-dual of the manifold E_{7}/F_{4} is the Thom spectrum of its stable normal bundle. However, K̃O(E_{7}/F_{4})_{(3)} = 0, since E_{7}/F_{4} has no cells whose dimension is a multiple of 4. Thus the bottom class splits off the Thom spectrum of the stable normal bundle, and dually the top cell stably splits off the manifold itself. □

The following proof is due to Pete Bousfield.

Proof of Theorem 4.8. In [17], Bousfield utilizes a functor Φ from spaces to spectra, which he had introduced in earlier papers. A map f induces an isomorphism in v_{∗}(−) if and only if Φ(f) is an equivalence. Let X = E_{7}/F_{4}, and consider the commutative diagram

Φ(F_{4}) ⟶ Φ(E_{7}) ⟶ Φ(X)
↓  ↓  ↓
Φ((F_{4})_{K/p}) ⟶ Φ((E_{7})_{K/p}) ⟶ Φ(X_{K/p})

Since Φ preserves fibrations, the top row is a fibration, and since [17, 7.8] states that H-spaces are K/p_{∗}-durable, the first two vertical arrows are equivalences. We will be done by the 5-lemma once we show that the bottom row is a fibration.

By [17, 6.3], K^{∗}(G; Ẑ_{p}) ≈ Λ̂(P_{G}), where P_{G} = P K^{1}(G; Ẑ_{p})), for G = F_{4} or E_{7}, and by [17, 8.1] Φ(G_{K/p}) ≃ ΦG is a KẐ_{p}^{∗}-Moore spectrum M(P_{G}/ψ^{p}, 1), where P_{G}/ψ^{p} is the quotient by the injective action of the Adams operation. Similarly, by Theorem 4.7,

K^{∗}(X_{K/p}; Ẑ_{p}) ≈ K^{∗}(X; Ẑ_{p}) ≈ Λ̂(M_{3}),

and, since X_{K/p}, being K/p-local, is certainly K/p_{∗}-durable, we can apply [17, 8.1] to obtain Φ(X_{K/p}) ≃ M(M_{3}/ψ^{p}). There is a short exact sequence of Adams modules, (e.g. from [34])

0 → M_{3} → P K^{1}(E_{7}) → P K^{1}(F_{4}) → 0

and hence a fiber sequence

M(P_{F_{4}}/ψ^{p}, 1) → M(P_{E_{7}}/ψ^{p}, 1) → M(M_{3}/ψ^{p}, 1)

which is the bottom row of the commutative diagram considered above, showing that it is a fibration, as desired. □

5. E_{2} of periodic UNSS of ΩE_{7}/Sp(2)

In this long section, we calculate the periodic UNSS of Y_{7} := ΩE_{7}/Sp(2). In Section 7, we perform the transition from these results to v_{1}^{−1}π_{∗}(E_{7}).

We begin by recalling the following result of Harper, which we used in [10].

Proposition 5.1. ([27, 4.4.1]) There is a 3-equivalence

F_{4} ≈ K × B(11, 15),

where K is a finite mod 3 H-space satisfying

H^{∗}(K; F_{3}) = Λ(x_{3}, x_{7}) ⊗ F_{3}[x_{8}]/(x_{8}^{3}),

with x_{7} = P^{1}x_{3} and x_{8} = βx_{7}. Also, B(11, 15) is an S^{11}-bundle over S^{15} with P^{1}x_{11} = x_{15}. Moreover, there is a fibration B(3, 7) → K → W, where W is the Cayley plane, and a fibration S^{7} → ΩW → ΩS^{23}.

Because of the torsion in H^{∗}(K; Z), and hence in H^{∗}(E_{7}; Z), we will work with loop spaces, and use the following result of Hamanaka and Hara ([26]).

Proposition 5.2. The mod 3 homology as Hopf algebras over the Steenrod algebra satisfies

H_{∗}(ΩF_{4}) ≈ F_{3}[t_{2}, t_{6}, t_{10}, t_{14}, t_{22}]/(t_{2}^{3})
H_{∗}(ΩE_{7}) ≈ F_{3}[t_{2}, t_{6}, t_{10}, t_{14}, t_{18}, t_{22}, t_{26}, t_{34}]/(t_{2}^{3}),

with the only nonzero reduced coproducts being

\overline{ϕ}(t_{6}) = −t_{2}^{2} ⊗ t_{2} − t_{2} ⊗ t_{2}^{2}

and

\overline{ϕ}(t_{18}) = t_{2}^{2}t_{6}^{2} ⊗ t_{2} + t_{2}t_{6}^{2} ⊗ t_{2}^{2} − t_{6}^{2} ⊗ t_{6} − t_{2}^{2}t_{6} ⊗ t_{2}t_{6}
−t_{2}t_{6} ⊗ t_{2}^{2}t_{6} − t_{6} ⊗ t_{6}^{2} + t_{2}^{2} ⊗ t_{2}t_{6}^{2} + t_{2} ⊗ t_{2}^{2}t_{6}^{2}.

The only nonzero action of dual Steenrod operations P_{∗}^{3^{r}} are P_{∗}^{1}(t_{6}) = t_{2}, P_{∗}^{1}(t_{14}) = t_{10}, P_{∗}^{1}(t_{18}) = ϵt_{14}−t_{2}t_{6}^{2}, P_{∗}^{1}(t_{22}) = κt_{6}^{3}, P_{∗}^{1}(t_{26}) = ϵt_{22}, P_{∗}^{1}(t_{34}) = −ϵt_{10}^{3}, P_{∗}^{3}(t_{18}) = t_{6}, P_{∗}^{3}(t_{26}) = t_{14}, and P_{∗}^{3}(t_{34}) = t_{22}. Here ϵ = ±1 and κ = ±1.

Because of the relation t_{2}^{3} = 0 in H_{∗}(ΩE_{7}), Theorem 2.6 does not apply to X = ΩE_{7}. Instead, we will work with the space Y_{7} defined in the following theorem. We begin by noting (see [10, p.296]) that the space B(3, 7) which occurs in 5.1 is 3-equivalent to Sp(2).

Theorem 5.3. Let E_{7}/Sp(2) denote the quotient of the group inclusion Sp(2) → F_{4} → E_{7}, and let Y_{7} = ΩE_{7}/Sp(2). Then

H_{∗}(Y_{7}; G) ≈ Λ[x_{7}] ⊗ G[x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, x_{34}]

for G = Z/3 or Z_{(3)}.

Proof. There is a commutative diagram of fibrations

Y_{7}  → B(3, 7) → E_{7}
(5.4)  ↓  ↓  ↓^{=}
ΩE_{7}/F_{4} → F_{4} → E_{7},

and this, together with the fibration B(3, 7) → F_{4} → W × B(11, 15), which is a consequence of 5.1, implies there is a fibration

ΩW × ΩB(11, 15) → Y_{7} → ΩE_{7}/F_{4}. (5.5)

The last fibration in 5.1 determines H_{∗}(ΩW), and the Serre spectral sequence of (5.5) collapses, yielding the claim of the theorem. The collapsing is proved by observing that the only possible differential on one of the three polynomial generators is d_{17}(x_{18}) = ϵx_{7} ⊗ x_{10}, but this has ϵ = 0 by consideration of the map from (5.5) to the fibration

ΩW × ΩB(11, 15) → B(3, 7) → F_{4}.
□

We easily obtain the following consequence.

Corollary 5.6. BP_{∗}(Y_{7}) is a free commutative algebra on classes x_{7}, x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, and x_{34}, with x_{i} ∈ BP_{i}(Y_{7}).

Proof. By [1, 12.1], the rationalization of Y_{7} is homotopy equivalent to K(Q, 7) × K(Q, 10) × ⋯ × K(Q, 34). Any differentials in the Atiyah-Hirzebruch spectral sequence

Λ[x_{7}] ⊗ Z_{(3)}[x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, x_{34}] ⊗ BP_{∗} ⇒ BP_{∗}(Y_{7})

must be seen rationally, and hence must be zero. That x_{7}^{2} = 0 is deduced from the inclusion S^{7} → Y_{7}. □

By Theorem 2.6, the UNSS of Y_{7} can be calculated as the homology of the unstable cobar complex. This complex splits as the direct sum of the unstable cobar complex for S^{7} plus the even-dimensional complex. That is, we have

(5.7) E_{2}^{s,t}(Y_{7}) ≈ \begin{cases}E_{2}^{s,t}(S^{7}) & if t is odd \\ Ext_{V}^{s,t}(BP_{∗}⟨x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, x_{34}⟩) & if t is even \end{cases}

Our work in this section will go into computing

v_{1}^{−1} Ext_{V}^{s,t}(BP_{∗}⟨x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, x_{34}⟩).

This is the v_{1}-periodic Ext which forms the E_{2}-term of the v_{1}-periodic UNSS of Y_{7}. In Section 7, we will use Theorem 4.6 to show that this spectral sequence converges to v_{∗}(Y_{7}). Throughout the remainder of the paper, E_{2} and Ext_{V} will always refer to their v_{1}-periodic versions, unless explicitly stated to the contrary.

To compute the homology of the unstable cobar complex of Y_{7}, we will utilize exact sequences in Ext_{V}(−) induced by the injective extension sequences

A(26) → A(26, 34) → A(34), (5.8)

A(18) → A(18, 26, 34) → A(26, 34), (5.9)

and

A(10, 14) ⊗ A(22) → BP_{ev}(Y_{7}) → A(18, 26, 34). (5.10)

Each of these A(−) is the subquotient of BP_{∗}(Y_{7}) on the generators of the indicated dimensions. Each has an induced Γ-coaction. The sequence (5.10) is closely related to the fibration

F_{4} → E_{7} → E_{7}/F_{4},

with F_{4} ≈ B(11, 15) × K.

By [9, 4.3], each of these three injective extension sequences yields a long exact sequence when ordinary (unlocalized) Ext_{V}(Q(−)) is applied, and these Ext-groups are the homology of the associated unstable cobar complexes. The v_{1}-periodic E_{2}-term is the direct limit of a direct system of v_{1}-power morphisms, and these commute with the morphisms in the exact sequences just described. Since the direct limit of exact sequences is exact, we obtain that there is an exact sequence of v_{1}-periodic E_{2}-terms. As observed after Theorem 2.14, we can still work with the unstable cobar complex, as long as we restrict attention to v_{1}-periodic classes. We will abbreviate Ext_{V}(Q(A(n_{1}, ⋯, n_{k}))) as E_{2}(n_{1}, ⋯, n_{k}), and the associated unstable cobar complex as C(n_{1}, ⋯, n_{k}).

In order to analyze ∂ in the long exact Ext sequences, we will need the following crucial result about the Γ-coaction.

Proposition 5.11. If M is a Γ-comodule which as a BP_{∗}-module is free on x_{10}, x_{14}, x_{18}, x_{26}, and x_{34}, and if

ψ(x_{34}) = 1 ⊗ x_{34} + α_{2} ⊗ x_{26} + T_{1} ⊗ x_{18} + T_{2} ⊗ x_{14} + T_{3} ⊗ x_{10}
ψ(x_{26}) = 1 ⊗ x_{26} + α_{2} ⊗ x_{18} + T_{4} ⊗ x_{14} + T_{5} ⊗ x_{10}
ψ(x_{18}) = 1 ⊗ x_{18} + α_{1} ⊗ x_{14} + T_{6} ⊗ x_{10}
ψ(x_{14}) = 1 ⊗ x_{14} + α_{1} ⊗ x_{10},

then T_{6} = \frac{1}{2}h^{2}, and, mod terms that desuspend lower than does the indicated term, T_{1} ≡ \frac{1}{2}v^{2}h^{2}, T_{2} ≡ −5vh^{4}, T_{3} ≡ \frac{1}{4}vh^{5}, T_{4} ≡ h^{3}, and T_{5} ≡ \frac{1}{4}vh^{3}.

This proposition will be applied when M is a quotient of BP_{∗}(Y_{7}). The α_{2}- terms in ψ(x_{26}) and ψ(x_{34}) are present there by Proposition 4.3, since α_{2} is the cycle which detects the homotopy class α_{2}. We will see after the proof that our application of this proposition to computing the homology of the unstable cobar complex would not be affected if a unit coefficient were present on α_{2}. Similarly, the α_{1}-terms in ψ(x_{18}) and ψ(x_{14}) are present because of P_{∗}^{1}(t_{18}) and P_{∗}^{1}(t_{14}) in 5.2, and the homology application would not be affected if they were multiplied by a unit.

Proof. We begin with the determination of T_{1}. Using also that ψx_{26} = 1 ⊗ x_{26} + α_{2}⊗x_{18} mod lower terms, the coassociativity formula (ψ⊗1)ψ(x_{34}) = (1⊗ψ)ψ(x_{34}) implies that ψ(α_{2}) = α_{2} ⊗ 1 + 1 ⊗ α_{2} and ψ(T_{1}) = T_{1} ⊗ 1 + 1 ⊗ T_{1} + α_{2} ⊗ α_{2}. Now, α_{2} is given in Lemma 2.19, and one can verify that it is primitive. Let \overline{ψ} be the reduced coproduct in BP_{∗}BP, defined by ψ(y) = y ⊗ 1 + 1 ⊗ y + \overline{ψ}(y). We use the condition that \overline{ψ}(T_{1}) = α_{2} ⊗ α_{2} to find T_{1}.

First, using Lemma 2.19 and Lemma 2.20(1), we compute

α_{2} ⊗ α_{2} = (2vh − 3h^{2}) ⊗ (2vh − 3h^{2})
= 4vh ⊗ vh − 6h^{2} ⊗ vh − 6vh ⊗ h^{2} + 9h^{2} ⊗ h^{2}
= 4v^{2}h ⊗ h − 18vh^{2} ⊗ h + 18h^{3} ⊗ h − 6vh ⊗ h^{2} + 9h^{2} ⊗ h^{2}.

Now T_{1} must be a combination of the following five terms, whose \overline{ψ} are listed.

(5.12)  h^{4} ↦ 4h^{3} ⊗ h + 6h^{2} ⊗ h^{2} + 4h ⊗ h^{3}.
vh^{3} ↦ 3vh^{2} ⊗ h + 3vh ⊗ h^{2} + v ⊗ h^{3} − 1 ⊗ vh^{3}
= 3vh^{2} ⊗ h + 3vh ⊗ h^{2} + 3h ⊗ h^{3}.
v^{2}h^{2} ↦ 2v^{2}h ⊗ h + v^{2} ⊗ h^{2} − 1 ⊗ v^{2}h^{2}
= 2v^{2}h ⊗ h + 6vh ⊗ h^{2} − 9h^{2} ⊗ h^{2}.
v^{3}h ↦ v^{3} ⊗ h − 1 ⊗ v^{3}h = 9v^{2}h ⊗ h − 27vh^{2} ⊗ h + 27h^{3} ⊗ h.
h_{2} ↦ 4h^{3} ⊗ h + 6h^{2} ⊗ h^{2} + 3h ⊗ h^{3} − vh ⊗ h^{2} − vh^{2} ⊗ h.

We solve a system of linear equations for the coefficients of these five terms, to see what combination T_{1} can have \overline{ψ}(T_{1}) = α_{2} ⊗ α_{2}, as required. We find that the desired term T_{1} is given by

T_{1} = \frac{9}{2}h^{4} − 6vh^{3} + 2v^{2}h^{2} + c_{1}(−3h^{4} + vh^{3} + 3h_{2})
+ c_{2}(−\frac{27}{4}h^{4} + 9vh^{3} − \frac{9}{2}v^{2}h^{2} + v^{3}h),

with c_{1} and c_{2} in Z_{(3)}. Replacing 3h by v − η(v) at several places, this simplifies to

T_{1} = \frac{1}{2}v^{2}h^{2} + L,

where L desuspends to S^{3}.

The other T_{i}’s are determined similarly. Coassociativity implies

\overline{ψ}(T_{6}) = α_{1} ⊗ α_{1}
\overline{ψ}(T_{4}) = α_{2} ⊗ α_{1}
(5.13)  \overline{ψ}(T_{5}) = α_{2} ⊗ T_{6} + T_{4} ⊗ α_{1}
\overline{ψ}(T_{2}) = α_{2} ⊗ T_{4} + T_{1} ⊗ α_{1}
\overline{ψ}(T_{3}) = α_{2} ⊗ T_{5} + T_{1} ⊗ T_{6} + T_{2} ⊗ α_{1}.

That T_{6} must equal \frac{1}{2}h^{2} is easily determined (since α_{1} = −h). To determine T_{4}, we write α_{2} ⊗ α_{1} = −2vh ⊗ h + 3h^{2} ⊗ h, and note that \overline{ψ} acts as follows:

h^{3} ↦ 3h^{2} ⊗ h + 3h ⊗ h^{2}
vh^{2} ↦ 2vh ⊗ h + 3h ⊗ h^{2}
v^{2}h ↦ 6vh ⊗ h − 9h^{2} ⊗ h

Solving a system of equations for the coefficients of h^{3}, vh^{2}, and v^{2}h yields

T_{4} = h^{3} − vh^{2} + c(3h^{3} − 3vh^{2} + v^{2}h), (5.14)

with c ∈ Z_{(3)}. All terms except the first are defined on S^{5}, and so T_{4} is as claimed.

We must have \overline{ψ}(T_{5}) = vh ⊗ h^{2} − \frac{3}{2}h^{2} ⊗ h^{2} − h^{3} ⊗ h + vh^{2} ⊗ h. The terms of which T_{5} is a linear combination are the same as those in T_{4}, which were listed with their \overline{ψ}(−) in (5.12). Solving this system of equations yields that the combination whose \overline{ψ}(−) is that required of T_{5} can be

−\frac{1}{4}h^{4} + \frac{1}{3}vh^{3} + c_{1}(−h^{4} + \frac{1}{3}vh^{3} + h_{2}) + c_{2}(−\frac{27}{4}h^{4} + 9vh^{3} − \frac{9}{2}v^{2}h^{2} + v^{3}h),

for any c_{1} and c_{2} in Z_{(3)}. However, fractions with 3 in the denominator do not lie in Z_{(3)}. The only way to prevent this is to specify that c_{1} must be of the form −1 + 3k, with k ∈ Z_{(3)}. This yields

T_{5} = \frac{3}{4}h^{4} − h_{2} + k(−3h^{4} + vh^{3} + 3h_{2}) + L,

where L (the c_{2}-term) desuspends to S^{3}. In two places, we replace 3h^{4} by vh^{3}−h^{3}v, yielding

T_{5} = \frac{1}{4}vh^{3} + L′, (5.15)

as desired. In our determination of T_{5}, we should also take into account the homogeneous part of (5.14), as it contributes to the T_{4} ⊗ α_{1}-term of \overline{ψ}(T_{5}). When the resulting equations are solved, we obtain an additional homogeneous part of T_{5}, equal to

c′(\frac{3}{4}h^{4} − vh^{3} + \frac{1}{2}v^{2}h^{2}) = c′(−\frac{3}{4}vh^{3} − \frac{3}{4}h^{3}v + \frac{1}{2}v^{2}h^{2}),

which desuspends farther than the leading term of (5.15). Thus T_{5} is as claimed.

Similarly, by (5.13) we must have (mod homogeneous terms that will be considered below)

\overline{ψ}(T_{2}) = (2vh − 3h^{2}) ⊗ (h^{3} − vh^{2}) − (\frac{9}{2}h^{4} − 6vh^{3} + 2v^{2}h^{2}) ⊗ h
= 2vh ⊗ h^{3} − 3h^{2} ⊗ h^{3} − 2v^{2}h ⊗ h^{2} + 9vh^{2} ⊗ h^{2} − 9h^{3} ⊗ h^{2}
−\frac{9}{2}h^{4} ⊗ h + 6vh^{3} ⊗ h − 2v^{2}h^{2} ⊗ h.

The terms that can comprise T_{2} are listed below, with their \overline{ψ}.

h^{5} ↦ 5h ⊗ h^{4} + 10h^{2} ⊗ h^{3} + 10h^{3} ⊗ h^{2} + 5h^{4} ⊗ h
vh^{4} ↦ 3h ⊗ h^{4} + 4vh ⊗ h^{3} + 6vh^{2} ⊗ h^{2} + 4vh^{3} ⊗ h
v^{2}h^{3} ↦ −9h^{2} ⊗ h^{3} + 6vh ⊗ h^{3} + 3v^{2}h ⊗ h^{2} + 3v^{2}h^{2} ⊗ h
v^{3}h^{2} ↦ 27h^{3} ⊗ h^{2} − 27vh^{2} ⊗ h^{2} + 9v^{2}h ⊗ h^{2} + 2v^{3}h ⊗ h
v^{4}h ↦ −81h^{4} ⊗ h + 108vh^{3} ⊗ h − 54v^{2}h^{2} ⊗ h + 12v^{3}h ⊗ h
vh_{2} ↦ 3vh ⊗ h^{3} + 6vh^{2} ⊗ h^{2} + 4vh^{3} ⊗ h − v^{2}h ⊗ h^{2} − v^{2}h^{2} ⊗ h + 3h_{1} ⊗ h_{2}
h_{1}h_{2} ↦ 3h ⊗ h^{4} + 9h^{2} ⊗ h^{3} + 10h^{3} ⊗ h^{2} + 4h^{4} ⊗ h − vh ⊗ h^{3}
−2vh^{2} ⊗ h^{2} − vh^{3} ⊗ h + h_{1} ⊗ h_{2} + h_{2} ⊗ h_{1}
v_{2}h_{1} ↦ 24h^{4} ⊗ h − 35vh^{3} ⊗ h + 18v^{2}h^{2} ⊗ h − 4v^{3}h ⊗ h + 3h_{2} ⊗ h_{1}

We solve a system of equations to find the combination of these terms having \overline{ψ} as desired. We obtain

T_{2} = −\frac{9}{10}h^{5} + \frac{3}{2}vh^{4} − \frac{2}{3}v^{2}h^{3} + c_{1}(\frac{81}{5}h^{5} − 27vh^{4} + 18v^{2}h^{3} − 6v^{3}h^{2} + v^{4}h)
+c_{2}(−\frac{12}{5}h^{5} + 7vh^{4} − \frac{17}{3}v^{2}h^{3} + 2v^{3}h^{2} + vh_{2} − 3h_{1}h_{2} + v_{2}h_{1}).

As in the previous case, in order to prevent 3 in a denominator, we choose c_{2} = −1 + 3c. This yields

T_{2} = \frac{3}{2}h^{5} − \frac{11}{2}vh^{4} + 5v^{2}h^{3} − 2v^{3}h^{2} − vh_{2} + 3h_{1}h_{2} − v_{2}h_{1}

plus two homogeneous terms which are defined on S^{7}. The first two terms in T_{2} combine to \frac{1}{2}vh^{4} − \frac{1}{2}h^{4}v − \frac{11}{2}vh^{4}, and so, mod terms that are defined on S^{7}, we have T_{2} ≡ −5vh^{4}, as claimed. We have omitted here consideration of homogeneous parts of T_{4} and T_{1} already obtained. These yield additional homogeneous terms in T_{2} which are, in fact, defined on S^{5}.

Finally we apply a similar method to determine T_{3}. It is again a matter of solving a system of linear equations for the coefficients of the monomials that can comprise T_{3}. We list the terms involved for the convenience of the reader, who can quite easily check that our claimed T_{3} does indeed have the required coproduct. The lead term of this T_{3} will play an important role in our subsequent calculations. Indeed, it caused the answer for v_{∗}(E_{7}) to turn out differently than the author had anticipated.

Momentarily ignoring some homogeneous parts, T_{3} must satisfy

\overline{ψ}(T_{3}) = (2vh − 3h^{2}) ⊗ (\frac{3}{4}h^{4} − h_{2}) + (\frac{9}{2}h^{4} − 6vh^{3} + 2v^{2}h^{2}) ⊗ \frac{1}{2}h^{2}
+(\frac{3}{2}h^{5} − \frac{11}{2}vh^{4} + 5v^{2}h^{3} − 2v^{3}h^{2} − vh_{2} + 3h_{1}h_{2} − v_{2}h_{1}) ⊗ (−h).
= −\frac{9}{4}h^{2} ⊗ h^{4} + \frac{9}{4}h^{4} ⊗ h^{2} − \frac{3}{2}h^{5} ⊗ h + \frac{3}{2}vh ⊗ h^{4} − 3vh^{3} ⊗ h^{2}
+\frac{11}{2}vh^{4} ⊗ h + v^{2}h^{2} ⊗ h^{2} − 5v^{2}h^{3} ⊗ h + 2v^{3}h^{2} ⊗ h + 3h^{2} ⊗ h_{2}

−3hh_{2} ⊗ h − 2vh ⊗ h_{2} + vh_{2} ⊗ h + v_{2}h ⊗ h (5.16)

We list the terms that can comprise T_{3} along with their coproducts.

h^{6} ↦ 6h ⊗ h^{5} + 15h^{2} ⊗ h^{4} + 20h^{3} ⊗ h^{3} + 15h^{4} ⊗ h^{2} + 6h^{5} ⊗ h
vh^{5} ↦ 3h ⊗ h^{5} + 5vh ⊗ h^{4} + 10vh^{2} ⊗ h^{3} + 10vh^{3} ⊗ h^{2} + 5vh^{4} ⊗ h
v^{2}h^{4} ↦ −9h^{2} ⊗ h^{4} + 6vh ⊗ h^{4} + 4v^{2}h ⊗ h^{3} + 6v^{2}h^{2} ⊗ h^{2} + 4v^{2}h^{3} ⊗ h
v^{3}h^{3} ↦ 27h^{3} ⊗ h^{3} − 27vh^{2} ⊗ h^{3} + 9v^{2}h ⊗ h^{3} + 3v^{3}h ⊗ h^{2} + 3v^{3}h^{2} ⊗ h
v^{4}h^{2} ↦ −81h^{4} ⊗ h^{2} + 108vh^{3} ⊗ h^{2} − 54v^{2}h^{2} ⊗ h^{2} + 12v^{3}h ⊗ h^{2} + 2v^{4}h ⊗ h
v^{5}h ↦ 243h^{5} ⊗ h − 405vh^{4} ⊗ h + 270v^{2}h^{3} ⊗ h − 90v^{3}h^{2} ⊗ h + 15v^{4}h ⊗ h
v^{2}h_{2} ↦ 3v^{2}h ⊗ h^{3} + 6v^{2}h^{2} ⊗ h^{2} + 4v^{2}h^{3} ⊗ h − v^{3}h ⊗ h^{2} − v^{3}h^{2} ⊗ h
−9h^{2} ⊗ h_{2} + 6vh ⊗ h_{2}
vhh_{2} ↦ 3vh ⊗ h^{4} + 9vh^{2} ⊗ h^{3} + 10vh^{3} ⊗ h^{2} + 4vh^{4} ⊗ h − v^{2}h ⊗ h^{3}
−2v^{2}h^{2} ⊗ h^{2} − v^{2}h^{3} ⊗ h + 3h ⊗ hh_{2} + vh ⊗ h_{2} + vh_{2} ⊗ h
h^{2}h_{2} ↦ 3h ⊗ h^{5} + 12h^{2} ⊗ h^{4} + 19h^{3} ⊗ h^{3} + 14h^{4} ⊗ h^{2} + 4h^{5} ⊗ h − vh ⊗ h^{4}
−3vh^{2} ⊗ h^{3} − 3vh^{3} ⊗ h^{2} − vh^{4} ⊗ h + 2h ⊗ hh_{2} + h^{2} ⊗ h_{2}
+2hh_{2} ⊗ h + h_{2} ⊗ h^{2}
v_{2}h^{2} ↦ 24h^{4} ⊗ h^{2} − 35vh^{3} ⊗ h^{2} + 18v^{2}h^{2} ⊗ h^{2} − 4v^{3}h ⊗ h^{2} + 3h_{2} ⊗ h^{2}
+2v_{2}h ⊗ h

vv_{2}h ↦ −72h^{5} ⊗ h + 129vh^{4} ⊗ h − 89v^{2}h^{3} ⊗ h + 30v^{3}h^{2} ⊗ h − 4v^{4}h ⊗ h

−9hh_{2} ⊗ h + 3vh_{2} ⊗ h + 3v_{2}h ⊗ h

The solution of the resulting system of linear equations is

(5.17) T_{3} = \frac{3}{4}h^{6} − \frac{1}{2}v^{2}h^{4} + \frac{1}{2}v^{3}h^{3} − \frac{1}{2}v^{2}h_{2} + vhh_{2} − \frac{3}{2}h^{2}h_{2} + \frac{1}{2}v_{2}h^{2}
+c_{1}(−\frac{81}{2}h^{6} + 81vh^{5} − \frac{135}{2}v^{2}h^{4} + 30v^{3}h^{3} − \frac{15}{2}v^{4}h^{2} + v^{5}h)
+c_{2}(9h^{6} − \frac{45}{2}vh^{5} + 21v^{2}h^{4} − \frac{59}{6}v^{3}h^{3} + 2v^{4}h^{2} + \frac{1}{2}v^{2}h_{2}
−3vhh_{2} + \frac{9}{2}h^{2}h_{2} − \frac{3}{2}v_{2}h^{2} + vv_{2}h).

The first term is rewritten as \frac{1}{4}(vh^{5} −h^{5}v), in order to see it with a unit coefficient. All other terms desuspend to S^{9}.

The terms T_{1}, T_{5}, and T_{2} which appear in the equation (5.13) for \overline{ψ}(T_{3}) which gave rise to the system of equations which we just solved have homogeneous parts whose coefficients we do not know. For example, T_{5} includes a summand of c(−3h^{4}+ vh^{3} + 3h_{2}). Thus added on to the RHS of (5.16) must be α_{2} ⊗ c(−3h^{4} + vh^{3} + 3h^{2}) and 7 other homogeneous parts arising similarly. For each of these we solve a system of equations similar to the one just solved, but with the RHS equal to the appropriate homogeneous term. These give homogeneous summands to T_{3}. All resulting terms desuspend to S^{7}, and so may be ignored. We spare the reader the details. □

The terms α_{2} ⊗ x_{26}, α_{2} ⊗ x_{18}, α_{1} ⊗ x_{14}, and α_{1} ⊗ x_{10} appear in the hypothesis of Proposition 5.11 because of attaching maps in ΩE_{7}. One might think that care is required as to the coefficients (±1) of the α_{2} and α_{1} in Proposition 5.11. However, this is not the case. For if the four terms listed at the beginning of this paragraph are multiplied by units u_{1}, u_{2}, u_{3}, and u_{4}, respectively, then the terms T_{1} to T_{6} which are determined in Proposition 5.11 are multiplied by units u_{1}u_{2}, u_{1}u_{2}u_{3}, u_{1}u_{2}u_{3}u_{4}, u_{2}u_{3}, u_{2}u_{3}u_{4}, and u_{3}u_{4}, respectively. This can be seen by consideration of the first part of the proof of 5.11. For example, we would have \overline{ψ}(T_{1}) = u_{1}α_{2} ⊗ u_{2}α_{2}.

The terms α_{1}, α_{2}, and T_{i} in 5.11 will be used in the proofs of the theorems throughout the remainder of this section to determine boundary morphisms in exact sequences, and in pulling back terms whose boundary is 0. If units u_{i} were present as we are discussing here, it will only have the effect of multiplying boundaries and pullbacks by unit amounts. The point is that all terms in a boundary will be multiplied by the same unit, so that cancellation due to different units cannot take place. For example, suppose that a term h^{I_{1}}x_{34} pulled back to h^{I_{1}}x_{34} + h^{I_{2}}x_{26} in the case where all u_{i} = 1. Then, with units u_{i} present, h^{I_{1}}x_{34} pulls back to h^{I_{1}}x_{34}+u_{1}h^{I_{2}}x_{26}, and the boundary sends this to h^{I_{1}} ⊗u_{1}u_{2}T_{1}x_{18}+u_{1}h^{I_{2}} ⊗u_{2}α_{2}x_{18}, which is just u_{1}u_{2} times what it would have been. These uniform units do not affect whether terms are zero, and hence can be ignored.

Now we can compute E_{2}^{s,2j}(Y_{7}), dividing into cases depending upon the parity and mod 9 value of j. These will be delineated in Theorems 5.18, 5.23, 5.29, 5.32, 5.34, and 6.1. Note that the exact sequences in E_{2} induced by (5.8), (5.9), and (5.10), together with (5.19), imply that if t is even, then E_{2}^{s,t}(Y_{7}) = 0 unless s = 1 or 2.

The first case is as follows.

Theorem 5.18. If j is odd, and j ≡ 1 or 7 mod 9, then

E_{2}^{1,2j}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}) ≈ Z/3 ⊕ Z/3^{min(8,ν(j−43)+5)}.

Proof. Let j be as in the theorem, and ν = ν(j − 7). Formally, we obtain the result by computing first the exact sequence in E_{2}(−) associated to (5.8), then that associated to (5.9), and then that associated to (5.10). We know from (2.5) and 2.15 that

(5.19) E_{2}^{s,2j}(2m) ≈ \begin{cases}Z/3^{min(m,ν(j−m)+1)} & if j ≡ m mod 2, and s = 1 or 2 \\ 0, & otherwise \end{cases}

and we know from [13, 2.4] how to compute E_{2}(10, 14) from E_{2}(10) and E_{2}(14). These are the building blocks, but the glue is the boundary morphisms in the exact sequences, and computing these requires much care.

A convenient way to picture the calculations is by Diagram 5.20, which we think of as resembling an Adams spectral sequence chart.

Diagram 5.20.

[figure: ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; 2; 2; 4; 4; 5; 5; 6; 6; 7; 7; 34; 26; 18; 14; 10; 22; ν = 1; ν = 3; ν = 4; ν = 5; ν ≥ 6]

Each • represents a Z/3, and each integer e represents a Z/3^{e}. These groups correspond to E_{2}(2m), where 2m is the integer indicated on the left side of the diagram. The vertical lines indicate nontrivial extensions (multiplication by 3). These are true because of the α_{1} and α_{2} attaching maps and Theorem 3.1. The positioning of the 22-class is due to (5.5), i.e., that it is split away from the 10-, 14-, and 18-classes.

For example, the diagram for the case ν = 3 means that if ν(j − 7) = 3 the boundary morphisms in (5.8) and (5.9) are 0, yielding E_{2}^{1,2j}(18, 26, 34) ≈ E_{2}^{2,2j}(18, 26, 34) ≈ Z/3^{4}, while in (5.10)

E_{2}^{1,2j}(18, 26, 34)⟶^{∂}E_{2}^{2,2j}(22) ⊕ E_{2}^{2,2j}(10, 14) ≈ Z/3 ⊕ Z/3^{5}

is 0 into the first summand and has image of order 3^{2} in the second summand. Using either the exact sequence (5.10) or the diagram, this implies that in this case

E_{2}^{1,2j}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}) ≈ Z/3 ⊕ Z/3^{7}.

The case ν = 2 omitted from Diagram 5.20 has groups of order 3^{3}, i.e. labeled “3,” on the 14-cell, and otherwise has the same groups as do the other values of ν. If (j − 7)/18 ≡ 1 mod 3, then it has a differential like that in the case ν = 1, while if (j − 7)/18 ≡ 2 mod 3, then it has no nonzero differentials.

Of course, we still have to verify that the differentials are as claimed in Diagram 5.20 and the above paragraph describing the case ν = 2. The reader can easily verify that this will imply Theorem 5.18. The Z/3 on the 22-cell splits for algebraic reasons.

We analyze the differentials by the methods used extensively in [23] and [20], involving the unstable cobar complex. One convention is that we often omit writing powers of v_{1} on the left; they can always be determined by consideration of total degree. The boundary E_{2}^{1,2j}(34)⟶^{∂}E_{2}^{2,2j}(26) sends the generator hι_{34} to h ⊗ α_{2}ι_{26} = h ⊗ (hv + vh)ι_{26}. This is obtained from ψ(x_{34}) = 1 ⊗ x_{34} + α_{2} ⊗ x_{26} in 5.11, and from 2.19. The relationship of ∂ with the coaction is standard; see, e.g., [20, 2.7].

We use 2.20 to write h ⊗ vh = vh ⊗ h − 3h^{2} ⊗ h, and 3h^{2} ⊗ h = h^{2} ⊗ (v − ηv) = 0. Also, h ⊗ hv is defined on S^{1} and hence is 0 in E_{2}(−). So the image of ∂ equals v^{pwr}h ⊗ hι_{26}. By 2.18(second part of (2)), this cycle equals d((h^{3} + L)ι_{26}), with ν = ν(j − 13) = 1 in 2.18. Here we have omitted a unit coefficient, which will be done routinely unless the coefficient plays a significant role. Here L desuspends lower than the associated term, in this case h^{3}, a notation that will be employed frequently, with the L’s sometimes adorned with primes to distinguish them from one another.

Thus the generator hι_{34} pulls back to hι_{34} + (h^{3} + L)ι_{26} in E_{2}^{1,2j}(26, 34), and ∂ in (5.9) sends this to (h ⊗ (\frac{1}{2}v^{2}h^{2} + L′) + (h^{3} + L) ⊗ α_{2})ι_{18} ∈ E_{2}^{2,2j}(18). Here we have used 5.11. The leading term here is

\frac{1}{2}h ⊗ v^{2}h^{2} = \frac{1}{2}(ηv)^{2}h ⊗ h^{2} = \frac{1}{2}(v − 3h)^{2}h ⊗ h^{2},

which has leading term \frac{1}{2}h ⊗ h^{2}. Note how v’s on the left are absorbed into other unstated v’s. By 2.18(2), this equals d((h^{3} + L′′)ι_{18}) (omitting unit coefficients), and so our generator pulls back to

z ≡ hι_{34} + h^{3}ι_{26} + h^{3}ι_{18} ∈ E_{2}^{1,2j}(18, 26, 34).

Here, and subsequently, “≡” will mean “mod L,” with the lower terms varying from term to term.

We analyze the two components of ∂(z) in the exact sequence of E_{2} derived from (5.10). We begin by showing that the component ∂_{2} into E_{2}^{2,2j}(22) is 0. We have ∂_{2}(z) = (h ⊗ (h^{3} + L) + h^{3} ⊗ h)ι_{22}. Here we use the α_{1} attaching map from 22 to 26 in ΩE_{7}, which causes the ⊗hι_{22}. The ⊗(h^{3} +L) is obtained by the same calculation that gave T_{4} in Proposition 5.11. But these terms don’t even matter very much, for such terms desuspend far below S^{22}, and hence are 0 in E_{2}^{2,2j}(22) ≈ Z/3. Here we use a fact that we will use frequently, essentially from 2.15(5), that if E_{2}^{2,2j}(2n + ϵ) ≈ Z/p, then an element in it which is in the image of the double desuspension is 0.

Similar, although much more delicate, considerations apply to obtaining the other component

∂_{1} : E_{2}^{1,2j}(18, 26, 34) → E_{2}^{2,2j}(10, 14). (5.21)

First we determine the composite when ∂_{1} is followed (by ρ) into E_{2}^{2,2j}(14). Using Proposition 5.11 and the usual relationship between the coaction and the boundary morphism, we obtain

ρ∂_{1}(z) ≡ (h ⊗ vh^{4} + h^{3} ⊗ h^{3} + h^{3} ⊗ h)ι_{14}. (5.22)

Here all terms except h ⊗ vh^{4} desuspend to S^{7}, while, mod S^{7}, h ⊗ vh^{4} ≡ h ⊗ h^{4}. If ν ≥ 3, then, by 2.18(1), this element has order 3^{min(4,ν−2)} in E_{2}^{2,2j}(14), and this is as claimed in Diagram 5.20, with the arrows above the lowest one being a consequence of the lowest one and the extensions.

If ν = 1, then by 2.18(2) and (5.22), ρ∂_{1}(z) = d((h^{6} + L)ι_{14}). Thus z pulls back to

z′ ≡ hι_{34} + h^{3}ι_{26} + h^{3}ι_{18} + h^{6}ι_{14} ∈ C(14, 18, 26, 34).

Using 5.11, this satisfies

∂(z′) ≡ (h ⊗ vh^{5} + h^{3} ⊗ vh^{3} + h^{3} ⊗ h^{2} + h^{6} ⊗ h)ι_{10}.

Here there can be “lower” terms associated with the factor on either side of the tensor sign, omitted v’s occur only on the left, and, as usual, unit coefficients are omitted. All terms here except the first desuspend to S^{9}, while that term generates E_{2}^{2,2j}(10), so the image of ∂_{1} in this case has order 3 in E_{2}^{2,2j}(10, 14), as claimed.

Finally we consider the delicate case when ν = 2. In this case, there are two terms with the potential to cancel, and so we must keep track of unit coefficients. We write j = 7 + 18c, with c ≢ 0 mod 3. As before, E_{2}^{1,2j}(18, 26, 34) is generated by z ≡ hι_{34} + h^{3}ι_{26} + h^{3}ι_{18}. The unit coefficients of the second and third terms will not be important, and so are omitted. The leading term of ρ∂_{1}(z) in E_{2}^{2,2j}(14) is, by 5.11, h ⊗ (−5vh^{4})ι_{14} ≡ h ⊗ h^{4}ι_{14}, where we have used that −5 ≡ 1 mod 3. By Lemma 2.21(3), d(h^{7}) ≡ −9ch ⊗ h^{6} in this stem. ((ℓ + n + 1) of the lemma multiplied by 2(p − 1) equals 2j − 14.) Thus, since 9h^{6} ≡ h^{4}, we obtain ρ∂_{1}(z) = d((−\frac{1}{c}h^{7} + L)ι_{14}), and so z pulls back to

z′ ≡ hι_{34} + h^{3}ι_{26} + h^{3}ι_{18} + \frac{1}{c}h^{7}ι_{14}.

This satisfies

∂(z′) ≡ (h ⊗ \frac{1}{4}vh^{5} + uh^{3} ⊗ vh^{3} + u′h^{3} ⊗ h^{2} + \frac{1}{c}h^{7} ⊗ (−h))ι_{10}

with u and u′ units in Z_{(3)}. The middle terms desuspend, while the first and last combine, using 2.21(2), to give \frac{1}{4} + \frac{1}{c} times the generator of E_{2}^{2,2j}(10) ≈ Z/3. This is nonzero if c ≡ 1 mod 3, and 0 if c ≡ 2 mod 3, as claimed in the paragraph earlier in the proof which described the case ν = 2. □

The statement and proof for the case j ≡ 4 mod 9 are quite similar to the cases just completed.

Theorem 5.23. If j is odd, and j ≡ 4 mod 9, then

E_{2}^{1,2j}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}) ≈ Z/3 ⊕ Z/3^{min(14,ν(j−13−4·3^{8})+5)}.

Proof. Let j be as in the theorem, and ν = ν(j − 13). As in the previous theorem, the way in which the result stated in the theorem is obtained is most conveniently expressed in a diagram.

Diagram 5.24.

[figure: ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ^{E_{2}^{2,2j} E_{2}^{1,2j} • • •  • • • 2  2 •  •}; ν + 1; ν + 1; 10; 10; 11; 11; 12; 12; 13; 13; 34; 26; 18; 14; 10; 22; ν ≤ 7; ν = 9; ν = 10; ν = 11; ν ≥ 12]

The omitted case ν = 8 is like the case ν ≤ 7 if (j − 13)/(2 · 3^{8}) ≡ 1 mod 3, while it has all differentials 0 if (j − 13)/(2 · 3^{8}) ≡ 2 mod 3. In most cases, the Z/3 from the 22-class splits for algebraic reasons. The splitting in the cases when ν ≥ 11 require a bit of care, which will be dealt with later in the proof. The boundary in the diagram in these cases is meant to be hitting the sum of the classes on the 22 and the 18.

We begin with the case ν ≤ 7. We start as in the proof of 5.18, but this time the boundary of hι_{34} in E_{2}^{2,2j}(26) is d((h^{ν+2} + L′)ι_{26}), by 2.18(2). (In the proof of 5.18, we had ν(j − 13) = 1.) Thus the generator of E_{2}^{1,2j}(26, 34) equals, mod lower terms, hι_{34} + h^{ν+2}ι_{26}. The next term is found by writing

(h ⊗ (\frac{1}{2}v^{2}h^{2} + L) + (h^{ν+2} + L′) ⊗ (2vh − 3h^{2}))ι_{18} (5.25)

as a boundary in the unstable cobar complex. The first term will dominate if ν ≤ 4, while the second term will dominate if 4 ≤ ν ≤ 7. (If ν = 4, there could be cancellation that would cause it to desuspend even lower, but that won’t affect the final result.) We obtain that

z ≡ hι_{34} + h^{ν+2}ι_{26} + h^{max(3,ν−1)}ι_{18}

generates E_{2}^{1,2j}(18, 26, 34). The h^{ν−1}ι_{18} when ν ≥ 4 is obtained since

h^{ν+2} ⊗ vh ≡ h^{ν} ⊗ h ≡ h ⊗ h^{ν−2} = d(h^{ν−1}),

using 2.21(1,2,3). Now

ρ∂_{1}(z) ≡ (h ⊗ vh^{4} + h^{ν+2} ⊗ h^{3} + h^{max(3,ν−1)} ⊗ h)ι_{14},

which has leading term a multiple, k, of h ⊗ h^{4}. By 2.18(2) this is d(kh^{6}ι_{14}) since ν(2j − 14) = 1, and so z pulls back to z′ ≡ z − kh^{6}ι_{14} in C(14, 18, 26, 34). The leading term of ∂(z′) in E_{2}^{2,2j}(10) is h ⊗ vh^{5}, which is a generator. It is also important to know here that E_{2}^{1,2j}(18, 26, 34)⟶^{∂}E_{2}^{2,2j}(22) is 0, for if it were nonzero then E_{2}^{2,2j}(Y_{7}) would be cyclic. The leading term of this ∂ is h^{ν+2} ⊗ hι_{22} which desuspends and hence is 0 in E_{2}.

Next we consider the case ν = 8, in which we have to keep track of unit coefficients because of the possibility of two cancelling terms. Let j − 13 = 2 · 3^{8}c, with c ≢ 0 mod 3. By 2.21(3), we have d(h^{10}ι_{26}) ≡ −3^{8}ch ⊗ h^{9}ι_{26} ≡ −ch ⊗ hι_{26}, where the second step utilizes 3^{8}h^{8} = (v − ηv)^{8}. Then E_{2}^{1,2j}(34)⟶^{∂}E_{2}^{2,2j}(26) sends the generator to h ⊗ (−h)ι_{26} = d((\frac{1}{c}h^{10} + L)ι_{26}), and so the generator pulls back to z ≡ hι_{34} − \frac{1}{c}h^{10}ι_{26}. The leading term of ∂(z) in E_{2}^{2,2j}(18) is −\frac{1}{c}h^{10} ⊗2vhι_{18}, which, using 2.21(1,2), is equivalent to −\frac{2}{c}h^{8} ⊗ hι_{18} ≡ \frac{2}{c}h ⊗ h^{6}ι_{18}. By 2.21(3),

d(h^{7}ι_{18}) ≡ −\frac{1}{2}(j − 9)h ⊗ h^{6}ι_{18} = −(2 + 3^{8}c)h ⊗ h^{6}ι_{18} ≡ −c∂(z).

Thus z pulls back to z′ ≡ z + \frac{1}{c}h^{7}ι_{18}. The leading term of ρ∂_{1}(z′) in E_{2}^{2,2j}(14) is \frac{1}{c}h^{7} ⊗ (−h)ι_{14}. By 2.21(3) again,

d(h^{7}ι_{14}) ≡ −\frac{1}{2}(j − 7)h ⊗ h^{6}ι_{14} ≡ −h ⊗ 3h^{6}ι_{14} ≡ −h ⊗ h^{5}ι_{14} ≡ −c∂(z′),

where we have used 2.21(1) at the last step. Thus z′ pulls back to z′′ ≡ z′ + \frac{1}{c}h^{7}ι_{14} in C(14, 18, 26, 34). There are two leading terms in ∂(z′′) ∈ E_{2}^{2,2j}(10). These are h ⊗ \frac{1}{4}vh^{5} and \frac{1}{c}h^{7} ⊗ (−h). They combine to give \frac{1}{4} + \frac{1}{c} times a generator, and this is 0 if c ≡ 2 mod 3, and nonzero if c ≡ 1 mod 3, as claimed. The ∂ into the 22-part is 0 as in the case ν ≤ 7.

If 9 ≤ ν ≤ 10, the situation is much easier. Similarly to the previous cases, but ignoring units, the generator of E_{2}^{1,2j}(18, 26, 34) is z ≡ hι_{34}+h^{ν+2}ι_{26}+h^{ν−1}ι_{18}. The leading term of ρ∂_{1}(z) in E_{2}^{2,2j}(14) is h^{ν−1} ⊗ hι_{14}, which is a generator if ν = 10, and is 3 times the generator if ν = 9. The boundary E_{2}^{1,2j}(18, 26, 34)⟶^{∂_{2}}E_{2}^{2,2j}(22) is 0 because its leading term is h^{ν+2} ⊗ hι_{22} which is 0 in E_{2} for ν ≤ 10.

When ν = 11, the boundary from E_{2}^{1,2j}(26, 34) to E_{2}^{2,2j}(18) is now nonzero. Indeed, its image, given in (5.25), has leading term

h^{13} ⊗ 2vhι_{18} ≡ 2h^{11} ⊗ hι_{18} ≡ h ⊗ h^{9}ι_{18},

which is a generator. Here we have used 2.21(1) and 2.21(2). The boundary from E_{2}^{1,2j}(26, 34) to E_{2}^{2,2j}(22) is also nonzero since the generator z ≡ hι_{34} + h^{13}ι_{26} satisfies ∂(z) ≡ h^{13} ⊗ hι_{22}, and this is a generator. The chart would then suggest (accurately) that the boundary hits into the sum of the two classes, and the extension is also into this sum. One way to formalize this uses the exact sequence

E_{2}^{1,2j}(26, 34)⟶^{∂}E_{2}^{2,2j}(10, 14, 18, 22) → E_{2}^{2,2j}(Y_{7}) → E_{2}^{2,2j}(26, 34). (5.26)

The first and last groups are Z/3^{13}, while the second is Z/3 ⊕ Z/3^{4}. The boundary ∂ hits the sum of the two generators. There is a cycle representative z in E_{2}^{2,2j}(Y_{7}) which projects to an element of order 3 in E_{2}^{2,2j}(26, 34) and satisfies that 3 times this generator is the image of the sum of the two generators of E_{2}^{2,2j}(10, 14, 18, 22). This implies E_{2}^{2}(Y_{7}) ≈ Z/3 ⊕ Z/3^{13}.

Actually, a little bit more care is required here with regard to coefficients of the generators. It is conceivable that the boundary could hit the sum of generators but the extension be into their difference, and then the extension group would be cyclic of order 3^{14}. What really happens is that, if c is defined as before by j−13 = 2·3^{11}c, then a generator z ≡ hι_{34} − \frac{1}{c}h^{13}ι_{26} satisfies

∂(z) ≡ \frac{1}{c}h^{13} ⊗ hι_{22} − \frac{1}{c}h^{13} ⊗ 2vhι_{18}, (5.27)

while on the other hand, the argument of Theorem 3.1 shows that the element d(3^{j−27}h^{j−13})ι_{26} of order 3 extends to a cycle z′ in C(18, 22, 26) such that, mod classes that desuspend farther, 3z′ is homologous to

3^{j−26}h^{j−13} ⊗ (−h)ι_{22} + 3^{j−26}h^{j−13} ⊗ 2vhι_{18}. (5.28)

The classes in (5.27) and (5.28) are clearly unit multiples of one another. In each case, we use h^{13} ⊗ vh ≡ h^{11} ⊗ h to see that the second term is a generator.

This completes the case ν = 11. The case ν ≥ 12 is very similar. Actually it is a bit easier, for the consideration of the previous paragraph need not be addressed, since the initial differential hits into a cyclic group. □

The case j ≡ 0 mod 3 introduces no new ideas.

Theorem 5.29. If j is odd, and j ≡ 0 mod 3, then

E_{2}^{1,2j}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}) ≈ Z/3 ⊕ Z/3^{min(10,ν(j−9−2·3^{5})+4)}.

Proof. Let j be as in the theorem, and ν = ν(j − 9). We will show that the diagram encapsulating the exact sequences of (5.8), (5.9), and (5.10) is as depicted in Diagram 5.30 for certain values of ν. This diagram, together with the subsequent discussion of what happens for values of ν not included in the diagram, implies Theorem 5.29.

Diagram 5.30.

[figure: 34; 26; 18; 22; 14; 10; •; •; 8; •; •; •; •; •; 8; •; •; •; •; •; 7; •; •; •; •; •; 7; •; •; •; •; •; ν + 1; •; •; •; •; •; ν + 1; •; •; •; E_{2}^{2,2j}; E_{2}^{2,2j}; E_{2}^{2,2j}; E_{2}^{1,2j}; E_{2}^{1,2j}; E_{2}^{1,2j}; ν = 7; ν = 6; ν ≤ 4]

If ν ≥ 8, then the group corresponding to the 18-cell has order 3^{9}, and by [13, 2.4] there is a nonzero boundary morphism from E_{2}^{1,2j}(26) to E_{2}^{2,2j}(18) (hitting the element of order 3, of course), and three other boundary morphisms (one below and two above it) follow from it by the extensions in a diagram similar to that of 5.30.

We will show below that the case ν = 5 is like the case ν ≤ 4 if (j −9)/(2·3^{5}) ≡ 2 mod 3, while it has no differentials if (j−9)/(2·3^{5}) ≡ 1 mod 3. But first we establish that the cases in Diagram 5.30 are as depicted.

Let ν = 7. The nonzero differential from E_{2}^{1,2j}(34) ≈ Z/3 to E_{2}^{2,2j}(18) ≈ Z/3^{8} is established similarly to that in the case ν = 11 in the preceding theorem. Indeed, the boundary E_{2}^{1,2j}(34)⟶^{∂}E_{2}^{2,2j}(26) sends the generator to

h⊗α_{2}ι_{26} = h⊗(2vh−3h^{2})ι_{26} = (2vh⊗h−6h^{2} ⊗h−3h⊗h^{2})ι_{26} = d(vh^{2} −2h^{3})ι_{26},

and so it pulls back to hι_{34} + (2h^{3} − vh^{2})ι_{26}. This in turn has boundary (h ⊗ (\frac{1}{2}v^{2}h^{2} + L) + (2h^{3} − vh^{2}) ⊗ α_{2})ι_{18}, whose leading term \frac{1}{2}h ⊗ v^{2}h^{2}ι_{18} ≡ \frac{1}{2}h ⊗ h^{2}ι_{18} has order 3 by Theorem 2.18(1). The two differentials above this differential then follow from the extensions. They could also be obtained by the method of pulling back cycles that we have been using.

When ν ≤ 6, the generator of E_{2}^{1,2j}(18, 26, 34) is z ≡ hι_{34} + h^{3}ι_{26} + h^{ν+3}ι_{18}. To obtain the last term, we used 2.21(3) to write h ⊗ h^{2}ι_{18} as d(h^{ν+3}ι_{18}) mod lower terms. The leading term of ρ∂_{1}(z) in E_{2}^{2,2j}(14) is h^{ν+3} ⊗ hι_{14}, which is a generator if ν = 6. If ν < 6, then this is d((h^{ν+2} + L)ι_{14}), and so z pulls back to z′ ≡ z + h^{ν+2}ι_{14}. Then ∂(z′) in E_{2}^{2,2j}(10) is

(h ⊗ vh^{5} + h^{3} ⊗ vh^{3} + h^{ν+3} ⊗ h^{2} + h^{ν+2} ⊗ h)ι_{10}, (5.31)

using Proposition 5.11. If ν < 5, then the first term is the leading term, and it is a generator. If ν = 5, we must keep track of unit coefficients, since the first and last terms have the same excess.

Let j − 9 = 2 · 3^{5}c, with c ≢ 0 mod 3. We start with hι_{34}. The next term (h^{3}ι_{26}) is insignificant. The leading term of the image under ⟶^{∂}E_{2}^{2,2j}(18) is h ⊗ \frac{1}{2}v^{2}h^{2} ≡ \frac{1}{2}h ⊗ h^{2}. Incorporating coefficients into the analysis of the previous paragraph, 2.21(3) actually says that d(h^{8}ι_{18}) ≡ −3^{5}ch ⊗ h^{7}ι_{18} ≡ −ch ⊗ h^{2}ι_{18}, and so z is actually equivalent to hι_{34} + h^{3}ι_{26} + \frac{1}{2c}h^{8}ι_{18}. The leading term of ρ∂_{1}(z) is

\frac{1}{2c}h^{8} ⊗ (−h)ι_{14} ≡ \frac{1}{2c}h ⊗ h^{6}ι_{14} ≡ −d(\frac{1}{2c}h^{7}ι_{14}),

since by 2.21(3) d(h^{7}ι_{14}) ≡ −\frac{1}{2}(2 · 3^{5} + 2)h ⊗ h^{6}ι_{14}. Thus the refined form of z′ has significant terms hι_{34} + \frac{1}{2c}h^{7}ι_{14}, and so the leading terms of ∂(z′) are

(h ⊗ \frac{1}{4}vh^{5} + \frac{1}{2c}h^{7} ⊗ (−h))ι_{10} ≡ (\frac{1}{4}h ⊗ h^{5} + \frac{1}{2c}h ⊗ h^{5})ι_{10},

and this is 0 in E_{2} if c ≡ 1 mod 3, and is a generator if c ≡ 2 mod 3.

The boundary into E_{2}^{2,2j}(22) is (h ⊗ h^{3} + h^{3} ⊗ h)ι_{22}, which is 0 when the group is isomorphic to Z/3. □

The next result also follows by the methods already employed. Note however the excluded case, which requires major refinements, deferred to the next section.

Theorem 5.32. If j is odd, and j ≡ 5 or 8 mod 9, but ν(j − 17) ≠ 13, then

E_{2}^{1,2j}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}) ≈ Z/3^{2} ⊕ Z/3^{min(17,ν(j−17)+4)}.

Proof. The proof when j ≡ 5 is particularly simple. The result here is just that E_{2}^{1,2j}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}) ≈ Z/3^{2} ⊕ Z/3^{5}. It is most conveniently seen with charts such as those of the earlier proofs in this section. In this case, the two main towers have groups of exponent 2, 1, 1, 1, and min(ν(j − 5) + 1, 5), reading from bottom to top. These are the groups corresponding to generators of dimensions 34, 26, 18, 14, and 10, respectively. There is also a group of exponent 2 from the 22-class, and it extends cyclically above the lowest 1.

We will show that the boundary is nonzero from enough of the bottom groups of the E_{2}^{1,2j}-tower to just kill the group of exponent min(ν(j − 5) + 1, 5) at the top of the E_{2}^{2,2j}-tower. That leaves a Z/3^{5} in each tower, and the Z/3^{2} coming from the 22-class cannot be involved in differentials and must split off for algebraic reasons.

Let ν = ν(j − 5). To see these boundary morphisms, we show that the element at the top of the E_{2}^{2,2j}-tower (i.e., the element of order 3 in E_{2}^{2,2j}(10)) is hit by the Z/3 on 14 if ν ≥ 4, by the Z/3 on 18 if ν = 3, and by the Z/3 on 26 if ν = 2. Other differentials are seen from the cyclic extensions by reading down the towers. The differential when ν ≥ 4 was proved in [13, 2.4]. The differential when ν = 3 is seen by pulling the generator of E_{2}^{1,2j}(18) back to z ≡ hι_{18} + h^{2}ι_{14} and then using 5.11 to obtain ∂(z) ≡ h ⊗ \frac{1}{2}v^{2}h^{2}ι_{10}. The v^{2} can be moved to the left using 2.20(1), and by 2.18(1) \frac{1}{2}h ⊗ h^{2}ι_{10} is an element of order 3 in E_{2}^{2,2j}(10) ≈ Z/3^{4}. The case ν = 2 is similar, with the leading term of ∂(hι_{26} + h^{2}ι_{18} + h^{4}ι_{14}) being h ⊗ vh^{3}ι_{10}, which has order 3 in E_{2}^{2,2j}(10) ≈ Z/3^{3}. This completes the proof when j ≡ 5 mod 9.

Now suppose j ≡ 8 mod 9, and let ν = ν(j − 17). The picture is similar to that just described, with groups of exponent min(17, ν + 1), 1, 1, 1, and 2, from bottom to top, and a group of exponent 2 extending just above the lowest 1. These groups correspond to generators of dimensions 34, 26, 18, 14, 10, and 22, as in the case j ≡ 5 just considered. The claim is that differentials from the E_{2}^{1,2j}-tower kill all but the bottom 3^{17} elements in the E_{2}^{2,2j}-tower if ν ≥ 14, and that they kill the top Z/3^{2} if ν ≤ 12. Actually, when ν ≥ 14, the initial element hit also involves a summand in the 22-summand, but these elements hit are just the appropriate 3-power times the element at the bottom of the E_{2}^{2,2j}-tower. The E_{2}^{2,2j}(22) ≈ Z/3^{2} is a split summand in E_{2}^{2,2j}(Y_{7}) even though it may be a summand of a class hit by a boundary. We will illustrate this carefully in the case ν = 15 below.

When ν ≥ 16, the differential from the bottom of the tower into the class on the 26-class follows from [13, 2.4]. Of course, the remaining differentials in this case follow from the extensions.

When ν = 15, the generator of E_{2}^{1,2j}(34) has leading term h^{16}ι_{34} by 2.15(2), and this pulls back to z ≡ h^{16}ι_{34} + uh^{13}ι_{26}, where u is a unit in Z_{(3)}. Usually we don’t bother to list these unit coefficients, and here the value of u will not be important, but because cancellation issues will come into play, we feel that the unit should at least be given lip service. The leading term of ∂(z) in E_{2}^{2,2j}(18) ⊕ E_{2}^{2,2j}(22) is

uh^{13} ⊗ 2vhι_{18} + uh^{13} ⊗ (−h)ι_{22}. (5.33)

Using 2.21, each of these terms is a generator of its summand. On the other hand, as in the case ν = 11 of the proof of 5.23, there is a cycle z′ in C(18, 22, 26) which restricts to a generator of E_{2}^{2,2j}(26), and has 3z′ homologous to a unit times (5.33). To clarify the splitting, that E_{2}^{2,2j}(22) ≈ Z/3^{2} splits as a direct summand of E_{2}^{2,2j}(Y_{7}), we again use the exact sequence (5.26). The argument following (5.26) applies verbatim, with Z/3^{13} and Z/3 replaced by Z/3^{17} and Z/3^{2}, respectively.

The case ν = 14 is similar, but involves a 2-step extension process. In the diagram of the type 5.30, E_{2}^{2,2j}(26) extends into E_{2}^{2,2j}(22) ≈ Z/3^{2} and into a Z/3^{2} built from E_{2}^{2,2j}(18) and E_{2}^{2,2j}(14).^{1} The boundary hits into an element of order 3 in each of these summands, which in the case of the second summand means that it hits a generator of E_{2}^{2,2j}(14). In order to know that the splitting is as claimed, we must verify that the element hit is 3^{2} times a generator of E_{2}^{2,2j}(26). This is the same sort of verification that we have been making in some other cases, i.e. that the boundary and the extension involve classes that are unit multiples of one another, but here the extension is a 2-step process.

Boundary: The generator of E_{2}^{1,2j}(34) pulls back to z ≡ h^{15}ι_{34} + uh^{12}ι_{26}, with u a unit. The component of the boundary of this in E_{2}^{2,2j}(22) is uh^{12} ⊗ (−h)ι_{22}.

1E_{2}^{2,2j}(10) is in the image of ∂, and hence does not figure into the extension question being considered here.

On the other hand, the boundary into E_{2}^{2,2j}(18) satisfies

∂(z) ≡ uh^{12} ⊗ vhι_{18} ≡ 2uh^{10} ⊗ hι_{18} ≡ −2uh ⊗ h^{8}ι_{18} = d(2uh^{9}ι_{18}).

Here we have used the three parts of 2.21, with the last step using that 2j − 18 = 2(2 · 3^{14}c + 8), and so \frac{1}{4}(2j − 18) ≡ 1 mod 3. Thus z pulls back to z′ ≡ z − 2uh^{9}ι_{18}, and the leading term of ∂(z′) is 2uh^{9} ⊗ hι_{14}.

Extension: Similarly to (5.28), d(h^{14})ι_{26} is an element of order 3 in E_{2}^{2,2j}(26), and it extends to a cycle z′ in C(18, 22, 26) such that, mod lower classes, 3z′ is homologous to

h^{13} ⊗ (−h)ι_{22} + h^{13} ⊗ 2vhι_{18}.

To evaluate 3^{2}z′, we use the second 3 to reduce each h^{13} to h^{12}. The second term becomes

2h^{10} ⊗ hι_{18} ≡ −2h ⊗ h^{8}ι_{18} ≡ d(2h^{9})ι_{18} ≡ −2h^{9} ⊗ (−h)ι_{14}.

Here we have applied (3.2) at the last step.

Thus we have a unit times h^{12} ⊗(−h)ι_{22} +2h^{9} ⊗hι_{14} as the leading term of both the image of the boundary, and the 3^{2}-multiple of the generator.

The case ν ≤ 12 is much easier. The generator of E_{2}^{1,2j}(34) pulls back to z ≡ h^{ν+1}ι_{34} +h^{ν−2}ι_{26} +h^{ν−5}ι_{18} +h^{ν−6}ι_{14} and this satisfies ∂(z) ≡ h^{ν+1} ⊗vh^{5}ι_{10}, which is a generator since it does not desuspend. □

The final case differs from the others in that E_{2}^{1,2j}(Y_{7}) and E_{2}^{2,2j}(Y_{7}) are not isomorphic.

Theorem 5.34. Assume j is odd and j ≡ 2 mod 9. Then E_{2}^{2,2j}(Y_{7}) ≈ Z/3^{2} ⊕ Z/3^{min(13,ν(j−11)+4)}, while

E_{2}^{1,2j}(Y_{7}) ≈ \begin{cases}Z/3^{3} ⊕ Z/3^{5} & if ν(j − 11) = 2 \\ Z/3^{4} ⊕ Z/3^{min(11,ν(j−11)+2)} & if ν(j − 11) > 2. \end{cases}

Proof. Let j be as in the theorem, and ν = ν(j − 11). The picture when ν < 8 is as in Diagram 5.35.

Diagram 5.35.

[figure: 34; 26; 18; 22; 14; 10; 2; ν + 1; •; 2; •; •; 2; ν + 1; •; 2; •; •; E_{2}^{2,2j}; E_{2}^{1,2j}; ν < 8]

The indicated boundary is seen by pulling back the generator α_{m/2}ι_{34} to a cycle z on C(14, 18, 22, 26, 34), and then obtaining α_{m/2} ⊗ vh^{5}ι_{10} as the leading term of ∂(z). This generates E_{2}^{2,2j}(10). This generator α_{m/2} is as described in 2.15(1). The slash does not mean division; this notation was introduced in papers preceding [8], where it was first applied unstably.

The boundary from C(26, 34) into the large group E_{2}^{2,2j}(22) has leading term α_{m/2} ⊗ h^{3}ι_{22}, which is 0 if ν < 8. Here we have m = \frac{1}{2}(j − 17), and we use the argument of 5.11 to see the factor on the RHS of the ⊗. (Because of α_{2} and α_{1} attaching maps, going from 34 to 22 is like going from 26 to 14, with coefficient T_{4} ≡ h^{3} in Proposition 5.11.) The claimed splitting when ν < 8 follows for algebraic reasons from Diagram 5.35.

If ν = 8, then α_{m/2} ⊗ h^{3}ι_{22} has order 3 in E_{2}^{2,2j}(22), by 2.18(1). If we let g denote a generator of E_{2}^{2,2j}(34), then similarly to Diagram 5.35, 3^{3}g = a + b, where a is detected on the 22-class, and b on the 18-class. We have relations in E_{2}^{2,2j}(Y_{7}) 3^{9}a, 3^{4}b, and (from the boundary) 3^{2}b + 3^{8}ua, with u a unit in Z_{(3)}. The quotient group is easily seen to be Z/3^{12} ⊕ Z/3^{2}, with generators g and (1 − 3^{6}u)a − 3^{3}g. The case ν = 9 is extremely similar.

If ν ≥ 10, then E_{2}^{2,2j}(22) ≈ Z/3^{11}, and the component of the boundary into this part hits 3^{8} times the generator, as before. But this implies now that the class on E_{2}^{1,2j}(26) hits the element of order 3 in E_{2}^{2,2j}(22). Whereas in the cases ν = 8 and 9, the hitting into the 22-part was without much consequence, because it just adjoined another summand to the classes on the 10-cell which were being hit, the boundary described in the preceding sentence causes one less element in the kernel and cokernel. In the sort of description given in the previous paragraph, the relation 3^{9}a is changed to 3^{11}a. Now we have

3^{13}g = 3^{10}a = −\frac{1}{u}3^{4}b = 0,

and the claimed splitting follows. □

6. The final case

In this section, we establish the final and most difficult case of E_{2}^{1,2j}(Y_{7}), with ν(j − 17) = 13. We will explain why we cannot say for exactly which such values of j the maximal order is achieved.

Theorem 6.1. If j is odd, and ν(j − 17) = 13, then for δ equal to one of the numbers 2, 5, or 8,

E_{2}^{1,2j}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}) ≈ Z/3^{2} ⊕ Z/3^{min(19,ν(j−17−2δ·3^{13})+4)}.

The methods of this paper do not allow us to determine which of the three numbers equals δ.

Proof. Let j − 17 = 2 · 3^{13}c, with c ≢ 0 mod 3. The proof begins just like that of the case j ≡ 8 mod 9 in Theorem 5.32. In the diagram of the type that we have been using, the main tower has groups of exponent 14, 1, 1, 1, and 2, reading from bottom to top, and a group of exponent 2 extending above the lowest 1. We choose as the generator of E_{2}^{1,2j}(34) the element −α_{m/14}ι_{34}, where m = \frac{1}{2}(j − 17). We use (2.16) to write it as (h^{14} + L)ι_{34}, with L defined on S^{27}. (We choose the minus on α to remove the minus signs in (2.16) and (2.17).)

The boundary E_{2}^{1,2j}(34)⟶^{∂}E_{2}^{2,2j}(26) sends this generator to a class congruent mod lower terms to

h^{14} ⊗ 2vhι_{26} ≡ 2h^{12} ⊗ hι_{26} ≡ −2h ⊗ h^{10}ι_{26} = d((h^{11} + L′)ι_{26}).

Here we use all three parts of 2.21, with the last step using that j − 13 = 2c3^{13} + 4, and so \frac{1}{2}(j − 13) ≡ 2 mod a high power of 3. Thus the generator pulls back to z ≡ h^{14}ι_{34} − h^{11}ι_{26}.

Next we consider ∂(z) in both E_{2}^{2,2j}(22) and in E_{2}^{2,2j}(18). The former has leading term (h^{14} ⊗ h^{3} − h^{11} ⊗ (−h))ι_{22}. This desuspends to S^{19} and hence is 0 in E_{2}^{2,2j}(22) ≈ Z/3^{2}. Since the 22-cell factor is split from C(10, 14), we do not need to write this as a boundary and append to z. In C^{2}(18), we have

∂(z) ≡ (h^{14} ⊗ \frac{1}{2}v^{2}h^{2} − h^{11} ⊗ 2vh)ι_{18} ≡ −2h^{9} ⊗ hι_{18} ≡ 2h ⊗ h^{7}ι_{18} = d(−\frac{1}{2}h^{8}ι_{18}),

similarly to the previous paragraph. Thus z pulls back to z′ ≡ z + \frac{1}{2}h^{8}ι_{18}.

The leading term of ∂(z′) in E_{2}^{2,2j}(14) is \frac{1}{2}h^{8} ⊗ (−h)ι_{14} ≡ \frac{1}{2}h ⊗ h^{6}ι_{14}, and so ∂(z′) = d((−\frac{1}{10}h^{7} + L)ι_{14}), since \frac{1}{2}(j − 7) ≡ 5 mod a high power of 3. As we will be working at most mod 9, we replace the 10 by 1. Thus z′ pulls back to

z′′ ≡ −α_{m/14}ι_{34} − h^{11}ι_{26} + \frac{1}{2}h^{8}ι_{18} + h^{7}ι_{14}. (6.2)

Now we use (2.17) for α_{m/14}, and obtain terms in ∂(z′′) due to the first and last terms of (6.2):

∂(z′′) ≡ (ch ⊗ \frac{1}{4}vh^{5} + h^{7} ⊗ (−h))ι_{10} ≡ (\frac{1}{4}c + 1)h ⊗ h^{5}ι_{10}. (6.3)

This is a generator if c ≡ 1 mod 3, in which case the diagram described at the beginning of the proof has differential from the generator of E_{2}^{2,2j}(34) and 3 times the generator killing E_{2}^{2,2j}(10), yielding Z/3^{2} ⊕ Z/3^{17} as the groups E_{2}^{1,2j}(Y_{7}) and E_{2}^{2,2j}(Y_{7}), as claimed in this case. The splitting is true for algebraic reasons.

If c ≡ 2 mod 3, then ∂(z′′) is not a generator of E_{2}^{2,2j}(10) ≈ Z/9, but it might be 3 times the generator. This requires second-order information throughout the entire analysis above. This is something that we have not had to do in past applications. In particular, we need finer information in all three parts of Lemma 2.21, in both descriptions of α_{m/e} in Theorem 2.15(2), and in Proposition 5.11.

We now write c = 3k + 2. The cycle z′′ above can be written as

(6.4)  z′′ = −α_{m/14}ι_{34} + (−h^{11} + A_{10} + L_{10})ι_{26} + (\frac{1}{2}h^{8} + A_{7} + L_{7})ι_{18}
+ (h^{7} + A_{6} + L_{6})ι_{14},

where A_{i} has excess exactly i, and L_{i} has excess less than i. When we evaluate ∂(z′′), the terms of excess 5 will cancel out as in (6.3) with c ≡ 2, and so we can desuspend ∂(z′′) to S^{9}. Our differential into E_{2}^{2,2j}(10) is equal to 3 times the generator if and only if the desuspension of ∂(z′′) yields a generator of E_{2}^{2,2j−1}(S^{9}).

Let B_{4} be the terms of excess exactly 4 in T_{3} of Proposition 5.11. The terms of excess 4 or 5 in ∂(z′′) are

−α_{m/14} ⊗ (\frac{1}{4}vh^{5} + B_{4}) + \frac{1}{2}h^{8} ⊗ \frac{1}{2}h^{2} + (h^{7} + A_{6}) ⊗ (−h). (6.5)

Note how certain terms such as ∂A_{10} and ∂A_{7} were dropped because they yield terms whose excess is less than 4.

By an analysis similar to [13, 2.11(5)] we have, when p = 3 and c ≢ 0 mod 3,

α_{c3^{e−1}/e} ≡ −chv^{c3^{e−1}−1} + \frac{3}{2}ch^{2}v^{c3^{e−1}−2} − 3ch^{3}v^{c3^{e−1}−3} mod 9.

Let h^{7} ⊗ (−h) = h ⊗ h^{5} + C_{4} + L_{4}, where C_{4} has excess 4, and L_{4} excess less than 4. Omitting terms of excess less than 4, (6.5) becomes

(6.6) ((3k + 2)(h + 3h^{2}) − 3h^{2} + 6h^{3}) ⊗ (\frac{1}{4}vh^{5} + B_{4})

+ \frac{1}{4}h^{8} ⊗ h^{2} + h ⊗ h^{5} + C_{4} − A_{6} ⊗ h,

where the (h + 3h^{2}) comes from hv^{c3^{e−1}−1} = (v − 3h)^{c3^{e−1}−1}h. Now write h ⊗ vh^{5} as h ⊗ h^{5} − 3h^{2} ⊗ h^{5}. Using coefficients of 3 to reduce the excess of terms on the right side of the ⊗, we can rewrite (6.6) in excess 4 as

\frac{3}{4}kh ⊗ h^{5} + \frac{3}{2}(h + \frac{1}{2}h^{2} + h^{3}) ⊗ h^{5} + 2h ⊗ B_{4} + \frac{1}{4}h^{8} ⊗ h^{2} + C_{4} − A_{6} ⊗ h, (6.7)

where the \frac{3}{2}h ⊗ h^{5} comes from the 2h ⊗ \frac{1}{4}h^{5} and h ⊗ h^{5} in (6.6). Let D_{4} = \frac{3}{2}(h + \frac{1}{2}h^{2} + h^{3}) ⊗ h^{5} + \frac{1}{4}h^{8} ⊗ h^{2} + C_{4}, a specific class of excess 4, independent of the value of k and of any choices of the sort that we are about to mention.

The term B_{4} is the terms of excess 4 in (5.17). It could also have included any terms of excess 4 in the homogeneous part of T_{3} discussed in the paragraph after (5.17), but as discussed there, this homogeneous part has excess less than 4. Then B_{4} contains a term −\frac{1}{2}v^{2}h^{4} which appears in (5.17), and it could contain a term 9c_{2}h^{6} if c_{2} ≢ 0 mod 3. However, because of a term with coefficient \frac{59}{6} which has c_{2} as coefficient, we can infer that c_{2} ≡ 0 mod 3. Thus B_{4} = −\frac{1}{2}v^{2}h^{4}, and so we can let D′_{4} = D_{4} − h ⊗ v^{2}h^{4}, still a specific element of excess 4, and we have

\frac{1}{4}kh ⊗ h^{4} + D′_{4} − A_{6} ⊗ h (6.8)

as our new expression for ∂(z′′) mod L.

Next we study A_{6}. To find it, we apply ⟶^{∂}E_{2}^{2,2j}(14) to the sum z_{3} = X_{1} + X_{2} + X_{3} of the first three terms of (6.4), and write the result as d(A_{6}). The terms in ∂(X_{1}) will have excess less than 5, and so may be omitted from the analysis. There is one term, \frac{1}{2}h^{8} ⊗(−h), of excess 6, which accounts for the h^{7} in (6.2). There are a number of terms of excess 5, which contribute toward A_{6}. In particular, note that d(h^{6}) ≡ h ⊗ h^{5}, and so each occurrence of h ⊗ h^{5} in ∂(z_{3}) affects the coefficient of h^{6} in A_{6}. The leading part of ∂(X_{2}) is −h^{11} ⊗ T_{4}, where T_{4} is as in 5.11. The full form of T_{4} is given in (5.14) and involves a homogeneous part whose coefficient c we do not know. Two parts of this homogeneous part have a factor of 3, which can be used to reduce the excess, but ch^{11} ⊗ v^{2}h ≡ ch^{7} ⊗ h ≡ −ch ⊗ h^{5} will cause a ch^{6}-term in A_{6}, and hence a ch ⊗ h^{4} in (6.8). Thus the coefficient of h ⊗ h^{4} in (6.8) is k + D + c ∈ Z/3, where D is something which we could compute if we really needed to. Note also that for our purposes (6.8) lies in Z/3 generated by h ⊗ h^{4}. The coefficient c has a value; we just don’t know how to find it. Therefore, there is one value of k in Z/3 for which (6.8) is 0. (The diligent reader can check that such considerations cannot affect earlier parts of the argument.) Thus the differential into E_{2}^{2,2j}(10) is 0 if and only if k, defined by j − 17 = 2(3k + 2)3^{13}, has this value mod 3. Letting δ = 3k + 2 mod 9, this establishes the theorem. □

7. Periodic homotopy of E_{7}

In this section we use the results for E_{2}^{s,2j}(Y_{7}) already achieved to deduce that v_{∗}(E_{7}) is as claimed in Theorem 1.1. The first result almost finalizes v_{∗}(Y_{7}), given the results for E_{2}^{s,2j}(Y_{7}) determined in the previous two sections.

Theorem 7.1. The v_{1}-periodic UNSS of Y_{7} converges to v_{∗}(Y_{7}). If j is odd, then v_{2j+1}(Y_{7}) = 0, v_{2j}(Y_{7}) ≈ v_{2j}(S^{7}), v_{2j−2}(Y_{7}) ≈ E_{2}^{2,2j}(Y_{7}), and there is an exact sequence

0 → v_{2j−1}(S^{7}) → v_{2j−1}(Y_{7}) → E_{2}^{1,2j}(Y_{7}) → 0.

Proof. One thing that we have to worry about in proving convergence of the v_{1}-periodic UNSS is to rule out the possibility of a v_{1}-periodic homotopy class which is not seen in v_{1}-periodic E_{2}. This could come about by having a sequence of homotopy classes related by a filtration-increasing v_{1}-multiplication. We must also rule out the existence of elements in v_{1}-periodic E_{∞} which do not correspond to elements of v_{1}-periodic homotopy. This could come about from a sequence of E_{2}-classes related by filtration-preserving v_{1}-periodicity in E_{2}, which support arbitrarily large differentials into a sequence of classes related by filtration-increasing v_{1}-multiplications. The way that we will show that these things cannot happen for Y_{7} is to note that Y_{7} is built by fibrations from spaces where we have already established convergence.

In (5.7), it was noted how the v_{1}-periodic UNSS of Y_{7} splits into the part from S^{7} and the part from even-dimensional classes. As all of this is confined to filtrations 1 and 2, we obtain the following schematic picture for E_{2}^{s,t}(Y_{7}), which must necessarily equal E_{∞}.

s = 2  ev  S^{7}
s = 1  ev  S^{7}
t − s = 2j − 2 2j − 1 2j 2j + 1 j odd

Here a box labeled S^{7} means the corresponding group E_{2}^{s,t}(S^{7}), while a box labeled “ev” (for “even”) means the corresponding group E_{2}^{s,t}(10, 14, 18, 22, 26, 34), as computed in Section 5. This E_{2} calculation is consistent with the fibrations (5.5) and S^{7} → ΩW → ΩS^{23} of Proposition 5.1.

For X = ΩS^{23}, ΩB(11, 15), or ΩE_{7}/F_{4}, the v_{1}-periodic UNSS collapses to isomorphisms, if j is odd,

v_{1}^{−1}π_{2j+ϵ}(X) ≈ \begin{cases}0 & if ϵ = 0 or 1 \\ E_{2}^{2,2j}(X) & if ϵ = −2 \\ E_{2}^{1,2j}(X) & if ϵ = −1. \end{cases}

This is true for ΩS^{23} by [9, 6.1], for ΩB(11, 15) by the fibration

ΩS^{11} → ΩB(11, 15) → ΩS^{15},

and for ΩE_{7}/F_{4} by Theorem 4.6. (Although 4.6 dealt with convergence for E_{7}/F_{4}, the methods of Section 5 show that the calculation for E_{2}(ΩE_{7}/F_{4}) is just that for E_{2}(E_{7}/F_{4}) shifted back by 1 dimension, and of course the same is true of v_{1}-periodic homotopy groups.)

Let j be odd. We can use a Five Lemma argument once we establish that, for ϵ = 1 or 2, there are morphisms v_{2j−ϵ}(−) → E_{2}^{ϵ,2j}(−) for these spaces. To see that such morphisms exist, we note that since compact Lie groups and spheres have H-space exponents ([28]), the spaces with which we deal here all have H-space exponents. By [21], this implies that each v_{1}-periodic homotopy group is a direct summand of some actual homotopy group, and then we can take the morphism from homotopy to homotopy mod filtration greater than ϵ, which is (unlocalized) E_{∞}^{ϵ}, then to (unlocalized) E_{2}^{ϵ} as the kernel of the differentials, and then to v_{1}-periodic E_{2}^{ϵ}. This argument is similar to that used in [19].

Thus, letting X = E_{7}/F_{4} and B = B(11, 15), there is a commutative diagram of exact sequences

v_{2j−1}(ΩX) → v_{2j−2}(ΩS^{23} × ΩB) → v_{2j−2}(Y_{7}) → v_{2j−2}(ΩX) → 0
≈↓ ≈↓ ↓ ≈↓
E_{2}^{1,2j}(ΩX) → E_{2}^{2,2j}(ΩS^{23} × ΩB) → E_{2}^{2,2j}(Y_{7}) → E_{2}^{2,2j}(ΩX) → 0

which implies that v_{2j−2}(Y_{7}) → E_{2}^{2,2j}(Y_{7}) is an isomorphism.

Similarly, there is a commutative diagram with exact rows and the first column exact

0 ↓

E_{2}^{2,2j+1}(S^{7})  ≈ E_{2}^{2,2j+1}(Y_{7})
↓  ↓
0 → v_{2j−1}(ΩW × ΩB) → v_{2j−1}(Y_{7}) → v_{2j−1}(ΩX) → v_{2j−2}(ΩW × ΩB)
↓ ↓ ≈↓ ≈↓
0 → E_{2}^{1,2j}(ΩS^{23} × ΩB) → E_{2}^{1,2j}(Y_{7}) → E_{2}^{1,2j}(ΩX) → E_{2}^{2,2j}(ΩS^{23} × ΩB)

↓ 0

which implies that the second column fits into a short exact sequence.

The portion of the theorem about v_{2j+1}(Y_{7}) and v_{2j}(Y_{7}) is immediate from the exact sequence in v_{∗}(−) associated to the fibration (5.5). □

We restate the following result from [13, 1.3(1)].

Lemma 7.2. The projection map B(3, 7) → S^{7} induces an isomorphism in v_{2j−1}(−) unless j is odd and j ≡ 21 mod 27, in which case it is a surjection Z/3^{4} → Z/3^{3}. The isomorphic groups are 0 if j is even, while if j is odd, they are cyclic of order 3^{min(3,1+ν(j−3))}.

The next result, combined with the above results and Theorems 5.18, 5.23, 5.29, 5.32, and 6.1 gives v_{∗}(E_{7}) for most values of ∗.

Theorem 7.3. (a) If j is odd, j ≢ 2 mod 9, and j ≢ 21 mod 27, then the exact sequence of the fibration Y_{7} → B(3, 7) → E_{7} breaks up into isomorphisms

v_{2j}(Y_{7})⟶^{≈}v_{2j}(B(3, 7)) and v_{2j−1}(E_{7})⟶^{≈}v_{2j−2}(Y_{7})

and a short exact sequence

0 → v_{2j}(E_{7}) → v_{2j−1}(Y_{7})⟶^{ϕ}v_{2j−1}(B(3, 7)) → 0.

If E_{2}^{1,2j}(Y_{7}) ≈ Z/3^{e_{1}} ⊕ Z/3^{m}, with 1 ≤ e_{1} ≤ 2, is as given in Theorems 5.18, 5.23, 5.29, 5.32, and 6.1, and v_{2j−1}S^{7} ≈ v_{2j−1}(B(3, 7)) ≈ Z/3^{e_{2}} is as in 7.2, then

v_{2j−1}(Y_{7}) ≈ Z/3^{e_{1}+e_{2}} ⊕ Z/3^{m}, (7.4)

and ϕ sends the first summand onto Z/3^{e_{2}}.

(b) If j is even, then v_{2j}E_{7} = v_{2j−1}E_{7} = 0.

Note that even if ϕ sent the second summand nontrivially, its kernel would still be Z/3^{e_{1}} ⊕ Z/3^{m}, since m ≥ e_{1} + e_{2}. Thus if j is as in Theorem 7.3(a.), there are abstract isomorphisms v_{2j}(E_{7}) ≈ E_{2}^{1,2j}(Y_{7}) and v_{2j−1}(E_{7}) ≈ E_{2}^{2,2j}(Y_{7}), with E_{2}^{s,2j}(Y_{7}) as given in Theorems 5.18, 5.23, 5.29, 5.32, and 6.1. This implies Theorem 1.1 in these cases.

Proof of Theorem 7.3. There is a commutative diagram of fibrations

ΩW → B(3, 7) → K
(7.5)  ↓  ↓^{=}  ↓
Y_{7}  → B(3, 7) → E_{7}

where the last map is the composite K → F_{4} → E_{7}. Since by [10, 2.10(i)] the composite S^{7} → ΩW → B(3, 7) → S^{7} has degree 3, we deduce the same of the composite S^{7} → Y_{7} → B(3, 7) → S^{7}. We already know that v_{2j}S^{7} → v_{2j}Y_{7} is an isomorphism, and v_{2j}B(3, 7) → v_{2j}S^{7} is multiplication by 3 on isomorphic groups. It follows that v_{2j}Y_{7} → v_{2j}B(3, 7) is an isomorphism.

There is a commutative diagram of fibrations

S^{7} → ΩW → ΩS^{23}
(7.6) ↓ ↓
S^{7} → Y_{7}  .

The cyclic extension in v_{2j−1}(ΩW) was established in [10, pp.294-5]. This implies the nontrivial extension in v_{2j−1}Y_{7} claimed in the theorem from the Z/3^{e_{1}} on the 22-class in E_{2}^{1,2j}(Y_{7}) to v_{2j−1}(S^{7}) in the exact sequence of Theorem 7.1.

There cannot be an extension in v_{2j−1}Y_{7} from the Z/3^{m}-summand of E_{2}^{1,2j}Y_{7} because of the splitting F_{4} = K × B(11, 15). The element of order 3 in the large summand of v_{2j−1}Y_{7} comes from B(11, 15), while the S^{7} lies in K. This is made explicit in the commutative diagram of fibrations

ΩB(11, 15) × ΩW → B(3, 7) → F_{4}
(7.7) ↓ ↓ ↓
Y_{7}  → B(3, 7) → E_{7}

That ϕ sends the first summand of (7.4) onto v_{2j−1}B(3, 7) follows from the diagram (7.7) and the surjectivity of v_{2j−1}(ΩW) → v_{2j−1}B(3, 7) established in [10, pp.297-8]. □

One of the cases omitted in the previous theorem is covered in the following result, the proof of which is very similar.

Theorem 7.8. If j is odd and j ≡ 21 mod 27, then the exact sequence (with B = B(3, 7))

0 → v_{2j}Y_{7}⟶^{ϕ_{1}}v_{2j}B → v_{2j}E_{7} → v_{2j−1}Y_{7}⟶^{ϕ_{2}}v_{2j−1}B → v_{2j−1}E_{7} → v_{2j−2}Y_{7} → 0

has ϕ_{1} an injection Z/3^{3} ↪ Z/3^{4}, and ϕ_{2} a surjection from the first summand in Z/3^{4} ⊕ Z/3^{m} → Z/3^{4}. Moreover,

v_{2j}E_{7} ≈ coker ϕ_{1} ⊕ ker ϕ_{2} ≈ Z/3 ⊕ Z/3^{m}.

Proof. Similarly to the previous proof, the morphism ϕ_{1} follows from [13, 2.5] and [10, 2.10(i)], the structure of v_{2j−1}Y_{7} follows from (7.6), and the morphism ϕ_{2} follows from (7.7). The Z/3^{m} in v_{2j−1}Y_{7} cannot extend cyclically with coker ϕ_{1} in v_{2j}E_{7} because the element of order 3 in Z/3^{m} lies in v_{2j−1}ΩB(11, 15), while coker ϕ_{1} lies in v_{2j}K, and these cannot be related by a ·3-extension due to the splitting F_{4} = K × B(11, 15). □

We begin working toward determination of v_{2j−ϵ}E_{7} when j ≡ 2 mod 9 with the following proposition.

Proposition 7.9. If j is odd and j ≡ 2 mod 9, then the exact sequence of the fibration Y_{7} → B(3, 7) → E_{7} yields

• v_{2j}Y_{7} → v_{2j}B(3, 7) is an isomorphism of Z/3’s;

• v_{2j−1}E_{7} → v_{2j−2}Y_{7} is an isomorphism;

• v_{2j}E_{7} ≈ ker(v_{2j−1}Y_{7}⟶^{ϕ}v_{2j−1}B(3, 7) ≈ Z/3).

Proof. Surjectivity of v_{2j−1}Y_{7} → v_{2j−1}B(3, 7) follows from (7.7), while v_{2j}Y_{7} → v_{2j}B(3, 7) is bijective as in the proof of 7.3. □

By 7.9, 7.1, and 5.34, v_{2j−1}E_{7} is seen to be as claimed in Theorem 1.1 when j ≡ 2. It remains to determine v_{2j−1}Y_{7} and ϕ, from which v_{2j}E_{7} follows.

Theorem 7.10. Let j be odd, and ν = ν(j − 11). If 2 ≤ ν ≤ 9, then

v_{2j−1}Y_{7} ≈ Z/3^{ν+3} ⊕ Z/3^{4}

and ϕ sends Z/3^{4} nontrivially. Thus ker ϕ ≈ Z/3^{ν+3}⊕Z/3^{3}, regardless of ϕ | Z/3^{ν+3}.

Proof. Similarly to the proof of Theorem 7.3, the extension in

Z/3 ≈ v_{2j−1}S^{7} → v_{2j−1}Y_{7} → E_{2}^{1,2j}Y_{7} ≈ Z/3^{ν+2} ⊕ Z/3^{4}

is nontrivial from the first summand. From [10, 2.12], v_{2j−1}(ΩW) → v_{2j−1}B(3, 7) is a surjection Z/3^{ν+2} → Z/3, and from (7.5) it factors as

v_{2j−1}(ΩW) → v_{2j−1}Y_{7}⟶^{ϕ}v_{2j−1}B(3, 7).

From (7.6), v_{2j−1}(ΩW) → v_{2j−1}Y_{7} is an injection Z/3^{ν+2} → Z/3^{ν+3} ⊕ Z/3^{4}, since the element of order 3 in v_{2j−1}(ΩW), which comes from v_{2j−1}(S^{7}), maps nontrivially. The result now follows from elementary algebra. □

The same ingredients imply the following result.

Theorem 7.11. If j is odd and ν(j − 11) ≥ 10, but j ≢ 11 + 2 · 3^{10} mod 2 · 3^{11}, then

v_{2j−1}Y_{7} ≈ Z/3^{12} ⊕ Z/3^{4}

and ϕ is surjective in Proposition 7.9.

We cannot deduce from this which summand(s) of v_{2j−1}Y_{7} maps nontrivially under ϕ, and so we cannot tell whether ker ϕ is Z/3^{12} ⊕ Z/3^{3} or Z/3^{11} ⊕ Z/3^{4}. We suspect that Z/3^{4} maps across, which would imply the first splitting.

Finally we have the following result in the exceptional case. In order to keep the statement of Theorem 1.1 readable, we did not distinguish there between this case, in which we know the precise structure of v_{2j−1}E_{7}, and the case of Theorem 7.11, where we do not.

220 D. Davis

Theorem 7.12. If j ≡ 11 + 2 · 3^{10} mod 2 · 3^{11}, then

v_{2j}E_{7} ≈ Z/3^{12} ⊕ Z/3^{3}.

Proof. As in the proof of 7.10, v_{2j−1}Y_{7} ≈ Z/3^{12} ⊕ Z/3^{4}. It was shown in [10, 2.12] that v_{2j−1}(ΩW) → v_{2j−1}B(3, 7) is 0 if j ≡ 11 + 2 · 3^{10} mod 2 · 3^{11}.

Let G denote the fiber of K → E_{7}. There is a commutative diagram of fibrations

ΩW ⟶^{=} ΩW
↓  ↓
Y_{7} → B(3, 7) → E_{7}
↓ ↓ ↓
G → K → E_{7}.

It follows from the Serre spectral sequence of the fibration ΩW → Y_{7} → G that

BP_{∗}(G) ≈ BP_{∗}[x_{10}, x_{14}, x_{18}, x_{26}, x_{34}],

and so charts for v_{∗}G are like charts for E_{2}^{s,2j}(Y_{7}) without the part on the 22-class. The chart for v_{2j−1}G and v_{2j−2}G whenever j ≡ 2 mod 9 is like Diagram 5.35 without the ν + 1. In particular, v_{2j−1}G is cyclic with generator on the 26-class. The proof of Theorem 5.34 in the case ν ≥ 10, where it says that the class on E_{2}^{1,2j}(26) hits the element of order 3 in E_{2}^{2,2j}(22), implies that v_{2j−1}G → v_{2j−1}K sends the generator to the element of order 3^{2}. Now it follows from the following commutative diagram with exact rows that ϕ is surjective on the Z/3^{4} summand.

Z/3^{11}  Z/3^{12} ⊕ Z/3^{4}  Z/3^{5}
0 → v_{2j}W →  v_{2j−1}Y_{7}  → v_{2j−1}G → v_{2j−1}W
↓^{ϕ}  ↓  ↓^{=}
v_{2j−1}B  → v_{2j−1}K → v_{2j−1}W
Z/3  Z/3^{12}  Z/3^{11}
□

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Lehigh University, Bethlehem, PA 18015

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