ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n))

DONALD M. DAVIS

1. Discussion of main theorem

If p is a prime number, then the p-exponent of a topological 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}. Our main theorem is a lower bound for exp_{p}(SU(n)) if p is an odd prime.

Theorem 1.1. If p is an odd prime, then

exp_{p}(SU(n)) ≥ n − 1 + [\frac{n + 2p − 3}{p^{2}}] + [\frac{n + p^{2} − p − 1}{p^{3}}].

In other words, we show that, for some i, π_{i}(SU(n)) contains an element of order p^{e} with e ≥ n − 1 − [\frac{n+2p−3}{p^{2}}] + [\frac{n+p^{2}−p−1}{p^{3}}].

Computing homotopy groups is a notoriously difficult problem. The groups π_{i}(S^{n}) are known only for relatively small values of i − n, and quickly become quite complicated, but the celebrated exponent theorem of [7] showed exp_{p}(S^{2n+1}) = n for all odd primes p. Groups π_{i}(SU(n)) have been computed for small values of i − n by many topologists, with all known results compiled in [12]. It was not until the work of the author in [9] that information about all homotopy groups of all SU(n) was obtained. There it was shown that

exp_{p}(SU(n)) ≥ n − 1. (1.2)

Theorem 1.1 of this paper improves upon that result. In this introductory section, we discuss the extent to which Theorem 1.1 might be nearly optimal.

Theorem 1.1 is proved by computing certain v_{1}-periodic homotopy groups of SU(n). We will review the basic properties of v_{1}-periodic homotopy groups in Section 2. Here

1991 Mathematics Subject Classification. 55T15.

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

2 D. DAVIS

we just point out that for each prime p there is a v_{1}-periodic homotopy theory, and the prime is not usually written explicitly in the notation. The v_{1}-periodic homotopy groups yield lower bounds for p-exponents since each v_{1}-periodic homotopy group of SU(n) occurs as a direct summand of some actual homotopy group of SU(n). It is also important to note from [9] that v_{1}^{−1}π_{2k}(SU(n)) is cyclic, and |v_{1}^{−1}π_{2k−1}(SU(n))| = |v_{1}^{−1}π_{2k}(SU(n))|.

It is possible that our methods might allow us to compute, for each integer n, the exact exponent of the order of the largest v_{1}-periodic group of SU(n). We denote this number by exp_{p}(v_{1}^{−1}SU(n)). As observed above, we have

exp_{p}(SU(n)) ≥ exp_{p}(v_{1}^{−1}SU(n)). (1.3)

This number exp_{p}(v_{1}^{−1}SU(n)) will depend intimately on the precise form of n. In Theorem 1.1 we have sacrificed optimality for simplicity. It seems likely that Theorem 1.1 is quite close to the exact value of exp_{p}(v_{1}^{−1}SU(n)).

A refined version of Theorem 1.1 is stated as Theorem 1.4 below, the proof of which will occupy most of our effort. Throughout the paper, ν( ) denotes the exponent of p in an integer.

Theorem 1.4. Given n and an odd prime p, define integers N, i, j, and k by

n = N + 1 + (p − 1)i with 1 ≤ N ≤ p − 1 (1.5)

and

i = jp + k with 1 ≤ k ≤ p. (1.6)

For 0 ≤ t < i, define

S′_{t} = |{s : t ≤ s < i, s ≡ N mod p}|

and

S′′_{t} = |{s : t < s < i, s ≡ kp + N mod p^{2}}|.

If N = k, define

S′′′_{t} = |{s : t < s < i, s ≡ jp^{2} + Np + N mod p^{3}}|

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 3

and

δ_{t} = \begin{cases}1 & if t ≡ Np + N mod p^{2} \\ 0 & otherwise. \end{cases}

Let

S_{t} = \begin{cases}S′_{t} + S′′_{t} & if N ≠ k \\ S′_{t} + S′′_{t} + S′′′_{t} + δ_{t} & if N = k. \end{cases}

Let

ϵ = \begin{cases}1 & if N ≤ k \\ 0 & if N > k. \end{cases} (1.7)

We define n to be p-clean if for all D ≥ ϵ

ν(i − j − D) < N − k + pD,

p-cleaner if for all D ≥ ϵ

ν(i − j − D) < N − k + ϵ + (p − 1)D − ν((D − ϵ)!) + S_{j+ϵ} − S_{j+D}, (1.8)

and p-cleanest if N = k and for all D ≥ 1

ν(i − j − D) < (p − 1)D − ν((D − 1)!) + S_{j+1} − S_{j+D}.

Let r = ν(i − j). Then exp_{p}(v_{1}^{−1}SU(n)) ≥

n + S_{j+ϵ}  if n is not p-clean
n − 1 + ν((i − j) ⋯ (i − j − r)) + S_{j+r+1} if N = k and n is p-cleaner
n + ν((i − j) ⋯ (i − j − r)) + S_{j+r+1}  if N = k and n is p-cleanest
n − 1 + S_{j+ϵ}  otherwise.

(1.9)

All the notation of Theorem 1.4 will pervade the paper. The variants on n being pclean have to do with the avoidance of associated numbers having large p-exponents. Note also that p-cleanest implies p cleaner, and p-cleaner implies p-clean.

The bridge from Theorem 1.4 to Theorem 1.1 is provided by the following result, which will be proved at the end of Section 2.

Theorem 1.10. Assume the notation of Theorem 1.4. Then

S′_{j+ϵ} ≥ [(n + 2p − 3)/p^{2}] − 1 (1.11)

4 D. DAVIS

with equality occurring if and only if k = N and j ≡ N or N + 1 mod p, while

S′′_{j+ϵ} ≥ [(n + p^{2} − p − 1)/p^{3}] − 1 (1.12)

with equality occurring if and only if k = N and j ≡ kp + N or kp + N − 1 mod p^{2}.

Now we provide the easy steps across the bridge from Theorem 1.4 to Theorem 1.1.

Proof of Theorem 1.1. We use (1.3), (1.9), and Theorem 1.10. If k ≠ N, we have

exp_{p}(SU(n)) ≥ n − 1 + S′_{j+ϵ} + S′′_{j+ϵ} ≥ n − 1 + [\frac{n+2p−3}{p^{2}}] + [\frac{n+p^{2}−p−1}{p^{3}}].

We now consider the case in which N = k. If i − j − 1 ≥ 2p^{3}, then S′′′_{j+1} ≥ 2, which makes up for the 2 that we might possibly be missing if equality holds in both (1.11) and (1.12). If i − j − 1 < 2p^{3}, then either n is p-cleanest or else p = 3 and i − j = 10, 19, 28, 37, or 46. If i − j − 1 < 2p^{3} and n is p-cleanest, then either equality holds in at most one of (1.11) or (1.12), in which case the desired result follows, or else j ≡ kp + N mod p^{2}. In this case, ν(i − j) ≥ 2, which implies the desired result. Finally, if p = 3 and i − j = 10, 19, 28, 36, or 46, one can check that equality holds in neither (1.11) nor (1.12), and so again the result follows.

In [9], an explicit, but not very tractable, formula for exp_{p}(v_{1}^{−1}SU(n)) was given when p is odd. This was

exp_{p}(v_{1}^{−1}SU(n)) = max{e_{p}(k, n) : k ≥ n}, (1.13)

where

e_{p}(k, n) = min\{ν(∑_{i=1}^{j}(−1)^{i}\binom{j}{i}i^{k}) : n ≤ j ≤ k\}. (1.14)

In [9], following work of [8], the Little Fermat Theorem was applied to (1.13) to deduce (1.2), a precursor of Theorem 1.1.

The formula given in (1.13) and (1.14) has the positive feature of being writable on one line, but the negative feature of not offering much insight into the actual values. These cyclic groups Z/p^{e_{p}(k,n)} were first determined by Bendersky in [2] as the 1-line groups of the unstable Novikov spectral sequence (UNSS) for SU(n). In [9], the relationship of the UNSS with v_{1}-periodic homotopy groups was first noted. The calculation in [2] was by downward induction on n. In [11], the UNSS for SU(n) was

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 5

computed by increasing induction on n, with complete and tractable results obtained, provided n ≤ p^{2} −p+1. A comparison of the results of these two computations yields results in number theory, proved via algebraic topology. The methods of this paper are an extension of those of [11]. We could compute most of the v_{1}-periodic groups of SU(n), but displaying them in a comprehensible form becomes a serious bookkeeping problem. So instead we concentrate our attention on the largest groups.

As discussed in [9], we can use a computer to compute the RHS of (1.13) for small values of n. This uses the periodicity in k of (1.14) proved in [8]. In the following table, we compare the computer result for exp_{p}(v_{1}^{−1}SU(n)) with our estimates of Theorems 1.4 and 1.1 when p = 3 and n ≤ 45. The few cases in which Theorem 1.4 fails to be sharp are due to lack of control of coefficients in Lemma 3.6; a huge amount of analysis would be required in order to sharpen this more finely. The cases in which Theorem 1.4 is stronger than Theorem 1.1 are due to approximations made in the proof and application of Theorem 1.10; they are due to our desire for a tractable formula. We remark that the computer value when n = 34 is 1 greater than that presented in [9]; our theoretical results here led to the discovery of this correction.

n N i k j exp_{p}(v_{1}^{−1}SU(n)) Thm 1.4 Thm 1.1
3 2 0 0 0  2  2  2
4 1 1 1 0  4  4  3
5 2 1 1 0  4  4  4
6 1 2 2 0  6  6  6
7 2 2 2 0  7  7  7
8 1 3 3 0  8  8  8
9 2 3 3 0  10  10  9
10 1 4 1 1  11  11  10
11 2 4 1 1  12  12  11
12 1 5 2 1  12  12  12
13 2 5 2 1  14  14  13
14 1 6 3 1  14  14  14
15 2 6 3 1  16  16  16
16 1 7 1 2  18  18  17
17 2 7 1 2  19  19  18
18 1 8 2 2  20  20  19
19 2 8 2 2  21  21  20
20 1 9 3 2  22  22  21
21 2 9 3 2  22  22  22
22 1 10 1 3  25  25  24
23 2 10 1 3  26  25  25
24 1 11 2 3  28  27  27
25 2 11 2 3  29  29  28
26 1 12 3 3  30  29  29
27 2 12 3 3  31  30  30
28 1 13 1 4  32  32  31
29 2 13 1 4  34  33  32
30 1 14 2 4  34  34  33
31 2 14 2 4  34  34  34
32 1 15 3 4  35  35  35
33 2 15 3 4  37  37  37
34 1 16 1 5  38  38  38
35 2 16 1 5  39  39  39
36 1 17 2 5  41  41  40
37 2 17 2 5  42  42  41
38 1 18 3 5  43  43  42
39 2 18 3 5  43  43  43
40 1 19 1 6  45  45  44
41 2 19 1 6  45  45  45
42 1 20 2 6  47  47  47
43 2 20 2 6  50  50  48
44 1 21 3 6  51  50  49
45 2 21 3 6  52  51  50

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 7

An optimistic conjecture would be that the elements of largest order in π_{∗}(SU(n))_{(p)} are v_{1}-periodic, i.e., that

exp_{p}(SU(n)) = exp_{p}(v_{1}^{−1}SU(n)).

The justification for such a conjecture would be its validity for spheres and the fact that v_{1}-periodic homotopy groups are related to K-theory, as are the spaces SU(n). The only cases in which the precise value of exp_{p}(SU(n)) has been established are the following.

Proposition 1.15. If p is an odd prime, then

exp_{p}(SU(n)) = \begin{cases}n − 1 & if n ≤ p \\ n & if n = p + 1. \end{cases}

This is an immediate consequence of the following facts.

(1) Localized at an odd prime p, there are homotopy equivalences

SU(n) ≃ ∏_{i=2}^{n} S^{2i−1} if n ≤ p, and

SU(p + 1) ≃ ∏_{i=3}^{p} S^{2i−1} × B(3, 2p + 1),

where B(3, 2p + 1) is an S^{3}-bundle over S^{2p+1} with bottom attaching map α_{1}. ([14])

(2) If p is an odd prime, then exp_{p}(S^{2i+1}) = i. ([7])

(3) If p is an odd prime, then exp_{p}(B(3, 2p + 1)) = p + 1. ([6])

Bounds for odd-primary p-exponents for other classical groups follow from Theorem 1.1 and the p-equivalences of Harris ([13]). Indeed, we obtain

Corollary 1.16. If p is an odd prime, then

exp_{p}(SO(2n + 1)) = exp_{p}(Sp(n)) ≥ 2n − 1 + [\frac{2n + 2p − 3}{p^{2}}] + [\frac{2n + p^{2} − p − 1}{p^{3}}]

and

exp_{p}(SO(2n + 2)) = max{2n + 1, exp_{p}(SO(2n + 1))}.

8 D. DAVIS

Proof. The first “=” is immediate from Harris’ p-equivalence

SO(2n + 1) ≃ Sp(n).

The inequality follows from Theorem 1.1 and Harris’ p-equivalence

Sp(n) × (SU(2n)/Sp(n)) ≃ SU(2n).

Actually here we must resort to part of the proof of Theorem 1.1 to observe that the summand of the computed order in v_{1}^{−1}π_{∗}(SU(2n)) comes from the Sp(n) factor. It seems very likely that (for odd p) exp_{p}(SU(2n)) = exp_{p}(Sp(n)), but this is not something that we can prove.

The last “=” follows from Harris’ p-equivalence

SO(2n + 1) × S^{2n+2} ≃ SO(2n + 2)

and the p-equivalence ΩS^{2n+2} ≃ S^{2n+1} × ΩS^{4n+3}. Then [7] shows that exp_{p}(S^{2n+2}) = 2n + 1.

In [9, 5.7,5.8], the upper bound

exp_{p}(v_{1}^{−1}SU(n)) < (n − 1)(1 + \frac{1}{p−1} + \frac{1}{(p−1)^{2}} )

was established. We can give the following upper bound for exp_{p}(SU(n)), but we feel that it is rather weak.

Proposition 1.17. exp_{p}(SU(n)) ≤ \frac{1}{2(p−1)}(n + \frac{p−3}{2})^{2}.

Proof. The p-local splitting of SU(n) proved in [14] and stated in Proposition 2.11, along with the bound for spheres ([7]), implies that if n = N + 1 + (p − 1)i with 1 ≤ N ≤ p − 1, then

exp_{p}(SU(n)) ≤ ∑_{t=0}^{i} N + (p − 1)t.

This can be simplified to \frac{1}{2(p−1)}(n − N + p − 2)(n + N − 1), and this is less than or equal to its value when N = \frac{p−1}{2}.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 9

When p = 2, a result similar to Theorem 1.1 could probably be obtained by a similar analysis, although it will be more complicated due to d_{3}-differentials and exotic extensions in the UNSS. For Sp(n) and SO(n), Harris’s p-equivalences do not hold for p = 2, and so there will be nothing like Corollary 1.16. A non-tractable bound for exp_{2}(Sp(n)), similar to that for exp_{p}(SU(n)) given in (1.13) and (1.14), was obtained in [5], but our goal in this paper is tractable bounds, such as that of Theorem 1.1.

The author would like to thank Martin Bendersky and Huajian Yang for their insights into this work, and the University of Oxford for support while some of this work was performed.

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

In this section, we mostly review known results. Although some of these results are also true when p = 2, it will simplify exposition to assume that p is an odd prime.

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

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

where q = 2p−2, a notation that will be used consistently throughout this paper, and M^{t}(n) denotes the Moore space S^{t−1}∪_{n}e^{t}. Here the direct limit is taken over increasing values of e and k using Adams maps M^{t+qp^{e}}(p^{e}) → M^{t}(p^{e}) and canonical maps M^{t}(p^{e+1}) → M^{t}(p^{e}). Roughly speaking, the v_{1}-periodic homotopy groups describe the portion of the actual homotopy groups which are detected by K-theory. This definition was given in [10], where their relationship with actual homotopy groups of spherically resolved spaces was established.

A space is said to be spherically resolved if it can be built from odd-dimensional spheres by a finite number of fibrations. Then SU(n) is spherically resolved from S^{3}, S^{5}, …, S^{2n−1} using the fibrations

SU(i − 1) → SU(i) → S^{2i−1}, (2.2)

starting with SU(1) = ∗. It was shown in [10] that if X is spherically resolved, then

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

10 D. DAVIS

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

Next we review the unstable cobar complex, which can be used to compute the UNSS for spherically resolved spaces. 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 ⊗ M. Tensor products are always over BP_{∗}. The stable cobar complex SC^{∗}(M) is defined by

SC^{s}(M) = Γ ⊗ Γ ⊗ ⋯ ⊗ Γ ⊗ M,

with s copies of Γ, and differential d given by

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

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

Definition 2.3. [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 ⊗ M,

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

This unstable condition will pervade our computations. Usually we will have |m| = 2d + 1, for which the condition becomes ∑ i_{j} ≤ d.

Define UC^{0}(M) = M, and UC^{s}(M) = U(UC^{s−1}(M)). If M is a Γ-comodule, then the differential d of the stable cobar complex of M induces a differential on

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 11

the subcomplex UC^{∗}(M). 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. We will write UC^{∗}(X) for the complex UC^{∗}(P (BP_{∗}X)), whose homology is E_{2}(X).

In [3], the v_{1}-periodic UNSS was defined and shown to satisfy the following very nice property.

Theorem 2.4. If p is odd and X is spherically resolved, the v_{1}-periodic UNSS of X satisfies

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

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

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

We will use the unstable cobar complex for the unlocalized UNSS, but, as we are dealing exclusively with v_{1}-periodic classes, we can, in effect, act as if it satisfies the first two properties of Theorem 2.4.

The v_{1}-periodic UNSS constructed in [3] mimicked the original, less satisfactory, definition of v_{1}^{−1}π_{∗}(X) given in [9]. Alternatively, one can mimic the definition in 2.1 and obtain a v_{1}-periodic UNSS which has a natural unstable cobar complex. This complex is obtained from the ordinary unstable cobar complex by tensoring with Q/Z_{(p)} and localizing over powers of v_{1}. Filtrations in this spectral sequence are 1 lower than those of 2.4. However, this cobar complex does not appear to be as convenient as the ordinary unstable cobar complex for the computations required in the study of SU(n). We may return to that complex in a subsequent paper.

12 D. DAVIS

We return now to properties of the unstable cobar complex which will be useful in our calculations. 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^{p^{i}} and exp(log x) = log(exp x) = x. Here {m_{i}} is a set of polynomial generators for H_{∗}BP with |m_{i}| = |v_{i}| and

v_{n} = pm_{n} − ∑_{j=1}^{n−1} m_{j}v_{n−j}^{p^{j}}.

The following lemma of Bendersky ([2]) is useful.

Lemma 2.5. 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 2(k − j). Note that ψ(x_{2k+1}) involves only those x_{2j+1} for which j ≡ k mod p − 1.

We will need the following explicit computation, which we prove at the end of this section.

Proposition 2.6. 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}.

All terms in ∑_{i}^{F^{∗}} h_{i} are multiples of terms h_{1}^{e_{1}} ⋯ h_{r}^{e_{r}}v, with v ∈ BP_{∗} and ∑ e_{i} ≤ \frac{1}{2}|v| + 1.

The coaction formula of 2.5 will be extremely important, as it determines the boundary homomorphism in the exact sequence associated to the fibration (2.2). 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}). (2.7)

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.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 13

We will make frequent use of the following result for the spheres. Most of the 1-line part was proved in [4], with (2.10) being from [6, 2.11], while the 2-line part is from [1]. 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 v_{1}^{−1}E_{2}^{s,t}(S^{2n+1}) are

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

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

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

α_{m/e} ≡ −v_{1}^{m−e}h_{1}^{e} mod S^{2e−1} (2.9)

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

α_{m/e} ≡ −sv_{1}^{m−1}h_{1} mod p. (2.10)

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

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

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

The precise form of the classes on the 2-line in (3) and (4) differs here from previous treatments such as [1] and [11], which were always written mod boundaries or classes which desuspend. The classes are easily seen to be precise by the method employed in the proof of Lemma 4.6 below.

Our final preliminary is to introduce the spaces which are the factors in the decomposition of [14] of the p-localization of SU(n) and its quotients as a product of p − 1 spaces.

Proposition 2.11. ([14]) 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};

14 D. DAVIS

(3) H^{∗}(X_{j}^{i}(N)) 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 2.5. Theorem 2.4 applies to the spaces X_{j}^{i}(N), and, because of the sparseness results for spheres given in Theorem 2.8, the following is immediate.

Proposition 2.12. If 1 ≤ N ≤ p − 1 and s = 1 or 2, then

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

if

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

Combining this with Theorem 2.4, we see that it suffices to show that if n = N + 1 + (p − 1)i, there is an integer m such that ν_{p}(|E_{2}^{1,2N+1+qm}(X_{0}^{i}(N))|) is at least as large as (1.9).

We close this section with the proofs of Proposition 2.6 and Theorem 1.10.

Proof of Proposition 2.6. The first part was proved as [11, 2.4]. From that proof, we have

\begin{matrix}∑ & F^{∗} \\ i \end{matrix} h_{i} = ∑_{i≥0}( ∑_{j,k≥0} h_{k}^{p^{j}}m_{j})^{i+1}b_{i}.

Here b_{i} ∈ H_{2i}(BP), and by writing m_{j} and b_{i} on the right, we mean that we are multiplying by their conjugates, which becomes the right action of BP_{∗} on BP_{∗}BP after binomial coefficients are used to write these elements as elements of BP_{∗} ⊂ H_{∗}BP. It turns out that b_{i} = 0 unless i is a multiple of p − 1, but that is not relevant to the proof.

Since p^{j} − \frac{1}{2}|m_{j}| = 1, each term in (∑ h_{k}^{p^{j}}m_{j})^{i+1} is a multiple of some h_{1}^{e_{1}} ⋯ h_{r}^{e_{r}}m with m ∈ H_{∗}BP and ∑ e_{i} − \frac{1}{2}|m| = i + 1. Then ∑ e_{i} − \frac{1}{2}|mb_{i}| = 1.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 15

Proof of Theorem 1.10. We begin by showing that if N > k, then the two inequalities are true without the −1 on the RHS. For the first,

S′_{j} = [\frac{i − N − 1}{p}] − [\frac{j − N − 1}{p}] = j + [\frac{k − N − 1}{p}] − [\frac{j − N − 1}{p}].

The putative lower bound is [((k + 2)(p − 1) + p^{2}j + N − pj)/p^{2}], and so the desired inequality is equivalent to

[\frac{k − N − 1}{p}] ≥ [\frac{(k + 2)(p − 1) + N − pj}{p^{2}} ] + [\frac{j − N − 1}{p}]. (2.13)

Since 1 ≤ k < N ≤ p − 1, the LHS of (2.13) equals −1. Increasing j by p does not change either side of this inequality, and so it suffices to consider j between N + 1 and N + p, inclusive, so that [(j − N − 1)/p] = 0. The RHS of (2.13) satisfies

RHS ≤ [\frac{(k + 2)(p − 1) + N − p(N + 1)}{p^{2}}] = [\frac{(k + 2 − N)(p − 1) − p}{p^{2}}] < 0,

since k + 2 − N ≤ 1. Thus LHS ≥ RHS, as desired.

Next we prove that if N > k, (1.12) is true without the −1 on the RHS. We have

S′′_{j} = [\frac{pj + k − 1 − kp − N}{p^{2}} ] − [\frac{j − kp − N}{p^{2}} ],

while the putative lower bound is [(N + 1 + (p − 1)(pj + k) + p^{2} − p)/p^{3}]. Increasing j by p^{2} increases both sides by p − 1. Thus we may assume that j = kp + N + Δ for 0 ≤ Δ < p^{2}, so that [(j − kp − N)/p^{2}] = 0. After k is subtracted from both sides, the desired inequality becomes, with D = N − k,

[\frac{(D + Δ)p − D − 1}{p^{2}}] ≥ [\frac{(D + Δ + 1)(p^{2} − p + 1) − 1}{p^{3}}]. (2.14)

Since 0 < D < p − 1, the LHS of (2.14) equals [(D + Δ − 1)/p], and so letting s = D + Δ reduces (2.14) to

[\frac{s − 1}{p}] ≥ [\frac{s(p^{2} − p + 1)}{p^{3}}] (2.15)

with s > 0. For 1 ≤ s ≤ p, both sides of (2.15) are 0. Since increasing s by p increases the LHS of (2.15) by 1, and increases the RHS of (2.15) by no more than 1, this establishes (2.15), and hence the desired result.

Now assume N ≤ k. Then (1.11) becomes

j + [\frac{k − N − 1}{p}] − [\frac{j − N}{p} ] ≥ [\frac{N + 1 + (p − 1)(jp + k) + 2p − 3}{p^{2}}] − 1.

We subtract j from both sides, and note that increasing j by p changes both sides by the same amount, so that it suffices to consider N ≤ j < N + p, and the desired inequality is

[\frac{k − N − 1}{p}] ≥ [\frac{(k − N + 2)(p − 1) − p(j − N)}{p^{2}}] − 1.

For 0 ≤ k − N < p and 0 ≤ j − N < p, this is always satisfied, with equality if and only if k − N = 0 and j − N = 0 or 1, as claimed.

Similarly, if N ≤ k, then (1.12) becomes

[\frac{pj + k − 1 − kp − N}{p^{2}} ] − [\frac{j − kp − N + 1}{p^{2}}] ≥ [\frac{N + 1 + (p − 1)(pj + k + p)}{p^{3}}] − 1,

which is unaffected when j is increased by p^{2}. We let D = k − N and j = kp + N − 1 + ap + b for 0 ≤ a, b, D < p. The inequality simplifies to

[\frac{(b − D − 1)p + D − 1}{p^{2}}] ≥ [\frac{(b − D)(p^{2} − p) − D − ap^{2} + 1}{p^{3}}] − 1,

which is always satisfied, with equality if and only if D = 0, a = 0, and b = 0 or 1.

3. The cellular spectral sequence

In this section, we sketch the proof of Theorem 1.4 when n is p-cleaner and N ≠ k by showing, modulo two proofs postponed until Section 4, that for suitably chosen m

ν_{p}(|E_{2}^{1,2N+1+qm}(X_{0}^{i}(N))|) ≥ n − 1 + S_{j+ϵ}. (3.1)

The relations (1.5) and (1.6) which define N, i, j, and k will be used throughout the rest of the paper. Also ϵ will always be as in (1.7). We will usually think of N and m as being fixed, so that they may be omitted from some notation. The expression E_{2}^{s}(−) will always mean E_{2}^{s,2N+1+mq}(−), and we let X_{ℓ}^{i} = X_{ℓ}^{i}(N). Note that X_{ℓ}^{ℓ} = S^{2N+1+qℓ}.

We can organize the computation of E_{2}^{s}(X_{0}^{i}) as a cellular spectral sequence (CSS) with E_{1}^{s,ℓ} = E_{2}^{s}(X_{ℓ}^{ℓ}) for 1 ≤ s ≤ 2 and 0 ≤ ℓ ≤ i, and d_{r} : E_{r}^{1,ℓ} → E_{r}^{2,ℓ−r} induced

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 17

by pulling the class in E_{2}^{1}(X_{ℓ}^{ℓ}) back to E_{2}^{1}(X_{ℓ−r+1}^{ℓ}) and then applying ⟶^{∂}E_{2}^{2}(X_{ℓ−r}^{ℓ−r}). We will usually omit the subscripts of differentials. This subscript just equals the difference of the second superscripts. It serves no major purpose, although we should keep in mind that short differentials take place before longer ones. Our desired E_{2}^{s}(X_{0}^{i}) is filtered with subquotients E_{∞}^{s,ℓ}, 0 ≤ ℓ ≤ i. Each group E_{2}^{1,t}(X_{0}^{i}) is cyclic, since, if nonzero, it is isomorphic to E_{2}^{1,t}(SU(n)), which is cyclic by [2]. Thus our desired p-exponent is given by the sum

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

Note that for s = 1 or 2

E_{1}^{s,ℓ} = Z/p^{min(N+(p−1)ℓ, ν(m−ℓ)+1)}.

This is a consequence of Theorem 2.8(1). Perhaps “cellular” is not the best name for this spectral sequence, since each new sphere doubles the number of cells of the CW -complex SU(n), but we hope that the name suggests that it is measuring the effect of building SU(n) one sphere at a time.

We are computing the CSS only in certain degrees that yield large groups. In [11] the entire CSS was computed for SU(n) with n ≤ p^{2} − p + 1, but to do this for arbitrary n would cause too many cases to have to be considered. We will always choose m so that ν(m − i) is large, although not arbitrarily large. The largest groups E_{2}^{1,2N+1+qm}(X_{0}^{i}) are obtained when m is chosen in this way. There are variations in the exact form of the CSS depending upon relationships among j, k, and N. In this section, we will establish (3.1) under the hypothesis that N ≠ k and n is p-cleaner, relegating two proofs to Section 4. Then in Sections 5 and 6 we will discuss the modifications required when these hypotheses are not satisfied.

The following result, which will be proved in Section 4, is essential to our bounds on the size of the v_{1}-periodic homotopy groups. The reader should keep in mind that we are only dealing with groups of total degree 2N + 1 + mq with ν(m − i) large. Here we do not need to make any assumptions about n, N, and k.

Proposition 3.2. All differentials E^{1,b} → E^{2,a} are 0 if b < i and a ≥ j + ϵ.

18 D. DAVIS

This says that there are no differentials in a broad band in the CSS. Indeed, the only possible nonzero differentials either emanate from the top group, E^{1,i}, or else hit into the bottom band, E^{2,t} with t < j + ϵ. Recall j = [\frac{i−1}{p}].

On the other hand, it will usually be the case that all groups E_{1}^{2,a} with a < j + ϵ are killed by differentials. There is an exception to this when N = k and a = j, but aside from that, the cases in which this does not happen are so rare and so difficult to describe that we shall not attempt to describe them explicitly in this paper. We are making a small sacrifice in optimality here for a big gain in simplicity of statement. In Section 4 we will describe the general pattern of differentials by which the groups E_{1}^{2,a} with a < j + ϵ are killed. In Example 6.15, we give an example in which a group E_{∞}^{2,a} with a < j is nonzero.

Now we can see how (3.1) is obtained when N ≠ k and n is p-cleaner. As just noted, all of the groups E^{2,a} with a < j + ϵ are usually killed by differentials from E^{1,ℓ} with ℓ < i, while by Proposition 3.2 no other groups E^{2,a} are killed by such differentials.

We wish to choose m so that \overline{e} := ν(m − i) + 1 is as large as possible with the property that the generator α_{(m−i)/\overline{e}} of E_{2}^{1}(X_{i}^{i}) pulls back to a class z ∈ E_{2}^{1}(X_{j+ϵ}^{i}). This implies that differentials E^{1,i} → E^{2,a} are 0 if a ≥ j + ϵ. Later in this section we shall determine this \overline{e}.

Then ∑_{ℓ≤i} ν(|E_{∞}^{1,ℓ}|) = A − B, where A = ∑_{ℓ≤i} ν(|E_{1}^{1,ℓ}|) and B is the sum of the exponents of the orders of the groups killed by differentials. Since ν(m − i) is large, ν(m − ℓ) = ν(i − ℓ) for 0 ≤ ℓ < i, and hence, for d > 0, we have ν(|E_{1}^{1,i−d}|) = min(1 + ν(d), N + (p − 1)(i − d)). We may assume without loss of generality that this will always equal 1 + ν(d), for in the rare case in which N + (p − 1)(i − d) < 1 + ν(d), both A and B will be modified in the same way. Thus A = \overline{e} + i + ν(i!), and

B ≤ ∑_{ℓ=0}^{j−1+ϵ} ν(|E_{1}^{2,ℓ}|) = ∑_{ℓ=0}^{j−1+ϵ}(1 + ν(i − ℓ)) = j + ϵ + ν(i!) − ν((i − j − ϵ)!).

We obtain

∑_{ℓ=0}^{i} ν(|E_{∞}^{1,ℓ}|) = A − B ≥ \overline{e} + i − j − ϵ + ν((i − j − ϵ)!). (3.3)

In order to explain how we obtain an estimate of the threshold value \overline{e} of ν(m − i) + 1, we introduce some notation and conventions for working in the unstable cobar complex. Most of these conventions were employed in [11].

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 19

Convention 3.4. (1) We are dealing with UC^{∗}(X_{0}^{i}(N)), where 1 ≤ N ≤ p−1, in total degree 2N +1+qm, with ν(m−i) large. Also, i = jp+k with 1 ≤ k ≤ p. We write X_{a}^{b} for subquotients of X_{0}^{i}(N).

(2) We write y_{ℓ} for the generator x_{2N+qℓ+1} of BP_{∗}(X_{0}^{i}(N)).

(3) We write h for h_{1} and v for v_{1}.

(4) We often omit powers of v on the left side of terms of the cobar complex. In particular, h^{e}y_{ℓ} means v^{m−e−ℓ}h^{e}y_{ℓ}. This can be done because we are in fixed degree 2N + 1 + qm.

(5) If ∑ c_{ℓ}y_{ℓ} and ∑ d_{ℓ}y_{ℓ} are classes in UC^{s}(X), we write

∑ c_{ℓ}y_{ℓ} ≡ ∑ d_{ℓ}y_{ℓ} mod L

if, for each ℓ, c_{ℓ} − d_{ℓ} is defined on a lower sphere than is d_{ℓ}. The “L” stands for “lower sphere.” This will usually be used in the context of d_{ℓ} being the “leading term” of c_{ℓ}, i.e., the term which desuspends least far. Note that the dimension of the lower sphere varies from term to term in this sum.

The following definition will be extremely important.

Definition 3.5. The excess exc(γ) of an element γ of Γ̄^{s} is defined to be the smallest n such that γι_{2n+1} is an element of UC^{s}(M).

This means that if γ = γ_{1} ⊗ ⋯ ⊗ γ_{s}, then for 1 ≤ i ≤ s, γ_{i} ⊗ (γ_{i+1} ⊗ ⋯ ⊗ γ_{s}ι_{2n+1}) must satisfy 2.3. In Lemma 4.2, we give a formula for excess of certain monomials when s = 2.

One of our main results for pulling back is given in the following lemma, which will be proved in Section 4.

Lemma 3.6. Assume that N ≠ k, ℓ ≥ j + ϵ, and

(i − ℓ)(p − 1) − ν((i − ℓ − 1)!) + S_{ℓ} < e_{i} ≤ i(p − 2) + ℓ + N − ν((i − ℓ)!) + S_{ℓ}. (3.7)

Let z be a cycle in UC^{1}(X_{ℓ+1}^{i}) satisfying

z = ∑_{t=ℓ+1}^{i}(c_{t}h^{e_{t}} + L_{t})y_{t} (3.8)

20 D. DAVIS

with

e_{t} = e_{i} − (i − t)(p − 2) + ν((i − t)!) − S_{t}, (3.9)

c_{t} ∈ Z_{(p)}, and exc(L_{t}) < e_{t}. Assume also that if ℓ + 1 ≡ kp + N mod p^{2} and c_{ℓ+2} ≢ 0 mod p, then c_{ℓ+1} ≡ c_{ℓ+2} mod p. Then z pulls back to a cycle z_{1} in UC^{1}(X_{ℓ}^{i}) satisfying z_{1} = z + (c_{ℓ}h^{e_{ℓ}} + L_{ℓ})y_{ℓ}, where e_{ℓ} satisfies (3.9) and exc(L_{ℓ}) < e_{ℓ}. Moreover, if ℓ ≡ kp + N mod p^{2} and c_{ℓ+1} ≢ 0 mod p, then c_{ℓ} ≡ c_{ℓ+1} mod p in z_{1}.

Note that this result is self-perpetuating as long as e_{i} satisfies (3.7). The requirement that n be p-cleaner is needed in order that the first inequality in (3.7) is satisfied for all ℓ ≥ j + ϵ. The upper bound on e_{i} in (3.7) guarantees that z_{1} satisfies the unstable condition.

Now we can easily obtain our estimate for \overline{e} of (3.3). We wish to choose e_{i} so that e_{j+ϵ}, defined by (3.9), satisfies e_{j+ϵ} = N + (j + ϵ)(p − 1), since this is the largest value for which h^{e_{j+ϵ}}y_{j+ϵ} satisfies the unstable condition of 2.3. Thus by Lemma 3.6 there is a cycle z in UC^{1}(X_{j+ϵ}^{i}) satisfying

z = ∑_{t=j+ϵ}^{i}(c_{t}h^{e_{t}} + L_{t})y_{t}

with

e_{i} = N + (j + ϵ)(p − 1) + (i − j − ϵ)(p − 2) − ν((i − j − ϵ)!) + S_{j+ϵ} (3.10)

The RHS of (3.10) is our value for \overline{e}. Substituting this into (3.3) yields (3.1) when N ≠ k and n is p-cleaner, modulo the proofs of 3.2 and 3.6. These proofs will form the main content of Section 4.

4. Proofs of claims about the CSS

In this section we establish the claims about the CSS made in Section 3, hence establishing (3.1) when N ≠ k and n is p-cleaner. That is, we prove Proposition 3.2 and Lemma 3.6. We also discuss the pattern of differentials which usually kill the groups E_{1}^{2,a} when a < j + ϵ.

We begin with some more general background regarding the unstable cobar complex. These general results do not assume any of the special notation and conventions of Section 3. The following basic formulas were used or proved in [6].

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 21

Lemma 4.1. (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} + ∑ a_{i}v_{1}^{p+1−i}p^{i}h_{1}^{i}, where a_{i} ∈ Z.

(3) d(v_{1}^{n}) = 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}). For example, it is used in establishing the following formula about excess, which was defined in 3.5.

Lemma 4.2. If a ≤ b and a ≤ d, then

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

Proof. The effect of v^{e} on the excess is clear, and so we will assume e = 0. Using Lemma 4.1(1), we can write the given monomial as

(v − ηv)^{a}h^{b−a} ⊗ v^{c}h^{d} = v^{a}h^{b−a} ⊗ v^{c}h^{d} + ⋯ ± h^{b−a} ⊗ v^{a+c}h^{d},

which has excess

max(max(b − a − (p − 1)(c + d), d), max(b − a − (p − 1)(a + c + d), d))

= max(b − a − (p − 1)(c + d), d).

On the other hand, it can also be written as

h^{b} ⊗ v^{c}(v − ηv)^{a}h^{d−a} = h^{b} ⊗ v^{a+c}h^{d−a} + ⋯ ± h^{b} ⊗ v^{c}h^{d−a}v^{a},

which has excess

max(max(b − (p − 1)(c + d), d − a), max(b − (p − 1)(c + d), d − pa))
= max(b − (p − 1)(c + d), d − a).

The excess of the given monomial is the minimum of the excesses of these two ways of writing it. Letting B = b − (p − 1)(c + d), this is

min(max(B − a, d), max(B, d − a)) = max(B, d) − min(|B − d|, a),

as claimed.

22 D. DAVIS

We will need the following fact, which follows from Lemma 4.1(2). Lemma 4.3. In E_{2}^{1}(S^{3}) h_{1}^{p}v_{1} ≡ v_{1}^{p}h_{1} mod S^{1}.

The following result, proved in [11, 2.12], will be very useful. Lemma 4.4. 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} if n > 1.

The following lemma, proved in [11, 2.13], will also be used many times. It is the one place where the number of v_{1}’s on the left is important. Lemma 4.5. d(v_{1}^{ℓ}h_{1}^{n+1}) ≡ −(ℓ + n + 1)v_{1}^{ℓ}h_{1} ⊗ h_{1}^{n} mod S^{2n−1}.

A related result about the unstable cobar complex for spheres is given in the next lemma. Lemma 4.6. Let ν = ν(σ), and let

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

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

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

where u is a unit in Z_{(p)}, and exc(L′) < e + ν − p + 2. The same conclusion holds for z = ϵv^{σ−e+p−2}h ⊗ h^{e−p+1} + L. Proof. If ϵ ≡ 0 mod p, then z is defined on S^{2e−2p+1}. Since, by 2.8(1,5) or [1, 3.3], the iterated suspension

E_{2}^{2,2e−2p+1+σq}(S^{2e−2p+1}) → E_{2}^{2,2e−2p+2ν+3+σq}(S^{2e−2p+2ν+3})

is the 0-homomorphism, z is a boundary on S^{2e−2p+2ν+3}, as desired.

If ϵ ≢ 0 mod p, we claim that, with E = e + ν − p + 2,

(4.7)  p^{σ−E}h^{σ} = (v − ηv)^{σ−E}h^{E}
= v^{σ−E}h^{E} + L′

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 23

with exc(L′) < E, while, if σ = sp^{ν},

(4.8)  d(p^{σ−E}h^{σ}) = sp^{σ−E+ν}h ⊗ h^{σ−1} + L_{1}
= sh ⊗ (v − ηv)^{σ−E+ν}h^{E−ν−1} + L_{1}
(4.9)  = s(v − ph)^{σ−E+ν}h ⊗ h^{E−ν−1} + L_{2}
(4.10)  = sv^{σ−E+ν}h ⊗ h^{E−ν−1} + L_{3}
(4.11)  = sv^{σ−e−1}h^{e} ⊗ h + L_{4},

where exc(L_{i}) < E − ν − 1 for 1 ≤ i ≤ 4. Then, if us ≡ 1 mod p,

d(uϵ(v^{σ−E}h^{E} + L′)) = ϵv^{σ−e−1}h^{e} ⊗ h + uϵL_{4} = z + (uϵL_{4} − L)

with uϵL_{4} − L a cycle of excess less than e − p + 1, and so, as in the first part of the lemma, uϵL_{4} − L = d(L′′) with exc(L′′) < E, completing the proof, modulo verification of the numbered equations above.

At (4.7), we merely use 4.1(1), and at (4.8) we just need

d(h^{σ}) = ∑_{j=1}^{σ−1} \binom{σ}{j}h^{j} ⊗ h^{σ−j}.

The term with j = 1 has the largest excess. At (4.9), we have expanded (v−ηv)^{σ−E+ν} and incorporated all terms except the first into L_{2}, since η(v)h^{E−ν−1} = h^{E−ν−1}v has lower excess than h^{E−ν−1}. The term v^{σ−E+ν} is then moved to the left of the ⊗, where it becomes (ηv)^{σ−E+ν}, which we expand, using 4.1(1). In (4.10), we note that when (4.9) is expanded, all terms except the first have lower excess, since, if j ≥ 1,

(ph)^{j}h ⊗ h^{E−ν−1} = (v − ηv)^{j−1}h^{2} ⊗ (v − ηv)h^{E−ν−2}.

Finally, at (4.11) we use Lemma 4.4 and the definition of E.

The last sentence of the lemma follows from the above analysis and Lemma 4.4.

Another result of the same sort is the following.

Lemma 4.12. If x ∈ UC^{2,2n+1+σq}(S^{2n+1}) is an unstable cycle such that {x} has order p^{j} in E_{2}^{2,2n+1+σq}(S^{2n+1}), then x = d(uv^{σ−n−j}h^{n+j} + L), with u a unit in Z_{(p)} and exc(L) < n + j. Moreover, if e < j, then exc(p^{e}L) < n + j − e.

24 D. DAVIS

Proof. By Theorem 2.8(4), there is a unit u in Z_{(p)} such that

x − ud(p^{σ−n−j}h^{σ}) = d(L_{1})

with exc(L_{1}) ≤ n. We have

p^{σ−n−j}h^{σ} = (v − ηv)^{σ−n−j}h^{n+j} = v^{σ−n−j}h^{n+j} + L_{2},

with exc(L_{2}) < n + j. The result follows, with L = uL_{2} + L_{1}. The last claim of the lemma follows from the explicit form of L_{2}; we can say nothing about multiplication by p^{e} decreasing the excess of L_{1}.

Corollary 4.13. If x ∈ UC^{2,2n+1+σq}(S^{2n+1}) is an unstable cycle with exc(x) ≤ e, then x = d(uv^{σ−t}h^{t} + L) for some t ≤ e + ν(σ) + 1, u a unit, and exc(L) < t.

Proof. The class {x} is defined in UC^{2,2e+1+σq}(S^{2e+1}), where it has order p^{j} for some j ≤ ν(σ) + 1. The result now follows from Lemma 4.12.

The following lemma will be very useful.

Lemma 4.14. The boundary ∂ : E_{2}^{1}(X_{t}^{t}) → E_{2}^{2}(X_{0}^{t−1}) satisfies

∂((c_{t}h^{e_{t}} + L_{t})y_{t}) = (c_{t}h^{e_{t}} + L_{t}) ⊗ ∑_{ℓ<t} Hy_{ℓ},

where

H = (1 + h − hv + hv^{2} − \binom{p−1}{2}h^{2}v + ⋯)_{(t−ℓ)(p−1)}^{N+ℓ(p−1)}
= \binom{N+ℓ(p−1)}{t−ℓ}(h^{t−ℓ} + ∑_{r} a_{r}h(r)v(r)),

where a_{r} is an integer, h(r) is a product of h_{i}’s, and v(r) ∈ \overline{BP}_{∗}, with exc(h(r)v(r)) ≤ t − ℓ.

Proof. The first two lines follow from (2.7), 2.6, and 2.5. We use the last part of Proposition 2.6 to see that the terms are of the form

\frac{(ℓ(p − 1) + N)!}{γ_{1}! ⋯ γ_{r}!(ℓ(p − 1) + N − ∑ γ_{i}d_{i})!} ∏(h(i)v(i))^{γ_{i}} (4.15)

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 25

with exc(h(i)v(i)) ≤ 1, d_{i} = \frac{1}{q}|h(i)v(i)|, and ∑ γ_{i}d_{i} = t − ℓ. The result follows, with h(r)v(r) = ∏(h(i)v(i))^{γ_{i}}, once we note that the coefficient in (4.15) equals

\frac{(ℓ(p − 1) + N)!}{(ℓ(p − 1) + N − ∑ γ_{i}d_{i})!(∑ γ_{i}d_{i})!} · \frac{(∑ γ_{i})!}{γ_{1}! ⋯ γ_{r}!} · \frac{(∑ γ_{i}d_{i})!}{(∑ γ_{i})!},

of which the first fraction equals \binom{ℓ(p−1)+N}{t−ℓ} and the other two are integers.

Now we begin to discuss more specifically the claims of Section 3. The notation of Theorem 1.4 and Convention 3.4 are in effect for the rest of the paper. We begin by proving the important pulling back lemma, Lemma 3.6.

Proof of Lemma 3.6. Let S_{t}, S′_{t}, and S′′_{t} be as in Theorem 1.4. Note that

S′_{t} = \begin{cases}S′_{t+1} + 1 & if t ≡ N mod p \\ S′_{t+1} & if t ≢ N mod p \end{cases}

and

S′′_{t} = \begin{cases}S′′_{t+1} + 1 & if t + 1 ≡ kp + N mod p^{2} \\ S′′_{t+1} & if t + 1 ≢ kp + N mod p^{2}. \end{cases}

Let

e = e_{i} − (i − ℓ)(p − 2) + ν((i − ℓ − 1)!) − S_{ℓ} − 1. (4.16)

We will show below that for every term (c_{t}h^{e_{t}} + L_{t})y_{t} in (3.8), ∂((c_{t}h^{e_{t}} + L_{t})y_{t}) in UC^{2}(X_{ℓ}^{ℓ}) is a sum of terms T satisfying exc(T) ≤ e, and so by Lemma 4.13, ∂(z) = d(ch^{e+ν+1} + L)y_{ℓ}, where ν = ν(m − ℓ) and exc(L) < e + ν + 1. Since ν(m − i) is large, we have ν(m − ℓ) = ν(i − ℓ), and so e + ν + 1 is equal to the RHS of (3.9) with t = ℓ. The conclusion of Lemma 3.6 (about the cycle z_{1}) is now immediate, using z_{1} = z − (ch^{e+ν+1} + L).

Now we establish this claim about the terms T in ∂((c_{t}h^{e_{t}} + L_{t})y_{t}), beginning with the case ℓ + 1 ≢ kp + N mod p^{2}. Let X_{ℓ+1} satisfy X_{ℓ+1} = c_{ℓ+1}h^{e_{ℓ+1}} or exc(X_{ℓ+1}) < e_{ℓ+1}. Here e_{ℓ+1} is as in (3.9). Let

δ = \begin{cases}1 & if ℓ ≡ N mod p \\ 0 & otherwise. \end{cases}

26 D. DAVIS

Then we have

exc(∂(X_{ℓ+1}y_{ℓ+1})) = exc(X_{ℓ+1} ⊗ (N + ℓ(p − 1))h)
≤ e_{ℓ+1} − (p − 1) − δ
= e_{i} − (i − ℓ − 1)(p − 2) + ν((i − ℓ − 1)!)
−S_{ℓ+1} − (p − 1) − δ
= e_{i} − (i − ℓ)(p − 2) + ν((i − ℓ − 1)!) − S_{ℓ} − 1,

as desired. The δ occurs in the first step here because either exc(X_{ℓ+1}) < e_{ℓ+1} or else X_{ℓ+1} is a multiple of h^{e_{ℓ+1}}, the excess of which is decreased by 1 when it is multiplied by p.

All other terms in ∂(z) satisfy (with t > ℓ + 1)

exc(X_{t} ⊗ a_{r}h(r)v(r)) ≤ max{e_{t} − (t − ℓ)(p − 1), t − ℓ}. (4.17)

We have

e_{t} − (t − ℓ)(p − 1) = e_{i} − (i − t)(p − 2) + ν((i − t)!) − S_{t} − (t − ℓ)(p − 1)
≤ e_{i} − (i − ℓ)(p − 2) + ν((i − ℓ − 1)!) − S_{t} − (t − ℓ)
≤ e_{i} − (i − ℓ)(p − 2) + ν((i − ℓ − 1)!) − S_{ℓ} − 1.

The last inequality is true because we assume ℓ + 1 ≢ kp + N mod p^{2}. We also have

t − ℓ ≤ i − ℓ ≤ e_{i} − (i − ℓ)(p − 2) + ν((i − ℓ − 1)!) − S_{ℓ} − 1, (4.18)

with the latter inequality implied by (3.7).

Next we verify the claim about excess when ℓ + 1 ≡ kp + N mod p^{2}. Now we have S′_{ℓ+1} = S′_{ℓ+2} + 1 and S′′_{ℓ} = S′′_{ℓ+1} + 1, which cause the inequalities proved above in the case ℓ + 1 ≢ kp + N to fail for exc((N + ℓ(p − 1))c_{ℓ+1}h^{e_{ℓ+1}} ⊗ h) if c_{ℓ+1} is a unit, and for exc(c_{t}h^{e_{t}} ⊗ a_{r}h(r)v(r)) if t = ℓ + 2, and c_{ℓ+2} and a_{r} are units. The special hypothesis in the lemma says that we are computing

∂(uh^{ê−p+1}y_{ℓ+1} + uhêy_{ℓ+2}),

with

ê = e_{i} − (i − ℓ − 2)(p − 2) + ν((i − ℓ − 2)!) − S_{ℓ+2}

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 27

and u a unit in Z_{(p)}. We write ℓ + 1 = Kp + N, with K ≡ k mod p, and, working mod terms of excess less than ê − 2(p − 1), find that

∂(uh^{ê−p+1}y_{ℓ+1} + uhêy_{ℓ+2})
≡ u((Kp + N − 1)(p − 1) + N)(h^{ê−p+1} ⊗ h
+hê ⊗ (\frac{(Kp+N−1)(p−1)+N−1}{2}h^{2} − hv))y_{Kp+N−1}
(4.19) ≡ u(h^{ê−p+1} ⊗ h + \frac{p(N−1−K+Kp)}{2}hê ⊗ h^{2} − hê ⊗ (vh − ph^{2}))y_{Kp+N−1}.

In the first ≡, we have argued as in the proof of Lemma 4.14, and in the second ≡ we have used Lemma 4.1(1). In (4.19), the two terms with phê ⊗ h^{2} can, by Lemma 4.1(1), be written as h^{ê−1} ⊗ h^{2}, which has excess ê − 1 − 2(p − 1), and we can use Lemma 4.4 to write hê ⊗ vh as (v^{p}h^{ê−p+1} + L) ⊗ h with exc(L) < ê− p + 1. The first term cancels the initial h^{ê−p+1} ⊗ h, and exc(L ⊗ h) ≤ ê − 1 − 2(p − 1). Thus (4.19) has excess

≤ ê − 2(p − 1) − 1
= e_{i} − (i − ℓ − 2)(p − 2) + ν((i − ℓ − 2)!) − S_{ℓ+2} − 2(p − 2) − 3
≤ e_{i} − (i − ℓ)(p − 2) + ν((i − ℓ − 1)!) − (S_{ℓ} − 2) − 3,

as claimed in (4.16). It is our lack of knowledge of the precise excess here that is the problem in the examples in the table in Section 1 where Theorem 1.4 does not agree with the actual value of exp_{3}(v_{1}^{−1}SU(n)).

Next we establish the technical condition in the last sentence of Lemma 3.6 when ℓ ≡ kp + N mod p^{2}. Excess considerations imply that the only term which we need to consider is c_{ℓ+1}h^{e_{ℓ+1}}y_{ℓ+1}, with c_{ℓ+1} a unit in Z_{(p)}. We will show that when this is pulled back to UC^{1}(X_{ℓ}^{ℓ+1}), the new term added is c_{ℓ+1}h^{e_{ℓ+1}−(p−1)}y_{ℓ}, mod L.

28 D. DAVIS

We have, mod L,

∂(c_{ℓ+1}h^{e_{ℓ+1}}y_{ℓ+1}) = (N + ℓ(p − 1))c_{ℓ+1}h^{e_{ℓ+1}} ⊗ hy_{ℓ}
≡ (N − k)pc_{ℓ+1}h^{e_{ℓ+1}} ⊗ hy_{ℓ}
≡ (k − N)pc_{ℓ+1}h ⊗ h^{e_{ℓ+1}−p+1}y_{ℓ}
≡ (k − N)c_{ℓ+1}h ⊗ h^{e_{ℓ+1}−p}y_{ℓ}
(4.20)  = \frac{N−k}{m−ℓ}c_{ℓ+1}d((h^{e_{ℓ+1}−p+1} + L′)y_{ℓ})

with exc(L′) < e_{ℓ+1} − p + 1. Here the first “=” follows from the method of Lemma 4.14, the first “≡” follows from the assumption that ℓ ≡ kp + N mod p^{2}, the next “≡” uses Lemma 4.1(1) to replace ph by v − η(v), the next “≡” uses Lemma 4.4, while the final “≡” uses Lemma 4.5 to control the coefficient and either Lemma 4.6 or Lemma 4.12 for the L′-part. Since m ≡ k and ℓ ≡ N mod p, the fraction in the last line of (4.20) is −1 mod p, and so, as claimed in the last line of Lemma 3.8, there is a cycle z_{1} satisfying

z_{1} ≡ c_{ℓ+1}h^{e_{ℓ+1}}y_{ℓ+1} + c_{ℓ+1}h^{e_{ℓ+1}−p+1}y_{ℓ} mod L.

This completes the proof of Lemma 3.6.

Next we prove Proposition 3.2.

Proof of Proposition 3.2. We will show that there is a cycle

∑_{ℓ=j+ϵ}^{b}(u_{ℓ}h^{e_{ℓ}} + L_{ℓ})y_{ℓ} (4.21)

with exc(L_{ℓ}) < e_{ℓ}, u_{ℓ} a unit in Z_{(p)}, e_{ℓ} ≤ (p − 1)ℓ + N, and (u_{b}h^{e_{b}} + L_{b})y_{b} a generator of E_{2}^{1}(X_{b}^{b}). The numbers e_{ℓ} will be less than or equal to the numbers a_{ℓ} defined in the following lemma. Note that the conclusion of the lemma implies that the cycle (4.21) satisfies the unstable condition.

Lemma 4.22. Suppose i, j, k, N, and ϵ are as in Theorem 1.4, and b < i. For ℓ going from b down to j + ϵ, define numbers M_{ℓ} and a_{ℓ} iteratively by M_{b} = 0, and

a_{ℓ} = M_{ℓ} + ν(i − ℓ) + 1

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 29

with

M_{ℓ} = max\{t − ℓ − ν(\binom{ℓ(p−1)+N}{t−ℓ}), a_{t} − (t − ℓ)(p − 1) : ℓ < t ≤ b\}. (4.23)

Then a_{ℓ} ≤ (p − 1)ℓ + N for j + ϵ ≤ ℓ ≤ b.

Proof. We first verify that the lemma is true when M_{ℓ} = t − ℓ − ν(\binom{ℓ(p−1)+N}{t−ℓ}). The desired inequality in this case is

t − ℓ − ν(\binom{ℓ(p−1)+N}{t−ℓ}) + ν(i − ℓ) + 1 ≤ ℓ(p − 1) + N.

Using Lemma 4.25, this inequality will follow from

t + ν((i − t)\binom{ℓp+N−t}{i−t}) + 1 ≤ ℓp + N,

which is implied by Lemma 4.26, since log_{p}(α) + 1 ≤ α.

Now we illustrate the proof in the general case by assuming that for ℓ < ℓ_{1} < ℓ_{2} < t, we have

a_{ℓ} = a_{ℓ_{1}} − (ℓ_{1} − ℓ)(p − 1) + ν(i − ℓ) + 1,
a_{ℓ_{1}} = a_{ℓ_{2}} − (ℓ_{2} − ℓ_{1})(p − 1) + ν(i − ℓ_{1}) + 1,
a_{ℓ_{2}} = t − ℓ_{2} − ν(\binom{ℓ_{2}(p−1)+N}{t−ℓ_{2}}) + ν(i − ℓ_{2}) + 1.

The desired inequality is then easily simplified to the following.

t + ν(i − ℓ) + ν(i − ℓ_{1}) + ν(i − ℓ_{2}) + 3 ≤ ℓ_{2}p + N + ν(\binom{ℓ_{2}(p−1)+N}{t−ℓ_{2}}). (4.24)

We write the final binomial coefficient as \frac{(ℓ_{2}(p−1)+N)···(t−ℓ_{2}+1)}{(ℓ_{2}p+N−t)!}, and observe that (i−ℓ), (i − ℓ_{1}), and (i − ℓ_{2}) are all factors of the numerator. Since

ℓ_{2}p + N ≥ (ℓ + 2)p + N ≥ (j + ϵ + 2)p + N ≥ i + 2p > t + 2p,

the only way that (4.24) might fail is if one of ν(i − ℓ), ν(i − ℓ_{1}), or ν(i − ℓ_{2}) is large. If more than one of these is large, then (ℓ_{2} − ℓ) is very large, and this will cause ℓ_{2}p + N − t to totally dominate all the ν(−)’s.

We assume that it is ν(i − ℓ_{1}) which is large, with the other cases being handled similarly. Similarly to Lemma 4.25, we have

ν(i − ℓ_{1}) − ν(\binom{ℓ_{2}(p−1)+N}{t−ℓ_{2}}) ≤ ν((i − t + ℓ_{2} − ℓ_{1})\binom{ℓ_{2}p+N−t}{i−t+ℓ_{2}−ℓ_{1}}),

30 D. DAVIS

and so, using also Lemma 4.26, the desired inequality, (4.24), follows from

ν(i − ℓ) + ν(i − ℓ_{2}) + [log_{p}(ℓ_{2}p + N − t)] + 3 ≤ ℓ_{2}p + N − t,

which is clear from the preceding discussion. Indeed, it it true that

ℓ_{2}p + N − t ≥ p^{ν(i−ℓ)} + p^{ν(i−ℓ_{2})}.

A final case deserving mention is the case ℓ = b. Here we require ν(i − b) + 1 ≤ (p − 1)b + N. For b ≥ j + ϵ, this is implied by the fact that ν(x) < x.

Now we complete the proof of Proposition 3.2. We are trying to establish existence of the cycle (4.21). Assume that we have a cycle

z_{ℓ+1} = ∑_{t=ℓ+1}^{b}(u_{t}h^{e_{t}} + L_{t})y_{t}

with exc(L_{t}) < e_{t} ≤ a_{t}, where a_{t} is as defined in Lemma 4.22. Then ∂(z_{ℓ+1}) is evaluated as in Lemma 4.14. By Lemma 4.2 and the last part of Lemma 4.12, exc(∂(z_{ℓ+1})) ≤ M_{ℓ}, where M_{ℓ} is as in (4.23). Then since a_{ℓ} = M_{ℓ}+ν(i−ℓ)+1, Lemma 4.12 implies that this ∂(z_{ℓ+1}) can be written as d(u_{ℓ}h^{e_{ℓ}} + L) with exc(L) < e_{ℓ} ≤ a_{ℓ}. Letting z_{ℓ} = z_{ℓ+1} − u_{ℓ}h^{e_{ℓ}} − L extends the induction.

The following two combinatorial lemmas were used above.

Lemma 4.25. Assume that ℓ < t < i < ℓp + N. Then

ν(i − ℓ) − ν(\binom{ℓ(p−1)+N}{t−ℓ}) ≤ ν((i − t)\binom{ℓp+N−t}{i−t}).

Equality holds if p^{ν(i−ℓ)} > max{ℓp + N − i, t − ℓ}.

Proof. Note that t − ℓ + 1 ≤ i − ℓ ≤ ℓ(p − 1) + N. Thus

ν(i − ℓ) − ν(\binom{ℓ(p−1)+N}{t−ℓ})
= ν(\frac{(ℓp − t + N)!}{(ℓ(p − 1) + N) ⋯ (i − ℓ + 1) · (i − ℓ − 1) ⋯ (t − ℓ + 1)})
= ν(\frac{(ℓp−t+N)!}{(i−t−1)!(ℓ+N−i)!} \frac{(ℓp+N−i)!}{(ℓp−ℓ+N)···(i−ℓ+1)} \frac{(i−t−1)!}{(i−ℓ−1)···(t−ℓ+1)})
= ν((i − t)\binom{ℓp+N−t}{i−t}) − ν(\binom{ℓ(p−1)+N}{i−ℓ}) − ν(\binom{i−ℓ−1}{t−ℓ}).

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 31

The condition p^{ν(i−ℓ)} > max\{ℓp + N − i, t − ℓ\} guarantees that ν(\binom{ℓ(p−1)+N}{i−ℓ}) = 0 and ν(\binom{i−ℓ−1}{t−ℓ}) = 0 by the standard result relating the mod p value of binomial coefficients to the p-ary expansions of the numbers.

Lemma 4.26. If 0 < i ≤ b, then ν_{p}(i\binom{b}{i}) ≤ log_{p}(b).

Proof. Assume p^{j} ≤ b < p^{j+1}. We show

ν(\frac{b ⋯ (b − i + 1)}{(i − 1)!}) ≤ j.

We have max{ν_{p}(k) : b − i < k ≤ b} ≤ j. The i − 1 numbers ν_{p}(k) other than the one which achieves the maximum are less than or equal to corresponding numbers ν(1), …, ν(i − 1) in the denominator.

We close this section by discussing the pattern of differentials which usually kill all elements in E_{1}^{2,a} for a < j + ϵ. As discussed in Section 3, these differentials are not essential to Theorem 1.1. That theorem assumed that all such elements are killed. If some are not, then our theorem becomes weaker than it might be; however, the author believes that the theorem as stated is the correct compromise between tractability and optimality.

The main differentials are

(4.27) d_{ap+N−a} : E_{ap+N−a}^{1,ap+N} → E_{ap+N−a}^{2,a} is usually surjective.

The cases in which this differential is not surjective occur when there is a value of t slightly greater than a for which ν(i − t) is very large. In such a case, there can be a shorter nonzero differential E^{1,ap+N} → E^{2,t}. If this happens, there will usually be a nonzero differential from E^{1,ap+N−Δ} to E^{2,a} for some small positive integer Δ.

The surjectivity of the differential in (4.27) requires that if |E_{1}^{2,a}| > |E_{1}^{1,ap+N}|, then there are shorter differentials which kill some of the multiples of the generator of E_{1}^{2,a}. This will indeed be the case, with most of the killing done by E^{1,ap+N−Δ} for small positive values of Δ.

We illustrate with the case p = 5, i = 5^{5} + 2, and N = 4. For 0 ≤ t ≤ 5^{5} + 1, we have

ν(|E_{1}^{1,t}|) = ν(|E_{1}^{2,t}|) = \begin{cases}1 & if t ≢ 2 mod 5 \\ 2 & if t ≡ 2 mod 5 but t ≢ 2 mod 5^{2} \\ 3 & if t ≡ 2 mod 5^{2} but t ≢ 2 mod 5^{3} \\ 4 & if t ≡ 2 mod 5^{3} but t ≢ 2 mod 5^{4} \\ 5 & if t = α5^{4} + 2 with 1 ≤ α ≤ 4 \\ 6 & if t = 2. \end{cases}

We are not concerned here with E_{1}^{2,i}.

If 0 ≤ a < 5^{4} and a ≠ 1, the pattern of differentials into E^{2,a} is as follows. If ν(|E_{1}^{2,a}|) > 2, then d_{4a+2} : E^{1,5a+2} → E^{2,a} hits all multiples of p^{2}. If ν(|E_{1}^{2,a}|) > 1, then a nonzero d_{4a+3} from E^{1,5a+3} hits p times the generator of E_{1}^{2,a}. Finally, d_{4a+4} : E_{4a+4}^{1,5a+4} → E_{4a+4}^{2,a} is surjective for all a satisfying 0 ≤ a < 5^{4} and a ≠ 1.

The large group E^{2,2} causes the unusual behavior when a = 1. The group E^{2,2} is hit by E^{1,9}, E^{1,11}, E^{1,12} ≈ Z/5^{2}, E^{1,13}, and E^{1,14}, each differential, starting with the shortest, hitting the element of order p in the remaining group E_{r}^{2,a}. (The differential from E^{1,12} also hits elements of order p^{2}.) Finally E^{1,8} → E^{2,1} is surjective. So in the notation of (4.27), a large group when t = 2 blocked the expected nonzero differential E^{1,9} → E^{2,1}.

Since the claims surrounding (4.27) are not necessary to our main theorem, we need not provide complete justification for them. We just outline the background for the basic differential. We use (2.10) to represent the generator of E_{1}^{1,ap+N}, mod p, by hy_{ap+N}. Under favorable conditions, this pulls back to a cycle z in UC^{1}(X_{a+1}^{ap+N}) satisfying, mod L,

∂(z) ≡ \binom{a(p−1)+N}{ap+N−a}h ⊗ h^{ap+N−a}y_{a} ∈ UC^{2}(X_{a}^{a}).

This class is a generator of E_{2}^{2}(X_{a}^{a}). Here we have used (2.7), 2.6, and 2.5.

5. Results when N = k and n is p-cleaner

In this section, we establish the second and third cases of (1.9) by proving that if N = k and n is p-cleaner, then

∑_{ℓ=0}^{i} ν(|E_{∞}^{1,ℓ}|) ≥ n − 1 + ν((i − j) ⋯ (i − j − r)) + S_{j+r+1},

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 33

and the estimate can be improved by 1 if n is p-cleanest. We continue to employ freely all the notation of Theorem 1.4.

We still have Proposition 3.2 and that, for all practical purposes, E_{∞}^{2,t} = 0 for all t < j. Most of this section will be devoted to establishing the following result.

Theorem 5.1. Assume the notation of Theorem 1.4, with N = k. Assume that for all D ≥ 1

ν(i − j − D) ≤ (p − 1)D − ν((D − 1)!) + S_{j+1} − S_{j+D}. (5.2)

Let r = ν(i − j). Either (1) for some m with

ν(m − i) + 1 = N + j + r + 1 + i(p − 2) − ν((i − j − r − 1)!) + S_{j+r+1}, (5.3)

the differentials E^{1,i} → E^{2,j+t} are 0 for t > r, while for 0 ≤ t ≤ r, they have image of order ≤ p, and, if equality never occurs in (5.2), then at most r of them are nonzero; or

(2) if

ν(m − i) + 1 = N + j + r + 2 + i(p − 2) − ν((i − j − r − 1)!) + S_{j+r+1},

then the differentials E^{1,i} → E^{2,j+t} are 0 for t > r, while for 0 ≤ t ≤ r, they have image of order ≤ p; or

(3) r = 0 and if

ν(m − i) + 1 = N + j + 2 + i(p − 2) − ν((i − j − 1)!) + S_{j+1},

then, for t ≥ 0, the differential E^{1,i} → E^{2,j+t} is possibly nonzero only for t = 2 or 3 (but not both) and, if nonzero, its image has order p.

Thus, with R denoting the RHS of (5.3), we have, similarly to (3.3),

∑_{t=0}^{i}(|E_{∞}^{1,t}|) ≥ R + i + ν(i!) − (j + ν(i ⋯ (i − j + 1)) + r + 1)
(5.4)  = n − 1 + ν((i − j) ⋯ (i − j − r)) + S_{j+r+1},

and if equality never occurs in (5.2), then the RHS of (5.4) can be increased by 1. This completes the proof of the second and third cases of (1.9).

In order to prove Theorem 5.1, we will need the following analogue of Lemma 3.6.

Lemma 5.5. Assume the notation of Theorem 1.4 with N = k, and that

(i − ℓ)(p − 1) − ν((i − ℓ − 1)!) + S_{ℓ} < e_{i} ≤ i(p − 2) + ℓ + N − ν((i − ℓ)!) + S_{ℓ}. (5.6)

Let z be a cycle in UC^{1}(X_{ℓ+1}^{i}) satisfying

z = ∑_{t=ℓ+1}^{i}(c_{t}h^{e_{t}} + L_{t})y_{t}

with

e_{t} = e_{i} − (i − t)(p − 2) + ν((i − t)!) − S_{t}, (5.7)

c_{t} ∈ Z_{(p)}, ℓ ≥ j, and exc(L_{t}) < e_{t}. Assume also that if j ≢ N mod p and ℓ + 1 ≡ jp^{2} + Np + N mod p^{3} and c_{ℓ+2} ≢ 0 mod p, then c_{ℓ+1} ≡ c_{ℓ+2} mod p. Then z pulls back to a cycle z_{1} in UC^{1}(X_{ℓ}^{i}) satisfying z_{1} = z + (c_{ℓ}h^{e_{ℓ}} + L_{ℓ})y_{ℓ}, where e_{ℓ} satisfies (5.7) and exc(L_{ℓ}) < e_{ℓ}. Moreover, if j ≢ N mod p and ℓ ≡ jp^{2} + Np + N mod p^{3} and c_{ℓ+1} ≢ 0 mod p, then c_{ℓ} ≡ c_{ℓ+1} mod p in z_{1}.

As in Lemma 3.6, the last sentence is needed to make this lemma self-perpetuating.

Proof. When ℓ, ℓ + 1 ≢ Np + N mod p^{2}, the proof is the same as that of Lemma 3.6 in Section 4.

We will analyze two steps in pulling back (h^{e} + L)y_{ℓ+1} when

ℓ = αp^{2} + Np + N,

a condition which will be assumed through the remainder of this proof. Let b = ν(i − ℓ). We will show that (h^{e} + L)y_{ℓ+1} pulls back, mod L, to

(5.8) h^{e}y_{ℓ+1} + ch^{e−p+b}y_{ℓ} + \begin{cases}uh^{e−2p+b+2}y_{ℓ−1} & if α ≢ j mod p \\ c′h^{e−2p+b+1}y_{ℓ−1} & if α ≡ j mod p. \end{cases}

Here u denotes a unit in Z_{(p)}, c′ ∈ Z_{(p)} is unrestricted, and c ∈ Z_{(p)} is a unit if and only if α ≢ N mod p, while c ≡ 1 mod p if α ≡ j ≢ N mod p.

Note that ν(i − ℓ) ≥ 1, and ν(i − ℓ) ≥ 2 if and only if j ≡ N mod p. In (5.7),

e_{ℓ} − e_{ℓ+1} = −p + 2 + ν(i − ℓ) − (S_{ℓ} − S_{ℓ+1}) = −p + ν(i − ℓ),

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 35

and this is consistent with the exponent e − p + ν(i − ℓ) of h on y_{ℓ} in (5.8). Similarly,

e_{ℓ−1} − e_{ℓ+1} = −2(p − 2) + ν(i − ℓ) − (S_{ℓ−1} − S_{ℓ+1})
= −2p + ν(i − ℓ) + \begin{cases}1 & if α ≡ j mod p \\ 2 & if α ≢ j mod p \end{cases}

is consistent with the exponents of h on y_{ℓ−1} in (5.8).

The actual proof of this lemma should perhaps be carried out more along the lines of that of Lemma 3.6. However, the guts of the argument is in (5.8). Some work is required to see that terms on y_{t} for t > ℓ + 1 cannot mess up the pulling back claimed in (5.8), but this can be accomplished as in the proof of (4.17) in 3.6, using the LHS of (5.6). The part about c ≡ 1 in (5.8) when α ≡ j ≢ N translates to the last sentence of Lemma 5.5. This specific information is required to obtain the exponent of h on y_{ℓ−1} in this case, which is the content of the assumption after (5.7). The RHS of (5.6) is required so that z_{1} satisfies the unstable condition.

Now we prove (5.8). We consider first the case j ≢ N mod p. Then ℓ(p−1)+N = p^{2}(αp + N − α) and i − ℓ = p(j − N − αp), and so

∂((h^{e} + L)y_{ℓ+1}) = p^{2}(αp + N − α)(h^{e} + L) ⊗ hy_{ℓ}
= −p^{2}(αp + N − α)(h ⊗ h^{e−p+1} + L_{1})y_{ℓ}
= p\frac{αp + N − α}{j − N − αp}d(h^{e−p+2} + L_{2})y_{ℓ}
= \frac{αp + N − α}{j − N − αp}d((h^{e−p+1} + L′)y_{ℓ}),

where in each case the L has excess less than that of the accompanying term. Here we have used Lemma 4.4 at the second step, Lemma 4.5 at the third step, and Lemma 4.12 (applied to p(h⊗h^{e−p+1}+L_{1})ι_{2(e−p)+1}) at the last step to know that the p reduces the excess of L_{2}.

Thus we can pull back to a cycle

z′ = (h^{e} + L)y_{ℓ+1} + (\frac{αp + N − α}{αp + N − j}h^{e−p+1} + L′)y_{ℓ}.

(Note that the fraction here is 1 mod p if α ≡ j mod p, and is a unit if α ≢ N mod p.) Now we have, omitting a factor (ℓ − 1)(p − 1) + N, which is 1 mod p,

∂(z′) = ((h^{e} + L) ⊗ (\frac{p(αp^{2}+Np−αp−1)}{2}h^{2} − hv) + (\frac{αp+N−α}{αp+N−j}h^{e−p+1} + L′) ⊗ h)y_{ℓ−1}
= (βh^{e−1} ⊗ h^{2} + \frac{j−α}{αp+N−j} h^{e−p+1} ⊗ h + L′′)y_{ℓ−1}.

Here we have used, as usual, hv = vh − ph^{2} and h^{p}v ≡ v^{p}h, and then combined the (h^{e−1} ⊗ h^{2})-terms and the (h^{e−p+1} ⊗ h)-terms. If α ≢ j, then the leading term is a unit times h^{e−p+1} ⊗ h, and so the claim follows from Lemma 4.6, while if α ≡ j, then all terms have excess ≤ e − 2p + 1, and so the claim follows from Corollary 4.13.

Now assume j ≡ N mod p, so that b := ν(i − ℓ) ≥ 2. As before,

∂((h^{e} + L)y_{ℓ+1}) = (h^{e} + L) ⊗ p^{2}(αp + N − α)hy_{ℓ}.

If α ≢ N mod p, this is d((uh^{e−p+b} + L′)y_{ℓ}) by Lemma 4.12. We need here that exc(p^{2}L) < e − 2, which will be true if L was obtained as in Lemma 4.12, using also that, in the notation of 4.12, ν(σ) ≥ 2. Then ∂((h^{e} + L)y_{ℓ+1} − (uh^{e−p+b} + L′)y_{ℓ}) is dominated by the term ∂(uh^{e−p+b}y_{ℓ}), which is easily obtained.

If, on the other hand, α ≡ N mod p, then all terms T in ∂((h^{e} + L)y_{ℓ+1}) satisfy exc(T) ≤ e − 2 − p. Here we need again that L has been obtained as in Lemma 4.12, and we only need to decrease the excess of L by 2. This ∂((h^{e} + L)y_{ℓ+1}) is d((ch^{e−p+b} + L′)y_{ℓ}) with c ≡ 0 mod p, and from here we proceed similarly to the previous paragraph.

Proof of Theorem 5.1. We choose e_{i} equal to the RHS of (5.3), and use Lemma 5.5 to deduce that h^{e_{i}}y_{i} pulls back to a cycle

z_{j+r+1} ≡ h^{e_{i}}y_{i} + ⋯ + c_{j+r+1}h^{(j+r+1)(p−1)+N}y_{j+r+1} mod L (5.9)

with c_{j+r+1} ∈ Z_{(p)}. The first part of (5.6) for ℓ ≥ j + r + 1 is implied by (5.2), and the second part is automatic.

First we consider the case r = 0 and c_{j+1} a unit. Then, mod L

∂(c_{j+1}h^{(j+1)(p−1)+N}y_{j+1}) ≡ uh^{(j+1)(p−1)+N} ⊗ hy_{j} ≡ −uh ⊗ h^{j(p−1)+N}y_{j}

with u a unit. However, this can be cancelled if equality never occurs in (5.2). We refer back to Theorem 2.8(2), which says that the initial term α_{(m−i)(p−1)/e_{i}}y_{i} (which

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 37

we have been representing as h^{e_{i}}y_{i} mod L) is congruent mod p to −sv^{(m−i)(p−1)−1}hy_{i}, where

(m − i)(p − 1) ≡ sp^{e_{i}−1} mod p^{e_{i}}.

This yields a contribution −sh ⊗ h^{i−j}y_{j} to ∂(z_{j+1}). Note that the binomial coefficient here is \binom{j(p−1)+N}{i−j} = \binom{j(p−1)+N}{j(p−1)+N} = 1, and, because i − j = j(p − 1) + N, the class of h ⊗ h^{i−j}y_{j} generates E_{2}^{2}(S^{jq+2N+1}). We obtain

∂(z_{j+1}) = (−s − u)h ⊗ h^{j(p−1)+N}y_{j}.

We can choose m so that s ≡ −u mod p. Here it is important to note that the value of the unit u does not depend on this s. In our pulling back, we use (2.9) to view the initial term as −v^{(m−i)(p−1)−e_{i}}h^{e_{i}}y_{i}, independent of s. The assumption that equality never occurs in (5.2) implies that the LHS of (5.6) can be made 1 stronger, so that, in ∂(z) which is relevant at each stage of pulling back, the term α ⊗ h^{i−ℓ} will have excess strictly less than the bound being used for the excess of terms such as ch^{e_{ℓ+1}} ⊗ h. This will prevent the value of s from contributing to the coefficient u. Since E_{1}^{2,j} ≈ Z/p in the case under consideration, we obtain that if m is chosen so that s ≡ −u mod p, then d_{i−j} = 0 : E^{1,i} → E^{2,j}, so that the last clause of Theorem 5.1(1) is satisfied. If equality does occur in (5.2), then we cannot assert that this cancellation can be achieved, but Theorem 5.1(1) is still satisfied. We had not discussed this kind of cancellation when N ≠ k and n is p-cleaner because h ⊗ h^{i−j−ϵ}y_{j+ϵ} is a boundary mod L in these cases.

Next we consider the case where r = 0 and c_{j+1} ≡ 0 mod p. Then the last term of (5.9), c_{j+1}h^{(j+1)(p−1)+N}y_{j+1}, will still be unstable if the exponent of h is increased by 1. It will usually be the case that the whole sum (5.9) will still be unstable if each term is multiplied by h (and divided by v to keep the degree unchanged). To see this, we note that in (5.7) e_{t+1} − e_{t} ≤ p − 1, with equality if and only if

t ≡ Np + N mod p^{2} and N ≢ j mod p
(5.10)  or t ≡ jp^{2} + Np + N − 1 mod p^{3}.

If neither of the conditions in (5.10) is satisfied when t = j + 1, then (5.9) remains unstable after being multiplied by h, since all terms e_{t} with t > j + 1 satisfy e_{t} < t(p − 1) + N. In this case, for any m with ν(m − i) + 1 exactly 1 greater than the

38 D. DAVIS

value stated in (5.3), d : E^{1,i} → E^{2,j} will be possibly nonzero due to the −sh ⊗ h^{i−j} term discussed in the two preceding paragraphs, and Theorem 5.1(2) is satisfied in this case. Since we do not assert cancellation here, we do not need the assumption of no equality in (5.2).

If, however, j + 1 ≡ Np + N mod p^{2} with N ≢ j mod p, then it is possible that the penultimate term of (5.9) is uh^{(j+2)(p−1)+N}y_{j+2} with u a unit. (This would be the case where ℓ = j + 1, c ≡ 0 mod p, and b = 1 in (5.8).) All preceding terms will be of the form c_{t}h^{e_{t}}y_{t} with e_{t} < t(p−1)+N. If e_{i} is 1 greater than the value on the RHS of (5.3), h^{e_{i}}y_{i} will pull back to a cycle w in UC^{1}(X_{j+3}^{i}) with exponents of h on the leading terms 1 greater than those of (5.9), and there may be a nonzero differential E^{1,i} → E^{2,j+2} due to

∂(h^{(j+3)(p−1)+N}y_{j+3}) ≡ u′h^{(j+3)(p−1)+N} ⊗ hy_{j+2}.

Then pw pulls back to a cycle whose leading terms equal those of (5.9), and the differential into E^{2,j} is 0. Indeed, the component from y_{j+1} is 0 due to c_{j+1} ≡ 0, while the component from y_{i} is 0 since w has been multiplied by p. Thus we obtain Theorem 5.1(3) with t = 2.

When j + 1 ≡ jp^{2} + Np + N − 1 mod p^{3}, the situation is somewhat similar. If j ≡ N, then in (5.8) with ℓ − 1 = j + 1, we have b > 1 and c is not a unit, and we deduce that all terms in (5.9) remain unstable after the exponents of h are increased by 1, and so Theorem 5.1(2) is satisfied. If j ≢ N, then b = 1 and c is a unit in (5.8). In this case, the last three terms of (5.9) could be

uh^{(j+3)(p−1)+N}y_{j+3} + u′h^{(j+2)(p−1)+N}y_{j+2} + ch^{(j+1)(p−1)+N}y_{j+1}

with u and u′ units, but c ≡ 0 mod p, and e_{t} < t(p − 1) + N if t > j + 3. Hence, if e_{i} is 1 greater than the value on the RHS of (5.3), h^{e_{i}}y_{i} will pull back to a cycle w in UC^{1}(X_{j+4}^{i}) with exponents of h on the leading terms 1 greater than those of (5.9), and there may be a nonzero differential E^{1,i} → E^{2,j+3} due to

∂(h^{(j+4)(p−1)+N}y_{j+4}) ≡ u′′h^{(j+4)(p−1)+N} ⊗ hy_{j+3}.

Then pw pulls back to a cycle whose leading terms equal those of (5.9), and, as before, the differential into E^{2,j} is 0. Thus we obtain Theorem 5.1(3) with t = 3.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 39

Assume next that 0 < r < p and c_{j+r+1} is a unit. Then, mod L,

∂(z_{j+r+1}) ≡ uh ⊗ h^{(j+r)(p−1)+N}y_{j+r},

and so

d_{i−j−r} : E_{1}^{1,i} = E_{i−j−r}^{1,i} → E_{i−j−r}^{2,j+r} ≈ E_{1}^{2,j+r} ≈ Z/p

is surjective. Then by Lemma 4.6

∂(pz_{j+r+1}) = d((u′h^{(j+r)(p−1)+N} + L_{j+r})y_{j+r}),

and so pz_{j+r+1} pulls back to a cycle w_{j+r} satisfying

w_{j+r} ≡ pz_{j+r+1} − u′h^{(j+r)(p−1)+N}y_{j+r} mod L.

This continues, yielding, for 1 ≤ t ≤ r, surjective differentials

d_{i−j−t} : E_{i−j−t}^{1,i} → E_{i−j−t}^{2,j+t} = E_{1}^{2,j+t} ≈ Z/p

and classes

w_{j+t} ≡ pw_{j+t+1} − uh^{(j+t)(p−1)+N}y_{j+t} mod L.

Finally, ∂(w_{j+1}) contains significant contributions from its top (p^{r}α_{(m−i)(p−1)/e_{i}}y_{i}) and bottom (uh^{(j+1)(p−1)+N}y_{j+1}) classes. Similarly to the case above with r = 0 and c_{j+1} a unit, the top hits −p^{r}sh ⊗ h^{i−j}y_{j} if (m − i)(p − 1) ≡ sp^{e_{i}−1} mod p^{e_{i}}, while the bottom hits

u(j(p − 1) + N)h^{(j+1)(p−1)+N} ⊗ hy_{j} ≡ −u(i − j)h ⊗ h^{j(p−1)+N}y_{j}.

Since ν(i − j) = r and i − j = j(p − 1) + N, we find that if m is chosen so that s has a certain mod p value, then the two contributions cancel in E^{2,p} ≈ Z/p^{r+1}, and so d_{i−j} = 0. Hence here we have Theorem 5.1(1). Again we need the assumption of no equality in (5.2) in order to assert that the coefficients obtained in pulling back do not depend on s.

If r ≥ p, then in the procedure of the previous paragraph, there will be at least one value of t for which j + t ≡ N mod p. This will have two effects: (a) |E_{1}^{2,j+t}| = p^{1+ν(i−j−t)} ≥ p^{2}, and (b) the coefficient \binom{(j+t)(p−1)+N}{1} which occurs in ∂(w_{j+t+1}) is a multiple of p. Since i − j − t = j(p − 1) + N − t and (j + t)(p − 1) + N have the same

40 D. DAVIS

p-exponent for the values of t being considered, these effects will effectively cancel. Indeed, if (j + t)(p − 1) + N = up^{ν}, then, mod L,

∂(w_{j+t+1}) ≡ \binom{(j+t)(p−1)+N}{1}h^{(j+t+1)(p−1)+N} ⊗ hy_{j+t}
≡ −up^{ν}h ⊗ h^{(j+t)(p−1)+N}y_{j+t}

has order p in E_{1}^{2,j+t}. Thus d : E_{i−j−t}^{1,i} → E_{i−j−t}^{2,j+t} has image of order p, and pw_{j+t+1} extends as in the previous paragraph.

If ν(c_{j+r+1}) = d > 0, then the first d of the differentials described in the two previous paragraphs will usually be 0, and then the pattern described there will begin. For example, if d = 2, then p^{2}h^{(j+r+1)(p−1)+N}y_{j+r+1} pulls back to a class congruent, mod L and ignoring unit coefficients, to

z ≡ p^{2}h^{(j+r+1)(p−1)+N}y_{j+r+1} + ph^{(j+r)(p−1)+N}y_{j+r} + h^{(j+r−1)(p−1)+N}y_{j+r−1},

and

∂(z) ≡ h ⊗ h^{(j+r−2)(p−1)+N}y_{j+r−2} ≠ 0 ∈ E^{2,j+r−2}.

It could happen that p^{d}h^{(j+r+1)(p−1)+N}y_{j+r+1} is preceded in (5.9) by a term such as uh^{(j+r+2)(p−1)+N}y_{j+r+2}, which will dominate in the calculation of ∂(z_{j+r+1}). That could make it so that these first differentials are not zero, but we still obtain the pattern and results of the previous two paragraphs.

6. Results when n is not p-cleaner

In this section, we prove the remaining cases of (1.9). That is, we prove

Theorem 6.1. If n is not p-clean, then

∑_{ℓ=0}^{i} ν(|E_{∞}^{1,ℓ}|) ≥ n + S_{j+ϵ},

while if n is p-clean but not p-cleaner, then

∑_{ℓ=0}^{i} ν(|E_{∞}^{1,ℓ}|) ≥ n − 1 + S_{j+ϵ}.

The following result gives the description of the CSS (in the degree specified in Convention 3.4(1)) in this case.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 41

Lemma 6.2. Assume n is not p-cleaner. Let D be the largest integer satisfying ϵ ≤ D < i − j − ϵ such that

ν(i − j − D) ≥ N − k + ϵ + D(p − 1) − ν((D − ϵ)!) + S_{j+ϵ} − S_{j+D}, (6.3)

and then choose the smallest Δ ≥ 0 such that

Δ + ν(Δ!) − (S_{j+D−Δ} − S_{j+D}) ≥ N − k + Dp − ν(i − j − D). (6.4)

Such a Δ exists satisfying Δ ≤ D.

For j +ϵ ≤ ℓ ≤ j +D−Δ, the differential d_{i−ℓ} is possibly nonzero. (Here and in the rest of this lemma, all differentials emanate from E^{1,i}.) However, if ℓ ≡ N mod p, then either d_{i−ℓ} is not surjective or else d_{i−ℓ−δ} = 0 for 1 ≤ δ ≤ p. Similarly, if k ≠ N and ℓ ≡ kp + N − 1 mod p^{2}, then either d_{i−ℓ} is not surjective or else d_{i−ℓ−δ} = 0 for 1 ≤ δ ≤ p + 1.

In addition, if k = N and ℓ = αp^{2} + Np + N, then either

| coker(d_{i−ℓ})| \begin{cases}> p^{2} & if α ≡ N ≡ j mod p \\ = p^{2} & otherwise \end{cases}

and d_{i−ℓ+1} = 0 if α ≡ j ≢ N mod p, or else d_{i−ℓ−δ} = 0 for 1 ≤ δ ≤ p.

If ν(Δ) > 0 (or Δ = 0), then ν(E_{1}^{2,j+D−Δ}) > 1, and the order of the image of E^{1,i} → E^{2,j+D−Δ} is less than or equal to p^{γ+1}, where γ is the difference of the two sides of (6.4).

There will be no other differentials into groups E^{2,ℓ} with ℓ ≥ j + ϵ, provided e_{i} = ν(m − i) + 1 satisfies

e_{i} ≤ j + k + 1 − D(p − 1) + i(p − 2) − ν((i − j − D − 1)!) + S_{j+D}. (6.5)

If Δ = 0, then either there exists such an m for which the image of the differential into E^{2,j+D} has order less than or equal to p^{γ}, or else e_{i} can be made 1 larger than the RHS of (6.5) and still have no additional differentials.

For example, if N = 5, k = 2, and j = (p^{p^{5}} − 1 + p^{4} − 1)/(p − 1), then D = p^{4} and Δ = 3.

Before proving this lemma, we deduce Theorem 6.1 from it.

Proof of Theorem 6.1. Lemma 6.2 shows that if n is not p-cleaner, then, analogously to (3.3), we have

∑_{ℓ=0}^{i} ν(|E_{∞}^{1,ℓ}|) = A − B (6.6)

with

A = i + ν(i!) + RHS of (6.5)

and

B ≤ j + D − Δ + ν(i ⋯ (i − j − D + Δ + 1)) − (S_{j+ϵ} − S_{j+D−Δ}) + γ + 1,

where γ is as in the lemma. The S_{j+ϵ} − S_{j+D−Δ} in B is due to the cases in Lemma 6.2 in which d_{i−t} might not be surjective for j + ϵ ≤ t < j + D − Δ. Simplifying somewhat gives the first step of

∑_{ℓ=0}^{i} ν(|E_{∞}^{1,ℓ}|) ≥ i(p − 1) + k − Dp + Δ + S_{j+D} + S_{j+ϵ} − S_{j+D−Δ} − γ
+ν((i − j − D + Δ) ⋯ (i − j − D))
= i(p − 1) + N + S_{j+ϵ}
= n − 1 + S_{j+ϵ},

while substituting the definition of γ gives the second step, along with the observation that ν(i − j − D + t) = ν(t) for t ≠ 0 in the range under consideration.

Clearly Δ = 0 satisfies (6.4) if and only if n is not p-clean. The final portion of Lemma 6.2 says that in this case either B may be decreased by 1 or A may be increased by 1 in (6.6).

Proof of Lemma 6.2. It is immediate from (6.3) that inequality (6.4) is satisfied for Δ = D. This establishes the claim that the smallest such Δ is less than or equal to D.

Inequality (6.3) is the antithesis of (1.8), which was used in the proof of Lemma 3.6 to guarantee that (3.7) could be satisfied. If D is chosen as in (6.3), then the cycle z of (3.8) with ℓ = j + D yields a term T in ∂(z) of the form

T = \binom{(j+D)(p−1)+N}{i−j−D}h^{e_{i}} ⊗ h^{i−j−D}y_{j+D}.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 43

The binomial coefficient here is a unit, since

(j + D)(p − 1) + N = (i − j − D) + (Dp + N − k),

so if ν(i − j − D) is a large number L, the coefficient is \binom{αp^{L}+d}{αp^{L}}, with d << p^{L}; in fact, d is not much larger than L. The excess of T is at least i − j − D, and, by the analysis of (4.18), this does not satisfy exc(T) ≤ e with e as in (4.16). Thus Lemma 3.6 must be modified for subsequent pulling back.

In this situation, e_{i} can be increased over its value in (3.10). Recall that this value of e in (3.10) was obtained as the largest value that would allow pulling back to a class with e_{j} ≤ j(p − 1) + N. In our new situation, we choose the largest e_{i} that will allow pulling back to a class with e_{j+D+1} ≤ i − j − D + p − 1. This is appropriate since

∂(h^{i−j−D+p−1}y_{j+D+1}) = \binom{(j+D)(p−1)+N}{1}h^{i−j−D+p−1} ⊗ hy_{j+D}
≡ \binom{(j+D)(p−1)+N}{1}h ⊗ h^{i−j−D}y_{j+D},

whose excess is equal to or less than that of the term T, obtained independently of the value of e_{i}. The same pullback analysis which yielded (3.10) now yields

e_{i} = i − j − D + p − 1 + (i − j − D − 1)(p − 2) − ν((i − j − D − 1)!) + S_{j+D}
(6.7) = j + k + 1 − D(p − 1) + i(p − 2) − ν((i − j − D − 1)!) + S_{j+D},

which is (6.5).

With e_{i} chosen in this way, and e_{t} satisfying (3.9), we have a cycle

z = ∑_{t=j+D+1}^{i}(c_{t}h^{e_{t}} + L_{t})y_{t} (6.8)

with exc(L_{t}) < e_{t}. By the analysis in the proof of Lemma 3.6, together with the observations so far in this proof, all terms T in ∂(z) have exc(T) ≤ i − j − D. If Δ = 0, then, by (6.4),

i − j − D = (j + D)(p − 1) + k − Dp ≥ (j + D)(p − 1) + N − ν(i − j − D), (6.9)

and so {∂(z)} is possibly nonzero in E_{2}^{2}(S^{(j+D)(p−1)+N}). Moreover, its order is less than or equal to p^{γ+1}, where γ is the difference in (6.4).

44 D. DAVIS

In fact, we can do 1 better than this. If equality is obtained in (6.9), and in (6.8) we have c_{j+D+1} a unit, e_{j+D+1} = i − j − D + p − 1, and N ≠ k (so that \binom{(j+D)(p−1)+N}{1} is a unit), then, for some m with ν(m − i) + 1 equal to (6.7), the two terms in

∂(z) ≡ (uα_{m−i/e_{i}} ⊗ h^{i−j−D} + \binom{(j+D)(p−1)+N}{1}c_{j+D+1}h^{i−j−D+p−1} ⊗ h)y_{j+D}, (6.10)

will be cancelling elements of order p in E^{2,j+D}, and so here we have γ = 0 and im(d_{i−j−D}) = 0. The coefficient u = \binom{(j+D)(p−1)+N}{i−j−D} is a unit since the top part equals i − j − D + N − k + Dp, and ν(i − j − D) ≥ N − k + Dp, so the binomial coefficient is of the form \binom{cp^{d}+d}{cp^{d}}, which is a unit.

If one of the hypotheses at the beginning of this paragraph fails, then we will not get the cancellation and will still have the nonzero differential due to α_{m−i/e_{i}} ⊗h^{i−j−D}, but now we can increase e_{i} by 1 over its value in (6.7) and still have all terms unstable. The argument in this case (Δ = 0) continues as in the paragraph, soon to follow, which begins “If p^{R} is the order.”

On the other hand, if Δ > 0, then (6.9) does not hold, and so by Corollary 4.13 ∂(z) = d(uh^{e} + L)y_{j+D} with e ≤ i − j − D + ν(i − j − D) + 1 and exc(L) < e. Thus z pulls back to the cycle z′ = z − (uh^{e} + L)y_{j+D} ∈ UC(X_{j+D}^{i}).

Similarly to the procedure in the proof of Lemma 3.6, this cycle z′ pulls back to

z′ + ∑_{t≥1}(c_{j+D−t}h^{e_{j+D−t}} + L_{j+D−t})y_{j+D−t}

with

e_{j+D−t} ≤ i − j − D + ν(i − j − D) + 1 − (p − 2)t
+ν((i − j − D + 1) ⋯ (i − j − D + t)) − S_{j+D−t} + S_{j+D}

and exc(L_{j+D−t}) < e_{j+D−t}, as long as e_{j+D−t} ≤ (j + D − t)(p − 1) + N. Using the fact that ν((i − j − D + 1) ⋯ (i − j − D + t)) = ν(t!) since ν(i − j − D) is large, this last inequality simplifies to

t + ν(t!) + ν(i − j − D) < N − k + Dp + S_{j+D−t} − S_{j+D}.

The smallest t for which this fails will be the Δ of (6.4). One can verify that in this pulling back from j + D to j + D − Δ, the factor on the right side of the ⊗ cannot be the determining factor for excess, i.e., that the analogue of (4.18) is satisfied.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 45

Next we establish the claimed possible differentials into E^{2,j+D−Δ} when Δ > 0. We have a cycle

z̃ = ∑_{t=j+D−Δ+1}^{i}(c_{t}h^{e_{t}} + L_{t})y_{t} (6.11)

as described in the two previous paragraphs. The leading term in ∂(z̃) will be

T = c_{j+D−Δ+1}\binom{(j+D−Δ)(p−1)+N}{1}h^{e_{j+D−Δ+1}} ⊗ hy_{j+D−Δ}.

The binomial coefficient will be a unit, since, if it were not, the LHS of (6.4) would be no larger for Δ than it is for Δ − 1. For the same reason, S_{j+D−Δ} = S_{j+D−Δ+1}, a fact which will be used later in this paragraph. If c_{j+D−Δ+1} is a unit and e_{j+D−Δ+1} has its maximum possible value of

i − j − D + ν(i − j − D) + 1 − (p − 2)(Δ − 1) + ν((Δ − 1)!) − S_{j+D−Δ+1} + S_{j+D},

then, using Lemma 4.4, T is a unit multiple of h ⊗ h^{α}y_{j+D−Δ}, where

α = (p − 1)j + k − D + ν(i − j − D) − (p − 2)Δ + ν((Δ − 1)!) − S_{j+D−Δ+1} + S_{j+D}.

By Theorem 2.8(4), the order of the subgroup of E_{2}^{2}(S^{q(j+D−Δ)+2N+1}) generated by h ⊗ h^{α} is

α − ((j + D − Δ)(p − 1) + N) + ν(i − j − D + Δ) + 1 = γ + 1,

where γ is, as in the lemma, the difference of the two sides of (6.4). This establishes the claim of the lemma that the order of the image of the differential into E^{2,j+D−Δ} is less than or equal to p^{γ+1}. (If c_{j+D−Δ+1} is not a unit or e_{j+D−Δ+1} does not have its maximum value, then the order of the image of the differential will be less than p^{γ+1}.)

If p^{R} is the order of the image of the differential from E^{1,i} to E^{2,j+D−Δ}, and z̃ is as in (6.11), then ∂(p^{R}z̃) is a cycle of excess (j + D − Δ)(p − 1) + N − ν(j + D − Δ) − 1, and so by Corollary 4.13, it is d(u′h^{(j+D−Δ)(p−1)+N} +L)y_{j+D−Δ}, where u′ is a unit and L has lower excess. Thus there is a cycle

z_{j+D−Δ} = p^{R}z̃ + (uh^{(j+D−Δ)(p−1)+N} + L)y_{j+D−Δ} ∈ UC^{1}(X_{j+D−Δ}^{i}).

This begins an induction which we would like to use to give us classes z_{ℓ} in UC^{1}(X_{ℓ}^{i}) for ℓ = j +D−Δ down to j +ϵ satisfying certain properties. The anticipated behavior

46 D. DAVIS

would be that differentials E^{1,i} → E^{2,ℓ} would be surjective except in the following cases:

• If ℓ ≡ N mod p, then

| coker(d_{i−ℓ})| \begin{cases}= p & if k ≠ N or ℓ ≢ kp + N mod p^{2} \\ > p^{2} & if k = N ≡ j and ℓ ≡ Np^{2} + Np + N mod p^{3} \\ = p^{2} & otherwise; \end{cases}

• If ℓ ≡ kp+N mod p^{2}, then d_{i−ℓ+1} is not surjective if k ≠ N or ℓ ≡ jp^{2}+Np+N

mod p^{3} and j ≢ N mod p.

In each case, if a class z_{ℓ} supports a differential whose image has order p^{s}, then p^{s}z_{ℓ} pulls back to some z_{ℓ−1}.

Unfortunately, when we pass an ℓ which is congruent to kp + N mod p^{2}, we may lose some control. Thus we cannot assert that differentials necessarily behave as we expect that they usually will, causing the lack of precision in Lemma 6.2. The following result will be adequate for our purposes; indeed it completes the proof of Lemma 6.2 and hence of Theorem 1.1. See Remark 6.14 for more on how Proposition 6.12 translates into Lemma 6.2.

Proposition 6.12. In the notation of Lemma 6.2, assume ℓ < j + D − Δ. Assume that z_{ℓ+1} ∈ UC^{1}(X_{ℓ+1}^{i}) is a cycle of the form

z_{ℓ+1} = (u_{ℓ+1}h^{(ℓ+1)(p−1)+N} + L_{ℓ+1})y_{ℓ+1} + ∑_{t=ℓ+2}^{i} T_{t}y_{t}

with exc(T_{t}) ≤ t(p − 1) + N for ℓ + 2 ≤ t ≤ i. Here and throughout the statement and proof of this proposition, u is always a unit, and exc(L) is less than that of the accompanying power of h. Then

(1) If ℓ ≥ j + ϵ and ℓ ≢ N mod p and ℓ ≢ kp + N − 1 mod p^{2}, then d_{i−ℓ} : E^{1,i} → E^{2,ℓ} is surjective, and p^{ν(i−ℓ)+1}z_{ℓ+1} pulls back to a cycle

z_{ℓ} = p^{ν(i−ℓ)+1}z_{ℓ+1} + (u_{ℓ}h^{ℓ(p−1)+N} + L_{ℓ})y_{ℓ} ∈ UC^{1}(X_{ℓ}^{i}).

(2) Assume ℓ − 1 ≥ j + ϵ and ℓ = ap + N. Let c = ν(a(p − 1) + N), b = ν(i − ℓ), and s = max(b − c, 0). Then | im(d_{i−ℓ})| = p^{s}. If it is not the case that

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 47

b = c < ν(jp − a + k), then d_{i−ℓ+1} is surjective, and p^{s+1}z_{ℓ+1} pulls back to a cycle

z_{ℓ−1} = p^{s+1}z_{ℓ+1} + (c_{ℓ}h^{ℓ(p−1)+N} + L_{ℓ})y_{ℓ}
+(u_{ℓ−1}h^{(ℓ−1)(p−1)+N} + L_{ℓ−1})y_{ℓ−1} ∈ UC^{1}(X_{ℓ−1}^{i}).

If b = c < ν(jp − a + k), then d_{i−ℓ+1} = 0 is not surjective. If τ is the smallest positive integer with ℓ − τ ≥ j + ϵ such that d_{i−ℓ+τ} : E^{1,i} → E^{2,ℓ−τ} is nonzero, then, if p^{s} is the order of the image of this d_{i−ℓ+τ}, p^{s}z_{ℓ+1} pulls back to a cycle z_{ℓ−τ} ∈ UC^{1}(X_{ℓ−τ}^{i}) of the form

z_{ℓ−τ} = p^{s}z_{ℓ+1} + ∑_{t=0}^{τ}(c_{t}h^{(ℓ−t)(p−1)+N} + L_{ℓ−t})y_{ℓ−t}

with c_{t} ∈ Z_{(p)} and c_{τ} a unit.

Proof. (1) Mod L,

∂(z_{ℓ+1}) ≡ u_{ℓ+1}\binom{ℓ(p−1)+N}{1}h^{(ℓ+1)(p−1)+N} ⊗ hy_{ℓ}
≡ u′h ⊗ h^{ℓ(p−1)+N}y_{ℓ}

by Lemma 4.4. By Theorem 2.8, this generates E_{2}^{2}(S^{ℓ(p−1)+N}). Thus the differential is surjective. By Lemma 4.6, ∂(p^{ν(i−ℓ)+1}z_{ℓ+1}) can be written as d(uh^{ℓ(p−1)+N} + L_{ℓ})y_{ℓ}, and so the pulling back here follows as usual. The hypothesis ℓ ≢ kp + N − 1 is necessary to insure that ∂(T_{ℓ+2}y_{ℓ+2}) is not significant.

(2) The calculation here is similar to (4.19). Assume without loss of generality that the unit attached to h^{(ℓ+1)(p−1)+N}y_{ℓ+1} is 1. Write i − ℓ = up^{b} and a(p − 1) + N = u′p^{c}. Then, mod L,

∂z_{ℓ+1} ≡ p(a(p − 1) + N)h^{(ℓ+1)(p−1)+N} ⊗ hy_{ℓ}
≡ −u′p^{c}h ⊗ h^{ℓ(p−1)+N−1}y_{ℓ}.

If c ≥ b, then this is \frac{u′}{u} p^{c−b}d((h^{ℓ(p−1)+N} + L)y_{ℓ}), and so z_{ℓ+1} pulls back to

z_{ℓ} = z_{ℓ+1} + (−\frac{u′}{u} p^{c−b}h^{ℓ(p−1)+N} + L)y_{ℓ}.

Omitting a factor ((ℓ − 1)(p − 1) + N) ≡ 1 mod p, we have

∂z_{ℓ} ≡ (h^{(ℓ+1)(p−1)+N} ⊗ (\frac{(ℓ−1)(p−1)+N−1}{2}h^{2} − hv)
−\frac{u′}{u} p^{c−b}h^{ℓ(p−1)+N} ⊗ h)y_{ℓ−1}.

We note that the coefficient of h^{2} is a multiple of p, and replace hv by vh−ph^{2}. Then we use h^{p}v ≡ v^{p}h to obtain a term −h^{ℓ(p−1)+N} ⊗ h. If c > b, this term −h^{ℓ(p−1)+N} ⊗ h will be the leading term, giving the surjective differential into E^{2,ℓ−1}, and the claimed pullback. If c = b, two leading terms combine to yield

∂z_{ℓ} ≡ (−1 − \frac{u′}{u})h^{ℓ(p−1)+N} ⊗ hy_{ℓ−1}. (6.13)

Since p^{b}(u+u′) = jp−a+k, the coefficient (−1− \frac{u′}{u}) is a unit if it is not the case that b < ν(jp − a + k), and when the coefficient is a unit, (6.13) has higher excess than the ph^{2}-terms. Thus ∂z_{ℓ} generates E_{2}^{2}(S^{(ℓ−1)(p−1)+N}), and the argument is finished as in the previous case. If b < ν(jp − a + k), then the coefficient in (6.13) is a multiple of p, and so this term is not a generator, and d_{i−ℓ+1} is not surjective. We continue pulling z_{ℓ} back as far as possible, to z_{ℓ−τ+1} for some τ ≥ 1. If im(d_{i−ℓ+τ}) has order p^{s} with s > 0, then by Lemma 4.12

∂(p^{s}z_{ℓ−τ+1}) = d(uh^{(ℓ−τ)(p−1)+N} + L)y_{ℓ−τ},

and the pullback is completed as usual.

If b > c, ∂(z_{ℓ+1}) ≡ −u′p^{c+1}h⊗h^{ℓ(p−1)+N}y_{ℓ} has order p^{b−c} in E^{2,ℓ}, p^{b−c}z_{ℓ+1} pulls back to

z_{ℓ} ≡ p^{b−c}z_{ℓ+1} − \frac{u′}{u} h^{ℓ(p−1)+N}y_{ℓ},

and ∂(z_{ℓ}) ≡ u′′h^{ℓ(p−1)+N} ⊗ hy_{ℓ−1}, which is a generator of E^{2,ℓ−1} ≈ Z/p. The pullback of pz_{ℓ} is achieved as usual.

Remark 6.14. The translation from Proposition 6.12 to Lemma 6.2 is straightforward. Assume ℓ = ap + N.

If k ≠ N, then b = 0 in 6.12, and so d_{i−ℓ} = 0. If also a ≡ k, then c = 0 and ν(jp − a + k) > 0, so that d_{i−ℓ+1} is not surjective.

ELEMENTS OF LARGE ORDER IN π_{∗}(SU(n)) 49

Now assume k = N. If a ≢ N, then c = 0 and b > 0 so that

ν(| coker(d_{i−ℓ})|) = (b + 1) − b = 1.

Now assume also that a = αp + N. If N ≢ j, then b = 1 and c ≥ 1, so that

| coker(d_{i−ℓ})| = |E^{2,ℓ}| = p^{2}.

If N ≡ j and α ≢ N, then b ≥ 2 and c = 1, and so

ν(| coker(d_{i−ℓ})|) = (b + 1) − (b − 1) = 2.

Similarly, if N ≡ j and α ≡ N, then b ≥ 2, c ≥ 2, and | coker(d_{i−ℓ})| ≥ p^{3}. Finally, if N ≢ j and α ≡ j, then b = c = 1 < ν(jp − a + k), and so d_{i−ℓ+1} is not surjective.

The scenario in Lemma 6.2 in which d_{i−ℓ−δ} = 0 for 1 ≤ δ ≤ p or p + 1 is due to the possibility in Proposition 6.12 that d_{i−(ℓ+p)+τ} could be 0 for τ < p.

We close with an example, promised earlier, in which there is a nonzero group E_{∞}^{2,a} with a < j. Recall that our results are predicated on these groups being 0. When these groups are nonzero, it just means that our Theorems 1.1 and 1.4 are a bit weaker than they might be. These cases were omitted because they are very rare and difficult to describe.

Example 6.15. Let p = 3, i = 3^{9} + 10, and N = 2. (Thus n = 2 · 3^{9} + 23, j = 3^{8} + 3, and k = 1.) We claim that E_{∞}^{2,7} ≈ Z/3. This was suggested by our theory, and was verified by computer calculation using (1.13).

From our point of view, the thing that causes this anomaly is the large group E_{1}^{2,10} ≈ Z/3^{10}. It is killed by differentials from some of the groups E^{1,t} with 23 ≤ t ≤ 32. In the usual pattern, there would have been nonzero differentials E^{1,23} → E^{2,7}, E^{1,26} → E^{2,9}, and E^{1,29} → E^{2,9}. However, what actually happens is that there are shorter nonzero differentials

Z/9 ≈ E_{13}^{1,22} → E_{13}^{2,9}, E_{14}^{1,22} → E_{14}^{2,8}, E^{1,20} → E^{2,7} ≈ Z/9, Z/27 ≈ E^{1,19} → E^{2,6}.

Groups whose order is not indicated are Z/3.

Without going into much detail, we mention that the differential E^{1,20} → E^{2,7} is obtained by pulling hy_{20} back to

z ≡ hy_{20} + ⋯ + h^{20}y_{10} + h^{19}y_{9} + h^{16}y_{8}.

Here we have omitted unit coefficients. The terms in ⋯ have small enough powers of h as to be insignificant. The h^{20}y_{10} comes from writing h ⊗ h^{10}y_{10} as d(h^{20}y_{10}). The leading term in ∂(z) is h^{19} ⊗ hvy_{7} ≡ h^{17} ⊗ hy_{7} ≡ h ⊗ h^{15}y_{7}, which is the element of order 3 in E_{1}^{2,7}.

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