COMPOSITIONS IN THE v_{1}-PERIODIC HOMOTOPY GROUPS OF SPHERES

MARTIN BENDERSKY AND DONALD M. DAVIS

Abstract. Let ρ_{i} ∈ π_{n+8i−1}(S^{n}) denote an element which suspends to a generator of the image of the stable J- homomorphism. We determine the image of the composite ρ_{j} ∘ ρ_{k} in v_{1}-periodic homotopy v_{1}^{−1}π_{n+8i+8j−2}(S^{n}). The method is to use Adams operations to compute the 1-line of an unstable homotopy spectral sequence constructed by Bendersky and Thompson.

1. Main theorem

The p-primary v_{1}-periodic homotopy groups of a space X, denoted v_{1}^{−1}π_{∗}(X; p), are a localization of the portion of π_{∗}X_{(p)} detected by K-theory.([14]) The v_{1}-periodic homotopy groups of spheres contain the image of the J-homomorphism.([18]) Until the last two sections of this paper, we will deal with 2-primary homotopy theory, let ν(−) denote the exponent of 2 in an integer, and let v_{1}^{−1}π_{∗}(X) = v_{1}^{−1}π_{∗}(X; 2).

We need the following known result.

Proposition 1.1. i.) v_{1}^{−1}π_{2n+8i−1}(S^{2n+1}) ≈ Z/2^{min(n,ν(i)+4)};

ii.) there are morphisms v_{1}^{−1} : π_{∗}(S^{2n+1}) → v_{1}^{−1}π_{∗}(S^{2n+1}), ∗ > 2n + 1, which are split surjections for ∗ = 2n + 8i − 1 with 4i − ν(i) ≥ n + 8.

Proof. i.) See Theorem 2.1 and the accompanying diagrams, 2.2 and 2.3. (ii.) In [14, 1.7] and [12, 2.4] it is shown that if p^{e} : Ω^{n}X → Ω^{n}X is null homotopic, then there is a natural morphism v_{1}^{−1} : π_{j}(X) → v_{1}^{−1}π_{j}(X) for j > n. In [22], it is noted that 2^{3n/2+1} is null homotopic on Ω^{2n}S^{2n+1}⟨2n + 1⟩. Thus v_{1}^{−1} is defined on π_{∗}(S^{2n+1}) for ∗ > 2n + 1. The split surjection follows from [18, 1.3,1.5].

Date: March 7, 2000.

1991 Mathematics Subject Classification. 55Q40,55Q50.

Key words and phrases. homotopy groups of spheres.

2 BENDERSKY AND DAVIS

In this paper, we determine the v_{1}-periodic homotopy class of the composition

S^{N+8j+8k−2} ⟶^{ρ_{k}} S^{N+8j−1} ⟶^{ρ_{j}} S^{N},  (1.2)

where ρ_{i} denotes a map (or its homotopy class) which suspends to a generator of the image of the stable J-homomorphism in the 2-primary (8i − 1)-stem. That is, we will determine v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) in v_{1}^{−1}π_{N+8j+8k−2}(S^{N}), and will sometimes say that ρ_{j} ∘ρ_{k} is a certain multiple of the generator in the v_{1}-periodic summand to mean that v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) equals this multiple.

These compositions were first considered in [19] and were determined in [20] when ν(k) ≤ 4j − 5. Although our main theorem, 1.3, will be stated when N in (1.2) is an arbitrary integer, most of our discussion will, for simplicity, center around odd values of N; the minor interpolation required for even values of N can be obtained by the same methods, using results from [18].

Let ϕ(n) denote the number of positive integers i satisfying i ≤ n and i ≡ 0, 1, 2, 4

mod 8, and let δ = \begin{cases}3 & if ν(j) + 4j > ν(k) \\ 2 & if ν(j) + 4j ≤ ν(k) \end{cases}. Our main theorem is

Theorem 1.3. Let m_{j,k} = max(ν(j) + 4, ν(k) + 4 − 4j), and let N satisfy

ϕ(N + δ) ≥ m_{j,k} + 3.  (1.4)

Then^{1} v_{1}^{−1}(ρ_{j} ∘ρ_{k}) is 2^{ϕ(N+3)−(m_{j,k}+3)+ν(e(j,k))} times a generator of v_{1}^{−1}π_{N+8j+8k−2}(S^{N}), where

e(j, k) = \begin{cases}\frac{1}{k} (∑_{i≥1} 80^{i−1}(\binom{j}{i} + (2^{4j} − 1)\binom{j+k}{i})) & if ν(k) = 4j + ν(j) \\ 0 & if ν(k) ≠ 4j + ν(j). \end{cases}

An integer N satisfies (1.4) if and only if the composite (1.2) is defined; an explicit version is given by:

N ≥ 2m_{j,k} + 4 − δ + \begin{cases}0 & if m_{j,k} ≡ 0, 3 mod 4 \\ 1 & if m_{j,k} ≡ 2 mod 4 \\ 2 & if m_{j,k} ≡ 1 mod 4. \end{cases}  (1.5)

Our theorem may be summarized by saying that

1except in the (rare) cases specified in Proposition 2.14, in which cases we can only assert that v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) is a multiple of the specified element

COMPOSITIONS IN HOMOTOPY OF SPHERES 3

Remark 1.6. ρ_{j} ∘ ρ_{k} generates the v_{1}-periodic summand the first time it is defined, with two exceptions:

(i) If ν(k) > ν(j) + 4j and ν(k) ≡ 1, 2 mod 4, then ρ_{j} ∘ ρ_{k} is 2 times a generator. (ii) If ν(k) = ν(j) + 4j, then ρ_{j} ∘ ρ_{k} is e(j, k) or 2e(j, k) times a generator, and e(j, k) is even, since, when ν(k) = 4j + ν(j), the mod 8 value of e(j, k) is that of (2^{4j}j − k)/k, which is even. We hope that Remark 1.6 illuminates the technical content of Theorem 1.3. Once we know ρ_{j} ∘ ρ_{k} on the smallest sphere on which it is defined, its value on larger spheres is easily determined from the well-known effect of the suspension homomorphism on the unstable v_{1}-periodic homotopy groups of spheres.

In [20], the noncommutativity of these compositions when ν(j) ≠ ν(k) is emphasized; here this is seen from the fact that m_{j,k} ≠ m_{k,j} in these cases. The results of [20] are a subset of our cases in which ν(k) < 4j + ν(j); [20] does not see the change that occurs when ν(k) = ν(j) + 4j.

It is immediate to read off from Theorem 1.3 that (provided ν(k) ≠ 4j + ν(j)) v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) = 0 in v_{1}^{−1}π_{N+8j+8k−2}(S^{N}) if and only if

ϕ(N + δ) ≥ m_{j,k} + ν(j + k) + 7.  (1.7)

However, we cannot conclude that ρ_{j} ∘ρ_{k} is 0 when (1.7) is satisfied, for it might have a nontrivial v_{1}-torsion component. The composite ρ_{1} ∘ ρ_{1} is an extreme example of this, which will be discussed at the end of Section 2.

In [8] the first author and Thompson constructed a spectral sequence E_{r}^{s,t}(X) converging to the p-primary homotopy groups of the K-completion X_{K}^{∧} of a space X. The second author wishes to call this the BTSS. Our theorems are proved by using Adams operations in K-theory to compute what would be the 1-line of the BTSS of B(ρ_{j}), which is defined to be the total space of a fibration

S^{2n+1} → B(ρ_{j}) → S^{2n+8j+1}  (1.8)

with attaching map ρ_{j}, if such a fibration existed. Our algebraic calculation has the desired homotopy-theoretic implication, even if the fibration does not exist.

The odd-primary analogue of the composition (1.2) was considered in [4] and [17], and results obtained in a large family of cases. In Section 6 we give complete results at the odd primes, following exactly the same methods that we employ at the prime

4 BENDERSKY AND DAVIS

2. The odd-primary case is slightly simpler because the d_{3}-differentials in the BTSS which make the 2-primary case complicated to state are not present at the odd primes.

In Section 7, we determine compositions of elements which suspend to elements of stable ImJ which are not necessarily generators. This generalizes our main theorem, 1.3, and is not implied by it, since the multiples of generators are defined on smaller spheres than are the generators.

In [12], homotopy-theoretic methods were used to determine the 2-primary v_{1}- periodic homotopy groups of F_{4}/G_{2}, which, by [15], is an S^{15}-bundle over S^{23} with attaching map ρ_{1}. The results obtained there are not consistent with those obtained in this paper for such a sphere bundle. The reason for the discrepancy is apparently a mistake in an aspect of the proof in [12] which was also used in computing the v_{1}-periodic homotopy groups of G_{2} in [15]. The results for G_{2} in [15] are correct, but the proof requires some modification. These matters will be discussed in Section 5.

We would like to thank Pete Bousfield for a detailed reading of a portion of this paper, and for allowing us to incorporate work which has not yet been published. Also, we would like to thank Johns Hopkins University and their JAMI Program for supporting both authors during the development and writing of this paper.

2. Outline of proof

In this section, we outline the proof of Theorem 1.3. Indeed, in this section we reduce the proof of this theorem to a certain calculation in the BTSS. This calculation is then performed in Section 4. The required relationship of the BTSS with compositions is established in Section 3.

We begin by discussing the relevant features of the spectral sequence for odddimensional spheres.

Theorem 2.1. Let s = 1 or 2, j ≥ 1, and n > 2.

1. E_{2}^{s,2n+8j+1}(S^{2n+1}) ≈ Z/2^{min(n,ν(j)+4)}.

2. The double suspension

E_{2}^{s,2n+8j−1}(S^{2n−1}) → E_{2}^{s,2n+8j+1}(S^{2n+1}) is multiplication by 2 of isomorphic groups if s = 2 and n > ν(j) + 4, and is injective otherwise.

COMPOSITIONS IN HOMOTOPY OF SPHERES 5

3. There is an isomorphism

v_{1}^{−1}π_{∗}(S^{2n+1}) ≈ v_{1}^{−1}π_{∗}((S^{2n+1})_{K}^{∧})

and an exact sequence 0 → K_{1} ⊕ K_{2} → v_{1}^{−1}π_{2n+8j+1−s}(S^{2n+1}) → E_{2}^{s,2n+8j+1}(S^{2n+1}) → C → 0,

with

C = \begin{cases}Z/2 & if n ≡ 1, 2 mod 4 and (s = 2 or n ≤ ν(j) + 4) \\ 0 & otherwise \end{cases}
K_{1} = \begin{cases}Z/2 & if n ≡ 1, 2 mod 4 and s = 2 and n > ν(j) + 4 \\ 0 & otherwise \end{cases}
K_{2} = \begin{cases}Z/2 & if n ≡ 0, 3 mod 4 and s = 1 \\ 0 & otherwise. \end{cases}

The extension is nontrivial into K_{1}, and K_{2} is split.

4. The localization π_{i}((S^{2n+1})_{K}^{∧}) → v_{1}^{−1}π_{i}((S^{2n+1})_{K}^{∧}) is bijective if i > 2n + 1.

The reader may find the following charts of the BTSS of S^{2n+1} helpful in understanding this theorem, and also, perhaps, the proof of Theorem 1.3 presented later in this section. These charts are similar to those presented in [7, p.488] and [3, p.58]. The primary difference is that those presented E_{2}, while these present E_{∞} when n ≡ 0, 3 mod 4, and a subquotient of E_{3} after most of the d_{3}-differentials have been taken into account when n ≡ 1, 2 mod 4. That the charts of [7] and [3] were of the v_{1}-periodic UNSS while the ones here are of the BTSS is inconsequential, since the two spectral sequences are isomorphic in dimension > 2n + 1.

The horizontal coordinate t − s corresponds to the homotopy group, dots are Z/2, integers are cyclic groups of the indicated order, diagonal lines of slope 1 are multiplication by η, and dotted vertical lines are extensions (multiplication by 2). In these charts ν = min(n, ν(j) + 4), and the dotted d_{3}-differential occurs iff ν = n.

Diagram 2.2.

[figure: 4; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 2^{ν}; 2^{ν}; t − s = 2n + 8j − 7+; 1; 2; 3; 4; 5; 6; 7; 8; S^{2n+1}, n ≡ 0, 3 mod 4; s = 1; 2; 3; 4; 4; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·]

Diagram 2.3.

[figure: 4; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 2^{ν}; 2^{ν}; t − s = 2n + 8j − 7+; 1; 2; 3; 4; 5; 6; 7; 8; S^{2n+1}, n ≡ 1, 2 mod 4; s = 1; 2; 3; 4; 8; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·]

Proof of Theorem 2.1. There is also a v_{1}-periodic BTSS, denoted v_{1}^{−1}E_{r}^{s,t}(X), with the localization performed as in [3]; it is isomorphic to the (unlocalized) BTSS in t − s > dim(X). In [8, 5.2] it is proved that, for any prime p, the mod p v_{1}-periodic BTSS of S^{2n+1} is isomorphic to the mod p v_{1}-periodic unstable Novikov spectral sequence

COMPOSITIONS IN HOMOTOPY OF SPHERES 7

(UNSS). (The BTSS is an unstable Adams spectral sequence based on periodic Ktheory, while the UNSS is an unstable Adams spectral sequence based on BP -theory.) A Bockstein spectral sequence argument implies that the two v_{1}-periodic spectral sequences are isomorphic (integrally). Note that [8, 5.2] just deals with E_{2}, but since there is a morphism of spectral sequences which is an isomorphism of E_{2}-terms, it is an isomorphism of spectral sequences. Thus parts 1 and 2 follow from computations made in [3] of the 2-primary v_{1}-periodic UNSS.

The first part of part 3 follows since the periodic UNSS converges to v_{1}^{−1}π_{∗}(S^{2n+1}), while the periodic BTSS converges to v_{1}^{−1}π_{∗}((S^{2n+1})_{K}^{∧}), and the spectral sequences are isomorphic. The second part of part 3 follows from results about the periodic UNSS in [3], as depicted in our charts here. Cases of C = Z/2 correspond to a nonzero d_{3}-differential on E_{2}^{s}, and cases of K_{1} = Z/2 correspond to a nontrivial extension into E_{2}^{s+2}. The K_{2} = Z/2 when n ≡ 0, 3 is the element in filtration 3. Part 4 follows from the isomorphism of the BTSS and periodic BTSS above dim(X).

If K_{∗}X is a free commutative algebra, then E_{2}^{s,t}(X) ≈ Ext_{U}^{s}(P K_{∗}S^{t}, P K_{∗}X), where U is an abelian category of unstable K_{∗}K-comodules.([8, 4.9]) We abbreviate Ext_{U}(P K_{∗}S^{t}, M) as Ext_{U}^{s,t}(M). Let α_{i/e} = −d(v_{1}^{i})/2^{e} ∈ K_{2i}(K) be as in [8, §5].

Definition 2.4. Assume n ≥ ν(j) + 4 and e ≥ 0. Let ϕ_{j} = α_{4j/ν(j)+4}. Let M_{n,j,e} be the U-object which is a free K_{∗}-module on generators g_{n} and g_{n+4j} with |g_{i}| = 2i + 1 and coaction

ψ(g_{n}) = 1 ⊗ g_{n}, ψ(g_{n+4j}) = 1 ⊗ g_{n+4j} + 2^{e}ϕ_{j} ⊗ g_{n}.

Proposition 2.5. With the notation and hypotheses of Definition 2.4, then 1. ϕ_{j}ι_{2n+1} generates E_{2}^{1,2n+1+8j}(S^{2n+1}), and

2. If the fibration (1.8) exists, then P K_{1}(B(ρ_{j})) ≈ M_{n,j,0}.

Proof. 1. In the BP -based UNSS, the generator of E_{2}^{1,2n+1+8j}(S^{2n+1}) described in [5, 9.10]^{2} is ±d(v_{1}^{4j} + 2^{ν(j)+3}v_{1}^{4j−3}v_{2})/2^{ν(j)+4}. In the K-based UNSS, the correction term

2Actually, [5, 9.10] contains a misprint in the exponent; the result here is what it should have said.

8 BENDERSKY AND DAVIS

2^{ν(j)+3}v_{1}^{4j−3}v_{2} is not needed to obtain maximal divisibility. To see this, we note that

K_{∗}K ≈ v_{1}^{−1}BP_{∗}BP/(v_{i}, η_{R}v_{i} : i > 1).

The relation η_{R}v_{2} implies that v_{1}^{2}h_{1} + v_{1}h_{1}^{2} is divisible by 2. On the other hand, d(v_{1}^{4j})/2^{ν(j)+3} = ((v_{1}−2h_{1})^{4j}−v_{1}^{4j})/2^{ν(j)+3} ≡ v_{1}^{4j−3}(v_{1}^{2}h_{1}+v_{1}h_{1}^{2}) mod 2, and hence this is divisible by 2 in K_{∗}K.

2. If e is as in (2.8), then 2^{e}ϕ_{j} survives the BTSS to an element 2^{e}ρ_{j} ∈ v_{1}^{−1}π_{2n+8j}(S^{2n+1}). To see this, note that e = 1 iff C = Z/2 in Theorem 2.1 when s = 1 iff there is a nonzero differential in the BTSS emanating from E_{3}^{1,2n+1+8j}(S^{2n+1}). It is immediate that if (1.8) exists, then P K_{1}(B(ρ_{j})) ≈ M_{n,j,0} because the K_{∗}K-coaction detects the attaching map.

Regardless of whether or not the fibration exists, algebraic analysis yields information about topological compositions by the following result, whose proof appears in Section 3.

Theorem 2.6. If n ≥ ν(j) + 4 and e ≥ 0, there are exact sequences

0 → E_{2}^{1,t}(S^{2n+1}) → Ext_{U}^{1,t}(M_{n,j,e}) → E_{2}^{1,t}(S^{2n+1+8j}) ⟶^{∂_{e}} E_{2}^{2,t}(S^{2n+1}), (2.7)

where ∂_{e}(xι_{2n+1+8j}) = x ⊗ 2^{e}ϕ_{j}ι_{2n+1}. Let i : π_{∗}(S^{N}) → π_{∗}((S^{N})_{K}^{∧}) denote the completion. If xι_{2n+1+8j} represents i(Σ^{2}τ) with τ ∈ π_{t−3}(S^{2n+8j−1}), and if

e = \begin{cases}1 & if n ≡ 1, 2 mod 4, and n = ν(j) + 4 \\ 0 & otherwise, \end{cases}  (2.8)

then ∂_{e}(xι_{2n+1+8j}) represents the composite i(2^{e}ρ_{j}∘Στ) ∈ π_{t−2}(S^{2n+1})_{K}^{∧} ≈ v_{1}^{−1}π_{t−2}(S^{2n+1}).

The compositions we have to compute lie in filtration 2 in the BTSS. Theorem 2.6 implies that the order of Ext_{U}^{1,t}(M_{n,j,e}) determines the order of the composition class. The following result is an algebraic analogue of [6, 1.1] and [6, 2.2] which reduces the calculation of the 1-line of the BTSS to a calculation involving Adams operations.

Theorem 2.9. For any positive integers n, j, and k, there is an isomorphism

Ext_{U}^{1,2n+8j+8k+1}(M_{n,j,0}) ≈ ⟨G, G′ : R_{2}, R′_{2}, R_{3}, R′_{3}⟩^{#}, (2.10)

COMPOSITIONS IN HOMOTOPY OF SPHERES 9

where

R_{t} = (t^{n+4j+4k} − t^{n})G + t^{n}(t^{4j} − 1)/2^{−ν(j)−4}G′
R′_{t} = (t^{n+4j+4k} − t^{n+4j})G′,

and (−)^{#} denotes the Pontryagin dual.

Proof. Because the argument is so similar to that of [6, 1.1], we merely sketch. We begin with a result similar to [6, 2.2] for a K_{∗}K-module M which is free as a K_{∗}- module. The proof is essentially the same as that of [6, 2.2], and is omitted.

We say that s⊗y ∈ K_{∗}K⊗M is unstable if s is in the K_{∗}-span of {h^{I} : 2 ∑ i_{j} < |y|}.

Proposition 2.11. An element s ⊗ y ∈ K_{∗}K ⊗ M_{2d+1} ⊗ Q is (integral and ) unstable if and only if t^{d}⟨ψ^{t}, s⟩ ∈ Z_{(2)} for all integers t.

We define U(M) ⊂ K_{∗}K ⊗ M to be the span of unstable elements. As in [6], we have

Ext_{U}^{1,2t+1}(M) ≈ ker(d̄: M_{2t+1} ⊗ Q/Z → K_{∗}K ⊗ M ⊗ Q/U(M)), where d̄ is induced by the boundary d. We apply 2.11 to determine ker(d̄) when M = M_{n,j,0} and t = n + 4j + 4k. A basis for M in this grading is {v_{1}^{4j+4k}g_{n}, v_{1}^{4k}g_{n+4j}} with

d̄(v_{1}^{4j+4k}g_{n}) = ((η_{R}v_{1})^{4j+4k} − v_{1}^{4j+4k}) ⊗ g_{n}
d̄(v_{1}^{4k}g_{n+4j}) = ((η_{R}v_{1})^{4k} − v_{1}^{4k}) ⊗ g_{n+4j} + v_{1}^{4k}((η_{R}v_{1})^{4j} − v_{1}^{4j})2^{−ν(j)−4} ⊗ g_{n}.

By [9, p.676] ⟨ψ^{t}, η_{R}(v_{1}^{i})⟩ = t^{i}v_{1}^{i} and ⟨ψ^{t}, v_{1}^{i}⟩ = v_{1}^{i}, and hence, by 2.11, ker(d̄) is the intersection over all integers t of the kernel of the morphism of free Q/Z_{(2)}-modules with matrix

A_{t} = (\begin{matrix}t^{n}(t^{4j+4k} − 1) & t^{n}(t^{4j} − 1)2^{−ν(j)−4} \\ 0 & t^{n+4j}(t^{4k} − 1) \end{matrix})

By [6, 2.5], this intersection is Pontryagin dual to the abelian group presented by all matrices A_{t} stacked. By an argument, presented in the next paragraph, similar to that of [6, 3.9], the relations from A_{−1}, A_{2} and A_{3} imply all other relations, but A_{−1} = 0, completing the proof of Theorem 2.9.

Although the rows of A_{t} may not be t^{n+4j+4k} − ψ^{t} acting on P K^{1}(X) for a topological space X, they behave as if they were:

10 BENDERSKY AND DAVIS

i: Analogous to [2, 5.1], we verify directly that A_{t} ≡ A_{t+m^{e}} mod

m^{e}/2^{ν(j)+4}; ii: Transformations ψ̃^{t} defined by ψ̃^{t}(x_{1}) = t^{n}x_{1}−t^{n}(t^{4j}−1)2^{−ν(j)−4}x_{2},

ψ̃^{t}(x_{2}) = t^{n+4j}x_{2} satisfy ψ̃^{s}ψ̃^{t} = ψ̃^{st}. Let s be odd. Use [1, 2.9] to write s = (−1)^{ϵ}3^{ℓ} + c2^{N} with ϵ = 0 or 1, c ∈ Z, and N sufficiently large. By (i), mod 2^{N−ν(j)−4}, A_{s} ≡ A_{(−1)^{ϵ}3^{ℓ}} = (−1)^{ϵn}A_{3^{ℓ}}. The relations in A_{3} say that ψ̃^{3} = 3^{n+4j+4k} on the group presented by A_{3}. Thus by (ii), ψ̃^{3^{ℓ}} = (3^{n+4j+4k})^{ℓ} on this group. Thus the rows of A_{3^{ℓ}}, and hence of A_{s}, are consequences of those of A_{3}. Similarly, using (ii), the relations in A_{2s} are a consequence of those in A_{2} and those in A_{s}.

In Section 4, we shall calculate the right hand side of (2.10), obtaining the following result.

Theorem 2.12. Let m = m_{j,k} = max(ν(j) + 4, ν(k) + 4 − 4j). Then

Ext_{U}^{1,2m+8j+8k+1}(M_{m,j,0}) ≈ Z/2^{ν(k)+4+f(j,k)},

where

f(j, k) = \begin{cases}0 & if ν(k) ≠ 4j + ν(j) \\ min(ν(e(j, k)), ν(j) + 4) & if ν(k) = 4j + ν(j), \end{cases}

with e(j, k) as in Theorem 1.3.

Proof of Theorem 1.3. We first assume ν(k) < 4j + ν(j). We will first prove that, under this assumption, v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) generates v_{1}^{−1}π_{N+8j+8k−2}(S^{N}) when

N = 2m_{j,k} + \begin{cases}1 & if m_{j,k} ≡ 0, 3 mod 4 \\ 2 & if m_{j,k} ≡ 2 mod 4 \\ 3 & if m_{j,k} ≡ 1 mod 4 \end{cases}  (2.13)

is the smallest integer such that ϕ(N + 3) ≥ m_{j,k} + 3.

The exact sequence (2.7) with n = m_{j,k}, e = 0, and t = 2n + 8j + 8k + 1 is

0 → Z/2^{ν(j+k)+4} → Z/2^{ν(k)+4} → Z/2^{ν(k)+4} ⟶^{∂_{0}} Z/2^{ν(j+k)+4}

unless ν(k) = ν(j), in which case ν(j + k) should be replaced by ν(j) twice in the exact sequence. This easily implies that ∂_{0} is surjective (e.g., by Euler characteristic). We emphasize that it is here that the calculation of Ext_{U}(M_{n,j,0}) is being used.

COMPOSITIONS IN HOMOTOPY OF SPHERES 11

Now let e be as in (2.8), with n and t as in the previous paragraph. By the previous paragraph, ∂_{e} in (2.7) is multiplication by 2^{e}. We apply Theorem 2.6 with xι_{2n+1+8j} representing 2^{a} times a generator of E_{2}^{1,t}(S^{2n+1+8j}), subject to the condition that this element is the double suspension of a homotopy class which suspends to 2^{a}ρ_{k}. This is possible since n + 4j ≥ ν(k) + 4, which implies that E_{2}^{1,2n+8j+8k+1}(S^{2n+1+8j}) surjects under stabilization. Using Theorem 2.1, we find that this can be done (i.e., 2^{a} times a generator of E_{2}^{1,2n+8j+8k+1}(S^{2n+1+8j}) is the double suspension of a permanent cycle) if

a = \begin{cases}2 & if ν(j) + 4j ≤ ν(k) and m_{j,k} ≡ 2, 3 mod 4 \\ 1 & if ν(j) + 4j ≤ ν(k) and m_{j,k} ≡ 0, 1 mod 4 \\ 1 & if ν(j) + 4j = ν(k) + 1 and m_{j,k} ≡ 2, 3 mod 4 \\ 0 & otherwise. \end{cases}

We deduce from 2.6 that v_{1}^{−1}(2^{e}ρ_{j} ∘ 2^{a}ρ_{k}) is represented by 2^{e+a} times a generator of v_{1}^{−1}E_{2}^{2,2m_{j,k}+8j+8k+1}(S^{2m_{j,k}+1}).

If m_{j,k} ≡ 0, 3 mod 4, then (by 2.1.3) the E_{2}^{2}-generator represents the generator of v_{1}^{−1}π_{2m_{j,k}+8j+8k−1}(S^{2m_{j,k}+1}), and so, since we can divide by 2^{e+a} in this group of order divisible by 2^{4}, v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) generates the homotopy group in S^{N} with N = 2m_{j,k} + 1 in this case, as desired. If m_{j,k} ≡ 1 mod 4, then the E_{2}-generator does not yield a homotopy class, but by 2.1 (or see [7, p.483]) its double suspension generates v_{1}^{−1}π_{N+8j+8k−2}(S^{N}) with N = 2m_{j,k} + 3, and so we deduce that v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) generates this group in this case, as desired. Similar considerations apply when m_{j,k} ≡ 2 mod 4, except that only a single suspension is required in order that the E_{2} generator yield a homotopy generator (This can be deduced from [16]), and so ρ_{j} ∘ ρ_{k} generates in S^{N} with N = 2m_{j,k} + 2 in this case.

The case of an arbitrary value of N in Theorem 1.3 now follows from the fact that

v_{1}^{−1}π_{N+8j+8k−3}(S^{N−1}) ⟶^{Σ} v_{1}^{−1}π_{N+8j+8k−2}(S^{N}) is surjective if N + 3 ≢ 0, 1, 2, 4 mod 8,

and hits the multiples of 2 if N +3 ≡ 0, 1, 2, 4 mod 8. This is probably best seen in the table at the top of page 483 of [7], with an obvious intermediate tower interpolated when N ≡ 6 mod 8.

Now assume ν(k) > 4j + ν(j). The entire argument above goes through, except that S^{N+8j+8k−2} ⟶^{ρ_{k}} S^{N+8j−1} is not defined. One more suspension is required. That is why, in this case, N in (1.5) must be 1 greater than its value in (2.13). The

12 BENDERSKY AND DAVIS

statement in the previous paragraph about the effect on one suspension on the v_{1}- periodic (8i − 2)-stem implies the minor modifications required in the proof in this case.

The situation when ν(k) = ν(j) + 4j is exactly the same except that ∂_{0} in (2.7) is multiplied by e(j, k) because of Theorem 2.12 and the Euler characteristic argument. There is, however, a problem that if e + a is as in the preceding paragraphs and if 2^{e+a}e(j, k) is 0 in Z/2^{ν(j)+4}, we cannot divide by 2^{e+a} to make an assertion about the image of ρ_{j} ∘ ρ_{k}. This causes the exceptional case in Theorem 1.3.

Proposition 2.14. Let

E = \begin{cases}1 & if ν(j) ≡ 0 mod 4 \\ 2 & if ν(j) ≡ 1, 3 mod 4 \\ 3 & if ν(j) ≡ 2 mod 4. \end{cases}

If ν(k) = ν(j) + 4j and ν(e(j, k)) ≥ ν(j) + 4 − E, we can only assert in Theorem 1.3 that v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) is a multiple of 2^{ϕ(N+3)−(m_{j,k}+3)+ν(e(j,k))} (and not that it equals this 2-power times a generator).

We close this section by discussing the special case ρ_{1} ∘ ρ_{1}. Theorem 1.3 and (1.7) say v_{1}^{−1}(ρ_{1} ∘ ρ_{1}) is a generator of v_{1}^{−1}π_{23}(S^{9}) and is 0 in v_{1}^{−1}π_{N+14}(S^{N}) for N ≥ 21. Yet ρ_{1} ∘ ρ_{1} is nonzero in π_{N+14}(S^{N}) for all N ≥ 9.

An explanation for this is given by noting that the v_{1}-localization for S^{2n+1} can be obtained as the composite (see [18] or [12])

π_{2n+8i−1}(S^{2n+1}) ⟶^{s_{∗}} J_{2n+8i−1}(Σ^{2n+1}P^{2n}) ⟶^{v} v_{1}^{−1}J_{2n+8i−1}(Σ^{2n+1}P^{2n}) ≈ v_{1}^{−1}π_{2n+8i−1}(S^{2n+1}).

By [18, 1.5], s_{∗} is surjective for 4i − ν(i) ≥ n + 8. The morphism v is bijective if 4i−2 ≥ n; for smaller values of i, v_{1}^{−1}J_{2n+8i−1}(Σ^{2n+1}P^{2n}) involves elements of negative filtration, which are not present in J_{2n+8i−1}(Σ^{2n+1}P^{2n}). The class ρ_{1} ∘ ρ_{1} is present and nonzero in filtration 2 in J_{2n+15}(Σ^{2n+1}P^{2n}) for n ≥ 4, but for n ≥ 10 it is killed by a d_{3}-differential from filtration −1 in v_{1}^{−1}J_{2n+8i}(Σ^{2n+1}P^{2n}).

This is the only case of ρ_{j} ∘ ρ_{k} for which this happens—that ρ_{j} ∘ ρ_{k} is nonzero in J_{2n+8j+8k−1}(Σ^{2n+1}P^{2n}) while 0 in v_{1}^{−1}J_{2n+8j+8k−1}(Σ^{2n+1}P^{2n}). To avoid such a situation, we need (roughly)

4(j + k) − 2 ≥ m_{j,k} + ν(j + k) + 5, which is satisfied unless j = k = 1.

3. Compositions in the BTSS

In this section, we prove Theorem 2.6. We begin with some background on pairings in the BTSS.

Let {E_{r}^{s,t}X } be the homotopy spectral sequence of an augmented cosimplicial space X. ([11]) For a space X, a K-based cosimplicial space KX is constructed in [8]. The BTSS of X, {E_{r}^{s,t}(X)}, is the homotopy spectral sequence of KX.

We wish to show that the Yoneda pairing of E_{2}-terms of the BTSS induces a pairing of spectral sequences which survives to the composition pairing of homotopy groups. One problem is that the BTSS converges to the homotopy groups of the K-completion of X rather than those of X itself. More importantly, the cosimplicial space used to construct the spectral sequence in [8] is not termwise abelian. The required generalization of the construction in [11] will appear in [10]. In order to understand why one needs this generalization, we first review the constructions and results of [11].

If X is a simply-connected space and R is a commutative ring, then RX is the cosimplicial space obtained by resolving X with respect to R. The R-completion of X, X_{R}^{∧}, is defined to be Tot(RX). The Bousfield-Kan spectral sequence, {E_{r}RX}, converges to the homotopy groups of X_{R}^{∧}.

In [11, §10], an action

E_{r}^{s,t+m}RX ⊗ E_{r}^{s′,t′}RS^{m} ⟶^{∗} E_{r}^{s+s′,t+t′}RX

is constructed for t − s ≥ 1. This action survives to a pairing

π_{t+m−s}X_{R}^{∧} ⊗ π_{t′−s′}(S^{m})_{R}^{∧} ⟶^{∗} π_{t+t′−s−s′}X_{R}^{∧}

for t − s ≥ 1. In [11, 10.2], it is shown that this pairing is compatible with the composition pairing in homotopy, in the sense that, with i : X → X_{R}^{∧} the natural map, there is a commutative diagram

π_{t+m}X ⊗ π_{t′}S^{m}  ⟶^{∘} π_{t+t′}X
↓^{i_{∗}⊗i_{∗}}  ↓^{i_{∗}}
π_{t+m}X_{R}^{∧} ⊗ π_{t′}(S^{m})_{R}^{∧} ⟶^{∗} π_{t+t′}X_{R}^{∧}

14 BENDERSKY AND DAVIS

In [11, §18], this pairing is shown to correspond to the Yoneda preoduct.

The results of [11] do not immediately apply to our situation. The pairing ∗ depends on a map c, constructed in [11, 9.1], between cosimplicial spaces. That c is a map of cosimplicial spaces uses the fact that RX is termwise abelian. Since the cosimplicial space in [8] is not termwise abelian, we cannot use the results of [11]. However, Bousfield has shown in [10] that there is a pairing

E_{r}^{s,t+m}X ⊗ E_{r}^{s′,t′}S^{m} ⟶^{∗} E_{r}^{s+s′,t+t′}X

(with t − s ≥ 1 and r ≥ 2) which generalizes the pairing in [11] to the K-theory situation. Specializing to odd spheres, we have the following result.

Theorem 3.1. The ∗-pairing is compatible with the ∘-pairing in the sense that the following diagram commutes.

π_{t+2m+1}S^{2n+1} ⊗ π_{t′}S^{2m+1}  ⟶^{∘} π_{t+t′}S^{2n+1} ⟶^{v_{1}^{−1}} v_{1}^{−1}π_{t+t′}S^{2n+1}
i⊗i↓  ^{i}↓  ^{i}↓
π_{t+2m+1}(S^{2n+1})_{K}^{∧} ⊗ π_{t′}(S^{2m+1})_{K}^{∧} ⟶^{∗} π_{t+t′}(S^{2n+1})_{K}^{∧} ⟶^{≈} v_{1}^{−1}π_{t+t′}(S^{2n+1})_{K}^{∧}

We apply this theorem to the element 2^{e}ρ_{j} ⊗ Στ of Theorem 2.6, and deduce that the element i(2^{e}ρ_{j} ∘ Στ) in the conclusion of that theorem equals

i(2^{e}ρ_{j}) ∗ i(Στ).  (3.2)

Now we interpret the pairing in the BTSS in terms of the usual pairing on Ext. This follows since K_{∗}S^{2n+1} is isomorphic to a free commutative algebra, so that, as noted prior to 2.4, E_{2}^{s,t}(S^{2n+1}) ≈ Ext_{U}^{s}(K_{∗}S^{t}, K_{∗}S^{2n+1}).

Lemma 3.3. The ∗ pairing corresponds to the Yoneda product

Ext_{U}^{s}(K_{∗}S^{t+2m+1}, K_{∗}S^{2n+1}) ⊗ Ext_{U}^{s′}(K_{∗}S^{t′}, K_{∗}S^{2m+1})

→ Ext_{U}^{s}(K_{∗}S^{t+2m+1}, K_{∗}S^{2n+1}) ⊗ Ext_{U}^{s′}(K_{∗}S^{t+t′}, K_{∗}S^{t+2m+1})
→ Ext_{U}^{s+s′}(K_{∗}S^{t+t′}, K_{∗}S^{2n+1}).

Proof. This follows as in [11, §18], using the cosimplicial pairings constructed in [10].

COMPOSITIONS IN HOMOTOPY OF SPHERES 15

In the notation of Theorem 2.6, the homotopy pair (3.2) comes from the E_{2} pair 2^{e}ϕ_{j}ι ∗ xι. The value of e given in (2.8) is required in order that 2^{e}ϕ_{j} is a permanent cycle in the BTSS. By Lemma 3.3, this ∗ product of E_{2} classes is the Yoneda product of Ext elements. The relationship of this Yoneda product with the boundary in the exact sequence (2.7) is standard, and is formalized and proved in [21, 2.3.4].

Indeed the sequence (2.7) is induced from the short exact sequence in U

0 → K_{∗}S^{2n+1} → M_{n,j,e} → K_{∗}S^{2n+1+8j} → 0,  (3.4)

and [21, 2.3.4] asserts that the boundary morphism in (2.7) is given by Yoneda product with the element of Ext^{1} corresponding to the extension (3.4). Our expression of this boundary as x ⊗ 2^{e}ϕ_{j}ι is easily seen by the usual way of seeing such a boundary morphism by diagram-chasing.

4. The calculation

In this section we prove Theorem 2.12 by computing the right hand side of (2.10) with n = m_{j,k}. We begin by letting n be arbitrary. We remove terms which are divisible by 2^{n+4j+4k}, for, as we shall see, the groups will be of order smaller than this. The four relations, after multiplying by unit factors, are

2^{n}G − 2^{n−4−ν(j)}(2^{4j} − 1)G′

2^{n+4j}G′
(3^{4(j+k)} − 1)G + (3^{4j} − 1)2^{−4−ν(j)}G′
(3^{4k} − 1)G′

We use that ν(3^{i} − 1) = ν(i) + 2 if i is even, and note that the coefficient of G′ in the third relation is a unit. We use the third relation to eliminate G′. With the second and fourth relations combined, we have two relations (on G)

2^{n} + 2^{n}(2^{4j} − 1)(3^{4j+4k} − 1)/(3^{4j} − 1)  (4.1)
2^{ν(j+k)+4+min(n+4j,ν(k)+4).}  (4.2)

Multiplying (4.1) by the unit (3^{4j} − 1)/2^{ν(4j)+4} yields 2^{n−4−ν(j)}(3^{4j} − 1 + (2^{4j} − 1)(3^{4j+4k} − 1)).

16 BENDERSKY AND DAVIS

Expanding 3^{4i} − 1 = (80 + 1)^{i} − 1 brings this to the form

5 · 2^{n−ν(j)} ∑_{i≥1} 80^{i−1}(\binom{j}{i} + (2^{4j} − 1)\binom{j+k}{i}).  (4.3)

The 2-exponent of our desired group is the smaller of the exponents of (4.2) and (4.3) when n = m_{j,k}. Considering several cases, one verifies that if ν(k) ≠ 4j + ν(j), then the exponent of (4.3) is ν(k)+4, while that of (4.2) is larger. If ν(k) = 4j +ν(j), then the exponent of (4.2) is ν(j)+ν(k)+8, while that of (4.3) is ν(e(j, k))+ν(k)+4, as claimed.

5. Discussion of previous work on G_{2} and F_{4}/G_{2}

In an attempt to determine v_{1}^{−1}π_{∗}(F_{4}/G_{2}), the second author proposed a proof in [12, 8.15] that v_{1}^{−1}(ρ_{1}∘ρ_{k}) is 8 times a generator of v_{1}^{−1}π_{8k+21}(S^{15}) ≈ Z/16. According to our Theorem 1.3, its actual coefficient is

\begin{cases}8 & if ν(k) < 4 \\ 0 & if ν(k) = 4 \\ 2^{7−ν(k)} & if 5 ≤ ν(k) ≤ 6. \end{cases}

The composite is not defined on S^{15} if ν(k) > 6.

The proof of [12, 8.15] relied on the existence of a certain map

Ω^{∞}(Σ^{2n+1}P^{2n} ∧ J) → (S^{2n+1})_{K}  (5.1)

when n = 7. Such a map had been produced when n = 2 in [15, 4.6]. It appears that the maps (5.1) must not exist, because of the contradiction that they imply for ρ_{1} ∘ ρ_{k} when ν(k) ≥ 4, and many other similar composites. The method of [12] was similar to that employed by Mahowald and Thompson in [20], except that [20] did not use maps (5.1). If maps (5.1) exist, this method applied to many composites ρ_{j} ∘ ρ_{k} disagrees with Theorem 1.3. The results of [20] about ρ_{j} ∘ ρ_{k} did not deal with cases where existence of (5.1) yields results that contradict our Theorem 1.3.

One aspect of the proof of [15, 4.6] seems incomplete. This is apparently the cause of the problem. A diagram

Ω^{∞}(Σ^{2n+1}P ∧ J) ⟶ Ω^{∞}(Σ^{2n+1}P_{2n+1} ∧ J)
↓  ↓
Ω^{∞}(Σ^{∞}S^{2n+1})_{K} ⟶ Ω^{∞}(Σ^{2n+1}P_{2n+1})_{K}

COMPOSITIONS IN HOMOTOPY OF SPHERES 17

is constructed. Its commutativity was tacitly assumed. The bottom map was induced from the K-localization of the Snaith map QS^{2n+1} → QΣ^{2n+1}P_{2n+1}. It is probably not an infinite loop map. So there seems no clear way to establish commutativity of this diagram.

The (apparently false) result [15, 4.6] was used in computing a family of higher differentials in an Adams-type spectral sequence converging to v_{1}^{−1}π_{∗}(G_{2}). There were two possibilities for this family of differentials, yielding two possible results for v_{1}^{−1}π_{∗}(G_{2}). Another approach to v_{1}^{−1}π_{∗}(G_{2}) is via the BTSS. Our expertise with the BTSS is not yet at the level where we can determine, strictly within BTSS technology, the entire BTSS for G_{2}. However, we can use [6, 1.1] to calculate the 1-line, and the result of this calculation is consistent with the picture of v_{1}^{−1}π_{∗}(G_{2}) claimed in [15], and is not consistent with the alternate picture. Thus we assert that [15, 1.3], the explicit listing of the groups v_{1}^{−1}π_{i}(G_{2}), is correct, even though the argument there apparently had a gap.

Indeed, we have that

E_{2}^{1,2m+1}(G_{2}) ≈ (P K^{1}(G_{2})/ im(ψ^{2}, ψ^{3} − 3^{m}))^{#}.

Using software LiE as in [13], we find that P K^{1}(G_{2})_{(2)} has basis {w_{1}, w_{2}} satisfying ψ^{k}(w_{1}) = kw_{1} + \frac{1}{2}(k − k^{5})w_{2} and ψ^{k}(w_{2}) = k^{5}w_{2}. This agrees with results of [23, 2.5]. From this, we obtain

E_{2}^{1,4i+3}(G_{2}) ≈ Z/2^{3+min(ν(i−2),3)}.

These groups are isomorphic to the groups v_{1}^{−1}π_{4i+2}(G_{2}) asserted in [15, 1.3]. A picture of v_{1}^{−1}π_{∗}(G_{2}) is presented on [15, p.666]. The two possibilities there are whether a differential d is nonzero for even values of k or for odd values of k. In either case, the fact that the element in the top of the tower is divisible by η^{2} implies that in the BTSS there must be an extension from the 1-line group to a Z/2 on the 3-line. In order to make the E_{2}-group and the homotopy group have the same order, there must be a compensating d_{3}-differential on the generator of the 1-line group. On the other hand, there is no way to create a BTSS picture with 1-line as computed which yields the picture of [15, p.666] with the opposite pattern for d.

18 BENDERSKY AND DAVIS

6. The odd-primary analogue

The exact same methods which were applied at the prime 2 yield similar results when applied at an odd prime p. In this section, p is an odd prime, q = 2(p − 1), and ν(−) is the exponent of p in an integer. Let ρ_{i} denote a map (or its homotopy class) which suspends to a generator of the image of the stable J-homomorphism in the p-primary (qi − 1)-stem.

It was shown in [4] that v_{1}^{−1}π_{2n+qi−1}(S^{2n+1}) ≈ Z/p^{min(n,ν(i)+1)}, and

π_{2n+qi−1}(S^{2n+1}) ⟶^{v_{1}^{−1}} v_{1}^{−1}π_{2n+qi−1}(S^{2n+1})

is split surjective if i − ν(i) ≥ n + 1. Our theorem at the odd prime p, analogous to Theorem 1.3 is

Theorem 6.1. Let n_{j,k} = max(ν(j) + 1, ν(k) + 1 − (p − 1)j). Then, by [5], S^{2n_{j,k}+1} is the smallest odd sphere on which ρ_{j} ∘ ρ_{k} is defined. If n ≥ n_{j,k}, then v_{1}^{−1}(ρ_{j} ∘ ρ_{k}) is 2^{n−n_{j,k}+ν(e(j,k))} times a generator of v_{1}^{−1}π_{2n+qj+qk−1}(S^{2n+1}), where

e(j, k) = \begin{cases}1 & if ν(k) ≠ (p − 1)j + ν(j) \\ \frac{1}{k} ∑_{i≥1}(αp)^{i−1}(\binom{j}{i} + (p^{(p−1)j} − 1)\binom{j+k}{i}) & if ν(k) = (p − 1)j + ν(j), \end{cases}

where α ≢ 0 mod p satisfies αp + 1 = r^{p−1} with r a generator of (Z/p^{2})^{×}.

Again it is true that ρ_{j} ∘ ρ_{k} generates the v_{1}-periodic summand the first time it is defined, provided ν(k) ≠ (p − 1)j + ν(j). Also note that one can easily read off when the v_{1}-periodic component of ρ_{j} ∘ ρ_{k} is 0, but cannot usually deduce that ρ_{j} ∘ ρ_{k} is in fact 0. The proof of Theorem 6.1 is a direct adaptation of our proof in the 2-primary case, and is omitted.

In [4, 4.1], the first author used the UNSS to prove Theorem 6.1 when ν(k) < (p − 1)j + ν(j).

7. Compositions of non-generators

In this section, we show how our main theorem generalizes to compositions of any elements which suspend to elements in the image of J, not necessarily the generators. Results for multiples of the generators are not usually implied by results for the

COMPOSITIONS IN HOMOTOPY OF SPHERES 19

generators because the multiples are defined on smaller spheres than are the generators, and the double suspension homomorphism is often not injective on the unstable homotopy groups where these compositions lie.

For simplicity, we just treat the odd-primary case here, and we do not deal with borderline cases which cause coefficients such as e(j, k) of Theorem 6.1 to arise. We just write what the composite is on the smallest sphere on which it is defined. Results for these composites on larger spheres are, of course, then determined since the double suspension induces multiplication by 2.

We use the notation that α_{j/e} is an element of order p^{e} in the (qj −1)-stem. This is defined provided e ≤ ν(j) + 1. Then ρ_{j} of Section 6 is α_{j/ν(j)+1}. The following result generalizes Theorem 6.1. Composites of this sort were considered in a limited range of cases in [4] and [17].

Theorem 7.1. The smallest odd sphere S^{2n_{0}+1} on which α_{j/e} ∘ α_{k/f} is defined has n_{0} = min(e, f − (p − 1)j + 1). The composite v_{1}^{−1}(α_{j/e} ∘ α_{k/f}) equals p^{c_{1}+c_{2}} times a generator of v_{1}^{−1}π_{qj+qk+2n_{0}−1}(S^{2n_{0}+1}), where

c_{1} = max(0, min(ν(k) − (p − 1)j, ν(j)) + 1 − e)
c_{2} = max(0, min(ν(k) + 1, e + (p − 1)j − 1) − f),

provided ν(k) ≠ ν(j) + (p − 1)j.

Note that if α_{j/e} = ρ_{j}, then c_{1} = 0, and if α_{k/f} = ρ_{k}, then c_{2} = 0, and so the result agrees with Theorem 6.1, which states that the composite of elements which suspend to generators of ImJ is a generator the first time it is defined.

The proof is similar to that of Theorem 6.1, which was similar to that of Theorem 1.3. We define an unstable K_{∗}K-comodule M which depends on n_{0}, j, and e, and fits into an exact sequence

0 → E_{2}^{1,t}(S^{2n_{0}+1}) → Ext_{U}^{1,t}(M) → E_{2}^{1,t}(S^{2n_{0}+1+qj}) ⟶^{∂} E_{2}^{2,t}(S^{2n_{0}+1}).

We work with t_{0} = 2n_{0}+1+qj +qk. Then α_{k/f} is represented by p^{c_{2}} times a generator of E_{2}^{1,t_{0}}(S^{2n_{0}+1+qj}). Similarly to Theorem 2.12, we compute the order of Ext_{U}^{1,t_{0}}(M). The group is not always cyclic. It turns out that

| Ext_{U}^{1,t_{0}}(M)|/|E_{2}^{1,t_{0}}(S^{2n_{0}+1+qj})| = p^{c_{1}}.

It is elementary to conclude that ∂ sends a generator to p^{c_{1}} times a generator. By an analogue of Theorem 2.6, ∂ sends α_{k/f} to α_{j/e} ∘ α_{k/f}, yielding Theorem 7.1.

References

[1] J. F. Adams, On the groups J(X)-II, Topology 3 (1965) 137-171.

[2] , On the groups J(X)-III, Topology 3 (1965) 193-222.

[3] M. Bendersky, The v_{1}-periodic unstable Novikov spectral sequence, Topology 31 (1992) 47-64.

[4] , Unstable towers in the odd primary homotopy groups of spheres, Trans Amer Math Soc 287 (1985) 529-542.

[5] M. Bendersky, E.B. Curtis, and H.R. Miller, The unstable Adams spectral sequence for generalized homology, Topology 17 (1978) 229-248.

[6] M. Bendersky and D. M. Davis, The 1-line of the K-theory Bousfield-Kan spectral sequence for Spin(2n + 1), to appear.

[7] , 2-primary v_{1}-periodic homotopy groups of SU(n), Amer Jour Math 114 (1991) 465-494.

[8] M. Bendersky and R. D. Thompson, The Bousfield-Kan spectral sequence for periodic homology theories, to appear in Amer Jour Math.

[9] J. M. Boardman, Stable cohomology operations in generalized cohomology theories, Handbook of Algebraic Topology, Elsevier (1995) 585-686.

[10] A. K. Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, in preparation.

[11] A. K. Bousfield and D. M. Kan, Pairings and products in the homotopy spectral sequence, Trans Amer Math Soc 177 (1973) 319-343.

[12] D. M. Davis, Computing v_{1}-periodic homotopy groups of spheres and some compact Lie groups, Handbook of Algebraic Topology, Elsevier (1995) 1001-1056.

[13] , From representation theory to homotopy groups, to appear.

[14] D. M. Davis and M. Mahowald, Some remarks on v_{1}-periodic homotopy groups, London Math Society Lecture Notes, 176 (1992) 55-72.

[15] , Three contributions to the homotopy theory of the exceptional Lie groups G_{2} and F_{4}, Jour Math Soc Japan 43 (1991) 661-671.

[16] , v_{1}-periodic homotopy of Sp(2), Sp(3), and S^{2n}, Springer Lecture Notes 1418 (1990) 219-237.

[17] B. Gray, Unstable families related to the image of J, Math Proc Camb Phil Soc 96 (1984) 95-113.

[18] M. Mahowald,The image of J in the EHP sequence, Annals of Math 116 (1982)

65-112.

[19] , Descriptive homotopy of the image of J, Proc of Int Conf on Manifolds, Tokyo Univ Press, (1976) 245-253.

[20] M. Mahowald and R. Thompson, Unstable compositions related to the image of J, Proc Amer Math Soc 102 (1988) 431-436.

[21] D. R. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press (1986).

[22] P. Selick, Space exponents for loop spaces of spheres, Fields Institute Comm 19 (1998) 279-283.

[23] T. Watanabe, Adams operations in the connective K-theory of compact Lie groups, Osaka Jour Math 23 (1986) 617-632.

Hunter College, CUNY, NY, NY 10021

E-mail address: mbenders@shiva.hunter.cuny.edu

Lehigh University, Bethlehem, PA 18015

E-mail address: dmd1@lehigh.edu