CHAPTER 1

Computing v_{1}-periodic homotopy groups of spheres and some compact Lie groups

Donald M. Davis

Lehigh University Bethlehem, PA 18015 dmd1@lehigh.edu

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Definition of v_{1}-periodic homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. The isomorphism v_{1}^{−1}π_{∗}(S^{2n+1}) ≈ v_{1}^{−1}π_{∗−2n−1}^{s}(B^{qn}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4. J-homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5. The v_{1}-periodic homotopy groups of spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6. The v_{1}-periodic UNSS for spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7. v_{1}-periodic homotopy groups of SU(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 8. v_{1}-periodic homotopy groups of some Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

HANDBOOK OF ALGEBRAIC TOPOLOGY Edited by I.M. James \begin{matrix}c \\ ◯ \end{matrix} 1995 Elsevier Science B.V. All rights reserved

2  Donald M. Davis  Chapter 1

1. Introduction

In this paper, we present an account of the principal methods which have been used to compute the v_{1}-periodic homotopy groups of spheres and many compact simple Lie groups. The two main tools have been J-homology and the unstable Novikov spectral sequence (UNSS), and we shall strive to present all requisite background on both of these.

The v_{1}-periodic homotopy groups of a space X, denoted v_{1}^{−1}π_{∗}(X; p), are a certain localization of the actual p-local homotopy groups π_{∗}(X)_{(p)}. We shall drop the p from the notation except where it seems necessary. Very roughly, v_{1}^{−1}π_{∗}(X) is a periodic version of the portion of π_{∗}(X) detectable by real and complex K-theory and their operations. It is the first of a hierarchy of theories, v_{n}^{−1}π_{∗}(X), which should account for all of π_{∗}(X). Each group v_{1}^{−1}π_{i}(X) is a direct summand of some actual group π_{i+L}(X), at least if X has an H-space exponent and v_{1}^{−1}π_{i}(X) is a finitely-generated abelian group, which is the case in all examples discussed here.

The v_{1}-periodic homotopy groups are important because for spaces such as spheres and compact simple Lie groups they give a significant portion of the actual homotopy groups, and yet are often completely calculable. The goal of this paper is to explain how those calculations can be made. One might hope that these methods can be adapted to learning about v_{n}-periodic homotopy groups for n > 1.

One application of v_{1}-periodic homotopy groups is to obtain lower bounds for the exponents of spaces. The p-exponent of X is the largest e such that some homotopy group of X contains an element of order p^{e}. We can frequently determine the largest p-torsion summand in v_{1}^{−1}π_{∗}(X). Such a summand must also exist in some π_{i}(X), although we cannot usually specify which π_{i}(X). Thus we obtain lower bounds for the p-exponents of spaces, which we conjecture to be sharp in many of the cases studied here. It is known to be sharp for S^{2n+1} if p is odd. See Corollaries 7.8 and 8.9 for estimates of the p-exponent of SU(n) when p is odd.

Another application, which we will not discuss in this paper, is to James numbers, which are an outgrowth of work on vector fields in the 1950’s. It is proved in [23] that, for sufficiently large values of the parameters, the unstable James numbers equal the stable James numbers, and these equal a certain value which had been conjectured by a number of workers.

In Section 2, we present the definition and basic properties of the v_{1}-periodic homotopy groups. In Section 3, we describe the reduction of the calculation of the unstable groups v_{1}^{−1}π_{∗}(S^{2n+1}) to the calculation of stable groups v_{1}^{−1}π_{∗}^{s}(B^{qn}). Here B^{qn} is a space which will be defined in Section 3; when p = 2, it is the real projective space P^{2n}. Here we have begun the use, continued throughout the paper, of q as 2p − 2. The original proof of this result, 3.1, appeared in [40] and [56]; it involved delicate arguments involving the lambda algebra. We present a new proof, due to Langsetmo and Thompson, which involves completely different techniques, primarily K-theoretic.

In Section 4, we explain how to compute J_{∗}(B^{qn}), while in Section 5, we sketch the proof that if X is a spectrum, then v_{1}^{−1}π_{∗}(X) ≈ v_{1}^{−1}J_{∗}(X). Combining the results of Sections 3, 4, and 5 yields a nice complete result for v_{1}^{−1}π_{∗}(S^{2n+1}), which can be summarized as

v_{1}^{−1}π_{2n+1+i}(S^{2n+1}) ≈ v_{1}^{−1}π_{i}^{s}(B^{qn}) ≈ v_{1}^{−1}J_{i}(B^{qn})

and, if p is odd,

≈ \{ \begin{matrix}Z/p^{min(n,ν_{p}(a)+1)} & if i = qa − 2 or qa − 1 \\ 0 & if i ≢ −1 or −2 mod q. \end{matrix}

We will use Z/n and Z_{n} interchangeably, and let ν_{p}(n) denote the exponent of p in n. The subscript p of ν will sometimes be omitted if it is clear from the context. The final result for v_{1}^{−1}π_{∗}(S^{2n+1}) when p = 2 is more complicated; see Theorem 4.2.

In Section 6, we sketch the formation of the UNSS and its v_{1}-localization, and for S^{2n+1} we compute the entire v_{1}-localized UNSS and part of the unlocalized UNSS. In Section 7, we discuss the computation of the v_{1}-periodic UNSS and v_{1}-periodic homotopy groups in general for spherically resolved spaces and specifically for the special unitary groups SU(n). This is considerably easier at the odd primes than at the prime 2. The following key result of Bendersky ([4]) will be proved by observing that the homotopy-theoretic calculation and UNSS calculation agree for S^{2n+1}.

If p is odd, and X is built by fibrations from finitely many odd-dimensional spheres, then v_{1}^{−1}E_{2}^{s,t}(X) = 0 in the v_{1}-periodic UNSS unless s = 1 or 2 and t is odd, in which case

v_{1}^{−1}π_{i}(X; p) ≈ \{ \begin{matrix}v_{1}^{−1}E_{2}^{1,i+1}(X) & if i is even \\ v_{1}^{−1}E_{2}^{2,i+2}(X) & if i is odd. \end{matrix}

In Section 7 we also review the computation of E_{2}^{1}(SU(n)) in [6] and combine it with Theorem 1.1 to obtain the following result, which was the main result of [23].

Let ν_{p}(m) denote the exponent of p in m, and define integers a(k, j) and e_{p}(k, n) by

(e^{x} − 1)^{j} = ∑_{k≥j} a(k, j)\frac{x^{k}}{k!} ,

and

e_{p}(k, n) = min{ν_{p}(a(k, j)) : n ≤ j ≤ k}.

If p is odd, then v_{1}^{−1}π_{2k}(SU(n)) ≈ Z/p^{e_{p}(k,n)}, and v_{1}^{−1}π_{2k−1}(SU(n)) is an abelian group of order p^{e_{p}(k,n)}, although not always cyclic.

In Section 8, we illustrate the two principal methods used in computing v_{1}- periodic homotopy groups of the exceptional Lie groups, focusing on v_{1}^{−1}π_{∗}(G_{2}; 5) for UNSS methods, and on v_{1}^{−1}π_{∗}(F_{4}/G_{2}; 2) for homotopy (J-homology) methods. We also discuss the recent thesis of Yang ([58]), which gives formulas more tractable than that of Definition 1.2 for the numbers e_{p}(k, n) which appear in Theorem 1.3, provided n ≤ p^{2} − p.

2. Definition of v_{1}-periodic homotopy groups

In this section, we present the definition and basic properties of the v_{1}-periodic homotopy groups. We work toward the definition of v_{1}^{−1}π_{∗}(X) by recalling the definition of v_{1}^{−1}π_{∗}(X; Z/p^{e}). Let M^{n}(k) denote the Moore space S^{n−1} ∪_{k} e^{n}. The mod k homotopy group π_{n}(X; Z/k) is defined to be the set of homotopy classes [M^{n}(k), X]. With the prime p implicit, and q = 2(p − 1), let

s(e) = \{ \begin{matrix}p^{e−1}q & if p is odd \\ max(8, 2^{e−1}) & if p = 2. \end{matrix}  (2.1)

Let A : M^{n+s(e)}(p^{e}) → M^{n}(p^{e}) denote a map, as introduced by Adams in [1], which induces an isomorphism in K-theory. Such a map exists provided n ≥ 2e + 3. ([28, 2.11]) Then v_{1}^{−1}π_{i}(X; Z/p^{e}) is defined to be dirlim_{N} [M^{i+Ns(e)}(p^{e}), X], where the maps A are used to define the direct system. The map is what Hopkins and Smith would call a v_{1}-map, and they showed in [35] that any two v_{n}-maps of a finite complex which admits such maps become homotopic after a finite number of iterations (of suspensions of the same map), and hence v_{1}^{−1}π_{∗}(X; Z/p^{e}) does not depend upon the choice of the map A. Note that although v_{1}^{−1}π_{∗}(X; Z/p^{e}) is a theory yielding information about the unstable homotopy groups of X, the maps A which define the direct system may be assumed to be stable maps, since the direct limit only cares about large values of i + Ns(e). Note also that the groups v_{1}^{−1}π_{i}(X; Z/p^{e}) are defined for all integers i and satisfy v_{1}^{−1}π_{i}(X; Z/p^{e}) ≈ v_{1}^{−1}π_{i+s(e)}(X; Z/p^{e}).

There is a canonical map ρ : M^{n}(p^{e+1}) → M^{n}(p^{e}) which has degree p on the top cell, and degree 1 on the bottom cell. It satisfies the following compatibility with Adams maps.

[34, p. 633] If A : M^{n+s(e)}(p^{e}) → M^{n}(p^{e}) and

A′ : M^{n+s(e+1)}(p^{e+1}) → M^{n}(p^{e+1})

are v_{1}-maps, then there exists k so that the following diagram commutes.

M^{n+ks(e+1)}(p^{e+1}) ⟶^{ρ} M^{n+ks(e+1)}(p^{e})
↓_{A^{′k}}  ↓_{A^{kp′}}
M^{n}(p^{e+1})  ⟶^{ρ}  M^{n}(p^{e})

Here p′ = p unless p = 2 and e < 4, in which case p′ = 1.

Thus, after sufficient iteration of the Adams maps, there are morphisms ρ^{∗} between the direct systems used in defining v_{1}^{−1}π_{∗}(X; Z/p^{e}) for varying e, and passing to direct limits, we obtain a direct system

v_{1}^{−1}π_{∗}(X; Z/p^{e}) ⟶^{ρ^{∗}} v_{1}^{−1}π_{∗}(X; Z/p^{e+1}) ⟶^{ρ^{∗}} · · · .  (2.2)

The following definition was given in [28], following less satisfactory definitions in [30] and [23].

For any space X and any integer i,

v_{1}^{−1}π_{i}(X) = dirlim_{e} v_{1}^{−1}π_{i+1}(X; Z/p^{e}),

using the direct system in (2.2).

The reason for the use of the (i+1)st mod-p^{e} periodic homotopy groups in defining the ith (integral) periodic groups is that the maps ρ of Moore spaces have degree 1 on the bottom cells, but mod-p^{e} homotopy groups are indexed by the dimension of the top cell.

The mod-p^{e} periodic homotopy groups have received more attention in the literature, especially when e = 1. For spaces with H-space exponents, there is a close relationship between the integral periodic groups and the mod-p^{e} groups, which we recall after giving the relevant definition.

A space X has H-space exponent p^{e} if for some positive integer L the p^{e}-power map δ^{L}X → δ^{L}X is null-homotopic.

By [20] and [36], spheres and compact Lie groups have H-space exponents.

(i) [28, 1.7] If X has H-space exponent p^{e}, then there is a split short exact sequence

0 → v_{1}^{−1}π_{i}(X) → v_{1}^{−1}π_{i}(X; Z/p^{e}) → v_{1}^{−1}π_{i−1}(X) → 0.

(ii) On the category of spaces with H-space exponents, there is a natural transformation π_{∗}(−)_{(p)} → v_{1}^{−1}π_{∗}(−; p).

(iii) If X has an H-space exponent, and v_{1}^{−1}π_{i}(X) is a finitely generated abelian group, then v_{1}^{−1}π_{i}(X) is a direct summand of π_{i+L}(X) for some non-negative integer L.

The proof of part ii utilizes the fibration

map_{∗}(M^{n+1}(p^{e}), X) → δ^{n}X ⟶^{p^{e}} δ^{n}X,

where map_{∗}(−, −) denotes the space of pointed maps. If the second map is nullhomotopic, then the first map admits a section s. The natural transformation is induced by

δ^{n}X ⟶^{s} map_{∗}(M^{n+1}(p^{e}), X) → dirlim_{k} map_{∗}(M^{n+1+ks(e)}(p^{e}), X).

Techniques of [28] imply naturality of this construction. To prove part iii, we note that the map s allows v_{1}^{−1}π_{i}(X) to be written as dirlim_{k} π_{i+ks(e)}(X), which is a direct summand of one of the groups in the direct system, provided the direct limit is finitely generated.

If X is a spectrum, then v_{1}^{−1}π_{∗}(X) is defined in exactly the same way as for spaces, that is, as in Definition 2.2. If X is a space, then stable groups, v_{1}^{−1}π_{∗}^{s}(X), can be defined either as v_{1}^{−1}π_{∗}(∈^{∞} X), where ∈^{∞} X denotes the suspension spectrum of X, or as v_{1}^{−1}π_{∗}(QX), where QX = δ^{∞} ∈^{∞} X is the associated infinite loop space.

3. The isomorphism v_{1}^{−1}π_{∗}(S^{2n+1}) ≈ v_{1}^{−1}π_{∗−2n−1}^{s}(B^{qn})

In this section, we sketch a new proof, due to Thompson and Langsetmo ([39, 4.2]), of the following crucial result.

There is a map

δ^{2n+1}S^{2n+1} → QB^{qn}  (3.1)

which induces an isomorphism in v_{1}^{−1}π_{∗}(−).

Here QX = δ^{∞} ∈^{∞} X, and B^{qn} is the qn-skeleton of the p-localization of the classifying space B ∈_{p} of the symmetric group ∈_{p} on p letters. Note that if p = 2, then B^{qn} is the real projective space RP^{2n}. The original proof, from [40] when p = 2 and [56] when p is odd, involved delicate arguments involving the lambda algebra and unstable Adams spectral sequences. We feel that the following argument, primarily K-theoretic, will speak to a broader cross-section of readers. The following elementary result shows that it is enough to show that the map (3.1) induces an iso in v_{1}-periodic mod p homotopy.

If a map X → Y induces an isomorphism in v_{1}^{−1}π_{∗}(−; Z/p), then it induces an isomorphism in v_{1}^{−1}π_{∗}(−; p).

Proof. There are cofibrations of Moore spaces which induce natural exact sequences

→ v_{1}^{−1}π_{n}(X; Z/p^{e}) ⟶^{ρ^{∗}} v_{1}^{−1}π_{n}(X; Z/p^{e+1}) →
v_{1}^{−1}π_{n}(X; Z/p) → v_{1}^{−1}π_{n−1}(X; Z/p^{e}) → .

Induction on e using the 5-lemma implies that there are isomorphisms

v_{1}^{−1}π_{∗}(X; Z/p^{e}) → v_{1}^{−1}π_{∗}(Y; Z/p^{e})

for all e, compatible with the maps ρ^{∗} which define the direct system (2.2). The desired isomorphism of the direct limits is immediate.

The construction of the map (3.1) takes us far afield, and is not used elsewhere in the computations. For completeness, we wish to say something about it, but we will be extremely sketchy. The map is due to Snaith. Work of many mathematicians, especially Peter May, is important in the construction. However, we shall just refer the reader to [37], where the proof of naturality of these maps is given, along with references to the earlier work.

i. There are maps

s_{n} : δ^{2n+1}S^{2n+1} → QB^{qn}

which are compatible with respect to inclusion maps as n increases, and such that the adjoint map

∈^{∞} δ^{2n+1}S^{2n+1} →∈^{∞} B^{qn}

is the projection onto a summand in a decomposition of ∈^{∞} δ^{2n+1}S^{2n+1} as a wedge of spectra.

ii. There is a map QS^{2n+1} ⟶^{g} Q ∈^{2n+1} B_{(n+1)q−1} whose fiber, F, satisfies

v_{1}^{−1}π_{∗}(δ^{2n+1}F; Z/p) ≈ v_{1}^{−1}π_{∗}(QB^{qn}; Z/p).

Sketch of proof. Let C_{N}(k) denote the space of ordered k-tuples of disjoint little cubes in I^{N}. If X is a based space, let C_{N}X denote the space of finite collections of disjoint little cubes of I^{N} labeled with points of X. More formally,

C_{N}X = ∐_{k≥1} C_{N}(k) ×_{∈_{k}} X^{k}/ ∼,

where

[(c_{1}, …, c_{k}), (x_{1}, …, x_{k−1}, ∗)] ∼ [(c_{1}, …, c_{k−1}), (x_{1}, …, x_{k−1})].

There are natural maps, due to May,

C_{N}X → δ^{N} ∈^{N} X,  (3.2)

which are weak equivalences if X is connected.

The space C_{N}X is filtered by defining F_{m}(C_{N}X) to be the subspace of m or fewer little labeled cubes. The successive quotients are defined by

D_{N,m}X = F_{m}(C_{N}X)/F_{m−1}(C_{N}X) ≈ C_{N}(m)^{+} ∧_{∈_{m}} X^{[m]},

where X^{[m]} denotes the m-fold smash product. Snaith proved that if X is pathconnected, there is a weak equivalence of suspension spectra

∈^{∞} C_{N}X ≃ ⋁_{m≥1} ∈^{∞} D_{N,m}X.  (3.3)

Let N = 2n + 1 and X = S^{0}. Stabilize the equivalence of (3.2), and project onto the summand of (3.3) with m = p to get

∈^{∞} δ^{2n+1}S^{2n+1} →∈^{∞} C_{2n+1}(p)/ ∈_{p} .

The identification of C_{2n+1}(p)/ ∈_{p} as the nq-skeleton of B ∈_{p} after localization at p was obtained by Fred Cohen in [19, p. 246].

Note that (3.2) and (3.3) yield a map δ^{N} ∈^{N} X → QD_{N,m}X. The map g of ii) is obtained from the case N = ∞ and X = S^{2n+1} as the composite

QS^{2n+1} → QD_{∞,p}S^{2n+1} = Q(B ∈_{p}^{+} ∧_{∈_{p}}(S^{2n+1})^{[p]}) ≃ Q ∈^{2n+1} B_{(n+1)q−1}.

Here we have used results of [45] for the last equivalence.

The compatible maps s_{n} of 3.3(i) combine to yield a map s′ : QS^{0} → QB^{∞}, and one can show that the right square commutes in the diagram of fibrations below.

\begin{aligned}δ^{2n+1}F & → QS^{0} ⟶^{δ^{2n+1}g} & QB_{(n+1)q−1} \\ ↓^{t} & ↓_{s′} & ↓^{=} \\ QB^{qn} & → QB^{∞} → & QB_{(n+1)q−1} \end{aligned}

The map t then follows, and will induce an isomorphism in v_{1}^{−1}π_{∗}(−; Z/p), as asserted in Theorem 3.3(ii), once we know that s′ does. Kahn and Priddy showed that there is an infinite loop map λ : QB^{∞} → QS^{0} such that λ ∘ s′ induces an isomorphism in π_{j}(−) for j > 0. It is easily verified using methods of the next two sections that the associated stable map B^{∞} → S^{0} induces an isomorphism in v_{1}^{−1}π_{∗}(−; Z/p). Hence so does s′.

Throughout this section, let K_{∗}(−) denote mod p K-homology, and M^{k} denote the mod p Moore space M^{k}(p). The following two theorems, whose proofs occupy most of the rest of this section, imply that j : S^{2n+1} → F induces an iso in v_{1}^{−1}π_{∗}(−; Z/p).

[16, 14.4] If k ≥ 2, and ϕ : X → Y is a map of k-connected spaces such that δ^{k}ϕ is a K_{∗}-equivalence, then ϕ induces an isomorphism in v_{1}^{−1}π_{∗}(−; Z/p).

[39] Let F be as in Theorem 3.3ii. The map S^{2n+1} → QS^{2n+1} lifts to a map j : S^{2n+1} → F such that δ^{2}j is a K_{∗}-equivalence.

The map j of Theorem 3.5 can be chosen so that t ∘ δ^{2n+1}j = s_{n}, where t is the map in the proof of Theorem 3.3ii which induces the isomorphism in v_{1}^{−1}π_{∗}(−; Z/p). Theorem 3.1 is now proved by applying Lemma 3.2 to s_{n}.

Bousfield localization is involved in the proof of Theorem 3.4 and several other topics later in the paper, and so we review the necessary material. If E is a spectrum, then a space (or spectrum) X is E_{∗}-local if every E_{∗}-equivalence Y → W induces a bijection [W, X] → [Y, X]. The E_{∗}-localization of X is an E_{∗}-local space (or spectrum) X_{E} together with an E_{∗}-equivalence X → X_{E}. The existence and uniqueness of these localizations were established in [18] and [17]. We will need the following result of Bousfield, the proof of which is easily obtained using the ideas at the beginning of Section 5.

[18, 4.8] A spectrum is K_{∗}-local if and only if its mod p homotopy groups are periodic under the action of the Adams map.

We will omit the proof of Theorem 3.4, as it requires many peripheral ideas. Instead, we will sketch a proof of the following result, which, although weaker than 3.4, has the same flavor, and predated it. An alternate proof of Theorem 3.1 can be given by using Theorem 3.7 and strengthening Theorem 3.5 to show that K_{∗}(δ^{3}j) is bijective. As the calculations for δ^{3} seem significantly more difficult than for δ^{2}, we omit that approach.

[57] Let p be an odd prime, and let X and Y be 3-connected spaces. Suppose that f : X → Y is a map such that K_{∗}(δ^{k}f) is an isomorphism for k = 0, 1, 2, and 3. Then f induces an isomorphism in v_{1}^{−1}π_{∗}(−; Z/p).

Sketch of proof. If F → E → B is a principal fibration, there is a bar spectral sequence converging to K_{∗}(B) with E_{s,t}^{2} ≈ T or_{s,t}^{K_{∗}F} (K_{∗}E, K_{∗}). (See [55] for a discussion of this spectral sequence.)

Applied to the commutative diagram of principal fiber sequences

\begin{matrix}δ^{3}X & ⟶^{p} δ^{3}X → & map_{∗}(M^{3}, X) \\ ↓^{δ^{3}f} & ↓^{δ^{3}f} & ↓^{f′} \\ δ^{3}Y & ⟶^{p} δ^{3}Y → & map_{∗}(M^{3}, Y), \end{matrix}

the spectral sequence and the hypothesis of the theorem imply that f′ induces an isomorphism in K_{∗}(−). Hence there is an equivalence of the K_{∗}-localizations

(map_{∗}(M^{3}, X))_{K} ⟶^{f′_{K}} (map_{∗}(M^{3}, Y))_{K}.

Let V (X) denote the mapping telescope of

map_{∗}(M^{3}, X) ⟶^{A^{∗}} map_{∗}(M^{3+q}, X) ⟶^{A^{∗}} · · · .

Then V (X) is K_{∗}-local, since it is δ^{∞} of a periodic spectrum which is K_{∗}-local by Theorem 3.6. This implies that there are maps i′ making the following diagram commute.

\begin{aligned}map_{∗}(M^{3}, X) & → (map_{∗}(M^{3}, X))_{K} ⟶^{i′} & V (X) \\ ↓^{f′} & ↓^{f′_{K}} & ↓^{V (f)} \\ map_{∗}(M^{3}, Y) & → (map_{∗}(M^{3}, Y))_{K} ⟶^{i′} & V (Y) \end{aligned}

Since v_{1}^{−1}π_{∗}(X; Z/p) ≈ π_{∗}(V (X); Z/p), the desired isomorphism is a consequence of the following construction of an inverse to

V (f)_{∗} : π_{∗}(V (X); Z/p) → π_{∗}(V (Y); Z/p).

An element α ∈ π_{k}(V (Y); Z/p) can be represented by a map

M^{k} → map_{∗}(M^{3+qpj}, Y),

or, adjointing, by a map M^{k+qpj} → map_{∗}(M^{3}, Y). Here some care is required to see that we can switch the Moore space factor on which the map A is performed. The element which corresponds to α is the composite

M^{k+qpj} → map_{∗}(M^{3}, Y) → (map_{∗}(M^{3}, Y))_{K}
⟶^{(f′_{K})^{−1}} (map_{∗}(M^{3}, X))_{K} ⟶^{i′} V (X).

The proof of Theorem 3.5 involves a good bit of delicate computation. The hardest part is the determination of K_{∗}(δ^{2}F) as a Hopf algebra. In order to conveniently obtain the coalgebra structure of K_{∗}(δ^{2}F), we proceed in two steps. We first calculate the algebra K_{∗}(δ^{3}F), using the bar spectral sequence associated to the principal fibration

δ^{4}QS^{2n+1} → δ^{4}Q ∈^{2n+1} B_{q(n+1)−1} → δ^{3}F.

This spectral sequence is calculated in [38], obtaining an algebra isomorphism

K_{∗}(δ^{3}F) ≈ P [y_{1}, y_{2}] ⊗ E[z].  (3.4)

Here y_{i} (resp. z) has bidegree (1, 1) (resp. (0, 1)) in the spectral sequence, and hence even (resp. odd) degree in K_{∗}(δ^{3}F). The calculation of this spectral sequence requires some preliminary computation regarding the algebra structure of K_{∗}(δ^{2}F), and this requires major input from [44].

Now we calculate the bar spectral sequence associated to the principal fibration

δ^{3}F → ∗ → δ^{2}F.

This spectral sequence, with E_{2} ≈ T or^{P [y_{1},y_{2}]⊗E[z]}(K_{∗}, K_{∗}), collapses to yield an isomorphism of Hopf algebras

K_{∗}(δ^{2}F) ≈ E[a_{1}, a_{2}] ⊗ ·[b],  (3.5)

where · denotes the divided polynomial algebra over Z_{p}. The coproduct has a_{1}, a_{2}, and γ_{1}(b) as the primitives, and ψ(γ_{i}(b)) = ∑ γ_{j}(b) ⊗ γ_{i−j}(b).

Dualizing eq. (3.5) yields an isomorphism of algebras

K^{∗}(δ^{2}F) ≈ E[α_{1}, α_{2}] ⊗ P [β],

with |α_{i}| odd and |β| even. This matches nicely with the following result from [52, 3.8].

For any prime p, there is an isomorphism of algebras

K^{∗}(δ^{2}S^{2n+1}) ≈ E[u_{0}, u_{1}] ⊗ P [w],

where E and P denote exterior and polynomial algebras over K_{∗}.

We will use the Atiyah-Hirzebruch spectral sequence to show that the map

δ^{2}S^{2n+1} ⟶^{δ^{2}j} δ^{2}F

of Theorem 3.5 sends the generators of the isomorphic K^{∗}(−)-algebras across. This will imply the second half of Theorem 3.5.

In [52], it is shown how the generators u_{0}, w, and u_{1} of K^{∗}(δ^{2}S^{2n+1}) arise in the Atiyah-Hirzebruch spectral sequence whose E_{2}-term is H^{∗}(δ^{2}S^{2n+1}; K_{∗}). Indeed, they arise from the bottom three cohomology classes, of grading 2n − 1, 2pn − 2, and 2pn − 1, respectively. The map δ^{2}j induces an isomorphism in H^{i}(−; Z_{p}) for i < 2pn−1+min(q, 2n−2), and so it maps onto the three generators of K^{∗}(δ^{2}S^{2n+1}).

4. J-homology

In this section, we show how to compute J_{∗}(B^{qn}). When combined with Theorems 3.1 and 5.1, this gives an explicit computation of v_{1}^{−1}π_{∗}(S^{2n+1}), which we state at the end of this section as Theorem 4.2. This will be extremely important in our calculation of v_{1}^{−1}π_{∗}(Y) for other spaces Y. We begin with the case p odd, where the results are somewhat simpler to state. Historically it worked in the other order, with Mahowald’s 2-primary results in [40] preceding Thompson’s odd-primary work in [56].

Let p be an odd prime. We follow quite closely the exposition in [24] and [56, §3]. The spectrum bu_{(p)} splits as a wedge of spectra ∈^{2i} ℓ satisfying H^{∗}(ℓ; Z_{p}) ≈ A//E, where A is the mod p Steenrod algebra, and E is the exterior subalgebra generated by Q_{0} = β and Q_{1} = P^{1}β − βP^{1}. The spectrum ℓ is sometimes written BP ⟨1⟩. Then ℓ_{∗} = π_{∗}(ℓ) is calculated from the Adams spectral sequence (ASS) with

E_{2}^{s,t} ≈ Ext_{A}^{s,t}(H^{∗}ℓ, Z_{p}) ≈ Ext_{E}^{s,t}(Z_{p}, Z_{p}) ≈ Z_{p}[a_{0}, a_{1}],  (4.1)

where a_{i} has bigrading (1, iq + 1). Here we have used the change-of-rings theorem in the middle step. There are no possible differentials in the spectral sequence, and since multiplication by a_{0} corresponds to multiplication by p in homotopy, we find that π_{∗}(ℓ) is a polynomial algebra over Z_{(p)} on a class of grading q. Using the ring structure of ℓ, one easily sees that there is a cofibration

∈^{q} ℓ → ℓ → HZ_{(p)}.

Let k be a (p − 1)st root of unity mod p but not mod p^{2}, and let ψ^{k} denote the Adams operation. The map ψ^{k} −1 : ℓ → ℓ lifts to a map θ : ℓ →∈^{q} ℓ. The connective J-spectrum, J, is defined to be the fiber of θ. The homotopy exact sequence of θ easily implies that

π_{i}(J) ≈ \{ \begin{matrix}Z_{(p)} & if i = 0 \\ Z/p^{ν_{p}(j)+1} & if i = qj − 1 with j > 0 \\ 0 & otherwise. \end{matrix}

The image of the classical J-homomorphism is mapped isomorphically onto these groups by the map S^{0} → J; this is the reason for the name of the spectrum.

We now proceed toward the calculation of J_{∗}(B^{qn}). We let B be the p-localization of the suspension spectrum of B ∈_{p}. Then, with coefficients always in Z_{p}, the only nonzero groups H^{i}(B) occur when i ≡ 0 or −1 mod q, and i > 0. These groups are cyclic of order p with generator x_{i} satisfying Q_{0}x_{aq−1} = x_{aq} and Q_{1}x_{aq−1} = x_{(a+1)q}. We will work with the skeleta B^{qn} and the quotients B_{q(n+1)−1} = B/B^{qn}; these are suspension spectra of the spaces which appeared in Theorem 3.3.

There is a p-local map B → S^{0} constructed by Kahn and Priddy. If R denotes its cofiber, there is a filtration of the E-module H^{∗}R with subquotients ∈^{qi} E//E_{0} for i ≥ 0. Here E_{0} is the exterior subalgebra of E generated by Q_{0}. Since Ext_{E}(E//E_{0}) ≈ Ext_{E_{0}}(Z_{p}) ≈ Z_{p}[a_{0}], we find that Ext_{E}(H^{∗}R) has a “spike,” consisting of the powers of a_{0}, for each nonnegative value of t − s which is a multiple of q. Here we have begun a practice of omitting Z_{p} from the second variable of Ext_{B}(−, −) if B is a subalgebra of the Steenrod algebra. The action of Ext_{E}(Z_{p}) on Ext_{E}(H^{∗}R) has a_{1} always acting nontrivially.

We draw ASS pictures with coordinates (t − s, s), so that horizontal component refers to homotopy group. A chart for the ASS of R ∧ ℓ is given in fig. 1.

The short exact sequence

0 → H^{∗}(∈ B) → H^{∗}(R) → H^{∗}(S^{0}) → 0

induces an exact sequence

→ Ext_{E}^{s,t}(H^{∗}S^{0}) ⟶^{i_{∗}} Ext_{E}^{s,t}(H^{∗}R) → Ext_{E}^{s,t}(H^{∗}(∈ B)) → Ext_{E}^{s+1,t}(H^{∗}S^{0}) → .

(4.2)

These morphisms are Ext_{E}(Z_{p})-module maps, and the action of a_{1} implies that i_{∗}

Figure 1. ASS for ℓ_{∗}(R)

[figure: t − s =; 0; q; 2q; ⋯]

is injective. Thus there are elements x_{iq−1} ∈ Ext_{E}^{0,iq−1}(H^{∗}B) for i > 0 such that

Ext_{E}^{s,t}(H^{∗}B) = \{ \begin{matrix}Z_{p} & if t − s = iq − 1, i > 0, 0 ≤ s < i \\ 0 & otherwise, \end{matrix}

with generators a_{0}^{s}x_{iq−1}. Hence

ℓ_{i}(B) ≈ \{ \begin{matrix}Z/p^{(i+1)/q} & if i ≡ −1 mod q, and i > 0 \\ 0 & otherwise. \end{matrix}

There is an isomorphism of E-modules H^{∗}(B_{q(n+1)−1}) ≈∈^{qn} H^{∗}(B), and so

ℓ_{∗}(B_{q(n+1)−1}) ≈ ℓ_{∗}(∈^{qn} B).

The Ext calculation easily implies that the morphism ℓ_{∗}(B) → ℓ_{∗}(B_{q(n+1)−1}) induced by the collapse map is surjective, and so the exact sequence of the cofibration B^{qn} → B → B_{q(n+1)−1} implies that

ℓ_{i}(B^{qn}) ≈ \{ \begin{matrix}Z/p^{min((i+1)/q,n)} & if i ≡ −1 mod q, and i > 0 \\ 0 & otherwise. \end{matrix}

The ASS chart for B^{4q} is illustrated in fig. 2.

The map S^{0} → R implies that θ_{∗} : ℓ_{qj}(X) → ℓ_{(q−1)j}(X) is multiplication by the same number for X = R as it was for X = S^{0}. Thus it is multiplication by p^{ν_{p}(j)+1}. Now the map R →∈ B implies that θ_{∗} : ℓ_{qj−1}(B) → ℓ_{(q−1)j−1}(B) is multiplication by p^{ν_{p}(j)+1}. The maps B^{qn} → B → B_{q(n+1)−1} imply that the same is true in B^{qn} and B_{q(n+1)−1}. We obtain

J_{i}(B^{qn}) ≈ \{ \begin{matrix}Z/p^{min(n,ν_{p}(j)+1)} & if i = jq − 1, j > 0 \\ Z/p^{min(n,ν_{p}(j)+1)} & if i = jq − 2, j > n \\ Z/p^{min(n−1,ν_{p}(j))} & if i = jq − 2, 0 < j ≤ n \\ 0 & otherwise. \end{matrix}  (4.3)

Figure 2. ASS for ℓ_{∗}(B^{4q}) for t − s < 6q

[figure: q − 1; 2q − 1; 3q − 1; 4q − 1; 5q − 1; 6q − 1]

This is illustrated in fig. 3, which is not quite an ASS chart. It is a combination of the charts for ℓ_{∗}(B^{qn}) and ℓ_{∗−q+1}(B^{qn}) and the homomorphism θ_{∗} between them, which is represented by lines of negative slope. The exact sequence

0 → coker(θ_{∗+1}) → J_{∗}(B^{qn}) → ker(θ_{∗}) → 0

says that elements which are not involved in these boundary morphisms comprise J_{∗}(B^{qn}). There are several reasons for our having elevated the filtrations of ℓ_{∗−q+1}(B^{qn}) by 1 in this chart. One is that it makes all the boundary morphisms go up, so that it looks like an ASS chart. Another is that (by [40]) there is a resolution of B^{qn} ∧J (which is not an Adams resolution) for which the homotopy exact couple is depicted by this chart. A third is that if J_{1} is defined to be the fiber of J → HZ_{2}, then the ASS chart for B^{qn} ∧ J_{1} will agree with this chart in filtration greater than 1. See [13, §6] for an elaboration on this.

Figure 3. Beginning of chart for J_{∗}(B^{4q})

[figure: q − 1; 2q − 1; 3q − 1; 4q − 1; 5q − 1; 6q − 1]

If X is a space or spectrum, then v_{1}^{−1}J_{i}(X) is defined analogously to Definition 2.2 to be dirlim_{e,k} [M^{i+1+ks(e)}(p^{e}), X ∧J]. Since J is a stable object, we can S-dualize the Moore space, obtaining dirlim_{e,k} J_{i+1+ks(e)}(X ∧ M(p^{e})), where the Moore spectrum M(n) has cells of degree 0 and 1. The “+1” in this J-group is present due to the maps M(p^{e}) → M(p^{e+1}) having degree 1 on the 1-cell.

One can often compute v_{1}^{−1}J_{∗}(X) directly from J_{∗}(X) without having to worry about the “∧M(p^{e})”. This can be done by extending the periodic behavior which occurs in positive filtration down into negative filtrations and negative stems. For example (cf. fig. 3), a chart for v_{1}^{−1}J_{∗}(B^{nq}) has, for all integers a, adjacent towers of height n in aq − 2 and aq − 1 with d_{ν(a)+1}-differential. If ν(a) + 1 ≥ n, then the differential is 0. This interpretation of v_{1}^{−1}J_{∗}(−) can be justified using the following result.

Let p be an odd prime, and let KU be the spectrum for periodic K-theory localized at p. Let k be a (p − 1)st root of unity mod p but not mod p^{2}, and let Ad denote the

fiber of KU ⟶^{ψ^{k}−1} KU. Then v_{1}^{−1}J_{∗}(−) ≈ Ad_{∗}(−).

This follows from the fact that if v_{1}^{−1}ℓ_{∗}(−) is defined as dirlim_{e,k} ℓ_{i+1+ks(e)}(X ∧ M(p^{e})), then v_{1}^{−1}ℓ_{∗}(−) ≈ KU_{∗}(−), which is a consequence of the fact that A ∧ ι = v ∧ 1_{M} :∈^{q} M → M ∧ ℓ, where A :∈^{q} M → M and v : S^{q} → ℓ.

When p = 2, the results are a bit messier to state and picture. If bsp denotes the 2-local connected δ-spectrum whose (8k)th space is BSp[8k], then ∈^{4} bsp ≃ bo[4], the spectrum formed from bo by killing π_{i}(−) for i < 4. The map ψ^{3} − 1 : bo → bo lifts to a map θ : bo →∈^{4} bsp, and J is defined to be the fiber of θ.

Let A_{1} denote the subalgebra of the mod 2 Steenrod algebra A generated by Sq^{1} and Sq^{2}. Then H^{∗}bo ≈ A//A_{1} and H^{∗}bsp ≈ A ⊗_{A_{1}} N, where N = ⟨1, Sq^{2}, Sq^{3}⟩. Hence, using the change of rings theorem, the E_{2}-term of the ASS converging to π_{∗}(bo) is Ext_{A_{1}}(Z_{2}), while that for bsp is Ext_{A_{1}}(N). These are easily computed to begin as in fig. 4, with each chart acted on freely by an element in (t − s, s) = (8, 4). Positively sloping diagonal lines indicate the action of h_{1} ∈ Ext_{A_{1}}^{1,2}(Z_{2}). It corresponds to the Hopf map η in homotopy.

Figure 4. Part of ASS for bo and bsp

[figure: 0; 4; 0; 4; bo; bsp]

There are no possible differentials in these ASS’s, and so we obtain

π_{i}(bo) ≈ \{ \begin{matrix}Z_{(2)} & if i ≡ 0 mod 4, i ≥ 0 \\ Z_{2} & if i ≡ 1, 2 mod 8, i > 0 \\ 0 & otherwise, \end{matrix}

in accordance with Bott periodicity.

From Adams’ work, we have θ_{∗} : π_{4j}(bo) → π_{4j}(∈^{4} bsp) hitting all multiples of 2^{ν_{2}(j)+3}, while θ_{∗} is 0 on the Z_{2}’s. This yields π_{i}(J) = 0 if i < 0, while for i ≥ 0

π_{i}(J) ≈ \{ \begin{matrix}Z_{(2)} & if i = 0 \\ Z/2^{ν_{2}(i)+1} & if i ≡ 3 mod 4 \\ Z_{2} & if i ≡ 0, 2 mod 8, i > 0 \\ Z_{2} ⊕ Z_{2} & if i ≡ 1 mod 8, i > 1 \\ 0 & if i ≡ 4, 5, 6 mod 8. \end{matrix}  (4.4)

From Adams’ work and the confirmation of the Adams Conjecture, it is known that π_{∗}(S^{0}) → π_{∗}(J) sends the image of the classical J-homomorphism plus Adams’ elements μ_{j} and ημ_{j} isomorphically onto π_{∗}(J). A chart for π_{i}(J) with i ≤ 18 is given in fig. 5. Here the elements coming from bo are indicated by •’s, while those from ∈^{4} bsp are indicated by ∘’s.

Figure 5. 2-primary π_{i}(J), i ≤ 18

[figure: 0; 3; 7; 11; 15]

The first dotted η-extension can be deduced from the fact that θ^{∗} : H^{4}(∈^{4} bsp) → H^{4}(bo) hits Sq^{4}, together with the relation η^{3} = 4ν. This η-action is then pushed along by periodicity. Another argument which is frequently useful for deducing η-extensions such as this involves Toda brackets. The generator of π_{8i+4}(bo) is obtained from the element α ∈ π_{8i+2}(bo) as ⟨α, η, 2⟩. Clearly α pulls back to π_{∗}(J), and if αη were 0 here, then the bracket could also be formed in J. However, the boundary morphism on π_{8i+4}(bo) implies that this bracket cannot be formed, and so αη must be nonzero in J. Moreover, it must be the element β such that 2β is δ(⟨α, η, 2⟩).

We will rename B^{qn} as P^{2n} when p = 2, with P denoting the suspension spectrum of RP^{∞}. As in the odd primary case, there is a map λ : P → S^{0} with nontrivial cohomology operations in its mapping cone R. This 2-primary map λ can be viewed more geometrically than its odd-primary analogue, as an amalgamation of composites

P^{n} → SO(n + 1) ⟶^{J} δ^{n}S^{n}.

With A_{0} denoting the exterior subalgebra of A generated by Sq^{1}, H^{∗}R can be filtered as an A_{1}-module with subquotients ∈^{4i} A_{1}//A_{0} for i ≥ 0, and so Ext_{A}(H^{∗}(R ∧ bo)) consists of h_{0}-spikes rising from each position (t − s, s) = (4i, 0) for i ≥ 0. Here h_{0} is the element of Ext_{A}^{1,1}(Z_{2}) or Ext_{A_{1}}^{1,1}(Z_{2}) corresponding to Sq^{1} and to multiplication by 2 in homotopy. Also, we begin a practice of using without comment the relation

Ext_{A}(H^{∗}(X ∧ bo)) ≈ Ext_{A}(H^{∗}X ⊗ A//A_{1}) ≈ Ext_{A_{1}}(H^{∗}X).

Thus the nonzero groups bo_{i}(R) occur only when i ≡ 0 mod 4 and i ≥ 0, and these groups are Z_{(2)}. We also need

bsp_{i}(R) ≈ bo_{i}(R ∧ (S^{0} ∪_{η} e^{2} ∪_{2} e^{3})) ≈ \{ \begin{matrix}Z_{(2)} & if i ≡ 0 mod 4, i ≥ 0 \\ Z_{2} & if i ≡ 2 mod 4, i > 0 \\ 0 & otherwise. \end{matrix}

The Z_{2}’s are obtained from the exact sequence in bo_{∗}(−) associated to the cofibration

R ∧ S^{2} → R ∧ (S^{2} ∪_{2} e^{3}) → R ∧ S^{3} ⟶^{2} .

Analogous to (4.2) is an exact sequence which allows us to compute Ext_{A_{1}}(H^{∗}P) from Ext_{A_{1}}(H^{∗}S^{0}) and Ext_{A_{1}}(H^{∗}R). This is most easily seen in the chart of fig. 6, in which •’s are from Ext_{A_{1}}(H^{∗}S^{0}), and ∘’s are from Ext_{A_{1}}(H^{∗}R).

The groups are read off from this as

bo_{i}(P) ≈ \{ \begin{matrix}Z/2^{4j+3} & i = 8j + 3, j ≥ 0 \\ Z/2^{4j} & i = 8j − 1, j > 0 \\ Z_{2} & i = 8j + 1 or 8j + 2, j ≥ 0 \\ 0 & otherwise. \end{matrix}

Figure 6. Ext_{A_{1}}(H^{∗}P), from Ext_{A_{1}}(H^{∗}S^{0}) and Ext_{A_{1}}(H^{∗}R), t − s < 15

[figure: 3; 7; 11]

Next on the agenda is bsp_{∗}(P), which is computed from bsp_{∗}(S^{0}) (in •) and bsp_{∗}(R) (in ∘) as in fig. 7.

Figure 7. bsp_{∗}(P), from bsp_{∗}(S^{0}) and bsp_{∗}(R), ∗ ≤ 11

[figure: 1; 3; 7; 11]

Note that in positive filtration bsp_{∗}(P) looks like bo_{∗}(∈^{4} P) pushed up by 1 filtration. The explanation for this is the short exact sequence of A_{1}-modules

0 →∈^{5} Z_{2} → A_{1}//A_{0} → N → 0,  (4.5)

where N = ⟨1, Sq^{2}, Sq^{3}⟩, as before. If this is tensored with any A_{0}-free A_{1}-module M, such as P, then the exact Ext_{A_{1}}-sequence reduces to isomorphisms

Ext_{A_{1}}^{s−1,t}(∈^{5} M) → Ext_{A_{1}}^{s,t}(N ⊗ M)

when s > 1. When M = P, an iso is also obtained when s = 1.

The isomorphism of A_{1}-modules H^{∗}(P_{4n+1}) ≈ H^{∗}(∈^{4n} P) allows one to immediately obtain bo_{∗}(P_{4n+1}) and bsp_{∗}(P_{4n+1}) from the above calculations. One way of determining bo_{∗}(P_{4n+3}) and bsp_{∗}(P_{4n+3}) is from the short exact sequence of A_{1}-modules

0 → H^{∗}(∈^{4n+4} Z_{2}) → H^{∗}(P_{4n+3}) → H^{∗}(∈^{4n+3} R) → 0.

This yields as bsp_{∗}(P_{4n+3}) a chart which begins as in fig. 8, while bo_{∗}(P_{4n+7}) ≈ bo_{∗}(∈^{4} P_{4n+3}) is obtained from this chart by deleting all classes in filtration 0.

Figure 8. bsp_{∗}(P_{4n+3}), ∗ ≤ 15 [figure: 4n+; 3; 7; 11; 15]

Next we compute bo_{∗}(P^{2m}) and bsp_{∗}(P^{2m}) using the exact Ext_{A_{1}}-sequence corresponding to the short exact sequence

0 → H^{∗}(P_{2m+1}) → H^{∗}(P) → H^{∗}(P^{2m}) → 0.

For example, this yields the calculation of bo_{∗}(P^{8n}) indicated in fig. 9, where bo_{∗}(P) is in •’s, while bo_{∗}(P_{8n+1}) is in ∘’s.

Next we form J_{∗}(P^{2m}) from bo_{∗}(P^{2m}) and (∈^{3} bsp)_{∗}(P^{2m}), with filtrations of the latter pushed up by 1, similarly to the odd primary case. The boundary morphism bo_{4i−1}(P^{2m}) → (∈^{4} bsp)_{4i−1}(P^{2m}) is pictured by a differential in the chart, and, for the same reason as in the odd-primary case, its value is the same as in bo_{4i−1}(S^{0}) → (∈^{4} bsp)_{4i−1}(S^{0}), namely a nonzero d_{ν(i)+1} wherever possible. If m ≥ k, then the charts for J_{∗}(P^{2m}) and J_{∗}(P^{2k}) are isomorphic through dimension 2k − 1. This is illustrated in fig. 10 for k = 8.

For ∗ ≥ 2m − 1, the form of the chart for J_{∗}(P^{2m}) depends upon the mod 4 value of m. The last of the filtration-1 Z_{2}’s occurs in ∗ = 2m or 2m + 2. The chart near ∗ = 2m is indicated in fig. 11. Note how the bsp-part is like the bo-part shifted

Section 4  Computing periodic homotopy groups  21

Figure 9. bo_{∗}(P^{8n}), from bo_{∗}(P) and bo_{∗}(P_{8n+1})

[figure: 1; 3; 7; 11; 8n − 1; 8n + 7]

Figure 10. J_{∗}(P^{2m}) in ∗ ≤ 15, provided m ≥ 8

[figure: 1; 3; 7; 11; 15]

one unit left and two units down. Differentials d_{r} with r > 1 are omitted from this chart; they occur on the towers in 8n − 1 and 8n + 7.

Figure 11. J_{∗}(P^{2m}) where it starts to ascend

[figure: 8n − 1; 8n + 3; 8n + 3; 8n + 11; 8n + 7; P^{8n}; P^{8n+2}; P^{8n+4}; P^{8n+6}; 4n]

To obtain v_{1}^{−1}J_{∗}(P^{2m}) from J_{∗}(P^{2m}), one removes the filtration-1 Z_{2}’s, and extends into negative filtration the periodic behavior which is present in the towers to the right of dimension 2m. The justification for this is similar to that in the oddprimary case, namely Proposition 4.1. For example, if i is any integer, v_{1}^{−1}J_{∗}(P^{8n+4}) for 8i + 6 ≤ ∗ ≤ 8i + 13 looks like the portion of fig. 11 for P^{8n+4} between 8n + 6 and 8n + 13, with a d_{ν(4i+4)}-differential on the tower in 8i + 7. One might find it easier to compute v_{1}^{−1}J_{∗}(P^{2m}) directly without bothering to first compute the non-periodic J; however, one sometimes needs the non-periodic J-groups.

We combine the results of this section with Theorem 3.1 and Theorem 5.1 to obtain the following extremely important result, Theorem 4.2.

If p is odd, then

v_{1}^{−1}π_{2n+1+i}(S^{2n+1}; p) ≈ \{ \begin{matrix}Z/p^{min(n,ν_{p}(a)+1)} & if i = qa − 2 or qa − 1 \\ 0 & if i ≢ −1 or −2 mod q. \end{matrix}

If n ≡ 1 or 2 mod 4, then

v_{1}^{−1}π_{2n+1+i}(S^{2n+1}; 2) ≈ \{ \begin{matrix}Z_{2} & if i ≡ 0, 5 mod 8 \\ Z_{2} ⊕ Z_{2} & if i ≡ 1, 4 mod 8 \\ Z_{2} ⊕ Z/2^{min(3,n+1)} & if i ≡ 2, 3 mod 8 \\ Z/2^{min(n−1,ν_{2}(j)+4)} & if i = 8j − 2 or 8j − 1. \end{matrix}

If n ≡ 0 or 3 mod 4, then

v_{1}^{−1}π_{2n+1+i}(S^{2n+1}; 2) ≈ \{ \begin{matrix}Z_{2} ⊕ Z_{2} ⊕ Z_{2} & if i ≡ 0, 1 mod 8 \\ Z_{8} ⊕ Z_{2} & if i ≡ 2 mod 8 \\ Z_{8} & if i ≡ 3 mod 8 \\ 0 & if i ≡ 4, 5 mod 8 \\ Z/2^{min(n,ν_{2}(j)+4)} & if i = 8j − 2 \\ Z_{2} ⊕ Z/2^{min(n,ν_{2}(j)+4)} & if i = 8j − 1. \end{matrix}

5. The v_{1}-periodic homotopy groups of spectra

In this section, we sketch three proofs of the following central result.

If X is a spectrum, then v_{1}^{−1}π_{∗}(X) ≈ v_{1}^{−1}J_{∗}(X).

This result was first stated, at least for mod p v_{1}-periodic homotopy groups, in [56].

Theorem 5.1 is a consequence of the following result, which is the special case where X is the mod p Moore spectrum M = S^{0} ∪_{p} e^{1}.

Let v_{1}^{−1}M denote the mapping telescope of

M →∈^{−s} M →∈^{−2s} M → ⋯,

where s = 8 if p = 2 and s = q if p is odd, and the maps are all suspensions of an Adams map A. Then the Hurewicz morphism

π_{∗}(v_{1}^{−1}M) → J_{∗}(v_{1}^{−1}M)

is an isomorphism.

This theorem implies that for any spectrum X, the map

X ∧ v_{1}^{−1}M → X ∧ v_{1}^{−1}M ∧ J

is an equivalence, which, after dualizing the Moore spectra, implies that Theorem 5.1 is true with mod p coefficients. The general case of the theorem then follows from Lemma 3.2.

As an aside, we note that these results are equivalent to the validity of Ravenel’s Telescope Conjecture ([53]) when n = 1. This result, for which the analogue with n = 2 has been shown to be false, can be stated in the following way.

The v_{1}-telescope equals the K_{∗}-localization, i.e., v_{1}^{−1}M = M_{K}.

Proof. Since the Adams maps induce isomorphisms in K_{∗}(−), the inclusion M → v_{1}^{−1}M is a K_{∗}-equivalence. Since v_{1}^{−1}M ≃ v_{1}^{−1}M ∧ J ≃ M ∧ v_{1}^{−1}J, and, similarly to Proposition 4.1, there is a cofibration

v_{1}^{−1}J → KO → KO,

it follows readily that v_{1}^{−1}M is K_{∗}-local.

The remainder of this section is concerned with proofs of Theorem 5.2. Three distinct proofs have been given, although each is too complicated to present in detail here. We sketch each, relegating details to the original papers.

The first proof, when p = 2, was given by Mahowald in [40], although he was offering sketches of this proof as early as 1970. The odd-primary analogue was given in [24]. A sketch of Mahowald’s proof, involving bo-resolutions, follows.

Using self-duality of M, it suffices to show that

dirlim_{i} [∈^{k+8i} M, S^{0}] → dirlim_{i} [∈^{k+8i} M, J]  (5.1)

is an isomorphism. The target groups are easily determined by the methods of the preceding section to be given by two sequences of “lightning flashes” as in fig. 12. It is easily seen, for example from the upper edge of Adams spectral sequence, that these elements all come from actual stable homotopy classes, i.e., the morphism (5.1) is surjective.

Figure 12. dirlim_{i} [∈^{k+8i} M, J]

[figure: k ≡; −2; 0; 2; 3; mod 8]

The injectivity of the morphism (5.1) will be proved by showing that if ∈^{k} M ⟶^{f} S^{0} → J is trivial, then for i sufficiently large, ∈^{k+8i} M ⟶^{A^{i}} ∈^{k} M ⟶^{f} S^{0} is trivial. This will be done using bo-resolutions.

Let \overline{bo} denote the cofiber of the inclusion S^{0} → bo. There is a tower of (co)fibrations

S^{0}←  ∈^{−1} \overline{bo}  ← ∈^{−2} \overline{bo} ∧ \overline{bo} ←
↓  ↓  ↓
bo  ∈^{−1} \overline{bo} ∧ bo  ∈^{−2} \overline{bo} ∧ \overline{bo} ∧ bo

The homotopy exact couple of this tower gives the bo-ASS for S^{0}. It was proved in [25], following [41], that the E_{2}-term of this spectral sequence vanishes above a line of slope 1/5. That is, E_{2}^{s,t}(S^{0}) = 0 if s > \frac{1}{5}(t − s) + 3. One can show that \overline{bo} ∧ bo ≃∈^{4} bsp ∨ W, where W can be written explicitly, and the map

∈^{−1} bo ⟶^{δ} ∈^{−1} \overline{bo} ∧ bo →∈^{3} bsp

may be used as the map whose fiber is J. Here δ induces the lowest d_{1} in the bo-ASS, and the second map collapses W.

Let E_{s} = (∈^{−1} \overline{bo})^{∧s}, the sth stage of the tower. By explicit calculation it can be shown that, if s > 1 or if s = 1 and the map is detected entirely in the W -part, a map X → E_{s} of Adams (HZ/2) filtration greater than 1 can be varied so that its projection to E_{s−1} is unchanged, while the new map lifts to E_{s+1}. Originally it was thought that this was true for maps of Adams filtration greater than 0, but a complication was noted in [27].

Now suppose that ∈^{k} M ⟶^{f} S^{0} → J is trivial. Then f lifts to a map into E_{1} whose projection into ∈^{3} bsp is trivial. The Adams map A can be written as the composite of two maps, each of HZ/2-Adams filtration greater than 1. Thus, by the result of the previous paragraph, f ∘A^{i} lifts to E_{2i+1}. If i is chosen large enough that 2i + 1 > \frac{1}{5}(k + 8i + 1) + 3, then this map ∈^{k+8i} M → S^{0} will have bo-filtration so large that all such maps are trivial by the vanishing line result, completing the proof.

The first proof of Theorem 5.2 for p odd was given by Haynes Miller. A proof analogous to his for p = 2 has not been achieved; [32] was a step in that direction. Miller’s work did not involve the spectrum J. Instead, in [46], he defined a localized ASS for M, and computed its E_{2}-term. Then, in [47], using a clever comparison with the BP -based Novikov spectral sequence, he computed the differentials in the ASS, obtaining the following result.

If p is odd, then π_{∗}(v_{1}^{−1}M) is free over Z_{p}[v_{1}^{±1}] on two classes, namely [S^{0} ↪

M → v_{1}^{−1}M] and [S^{q−1} ⟶^{α_{1}} M → v_{1}^{−1}M].

The methods of Section 4 show easily that these map isomorphically to v_{1}^{−1}J_{∗}(M).

We provide a little more detail about Miller’s calculations. In [46] he obtained as an E_{2}-term for the localized ASS

v_{1}^{−1}E[h_{i,0} : i ≥ 1] ⊗ P [b_{i,0} : i ≥ 1],  (5.2)

where h_{i,0} corresponds to [ξ_{i}] and has bigrading (1, 2(p^{i} −1)), while b_{i,0} corresponds to ∑ \frac{1}{p} \binom{p}{j}[ξ_{i}^{j}|ξ_{i}^{p−j}] and has bigrading (2, 2p(p^{i} −1)). The first step in obtaining this is to use a change-of-rings theorem to write the E_{2}-term as v_{1}^{−1}Cotor_{A(1)_{∗}}(Z_{p}, Z_{p}), where A(1)_{∗} is the quotient A_{∗}/(τ_{0}). This is then shown to be isomorphic to

Z_{p}[v_{1}^{±1}] ⊗ Cotor_{P (1)}(Z_{p}, Z_{p}),

where P (1) = Z_{p}[ξ_{1}, ξ_{2}, …]/(ξ_{1}^{p}, ξ_{2}^{p}, …), and this yields eq. (5.2).

In [47], the differential d_{2}(h_{i,0}) = v_{1}b_{i−1,0} is established in the localized ASS. This leaves Z_{p}[v_{1}^{±1}] ⊗ E[h_{1,0}] as E_{3} = E_{∞}, and this is easily translated into Proposition 5.4. Miller first established this differential in an algebraic spectral sequence converging to the E_{2}-term of the BP -based Novikov spectral sequence, and then showed that this implies the desired differential in the ASS by a comparison theorem.

Somewhat later, Crabb and Knapp ([21]) gave a proof of Theorem 5.1 for finite spectra X which was much less computational than those just discussed. Their proof utilized the solution of the Adams conjecture, and some refinements thereof. They let Ad^{∗}(−) be the generalized cohomology theory corresponding to the fiber of ψ^{k} − 1 : KO → KO. By Proposition 4.1, this is just our v_{1}^{−1}J^{∗}. They prove the following result about stable cohomotopy, which by S-duality is equivalent to Theorem 5.1 for finite spectra.

If X is a finite spectrum, then the Hurewicz morphism

v_{1}^{−1}π_{s}^{∗}(X; Z/p^{e}) ⟶^{h} Ad^{∗}(X; Z/p^{e})

is bijective.

Their main weapon is a result of May and Tornehave which says that if A^{∗}(−) is the connective theory associated to Ad^{∗}(−), and j is the morphism given by a solution of the Adams conjecture, then the composite

A^{0}(X) ⟶^{j} π_{s}^{0}(X) ⟶^{h} A^{0}(X)

is bijective for a connected space X. This is used to show that, for k sufficiently large, there is a stable Adams map ∈^{ks(e)} M(p^{e}) ⟶^{A_{e}} M(p^{e}) which is in the image under j from A^{−ks(e)}(M(p^{e}); Z/p^{e}). This is then used to show that for any element x of π_{s}^{n}(X; Z/p^{e}), for L sufficiently large, A_{e}^{L}x is in the image of the morphism j, and this easily implies injectivity in Theorem 5.5. Care is required throughout in distinguishing stable maps from actual maps.

6. The v_{1}-periodic unstable Novikov spectral sequence for spheres

In this section, we review the basic properties of the unstable Novikov spectral sequence (UNSS) based on the Brown-Peterson spectrum BP, and sketch the determination of the 1- and 2-lines of this spectral sequence when applied to S^{2n+1}. Then we show how the v_{1}-periodic UNSS is defined, and compute it completely for S^{2n+1}.

The spectrum BP associated to the prime p is a commutative ring spectrum satisfying BP_{∗} = π_{∗}(BP) = Z_{(p)}[v_{1}, v_{2}, …] and BP_{∗}(BP) = BP_{∗}[h_{1}, h_{2}, …], with |v_{i}| = |h_{i}| = 2p^{i}−2. The generators v_{i} are those of Hazewinkel, while h_{i} is conjugate to Quillen’s generator t_{i}. We shall often abbreviate BP_{∗}BP as ·.

We will make frequent use of the right unit η_{R} : BP_{∗} → BP_{∗}BP. η_{R}(v_{1}) = v_{1} − ph_{1}, and

η_{R}(v_{2}) = v_{2} − ph_{2} + (p^{p−1} − 1)h_{1}^{p}v_{1} + (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.

In writing h_{1}^{p}v_{1} here, we have begun the practice of writing η_{R}(v)h as hv. Thus h_{1}^{p}v_{1} ≠ v_{1}h_{1}^{p}. Proposition 6.1 is easily derived from formulas relating v_{i} to m_{i}, and for η_{R}(m_{i}). See [14, 2.6], where the following formula for the comultiplication ↓: BP_{∗}BP → BP_{∗}BP ⊗ BP_{∗}BP is also computed. All tensor products in this and subsequent sections are over BP_{∗}.

↓ (h_{1}) = h_{1} ⊗ 1 + 1 ⊗ h_{1}, and

↓ (h_{2}) = h_{2} ⊗ 1 + 1 ⊗ h_{2} + ∑_{i=1}^{p−1} \frac{1}{p} \binom{p}{i}h_{1}^{i} ⊗ h_{1}^{p−i}v_{1} + h_{1}^{p} ⊗ h_{1}.

Let BP_{n} denote the nth space in the δ-spectrum for BP. If X is a space, then a space BP (X) is defined as lim_{n} δ^{n}(BP_{n} ∧ X). Define D^{1}(X) to be the fiber of the unit map X → BP (X), and inductively define D^{s}(X) to be the fiber of D^{s−1}(X) → D^{s−1}(BP (X)). This gives rise to a tower of fibrations

⋯ → D^{2}(X) → D^{1}(X) → X.

The homotopy exact couple of this tower is the UNSS of X; if X is simply connected, it converges to the localization at p of π_{∗}(X).

In general, computing this spectral sequence can be extremely difficult, but if BP_{∗}X is free as a BP_{∗}-module, and cofree as a coalgebra, then it becomes somewhat tractable. Indeed, in such a case

E_{2}^{s,t}(X) ≈ Ext_{U}^{s}(A_{t}, P (BP_{∗}X)),  (6.1)

where A_{t} denotes a free BP_{∗}-module on a generator of degree t, P (−) denotes the primitives in a coalgebra, and U denotes the category of unstable ·-comodules. We sketch a definition of the category U and the proof of eq. (6.1), referring the reader to [6, p. 744] or [8, §7] for more details. If M is a free BP_{∗}-module, then U(M) is defined to be the BP_{∗}-submodule of · ⊗ M spanned by all elements of the form h^{I} ⊗ m satisfying the unstable condition

2(i_{1} + i_{2} + ⋯) < |m|,  (6.2)

where h^{I} = h_{1}^{i_{1}}h_{2}^{i_{2}} · · ·. If M is not BP_{∗}-free, then U(M) is defined as coker(U(F_{1}) → U(F_{0})), where F_{0} and F_{1} are free BP_{∗}-modules with M = coker(F_{1} → F_{0}). We define U^{s}(M) by iterating U(−). The category U consists of BP_{∗}-modules equipped with morphisms M ⟶^{ψ} U(M), U(M) ⟶^{δ} U^{2}(M), and U(M) ⟶^{ϵ} M satisfying certain properties. The unstable condition (6.2) is analogous to the one for unstable right modules over the Steenrod algebra, but its proof relies on deep work of Ravenel and Wilson in [54].

The category U is abelian. We abbreviate Ext_{U}^{s}(A_{t}, N) to Ext_{U}^{s,t}(N). These groups may be calculated as the homology groups of the unstable cobar complex \overline{C}^{∗,∗}(N), defined by \overline{C}^{s,t}(N) = U^{s}(N)_{t}, with boundary \overline{C}^{s} ⟶^{d} \overline{C}^{s+1} defined by

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

where ↓ (γ_{j}) = ∑ γ′_{j} ⊗ γ′′_{j} and ψ(m) = ∑ γ′ ⊗ m′′.

We will use a reduced complex C^{∗,∗}(N), which is chain equivalent to \overline{C}^{∗,∗}(N). This is obtained from Ũ(N) = ker(U(N) ⟶^{ϵ} N) and its iterates Ũ^{s} by C^{s,t}(N) = Ũ^{s}(N)_{t}. Finally, we illustrate how d(v) = η_{R}(v) − v for v ∈ BP_{∗} comes into play. Suppose ↓ (h) = h ⊗ 1 + 1 ⊗ h + ∑ h′ ⊗ h′′ and ψ(I) = 1 ⊗ I, and let v, v′ ∈ BP_{∗}. Then

d([vh]v′I) = [1|vh]v′I − [vh|1]v′I − [v|h]v′I − ∑[vh′|h′′]v′I + [vh|v′]I
= [η_{R}(v) − v|h]v′I − ∑[vh′|h′′]v′I − [vh|η_{R}(v′) − v′]I

We abbreviate C^{∗,∗}(BP_{∗}(X)) to C^{∗,∗}(X).

The first result about the UNSS, both historically and pedagogically, is the following, which appeared in [8, 9.12]. We repeat their proof because it gives a good first example of working with the unstable cobar complex. Recall that q = 2(p − 1).

Let p be an odd prime. If k > 0, then

E_{2}^{1,2n+1+kq}(S^{2n+1}) ≈ Z/p^{min(n,ν_{p}(k)+1)}.

If t ≢ 2n + 1 mod q, or if t < 2n + 1, then E_{2}^{s,t}(S^{2n+1}) = 0.

Proof. Since |v_{i}| and |h_{i}| are divisible by q, all nonzero elements in BP_{∗}(S^{2n+1}) have degree congruent to 2n + 1 mod q, and so the only possible nonzero elements in E_{2}^{s,t}(S^{2n+1}) occur when t ≡ 2n + 1 mod q, and t ≥ 2n + 1. There is an injective chain map

C^{∗,∗}(S^{2n−1}) → C^{∗,∗+2}(S^{2n+1})

defined by A⊗ι_{2n−1} ↦ A⊗ι_{2n+1}, corresponding to the double suspension homomorphism of homotopy groups. Since the boundaries in C^{1}(S^{2n−1}) are sent bijectively to those in C^{1}(S^{2n+1}), the morphism E_{2}^{1,t}(S^{2n−1}) → E_{2}^{1,t+2}(S^{2n+1}) is injective.

We quote a result, originally due to Novikov (but see [51, §5.3] for the proof), about the stable groups: if n is sufficiently large, then

E_{2}^{1,2n+1+kq}(S^{2n+1}) ≈ Z/p^{ν_{p}(k)+1}

with generator d(v_{1}^{k})ι_{2n+1}/p^{ν_{p}(k)+1}. We will prove Theorem 6.3 by showing that if n ≤ ν_{p}(k) + 1, then d(v_{1}^{k})/p^{n} is defined on S^{2n+1}, but not on S^{2n−1}.

We begin by observing

d(v_{1}^{k})/p^{n} = ((η_{R}(v_{1}))^{k} − v_{1}^{k})/p^{n} = ((v_{1} − ph_{1})^{k} − v_{1}^{k})/p^{n}
= ∑_{j=1}^{k}(−1)^{j}\binom{k}{j}p^{j−n}v_{1}^{k−j}h_{1}^{j}.  (6.3)

Note that the coefficients \binom{k}{j}p^{j−n} have nonnegative powers of p, since ν_{p}(\binom{k}{j})+j ≥ ν_{p}(k) + 1 ≥ n for j ≥ 1. Now we work mod terms that are defined on S^{2n−1}. This allows us to ignore terms in the sum (6.3) for j < n. For the other terms, we write p^{j−n}h_{1}^{j} as (v_{1} − η_{R}v_{1})^{j−n}h_{1}^{n}, and note that when this is expanded by the binomial theorem, all terms except v_{1}^{j−n}h_{1}^{n} may be ignored, since (η_{R}v_{1})^{i}h_{1}^{n}ι_{2n−1} = h_{1}^{n}v_{1}^{i}ι_{2n−1} satisfies eq. (6.2) when i > 0. Thus the sum (6.3) reduces to

∑_{j=n}^{k}(−1)^{j}\binom{k}{j}v_{1}^{k−n}h_{1}^{n} = − ∑_{j=0}^{n−1}(−1)^{j}\binom{k}{j}v_{1}^{k−n}h_{1}^{n},

since ∑_{j=0}^{k}(−1)^{j}\binom{k}{j} = 0. If j > 0 in the right-hand sum, then \binom{k}{j} is divisible by p, and then ph_{1}^{n} can be written as v_{1}h_{1}^{n−1} − h_{1}^{n−1}v_{1}, so that the term is defined on S^{2n−1}. Thus, mod S^{2n−1}, (6.3) reduces to −v_{1}^{k−n}h_{1}^{n}. This class is not defined on S^{2n−1}.

If p = 2, a similar argument establishes the following result.

If p = 2, then, for u > 0,

E_{2}^{1,2n+1+u}(S^{2n+1}) ≈ \{ \begin{matrix}0 & u odd \\ Z/2 & ν_{2}(u) = 1 \\ Z/4 & u = 4 \\ Z/2^{min(n,ν_{2}(u)+1)} & u ≡ 0 mod 4, and u > 4. \end{matrix}

If u = 2k in the three nonzero cases, then the generators are, respectively, d(v_{1}^{k})/2, d(v_{1}^{2})/4, and d(v_{1}^{k} + 2^{ν_{2}(k)+1}v_{1}^{k−3}v_{2})/2^{ν_{2}(k)+2}.

When p is odd, the element −(d(v_{1}^{k})/p^{j})ι_{2n+1} ∈ E_{2}^{1,2n+1+kq}(S^{2n+1}) is denoted α_{k/j}. If j = 1, this will frequently be shortened to α_{k}. We note the following from the proof of Theorem 6.3.

If n ≥ j, then α_{k/j}ι_{2n+1} = v_{1}^{k−j}h_{1}^{j}ι_{2n+1} mod terms defined on S^{2j−1}.

Next we cull from [5] information about unstable elements in E_{2}^{2,∗}(S^{2n+1}), which form a subgroup which we shall denote by Ẽ_{2}^{2,∗}(S^{2n+1}). By “unstable,” we mean an element in the kernel of the iterated suspension. The main theorem of [5] is the following.

Let p be an odd prime, and let t = ν_{p}(a). Then

Ẽ_{2}^{2,qa+2n+1}(S^{2n+1}) ≈ \{ \begin{matrix}Z/p^{n} & if n ≤ t + 1 \\ Z/p^{t+1} & if t + 1 ≤ n < a − t \\ Z/p^{a−n} & if a − t − 1 ≤ n < a. \end{matrix}

The homomorphism Ẽ_{2}^{2,qa+2n−1}(S^{2n−1}) ⟶^{∈^{2}} Ẽ_{2}^{2,qa+2n+1}(S^{2n+1}) is

\{ \begin{matrix}injective & if n ≤ t + 1 \\ ·p & if t + 1 < n < a − t \\ surjective & if a − t ≤ n < a. \end{matrix}

Let m = min(n, a − t − 1). Then p^{j} times the generator of Ẽ_{2}^{2,qa+2n+1}(S^{2n+1}) is h_{1} ⊗ v_{1}^{a−m+j−1}h_{1}^{m−j}ι_{2n+1} mod terms defined on S^{2(m−j)−1}.

This is illustrated in the chart below, where we list just leading terms, an element connected to one just below it by a vertical line is p times that element, and elements at the same horizontal level are related by the iterated double suspension homomorphism. We omit the subscript from h_{1} and v_{1}, and the ⊗.

S^{3}  S^{5}  S^{2t+3}  S^{2t+5}
hv^{a−2}h hv^{a−2}h  ⋯  hv^{a−2}h
|  |
hv^{a−3}h^{2} ⋯ hv^{a−3}h^{2} hv^{a−3}h^{2}
|  |
...  ⋮  ⋮  ...
|  |
hv^{a−t−2}h^{t+1} hv^{a−t−2}h^{t+1}
|
hv^{a−t−3}h^{t+2}
...
S^{2(a−t)−1}  S^{2(a−t)+1}  S^{2a−1}
...
hv^{2t}h^{a−2t−1}
|
hv^{2t−1}h^{a−2t} hv^{2t−1}h^{a−2t}
|  |
...  ⋮  ⋮  ...
hv^{t}h^{a−t−1} hv^{t}h^{a−t−1}  ⋯  hv^{t}h^{a−t−1}

The proof of Theorem 6.6 requires results about Hopf invariants which we will address shortly. We begin by describing a plausibility argument for it using only elementary ideas about the unstable cobar complex. We continue to omit the subscript of h_{1} and v_{1}.

(i) The lead term h ⊗ v^{a−n+j−1}h^{n−j} does not pull back to S^{2(n−j)−1} because h^{n−j}ι_{2(n−j)−1} does not satisfy the unstable condition.

(ii) By Proposition 6.1, ph = v − η_{R}(v), a fact which we will begin to use frequently. It implies that

p · h ⊗ v^{a−n+j−1}h^{n−j} = h ⊗ v^{a−n+j}h^{n−j−1} − h ⊗ v^{a−n+j−1}h^{n−j−1}v.

The second term desuspends below S^{2(n−j)−1}, and the first term is the next term up the unstable tower.

(iii) We show that ∈^{2} applied to the element of order p on S^{2n+1} is a boundary when n ≥ t + 1. To do this, we give a more precise description of this element of order p as d(v^{a−n−1}h^{n+1})ι_{2n+1}. This clearly double suspends to the boundary d(v^{a−n−1}h^{n+1}ι_{2n+3}). Note how we had to wait until S^{2n+3} in order to put the ι inside the d(−), since h_{n+1}ι_{2n+1} does not satisfy the unstable condition. It remains to show that the lead term is correct, which is the content of the following proposition.

If t = ν_{p}(a), then

d(v^{a−n−1}h^{n+1}) ≡ h ⊗ v^{a+t−n−1}h^{n−t}

mod terms defined on S^{2(n−t−1)+1}.

Proof. Replacing v by ph + η_{R}(v) implies

v^{a−n−1}h^{n+1} ≡ p^{a−n−1}h^{a} (mod S^{2n+1}).

Since boundaries on S^{2n+1} desuspend to S^{2(n−t)−1}, we obtain

d(v^{a−n−1}h^{n+1}) ≡ p^{a−n−1}d(h^{a})

mod the indeterminacy stated in the proposition. Now d(h^{a}) = ∑ \binom{a}{j}h^{j} ⊗ h^{a−j}, and, since ν_{p}(\binom{a}{j}) ≥ t + 2 − j for j > 1, we find

p^{a−n−1}d(h^{a}) ≡ sp^{a−n−1+t}h ⊗ h^{a−1} ≡ sh ⊗ v^{a+t−n−1}h^{n−t}

mod terms defined on S^{2(n−t−1)+1}. Here a = sp^{t} with s not a multiple of p, and we have freely replaced ph by v − η_{R}(v).

The main detail in the proof of Theorem 6.6 which is lacking in the plausibility argument above is an argument for why these are the only unstable elements on the 2-line. For this, we need the following result, which will also be useful in other contexts.

i. There is an unstable ·-comodule W (n) and an exact sequence

⟶^{P_{2}} E_{2}^{s,t−1}(S^{2n−1}) ⟶^{∈^{2}} E_{2}^{s,t+1}(S^{2n+1}) ⟶^{H_{2}} Ext_{U}^{s−1,t−1}(W (n)) ⟶^{P_{2}} E_{2}^{s+1,t−1}(S^{2n−1}).

(6.4)

ii. W (n) is a free module over BP_{∗}/p on classes x_{2p^{i}n−1} for i > 0 with coaction

ψ(x_{2p^{k}n−1}) = ∑_{i} p^{k−i}h_{k−i}^{np^{i}} ⊗ x_{2p^{i}n−1}.

iii. Ext_{U}^{0}(W (n)) ≈ Z_{p}[v_{1}]x_{2pn−1}. iv. If z ∈ Ext_{U}(W (n)) is represented by ∑ γ_{k} ⊗ x_{2p^{k}n−1}, then

P_{2}(z) = d(∑ γ_{k} ⊗ p^{k−1}h_{k}^{n}) ⊗ ι_{2n−1}.

v. Every element x ∈ E_{2}^{s}(S^{2n+1}) may be represented, mod terms which desuspend to S^{2n−1}, by a cycle of the form ∑ γ_{k} ⊗ p^{k−1}h_{k}^{n} ⊗ ι_{2n+1}, with γ_{k} ∈ C^{∗}(A_{2p^{k}n−1} ⊗ Z_{p}). Then H_{2}(x) = ∑ γ_{k} ⊗ x_{2p^{k}n−1}.

Recall in (v.) that A_{t} is the free BP_{∗}-module on a generator of degree t, and that C^{∗}(−) denotes the reduced unstable cobar complex.

We provide a bare outline of the proof, beginning with the construction from [9]. There is a nonabelian category G of unstable ·-coalgebras, and a notion of Ext_{G} such that if BP_{∗}X is a free BP_{∗}-module of finite type, then E_{2}(X) (of the UNSS) is Ext_{G}(BP_{∗}X). Letting P_{G}(−) denote the primitives in G, one finds that if M is an object of G, then P_{G}(M) is in the category U. By considering an appropriate double complex, one can construct a composite functor spectral sequence converging to Ext_{G}(M) with

E_{2}^{p,q} ≈ Ext_{U}^{p}(R^{q}P_{G}(M)).

Here R^{q}P_{G} denotes the qth right derived functor of P_{G}. If M satisfies R^{q}P_{G}M = 0 for q > 1, then the spectral sequence has only two nonzero columns, and reduces to an exact sequence

→ Ext_{U}^{s}(P_{G}M) → Ext_{G}^{s}(M) → Ext_{U}^{s−1}(R^{1}P_{G}M) → .  (6.5)

This will be the case when M = BP_{∗}(δS^{2n+1}). One verifies that P_{G}BP_{∗}(δS^{2n+1}) ≈ A_{2n}, and that R^{1}P_{G}BP_{∗}(δS^{2n+1}) is the comodule W (n) described in Theorem 6.8, and the exact sequence (6.5) reduces to the sequence (6.4) in this case. The descriptions of the morphisms in (iv.) and (v.) are obtained in [3] using an alternate construction of the exact sequence. Part (iii.) is proved in [5, p. 535] by studying explicit cycles.

Now we complete our observations on the proof of Theorem 6.6. If x is any nonzero unstable element on the 2-line, then there must be k and n so that

∈^{2k} x ≠ 0 ∈ ker(E_{2}^{2,∗}(S^{2n−1}) ⟶^{∈^{2}} E_{2}^{2,∗+2}(S^{2n+1})).

Then by Theorem 6.8, there must be an element v_{1}^{e}x_{2pn−1} ∈ Ext_{U}^{0}(W (n)) such that

P_{2}(v_{1}^{e}x_{2pn−1}) = d(v_{1}^{e}h_{1}^{n})ι_{2n−1} =∈^{2k} x.

But this is exactly the description of the unstable elements on the 2-line which was given in the third part of our plausibility argument for Theorem 6.6.

When p = 2, the discussion above about the unstable elements on the 2-line goes through almost without change, as described in [10, pp. 482-484]. The result is that if n ≤ a − ν_{2}(a) − 2, then

Ẽ_{2}^{2,2n+1+2a}(S^{2n+1}) ≈ \{ \begin{matrix}Z/2 & if a is odd \\ Z/2^{min(ν_{2}(a)+2,n)} & if a is even. \end{matrix}

The orders when a is even are 1 larger than in the odd primary case because v_{2} can be used to obtain 1 additional desuspension.

Now we construct the v_{1}-periodic UNSS, following [4] for the most part. In [7] a UNSS converging to map_{∗}(Y, X) was constructed. If Y = M^{n}(p^{e}), the E_{2}-term is the homology of C^{∗}(P (BP_{∗}(X))) ⊗ Z/p^{e}. The Adams map A induces

UNSS(map_{∗}(M^{n}(p^{e}), X)) ⟶^{A^{∗}} UNSS(map_{∗}(M^{n+s(e)}(p^{e}), X)),

where s(e) is as in eq. (2.1). By [35], on E_{2} this is just multiplication by a power of v_{1} after iterating sufficiently. As in our definition of v_{1}^{−1}π_{∗}(−), we define the v_{1}-periodic UNSS of X by

v_{1}^{−1}E_{r}^{∗,∗}(X) = dirlim_{e,k} E_{r}^{∗,∗+1}(map_{∗}(M^{ks(e)}(p^{e}), X)).

The direct system over e utilizes the maps ρ : M^{n}(p^{e+1}) → M^{n}(p^{e}) used in Section 1, and the shift of one dimension is done for the same reason as in our definition of v_{1}^{−1}π_{∗}(−). Similarly to Proposition 2.4, we have

On the category of spaces with H-space exponents, there is a natural transformation from the UNSS to the v_{1}-periodic UNSS.

One of the main theorems of [4] is the following determination of the v_{1}-periodic UNSS of S^{2n+1}.

Let p be an odd prime. The v_{1}-periodic UNSS of S^{2n+1} collapses from E_{2} and satisfies

v_{1}^{−1}E_{2}^{s,2n+1+u}(S^{2n+1}) ≈ \{ \begin{matrix}Z/p^{min(n,ν_{p}(a)+1)} & if s = 1 or 2, and u = qa \\ 0 & otherwise. \end{matrix}

The morphism E_{2}^{s,t}(S^{2n+1}) → v_{1}^{−1}E_{2}^{s,t}(S^{2n+1}) is an isomorphism if s = 1 and t > 2n + 1, while for s = 2 it sends the unstable towers injectively, and bijectively unless n ≥ a − ν_{p}(a) − 1, where t = 2n + 1 + qa.

Proof. It is readily verified that the elements on the 1- and 2-lines described in Theorems 6.3 and 6.6 form v_{1}-periodic families. The main content of this theorem is that there is nothing else which is v_{1}-periodic.

In order to prove this, we use a v_{1}-periodic version of the double suspension sequence (6.4). It is proved in [4] that the morphisms of (6.4) behave nicely with respect to v_{1}-action, yielding an exact sequence

⟶^{P_{2}} v_{1}^{−1}E_{2}^{s,t−1}(S^{2n−1}) ⟶^{∈^{2}} v_{1}^{−1}E_{2}^{s,t+1}(S^{2n+1}) ⟶^{H_{2}} v_{1}^{−1}Ext_{U}^{s−1,t−1}(W (n)) ⟶^{P_{2}} .

(6.6)

By Theorem 6.8(ii.), there is a spectral sequence converging to v_{1}^{−1}Ext_{U}(W (n)) with

E_{1}^{s,t} ≈ ⨁_{i≥1} v_{1}^{−1}Ext_{U}^{s,t}(A_{2p^{i}n−1} ⊗ Z_{p}).  (6.7)

There is a short exact sequence given by the universal coefficient theorem

0 → v_{1}^{−1}E_{2}^{s,t}(S^{n}) ⊗ Z_{p} → v_{1}^{−1}Ext_{U}^{s,t}(A_{n} ⊗ Z_{p}) → T or(v_{1}^{−1}E_{2}^{s−1,t}(S^{n}), Z_{p}) → 0. (6.8)

We will use eqs. (6.6), (6.7), and (6.8) to show inductively that there are no unexpected elements in v_{1}^{−1}E_{2}(S^{2n+1}). But first we show how the known elements fit into this framework. By (6.8), each summand in v_{1}^{−1}E_{2}^{1}(S^{n}) gives two Z_{p}’s, called stable, in v_{1}^{−1}Ext_{U}(A_{n} ⊗ Z_{p}), and similarly each summand in v_{1}^{−1}E_{2}^{2}(S^{n}) gives two summands in v_{1}^{−1}Ext_{U}(A_{n} ⊗ Z_{p}), called unstable. We claim that

v_{1}^{−1}Ext_{U}^{s,t}(W (n)) ≈ \{ \begin{matrix}Z_{p} & if s = 0 or 1, and t ≡ 2n − 1 mod q \\ 0 & otherwise. \end{matrix}  (6.9)

In [4, p. 57], the relationship between (6.9) and (6.7) is discussed: in the spectral sequence (6.7), stable classes from the (i+1)-summand hit unstable classes from the i-summand, yielding in v_{1}^{−1}E_{∞} only the stable classes from the 1-summand. These are the elements described in eq. (6.9). On the other hand, in the exact sequence (6.6), let t = 2n + kq with e = ν_{p}(k). If e < n − 1, then ∈^{2} is Z/p^{e} ⟶^{·p} Z/p^{e} when s = 2, yielding the elements in v_{1}^{−1}Ext_{U}^{s,t−1}(W (n)) for s = 0 and 1, while if e ≥ n − 1, then for s = 1 and 2, ∈^{2} is Z/p^{n−1} ↪ Z/p^{n}, also yielding elements in v_{1}^{−1}Ext_{U}^{s,t−1}(W (n)) for s = 0 and 1.

It seems useful for this proof to have one more bit of input, namely the result for the v_{1}-periodic stable Novikov spectral sequence, which can be defined as a direct limit over e of stable Novikov spectral sequences of M(p^{e}).

[13, §2] There is a v_{1}-periodic stable Novikov spectral sequence for S^{0}, satisfying

v_{1}^{−1}E_{r}^{s,t}(S^{0}) = dirlim_{n} v_{1}^{−1}E_{r}^{s,t+2n+1}(S^{2n+1}),

and

v_{1}^{−1}E_{2}^{s,t}(S^{0}) ≈ \{ \begin{matrix}Z/p^{ν_{p}(t)+1} & if s = 1 and t ≡ 0 mod q \\ 0 & otherwise. \end{matrix}

We prove by induction on s that, for all n, v_{1}^{−1}E_{2}^{s}(S^{2n+1}), v_{1}^{−1}E_{2}^{s+1}(S^{2n+1}), and v_{1}^{−1}Ext_{U}^{s}(W (n)) contain only the elements described in Theorem 6.10 and eq. (6.9). This is easily seen to be true when s = 0, where we know all groups completely. Assume it is true for all s < σ. Then v_{1}^{−1}E_{2}^{σ}(S^{2n+1}) contains no unexpected elements because σ = (σ − 1) + 1. If v_{1}^{−1}E_{2}^{σ+1}(S^{2n+1}) contains an unexpected element, then some v_{1}^{−1}E_{2}^{σ+1}(S^{2L+1}) must contain one in ker(∈^{2}) by Theorem 6.11. Such an element must be P_{2}(x), where x is an unexpected element of v_{1}^{−1}Ext_{U}^{σ−1}(W (n)), but no such element exists by our induction hypothesis. Finally, v_{1}^{−1}Ext_{U}^{σ}(W (n)) contains no unexpected elements by (6.7) and (6.8) since, as just established, v_{1}^{−1}E_{2}^{σ}(S^{2m+1}) and v_{1}^{−1}E_{2}^{σ+1}(S^{2m+1}) contain no unexpected elements for any m.

The UNSS and v_{1}-periodic UNSS are considerably more complicated at the prime 2 than at the odd primes, but the v_{1}-periodic UNSS of S^{2n+1} is still completely understood. We shall not discuss it in detail because most of our applications in this paper will be at the odd primes. The reader desiring more detail is referred to [4], which gives a chart with UNSS names of the elements.

We reproduce in figs. 13 and 14 the charts from [10, p. 488] of the v_{1}-periodic UNSS of S^{2n+1} at the prime 2. Here “3” means Z/2^{3}, and “ν” means Z/2^{ν}, where

ν = min(ν_{2}(8k + 8) + 1, n).

Differentials emanating from a summand of order greater than 2 are nonzero only on a generator of the summand. Note how Z/8 in periodic homotopy is obtained as an extension by the Z_{2} in filtration 3 of the elements in the 1-line group which are divisible by 2 in a Z/8.

Figure 14 applies to S^{2n+1} when n ≡ 1 or 2 mod 4, with n > 2. The reader is referred to [10, p.487] for the minor changes required when n ≤ 2. In fig. 14, the dotted differential is present if and only if ν = n. In both charts, the left η-action on the Z/2^{ν} on line 1, which is usually indicated by positively sloping solid lines, is indicated by the dotted line if n < ν(8k + 8) + 1.

The argument establishing these charts appears in [4]. Note that v_{1}^{−1}E_{4}^{s,∗}(S^{2n+1}) = 0 if s > 4, and hence no higher differentials are possible.

7. v_{1}-periodic homotopy groups of SU(n)

In this section we show how the v_{1}-periodic UNSS determines the v_{1}-periodic homotopy groups of spherically resolved spaces. This relationship is particular nice when localized at an odd prime, and it is this case on which we focus most of our attention. At the end of the section, we discuss the changes required when p = 2. After proving the general result for spherically resolved spaces, we specialize to SU(n), where the result is, in some sense, explicit.

It is not clear that the v_{1}-periodic UNSS of a space X must converge to the p-primary v_{1}-periodic homotopy groups of X. There might be periodic homotopy classes which are not detected in the periodic UNSS because multiplication by v_{1} repeatedly increases BP filtration. It is also possible that a v_{1}-periodic family in E_{2} might support arbitrarily long differentials in the unlocalized spectral sequence, in which case it would exist in all v_{1}^{−1}E_{r}, but would not represent an element of periodic homotopy. We now show that neither of these anomalies can occur

Figure 13. v_{1}^{−1}E_{2}^{s,2n+1+j}(S^{2n+1}), n ≡ 0, 3 mod 4, p = 2

[figure: 3; 3; ν; ν; j − s = 8k+; 1; 3; 5; 7; s = 1; s = 4]

Figure 14. v_{1}^{−1}E_{2}^{s,2n+1+j}(S^{2n+1}), n ≡ 1, 2 mod 4, n > 2, p = 2

[figure: 3; 3; ν; ν; j − s = 8k+; 1; 3; 5; 7; s = 1; s = 4]

for a spherically resolved space, essentially because they cannot happen for an odd sphere, where the v_{1}-periodic homotopy groups are known to agree with the v_{1}^{−1}E_{2}- term.

A space X is spherically resolved if there are spaces X_{0}, …, X_{L}, with X_{0} = ∗, X_{L} = X, and fibrations

X_{i−1} → X_{i} → S^{n_{i}}  (7.1)

with n_{i} odd, and algebra isomorphisms

H^{∗}(X_{i}) ≈ H^{∗}(X_{i−1}) ⊗ H^{∗}(S^{n_{i}}).

The following result was stated as Theorem 1.1. It was proved in [4].

If p is odd, and X is spherically resolved, then v_{1}^{−1}E_{2}^{s,t}(X) = 0 unless s = 1 or 2, and t is odd. The v_{1}-periodic UNSS collapses to the isomorphisms

v_{1}^{−1}π_{i}(X) ≈ \{ \begin{matrix}v_{1}^{−1}E_{2}^{1,i+1}(X) & if i is even \\ v_{1}^{−1}E_{2}^{2,i+2}(X) & if i is odd. \end{matrix}

Proof. Each algebra BP^{∗}(X_{i}) is free, and so eq. (6.1) applies to give E_{2}(X_{i}) ≈ Ext_{U}(M(x_{n_{1}}, …, x_{n_{i}})), where M(−) denotes a free BP_{∗}-module on the indicated generators. There are short exact sequences in U

0 → M(x_{n_{1}}, …, x_{n_{i−1}}) → M(x_{n_{1}}, …, x_{n_{i}}) → M(x_{n_{i}}) → 0,

and hence long exact sequences

→ E_{2}^{s,t}(X_{i−1}) → E_{2}^{s,t}(X_{i}) → E_{2}^{s,t}(S^{n_{i}}) → E_{2}^{s+1,t}(X_{i−1}) → .  (7.2)

These exact sequences are compatible with the direct system of v_{1}-maps whose limit is the v_{1}-periodic groups. Thus there is a v_{1}-periodic version of (7.2), and hence by Theorem 6.10 and induction on i, v_{1}^{−1}E_{2}^{s,t}(X) = 0 unless s = 1 or 2, and t is odd. Thus v_{1}^{−1}E_{2}(X) ≈ v_{1}^{−1}E_{∞}(X).

If u > max{n_{i}}, s = 1 or 2, and s + u is odd, there are natural edge morphisms π_{u}(X) → E_{2}^{s,s+u}(X), and these are compatible with the direct system of v_{1}-maps, giving morphisms v_{1}^{−1}π_{u}(X) → v_{1}^{−1}E_{2}^{s,s+u}(X). These yield a commutative diagram of exact sequences

0 → v_{1}^{−1}π_{2k}(X_{i−1}) →  v_{1}^{−1}π_{2k}(X_{i})  → v_{1}^{−1}π_{2k}(S^{n_{i}}) →
↓^{ϕ_{i−1}}  ↓^{ϕ_{i}}  ↓^{ψ}
0 →v_{1}^{−1}E_{2}^{1,2k+1}(X_{i−1}) → v_{1}^{−1}E_{2}^{1,2k+1}(X_{i}) → v_{1}^{−1}E_{2}^{1,2k+1}(S^{n_{i}}) →
→ v_{1}^{−1}π_{2k−1}(X_{i−1}) →  v_{1}^{−1}π_{2k−1}(X_{i})  → v_{1}^{−1}π_{2k−1}(S^{n_{i}}) → 0
↓^{ϕ′_{i−1}}  ↓^{ϕ′_{i}}  ↓^{ψ′}
→ v_{1}^{−1}E_{2}^{2,2k+1}(X_{i−1}) → v_{1}^{−1}E_{2}^{2,2k+1}(X_{i}) → v_{1}^{−1}E_{2}^{2,2k+1}(S^{n_{i}})→ 0.

(7.3)

The zeros at the ends of the v_{1}^{−1}E_{2}-sequence follow from the previous paragraph. The zero morphism coming into v_{1}^{−1}π_{2k}(X_{i−1}) follows from v_{1}^{−1}E_{2}^{0}(S^{n_{i}}) = 0 and ϕ_{i−1} being an isomorphism, which is inductively known. The zero morphism coming out from v_{1}^{−1}π_{2k−1}(S^{n_{i}}) follows since anything in the image must have filtration ≥ 2, but, by induction, v_{1}^{−1}π_{2k−2}(X_{i−1}) is 0 above filtration 1.

Comparison of Theorems 4.2 and 6.10 shows that the groups related by ψ and by ψ′ are isomorphic, and it is easy to see that ψ and ψ′ induce the isomorphisms. (See [23] for reasons.) Since X_{1} is a sphere, this comparison also shows that ϕ_{1} and ϕ′_{1} are isomorphisms, which starts the induction. Thus all ϕ_{i} and ϕ′_{i} are isomorphisms by induction and the 5-lemma.

Theorem 7.2 is a nice result, but it still leaves the formidable task of calculating v_{1}^{−1}E_{2}(X). In [6], Bendersky proved the following seminal result, whose proof we will discuss throughout much of the remainder of this section.

If k ≥ n, then in the UNSS E_{2}^{1,2k+1}(SU(n)) ≈ Z/p^{e_{p}(k,n)}, where e_{p}(k, n) is as defined in Definition 1.2, and p is any prime.

This allows us to easily deduce Theorem 1.3, now demoted to corollary status, which we restate for the convenience of the reader.

If p is odd, then v_{1}^{−1}π_{2k}(SU(n)) ≈ Z/p^{e_{p}(k,n)}, and v_{1}^{−1}π_{2k−1}(SU(n)) is an abelian group of the same order.

Proof of corollary. The first part of the corollary is a straightforward application of Theorems 7.2 and 7.3, once we know that the groups in Theorem 7.3 are v_{1}- periodic. This can be seen by observing how they arise, from exact sequences built from spheres, where the classes are all v_{1}-periodic. In [23], a slightly different proof of this part of the corollary was given, before the v_{1}-periodic UNSS had been hatched.

The exact sequence like the top row of (7.3) for the fibration

SU(n − 1) → SU(n) → S^{2n−1}  (7.4)

implies, by induction on n, that |v_{1}^{−1}π_{2k−1}(SU(n))| = |v_{1}^{−1}π_{2k}(SU(n))|. Indeed, the orders are equal for S^{2n−1} by Theorem 6.10, and so if they are equal for SU(n − 1), then they will be equal for SU(n), since the alternating sum of the exponents of p in an exact sequence is 0. The fact that SU(2) = S^{3} starts the induction.

In [23], an example of a noncyclic group v_{1}^{−1}π_{2k−1}(SU(n); 3) was given, and in [13] it was shown that v_{1}^{−1}π_{2k−1}(SU(n); 2) will often have many summands (in addition to a regular pattern of Z_{2}’s).

In order to prove Theorem 7.3, it is convenient to work with the UNSS based on MU, rather than BP. This allows us to work with the ordinary exponential series, rather than its p-typical analogue. The facts about MU that we need are summarized in the following result.

i. MU_{∗}(MU) is a polynomial algebra over MU_{∗} with generators B_{i} of grading 2i for i > 0. There are elements β_{i} ∈ MU_{2i}(CP^{∞}) which form a basis for MU_{∗}(CP^{∞}) as an MU_{∗}-module.

ii. Let B denote the formal sum 1 + ∑_{i>0} B_{i}. The coaction

MU_{∗}(CP^{∞}) ⟶^{ψ} MU_{∗}MU ⊗_{MU_{∗}} MU_{∗}(CP^{∞})

satisfies

ψ(β_{n}) = ∑_{j}(B^{j})_{n−j} ⊗ β_{j}.

Here (B^{j})_{n−j} denotes the component in grading 2(n − j) of the jth power of the formal sum B.

iii. There is a ring homomorphism \overline{e} : MU_{∗}(MU) ⊗ Q → Q satisfying

• \overline{e}(B_{i}) = 1/(i + 1).

• \overline{e}(η_{R}(a)) = 0 if a ∈ MU_{i} with i > 0.

• \overline{e} induces an injection E_{2}^{1,2n+1+2k}(S^{2n+1}) → Q/Z.

iv. The BP -based UNSS is the p-localization of the MU-based UNSS.

Proof. Part (i.) is standard (e.g., [2]), while part (ii.) is [2, 11.4]. Part (iii.) is from [6]. There are elements m_{i} ∈ MU_{2i} ⊗ Q such that MU_{∗}MU ⊗ Q is a polynomial algebra over Q on all m_{i} and η_{R}(m_{i}). One defines \overline{e} to be the ring homomorphism which sends m_{i} to 1/(i + 1) and η_{R}(m_{i}) to 0. The second property in (iii.) is clear, and the first follows by conjugating [2, 9.4] to obtain

m_{n} = ∑ η_{R}(m_{i})(B^{i+1})_{n−i},

and then applying \overline{e} to obtain \overline{e}(m_{n}) = \overline{e}(B_{n}).

One way to see the third property is to localize at p and pass to BP. Then m_{p^{i}−1}^{MU} passes to m_{i}^{BP}, and so our \overline{e} passes to that of [6, 4.3]. It is shown on [6, p.751] that \overline{e}_{BP} sends the p-local 1-line injectively, and this works for all p.

Now we can state a general theorem which incorporates most of the work in proving Theorem 7.3. This theorem was stated without proof in [11, 3.10], where it was applied to X = Sp(n), p = 2. We will outline the proof, which is a direct generalization of [6], later in this section.

Suppose X is spherically resolved as in Definition 7.1 with n_{1} < n_{2} < ⋯, and L possibly infinite. Then MU_{∗}(X_{k}) is an exterior algebra over MU_{∗} on classes y_{1}, …, y_{k}, with |y_{i}| = n_{i}. Let

\overline{E}^{1,t}(X_{k}) = ker(E_{2}^{1,t}(X_{k}) → E_{2}^{1,t}(X_{L})).

Let γ_{k,j} ∈ MU_{∗}(MU) be defined in terms of the coaction in MU_{∗}(X_{i}) by

ψ(y_{k}) = ∑_{j=1}^{k} γ_{k,j} ⊗ y_{j},  (7.5)

and let b_{k,j} = \overline{e}(γ_{k,j}) ∈ Q. Then the matrix B = (b_{k,j}) is lower triangular with 1’s on the diagonal. Let C = (c_{k,j}) be the inverse of B, and let

ω_{k}(m) = l.c.m.{den(c_{k,j}) : m ≤ j ≤ k}.

Then coker(\overline{E}^{1,n_{k}}(X_{m−1}) → \overline{E}^{1,n_{k}}(X_{k−1})) is cyclic of order ω_{k}(m).

Now we specialize to SU(n), where we need the following result.

In the UNSS (i) E_{2}^{s,t}(SU) = 0 if s > 0. (ii) E_{2}^{1,2k+1}(SU(n)) = 0 if n > k. (iii) E_{2}^{1,2k+1}(SU(k)) ≈ Z/k!. (iv) If i < j ≤ k, the inclusion SU(i) → SU(j) induces an injection in E_{2}^{1,2k+1}.

(v) SU is spherically resolved as in Theorem 7.6 with X_{k} = SU(k + 1), n_{k} = 2k + 1, and if the MU-coaction on SU is as in eq. (7.5), then

∑_{k≥j} \overline{e}(γ_{k,j})x^{k} = (− log(1 − x))^{j}.  (7.6)

Proof. Part i is a nontrivial consequence of the fact that SU is an H-space with torsion-free homology and homotopy. See [6, 3.1]. The coalgebra MU_{∗}(SU(n)) is cofree with primitives isomorphic to MU_{∗}(∈ CP^{n−1}). Hence by (6.1)

E_{2}(SU(n)) ≈ Ext_{U}(MU_{∗}(∈ CP^{n−1})),

and so the fibrations (7.4) induce exact sequences in E_{2}. Part ii follows from part i and the exact sequence, and part iv is also immediate from the exact sequence.

Let y_{2k+1} ∈ MU_{2k+1}(SU(n)) be the generator corresponding to

∈ β_{k} ∈ MU_{2k+1}(∈ CP^{n−1})

for k < n. Part iii is proved on [6, p.748] by showing that the generator of E_{2}^{0,2k+1}(SU) is of the form k!y_{2k+1}+ lower terms, so that

E_{2}^{0,2k+1}(SU(k + 1)) → E_{2}^{0,2k+1}(S^{2k+1})

sends the generator of one Z to k! times the generator of the other Z. This implies part iii.

By Proposition 7.5(ii.) and the relationship of MU_{∗}(SU(n)) with MU_{∗}(∈ CP^{n−1}) noted above, we find

ψ(y_{2k+1}) = ∑(B^{j})_{k−j} ⊗ y_{2j+1}.

By Proposition 7.5(iii.), we have ∑ \overline{e}(B_{i})x^{i+1} = − log(1−x). These facts yield part v.

Now we can prove Theorem 7.3. If f(x) is a power series with constant term 0, let [f(x)] denote the infinite matrix whose entries a_{k,j} satisfy f(x)^{j} = ∑_{k} a_{k,j}x^{k}. One easily verifies that [g(x)][f(x)] = [f(g(x))]. Hence the inverse of the matrix [− log(1 − x)] is [1 − e^{−x}]. This observation, with Theorem 7.6 and Proposition 7.7, implies that there is a short exact sequence

0 → E_{2}^{1,2k+1}(SU(n)) → E_{2}^{1,2k+1}(SU(k)) → Z/ω_{k}(n) → 0

with middle group Z/k! and

ω_{k}(n) = l.c.m.{den(coef(x^{k}, (1 − e^{−x})^{j})) : n ≤ j ≤ k}.

Thus E_{2}^{1,2k+1}(SU(n)) is cyclic of order

k! / l.c.m.{den(coef(x^{k}, (e^{x} − 1)^{j})) : n ≤ j ≤ k}
= gcd\{coef(\frac{x^{k}}{k!} , (e^{x} − 1)^{j}) : n ≤ j ≤ k\}.

Looking at exponents of p yields Theorem 7.3 by Proposition 7.5(iv.).

It remains to prove Theorem 7.6, the notation of which we employ without comment. Define b_{k,j}(m) for m ≤ k recursively by b_{k,j}(k) = b_{k,j} and

b_{k,j}(m) = b_{k,j}(m + 1) − b_{k,m}(m + 1)b_{m,j}.  (7.7)

We begin by noting that if row reduction is performed on (B|I) so as to get at each step one more diagonal of 0’s below the main diagonal of B, we find that the entries c_{k,j} of B^{−1} satisfy

c_{k,j} = \{ \begin{matrix}0 & if j > k \\ 1 & if j = k \\ −b_{k,j}(j + 1) & if j < k. \end{matrix}

Then

ω_{k}(m) = max(ord(b_{k,j}(j + 1)) : m ≤ j ≤ k).  (7.8)

Here and throughout this proof, ord(−) refers to order in Q/Z.

Fix k, and let τ(m) = |coker(\overline{E}^{1,n_{k}}(X_{m−1}) → \overline{E}^{1,n_{k}}(X_{m}))|. We drop the subscript k from eqs. (7.7) and (7.8). The fibration X_{k−1} → X_{k} → S^{n_{k}} implies that the generator g(k − 1) of \overline{E}^{1,n_{k}}(X_{k−1}) is d(y_{k}) = ∑_{j<k} γ_{k,j} ⊗ y_{j}. Let a_{j}(k − 1) = \overline{e}(γ_{k,j}) = b_{k,j}. By Proposition 7.5(iii), τ(k − 1) = ord(a_{k−1}(k − 1)). Then there is α ∈ MU_{∗} so that τ(k−1)g(k−1)+d(αy_{k−1}) pulls back to a generator of \overline{E}^{1,n_{k}}(X_{k−2}). Write this generator as ∑ γ_{j}(k − 2)y_{j}. Then

a_{j}(k − 2) := \overline{e}(γ_{j}(k − 2)) = τ(k − 1)a_{j}(k − 1) + \overline{e}(α)b_{k−1,j}
= τ(k − 1)(a_{j}(k − 1) − a_{k−1}(k − 1)b_{k−1,j}).

Here we have used that \overline{e}(α) = −\overline{e}(d(α)), which follows from Proposition 7.5(iii).

This procedure can be continued until we obtain a generator ∑ γ_{j}(m − 1)y_{j} of \overline{E}^{1,n_{k}}(X_{m−1}) with a_{j}(m − 1) := \overline{e}(γ_{j}(m − 1)) satisfying

a_{j}(m − 1) = τ(m)(a_{j}(m) − a_{m}(m)b_{m,j})

and τ(m) = ord(a_{m}(m)). Now we prove by downward induction on m that a_{j}(m − 1) = ω(m)b_{j}(m), the case m = k being trivial.

a_{j}(m − 1) = τ(m) (a_{j}(m) − a_{m}(m)b_{m,j})
= τ(m) ω(m + 1) (b_{j}(m + 1) − b_{m}(m + 1)b_{m,j})
= ord(a_{m}(m)) ω(m + 1) b_{j}(m)
= ord(ω(m + 1)b_{m}(m + 1)) ω(m + 1) b_{j}(m)
= max(ω(m + 1), ord(b_{m}(m + 1))) b_{j}(m)
= ω(m)b_{j}(m)

The portion of this string after the first line and not including the last factor shows that τ(m)ω(m + 1) = ω(m), and hence ω(m) = τ(m) ⋯ τ(k − 1), which is a restatement of the desired conclusion of Theorem 7.6.

Since each v_{1}^{−1}π_{i}(SU(n)) occurs as a direct summand of some actual homotopy group of SU(n), the following result about the p-exponent of SU(n) is immediate from Theorem 1.3.

Let exp_{p}(X) denote the largest e such that, for some i, π_{i}(X) has an element of order p^{e}. Then, if p is odd,

exp_{p}(SU(n)) ≥ e_{p}(n),

where e_{p}(n) = max{e_{p}(k, n) : k ≥ n}.

We conjecture that this bound is sharp, the main evidence being that the analogous statement is true for odd spheres, by [20]. The numbers e_{p}(k, n) are explicit in Definition 1.2, and this formula can be used with a computer for specific calculations. However, this formula is not very tractable. In Section 8, we sketch how simple formulas for v_{1}^{−1}π_{2k}(SU(n)) can be obtained for n ≤ p^{2} − p without using Theorem 1.3, but rather by studying the exact sequences of UNSS E_{2}-terms.

Theorem 7.6 is valid when p = 2, but Theorems 7.2 and 1.3 must be modified, due to the more complicated form of the v_{1}-periodic UNSS for S^{2n+1} when p = 2, as discussed at the end of Section 6. We state below the 2-primary version of Theorem 7.2 which was proved in [4], but we refer the reader to [10] for the 2-primary analogue of Theorem 1.3.

[6] Suppose X is spherically resolved and p = 2. Then

• v_{1}^{−1}E_{2}(X) is generated as an η-module by elements with s = 1 or 2. Here η has (s, t) = (1, 2).

• η acts freely on elements with s > 2.

• v_{1}^{−1}E_{4}(X) = v_{1}^{−1}E_{∞}(X), and v_{1}^{−1}E_{4}^{s}(X) = 0 if s > 4.

• If the groups v_{1}^{−1}E_{2}^{1,t}(X) are cyclic, then the v_{1}-periodic UNSS converges to v_{1}^{−1}π_{∗}(X).

8. v_{1}-periodic homotopy groups of some Lie groups

In this section we focus on two examples. One uses UNSS methods to determine v_{1}^{−1}π_{∗}(G_{2}; 5), while the other uses ASS methods to determine v_{1}^{−1}π_{∗}(F_{4}/G_{2}; 2). Here G_{2} and F_{4} are the two simplest exceptional Lie groups. The first example is just one of many discussed in [14]. We also discuss how these UNSS methods can be used to give tractable formulas for v_{1}^{−1}π_{∗}(SU(n); p) when p is odd and n ≤ p^{2} − p. We close by summarizing the status of the program, initially proposed by Mimura, of computing the v_{1}-periodic homotopy groups of all compact simple Lie groups.

Our first theorem concerns the v_{1}-periodic homotopy groups of certain sphere bundles over spheres, which appear frequently as direct factors of compact simple Lie groups localized at p, according to the decompositions given in [49].

Let p be an odd prime, and let B_{1}(p) denote an S^{3}-bundle over S^{2p+1} with attaching map α_{1}. Then the only nonzero v_{1}-periodic homotopy groups of B_{1}(p) are

v_{1}^{−1}π_{2p+qm−1}(B_{1}(p)) ≈ v_{1}^{−1}π_{2p+qm}(B_{1}(p)) ≈ Z/p^{min(p+1,1+ν_{p}(m−p^{p−1}))}.

This is the case k = 1 of [14, 2.1]. The spaces B_{1}(p) were called B(3, 2p+1) in [14]. The following result follows immediately from the 5-local equivalence G_{2} ≃ B_{1}(5).

v_{1}^{−1}π_{i}(G_{2}; 5) ≈ \{ \begin{matrix}Z/5^{min(6,1+ν_{5}(i−5009))} & if i ≡ 1 mod 8 \\ Z/5^{min(6,1+ν_{5}(i−5010))} & if i ≡ 2 mod 8 \\ 0 & otherwise \end{matrix}

The following result is the central part of the proof of Theorem 8.1. Indeed, this theorem, 8.3, along with Theorem 6.3, gives the order of each group E_{2}^{1,t}(B_{1}(p)), and Theorem 8.5 shows the group is cyclic. Then Theorem 7.2 shows that this gives v_{1}^{−1}π_{∗}(B_{1}(p)) when ∗ is even, and the proof of Corollary 7.4 shows that

|v_{1}^{−1}π_{2k−1}(B_{1}(p))| = |v_{1}^{−1}π_{2k}(B_{1}(p))|.

Finally, v_{1}^{−1}π_{∗}(B_{1}(p)) is shown to be cyclic when ∗ is odd in Theorem 8.6.

In the exact sequence

0 → E_{2}^{1,qm+2p+1}(S^{3}) ⟶^{i_{∗}} E_{2}^{1,qm+2p+1}(B_{1}(p))
⟶^{j_{∗}} E_{2}^{1,qm+2p+1}(S^{2p+1}) ⟶^{∂} E_{2}^{2,qm+2p+1}(S^{3}),

the morphism ∂ is a surjection to Z/p unless

ν_{p}(m) ≥ p − 1 and m/p^{p−1} ≡ 1 mod p,

in which case it is 0.

We will use the double suspension Hopf invariant H_{2} discussed in Theorem 6.8. We denote by H′ the morphism

H′ : E_{2}^{2}(S^{2n+1}) ⟶^{H_{2}} Ext_{U}^{1}(W (n)) → E_{2}^{1}(M),

obtained by following H_{2} by the stabilization. Here

E_{2}^{s}(M) ≈ Ext_{BP_{∗}BP}^{s} (BP_{∗}, BP_{∗}/p)

denotes the E_{2}-term of the stable NSS for the mod p Moore spectrum M. We will eventually need the following facts about E_{2}(M).

(i) E_{2}(M) is commutative.

(ii) v_{1}^{k}h_{1} ≠ 0 ∈ E_{2}^{1}(M).

(iii) If x ∈ E_{2}(M), then v_{2}x = xv_{2} = 0.

(iv) v_{1}h_{1}^{k(p−1)+1} = v_{1}^{k(p−1)+1}h_{1} in E_{2}(M).

(v) If s ≢ 0 mod p, then α_{sp^{e−1}/e} = −sv_{1}^{sp^{e−1}−1}h_{1} in E_{2}(M).

Proof. Part i is well-known, part ii is [51, p 157], and part iii follows from [48, 2.10]. Part iv follows from Proposition 6.1, part iii of this lemma, and the fact that ph_{1} = 0 ∈ E_{2}(M). To prove part v, we use 6.1 to expand v_{1}^{sp^{e−1}}, obtaining

α_{sp^{e−1}/e} = \frac{1}{p^{e}} (η_{R}(v_{1}^{sp^{e−1}}) − (ph_{1} + η_{R}(v_{1}))^{sp^{e−1}})

= − ∑_{j=1}^{sp^{e−1}} \binom{sp^{e−1}}{j}p^{j−e}h_{1}^{j}v_{1}^{sp^{e−1}−j}.

All terms except j = 1 are divisible by p, and hence are 0. To insure that terms with j large are p times an admissible element, write p^{j−e}h_{1}^{j} as p(v_{1} −η_{R}(v_{1}))^{j−e−1}h^{e+1}.

Now we begin the proof of Theorem 8.3. We begin with the case ν(m) < p − 1. In this case,

∂(gen) = α_{m/ν(m)+1} ⊗ α_{1}ι_{3} ≡ −α_{m/ν(m)+1} ⊗ h_{1}ι_{3},  (8.1)

mod terms that desuspend to S^{1}. Here we have used [6, 4.9] and Proposition 6.5. By Proposition 6.5, the assumption that ν(m) < p − 1 implies that α_{m/ν(m)+1} is defined on S^{2p−1}, and hence Theorem 6.8.v implies that

H′(∂(gen)) = −α_{m/ν(m)+1} ≠ 0,

where the last step uses parts v and ii of Lemma 8.4. Thus ∂ ≠ 0 in this case, as claimed.

Now we complete the proof of Theorem 8.3 by considering the case ν(m) ≥ p−1. We let s = m/p^{ν(m)} and

ϵ = \{ \begin{matrix}0 & if ν(m) > p − 1 \\ 1 & if ν(m) = p − 1. \end{matrix}

We will establish the following string of equations in the next paragraph, and then we will further analyze whether these terms are 0 by studying their Hopf invariant. The following string is valid mod terms which desuspend to S^{1}.

∂(gen) = α_{m/p} ⊗ α_{1}ι_{3}
= p^{−p}(η_{R}(v_{1}^{m}) − (ph_{1} + η_{R}(v_{1}))^{m}) ⊗ α_{1}ι_{3}  (8.2)
= − ∑_{j=1}^{m} \binom{m}{j}p^{j−p}h_{1}^{j} ⊗ v_{1}^{m−j}α_{1}ι_{3}
= −p^{m−p}h_{1}^{m} ⊗ α_{1}ι_{3} − ϵsh_{1} ⊗ v_{1}^{m−1}α_{1}ι_{3}  (8.3)
= −v_{1}^{m−p}h_{1}^{p} ⊗ α_{1}ι_{3} − v_{1}^{m−p−1}h_{1}^{p} ⊗ v_{1}h_{1}ι_{3} + ϵsh_{1} ⊗ v_{1}^{m−1}h_{1}ι_{3} (8.4)
= A + B + C,  (8.5)

where A, B, and C denote the three terms in the preceding line.

Line (8.2) follows from Propositions 6.5 and 6.1. Line (8.3) has been obtained by observing that in the sum all terms desuspend to S^{1} except j = m and, if ν(m) = p − 1, j = 1. To see this, we observe that we need to have a p to make α_{1}ι_{3} desuspend. This factor will be present unless j = 1 and ν(m) = p − 1. The requirement that j be ≤ \frac{1}{2} times the degree of the symbols following h_{1}^{j} will only be a problem for large values of j. When j is large, write the term as

\binom{m}{j}p(v_{1} − η_{R}(v_{1}))^{j−p−1}h_{1}^{p+1} ⊗ v_{1}^{m−j}α_{1}ι_{3}.

Since p times anything which is defined on S^{3} desuspends to S^{1}, this desuspends to S^{1} provided p + 1 ≤ (p − 1)(m − j + 1) + 1, which simplifies to 1 ≤ (p − 1)(m − j), i.e., j < m. To obtain (8.4), we have rewritten the first term of eq. (8.3) as

−p(v_{1} − η_{R}(v_{1}))^{m−p−1}h_{1}^{p+1} ⊗ α_{1}ι_{3},

observed that when this is expanded, all terms except the first desuspend, and in that first term we write ph_{1} = v_{1} − η_{R}(v_{1}).

We note first that, by Proposition 6.2, A is d(h_{2}) mod S^{1}, and so H′(A) = 0. We can evaluate the Hopf invariant of B and C by Theorem 6.8.v; using Lemma 8.4, we obtain

H′(B + C) = (−1 + ϵs)v_{1}^{m−1}h_{1}.

Hence H′(∂(gen)) = 0 if and only if −1 + ϵs ≡ 0 mod p. Since H′ is injective on E_{2}^{2}(S^{3}), this completes the proof of Theorem 8.3.

Now we settle the extension in the exact sequence of Theorem 8.3.

The groups E_{2}^{1,qm+2p+1}(B_{1}(p)) in Theorem 8.3 are cyclic.

Proof. We will show that whenever ker(∂) ≠ 0 in the exact sequence of 8.3, there is an element z ∈ E_{2}^{1,qm+2p+1}(B_{1}(p)) such that j_{∗}(z) = α_{m}ι_{2p+1}, the element of order p, and pz = i_{∗}(gen). Since ∂(α_{m}ι_{2p+1}) = 0 in these cases, there is wι_{3} ∈ C^{1,qm+2p+1}(S^{3}) such that d(wι_{3}) = α_{m} ⊗ α_{1}ι_{3}. Let

z = α_{m}ι_{2p+1} − wι_{3}.

Then z is a cycle, since d(z) = α_{m} ⊗ α_{1}ι_{3} − α_{m} ⊗ α_{1}ι_{3}, and clearly j_{∗}(z) is as required. Since pα_{m} = d(v_{1}^{m}), we have

pz − d(v_{1}^{m}ι_{2p+1}) = d(v_{1}^{m})ι_{2p+1} − pwι_{3} − d(v_{1}^{m})ι_{2p+1} + v_{1}^{m}α_{1}ι_{3}
= i_{∗}((v_{1}^{m}α_{1} − pw)ι_{3})

We show (v_{1}^{m}α_{1} −pw)ι_{3} ≠ 0 ∈ E_{2}^{1,qm+2p+1}(S^{3}) by noting from [7, §7] that the Hopf invariant

H_{2} : E_{2}^{1}(S^{3}) → Ext^{0}(W (1))

factors through the mod p reduction of the unstable cobar complex. Thus

H_{2}((v_{1}^{m}α_{1} − pw)ι_{3}) = H_{2}(v_{1}^{m}h_{1}ι_{3}) = v_{1}^{m} ≠ 0.

The second “=” uses Theorem 6.8v, and the “≠” uses 6.8iii.

The following result completes the proof of Theorem 8.1 according to the outline given after Corollary 8.2.

In Theorem 8.1, the group v_{1}^{−1}π_{qm+2p−1}(B_{1}(p)) is cyclic.

Proof. We use the exact sequence in v_{1}^{−1}π_{∗}(−) of the fibration which defines B_{1}(p). The cyclicity follows from that of v_{1}^{−1}π_{qm+2p−1}(S^{2p+1}) unless

∂ = 0 : v_{1}^{−1}π_{qm+2p}(S^{2p+1}) → v_{1}^{−1}π_{qm+2p−1}(S^{3}).  (8.6)

If eq. (8.6) is satisfied, then

∘α_{1} ≠ 0 : v_{1}^{−1}π_{qm+2}(S^{2p+1}) → v_{1}^{−1}π_{qm+2p−1}(S^{2p+1})

by [23, 6.2], and

∂ : v_{1}^{−1}π_{qm+2}(S^{2p+1}) → v_{1}^{−1}π_{qm+1}(S^{3})

is an isomorphism of Z/p’s by Theorem 8.3. Let G denote a generator of v_{1}^{−1}π_{qm+2}(S^{2p+1}), and let

Y ∈ v_{1}^{−1}π_{qm+2p−1}(B_{1}(p))

project to G ∘ α_{1}. By [50, 2.1],

pY = i_{∗}(⟨∂G, α_{1}, p⟩) = ∂(G) ∘ v_{1} ≠ 0.

It is shown in [49] that B_{1}(p) is a direct factor of SU(n)_{(p)} if p < n < 2p, and hence v_{1}^{−1}π_{i}(SU(n); p) is given by Theorem 8.1 if p < n < 2p and i ≡ 1 or 2 mod q. This yields the following number theoretic result.

If p is an odd prime, k ≡ 1 mod p − 1, and p < n < 2p, then the number e_{p}(k, n) defined in 1.2 equals min(p, ν_{p}(k − p − p^{p} + p^{p−1})) + 1.

The author has been unable to prove this result without the UNSS. In fact, the only tractable result for e_{p}(k, n) which follows easily from Definition 1.2 seems to be that if n ≤ p and k ≡ n − 1 mod p − 1, then e_{p}(k, n) ≥ min(n − 1, ν_{p}(k − n + 1) + 1), which is proved using the Little Fermat Theorem as in [22, p.792].

Using methods similar to those in our proof of Theorem 8.1, Yang ([58]) has proved the following tractable result for v_{1}^{−1}π_{∗}(SU(n); p) when p is odd and n ≤ p^{2} − p. Of course this can also be interpreted as a theorem about the numbers e_{p}(k, n). We emphasize that the proof of Theorem 8.8 does not involve the use of Theorem 1.3.

Suppose p is odd, k = c + (p − 1)d with 1 ≤ c < p, and

c + (p − 1)b + 1 ≤ n ≤ c + (p − 1)(b + 1)

with 0 ≤ b ≤ p − 1. Define j by 1 ≤ j ≤ p and d ≡ j mod p. Then v_{1}^{−1}π_{2k}(SU(n); p) is cyclic of order p^{e}, with

e = \{ \begin{matrix}min(c + (p − 1)j, b + ν(d − j) + 1) \\ if b < c and 1 ≤ j ≤ b \\ min(c + (p − 1)j + 1, b + ν(d − j + (−1)^{j}j\binom{c}{j}p^{j(p−1)})) \\ if c ≤ b and 1 ≤ j ≤ b \\ min(c, b + 1 + ν(d)) \\ if b < c and b < j ≤ p \\ b if c ≤ b and b < j ≤ p \end{matrix}

One can easily read off from Theorem 8.8 the precise value of the numbers e_{p}(n) which appeared in Corollary 7.8, yielding the following result for the p-exponent of the space SU(n).

If p is odd and n ≤ p^{2} − p, then

exp_{p}(SU(n)) ≥ e_{p}(n) = \{ \begin{matrix}n & if i(p − 1) + 2 ≤ n ≤ ip + 1 for some i \\ n − 1 & otherwise. \end{matrix}

When p = 2, UNSS methods of computing v_{1}^{−1}π_{∗}(−; p) become more complicated because of the η-towers. Then ASS techniques become more useful, as they did for v_{1}^{−1}π_{∗}(G_{2}; 2) in [29]. Here we show how to use ASS methods to determine v_{1}^{−1}π_{∗}(F_{4}/G_{2}; 2). It is hoped that the result of the calculations of v_{1}^{−1}π_{∗}(G_{2}; 2) and v_{1}^{−1}π_{∗}(F_{4}/G_{2}; 2) might be combined to yield v_{1}^{−1}π_{∗}(F_{4}; 2), but this involves one difficulty not yet resolved. Our main reason for including this example is to give a new illustration of this method.

The proof of the following theorem will consume most of the remainder of this paper. If G denotes an abelian group, then mG denotes the direct sum of m copies of G.

Let G(n) denote some group of order n.

v_{1}^{−1}π_{i}(F_{4}/G_{2}; 2) ≈ \{ \begin{matrix}4Z_{2} & i ≡ 0 mod 8 \\ Z_{8} ⊕ Z_{8} ⊕ Z_{2} & i ≡ 1 mod 8 \\ Z/64 & i ≡ 2 mod 8 \\ 0 & i ≡ 3 or 4 mod 8 \\ G(2^{min(15,ν(i−21)+4)}) & i ≡ 5 mod 8 \\ Z_{2} ⊕ Z_{2} ⊕ Z/2^{min(15,ν(i−22)+4)} & i ≡ 6 mod 8 \\ 5Z_{2} & i ≡ 7 mod 8 \end{matrix}

There is a fibration

S^{15} ⟶^{i} F_{4}/G_{2} ⟶^{p} S^{23},  (8.7)

derived in [29, 1.1]. Here and throughout this proof all spaces and spectra are localized at 2.

In [31], it was shown that for any spherically resolved space Y, there is a finite torsion spectrum X satisfying v_{1}^{−1}π_{∗}(Y) ≈ v_{1}^{−1}J_{∗}(X). In our case, we have

There is a spectrum X such that

(i.) v_{1}^{−1}J_{∗}(X) ≈ v_{1}^{−1}π_{∗}(F_{4}/G_{2}), and

(ii.) there is a cofibration

∈^{15} P^{14} → X →∈^{23+L} P^{22},  (8.8)

where L equals 0 or a large 2-power.

We present in fig. 15 a chart which depicts an initial part of the ASS for v_{1}^{−1}J_{∗}(X) if X is as in Proposition 8.11 and L = 0. It depicts the direct sum of the spectral sequences for v_{1}^{−1}J_{∗}(∈^{15} P^{14}) and v_{1}^{−1}J_{∗}(∈^{23} P^{22}), together with one differential, which will be established in Proposition 8.13. The •’s are elements from P^{14}, while the ∘’s are from P^{22}. Charts such as these for v_{1}^{−1}J_{∗}(P^{n}) were derived in Section 4.

To see that fig. 15 is also valid when L in Proposition 8.11 is a large 2-power, we use the following result.

If L is a large 2-power, then the attaching map

∈^{22+L} P^{22} → v_{1}^{−1} ∈^{15} P^{14}

in Proposition 8.11 has filtration L/2 + 1.

Using results of [40], this implies that a resolution of v_{1}^{−1}X can be formed from v_{1}^{−1} ∈^{15} P^{14} and ϕ^{L/2}v_{1}^{−1} ∈^{23+L} P^{22}, where ϕ^{j} increases filtrations by j, and this yields fig. 15.

Proof of Lemma 8.12. Under S-duality, the generator corresponds to an element of v_{1}^{−1}J_{L+7}(P^{14} ∧ D(P^{22})). This group is isomorphic to

v_{1}^{−1}J_{L+6}(P^{14} ∧ P_{−23}^{−2}).  (8.9)

Using methods of [26], one can show that the relevant chart is as in fig. 16, where the class indicated with a bigger • is the generator of (8.9), and has filtration L/2 + 1.

Borel ([15]) showed that Sq^{8}(x_{15}) = x_{23} in H^{∗}(F_{4}; Z_{2}). This implies that the attaching map in F_{4}/G_{2} and in X is the Hopf map σ, and hence corresponds to the generator of (8.9). Thus the attaching map has filtration L/2 + 1.

We can now establish the d_{2}-differentials in fig. 15.

There are d_{2}-differentials as indicated in fig. 15.

Proof. This follows from the σ attaching map just observed, together with the observation that the first of the pair of elements that are related by the differential in fig. 15 are v_{1}-periodic versions of η_{23} and ησ_{15}.

Figure 15. Initial chart for v_{1}^{−1}π_{∗}(F_{4}/G_{2})

[figure: d; d = d_{ν(k)+2}; d′; d′ = d_{ν(k+1)+2}; 8k + 26; 8k + 21]

Figure 16. The generator of v_{1}^{−1}J_{L+6}(P^{14} ∧ P_{−23}^{−2})

[figure]

This d_{2}-differential implies there is a nontrivial extension in v_{1}^{−1}π_{8k+26}(F_{4}/G_{2}) as follows.

v_{1}^{−1}π_{8k+26}(F_{4}/G_{2}) ≈ Z/64.

Proof. This follows from fig. 15 and a standard Toda bracket argument ([50, 2.1]), which in this situation says the following. Let A be the element supporting the higher of the two d_{2}-differentials, and let D be the lowest • in 8k + 26. Let ∂ denote the boundary morphism in the exact sequence in v_{1}^{−1}π_{∗}(−) associated to the fibration (8.7). Then D lies in the Toda bracket ⟨∂(A), η, 2⟩, and so there exists an element E ∈ v_{1}^{−1}π_{8k+26}(F_{4}/G_{2}) such that p_{∗}(E) = A ∘ η and i_{∗}(D) = 2E.

As indicated in fig. 15, there are d_{ν(k)+2}-differentials between ∘-towers in 8k + 22 and 8k + 21, and there are d_{ν(k+1)+2}-differentials between •-towers in 8k + 22 and 8k+21. This follows just from standard J_{∗}(−)-considerations. But there may also be differentials from the ∘-tower in 8k+22 to the •-tower in 8k+21. These differentials from ∘ to • are determined from the homomorphism

v_{1}^{−1}π_{8k+22}(S^{23}) ⟶^{∂} v_{1}^{−1}π_{8k+21}(S^{15}),  (8.10)

which is evaluated in the following result.

The image of the homomorphism (8.10) consists of all multiples of 8 if ν(k) ≤ 7, and is 0 if ν(k) > 7.

The following result plays a central role in the proof of Proposition 8.15.

Let (S^{15})_{K} denote the K_{∗}-localization as constructed in [42]. There is a commutative diagram

δS^{23}  ⟶^{∂} S^{15}
↓^{ℓ}  ↓^{e}
δ^{∞}(∈^{15} P^{14} ∧ J) ⟶^{h} (S^{15})_{K}

in which ∂ is obtained from the fiber sequence (8.7), e is the localization, and h induces an isomorphism in π_{j}(−) for j = 22 and j ≥ 28.

Proof. The map h is constructed as in [29, pp. 669-670], using results of [42]. It induces an isomorphism in π_{j}(−) for many other small values of j, but we only care about values of j which are positive multiples of 22. The obstructions to its being an isomorphism for all small values of j are Z_{2}-classes in filtration 1 in J_{j}(∈^{15} P^{14}) for j = 19, 23, and 27. The map ℓ is obtained by obstruction theory, since δS^{23} has cells only in dimensions which are positive multiples of 22.

Now we prove Proposition 8.15. Let ℓ be as in Proposition 8.16. The morphism π_{∗}(ℓ) can be factored as

π_{∗}(δS^{23}) → π_{∗}^{s}(δS^{23}) → J_{∗}(∈^{15} P^{14}).

There is a splitting

π_{∗}^{s}(δS^{23}) ≈ ⨁_{i>0} π_{∗}^{s}(S^{22i}).

We will use the method of [43] to deduce that

π_{8k+21}^{s}(S^{22}) → J_{8k+21}(∈^{15} P^{14})

sends the v_{1}-periodic generator ρ_{k} to 8 times the generator. Indeed, the stable map S^{22} →∈^{15} P^{14} ∧ J which induces the morphism factors through ∈^{15} P^{8} ∧ J, from which it projects nontrivially to ∈^{15} P_{7}^{8} ∧ J. We then use [43, 2.8] to deduce that ρ_{k} goes to the nonzero element of J_{8k+21}(∈^{15} P_{7}^{8}). This implies that its image in J_{8k+21}(∈^{15} P^{8}) is the generator, and this maps to 8 times the generator of J_{8k+21}(∈^{15} P^{14}).

The composite of v_{1}-periodic summands of

π_{8k+21}(δS^{23}) → π_{8k+21}^{s}(δS^{23}) → π_{8k+21}^{s}(S^{22})  (8.11)

is bijective if ν(k) ≤ 7, but is not surjective if ν(k) > 7. Thus when the composite (8.11) is followed into J_{8k+21}(∈^{15} P^{14}), the image of a v_{1}-periodic generator is 8 times the generator if ν(k) ≤ 7 and 0 ∈ Z/16 if ν(k) > 7. Once we observe that, in the diagram of Proposition 8.16, h induces an isomorphism in π_{8k+21}(−) and e sends the v_{1}-periodic summand isomorphically, we obtain the desired conclusion of Proposition 8.15.

The differentials implied by Proposition 8.15 have an interesting and unexpected implication about fig. 15. Since d_{r} from the ∘-tower in 8k + 22 hits the top ∘ in 8k + 21 and the • just above it with the same r, and since d_{r} respects the action of h_{0}, there must be an h_{0}-extension between these classes in 8k + 21. If L = 0 in (8.8), then this extension can only be accounted for by a failure of the map (8.8) to induce a split short exact sequence of A_{1}-modules in cohomology. Indeed, we have

If L = 0 in (8.8), then there is a splitting of A_{1}-modules

H^{∗}X ≈ H^{∗}(∈^{15} P^{30}) ⊕ H^{∗}(∈^{23} P^{6}).

This splitting is caused by having Sq^{2} ≠ 0 on the class in H^{∗}X corresponding to the top cell of H^{∗}(∈^{15} P^{14}), i.e.,

Sq^{2} : H^{29}X → H^{31}X

is an isomorphism. This is the only way to account for the h_{0}-extension in fig. 15. Proposition 8.17 implies that the h_{0}-extensions are present in the chart for values of k (ν(k) > 7) where they cannot be deduced from differentials. It also implies that there is an h_{0}-extension on the top ∘ in 8k + 22. If L > 0 in (8.8), the same conclusion about the charts can be deduced from a more complicated analysis.

It causes fig. 15 to take the form of fig. 17.

Figure 17. Final chart for v_{1}^{−1}π_{∗}(F_{4}/G_{2})

[figure: k odd; 8k + 26; 8k + 21; k even; d; d = d_{ν(k)+2}; 8k + 26; 8k + 21]

We can read off almost all of Theorem 8.10 from fig. 17. We must show that d_{6} is 0 on the ∘’s near the bottom in 8k + 23 and 8k + 24. This is done by the argument used to prove Proposition 8.15. The classes involved are present in all spaces in the diagram in Proposition 8.16, but they are not mapped across by ℓ_{∗}, since it factors through π_{∗}^{s}(δS^{23}).

All that remains is the verification of the abelian group structure. Most of the extensions are trivial due to the relation 2η = 0. The extension in 8k + 22 when k is odd was present before the exotic extension was deduced, and remains true. The cyclicity of this 2^{7} summand can also be deduced by consideration of the kernel of the homomorphism in the fibration which defines J, but that seems unnecessary. Note that no claim is made about the group structure in 8k + 21.

Section References  Computing periodic homotopy groups  55

In 1989, Mimura suggested to the author that he try to calculate v_{1}^{−1}π_{∗}(G) for all compact simple Lie groups G. If p is odd, and G = Sp(n) or SO(n), then the result follows from Theorem 1.3 and [33]. With great effort, v_{1}^{−1}π_{∗}(Sp(n); 2) was calculated in [13]. The result involves a surpising pattern of differentials among Z_{2}’s from the various spheres which build Sp(n), resulting in [log_{2}(4n/3)] copies of Z_{2} in certain v_{1}^{−1}π_{i}(Sp(n)). Of the classical groups, only v_{1}^{−1}π_{∗}(SO(n); 2) remains. All torsion-free exceptional Lie groups were handled in [14], using the UNSS. In [29] and [12], the torsion cases (G_{2}, 2), (F_{4}, 3), and (E_{6}, 3) were handled. Remaining then are seven cases of (G, p) yet to be calculated. At least a few of these should lend themselves to the methods of this paper.

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