v_{1}-PERIODIC HOMOTOPY GROUPS OF SO(n)

MARTIN BENDERSKY AND DONALD M. DAVIS

Abstract. We compute the 2-primary v_{1}-periodic homotopy groups of the special orthogonal groups SO(n). The method is to calculate the Bendersky-Thompson spectral sequence, a K_{∗}-based unstable homotopy spectral sequence, of Spin(n). The E_{2}-term is an Ext group in a category of Adams modules. Most of the differentials in the spectral sequence are determined by naturality from those in the spheres.

The resulting groups consist of two main parts. One is summands whose order depends on the minimal exponent of 2 in several sums of binomial coefficients times powers. The other is a sum of roughly [log_{2}(2n/3)] copies of Z/2.

As the spectral sequence converges to the v_{1}-periodic homotopy groups of the K-completion of a space, one important part of the proof is that the natural map from Spin(n) to its K-completion induces an isomorphism in v_{1}-periodic homotopy groups.

1. Introduction

The p-primary v_{1}-periodic homotopy groups of a topological space X, denoted v_{1}^{−1}π_{∗}(X; p), are a localization of the portion of the actual homotopy groups detected by K-theory. The study of these groups was first suggested in Mahowald’s 1982 paper [32], although it was not until the 1991 paper [28] that a satisfactory definition appeared. In the 1989 paper [27], mod 2 v_{1}-periodic homotopy groups of SO(n) were computed for some small values of n, but the mod 2 groups do not contain the information about higher 2-torsion in v_{1}^{−1}π_{∗}(X; 2). In 1988, the second author suggested to Mahowald that they try to compute v_{1}^{−1}π_{∗}(SO(n); 2), to which Mahowald

Date: July 18, 2002. 1991 Mathematics Subject Classification. 55Q52,55T15,57T20. Key words and phrases. homotopy groups, orthogonal groups, Adams spectral

sequences, K-theory.

Both authors thank the Johns Hopkins University JAMI program, which supported them during their work on this in Spring 2000. The second author thanks the Reidler Foundation for support on this work during Summer 2002.

2 BENDERSKY AND DAVIS

wisely responded that it would be worthwhile to try the easier cases SU(n) and Sp(n) first.

It quickly became apparent that odd-primary groups were easier than 2-primary, and the second author determined v_{1}^{−1}π_{∗}(SU(n); p) for odd primes p in 1989, published in [23]. From these results, one easily reads off the groups v_{1}^{−1}π_{∗}(SO(n); p) for p odd. In 1989, Mimura suggested to the second author that the computation of v_{1}^{−1}π_{∗}(X; p) for all compact simple Lie groups X and all primes p would be an interesting project. Thanks to a new approach to odd primary v_{1}-periodic homotopy groups introduced in the 1999 Bousfield paper [15], the determination of v_{1}^{−1}π_{∗}(X; p) for all compact simple Lie groups X and all odd primes p was completed in [22].

In this paper, we determine the long-sought groups v_{1}^{−1}π_{∗}(SO(n); 2). This leaves v_{1}^{−1}π_{∗}(X; 2) for X the exceptional Lie groups E_{7} and E_{8} as the only cases remaining to complete Mimura’s challenge, with the E_{6} case having been completed very recently in [25] together with unpublished work of Bousfield.

Our method is to compute the v_{1}-periodic Bendersky-Thompson spectral sequence (BTSS) of Spin(n). This spectral sequence ([12]) is a K-based version of a spectral sequence of Bousfield and Kan ([20]), and, for a collection of spaces which includes S^{n}, ΩS^{n}, and simply-connected finite H-spaces X for which H_{∗}(X; Q) is associative, converges to v_{1}^{−1}π_{∗}(X^{∧}), where X^{∧} is the K-completion of X, which we will define in Section 10. (From now on, all work will be 2-primary, and we write v_{1}^{−1}π_{∗}(X) for v_{1}^{−1}π_{∗}(X; 2).) We say that X satisfies the Completion Telescope Property (CTP) if the natural map X → X^{∧} induces an isomorphism in v_{1}^{−1}π_{∗}(−). We will prove in Theorem 2.13 that Spin(n) satisfies the CTP. Thus, since Spin(n) is the simplyconnected cover of SO(n), a complete computation of the BTSS of Spin(n), including differentials and extensions, yields v_{1}^{−1}π_{∗}(SO(n)).

A major advance in the understanding of the BTSS was made in [9], where it was shown that for spaces X for which K_{∗}(X) is a nice exterior algebra, E_{2}^{s,t}(X) can be computed directly from the Adams module K^{∗}(X) as Ext_{A}^{s}(QK^{1}(X; Z_{2}^{∧})/ im(ψ^{2}), K^{1}(S^{t})). The d_{3}-differentials in the BTSS of Spin(n) are determined from the known behavior of d_{3} in the BTSS of spheres, using naturality.

In this introductory section, we describe the result for SO(8a ± 1). The similar results for SO(n) for other mod 8 congruences of n are described in Section 3, which

PERIODIC HOMOTOPY GROUPS OF SO(N) 3

also includes discussion of the morphisms v_{1}^{−1}π_{∗}(SO(n)) → v_{1}^{−1}π_{∗}(SO(n + 1)) and some numerical examples.

The following numbers, which are closely related to the numbers of [8, 1.5], play an important role.

Definition 1.1. Let n ≥ 3 and let m be an odd integer satisfying m ≥ 2n.

eSp(m, n) = min\{ν(∑_{k}(−1)^{k}\binom{j}{k}k^{m}) : j > 2n\}.
P_{1}(m, n) = ∑_{odd k≥1} k^{m}( ∑_{i=0}^{n−1−k} \binom{2n−1}{i} − 2 ∑_{t≥0} \binom{2n}{n−2−k−4t}).
P_{2}(m, n) = \begin{cases}2^{n−1} & if n < ν(m + 1) + 3 \\ T (m, n)/2^{ν(m+1)+2} & if ν(m + 1) + 3 ≤ n, \end{cases}

where T (m, n) =

(2^{2n−1}−3^{m+1}+1) ∑_{odd k≥1} k^{m} ∑_{t≥0} \binom{2n}{n−2−k−4t}−3·2^{2n−2} ∑_{odd k≥1} k^{m} ∑_{t≥0} \binom{2n−1}{n−2−k−3t}.

Our P_{i}(m, n) equals R_{i}(m, n − 1) of [8, 1.5]. The reason for the change is to make the formulas nicer for Spin(2n). The apparent difference between P_{2}(m, n) here and R_{2}(m, n − 1) of [8, 1.5] will be explained in Remark 3.2.

Let ν(−) denote the exponent of 2 in an integer.

Theorem 1.2. Let 2n + 1 = 8a ± 1. Let ℓ = [log_{2}(\frac{4}{3}(n − 1))]. In the notation of 1.1, let

e_{1}(m) = \begin{cases}n & if n < 2 + ν(m + 1) \\ min(eSp(m, n), ν(P_{1}(m, n + 1)), ν(P_{2}(m, n + 1))) & otherwise \end{cases}

and e_{2}(m) = min(2 + ν(m + 1), n).^{1} Let G(t) denote an abelian group of order t, and dZ_{2} a Z_{2}-vector space of dimension d. Then v_{1}^{−1}π_{8k+r}(SO(8a ± 1)) ≈

v_{1}^{−1}π_{8k+r}(Spin(8a ± 1)) ≈ \begin{cases}G(2^{e_{1}(4k−1)+e_{2}(4k−1)}) & r = −3 \\ Z/2^{e_{1}(4k−1)} ⊕ Z/2^{e_{2}(4k−1)} ⊕ ℓZ_{2} & r = −2 \\ G(2^{2ℓ+2}) & r = −1 \\ G(2^{ℓ+4}) & r = 0 \\ G(2^{e_{1}(4k+1)+5}) & r = 1 \\ Z/2^{e_{1}(4k+1)} ⊕ Z/8 & r = 2 \\ 0 & r = 3, 4 \end{cases}

The G(−) when r = −3 has exactly ℓ summands. The G(−) groups when r = −1 and 0 are extensions of two Z_{2}-vector spaces.

The reader may get a better feeling about where these groups come from and how they are related to one another in Diagram 1.3, which pictures a stage of the BTSS of Spin(8a±1). As usual with charts of Adams spectral sequence type, position (t−s, s) depicts E_{r}^{s,t}(X) for appropriate r, differentials d_{r} are homomorphisms from E_{r}^{s,t} to E_{r}^{s+r,t+r−1}, and E_{∞}^{∗,∗+i} is an associated graded for π_{i}(X) (here v_{1}^{−1}π_{i}(X^{∧})). A small dot represents an element of order 2, while a big • denotes a Z_{2}-vector space of dimension ℓ. We sometimes call elements in this vector space “log-classes” because ℓ = [log_{2}(−)]. Small labels D, 1, and 4a − 3 next to dots refer to names of elements which will be important later when we derive this chart. In position (2m, 1) with 2m = 8k ± 2, we have summands C_{1} = Z/2^{e_{1}(m)}, while 8 represents Z/8 and C_{2} = Z/2^{e_{2}(m)}. The letter G in position (x, 2) denotes a group of same order as the neighboring group C_{1} ⊕ C_{2} or C_{1} ⊕ Z/8 in position (x + 1, 1), but we don’t know the group structure of G.

Lines of slope 1 connecting dots are the action of h_{1} on E_{2}. This corresponds to the action of the Hopf map η on homotopy groups. We call these eta towers. This action was defined in [9, 3.6], where it was shown that it acts bijectively on groups of filtration ≥ 2. This was shown in a slightly different context in [5]. Multiple lines of slope 1 between big •’s indicate nontrivial action of h_{1} on ℓ linearly independent elements. This carries the implication that G has at least ℓ summands, but we will show in 11.3 that G has exactly ℓ summands. Lines of slope −3 are d_{3}-differentials,

1If 8a + 1 = 9 and m ≡ 3 mod 4, there is an anomaly discussed in [8, 4.21]. In this case, we have e_{1}(m) = min(ν(m − 7) + 2, 8) and e_{2}(m) = 3.

PERIODIC HOMOTOPY GROUPS OF SO(N) 5

which imply that the elements that they connect do not survive the spectral sequence. If the chart does not depict h_{1}x for an element x in filtration ≥ 2, it is because the omitted elements are involved in d_{3}-differentials implied by the chart.

The lack of depicted h_{1}-action on certain summands in filtration 1 carries no implication about whether or not h_{1} is nonzero on their generators. The determination of this requires careful analysis, which is stated in Proposition 1.4 and proved in Section 7. Indeed, the chart in Diagram 1.3 depicts the BTSS prior to the consideration of h_{1}-action, d_{3}-differentials, and extensions on the summands in position (8k + 2, 1).

Diagram 1.3. A stage of the BTSS of Spin(8a ± 1)

[figure: t − s = 8k+; −3; −2; 0; 2; 4; s = 1; 2; 3; 4; C_{1} ⊕ 8; C_{1} ⊕ C_{2}; G; G; 1; D; 1; D; D; 4a−3; 4a−3; D]

In this chart, the 1-line is from 3.1, the 2-line from 5.2, the eta towers from 5.14, and d_{3}-differentials on them from 6.2. The fine tuning described in the following result is proved in Section 7.

Proposition 1.4. The BTSS of Spin(8a ± 1) is as pictured in Diagram 1.3, with the following modifications and clarifications.

1. d_{3} is 0 on E_{2}^{1,8k+1};

2. d_{3} : E_{2}^{1,8k+3} → E_{2}^{4,8k+5} is nonzero on both summands;

3. there are nontrivial extensions (·2) from the two summands of E_{2}^{1,8k+3} to the two of E_{2}^{3,8k+5}.

Theorem 1.2 follows immediately from 1.3 and 1.4. Some other extensions are ruled out since 2η = 0 in π_{∗}(−), while others are left undetermined as discussed in Remark 3.9.

The rest of the paper is organized as follows. We begin in Section 2 by proving Theorem 2.13, that Spin(n) satisfies the CTP. This utilizes an analysis of the BTSS

PERIODIC HOMOTOPY GROUPS OF SO(N) 7

of the classifying spaces BSpin(n). In order to accomplish this, we utilize a general result, 2.2, about fibrations which induce a relatively injective extension sequence in K-homology. The proof of this result is not needed in the rest of the paper, and is presented in Section 10.

As suggested above, the form of v_{1}^{−1}π_{∗}(Spin(n)) depends on the mod 8 value of n. Results similar to those listed above for Spin(8a ± 1) will be collected in Section 3, which will also give some explicit numbers. The next four sections go through the details of the computation in the following order. In Section 4 we compute the 1-line, by extending the methods of [8], which computed the 1-line for Spin(2n + 1). We explain in 5.2 the simple reason why E_{2}^{2,4k+3}(Spin(n)) has essentially the same order as E_{2}^{1,4k+3}(Spin(n)). We compute the 1-line group explicitly as a direct sum of two or three summands; however, we do not know the group structure of the 2-line groups.

As noted above, in filtration greater than 2, elements are arranged in what we call eta towers. These are elements of the group η_{i}(X), defined in 1.5.

Definition 1.5. The group η_{i}(X) is defined to be the direct limit of the groups E_{2}^{s,2s+2i+1}(X),^{2} using h_{1} : E_{2}^{s,t}(X) → E_{2}^{s+1,t+2}(X).

The natural morphism E_{2}^{s,2s+2i+1}(X) → η_{i}(X) is an isomorphism for s ≥ 3. In Section 5, we compute the eta towers in Spin(n), and in Section 6 we compute the d_{3}-differentials on the eta towers. In Section 7, we compute the d_{3}-differential on the 1-line groups and the extensions in the BTSS.

In Section 8, we prove some combinatorial results needed earlier in the paper. In Section 9, we compare our results with those obtained by a J-homology approach such as was employed in [26] and [27].

The method used to compute Ext_{A} in Sections 5, 6, and 7 is that developed in [9]. In Section 11, we describe an alternate way of computing Ext_{A}, involving an explicit small resolution. It has several advantages over the previous method: (a) it describes eta-towers in a way which does not involve an extension in a short exact sequence (compare 5.1 and 11.3); (b) it gives a different proof of a formula for h_{1}-action and

2We use this notation here because eta towers for Spin(n) occur only when t−2s is odd. If one is dealing with situations in which eta towers occur in both parities of t − 2s, then defining η_{i}(X) as the limit of E_{2}^{s,2s+i}(X) would be more sensible.

8 BENDERSKY AND DAVIS

extends that formula to other situations (compare 7.2 with 11.5 and 11.18); and (c) it gives a new interpretation of the 2-line groups, which shows exactly their number of summands, and lends hope to their complete calculation. (See discussion after 11.3).

2. The BTSS of BSpin(n) and the CTP

In this section, we prove, by induction on n, that Spin(n) satisfies the CTP for all n. In order to accomplish this, we consider also the BTSS and CTP for the classifying spaces BSpin(n).

We begin by recalling an important definition.

Definition 2.1. Let A be a commutative ring with unit. Let S be the category of coassociative cocommutative coalgebras over A which are free positively-graded Amodules of finite type. A relatively injective extension sequence is a sequence of maps in S

C′ ⟶^{f} C ⟶^{g} C′′

such that • g is a split epimorphism of A-modules;

• the map f is the inclusion C□_{C′′}A → C;

• C is a relatively injective C′′-comodule, which means that it is a direct summand (over K_{∗}) of a C′′-comodule of the form C′′ ⊗ N for some K_{∗}-module N.([39, p.321])

This definition is that of [17, 2.1], modified to relatively injective comodules rather than injective ones. The following result will be proved in Section 10.

Theorem 2.2. Let F ⟶^{j} E ⟶^{h} B be a fibration. Suppose that, for X = F, E, and B, the BTSS converges to v_{1}^{−1}π_{∗}(X^{∧}), and that the induced sequence of K_{∗}-coalgebras

K_{∗}(F) → K_{∗}(E) → K_{∗}(B)

is a relatively injective extension sequence. Then

(i) the induced maps of K-completions F^{∧} → E^{∧} ⟶^{h^{∧}} B^{∧} form a fibration,

(ii) there is an exact sequence

⋯ → E_{2}^{s}(F) → E_{2}^{s}(E) → E_{2}^{s}(B) ⟶^{∂} E_{2}^{s+1}(F) → ⋯, and

PERIODIC HOMOTOPY GROUPS OF SO(N) 9

(iii) ∂ commutes with differentials in the BTSS.

The following result is quite easy.

Proposition 2.3. The fibration Spin(2n − 1) → Spin(2n) → S^{2n−1} induces a relatively injective extension sequence in K_{∗}(−).

Proof. Using [29] or [37], we easily see that the K^{∗}(−)-algebras of the fibration are polynomial algebras on indecomposables which form a short exact sequence. Indeed, in notation which will be prevalent in the last half of this paper, the generators are

⟨x_{1}, …, x_{n−2}, D⟩ ⟵^{i^{∗}} ⟨x_{1}, …, x_{n−2}, D_{+}, D_{−}⟩ ⟵^{p^{∗}} ⟨g⟩ (2.4)

with i^{∗}(x_{j}) = x_{j}, i^{∗}(D_{±}) = D, and p^{∗}(g) = D_{+} − D_{−}. Dualizing this result (e.g., [2, 1.4]) says that K_{∗}(−) is a relatively injective extension sequence.

The following similar result is somewhat more delicate.

Proposition 2.5. There is a relatively injective extension sequence of coalgebras

K_{∗}(S^{2n}) → K_{∗}(BSpin(2n)) → K_{∗}(BSpin(2n + 1))

induced by the fibration

S^{2n} → BSpin(2n) → BSpin(2n + 1).  (2.6)

Proof. We begin by showing that

R(Spin(2n + 1)) ⟶^{i^{∗}} R(Spin(2n)) ⟶^{ϕ} K^{0}(S^{2n})  (2.7)

is a projective extension sequence. This means that i^{∗} is a split (over Z) monomorphism,

R(Spin(2n)) ⊗_{im(i^{∗})} Z \underrightarrow{_{ϕ}} K^{0}(S^{2n})

is an isomorphism, and R(Spin(2n)) is a projective R(Spin(2n + 1))-module. (See [36] for the analogue over a field.)

From [30] or [14], we have

R(Spin(2n + 1)) = Z[ρ_{2n+1}, λ^{2}ρ_{2n+1}, …, λ^{n−1}ρ_{2n+1}, Δ]
R(Spin(2n)) = Z[ρ_{2n}, λ^{2}ρ_{2n}, …, λ^{n−2}ρ_{2n}, Δ_{+}, Δ_{−}].

10 BENDERSKY AND DAVIS

For 1 ≤ j ≤ n − 2, i^{∗}(λ^{j}ρ_{2n+1}) = λ^{j}ρ_{2n} + λ^{j−1}ρ_{2n}. Also, i^{∗}(Δ) = Δ_{+} + Δ_{−}, and

i^{∗}(λ^{n−1}ρ_{2n+1}) = Δ_{+} · Δ_{−} + λ^{n−2}ρ_{2n} − λ^{n−3}ρ_{2n} − λ^{n−5}ρ_{2n} − · · · . The morphism ϕ is the composite R(Spin(2n)) → K^{0}(BSpin(2n)) ⟶^{j^{∗}} K^{0}(S^{2n}) with j the map in (2.6), and satisfies ϕ(λ^{i}ρ_{2n}) = 0 and ϕ(Δ_{±}) = ±θ, where K^{0}(S^{2n}) = Z[θ]/θ^{2}. After an obvious change of basis, (2.7) becomes

Z[g′_{1}, …, g′_{n−1}, Δ] ⟶^{i^{∗}} Z[g_{1}, …, g_{n−2}, Δ_{+}, Δ_{−}] ⟶^{ϕ} Z[θ]/θ^{2}

with i^{∗}(g′_{i}) = g_{i} for i ≤ n − 2, i^{∗}(g′_{n−1}) = Δ_{+} · Δ_{−}, ϕ(g_{i}) = 0, and i^{∗}(Δ) and ϕ(Δ_{±}) as above.

The first two properties of projective extension sequence are clearly satisfied. To see the projectivity, we observe that Z[{g_{i}}, x, y] is a free Z[{g_{i}}, xy, x + y]-module on 1 and x. This can be achieved by noting that the change-of-basis matrix relating (xy)^{n}, x(xy)^{n−1}(x+y), (xy)^{n−1}(x+y)^{2}, x(xy)^{n−2}(x+y)^{3}, (xy)^{n−2}(x+y)^{4}, … to

x^{n}y^{n}, x^{n+1}y^{n−1}, x^{n−1}y^{n+1}, x^{n+2}y^{n−2}, x^{n−2}y^{n+2}, …

is triangular, and similarly in odd degree.

In [2, 1.2], the following result is proved.

Theorem 2.8. ([2]) If G is a connected compact Lie group, then K_{1}(BG) = 0, and there are natural isomorphisms

K_{0}(BG) ≈ Hom(K^{0}(BG), Z) ≈ Hom(R(G), Z),

where Hom(−, −) refers to continuous homomorphisms. Here R(G) has the I(G)- adic topology, and Z is discrete.

As Anderson remarks on [2, p.5], the effect of this is given in the following corollary.

Corollary 2.9. If {λ_{i}} is a set of irreducible representations of G and ρ_{i} = λ_{i} − dim(λ_{i}), let ρ^{E} = ρ_{1}^{e_{1}} ⋯ ρ_{k}^{e_{k}} ∈ R(G). Let ϕ^{E} ∈ Hom(R(G), Z) be dual to ρ^{E} in the basis of ρ^{E}’s. Then K_{0}(BG) is free abelian with basis {ϕ^{E}}, and its coalgebra structure is given by

ψ(ϕ^{E}) = ∑_{F ≤E} ϕ^{F} ⊗ ϕ^{E−F} .

PERIODIC HOMOTOPY GROUPS OF SO(N) 11

We apply Hom(−, Z) to the objects of (2.7), obtaining the sequence of 2.5, which is 0 in odd gradings. This duality applied to a projective extension sequence yields a relatively injective extension sequence. As described in Corollary 2.9, the effect of Hom is to make the dualization act as if R(G) were finite dimensional. For the tensor/cotensor criterion, we use [34, 3.2.2]. We remark that we need to use “relatively injective” because these Z_{(p)}-modules lack the divisibility to be injective, but projectivity does not have this problem. This completes the proof of Proposition 2.5.

The above proposition is relevant for Spin(n) because of the following result.

Proposition 2.10. i. There is an isomorphism of BTSS’s

E_{r}^{s,t}(BSpin(n)) ≈ E_{r}^{s,t−1}(Spin(n)), r ≥ 2;

ii. the map of K-completions

Spin(n)^{∧} → Ω(BSpin(n)^{∧})  (2.11)

induces an isomorphism in v_{1}^{−1}π_{∗}(−); iii. Spin(n) satisfies the CTP if and only if BSpin(n) does.

Proof. We first establish the isomorphisms, for G = Spin(n), E_{2}^{s,t}(BG) ≈ Ext_{G}^{s,t}(K_{∗}BG) ≈ Ext_{U}^{s,t}(P K_{∗}BG) ≈ Ext_{U}^{s,t−1}(P K_{∗}G) ≈ E_{2}^{s,t−1}(G). Here G and U are the categories of unstable K_{∗}K-coalgebras and unstable K_{∗}Kcomodules discussed in Section 10 and in [12]. Our convention is to omit writing K_{∗} as the first component of Ext groups.

The first isomorphism is [12, 4.3]. The fourth isomorphism is [12, 4.9], and the second follows similarly. To deduce the third isomorphism, first note that, for X = BG, the map ΣΩX → X induces K_{∗−1}(ΩX) → K_{∗}(X), which on primitives is an isomorphism in U. The desired isomorphism then follows from the isomorphism

U(A[2n]) ≈ σU(A[2n − 1]).  (2.12)

Here U is the functor from free K_{∗}-modules to unstable Γ-comodules defined in [12, §4], A[t] is the free K_{∗}-module on a generator of grading t, σ is suspension, and (2.12) follows from [12, 4.5].

12 BENDERSKY AND DAVIS

To prove (i) for all r, we note that the isomorphism of E_{2}-terms is induced by a map of towers. To see this, we use the following natural map of augmented cosimplicial spaces, where K(X) = Ω^{∞}(K ∧ Σ^{∞}X). We take X = BSpin(n); however, the argument works in much greater generality.

[figure: ΩX; ΩKX; ΩK(KX); ΩK^{3}X; ⋯; ΩX; KΩX; K(K(ΩX)); K^{3}ΩX; ⋯]

Applying π_{∗}(−) and taking homology of the alternating sum to the first row yields E_{2}^{∗,∗}(ΩX), and doing this to the second yields E_{2}^{∗,∗−1}(X). The induced morphism in homology is the E_{2} isomorphism observed above. But these cosimplicial spaces give rise, by filtering the Tot construction, to the towers that define the entire spectral sequence, and so the morphism induces a morphism of spectral sequences, which is then an isomorphism. See the first few pages of Section 10, or [12], for more details regarding the Tot construction and the BTSS. By [9, 5.1], the spectral sequences converge, respectively, to v_{1}^{−1}π_{∗}(Spin(n)^{∧}) and v_{1}^{−1}π_{∗}(Ω(BSpin(n)^{∧})), which is consequently an isomorphism.

Part iii follows from part ii and the commutative diagram

Spin(n) ⟶^{≃} Ω BSpin(n)
ι_{Spin}↓  ^{Ωι_{BSpin}}↓
Spin(n)^{∧} ⟶ Ω(BSpin(n)^{∧})

Now we can prove the main theorem of this section.

Theorem 2.13. For each n, the natural map Spin(n) → Spin(n)^{∧} induces an isomorphism in v_{1}^{−1}π_{∗}(−); i.e., Spin(n) satisfies the CTP.

Proof. It was proved in [9] that S^{n} satisfies the CTP. Since Spin(3) = S^{3}, this will initiate the induction. The induction steps are immediate from Propositions 2.14 and 2.10.iii.

PERIODIC HOMOTOPY GROUPS OF SO(N) 13

Proposition 2.14. a. If Spin(2n − 1) satisfies the CTP, then so does Spin(2n). b. If BSpin(2n) satisfies the CTP, then so does BSpin(2n + 1).

Proof. It was proved in [9, 5.8,5.11,5.12] that if F → E → B induces a relatively injective extension sequence in K_{∗}(−), and two of the spaces satisfy the CTP, then so does the third. The proposition then follows from 2.3 and 2.5.

3. Listing of results

In this section, we state the results for the explicit form of the BTSS of Spin(N) for the values of N not covered in Section 1. The proofs of these statements occupy the next four sections of the paper. Indeed, results for the 1-line are proved in Section 4, the 2-line in 5.2, the eta towers in 5.14, 5.16, and 5.22, and d_{3} on the eta towers in 6.2. Finally, d_{3} on the 1-line and the extensions are in Section 7, with explicit references there to the theorems of this section whose proofs are being completed. At the end of this section, we also describe homomorphisms induced by inclusion maps, and give explicit numerical examples.

We begin by recalling from [8] the determination of the 1-line groups of Spin(N) when N is odd, with a refinement established in Remark 3.2.

Theorem 3.1. ([8, 1.5], 3.2) If n ≥ 6, and m is odd, then

E_{2}^{1,2m+1}(Spin(2n+1)) = \begin{cases}Z/2^{n} ⊕ Z/2^{n} & n ≤ 2 + ν(m + 1) \\ Z/2^{R(m,2n+1)} ⊕ Z/2^{2+ν(m+1)} & otherwise, \end{cases}

where

R(m, 2n + 1) = min(eSp(m, n), ν(P_{1}(m, n + 1)), ν(P_{2}(m, n + 1))) with eSp, P_{1}, and P_{2} as in 1.1. If m is even, then E_{2}^{1,2m+1}(Spin(2n+1)) ≈ Z/2⊕Z/2.

Remark 3.2. The simple form of E_{2}^{1,2m+1}(Spin(2n + 1)) when n ≤ 2 + ν(m + 1) was not observed in [8]. The group of [8, 1.5] is obtained by elementary reductions applied to [8, 3.18]. By 8.1, the first relation of [8, 3.18] is of the form A_{1}2^{n+1}ξ_{1} for A_{i} ∈ Z. By [8, 3.18],^{3} the second and third relations of [8, 3.18] are of the form

3The statement in [8, 3.18] implied divisibility by 2^{n} here, but the argument implied divisibility by 2^{n+1}, which is what we need.

14 BENDERSKY AND DAVIS

A_{2}2^{n+1}ξ_{1} − 2^{n+1}D and A_{3}2^{n+1}ξ_{1} − 2^{n}D, while by 8.11 the fourth relation is of the form u2^{n}ξ_{1} − A_{4}2^{n}D, with u an odd integer. It is elementary to check that, localized at 2, such a group is Z/2^{n} ⊕ Z/2^{n}.

The following result will be proved in Section 4.

Theorem 3.3. Let n = 4 or n ≥ 6. If m is even, then E_{2}^{1,2m+1}(Spin(2n)) is isomorphic to

Z_{2} ⊕ \begin{cases}Z/2^{min(n−1,ν(m+1−n)+2)} & if n odd \\ Z_{2} ⊕ Z_{2} & if n even. \end{cases}

If m is odd, then E_{2}^{1,2m+1}(Spin(2n)) is isomorphic to

Z/2^{R(m,2n)}⊕\begin{cases}Z/2^{min(n−1,ν(m+1)+2)} & if n odd \\ Z/2^{min(n−1,ν′(m+1,n)+2)} ⊕ Z/2^{min(ν(n)+1,ν(m+1)+2)} & if n even. \end{cases}

Here

ν′(m + 1, n) = \begin{cases}ν(m + 1) & if ν(m + 1) < ν(n) \\ ν(m + 1 − n) + 1 & if ν(m + 1) = ν(n) \\ ν(m + 1) + 1 & if ν(m + 1) > ν(n) \end{cases}

and

R(m, 2n) = min(eSp(m, n), ν(P_{1}(m, n)), ν(P′_{2}(m, n)), ν(P_{3}(m, n))), where eSp, P_{1}, and P_{2} are as in 1.1, while

P′_{2}(m, n) = \begin{cases}2P_{2}(m, n) & if ν(m + 1 − n) ≥ n − 3 \\ n & if ν(m + 1) ≥ n − 3 and n even \\ P_{2}(m, n) & otherwise, and \end{cases}
P_{3}(m, n) = \frac{1}{n} ∑_{k odd} k^{m+1}\binom{2n}{n−k}.

If n − 3 ≤ ν(m + 1), then

R(m, 2n) = \begin{cases}n − 1 & if n odd \\ n & if n even. \end{cases}

The sum which defines P_{3} begins a convention, used throughout the paper, that summation variables are always nonnegative. Sometimes this is included in the summation adornments, but even if it is not explicitly stated, it is implicitly assumed. Another notational convention was initiated in 1.1. This is that our formulas related

PERIODIC HOMOTOPY GROUPS OF SO(N) 15

to E_{2}^{1,2m+1}(Spin(N)) are only applicable when m > N. The BTSS is periodic, and so it suffices to specify the groups in this range. Thus, for example, the integers R(m, N) of 3.1 and 3.3 are defined by the given formulas only for m > N.

We remark that it is true for dimensional reasons that E_{2}^{1,2m}(Spin(n)) = 0. Also, for the omitted cases, since Spin(4) ≈ S^{3} × S^{3}, its v_{1}-periodic homotopy groups follow from [32] or [24, 4.2], while the 1-lines of Spin(6) and Spin(10) have arithmetic anomalies and are covered in Proposition 4.33. Although they are quite rare, there are cases in which the value of R(m, N) is determined by eSp(m, N), and cases in which the value of R(m, 2n) is determined by P_{3}(m, n).

Now we describe the entire BTSS of Spin(N), divided into cases by the mod 8 value of N. We begin with the case N = 8a.

Theorem 3.4. The BTSS for Spin(8a) is the direct sum of the BTSS of S^{8a−1} given in Diagram 3.6 and the BTSS of Spin(8a − 1), as given in 1.3 and 1.4, except that if s = 1 or 2, the short exact sequence

0 → E_{2}^{s,8k−1}(Spin(8a − 1)) ⟶^{i_{∗}} E_{2}^{s,8k−1}(Spin(8a)) ⟶^{p_{∗}} E_{2}^{s,8k−1}(S^{8a−1}) → 0

(3.5)

is not always split. If s = 2, no claim is made about the structure of the groups. If s = 1, it splits if ν(k) < ν(a), but does not in the remaining cases:

• if ν(k) = ν(a) and ν(k − a) < 4a − 5, it is

0 → Z/2^{e_{1}}⊕Z/2^{ν(k)+4} ⟶^{i_{∗}} Z/2^{e_{1}}⊕Z/2^{ν(k−a)+5}⊕Z/2^{ν(k)+3} ⟶^{p_{∗}} Z/2^{ν(k−a)+4} → 0

in which p_{∗} sends the second summand surjectively and the third summand injectively, and i_{∗}(g_{2}) = g_{3} − 2^{ν(k−a)−ν(k)+1}g_{2}; in particular, the initial summands for Spin(8a − 1) and Spin(8a) are equal in this case; here we initiate a custom of letting g_{i} denote a generator of the ith summand;

• if ν(k − a) ≥ 4a − 5, it is

0 → Z/2^{e_{1}}⊕Z/2^{ν(k)+4} ⟶^{i_{∗}} Z^{e_{1}+1}⊕Z/2^{4a−1}⊕Z/2^{ν(k)+3} ⟶^{p_{∗}} Z/2^{4a−1} → 0

in which p_{∗} sends the first summand surjectively, the second summand to multiples of 2, and the third summand injectively, while

16 BENDERSKY AND DAVIS

i_{∗}(g_{1}) = 2g_{1}−g_{2} and i_{∗} sends g_{2} injectively to the second summand plus surjectively to the third summand;

• if ν(a) < ν(k) < 4a − 5, then it is

0 → Z/2^{e_{1}}⊕Z/2^{ν(k)+4} ⟶^{i_{∗}} Z/2^{e_{1}}⊕Z/2^{ν(k)+5}⊕Z/2^{ν(a)+3} ⟶^{p_{∗}} Z/2^{ν(a)+4} → 0

in which p_{∗} sends the second summand surjectively and the third summand injectively, and i_{∗}(g_{2}) = g_{3} − 2g_{2};

• if ν(k) ≥ 4a − 5, it is

0 → Z/2^{4a−1}⊕Z/2^{4a−1} ⟶^{i_{∗}} Z/2^{4a−1}⊕Z/2^{4a−1}⊕Z/2^{ν(a)+3} ⟶^{p_{∗}} Z/2^{ν(a)+4} → 0

in which p_{∗} sends the first summand surjectively and the third summand injectively.

Here we are using the following diagram for the BTSS of S^{8a−1}, taken from [7, p.488], in which C denotes Z/2^{min(ν(k−a)+4,4a−1)}, and 8 denotes Z/8.

Diagram 3.6. BTSS of S^{8a−1}

[figure: t − s = 8k+; −2; 0; 2; 4; s = 1; 2; 3; 4; C; C; 8; 8]

Next we describe the BTSS for Spin(8a + 3) and Spin(8a + 5), except for the group structure of some 2-line groups. We begin with a picture of a certain stage of the BTSS, and then describe the result in a theorem.

Diagram 3.7. A stage of the BTSS for Spin(8a + 4 ± 1)

[figure: t − s = 8k+; −3; −2; 0; 2; 4; s = 1; 2; 3; 4; C_{1} ⊕ 8; C_{1} ⊕ C_{2}; G; G; 4a−1; 4a−1; 1; 1; D; D; D; D; D; D]

Theorem 3.8. Let 2n + 1 = 8a + 4 ± 1. The BTSS of Spin(2n + 1) is as depicted in Diagram 3.7, with the following additions and interpretations.

• The 1-line groups are as given in 3.1.

• A G in position (x, 2) represents an abelian group of the same order as the group in position (x + 1, 1).

• All elements x in filtration 2 are acted on freely by η in E_{2}. When the elements η^{i}x for i > 0 are not depicted, it means that they support d_{3}-differentials inferred from d_{3}(x).

• The big •’s represent a vector space of dimension ℓ = [log_{2}(4(n − 1)/3)] + δ_{α(n−1),1}, as specified in 5.14. Multiple lines indicate d_{3}-differentials or eta-actions acting bijectively on these vector spaces. The groups G in position (8k − 1 ± 2, 2) have exactly ℓ summands.

• In addition to the differentials pictured, d_{3} on the generator of the C_{1} summand in position (8k + 2, 1) is nonzero, while d_{3} from

18 BENDERSKY AND DAVIS

position (8k − 2, 1) hits the class D. This differential is on the C_{2}-summand if ν(k) + 3 < n; otherwise, C_{1} ≈ C_{2} ≈ Z/2^{n} and the differential is from C_{1}. Other than these, there are no more nonzero differentials.

• There is a nontrivial extension (multiplication by 2) in dimension 8k − 2 from C_{2} to D.

Remark 3.9. We can make some other general statements about extensions in Diagram 3.7. For the summands x in the G in (8k − 3, 2) which are of order 2, there must be an extension (·2) from (8k − 1, 2) to h_{1}^{2}x in (8k − 1, 4). For summands y in (8k − 1, 2) which do not extend into (8k − 1, 4), there must be an extension from G in (8k + 1, 2) to h_{1}^{2}y in (8k + 1, 4). These follow because the homotopy groups of the mod-2 Moore space imply that if α ∈ π_{n}(X) satisfies 2α = 0, then η^{2}α is divisible by 2.

We easily read off the groups as follows.

Corollary 3.10. Let 2n + 1 = 8a + 4 ± 1, ℓ = log_{2}(4(n − 1)/3)] + δ_{α(n−1),1}, and let e_{1}(m)and e_{2}(m) be defined by the formulas of 1.2 unless m ≡ 3 mod 4 and n < 2 + ν(m + 1), in which case e_{1}(m) = n − 1 and e_{2}(m) = n + 1. Then

v_{1}^{−1}π_{8k+r}(Spin(2n + 1)) ≈ \begin{cases}G(2^{e_{1}(4k−1)+e_{2}(4k−1)}) ⊕ Z_{2} & r = −3 \\ Z/2^{e_{1}(4k−1)} ⊕ Z/2^{e_{2}(4k−1)} ⊕ ℓZ_{2} & r = −2 \\ G(2^{2ℓ+1}) & r = −1 \\ (ℓ + 2)Z_{2} & r = 0 \\ G(2^{e_{1}(4k+1)+4}) & r = 1 \\ Z/2^{e_{1}(4k+1)} ⊕ Z/8 & r = 2 \\ Z_{2} ⊕ Z_{2} & r = 3, 4 \end{cases}

The G(−) when r = −3 has exactly ℓ summands. The G(−) group when r = −1 is an extension of two Z_{2}-vector spaces.

The result for Spin(4a + 2) is given as follows.

Theorem 3.11. A chart for the BTSS of Spin(4a + 2) is as in Diagram 3.12. The 1-line groups are as given in Theorem 3.3. A group labeled G in position (x, 2) has the same order as the group in (x + 1, 1). The group labeled C′ in (4k − 1, 2) is

PERIODIC HOMOTOPY GROUPS OF SO(N) 19

cyclic of exponent 1 greater than that of the cyclic group C in (4k, 1). Each big • represents a Z_{2}-vector space of dimension [log_{2}(\frac{4}{3}(2a−3))]. The d_{3}-differential on the generator of C_{1} in (8k + 2, 1) is nonzero if and only if R(4k + 1, 4a + 2) of 3.3 equals R(4k + 1, 4a + 3) of 3.1. All other d_{3}-differentials are 0, except those indicated in the chart. The extension from C′ into (8k − 1, 4) is trivial.

Diagram 3.12. The BTSS of Spin(4a + 2)

[figure: t − s = 8k+; −3; −2; 0; 2; 4; s = 1; 2; 3; 4; C_{1} ⊕ 8; C_{1} ⊕ C_{2}; G; G; C′; C′; 1; C; 1; C; 4a−1]

The groups which are the result of this chart are given in the following result.

Corollary 3.13. Let ℓ = [log_{2}(\frac{4}{3}(2a − 3)], R(−, −) be as in 3.3,^{4}

e_{1}(m) = \begin{cases}R(m, 4a + 2) & m ≡ 3 mod 4 \\ min(R(m, 4a + 2), R(m, 4a + 3) − 1) & m ≡ 1 mod 4 \end{cases}
e_{2}(m) = min(2a, ν(m + 1) + 2)
e_{3}(m) = min(2a, ν(m − 2a) + 2).

4If a = 2 and m ≡ 3 mod 4, the same anomaly occurs as in 1.2. Thus e_{1}(m) = min(ν(m − 7) + 2, 8) and e_{2}(m) = 3.

Let G(t) denote an abelian group of order t, and dZ_{2} a Z_{2}-vector space of dimension d. Then

v_{1}^{−1}π_{8k+r}(Spin(4a + 2)) ≈ \begin{cases}G(2^{e_{1}(4k−1)+e_{2}(4k−1)}) & r = −3 \\ Z/2^{e_{1}(4k−1)} ⊕ Z/2^{e_{2}(4k−1)} ⊕ ℓZ_{2} & r = −2 \\ Z/2^{e_{3}(4k)+1} ⊕ G(2^{2ℓ}) & r = −1 \\ Z/2^{e_{3}(4k)} ⊕ (ℓ + 2)Z_{2} & r = 0 \\ G(2^{e_{1}(4k+1)+5}) & r = 1 \\ Z/2^{e_{1}(4k+1)+1} ⊕ Z/8 & r = 2 \\ Z/2^{e_{3}(4k+2)} & r = 3, 4 \end{cases}

The G(−) when r = −3 has exactly ℓ summands. The G(−) group when r = −1 is an extension of two Z_{2}-vector spaces.

From Table 3.22 we can see, for example, that d_{3} is nonzero on E_{2}^{1,8k+3}(Spin(18)) iff k ≡ 2 mod 8 or k ≡ 259 mod 512 and that d_{3} ≠ 0 on E_{2}^{1,8k+3}(Spin(22) iff k ≡ 3 mod 2^{7} or k ≡ 4 + 2^{9} mod 2^{12}.

Next we describe the BTSS of Spin(8a + 4).

Theorem 3.14. The BTSS for Spin(8a + 4) is the direct sum of the BTSS of S^{8a+3} given in Diagram 3.16 and the BTSS of Spin(8a + 3) as described in Theorem 3.8 except that the short exact sequence

0 → E_{2}^{s,4ℓ−1}(Spin(8a + 3)) ⟶^{i_{∗}} E_{2}^{s,4ℓ−1}(Spin(8a + 4)) ⟶^{p_{∗}} E_{2}^{s,4ℓ−1}(S^{8a+3}) → 0

(3.15)

is not always split when s = 1 and 2. No claim is made about the group structure when s = 2. If s = 1 and ℓ = 2k + 1, then

• if ν(k − a) ≤ 4a − 4, the sequence is 0 → Z/2^{e_{1}} ⊕ Z/8 → Z/2^{e_{1}} ⊕ Z/2^{ν(k−a)+5} ⊕ Z/4 → Z/2^{ν(k−a)+4} → 0

with the Z/8 mapping injectively to the second summand and surjectively to the third, while these summands map, respectively, surjectively and injectively to the Z/2^{ν(k−a)+4}. Note that the two Z/2^{e_{1}} summands are of equal order. The d_{3}-differential from E_{2}^{1,8k+3}(Spin(8a + 4)) is nonzero on only the first summand.

• if ν(k − a) > 4a − 4, the sequence is 0 → Z/2^{e_{1}} ⊕ Z/8 → Z/2^{e_{1}+1} ⊕ Z/2^{4a+1} ⊕ Z/4 → Z/2^{4a+1} → 0

PERIODIC HOMOTOPY GROUPS OF SO(N) 21

with i_{∗} sending the first generator to (2, 1, 0) and the second to (0, 2^{4a−2}, 1), while p_{∗} sends the the three generators, respectively, to 1, −2, and 2^{4a−1}. The d_{3}-differential from E_{2}^{1,8k+3}(Spin(8a+4)) is nonzero on just the first and second summands.

If s = 1 and ℓ = 2k, then • if ν(k) ≤ 4a − 4, the sequence is 0 → Z/2^{e_{1}} ⊕ Z/2^{ν(k)+4} → Z/2^{e_{1}} ⊕ Z/2^{ν(k)+5} ⊕ Z/4 → Z/8 → 0 with the Z/2^{ν(k)+4} mapping injectively to the second summand and surjectively to the third, while these summands map, respectively, surjectively and injectively to the Z/8. The d_{3}-differential from E_{2}^{1,8k−1}(Spin(8a+4)) is nonzero on just the second and third summands.

• if ν(k) > 4a − 4, the sequence is 0 → Z/2^{4a+1} ⊕ Z/2^{4a+1} → Z/2^{4a+2} ⊕ Z/2^{4a+1} ⊕ Z/4 → Z/8 → 0

with i_{∗} sending the first generator to (2, 0, 1) and the second summand bijectively to the second summand, while p_{∗} sends the first summand surjectively and the third summand injectively. The d_{3}- differential from E_{2}^{1,8k−1}(Spin(8a + 4)) is nonzero on just the first and third summands.

Here we are using the following diagram for the BTSS of S^{8a+3}, taken from [7, p.488], in which C′ denotes Z/2^{min(ν(k−a)+4,4a+1)}, and 8 denotes Z/8. The dotted d_{3}-differential is present iff 4a + 1 ≤ ν(k − a) + 4.

Diagram 3.16. BTSS of S^{8a+3}

[figure: t − s = 8k+; −2; 0; 2; 4; s = 1; 2; 3; 4; 8; 8; C′; C′; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·]

The morphisms in v_{1}^{−1}π_{∗}(−) induced by inclusion maps Spin(n) → Spin(n+1) can, for the most part, be determined from the above charts together with knowledge of the induced morphism of 1-line groups and eta towers. For the 1-line groups, the results are presented in Proposition 3.17. These are easily read off using the names of generators described in Section 4. Some details of the proof will be given in that section.

Proposition 3.17. Let E_{2m}(N) = E_{2}^{1,2m+1}(Spin(N)), and consider the sequence

E_{2m}(4a+1) ⟶^{i_{1}} E_{2m}(4a+2) ⟶^{i_{2}} E_{2m}(4a+3) ⟶^{i_{3}} E_{2m}(4a+4) ⟶^{i_{4}} E_{2m}(4a+5),

beginning with E_{2m}(11).

a. Let m be even. All groups have a summand Z_{2} generated by ξ_{1}, which maps across. In addition, there are summands

Z_{2} ⟶^{i_{1}} Z/2^{ν} ⟶^{i_{2}} Z_{2} ⟶^{i_{3}} Z_{2} ⊕ Z_{2} ⟶^{i_{4}} Z_{2}

satisfying i_{1} is injective, i_{2} = 0, i_{3} hits the sum of the generators, and i_{4} sends both generators nontrivially.

b. Let m be odd. Each group E_{2m}(4a + d) has an initial summand, the orders of which increase with d. These map injectively to one another, plus possibly also to a summand in the second component. In addition, there are summands as described below.

PERIODIC HOMOTOPY GROUPS OF SO(N) 23

If 2a < ν(m + 1) + 1, then the summands are

Z/2^{2a} ⟶^{i_{1}} Z/2^{2a} ⟶^{i_{2}} Z/2^{2a+1} ⟶^{i_{3}} Z/2^{2a+1} ⊕ Z/2^{e} ⟶^{i_{4}} Z/2^{2a+2}, (3.18)

satisfying i_{1} is bijective, i_{2} hits multiples of 4, i_{3} is bijective into the first summand, i_{4} sends the first summand to multiples of 4, and the second summand injectively.

If 2a > ν(m + 1) + 1, then, with ν = ν(m + 1) + 2, the summands are

Z/2^{ν} ⟶^{i_{1}} Z/2^{ν} ⟶^{i_{2}} Z/2^{ν} ⟶^{i_{3}} Z/2^{ν+ϵ} ⊕ Z/2^{e} ⟶^{i_{4}} Z/2^{ν},

where ϵ and e can be determined from Theorem 3.3. Then i_{1} is bijective, i_{2} hits multiples of 2, the first component of i_{3} is injective, and i_{4} sends the second summand injectively. If ν(m + 1) < ν(2a + 2), then ϵ = 0, e = ν, and i_{4} is multiplication by 2 on the first summand. If ν(m + 1) ≥ ν(2a + 2), then ϵ > 0, e = ν − 1, the second component of i_{3} is surjective, and i_{4} is multiplication by 2 on the first summand. Some more detailed information is given in Theorems 3.4 and 3.14.

If 2a = ν(m + 1) + 1, the groups and morphisms are as in (3.18) except that the last group is Z/2^{2a+1} and i_{4} sends the first summand to multiples of 2.

The anomalies in the 1-lines of Spin(6), Spin(9), and Spin(10) cause the following changes in the morphisms involving them. We will sketch portions of the proof in Section 4 along with the sketch for Proposition 3.17.

Proposition 3.19. Let m = 2k + 1 with k odd, and E_{2m}(n) = E_{2}^{1,2m+1}(Spin(n)). Then E_{2m}(5) → E_{2m}(6) and E_{2m}(9) → E_{2m}(10) are bijective, and

E_{2m}(6) ⟶^{j_{1}} E_{2m}(7) ⟶^{j_{2}} E_{2m}(8) ⟶^{j_{3}} E_{2m}(9) ⟶^{j_{4}} E_{2m}(11)

is

Z_{16} ⟶^{j_{1}} Z_{8}⊕Z_{8} ⟶^{j_{2}} Z_{8}⊕Z_{8}⊕Z_{8} ⟶^{j_{3}} Z_{8}⊕Z/2^{ν_{1}} ⟶^{j_{4}} \begin{cases}Z_{32} ⊕ Z_{16} & k ≡ 1 (4) \\ Z/2^{ν_{2}} ⊕ Z_{32} & k ≡ 3 (4) \end{cases}

where ν_{1} = min(8, ν(k − 3) + 3) and ν_{2} = min(8, ν(k − 19) + 3). We have j_{1}(g) = 2g_{1} + g_{2}, j_{2}(g_{1}) = g_{1}, j_{2}(g_{2}) = g_{2} + g_{3}, j_{3}(g_{1}) = g_{1}, and j_{3} sends the second and third summands injectively into the second summand plus an even component in the first summand. If k ≡ 1 mod 4, then j_{4}(g_{1}) = 4g_{1} + 4g_{2} and j_{4}(g_{2}) = 4g_{1} + 2g_{2}. If k ≡ 3 mod 4, j_{4} sends the first summand injectively into both summands, while the

24 BENDERSKY AND DAVIS

component of j_{4} from the second summand to the first (resp. second) summand has kernel consisting of elements of order 2 (resp. 8).

The eta towers which occur in big blocks (big •) in the BTSS charts of this section are described most explicitly in Table 6.1 using (5.7) and (5.8). We tabulate in Table 3.20 for a range of values of n the integers j for which x_{j} ∈ η_{i}(Spin(2n + 1)) = η_{i}(Spin(2n + 2)). These depend on just the parity of i. The BTSS charts show that elements in filtration 2, 3, and 4 in η_{ev}(Spin(N)) (resp. η_{od}(Spin(N))) survive to elements of v_{1}^{−1}π_{8k+r}(Spin(N)) for −1 ≤ r ≤ 1 (resp. −3 ≤ r ≤ −1).

Table 3.20.

n η_{ev}(Spin(2n + 1)) η_{od}(Spin(2n + 1))
11  5, 6, 8  8, 9, 10
12  6, 7, 8  8, 10, 11
13  6, 7, 8, 12  8, 10, 11, 12
14  7, 8, 10, 12  8, 10, 12, 13
15  7, 8, 10, 12  8, 12, 13, 14
16  8, 9, 10, 12  8, 12, 14, 15
17 8, 9, 10, 12, 16  8, 12, 14, 15, 16
18  9, 10, 12, 16  12, 14, 16, 17
19  9, 10, 12, 16  12, 16, 17, 18
20 10, 11, 12, 16  12, 16, 18, 19
21 10, 11, 12, 16  16, 18, 19, 20

From 5.14, we can easily verify the following proposition.

Proposition 3.21. Let b(j) = \begin{cases}2j + 3 & if j = 2^{i} or 3 · 2^{i} \\ 4j + 5 − 2^{ν(j)+3} & otherwise \end{cases} . Then

x_{j} ∈ η_{ev}(Spin(N)) iff b(j) ≤ N ≤ 4j + 4

and

x_{j} ∈ η_{od}(Spin(N)) iff 2j + 3 ≤ N ≤ \begin{cases}4j + 4 & if j = 2^{i} \\ 2j + 2 + 2^{ν(j)+2} & otherwise. \end{cases}

Explicit values of 1-line exponents for Spin(11) and Spin(13) were presented in [8, 1.6]. There was one mistake in that table. The formula for e_{1}(2n + 1) when n = 5 and m ≡ 7 mod 8 should have been min(8, ν(m − 39) + 2). The cause of this mistake was just overlooking one of the two numbers whose minimum equalled e_{1}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 25

In 3.1 and 3.3, we give formulas for the largest exponent, R(m, N), in E_{2}^{1,2m+1}(Spin(N)) when m is odd. In Table 3.22, we give explicit values for R(2k + 1, 2n − 1) for 8 ≤ n ≤ 13. We have verified that in this range R(m, 2n) = R(m, 2n − 1) when n is odd. It is proved in 3.4 and 3.14 that if n is even, then R(m, 2n) = R(m, 2n−1) unless ν(m+1−n) ≥ n−3 or ν(m+1) ≥ n−3, in which case R(m, 2n) = R(m, 2n−1)+1.

Table 3.22. R(2k + 1, 2n − 1)

n k ≡ 15 mod 16 n k ≡ 0 mod 4
8 max(7, 11 − ν(k + 1))  8 min(13, ν(k − 4) + 10)
9 max(8, 13 − ν(k + 1))  9 min(14, ν(k − 4) + 10)
10 max(9, 15 − ν(k + 1))  10 min(25, ν(k − 8 − 2^{10} − 2^{12}) + 12)
11 max(10, 17 − ν(k + 1))  11 min(26, ν(k − 8 − 2^{10}) + 13)
12 max(11, 19 − ν(k + 1))  12 min(26, ν(k − 8) + 17)
13 max(12, 21 − ν(k + 1))  13 min(26, ν(k − 8) + 17)
n  k ≡ 1 mod 8  n  k ≡ 5 mod 8
8 11  8 min(15, ν(k − 5) + 9)
9 11  9 min(15, ν(k − 5) + 9)
10 13  10 min(16, ν(k − 21) + 11)
11 min(24, ν(k − 9) + 15)  11 16
12 min(27, ν(k − 9 − 2^{10}) + 15)  12 17
13 min(28, ν(k − 9 − 2^{10} − 2^{11}) + 16) 13 18
n  k ≡ 2 mod 8  n  k ≡ 6 mod 8
8 12  8 min(18, ν(k − 6) + 10)
9 12  9 min(20, ν(k − 6 − 2^{9}) + 10)
10 14  10 min(21, ν(k − 6 − 2^{8}) + 12)
11 15  11 min(21, ν(k − 6) + 13)
12 min(29, ν(k − 10) + 18)  12 21
13 min(29, ν(k − 10) + 18)  13 21
n  k ≡ 3 mod 8  n  k ≡ 7 mod 16
8 9  8 8
9 10  9 min(18, ν(k − 7 − 2^{10}) + 7)
10 11  10 min(19, ν(k − 7 − 2^{9}) + 9)
11 14  11 min(20, ν(k − 7 − 2^{8}) + 11)
12 16  12 min(21, ν(k − 7 − 2^{7}) + 13)
13 min(30, ν(k − 11 − 2^{14}) + 16)  13 min(22, ν(k − 7 − 2^{6}) + 15)

26 BENDERSKY AND DAVIS

4. The 1-line of Spin(2n)

In this section we prove Theorem 3.3 regarding E_{2}^{1}(Spin(2n)). We begin with the following adaptation of [8, 3.4,3.10] to Spin(2n).

Proposition 4.1. The abelian group of indecomposables QK^{1}(Spin(2n)) has generators ξ_{i}, i ≥ 1, D_{+}, and D_{−}, and relations T_{n−1}, T_{n}, R_{n+1}, …, R_{2n−1}, S_{j}, j ≥ 2n defined by:

T_{n−1} : 2^{n−1}(D_{+} + D_{−}) + ∑_{\begin{matrix}j odd \\ j≥1 \end{matrix}} ∑_{k}(−1)^{k}\binom{2n}{n−j−k}ξ_{k};
T_{n} : ∑_{k}(−1)^{k}\binom{2n}{n−k}ξ_{k} + 2 ∑_{j≥1}(−1)^{j} ∑_{k}(−1)^{k}\binom{2n}{n−j−k}ξ_{k};
R_{j} : ∑_{k}(−1)^{k}\binom{2n}{j−k}ξ_{k} − ∑_{k}(−1)^{k}\binom{2n}{2n−j−k}ξ_{k};
S_{j} : ∑_{k}(−1)^{k}\binom{j}{k}ξ_{k}.

Adams operations satisfy ψ^{t}ξ_{k} = ξ_{kt} for t > 0, ψ^{−1}ξ_{k} = −ξ_{k}, and

ψ^{t}(D_{+} − D_{−}) = t^{n−1}(D_{+} − D_{−}).  (4.2)

Each relation with subscript j expresses ξ_{j} in terms of ξ_{i} with i < j and D_{±}. Thus QK^{1}(Spin(2n)) is a free abelian group with basis ξ_{1}, …, ξ_{n−2}, D_{+}, D_{−}. Formulas for ψ^{t}(D_{±}) will be obtained in the proof of 4.9.

Proof of Proposition 4.1. Naylor ([37, p.151]) describes the use of Hodgkin’s theorem ([29]) and the representation ring R(Spin(2n)) to determine QK^{1}(Spin(2n)). Unfortunately, his description contains many typographical errors. Husemoller ([30, p.189]) has correct versions of the results about R(Spin(2n)), and proofs.

Similarly to [8, 3.1], the morphism j^{∗} : R(SU(2n)) → R(Spin(2n)) satisfies

j^{∗}(μ_{i}) = j^{∗}(μ_{2n−i}),  (4.3)

where μ_{i} is the ith exterior power of the canonical representation. Then R(Spin(2n)) has fundamental representations j^{∗}μ_{1}, …, j^{∗}μ_{n−2}, Δ_{+}, Δ_{−} with relations

Δ_{+} ⊗ Δ_{−} = j^{∗}μ_{n−1} + j^{∗}μ_{n−3} + j^{∗}μ_{n−5} + ⋯ (4.4)
Δ_{+} ⊗ Δ_{+} + Δ_{−} ⊗ Δ_{−} = j^{∗}μ_{n} + 2(j^{∗}μ_{n−2} + j^{∗}μ_{n−4} + ⋯).(4.5)

PERIODIC HOMOTOPY GROUPS OF SO(N) 27

Hodgkin associates to each fundamental representation θ of G a primitive element β(θ) of K^{−1}(G). We denote by β̃(θ) the element of K^{1}(G) which corresponds to this under Bott periodicity. Then β̃(θ⊗τ) = dim(θ)β̃(τ)+dim(τ)β̃(θ). In QK^{1}(Spin(2n)), we let D_{±} = β̃(Δ_{±}) and B_{i} = β̃(j^{∗}μ_{i}). We obtain from (4.4) and (4.5),

2^{n−1}(D_{+} + D_{−}) = B_{n−1} + B_{n−3} + B_{n−5} + ⋯  (4.6)
2^{n}(D_{+} + D_{−}) = B_{n} + 2(B_{n−2} + B_{n−4} + ⋯).  (4.7)

Under the isomorphism QK^{1}(SU(2n)) ≈ K̃^{0}(CP^{2n−1}), let ξ′_{i} correspond to ξ^{i} − 1, with ξ the Hopf bundle, and let ξ_{i} = j^{∗}(ξ′_{i}) ∈ QK^{1}(Spin(2n)). From [8, 3.2], we have B_{j} = ∑(−1)^{k+1}\binom{2n}{j−k}ξ_{k}. Now T_{n−1} follows from (4.6), T_{n} is obtained from (4.7) and (4.6), R_{i} follows from (4.3), and S_{j} is a consequence of (ξ − 1)^{j} = 0 in K̃^{0}(CP^{2n−1}) for j ≥ 2n.

The formula for ψ^{t}ξ_{k} follows from ψ^{t}ξ = ξ^{t} in K̃^{0}(CP^{2n−1}). The formula for ψ^{−1}ξ_{k} follows from [8, 3.17]. The formula for ψ^{t}(D_{+} − D_{−}) follows from the short exact sequence

0 → QK^{1}(S^{2n−1}) ⟶^{p^{∗}} QK^{1}(Spin(2n)) ⟶^{i^{∗}} QK^{1}(Spin(2n − 1)) → 0

(4.8)

with p^{∗}(gen) = D_{+} − D_{−}.

From Proposition 4.1 and [8, 3.18], we deduce the following result.

28 BENDERSKY AND DAVIS

Proposition 4.9. The Pontryagin dual of the abelian group E_{2}^{1,2m+1}(Spin(2n)) is generated by ξ_{1}, D_{+}, and D_{−} subject to the following relations:

( ∑_{k odd} \binom{j}{k}k^{m})ξ_{1}, j ≥ 2n;  (4.10)
( ∑_{k odd}(\binom{2n}{j−k} − \binom{2n}{j+k})k^{m})ξ_{1}, n + 1 ≤ j ≤ 2n − 1;  (4.11)
(\frac{1}{n} ∑_{k odd} k^{m+1}\binom{2n}{n−k})ξ_{1};  (4.12)
(1 + (−1)^{m})ξ_{1};  (4.13)
2^{n−1}(D_{+} + D_{−}) − ( ∑_{\begin{matrix}k odd \\ j odd≥1 \end{matrix}} \binom{2n}{n−j−k}k^{m})ξ_{1};  (4.14)
2^{n−1}(D_{+} − D_{−});  (4.15)
(3^{n−1} − 3^{m})(D_{+} − D_{−});  (4.16)
( ∑_{k odd} k^{m} ∑_{t≥0} \binom{2n}{n−2−k−4t})ξ_{1} − 2^{n−1}D_{+};  (4.17)

−(2^{n−1} ∑_{k odd} k^{m} ∑_{t≥0} \binom{2n−1}{n−2−k−3t})ξ_{1} + (\frac{1}{6}(2^{2n−1} + 1 + 3^{n}) − 3^{m})D_{+}

+\frac{1}{6}(2^{2n−1} + 1 − 3^{n})D_{−};  (4.18)
(1 + (−1)^{m})D_{±} if n even;  (4.19)
D_{+} + (−1)^{m}D_{−} if n odd.  (4.20)

Proof. By [8, 1.1], E_{2}^{1,2m+1}(Spin(2n))^{#} is the quotient of QK^{1}(Spin(2n)) by the image of ψ^{2} and ψ^{r} − r^{m} for all odd r, although it suffices to consider just r = −1 and 3. As in [8, 3.15], ψ^{2}ξ_{m} = ξ_{2m} allows us to remove all ξ_{ev}, and ψ^{r}ξ_{1} = ξ_{r} allows us to equate all ξ_{r} with r odd to r^{m}ξ_{1}. This reduces the generating set to ξ_{1}, D_{+}, and D_{−}, and the relations of Proposition 4.1 become (4.10), (4.11), (4.12), and (4.14). For (4.12), we first obtain

( ∑_{k odd} k^{m}(\binom{2n}{n−k} + 2 ∑_{j≥1}(−1)^{j}\binom{2n}{n−j−k}))ξ_{1},

but then when each \binom{2n}{i} in this expression is expressed as \binom{2n−1}{i} + \binom{2n−1}{i−1}, all terms in the alternating sum cancel out except

∑_{k} k^{m}(\binom{2n−1}{n−k} − \binom{2n−1}{n−k−1}) = \frac{1}{n} ∑_{k} k^{m+1}\binom{2n}{n−k}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 29

The relation (4.13) is from ψ^{−1}ξ_{1}, and (4.15) and (4.16) are from ψ^{t}(D_{+} − D_{−}) as given in 4.1.

Suppose that we know ψ^{t}(D) = ∑ b_{j}ξ_{j} + cD in QK^{1}(Spin(2n − 1)). We claim that it follows that

ψ^{t}(D_{+} + D_{−}) = 2 ∑ b_{j}ξ_{j} + c(D_{+} + D_{−})

in QK^{1}(Spin(2n)). By adding and subtracting this with (4.2), we obtain

ψ^{t}(D_{±}) = ∑ b_{j}ξ_{j} + \frac{1}{2}(c + t^{n−1})D_{±} + \frac{1}{2}(c − t^{n−1})D_{∓}. (4.21)

Using this, (4.17) and (4.18) follow from [8, (3.20),(3.21)].

The above “claim” follows from the short exact sequence (4.8) in which i^{∗}(D_{±}) = D, once we observe that D_{+}−D_{−} cannot appear in ψ^{t}(D_{+}+D_{−}). This follows by working in R(Spin(2n)). Recall that ψ^{t} in QK^{1}(G) corresponds to (−1)^{t+1}λ^{t} in I/I^{2}, where I is the augmentation ideal of the representation ring. The representation ring of the maximal torus is Z[α_{1}^{±1/2}, …, α_{n}^{±1/2}], and here all the λ_{i}(ρ)’s and Δ_{+} + Δ_{−} are invariant under α ↦ α^{−1}, while Δ_{+} − Δ_{−} is not. (See [30].) So the exterior powers of these invariant classes will also be invariant, and hence cannot contain Δ_{+} − Δ_{−} as a summand.

Finally, (4.19) follows from ψ^{−1} = −1 in QK^{1}(Spin(2n)) if n is even, which follows from (4.8) and [8, 3.17], and (4.20) follows from ψ^{−1}(D_{±}) = −D_{∓} in QK^{1}(Spin(2n)) if n is odd, which is a consequence of (4.21) with c = −1.

Now we can prove Theorem 3.3.

Proof of Theorem 3.3. We begin by observing that all coefficients in relations (4.10) through (4.19) are even, with the single exception of (4.17) when n = 3, which accounts for the anomaly for Spin(6) mentioned following Theorem 3.3. Although sometimes a bit of argument is required, these all follow from the facts that \binom{a}{b} is even if a is even and b odd, that \binom{2a}{2b} ≡ \binom{a}{b} mod 2, that ∑_{i} \binom{n}{i} = 2^{n}, and that \binom{n}{i} = \binom{n}{n−i}.

For example, the coefficient of ξ_{1} in (4.17) is congruent to ∑_{k odd} ∑_{t≥0} \binom{2n}{n−2−k−4t}. If n is even, then all terms are even. If n = 2a + 1, then writing k = 2b + 1, the sum is congruent to ∑_{b≥0} ∑_{t≥0} c_{a,b,t} with c_{a,b,t} = \binom{2a+1}{a−1−b−2t}. If a is even, then

30 BENDERSKY AND DAVIS

c_{a,2b′,t} ≡ c_{a,2b′+1,t}, and hence the sum is even. If a is odd, then c_{a,2b′,t} ≡ c_{a,2b′−1,t} for b′ > 0, c_{2d+1,0,d−2A} ≡ c_{2d+1,0,d−2A−1}, and c_{2d+1,0,0} is even unless d = 0. This is the anomaly for Spin(6)—that \binom{2d}{d} is even unless d = 0.

Now the case m even and n even of 3.3, that E_{2}^{1,2m+1}(Spin(2n)) ≈ Z_{2} ⊕ Z_{2} ⊕ Z_{2}, with generators ξ_{1}, D_{+}, and D_{−}, is immediate, since (4.13) and (4.19) give these relations, and there are no relations involving odd multiples of these generators.

The case m even and n odd, that E_{2}^{1,2m+1}(Spin(2n)) ≈ Z_{2}⊕Z/2^{min(n−1,ν(m+1−n)+2)} with generators ξ_{1} and D_{+}, is also easy. We use here, and throughout, that

ν(3^{e} − 1) = \begin{cases}1 & if e odd \\ ν(e) + 2 & if e even. \end{cases}

The Z_{2} comes from (4.13). Replace 2ξ_{1} by 0, and use (4.20) to replace D_{−} by −D_{+}. The other relations reduce to 2^{n−1}D_{+} from (4.17) and −3^{n−1}(3^{m+1−n} − 1)D_{+} from (4.18).

When m is odd, (4.13) gives no information, and so the analysis becomes more complicated. From (4.10), we have 2^{eSp(m,n)}ξ_{1} = 0. We shall prove in Lemma 8.3 that (4.11) gives no additional information. For Spin(2n + 1), we proved the analogous result in [8, 3.6] using topology (two ways of computing E_{2}^{1,2m+1}(Sp(n))). For Spin(2n) it seems that we must resort to combinatorics.

We also need the following estimate, which we prove at the beginning of Section 8.

Lemma 4.22. The expressions of Theorem 3.3 satisfy, if m is odd and n ≥ 6, min(eSp(m, n), ν(P_{1}(m, n)), ν(P_{3}(m, n)) ≥ n.

We now obtain 3.3 in the case when m is odd and n is odd. The generators will be ξ_{1} and D_{+} + cξ_{1} for appropriate c. The relations (4.10) and (4.11) give 2^{eSp(m,n)}ξ_{1}, as just explained, while (4.12) gives 2^{ν(P_{3}(m,n))}ξ_{1}. The relation (4.20) allows us to replace D_{−} by D_{+}. The other relations become

2^{n}D_{+} − Y_{1}ξ_{1}  (4.23)
Y_{2}ξ_{1} − 2^{n−1}D_{+}  (4.24)
−3 · 2^{n−1}Y_{3}ξ_{1} + (2^{2n−1} + 1 − 3^{m+1})D_{+},  (4.25)

where Y_{1}, Y_{2}, and Y_{3} refer to the sums (just the ∑-part) in (4.14), (4.17), and (4.18). Replacing (4.23) by (4.23)+2(4.24) yields P_{1}(m, n)ξ_{1}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 31

It was proved in [8, 3.18] that^{5}

ν(Y_{2}) ≥ n for n ≥ 6.  (4.26)

Thus the smallest 2-exponent in (4.24) and (4.25) is M := min(n − 1, ν(m + 1) + 2), and by 4.22, this is smaller than the exponents which have occurred earlier in the analysis. The summand with generator D_{+} + cξ_{1} is obtained by dividing whichever of (4.24) or (4.25) has the smallest exponent by 2^{M}, so that its coefficient of D_{+} is odd. Subtracting an appropriate multiple of this relation from the other yields the final relation, P_{2}(m, n)ξ_{1}.

Finally, we consider the case m odd and n even. We obtain 2^{eSp(m,n)}ξ_{1} and 2^{ν(P_{3}(m,n))}ξ_{1} as in the previous case. Let M = min(n − 1, ν(m + 1 − n) + 2), and let Y_{1}, Y_{2}, and Y_{3} be as above. By (8.2), ν(Y_{1}) ≥ n, while Y_{3} is odd by 8.11. We also use (4.26). The relations are

2^{n−1}(D_{+} + D_{−}) − Y_{1}ξ_{1},  (4.27)
2^{M}(D_{+} − D_{−}),  (4.28)
Y_{2}ξ_{1} − 2^{n−1}D_{+},  (4.29)
−3 · 2^{n−1}Y_{3}ξ_{1} + (2^{2n−2} + \frac{1}{2}(3^{n} − 1) − (3^{m+1} − 1))D_{+}
+(2^{2n−2} − \frac{1}{2}(3^{n} − 1))D_{−}.  (4.30)

Replace (4.27) by (4.27)+2^{n−1−M}(4.28)+2(4.29) to get P_{1}(m, n)ξ_{1}. Now we divide into subcases.

If ν(m + 1) < ν(n), the smallest 2-exponent in any coefficient is ν(m + 1) + 2 in (4.28). This gives a summand Z/2^{ν(m+1)+2} generated by D_{+} −D_{−}. Use this to replace 2^{ν(m+1)+2}D_{−} by 2^{ν(m+1)+2}D_{+} in (4.30), which becomes

− 3 · 2^{n−1}Y_{3}ξ_{1} + (2^{2n−1} − 3^{m+1} + 1)D_{+}.  (4.31)

We get a second Z/2^{ν(m+1)+2} summand from (4.31)/2^{ν(m+1)+2}. Add a multiple of (4.31) to (4.29) to get the final relation, P_{2}(m, n)ξ_{1}. The generators of the respective summands in 3.3 are ξ_{1}, D_{+} + cξ_{1}, and D_{+} − D_{−}.

If ν(m + 1) ≥ ν(n), the smallest 2-exponent in any coefficient is ν(n) + 1 in D_{−} (or D_{+}) in (4.30). We get a summand Z/2^{ν(n)+1} with generator (4.30)/2^{ν(n)+1}. Replace

5There it was stated and proved that ν(Y_{2}) ≥ n − 1, but the same argument establishes this stronger result, which we will need.

32 BENDERSKY AND DAVIS

(4.28) by

(2^{2n−2} − \frac{1}{2}(3^{n} − 1))2^{−ν(n)−1}(4.28) + 2^{M−ν(n)−1}(4.30), obtaining

− 3 · 2^{n−1+M−ν(n)−1}Y_{3}ξ_{1} + 2^{M−ν(n)−1}(2^{2n−1} − 3^{m+1} + 1)D_{+}.(4.32)

The smallest 2-exponent in the remaining coefficients ((4.29) or (4.32)) is

min(ν(m + 1 − n) + 3, n − 1) if ν(m + 1) = ν(n)
min(ν(m + 1) + 3, n − 1)  if ν(m + 1) > ν(n),

with the ν(−) + 3 coming from D_{+} in (4.32), and n − 1 coming from D_{+} in (4.29). We get a summand of this 2-exponent generated by the relevant relation divided by its 2-power. Add a multiple of this relation to the other to get P′_{2}(m, n)ξ_{1}. The respective generators in 3.3 in this case are ξ_{1}, D_{+} + cξ_{1}, and D_{−} + uD_{+} + c′ξ_{1}.

That R(m, 2n) = n − 1 when n − 1 ≤ ν(m + 1) + 2 and n is odd follows as in Remark 3.2. If n is even and ν(m + 1) ≥ n − 3, then M = ν(n) + 2 and in the above argument the relation P′_{2}(m, n) has its 2-divisibility determined by 3 · 2^{n}Y_{3}, which equals a unit times 2^{n} by 8.11.

Here is a sketch of proof that was postponed in Section 3.

Proof of Proposition 3.17. These groups are Pontryagin dual to the groups that we computed as quotients of the K-groups. For Spin(2b) ⟶^{j_{0}} Spin(2b + 1) ⟶^{j_{1}} Spin(2b + 2), we have j_{1}^{∗}(D_{±}) = D and j_{0}^{∗}(D) = D_{+} + D_{−}. We focus on the hardest case, i_{4} when 2a > ν(m + 1) ≥ ν(n). Here n = 2a + 2.

We have ν = ν(m + 1) + 2, e = ν(n) + 1, and ϵ = ν(m + 1 − n) + 1 − ν(n). The image of the generator of Z/2^{ν} under i_{4}^{#} is

D_{+} + D_{−} + cξ_{1} ∈ Z/2^{ν+ϵ} ⊕ Z/2^{e},

where the respective generators of these summands are g_{1} = D_{+} + c′ξ_{1} and g_{2} = D_{+} −D_{−} +2αD_{+} +c′′ξ_{1} with α = (2^{2n−1} −3^{m+1} +1)/(−2^{2n−1} +3^{n} −1). All coefficients of ξ_{1} are sufficiently 2-divisible that they can be ignored. We obtain

i_{4}^{#}(gen) = −g_{2} + (2α + 2)g_{1}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 33

Dualizing, this says that the second summand injects under i_{4}, while the kernel of the morphism from the first summand has 2-exponent 1 + ν(α + 1) = 2 + ν(m + 1 − n) − ν(n) = ϵ + 1 and hence this morphism maps onto multiples of 2.

The cases of Spin(6) and Spin(10) were omitted from 3.3 because of arithmetic anomalies. We handle those cases now.

Proposition 4.33.

E_{2}^{1,2m+1}(Spin(6)) ≈ \begin{cases}Z/2^{4} & if m ≡ 3 mod 4 \\ Z/2^{3} & if m ≢ 3 mod 4. \end{cases}

E_{2}^{1,2m+1}(Spin(10)) ≈ \begin{cases}Z_{2} ⊕ Z/2^{min(4,ν(m−4)+2)} & if m even \\ Z/8 ⊕ Z/2^{min(ν(m−5)+2,6)} & if m ≡ 1 mod 4 \\ Z/8 ⊕ Z/2^{min(ν(m−7)+2,8)} & if m ≡ 3 mod 4. \end{cases}

Proof. We first consider Spin(6). Proposition 4.9 is still valid. As remarked previously, the subsequent argument fails because the coefficient of ξ_{1} in (4.17) is odd when n = 3. Use (4.17) to replace ξ_{1} by 4D_{+}, and use (4.20) to replace D_{−} by (−1)^{m+1}D_{+}. The smallest resulting relation on D_{+} is 8D_{+} from (4.13) if m is even, while if m is odd, we have 16D_{+} from (4.14) and \frac{1}{3}(3^{m+1} − 1 + 16)D_{+} from (4.18), yielding the result.

For Spin(10), the case m even follows as in Theorem 3.3. The anomaly is the same as occurred for Spin(9) in [8, 4.21]. Here it occurs as (4.26), where Y_{2} = 45. Since (4.20) makes D_{+} = D_{−}, the relations become the same as in Spin(9).

We close the section with another postponed proof.

Proof of Proposition 3.19. We compute the duals of the morphisms, using 4.33, 3.1, and 3.3 for the groups. We use ξ_{1} and D as the generators of E_{2m}(7)^{#}. They map to ξ_{1} and D_{+} + D_{−} in E_{2m}(6)^{#}, which is generated by D_{+} and has relations ξ_{1} = 4D_{+} and D_{−} = D_{+}.

With u = (3^{m} − 3)/8, we use ξ_{1} and D + (8 + 3u)ξ_{1} as generators of E_{2m}(11)^{#} and (6+u)ξ_{1}+2D and D as generators of E_{2m}(9)^{#}. The morphism E_{2m}(11)^{#} ⟶^{j_{4}^{#}} E_{2m}(9)^{#} is easily determined using j_{4}^{#}(D) = 2D. At the end, we need ν(u + 1) = 1.

34 BENDERSKY AND DAVIS

5. Eta towers

In this section, we compute the groups E_{2}^{s,t}(Spin(n)) for s > 2 (and s = 2 if s + n is even). These are called eta-towers because of the isomorphism, E_{2}^{s,t} ⟶^{h_{1}} E_{2}^{s+1,t+2}, which on classes which survive to homotopy is related to composition with the Hopf map S^{t−s+1} ⟶^{η} S^{t−s}. Notation for the eta-towers was initiated in 1.5. We begin with the closely-related computation for Sp(n).

We first recall the following key results from [9, 1.1,3.1].

Theorem 5.1. ([9]) a. If X is a simply-connected finite H-space with H_{∗}(X; Q) associative, then the E_{2} term of its BTSS satisfies

E_{2}^{s,t}(X) ≈ \begin{cases}Ext_{A}^{s,t}(QK^{1}(X; Z_{2}^{∧})/ im(ψ^{2})) & if t is odd \\ 0 & if t is even. \end{cases}

b. Let M be a finite stable 2-adic Adams module with ψ^{−1} = −1 and let M_{2} = ker(2|M) and θ = ψ^{3} − 1. Then there are split short exact sequences

0 → coker(θ|M/2) → Ext_{A}^{s,2b+1}(M)^{#} → ker(θ|M_{2}) → 0 if s + b is odd and s > 2, and

0 → coker(θ|M_{2}) → Ext_{A}^{s,2b+1}(M)^{#} → ker(θ|M/2) → 0 if s + b is even and s > 1.

These hypotheses apply to all cases considered here except Spin(4a + 2), where ψ^{−1} ≠ −1; in (5.17) we will present a modified version of 5.1b which will apply in that case.

The following result from [9, 3.1] is also useful.

Proposition 5.2. If M is a finite stable 2-adic Adams module with ψ^{−1} = −1 and n is odd, there is a split short exact sequence 0 → coker(ψ^{3} − 1|M/2) → Ext_{A}^{2,2n+1}(M)^{#} → ker((ψ^{3} − 3^{n})|M) → 0. Also, |ker((ψ^{3} − 3^{n})|M)| = | Ext_{A}^{1,2n+1}(M)|.

An analogue when it is not true that ψ^{−1} = −1 is given in [9, 3.10], and is similar to (5.17).

PERIODIC HOMOTOPY GROUPS OF SO(N) 35

Because of the following elementary proposition, the functors of Theorem 5.1.b depend only on QK^{1}(X; Z/2).

Proposition 5.3. Let Q be a torsion-free 2-adic Adams module with ψ^{2} injective (viz. Q = QK^{1}(X; Z_{2}^{∧}) with X as above), and let M = Q/ im(ψ^{2}). There is an isomorphism of stable Adams modules

\frac{1}{2}ψ^{2} : ker(ψ^{2}|Q/2) → M_{2}.

Proof. Let K = ker(ψ^{2}|Q/2), and apply the Snake Lemma to the commutative diagram

0 ⟶ K
↓  ↓
0 ⟶ Q ⟶^{2} Q ⟶ Q/2 ⟶ 0
ψ^{2}↓  ^{ψ^{2}}↓  ^{ψ^{2}}↓
0 ⟶ Q ⟶^{2} Q ⟶ Q/2 ⟶ 0
↓  ↓
M ⟶^{2} M

We have the following result for QK^{1}(Sp(n); Z/2).

Proposition 5.4. Let ξ_{i} = ξ^{i} − 1 be generators for QK^{1}(Sp(n); Z/2) as used in [8, 3.4]. Let

x_{i} = ∑_{j≥0} \binom{i}{j}ξ_{i−2j}.

Then {x_{1}, …, x_{n}} is a basis for QK^{1}(Sp(n); Z/2) which satisfies

• If i : Sp(n−1) → Sp(n) denotes the inclusion map, then i^{∗}(x_{n}) = 0 and i^{∗}(x_{j}) = x_{j} for j < n.

• ψ^{2}(x_{i}) = x_{2i}, and is 0 if 2i > n.

• ψ^{3}(x_{i}) = ∑_{j≥0} \binom{i}{j}x_{i+2j}.

36 BENDERSKY AND DAVIS

Proof. Since it was shown in [8, 3.4] that {ξ_{1}, …, ξ_{n}} forms a basis, it is immediate that {x_{1}, …, x_{n}} does as well. To prove the result about i^{∗}(x_{n}), we observe from [8, 3.4] that

i^{∗}(ξ_{n}) = ∑_{k<n}(\binom{2n−1}{n−k} + \binom{2n−1}{n−1−k})ξ_{k} = ∑_{k<n} \binom{2n}{n−k}ξ_{k}.

Thus

i^{∗}(x_{n}) = i^{∗}(ξ_{n}) + ∑_{j>0} \binom{n}{j}ξ_{n−2j} = ∑_{j>0} \binom{2n}{n−(n−2j)}ξ_{n−2j} + ∑_{j>0} \binom{n}{j}ξ_{n−2j} = 0.

Since ψ^{2}(ξ_{i}) = ξ_{2i}, we have

ψ^{2}(x_{i}) = ∑_{j≥0} \binom{i}{j}ξ_{2i−4j} = ∑_{j≥0} \binom{2i}{2j}ξ_{2i−4j} = ∑_{k≥0} \binom{2i}{k}ξ_{2i−2k} = x_{2i}.

Here we have used several elementary facts about binomial coefficients mod 2. The proof of the ψ^{3} formula involves a more elaborate combinatorial argument, which is relegated to Proposition 8.6.

Remark 5.5. Because of the elaborate computer-dependent proof of the ψ^{3}-formula, some may prefer the following simpler argument. This argument just proves

ψ^{3}(x_{i}) ≡ x_{i} + x_{i+2^{ν(i)+1}} mod ⟨x_{j} : j > i + 2^{ν(i)+1}⟩,

but that is all that we really need.

First, by using ψ^{2}ψ^{3} = ψ^{3}ψ^{2} and the formula for ψ^{2}(x_{i}), it suffices to prove that if i is odd, then ψ^{3}(x_{n}) = x_{n} + x_{n+2} in QK^{1}(Sp(n + 2)/Sp(n − 1)); Z/2). The definition of x_{i} and formula for ψ^{3}(ξ_{i}) easily imply that ψ^{3}x_{i} can only involve x_{j} with j ≡ i mod 2. Thus, if the claimed formula is not true, then ψ^{3} −1 = 0 in QK^{1}(Sp(n+2)/Sp(n− 1)); Z/2). It was shown in [11, 2.7] that ψ^{3} − 1 ≠ 0 in ku_{∗}(Q_{n}^{n+2}) ⊗ Z/2, hence in P K_{∗}(Sp(n + 2)/Sp(n − 1); Z/2), and hence in QK^{1}(Sp(n + 2)/Sp(n − 1); Z/2) by duality.

Using 5.4, we easily derive the following description of the eta-towers for Sp(n).

Proposition 5.6. For X = Sp(n), the split SES’s of 5.1b become 0 → ⟨x_{1}⟩ → E_{2}^{s,2b+1}(Sp(n))^{#} → K[[\frac{n}{2}] + 1, n] → 0

PERIODIC HOMOTOPY GROUPS OF SO(N) 37

if s + b is odd, and

0 → C[[\frac{n}{2}] + 1, n] → E_{2}^{s,2b+1}(Sp(n))^{#} → ⟨x_{n^{∗}}⟩ → 0

if s + b is even. Here n^{∗} is the largest odd integer satisfying n^{∗} ≤ n,

K[a, b] = ⟨x_{i} : a ≤ i ≤ b, i + 2^{ν(i)+1} > b⟩,  (5.7)

and

C[a, b] = ⟨x_{i} : a ≤ i ≤ b, i − 2^{ν(i)+1} < a⟩.  (5.8)

Proof. With M = QK^{1}(Sp(n); Z/2)/ im(ψ^{2}) in 5.1b, we have, using 5.4, M/2 = ⟨x_{i} : i odd, 1 ≤ i ≤ n⟩ and θ(x_{i}) ≡ x_{i+2} mod H (which we will use to mean mod terms with larger subscripts). Thus coker(θ|M/2) ≈ ⟨x_{1}⟩ and ker(θ|M/2) = ⟨x_{n^{∗}}⟩.

Let S_{n} = \{i : [\frac{n}{2}] + 1 ≤ i ≤ n\} and, for e ≥ 0, let S_{n}(e) = \{i ∈ S_{n} : ν(i) = e\}. By 5.3, M_{2} ≈ ⟨x_{i} : i ∈ S_{n}⟩. Let M_{2}(e) = ⟨x_{i} : i ∈ S_{n}(e)⟩. Then θ induces automorphisms of each M_{2}(e) given by θ(x_{2^{e}u}) ≡ x_{2^{e}(u+2)} mod H, with u odd, and so, similarly to the previous paragraph

ker(θ|M_{2}) ≈ ⟨x_{i} : i maximal in some S_{n}(e)⟩

and

coker(θ|M_{2}) ≈ ⟨x_{i} : i minimal in some S_{n}(e)⟩.

This result is, for all intents and purposes, dual to [11, 1.8]. We illustrate with the case n = 10, which is depicted in [11, 1.13]. Columns 2 and 4 (resp. 3) in [11, 1.13] correspond to E_{2}^{s,2b+1}(Sp(10)) with s + b odd (resp. even), as can be seen by comparison with [11, 1.7]. The boundary pattern from u to (u − 2)′ in [11, 1.13] is dual to our formula for θ mod H in M/2, and the elements 1 and 9′ that survive correspond to our ⟨x_{1}⟩ and ⟨x_{n^{∗}}⟩. Note that the computations should be dual, since [11] is depicting E_{2}, while we are computing E_{2}^{#}.

The comparison of our θ|M_{2} with the boundaries in [11, 1.13] is complicated slightly by different ways of filtering elements in the two approaches. The two approaches would agree if the elements in 5_{u} and 6′ in [11, 1.13] were interchanged, and also the

38 BENDERSKY AND DAVIS

elements 7_{u} and 10′. If this change were made, then the bottom part of columns 2 and 3 of [11, 1.13] would be

6′_{u} ← 10_{u}
7′_{u} ← 9_{u}
8′_{u}  6_{u}
9′_{u}  7_{u}
10′_{u}  8_{u},

which is dual to our θ|M_{2}.

As in [11, 1.14], we conclude that the number of eta-towers in E_{2}^{s,2b+1}(Sp(n)) is 1 + [log_{2}(4n/3)], for either parity of s + b.

Now we perform a similar analysis for Spin(2n + 1).

Proposition 5.9. QK^{1}(Spin(2n + 1); Z/2) has basis {x_{1}, …, x_{n−1}, D}, with ψ^{2}(x_{i}) and ψ^{3}(x_{i}) as in QK^{1}(Sp(n − 1); Z/2), ψ^{2}(D) = x_{n−1}, and ψ^{3}(D) = D.

Proof. By [8, pf of 3.1], there is an Adams-module morphism

QK^{1}(Sp(n)) ⟶^{ϕ} QK^{1}(Spin(2n + 1)),

and by [8, 3.10] QK^{1}(Spin(2n + 1)) has integral basis {ξ_{1}, …, ξ_{n−1}, D}, where ξ_{i} = ϕ(ξ_{i}) and

2^{n+1}D = (−1)^{n+1}ϕ(ξ_{n}) + ∑_{k=1}^{n−1}(−1)^{k+1}ξ_{k} ∑_{j=0}^{n−k} \binom{2n+1}{j}. (5.10)

Reduce mod 2 and change to the basis {x_{i}} of 5.4. The mod 2 reduction of (5.10) is

0 = ϕ(ξ_{n}) + ∑_{j>0} ξ_{n−2j}\binom{n}{j};

i.e., ϕ(x_{n}) = 0. Thus if, for i < n, ψ^{k}(x_{i}) = ∑_{j=1}^{n} α_{j}x_{j} in QK^{1}(Sp(n); Z/2), where coefficients α_{j} are as in 5.4, then, applying ϕ, we obtain ψ^{k}(x_{i}) = ∑_{j=1}^{n−1} α_{j}x_{j} in QK^{1}(Spin(2n + 1); Z/2), which is exactly the formula in QK^{1}(Sp(n − 1); Z/2).

By [8, (3.20)] we obtain the integral formula

ψ^{2}(D) = ∑_{k<n}(−1)^{k}ξ_{k} ∑_{t≥0} \binom{2n+2}{n−1−k−4t} + 2^{n}D.  (5.11)

PERIODIC HOMOTOPY GROUPS OF SO(N) 39

Reduction mod 2 yields

ψ^{2}(D) ≡ ∑_{i≥0} ξ_{n−1−2i} ∑_{t≥0} \binom{n+1}{i−2t} ≡ ∑_{i≥0} ξ_{n−1−2i}\binom{n−1}{i} = x_{n−1}. (5.12)

The middle congruence here is obtained by consideration of (1 + x)^{−2}(1 + x)^{n+1} = (1 + x)^{n−1}.

An equation which appears in [8] just after (4.10) says that

ψ^{3}(D) = \frac{1}{3}(2^{2n+1} + 1)D + 2^{n} ∑_{k=1}^{n−1}(−1)^{k}ξ_{k} ∑_{t≥0} \binom{2n+1}{n−1−k−3t}, (5.13)

which reduces mod 2 to ψ^{3}(D) ≡ D.

From this we obtain the following description of the eta-towers for Spin(2n + 1).

Proposition 5.14. For X = Spin(2n + 1), the split SES’s of 5.1b become

0 → ⟨x_{1}, D⟩ → E_{2}^{s,2b+1}(Spin(2n + 1))^{#} → K[[\frac{n}{2}], n − 1] → 0 if s + b is odd, and

0 → C[[\frac{n}{2}], n − 1] → E_{2}^{s,2b+1}(Spin(2n + 1))^{#} → ⟨x_{n^{∗∗}}, D⟩ → 0 if s + b is even. Here n^{∗∗} is the largest odd integer satisfying n^{∗∗} < n − 1, while K[[\frac{n}{2}], n − 1] and C[[\frac{n}{2}], n − 1] are as in (5.7) and (5.8). If n = 2a + 1 is odd, then x_{a} should be replaced by x_{a} + D in C[[\frac{n}{2}], n − 1]. If n = 2^{e} + 1, then x_{a} should be replaced by x_{a} + D in K[[\frac{n}{2}], n − 1].

The number of eta-towers in E_{2}^{s,2b+1}(Spin(2n+1)) is 2+[log_{2}(4(n−1)/3)]+δ_{α(n−1),1}, where δ is the Kronecker delta and α(m) denotes the number of 1’s in the binary expansion of m.

Proof. With M = QK^{1}(Spin(2n+1); Z/2)/ im(ψ^{2}) in 5.1b, we have, using 5.9, M/2 = ⟨D, x_{i} : i odd , 1 ≤ i < n − 1⟩ and θ(x_{i}) = x_{i+2} and θ(D) = 0. Thus coker(θ|M/2) ≈ ⟨x_{1}, D⟩ and ker(θ|M/2) = ⟨x_{n^{∗∗}}, D⟩.

Let S_{n} = \{i : [\frac{n}{2}] ≤ i ≤ n − 1\} and, for e ≥ 0, let S_{n}(e) = \{i ∈ S_{n} : ν(i) = e\}. By 5.3, M_{2} ≈ ⟨x′_{i} : i ∈ S_{n}⟩, where x′_{i} = x_{i} unless n = 2a + 1 and i = a, in which case x′_{i} = x_{i} + D. Let M_{2}(e) = ⟨x′_{i} : i ∈ S_{n}(e)⟩. Then θ induces automorphisms of each M_{2}(e) given by θ(x′_{2^{e}u}) = x′_{2^{e}(u+2)} mod H, and so the result follows similarly to

40 BENDERSKY AND DAVIS

the proof of 5.6. If n = 2a + 1, then a is maximal in some S_{n}(e) if and only if a is a 2-power.

One way to make the asserted count of the number of eta-towers is by comparison with Sp(n − 1). The number of eta-towers is 2 plus the number of values of e which occur as ν(i) for some i ∈ [[\frac{n}{2}], n − 1], whereas it was shown in [11, 1.14] that the number of values of ν(i) in [[\frac{n−1}{2}] + 1, n − 1] is [log_{2}(4(n − 1)/3)]. The two intervals are the same if n is even, while if n = 2a + 1, the interval considered here contains a as an additional element. This a will give an additional value of ν(−) if and only if it is a 2-power.

The above result can also be deduced from 11.3.

The following result will be useful for Spin(4a), in which ψ^{−1} = −1, but less useful for Spin(4a + 2).

Proposition 5.15. QK^{1}(Spin(2n); Z/2) has basis {x_{1}, …, x_{n−2}, D_{+}, D_{−}} with ψ^{2}x_{i}

and ψ^{3}x_{i} as in Sp(n − 2), ψ^{2}(D_{±}) = x_{n−2}, and ψ^{3}(D_{±}) = \begin{cases}D_{±} & if n even \\ D_{∓} & if n odd. \end{cases}

Proof. The group and much of the information about ψ^{k} follows from (4.8) and 5.9. The observation in the proof of 4.9 about invariants implies that D_{+} − D_{−} cannot appear in ψ^{k}(x_{i}). The formula for ψ^{k}(D_{±}) follows from (4.21), (5.12), and (5.13).

The following result together with 5.14 gives the eta-towers for Spin(4a).

Proposition 5.16. If n is even, and s > 2 (or s = 2 and b even), then there is a short exact sequence 0 → Z_{2} ⊕ Z_{2} → E_{2}^{s,2b+1}(Spin(2n))^{#} → E_{2}^{s,2b+1}(Spin(2n − 1))^{#} → 0, where the two classes in the kernel are both D_{+} + D_{−}, one in the ker-part and one in the coker-part in 5.1b.

PERIODIC HOMOTOPY GROUPS OF SO(N) 41

Proof. One way of interpreting this result is to first note that the SES (4.8) remains short exact after modding out by im(ψ^{2}), and hence, by 5.1a, induces an exact sequence

→ E_{2}^{s,2b+1}(S^{2n−1})^{#} → E_{2}^{s,2b+1}(Spin(2n))^{#} → E_{2}^{s,2b+1}(Spin(2n − 1))^{#}
⟶^{δ} E_{2}^{s+1,2b+1}(S^{2n−1})^{#} → .

This exactness also follows from 2.2 and 2.3. We are, in effect, claiming that δ = 0 when n is even, so that this sequence is short exact.

The direct computation for Spin(2n) is extremely similar to the proof of 5.14, performed for Spin(2n−1). The M/2-part, comprising the first paragraph, is changed only by replacing D by both D_{+} and D_{−}. The M_{2}-part, comprising the second paragraph, has an extra D_{+} + D_{−} in M_{2}, and it appears in both ker(θ) and coker(θ).

For Spin(2n) with n odd, ψ^{−1} ≠ −1 and so 5.1b does not apply. The generalization is given by [9, 3.8], which states that there is a short exact sequence

0 → coker(θ_{b}|Q_{s+b}) → E_{2}^{s,2b+1}(Spin(2n))^{#} → ker(θ_{b}|Q_{s+b−1}) → 0,

(5.17)

where

Q_{m} = \frac{ker((1 − (−1)^{m}ψ^{−1})|M)}{im((1 + (−1)^{m}ψ^{−1})|M)}

with M = QK^{1}(Spin(2n))/ im(ψ^{2}), and θ_{b} = ψ^{3} − 3^{b}. For this, we need more than just mod-2 K-theory.

We use integral classes \overline{x}_{i} which reduce mod 2 to the classes x_{i} of Proposition 5.4, and satisfy the same restriction formula as x_{i}. By [8, 3.4], these classes must be defined by

\overline{x}_{i} = ∑_{j=0}^{i−1}(−1)^{j}ξ_{i−j}(\binom{2i−1}{j} − \binom{2i−1}{j−1}).  (5.18)

Proposition 5.19. If n is odd, QK^{1}(Spin(2n)) has basis \{\overline{x}_{1}, …, \overline{x}_{n−2}, D_{+}, D_{−}\} with ψ^{−1}(\overline{x}_{i}) = −\overline{x}_{i}, ψ^{−1}(D_{±}) = −D_{∓},

ψ^{2}(\overline{x}_{i}) = ∑_{j=1}^{n−2} α_{i,j}\overline{x}_{j} + β_{j}2^{n−1}(D_{+} + D_{−}) with α_{i,j} \begin{cases}odd & j = 2i \\ even & j ≠ 2i \end{cases}
ψ^{2}(D_{+}) = ∑_{j=1}^{n−2} γ_{i,j}\overline{x}_{j} + 2^{n−1}D_{+} with γ_{i,j} \begin{cases}= −1 & j = n − 2 \\ even & j ≠ n − 2 \end{cases}
ψ^{2}(D_{+} − D_{−}) = 2^{n−1}(D_{+} − D_{−}).  (5.20)

Proof. The basis was derived in 4.1 and 4.9, and (5.20) is just (4.2). The mod 2 reduction of the coefficients is immediate from 5.15. That the D_{±}-part of ψ^{2}(\overline{x}_{i}) involves only D_{+} + D_{−} is in the proof of 4.9. That the coefficient of D_{+} + D_{−} in ψ^{2}(\overline{x}_{i}) is divisible by 2^{n−1} is a consequence of (4.8) and the fact that the coefficient of D in ψ^{2}(ξ_{i}) in QK^{1}(Spin(2n − 1)) is divisible by 2^{n}. This “fact” follows from [8, 3.10], which says that in QK^{1}(Spin(2n − 1)) we have

ξ_{n−1} = (−2)^{n}D + ∑_{j=1}^{n−2} c_{j}ξ_{j}.  (5.21)

The algorithm for ψ^{2}ξ_{i} begins by expressing it as ξ_{2i}, and if 2i > n − 1, then relations identical to S_{j} of Proposition 4.1 are used to express each ξ_{j} in terms of lower ξ’s until it gets down to ξ_{1}, …, ξ_{n−1}, and then (5.21) is used to eliminate ξ_{n−1}, obtaining a coefficient of D divisible by 2^{n}.

That the coefficients of D_{+} and D_{−} in ψ^{2}(D_{+}) are 2^{n−1} and 0, respectively, was derived in (4.17). Finally, that the coefficient of \overline{x}_{n−2} is −1 follows from (5.11) and the argument in the proof of 4.9.

Now we can obtain our final result enumerating eta-towers.

Proposition 5.22. If n is odd and s > 2, then the morphism

E_{2}^{s,2b+1}(Spin(2n))^{#} ⟶^{i_{∗}^{#}} E_{2}^{s,2b+1}(Spin(2n − 1))^{#}

satisfies

ker(i_{∗}^{#}) = \begin{cases}⟨x_{n−2} ∼ 2^{n−1}D_{+}⟩ & if s + b even \\ 0 & if s + b odd \end{cases}

PERIODIC HOMOTOPY GROUPS OF SO(N) 43

and

coker(i_{∗}^{#}) ≈ \begin{cases}⟨x_{n−4}, D⟩ & if s + b even \\ ⟨D⟩ & if s + b odd. \end{cases}

Thus, when n is odd, the number of eta-towers in Spin(2n) is 1 less than that in Spin(2n − 1), which was determined in 5.14.

Proof. We use (5.17) and begin by determining the groups Q_{m}. We obtain that Q_{od} has generators \overline{x}_{i} for 1 ≤ i ≤ n − 2, D_{+} + D_{−}, and 2^{n−1}D_{+} with relations 2\overline{x}_{i} if i is odd, \overline{x}_{i} if i is even, D_{+} + D_{−}, and \overline{x}_{n−2} + 2^{n−1}D_{+}. The quotient Q_{od} is a Z_{2}-vector space with basis \{x_{i} : i odd, 1 ≤ i ≤ n − 4, \overline{x}_{n−2} ∼ 2^{n−1}D_{+}\}. Similarly, we find that Q_{ev} is a Z_{2}-vector space with basis \{\frac{1}{2}ψ^{2}(\overline{x}_{i}) : [\frac{n}{2}] ≤ i ≤ n − 2\}. We use 5.3 to think of this ψ^{3}-module as ⟨x_{[n/2]}, …, x_{n−2}⟩.

Similarly to our previous cases, θ = ψ^{3} − 1 satisfies θ(\overline{x}_{2^{e}u}) ≡ \overline{x}_{2^{e}(u+2)} mod H, if u is odd. Thus (5.17) becomes

0 → ⟨x_{1}⟩ → E_{2}^{s,2b+1}(Spin(2n))^{#} → K[[\frac{n}{2}], n − 2] → 0 if s + b is odd, and

0 → C[[\frac{n}{2}], n − 2] → E_{2}^{s,2b+1}(Spin(2n))^{#} → ⟨x_{n−2} ∼ 2^{n−1}D_{+}⟩ → 0 if s + b is even. The morphism i_{∗}^{#} of the proposition sends the K[−, −] and C[−, −] parts bijectively, and also x_{1} maps across. This, with 5.14, yields the claim. Note that n^{∗∗} becomes n − 4 here.

6. d_{3} on eta towers

Since η^{4} = 0 in homotopy, d_{3}-differentials must annihilate all eta-towers, except for a few elements at the bottom of the target tower. In this section, we determine the d_{3}-differential on the eta towers.

The group η_{i}(X) of eta-towers passing through E_{2}^{s,2(s+i)+1}(X) (s > 2) was defined in Definition 1.5. Note that d_{3} is a homomorphism from η_{i}(X) to η_{i−2}(X). As customary with Adams-type spectral sequence charts, we place E_{2}^{s,t} in position (x, y) = (t−s, s), so that (assuming convergence) π_{i}(X) has associated graded E_{∞}^{∗,∗+i}. Then η_{i}(X) is a tower of elements whose position satisfies x − y = 2i + 1.

44 BENDERSKY AND DAVIS

It will be convenient to classify the eta-towers determined in Section 5 as “unstable” or “stable” depending upon whether or not they are of the form ψ^{2}(x)/2. Thus the unstable classes in Spin(n) come from M_{2} if n ≢ 2 mod 4, and from Q_{ev} if n ≡ 2 mod 4. We will abbreviate η_{i}(Spin(n)) as η_{i}(n). We tabulate the elements found in Propositions 5.14, 5.16, and 5.22 in the following table.

Table 6.1. This table describes all eta towers.

η_{i}(4a − 2)  η_{i}(4a − 1)  η_{i}(4a)  η_{i}(4a + 1)
i even,  x_{2a−3}  x_{2a−3}  x_{2a−3}  x_{2a−3}
stable  D  D_{+}, D_{+} − D_{−}  D
i even, C[a − 1, 2a − 3] C[a − 1, 2a − 2] C[a − 1, 2a − 2] C[a, 2a − 1]
unstable  D_{+} − D_{−}
i odd,  x_{1}  x_{1}  x_{1}  x_{1}
stable  D  D_{+}, D_{+} − D_{−}  D
i odd, K[a − 1, 2a − 3] K[a − 1, 2a − 2] K[a − 1, 2a − 2] K[a, 2a − 1]
unstable  D_{+} − D_{−}

Now we can state the main theorem of this section.

Theorem 6.2. For the eta towers as described in Table 6.1,

d_{3} : η_{i}(4a + ϵ) → η_{i−2}(4a + ϵ), with − 2 ≤ ϵ ≤ 1,

sends the following eta towers nontrivially to eta towers with the same name.

i even, stable:

x_{2a−3} if i ≡ 0 mod 4;

D, D_{+}, D_{+} − D_{−} if i ≡ 2a mod 4;

i even, unstable:

all classes if i ≡ 0 mod 4; i odd, stable:

x_{1} if i ≡ 1 mod 4;

D, D_{+}, D_{+} − D_{−} if i ≡ 2a + 1 mod 4;

i odd, unstable:

all classes if i ≡ 3 mod 4.

PERIODIC HOMOTOPY GROUPS OF SO(N) 45

This theorem will be proved by comparing with known d_{3}’s in the BTSS of spheres. It is immediate from 2.2, 2.3, 2.5, and 2.10.i that there are exact sequences

→ E_{2}^{s,t}(Spin(2n − 1)) ⟶^{j_{∗}} E_{2}^{s,t}(Spin(2n)) ⟶^{p_{∗}} E_{2}^{s,t}(S^{2n−1}) ⟶^{δ} E_{2}^{s+1,t}(Spin(2n − 1)) →

(6.3)

and → E_{2}^{s,t+1}(S^{2n}) → E_{2}^{s,t}(Spin(2n)) → E_{2}^{s,t}(Spin(2n + 1)) → E_{2}^{s+1,t+1}(S^{2n}) →

(6.4)

in which all morphisms respect differentials in the BTSS.

The behavior of d_{3} in the BTSS of the odd spheres is stated in 6.5. In [12], it was shown that the BTSS for odd spheres agrees with the v_{1}-periodic UNSS, which was computed in [5] and also described in [7, p.488].

Proposition 6.5. The groups η_{i}(S^{2n+1}) equal Z_{2} ⊕ Z_{2} with one tower beginning in filtration 1 and called stable, and the other tower beginning in filtration 2 and called unstable. (The lowest class in each tower may have order greater than 2.) The stable towers map to one another under double suspension, while the unstable towers map to 0 under double suspension (except perhaps on their lowest class). The differential d_{3} : η_{i}(S^{2n+1}) → η_{i−2}(S^{2n+1}) is nonzero in the following cases:

n i type  i mod 4
ev ev stable  n + 2
ev ev unstable  0
ev od stable  n + 1
ev od unstable  1
od ev stable  n + 1
od ev unstable  0
od od stable  n + 2
od od unstable  3

As for the even spheres, we have the following.

Proposition 6.6. In the EHP sequence

⟶^{P} E_{2}^{s,t}(S^{2n−1}) ⟶^{E} E_{2}^{s,t+1}(S^{2n}) ⟶^{H} E_{2}^{s−1,t}(S^{4n−1}) ⟶^{P} E_{2}^{s+1,t}(S^{2n−1}) (6.7)

of [4, 5.4] and [6, 7.1(ii)], the homomorphism P is 0 on eta-towers if n is even, while if n is odd, P sends the stable eta-towers of S^{4n−1} to the unstable eta-towers of S^{2n−1},

46 BENDERSKY AND DAVIS

and sends the unstable eta-towers of S^{4n−1} to 0. The d_{3}-differentials on the eta-towers of S^{2n} agree with those of S^{4n−1} (excluding the stable ones when n is odd) and S^{2n−1} (excluding the unstable ones when n is odd) in (6.7).

Proof. We use the determination of v_{1}^{−1}π_{∗}(S^{2n}) given in [26]. For a pair of eta towers in an odd sphere with d_{3}(A) = B, the bottom few elements in the eta-tower B survive to periodic homotopy classes represented from the classical Adams spectral sequence viewpoint utilized in [26] by classes connected by diagonal lines near the top or bottom of vertical towers such as that pictured below. The ones at the top are stable and the ones at the bottom are unstable.

[figure]

The exact sequence (6.7) is depicted (for S^{13} → S^{14}, which is representative of any odd value of n) on the left side of [26, p.233]. The boundary P from the stable classes on the large sphere to the unstable classes on the large sphere is apparent. Similarly, the left diagram on [26, p.235] shows that the boundary morphism P is 0 on eta towers when n is even in (6.7).

Proof of Theorem 6.2. Case 1: x_{1}.

The classes x_{1} are, of course, compatible under restriction. They pull back to Spin(7), and the bottom of the target eta-tower survives to give the Z_{2}’s in π_{8i}(Spin) and π_{8i+1}(Spin).^{6} One way to see the differential in Spin(7) is to use the 2-primary

6This statement seems slightly inconsistent with the result of [27], which stated that these Z_{2}’s pull back to Spin(6). The cause of the apparent discrepancy is that x_{1} is in the image of η_{od}(Spin(6)) ⟶^{η_{od}(i)} η_{od}(Spin(7)), but it has a different name in Spin(6). This seems to be related to the anomaly for the 1-line group of Spin(6)

discussed in Proposition 4.33. To see this, we let M_{n} = QK^{1}(Spin(n))/ im(ψ^{2}) and Q(M) = ker(1 + ψ^{−1})/ im(1 − ψ^{−1}) on M. Then the above morphism η_{od}(i) is dual to Q(M_{7}) → Q(M_{6}). (Actually it is dual to coker(ψ^{3} −1) applied to these modules, but ψ^{3} − 1 turns out to be 0 here.) We find that Q(M_{7}) ≈ ⟨x_{1}, D : 2x_{1}, 2D⟩ and Q(M_{6}) is the subspace of ⟨x_{1}, D_{+}, D_{−} : 2x_{1}, x_{1} − 4D_{+}, D_{+} + D_{−}⟩ generated by x_{1} and 2(D_{+} − D_{−}), and the morphism sends x_{1} ↦ x_{1} and D ↦ D_{+} + D_{−}. The generator of Q(M_{6}) is most naturally thought of as 2(D_{+} − D_{−}), but the relations make it equivalent to x_{1}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 47

splitting Spin(7) ≃ G_{2} × S^{7} to see that the two eta towers in η_{od}(Spin(7)) which emanate from filtration 1 both have d_{3} : η_{i}(Spin(7)) → η_{i−2}(Spin(7)) nonzero for i ≡ 1 mod 4. The splitting cited here was proved in [35, 9.1], while the claims about d_{3} in G_{2} and S^{7} were proved, respectively, in [9, 4.8] and references cited just before 6.5. Case 2: D_{+} − D_{−} in η_{i}(4a).

Dualizing the exact sequence in the proof of 5.16, we obtain that the four families of eta towers in Spin(4a) dual to D_{+} − D_{−} map isomorphically to the eta towers of S^{4a−1}. The pattern of d_{3}-differentials on these towers in Spin(4a) must be the same as in S^{4a−1}, which was given in 6.5, with n of 6.5 replaced by 2a − 1. Case 3: D in η_{i}(4a + 1).

We use the exact sequence (6.3) with n = 2a + 1 and t = 2b + 1. For either parity of s + b, the element D ∈ E_{2}^{s,2b+1}(Spin(4a + 1))^{#} is obtained from Q = ker(1 + ψ^{−1})/ im(1−ψ^{−1}). In one parity, it is as an element of ker((ψ^{3}−1)|Q), and in the other as an element of coker((ψ^{3} −1)|Q). In either case, δ^{#}(D) in E_{2}^{s,2b+1}(S^{4a+1}) is obtained by pulling D back to D_{+} ∈ QK^{1}(Spin(4a + 2))/ im(ψ^{2}), applying 1 + ψ^{−1} to that, obtaining D_{+}−D_{−}, and pulling that back to an element in P K^{1}(S^{4a+1})/ im(ψ^{2}), which will be in ker(1−ψ^{−1}). This element can be chosen to be the generator of P K^{1}(S^{4a+1}). Thus δ^{#}(D) = g, and dually we obtain that δ sends the dual class, that we are also calling D, to the stable class in E_{2}^{1,2b+1}(S^{4a+1}). Thus, for D ∈ η_{b−s−1}(Spin(4a + 1)), d_{3}(D) is nonzero if and only if d_{3} is nonzero on the stable class of η_{b−s}(S^{4a+1}), and from 6.5 we see that this happens if b − s − 1 ≡ 2a or 2a + 1 mod 4. Case 4: D_{+} in η_{i}(4a), and D in η_{i}(4a − 1).

The morphisms η_{i}(4a − 1) → η_{i}(4a) → η_{i}(4a + 1) are dual to P K^{1}(Spin(4a+1))/(2, ψ^{2}) ⟶^{i_{1}^{∗}} P K^{1}(Spin(4a))/(2, ψ^{2}) ⟶^{i_{0}^{∗}} P K^{1}(Spin(4a−1))/(2, ψ^{2}). These satisfy i_{1}^{∗}(D) = D_{+} + D_{−} and i_{0}^{∗}(D_{±}) = D. Thus d_{3} on the D_{+}-towers in Spin(4a) agrees with that on the D towers in Spin(4a + 1), and (since D_{+} + D_{−} ≡ D_{+}−D_{−} mod 2) d_{3} on the D-towers in Spin(4a−1) agrees with that on the (D_{+}−D_{−})- towers in Spin(4a). Case 5: All classes x_{2a−3}.

48 BENDERSKY AND DAVIS

We use the exact sequence (6.3) with n = 2a−1 and t = 2b+1. In QK^{1}(Spin(4a− 2))/ im(ψ^{2}, 1 − ψ^{−1}), we have x_{2a−3} ∼ 2^{2a−2}D_{+} ∼ p^{∗}(2^{2a−3}g), using 5.19 and (2.4). Since 2^{2a−3}g generates ker(1 + ψ^{−1})|QK^{1}(S^{4a−3})/ im(ψ^{2}), we deduce that

ker(θ|Q_{od}(S^{4a−3})) ⟶^{p^{∗}} ker(θ|Q_{od}(Spin(4a − 2)))

sends 2^{2a−3}g to x_{2a−3}. Then (5.17) says that dually p_{∗} : η_{ev}(Spin(4a−2)) → η_{ev}(S^{4a−3}) sends the class we call x_{2a−3} to the unstable class. Now the fact that d_{3} is nonzero on the unstable class in η_{i}(S^{4a−3}) if i ≡ 0 mod 4 implies the same for x_{2a−3} in η_{i}(Spin(4a−2)). That d_{3} behaves in the same way on x_{2a−3} in Spin(4a−1), Spin(4a), and Spin(4a + 1) follows by naturality. Case 6: All elements in C[α, β] and K[α, β].

Using Proposition 6.6, the sequences (6.4) and (6.7) combine to

→ η_{i}^{st}(S^{4a−3}) ⊕ η_{i+1}^{un}(S^{8a−5}) → η_{i}(Spin(4a − 2))  (6.8)
⟶^{ϕ_{i,4a−2}} η_{i}(Spin(4a − 1)) → η_{i−1}^{st}(S^{4a−3}) ⊕ η_{i}^{un}(S^{8a−5}) →,

where st and un refer to stable and unstable classes, respectively, and

→ η_{i}(S^{4a−1}) ⊕ η_{i+1}(S^{8a−1}) → η_{i}(Spin(4a))  (6.9)
⟶^{ϕ_{i,4a}} η_{i}(Spin(4a + 1)) → η_{i−1}(S^{4a−1}) ⊕ η_{i}(S^{8a−1}) → .

The morphisms ϕ_{i,4a−ϵ} in the above exact sequences are closely related to natural morphisms Ĉ_{i,4a−ϵ} and K̂_{i,4a−ϵ} defined using Definitions 5.8 and 5.7.

Definition 6.10. Let C_{4a−2} = C[a − 1, 2a − 3] and C_{4a} = C[a − 1, 2a − 2], and Ĉ_{i,2b} : C_{2b} → C_{2b+2} the morphism obtained from η_{i}(2b) → η_{i}(2b + 2) in Table 6.1.

Make a similar definition with all C’s replaced by K’s. Note that Ĉ_{i,2b} does not depend upon the value of i, but we will need to keep track of the value of i as it relates to η_{i}(−). The exact sequences (6.8) and (6.9) can be interpreted as the following short exact sequences, which preserve d_{3}-differentials. Here d_{st} and d_{un} refer to D_{+} − D_{−} in Table 6.1 in stable and unstable boxes. The value of i associated to the elements D and d agrees with that of the accompanying (co)ker(Ĉ or K̂).

0 → ⟨D⟩ ⊕ coker(K̂_{2k+1,4a−2}) → η_{2k}^{st}(S^{4a−3}) ⊕ η_{2k+1}^{un}(S^{8a−5}) → ker(Ĉ_{2k,4a−2}) → 0 (6.11)

0 → ⟨D⟩ ⊕ coker(Ĉ_{2k,4a−2}) → η_{2k−1}^{st}(S^{4a−3}) ⊕ η_{2k}^{un}(S^{8a−5}) → ker(K̂_{2k−1,4a−2}) → 0 (6.12)

0 → coker(K̂_{2k+1,4a}) → η_{2k}(S^{4a−1}) ⊕ η_{2k+1}(S^{8a−1}) → ⟨d_{st}, d_{un}⟩ ⊕ ker(Ĉ_{2k,4a}) → 0 (6.13)

0 → coker(Ĉ_{2k,4a}) → η_{2k−1}(S^{4a−1}) ⊕ η_{2k}(S^{8a−1}) → ⟨d_{st}, d_{un}⟩ ⊕ ker(K̂_{2k−1,4a}) → 0

(6.14)

The d_{3}-differentials on the eta-towers in the spheres were tabulated in Theorem 6.5. The d_{3}-differentials on the classes D in (6.11) and (6.12) were established in Case 4, and the d_{3}-differentials on the classes d_{st} and d_{un} in (6.13) and (6.14) were established in Case 2. In the short exact sequences (6.11)–(6.14), the classes D and d will have to match up with classes in spheres with agreeing d_{3}. Then d_{3} on (co)ker(Ĉ or K̂) must agree with that on the remaining classes in the spheres.

For example, for the class D in (6.11), d_{3}(D) ≠ 0 if 2k + 1 ≡ 2a + 1 mod 4. Also, in η_{2k}^{st}(S^{4a−3}), d_{3} ≠ 0 if 2k ≡ 2a mod 4, while in η_{2k+1}^{un}(S^{8a−5}), d_{3} ≠ 0 if 2k + 1 ≡ 3 mod 4. Although this may not always imply that the D-class maps to the stable class on S^{4a−3}, (6.11) does imply that coker(K̂_{2k+1,4a−2}) ⊕ ker(Ĉ_{2k,4a−2}) does have just one nonzero element, and d_{3} is nonzero on it iff 2k + 1 ≡ 3 mod 4. This yields the first of the four cases of Proposition 6.15.

As another example, in (6.13) the d_{st} and d_{un} have d_{3} ≠ 0 when 2k ≡ 2a and 0, as do the two classes in η_{2k}(S^{4a−1}), while the two in η_{2k+1}(S^{8a−1}) have d_{3} ≠ 0 when 2k+1 ≡ 4a+1 and 3; i.e., 2k ≡ 0, 2, all mod 4. This yields the third case of 6.15. The second and fourth cases follow similarly from (6.12) and (6.13), respectively, yielding the following result.

Proposition 6.15. For ϵ = 0 or 2, let z_{ϵ} denote an element on which d_{3} is nonzero if 2k ≡ ϵ mod 4. Then

coker(K̂_{2k+1,4a−2}) ⊕ ker(Ĉ_{2k,4a−2}) ≈ ⟨z_{2}⟩
coker(Ĉ_{2k,4a−2}) ⊕ ker(K̂_{2k−1,4a−2}) ≈ ⟨z_{0}⟩
coker(K̂_{2k+1,4a}) ⊕ ker(Ĉ_{2k,4a}) ≈ ⟨z_{0}, z_{2}⟩
coker(Ĉ_{2k,4a}) ⊕ ker(K̂_{2k−1,4a}) ≈ ⟨z_{0}, z_{0}⟩

50 BENDERSKY AND DAVIS

Suppose now that x_{j} ∈ C[α, β] ⊂ η_{i}(n). We will show in Proposition 6.16 that x_{j} is in the appropriate C[α, β] groups for an interval of values of n. At the beginning of that interval, it is in some coker(Ĉ_{i,2b}), and at the end of the interval, it is in some ker(Ĉ_{i,2b′}). We will see in Corollary 6.17 that at least one of these is of the first, second, or fourth type in Proposition 6.15, for which d_{3} is determined by the proposition, and, indeed, is shown to be nonzero when i ≡ 0 mod 4, as claimed. The same behavior (d_{3}(x_{j}) ≠ 0 if i ≡ 0 mod 4) for all n in this interval follows by naturality. A similar argument will be performed for elements of K[α, β].

The following proposition refers to the notation established in Definition 6.10.

Proposition 6.16.

x_{j} ∈ K_{2b} iff j + 2 ≤ b ≤ \begin{cases}2j + 2 & if j is a 2-power \\ j + 2^{ν(j)+1} + 1 & if j is not a 2-power. \end{cases}

x_{j} ∈ C_{2b} iff f(j) ≤ b ≤ 2j + 2, where f(j) = \begin{cases}j + 2 & if j = 2^{e} or 3 · 2^{e} \\ 2j − 2^{ν(j)+2} + 3 & otherwise. \end{cases}

Proof. This follows easily from the definitions. We illustrate with x_{j} ∈ C_{2b} when j is not 2^{e} or 3 · 2^{e}.

C_{4j+4} = ⟨x_{i} : j ≤ i ≤ 2j and i − 2^{ν(i)+1} < j⟩
C_{4j+6} = ⟨x_{i} : j + 1 ≤ i ≤ 2j + 1 and i − 2^{ν(i)+1} < j + 1⟩

Clearly x_{j} is in C_{4j+4} and not in C_{4j+6}.

C_{4j−2^{ν(j)+3}+4} = ⟨x_{i} : j − 2^{ν(j)+1} ≤ i ≤ 2j − 2^{ν(j)+2} and i − 2^{ν(i)+1} < j − 2^{ν(j)+1}⟩
C_{4j−2^{ν(j)+3}+6} = ⟨x_{i} : j − 2^{ν(j)+1} + 1 ≤ i ≤ 2j − 2^{ν(j)+2} + 1
and i − 2^{ν(i)+1} < j − 2^{ν(j)+1} + 1⟩

Clearly x_{j} ∉ C_{4j−2^{ν(j)+3}+4}, while x_{j} ∈ C_{4j−2^{ν(j)+3}+6} iff j = 2^{e} + A · 2^{e+1} with 2A ≥ 3, which is true since j ≠ 2^{e} or 3 · 2^{e}. The same sort of argument shows that x_{j} is in C_{2b} for intermediate values of b, i.e. between 2j − 2^{ν(j)+2} + 3 and 2j + 2.

From this, we can read off the following corollary.

Corollary 6.17. 1. x_{j} ∈ coker(K̂_{i,2j+2}) if j is even;

PERIODIC HOMOTOPY GROUPS OF SO(N) 51

2. x_{j} ∈ ker(K̂_{i,2j+6}) if j is odd;

3. x_{j} ∈ coker(Ĉ_{i,2j+2}) if j = 2^{e} or 3 · 2^{e};

4. x_{j} ∈ coker(Ĉ_{i,4j+4−2^{ν(j)+3}}) if j ≠ 2^{e} or 3 · 2^{e}.

Proof. Just use Proposition 6.16. For example, if j ≠ 2^{e} or 3 · 2^{e}, then x_{j} ∈ C_{2(2j+3−2^{ν(j)+2})} but x_{j} ∉ C_{2(2j+2−2^{ν(j)+2})}.

Case 6 now follows from Corollary 6.17 and Proposition 6.15. Indeed, if x_{j} is as in Corollary 6.17.1, then it is of the first type in Proposition 6.15 with i = 2k +1, and so d_{3}(x_{j}) ≠ 0 iff i ≡ 3 mod 4. If x_{j} is as in 6.17.2, then it is of the fourth type in 6.15 with i = 2k − 1, and again d_{3}(x_{j}) ≠ 0 iff i ≡ 3 mod 4. Since these two cases comprise all values of j, we find that for all j there exists n such that x_{j} ∈ K[α, β] ⊂ η_{i}(Spin(n)) has d_{3}(x_{j}) ≠ 0 iff i ≡ 3 mod 4, and by naturality this holds for all values of n for which x_{j} ∈ K[α, β] ⊂ η_{i}(Spin(n)).

Similarly, if x_{j} is as in 6.17.3, then it is of the second type in 6.15 with i = 2k, and so d_{3}(x_{j}) ≠ 0 iff i ≡ 0 mod 4, while if x_{j} is as in 6.17.4, then it is of the fourth type in 6.15, and the same conclusion follows. Since these two types comprise all values of j, we conclude by naturality that whenever x_{j} ∈ C[α, β] ⊂ η_{i}(Spin(n)), then d_{3}(x_{j}) ≠ 0 iff i ≡ 0 mod 4.

This completes the proof of Theorem 6.2.

7. Fine tuning

In this section, we determine the d_{3}-differential on the 1-line and most of the extensions (exotic multiplication by 2) in the BTSS of Spin(N). For the most part, we are carrying out proofs of results stated in Section 3. This section also contains an important result, 7.2, regarding computing h_{1} on the 1-line.

We begin with the case N = 8a ± 1, where the results were stated in 1.4.

Proof of 1.4.2. Because both d_{3} : E_{2}^{2,8k+5} → E_{2}^{5,8k+7} and h_{1} : E_{2}^{4,8k+5} → E_{2}^{5,8k+7} are bijective, it is equivalent to prove that for Spin(8a ± 1), h_{1} : E_{2}^{1,8k+3} → E_{2}^{2,8k+5} is nonzero on both summands. We use the isomorphism of E_{2} with Ext_{A} of 5.1.a.

Letting X = Spin(8a ± 1) and N = (QK^{1}(X)/ im(ψ^{2}))^{#}, we obtain, similarly to [9, 3.6], a commutative diagram of exact sequences

\begin{aligned}& & 0 & ⟶ & E_{2}^{1,8k+3}(X) & ⟶ & N & ⟶^{θ^{#}} & N \\ & & ↓ & & ^{h_{1}·}↓ & & ^{ρ_{2}}↓ & & ^{ρ_{2}}↓ \\ N_{2} & ⟶^{θ^{#}} & N_{2} & ⟶ & E_{2}^{2,8k+5}(X) & ⟶ & N/2 & ⟶^{θ^{#}} & N/2 \end{aligned} (7.1)

Here N_{2} = ker(2|N) and θ = ψ^{3} − 3^{4k+1}. The effect of h_{1}· on elements of E_{2}^{1}(X) corresponding to elements of ker(θ^{#}|N) which are not divisible by 2 in N is clear from the diagram. However, for other elements we need the following result, which we prove after completing the proof of 1.4.2. Another approach to this result, with various extensions of the formula, is given in Theorems 11.5 and 11.18.

Proposition 7.2. In (7.1), if x ∈ ker(ρ_{2}) ∩ ker(θ^{#}), then the corresponding element of E_{2}^{1}(X) satisfies h_{1} · x = θ^{#}(x/2), which is well-defined as an element of coker(θ^{#}|N_{2}).

Note that this x/2 is not an element of N_{2}; it is in N.

Now diagram chasing on (7.1) implies that h_{1}· is injective on E_{2}^{1,8k+3}(X)/2, as desired. Indeed, suppose h_{1} · x = 0. Then x corresponds to an element x ∈ N satisfying θ^{#}(x) = 0 and x = 2y. By Proposition 7.2, h_{1} · x = θ^{#}(y) considered as an element of coker(θ^{#}|N_{2}). Since this is assumed to be 0, we deduce θ^{#}(y) = θ^{#}(z) with 2z = 0. Then y − z ∈ ker(θ^{#}) and so it pulls back to an element y′ ∈ E_{2}^{1,8k+3}(X) satisfying 2y′ = x. Hence x = 0 ∈ E_{2}^{1,8k+3}(X)/2.

Proof of Proposition 7.2. Using the exact sequence in A of [9, after 3.3],

0 → U(M) ⟶^{θ} U(M) ⟶^{p} M → 0,

(7.1) is, with M = QK^{1}(X)/ im(ψ^{2}), S = QK^{1}(S^{8k+3}), and S′ = QK^{1}(S^{8k+5}),

⟶^{θ^{∗}} Ext_{A}^{0}(U(M), S) ⟶^{δ} Ext_{A}^{1}(M, S) ⟶^{p^{∗}} Ext_{A}^{1}(U(M), S) ⟶^{θ^{∗}}
h_{1}↓  ^{h_{1}}↓  ^{h_{1}}↓
⟶^{θ^{∗}} Ext_{A}^{1}(U(M), S′) ⟶^{δ′} Ext_{A}^{2}(M, S′) ⟶^{p^{∗}} Ext_{A}^{2}(U(M), S′) ⟶^{θ^{∗}}

(7.3)

Here the vertical maps are Yoneda product with an element h_{1} ∈ Ext_{A}(S, S′) described in [9, 3.6]. Also, U : GInv → A is left adjoint to the forgetful functor, where

PERIODIC HOMOTOPY GROUPS OF SO(N) 53

GInv denotes the category of 2-profinite abelian groups with involution ψ^{−1}, as in [9, §3]. The existence of U, its adjointness and exactness, and the fundamental SES above all follow by Pontrjagin duality from the analogous results for the functor \overline{U} in [18, pp 145-6]. Here we use that GInv is Pontrjagin dual to the category of 2-torsion abelian groups with involution, and our category A of stable 2-adic Adams modules is dual to the category of stable 2-torsion Adams modules, as noted in [15, 10.2]. The properties proved for \overline{U} restrict to properties on the 2-torsion subcategories which dualize to the properties of U that we need. For example, using results of [9, §3], we have

Ext_{A}^{2}(U(M), S′) ≈ Ext_{GInv}^{2}(M, S′) ≈ Ext_{Inv}^{1}(Z_{(2)}^{(+)}, M^{#}) ≈ M^{#}/2.

If ⋯ → P_{1} → P_{0} → M → 0 is a projective resolution in GInv, then ⋯ → U(P_{1}) → U(P_{0}) → U(M) → 0 is a projective resolution in A. This is true because of the left adjointness and exactness of U.

If ⋯ → P_{1} → P_{0} → M → 0 is a projective resolution in A, then it is also a projective resolution in GInv. This is true because free objects in A are also free in GInv, and a module is projective iff it is a direct summand of a free module.

Combining these, we obtain that if ⋯ → P_{1} → P_{0} → M → 0 is a projective resolution in A, then there is a SES of projective resolutions in A

↓  ↓  ↓
0 ⟶ U(P_{1}) ⟶^{θ} U(P_{1}) ⟶^{p} P_{1} ⟶ 0
↓  ↓  ↓
0 ⟶ U(P_{0}) ⟶^{θ} U(P_{0}) ⟶^{p} P_{0} ⟶ 0
↓  ↓  ↓
0 ⟶ U(M) ⟶^{θ} U(M) ⟶^{p} M ⟶ 0
↓  ↓  ↓
0  0  0

54 BENDERSKY AND DAVIS

which yields the exact sequences of (7.3) in the usual way. In particular, the following derived diagram of SESs will be useful to us.

0 ⟶ Hom_{A}(P_{1}, S′) ⟶^{p^{∗}} Hom_{A}(U(P_{1}), S′) ⟶^{θ^{∗}} Hom_{A}(U(P_{1}), S′) ⟶ 0
d^{∗}↓  ^{d^{∗}}↓  ^{d^{∗}}↓
0 ⟶ Hom_{A}(P_{2}, S′) ⟶^{p^{∗}} Hom_{A}(U(P_{2}), S′) ⟶^{θ^{∗}} Hom_{A}(U(P_{2}), S′) ⟶ 0
h_{1}↑  ^{h_{1}}↑  ^{h_{1}}↑
0 ⟶ Hom_{A}(P_{1}, S) ⟶^{p^{∗}} Hom_{A}(U(P_{1}), S) ⟶^{θ^{∗}} Hom_{A}(U(P_{1}), S) ⟶ 0

(7.4)

Let \overline{x} ∈ Ext_{A}^{1}(M, S) in (7.3) map to the given element x ∈ ker(θ^{∗})∩ker(h_{1}), and let \overline{z} ∈ Hom_{A}(P_{1}, S) and z ∈ Hom_{A}(U(P_{1}), S) be representative cycles with p^{∗}(\overline{z}) = z. Then, by the definition of δ′, δ′(y) = h_{1}(\overline{x}) in (7.3) if and only if in (7.4) there is w ∈ Hom_{A}(U(P_{1}), S′) such that θ^{∗}(w) represents y and d^{∗}(w) = p^{∗}h_{1}(\overline{z}). Proposition 7.5 below will imply Proposition 7.2, since we have

p^{∗}h_{1}\overline{z} = h_{1}p^{∗}\overline{z} = h_{1}z = d^{∗}(v(z/2)),

and so w = v(z/2) works and hence δ′\{θ^{∗}(v(z/2))\} = h_{1}\overline{x}.

Proposition 7.5. If → F_{2} ⟶^{d_{2}} F_{1} → F_{0} → M → 0 is part of a free resolution in A, there are natural isomorphisms

Hom_{A}(F_{1}, S) ⟶^{v} Hom_{A}(F_{1}, S′)

satisfying d_{2}^{∗}(v(z)) = 2h_{1}z.

Proof. A free object of A is of the form Γ ⊗ N, where Γ is a free object in A on one generator as in [9, 2.2], and N is a free K_{∗}-module. This Γ is a free K_{∗}-module on elements ξ_{k} with k odd and positive, with ψ^{k}ξ_{1} = ξ_{k}. There are isomorphisms for all K_{∗}-modules N and A-objects L

Hom_{K_{∗}}(N, L) ⟶^{ι} Hom_{A}(Γ ⊗ N, L)

defined by ι(ϕ)(ξ_{k} ⊗ n) = ψ^{k}(ϕ(n)). Note that ι^{−1} is just given by restricting to ξ_{1} ⊗ N. The morphism v is the composite

Hom_{A}(Γ⊗N, S) ⟶^{ι^{−1}} Hom_{K_{∗}}(N, S) ⟶^{j_{∗}} Hom_{K_{∗}}(N, S′) ⟶^{ι} Hom_{A}(Γ⊗N, S′),

PERIODIC HOMOTOPY GROUPS OF SO(N) 55

where j : S → S′ is the identity map of Z_{2}^{∧}. Recall that S = QK^{1}(S^{2m+1}) and S′ = QK^{1}(S^{2m+3}), where m = 4k + 1.

Naturality for A-morphisms f : Γ⊗N_{1} → Γ⊗N_{2} follows since, for θ ∈ Hom_{K_{∗}}(N_{2}, S), both vf^{∗} and f^{∗}v send ι(θ) to the element of Hom_{A}(Γ ⊗ N_{1}, S′) which sends x to ∑ c_{i}k_{i}^{m+1}θ(n_{i}) if f(x) = ∑ c_{i}ξ_{k_{i}} ⊗ n_{i}.

To define h_{1}, we use the resolution of S which begins

0 ← S ⟵^{ϵ} Γ = C_{0} ⟵^{d_{1}} Γ ⊗ Γ = C_{1}

with ϵ(ξ_{k}) = k^{m} and d_{1}(ξ_{k} ⊗ ξ_{ℓ}) = ξ_{kℓ} − ℓ^{m}ξ_{k}. Since Ext_{A}^{1}(S, S′) = Z/2, its nonzero element h_{1} satisfies 2h_{1} = ϵ′ ∘ d_{1}, where ϵ′ : Γ → S′ is the generator, satisfying ϵ′(ξ_{k}) = k^{m+1}. Note that ϵ′(d_{1}(ξ_{k} ⊗ ξ_{ℓ})) = k^{m+1}ℓ^{m}(ℓ − 1) is even.

If τ ∈ Hom_{A}(F_{1}, S) is a cocycle, then h_{1}{τ} is the class of h_{1}τ_{1} in the diagram

F_{1} ⟵^{d_{2}} F_{2}
↙ ^{τ} ↓^{τ_{0}} ↓^{τ_{1}}
S ⟵^{ϵ} C_{0} ⟵^{d_{1}} C_{1}
↓^{h_{1}}
S′

Let F_{1} = Γ ⊗ R with R = ker(F_{0} → M). Then τ(ξ_{k} ⊗ r) = k^{m}τ(r) and τ_{0}(ξ_{k} ⊗ r) = τ(r)ξ_{k}. By the definition of v and ϵ′, we have

vτ(ξ_{k} ⊗ r) = k^{m+1}τ(r) = ϵ′τ_{0}(ξ_{k} ⊗ r).

Thus

d_{2}^{∗}(v(τ)) = ϵ′τ_{0}d_{2} = ϵ′d_{1}τ_{1} = 2h_{1}τ_{1} = 2h_{1}{τ},

as desired.

The following proof is easier.

Proof of 1.4.1. At first glance, this seems obvious from Diagram 1.3, using the pictured η-action and d_{3} from the 2-line. However, what we must rule out is that one of the Z_{2}’s in E_{2}^{2,8k+3} (labeled 1 or D) supports a nonzero d_{3}-differential into the logclasses. The class x_{1} is in the image from E_{2}^{1,8k+1}(Spin(7)), where it does not support a differential, and hence d_{3}(x_{1}) = 0 in Spin(8a±1). The D-class in Spin(8a+1) maps

56 BENDERSKY AND DAVIS

to 0 in Spin(8a + 3), while the log-classes inject. Thus there can be no differential from D to a log class in Spin(8a + 1).

The D-class and log-classes in Spin(8a − 1) inject into Spin(8a). Then D ↦ 0 in Spin(8a + 1) (see 6.1), while all but one of the log classes (x_{4a−3}) map across to log classes in Spin(8a + 1). The only possible differential involving D in Spin(8a) and Spin(8a − 1) is to have d_{3}(D) = x_{4a−3}. However, D in E_{2}^{1,8k+1}(Spin(8a)) is in the image from E_{2}^{1,8k+1}(S^{8a−1}) in (6.4), and d_{3} on this class in BTSS(S^{8a−1}) is 0. Hence the same is true on D in BTSS(Spin(8a)) and BTSS(Spin(8a − 1)).

Now we settle the extension questions in the BTSS of Spin(8a ± 1).

Proof of Proposition 1.4.3. The groups C_{1} in E_{2}^{1,8k+3}(Spin(n)) inject as n increases and the classes x_{1} all correspond as n increases from 7. Thus it suffices to verify the nontrivial extension in the BTSS of Spin(7). Localized at 2, Spin(7) ≃ G_{2} × S^{7}. By [9, 4.8], the BTSS of G_{2} has a nontrivial extension from filtration 1 to filtration 3 in dimension 8k + 2, and by [7, p.488], the same is true of S^{7}. Analysis of the short exact sequence in QK^{1}(−) for the fibration G_{2} → Spin(7) → S^{7} shows that the C_{1}-summand in E_{2}^{1,8k+3}(Spin(7)) and the x_{1}-summand in E_{2}^{3,8k+5}(Spin(7)) are both in the image from E_{2}(G_{2}), and so the extension in Spin(7) follows from that in G_{2}.

The extension from the Z/8 to the class D follows from an analysis of

E_{2}^{s,t}(Spin(8a − 1)) ⟶^{i_{3}} E_{2}^{s,t}(Spin(8a)) ⟶^{i_{4}} E_{2}^{s,t}(Spin(8a + 1)).

(7.6)

If s = 1, t = 8k + 3, then from 3.17, (7.6) is (ignoring C_{1}-summands)

Z/8 ⟶^{i_{3}} Z/8 ⊕ Z/8 ⟶^{i_{4}} Z/8,

with i_{3} injecting to the first summand, and i_{4} sending just the second summand across. From Table 6.1 and the analysis surrounding it, if s = 3 and t = 8k + 5, (7.6) is, on stable classes and ignoring the x_{1}-class,

Z_{2} ⟶^{i_{3}} Z_{2} ⊕ Z_{2} ⟶^{i_{4}} Z_{2}

with i_{3} mapping to the first summand, and i_{4} sending just the second summand across. We wish to show that the first (resp. second) summand in E_{2}^{1,8k+3}(Spin(8a)) has a nontrivial extension into the first (resp. second) summand of E_{2}^{3,8k+5}(Spin(8a)), for then the extension in Spin(8a − 1) (resp. Spin(8a + 1)) follows by naturality.

PERIODIC HOMOTOPY GROUPS OF SO(N) 57

For the second summand, we use (6.3) with n = 4a. The summands of concern map isomorphically to E_{2}(S^{8a−1}), and the extension there is known. (See, e.g., [7, p.488].) For the first summand, we use (6.4) with n = 4a. The summands of concern are in the image of E_{2}^{s,t}(S^{8a−1}) → E_{2}^{s,t+1}(S^{8a}) → E_{2}^{s,t}(Spin(8a)), and again the extension in E_{2}(S^{8a−1}) is known.

Now we prove the claims made earlier for Spin(8a).

Proof of Theorem 3.4. Most of the information about the 1-line groups was established in 3.1, 3.3, and 3.17. That the initial summand in E_{2}^{1,8k−1}(Spin(8a)) has the same order as that of E_{2}^{1,8k−1}(Spin(8a − 1)) when ν(k) < 4a − 5 and ν(k − a) < 4a − 5 follows from the fact that, in an exact sequence, e.g., (3.5), the order of a group can be no greater than the product of the orders of the groups on both sides of it. If ν(k) ≥ 4a − 5, the orders of the initial 1-line summands are known by 3.1 and 3.3. When ν(k − a) ≥ 4a − 5, naïve consideration of (3.5) does not allow us to settle whether the orders are equal or differ by 1. So we must resort to combinatorics to determine it. In section 8 we prove the following result, from which it follows that the 1-line morphism is as claimed in this case. Note that k and a in the lemma correspond to 2k and 2a in the above discussion.

Lemma 7.7. If ν(k − a) > 2a − 5, then R(2k − 1, 4a − 1) + 1 = R(2k − 1, 4a) = ν((4a − 3)!). Here R(−, −) is as in 3.1 and 3.3.

For the d_{3}-differentials on the 1-line, we use the exact sequence (6.3) with n = 4a. By 5.16, the eta towers of Spin(8a) are the direct sum of those of Spin(8a − 1) and of S^{8a−1}. Since the morphisms commute with d_{3}-differentials and extensions, the only thing that we have to worry about is that there could be a d_{3}-differential from a class in E_{2}^{1,8k+1}(Spin(8a)) which maps nontrivially to E_{2}^{1,8k+1}(S^{8a−1}) hitting a class in E_{2}^{4,8k+3}(Spin(8a)) in the image from Spin(8a − 1).

In the notation of 6.1, the class in E_{2}^{1,8k+1} which concerns us is δ := D_{+} − D_{−}. It maps nontrivially to Spin(8a + 1), but goes to 0 in Spin(8a + 2). Its image under d_{3} must map to 0 in E_{2}^{4,8k+3}(Spin(8a + 2)). There is one nonzero class which does so, x_{4a−3} ∈ K[2a − 1, 4a − 2]. If d_{3}(δ) = x_{4a−3} in BTSS(Spin(8a)), then it must be the

58 BENDERSKY AND DAVIS

case that h_{1}^{2}(δ) = δ +x_{4a−3} in E_{2}(Spin(8a)). This is true because the bases of the etatowers have been chosen to match up under d_{3}; i.e., d_{3} : E_{2}^{3,8k+5} → E_{2}^{6,8k+7}(Spin(8a)) satisfies d_{3}(δ) = 0 and d_{3}(x_{4a−3}) = x_{4a−3}. We use h_{1}^{2} rather than just h_{1} in order to get fully into the region of the eta-towers.

Let \overline{x}_{i} in QK^{1}(Sp(n)) and QK^{1}(Spin(2n)) be defined as in 5.18. Using 4.1, we obtain, similarly to 5.9, the following useful result.

Proposition 7.8. There are bases \{\overline{x}_{1}, …, \overline{x}_{n}\} and \{\overline{x}_{1}, …, \overline{x}_{n−2}, D_{+}, D_{−}\} of QK^{1}(Sp(n)) and QK^{1}(Spin(2n)), respectively, such that

• Under the inclusion map Sp(n − 1) → Sp(n), \overline{x}_{i} ↦ \overline{x}_{i} if i < n, while \overline{x}_{n} ↦ 0;

• There is an A-morphism, QK^{1}(Sp(n)) ⟶^{ϕ} QK^{1}(Spin(2n)) such that ϕ(\overline{x}_{i}) = \overline{x}_{i} for i ≤ n − 2, ϕ(\overline{x}_{n−1}) = 2^{n−1}(D_{+} + D_{−}) + ∑ α_{i}\overline{x}_{i} with α_{i} even, and ϕ(\overline{x}_{n}) = β_{n−1}ϕ(\overline{x}_{n−1}) + ∑ β_{i}\overline{x}_{i} with β_{i} even.

Dualizing, we obtain a morphism

P K_{1}(Spin(2n)) ⟶^{ϕ̂} P K_{1}(Sp(n))

of K_{∗}K-comodules whose mod-2 reduction factors through P K_{1}(Sp(n − 2)) ⊗ Z_{2}. Thus there is a morphism

E_{2}(Spin(2n); Z_{2}) → E_{2}(Sp(n − 2); Z_{2})

which, when followed into E_{2}(Sp(n); Z_{2}), is the mod 2 reduction of ϕ̂_{∗}. Reduction mod 2 induces a morphism

E_{2}(Spin(2n)) ⟶^{ρ} E_{2}(Spin(2n); Z_{2}),

which sends eta-towers injectively, since they are of order 2.

In our case, n = 4a, the composite E_{2}(Spin(8a)) → E_{2}(Sp(4a − 2); Z_{2}) on the eta-towers is K[2a − 1, 4a − 2] → K[2a, 4a − 2], in the notation of 5.6. In particular, x_{4a−3} is mapped nontrivially. However, the 1-line class δ defined above maps to 0 in E_{2}(Sp(4a − 2); Z_{2}). Since this morphism respects h_{1}-action, we deduce that h_{1}^{2}δ cannot equal δ + x_{4a−3}.

Now we prove the results claimed for Spin(8a + 3). The argument works verbatim for Spin(8a + 5).

PERIODIC HOMOTOPY GROUPS OF SO(N) 59

Proof of 3.8. Diagram 3.7 is a consequence of 5.14, 6.2, [8, 1.5], and, for the G-groups, [9, 3.1] and the fact that the kernel and cokernel of a morphism of finite abelian groups have the same order. The extension in dimension 8k − 2 is deduced from (6.4) as follows.

By 3.17, E_{2}^{1,8k−1}(Spin(8a + 2)) ⟶^{i_{∗}} E_{2}^{1,8k−1}(Spin(8a + 3)) has kernel given by the element of order 2 in the C_{2}-summand. The element which hits this class in (6.4) supports a d_{3}-differential in the BTSS of S^{8a+2}. This can be seen by noting that the element pulls back to S^{8a+1} and the differential there follows from 6.5 (stable class with n ≡ 0 mod 4 and i ≡ 3 mod 4). This implies that in the homotopy exact sequence corresponding to (6.4) the image of the element of ker(i_{∗}) is the element of E_{2}^{3,8k+1}(Spin(8a + 3)) which maps in (6.4) to the element of E_{2}^{4,8k+1}(S^{8a+2}) hit by the d_{3}-differential. By 6.1, this element of E_{2}^{3,8k+1}(Spin(8a + 3)) is D.

It remains to determine d_{3} on E_{2}^{1,8k±2+1}. This is done similarly to the way it was done for Spin(8a ± 1), using the action of h_{1}, but here it is more delicate, because some of the elements in the target of h_{1} support d_{3}-differentials and others do not. If g is a generator of a summand of E_{2}^{1,8k±2+1}, then d_{3}(g) ≠ 0 iff h_{1}g equals an element which supports a nonzero differential. Thus this last remaining part of Theorem 3.8 will follow from the following result.

Proposition 7.9. For Spin(8a + 3), in h_{1} : E_{2}^{1,8k−1} → E_{2}^{2,8k+1}, D is a summand of h_{1}(g_{2}) iff ν(k)+4 ≤ n, and is a summand of h_{1}(g_{1}) iff n < ν(k)+4. In h_{1} : E_{2}^{1,8k+3} → E_{2}^{2,8k+5}, h_{1}(g_{1}) contains nonzero summands other than D, while h_{1}(g_{2}) does not.

Proof. We will work with the dual h_{1}^{#} of h_{1}. With M = QK^{1}(Spin(8a + 3))/ im(ψ^{2}), the dual of (7.1) is the following commutative diagram of exact sequences, in which θ = ψ^{3} − 3^{2ℓ+1}.

\begin{aligned}& & 0 & ⟵ & E_{2}^{1,4ℓ+3}^{#} & ⟵^{ρ} & M & ⟵^{θ} & M \\ & & ↑ & & ^{h_{1}^{#}}↑ & & ^{i}↑ & & ^{i}↑ \\ M/2 & ⟵^{θ} & M/2 & ⟵ & E_{2}^{2,4ℓ+5}^{#} & ⟵ & M_{2} & ⟵ & M_{2} \end{aligned} (7.10)

Dual to Proposition 7.2 (or using 11.5), we have the following interpretation of h_{1}^{#}. Suppose x ∈ M/2 satisfies θ(x) = 0 ∈ M/2. Represent x by x̄ ∈ M. Then θ(x̄) = 2y for some y ∈ M. If x̂ ∈ E_{2}^{2,4ℓ+5}^{#} maps to x, then h_{1}^{#}(x̂) = ρ(y). One easily verifies

60 BENDERSKY AND DAVIS

that this is well-defined. In (7.10), the elements x_{4a−1} and D of E_{2}^{2,4ℓ+5}^{#} come from M/2, while the log-classes, represented by the big • in Diagram 3.7, come from M_{2}.

We consider first the case where 4ℓ + 3 = 8k − 1. Since only the class D in E_{2}^{2,8k+1} supports a nonzero d_{3}, we need E_{2}^{1,8k−1}^{#}/(h_{1}^{#}(D)), and this is obtained by adjoining to the four relations of [8, 3.18] which yield E_{2}^{1,8k−1}^{#} the additional relation (ψ^{3} − 3^{4k−1})(D)/2. Since the relation [8, (3.21)] is (ψ^{3} − 3^{4k−1})(D), it means that this fourth relation of [8, 3.18] is divided by 2. Using 8.1 for the first, [8, 3.18] for the second and third, and 8.11 for the fourth, the relations which yield E_{2}^{1,8k−1}^{#} are, with n = 4a + 1, A_{1}2^{n}ξ_{1}, A_{2}2^{n}ξ_{1} − 2^{n+1}D, A_{3}2^{n}ξ_{1} − 2^{n}D, and u2^{n}ξ_{1} + 2^{ν}D, with ν = ν(k) + 4, A_{i} integers, and u an odd integer by 8.11. If ν ≤ n, then one summand of the group presented is Z/2^{ν} and the other is obtained by subtracting multiples of the fourth relation from the others to remove D, and observing the smallest exponent of 2; if the fourth relation is divided by 2, the Z/2^{ν}-summand has order divided by 2, while the other is unchanged. If ν > n, then one summand is Z/2^{n} generated by 2^{−n} times the last relation, and the other summand is Z/2^{n} obtained from 2^{−n} times the third relation; if the fourth relation is divided by 2, then the first of these Z/2^{n}’s becomes Z/2^{n−1}, while the second is unchanged. Thus h_{1}^{#} hits the element of order 2 in the C_{2}- (resp. C_{1}-)summand if ν ≤ n (resp. ν > n); dually h_{1} is nonzero on the stated summand.

The case where 4ℓ + 3 = 8k + 3 is handled similarly. In this case, all the elements in E_{2}^{2,8k+5} except D support a nonzero d_{3}. Thus we wish to mod out E_{2}^{1,8k+3}^{#} by the image under h_{1}^{#} of all elements except D. This is accomplished by dividing the first three of the four relations in [8, 3.18] by 2. The relations have the same form as the four of the previous paragraph, except now ν = 3. Since ν < n, the Z/8 summand will be unchanged if the first three relations are divided by 2, but the other summand will be divided by 2. Thus h_{1}^{#} hits the element of order 2 in C_{1}; dually h_{1} is nonzero on the C_{1}-summand, as claimed.

Now we prove the results stated earlier for Spin(4a + 2).

Proof of Theorem 3.11. The eta towers and d_{3} between them were established in 6.1 and 6.2. When s = 2, (5.17) must be modified according to [9, 3.8]; the Q_{s+b−1} in (5.17) must be replaced by coker(1 − (−1)^{b}ψ^{−1}).

PERIODIC HOMOTOPY GROUPS OF SO(N) 61

For E_{2}^{2,2b+1}(Spin(4a + 2)), we compare with the short exact sequences at the end of the proof of 5.22. For either parity of b, the left part of the SES is the same as it is when s > 2, which is the case described there. This accounts for the class labeled 1 in (8k + 1, 2), while in (8k − 3, 2) this class is not depicted because it supports a d_{3}-differential. It also accounts for the big •’s in (8k − 1, 2) and (8k + 3, 2); these represent the group C[[\frac{n}{2}], n − 2] with n = 2a + 1.

The quotient part of the SESs must have the same order as the groups E_{2}^{1,2b+1}(Spin(4a+ 2)), because one is the cokernel and the other the kernel of the same endomorphism of a finite abelian group, namely θ_{b} on coker(1 − (−1)^{b}ψ^{−1}). If b is odd, this is just represented in our chart by the groups labeled G, the group structure of which we do not attempt to determine. The cyclicity of this group when b is even requires the following calculation.

Let b = 2c. Using 5.19, we obtain a description of coker(1 − ψ^{−1})|QK^{1}(Spin(4a + 2))/ im(ψ^{2}) as

⟨x_{1}, x_{3}, …, x_{2a−3}, x_{2a−1}, D_{+} : 2x_{i}, x_{2a−1} − 2^{2a}D_{+}⟩,

with θ(x_{i}) ≡ x_{i+2} mod ⟨x_{j} : j > i + 2⟩, and θ(D_{+}) = (3^{2a} − 3^{2c})D_{+}, using 4.1 for ψ^{3}(D_{+}). If 3+ν(a−c) ≤ 2a, then ker(θ) ≈ Z/2^{4+ν(a−c)} generated by 2^{2a−3−ν(a−c)}D_{+}− x_{2a−3}, while if 3 + ν(a − c) > 2a, then ker(θ) ≈ Z/2^{2a+1} generated by D_{+}.

The extension from C_{1} in 8k + 2 follows by naturality from Spin(4a + 1). One way to establish the d_{3} from the C′ in (8k + 3, 2) to h_{1}^{3}g_{C′} is to use that

E_{2}^{2}(Spin(4a + 2)) → E_{2}^{2}(S^{4a+1})

sends the C′ and h_{1}^{3}g_{C′} surjectively, and the d_{3} is present in S^{4a+1} by [7, p.488]. The extension from C′ into (8k − 1, 4) is trivial since the morphism

E_{2}^{2,8k+2}(S^{4a+2}) → E_{2}^{2,8k+1}(Spin(4a + 2))

of (6.4) sends one summand injectively onto the multiples of 2 in the C′-summand, and the extension on this summand in the BTSS of S^{4a+2} is trivial, by comparison with the computation of v_{1}^{−1}π_{∗}(S^{2n}) in [26].

That d_{3} = 0 on E_{2}^{1,8k+1} follows for the class labeled 1 by naturality from Spin(4a+1), and for the group labeled C by pushing into Sp(2a − 1) ⊗ Z_{2}, similarly to the proof of 3.4. As in that proof, d_{3} ≠ 0 iff h_{1} ≠ 0. We must use h_{1} because the morphism is

62 BENDERSKY AND DAVIS

only algebraic. The C-group maps to 0, but the log classes which form the putative target under h_{1} map bijectively. The groups are both K[a, 2a − 1] in the notation of 5.7.

To determine d_{3} on E_{2}^{1,8k+3}, we first observe that

E_{2}^{4,8k+5}(Spin(4a + 2)) → E_{2}^{4,8k+5}(Spin(4a + 3))

is injective. This can be seen in 6.1, where we have i even and the a in that table corresponds to our a+1 here. Note that C[a−1, 2a−3] → C[a−1, 2a−2] is injective. Thus the generators of E_{2}^{1,8k+3}(Spin(4a + 2)) support nonzero d_{3} iff their image in Spin(4a + 3) does. By 3.17, the Z/8 maps by ·2, so its image does not support a nonzero differential. The condition stated in the theorem that R(4k + 1, 4a + 2) equals R(4k + 1, 4a + 3) exactly says that the C_{1}-summand in E_{2}^{1,8k+3}(Spin(4a + 2)) maps onto that of E_{2}^{1,8k+3}(Spin(4a + 3)). Since it was shown in 3.8 and 1.4 that C_{1} in E_{2}^{1,8k+3}(Spin(4a + 3)) supports a nonzero d_{3}, the claim follows.

Now we prove the claims made earlier for Spin(8a + 4).

Proof of Theorem 3.14. The claims about the 1-line groups and homomorphisms follow as in the proof of 3.4.

In the first of the four cases of 3.14, d_{3} is nonzero on the first summand of E_{2}^{1,8k+3}(Spin(8a+3)), and its possible targets map injectively to E_{2}^{4,8k+5}(Spin(8a+4)) by Table 6.1, implying d_{3} ≠ 0 on the first summand of E_{2}^{1,8k+3}(Spin(8a + 4)). That d_{3} = 0 on the third summand holds since this class is the image of a class in Spin(8a + 3) on which d_{3} = 0.

To see that d_{3} = 0 on the second summand, we must show that it does not hit one of the classes in the image from E_{2}^{4,8k+5}(Spin(8a + 3)). To do this, we show that h_{1} times this class does not equal an element of E_{2}^{2,8k+5}(Spin(8a + 4)) supporting a nonzero d_{3}. This is done by dualizing and using Diagram (7.10). We must show that the order of this summand in E_{2}^{1,8k+3}(Spin(8a + 4))^{#} is not decreased when the relations for the elements of E_{2}^{2,8k+5}(Spin(8a + 4))^{#} supporting nonzero d_{3}’s are divided by 2. The argument leading to (4.32) shows that the relation for Z/2^{ν(k−a)+5} in E_{2}^{1,8k+3}(Spin(8a+4))^{#} involves ψ^{2} and ψ^{3} −3^{4k+1} applied to D_{+} and D_{+} −D_{−}. These are not the relations that will be divided by 2, since D_{+}−D_{−} comes from E_{2}(S^{8a+3})^{#},

PERIODIC HOMOTOPY GROUPS OF SO(N) 63

while D_{+} ∈ E_{2}^{2,8k+5}(Spin(8a+4)) does not support a nonzero d_{3}, inasmuch as it comes from D in Diagram 3.7.

In the second of the four cases of 3.14, d_{3} is nonzero on the first summand because it maps onto an element of E_{2}^{1,8k+3}(S^{8a+3}) on which d_{3} ≠ 0. The nonzero d_{3} on the second summand is a consequence of its being the image of the first summand of E_{2}^{1,8k+3}(Spin(8a + 3)), on which d_{3} is nonzero into classes mapping injectively under i_{∗}. That d_{3} = 0 on the third summand is true because it is the image of a class (the second summand of E_{2}^{1,8k+3}(Spin(8a + 3))) on which d_{3} = 0.

In the third of the four cases of 3.14, d_{3} is zero on the first summand and nonzero on the third by naturality from Spin(8a + 3). The nonzero part requires the observation that E_{2}^{4,8k+1}(Spin(8a + 3)) → E_{2}^{4,8k+1}(Spin(8a + 4)) is injective by 6.1. Naturality from Spin(8a + 4) → S^{8a+3} implies d_{3} nonzero on the second summand.

Finally, in the fourth case, naturality from Spin(8a + 3) implies d_{3} is zero on the second summand of E_{2}^{1,8k−1}(Spin(8a + 4)) and nonzero on the third, while naturality from Spin(8a + 4) → S^{8a+3} implies it nonzero on the first.

8. Combinatorics

In this section we present some combinatorial arguments used earlier in the paper. We begin with the proof of Lemma 4.22. For the first part, we have the following

sharper result.

Proposition 8.1. For any nonnegative integers m and j, ∑_{k}(−1)^{k}\binom{j}{k}k^{m} is divisible by j!.

Note that the numbers whose minimal 2-exponent define eSp(m, n) are like the sum in 8.1 with j > 2n and without the terms having k even. These omitted terms will be divisible by 2^{m}, and we consider m to be large enough that these terms will not affect the divisibility. (e.g. m > n.) Proposition 8.1 is sharper than what is required for 4.22 since ν(j!) = j − α(j), where α(j) denotes the number of 1’s in the binary expansion of j, and j − α(j) ≥ n if j > 2n.

64 BENDERSKY AND DAVIS

Proof of Proposition 8.1. The proof is by induction on m and j. The result is trivially true if j = 1 or m = 0. We have ∑(−1)^{k}\binom{j}{k}k^{m+1} = j ∑(−1)^{k}\binom{j−1}{k−1}k^{m} = j ∑(−1)^{k}\binom{j}{k}k^{m}−j ∑(−1)^{k}\binom{j−1}{k}k^{m}. By the induction hypothesis, both terms are divisible by j!.

Next we prove the part of 4.22 which states ν(P_{1}(m, n)) ≥ n. The second double sum in P_{1}(m, n) is the same (with n here corresponding to n + 1 there) as the sum in [8, (3.20)], which was shown to be divisible by 2^{n+1} in [8, 3.18].^{7} Thus this second double sum in P_{1}(m, n), with its factor of 2, is divisible by 2^{n+1}. The first double sum in P_{1}(m, n) can be evaluated as

∑_{k} k^{m} ∑_{t≥0} \binom{2n}{n−1−k−2t} = ∑_{k} k^{m} ∑_{t≥0}(\binom{2n−1}{n−1−k−2t} + \binom{2n−1}{n−2−k−2t})
= ∑_{k} k^{m} ∑_{t≥0} \binom{2n−1}{n−1−k−t}.  (8.2)

This is divisible by 2^{n} by the divisibility of [8, (3.19)] proved in [8, 3.18].

The desired divisibility result for P_{3}(m, n) follows from Lemma 8.19, completing the proof of 4.22.

The following lemma was used in the proof of Theorem 3.3.

Lemma 8.3. Let m be a fixed odd positive integer, and define

A_{t} = min\{ν(∑_{k} \binom{j}{k}k^{m}) : j ≥ t\},
B_{t} = min\{ν(∑_{k}(\binom{t}{j−k} − \binom{t}{j+k})k^{m}) : [t/2] ≤ j < t\}.

Then A_{2n} = A_{2n+1} ≤ B_{2n+1} ≤ B_{2n}.

In fact, we conjecture that the four expressions are equal, but all we need is the weaker result stated in 8.3.

Proof. The equality of A_{2n+1} and A_{2n} was established in [10, 1.4], using a topological argument. That A_{2n+1} ≤ B_{2n+1} was established in [8, 3.6], using another topological argument.

7The statement in [8, 3.18] was divisibility by 2^{n}, but the argument implied divisibility by 2^{n+1}.

Let f(t, j) = ∑_{k}(\binom{t}{j−k} − \binom{t}{j+k})k^{m}. The following facts are elementary:

f(t + 1, j) = f(t, j) + f(t, j − 1)  (8.4)
f(2n, n) = 0  (8.5)

Choose minimal j ≥ n + 1 such that B_{2n} = ν(f(2n, j)). Using (8.5) in case j = n + 1, we have ν(f(2n, j)) < ν(f(2n, j − 1)). Thus, using (8.4) in the middle equality, we have

B_{2n+1} ≤ ν(f(2n + 1, j)) = ν(f(2n, j)) = B_{2n}.

The following result immediately implies the ψ^{3} part of Proposition 5.4.

Proposition 8.6. If bases {ξ_{i} : i ≥ 1} and {x_{i} : i ≥ 1} of a vector space over Z_{2} are related by

x_{i} = ∑_{j=0}^{[i/2]} \binom{i}{j}ξ_{i−2j},  (8.7)

then the endomorphism ψ^{3} defined by ψ^{3}(ξ_{i}) = ξ_{3i} satisfies

ψ^{3}(x_{i}) = ∑_{j≥0} \binom{i}{j}x_{i+2j}.  (8.8)

Proof. Substituting (8.7) into (8.8) shows that it suffices to prove the following equivalences mod 2, for positive integers i and m, and 0 ≤ ϵ ≤ 2:

∑_{j} \binom{i}{j}\binom{i+2j}{j−i+3m+ϵ} ≡ \begin{cases}\binom{i}{m} & if ϵ = 0 \\ 0 & if ϵ = 1, 2. \end{cases}

This is immediate from the following integral analogue, which we will prove.

∑_{j} \binom{i}{j}\binom{i+2j}{j−i+3m+ϵ} = ∑_{k} 2^{3k−ϵ}\binom{2m+2k}{2m−k+ϵ}\binom{i}{m+k}  (8.9)

Note that sums involving binomial coefficients are, unless specified to the contrary, taken over all values of the summation variable for which the terms are nonzero. The RHS of (8.9) has a possibly odd term only if ϵ = 0, the term with k = 0.

66 BENDERSKY AND DAVIS

We prove (8.9) by showing that both sides satisfy the same recurrence relation

(3i − 3m − ϵ)(3i − 3m − ϵ − 1)(3i − 3m − ϵ − 2)f(i)  (8.10)
= (49i^{2} − (87 + 22ϵ)i − 66im + (87 + 54ϵ)m + 81m^{2} + 44 + 29ϵ + 9ϵ^{2})if(i − 1)

−(17i + 15m − 28 + 5ϵ)i(i − 1)f(i − 2) − 5i(i − 1)(i − 2)f(i − 3) for 3i − 3m − 2 − ϵ > 0, with initial values

f(i) = \begin{cases}1 & if ϵ = 0 and i = m \\ 0 & if ϵ = 0 and i < m \\ 4(m + 1)(2m + 1) & if ϵ = 1 and i = m + 1 \\ 4(m + 1) & if ϵ = 2 and i = m + 1 \\ 0 & if ϵ ∈ \{1, 2\} and i < m + 1. \end{cases}

The initial values are easily verified. The equation (8.9) was discovered by computing the LHS of (8.9) for many values of i and m and observing the pattern of iterated differences. To prove (8.9), we used the software associated to the book [38] to find the recurrence relation satisfied by both sides of (8.9), and observing that they are the same recurrence relation. This software is a batch of Maple programs which can be downloaded from www.math.temple.edu/∼zeilberg. If the downloaded program zeil is run using as input the formula being summed on either side of (8.9), it will say that the recursion relation (8.10) is satisfied by the sum. Although the authors have not done so, this relation is simple enough that one could probably verify it by hand. The recurrence relation has been verified for several values of i and m, but the strongest verification of this relation is that this same relation was found for the disparate sums in the two sides of (8.9), which had been empirically observed to be equal by computing the value of the LHS in more than 100 cases, and using this to determine the RHS.

The following result was used in the proof of 3.2 and 7.9.

Proposition 8.11. For any positive integer n,

∑_{\begin{matrix}k odd \\ k>0 \end{matrix}} ∑_{t≥0} \binom{2n+1}{n−1−k−3t}

is odd.

PERIODIC HOMOTOPY GROUPS OF SO(N) 67

The proof requires several subsidiary results. Lemma 8.12. For n ≥ 0, the coefficient of x^{n} in (1 + x)^{2n+3}/(1 + x^{3}) is odd. Proof. The proof is by induction on n. The validity for n = 0 or 1 is elementary. Assume the result is true for n − 1. Working mod 2, the desired coefficient is

∑_{i≥0} \binom{2n+3}{n−3i} ≡ ∑_{i≥0}(\binom{2n+1}{n−3i} + \binom{2n+1}{n−3i−2})
= ∑_{\begin{matrix}j≢0 (3) \\ j≥0 \end{matrix}} \binom{2(n−1)+3}{n−1−j}
= ∑_{j≥0} \binom{2(n−1)+3}{n−1−j} − ∑_{i≥0} \binom{2(n−1)+3}{n−1−3i}.

The first sum on the last line equals 2^{n}, while the second sum is odd by the induction hypothesis.

Corollary 8.13. Let

g(i) = \begin{cases}1 & if i > 0, i ≢ 0 (3) \\ 0 & otherwise \end{cases} and h(n) = ∑_{j≥0} g(n − 2j)\binom{n}{j}.

Then h(n) is odd for n ≥ 1. Proof. We work mod 2. For ϵ = 0 or 1, let G_{ϵ}(n) ≡ g(2n + ϵ). Then

(x^{1−ϵ} + x^{2})/(1 + x^{3}) = ∑_{i≥0} G_{ϵ}(i)x^{i}.

Hence, using Lemma 8.12 at the last step, we have

h(2n + ϵ) = ∑_{j≥0} G_{ϵ}(n − j)\binom{2n+ϵ}{j}
≡ coef(x^{n}, (x^{1−ϵ} + x^{2})(1 + x)^{2n+ϵ}/(1 + x^{3}))
≡ coef(x^{n}, x^{1−ϵ}(1 + x)^{2n+1+2ϵ}/(1 + x^{3}))
= coef(x^{n+ϵ−1}, (1 + x)^{2(n+ϵ−1)+3}/(1 + x^{3}))
≡ 1.

Proposition 8.14. Suppose f(n, k) ∈ Z_{2} is defined for n ≥ 0 and k ∈ Z by

f(0, k) = \begin{cases}1 & k < 0, k ≢ 0 (3) \\ 0 & otherwise, \end{cases} f(n, k) = f(n−1, k−1)+f(n−1, k+1).

Then

∑_{\begin{matrix}k>0 \\ k odd \end{matrix}} f(n, k) = 1 for n ≥ 2.

Proof. We begin by using Corollary 8.13 to deduce that f(n, 0) = 1 for n ≥ 1. This is done by noting that f(0, −i) = g(i) of the corollary, and that f(n, 0) = ∑ f(0, −n + 2i)\binom{n}{i}, so that f(n, 0) = h(n) of the corollary.

For n ≥ 2, we have

∑_{\begin{matrix}k>0 \\ k odd \end{matrix}} f(n, k) = ∑_{\begin{matrix}k>0 \\ k odd \end{matrix}} f(n − 2, k − 2) + ∑_{\begin{matrix}k>0 \\ k odd \end{matrix}} f(n − 2, k + 2)
= f(n − 2, −1) + f(n − 2, 1) = f(n − 1, 0) = 1.

This looks like an induction proof, but it isn’t. At the second step, we are using that all except the initial terms appear twice and hence cancel. At the first and third steps we are using the recursive formula for f, and at the last step we use the result of the first paragraph.

Proof of Proposition 8.11. We continue to work mod 2. Define f̃(n, k) = ∑_{t≥0} \binom{2n+1}{n−1−k−3t}. We will show that f̃ satisfies the same formulas that define f of Proposition 8.14, and hence the desired result follows from the conclusion of 8.14.

We first note that f̃(0, k) = ∑_{t≥0} \binom{1}{−1−k−3t}, and this is 1 iff −1 − k ≥ 0 and −1 − k ≡ 0, 1 (3), which is true iff f(0, k) = 1. Finally, we have

f̃(n, k) = ∑_{t≥0}(\binom{2n−1}{n−1−k−3t} + \binom{2n−1}{n−3−k−3t})
= ∑_{t≥0}(\binom{2(n−1)+1}{(n−1)−1−(k−1)−3t} + \binom{2(n−1)+1}{(n−1)−1−(k+1)−3t})
= f̃(n − 1, k − 1) + f̃(n − 1, k + 1).

PERIODIC HOMOTOPY GROUPS OF SO(N) 69

Proof of Lemma 7.7. We are assuming that k and a are fixed integers satisfying ν(k− a) > 2a − 5. There is also the implicit assumption that k > 2a, as discussed after 3.3. Let g(j) = ∑_{i}(−1)^{i}\binom{j}{i}i^{2k−1}. We will prove, in notation of 3.3,

ν(P_{1}(2k − 1, 2a)) ≥ ν((4a − 3)!)  (8.15)
ν(P_{2}(2k − 1, 2a)) = ν((4a − 3)!) − 1  (8.16)
ν(P_{3}(2k − 1, 2a)) ≥ ν(4a − 2)!)  (8.17)
ν(g(j)) ≥ ν((4a − 2)!) for j ≥ 4a − 1. (8.18)

The lemma follows from these results and the definitions.

Proposition 8.1 immediately implies (8.18). The inequality (8.17) is implied by Lemma 8.19. The proofs of (8.15) and (8.16) will occupy the remainder of this section (after the proof of 8.19).

Lemma 8.19. If n is positive, then

∑(−1)^{i}i^{2n}\binom{2n}{n−i} = (−1)^{n}(2n)!/2,  (8.20)

while if d is positive and even, then ∑(−1)^{i}i^{2n+d}\binom{2n}{n−i} is divisible by (2n)!/2.

Proof. Both parts of the lemma utilize the following lemma, which is a standard combinatorial result (e.g. [21, pp. 243-245]). The coefficients in these polynomials are known as Eulerian numbers.

Lemma 8.21. There are polynomials

p_{n}(x) = ∑_{i=1}^{n} a_{i,n}x^{i}

satisfying 1. p_{1}x = x;

2. p_{n}(x) = x(1 − x)p′_{n−1}(x) + nxp_{n−1}(x), where p′ denotes the derivative;

3. ∑_{i≥1} i^{n}x^{i} = p_{n}(x)/(1 − x)^{n+1};

4. a_{i,n} = ia_{i,n−1} + (n − i + 1)a_{i−1,n−1};

5. a_{i,n} = a_{n+1−i,n};

6. p_{n}(1) = n!.

70 BENDERSKY AND DAVIS

Proof. If the polynomials p_{n} are defined by (1) and (2), then (4) is immediate and (6) is easily proved by induction on n. The symmetry property (5) is easily obtained from (4), while (3) is proved by induction on n using that ∑ i^{n}x^{i} = x(∑ i^{n−1}x^{i})′.

To prove (8.20), we note that, by 8.21.3, the left hand side of (8.20) is the coefficient of x^{n} in

(−1)^{n}(1 − x)^{2n} \frac{p_{2n}(x)}{(1 − x)^{2n+1}} = (−1)^{n}p_{2n}(x) ∑_{i≥0} x^{i}.

This coefficient equals

(−1)^{n}(a_{1,2n} + ⋯ + a_{n,2n}) = (−1)^{n} \frac{1}{2}p_{2n}(1) = (−1)^{n} \frac{1}{2}(2n)!, using parts 5 and 6 of 8.21.

The second part of 8.19 is proved by induction on d and n, with the case d = 0 being (8.20) and n = 1 being trivial. Let C(n, d) denote the coefficient of x^{n} in p_{2n+d}(x)/(1 − x)^{d+1}. By 8.21.3, we must prove ν(C(n, d)) ≥ ν((2n)!) − 1. We will prove that, for d ≥ 2 and n > 1,

C(n, d) = n^{2}C(n, d − 2) + 2n(2n − 1)C(n − 1, d),  (8.22)

from which our desired conclusion follows by induction. To prove (8.22), we calculate

C(n, d) = ∑_{i≥0} a_{n−i,2n+d}\binom{d+i}{d}
= ∑_{i≥0}((n − i)^{2}a_{n−i,2n+d−2} + (2(n^{2} − i^{2}) + 2dn − (d + 1)(2i + 1))a_{n−i−1,2n+d−2}
+(n + d + i + 1)^{2}a_{n−i−2,2n+d−2})\binom{d+i}{i}
= ∑_{i≥0} a_{n−i,2n+d−2}((n − i)^{2}\binom{d+i}{i} + (2(n^{2} − (i − 1)^{2}) + 2dn
−(d + 1)(2i − 1))\binom{d+i−1}{d} + (n + d + i − 1)^{2}\binom{d+i−2}{d})
= ∑_{i≥0} a_{n−i,2n+d−2}(n^{2}\binom{d−2+i}{d−2} + 2n(2n − 1)\binom{d+i−1}{d})
= n^{2}C(n, d − 2) + 2n(2n − 1)C(n − 1, d).

At the first step, we made two applications of 8.21.4; other steps were just algebraic manipulation. The equality of coefficients of a_{n−i,2n+d−2} in the next-to-last step can be verified by considering separately terms involving n^{2}, n^{1}, and n^{0}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 71

Next we prove (8.15). Referring to 1.1, we have P_{1}(m, n) = S_{1}(m, n) − S_{2}(m, n), where

S_{1}(m, n) = ∑_{odd ℓ} ℓ^{m} ∑_{i=0}^{n−1−ℓ} \binom{2n−1}{i}
S_{2}(m, n) = 2 ∑_{odd ℓ} ℓ^{m} ∑_{t≥0} \binom{2n}{n−2−ℓ−4t}.  (8.23)

We show that both S_{1} and S_{2} are sufficiently divisible. First note that S_{1} is the same as the sum in [8, (3.19)] with n replaced by n − 1. Using the alternate expression below the middle of [8, p.54], the required divisibility for S_{1}(2k − 1, n) follows from Lemma 8.24. The divisibility of S_{2}(2k − 1, n) is included in Lemma 8.31. These lemmas then will imply (8.15).

Lemma 8.24. If n is even, k > n, j ≥ 2, and ν(n) > ν(k), then

ν((\binom{n+1−j}{j} − \binom{n−1−j}{j−2}) ∑_{i≥j−1} 8^{i}\binom{k−1}{i}f_{j}(i)) ≥ 4j − 3 − α(n − 2),

where

f_{j}(i) := ∑_{t=0}^{j−2}(−1)^{t}\binom{2j−1}{t}(2j − 2t − 1)\binom{j−t}{2}^{i}.

The condition ν(n) > ν(k) here is much less restrictive than the condition ν(k − \frac{n}{2}) > n − 5 of (8.15).

The proof of 8.24 requires the following three lemmas.

Lemma 8.25. The expression f_{j}(i) of 8.24 equals

(2j − 1)! · 2^{−(j−1)} ∑_{|\overline{e}|=i−j+1} ∏_{ℓ=2}^{j} \binom{ℓ}{2}^{e_{ℓ}}.

Here |\overline{e}| = ∑_{ℓ} e_{ℓ}, and the sum is taken over all \overline{e} = (e_{2}, …, e_{j}) with the prescribed |\overline{e}| and e_{i} ≥ 0.

72 BENDERSKY AND DAVIS

Proof. The proof is very similar to that of [8, 4.23]. We show that, for i ≥ j − 1, the system

a_{0}\binom{2}{2} + a_{1}\binom{3}{2} + ⋯ + a_{j−2}\binom{j}{2} = 0
a_{0}\binom{2}{2}^{2} + a_{1}\binom{3}{2}^{2} + ⋯ + a_{j−2}\binom{j}{2}^{2} = 0
⋮
a_{0}\binom{2}{2}^{j−2} + a_{1}\binom{3}{2}^{j−2} + ⋯ + a_{j−2}\binom{j}{2}^{j−2} = 0
a_{0}\binom{2}{2}^{i} + a_{1}\binom{3}{2}^{i} + ⋯ + a_{j−2}\binom{j}{2}^{i} = (2j − 1)!2^{1−j} ∑_{|\overline{e}|=i−j+1} ∏_{ℓ=2}^{j} \binom{ℓ}{2}^{e_{ℓ}}

has solution a_{s−2} = (−1)^{j+s}\binom{2j−1}{j−s}(2s − 1) for 2 ≤ s ≤ j. The last equation is then the content of the lemma. That this solution (or any multiple of it) is the solution of all but the last equation was proved in [8, 4.23], but it seems convenient to prove them all together.

The Vandermonde method easily implies that

det ( x_{1} ⋯ x_{n} ; x_{1}^{2} ⋯ x_{n}^{2} ; ⋮ ; x_{1}^{n−1} ⋯ x_{n}^{n−1} ; x_{1}^{n+k} ⋯ x_{n}^{n+k}) = ∏_{i=1}^{n} x_{i} ∏_{1≤i<j≤n}(x_{j} − x_{i}) ∑_{|\overline{e}|=k} x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}}.

It follows easily by the method of [8, 4.23] (Cramer’s rule) that the solution of the system is as claimed.

In order to estimate the 2-exponent in the sum which occurs in 8.25, we need two more lemmas. The first deals with symmetric polynomials.

Lemma 8.26. Let \{y_{j}\} be a set of indeterminates. Let f_{i} = ∑_{j} y_{j}^{i}. If \overline{e} = (e_{1}, e_{2}, …), let |\overline{e}| = ∑ e_{j}, and for t ≥ 1, let p_{t} = ∑_{|\overline{e}|=t} ∏ y_{j}^{e_{j}} be the sum of all monomials of degree t. Then t!p_{t} is an integral polynomial in {f_{i}}.

Proof. For notational convenience, we rename the y_{i}’s as x_{j,k} with j ≥ 1 and k ≥ 1. Now f_{i} = ∑_{j,k} x_{j,k}^{i}. Let g\overline{m} = g_{m_{1},... ,m_{s}} denote the smallest symmetric polynomial in all the x_{j,k} containing the monomial

∏_{j=1}^{s} ∏_{k=1}^{m_{j}} x_{j,k}^{j}.

Thus, for each j, m_{j} of the indeterminates are raised to the jth power in it. We will show that ∏_{j} m_{j}! · g\overline{m} can be written as a polynomial in the f_{i} with integral coefficients. The lemma follows, since p_{t} is the sum of all g\overline{m} for which ∑ jm_{j} = t, and t! is divisible by each of the relevant coefficients ∏_{j} m_{j}!.

To prove the claim, we begin by showing that the coefficient of g\overline{m} in any f\overline{a} is a multiple of ∏ m_{j}!, and its coefficient in f\overline{m} equals ∏ m_{j}!. Here f\overline{a} = ∏ f_{j}^{a_{j}}. To establish this claim, we note that the desired coefficient is the number of ways of choosing one term from each factor of f\overline{a} so that the product of the selected terms is

x_{1,1} ⋯ x_{1,m_{1}}x_{2,1}^{2} ⋯ x_{2,m_{2}}^{2} · · · .

If the coefficient is nonzero, pick one way of making this choice. Let σ_{j} be a permutation of {(j, 1), …, (j, m_{j})}. Then choosing x_{j,σ_{j}(k)} instead of x_{j,k} yields distinct ways of making this choice. Thus the total number of ways of making the choice is divisible by ∏ m_{j}!. As for the coefficient of g\overline{m} in f\overline{m}, the only choices have the m_{j} x^{j}’s coming from the m_{j} f_{j}’s.

Now order the tuples \overline{m} for which ∑ jm_{j} = t in such a way that if ∑ m_{j} > ∑ m′_{j}, then \overline{m} precedes \overline{m}′. This order is not unique, but it does not matter how the tuples with equal ∑ m_{j} are ordered. Form a matrix A with these tuples in this order labeling the rows and columns. Let the entries in the \overline{m} column be the entries of the various g_{\overline{m}}′’s in the expansion of f\overline{m}. The matrix is lower triangular since if g_{\overline{m}}′ has nonzero coefficient in f\overline{m}, then either \overline{m}′ = \overline{m} or ∑ m′_{j} < ∑ m_{j}. By the claim proved in the previous paragraph, the \overline{m} row is divisible by ∏ m_{j}! and its diagonal entry equals ∏ m_{j}!.

The columns of A^{−1} give the unique way of writing each g\overline{m} in terms of the f^{\overline{m}′}’s. Let B be obtained from A by dividing the g\overline{m} row by ∏ m_{j}!. Then B is an integral triangular matrix with 1’s on its diagonal. Hence so is B^{−1}. But A^{−1} is obtained from B^{−1} by dividing the \overline{m} column by ∏ m_{j}!. The proposition follows.

74 BENDERSKY AND DAVIS

The other lemma needed in the proof of 8.24 is the following result about exponents of 2.

Lemma 8.27. For e ≥ 1 and j ≥ 2,

ν(∑_{k=2}^{j} \binom{k}{2}^{e}) = \begin{cases}0 & if j ≡ 2 mod 4 \\ ν(a) + 1 & if e is even and |j − 4a| ≤ 1 \\ ν(a) + 1 & if e = 1 and j = 4a \\ ν(a) + 2 & if e is odd and j = 4a ± 1 \\ ν(a) + 2 & if e > 1 is odd and j = 4a. \end{cases}

Proof. If e = 1, the sum equals \binom{j+1}{3}, from which the result follows easily.

Let e > 1. The proof is by induction on j. By consideration of the next term added, it is easy to see that validity for j = 4a − 1 implies validity for j = 4a, 4a + 1, and 4a + 2. We will prove, for t ≥ 2,

ν( ∑_{k=0}^{2^{t}−1} \binom{2^{t+1}b + k}{2}^{e}) = \begin{cases}t − 1 & if e even \\ t & if e odd. \end{cases}  (8.28)

Then the case b = 0 of (8.28) implies the lemma for 4a − 1 = 2^{t} − 1, while the lemma for 4a − 1 = 2^{t+1}b + 2^{t} − 1 with b > 0 follows from the case 4a − 1 = 2^{t+1}b − 1 of the lemma and (8.28).

It remains to prove (8.28). We prove it by induction on t, and assume it proven for t − 1. We work mod 2^{t+1}. Combining the summands for k = 2ℓ and 2ℓ + 1, the desired sum equals

∑_{ℓ=0}^{2^{t−1}−1}(2^{t}b+ℓ)^{e}((2^{t+1}b+2ℓ−1)^{e}+(2^{t+1}b+2ℓ+1)^{e}) ≡ ∑_{ℓ=0}^{2^{t−1}−1} ℓ^{e}((2ℓ−1)^{e}+(2ℓ+1)^{e}).

One can verify that the summands for ℓ = 2^{t−2} − s and ℓ = 2^{t−2} + s are congruent. By this symmetry, the desired sum becomes

2 ∑_{ℓ=0}^{2^{t−2}−1} ℓ^{e}((2ℓ − 1)^{e} + (2ℓ + 1)^{e}).

This sum, without the factor of 2, is, by induction, congruent mod 2^{t} to ±2^{t−2} if e is even and to 2^{t−1} if e is odd. Doubling this yields our claim.

Now we can prove Lemma 8.24.

PERIODIC HOMOTOPY GROUPS OF SO(N) 75

Proof of Lemma 8.24. Let g_{d}(j) = ∑_{|\overline{e}|=d} ∏_{ℓ=2}^{j} \binom{ℓ}{2}^{e_{ℓ}}. Since \binom{n+1−j}{j} − \binom{n−1−j}{j−2} = \frac{n}{j} \binom{n−1−j}{j−1} and \binom{k−1}{i} = \frac{i+1}{k} \binom{k}{i+1}, our desired inequality is implied by the statement that, for all i ≥ j − 1,

ν(\frac{n}{j} \binom{n−1−j}{j−1}) + 3i + ν(i + 1) − ν(k) + ν\binom{k}{i+1}  (8.29)
+j − α(2j − 1) + ν(g_{i−j+1}(j)) ≥ 4j − 3 − α(n − 2).

We have also used 8.25. Using the hypothesis that ν(n) > ν(k) and well-known formulas for ν\binom{n−1−j}{j−1} and α(2j − 1), and removing the nonnegative quantity ν\binom{k}{i+1}, (8.29) will be implied by 1 − ν(j) + α(j − 1) + α(n − 2j) − α(n − 1 − j) + 3i

+ν(i + 1) − α(j − 1) − 1 + ν(g_{i−j+1}(j)) ≥ 3j − 3 − α(n − 2).

Next we note that since α(n − 2j) + α(n − 2) − α(n − 1 − j) = ν\binom{2n−2−2j}{n−2j} ≥ 0, this inequality will be implied by

ν(g_{i−j+1}(j)) ≥ ν(j) − ν(i + 1) − 3(i − j + 1).  (8.30)

This inequality is true (0 ≥ 0) if i = j − 1. If i > j − 1, then it is certainly true unless ν(j) > 3. Let d = i − j + 1 > 0 and j = 4a. By Lemma 8.26, g_{d}(j) equals \frac{1}{d!} times an integral polynomial in S_{e}(j) := ∑_{k=2}^{j} \binom{k}{2}^{e} for various e > 0. By Lemma 8.27, ν(S_{e}(j)) ≥ ν(a) + 1. Thus ν(g_{d}(j)) ≥ ν(a) + 1 − ν(d!). Hence (8.30) follows from the observation that for d > 0

3d ≥ ν(d!) + 1.

Finally we prove (8.16). Reverting to the notation of 1.1, we will show that if n is even and ν(m+1−n) > n−4, then ν(P_{2}(m, n)) = ν((2n−3)!)−1 = 2n−5−α(n−2). The hypothesis implies that ν(m + 1) = ν(n) ≤ n − 3 (for n > 4). Now P_{2}(m, n) is the sum of two terms. The first has the same 2-divisibility as \frac{1}{2}S_{2}(m, n) of (8.23), while, by 8.11, the exponent of 2 in the second is 2n − 4 − ν(n). This latter is ≥ 2n − 5 − α(n − 2), with equality if and only if n is a 2-power. Thus (8.16) follows from Lemma 8.31, which will complete the proof of 7.7.

Lemma 8.31. If n is even and ν(m + 1 − n) > n − 4, then

ν(S_{2}(m, n)) \begin{cases}> 2n − 4 − α(n − 2) & if n is a 2-power \\ = 2n − 4 − α(n − 2) & if n is not a 2-power. \end{cases}

Proof. We use the expression of S_{2}(m, n) given (with minor notational modifications) in [8, (4.20)]. With f_{j}(i) as in 8.24, the lemma will follow from the statement that

ν(\binom{n−j−1}{j} ∑_{i≥j−1} 8^{i}\binom{(m−1)/2}{i}f_{j}(i)) ≥ 4j − 3 − α(n − 2)

with equality iff n is not a 2-power and j = (n − 2^{ν(n)})/2. With d = i − j + 1 and g_{d}(j) as in the proof of 8.24, this will follow from, for n even, 2 ≤ j < n/2, and d ≥ 0,

ν\binom{n−j−1}{j} + ν\binom{m−1}{2j−2+2d} + 3d − α(2j − 1) + ν(g_{d}(j)) + α(n − 2)

\begin{cases}= 1 & n = 2^{e}, j = 2^{e−2}, and d = 0 \\ > 1 & n = 2^{e}, j, d otherwise \\ = 0 & α(n) > 1, j = (n − 2^{ν(n)})/2, and d = 0 \\ > 0 & α(n) > 1, j, d otherwise. \end{cases}  (8.32)

Here we have used 8.25 to relate f_{j}(i) and g_{d}(j).

We use ν\binom{a}{b} = α(b) + α(a − b) − α(a), ν(a!) = a − α(a), α(a − 1) = α(a) − 1 + ν(a), α(2a) = α(a), and α(2^{e} − k) = e − α(k − 1) without comment. For our first simplification of the LHS of (8.32), we note that m − 1 = n − 2 + Δ with Δ highly 2-divisible. Then ν\binom{m−1}{2j−2+2d} = ν\binom{n−2}{2j−2+2d} unless 2j + 2d > n, in which case d > 0, and the inequalities of (8.32) are easily established, for ν\binom{m−1}{2j−2+2d} will be roughly ν(Δ). Thus the LHS of (8.32) becomes ν\binom{n−j−1}{j} + α(j − 1 + d) + α(n − 2j − 2d) + 3d − α(j) − ν(j) + ν(g_{d}(j)). (8.33)

If n = 2^{e}, then (8.33) equals e − 1 + 3d − ν(j) + ν(g_{d}(j)). Since ν(g_{d}(j)) ≥ 0 and g_{0}(j) = 1, and j < 2^{e−1}, the first two cases of (8.32) follow.

Next we consider the case where α(n) > 1 and d = 0. In this case, (8.33) becomes

α(n − 1 − 2j) − α(n − 1 − j) + α(j) − 1 + α(n − 2j)

= α(n − 1 − 2j) − α(2n − 1 − 2j) + α(2j) + α(n − 2j)
= ν(n − 1 − 2j, 2j, n − 2j),

PERIODIC HOMOTOPY GROUPS OF SO(N) 77

where the last expression denotes the exponent of 2 in a multinomial coefficient. This exponent is ≥ 0 and is 0 iff the binary expansions of n − 1 − 2j, 2j, and n − 2j are disjoint. One readily verifies that this is the case iff n = 2^{e} + A2^{e+1} with A > 0 and 2j = A2^{e+1}.

If d = 1, then (8.33) equals

ν\binom{n−j−1}{j} + 3 − ν(j) + α(n − 2j − 2) + ν(g_{1}(j)),

which could be ≤ 0 only if j ≡ 0 mod 4, in which case ν(g_{1}(j)) ≥ ν(j) − 1, by the argument at the end of the proof of 8.24. So (8.33) is positive in this case.

The case d = 2 is handled similarly. This time (8.33) equals

ν\binom{n−j−1}{j} + 7 − ν(j) − ν(j + 1) + α(n − 2j − 4) + ν(g_{2}(j)). This could possibly be ≤ 0 only if j or j + 1 is highly 2-divisible, in which case ν(g_{2}(j)) ≥ max(ν(j), ν(j + 1)) − 2.

Finally we consider the case d ≥ 3. This case is different because (8.33) has a term −ν(j + 2) which could be very negative without compensation from ν(g_{3}(j)), because of the way 8.27 comes out when j ≡ 2 mod 4. For any d, (8.33) equals ν\binom{n−j−1}{j} + 4d − 1 + α(n − 2j − 2d) − ν(j ⋯ (j + d − 1)) + ν(g_{d}(j)). (8.34)

This is positive unless, perhaps, ν(j + δ) is very large for some δ satisfying 1 < δ ≤ d − 1. If so, let j + δ = B2^{k+1} + 2^{k}, ϵ = d − δ, and n = 2(j + δ) + D. We have 0 < ϵ < d and D ≥ 0. Now (8.34) is ≥ α(B)−α(δ−1)+α(D+2δ−1)+3d−α(j+2δ+D−1)+α(D−2ϵ)+α(d−1). We drop the nonnegative term α(d − 1), replace j + δ by B2^{k+1} + 2^{k} in the fifth term, and add 0 in the guise of α(D+δ−1)−α(2D+2δ−1−2ϵ)−(2ϵ−1)+ν((2D+2δ−2ϵ) ⋯ (2D+2δ−2)), and replace this last ν(−) by ϵ, which it certainly exceeds. We obtain now that (8.34) is

≥ 3d − α(δ − 1) − (2ϵ − 1) + ν\binom{2D+2δ−1−2ϵ}{D+2δ−1} + α(B) + ϵ
− α(B2^{k+1} + 2^{k} + D + δ − 1) + α(D + δ − 1).  (8.35)

78 BENDERSKY AND DAVIS

Now we write D + δ − 1 = C2^{k+1} + E2^{k} + F with E = 0 or 1 and 0 ≤ F < 2^{k}. The sum of the last two terms of (8.35) is ≥ α(C) + E − α(B + C) − 1. Thus (8.34) is

≥ 3d − α(δ − 1) − ϵ + ν\binom{2D+2δ−1−2ϵ}{D+2δ−1} + E + ν\binom{B+C}{B}. The only negative terms are much smaller than 3d, completing the proof that (8.34) is positive. This completes the proof of 8.31.

9. Comparison with J-homology approach

In the late 1980’s, the second author and Mahowald attempted to compute the groups v_{1}^{−1}π_{∗}(SO(n); 2) by using charts for v_{1}^{−1}π_{∗}(S^{2m+1}; 2) derived from J-homology, and exact sequences of fibrations. In [27], this approach was applied to obtain mod 2 v_{1}-periodic homotopy groups^{8} of SO(n) for n = 5, 7, and 9, and in [26], it was used to compute v_{1}^{−1}π_{∗}(Sp(2); 2) and v_{1}^{−1}π_{∗}(Sp(3); 2). In this section, we use our BTSS results here to draw some conclusions about this J-homology approach to v_{1}-periodic homotopy groups.

The J-homology approach is simpler for SO(2n+1) than for SO(2n+2). The latter has more interacting towers than does the former. It seems very difficult, at best, to see from the J-homology approach that v_{4k−2}(SO(2n + 1)) consists of exactly two cyclic summands plus a certain number of Z/2’s associated to multiplications by the Hopf map η. (Here we have initiated the abbreviation v_{∗}(−) for v_{1}^{−1}π_{∗}(−), which we will utilize throughout this section.) On the other hand, this is readily apparent from the BTSS charts 1.3 and 3.7. The small third summand in v_{4k+2}(SO(4a)), described explicitly in Theorems 3.3, 3.4, and 3.14, results, in the J-homology approach, from some complicated interaction of the towers, but seems virtually impossible to deduce from that perspective. So we restrict our comparisons here to SO(2n + 1).

The J-homology approach builds a chart for v_{∗}(SO(2n+1)) from those of v_{∗}(SO(2n− 1)) and v_{∗}(V_{2n+1,2}) using the exact sequence associated to the fibration

SO(2n − 1) → SO(2n + 1) → V_{2n+1,2}.  (9.1)

A chart for V_{2n+1,2} can be obtained from the fibration S^{2n−1} → V_{2n+1,2} → S^{2n},

8Mod 2 does not mean (integral) 2-primary periodic homotopy groups. Mod 2 does not contain the important information about large 2-torsion summands.

PERIODIC HOMOTOPY GROUPS OF SO(N) 79

using charts of v_{∗}(S^{2n−1}) and v_{∗}(S^{2n}), such as those of [26]. We obtain as a chart for v_{∗}(V_{2n+1,2}) a sum over all integers k of Diagram 9.2. Our filtration convention is to use a filtration-preserving isomorphism

v_{∗}(S^{2n+1}) ≈ v_{1}^{−1}J_{∗}(Σ^{2n+1}P^{2n}).

This puts many elements in the chart for v_{∗}(S^{2n+1}) in filtration one less than their Adams filtration; e.g. η has filtration 0. The differentials between adjacent towers indicated in the diagram might not be quite accurate when they are near their maximum value. The indicated formula is for the differential in S^{4n−1}. The towers in V_{2n+1,2} are slightly taller than those of S^{4n−1}; we make no assertion about the differential in cases when it is 0 in S^{4n−1}. The big dots establish the coordinates for the two parts of the diagram.

Diagram 9.2. A summand of v_{∗}(V_{2n+1,2})

[figure: ∈ (2n + 8k, 4k); ⊕; ∈ (8k − 2, 4k + 1 − 2n); d_{ν(4k−2n)}; towers; are Z/2^{2n}; d_{ν(4k+2−2n)}]

The difficult part in computing v_{∗}(SO(2n + 1)) is the determination of the boundary morphism

v_{∗+1}(V_{2n+1,2}) ⟶^{∂_{∗}} v_{∗}(SO(2n − 1))

and the extensions in forming v_{∗}(SO(2n + 1)) from coker(∂_{∗}) and ker(∂_{∗−1}).

80 BENDERSKY AND DAVIS

In a 1988 e-mail to the second author, Mahowald wrote “In SO(n), there are two phenomena going on at the same time. The first deals with the ‘stable’ stuff in the sphere, and this just makes up the metastable homotopy of the stunted projective space like the Barratt-Mahowald theorem says. The unstable ‘S^{4n−1}’ which goes with each S^{2n} is busy making up the stable homotopy of SO. It does so in a way very similar to Sp.”

The “stable stuff” to which he refers is essentially the way that the left parts of Diagram 9.2 build up and go out, which is indeed very similar to the way in which v_{1}^{−1}J_{∗}(P_{2a+1}^{2b}) is built from v_{1}^{−1}J_{∗}(P_{2i−1}^{2i}) for a < i ≤ b. The “unstable” part is the way in which the right parts of Diagram 9.2 interact.

The “stable stuff” mainly builds the regular second summands of E_{2}^{1,4k−1}(Spin(2n+ 1)) (Theorem 3.1) together with the occasional d_{3}-differential on them and the occasional extension on them into E_{2}^{3}(Spin(2n + 1)), as described in 1.4 and 3.8. It also involves the eta towers which begin in filtration 1 in Diagrams 1.3 and 3.7. Throughout this section, we talk about the BTSS of Spin(2n + 1) and the (J_{∗}-approach) chart for SO(2n + 1), keeping in mind that v_{∗}(Spin(m)) ≈ v_{∗}(SO(m)).

When the differentials and extensions are taken into account, the morphisms of these second summands of v_{4k−2}(SO(2n + 1)) are

2n + 1 = 8a − 1 8a + 1 8a + 3 8a + 5 8a + 7
n ≤ ν(k) + 3 Z/2^{4a−1} ⟶^{·4} Z/2^{4a} ↪ Z/2^{4a+2} ⟶^{·4} Z/2^{4a+3} ⟶^{·4} Z/2^{4a+3}

(9.3)

ν(k) + 3 < n Z/2^{e} ⟶^{·2} Z/2^{e} ⟶^{≈} Z/2^{e} ⟶^{·2} Z/2^{e} ⟶^{·4} Z/2^{e},

(9.4)

where e = ν(k)+4. These groups and homomorphisms agree exactly with v_{1}^{−1}π_{4k−1}^{s}(P_{2n+1}^{4n+1}). This is consistent with, but probably not implied by, the Barratt-Mahowald theorem to which Mahowald alluded in his 1988 e-mail. The Barratt-Mahowald theorem ([3]) states that, if q < 2(m − 1) and m ≥ 13, then

π_{q}(SO(m)) ≈ π_{q}(SO(2m)) ⊕ π_{q+1}(V_{2m,m}),

i.e. that the homotopy sequence of the fibration ΩV_{2m,m} → SO(m) → SO(2m)

PERIODIC HOMOTOPY GROUPS OF SO(N) 81

splits in this range.

Because the Barratt-Mahowald theorem only makes a statement about homotopy groups in a limited range of dimensions, while v_{1}-periodic homotopy groups depend on all homotopy groups, one cannot really use it to draw a conclusion about

v_{1}^{−1}π_{∗+1}(V_{2m,m}) → v_{1}^{−1}π_{∗}(SO(m)).

Moreover, the relationship with v_{1}^{−1}π_{∗+1}^{s}(P_{m}^{2m−1}) is via the stable splitting map of James ([31]), V_{2m,m} ⟶^{j}QP_{m}^{2m−1}, which induces a homomorphism

v_{1}^{−1}π_{∗}(V_{2m,m}) ⟶^{j_{∗}} v_{1}^{−1}π_{∗}^{s}(P_{m}^{2m−1}) ≈ v_{1}^{−1}J_{∗}(P_{m}^{2m−1}).

Our observation is that, with m = 2n+1, for n ≥ 6 and all integers k, v_{1}^{−1}π_{4k−1}^{s}(P_{2n+1}^{4n+1}) is isomorphic to the regular summand of v_{1}^{−1}π_{4k−2}(SO(2n+1)), and both are mapped to from v_{1}^{−1}π_{4k−1}(V_{4n+2,2n+1}). Note that, by Proposition 11.4, it is apparently not true that v_{1}^{−1}π_{4k−2}^{s}(P_{2n+1}^{4n+1}) appears as a direct summand in v_{1}^{−1}π_{∗}(SO(2n + 1)); i.e. the splitting is valid in certain periodic homotopy groups but not others.

Building charts for v_{∗}(SO(2n + 1)) inductively using the fibrations (9.1) and the charts 9.2 is a complicated business. The pattern of differentials and extensions involving the interacting towers from the right side of Diagram 9.2 is particularly inscrutable. Another delicate matter is the pattern by which the short eta-towers ([figure] ) in Diagram 9.2 cancel out. We will use our BTSS work to show the way in which the regular (second) summands of v_{4k−2}(SO(2n+1)) (the ones described in the preceding paragraphs) are built, and the pattern of differentials among the eta-towers. Two complicating factors are that the charts for v_{∗}(SO(2n + 1)) are particularly crowded when n is small, and an anomaly for SO(9) noted in [8, 4.21].

The cases Spin(3) = S^{3}, Spin(5) = Sp(2), and Spin(7) ≃_{2} S^{7} × G_{2} have been dealt with thoroughly in [26] and [9]. A comparison of the J-homology approach and BTSS approach was useful for Spin(7) in [9]. We begin with Spin(9).

The BTSS of Spin(9) is essentially given in Diagram 1.3. The big • there represents Z_{2} ⊕ Z_{2} (e.g. by 5.14). The anomaly occurs in the 1-line, which is given in [8, 4.21] to be

Z/2^{min(ν(k−2)+4,8)} ⊕ Z/8 in t − s = 8k − 2
Z/2^{min(ν(k−1)+4,6)} ⊕ Z/8 in t − s = 8k + 2.

82 BENDERSKY AND DAVIS

In Diagram 9.5, we build a chart for v_{∗}(Spin(9)) from those of Spin(7) (in •) and V_{9,2} (in ∘). On the left side of Diagram 9.5, d_{1}-differentials, i.e. boundary morphisms in the exact sequence of (9.1), have been inserted, deducible from the action of Sq^{2} or from the proof of 6.2. The remaining classes are redrawn on the right side of the diagram, with some exotic h_{0}- and h_{1}-actions, deducible from Toda bracket considerations, inserted, together with one less obvious d_{1}. The higher differential between adjacent towers in 8k + 2 and 8k + 1 is the same as in S^{7}, while that between towers in 8k − 2 and 8k−3 is like that in S^{15} (which is related to S^{8} and hence to V_{9,2}) as far as it goes. As mentioned above, with the extra height obtained from the exotic extension, the differential may still be nonzero when it is zero in S^{15}, and BTSS methods seem to be the only way of determining this. The differential between these towers suggested in Diagram 9.5 is only schematic; depending on the specific value of k, it may be a short or long differential or 0. The E_{∞}-terms from the BTSS and chart approaches are easily seen to be consistent; i.e., they give the same groups v_{∗}(SO(9)).

Diagram 9.5. Periodic homotopy of SO(9)

[figure: s=4k−4; 8k − 3; 8k − 3; 8k; 8k; 8k + 2; 8k + 2]

If we perform a similar transition from SO(9) to SO(11), the situation for the towers, i.e., the differentials and extensions between 4k − 2 and 4k − 3, becomes more complicated. Indeed, it seems that they cannot be understood without the BTSS.

84 BENDERSKY AND DAVIS

We will not pursue that picture here. Instead, we will illustrate how the regular summands in v_{4k−2}(SO(2n + 1)), given in (9.3) and (9.4), are obtained from the Jhomology point of view. We will focus primarily on the situation when n ≤ ν(k) + 3, and, for this, we can consider the case k = 0 (since ν(0) = ∞), so we are looking at v_{−2}(SO(2n + 1)). These cases (n ≤ ν(k) + 3) are particularly nice because the first summand grows regularly, too. After differentials and extensions in the BTSS are taken into account, the pattern for the first summand of v_{8k−2}(SO(2n + 1)) is

2n + 1 = 8a − 1 8a + 1 8a + 3 8a + 5 8a + 7

Z/2^{4a−1} ↪ Z/2^{4a} ⟶^{≈} Z/2^{4a} ↪ Z/2^{4a+1} ↪ Z/2^{4a+3},

as is easily seen from Theorems 3.1, 1.2, and 3.8. We emphasize that this is only true when n ≤ ν(k) + 3.

We have just seen how the anomalous Z/8 in v_{−2}(SO(9)) has been obtained from SO(7). (According to the general pattern in (9.3), it should have been Z/16.) In forming v_{∗}(SO(2n + 1)) from V_{2n+1,2}, the chart for S^{4n−1} is involved; it has d_{ν(n)+1} between the towers in −2 and −3.

In Diagram 9.6, we show how the charts of SO(11), SO(13), SO(15), and SO(17) must be formed in the vicinity of v_{−2}(−). For the portion labeled SO(2n + 1), the • are the chart for SO(2n − 1) and the ∘ are the relevant part of V_{2n+1,2}. The resulting chart then becomes the • in the subsequent chart.

Diagram 9.6. Various v_{∗}(SO(2n + 1)) near ∗ = −2

[figure: SO(11); SO(13); SO(15); SO(17); −2; −2; −2; −2]

There is some irregular behavior for the extensions in the first two cases. In order to obtain v_{−2}(SO(11)) ≈ Z/2^{4} ⊕ Z/2^{6}, which we know it to be by the BTSS, the extension must be 4G = 2g_{32} + g_{8}, as indicated. Then g_{32} − 2G generates the Z/16. In the input to the chart for SO(13), we elevate the filtration of g_{32} − 2G by 1 unit. To obtain v_{−2}(SO(13)) ≈ Z/2^{5} ⊕ Z/2^{7}, the generator of the Z/2^{5} must be from the ∘ in im(η^{2}) in v_{8k−2}(V_{13,2}). In order to accommodate this in the chart, we raise the filtrations of the Z/16 by 1. After this, the pattern for forming v_{−2}(SO(2n + 1)), n ≥ 7, is quite regular. There are classes in low filtration in v_{−3}(−) which have been omitted from the charts. They will play the role of being hit by differentials in forming v_{∗}(SO(2n + 1)) for n ≥ 9.

86 BENDERSKY AND DAVIS

If it is not the case that n ≤ ν(k) + 3, then the pattern of forming the lower summand of Diagram 9.6, the one that maps as in (9.4), is similar except that the number of classes added to the bottom of the tower will cycle mod 4 as 0, 1, 2, 1, rather than always being 2. This is primarily due to the differential in v_{8k−2}(S^{4n−1}) being different than it was when k = 0. The pattern of growth of the upper summand of 9.6 when ν(k) + 3 < n is much more irregular at the bottom.

Finally we describe the way the eta towers are born and die from the J-homology perspective. This is somewhat similar to the situation for Sp(n) pictured in [11, 6.6].

The chart for S^{2n+1} has four short eta towers, which we label as s or u for stable/unstable and o or e for odd/even, where odd/even refers to the definition at the beginning of Section 6. The form of the chart is given in Diagram 9.7, where

p = \begin{cases}o & if n is odd \\ e & if n is even \end{cases}

and p′ ≠ p is the opposite.^{9} This can be seen by comparison with charts of [7, p.488]; the initial Z_{2} in filtration 1 in those charts corresponds to the part of Diagram 9.7 labeled sp, while the filtration-1 group labeled ν in [7, p.488] corresponds to the part of 9.7 labeled up′.

Diagram 9.7. Form of a chart for v_{∗}(S^{2n+1})

[figure: sp; sp′; up; up′]

The chart 9.2 for v_{∗}(V_{2n+1,2}) is formed from charts of S^{2n−1}, ΣS^{2n−1}, and S^{4n−1}, where ΣS^{2n−1} means the chart of S^{2n−1} pushed 1 unit to the right. Its eta towers are labeled by n_{t}^{p}, n̄_{t}^{p}, and n^{=}_{t}^{p}, where n is the integer in V_{2n+1}, the number of bars tells whether it comes from S^{2n−1} (no bars), ΣS^{2n−1} (one), or S^{4n−1} (two), p = o or e, and t = u or s, each of these corresponding to its label on the sphere from which it came. One can check that the labels on the two parts of Diagram 9.2 are as in Diagram 9.8.

9p stands for parity.

Diagram 9.8. Labels on v_{∗}(V_{2n+1,2})

[figure: n even; n odd; n_{s}^{e}; n̄_{s}^{e}; n_{s}^{o}; n̄_{s}^{o}; n̄_{u}^{e}; n̄_{u}^{o}; ^{n^{=}{e \atop u}}; n^{=}_{u}^{o}; n_{s}^{o}; n̄_{s}^{o}; n_{s}^{e}; n̄_{s}^{e}; n_{u}^{o}; n_{u}^{e}; ^{n^{=}{e \atop u}}; n^{=}_{u}^{o}]

The way v_{∗}(SO(7)) is formed from v_{∗}(SO(5)) and v_{∗}(V_{7,2}) is somewhat anomalous, and is consistent with the labeling given in Diagram 9.9 plus the same thing displaced by (−1, −2) units with parities reversed.

Diagram 9.9. Labeling a chart for v_{∗}(SO(7))

[figure: ¯3_{s}^{e}; 3_{u}^{o}; 3^{=}_{u}^{e}; ^{2^{=}{e \atop u}}]

We will abbreviate each little eta tower by a single dot. For the n^{=}_{u}^{p} classes, we will use the final dot ([figure]), while for the others we will use the middle dot ([figure]) or ([figure]). The left side of Diagram 9.5 would have its eta towers as indicated in Diagram 9.10.

Diagram 9.10. Eta towers in forming v_{∗}(SO(9))

[figure: 8k; 4k − 4; ^{4^{=}{o \atop u}}; ^{3^{=}{o \atop u}}; ^{4^{=}{e \atop u}}; ^{3^{=}{e \atop u}}; 4_{s}^{e} 2^{=}_{u}^{o}; ¯4_{u}^{o}; 3_{u}^{e}; 3_{u}^{o}; ¯4_{s}^{o}; ¯3_{s}^{e}; 4_{s}^{o} 2^{=}_{u}^{e} ¯3_{s}^{o}; ^{¯4_{u}^{e} ¯4_{s}^{e}}]

The stable classes D in η_{i}(2n+1) in Table 6.1 go to 0 in η_{i}(2n+3). They correspond to the classes with subscript s in Diagram 9.8. The way they are born and die is depicted in Diagram 9.11. Of course, this table should be extended down according to the same pattern; by periodicity, it would be equivalent to extend it to the right.

Diagram 9.11. How the stable classes come and go

[figure: 8_{s}^{o}; 9_{s}^{e}; 10_{s}^{o}; 11_{s}^{e}; ¯7_{s}^{e}; ¯8_{s}^{o}; ¯9_{s}^{e}; ^{\overline{10}{o \atop s}}; 4_{s}^{e}; 5_{s}^{o}; 6_{s}^{e}; 7_{s}^{o}; ¯3_{s}^{o}; ¯4_{s}^{e}; ¯5_{s}^{o}; ¯6_{s}^{e}; 4_{s}^{o}; 5_{s}^{e}; 6_{s}^{o}; 7_{s}^{e}; ¯3_{s}^{e}; ¯4_{s}^{o}; ¯5_{s}^{e}; ¯6_{s}^{o}; ⋮; 8k; 4k − 4]

If n_{s}^{p} ← (n + 1)_{s}^{p′} appears in this chart (perhaps with bars over n and n + 1), it means that the chart for v_{∗}(SO(2n + 1)) has [figure]with center dot in the position of n_{s}^{p},

PERIODIC HOMOTOPY GROUPS OF SO(N) 89

but in the exact sequence

v_{∗+1}(V_{2n+3,2}) → v_{∗}(SO(2n + 1)) → v_{∗}(SO(2n + 3))

it is hit by the classes from the “lightning flash” part of v_{∗}(V_{2n+3,2}) as depicted in Diagrams 9.2 and 9.8. The reason that the differentials look horizontal has to do with our denoting [figure]by its middle dot. A differential of the form

[figure]

would be indicated by a horizontal arrow since its middle dots are at the same height.^{10} That these differentials are as claimed can be deduced from the proof of Cases 3 and 4 of Theorem 6.2.

The classes labeled x_{2a−3} in Table 6.1 are born from v_{∗}(SO(4a − 1)) → v_{∗}(V_{4a−1,2}) and die from v_{∗+1}(V_{4a+3,2}) → v_{∗}(SO(4a + 1)). The classes in v_{∗}(V_{2m+1,2}) which cause their birth and death are analyzed in Case 5 of the proof of Theorem 6.2. The way in which the relevant portions of the charts of V_{4a−1,2} and V_{4a+3,2} are combined is pictured in Diagram 9.12. The classes (2a − 1)_{u}^{e} are present in v_{∗}(SO(4a − 1)) and v_{∗}(SO(4a + 1)).

Diagram 9.12. How the x_{2a−3} classes come and go

[figure: (2a + 1)_{u}^{o}; (2a − 1)_{u}^{e}; (2a − 1)_{u}^{o}]

Diagram 9.13 depicts the way in which all these eta towers come and go, representing each eta tower by its middle dot. Note that an element labeled (2a−1) which is hit

10The reason that the differentials from ¯4_{s}^{p} to ¯3_{s}^{p′} look diagonal in Diagram 9.10 and horizontal in Diagram 9.11 is that in forming v_{∗}(SO(7)) from v_{∗}(SO(5)) and v_{∗}(V_{7,2}), the dot that would be labeled ¯3_{s}^{p} becomes [figure], i.e. it is no longer in the middle.

90 BENDERSKY AND DAVIS

by an arrow from an element labeled (2a+1) is in v_{∗}(SO(4a−1)) and v_{∗}(SO(4a+1)), while an element from which an arrow emanates is not in any v_{∗}(SO(2n + 1)). The “stability” of these classes is mixed, as they are listed as stable in Table 6.1 but unstable regarding their relationship with v_{∗}(V_{2m+1,2}).

Diagram 9.13. Eta towers corresponding to x_{2a−3} classes

[figure: ⋮; 8k; 4k − 5; 3_{u}^{e}; 5_{u}^{e}; 7_{u}^{e}; 5_{u}^{o}; 7_{u}^{o}; 9_{u}^{o}]

We have accounted now for the killing of classes 3_{u}^{e}, 4_{s}^{o}, and 4_{s}^{e} in Diagram 9.10 of v_{∗}(SO(9)). The dot 3_{u}^{o} is not killed. It represents the Z_{2} in v_{8a}(SO(n)) and v_{8a+1}(SO(n)) for all n ≥ 7; i.e. it is the only Z_{2} pair that is stable in this sense, in accordance with Bott periodicity.

Finally we account for the coming and going of the unstable classes in η_{i}(2n + 1) in Table 6.1. These are the elements in C[a, b] and K[a, b] there. We denote by x_{i}^{o} (resp. x_{i}^{e}) an element of K[a, b] (resp. C[a, b]) with i satisfying Proposition 5.6. For η_{i}(2n − 1), the a and b here must be as in Table 6.1, i.e. a = [n/2] and b = n − 1. From the work in Section 6, we deduce the following result. When we say a class “dies after” Spin(2n + 1), we mean it is present in v_{∗}(Spin(2n + 1)) but goes to 0 in v_{∗}(Spin(2n + 3)). The class “from” which an element is born or dies is the class in v_{∗}(V_{2m+1,2}) which maps to or from it in the exact sequence associated to (9.1).

PERIODIC HOMOTOPY GROUPS OF SO(N) 91

Proposition 9.14. 1. For p = o or e, x_{2^{t}}^{p} is born on Spin(2^{t+1} + 3)

from \overline{\overline{(2^{t} + 1)}}_{u}^{p} and dies after Spin(2^{t+2} + 3) from \overline{(2^{t+1} + 2)}_{u}^{p}.

2. If a ≠ 2^{t}, then x_{a}^{o} is born on Spin(2a + 3) from \overline{\overline{(a + 1)}}_{u}^{o} and dies after Spin(2a + 2^{ν(4a)} + 1) from \overline{\overline{(a + 2^{ν(2a)} + 1)}}_{u}^{e}.

3. The class x_{3·2^{t}}^{e} is born on Spin(3 · 2^{t+1} + 3) from \overline{\overline{(3 · 2^{t} + 1)}}_{u}^{e} and dies after Spin(3 · 2^{t+2} + 3) from \overline{(3 · 2^{t+1} + 2)}_{u}^{e}.

4. If a ≠ 2^{t} or 3 · 2^{t}, then x_{a}^{e} is born on Spin(4a + 5 − 2^{ν(8a)}) from \overline{(2a + 2 − 2^{ν(4a)})}_{u}^{o} and dies after Spin(4a + 3) from \overline{(2a + 2)}_{u}^{e}.

This information is depicted through a range in Diagram 9.15, which, when combined with Diagrams 9.11 and 9.13 and the single stable eta tower 3_{u}^{o}, tells the role of all the eta towers from all V_{2n+1,2} with n ≥ 4 together with those of SO(7) in forming v_{∗}(SO(2n + 1)) for all n, from the J-homology point of view. These classes we call purely unstable, since they are labeled as unstable both in 6.1 and 9.2. We omit the subscript u in this table. The right half of Diagram 9.15 should really be beneath the left half.

Diagram 9.15. How the purely unstable classes come and go

[figure: ^{4^{=}^{o}}; ^{5^{=}^{o}}; ^{6^{=}^{o}}; ^{7^{=}^{o}}; ^{2^{=}^{o}}; ^{3^{=}^{o}}; ^{2^{=}^{e}}; ^{3^{=}^{e}}; ^{4^{=}^{e}}; ^{5^{=}^{e}}; ^{6^{=}^{e}}; ^{7^{=}^{e}}; ^{8^{=}^{e}}; ^{\overline{4}^{o}}; ^{\overline{6}^{o}}; ^{\overline{8}^{o}}; ^{\overline{10}^{o}}; ^{\overline{12}^{o}}; ^{\overline{14}^{o}}; ^{\overline{4}^{e}}; ^{\overline{6}^{e}}; ^{\overline{8}^{e}}; ^{\overline{10}^{e}}; ^{\overline{12}^{e}}; ^{\overline{14}^{e}}; ^{\overline{16}^{e}}; ^{8^{=}^{o}}; ^{9^{=}^{o}}; ^{10^{=}^{o}}; ^{11^{=}^{o}}; ^{12^{=}^{o}}; ^{13^{=}^{o}}; ^{\overline{16}^{o}}; ^{\overline{18}^{o}}; ^{\overline{20}^{o}}; ^{\overline{22}^{o}}; ^{\overline{24}^{o}}; ^{\overline{26}^{o}}; ^{9^{=}^{e}}; ^{10^{=}^{e}}; ^{11^{=}^{e}}; ^{12^{=}^{e}}; ^{13^{=}^{e}}; ^{14^{=}^{e}}; ^{\overline{18}^{e}}; ^{\overline{20}^{e}}; ^{\overline{22}^{e}}; ^{\overline{24}^{e}}; ^{\overline{26}^{e}}; ^{\overline{28}^{e}}]

PERIODIC HOMOTOPY GROUPS OF SO(N) 93

We summarize by giving in Diagram 9.16 a general description of the chart for v_{∗}(SO(2n + 1)), which is a sum of this chart over all integers k.

Diagram 9.16. General description of v_{∗}(SO(2n + 1))

[figure: t − s = 8k+ →; −3; −2; −1; 0; 1; 2; 3; 4; 5; s = 4k − 4n; s = 4k − 2n; s = 4k − n; s = 4k; n≡0,3 mod 4; n≡1,2 mod 4]

The tower in 8k ± 2 whose top is in filtration approximately 4k will not usually be formed from classes in consecutive filtrations. Its order is determined by^{11} the main summand of E_{2}^{1,8k±2+1}(Spin(2n + 1)), which is quite irregular (see e.g. Table 3.22). One of the biggest advantages of the BTSS approach compared with the J-homology approach is that the former sees the cyclicity and order of this group, while the latter does not. The filtrations in all the J-homology charts in this section, although

11after taking into account a possible d_{3}-differential and exotic extension

94 BENDERSKY AND DAVIS

suggestive of actual Adams filtrations, are not really meaningful. It can happen, as it did for SO(13) in Diagram 9.6, that 2 times a class will have filtration ≤ that of the class, as far as these charts are concerned. The order of the shorter, lower, tower in 8k ± 2 is more regular, but again the filtrations of the classes that comprise it will be complicated, at best.

The dashed lines in 8k − 1 ± 2 denote groups of approximately the same order as the sum of the two towers in 8k ± 2, but with no control over the group structure. So far, the BTSS is not much better in this regard, although 11.3 and 11.4 offer some hope for eventual knowledge about their group structure.

The one eta-pair which is present in v_{∗}(SO(2n+1)) for all n ≥ 3 appears in position (8k, 4k) and (8k+1, 4k+1). The boxes around height 4k−n denote a pair of eta-pairs, as indicated, in the center box if n ≡ 0, 3 mod 4, and in the right box if n ≡ 1, 2 mod 4. These are the classes described in Diagram 9.11. The single eta-pair around 4k−2n is that described in Diagram 9.13. The same eta-pair will be present in SO(4a − 1) and SO(4a + 1); then it disappears and a slightly lower one appears in SO(4a + 3). The long rectangle with the little eta-pairs in it denotes approximately 2[log_{2}(4n/3)] eta pairs in the indicated range of filtrations (roughly 4k − 4n to 4k − 2n). For a specific n they will correspond to elements in Diagram 9.15 having label ≤ n which are hit by an arrow from an element whose label in greater than n.

A comparison of Diagram 9.16 with Diagrams 1.3 and 3.7 can be illuminating. The top tower (resp. lower tower) in 9.16 corresponds to C_{1} (resp. C_{2} or 8). The top (resp. middle) eta pair in 9.16 corresponds to the eta tower labeled 1 (resp. 4a − 1 or 4a − 3). The dashed lines correspond to G, and extensions from it. The little boxes correspond to the eta towers labeled D. And the big rectangle corresponds to the two wide bands of eta towers in 1.3 and 3.7.

10. Proof of fibration theorem

In this section, we present a proof of Theorem 2.2. In [9, §5], a proof of parts (i) and (ii) of this theorem was presented which relied heavily on work of Bousfield, both in the preprint [19] and in preparation. Here we give a self-contained proof. It was pointed out in [9] that there is some confusion in [6, 4.3], which is very relevant to our work here. Although that theorem is stated for injective extension sequences, it

PERIODIC HOMOTOPY GROUPS OF SO(N) 95

is applied to sequences which are only relatively injective extension sequences. Here we will adapt methods of their proof to such sequences. To prove this theorem we compare the proof of [6, 4.3] with Mahowald’s resolution of the fiber construction in [32].

As in [6] we extend the category G of K_{∗}K-coalgebras to the naturally equivalent category G_{0} of connected K_{∗}K-coalgebras. The objects of G_{0} are of the form K_{∗} ⨁ M where M ∈ G. In particular, for a space X, K_{∗}(X) shall denote an object in G (the reduced K-homology of X) as well as an object in G_{0} (the unreduced K-homology of X).

Following the appendix in [17] we define a cosimplicial object X over a category C.

Definition 10.1. A cosimplicial object over C consists of

(i) for every integer n ≥ 0 an object X^{n} ∈ C (ii) for every pair of integers (i, n) with 0 ≤ i ≤ n coface and code-

generacy maps

d^{i} : X^{n−1} → X^{n} ∈ C
s^{i} : X^{n+1} → X^{n} ∈ C

satisfying the cosimplicial identities ([20, p.267], [17, p.487]).

In most of our applications, C will be the category of spaces or the category G_{0}.

If X is a topological space, let K(X) = Ω^{∞}(K ∧ Σ^{∞}X). Recall that π_{∗}(K(X)) ≈ K_{∗}(X). There is a cosimplicial space augmented by d^{0} : X → K(X)

(10.2)

[figure: KX = {X ⟶^{d^{0}} K(X); K(K(X)); ⋯ }]

In the notation of 10.1, X^{n} = K^{n+1}(X).

As in [12], we define T ot(KX) to be the total space associated to the cosimplicial complex KX = {K^{s+1}(X)} of 10.2, and the K-completion of X is defined to be

X^{∧} = T ot(KX).

A filtration T ot_{n}(KX) is defined on T ot(KX) as in [20, X.3.2].

96 BENDERSKY AND DAVIS

To prove 2.2, we use the tower under X^{∧} induced by this filtration of T ot(KX). The reference here is [20, X,§6]. The E_{1}-term associated to the tower of fibrations {T ot_{n}(KX)} is described in [20].

Lemma 10.3. ([20, X, 6.2]) E_{1}^{s,t} ≈ π_{t}K^{s+1}(X) ∩ ker(s^{0}) ∩ ⋯ ∩ ker(s^{s−1}), t ≥ s ≥ 0.

(Note: K^{s+1}(X) is denoted by \underline{X}^{s} in [20]). So E_{1} is the reduced cobar complex, and, by [20, p.283], d_{1} : E_{1} → E_{1} is the boundary in the reduced cobar complex

The spaces in the unaugmented complex (10.2) are the fibers that occur in the tower induced by the filtration of T ot(KX). After applying π_{∗} to the unaugmented complex, we have

(10.4)

[figure: π_{∗}(KX) = {K_{∗}(X); K_{∗}(K(X)); ⋯ }]

The cochain complex ch π_{∗}KX has (ch π_{∗}KX)^{n} = π_{∗}((KX)^{n}) = π_{∗}(K^{n+1}(X)) and δ = ∑(−1)^{i}d^{i}. It follows from [20, ch.X] that E_{2} = H^{∗}(ch π_{∗}KX).

In order to interpret E_{2} as an Ext_{G} (=Ext_{G_{0}}), we apply K_{∗} to (10.2), obtaining

[figure: K_{∗}(KX) = {K_{∗}(X) ⟶^{d^{0}} K_{∗}(K(X)); K_{∗}(K(K(X))); ⋯ }]

This is a cosimplicial object over the category G, augmented by K_{∗}(X) ⟶^{d^{0}} K_{∗}(K(X)). As usual, there is an extra codegeneracy, which implies H^{∗}(ch K_{∗}(KX)) = 0. So K_{∗}(KX) without the augmentation is a cosimplicial resolution of K_{∗}(X). Hence

Ext_{G}^{s}(K_{∗}, K_{∗}(X)) ≈ H^{∗}(Hom_{G}(K_{∗}, K_{∗}(K^{s+1}(X)))) ≈ H^{∗}(Hom_{U}(K_{∗}, P K_{∗}(K^{s+1}(X)))) ≈ H^{∗}(ch π_{∗}(KX)) ≈ E_{2}^{s}(X).

Here we have used the isomorphism

Hom_{G}(K_{∗}, K_{∗}(K(Y)) ≈ Hom_{U}(K_{∗}, P K_{∗}(K(Y))) ≈ K_{∗}(Y),(10.5)

where U is the category of K_{∗}K-comodules and P : G → U is the primitives functor. See [6] for details.

We will need the following standard result:

Proposition 10.6. Suppose we have a diagram of cosimplical resolutions of X over the category G

[figure: X ⟶^{d_{0}} Y^{0}; Y^{1}; ⋯; X ⟶^{d_{0}} X^{0}; X^{1}; ⋯; ↓ 1_{X}; ↓; ↓]

with X^{n} and Y^{n} relative injectives (e.g. objects of the form K_{∗}(K(Z))). Then the induced map Ext_{G}(K_{∗}, X) → Ext_{G}(K_{∗}, X) is the identity.

We now recall from [17, 3.7] the construction of the mapping cone M(f) of a cosimplicial map f : X → Y of cosimplicial objects over G_{0}.

Definition 10.7. M(f) is given by:

(i) M(f)^{0} = X^{0}, M(f)^{n} = X^{n} ⊗ Y^{n−1} ⊗ ⋯ ⊗ Y^{0} , n ≥ 1. (ii) d^{0} : M(f)^{n} → M(f)^{n+1} equals ((d^{0} ⊗ f)Δ) ⊗ id ⊗ ⋯ ⊗ id, where

Δ : X^{n} → X^{n} ⊗ X^{n} is the comultiplication. (iii) For 0 < i ≤ n, d^{i} : M(f)^{n} → M(f)^{n+1} equals d^{i} ⊗⋯⊗d^{1} ⊗((d^{0} ⊗

id)Δ) ⊗ id ⊗ ⋯ ⊗ id. (iv) d^{n+1} : M(f)^{n} = M(f)^{n} ⊗_{K_{∗}} K_{∗} → M(f)^{n+1} equals d^{n+1} ⊗ ⋯ ⊗

d^{1} ⊗ η, where η : K_{∗} → Y^{0} is the coaugmentation.

(v) s^{i} : M(f)^{n+1} → M(f)^{n} equals s^{i} ⊗ ⋯ ⊗ s^{0} ⊗ α ⊗ id ⊗ ⋯ ⊗ id, where α : Y^{n−1} → K_{∗} is the counit.

We adopt the following notation. For a space X, we shall let \overline{X}^{s} = K_{∗}(K^{s+1}(X)). This is the sth group in the cosimplicial resolution K_{∗}(KX). The map h of Theorem 2.2 induces a map of cosimplicial resolutions h : K_{∗}(KE) → K_{∗}(KB). Part (ii) of Theorem 2.2 is proven in [6] by studying the following sequence, which is an adaptation of [17] to the category G_{0}.

M(η) → M(K̃h) → M(α)  (10.8)

The mapping cones in (10.8) are given, as in [17, p.479], by:

M(η)^{0} = K_{∗} M(η)^{s} = K_{∗} ⊗ \overline{B}^{s−1} ⊗ ⋯ ⊗ \overline{B}^{0}
M(K̃h)^{0} = \overline{E}^{0} M(K̃h)^{s} = \overline{E}^{s} ⊗ \overline{B}^{s−1} ⋯ ⊗ \overline{B}^{0}
M(α)^{0} = \overline{E}^{0} M(α)^{s} = \overline{E}^{s} ⊗ K_{∗} ⊗ K_{∗} ⋯ ⊗ K_{∗}

98 BENDERSKY AND DAVIS

(⊗ = ⊗_{K_{∗}}).

There is a map k of cosimplicial resolutions of K_{∗}(F) ∈ G_{0}, k : K_{∗}(KF) → M(K̃h) with

k : \overline{F}^{s} = \overline{F}^{s} ⊗ K_{∗} ⊗ ⋯ ⊗ K_{∗} → \overline{E}^{s} ⊗ \overline{B}^{s−1} ⊗ ⋯ ⊗ \overline{B}^{0}

defined by k = j_{∗} ⊗ η ⊗ ⋯ ⊗ η, with j_{∗} induced by j : F → E, and η the coaugmentation. Here k is a map of cosimplicial complexes because h_{∗}j_{∗} = 0 on the kernel of the counit, using also the definition of the coface and codegeneracy maps in M(K̃h).

The following result is immediate from 10.6.

Corollary 10.9. The map k induces an isomorphism of Ext_{G}(K_{∗}, K_{∗}(F)).

The long exact sequence of E_{2}-terms in 2.2(ii) is obtained, as in [6, 4.3], by

• noting that applying P (−) to (10.8) yields a short exact sequence of cosimplicial objects over U;

• noting that applying Hom_{U}(K_{∗}, −) to this maintains the short exactness; thus Hom_{G}(K_{∗}, −) applied to (10.8) yields a SES;

• observing that H^{s}(ch Hom_{G}(K_{∗}, −)) applied to the three cosimplicial mapping cones of (10.8) yields, respectively, E_{2}^{s−1}(B), E_{2}^{s}(F), and E_{2}^{s}(E).

As explained in [39, pp 321-3] and [9, §5], relatively injective comodules yield the same exactness properties as injective comodules, provided we are working with free K_{∗}-modules.

Using that

Hom_{G}(K_{∗}, K_{∗}KX ⊗ K_{∗}KY) ≈ Hom_{U}(K_{∗}, P (K_{∗}KX ⊗ K_{∗}KY))

≈ Hom_{U}(K_{∗}, P K_{∗}KX) ⊕ Hom_{U}(K_{∗}, P K_{∗}KY) ≈ K_{∗}X ⊕ K_{∗}Y,

we find that the SES of chain complexes whose H^{∗}(−) yields

→ E_{2}^{s}(B) → E_{2}^{s+1}(F) → E_{2}^{s+1}(E) →

as above is 0 → \overline{B}^{s−1} ⊕ ⋯ ⊕ \overline{B}^{0} → \overline{E}^{s} ⊕ \overline{B}^{s−1} ⊕ ⋯ ⊕ \overline{B}^{0} → \overline{E}^{s} → 0.(10.10)

PERIODIC HOMOTOPY GROUPS OF SO(N) 99

For h : E → B as in 2.2, let R denote the chain complex with R^{s+1} = \overline{E}^{s} ⊕ \overline{B}^{s−1} and differential d : R^{s} → R^{s+1} given by d(x) = δ(x) ⊕ h_{∗}(x) if x ∈ \overline{E}^{s−1}, and d(y) = −δ(y) if y ∈ \overline{B}^{s−2}. To clarify why the sign is as indicated in the formula for the differential in R, first note that d(x) must be δ(x) + h_{∗}(x) because the boundary map E_{2}(E) → E_{2}(B) induced by the short exact sequence (2.15) is the map induced by h. The sign on d(y) is determined because d^{2} = 0. One easily verifies that the inclusion map from R to the middle chain complex of (10.10) is a chain map.

The chain complex R is motivated by Mahowald’s construction of the resolution of the fiber which we now recall. Following Mahowald ([32, p.77]), let X_{s}(h) be the fiber of the map T ot_{s}(KE) → T ot_{s−1}(KB) and F_{s+1}(h) the fiber of X_{s+1}(h) → X_{s}(h).

There is a commutative diagram of fibre sequences

ΩK^{s+1}B  → F_{s+1}(h) → K^{s+2}E  → K^{s+1}B
↓  ↓  ↓  ↓
ΩT ot_{s}(KB) → X_{s+1}(h) → T ot_{s+1}(KE) → T ot_{s}(KB)
↓ ↓ ↓ ↓
ΩT ot_{s−1}(KB) → X_{s}(h) → T ot_{s}(KE) → T ot_{s−1}(KB)

(10.11)

in which each lower vertical map is part of an obvious tower.

It is proved in [32, 3.3] (see also [33]) that

ΩK^{s+1}B → F_{s+1}(h) → K^{s+2}E  (10.12)

induces a short exact sequence in π_{∗}(−), and in fact that

F_{s+1}(h) = K^{s+2}E × ΩK^{s+1}B.

Hence there is a long exact sequence

⋯ → E_{2}^{s−1,t}(B) ⟶^{∂} E_{2}^{s,t}(F̃) → E_{2}^{s,t}(E) → ⋯, (10.13)

where E_{2}^{s,t}(F̃) is the E_{2} term of the spectral sequence associated to the tower {X_{s}(h)}. (Recall that E_{1}^{s,t} is π_{t−s} of the s-th stage of the tower. So (10.12) induces a long exact sequence with the indicated bidegree.) In particular, from diagram (10.11) the boundary map ∂ is a map of spectral sequences. By [33, Prop.3], the morphism E_{2}(E) → E_{2}(B) in (10.13) is induced by the map h. Hence the differential in the resolution associated to the tower {X(h)} is given by the differential in R.

100 BENDERSKY AND DAVIS

Let F̃ be holim of the tower {X(h)}. There is a map from the canonical tower for F to the tower {X(h)} which induces

\overline{F}^{s} ⟶^{j^{s}⊕0} \overline{E}^{s} ⊕ \overline{B}^{s−1}  (10.14)

in E_{1}. To see this, observe that the construction of X_{s}(h) is natural in h in the sense that a diagram

\begin{matrix}E & ⟶^{h} & B \\ ↓ & & ↓ \\ E′ & ⟶^{h′} & B′ \end{matrix}

induces a map {X_{s}(h)} → {X_{s}(h′)} In particular, the diagram

\begin{matrix}F & → & ∗ \\ ↓ j & & ↓ \\ E & ⟶^{h} & B \end{matrix}

shows that there is a map of spectral sequences as in (10.14). Note also that this map of towers induces a map

F^{∧} ⟶^{j} F̃  (10.15)

of their homotopy limits.

We now compare the two constructions. There is an obvious map z of the short exact sequence π_{∗}((10.12)) to the sequence (10.10), which induces a commutative diagram of exact sequences

\begin{aligned}& & & E_{2}(F) \\ & & & ↓ j_{∗} \\ → & E_{2}^{s−1}(B) & → & E_{2}^{s}(F̃) & → & E_{2}^{s}(E) & → ⋯ \\ & ↓ z_{B} & & ↓ z_{F} & & ∥ \\ ⋯ → & H^{s−1}(B̂) & → & H^{s}(F̂) & → & E_{2}^{s}(E) & → ⋯, \end{aligned} (10.16)

where B̂ and F̂ are the first two chain complexes of (10.10).^{12}

The morphisms in the top row of (10.16) are morphisms of spectral sequences. We shall show below that z_{B} is an isomorphism. Hence z_{F} is an isomorphism by the Five Lemma (but we do not yet know that it is a map of spectral sequences). The composite z_{F} ∘ j_{∗} is induced by the map k which was proven to be an isomorphism in Corollary 10.9. Hence j_{∗} is an isomorphism which we have shown commutes with differentials.

12Note that F̂ is not the same thing as F^{∧}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 101

This, together with the earlier observation that ∂ in (10.13) is a morphism of spectral sequences, implies parts (ii) and (iii) of Theorem 2.2.

Since the morphism j_{∗} of (10.16) is an isomorphism of spectral sequences, it follows from the convergence of the spectral sequences that (10.15) is an equivalence. Thus the fibration

F̃ → E^{∧} → B^{∧}

obtained from (10.11) becomes the fibration asserted in Theorem 2.2(i). We will be done once we prove the following result.

Lemma 10.17. z_{B} is an isomorphism.

Proof. Following [17], we study the fibration

∗ → B → B,

which satisfies the conditions of Theorem 2.2. Hence there is a diagram as in (10.16)

→ E_{2}^{s−1}(B) → E_{2}^{s}(P̃) → E_{2}^{s}(B) ⟶^{∂} ⋯
↓ z_{B} ↓ ∥
⋯ → H^{s−1}(B̂) → H^{s}(P̂) → E_{2}^{s}(B) → ⋯

where P̃ and P̂ are the spaces in (10.16) for the trivial fibration. The map ∂ in the top row is induced by the identity map, which implies E_{2}^{s}(P̃) is zero. An easy calculation using the definition of P̂ shows that H^{s}(P̂) = 0. Thus z_{B} is an isomorphism by the Five Lemma.

11. A small resolution for computing Ext_{A}

In this section, we introduce a small chain complex for computing Ext_{A}(M/ im(ψ^{2})) when M is algebraically spherically resolved (ASR). Some advantages of this approach to Ext_{A} are

• it gives a slightly different interpretation of eta-towers, one which does not involve an extension, which is useful for naturality arguments;

• it gives a somewhat more natural proof of the formula, 7.2, for the h_{1}-action on the 1-line, and generalizes this result in various ways;

102 BENDERSKY AND DAVIS

• it gives a new interpretation of the 2-line groups, which shows immediately their number of summands, and lends hope to their complete calculation.

The results of this section are not used in this paper, in part because they were discovered after most of the work had been done. They should, however, be useful in subsequent BTSS calculations.

To state the result, we introduce two functions from integer matrices to abelian groups.

Definition 11.1. If N is an integer matrix, then G(N) is the abelian group presented by N. If N is an m-by-n integer matrix, let R(N) denote the row space of N, S(N) the subspace of Z^{n} consisting of vectors v such that cv ∈ R(N) for some c ∈ Z, and Q(N) = S(N)/R(N).

Note that if rank(N) = n, then Q(N) = G(N). Another useful description of S(N) in the above definition is as all integral vectors qN obtained from vectors q of rational numbers. Then R(N) is the subset consisting of those qN for which q is integral.

Although this method may apply more broadly, we restrict it here to modules which are algebraically spherically resolved (ASR). This notion has been used in [9] and [13]. For purposes of this paper, we define it as follows.

Definition 11.2. An object M of A is ASR if there exist short exact sequences in A

0 → QK^{1}(S^{2n_{1}+1}) → M_{i} → M_{i−1} → 0

for 0 ≤ i ≤ k, with M_{−1} = 0 and M_{k} = M.

We begin with the case in which ψ^{−1} = −1 on a 2-adic Adams module M. The following result will be proved later in this section.

Theorem 11.3. Suppose M is ASR with ψ^{−1} = −1. Let B be any basis of M, and let m be an integer. Let Ψ denote the matrix of ψ^{2} on M with respect to B, and Θ_{m}

PERIODIC HOMOTOPY GROUPS OF SO(N) 103

the matrix of ψ^{3} − 3^{m}. Then

Ext_{A}^{s,2m+1}(M/ im(ψ^{2})) ≈ \begin{cases}0 & s = 0 \\ Q(1 + (−1)^{m} Ψ Θ_{m}) & s = 1 \\ G(\begin{matrix}Θ_{m} \\ Ψ \end{matrix}) ⊕ Q(2 Ψ Θ_{m}) & s = 2, m odd \\ G(\begin{matrix}Θ_{m} \\ Ψ \\ 2 \end{matrix}) ⊕ Q(2 Ψ Θ_{m}) & s + m odd , s ≥ 3 \\ Q(\begin{matrix}Ψ & Θ_{m} & 0 \\ 2 & 0 & Θ_{m} \\ 0 & 2 & −Ψ \end{matrix}) & s + m even, s ≥ 2 \end{cases}

The Pontryagin dual is given by

Ext_{A}^{s,2m+1}(M/ im(ψ^{2}))^{#} ≈ \begin{cases}0 & s = 0 \\ G (\begin{matrix}1 + (−1)^{m} \\ Ψ^{T} \\ Θ_{m}^{T} \end{matrix}) & s = 1 \\ Q(Θ_{m}^{T} Ψ^{T}) ⊕ G (\begin{matrix}2 \\ Ψ^{T} \\ Θ_{m}^{T} \end{matrix}) & s = 2, m odd \\ Q(Θ_{m}^{T} Ψ^{T} 2) ⊕ G (\begin{matrix}2 \\ Ψ^{T} \\ Θ_{m}^{T} \end{matrix}) & s + m odd, s ≥ 3 \\ Q (\begin{matrix}Ψ^{T} & 2 & 0 \\ Θ_{m}^{T} & 0 & 2 \\ 0 & Θ_{m}^{T} & −Ψ^{T} \end{matrix} ) & s + m even, s ≥ 2. \end{cases}

A basis-free form is given as follows, where θ_{m} = ψ^{3} − 3^{m}. We have

Ext_{A}^{s,2m+1}(M/ im(ψ^{2}))^{#} ≈

M/ im(1 + (−1)^{m}, ψ^{2}, θ_{m})  s = 1
ker(θ_{m}|M/2^{∞}) ∩ ker(ψ^{2}|M/2^{∞}) ⊕ M/ im(2, ψ^{2}, θ_{m}) s = 2, m odd
ker(θ_{m}|M/2) ∩ ker(ψ^{2}|M/2) ⊕ M/ im(2, ψ^{2}, θ_{m})  s + m odd, s ≥ 3
H(M/2 ⟶^{θ_{m}−ψ^{2}} M/2 ⊕ M/2 ⟶^{ψ^{2}+θ_{m}} M/2)  s + m even, s ≥ 2

Here H(−) refers to the homology of the short sequence.

104 BENDERSKY AND DAVIS

The description of part of Ext_{A}^{2,2m+1}(M/ im(ψ^{2})) as G\binom{Θ_{m}}{Ψ} when m is odd is particularly useful. Our previous description of this was as the Pontryagin dual of the kernel of θ_{m} on M/ im(ψ^{2}), and this was felt to be somewhat intractable. It seems quite possible that exploitation of this result might allow complete calculation of E_{2}^{2,2m+1}(Spin(n)), which has been the only gap in our complete knowledge. It seems even more likely that we could use this result to obtain complete information about the group structure of E_{2}^{2,2m+1}(SU(n)), both at p = 2 and at odd primes, where only the orders of the groups were determined in [7] and [23].

The fact that this new description of the 2-line reduces mod 2 to the eta towers, which are known, improves upon our previous understanding that h_{1} acts surjectively on the 2-line ([5, 5.4]), which had been used to give a lower bound on the number of summands of the 2-line group. Now we can say that the number of summands of the relevant part of the 2-line group equals the (known) number of eta towers. A relatively straightforward Maple row reduction of \binom{Θ_{m}}{Ψ} yields the following result.

Proposition 11.4. Let m be odd, and e = e(m, n) = ν(|E_{2}^{1,2m+1}(Spin(2n + 1))|), which is given in [8, 1.5] in terms of sums of binomial coefficients, and which is given explicitly for n ≤ 12 in 3.22. Then

E_{2}^{2,2m+1}(Spin(2n+1)) = Z_{2}⊕Z_{2}⊕\begin{cases}Z/2^{3} ⊕ Z/2^{e−3} & n = 3 \\ Z/2 ⊕ Z/2^{e−1} & n = 4 \\ Z/2 ⊕ Z/2^{2} ⊕ Z/2^{e−3} & n = 5 \\ Z/2^{3} ⊕ Z/2^{e−3} & n = 6, m ≡ 1 mod 4 \\ Z/2^{5} ⊕ Z/2^{e−5} & n = 6, m ≡ 3 mod 4 \\ Z/2 ⊕ Z/2^{6} ⊕ Z/2^{e−7} & n = 7, m ≡ 1 mod 4 \\ Z/2 ⊕ Z/2^{7} ⊕ Z/2^{e−8} & n = 8, m ≡ 3 mod 4 \end{cases}

The next result expresses the action of h_{1} in terms of the above descriptions of Ext_{A} and Ext_{A}^{#}. This includes a new proof of Lemma 7.2 and its implementation in 7.9, as well as generalizations. The extension of this result to modules which do not necessarily satisfy ψ^{−1} = −1 is given in 11.18.

Note that in all cases in which a submatrix 2 occurs in one of the matrices of 11.3, the group/summand depends only on M/2 and its Adams operations, in which case Θ_{m} ≡ Θ_{m+1}. The following description identifies Θ_{m} and Θ_{m+1} mod 2.

PERIODIC HOMOTOPY GROUPS OF SO(N) 105

Theorem 11.5. Under the identifications of Theorem 11.3, h_{1} : Ext^{s,2m+1} → Ext^{s+1,2m+3} and h_{1}^{#} : Ext^{s+1,2m+3}^{#} → Ext^{s,2m+1}^{#} satisfy

• if s ≥ 3 or s = 2 and m even, then h_{1} = 1 and h_{1}^{#} = 1;

• if s = 1 and m is even, then h_{1} is inclusion into the second summand, and h_{1}^{#} is the dual projection;

• if s = 2 and m is odd, then h_{1} = ρ_{2} ⊕ 1 and h_{1}^{#} = ι_{2} ⊕ 1, where ρ_{2} is reduction mod 2 and ι_{2} is inclusion into elements of order 2;

• if s = 1 and m is odd, then

h_{1} : Q(0 Ψ Θ_{m}) → Q (\begin{matrix}Ψ & Θ_{m+1} & 0 \\ 2 & 0 & Θ_{m+1} \\ 0 & 2 & −Ψ \end{matrix})  (11.6)

satisfies

h_{1}(\overline{0}, wΨ, wΘ_{m}) = (wΨ, wΘ_{m}, −3^{m}wΨ)  (11.7)
= (wΨ, \overline{0}, \frac{1}{2}wΨΘ_{m+1}) + (\overline{0}, wΘ_{m}, −\frac{1}{2}wΘ_{m}Ψ).
= (wΨ, wΘ_{m+1}, 0) + (0, 2 · 3^{m}w, −3^{m}wΨ)

Here w is a rational vector such that wΨ and wΘ_{m} are integral. The basis-free h_{1}^{#} in this case sends (x_{1}, x_{2}) to (ψ^{2}(x_{1})+θ_{m}(x_{2}))/2.

The different expressions in (11.7) can each be useful in different situations. In the description of Q(−) given after Definition 11.1, we have, if s = 1 and m is odd, h_{1} : Q(N_{1}) → Q(N_{2}) is given by

h_{1}(wN_{1}) = (0, \frac{1}{2}wΨ, \frac{1}{2}wΘ_{m})N_{2} = (w, 0, 3^{m}w)N_{2},  (11.8)

where N_{1} and N_{2} are the matrices in (11.6).

Before proceeding with the proof of Theorems 11.3 and 11.5, we illustrate them with M = P K^{1}(F_{4}/G_{2}), which was studied in [9]. We have

Ψ = (\begin{matrix}2^{7} & 0 \\ 120 & 2^{11} \end{matrix}) and Θ_{m} = (\begin{matrix}3^{7} − 3^{m} & 0 \\ 5 · 3^{7} & 3^{11} − 3^{m} \end{matrix}) .

Let m = 2k + 1 be odd, and define ν = ν(k − 3) + 3 and ν′ = ν(k − 5) + 3. Then N_{1} = (0 Ψ Θ_{m}) is, up to unit multiples,

(\begin{matrix}0 & 0 & 2^{7} & 0 & 2^{ν} & 0 \\ 0 & 0 & 2^{3} & 2^{11} & 1 & 2^{ν′} \end{matrix}) ,

106 BENDERSKY AND DAVIS

which pivots to

M′ = (\begin{matrix}0 & 0 & 2^{min′(7,3+ν)} & 2^{11+ν} & 0 & 2^{ν′+ν} \\ 0 & 0 & 2^{3} & 2^{11} & 1 & 2^{ν′} \end{matrix}) ,

where min′(a, b) = min(a, b) if a ≠ b, while min′(a, a) > a.

If k is even, then ν = ν′ = 3, and so Q(0 Ψ Θ_{m}) = Z/2^{6} with generator 1/2^{6} times the first row of M′, while if k ≡ 3 mod 4, Q(0 Ψ Θ_{m}) = Z/2^{7}, generated by 1/2^{7} times the first row of M′. The case k ≡ 1 mod 4 is more delicate, and will be omitted from this illustration.

We have

N_{2} = (\begin{matrix}Ψ & Θ_{m+1} & 0 \\ 2 & 0 & Θ_{m+1} \\ 0 & 2 & −Ψ \end{matrix}) = (\begin{matrix}2^{7} & 0 & 3^{7} − 3^{m+1} & 0 & 0 & 0 \\ 120 & 2^{11} & 5 · 3^{7} & 3^{11} − 3^{m+1} & 0 & 0 \\ 2 & 0 & 0 & 0 & 3^{7} − 3^{m+1} & 0 \\ 0 & 2 & 0 & 0 & 5 · 3^{7} & 3^{11} − 3^{m+1} \\ 0 & 0 & 2 & 0 & −2^{7} & 0 \\ 0 & 0 & 0 & 2 & −120 & −2^{11} \end{matrix}) .

Since m + 1 is even, ν(3^{7} − 3^{m+1}) = ν(3^{11} − 3^{m+1}) = 1. Let R_{i} denote the ith row of N_{2}. Then R_{1} is in the span of R_{3} and R_{5}, while 2R_{2} is in the span of R_{3}, R_{4}, R_{5}, and R_{6}. Because of the units in positions (2, 3) and (4, 5), we can deduce that Q(N_{2}) ≈ Z_{2} ⊕ Z_{2} generated by g_{1} = \frac{1}{2}R_{3} and g_{2} = \frac{1}{2}R_{6}.

In the fourth part of Theorem 11.5, let w be the vector whose components are the numbers by which the rows of N_{1} must be multiplied to give the generator of Q(N_{1}). Thus

w = \begin{cases}(5 · 3^{7}/2^{6} (3^{m} − 3^{7})/2^{6}) & if k even \\ (5 · 3^{7}/2^{7} (3^{m} − 3^{7})/2^{7}) & if k ≡ 3 mod 4, \end{cases}

and, mod 2,

g := (0 wΨ wΘ_{m}) ≡ \begin{cases}(0 0 1 0 0 1) & if k even \\ (0 0 1 0 0 0) & if k ≡ 3 mod 4. \end{cases}

From (11.8), we obtain, with the first equivalence mod 1,

h_{1}g ≡ \begin{cases}(0 0 \frac{1}{2} 0 0 \frac{1}{2})N_{2} = g_{1} + g_{2} & if k even \\ (0 0 \frac{1}{2} 0 0 0)N_{2} = g_{1} & if k ≡ 3 mod 4. \end{cases}

This provides an alternate argument for some d_{3}-differentials in the BTSS of F_{4}/G_{2} established by another method in the first paragraph of the proof of [9, 4.15].

PERIODIC HOMOTOPY GROUPS OF SO(N) 107

We begin now to work toward the proofs of 11.3 and 11.5. We will construct a free A-object Γ and a small resolution of an ASR object M (not assuming ψ^{−1} = −1) to which applying Hom_{A}(−, K^{1}(S^{2m+1})) yields the following.

Lemma 11.9. Assume M is an ASR A-object, and B any basis of M. Let Ψ^{k} and Θ_{m} denote the matrices of ψ^{k} and ψ^{3} − 3^{m}, respectively, with respect to B. Then Ext_{A}^{s,2m+1}(M/ im(ψ^{2})) is the homology of a sequence of free Z_{2}^{∧}-modules

C_{0} ⟶^{d_{1}} C_{1} ⟶^{d_{2}} C_{2} ⟶^{d_{3}} C_{3} → ⋯,

where the transposes of the matrices of d_{s} are given by

(Ψ^{−1} − (−1)^{m} Ψ^{2} Θ_{m})

for s = 1, and for s ≥ 2 by

Ψ^{−1} + (−1)^{s+m}  Ψ^{2}  Θ_{m}  0
0  −Ψ^{−1} + (−1)^{s+m}  0  Θ_{m}
0  0  −Ψ^{−1} + (−1)^{s+m}  −Ψ^{2}
0  0  0  Ψ^{−1} + (−1)^{s+m}

with the last row deleted if s = 2.

Note that, if rank(M) = n, then rank(C_{0}) = n, rank(C_{1}) = 3n, and rank(C_{s}) = 4n for s ≥ 2.

Proof of Theorem 11.3. Since M is ASR, d_{1} is injective and hence Ext^{0} = 0. Also since M is ASR and E_{2}(S^{2n+1}; Q) = 0, the rational homology of the sequence of 11.9 is 0, and hence the homology at C_{s} is given by dividing elements in im(d_{s}) as much as possible and using im(d_{s}) as the relations. If N is the matrix of d_{s}, then the columns of N are im(d_{s}), and so Q(N^{T}) measures the homology as just described. Since ψ^{−1} = −1, the matrices ±Ψ^{−1} + (−1)^{s+m} will be 0 or ±2. The desired homology is obtained by substituting these into the matrices displayed in Lemma 11.9 and applying Q. For example, if s = 2 and m is odd, the homology is

Q (\begin{matrix}−2 & Ψ & Θ_{m} & 0 \\ 0 & 0 & 0 & Θ_{m} \\ 0 & 0 & 0 & −Ψ \end{matrix}) ≈ Q(2 Ψ Θ_{m}) ⊕ Q (\begin{matrix}Θ_{m} \\ Ψ \end{matrix}) .

The desired result here follows from our remark about Q = G for matrices whose rank equals their number of columns. Other cases follow similarly.

108 BENDERSKY AND DAVIS

We obtain Ext_{A}^{s,2m+1}(M/ im(ψ^{2}))^{#} as the homology of the sequence

C_{0}^{#} ⟵^{d_{1}^{#}} C_{1}^{#} ⟵^{d_{2}^{#}} C_{2}^{#} ⟵^{d_{3}^{#}} ⋯,

where C_{s}^{#} = Hom(C_{s}, Q/Z) and the matrix of d_{s}^{#} is the matrix listed in 11.9. Since the cohomology of the sequence of Hom(C_{s}, Q) is acyclic, the cohomology exact sequence induced by

0 → Z → Q → Q/Z → 0

implies that Ext_{A}^{s,2m+1}(M/ im(ψ^{2}))^{#} is the homology at C_{s−1}^{∗} of the sequence

C_{0}^{∗} ⟵^{d_{1}^{∗}} C_{1}^{∗} ⟵^{d_{2}^{∗}} C_{2}^{∗} ⟵^{d_{3}^{∗}} ⋯,

where C_{s}^{∗} = Hom(C_{s}, Z) and the matrix of d_{s}^{∗} is that listed in 11.9. Thus

Ext^{1}(−)^{#} = coker(d_{1}^{∗}) = G (\begin{matrix}1 + (−1)^{m} \\ Ψ \\ Θ_{m} \end{matrix}) ,

while for s ≥ 2, Ext^{s}(−)^{#} is given by dividing elements in im(d_{s}^{∗}) as much as possible and using im(d_{s}^{∗}) as relations. Thus it is given by applying Q to the transposes of the matrices displayed in 11.9. This is as claimed in 11.3, once we replace Q by G for matrices whose rank equals their number of columns.

Finally we give the proof of the basis-free interpretation of Ext(−)^{#}. The case s = 1 and the second summand when s + m is odd are immediate since G(N) is the cokernel of the transformation with matrix N^{T}.

Next note that Q(Θ_{m}^{T} Ψ^{T}) is Pontryagin dual to G (\begin{matrix}Θ_{m} \\ Ψ \end{matrix}), which is coker(M ⊕ M ⟶^{θ_{m}^{T}⊕ψ^{T}} M). Hence it is

ker(M^{#} ⟶^{θ_{m}+ψ^{2}} M^{#} ⊕ M^{#}) = ker(θ_{m}|M^{#}) ∩ ker(ψ^{2}|M^{#}).

Since M is a free 2-primary module, M^{#} may be replaced by M/2^{∞}. Similarly,

Q(2 θ_{m}^{T} ψ^{T}) = ker(2|M/2^{∞}) ∩ ker(θ|M/2^{∞}) ∩ ker(ψ^{2}|M/2^{∞})
= ker(θ|M/2) ∩ ker(ψ^{2}|M/2)

since ker(2|M/2^{∞}) = M/2.

Finally, we need

Q (\begin{matrix}Ψ^{T} & 2 & 0 \\ Θ^{T} & 0 & 2 \\ 0 & Θ^{T} & −Ψ^{T} \end{matrix} ) ≈ H(M/2 ⟶^{θ_{m}−ψ^{2}} M/2 ⊕ M/2 ⟶^{ψ^{2}+θ_{m}} M/2).

To see this, first note that

Q (\begin{matrix}Ψ^{T} & 2 & 0 \\ Θ^{T} & 0 & 2 \\ 0 & 2Θ^{T} & −2Ψ^{T} \end{matrix} ) = Q (\begin{matrix}Ψ^{T} & 2 & 0 \\ Θ^{T} & 0 & 2 \\ Ψ^{T} Θ^{T} − Θ^{T} Ψ^{T} & 0 & 0 \end{matrix})
= Q (\begin{matrix}Ψ^{T} & 2 & 0 \\ Θ^{T} & 0 & 2 \end{matrix})
≈ ker(M/2 ⊕ M/2 ⟶^{ψ^{2}+θ_{m}} M/2)

with (v_{i}, w_{j}) ∈ M/2 ⊕ M/2 corresponding to the row

\frac{1}{2}((ψ^{2}e_{i}, 2e_{i}, 0) + (θ_{m}e_{j}, 0, 2e_{j})).

Then note that the homomorphism

Q (\begin{matrix}ψ^{T} & 2 & 0 \\ Θ^{T} & 0 & 2 \\ 0 & 2Θ^{T} & −2Ψ^{T} \end{matrix} ) → Q (\begin{matrix}Ψ^{T} & 2 & 0 \\ Θ^{T} & 0 & 2 \\ 0 & Θ^{T} & −Ψ^{T} \end{matrix} )

has kernel spanned by all elements (0 θ_{m}e_{j} −ψ^{2}e_{j}), which corresponds to (θ_{m}v_{j}, −ψ^{2}v_{j}) under the above correspondence, establishing the claim.

In order to prove 11.5 and 11.9, we need to describe the free A-resolution. We begin with the following description of the free objects.

Theorem 11.10. Define an object Γ in the category A of stable 2-adic Adams modules by letting S = {(i, j) : i ≥ 0, j ∈ {0, 1}} and Γ = (Z_{2}^{∧})^{S} with

ψ^{−1}(f)(i, j) = f(i, 1 − j)
(ψ^{3} − 1)(f)(i, j) = \begin{cases}f(i − 1, j) & i ≥ 1 \\ 0 & i = 0. \end{cases}

Then Γ is free on one generator.

Proof. [18, p.145] says that forgetting other odd operations on 2-torsion stable Adams modules (not 2-adic) is a categorical isomorphism. Indeed, it states that a 2-torsion object of A_{(2)} corresponds to an object in the category A_{(2)}^{3} of 2-torsion abelian groups with a locally nilpotent operator \overline{ψ^{3}} = ψ^{3} − 1 and a commuting involution ψ^{−1}. For a 2-torsion object G ∈ Inv, there is a universal A-object \overline{U}(G) corresponding to the object of A_{(2)}^{3} which is G ⊕ G ⊕ ⋯ with ψ^{−1} acting componentwise and

110 BENDERSKY AND DAVIS

\overline{ψ^{3}}(g_{1}, g_{2}, …) = (g_{2}, g_{3}, …). Here Inv and GInv are as in the proof of 7.2. (See also [9] and [18].)

The functor \overline{U} is right adjoint to the forgetful functor, and so it sends injectives to injectives. Recall that GInv (resp. A) is Pontryagin dual to the torsion subcategory of Inv (resp. A_{(2)}). Since Q/Z ⊕ Q/Z with ψ^{−1} reversing the factors is injective in Inv, applying \overline{U} to it yields an injective in A_{(2)}. Applying Pontryagin duality yields the desired projective objects in GInv and A.

The generator of Γ can be taken to be the element f_{0} defined by f_{0}(0, 0) = 1 and f_{0}(i, j) = 0 otherwise. A morphism Γ ⟶^{ϕ} N in A is determined by ϕ(f_{0}), which will be used implicitly.

We will prove the following result later in this section.

Theorem 11.11. Suppose M is an ASR 2-adic Adams module. Let B = {v_{1}, …, v_{n}} be any basis of M over Z_{2}^{∧}. Let F be a free A-module with basis {g_{1}, …, g_{n}}. Thus F is the sum of n copies of the object Γ described in Theorem 11.10.

There is a free A-resolution

0 ← M/ im(ψ^{2}) ⟵^{ϵ} R_{0} ⟵^{∂_{1}} R_{1} ⟵^{∂_{2}} R_{2} ⟵^{∂_{3}} ⋯, (11.12)

where

R_{s} = \begin{cases}F & s = 0 \\ F ⊕ F ⊕ F & s = 1 \\ F ⊕ F ⊕ F ⊕ F & s ≥ 2, \end{cases}

ϵ(g_{i}) = v_{i}, and the matrix of ∂_{1} is given by

(Ψ^{−1} − σ_{1} Ψ^{2} \overline{Ψ^{3}} − σ_{3}),

while, for s ≥ 2, that of ∂_{s} is

Ψ^{−1} + (−1)^{s}σ_{1}  Ψ^{2}  \overline{Ψ^{3}} − σ_{3}  0
0  −Ψ^{−1} + (−1)^{s}σ_{1}  0  \overline{Ψ^{3}} − σ_{3}
0  0  −Ψ^{−1} + (−1)^{s}σ_{1}  −Ψ^{2}
0  0  0  Ψ^{−1} + (−1)^{s}σ_{1}

with the last row deleted if s = 2, where • σ_{1} : Γ → Γ satisfies σ_{1}(f)(i, j) = f(i, 1 − j);

PERIODIC HOMOTOPY GROUPS OF SO(N) 111

• σ_{3} : Γ → Γ satisfies σ_{3}(f)(i, j) = f(i − 1, j) if i > 0, while σ_{3}(f)(0, j) = 0;

• σ_{j} : F → F does σ_{j} on each Γ summand;

• Ψ^{k} (resp. \overline{Ψ^{3}}) : F → F has matrix with respect to \{g_{j}\} the same as that of ψ^{k} (resp. ψ^{3} − 1) on M with respect to {v_{j}}.

Proof of Lemma 11.9. The complex (C_{s}, d_{s}) is obtained as (Hom_{A}(R_{s}, K^{1}S^{2m+1}), ∂_{s}^{∗}). We need merely to observe that the duals of σ_{1} and σ_{3} are (−1)^{m} and 3^{m} − 1, respectively. To see this, first note that Hom_{A}(Γ, K^{1}S^{2m+1}) is cyclic on generator γ satisfying

γ(f) = ∑_{i,j}(−1)^{mj}(3^{m} − 1)^{i}f(i, j).

Then

σ_{3}^{∗} : Hom_{A}(Γ, K^{1}S^{2m+1}) → Hom_{A}(Γ, K^{1}S^{2m+1})

satisfies

σ_{3}^{∗}(γ)(f) = γ(σ_{3}f)
= ∑_{i,j}(−1)^{mj}(3^{m} − 1)^{i}(σ_{3}f)(i, j)
= ∑_{i,j}(−1)^{mj}(3^{m} − 1)^{i}f(i − 1, j)
= (3^{m} − 1)γ(f),

and similarly for σ_{1}.

Proof of Theorem 11.5. We focus on the most difficult case, s = 1 and m odd. The Yoneda product with h_{1} is defined using the diagram

M/ im(ψ^{2}) ⟵^{ϵ}  F  ⟵^{∂_{1}} F ⊕ F ⊕ F ⟵^{∂_{2}} F ⊕ F ⊕ F ⊕ F
τ_{1}↓  ^{τ_{2}}↓
K^{1}S^{2m+1} ⟵^{ϵ′}  Γ  ⟵^{∂′_{1}}  Γ ⊕ Γ
↓^{h_{1}}
K^{1}S^{2m+3}

112 BENDERSKY AND DAVIS

where Γ is as in Theorems 11.10 and 11.11, ϵ′(f) = ∑(−1)^{j}(3^{m} − 1)^{i}f(i, j), and ∂′_{1} = (−1 − σ_{1} 3^{m} − 1 − σ_{3}). Note that there is no ψ^{2}-summand in the resolution of K^{1}S^{2m+1} since it is not a resolution of K^{1}S^{2m+1}/ im(ψ^{2}).

Similarly to [9, §3], h_{1} ∈ Ext_{A}^{1}(K^{1}S^{2m+1}, K^{1}S^{2m+3}) is the sole nonzero element and h_{1} is defined by

h_{1} = \frac{1}{2}ϵ′′ ∘ ∂′_{1},

where ϵ′′ : Γ → K^{1}S^{2m+3} satisfies ϵ′′(f) = ∑(3^{m+1} − 1)^{i}f(i, j). Thus

h_{1}(f, g) = − ∑_{i}(3^{m+1} − 1)^{i}(f(i, 0) + f(i, 1))  (11.13)

+ ∑_{i,j} \frac{1}{2}(3^{m+1} − 1)^{i}((3^{m} − 1)g(i, j) − g(i − 1, j))

= − ∑_{i}(3^{m+1} − 1)^{i}(f(i, 0) + f(i, 1)) − ∑_{i,j}(3^{m+1} − 1)^{i}3^{m}g(i, j).

The map τ_{1} is a lifting over ϵ′ of a map τ : F ⊕ F ⊕ F → K^{1}S^{2m+1}, and τ satisfies 2^{b}τ = ϕ ∘ ∂_{1} for some ϕ : F → K^{1}S^{2m+1} and some b ≥ 1. This latter is due to the characterization of cocycles that we have been using throughout, that some multiple of them equals a coboundary. This defines a vector

v = (ϕ(g_{1}), …, ϕ(g_{n})) ∈ Z_{2}^{∧}^{n},

where {g_{1}, …, g_{n}} is the basis of F used in Theorem 11.11. Next we note that

2^{b}τ = (\overline{0}, vΨ^{2}, v(Ψ^{3} − 3^{m})).

Here we are using the description of ∂_{1} given in Theorem 11.11 and the fact that ψ^{−1} = −1 in both M and K^{1}S^{2m+1}. The σ_{3} in the third component becomes 3^{m} − 1 in K^{1}S^{2m+1}, which is subtracted from the matrix \overline{Ψ^{3}}.

The lifting τ_{1} can be chosen to satisfy the same formula

2^{b}τ_{1} = (\overline{0}, vΨ^{2}, v(Ψ^{3} − 3^{m})).

The difference between this formula and the one for τ resides in the Adams operations in the target. Note that vΨ^{2} (and also v(Ψ^{3} − 3^{m})) is a 1-by-n matrix (α_{1}, …, α_{n}) with entries in Z_{2}^{∧}. Its meaning as a morphism Γ^{n} → Γ is the usual matrix of a linear transformation, while as a morphism from Γ^{n} → K^{1}S^{2m+1}, v(Ψ^{3} − 3^{m}) sends (f_{1}, …, f_{n}) to ∑_{i,j,k} α_{k}(3^{m} − 1)^{i}(−1)^{j}f_{k}(i, j), reflecting the action of ψ^{−1} and ψ^{3} − 1 on K^{1}S^{2m+1}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 113

Using the formula for ∂_{2} in 11.11, we find that

τ_{1}∂_{2} : F ⊕ F ⊕ F ⊕ F → Γ

is • 0 on the first summand,

• 2^{−b}vΨ^{2}(1 + σ_{1}) : Γ → Γ on the second summand,

• 2^{−b}v(Ψ^{3} − 3^{m})(1 + σ_{1}) on the third summand, and

• 2^{−b}v(Ψ^{2}(\overline{Ψ^{3}} − σ_{3}) − (Ψ^{3} − 3^{m})Ψ^{2}) on the fourth. The formula on the fourth summand simplifies to 2^{−b}vΨ^{2}(3^{m} − 1 − σ_{3}).

Now τ_{2} can be chosen as

• 0 on the first summand,

• −2^{−b}vΨ^{2} into the first summand on the second summand,

• −2^{−b}v(Ψ^{3} − 3^{m}) into the first summand on the third summand, and

• 2^{−b}vΨ^{2} into the second summand on the fourth summand. Here we have used that σ_{i} commutes with scalar multiplication. Following by (11.13) yields that the element h_{1}{τ} ∈ Ext_{A}^{2,2m+3}(M/ im(ψ^{2})) is represented by the map F ⊕ F ⊕ F ⊕ F → K^{1}S^{2m+3} defined by

• 0 on the first summand,

• 2^{−b}vΨ^{2} on the second summand,

• 2^{−b}v(Ψ^{3} − 3^{m}) on the third summand, and

• −3^{m}2^{−b}vΨ^{2} on the fourth summand. This yields the first equality of 11.7, with w = 2^{−b}v, while the second follows from −3^{m} = \frac{1}{2}(ψ^{3} − 3^{m+1} − (ψ^{3} − 3^{m})).

Finally, we prove the basis-free form of h_{1}^{#}. Let C = {C_{s}} and C^{∗} = {C_{s}^{∗}} = {Hom(C_{s}, Z_{2}^{∧})} be the complexes used in the proof of 11.3 involving θ_{m}, and let C̃ and C̃^{∗} be the analogous complexes involving θ_{m+1} instead of θ_{m}. We will use the Universal Coefficient Theorem and the formula (11.7) for

H_{1}(C) ⟶^{h_{1}} H_{2}(C̃)  (11.14)

to deduce the desired basis-free formula for

H_{1}(C̃^{∗}) ⟶^{h_{1}^{∗}} H_{0}(C^{∗}),  (11.15)

114 BENDERSKY AND DAVIS

which becomes h_{1}^{#} : H_{2}(C̃)^{#} → H_{1}(C)^{#} under the isomorphism used in the proof of 11.3. Note that the shift of indices is opposite to that of the usual UCT, since the boundary morphisms in the chain complex C whose homology is being considered increase the grading.

We consider the commutative diagram

Ext(H_{1}C, Z_{2}^{∧}) ⟶^{u} H_{0}(C^{∗})
(h_{1})^{∗}↑  ^{h_{1}^{∗}}↑
Ext(H_{2}C̃, Z_{2}^{∧}) \underrightarrow{_{u}} H_{1}(C̃^{∗})

(11.16)

where u and ũ are the homomorphisms of the UCT, which are isomorphisms here because C is assumed to be rationally acyclic. The (h_{1})^{∗} in the diagram is dual to (11.14). Recall that the UCT homomorphism u is induced by applying Hom(−, Z_{2}^{∧}) to C_{0} ⟶^{∂} B_{1}, noting that

Ext(H_{1}C, Z_{2}^{∧}) ≈ Hom(B_{1}, Z_{2}^{∧})/ Hom(Z_{1}, Z_{2}^{∧}).

Here and elsewhere B_{i} denotes the boundaries and Z_{i} the cycles in C_{i}.

The basis-free version of H_{1}(C̃^{∗}) is H(M/2 ⟶^{θ_{m+1}−ψ^{2}} M/2 ⊕ M/2 ⟶^{ψ^{2}+θ_{m+1}} M/2). Let (x_{j}, y_{k}) be a pair of basis vectors representing a cycle. If a sum of basis vectors is required, either a change of basis or an obvious adaptation of the argument will yield the result. It is often the case that only one of x_{j} and y_{k} is needed to represent a class. The argument in such a case is slightly easier.

Let V denote a free Z_{2}^{∧}-module of rank n. We consider

C̃_{1} = V ⊕V ⊕V \underrightarrow{_{∂_{2}}} C̃_{2} = V ⊕V ⊕V ⊕V, ( ˜∂_{2}) = (\begin{matrix}0 & 0 & 0 \\ Ψ^{T} & 2 & 0 \\ Θ_{m+1}^{T} & 0 & 2 \\ 0 & Θ_{m+1}^{T} & −Ψ^{T} \end{matrix} )

In the proof of 11.3, it is shown that (x_{j}, y_{k}) corresponds in H_{1}(C̃^{∗}) to \frac{1}{2}(r_{n+j} +r_{2n+k}) in Q( ˜∂_{2}), where r_{t} denotes the tth row of the matrix. Let B̃_{2} = im( ˜∂_{2}) denote the column space of ( ˜∂_{2}), and let c_{ℓ} denote the ℓth column. Under the isomorphism ũ of (11.16), \frac{1}{2}(r_{n+j} + r_{2n+k}) corresponds to the morphism B̃_{2} → Z_{2}^{∧} sending

c_{ℓ} ↦ \begin{cases}1 & if ℓ = n + j or 2n + k \\ 0 & if n + 1 ≤ ℓ ≤ 3n, ℓ ≠ n + j, ℓ ≠ 2n + k \\ \frac{1}{2}((˜∂_{2})_{n+j,ℓ} + (˜∂_{2})_{2n+k,ℓ})) & if 1 ≤ ℓ ≤ n. \end{cases}

PERIODIC HOMOTOPY GROUPS OF SO(N) 115

The latter element is an integer since ψ^{2}x_{j} + θ_{m+1}(y_{k}) ≡ 0 mod 2.

The formula for h_{1} already derived induces h_{1} : B_{1} → B̃_{2} sending

(0, vΨ, vΘ_{m}) ↦ (0, vΨ, 0, \frac{1}{2}vΨΘ_{m+1}) + (0, 0, vΘ_{m}, −\frac{1}{2}vΘ_{m}Ψ). These image vectors were previously viewed as columns of (˜∂_{2}). We deduce that

(h_{1})^{∗}ũ^{−1}(x_{j}, y_{k}) : B_{1} → Z_{2}^{∧}

sends (0, e_{ℓ}Ψ, e_{ℓ}Θ_{m}) to \frac{1}{2}(Ψ)_{ℓ,j} + \frac{1}{2}(Θ_{m})_{ℓ,k}, and so u(h_{1})^{∗}ũ^{−1}(x_{j}, y_{k}) sends e_{ℓ} to the ℓth component of \frac{1}{2}(ψ^{2}x_{j} +θ_{m}y_{k}). This is what is meant by \frac{1}{2}(ψ^{2}x_{j} +θ_{m}y_{k}) ∈ H_{0}(C^{∗}).

Proof of Theorem 11.11. We will prove that (11.12) is acyclic when M = QK^{1}(S^{2n+1}). It follows that (11.12) is acyclic when M is ASR by induction on the rank of M.

To see this, first note that there is a short exact sequence of 2-adic Adams modules

0 → QK^{1}S^{2n+1} → M → M′ → 0

with M′ ASR and rank(M′) <rank(M). Thus the sequence (11.12) is acyclic for M′ by the induction hypothesis. Since M′ is ASR, ker(ψ^{2}|M′) = 0, and so by the Snake Lemma, there is a short exact sequence

0 → QK^{1}S^{2n+1}/ im(ψ^{2}) → M/ im(ψ^{2}) → M′/ im(ψ^{2}) → 0. This SES is covered by a SES of the complexes of (11.12), and so there is a long exact sequence relating the homology groups of these complexes for QK^{1}S^{2n+1}, M, and M′. Since the complexes are acyclic for QK^{1}S^{2n+1} and M′, the same must be true for M.

Now we prove the acyclicity of (11.12) for M = QK^{1}S^{2n+1}. We will consider here the case when n is odd, so that ψ^{−1} = −1 in M; the case n even follows similarly. One easily verifies that composites are 0 in the sequence by multiplying the matrix forms of the boundary morphisms, using that σ_{j} commutes with Ψ^{k}.

We will now show exactness at R_{0} by showing ϵ is injective on R_{0}/ im(∂_{1}). Let f ∈ R_{0} = Γ. Define g_{0} ∈ Γ by

g_{0}(i, j) = \begin{cases}f(i, 1) & j = 0 \\ 0 & j = 1. \end{cases}

Then f_{1} := f + ∂_{1}(g_{0}, 0, 0) satisfies f_{1}(i, 1) = 0 for all i. Define g_{1} ∈ Γ by

g_{1}(i, j) = \begin{cases}∑_{k≥0}(3^{n} − 1)^{k}f_{1}(k + i + 1, 0) & j = 0 \\ 0 & j = 1. \end{cases}

Then f_{2} := f_{1} + ∂_{1}(0, 0, g_{1}) satisfies f_{2}(i, j) = 0 unless i = j = 0. If ϵ(f) = 0, then 0 = ϵ(f_{2}) = f_{2}(0, 0) and so f_{2}(0, 0) ≡ 0 mod 2^{n}. Hence f_{2} = ∂_{1}(0, h, 0) for some h ∈ Γ. Thus f ∈ im(∂_{1}).

A somewhat similar argument works for exactness at R_{i} for each i ≥ 0. By periodicity of the chain complex, we need only verify it at R_{1}, R_{2}, and R_{3}. We will perform the verification for R_{1}, and leave the similar argument at R_{2} and R_{3} to the reader.

We show ∂_{1} is injective on R_{1}/ im(∂_{2}). Let (f_{1}, f_{2}, f_{3}) ∈ R_{1}. For 1 ≤ ϵ ≤ 3, define g_{ϵ} ∈ Γ by

g_{ϵ}(i, j) = \begin{cases}−f_{ϵ}(i, 1) & j = 0 \\ 0 & j = 1, \end{cases}

and let

(f′_{1}, f′_{2}, f′_{3}) = (f_{1}, f_{2}, f_{3}) + ∂_{2}(g_{1}, g_{2}, g_{3}, 0).

One can easily check that f′_{ϵ}(i, 1) = 0 for all i.

Define g_{4} ∈ Γ by

g_{4}(i, j) = \begin{cases}∑_{k≥0}(3^{n} − 1)^{k}f′_{2}(k + i + 1, 0) & j = 0 \\ 0 & j = 1. \end{cases}

Then (f′′_{1}, f′′_{2}, f′′_{3}) := (f′_{1}, f′_{2}, f′_{3}) + ∂_{2}(0, 0, 0, g_{4}) satisfies f′′_{ϵ}(i, 1) = 0 for all i and f′′_{2}(i, 0) = 0 for i > 0. Note that (f′′_{1}, f′′_{2}, f′′_{3}) ≡ (f_{1}, f_{2}, f_{3}) in R_{1}/ im(∂_{2}). We compute

∂_{1}(f′′_{1}, f′′_{2}, f′′_{3})(i, j) = \begin{cases}−f′′_{1}(i, 0) + (3^{n} − 1)f′′_{3}(i, 0) − f′′_{3}(i − 1, 0) & j = 0, i > 0 \\ −f′′_{1}(i, 0) & j = 1 \\ −f′′_{1}(0, 0) + 2^{n}f′′_{2}(0, 0) + (3^{n} − 1)f′′_{3}(0, 0) & i = j = 0. \end{cases}

If ∂_{1}(f′′_{1}, f′′_{2}, f′′_{3}) = 0, then the (j = 1)-part implies f′′_{1}(i, 0) = 0 for all i. Now we obtain

f′′_{3}(0, 0) = (3^{n} − 1)f′′_{3}(1, 0) = (3^{n} − 1)^{2}f′′_{3}(2, 0) = · · · .

Since 3^{n} − 1 is even, this implies that f′′_{3}(0, 0) is infinitely 2-divisible, and hence is 0, and hence so are all f′′_{3}(i, 0). Finally, since 2^{n}f′′_{2}(0, 0) + (3^{n} − 1)f′′_{3}(0, 0) = 0, we deduce f′′_{2}(0, 0) = 0. Thus (f′′_{1}, f′′_{2}, f′′_{3}) = 0, as desired.

PERIODIC HOMOTOPY GROUPS OF SO(N) 117

Now we consider the generalization of the above work to the situation when ψ^{−1} is any involution, no longer assumed to equal −1. Lemma 11.9 and Theorem 11.11 were already done in this generality. The analogue of Theorem 11.3 is given below. It follows immediately from 11.9 and the UCT argument used in the paragraph containing (11.14). The simplifications which were made in 11.3 do not apply in the general case, nor does the “basis-free” version, which relied on the simplifications.

Theorem 11.17. Let M be as in Lemma 11.9. Then Ext_{A}^{s,2m+1}(M/ im(ψ^{2})) is obtained by applying the functor Q to the matrices displayed in 11.9, while the Pontryagin dual of these Ext groups are obtained by applying Q to the transposes of the matrices displayed in 11.9.

Finally, we give the generalization of Theorem 11.5. We restrict our attention to h_{1} and h_{1}^{#} between the 1-line and 2-line.

Theorem 11.18. Let M be ASR, N_{1} = (Ψ^{−1} − (−1)^{m} Ψ^{2} Θ_{m}) and

N_{2} = (\begin{matrix}Ψ^{−1} + (−1)^{m+1} & Ψ^{2} & Θ_{m+1} & 0 \\ 0 & −Ψ^{−1} + (−1)^{m+1} & 0 & Θ_{m+1} \\ 0 & 0 & −Ψ^{−1} + (−1)^{m+1} & −Ψ^{2}. \end{matrix})

Then

h_{1} : Ext_{A}^{1,2m+1}(M/ im(ψ^{2})) → Ext_{A}^{2,2m+3}(M/ im(ψ^{2}))

is the homomorphism h_{1} : Q(N_{1}) → Q(N_{2}) defined by

h_{1}(wN_{1}) = ((−1)^{m+1}w 0 3^{m}w)N_{2}.

Here w is a 1-by-n matrix of rational numbers such that wN_{1} is integral. The Pontyragin dual

Ext_{A}^{2,2m+3}(M/ im(ψ^{2}))^{#} ⟶^{h_{1}^{#}} Ext_{A}^{1,2m+1}(M/ im(ψ^{2}))^{#}

is the homomorphism h_{1}^{#} : Q(N_{2}^{T}) → Q(N_{1}^{T}) defined by

h_{1}^{#}(qN_{2}^{T}) = ((−1)^{m+1}q_{0} − 3^{m}q_{1})N_{1}^{T} ,

where q = (q_{1}, …, q_{4n}) is a 1-by-4n matrix of rationals such that qN_{2}^{T} is integral, q_{0} = (q_{1}, …, q_{3n}), and q_{1} = (q_{2n+1}, …, q_{4n}, 0, …, 0). Note that q_{0} and q_{1} are 1-by-3n matrices; there are n 0’s at the end of q_{1}.

118 BENDERSKY AND DAVIS

The reader can perform the simple verification that the formulas for h_{1} and h_{1}^{#} are well-defined; i.e., that integrality of wN_{1} implies that of ((−1)^{m+1}w 0 3^{m}w)N_{2}, and that integrality of qN_{2}^{T} implies that of ((−1)^{m+1}q_{0} − 3^{m}q_{1})N_{1}^{T}. The identity that makes this work appears later in the proof. Proof. We use the diagram at the beginning of the proof of Theorem 11.5, and will follow along that proof. We have now

ϵ′(f) = ∑(−1)^{mj}(3^{m} − 1)^{i}f(i, j),
∂′_{1} = ((−1)^{m} − σ_{1} 3^{m} − 1 − σ_{3}),
ϵ′′(f) = ∑(−1)^{(m+1)j}(3^{m+1} − 1)^{i}f(i, j),
h_{1}(f, g) = (−1)^{m} ∑_{i,j}(−1)^{(m+1)j}(3^{m+1} − 1)^{i}f(i, j)

−3^{m} ∑_{i,j}(−1)^{(m+1)j}(3^{m+1} − 1)^{i}g(i, j),

2^{b}τ = (v(Ψ^{−1} − (−1)^{m}), vΨ^{2}, v(Ψ^{3} − 3^{m})),
2^{b}τ_{1} = (v(Ψ^{−1} − (−1)^{m}), vΨ^{2}, v(Ψ^{3} − 3^{m})),
τ_{1}∂_{2} = 2^{−b}v(Ψ^{−1} − (−1)^{m})(Ψ^{−1} + σ_{1}) on first summand,

2^{−b}vΨ^{2}(−(−1)^{m} + σ_{1}) on second, 2^{−b}v((\overline{Ψ^{3}} − σ_{3})(−(−1)^{m} + σ_{1}) + (Ψ^{−1} − σ_{1})(3^{m} − 1 − σ_{3})) on third, 2^{−b}vΨ^{2}(3^{m} − 1 − σ_{3}) on fourth,

τ_{2} = −2^{−b}v(Ψ^{−1} − (−1)^{m}) into first summand on first summand,

−2^{−b}vΨ^{2} into first on second, −2^{−b}v(\overline{Ψ^{3}} − σ_{3}) into first on third, +2^{−b}v(Ψ^{−1} − σ_{1}) into second on third, 2^{−b}vΨ^{2} into second on fourth,

h_{1}{τ} = 2^{−b}v(−1)^{m+1}(Ψ^{−1} − (−1)^{m}) on first summand,

2^{−b}v(−1)^{m+1}Ψ^{2} on second, 2^{−b}v((−1)^{m+1}(Ψ^{3} − 3^{m+1}) − 3^{m}(Ψ^{−1} + (−1)^{m})) on third, −3^{m}2^{−b}vΨ^{2} on fourth.

With w = 2^{−b}v, this yields the claim for h_{1}.

PERIODIC HOMOTOPY GROUPS OF SO(N) 119

To prove the claim for h_{1}^{#}, we use (11.16). The element qN_{2}^{T} ∈ H_{1}(C̃^{∗}) corresponds under ũ to the morphism from the column space B̃_{2} of N_{2}^{T} into Z_{2}^{∧} which sends the jth column to the jth component of qN_{2}^{T}. The morphism H_{1}(C) ⟶^{h_{1}} H_{2}(C̃) was just seen to be given by h_{1}(wN_{1}) = ((−1)^{m+1}w 0 3^{m}w)N_{2}. This corresponds to a morphism B_{1} → B̃_{2} of boundaries in the chain complexes, or equivalently from the column space of N_{1}^{T} to that of N_{2}^{T} sending the jth column of N_{1}^{T} to

(−1)^{m+1}(jth column of N_{2}^{T}) + 3^{m}((2n + j)th column of N_{2}^{T}). Composing, we obtain that (h_{1})^{∗}ũ^{−1}(qN_{2}^{T}) is the morphism B_{1} → Z_{2}^{∧} sending the jth column of N_{1}^{T} to

(−1)^{m+1}(qN_{2}^{T})_{j} + 3^{m}(qN_{2}^{T})_{2n+j} = ((−1)^{m+1}q_{0}N_{1}^{T} − 3^{m}q_{1}N_{1}^{T})_{j}, (11.19)

where q_{0} and q_{1} are as in the statement of the result being proved, and subscripts on a vector denote the indicated component of the vector. To verify (11.19), we note that (q_{1}, …, q_{n}) is multiplied by (−1)^{m+1}((Ψ^{−1})^{T} + (−1)^{m+1}) on both sides, (q_{n+1}, …, q_{2n}) is multiplied by (−1)^{m+1}(Ψ^{2})^{T} on both sides, (q_{3n+1}, …, q_{4n}) is multiplied by −3^{m}(Ψ^{2})^{T} on both sides, while (q_{2n+1}, …, q_{3n}) is multiplied by (−1)^{m+1}Θ_{m+1}^{T}+ 3^{m}(−(Ψ^{−1})^{T} + (−1)^{m+1}) on the left side and by (−1)^{m+1}Θ_{m}^{T} − 3^{m}((Ψ^{−1})^{T} − (−1)^{m}) on the right side, and these are easily verified to be equal.

Hence h_{1}^{∗}(qN_{2}^{T}) = u(h_{1})^{∗}ũ^{−1}(qN_{2}^{T}) is the element ((−1)^{m+1}q_{0} − 3^{m}q_{1})N_{1}^{T}, as claimed.

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Hunter College, CUNY, NY, NY 10021

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

Lehigh University, Bethlehem, PA 18015

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