TRACTABLE FORMULAS FOR v_{1}-PERIODIC HOMOTOPY GROUPS OF SU(n) WHEN n ≤ p^{2} − p + 1

DONALD M. DAVIS AND HUAJIAN YANG

1. Statement of results

Let p be a fixed odd prime. The (p-local) v_{1}-periodic homotopy groups, v_{1}^{−1}π_{∗}(X), of a space X were defined in [9]. Roughly speaking, they tell the portion of the p-local homotopy groups of X detected by K-theory. They are the second of an envisioned sequence of periodic homotopy theories, rational homotopy being the first, which would give a global description of homotopy theory.

The v_{1}-periodic homotopy groups are particularly nice for spherically resolved spaces. A space is spherically resolved if it can be built from a finite number of spheres and their iterated loop spaces by fibrations. One property which will be important to us is that for a spherically resolved space X each v_{1}-periodic homotopy group v_{1}^{−1}π_{i}(X) is a direct summand of some actual homotopy group π_{L}(X). We will focus on the case where X is one of the special unitary groups SU(n), which are spherically resolved by the fibrations

SU(n − 1) → SU(n) → S^{2n−1}. (1.1)

Let ν(m) = ν_{p}(m) denote the exponent of p in m. Let

e_{p}(k, n) = min{ν_{p}(j!S(k, j)) : n ≤ j ≤ k}, (1.2)

where S(k, j) is the Stirling number of the second kind, which satisfies

j!S(k, j) = ∑_{i=1}^{j}(−1)^{j−i}\binom{j}{i}i^{k}. (1.3)

The following result was proved in [8].

1991 Mathematics Subject Classification. 55T15.

Key words and phrases. v_{1}-periodic homotopy groups, unitary groups, Stirling numbers, unstable Novikov spectral sequence.

Theorem 1.4. If k ≥ n, then v_{1}^{−1}π_{2k}(SU(n)) ≈ Z/p^{e_{p}(k,n)}, while v_{1}^{−1}π_{2k−1}(SU(n)) has order p^{e_{p}(k,n)}, but is not necessarily cyclic.

Periodicity in v_{1}^{−1}π_{∗}(SU(n)) would allow one to determine v_{1}^{−1}π_{2k}(SU(n)) for smaller or negative values of k from this, if one wished.

Theorem 1.4 at first glance appears to be all that one might want to know about v_{1}^{−1}π_{∗}(SU(n)). However, it has two drawbacks. One is that it only gives the orders of the odd groups, and not their actual structure, and the other has to do with the intractability of the formula (1.2).

The numbers e_{p}(k, n) which occur in Theorem 1.4 are in some sense given explicitly in (1.2), and some of them can be computed by a computer, but it seems to be very difficult to obtain useful general formulas for them from (1.2). By a careful study of binomial coefficients, Lundell ([10]) obtained results which agree with our Theorem 1.5 as long as the value of e_{p}(k, n) is less than p, but this condition is quite restrictive. The only other result was obtained in [7], where it was shown using Fermat’s Little Theorem that max{e_{p}(k, n) : k ∈ Z} ≥ n − 1. In [8], this was interpreted as a lower bound for the p-exponent of SU(n), which we mention later as (1.7).

The first main result of this paper is a tractable formula for v_{1}^{−1}π_{2k}(SU(n)), provided n ≤ p^{2} − p + 1.

Theorem 1.5. Suppose p is odd, k = N + (p − 1)m with 1 ≤ N < p, and

N + 1 + (p − 1)i ≤ n < N + 1 + (p − 1)(i + 1)

with 0 ≤ i ≤ p − 1. Define m̂ by 0 ≤ m̂ < p and m ≡ m̂ mod p. Then v_{1}^{−1}π_{2k}(SU(n)) ≈ Z/p^{e}, where e is equal to

i + 1  if i < N and i < m̂
i  if N ≤ i and (i < m̂ or m̂ = 0)
min(N + (p − 1)m̂, i + ν(m − m̂) + 1)  if i < N and m̂ ≤ i
min(N + (p − 1)m̂ + 1,
i + ν(m − m̂ + (−1)m̂m̂\binom{N}{m̂}p^{m̂(p−1)})) if N ≤ i and 1 ≤ m̂ ≤ i

The smallest n for which this theorem fails to give complete information is n = N + 1 + (p − 1)i with N = 1 and i = p.

FORMULAS FOR v_{1}-PERIODIC HOMOTOPY GROUPS 3

Table 1. Values of e when p = 11, N = 7, m̂ = 3

i
0 1 2 3 4 5 6 7 8 9 10
ν ≤ 28  1 2 3 ν + 4 ν + 5 ν + 6 ν + 7 ν + 7 ν + 8 ν + 9 ν + 10
ν = 29  1 2 3 33  34  35  36  36  37  38  38
m ≡ 3 + 6p^{30} mod p^{31} 1 2 3 34  35  36  37  38  38  38  38
ν = 30, m ≢ 3 + 6p^{30} 1 2 3 34  35  36  37  37  38  38  38
ν = 31  1 2 3 35  36  37  37  37  38  38  38
ν = 32  1 2 3 36  37  37  37  37  38  38  38
ν ≥ 33  1 2 3 37  37  37  37  37  38  38  38

For example, the results when p = 11, m̂ = 3, and N = 7 say that if m ≡ 3 mod 11 and 8 ≤ j < 18, then

v_{1}^{−1}π_{20m+14}(SU(10i + j)) ≈ Z/p^{e},

where e is as in Table 1, in which ν = ν(m − 3).

Many of the results of Theorem 1.5 were conjectured based on computer analysis of (1.2) when p = 7. However, the proof of Theorem 1.5 makes no use of formula (1.2); it is an independent calculation of v_{1}^{−1}π_{∗}(SU(n)). These separate calculations of the same homotopy group give a topological proof of a result in number theory, a tractable evaluation of

min{ν_{p}(j!S(k, j)) : n ≤ j ≤ k}

when n ≤ p^{2} − p + 1, as given in Theorem 1.5. For example, the following corollary is easily obtained from Theorem 1.5 and (1.2).

Corollary 1.6. Suppose p is an odd prime, 0 ≤ m̂ ≤ i < N < p, and i + ν_{p}(ℓ) + 2 ≤ N + (p − 1)m̂. Then

ν_{p}(S(N + (p − 1)m̂ + (p − 1)pℓ, N + (p − 1)i)) = ν_{p}(pℓ).

We view this merely as an amusing offshoot of our work.

An application of Theorem 1.5 to homotopy theory is a slightly improved lower bound for the p-exponent of SU(n). Recall that the p-exponent of a space X, denoted exp_{p}(X), is defined to be the largest e such that some homotopy group of X contains an element of order p^{e}. As noted at the outset, if X is spherically resolved, its pexponent will be at least as large as that of the order of any element of v_{1}^{−1}π_{∗}(X). It was shown in [8] that

exp_{p}(SU(n)) ≥ n − 1. (1.7)

This can be improved by 1 for certain values of n, as given in the following corollary.

Corollary 1.8. If p is odd, then exp_{p}(SU(n)) ≥ n if, for some i < p,

i(p − 1) + 2 ≤ n ≤ ip + 1.

Proof. We use Theorem 1.5 to determine max{e_{p}(k, n) : k ∈ Z}. In the notation of Theorem 1.5, this maximum will occur when n = N + 1 + (p − 1)i and m̂ = i. This maximum will equal n − 1 if i < N, and n if i ≥ N. The expression in the corollary is obtained as N + 1 + (p − 1)i for 1 ≤ N ≤ i.

Results for v_{1}^{−1}π_{∗}(SU(n)) when p = 2 are somewhat more complicated. In [5], a result of the type of Theorem 1.4 was obtained, and a tractable form for it was given for n ≤ 7 using computer calculations.

Our other main result is an explicit determination of the groups v_{2k−1}(SU(n)) when n ≤ p^{2} − p + 1. Recall that Theorem 1.4 only gave their order.

Theorem 1.9. Use the notation of Theorem 1.5, and let t = min{N, ν(m) + 1}. Then

v_{1}^{−1}π_{2k−1}(SU(n)) ≈ \begin{cases}Z/p^{e} & if i ≤ N \\ Z/p ⊕ Z/p^{e−1} & if i > N and m̂ ≠ 0 \\ Z/p^{t} ⊕ Z/p^{i−t} & if i > N and m̂ = 0 and t ≤ i − N \\ Z/p^{N} ⊕ Z/p^{i−N} & if i > N and m̂ = 0 and t ≥ i − N \end{cases}

A description of the conditions under which these groups are cyclic can be given without resorting to all the special notation of Theorem 1.5. The following result is easily obtained from Theorem 1.9.

Corollary 1.10. Let \overline{k} be defined by 1 ≤ \overline{k} ≤ p − 1 and \overline{k} ≡ k mod p − 1. Then v_{1}^{−1}π_{2k−1}(SU(n)) is cyclic if n < (\overline{k} + 1)p, and is the direct sum of two cyclic summands if

(\overline{k} + 1)p ≤ n ≤ p^{2} − p + 1.

FORMULAS FOR v_{1}-PERIODIC HOMOTOPY GROUPS 5

The proofs of both of our main theorems will involve a delicate analysis of the unstable Novikov spectral sequence (UNSS) based on BP. We let E_{2}^{s,t}(X) denote the E_{2}-term of this spectral sequence. The first part of Theorem 1.4 was an immediate consequence of the following result, the first part of which was proved in [3], and the second part in [8].

Theorem 1.11. (1) If k ≥ n, then E_{2}^{1,2k+1}(SU(n)) ≈ Z/p^{e_{p}(k,n)}.

(2) If k ≥ n, then v_{1}^{−1}π_{2k}(SU(n)) ≈ E_{2}^{1,2k+1}(SU(n)).

Theorem 1.5 is proved by computing E_{2}^{1,2k+1}(SU(n)) by increasing induction on n. By contrast, Theorem 1.11(1) was proved by computing E_{2}^{1,2k+1}(SU(n)) by downward induction on n, starting with E_{2}^{1,2k+1}(SU(k)) ≈ Z/k!. The methods of calculation of the UNSS used in proving Theorems 1.5 and 1.9 extend those of [6]; we think that the methods introduced here of calculating this spectral sequence for multicell complexes should be useful for other computations.

In Section 2, we present the requisite background on the UNSS. In Section 3, we outline the proof of Theorem 1.5, with details relegated to Section 4. In Section 5, we prove Theorem 1.9.

2. Background in the UNSS

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_{∗}[t_{1}, t_{2}, …], where t_{i} are Quillen’s generators. We have |v_{i}| = |t_{i}| = 2(p^{i} −1). Let c : BP_{∗}(BP) → BP_{∗}(BP) be the conjugation, and define h_{i} = c(t_{i}). Then Γ = BP_{∗}[h_{1}, h_{2}, …]. 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 ⊗_{BP_{∗}} M. The stable cobar complex C(T (M)) is defined as follows:

C^{s}(T (M)) = Γ ⊗_{BP_{∗}} Γ ⊗ ⋯ ⊗_{BP_{∗}} Γ ⊗_{BP_{∗}} M,

with s copies of Γ. We use the cobar notation, and write γ_{0} [γ_{1}| … |γ_{s}] m for γ_{0} ⊗ γ_{1} ⊗ … ⊗ γ_{s} ⊗ m. The differential d is given by

d(γ_{0}[γ_{1}| ⋯ |γ_{s}]m) = ∑ γ0^{′} [γ0^{′′}| ⋯ |γ_{s}] m

+∑_{j=1}^{s}(−1)^{j}γ_{0} [γ_{1}| ⋯ |γj^{′}|γj^{′′}| ⋯ |γ_{s}] m + (−1)^{s+1} ∑ γ_{0} [γ_{1}| ⋯ |γ_{s}|γ′] m′′ (2.1)

where ψ(γ_{i}) = ∑ γj^{′} ⊗ γj^{′′} for 0 ≤ j ≤ s and ψ_{M}(m) = ∑ γ′ ⊗ m′′.

The unstable cobar complex {C^{∗}(U(M)), d} is a subcomplex of {C^{∗}(T (M)), d}, consisting of terms satisfying an unstable condition, introduced in the following definition.

Definition 2.2. [4, p. 243] If M is a nonnegatively graded free left A-module, then U(M) is defined to be the BP_{∗}-span of

{h^{I} ⊗ m : 2(i_{1} + i_{2} + i_{3} + ⋯) < |m|} ⊂ Γ ⊗_{BP_{∗}} M,

where I = (i_{1}, i_{2}, …) is a sequence containing only finitely many nonzero i_{j}’s, and h^{I} = h_{1}^{i_{1}}h_{2}^{i_{2}} · · ·.

Define U^{s}(M) = U(U^{s−1}(M)), and C^{s}(U(M)) = U^{s}(M). If M is a Γ-comodule, then U(M) ⊂ T (M) = Γ ⊗ M as a sub-Γ-comodule, and so the differential d of the stable cobar complex {C^{s}(T (M)), d} induces a differential on {C^{s}(U(M))} which defines a complex {C^{s}(U(M)), d}. We will usually replace it by the chain-equivalent reduced complex obtained by replacing U(M) by ker(U(M)⟶^{ϵ}M). 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_{U}^{s,t}(M).

It was proved in [4] that, 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, and if the integral cohomology H^{∗}(X) is a free algebra, then

E_{2}^{s,t}(X) = Ext_{U}^{s,t}(P (BP_{∗}X)),

where P (BP_{∗}X) denotes the sub-Γ-comodule of BP_{∗}X consisting of the primitives under the coproduct. This is the UNSS for the space X.

The following basic formulas were used or proved in [6].

Lemma 2.3. (1) v_{1} = ph_{1} + η(v_{1}) and ψ(h_{1}) = h_{1} ⊗ 1 + 1 ⊗ h_{1};

(2) v_{2} = ph_{2} + (1 − p^{p−1})h_{1}^{p}v_{1} + η(v_{2}) − (p + 1)v_{1}^{p}h_{1} + ∑_{i=2}^{p} a_{i}v_{1}^{p+1−i}p^{i}h_{1}^{i}, where a_{i} ∈ Z; (3) d(v_{1}) = η(v_{1}^{n}) − v_{1}^{n} and

d(v^{a}h^{b}v^{c}) = (η(v^{a}) − v^{a}) ⊗ h^{b}v^{c} − v^{a}ψ̄(h^{b})v^{c} − v^{a}h^{b} ⊗ (η(v^{c}) − v^{c})

where ψ̄(h^{b}) = ψ(h^{b}) − h^{b} ⊗ 1 − 1 ⊗ h^{b}.

The first part of this lemma will be used very frequently in the context of replacing ph_{1} by v_{1} − η(v_{1}).

Let ∑^{F^{∗}} h_{i} = c(∑^{F} c(h_{i})) = c(∑^{F} t_{i}), where x +_{F} y is the formal group sum defined by x +_{F} y = exp(log x + log y) with log x = ∑_{i≥0} m_{i}x^{i+1}, exp x = ∑_{i≥0} b_{i}x^{i+1}, and exp(log x) = log(exp x) = x. Here {m_{i}} and {b_{i}} are two different polynomial generators sets for H_{∗}BP with |m_{i}| = |b_{i}| = 2(p^{i} − 1). Then the following lemma of Bendersky ([3]) is useful.

Lemma 2.4. The primitives P (BP_{∗}(SU(n)) form a free BP_{∗}-module generated by elements {x_{3}, x_{5}, x_{7}, ..., x_{2n−1}} with coaction given by

ψ(x_{2k+1}) = ∑_{j}(\begin{matrix}∑ & F^{∗} \\ i \end{matrix} h_{i})_{k−j}^{j} ⊗ x_{2j+1}.

The subscript k − j refers to the component in grading q(k − j). Here we have introduced the notation q = 2(p − 1), which will be used frequently throughout the paper.

We will need the following explicit computation.

Proposition 2.5. Mod terms of degree greater than 3q,

\begin{matrix}∑ & F^{∗} \\ i \end{matrix} h_{i} = 1 + h_{1} − h_{1}v_{1} + h_{1}v_{1}^{2} − \frac{p−1}{2}h_{1}^{2}v_{1}.

The only terms h_{1}^{j} which appear in ∑^{F^{∗}} h_{i} are 1 + h_{1}.

Proof. From

x = exp(log x)
= b_{0} log x + b_{1}(log x)^{2} + b_{2}(log x)^{3} + ⋯
= b_{0} ∑_{i≥0} m_{i}x^{p^{i}} + b_{1}(∑_{i≥0} m_{i}x^{p^{i}})^{2} + b_{2}(∑_{i≥0} m_{i}x^{p^{i}})^{3} + ⋯,

we deduce that the first nonzero b_{i}’s are b_{0} = 1, b_{p−1} = −m_{1}, b_{2p−2} = pm_{1}^{2}, and b_{3p−3} = \binom{p}{2} − p(2p − 1). Then, mod terms of degree greater than 3q,

\begin{matrix}∑ & F^{∗} \\ i \end{matrix} h_{i} = c(\begin{matrix}∑ & F \\ i \end{matrix} t_{i})
= c(∑_{i≥0} b_{i}( ∑_{j,k≥0} m_{j}t_{k}^{p^{j}})^{i+1})
= c(1 + m_{1} + t_{1} − m_{1}(1 + m_{1} + t_{1})^{p} + pm_{1}^{2}(1 + m_{1} + t_{1})^{2p−1}
+(\binom{p}{2} − p(2p − 1))m_{1}^{3})
= c(1 + m_{1} + t_{1} − m_{1}(1 + pm_{1} + pt_{1} + \binom{p}{2}(m_{1}^{2} + t_{1}^{2}) + p(p − 1)m_{1}t_{1})
+pm_{1}^{2}(1 + (2p − 1)(m_{1} + t_{1})) + (\binom{p}{2} − p(2p − 1))m_{1}^{3}))
= c(1 + t_{1} − pm_{1}t_{1} + p^{2}m_{1}^{2}t_{1} − \binom{p}{2}m_{1}t_{1}^{2}).

The desired result is now obtained using v_{1} = pm_{1} and c(v_{1}) = η(v_{1}).

The second statement follows since the only powers of t_{1} appearing in the expansion of ∑ b_{i}(∑ m_{j}t_{k}^{p^{j}})^{i+1} are 1 + t_{1}.

This coaction formula in SU(n) will be extremely important, as it determines the boundary homomorphism in the exact sequence associated to the fibration (1.1). Indeed, there is an exact sequence

→ E_{2}^{s,t}(SU(n − 1)) → E_{2}^{s,t}(SU(n)) → E_{2}^{s,t}(S^{2n−1})⟶^{∂}E_{2}^{s+1,t}(SU(n − 1)) →,

with

∂(y) = y ⊗ ψ(x_{2n−1}). (2.6)

This boundary formula is true since ψ gives the coboundary for the unstable cobar complex of SU(n). The term 1 ⊗ x_{2n−1} will not appear in our reduced complex.

We prefer to work with the v_{1}-periodic UNSS of [1]. This has the advantage that for X = SU(n) or S^{2n+1},

(2.7) E_{2}^{s,t}(X) = \begin{cases}v_{1}^{−1}π_{t−s}(X) & if s = 1 or 2, and t odd \\ 0 & otherwise. \end{cases}

We can still work with unstable cobar names for classes; we just view the periodic spectral sequence as a direct summand of the actual UNSS, after multiplication by sufficiently many v_{1}’s. So from now on, E_{2} refers to the periodic spectral sequence.

We will make frequent use of the following result for the spheres. The 1-line part was proved in [4], and the 2-line part in [2]. 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 2.8. (1) The only nonzero groups E_{2}^{s,t}(S^{2n+1}) are

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}.

(3) Suppose n ≤ ν(m) + 1. Then for 1 ≤ j ≤ n, there is an element of order p^{j} in E_{2}^{2,2n+1+qm}(S^{2n+1}) which equals v_{1}^{m−j−1}h_{1} ⊗ h_{1}^{j} mod S^{2j−1}.

(4) If ν(m) + 1 ≤ n, then d(v_{1}^{m−n−1}h_{1}^{n+1})ι_{2n+1} has order p in E_{2}^{2,2n+1+qm}(S^{2n+1}), and, for 0 ≤ j ≤ ν(m), p^{j} times a generator of E_{2}^{2,2n+1+qm}(S^{2n+1}) equals v_{1}^{m−n+j−1}h_{1} ⊗ h_{1}^{n−j} mod S^{2(n−j)−1}.

(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.

The formulas for leading terms of classes on the 2-line in parts 3 and 4 differ from those in [2], which have v_{1}^{e} on the right of the first h_{1}. We write them as we have because that is the way in which they will occur in our applications. To see that h_{1} ⊗ v_{1}^{e}h_{1}^{j} ≡ v_{1}^{e}h_{1} ⊗ h_{1}^{j} mod S^{2j−1}, we use Lemma 2.3(1) to write the difference as ∑_{i=1}^{e−1} \binom{e}{i}p^{i}h_{1}^{i+1}v_{1}^{e−i}h_{1}^{j}. The ith term is a multiple of p^{i−1}h_{1}^{i+1}v_{1}^{e−i}h_{1}^{j−1} = ∑ \binom{i−1}{j}v_{1}^{j}h_{1}^{2}v_{1}^{e−j}h_{1}^{j−1}, which is defined on S^{2j−1}. The following corollary encapsulates a common feature of parts 3 and 4 of the above theorem. We use the term “leading term” to mean the term which desuspends least far.

Corollary 2.9. If an element of E_{2}^{2}(S^{2n+1}) has order p^{f} and leading term h⊗h^{j}ι_{2n+1}, then

j + ν(|E_{2}^{2}(S^{2n+1})|) = f + n.

We will also need the following result about the 2-line for spheres, proved in [2].

Lemma 2.10. Let E_{2}^{s}(M) = Ext_{Γ}^{s}(BP_{∗}, BP_{∗}/p) denote the E_{2}-term of the stable Novikov spectral sequence for the mod p Moore spectrum M. There is a homomorphism

H′ : E_{2}^{s,t}(S^{2n+1}) → E_{2}^{s−1,t−2pn−1}(M)

satisfying H′(x) = 0 if and only if x double desuspends, and if x = z ⊗ h_{1}^{n}ι_{2n+1} with zι_{2pn−1} defined, then H′(x) = z.

We will need the following fact about E_{2}(M), proved in [6, 2.11].

Lemma 2.11. v_{1}^{k}h_{1} ≠ 0 ∈ E_{2}^{1}(M) and v_{1}h_{1}^{k(p−1)+1} = v_{1}^{k(p−1)+1}h_{1} in E_{2}^{1}(M). The second statement is also true unstably mod p.

The following result will be very useful.

Lemma 2.12. (1) h_{1}^{n+p−1} ⊗ h_{1}ι_{2n+1} ≡ −v_{1}^{p−1}h_{1} ⊗ h_{1}^{n}ι_{2n+1} mod S^{2n−1}. (2) h_{1}^{n+p−1} ⊗ h_{1}ι_{2n+1} double desuspends iff n = 1.

Proof. We use [6, 2.13], which says that

(h_{1}^{p} ⊗ h_{1}^{K(p−1)+n} + h_{1}^{K(p−1)+p+n−1} ⊗ h_{1})v_{1}^{K}ι_{2n+1}

is the sum of a boundary and a class which double desuspends, where n ≥ 1 and K ≥ 0. As noted in [6, 2.14], this gives immediately that h_{1}^{p}⊗h_{1}ι_{3} double desuspends, since 2 is a unit. It also gives

h_{1}^{n+p−1} ⊗ h_{1}ι_{2n+1} ≡ −h_{1}^{p} ⊗ h_{1}^{n}ι_{2n+1} mod S^{2n−1}.

Thus, if n > 1, H′(h_{1}^{n+p−1} ⊗ h_{1}ι_{2n+1}) = −h_{1}^{p} by Lemma 2.10. By Lemma 2.11, this equals −v_{1}^{p−1}h_{1} ≠ 0, so that h_{1}^{n+p−1} ⊗ h_{1}ι_{2n+1} does not double desuspend. However,

H′(h_{1}^{n+p−1} ⊗ h_{1}ι_{2n+1} + v_{1}^{p−1}h_{1} ⊗ h_{1}^{n}ι_{2n+1}) = −v_{1}^{p−1}h_{1} + v_{1}^{p−1}h_{1} = 0,

and so the sum double desuspends.

The following lemma will also be used many times. It is the one place where the number of v_{1}’s on the left is important. We begin a policy of writing h for h_{1}, and v for v_{1}.

Lemma 2.13. For j ≥ 0,

d(v^{ℓ}h^{n+1}v^{j}) ≡ −(ℓ + n + 1)v^{ℓ}h ⊗ h^{n}v^{j} + jv^{ℓ+1}h^{n} ⊗ v^{j−1}h mod S^{2n−1−qj}.

Proof. We evaluate the desired boundary using Lemma 2.3, expanding

d(v^{a}) = (v − ph)^{a} − v^{a},

to obtain

∑_{i=1}^{ℓ} \binom{ℓ}{i}v^{ℓ−i}(−ph)^{i} ⊗ h^{n+1}v^{j} − ∑_{i=1}^{n} \binom{n+1}{i}v^{ℓ}h^{i} ⊗ h^{n+1−i}v^{j}
− ∑_{i=1}^{j} \binom{j}{i}v^{ℓ}h^{n+1} ⊗ v^{j−i}(−ph)^{i}.

For each term in each sum, we study whether it satisfies the unstable condition on S^{2n−1−qj} (number of h’s ≤ \frac{1}{2} degree of stuff to the right of it) for both the part on the left side of the ⊗ and the part on the right side of the ⊗. In the first sum, h^{i} on the left of the ⊗ satisfies the unstable condition on any sphere since p^{i} with it can be used (via ph = v − ηv) to make the exponent of h small. Terms with i > 1 can use p^{2} to bring the h^{n+1} on the right side of ⊗ down to h^{n−1}; then h^{n−1}v^{j} is defined on S^{2n−1−qj}. The (i = 1)-term is, mod S^{2n−1−qj},

−ℓv^{ℓ−1}h ⊗ (v − ηv)h^{n}v^{j} ≡ −ℓv^{ℓ−1}(v − ph)h ⊗ h^{n}v^{j} ≡ −ℓv^{ℓ}h ⊗ h^{n}v^{j}.

All terms in the second sum with i > 1 are defined on S^{2n−1−qj}. In the third sum, terms with i > 1 can use p^{2} to bring h^{n+1} down to h^{n−1}, so that it satisfies the unstable condition. Remaining p’s can be used to bring h^{i} down to h^{2}, so that it satisfies the unstable condition. The term with i = 1 gives the second term in the lemma since ph^{n+1} = vh^{n} − h^{n}v.

The following corollary will be useful.

Corollary 2.14. If k + n ≢ 0 mod p, and z = (v^{k}h^{n+p−1} ⊗ h + pw_{0})ι_{2n+1} is a cycle with w_{0}ι_{2n+1} satisfying the unstable condition 2.2, then

z = d(\frac{1}{k+n+p}v^{k+p−1}h^{n+1}ι_{2n+1} + wι_{2n+1})

with wι_{2n+1} unstable.

Proof. Lemmas 2.12 and 2.13 imply that the leading term is correct. The group E_{2}^{2n+1+q(k+n+p)}(S^{2n+1}) has order p, and by Lemma 2.12 and Theorem 2.8(4), both of these leading terms yield nonzero classes. Since the other terms pw_{0} are p times an unstable class, they cannot cancel the leading term, and the tail terms on the second representation must be an unstable boundary.

Our final preliminary is to introduce the spaces which are the factors in the decomposition of [11] of the p-localization of SU(n) and its quotients as a product of p − 1 spaces.

Proposition 2.15. ([11]) There are spaces X_{j}^{i}(N) for 1 ≤ N ≤ p − 1 and 0 ≤ j ≤ i satisfying

(1) X_{j}^{j}(N) = S^{2N+1+jq};

(2) there is a fibration X_{j}^{i−1}(N) → X_{j}^{i}(N) → S^{2N+1+iq};

(3) H^{∗}(X_{j}^{i}) is an exterior algebra with a single generator of each dimension 2N + 1 + kq for j ≤ k ≤ i;

(4) SU(n)/SU(m) ≃_{(p)} ∏_{N=1}^{p−1}X_{d(m,N)+1}^{d(n,N)}(N), where d(n, N) = [\frac{n−1−N}{p−1} ].

The UNSS of X_{j}^{i}(N) has E_{2}-term Ext_{U}^{s,t}(P (BP_{∗}(X_{j}^{i}(N)))), where P (BP_{∗}(X_{j}^{i}(N))) has BP_{∗}-basis {x_{2N+jq+1}, x_{2N+(j+1)q+1}, …, x_{2N+iq+1}}, with coaction induced from Lemma 2.4. Because of the sparseness results for spheres given in (2.7) and Theorem 2.8, the following is immediate.

Proposition 2.16. (1) For s = 1 or 2, v_{1}^{−1}π_{2k+1−s}(X_{j}^{i}(N)) ≈ E_{2}^{s,2k+1}(X_{j}^{i}(N)),

and is 0 unless k ≡ N mod p − 1.

(2) In the notation of Theorems 1.5 and 1.9 with s = 1 or 2,

v_{1}^{−1}π_{2k+1−s}(SU(n)) ≈ v_{1}^{−1}π_{2k+1−s}(X_{0}^{i}(N)).

Thus our efforts in the remainder of the paper will be to show that for s = 1 or 2, E_{2}^{s,2N+1+qm}(X_{0}^{i}(N)) is as stated in Theorems 1.5 and 1.9. The fibrations whose exact sequence in E_{2}(−) will be studied are of the form

X_{k}^{j}(N) → X_{k}^{i}(N) → X_{j+1}^{i}(N)

for k ≤ j < i.

3. The cellular spectral sequence

In this section, we sketch the proof of Theorem 1.5 by showing (modulo details postponed until Section 4) that E_{2}^{1,2N+1+qm}(X_{0}^{i}(N)) ≈ Z/p^{e}, where e is as in that theorem. Here X_{0}^{i}(N) is the space introduced at the end of the previous section, and all notation of Theorem 1.5 is in effect; in particular, N < p and i < p.

Throughout the remainder of the paper, we consider N and m to be fixed, so that they may be omitted from some notation. The expression E_{2}^{s}(−) will always mean E_{2}^{s,2N+1+qm}(−), and we let X_{j}^{i} = X_{j}^{i}(N). Note that X_{j}^{j} = S^{2N+1+qj}. Our goal is to determine E_{2}^{s}(X_{0}^{i}) for s = 1 and 2.

We can organize this computation of E_{2}^{s}(X_{0}^{i}) as a (cellular) spectral sequence with E_{1}^{s,j} = E_{2}^{s}(X_{j}^{j}) for 1 ≤ s ≤ 2 and 0 ≤ j ≤ i, and d_{r} : E_{r}^{1,j} → E_{r}^{2,j−r} induced by pulling the class in E_{2}^{1}(X_{j}^{j}) back to E_{2}^{1}(X_{j−r+1}^{j}) and then applying ⟶^{∂}E_{2}^{2}(X_{j−r}^{j−r}). Our desired E_{2}^{s}(X_{0}^{i}) is filtered with subquotients E_{∞}^{s,j}, 0 ≤ j ≤ i. For s = 1, we know E_{2}^{1}(X_{0}^{i}) is cyclic by Theorem 1.11, and so we just need to compute the sum

ν(|E_{2}^{1}(X_{0}^{i})|) = ∑_{j=0}^{i} ν(|E_{∞}^{1,j}|). (3.1)

Note that for 1 ≤ s ≤ 2

E_{1}^{s,j} = \begin{cases}Z/p & if j ≠ m̂ \\ Z/p^{min(N+(p−1)j,ν(m−j)+1)} & if j = m̂. \end{cases}

Thus every differential involves at least one group which is Z/p or 0, and so it suffices to tell which differentials are nonzero.

The precise form of this spectral sequence falls into three different patterns, determined by (1) m̂ = 0, (2) N < m̂, and (3) N ≥ m̂ > 0. We begin by calculating the cellular spectral sequence when m̂ = 0, this being the easiest of the three cases. It is easiest to describe the spectral sequence when i = p − 1, the maximal value that we are considering, and then to obtain the spectral sequence for smaller values of i by restriction.

Theorem 3.2. If m̂ = 0, then in the cellular spectral sequence for X_{0}^{p−1},

E_{∞}^{s,j} = E_{1}^{s,j} = \begin{cases}Z/p^{min(N,ν(m)+1)} & s = 1, j = 0 \\ Z/p & s = 1, 1 ≤ j < N − ν(m) \\ Z/p & s = 1, N < j ≤ p − 1 \\ Z/p & s = 2, 1 ≤ j ≤ p − 1. \end{cases}

Table 2. Spectral sequence when p = 11, N = 6, m̂ = 0

j
0 1 2 3 4 5 6 7 8 9 10
ν(m) 1 2 1 1 1 1 /1 /1 1 1 1 1
2 3 1 1 1 /1 /1 /1 1 1 1 1
3 4 1 1 /1 /1 /1 /1 1 1 1 1
4 5 1 /1 /1 /1 /1 /1 1 1 1 1
≥ 5 6 /1 /1 /1 /1 /1 /1 1 1 1 1

If max(1, N − ν(m)) ≤ j ≤ N, then Z/p ≈ E_{j}^{1,j}⟶^{d_{j}}E_{j}^{2,0} is nonzero. Hence E_{∞}^{2,0} = 0, and E_{∞}^{1,j} = 0 if max(1, N − ν(m)) ≤ j ≤ N.

If i < p − 1, then E_{∞}^{1,j}(X_{0}^{i}) will equal the group described in Theorem 3.2 if j ≤ i, and will equal 0 if j > i. The same holds for E_{∞}^{2,j}(X_{0}^{i}) if j > 0 or i ≥ N; if i < N, then E_{∞}^{2,0}(X_{0}^{i}) = Z/p^{min(N−i,ν(m)+1)}. The reader can easily verify that the sum (3.1) when applied to 3.2 yields the desired result

ν(|E_{2}^{1}(X_{0}^{i})|) = \begin{cases}i & if N ≤ i \\ min(N, i + ν(m) + 1) & if i < N \end{cases}

for the case m̂ = 0.

The proof of Theorem 3.2 will be given in Section 4. In Table 2, we illustrate the spectral sequence in the case p = 11, N = 6. For each value of j from 0 to p − 1, we list

ν(|E_{1}^{1,j}|) = ν(|E_{2}^{1,13+20m}(S^{13+20j})|).

A slash (/) through a number indicates that it supports a differential. In this case (m̂ = 0), all differentials hit into E^{2,0}.

Next we describe the cellular spectral sequence when N < m̂.

Proposition 3.3. Let ν = ν(m − m̂). If N < m̂, then in the spectral sequence for X_{0}^{i},

E_{∞}^{1,j} = \begin{cases}Z/p & if 0 ≤ j < N \\ 0 & if j = N \\ Z/p & if N < j < m̂ \\ Z/p^{min(m̂(p−2)+N+2,ν+1)} & if j = m̂ \\ Z/p & if m̂ < j ≤ m̂(p − 1) + N + 1 − ν \\ 0 & if j > m̂ and j > m̂(p − 1) + N + 1 − ν, \end{cases}

provided j ≤ i.

It is then an easy exercise to verify that the sum (3.1) has the desired value

i + 1  if i < N
i  if N ≤ i < m̂
min(m̂(p − 1) + N + 1, i + ν(m − m̂)) if m̂ ≤ i.

Thus Theorem 1.5 in the case N < m̂ will be proved once we have proved Proposition 3.3.

In Table 3, we illustrate the spectral sequence in a particular case, namely p = 11, N = 2, and m̂ = 7. Let ν = ν(m − m̂) = ν(m − 7). For each value of ν and each j from 0 to 10, we list

ν(|E_{1}^{1,j}|) = ν(|E_{2}^{1,5+20m}(S^{5+20j})|).

Directly beneath this number, we list a number k if there is a nonzero differential E^{1,j} → E^{2,k}. Thus for example if ν = 68, there are nonzero differentials

E_{2}^{1,2} → E_{2}^{2,0}, E_{4}^{1,7} → E_{4}^{2,3}, E_{6}^{1,7} → E_{6}^{2,1}, E_{4}^{1,8} → E_{4}^{2,4}, E_{4}^{1,9} → E_{4}^{2,5}, E_{4}^{1,10} → E_{4}^{2,6}.

In this case, we have

E_{r}^{1,7} ≈ \begin{cases}Z/p^{69} & if 1 ≤ r ≤ 4 \\ Z/p^{68} & if 5 ≤ r ≤ 6 \\ Z/p^{67} & if 7 ≤ r ≤ ∞. \end{cases}

Proposition 3.3 is an immediate consequence of the following result, which describes the nonzero differentials in the spectral sequence when N < m̂. In Section 4 we will prove this result and its analogues in the other cases. We will frequently omit writing the subscript of the differential and the E-groups. It just equals the difference of the second superscripts.

Theorem 3.4. If N < m̂, the nonzero differentials in the spectral sequence converging to E_{2}(X_{0}^{i}) are those described below on groups E^{1,j} with j ≤ i.

(1) d_{N} ≠ 0 : E_{N}^{1,N} → E_{N}^{2,0}.

16 D. DAVIS AND H. YANG

Table 3. Spectral sequence when p = 11, N = 2, m̂ = 7

j

0 1 2 3 4 5 6 7 8 9 10
ν ≤ 63 1 1 1 1 1 1 1 ν + 1 1 1 1

0

ν = 64 1 1 1 1 1 1 1 65 1 1 1
0  1
65 1 1 1 1 1 1 1 66 1 1 1
0  1 3
66 1 1 1 1 1 1 1 67 1 1 1
0  1 3 4
67 1 1 1 1 1 1 1 68 1 1 1
0  1 3 4 5
68 1 1 1 1 1 1 1 69 1 1 1
0  1,3 4 5 6
69 1 1 1 1 1 1 1 70 1 1 1
0  1,3,4 5 6 7
70 1 1 1 1 1 1 1 71 1 1 1
0  1,3,4,5 6 7 7
ν ≥ 71 1 1 1 1 1 1 1 72 1 1 1
0  1,3-6 7 7 7

(2) For t > 1, there is a nonzero differential from x_{m̂(p−2)+N+1+t} to z_{t}, where

x_{k} = \begin{cases}element of order p^{k} in E_{1}^{1,m̂} & if k ≤ ν(|E_{1}^{1,m̂}|) \\ generator of E_{1}^{1,m̂+Δ} & if Δ = k − ν(|E_{1}^{1,m̂}|) > 0 \end{cases}

z_{t} = \begin{cases}generator of E_{1}^{2,t−1} & if 1 < t ≤ N \\ generator of E_{1}^{2,t} & if N < t < m̂ \\ element of order p^{t−m̂+1} in E_{1}^{2,m̂} & if m̂ ≤ t \end{cases}

We take a little liberty with notation here; although the elements x_{k} and z_{t} are defined as elements of E_{1}, the differentials generally involve the projections of these elements in E_{r} with r > 1.

Next we present the analogues of Proposition 3.3 and Theorem 3.4 when N ≥ m̂ > 0. The condition

m ≡ m̂ − (−1)m̂m̂\binom{N}{m̂}p^{m̂(p−1)} mod p^{m̂(p−1)+1} (3.5)

will be important here.

Proposition 3.6. If N ≥ m̂ > 0, then the nonzero groups E_{∞}^{1,j} in the spectral sequence converging to E_{2}^{1}(X_{0}^{i}) are the following groups which also satisfy j ≤ i. Here ν = ν(m − m̂).

Z/p  j < m̂
Z/p^{min(ν+1,N+m̂(p−2))} j = m̂
Z/p  ν ≤ m̂(p − 1) and m̂ < j < N
Z/p m̂(p − 1) < ν ≤ m̂(p − 2) + N − 2
and m̂ < j < m̂(p − 1) + N − ν
Z/p  j = N + 1 and not (3.5)
Z/p  j = N and (3.5)
Z/p  ν < m̂(p − 1) and N < j ≤ m̂(p − 1) + N + 1 − ν

As in the previous two cases, one can easily show that the sum (3.1) when applied to the groups of 3.6 yields the desired result, Theorem 1.5, when N ≥ m̂ > 0. Also as in the previous case, Proposition 3.6 is easily derived from a listing of the differentials in the cellular spectral sequence, which we now describe. The proof of the following result will be given in Section 4.

Theorem 3.7. If N ≥ m̂ > 0, the nonzero differentials in the spectral sequence converging to E_{2}(X_{0}^{i}) are those described below on groups E^{1,j} with j ≤ i.

j
0 1 2 3 4 5 6 7 8 9 10
ν ≤ 28  1 1 1 ν + 1 1 1 1 1 1 1 1

0

ν = 29  1 1 1 30 1 1 1 1 1 1 1
0  1
m ≡ 3 + 6p^{30} mod p^{31} 1 1 1 31 1 1 1 1 1 1 1
0 1 2
ν = 30, m ≢ 3 + 6p^{30} 1 1 1 31 1 1 1 1 1 1 1
0  1 2
ν = 31  1 1 1 32 1 1 1 1 1 1 1
0 1  2 3
ν = 32  1 1 1 33 1 1 1 1 1 1 1
0 1 2  3 3
ν = 33  1 1 1 34 1 1 1 1 1 1 1
0 1 2 3  3 3
ν = 34  1 1 1 35 1 1 1 1 1 1 1
0 1 2 3 3  3 3
ν = 35  1 1 1 36 1 1 1 1 1 1 1
0,1 2 3 3 3  3 3
ν ≥ 36  1 1 1 37 1 1 1 1 1 1 1
0,1,2 3 3 3 3  3 3

(1) If ν(m − m̂) < m̂(p − 1), then d_{N} ≠ 0 : E_{N}^{1,N} → E_{N}^{2,0}.

(2) If (3.5) is satisfied, then d_{N+1} ≠ 0 : E_{N+1}^{1,N+1} → E_{N+1}^{2,0}.

(3) If t ≥ 1 or if t = 0 and ν(m − m̂) ≥ m̂(p − 1) and (3.5) is not satisfied, then there is a nonzero differential from x_{m̂(p−2)+N+1+t} to z_{t}, where

x_{k} = \begin{cases}elt of order p^{k} in E_{1}^{1,m̂} & if k ≤ ν(|E_{1}^{1,m̂}|) \\ gen of E_{1}^{1,m̂+Δ} & if Δ = k − ν(|E_{1}^{1,m̂}|) > 0 and m̂ + Δ ≤ N \\ gen of E_{1}^{1,m̂+Δ+1} & if Δ = k − ν(|E_{1}^{1,m̂}|) > 0 and m̂ + Δ > N \end{cases}

z_{t} = \begin{cases}gen of E_{1}^{2,t} & if t < m̂ \\ elt of order p^{t−m̂+1} in E_{1}^{2,m̂} & if t ≥ m̂ \end{cases}

We close this section by illustrating 3.6 and 3.7 when p = 11, N = 7, and m̂ = 3 in Table 4. The conventions are as in Table 3. In particular, if in column j, a number k sits directly beneath a number e, then E_{1}^{1,j} ≈ Z/p^{e} and d ≠ 0 : E^{1,j} → E^{2,k}.

4. Determination of differentials in the cellular spectral sequence

In this section, we determine the differentials in the cellular spectral sequence, proving Theorems 3.2, 3.4, and 3.7, and thereby implying Theorem 1.5. We continue with the notational conventions of Section 3. In particular, N and m are fixed, and E_{2}^{s}(−) means E_{2}^{s,2N+1+qm}(−).

We will also take a number of liberties with notation, which we now explain. First, we omit v_{1}-powers on the left of most terms. All of our terms have degree 2N +1+qm, and so the number of v_{1}’s can be determined. This number of v_{1}’s does not affect most formulas significantly, although it is important in applications of Lemma 2.13. Second, we frequently omit units in Z_{(p)} which appear as coefficients in expressions. There will be some cases, giving rise to (3.5), where a comparison of two terms will be essential, and then we will of course keep track of coefficients, but usually we only need to know whether or not they are zero mod p. Third, we write x ≡ y mod L to mean that x − y desuspends to (or is defined on) lower spheres. “Lower than what” should be clear from the context. Sometimes we will omit writing “mod L.” Finally, we will often write y_{j} for x_{2N+1+qj} in BP_{∗}(X_{k}^{i}) and for ι_{2N+1+qj} in S^{2N+1+qj} = X_{j}^{j}.

We begin with some lemmas, the first of which is an elementary fact involving binomial coefficients.

Lemma 4.1. If 0 < a < p, and 0 ≤ b, c < p, then

ν(\binom{ap + b}{c}) = \begin{cases}1 & if b < c \\ 0 & if b ≥ c. \end{cases}

The following lemma is a basic technique that will be used often. We deal here and throughout this section with projection maps X_{a}^{j} → X_{j}^{j}.

Lemma 4.2. Suppose that (1) 0 ≤ a ≤ j < p, (2) either m̂ < a or m̂ > j, and (3) if a = 0, then j < N. Thinking of j and N as being fixed, let

(4.3) ϵ_{ℓ} = \begin{cases}1 & if ℓ ≤ N < j \\ 0 & otherwise. \end{cases}

Then there is an element z ∈ E_{2}^{1}(X_{a}^{j}) satisfying

z ≡ ∑_{ℓ=a}^{j} h^{j+1−ℓ−ϵ_{ℓ}}y_{ℓ} mod L (4.4)

which projects to a generator of E_{2}^{1}(X_{j}^{j}) ≈ Z/p. Hence d_{r} = 0 : E_{r}^{1,j} → E_{r}^{2,j−r} for r ≤ j − a.

Proof. The proof is by downward induction on a, and is true for a = j by parts 1 and 2 of Theorem 2.8. Assume that it is true for a certain a satisfying a > N + 1 if N < j, and also satisfying a − 1 > m̂. Let z_{a} denote the sum (4.4), which will have all ϵ_{ℓ} = 0. We will show below that

∂ : E_{2}^{1}(X_{a}^{j}) → E_{2}^{2}(X_{a−1}^{a−1})

satisfies

∂(z_{a}) ≡ ∑_{ℓ=a}^{j} \binom{N+(p−1)(a−1)}{ℓ−a+1}h^{j+1−ℓ} ⊗ h^{ℓ−a+1}y_{a−1}
(4.5)  ≡ h ⊗ h^{j−a+1}y_{a−1} ≡ d(h^{j−a+2}y_{a−1}) mod L.

Thus z_{a} − h^{j−a+2}y_{a−1} is a cycle in E_{2}^{1}(X_{a−1}^{j}), which extends the induction in this case (a > N + 1). (Remember that we are not usually worrying about unit coefficients.)

Now we prove the first ≡ in (4.5). By (2.6), Lemma 2.4, and Proposition 2.5, ∂(h^{j+1−ℓ}y_{ℓ}) = h^{j+1−ℓ} ⊗ Cy_{a−1}, where C is the component of

(1 + h − hv + hv^{2} − \binom{p−1}{2}h^{2}v)^{N+(a−1)(p−1)}

in degree q(ℓ − a + 1). We will see that terms of degree greater than 3q in the sum of 2.5 would not play an important role here. The term that appears in the sum of (4.5) is obtained from choosing (ℓ−a+1) h’s. If any other terms of 2.5 are chosen, the term will desuspend farther because of the v’s on the right. For terms in the sum of 2.5 of degree greater than 3q, only pure powers of h_{1} could yield terms which desuspend as far as the one already considered, and such terms were shown in Proposition 2.5 not to exist. Note that a term such as h_{2} occurring in 2.5 desuspends no farther than h_{1}, but it uses up many more degrees. If Ly_{ℓ} is a term that desuspends lower than h^{j+1−ℓ}y_{ℓ}, then L is effectively h^{t} for some t < j + 1 − ℓ, and so ∂(Ly_{ℓ}) will desuspend farther than h^{j+1−ℓ} ⊗ h^{ℓ−a+1}.

The second ≡ of (4.5) is true because the unstable condition easily implies that other terms desuspend farther than the (ℓ = j)-term. Also we need that the binomial coefficient is a unit. This follows from Lemma 4.1 and the hypothesis that we do not have a − 1 ≤ N < j. The third ≡ is true by Lemma 2.13, in which the coefficient of the first term is

\frac{1}{q}((2N + 1 + qm) − (2N + 1 + q(a − 1))) = m − (a − 1), (4.6)

which is not a multiple of p, since a − 1 > m̂.

The proof is similar when a − 1 ≤ N < j, except that \binom{N+(p−1)(a−1)}{j−a+1} is now p times a unit, by Lemma 4.1. Thus the last terms of (4.5) become ph ⊗ h^{j−a+1}y_{a−1} ≡ h ⊗ h^{j−a}y_{a−1} ≡ d(h^{j−a+1}y_{a−1}), where we have used ph = v − η(v) to “cancel” the ph, and then argued as before.

The condition that j +1 ≤ N if a = 0 is required in order that all terms satisfy the unstable condition, Definition 2.2. The term h^{j+1−a}y_{a} is closest to failing. It requires j + 1 − a ≤ N + a(p − 1), which is satisfied if a > 0 or j + 1 ≤ N.

The statement about d_{r}’s being 0 follows from the definition of the spectral sequence. If an element in E_{2}^{1}(X_{j}^{j}) pulls back to E_{2}^{1}(X_{a}^{j}), then it survives to E_{j−a}^{1,j}.

Lemma 4.2 must be modified as follows when a = m̂.

Lemma 4.7. Define ν = ν(m − m̂) and ϵ_{ℓ} as in (4.3). Suppose m̂ < j and

j + 1 + ν − ϵm̂ ≤ N + m̂p. (4.8)

Then, under the projection map,

h^{j+1−m̂−ϵm̂+ν}ym̂ + ∑_{ℓ=m̂+1}^{j} h^{j+1−ℓ−ϵ_{ℓ}}y_{ℓ} ∈ E_{2}^{1}(Xj ; m̂)

maps to a generator of E_{2}(X_{j}^{j}) ≈ Z/p. Hence d_{r} = 0 : E_{r}^{1,j} → E_{r}^{1,j−r} for r ≤ j − m̂.

The statement of this lemma initiates another abuse of notation which we will allow ourselves. The class here is really only a cycle mod L. We mean here only the sort of thing that was stated explicitly in Lemma 4.2.

Proof. The generator of E_{2}^{1}(X_{j}^{j}) pulls back to z ∈ E_{2}^{1}(X_{m̂+1}^{j}) as in Lemma 4.2, with

∂(z) ≡ \binom{N+(p−1)m̂}{j−m̂}h ⊗ h^{j−m̂}ym̂ ≡ h ⊗ h^{j−m̂−ϵm̂}ym̂.

When we try to apply Lemma 2.13 to write this as a boundary, the first coefficient is m − m̂ = sp^{ν}, where s is a unit is Z_{(p)}. We don’t worry about units, but the p^{ν} is missing from our term. To accommodate this, we note that, with e = j − m̂ − ϵm̂ and t + 1 + e = m − m̂,

v^{t}h ⊗ h^{e} ≡ v^{t−ν}h ⊗ v^{ν}h^{e} ≡ v^{t−ν}h ⊗ p^{ν}h^{e+ν}

mod L. Here the first ≡ follows from the remarks after Theorem 2.8, and the second follows from expanding p^{ν}h^{ν} = (v − η(v))^{ν}, noting that terms with η(v)’s will desuspend farther than other terms. We obtain

∂(z) ≡ p^{ν}h ⊗ h^{j−m̂−ϵm̂+ν}ym̂ ≡ d(h^{j+1−m̂−ϵm̂+ν}ym̂),

which satisfies the unstable condition by (4.8). The argument is completed as in Lemma 4.2.

A similar lemma tells how classes in E^{1,m̂} can be pulled back.

Lemma 4.9. Let ϵ_{ℓ} = 1 if ℓ ≤ N < m̂, and ϵ_{ℓ} = 0 otherwise. Suppose 0 ≤ a ≤ m̂,

that t − ϵ_{a} ≤ N + a, and that a(p − 2) + t − ϵ_{ℓ} > 1. Then h^{m̂(p−2)+t}ym̂ ∈ E_{2}^{1}(Xm̂ ; m̂) pulls

back, mod L, to a cycle ∑m̂ ; ℓ=a h^{ℓ(p−2)+t−ϵ_{ℓ}}y_{ℓ} ∈ E_{2}^{1}(Xm̂ ; a). Also, d_{r} = 0 : E_{r}^{1,m̂} → E_{r}^{1,m̂−r}

for r ≤ m̂ − a.

Proof. We assume that the formula is true for a (with class z_{a}), and will deduce it for a − 1. We assume at first that it is not true that a − 1 ≤ N < m̂, so that all ϵ’s are 0. We claim

∂(z_{a}) ≡ ∑_{ℓ=a}^{m̂} \binom{N+(a−1)(p−1)}{ℓ−a+1}h^{ℓ(p−2)+t} ⊗ h^{ℓ−a+1}y_{a−1}
(4.10)  ≡ h^{a(p−2)+t} ⊗ hy_{a−1} ≡ h ⊗ h^{(a−1)(p−2)+t−1}y_{a−1} ≡ d(h^{(a−1)(p−2)+t}y_{a−1}).

Then z_{a−1} ≡ z_{a} − h^{(a−1)(p−2)+t}y_{a−1} is a cycle, extending the induction. The claim about d_{r} = 0 is immediate, as in the proof of Lemma 4.2.

The first ≡ in (4.10) follows by the argument used for the first ≡ of (4.5). The second ≡ says that the term with ℓ = a desuspends least far of all terms, and has coefficient a unit mod p. The coefficient is a unit since a − 1 ≠ N. The measure of the size of the smallest sphere on which h^{i} ⊗ h^{j} is defined is

exc(h^{i} ⊗ h^{j}) = min(j, i − (p − 1)j).

For the terms in the sum of (4.10), this is largest when ℓ = a. The third ≡ follows from Lemma 2.12, and the fourth from Lemma 2.13, where the first coefficient is, similarly to (4.6), m − (a − 1) ≢ 0 mod p.

This completes the proof of the lemma as long as it is not true that a−1 ≤ N < m̂. If a − 1 = N, then the binomial coefficient \binom{N+(a−1)(p−1)}{1} is p times a unit, and this p can be used to cancel one h, as in the proof of Lemma 4.2; this is accommodated by ϵ = 1. If a − 1 < N, then the t in the exponent has already been decreased to t − 1, and subsequent steps will leave it there.

The condition t−ϵ_{a} ≤ N +a is necessary in order that all terms satisfy the unstable condition.

Now we are ready to establish the differentials claimed in Theorems 3.2, 3.4, and 3.7. The reader is encouraged to refer to Table 2 when considering this proof of Theorem 3.2.

Proof of Theorem 3.2. We emphasize that this is the case m̂ = 0. Let 1 ≤ j ≤ p−1, ν = ν(m), and ϵ_{ℓ} be as in (4.3). By Lemma 4.2, a generator of E_{2}^{1}(X_{j}^{j}) pulls back to z_{1} ≡ ∑_{ℓ=1}^{j} h^{j+1−ℓ−ϵ_{ℓ}}y_{ℓ}, and d_{r} = 0 : E_{r}^{1,j} → E_{r}^{2,r−j} for r < j. By Lemma 4.7, d_{j} = 0 : E_{j}^{1,j} → E_{j}^{2,0} if j < N − ν. If max(1, N − ν) ≤ j ≤ N, then, as in the proof of Lemma 4.7, ∂(z_{1}) ≡ h⊗h^{j}y_{0}, and by Theorem 2.8, this is an element of order p^{N+1−j} in E_{1}^{2,0} = E_{2}^{2}(S^{2N+1}). Thus E_{N+1}^{2,0} = 0, completing the proof.

The reader is encouraged to refer to Table 3 in the proof of Theorem 3.4 which follows.

Proof of Theorem 3.4. If j < N, then d_{r} = 0 on E_{r}^{1,j} for all r by Lemma 4.2. Also by Lemma 4.2, we have E_{j}^{1,j} = E_{1}^{1,j} if N ≤ j < m̂. We will show that

d_{N} ≠ 0 : E_{N}^{1,N} → E_{N}^{2,0}. (4.11)

Then E_{N+1}^{2,0} = 0 and hence d_{j} is 0 on E_{j}^{1,j} when j > N, since there is nothing for it to hit. To prove (4.11), we use Lemma 4.2 to pull the generator of E_{1}^{1,N} back to

z_{1} ≡ ∑_{ℓ=1}^{N} h^{N+1−ℓ}y_{ℓ} ∈ E_{2}^{1}(X_{1}^{N}),

which satisfies ∂(z_{1}) ≡ h ⊗ h^{N}y_{0} by part of (4.5). This is nonzero by Lemmas 2.10 and 2.11.

Next we consider the differentials from E^{1,m̂}. We first show that elements x = x_{m̂(p−2)+N+1+t} of order p^{m̂(p−2)+N+1+t} with N < t < m̂ support a nonzero differential to E^{2,t}. By Theorem 2.8(2), x is represented, mod L, by h^{m̂(p−2)+N+1+t}ym̂ ∈ E_{2}^{1}(S^{2N+1+qm̂}). By Lemma 4.9, x can be extended, mod L, to

x_{t+1} = ∑_{ℓ=t+1}^{m̂} h^{ℓ(p−2)+N+1+t}y_{ℓ} ∈ E_{2}^{1}(Xm̂ ; t+1),

which implies that differentials from x into E^{2,j} with j ≥ t + 1 are 0. Then

∂(x_{t+1}) ≡ (N + (p − 1)t)h^{(t+1)(p−2)+N+1+t} ⊗ hy_{t} ≢ 0 ∈ E_{2}^{2}(S^{2N+1+tq}). (4.12)

The first ≡ here is obtained similarly to the first two steps of (4.10), while the last step is from Lemma 2.12.

We have now completed the proof of the differential from x_{m̂(p−2)+N+1+t} to z_{t} asserted in Theorem 3.4 when t > N and m̂(p − 2) + N + 1 + t ≤ ν(|E_{1}^{1,m̂}|). If instead we have t ≤ N, but still m̂(p − 2) + N + 1 + t ≤ ν(|E_{1}^{1,m̂}|), then in Lemma 4.9 ϵ_{ℓ} = 1 for ℓ ≤ N, and so x extends to

x′ ≡ ∑_{ℓ=N+1}^{m̂} h^{ℓ(p−2)+N+1+t}y_{ℓ} + ∑_{ℓ=max(t,1)}^{N} h^{ℓ(p−2)+N+t}y_{ℓ}, (4.13)

provided also that p − 2 + N + t > 1. If t ≤ 1, this shows that d_{r}(x) = 0 for r < m̂, and hence x is a permanent cycle, since d2,0 ; m̂ = 0. If t > 1, then, by part of (4.10), ∂(x′) ≡ h^{t(p−2)+N+t} ⊗ hy_{t−1}, which is nonzero by Lemma 2.12. If the sum (4.13) cannot be extended down to ℓ = 1 because t is too negative, then we will be dealing with terms h^{e} ⊗ hy_{ℓ} with e < p which desuspend to S^{3} and are boundaries on larger spheres, which will be the case for us.

This completes the analysis of the differentials of Theorem 3.4 on E_{r}^{1,j} with j ≤ m̂. Now we will establish the differential on x = x_{m̂(p−2)+N+1+t} when

m̂(p − 2) + N + 1 + t > ν(|E_{1}^{1,m̂}|). (4.14)

Let e = ν(|E_{1}^{1,m̂}|). Then x ≡ hy_{m̂(p−1)+N+1+t−e}, and can be extended by Lemma 4.2 to

x′ ≡ hy_{m̂(p−1)+N+1+t−e} + ⋯ + h^{m̂(p−2)+N+1+t−e}y_{m̂+1}, (4.15)

with

∂(x′) ≡ h ⊗ h^{m̂(p−2)+N+1+t−e}ym̂. (4.16)

By Corollary 2.9, this equals the element of order p^{t−m̂+1} on S^{2N+1+qm̂}, establishing the claimed differential to z_{t} when t ≥ m̂ (so that the order is greater than 1).

Now we assume t < m̂. Let ν = ν(m − m̂). Recall that

e = ν(|E_{1}^{1,m̂}|) = min(ν + 1, N + m̂(p − 1)).

Combining t < m̂ with (4.14) yields e < N + m̂(p − 1), and hence e = ν + 1. By Lemma 4.7, the class x′ of (4.15) extends to

x′′ ≡ hy_{m̂(p−1)+N+t−ν} + ⋯ + h^{m̂(p−2)+N+t−ν}y_{m̂+1} + h^{m̂(p−2)+N+t+1}ym̂.

By Lemma 4.9, x′′ can be extended to x′′′ satisfying

x′′′ ≡ \begin{cases}x′′ + ∑_{ℓ=t+1}^{m̂−1} h^{ℓ(p−2)+N+t+1}y_{ℓ} & if t > N \\ x′′ + ∑_{ℓ=N+1}^{m̂−1} h^{ℓ(p−2)+N+t+1}y_{ℓ} + ∑_{ℓ=t}^{N} h^{ℓ(p−2)+N+t}y_{ℓ} & if t ≤ N. \end{cases}

Then

∂(x′′′) ≡ \begin{cases}h^{(t+1)(p−1)+N} ⊗ hy_{t} & if t > N \\ h^{t(p−1)+N} ⊗ hy_{t−1} & if t ≤ N. \end{cases}

These are nonzero by Lemma 2.12, establishing the final differentials of Theorem 3.4, namely those from the second type of x_{k} to the first two types of z_{t}.

The proof of Theorem 3.7 proceeds similarly, except that in a certain case we must keep track of unit coefficients because two terms are trying to have cancelling differentials. It is recommended that the reader consult with Table 4 while studying this proof. Proof of Theorem 3.7. We divide into cases determined by the value of j, where the differential emanates from E^{1,j}. The following six cases apply when m̂ < N. The minor modifications required when m̂ = N will be described at the end of the proof.

Case 1: j ≤ m̂. If j < m̂, then all differentials on E^{1,j} are 0 by Lemma 4.2. For j = m̂, we wish to show the differential asserted in 3.7 from x_{m̂(p−2)+N+1+t} to the generator of E_{1}^{2,t}. Here we must have t < m̂ in order that E_{1}^{1,m̂} has an element of the asserted order. By Lemma 4.9 and (4.10), this x_{m̂(p−2)+N+1+t} pulls back to

x′ ≡ h^{m̂(p−2)+N+1+t}ym̂ + ⋯ + h^{(t+1)(p−2)+N+1+t}y_{t+1},

which has ∂(x′) ≡ h^{N+(t+1)(p−1)} ⊗ hy_{t} ≢ 0, by Lemma 2.12.

Case 2: ν < m̂(p − 1) and m̂ < j ≤ N. Here and throughout the remainder of this proof, ν = ν(m − m̂). The claim here is that E_{1}^{1,j} consists of permanent cycles if j < N, while d_{N} ≠ 0 : E_{N}^{1,N} → E_{N}^{2,0}. To establish this, we begin by using Lemma 4.7 to pull the generator of E_{1}^{1,j} back to

x_{0} ≡ hy_{j} + ⋯ + h^{j−m̂}y_{m̂+1} + h^{j−m̂+ν+1}ym̂. (4.17)

Let k be the integer satisfying k(p − 1) ≤ ν < (k + 1)(p − 1). Note that k satisfies 0 ≤ k < m̂. Next we show that x_{0} pulls back to a cycle x′ in E_{2}^{1}(X_{1}^{j}) of the form

x′ ≡ x_{0} + ∑_{ℓ=m̂−k}^{m̂−1} h^{j−m̂+ν+1−(m̂−ℓ)(p−2)}y_{ℓ} + ∑_{ℓ=1}^{m̂−k−1} h^{j+1−ℓ}y_{ℓ}. (4.18)

(Actually, as will be discussed in the third bullet below, it is possible that one of the terms in this sum is incorrect, but this turns out to be inconsequential.)

To obtain (4.18), we note that if x_{0} has been pulled back to x′′ ∈ E_{2}^{1}(X_{ℓ+1}^{j}) with ℓ + 1 ≥ m̂ − k, with terms like those in (4.17) and the first sum of (4.18), then ∂(x′′) can be evaluated by applying (4.5) to the terms of (4.17), and (4.10) to the terms of (4.18). The leading terms will be h⊗h^{j−ℓ}y_{ℓ} and h⊗h^{j−m̂+ν−(m̂−ℓ)(p−2)}y_{ℓ}, respectively. Now there are three possibilities.

• If ℓ ≥ m̂ − k and it is not the case that ℓ = m̂ − k and ν = k(p − 1), then j − m̂ + ν − (m̂ − ℓ)(p − 2) > j − ℓ, and so the leading term of ∂(x′′) will be the second of those listed above. Thus, mod L, the extension of x′′ to E_{2}^{1}(X_{ℓ}^{j}) is obtained by extending the first sum of (4.18).

• If ℓ = m̂ − k − 1, then the leading term of ∂(x′′) will be the first of those listed above. Thus, mod L, the extension of x′′ to E_{2}^{1}(X_{ℓ}^{j}) contains the top term in the last sum of (4.18). From this point on, subsequent extensions will be determined as they were in (4.5), by the very top term, hy_{j}.

• If ν = k(p−1) and ℓ = m̂−k, these two leading terms will be equal. Actually, they may have different units as coefficients. If, when the units are taken into account, these terms do not sum to 0, then the extension of x′′ to E_{2}^{1}(X_{ℓ}^{j}) is obtained by extending the first sum of (4.18), just as it was in the first case. If, when the units are taken into account, these terms sum to 0 mod p, then the powers of p in this coefficient can be used to cancel some of the h’s in the y_{m̂−k}-term, rendering the expression (4.18) incorrect in this term. However, at the next step, i.e., in going to y_{m̂−k−1}, and in subsequent steps, the very top term will provide the leading term, just as it did in the second bullet above, and so this ambiguity in the y_{m̂−k}-term will be inconsequential.

If x′ is as in (4.18), then the leading term in ∂(x′) will be h ⊗ h^{j}y_{0} ∈ E_{2}^{1}(S^{2N+1}). This is a consequence of the assumption that ν < m̂(p − 1), using the same sort of excess comparisons that occurred in the three possibilities above. This ∂(x′) is 0 if j < N, since h ⊗ h^{j}ι_{2N+1} desuspends. Here and elsewhere, we use freely the fact that when E_{2}^{2}(S^{2N+1}) ≈ Z/p, then elements in it which desuspend are 0. If j = N, then h ⊗ h^{N}ι_{2N+1} ≠ 0 by Lemmas 2.10 and 2.11. This completes the proof of this case.

Case 3: ν = m̂(p − 1) and j < N. Here E_{1}^{1,j} consists of permanent cycles by the argument just completed. Indeed, in this case, the first sum of (4.18) extends down to ℓ = 0 with all terms satisfying the unstable condition.

Case 4: ν = m̂(p − 1) and j = N. This is the delicate case, requiring that we keep track of units. We will show that, under these hypotheses, d_{N} : E_{N}^{1,N} → E_{N}^{2,0} is zero if and only if (3.5) is satisfied.

As in (4.18), a generator of E_{1}^{1,N} pulls back to

hy_{N} + ⋯ + h^{N−m̂}y_{m̂+1} + h^{N+m̂(p−2)+1}ym̂ + ⋯ + h^{N+p−1}y_{1}, (4.19)

but now both the first and last terms hit a nonzero element of E_{2}^{2}(X_{0}^{0}) when ∂ is applied. We will show that if the coefficient of hy_{N} is chosen to be 1, and if m − m̂ = sp^{ν}, with s ≢ 0 mod p, then, for 0 ≤ k < m̂, the coefficient of h^{N+(m̂−k)(p−2)+1}y_{m̂−k} in (4.19) is, mod p,

u_{k} = (−1)^{k} \frac{1}{s}\binom{N − m̂ + k}{k}. (4.20)

Then E_{2}^{1}(X_{1}^{N})⟶^{∂}E_{2}^{2}(X_{0}^{0}) satisfies, mod L,

(4.21) ∂(hy_{N} + u_{m̂−1}h^{N+p−1}y_{1}) ≡ (h ⊗ h^{N} + u_{m̂−1}\binom{N}{1}h^{N+p−1} ⊗ h)y_{0}
≡ (1 − u_{m̂−1}N)h ⊗ h^{N}ι_{2N+1}.

We have used Lemma 2.12(2) at the last step.

When ∂ is applied to (4.19), modified so as to properly incorporate unit coefficients on all terms, the intermediate terms desuspend farther than the end terms, and the desired d_{N} is determined by (4.21). Incorporating (4.20), we find that the image of this d_{N} is

1 + (−1)m̂ \frac{1}{s}N\binom{N−1}{m̂−1} ∈ Z/p,

and this is 0 if and only if (3.5), as claimed.

It remains to prove (4.20), which we do by induction on k. For k = 0, we note that, similarly to the proof of Lemma 4.7, for any numbers c_{ℓ},

∂(hy_{N} + ∑_{ℓ=2}^{N−m̂} c_{ℓ}h^{ℓ}y_{N+1−ℓ}) ≡ \binom{N+m̂(p−1)}{N−m̂}h ⊗ h^{N−m̂}ym̂
≡ \frac{1}{s}(m − m̂)h ⊗ h^{N−m̂+ν}ym̂ ≡ −\frac{1}{s}d(h^{N−m̂+ν+1}ym̂),

and so adding \frac{1}{s}h^{N−m̂+ν+1}ym̂ extends the cycle, with u_{0} = \frac{1}{s}.

Assume now that u_{k} satisfies (4.20). Let z = hy_{N} + ⋯ + u_{k}h^{N+(m̂−k)(p−2)+1}y_{m̂−k}. Then, mod L, we have

∂(z) ≡ u_{k}\binom{N+(m̂−k−1)(p−1)}{1}h^{N+(m̂−k)(p−2)+1} ⊗ hy_{m̂−k−1}
≡ −u_{k}(N − m̂ + k + 1)h ⊗ h^{N+(m̂−k−1)(p−2)}y_{m̂−k−1}
≡ \frac{1}{m−m̂+k+1}u_{k}(N − m̂ + k + 1)d(h^{N+(m̂−k−1)(p−2)+1}y_{m̂−k−1})
≡ (−1)^{k} \frac{1}{s} \binom{N−m̂+k+1}{k+1}d(h^{N+(m̂−k−1)(p−2)+1}y_{m̂−k−1}).

Thus adding (−1)^{k+1} \frac{1}{s} \binom{N−m̂+k+1}{k+1}h^{N+(m̂−k−1)(p−2)+1}y_{m̂−k−1} to z yields a cycle mod L in E_{2}^{1}(X_{m̂−k−1}^{N}) which projects to z. The coefficient here is just what we have claimed is u_{k+1}, and so this extends the induction.

Case 5: ν > m̂(p − 1) and m̂ < j ≤ N. These are the second type of elements x_{k} in Theorem 3.7. As before, let e = ν(|E_{1}^{1,m̂}|) = min(ν + 1, N + m̂(p − 1)). Then, in the notation of Theorem 3.7, our j becomes m̂(p − 1) + N + 1 + t − e. We show in the next paragraph that if t ≥ m̂, then the differential from E^{1,j} hits the element of order p^{t−m̂+1} in E_{1}^{2,m̂}. Of course, this hit element should really be viewed as the element of order p in E_{j−m̂}^{2,m̂}, elements of order less than p^{t−m̂+1} already having been hit by shorter differentials.

By the first sentence of the proof of Lemma 4.7, a generator of E_{2}^{1}(X_{j}^{j}) pulls back to an element x ∈ E_{2}^{1}(X_{m̂+1}^{j}) satisfying ∂(x) ≡ h ⊗ h^{j−m̂}ym̂. This latter element has order p^{t−m̂+1} in E_{2}^{2}(S^{2N+1+qm̂}) by Corollary 2.9. Indeed, the equation of that corollary becomes

(m̂(p − 1) + N + 1 + t − e − m̂) + e = (t − m̂ + 1) + (N + m̂(p − 1)).

If, on the other hand, t < m̂, then by Lemma 4.7 the generator of E^{1,j} pulls back to a cycle

x′ ≡ ∑_{ℓ=m̂+1}^{j} h^{j+1−ℓ}y_{ℓ} + h^{m̂(p−2)+N+t−e+ν+2}ym̂ ∈ E_{2}^{1}(Xj ; m̂).

We can simplify this by noting that the hypotheses imply e = ν + 1. (If not, then e = m̂(p − 1) + N, so j = t + 1, contradicting j > m̂ > t.) Now by Lemma 4.9 we can pull x′ back to

x′′ ≡ x′ + ∑_{ℓ=t+1}^{m̂−1} h^{ℓ(p−2)+N+t+1}y_{ℓ} ∈ E_{2}^{1}(X_{t+1}^{j}).

As in (4.10), ∂(x′′) ≡ h^{(t+1)(p−1)+N} ⊗ hy_{t}, which is nonzero by Lemma 2.12. We observe that ∂(hy_{j}) cannot interfere here, since this yields a term which desuspends unless j = N and t = 0, and this is impossible since, if t = 0,

j = m̂(p − 1) + N + 1 − e = m̂(p − 1) + N − ν < N.

Case 6: j > N. In the notation of the theorem, we have

j = m̂ + Δ + 1 = m̂ + k − e + 1
(4.22) = m̂ + (m̂(p − 2) + N + 1 + t) − e + 1 = m̂(p − 1) + N + 2 + t − e.

The generator of E_{2}^{1}(X_{j}^{j}) can be pulled back as in Lemma 4.2 to a class x ∈ E_{2}^{1}(X_{m̂+1}^{j}) satisfying ∂(x) ≡ h ⊗ h^{j−m̂−1}ym̂, similarly to part of (4.5) except that ϵ = 1. Using Corollary 2.9 and (4.22), we obtain that the order of h⊗h^{j−m̂−1}ι_{2N+1+qm̂} is p^{t−m̂+1}, as claimed, provided t ≥ m̂. Note that if j = N +1, then the class which this differential would like to hit has already been killed by a differential from E^{1,N}.

If t < m̂, then by Lemma 4.7 a generator of E_{2}^{1}(X_{j}^{j}) pulls back to

x ≡ hy_{j} + ⋯ + h^{j−m̂−1}y_{m̂+1} + h^{j−m̂+ν}ym̂ ∈ E_{2}^{1}(Xj ; m̂).

By Lemma 4.9, x can be extended farther, to

x′ ≡ x + ∑_{ℓ=t+1}^{m̂−1} h^{ℓ(p−2)+N+2+t−e+ν}y_{ℓ} ∈ E_{2}^{1}(X_{t+1}^{j}),

with ∂(x′) ≡ h^{(t+1)(p−1)+N} ⊗ hy_{t} ≠ 0 by Lemma 2.12. We must worry here about possible cancellation from ∂(hy_{j}) ≡ ch ⊗ h^{j−t}y_{t}. This will desuspend, and hence be 0, if t > 0. Thus it remains to consider the possible cancellation when t = 0.

If ν > m̂(p − 1), then t must be greater than 0. This can be deduced using j > N, (4.22), and e = min(ν + 1, N + m̂(p − 1)).

If ν ≤ m̂(p − 1) and (3.5) is not satisfied, then it was proved in Cases 2 and 4 that E_{N+1}^{2,0} = 0, and so we need not worry about the case t = 0 here. Finally, we must establish that d_{N+1} ≠ 0 : E_{N+1}^{1,N+1} → E_{N+1}^{2,0} if (3.5) is satisfied. To prove this by the methods employed so far in this section is more delicate than we care to present. Instead, we deduce it using differentials already determined, together with results for cyclicity of certain E_{2}-groups which will be established in the next section.

So, we are assuming (3.5). In particular, ν = m̂(p − 1) and e = ν + 1. Earlier in Case 6, we verified that under this hypothesis d_{N+1} ≠ 0 : E_{N+1}^{1,N+2} → E_{N+1}^{2,1}. This says that the composite ρ ∘ ∂ ≠ 0 in (4.23), and that | coker(i_{2})| = p.

E_{2}^{1}(X_{2}^{N})⟶^{i_{1}}E_{2}^{1}(X_{2}^{N+1})⟶^{i_{2}}E_{2}^{1}(X_{2}^{N+2})⟶^{∂}E_{2}^{2}(X_{0}^{1})⟶^{ρ}E_{2}^{2}(X_{1}^{1}) (4.23)

By Proposition 5.2, ρ is a surjection from Z/p^{2} to Z/p. Thus ∂ is surjective, and hence ∂ ∘ i_{2} ≠ 0, hitting E_{2}^{2}(X_{0}^{0}) ⊂ E_{2}^{2}(X_{0}^{1}). On the other hand, the 0-differential from E_{N}^{1,N} to E_{N}^{2,0} established in Case 4 implies that the composite ∂ ∘ i_{2} ∘ i_{1} in (4.23) is 0. Thus ∂ ∘ i_{2} is nonzero on an element which projects nontrivially to E_{2}^{1}(X_{N+1}^{N+1}), and this establishes d_{N+1} ≠ 0 : E_{N+1}^{1,N+1} → E_{N+1}^{2,0}, as desired.

This completes the proof when m̂ < N. If m̂ = N, then Case 1 applies if j < m̂ or if j = m̂ and ν > m̂(p − 1), Case 2 applies if j = N and ν < m̂(p − 1), Case 4 applies if j = N and ν = m̂(p − 1), and Case 6 applies if j > N.

5. Group structure

In this section, we prove Theorem 1.9; we determine the group structure on the 2-line of the UNSS of SU(n) for n ≤ p^{2} − p + 1. We also prove a few subsidiary results about group structure of 1-line groups E_{2}^{1}(X_{a}^{b}).

The conventions of the previous sections are still in force. Thus p, N, and m are fixed, with 1 ≤ N < p. The second superscript of all E_{2}-groups is 2N + 1 + qm (unless specifically stated to the contrary), and m̂ is the least nonnegative residue of m mod p. Also, X_{a}^{b} is a direct factor of a complex Stiefel manifold (quotient of SU(n)’s) built from spheres 2N +1+qj for a ≤ j ≤ b. Also, we continue the practice of omitting v_{1}’s on the left and units in Z_{(p)} whenever they are unimportant.

Although Theorem 1.11 implies that E_{2}^{1}(X_{0}^{b}) is cyclic, it will not always be the case that E_{2}^{1}(X_{a}^{b}) is cyclic. The following result presents one case in which E_{2}^{1}(X_{a}^{b}) will be cyclic.

Proposition 5.1. The group E_{2}^{1}(X_{a}^{b}) is cyclic if there are no numbers j congruent to N mod p satisfying a ≤ j < b.

Proof. The proof is by induction on b. The result is true when b = a by Theorem 2.8(1). We assume it is true for X_{a}^{b}, and will deduce it for X_{a}^{b+1}, provided b ≢ N mod p. We consider the commutative diagram of exact sequences below.

0 → E_{2}^{1}(X_{a}^{b}) ⟶^{j_{∗}} E_{2}^{1}(X_{a}^{b+1}) ⟶^{ρ_{∗}} E_{2}^{1}(X_{b+1}^{b+1}) ⟶^{∂} E_{2}^{2}(X_{a}^{b})
↓^{f_{∗}}  ↓^{g_{∗}}  ↓^{=}
0 → E_{2}^{1}(X_{b}^{b}) ⟶^{j_{1∗}} E_{2}^{1}(X_{b}^{b+1}) ⟶ E_{2}^{1}(X_{b+1}^{b+1})

If j_{∗} is surjective, then E_{2}^{1}(X_{a}^{b+1}) is cyclic by the induction hypothesis, and so we are done. Thus we may assume that there is an element β such that ρ_{∗}(β) = α, where α has order p. Since b ≢ N mod p, the attaching map in X_{b}^{b+1} is nontrivial, and so [6, 2.23] implies that there is a generator γ of E_{2}^{1}(X_{b}^{b}) such that j_{1∗}(γ) = p · g_{∗}(β). Exactness implies that there is y such that j_{∗}(y) = pβ. Then j_{1∗}(f_{∗}(y) − γ) = 0, so that f_{∗}(y) = γ, and hence y is a generator of the cyclic group E_{2}^{1}(X_{a}^{b}). Thus the extension is cyclic.

Next we prove a result that implies the first case of Theorem 1.9.

Proposition 5.2. The group E_{2}^{2}(X_{a}^{b}) is cyclic if either b ≤ N or N < a ≤ b < p.

Proof. The proof is by induction on b and is true for b = a by 2.8(1). We assume the result for X_{a}^{b}, and consider the exact sequence

E_{2}^{2}(X_{a}^{b}) → E_{2}^{2}(X_{a}^{b+1}) ⟶^{ρ_{∗}} E_{2}^{2}(X_{b+1}^{b+1})⟶^{∂}0,

with b ≠ N. If

E_{2}^{2}(X_{b+1}^{b+1}) ≈ Z/p^{N+(b+1)(p−1)}, (5.3)

then by Theorems 3.4 and 3.7, E_{2}^{2}(X_{a}^{b}) = 0, and so we are done in this case, since ρ_{∗} will be iso. The easiest way to believe this claim that E_{2}^{2}(X_{a}^{b}) = 0 is to look at the last row of Tables 3 and 4. In the first one, it is the assertion that E^{1,7} kills E^{2,j} for 3 ≤ j ≤ 6, and in the second one, that E^{1,3} kills E^{2,j} for 0 ≤ j ≤ 2.

Thus we may assume that (5.3) is not true, and so d(h^{N+(b+1)(p−1)+1})y_{b+1} has order p in E_{2}^{2}(X_{b+1}^{b+1}), by Theorem 2.8(4). Since ∂ annihilates this class, there is w ∈ E_{1}^{2}(X_{a}^{b}) such that z = d(h^{N+(b+1)(p−1)+1})y_{b+1} − w is a cycle in E_{2}^{2}(X_{a}^{b+1}). We wish to show that pz is the image of a generator of E_{2}^{2}(X_{a}^{b}).

We have

d(h^{t}y) = d(h^{t})y + h^{t}d(y), (5.4)

where d(h^{t}) is as in Lemma 2.3(3). Note here that the minus sign which is attached to ψ(γ_{1}) in (2.1) has been incorporated into d(h^{t}) (which is really d(v^{s}h^{t})) in Lemmas 2.3(3) and 2.13. Now, mod L,

pz ≡ d(h^{N+(b+1)(p−1)}y_{b+1}) − (N + b(p − 1))h^{N+(b+1)(p−1)} ⊗ hy_{b} − pw.

Here we have used ph = v − η(v) and (5.4). The first term is a boundary, and so is ignored. Note that the exponent of h had to be brought down to N + (b + 1)(p − 1) in order that it be placed in front of y_{b+1}. Since b ≠ N, the coefficient of the second term is a unit, and so will be ignored. Since w was defined on X_{a}^{b}, pw desuspends below S^{N+b(p−1)}, and so may be incorporated into L. Thus, by Lemma 2.12(2), pz generates E_{2}^{2}(X_{b}^{b}) mod L, and so pz is the image of a generator of E_{2}^{2}(X_{a}^{b}), as desired.

Next we prove the second part of Theorem 1.9, which we restate as follows.

Theorem 5.5. Suppose i > N and m̂ > 0. If e is as in Theorem 1.5, then E_{2}^{2}(X_{0}^{i}) ≈ Z/p^{e−1} ⊕ Z/p.

Proof. In the exact sequence

E_{2}^{1}(X_{N+1}^{i})⟶^{∂}E_{2}^{2}(X_{0}^{N})⟶^{ϕ}E_{2}^{2}(X_{0}^{i})⟶^{ρ}E_{2}^{2}(X_{N+1}^{i}) → 0, (5.6)

the groups E_{2}^{2}(X_{0}^{N}) and E_{2}^{2}(X_{N+1}^{i}) ≈ Z/p^{a} are cyclic by Proposition 5.2. We will show that if x ∈ E_{2}^{2}(X_{0}^{i}) is such that ρ(x) is a generator, then p^{a}x = pϕ(g), with g a generator of E_{2}^{2}(X_{0}^{N}). The result is immediate from this, with x and ϕ(g) − p^{a−1}x generating the summands.

Case 1: N = m̂. The thing that distinguishes this case is that pg lives on the top cell of X_{0}^{N}, i.e., E_{2}^{2}(X_{0}^{N}) → E_{2}^{2}(X_{N}^{N}) sends pg nontrivially. Thus the extension question can be studied in the exact sequence

0 → E_{2}^{2}(X_{N}^{N}) → E_{2}^{2}(X_{N}^{i}) → E_{2}^{2}(X_{N+1}^{i}) → 0.

We let x also equal the image of a generator of E_{2}^{2}(X_{0}^{i}) in E_{2}^{2}(X_{N}^{i}). As in the proof of 5.2, we can write

p^{a−2}x = d(h^{N+(N+2)(p−1)+1})y_{N+2} + w_{0}y_{N+1} + w_{1}y_{N},

with each term unstable. Now we can use (5.4), (2.6), 2.4, and 2.5 to write

p^{a−1}x = d(h^{N+(N+2)(p−1)}y_{N+2}) + pw_{0}y_{N+1} + pw_{1}y_{N}

−h^{N+(N+2)(p−1)} ⊗ ((N + (N + 1)(p − 1))hy_{N+1} − Nphvy_{N} + \binom{Np}{2}h^{2}y_{N} ).

We ignore the boundary term and apply Corollary 2.14 to the terms on y_{N+1}. We also write hv = vh − ph^{2} and ignore the p^{2}w′y_{N} term (with w′y_{N} defined) which this yields. We obtain

p^{a−1}x = −\frac{(N+1)p−1}{m−N−1}d(h^{N+(N+1)(p−1)+1} + w)y_{N+1} − Nph^{N+(N+2)(p−1)} ⊗ (vh + uh^{2})y_{N}
+pw_{1}y_{N},

where wy_{N+1} is defined, and u is a unit. We multiply by another p and ignore terms which are p^{2} times an unstable class on y_{N}. We obtain

p^{a}x ≡ u′d(h^{N+(N+1)(p−1)} + pw)y_{N+1}
≡ u′(d((h^{N+(N+1)(p−1)} + pw)y_{N+1}) − Np(h^{N+(N+1)(p−1)} + pw) ⊗ hy_{N} )

with u′ a unit. Here we have applied (5.4) again. By Lemma 2.12, this is p times a generator of E_{2}^{2}(X_{N}^{N}).

Case 2: N > m̂ or N < i < m̂ or (N < m̂ ≤ i and i+ν(m−m̂) < m̂(p−1)+2N). In these cases, E_{∞}^{2,j} = E_{1}^{2,j} for j ≥ N − 1, and so in the short exact sequence

0 → coker(∂) → E_{2}^{2}(X_{0}^{i}) → E_{2}^{2}(X_{N+1}^{i}) → 0, (5.7)

the generator g of coker(∂) lives on y_{N}, and pg lives on y_{N−1}. Hence (5.7) maps to a short exact sequence

0 → E_{2}^{2}(X_{N−1}^{N})⟶^{ϕ′}E_{2}^{2}(X_{N−1}^{i}) → E_{2}^{2}(X_{N+1}^{i}) → 0,

and it suffices to show ϕ′(pg) = p^{a}x in this latter sequence.

The analysis is quite similar to Case 1. We write

p^{a−2}x = d(h^{N+(N+2)(p−1)+1})y_{N+2} + w_{0}y_{N+1} + w_{1}y_{N} + w_{2}y_{N−1}

with each term unstable. Then, ignoring terms which are p times an unstable class on y_{N−1}, and letting u, u′, etc., denote units in Z_{(p)}, we obtain

p^{a−1}x ≡ d(h^{N+(N+2)(p−1)}y_{N+2}) + pw_{0}y_{N+1} + pw_{1}y_{N}
−h^{N+(N+2)(p−1)} ⊗ (((N + 1)p − 1)hy_{N+1} − Nphvy_{N} + \binom{Np}{2}h^{2}y_{N}
+(uhv^{2} + u′h^{2}v)y_{N−1}).

Now we ignore boundaries and use 2.14, 2.3(1), and 2.3(2) to obtain

p^{a−1}x ≡ −\frac{(N+1)p−1}{m−N−1}d(h^{N+(N+1)(p−1)+1} + w)y_{N+1}
+Nph^{N+(N+1)(p−1)} ⊗ hy_{N} + pw_{1}y_{N} + u′′ph^{N+(N+2)(p−1)} ⊗ h^{2}y_{N}
−h^{N+(N+2)(p−1)} ⊗ (uhv^{2} + u′h^{2}v)y_{N−1}.

Now we multiply by p again, again omit p times unstable classes on y_{N−1}, and incorporate the two surrounding terms into w_{1}, obtaining

p^{a}x ≡ u′′′d(h^{N+(N+1)(p−1)} + pw)y_{N+1} + p^{2}w′_{1}y_{N}.

Now we apply (5.4) again, and omit writing the d(hy)-term, obtaining

p^{a}x ≡ −u′′′((h^{N+(N+1)(p−1)} + pw) ⊗ (Nphy_{N} − ((N − 1)p + 1)hvy_{N−1}
+\binom{(N−1)p+1}{2}h^{2}y_{N−1})) + p^{2}w′_{1}y_{N}.

We omit lots of terms which are p times an unstable class on y_{N−1}. We omit the −u′′′-factor, which can be considered as a factor of the entire expression. We also combine together several terms that are p^{2} times an unstable class on y_{N}. Finally, we apply Lemma 2.3(1) to hv. This yields

p^{a}x ≡ Np(h^{N+(N+1)(p−1)} ⊗ h + pw′′_{1})y_{N} − h^{N+(N+1)(p−1)} ⊗ vhy_{N−1}.

Now we apply 2.14 to the first term and 2.3(2) to the second term, which effectively cancels h^{p−1}v. This yields

p^{a}x ≡ \frac{N}{m−N} pd(h^{N+N(p−1)+1} + w_{3})y_{N} − h^{N+N(p−1)} ⊗ hy_{N−1}
≡ \frac{N}{m−N} d(h^{N+N(p−1)} + pw_{3})y_{N} − h^{N+N(p−1)} ⊗ hy_{N−1},

with w_{3}y_{N} defined. We apply (5.4) once again, and omit writing the term of the form d(hy), obtaining

p^{a}x ≡ −\frac{N}{m−N} ((N − 1)p + 1)(h^{N+N(p−1)} + pw_{3}) ⊗ hy_{N−1} − h^{N+N(p−1)} ⊗ hy_{N−1}
≡ −(\frac{N}{m−N} + 1)h^{N+N(p−1)} ⊗ hy_{N−1}.

Since \frac{N}{m−N} + 1 = \frac{m}{m−N}, this is nonzero by Lemma 2.12 since m ≢ 0 mod p by assumption.

Case 3: N < m̂ ≤ i and i+ν(m−m̂) ≥ m̂(p−1)+2N. In this case, Theorem 3.7 (with Table 3 again recommended to provide insight) implies (1) E_{∞}^{2,j} = 0 for j < N, and hence E_{2}^{2}(X_{0}^{N})/ im(∂) ≈ Z/p, and (2) E_{2}^{2}(X_{N+1}^{i}) → E_{2}^{2}(X_{N+1}^{i+1}) is multiplication by p of groups each isomorphic to Z/p^{a}. To see (2), note first that the generator of E_{2}^{2}(X_{N+1}^{i+1}) lives on y_{i+1}, and second that E_{2}^{1}(X_{i+1}^{i+1})⟶^{∂}E_{2}^{2}(X_{N+1}^{i}) hits the elements of order p. If i = p − 1, then we need to extend Theorem 3.7 to include the case i = p. This extension is straightforward; it is only Theorem 3.2 that must be modified when i = p.

Now the splitting follows from the following commutative diagram of exact sequences.

\begin{aligned}0 → E_{2}^{2}(X_{0}^{N})/ im(∂) → & E_{2}^{2}(X_{0}^{i}) & → E_{2}^{2}(X_{N+1}^{i}) → 0 \\ ↓^{=} & ↓^{ϕ} & ↓^{·p} \\ 0 → E_{2}^{2}(X_{0}^{N})/ im(∂) → & E_{2}^{2}(X_{0}^{i+1}) & → E_{2}^{2}(X_{N+1}^{i+1}) → 0 \end{aligned}

Indeed, suppose to the contrary that E_{2}^{2}(X_{0}^{i}) ≈ Z/p^{a+1} with generator G. Then the left square implies p^{a}ϕ(G) ≠ 0, and so we must have E_{2}^{2}(X_{0}^{i+1}) ≈ Z/p^{a+1} with generator ϕ(G). Then the composition around the bottom of the right square is surjective, but the composite around the top is not surjective.

Finally, we prove the third and fourth cases of Theorem 1.9, which we restate as follows. Again, we remind the reader that the conventions which were restated at the beginning of this section continue to be in effect.

Theorem 5.8. Suppose m̂ = 0, i > N, and t = min(N, ν(m) + 1). Then

E_{2}^{2}(X_{0}^{i}) ≈ E_{2}^{2}(X_{1}^{i}) ≈ E_{2}^{1}(X_{1}^{i}) ≈ \begin{cases}Z/p^{t} ⊕ Z/p^{i−t} & if t ≤ i − N \\ Z/p^{N} ⊕ Z/p^{i−N} & if t ≥ i − N \end{cases}

Proof. The first isomorphism is immediate from E_{∞}^{2,0} = 0 in Theorem 3.2, and is clearly depicted in Table 1. The second and third isomorphisms could be generalized to interesting results regarding isomorphisms of 1- and 2-line groups, and evaluation of noncyclic 1-line groups of complex Stiefel manifolds, but, in the interest of expediency, we will just deal with the case at hand.

The middle isomorphism of 5.8 is the composite

E_{2}^{1,2N+1+qm}(X_{1}^{i})⟶^{ϕ}E_{2}^{2,2N+1+qm+qL}(X_{1}^{i}) ≈ E_{2}^{2,2N+1+qm}(X_{1}^{i}),

with ν(L) sufficiently large. The second isomorphism here is due to the periodicity of the groups E_{2}^{2,∗}(X_{1}^{i}), while ϕ is defined by

ϕ(x) = α_{L/(N+m(p−1))} ⊗ x,

where α_{L/e} is as in Theorem 2.8. We need ν(L) ≥ N + m(p − 1) − 1. Note that ϕ(x) is defined unstably since α_{L/e} is defined on S^{2e+1}.

We will show that ϕ is injective, from which its bijectivity will follow since |E_{2}^{2}(X_{1}^{i})| = |E_{2}^{1}(X_{1}^{i})|. If x ≠ 0 ∈ E_{2}^{1}(X_{1}^{i}), choose the largest integer j such that the projection map ρ sends x to a nonzero element ρ_{∗}(x) of E_{2}^{1}(X_{j}^{i}). Then there is a nonzero element x_{1} ∈ E_{2}^{1}(X_{j}^{j}) such that g_{∗}(x_{1}) = ρ_{∗}(x), where g : X_{j}^{j} ↪ X_{j}^{i}. Let α = α_{L,N+m(p−1)}. Then g_{∗}(α ⊗ x_{1}) = ρ_{∗}(α ⊗ x). By Lemma 5.10 below, α ⊗ x_{1} ≠ 0. By Theorem 3.2,

g_{∗} : E_{2}^{2,2N+1+qm+qL}(X_{j}^{j}) → E_{2}^{2,2N+1+qm+qL}(X_{j}^{i}) (5.9)

is injective, and hence ρ_{∗}(α ⊗ x) = g_{∗}(α ⊗ x_{1}) ≠ 0. Thus α ⊗ x ≠ 0, establishing the desired injectivity. The injectivity of (5.9) is the crucial hypothesis in generalizing this result. The addition of qL to the second superscript doesn’t matter here by periodicity. The injectivity in (5.9) is the statement that E_{1}^{2,j} = E_{∞}^{2,j} for j > 0 in Theorem 3.2, or that in Table 1 the only elements hit by differentials have j = 0.

Now we prove the last isomorphism of (5.8). We use the following commutative diagram of short exact sequences.

\begin{aligned}0 → E_{2}^{1}(X_{0}^{N}) → & E_{2}^{1}(X_{0}^{i}) & → E_{2}^{1}(X_{N+1}^{i}) → 0 \\ ↓^{ϕ′} & ↓^{ϕ} & ↓^{=} \\ 0 → E_{2}^{1}(X_{1}^{N})⟶^{θ} & E_{2}^{1}(X_{1}^{i}) & → E_{2}^{1}(X_{N+1}^{i}) → 0 \end{aligned}

By Theorem 3.2, illustrated in Table 1, the diagram is

\begin{aligned}0 → Z/p^{N} → & Z/p^{i} & → Z/p^{i−N} → 0 \\ ↓^{·p^{t}} & ↓^{ϕ} & ↓^{=} \\ 0 → Z/p^{N} ⟶^{θ} & E & → Z/p^{i−N} → 0. \end{aligned}

The key observation here is that ϕ′ = ·p^{t}, which we now explain. By Theorem 3.2, there are no differentials in the cellular spectral sequence for X_{1}^{N}, but in the spectral sequence for X_{0}^{N}, there are differentials from E^{1,j} to E^{2,0} for N − t + 1 ≤ j ≤ N. Thus the generator of E_{2}^{1}(X_{0}^{i}) sits on y_{N−t}, which is where p^{t} times the generator of E_{2}^{1}(X_{1}^{i}) lives.

Now it is a matter of simple algebra to determine the structure of the abelian group E = E_{2}^{1}(X_{1}^{i}) in the diagram above. Let g denote a generator of E_{2}^{1}(X_{0}^{i}), and let h denote a generator of E_{2}^{1}(X_{1}^{N}). Then p^{i−N}ϕ(g) = p^{t}θ(h). If t = N, this implies p^{i−N}ϕ(g) = 0, and so E ≈ Z/p^{N} ⊕ Z/p^{i−N} with generators θ(h) and ϕ(g). If t < N and t ≤ i − N, then E ≈ Z/p^{t} ⊕ Z/p^{i−t} with generators θ(h) − p^{i−N−t}ϕ(g) and ϕ(g). If t < N and t ≥ i − N, then E ≈ Z/p^{N} ⊕ Z/p^{i−N} with generators θ(h) and ϕ(g) − p^{t−i+N}θ(h).

We close with a lemma which was used in the preceding proof. Previous conventions for N and m̂ apply.

Lemma 5.10. Suppose ν(L) ≥ N + m(p − 1) − 1, m̂ = 0, and 1 ≤ j < p. Then

α_{L,N+m(p−1)}⊗ : E_{2}^{1,2N+1+qm}(S^{2N+1+qj}) → E_{2}^{2,2N+1+qm+qL}(S^{2N+1+qj})

is bijective.

Proof. Since both groups have order p, it suffices to show α_{L/(N+m(p−1))}⊗α_{m−j}ι_{2N+1+qj} is nonzero in E_{2}. Let s = L−N−m(p−1). By Theorem 2.8(2), this class is equivalent, mod terms that desuspend, to

v^{s}h^{N+m(p−1)} ⊗ v^{m−j−1}hι_{2N+1+qj}.

By Lemma 2.13, this is equivalent to

\frac{1}{m−j} (d(v^{s−1}h^{N+m(p−1)+1}v^{m−j})ι_{2N+1+qj} + Lv^{s−1}h ⊗ h^{N+m(p−1)}v^{m−j}ι_{2N+1+qj}).

Since L is highly p-divisible, the second term is 0 in E_{2}. The first term is simplified by using the last part of Lemma 2.11 to replace (h^{p}v)^{m−j} by (v^{p}h)^{m−j}. Thus the desired term is a unit times d(v^{L−N+m−jp−1}h^{N+j(p−1)+1})ι_{2N+1+qj}, which is nonzero by Theorem 2.8(4).

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Lehigh University, Bethlehem, PA 18015 E-mail address: dmd1@lehigh.edu

South China Normal University, Guangzhou, P.R. China, 510631 Current address: Lehigh University, Bethlehem, PA 18015 E-mail address: hy02@lehigh.edu