v_{1}-periodic homotopy groups of Sp(n) ^{∗}

Martin Bendersky CUNY, Hunter College New York 10021 Donald M. Davis Lehigh University Bethlehem, PA 18015

Mark Mahowald Northwestern University Evanston, IL 60208 ^{†}

In this paper we calculate the 2-primary v_{1}-periodic homotopy groups of the symplectic groups Sp(n). The proof utilizes new methods of calculating the unstable Novikov spectral sequence. One corollary is that some homotopy group of Sp(n) contains an element of order 2^{2n−1}.

1 Main theorem

In this paper we calculate the 2-primary v_{1}-periodic homotopy groups of the symplectic groups Sp(n). The odd-primary v_{1}-periodic homotopy groups of Sp(n) were determined in [13]; they are an immediate consequence of the groups of SU(n). In Section 5, we apply our results to James numbers and exponents of actual homotopy groups.

The v_{1}-periodic homotopy groups of a space, whose definition we recall in the next paragraph, are a localization of the actual homotopy groups. Very roughly, they are the portion of the homotopy groups detectable by real and complex K-theory and their operations. A first attempt at computing the v_{1}-periodic homotopy groups of Sp(n) was made in [16]. The method there was to study the exact sequence of v_{1}-periodic homotopy groups associated to the fibration

∗AMS Subject Classification 55T15. Key words: v_{1}-periodic homotopy groups, symplectic groups, unstable Novikov spectral sequence, James numbers

†All authors were partially supported by the National Science Foundation. The second and third authors were also supported by the S.E.R.C.

Sp(n − 1) → Sp(n) → S^{4n−1}.  (1.1)

Difficulties in determining the homomorphisms in this exact sequence occurred in [16] as early as n = 3; indeed, there is a minor mistake in the calculation of v_{1}^{−1}π_{∗}(Sp(3)) in [16].^{1} We employ a similar method, but we use the unstable Novikov spectral sequence (UNSS) to lend precision to the calculations.

The p-primary v_{1}-periodic homotopy groups of any space X were defined in [15] by

v_{1}^{−1}π_{n}(X) = dirlim_{e} v_{1}^{−1}π_{n+1}(X; Z/p^{e}),  (1.2)

where

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

with the latter direct system being taken over Adams maps of the mod p^{e} Moore spaces. We say that X has an H-space exponent at p if, for some e and L, Ω^{L}X⟶^{p^{e}}Ω^{L}X is null-homotopic. For such spaces, which include spheres and compact Lie groups, v_{1}^{−1}π_{n}(X) is a direct summand of some π_{L}(X), provided v_{1}^{−1}π_{n}(X) is a finitelygenerated abelian group, which is true in the cases just mentioned. The importance of v_{1}^{−1}π_{∗}(X) is that it is often computable and yet gives significant information about actual homotopy groups. Much of the importance of our calculation of v_{1}^{−1}π_{∗}(Sp(n)) here is just that we were able to do it. It was an extremely challenging problem, requiring new methods of analysis of the UNSS.

Moreover, our result is new in a fundamental sense. The results for v_{1}^{−1}π_{∗}(SU(n)) proved in [13] and [9] yielded the same numbers that had come up in the work of Crabb and Knapp ([11]), although it was important to learn that all v_{1}-periodic homotopy classes were of this standard type. The same could be said for the large summands which we obtain in v_{1}^{−1}π_{∗}(Sp(n)); closely related groups have come up in the work of Morisugi ([26]) and Walker ([32]). But the way in which the Z_{2}’s from the various spheres contribute to Sp(n), as described in Proposition 1.8, is completely new, and should have further ramifications.

1The d_{2}-differential in the chart on [16, p. 228] should not be present. The image of Mimura and Toda’s boundary morphism cited there is in fact not v_{1}-periodic.

Let ϵ_{k} be defined by 0 ≤ ϵ_{k} ≤ 1 and ϵ_{k} ≡ k mod 2. Our results involve numbers defined for k > n by

e(k, n) = ϵ_{k−1} + min_{j>n} (ν (coef (\frac{t^{2k}}{(2k)!}, \frac{(e^{t} + e^{−t} − 2)^{j}}{2j} )) − ϵ_{k−j}) .  (1.3)

As usual, ν(−) = ν_{2}(−) denotes the exponent of 2. The numbers e(k, n) were called d_{2}^{A}(k, n) in [26]. They are periodic in k, and are extended to k ≤ n using this periodicity. This periodicity is proved similarly to [12, 3.15], using the following alternate form for the coefficients in (1.3).

\frac{1}{j} ∑_{i=0}^{j−1}(−1)^{i}\binom{2j}{i}(j − i)^{2k}  (1.4)

In Section 5, we will tabulate some values of the numbers e(k, n), which, along with the explicit calculations in [26], may shed some light on an unilluminating definition.

We now state our main result. We will denote by G(n) some abelian group satisfying |G(n)| = n. We make no claim about the precise structure of G(n), and, moreover, if G(n) occurs in two different formulas with the same n, we do not claim that the two groups are isomorphic. We denote by Z_{2} the cyclic group of order 2, and by nG the direct sum of n copies of the abelian group G.

Theorem 1.5 For 1 ≤ t ≤ 8 and any integer h, the 2-primary v_{1}-periodic homotopy group

v_{1}^{−1}π_{8h+t}(Sp(n)) = \{ \begin{matrix}G(2^{e(2h+1,n)}) & if t = 1 \\ Z/2^{e(2h+1,n)} & if t = 2 \\ Z_{2} & if t = 3 \\ Z_{2} ⊕ Z_{2} & if t = 4 \\ G(2^{1+e(2h+2,n)}) & if t = 5 \\ Z/2^{e(2h+2,n)} ⊕ [log_{2}(\frac{4n}{3})]Z_{2} & if t = 6 \\ G(2^{2[log_{2}(4n/3)]}) & if t = 7 \\ [log_{2}(\frac{4n}{3})]Z_{2} & if t = 8 \end{matrix}

The G(−)-group when t = 1, 5, or 7 has at least [log_{2}(4n/3)] direct summands. The group v_{1}^{−1}π_{8h+7}(Sp(n)) is an extension (possibly the trivial extension) of two Z_{2}-vector spaces of dimension [log_{2}(4n/3)].

One novel aspect of the result is the occurrence of arbitrarily many Z_{2}’s in a single v_{1}-periodic homotopy group, which had not occurred explicitly^{2} in previous examples such as SU(n). For the reader who wishes to obtain some feeling for these groups without studying the entire proof, we recommend reading the rest of this section, together with the portion of Section 5 which tabulates values of the numbers e(k, n). You might also take a look at Figure 4.4, which displays the final stage of the UNSS converging to v_{1}^{−1}π_{∗}(Sp(n)), and the recapitulation of the proof which precedes that figure.

The heart of the proof is the calculation of the E_{2}-term of the UNSS converging to v_{1}^{−1}π_{∗}(Sp(n)). This is done using the exact sequences of v_{1}-periodic UNSS’s induced by (1.1) to give a spectral sequence (SS) converging to v_{1}^{−1}E_{2}(Sp(n)) whose initial term is

⨁_{1≤i≤n} v_{1}^{−1}E_{2}^{s,t}(S^{4i−1}),  (1.6)

where v_{1}^{−1}E_{2} refers to the E_{2}-term of the v_{1}-periodic UNSS of [4]. We call this the cellular SS, or CSS. Strictly speaking, this isn’t quite accurate, since Sp(n) has more cells than the S^{4i−1}, but the name “cellular” seems to have the right flavor.

A complete description of v_{1}^{−1}E_{2}(S^{4i−1}) is given in [4, p. 58] and reproduced in our Figure 3.25. In later sections, we will make use of some of the technical names of classes given there, but to describe the result, we prefer the simpler form which we give in Figure 1.7. Note that the horizontal component in Figure 1.7 refers to the homotopy group, and not the stem, as it does in Figure 3.25. The letter i is part of the notation of all of these classes in v_{1}^{−1}E_{2}(S^{4i−1}) because we shall be combining the charts for various values of i. The letters i, i′, i_{u}, and i′_{u} each represent an η-tower; this is an infinite collection of elements {η^{j}x : j ≥ 0} of order 2, where η is the UNSS representative of the Hopf map. The action of the element η, which we will also denote by h_{1}, moves one position to the right, and one position up; the diagonal arrows in Figure 1.7 suggest this. The boxes which contain the elements U_{i} and i′_{u} also contain a second element, which is part of the η-tower labeled i − 1, and is implicit by the periodicity of the chart. The symbols S_{i} and U_{i} represent cyclic 2-groups of the same order. The letters suggest “stable” and “unstable.” The groups S_{i} and U_{i} are Z/8 if i ≡ k mod 2, while if i ≢ k mod 2, they are Z/2^{m} with m = min(ν(k − i + 1) + 3, n).

2But see Proposition 1.17.

Figure 1.7: v_{1}^{−1}E_{2}^{s,t}(S^{4i−1}) [figure: s = 1; 2; 3; t − s = 4k+; 1; 2; 3; 4; S_{i}; U_{i}; i; i′ i_{u}; i′_{u}]

These charts are superimposed as in (1.6), with d_{r}-differentials always going from E^{s,t} to E^{s+1,t}, and decreasing i by r. Thus they always go to the group one unit above and one to the left of the source class. There are d_{r}-differentials from S_{i}-groups to U_{i−r}-groups which cause v_{1}^{−1}E_{2}^{1,4k+3}(Sp(n)) to be cyclic of order 2^{e_{Sp}(k+1,n)}, as proved in [10]. Here e_{Sp}(k + 1, n) is as defined in (1.15). Since |U_{i}| = |S_{i}| and there are no differentials in the CSS going out from the groups U_{i}, the portion of the group v_{1}^{−1}E_{2}^{2,4k+3}(Sp(n)) built from the U_{i}’s will also have order 2^{e_{Sp}(k+1,n)}, but it will not necessarily be cyclic.

In Section 3, we will prove the following result, which establishes all differentials among the η-towers in the CSS.

Proposition 1.8 In the CSS of (1.6), the differentials among the η-towers depicted in Figure 1.7 satisfy, for all values of k,

d_{i}(2i) = i_{u}  if i = 2^{e} or 3 · 2^{e}, e ≥ 0  (1.9)
d_{i}((2i)′) = i′_{u}  if i = 2^{e}, e ≥ 0  (1.10)
d_{2^{e+1}}((2i + 3)2^{e}) = ((2i + 1)2^{e})′  if i ≥ 1, e ≥ 0 or i = e = 0 (1.11)
d_{2^{e+1}}(((2i + 3)2^{e})_{u}) = ((2i + 1)2^{e})′_{u} if i ≥ 1, e ≥ 0,  (1.12)

provided 2i ≤ n in (1.9) and (1.10), and (2i + 3)2^{e} ≤ n in (1.11) and (1.12).

The restrictions are present because in other cases one of the classes will have been involved in an earlier differential. For example, d_{5}(10) = 5_{u} is masked by d_{4}(10) = 6′ and d_{2}(5_{u}) = 3′_{u}.

The reader can easily verify that every class except “1” is involved in one of the differentials if n = ∞. This is consistent with Bott periodicity for Sp. However, if n satisfies i − r ≤ n < i, then a d_{r}-differential from a class numbered i to one numbered i − r cannot occur in Sp(n), and so it frees the (i − r)-class to survive the CSS. In Proposition 1.14, we enumerate the number of unstable η-towers that survive all the differentials in Sp(n).

We illustrate in Figure 1.13 the pattern of differentials among the η-towers in the CSS converging to v_{1}^{−1}E_{2}(Sp(10)). Classes in Figure 1.13 which are not involved in differentials survive to η-towers in v_{1}^{−1}E_{2}(Sp(10)). In Figure 1.16, we will show how the surviving classes in v_{1}^{−1}E_{2}(Sp(n)) are situated, and in Theorem 4.1 and Corollary 4.3 we will establish the d_{3}-differentials among them. Figure 4.4 shows a final picture of the UNSS for Sp(n). Theorem 1.5 is a consequence of this analysis.

Figure 1.13: Cellular SS for v_{1}^{−1}E_{2}(Sp(10))

1  2  3  4 1′ 3′ 5′ 7′ 9′  13579 ⟵ ⟵ ⟵ ⟵ 1′_{u} 2′_{u} 3′_{u} 4′_{u} 5′_{u} 6′_{u} 7′_{u}⟵ ⟵ ⟵ ⟵ ⟵ ⟵ ⟵ 2′ 4′ 5_{u} 8′ 7_{u} 10_{u} 9_{u} 1_{u} 2_{u} 3_{u} 4_{u} 6′ ⟵ ⟵ ⟵ ⟵ ⟵ 246810 10′ 6_{u} 8_{u} 8′_{u} 9′_{u} 10′_{u}

The following result shows how the [log_{2}(4n/3)] in Theorem 1.5 arises, as an enumeration of the number of η-towers in v_{1}^{−1}E_{2}(Sp(n)).

Proposition 1.14 The number of classes of the form i′_{u} in Sp(n) which are not hit by differentials in Proposition 1.8 is [log_{2}(4n/3)].

There is only one class, namely “1,” of the form “i” which does not support a differential. Consequently, there will also be 1 + [log_{2}(4n/3)] η-towers beginning in the box at height 2 and t − s = 4k + 3 which survive the CSS. To understand this, the reader may find it helpful to refer to Figure 1.13, where [log_{2}(40/3)] = 3.

Proof of Proposition 1.14. We must count the number of integers j ≤ n for which 2j > n if j is a 2-power, and j + 2^{ν(j)+1} > n if j is not a 2-power. There is one such 2-power.

Define

f(x, y) = ∑_{j∈S} x^{j} ∑_{n=j}^{j+2^{ν(j)+1}−1} y^{n},

where S is the set of positive integers which are not 2-powers. Our desired answer is 1 plus the coefficient of y^{n} in f(1, y). Collecting terms over common values of ν(j) yields

f(x, y) = ∑_{e≥0} x^{2^{e}}y^{2^{e}}(\frac{x^{2^{e+1}}y^{2^{e+1}}}{1 − x^{2^{e+1}}y^{2^{e+1}}} \frac{1 − y^{2^{e+1}}}{1 − y} ).

Setting x = 1 allows some cancellation in the latter product, from which we obtain

f(1, y) = \frac{∑ y^{3·2^{e}}}{1 − y} .

The coefficient of y^{n} in this is the number of 3 · 2^{e}’s which are equal to or less than n. This is 1 plus the number of 2^{e}’s with e ≥ 1 which are less than or equal to n/3. This number of 2^{e}’s is [log_{2}(n/3)], from which we obtain the answer claimed.

The result of the differentials of Proposition 1.8 is the following chart, Figure 1.16, for v_{1}^{−1}E_{2}(Sp(n)). Here a • represents a Z_{2} (of the type “1” when t − 2s ≡ 3 mod 4, and n′ or (n − 1)′ when t − 2s ≡ 1 mod 4), while “log” represents a Z_{2}-vector space of dimension [log_{2}(4n/3)]. For j = 1 and 2, we abbreviate e_{Sp}(2h + j, n) to e_{j} in the chart. Here and elsewhere e_{Sp}(k, n) is as in [10]; it is defined by

e_{Sp}(k, n) = min_{j>n} (ν_{2} (coef (\frac{t^{2k}}{(2k)!}, \frac{(e^{t} + e^{−t} − 2)^{j}}{2j} ))) .  (1.15)

These numbers are obtained from the numbers e(k, n) defined in (1.3) by removing the numbers ϵ_{k−1} and ϵ_{k−j}. As already noted, e_{Sp}(k, n) determines the size of the 1-line group of the UNSS. We shall show in Section 4 that the e_{k−1} which is added in forming e(k, n) corresponds to an extension to the 3-line of the SS, and the e_{k−j} which might be subtracted corresponds to a d_{3}-differential in the SS.

As in Theorem 1.5, G(2^{e}) indicates an abelian group of order 2^{e}. These 2-line groups are built from the U_{i}-groups of Figure 1.7. In Section 4, we will discuss how Theorem 1.5 is deduced from this chart, by inserting d_{3}-differentials and extensions.

Figure 1.16: v_{1}^{−1}E_{2}^{s,t}(Sp(n))

[figure: s = 1; 2; 3; 4; 5; t − s = 8h+; 1; 2; 3; 4; 5; 6; 7; 8; • log; • log; • log; • log; • log; • log; • log; • log; • log; • log; • log; • log; • log; • log; • G(2^{e_{1}}); • G(2^{e_{2}}); Z/2^{e_{1}}; Z/2^{e_{2}}; •; •]

Except for changes in values of e_{i}, this v_{1}^{−1}E_{2}(Sp(n)) has horizontal period 4; the period 8 is used in Figure 1.16 in order to display the η-towers more clearly, and for later use in inserting d_{3}-differentials, most of which have period 8. The only additional justification required for the chart is for the η-extension from the G in filtration 2 to the log in filtration 3. This is a consequence of [4, 5.4], which says that η : v_{1}^{−1}E_{2}^{2,t}(Sp(n)) → v_{1}^{−1}E_{2}^{3,t+2}(Sp(n)) is surjective. This η-extension implies that G(2^{e(k,n)}) has at least [log_{2}(4n/3)] summands for all values of k.

A similar analysis yields new and surprising information about v_{1}^{−1}π_{∗}(SU(2n + 1)). We recall from [9, 1.1] that v_{1}^{−1}π_{2k}(SU(2n + 1)) is a cyclic 2-group of exponent min{ν(coef(x^{k}/k!, (e^{x} − 1)^{j})) : 2n + 1 ≤ j ≤ k}, while v_{1}^{−1}π_{2k−1}(SU(2n + 1)) is an abelian group of the same order. We can now show that v_{1}^{−1}π_{2k−1}(SU(2n + 1)) will often have many summands.

Proposition 1.17 For every integer j, the abelian group v_{1}^{−1}π_{8j−1}(SU(2n + 1)) can be written as the direct sum of at least [log_{2}(4n/3)] summands.

Proof. Similarly to [10, §2], there is an exact sequence

→ v_{1}^{−1}Ext_{U}^{1,8j+1}(BP_{∗}(ΣHP^{n})) → v_{1}^{−1}E_{2}^{2,8j+1}(Sp(n)) → v_{1}^{−1}E_{2}^{2,8j+1}(SU(2n + 1)) → .

(1.18)

In the case n = ∞ of this sequence, v_{1}^{−1}Ext_{U}^{1,8j+1}(BP_{∗}(ΣHP^{∞})) sits between two 0 groups (using [10, 1.1]) and hence is 0. Thus the exact sequence

v_{1}^{−1}Ext_{U}^{0,8j+1}(BP_{∗}(S^{4n+5})) → v_{1}^{−1}Ext_{U}^{1,8j+1}(BP_{∗}(ΣHP^{n})) → v_{1}^{−1}Ext_{U}^{1,8j+1}(BP_{∗}(ΣHP^{n+1}))

implies that v_{1}^{−1}Ext_{U}^{1,8j+1}(BP_{∗}(ΣHP^{n})) is 0 for n ≥ 2j, is a quotient of Z (and hence cyclic) for n = 2j − 1, and is a subgroup of this cyclic group, and hence cyclic, for n < 2j − 1. By the calculation of v_{1}^{−1}E_{2}(Sp(n)), whose result is illustrated in Figure 1.16, v_{1}^{−1}E_{2}^{2,8j+1}(Sp(n)) is a direct sum of 1+[log_{2}(4n/3)] copies of Z_{2}. Thus by (1.18) v_{1}^{−1}E_{2}^{2,8j+1}(SU(2n + 1)) must have at least [log_{2}(4n/3)] direct summands.

Now we refer to the analysis in [9] beginning on page 485 which involves the space R_{m} = SU(2m + 1)/SU(2m − 1). This space was called Q_{m} in [9], but we have changed the name to avoid confusion with the quaternionic quasi-projective space Q^{n}, which will occur frequently in this paper. It was shown in [9] that v_{1}^{−1}π_{∗}(R_{m}) → v_{1}^{−1}E_{2}^{2,∗+2}(R_{m}) is bijective when ∗ is odd. Since by [9, 2.3a] v_{1}^{−1}π_{∗}(R_{m}) → v_{1}^{−1}E_{2}^{1,∗+1}(R_{m}) is bijective when ∗ is even, the 5-lemma and induction imply that

v_{1}^{−1}π_{∗}(SU(2n + 1)) → v_{1}^{−1}E_{2}^{2,2+∗}(SU(2n + 1))

is bijective when ∗ is odd. Here we have also used injectivity of v_{1}^{−1}π_{2j}(SU(2n−1)) → v_{1}^{−1}π_{2j}(SU(2n+1)) and the concentration of the v_{1}-periodic E_{2}-term of R_{m}, and hence SU(2n − 1), in filtrations 1 and 2.

The conclusions of the above two paragraphs imply the proposition.

In order to handle some of the d_{3}-differentials in the periodic UNSS for Sp(n), we will need a comparison with the periodic (stable) NSS for the quaternionic quasiprojective space Q^{n}. These spaces, discussed in [26] and [32], have one cell in each dimension 4i − 1 for 1 ≤ i ≤ n, and embed naturally in Sp(n). The analysis of this periodic NSS, which may be of independent interest, will be carried out in Section 2. Some of the techniques employed there will be used in the more delicate UNSS calculations which follow.

We close the introduction by presenting, for comparison, the easier calculation of v_{1}^{−1}π_{∗}(Sp). Note the shift of dimensions: Z/2^{∞} ⊂ v_{1}^{−1}π_{i}(Sp) if and only if Z ⊂ π_{i+1}(Sp).

Theorem 1.19 For any prime p,

v_{1}^{−1}π_{i}(SU) = \{ \begin{matrix}Z/p^{∞} & if i is even \\ 0 & if i is odd. \end{matrix}

If p = 2, then

v_{1}^{−1}π_{i}(Sp) = \{ \begin{matrix}Z/2^{∞} & if i ≡ 2 mod 4 \\ Z_{2} & if i ≡ 4 or 5 mod 8 \\ 0 & otherwise. \end{matrix}

Proof. As SU is similar and easier, we just prove it for Sp. The exact sequence

π_{n+1}(Sp)⟶^{2^{e}}π_{n+1}(Sp) → [M^{n+1}(2^{e}), Sp] → π_{n}(Sp)⟶^{2^{e}}π_{n}(Sp)

and Bott’s results for π_{∗}(Sp) show that [M^{n+1}(2^{e}), Sp] has summands, for n > 0, of Z/2^{e} if n ≡ 2 mod 4, Z_{2}’s coming from the bottom cell of the Moore space if n ≡ 4 or 5 mod 8, and Z_{2}’s coming from the top cell of the Moore space if n ≡ 3 or 4 mod 8. The Adams map induces an isomorphism

[M^{n+1}(2^{e}), Sp]⟶^{A^{∗}}[M^{n+2^{e+1}+1}(2^{e}), Sp],

and so v_{1}^{−1}π_{n}(Sp; Z/2^{e}) is the sum of the above summands. We consider the morphism

v_{1}^{−1}π_{n}(Sp; Z/2^{e})⟶^{ρ^{∗}}v_{1}^{−1}π_{n}(Sp; Z/2^{e+1})

induced by the map ρ which has degree 1 on the bottom cell and degree 2 on the top cell. This sends the summand Z/2^{e} injectively, is 0 on the Z_{2}’s due to the top cell of the Moore space, and sends the Z_{2}’s due to the bottom cell of the Moore space injectively. Clearly, the direct limit of (1.2) for X = Sp is as claimed in this theorem.

2 The periodic stable Novikov spectral sequence

The 2-primary v_{1}-periodic homotopy groups of a spectrum X are defined in [15] by

v_{1}^{−1}π_{i}(X) = dirlim_{e,L} π_{i}(X ∧ M^{−L2^{e}}(2^{e})).

Here M^{n}(k) is a mod k Moore spectrum with top cell in dimension n, and the direct system is over Adams maps of Moore spaces (for increasing L) and maps ρ′ defined below (for increasing e). This is compatible with the definition, (1.2), of v_{1}-periodic unstable homotopy groups since the Moore spectrum is self-dual. The v_{1}-periodic stable homotopy groups of a space X, denoted v_{1}^{−1}π_{∗}^{s}(X), are defined to be the v_{1}- periodic homotopy groups of the suspension spectrum of X.

Analogously, we define the v_{1}-periodic Novikov SS (NSS) of a space or spectrum X by

v_{1}^{−1}E_{r}^{s,t}(X) = dirlim_{L,e} E_{r}^{s−1,t−1}(X ∧ M^{−L2^{e}}(2^{e})),  (2.1)

where the SS on the right is the ordinary NSS. As described in [27, 4.4.1], this ordinary NSS satisfies

E_{2}^{s,t}(Y) = Ext_{BP_{∗}BP}^{s,t} (BP_{∗}, BP_{∗}(Y)).

The reader should be warned that the same symbol, E_{2}, is used for each of the spectral sequences in this paper. These include the cellular SS, the UNSS, and the (stable) NSS. Which SS is intended should be clear from the context. For the v_{1}-periodic SS’s, we use v_{1}^{−1}E_{2} instead of E_{2}.

There are compatible Adams maps M^{n}(2^{e}) → M^{n−2^{e−1}}(2^{e}) and canonical maps M^{n}(2^{e})⟶^{ρ′}M^{n}(2^{e+1}) among the Moore spaces which define the direct system. The maps ρ′ are dual to those of [15, 2.1] and those used in the proof of Theorem 1.19. The reason for the shift in s-filtration in (2.1) is that in the direct system of exact sequences based on the cellular decomposition of the Moore spaces, such as

→ E_{2}^{s−1,t−1}(X ∧ S^{−1})⟶^{α}E_{2}^{s−1,t−1}(X ∧ M^{0})⟶^{β}E_{2}^{s,t}(X) →,

it is the part mapped by β which persists in the direct system.

We sketch the computation of v_{1}^{−1}E_{r}^{s,t}(S^{0}). We begin by noting that η : E_{2}^{s,t} → E_{2}^{s+1,t+2} acts on the SS and satisfies 2η = 0. We first describe dirlim_{L} E_{2}^{s,t}(M^{−L2^{e}}(2^{e})), using results and methods from [27, pp. 199-200]. For all odd integers i, dirlim_{L} E_{2}^{s,t}(M^{−L2^{e}}(2^{e})) has cyclic summands in (s, t) = (0, i) and (1, i) of order 2 if i ≡ 1 mod 4, and of order 2^{min(ν(i+1)+1,e)} if i ≡ 3 mod 4. These summands are acted on freely by η, subject to 2η = 0. There are nonzero d_{3}-differentials from the generators in (0, i) and (1, i) with i ≡ 3 or 5 mod 8. They hit elements divisible by η^{3}, except that the differential from (0, k2^{e−1} + 3) hits the sum of the two elements in the box. These d_{3}-differentials act naturally on the entire η-tower. There is a nontrivial extension from the Z/4 in E_{∞}^{ϵ,i} for ϵ = 0 or 1, and i ≡ 3 mod 8, to the Z_{2} in E_{∞}^{ϵ+2,i+2} which is in the η-tower which begins in filtration ϵ.

Part of this dirlim_{L} E_{2}^{s,t}(M^{−L2^{e}}(2^{e})) is depicted in Figure 2.2. In this chart a

• represents a Z_{2}, while a letter or number represents a cyclic 2-group with that symbol as exponent. For example, a 3 denotes a Z/8.

Figure 2.2: Part of lim_{L} E_{2}^{s,t}(M^{−L2^{e}}(2^{e}))

[figure: e; e; 3; 3; 4; 4; +; +; s = 0; 1; 2; 3; 4; t − s = k2^{e−1}+; −2; −1; 0; 1; 2; 3; 4; 5; 6; 7; 8]

The map M(2^{e})⟶^{ρ′}M(2^{e+1}) which has degree 1 on the top cell and degree 2 on the bottom cell induces a morphism of the SS’s just described. The direct limit of these morphisms is, after a shift on one s-filtration, the v_{1}-periodic NSS for the sphere spectrum, according to (2.1). For i + 1 ≠ 0, the direct limit is determined by the morphisms for e ≥ ν(i + 1) + 1. These morphisms send the group in (0, i) and its η-tower isomorphically, while (unless i = 1) the second summand in (1, i) (the one not divisible by η) and its η-tower are mapped by multiplication by 2. The Z/2^{e}’s in (0, −1) and (1, −1) are both mapped injectively to Z/2^{e+1}. The morphism sends η times the generators in both (0, −1) and (1, −1) to 0, and it sends the other Z_{2} in (1, 1) to the sum of the two Z_{2}’s in the box. We remind the reader that (−, −) here refers to (s, t) before the shift of filtration.

From this information we can read off dirlim_{L,e} E_{2}^{s,t}(M^{−L2^{e}}(2^{e})), and then use (2.1) to yield the following result, a portion of which is displayed in Figure 2.4.

Theorem 2.3 The v_{1}-periodic NSS for S^{0} satisfies

a. v_{1}^{−1}E_{2}^{s,t}(S^{0}) has

• for every even integer i ≠ 0 a cyclic summand in (s, t) = (1, i) of order 2 if i ≡ 2 mod 4, and of order 2^{ν(i)+1} if i ≡ 0 mod 4. Each of these summands supports an η-tower.

• a Z/2^{∞} summand in (s, t) = (1, 0) and (2, 0).

• a Z_{2} in (2, 2) supporting an η-tower.

b. There are nonzero d_{3}-differentials on the classes in (1, i) for i ≡ 4 and 6 mod 8, and on the η-towers arising from them.

c. There is a nontrivial extension (multiplication by 2) from (1, i) to (3, i+2) if i ≡ 4 mod 8.

Let Q^{n} denote the quaternionic quasi-projective space with top cell of dimension 4n − 1. The standard cofibrations induce exact sequences

→ v_{1}^{−1}E_{2}^{s,t}(Q^{n−1}) → v_{1}^{−1}E_{2}^{s,t}(Q^{n}) → v_{1}^{−1}E_{2}^{s,t}(S^{4n−1}) → .  (2.5)

These can be spliced to give a cellular SS

⨁_{i≤n} v_{1}^{−1}E_{2}^{s,t}(S^{4i−1}) ⟹ v_{1}^{−1}E_{2}^{s,t}(Q^{n}).  (2.6)

Figure 2.4: Part of periodic NSS for S^{0}

[figure: ∞; ∞; 3; 4; s = 1; 2; 3; 4; t − s =; −3; −2; −1; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9]

Differentials in the SS (2.6) correspond to boundary morphisms in the exact sequences (2.5). We describe now the way in which they are related to the coaction

BP_{∗}(Q^{n})⟶^{ψ}Γ ⊗_{BP_{∗}} BP_{∗}(Q^{n})

and the reduced coaction \overline{ψ}(x) = ψ(x)−x⊗1−1⊗x. Here we introduce the common notation Γ = BP_{∗}BP.

The cobar complex for calculating the NSS is described, for example, in [27]. It satisfies

C^{s}(X) = Γ ⊗ ⋯ ⊗ Γ ⊗ BP_{∗}X,

with s factors of Γ. Here and throughout, tensor products are always over BP_{∗}. The formula for d : C^{s} → C^{s+1} is the usual alternating sum of comultiplications and coactions. It is a derivation and satisfies d(v) = η_{R}(v) − v for v ∈ BP_{∗}. This is illustrated as follows. Suppose g ∈ BP_{∗}(X) and v ∈ BP_{∗}. Suppose that the coaction satisfies ψ(g) = 1 ⊗ g + ∑ γ_{i} ⊗ g_{i}. Then

d([ ]vg) = [1]vg − [v]g − ∑[vγ_{i}]g_{i}
= [η_{R}(v) − v]g − vd(g).

The classes in the v_{1}-periodic NSS are all present in this cobar complex. To calculate

∂ : E_{2}^{s}(S^{4n−1}) → E_{2}^{s+1}(Q^{n−1})

on a class z represented by a cycle A⊗ι_{4n−1}, we first note that in the cobar complex for Q^{n}, A ⊗ \overline{ψ}(g_{n}) pulls back to an element γ_{n−1} ∈ C^{s+1}(Q^{n−1}). Here we have introduced generators g_{i} ∈ BP_{4i−1}(Q^{n}) which are compatible under inclusion maps, and reduce to the generator of BP_{4i−1}(S^{4i−1}) under the collapse map. More about the selection of these classes g_{i} will be prescribed in the proof of Proposition 2.7.

Assume that by varying it by boundaries you have pulled the class γ_{n−1} back to γ_{i} ∈ C^{s+1}(Q^{i}). Project γ_{i} into C^{s+1}(S^{4i−1}). If this class is not a boundary, then it represents ∂(z) in the SS. Otherwise, this class is d(B ⊗ι_{4i−1}) in C^{s+1}(S^{4i−1}) for some B ∈ Γ^{⊗s}. Then γ_{i} − d(B ⊗ g_{i}) pulls back to γ_{i−1} ∈ C^{s+1}(Q^{i−1}). This procedure of pulling back to smaller Q^{i}’s is continued as long as possible.

Explicit calculation of the coaction in BP_{∗}(Q^{n}) can be performed by a method similar to that of [10], but the formulas are extremely complicated. Instead we opt for calculating a homomorphic image of the coaction which contains the information relevant to our application.

The formula for the coaction is obtained using the Adams operation Ψ^{3} in bu_{∗}(−), but the route between them is somewhat tricky. We begin with the easy part, the evaluation of the operations. Recall that bu_{∗} is a polynomial algebra over Z_{(2)} on a single generator v of degree 2. Then bu_{∗}(Q^{n}) is a free bu_{∗}-module on classes g_{i} of degree 4i − 1, for 1 ≤ i ≤ n. An explicit choice of these generators will be made in the proof of Proposition 2.7.

Proposition 2.7 Generators g_{i} for bu_{∗}(Q^{n}) can be chosen so that

Ψ^{3}(g_{n}) = ∑_{i=0}^{n−1}(−3)^{−i}\binom{2n+i−1}{i}v^{2i}g_{n−i}.

Since the inclusion maps Q^{i} → Q^{n} send g_{j} to g_{j}, this proposition can be used to read off Ψ^{3}(g_{i}) in bu_{∗}(Q^{n}) for all i ≤ n.

Proof. The spectra Q^{n} and ΣHP_{−n}^{−2} are S-dual. Here HP_{−n}^{−2} is a spectrum which we think of as a shifted version of the suspension spectrum of the stunted quaternionic projective space HP_{A−n}^{A−2}, with A highly divisible. (A must be a multiple of the appropriate number of [31].)

It is standard that KU(HP^{n}) = Z[γ]/γ^{n+1} with Ψ^{3}(γ) = γ(γ + 3)^{2}. Let v ∈ bu^{−2}(S^{0}) denote the Bott element, which acts on any bu^{∗}(−). It satisfies Ψ^{3}(v) = 3v.

Let x_{i} ∈ bu^{4A−4n}(HP_{A−n}^{A−2}) be chosen to satisfy

v^{2A−2n}x_{i} = γ^{A−i} ∈ bu^{0}(HP_{A−n}^{A−2}) ≈ KU(HP_{A−n}^{A−2}).

Then

3^{2A−2n}v^{2A−2n}Ψ^{3}(x_{n}) = Ψ^{3}(v^{2A−2n}x_{n}) = Ψ^{3}(γ^{A−n}) = γ^{A−n}(γ + 3)^{2(A−n)}
= ∑ \binom{2(A−n)}{i}3^{2(A−n)−i}γ^{A−n+i}
= ∑ \binom{2A−2n}{i}3^{2A−2n−i}v^{2A−2n}x_{n−i}

We cancel 3^{2A−2n}v^{2A−2n}, and note that since A is highly divisible, \binom{2A−2n}{i} should be considered as \binom{−2n}{i} = (−1)^{i}\binom{2n+i−1}{i}. This yields the following formula in bu^{−4n}(HP_{−n}^{−2}).

Ψ^{3}(x_{n}) = ∑_{i=0}^{n−1}(−3)^{−i}\binom{2n+i−1}{i}x_{n−i}.

Letting v^{2(n−i)}g_{i} ∈ bu_{4n−1}(Q^{n}) be S-dual to x_{i} ∈ bu^{−4n}(HP_{−n}^{−2}), we obtain the desired formula in bu_{4n−1}(Q^{n}).

Using the \binom{−2n}{i}-form for the binomial coefficient immediately yields the following corollary.

Corollary 2.8 In bu/(2)_{∗}(Q^{n}),

Ψ^{3}(g_{n}) = g_{n} + v^{2^{ν(n)+2}}g_{n−2^{ν(n)+1}} + T,

where T ∈ ⟨v^{2i}g_{n−i} : i > 2^{ν(n)+1}⟩.

We will use Corollary 2.8 to obtain information about the BP -coaction of Q^{n}. The following result, well-known to experts, will be useful. In it, BP and bu denote the 2-local spectra, as they have throughout this paper.

Proposition 2.9 There is a multiplicative map Ψ_{BP}^{3} : BP → BP which is compatible with Ψ_{bu}^{3} : bu → bu under the usual reduction ρ : BP → bu.

The operation Ψ_{bu}^{3} here is the same as Ψ^{3} of 2.7 and 2.8; we just use the subscript to distinguish it from the BP -operation.

Proof. The map Ψ_{BP}^{3} was first defined in [2], and is also discussed in [33, §11]. It is defined by

Ψ^{3}(x) = \frac{1}{3}[3](x),  (2.10)

where x ∈ BP^{2}(CP^{∞}) denotes the standard generator, and [3](x) is the usual 3-series, defined from the formal group law (FGL). The map ρ induces a map of FGL’s to the 2-typical FGL of bu. Hence, if Ψ^{3} : bu → bu is defined from (2.10) using the 2-typical FGL for bu, it will be compatible with Ψ_{BP}^{3} . The usual formula for Ψ^{3} in bu, which was used in Proposition 2.7, is also derived from (2.10), but using the FGL defined by F (x, y) = x + y − vxy. This FGL is well-known to be equivalent to the 2-typical FGL (see [20, 2.3.1]), and hence the Ψ^{3} derived from it is the same map bu → bu, but just using a different choice of generator in bu^{2}(CP^{∞}). This establishes the desired compatibility.

The proof of the following result shows how the Adams operations in bu are used to give information about the BP -coaction. We recall that BP_{∗} = Z_{(2)}[v_{1}, v_{2}, …] and Γ = BP_{∗}BP = BP_{∗}[t_{1}, t_{2}, …], with |v_{i}| = |t_{i}| = 2(2^{i} − 1). In much of the paper, we will use the alternate generating set {h_{1}, h_{2}, …}, where h_{i} = c(t_{i}), with c denoting the conjugation.

Proposition 2.11 Let ϕ : BP_{∗}BP → Z_{2}[v] be the ring homomorphism defined by

ϕ(t_{i}) = ϕ(v_{i}) = \{ \begin{matrix}0 & if i > 1 \\ v & if i = 1. \end{matrix}  (2.12)

Let g_{i} ∈ BP_{4i−1}(Q^{n}) be the generator defined as in the proof of 2.7, using S-duality and the standard generators of BP^{∗}(HP^{m}). Suppose the coaction

BP_{∗}(Q^{n})⟶^{ψ}Γ ⊗_{BP_{∗}} BP_{∗}(Q^{n})

satisfies

ψ(g_{n}) = ∑ z_{i} ⊗ g_{i}.  (2.13)

Then

ϕ(z_{i}) = \{ \begin{matrix}0 & if n − 2^{ν(n)+1} < i < n \\ v^{2^{ν(n)+2}} & if i = n − 2^{ν(n)+1}. \end{matrix}

The reader should be careful to distinguish between the coaction ψ and the Adams operation Ψ^{3}.

Proof. The relationship between the coaction ψ and the Adams operation Ψ_{BP}^{3} is given by the following result of [1, p. 72]: If the BP -coaction is given by (2.13), then

Ψ_{BP}^{3} (g_{n}) = ∑ Ψ̂_{BP}^{3} (c(z_{i}))g_{i}.  (2.14)

Here, for z ∈ BP_{∗}BP, Ψ̂_{BP}^{3} (z) = ⟨Ψ_{BP}^{3} , z⟩ ∈ BP_{∗} denotes the Kronecker pairing. We will prove in the next paragraph that, for i ≥ 1,

Ψ̂_{BP}^{3} (h_{i}) ≡ Ψ̂_{BP}^{3} (v_{i}) ≡ Ψ̂_{BP}^{3} (η_{R}v_{i}) ≡ v_{i} mod 2.  (2.15)

Since Ψ̂_{BP}^{3} is multiplicative, this will imply that ρ_{∗}∘Ψ̂_{BP}^{3} ∘c = ϕ, where ρ : BP → bu/2 is the usual reduction. Thus we obtain

∑ ϕ(z_{i})g_{i} = ∑ ρ_{∗}(Ψ̂_{BP}^{3} (c(z_{i})))g_{i}
= ρ_{∗}(Ψ_{BP}^{3} (g_{n}))  (2.16)
= Ψ_{bu}^{3}(g_{n}),

and this last expression was evaluated in Corollary 2.8, yielding the claim of this proposition.

Now we prove (2.15). It is elementary that

Ψ̂_{BP}^{3} (v_{i}) = 3^{2^{i}−1}v_{i} ≡ v_{i} mod 2.  (2.17)

See, e.g., [33, p. 142], from which we also take the formula

Ψ̂_{BP}^{3} (∑_{i≥0}^{F^{∗}} h_{i}) = \frac{1}{3}[3](1).  (2.18)

Here we have also used ([33, 3.14]) that the conjugate formal sum on the left hand side of (2.18), defined by ∑^{F^{∗}} h_{i} = c ∑^{F} c(h_{i}), is the BP -analogue of b = ∑ b_{i} ∈ MU_{∗}MU. We next use that if x +_{F} y = ∑ a_{i,j}x^{i}y^{j}, then x +_{F^{∗}} y = ∑ η_{R}(a_{i,j})x^{i}y^{j}. Using also (2.17) and the formula Ψ̂_{BP}^{3} (η_{R}(−)) = Ψ̂_{BP}^{3} (−) of [33, 11.48iii], the left hand side of (2.18) equals

1 +_{F} ∑_{i>0}^{F} Ψ̂_{BP}^{3} (h_{i}).  (2.19)

On the other hand, the right hand side of (2.18) is congruent mod 2 to

[3](1) = 1 +_{F} ∑_{i>0}^{F} v_{i},  (2.20)

using [33, 3.17] and [3](x) = x +_{F} [2](x). Comparing terms of (2.19) and (2.20) one degree at a time yields Ψ̂_{BP}^{3} (h_{i}) ≡ v_{i}, as desired.

The following result will be useful in applying Proposition 2.11, since d(a) = η_{R}(a) − a for a ∈ BP_{∗}.

Proposition 2.21 The morphism ϕ defined in (2.12) sends η_{R}(a) − a to 0, for every a ∈ BP_{∗}.

Proof. By multiplicativity of η_{R}, it suffices to show ϕ(η_{R}(v_{i})) is nonzero if and only if i = 1. We modify the formal group law of BP by applying ϕ to the coefficients. Thus if x +_{F} y = ∑ a_{i,j}x^{i}y^{j}, then x +_{ϕ} y = ∑ ϕ(a_{i,j})x^{i}y^{j}. Applying ϕ to [28, Theorem 1], deleting terms which involve ϕ(v_{i}) or ϕ(t_{i}) for i > 1, and letting ϕ(η_{R}v_{j}) = δ_{j}v^{2^{j}−1} yields

∑_{j>0}^{ϕ} δ_{j}v^{2^{j}−1} +_{ϕ} ∑_{j>0}^{ϕ} v(δ_{j}v^{2^{j}−1})^{2} ≡ v +_{ϕ} v^{3} mod 2.

By induction on j, this implies δ_{1} = 1 and δ_{j} = 0 when j > 1.

With these preliminaries out of the way, we return to the detemination of the differentials in the SS (2.6). In this SS, shifted copies of Figure 2.4 (without d_{3}’s) are summed to give the initial term. There will be n copies of Z_{2} in each box [t − s, s] = [4j, 1] for all j ∈ Z, with η acting freely on them. We use square brackets to represent chart position to avoid confusion with (s, t) notation used earlier. Think of the square bracket as box position. This unusual notation will be followed throughout the remainder of the paper. It will usually be the case that an element in E_{r}^{s,t} will appear in position [t − s, s]. We label these Z_{2} summands 1, 2, …, n according to the value of i in the initial sum in (2.6). The value of j (in [4j, 1]) will not be incorporated into the notation, as its role is minimal. There will also be n Z_{2}’s in each box [4j − 1, 2] for j ∈ Z, with η-towers arising from each; these will be labeled i′ according to the summand of (2.6). In each box [4j −2, 1] there are summands of order 2^{ν(j−i)+3} for all values of i satisfying 1 ≤ i ≤ n. Finally, there is a Z/2^{∞} summand in each [4j − 3, 2] for 1 ≤ j ≤ n.

We will prove the following result about the differentials in the SS (2.6).

Proposition 2.22 In the CSS (2.6), there are differentials from the η-tower labeled i to the η-tower labeled (i − 2^{ν(i)+1})′, whenever i is not a 2-power and i ≤ n. These are the only nonzero differentials in this SS. Thus v_{1}^{−1}E_{2}^{∗,∗}(Q^{n}), the E_{2}-term of the v_{1}-periodic NSS converging to v_{1}^{−1}π_{∗}^{s}(Q^{n}), consists of

• [log_{2}(2n)] η-towers emanating from each position [4j, 1]. The labels of these are all 2^{e} satisfying 1 ≤ 2^{e} ≤ n.

• [log_{2}(2n)] η-towers emanating from each position [4j − 1, 2]. The labels of these will be primed integers. The only odd integer among these equals n or n − 1.

• An abelian group in [4j − 2, 1] of order 2^{S}, with S = 3n + ∑_{i=1}^{n} ν(j − i).

• A summand of Z/2^{∞} in [4j − 3, 2] for 1 ≤ j ≤ n.

If n = ∞, the analogous result holds, except that there are no elements of the second type, i.e., no η-towers arising from [4j − 1, 2].

We illustrate for Q^{10}. The differentials go 3 → 1′, 5 → 3′, 6 → 2′, 7 → 5′, 9 → 7′, and 10 → 6′. Thus 1, 2, 4, 8, 4′, 8′, 9′, and 10′ survive. The resulting v_{1}^{−1}E_{2}^{∗,∗}(Q^{10}) is pictured in Figure 2.23. The orders of the groups G_{1} and G_{2} are as in Proposition 2.22. They depend on h. This chart is assumed to have 2h ≥ n so that there are no Z/2^{∞}’s in filtration 2.

Before discussing the differentials and extensions in the v_{1}-periodic NSS for Q^{n}, we prove Proposition 2.22.

Proof of Proposition 2.22. To prove the differential on the η-tower, it suffices to prove it on the bottom element, since differentials commute with the action of η. It also suffices to prove it on the element labeled n in Q^{n}. The first main step is to show that g_{n} can be varied so that d([ ]g_{n}) has a particularly nice form, where d is the boundary in the cobar complex. Indeed, we show that there exist elements V_{i} ∈ BP_{∗} and A_{i} ∈ BP_{∗}BP such that if

g′_{n} = g_{n} + ∑_{i=1}^{2^{ν(n)+1}−1} V_{i}g_{n−i},  (2.24)

Figure 2.23: v_{1}^{−1}E_{2}^{s,t}(Q^{10}) if 2h > n

[figure: 1; 1; 2; 2; 4; 4; 8; 8; G_{1}; G_{2}; 4′; 4′; 8′; 8′; 9′; 9′; 10′; 10′; s = 1; 2; 3; 4; t − s = 8h+; 3; 4; 5; 6; 7; 8; 9; 10]

then

d([ ]g′_{n}) ≡ ∑_{i≥2^{ν(n)+1}}[A_{i}]g_{n−i} (mod 2).  (2.25)

The point here is that, mod 2, the first 2^{ν(n)+1} terms of d([ ]g′_{n}) are 0. It also follows from Proposition 2.21 that ϕ(g′_{n}) = ϕ(g_{n}).

If M is a Γ-comodule, we abbreviate Ext_{Γ}(BP_{∗}, M) as Ext(M). Recall that Ext^{1}(BP_{∗}) is the homology of the sequence

BP_{∗}⟶^{d}Γ⟶^{\overline{Δ}}Γ ⊗ Γ.  (2.26)

Recall that unadorned tensor products are always over BP_{∗}. Let Z = ker(\overline{Δ}) denote the cycles. Then Proposition 2.21 implies that the restriction to Z of the homomorphism ϕ defined in (2.12) factors through a homomorphism

Ext^{1}(BP_{∗}) → Z_{2}[v].

If all groups in (2.26) are reduced mod 2, the homology becomes Ext^{1}(BP_{∗}/(2)), and ϕ|Z can be factored as

Z → Ext^{1}(BP_{∗})⟶^{ρ}Ext^{1}(BP_{∗}/(2))⟶^{\overline{ϕ}}Z_{2}[v].

By [27, 5.3.13], \overline{ϕ} sends the image of ρ injectively.

Suppose that g′_{n} has been chosen similarly to (2.24) except that 2^{ν(n)+1} is replaced by some smaller number J, and that it satisfies

d([ ]g′_{n}) ≡ ∑_{i≥J}[A_{i}]g_{n−i} (mod 2)

with ϕ(A_{i}) = 0 if i < 2^{ν(n)+1}. Since ϕ(A_{J}) = 0, A_{J} is a boundary mod 2. That is, there exists V_{J} ∈ BP_{∗} with d(V_{J}) ≡ A_{J} mod 2 in (2.26). Moreover, V_{J} can be chosen to satisfy ϕ(V_{J}) = 0, since the only monomials of BP_{∗} with nonzero ϕ are v_{1}^{j}, and they have the property that d(v_{1}^{j}) = (v_{1} − 2h_{1})^{j} − v_{1}^{j} is divisible by 2, so that they may be removed from this mod 2 calculation. Subtracting V_{J}g_{n−J} from g′_{n} yields a new g′_{n} such that

d([ ]g′_{n}) ≡ ∑_{i>J}[A′_{i}]g_{n−i} (mod 2)

with ϕ(A′_{i}) = 0 if i < 2^{ν(n)+1}. To see ϕ(A′_{i}) = 0 we use the derivation property of d, Proposition 2.21, and ϕ(V_{J}) = 0. This extends the induction, establishing the existence of a generator as in (2.24) satisfying (2.25).

The differential which we wish to establish amounts to computing the homomorphism

v_{1}^{−1}E_{2}^{1,4k+4n+1}(S^{4n−1})⟶^{∂}v_{1}^{−1}E_{2}^{2,4k+4n+1}(Q^{n−1})⟶^{ρ}v_{1}^{−1}E_{2}^{2,4k+4n+1}(Q^{n−1}; Z_{2}), (2.27)

where ρ is reduction mod 2. It suffices to consider this reduction since the homomorphism ρ here is injective. The generator of v_{1}^{−1}E_{2}^{1,4k+4n+1}(S^{4n−1}) is the element usually called α_{2k+1}ι_{4n−1}. The element α_{2k+1} was called a_{2k+1} in [7, pp. 245-6], where it was shown that it is represented by v_{1}^{2k}h_{1}. The image ρ(∂(α_{2k+1}ι_{4n−1})) is determined from the mod 2 reduction of d([α_{2k+1}]g′_{n}) by the procedure outlined earlier. Since α_{2k+1} is a mod 2 cycle, this image equals v_{1}^{2k}h_{1} times the reduction of a class, d(g′_{n}), which pulls back to an element of v_{1}^{−1}E_{2}^{1,4n−1}(Q^{n−2^{ν(n)+1}}; Z_{2}) whose projection, A, into v_{1}^{−1}E_{2}^{1,4n−1}(S^{4n−2^{ν(n)+3}−1}; Z_{2}) maps nontrivially under \overline{ϕ}. By the calculation of v_{1}^{−1}E_{2}(S^{2n+1}) sketched earlier, v_{1}^{−1}E_{2}^{1,4n−1}(S^{4n−4d−1}) ≈ Z_{2}, and the mod 2 reduction of the nonzero element equals v_{1}^{2d−4}u by [27, 5.3.13c]. Here u = t_{1}^{4} + v_{1}t_{1}^{3} + v_{1}^{2}t_{1}^{2} + v_{1}t_{2} + v_{2}t_{1}, and so \overline{ϕ}(v_{1}^{2d−4}u) ≠ 0. Since \overline{ϕ} is an injection on the 1-line group, this implies that A ≡ v_{1}^{2^{ν(n)+2}−4}u mod boundaries. Thus the image of (2.27) is v_{1}^{2k}h_{1} ⊗ v_{1}^{2^{ν(n)+2}−4}u ⊗ g_{n−2^{ν(n)+1}}. The v_{1}’s can be shifted across the ⊗ mod 2, since η_{R}(v_{1}) − v_{1} = 2h_{1}, and so this element is the desired element (n − 2^{ν(n)+1})′, again using [27, 5.3.13c] to see that (n − 2^{ν(n)+1})′ is represented by v_{1}^{pwr}h_{1} ⊗ u ⊗ g.

The final ingredient in the proof of Proposition 2.22 is the following result, which implies that there are no differentials from the [4k + 2, 1]-box to the [4k + 1, 2]-box. The proof of this lemma involves methods foreign to the NSS, and is relegated to the end of Section 6.

Lemma 2.28 The boundary morphism

v_{1}^{−1}E_{2}^{1,4k+3}(S^{4n−1})⟶^{∂}v_{1}^{−1}E_{2}^{2,4k+3}(Q^{n−1})

is zero.

This completes the proof of Proposition 2.22, the determination of the E_{2}-term of the v_{1}-periodic NSS for Q^{n}. The analysis of the higher differentials in the SS is easier. In fact, the differentials are immediate from those in the spheres.

Proposition 2.29 In the v_{1}-periodic NSS for Q^{n}, whose E_{2}-term was described in Proposition 2.22, there are d_{3}-differentials from certain η-towers to the preceding ηtower with the same number as follows:

• from the η-tower labeled 1 beginning in [8h + 8, 1];

• from each η-tower labeled 2^{e} with 2 ≤ 2^{e} ≤ n beginning in [8h + 4, 1];

• from the η-tower labeled m′ with m odd beginning in [8h + 7, 2];

• from each η-tower labeled m′ with m even beginning in [8h + 3, 2].

There is also a d_{3}-differential from the group in [8h+6, 1] to the only odd primed class in [8h + 5, 4], and there are [log_{2}(n)] d_{3}-differentials from the group in [8h + 10, 1] to the classes labeled m′ with m even in [8h + 9, 4]. Thus v_{1}^{−1}E_{∞}(Q^{n}) consists of • [log_{2}(n)] η-towers truncated at η^{3} = 0 emanating from each position

[8h + 8, 1]. The labels of these are all 2^{e} satisfying 2 ≤ 2^{e} ≤ n.

• An η-tower truncated at η^{3} = 0 emanating from each position [8h+4, 1], labeled 1.

• [log_{2}(n)] η-towers truncated at η^{2} = 0 emanating from each position [8h + 7, 2]. The labels of these will be primed even integers.

• An η-tower truncated at η^{2} = 0 emanating from each position [8h+3, 2], labeled m′, where m = n − 1 or n and is odd.

• An abelian group in [4k−2, 1] of order 2^{S}, with S = 3n−D+∑_{i=1}^{n} ν(k− i), with D = 1 if k is even, and D = [log_{2}(n)] if k is odd.

• A summand of Z/2^{∞} in [4k − 3, 2] for 1 ≤ k ≤ n.

All elements in [4k + 2, 3] are divisible by 2 in v_{1}^{−1}π_{∗}^{s}(Q^{n}). This is accomplished by nontrivial extensions from the 1-line.

If n = ∞, a similar description holds without the primed classes, and with D = 0.

Proof. The differentials are all induced by naturality from the maps Q^{i} → S^{4i−1}. There can be no other differentials since all remaining classes are in filtration 1, 2, and 3, and t odd. The extensions can be seen from their presence in the spheres, or from the observation that if x ∈ π_{∗}^{s}(X) satisfies 2x = 0, then η^{2}x must be divisible by 2. This is true because there is a map from the mod 2 Moore spectrum M to X sending the bottom class of π_{∗}^{s}(M) to x, and the relation is true in π_{∗}^{s}(M).

We illustrate with a chart, Figure 2.30, for the d_{3}-differentials in Q^{10}, and then the much simpler chart, Figure 2.31, for v_{1}^{−1}E_{∞}(Q^{10}). The groups G_{1}, G_{2}, G′_{1}, and G′_{2} in Figures 2.30 and 2.31 are the appropriate abelian groups described in the propositions. The primed groups are smaller than the unprimed ones, as they are the kernels of d_{3}. The chart for v_{1}^{−1}E_{∞}(Q^{∞}) is like Figure 2.31, with the classes i′ and the short η-towers on them omitted, infinitely many short η-towers going up from [8h + 8, 1], namely one for each even 2-power, and the G′_{i}-groups replaced by G_{i}. These groups will be of infinite order.

Figure 2.30: Periodic NSS for Q^{10}

[figure: 1; 1; 2; 2; 4; 4; 8; 8; G_{1}; G_{2}; 4′; 4′; 8′; 8′; 9′; 9′; 10′; 10′; s = 1; 2; 3; 4; t − s = 8h+; 3; 4; 5; 6; 7; 8; 9; 10]

Figure 2.31: v_{1}^{−1}E_{∞}^{s,t}(Q^{10}) [figure: 1; 2; 4; 8; G′_{1}; G′_{2}; 4′; 8′; 9′; 10′; s = 1; 2; 3; t − s = 8h+; 3; 4; 5; 6; 7; 8; 9; 10]

3 Determination of v_{1}^{−1}E_{2}(Sp(n))

In this section, we prove Proposition 1.8. That is, we compute the differentials among the η-towers in (1.6). This yields v_{1}^{−1}E_{2}(Sp(n)), which we have depicted in Figure 1.16.

We discussed in the paragraph containing (1.6) the use of the exact sequence of UNSS’s and v_{1}-periodic UNSS’s associated to (1.1). For the UNSS, the existence of these exact sequences is proved similarly to the proof for SU(n) in [3], while for the periodic UNSS it follows since the exact sequences commute with the morphisms A′′ of [4, pp. 50-51] which define the direct systems whose direct limit is v_{1}^{−1}E_{2}(−).

We expand upon this a bit, in order to set the stage for the more complicated situation which follows. Indeed we review the definition and basic properties of the UNSS. We follow the exposition of [3, §2]. If X is a space, a space BP (X) is defined as lim_{n} Ω^{n}(BP_{n} ∧ X), where BP_{n} denotes the nth space in the Ω-spectrum for BP. Define D^{1}(X) to be the fibre of the unit map X → BP (X), and inductively define D^{s}(X) to be the fibre of D^{s−1}(X) → D^{s−1}(BP (X)). This gives rise to a tower of fibrations

⋯ → D^{2}(X) → D^{1}(X) → X.

The homotopy exact couple of this tower is the UNSS of X.

In general, computing this SS can be next to impossible, but if BP_{∗}(X) is free as a BP_{∗}-module and cofree as a coalgebra, then it becomes somewhat tractable. Indeed, in such a case

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

where M_{t} denotes a free BP_{∗}-module on a generator of dimension t, P (−) denotes the primitives in a coalgebra, and U denotes the category of unstable Γ-comodules. We sketch a definition of the category U, referring the reader to [3, p. 744] or [7, §7] for more details. Recall that Γ = BP_{∗}BP = BP_{∗}[h_{1}, h_{2}, …]. If M is a free BP_{∗}-module, then U(M) is the BP_{∗}-submodule of Γ ⊗_{BP_{∗}} M spanned by all elements of the form h^{I} ⊗ m satisfying 2(i_{1} + i_{2} + ⋯) < |m|, where h^{I} = h_{1}^{i_{1}}h_{2}^{i_{2}} · · ·. We shall not need the definition of U(M) when M is not BP_{∗}-free. We define U^{s}(M) by iterating U(−). The category U consists of BP_{∗}-modules M equipped with morphisms M⟶^{ψ}U(M), U(M)⟶^{δ}U^{2}(M), and U(M)⟶^{ϵ}M satisfying certain properties.

The category U is abelian. We abbreviate Ext_{U}^{s}(M_{t}, N) to Ext_{U}^{s,t}(N). These groups may be calculated as the homology groups of the unstable cobar complex C^{∗,∗}(N), defined by C^{s,t}(N) = U^{s}(N)_{t}, with boundary C^{s} → C^{s+1} defined as an alternating sum of Γ⟶^{\overline{Δ}}Γ ⊗ Γ and M⟶^{ψ}U(M).

The coalgebra BP_{∗}(Sp(n)) is cofree on the primitive elements e_{i} with |e_{i}| = 4i − 1 for 1 ≤ i ≤ n which are in the image from BP_{4i−1}(Q^{n}). Letting N_{n} denote the free BP_{∗}-module on {e_{1}, …, e_{n}}, it follows that the E_{2}-term of the UNSS, E_{2}^{s,t}(Sp(n)), is isomorphic to Ext_{U}^{s,t}(N_{n}). From the short exact sequence in U

0 → N_{n−1} → N_{n} → M_{4n−1} → 0,

we obtain the desired long exact sequence

→ E_{2}^{s,t}(Sp(n − 1)) → E_{2}^{s,t}(Sp(n)) → E_{2}^{s,t}(S^{4n−1}) → .  (3.1)

We could in principle obtain all differentials in the cellular SS which results from these exact sequences, or their v_{1}-periodic analogues, by using the coaction, similarly to the method used in the stable SS for E_{2}(Q^{n}) in Section 2. Indeed, for differentials from stable classes on the 1-line to stable classes on the 2-line, this method works fine. However, in order to more easily relate stable and unstable classes, we prefer to work with ΩSp(n), and this complicates the theory.

The following known result will be useful.

Proposition 3.2 ([24], [21]) There exist z_{2i−1} ∈ H_{4i−2}(ΩSp(n); Z) for 1 ≤ i ≤ n such that there is an isomorphism of algebras

H_{∗}(ΩSp(n); Z) ≈ Z[z_{1}, z_{3}, …, z_{2n−1}].

The coalgebra structure is given by

Δ̃(z_{i}) = 2 ∑_{j=1}^{i−1} z_{j} ⊗ z_{i−j},

where z_{2j} is defined inductively by

z_{2j} + ∑_{k=1}^{2j−1}(−1)^{k}z_{k}z_{2j−k} = 0.

The mod 2 reduction \overline{z}_{i} satisfies \overline{z}_{2^{e}(2j+1)} = \overline{z}_{2j+1}^{2^{e}}. The mod 2 dual Steenrod operations are given by

(\overline{z}_{2i−1})Sq = ∑_{j=1}^{2i−1} \binom{j−1}{2i−1−j}\overline{z}_{j}.

For h = BP or bu, there are elements g_{i} ∈ h_{4i−2}(ΩSp(n)) for 1 ≤ i ≤ n such that there is an isomorphism of algebras

h_{∗}(ΩSp(n)) ≈ h_{∗}[g_{1}, g_{2}, …, g_{n}].

The reduction from h_{∗}(−) to HZ_{2}_{∗}(−) sends g_{i} to z_{2i−1}.

Note that BP_{∗}(ΩSp(n)) is not a cofree coalgebra, and so the E_{2}-term of its UNSS cannot be handled by the methods which were used for Sp(n).

We will need the exact sequences of the UNSS and v_{1}-periodic UNSS of these loop spaces.

Proposition 3.3 There is an isomorphism of E_{2}-terms of the UNSS

E_{2}^{s,t−1}(ΩSp(n)) ≈ E_{2}^{s,t}(Sp(n)).  (3.4)

There are exact sequences

→ E_{2}^{s,t}(ΩSp(n − 1)) → E_{2}^{s,t}(ΩSp(n)) → E_{2}^{s,t}(ΩS^{4n−1}) → E_{2}^{s+1,t}(ΩSp(n − 1)) →

and

→ v_{1}^{−1}E_{2}^{s,t}(ΩSp(n−1)) → v_{1}^{−1}E_{2}^{s,t}(ΩSp(n)) → v_{1}^{−1}E_{2}^{s,t}(ΩS^{4n−1}) → v_{1}^{−1}E_{2}^{s+1,t}(ΩSp(n−1)). (3.5)

Proof. The exact sequences follow from (3.4) and (3.1). The proof of (3.4) is completely analogous to that for S^{2n+1} in [8, §6], but we review it, since the methods will be required later. Indeed, we will provide a rather comprehensive review of the Hopf ring structure of BP_{∗}(BP_{n}), since this plays an essential role in the proof.

There are two products on the spaces in the Ω-spectrum for BP. One is a map BP_{k} × BP_{k} → BP_{k} induced by loop multiplication. It induces the ∗-product in BP - homology. The other is a map BP_{k} × BP_{r} → BP_{k+r} induced by the ring spectrum structure. It induces the ∘-product in BP -homology.

There is a map x : CP^{∞} → BP_{2} representing a generator of BP^{2}(CP^{∞}). Dual to x^{n} is a class β_{n} ∈ BP_{2n}(CP^{∞}). We let b_{(i)} = x_{∗}(β_{2^{i}}) ∈ BP_{2^{i+1}}(BP_{2}). If I = (i_{0}, i_{1}, …), then b^{∘I} ∈ BP_{∗}(BP_{2|I|}) denotes the class b_{(0)}^{∘i_{0}} ∘ b_{(1)}^{∘i_{1}} ∘ · · ·. Here |I| = ∑ i_{j}.

If k ≤ 0, then π_{0}(BP_{k}) = BP^{k} = BP_{−k}, and so we have for each element v ∈ BP^{k} an element [v] ∈ BP_{0}(BP_{k}). It follows from [29] that the set of all b_{(i)} and all [v] generate BP_{∗}(BP_{∗}) under ∗ and ∘ products. Moreover, BP_{∗}(BP_{k}) is a polynomial algebra with respect to the ∗-product on a family of indecomposable elements which are identified in [29]. The stabilization map BP_{∗}(BP_{k})⟶^{σ_{∗}}Γ ⊗ M_{k} sends ∗-products to zero, and is injective on the indecomposables. We will frequently identify elements of BP_{∗}(BP_{∗}) with their stable images under σ_{∗}.

We first note that the class b_{(0)} is special. It corresponds to the suspension operator. Indeed, b_{(0)}^{∘n} stabilizes to 1 ⊗ ι_{2n} in Γ ⊗ M_{2n}. The class b_{(1)} lives in BP_{4}(BP_{2}) and stabilizes to h_{1} ⊗ ι_{2}. In general, b_{(i)}^{∘j} stabilizes to h_{i}^{j} ⊗ ι_{2j} mod (v_{1}, v_{2}, …)Γ, by [7, 8.3]. The class [v] stabilizes to η_{R}(v). Thus, for example, the class b_{(1)}^{∘2} ∘ b_{(0)} ∘ [v_{1}] stabilizes to η_{R}(v_{1})h_{1}^{2} ⊗ ι_{4} = h_{1}^{2} ⊗ v_{1}ι_{4}. Note that both classes have degree 10, and the Hopf ring class is defined on BP_{4} since [v_{1}] is defined on BP_{−2}. Also this class is in the image of the double suspension from BP_{8}(BP_{2}) since there is a factor of b_{(0)}.

Since the b_{(n)}’s and [v]’s generate BP_{∗}(BP_{k}) as a polynomial algebra, the ∗- indecomposables in BP_{∗}(BP_{k}) stabilize to the BP_{∗}-span of h^{I} ⊗ ι_{k} with 2|I| ≤ k. (The above examples justify this description for even k. It is shown in [29, 5.1] that the same description works for odd k.) By [29, 5.3], the primitives in BP_{∗}(BP_{k}) are the image of the suspension. Hence, since b_{(0)} is suppressed in the stable notation, we see that the stable image of the primitives is the BP_{∗}-span of h^{I} ⊗ ι_{k} with 2|I| < k. Whenever there is no chance of confusion, we prefer to use the stable names. So, for example, we will write ι_{2k}^{∗2} for the class (b_{(0)}^{∘k})^{∗2}, even though (b_{(0)}^{∘k})^{∗2} stabilizes to 0.

Having described the stable image of the primitives, we want to understand their names back in BP_{∗}(BP_{k}). For this, we use the fact ([29]) that BP_{∗}(BP_{k}) is a bipolynomial Hopf algebra. By [30], such Hopf algebras over Z_{(2)} can be written as a tensor product of the universal Hopf algebras B(x_{2n}) constructed in [22]. As an algebra, B(x_{2n}) is a polynomial algebra generated by classes x_{2^{s}n} with s ≥ 1. The primitives in the coalgebra are of the form

x_{2n}^{2^{s}} − 2(x_{4n})^{2^{s−1}} + ⋯ + (−2)^{s}x_{2^{s+1}n}  (3.6)

with s ≥ 0. For example, we have the primitive in BP_{4k}(BP_{2k}) given by 2h_{1}^{k} ⊗ ι_{2k} − 1⊗ι_{2k}^{∗2}, or in Hopf ring notation 2b_{(1)}^{∘k}−(b_{(0)}^{∘k})^{∗2}. We explain how this class is a primitive in the sense of the earlier description, namely that it is in the image of ∘b_{(0)}. The answer is given by a relation in the Hopf ring,

2b_{(1)} − (b_{(0)})^{∗2} = v_{1}b_{(0)} + [v_{1}] ∘ b_{(0)}^{∘2},

proved in [29, p. 261]. This suspends to the familiar formula, η_{R}(v_{1}) = v_{1} − 2h_{1}, for the right action on v_{1} and inductively gives the double desuspension of 2h_{1}^{k} ⊗ ι_{2k}. Up in BP_{4k}(BP_{2k}), 2b_{(1)}^{∘k} does not desuspend. This is prevented by the ∗-decomposable class in the formula.

We now generalize the above discussion to the case of a BP_{∗}-module M which is free on a set of generators {m_{i}}. Let BP (M) be the zeroth space of the Ωspectrum associated to the homology theory BP_{∗}(−) ⊗ M. Then G(M) is defined to be BP_{∗}(BP (M)). In particular, BP (M_{k}) = BP_{k}, and G(M_{k}) = BP_{∗}(BP_{k}). In terms of the basis, a non-canonical description of G(M) is given by G(M) = BP_{∗}(∏ BP_{|m_{i}|}).

The stabilization map G(M) → Γ ⊗ M is injective on the ∗-indecomposables, and U(M) and V (M) are defined to be the images of the primitives and indecomposables, respectively. As above, U(M) is the BP_{∗}-span of all h^{I} ⊗ m satisfying 2|I| < |m|, while V (M) is the BP_{∗}-span of all h^{I} ⊗ m satisfying 2|I| ≤ |m|. We will sometimes write \overline{U}(M) ⊂ G(M) for the primitives when we need the specific representations of elements in G(M) rather than their more familiar stable images.

Thus U(M) agrees with that which we considered earlier this section. Note that if M is concentrated in even degrees, then U(M) is properly contained in V (M), while U(σM) ≈ σV (M), where the suspension σ increases degrees by 1. As we did with U, we define V^{p}(M) by iterating the functor V.

The modules G(M) lie in the nonabelian category G of unstable BP_{∗}-coalgebras discussed in [7, §6]. It was shown there that if X is a simply-connected space with BP_{∗}(X) a free BP_{∗}-module, then the E_{2}-term of the UNSS of X is given by Ext_{G}(BP_{∗}, BP_{∗}(X)). As we will need some understanding of how this Ext is computed, we go into some detail about G-structure.

For a space X for which BP_{∗}(X) is a free BP_{∗}-module, we abbreviate G(BP_{∗}(X)) to G(X). We note that G(X) = BP_{∗}(BP (X)), for the space BP (X) defined earlier.

As in the case of G(S^{k}), we use a mix of stable and unstable notation. We will be primarily interested in X = ΩSp(n), for which BP_{∗}(ΩSp(n)) is given by Proposition 3.2. In G(ΩSp(n)), the BP_{∗}-module generators act as an indexing set for the factor of BP_{∗}(BP_{k}). For example, 1 ⊗ g_{2}^{2} is the bottom class in the factor of G(ΩSp(n)) corresponding to the generator g_{2}^{2} of BP_{∗}(ΩSp(n)), while 1 ⊗ g_{2}^{∗2} is the ∗-square of the bottom class in the factor corresponding to the generator g_{2}. The class 1 ⊗ g_{2}^{2} is primitive in the G-coproduct, while 1 ⊗ g_{2}^{∗2} is not. One has to keep in mind that G(X) does not see any of the structure of BP_{∗}(X).

The G-structure map η_{X} : BP_{∗}(X) → G(X) is obtained by applying BP_{∗}(−) to the Hurewicz map η : X → BP (X). The coaction ψ of BP_{∗}(X) is obtained from this as

BP_{∗}(X)⟶^{η_{X}}G(X) = BP_{∗}(BP (X))⟶^{σ_{∗}}Γ ⊗ BP_{∗}(X),  (3.7)

while the coproduct Δ is obtained as

BP_{∗}(X)⟶^{η_{X}}G(X) = BP_{∗}(BP (X))⟶^{Δ_{G}}G(X) ⊗ G(X)⟶^{ϵ⊗ϵ}BP_{∗}(X) ⊗ BP_{∗}(X). (3.8)

Here ϵ satisfies ϵ(1⊗g_{i}) = g_{i}, while ϵ applied to any other monomial is 0. The coaction and coproduct are related by the following commutative diagram.

BP_{∗}(X)  ⟶^{ψ}  Γ ⊗ BP_{∗}(X)
↓^{Δ}  ↓^{1⊗Δ}  (3.9)
BP_{∗}(X) ⊗ BP_{∗}(X) ⟶^{ψ⊗ψ} Γ ⊗ BP_{∗}X ⊗ Γ ⊗ BP_{∗}X⟶^{m_{Γ}}Γ ⊗ BP_{∗}X ⊗ BP_{∗}X

Now we proceed with the proof of (3.4). Let N = QBP_{∗}(ΩSp(n)), with Q denoting the indecomposable quotient. Thus σN is the module N_{n} considered earlier. Similarly to [5, 4.2] or [8, p. 387], there is a cosimplicial resolution

BP_{∗}(ΩSp(n))⟶^{ξ}G(N) \overrightarrow{→} G(V (N)) \begin{matrix}→ \\ → \\ → \end{matrix} G(V^{2}(N)) ⋯,  (3.10)

where the co-augmentation ξ is the composite

BP_{∗}(ΩSp(n)) ⟶^{η_{ΩSp(n)}} G(BP_{∗}(ΩSp(n))) → G(N).  (3.11)

The acyclicity follows just as it did for ΩS^{2n+1} in [8, p. 387].

We apply Hom_{G}(BP_{∗}, −) to (3.10), and taking the alternating sum of the coface maps yields a chain complex whose homology is Ext_{G}(BP_{∗}(ΩSp(n))), which by [7, 6.17] is E_{2}(ΩSp(n)). Since Hom_{G}(BP_{∗}, G(T)) ≈ T, this complex, whose homology equals E_{2}(ΩSp(n)), is isomorphic to the complex

N → V (N) → V^{2}(N) → · · · .  (3.12)

Since N is concentrated in even degrees, (3.12) is isomorphic to the complex

N → σ^{−1}U(σN) → σ^{−1}U^{2}(σN) → ⋯,

and the homology of this complex is σ^{−1}Ext_{U}(P (BP_{∗}(Sp(n)))), where P (−) denotes the primitives. By [7, p. 240], this homology is σ^{−1}E_{2}(Sp(n)). Thus

E_{2}^{s,t}(ΩSp(n)) ≈ E_{2}^{s,t+1}(Sp(n)),

as desired.

The following result is instructive.

Proposition 3.13 The morphism ξ of (3.10) is a ring homomorphism.

Proof. There is a commutative diagram

BP_{∗}(ΩSp(n)) ⟶^{η_{ΩSp(n)}} G(ΩSp(n)) ⟶^{σ_{∗}} Γ ⊗ BP_{∗}(ΩSp(n))
↘ ξ ↓^{G(ρ)} ↓^{Γ⊗ρ} (3.14)
G(N) ⟶^{σ_{∗}} Γ ⊗ N

If x_{1}x_{2} ∈ BP_{∗}(ΩSp(n)) is decomposable, then, using (3.7),

σ_{∗}ξ(x_{1}x_{2}) = (Γ ⊗ ρ)σ_{∗}η_{ΩSp(n)}(x_{1}x_{2}) = (Γ ⊗ ρ)ψ(x_{1})ψ(x_{2}) = 0,

since ρ annihilates decomposables. Hence

ξ(x_{1}x_{2}) ∈ ker(σ_{∗}) = I^{∗2},  (3.15)

where I is the augmentation ideal.

Next we observe that both maps in the composite (3.11) which defines ξ are morphisms in the category G, and hence so is ξ. In particular there is a commutative diagram

BP_{∗}(ΩSp(n))  ⟶^{ξ}  G(N)
\sqrt{Δ}  \sqrt{Δ}  (3.16)
BP_{∗}(ΩSp(n)) ⊗ BP_{∗}(ΩSp(n)) ⟶^{ξ⊗ξ} G(N) ⊗ G(N),

where \overline{Δ} is the reduced coproduct. This diagram implies that

\overline{Δ}(ξ(g_{i}^{2})) = 2g_{i} ⊗ g_{i}.

There are only two classes in G(N) satisfying \overline{Δ}(−) = 2g_{i} ⊗ g_{i}, namely g_{i}^{∗2} and 2h_{1}^{2i−1} ⊗ g_{i}. (To see that \overline{Δ}(2h_{1}^{2i−1} ⊗ g_{i}) = 2g_{i} ⊗ g_{i}, note that

0 = \overline{Δ}(2h_{1}^{2i−1} ⊗ g_{i} − 1 ⊗ g_{i}^{∗2}),

since this difference is a primitive.) By (3.15), ξ(g_{i}^{2}) is decomposable, which 2h_{1}^{2i−1}⊗g_{i} is not, and so ξ(g_{i}^{2}) must equal g_{i}^{∗2}.

We can now prove that ξ(g_{i}^{k}) = g_{i}^{∗k} by induction on k. The diagram (3.16) and induction hypothesis imply

\overline{Δ}(ξ(g_{i}^{k})) = ∑ \binom{k}{l}g_{i}^{∗l} ⊗ g_{i}^{∗(k−l)},

which leaves g_{i}^{∗k} as the only candidate for ξ(g_{i}^{k}). We deduce similarly that ξ(g_{i}^{k}g_{j}^{l}) = g_{i}^{∗k}g_{j}^{∗l}, and similarly for larger products.

The other result which we need before calculating the differentials concerns the interlocking EHP and double suspension sequences. Here H is the obstruction to desuspension, and H_{2} is the obstruction to double desuspension, both of which are described in [5].

Theorem 3.17 There is a commutative diagram.

v_{1}^{−1}E_{2}^{s}(S^{2n+1}) ⟶^{H_{2}} v_{1}^{−1}Ext_{U}^{s−1}(BP_{∗}/(2){x_{4n−1}})
↓^{H} ↓^{δ} (3.18)
v_{1}^{−1}E_{2}^{s}(S^{4n+1}) ⟵^{σ^{2}} v_{1}^{−1}E_{2}^{s}(S^{4n−1}),

where δ is the Bockstein homomorphism, H is the ordinary James-Hopf invariant, and H_{2} is a factorization of the double suspension Hopf invariant.

Before we begin the proof, we make a few remarks. Let W (n) denote, as in [5] and [8], the free BP_{∗}/(2)-module with basis {x_{2^{i}n−1} : i ≥ 2}. The morphism

v_{1}^{−1}Ext_{U}(BP_{∗}/(2){x_{4n−1}})⟶^{i_{∗}}v_{1}^{−1}Ext_{U}(W (n))

is shown on [4, p. 57] to send the stable classes bijectively, and the unstable classes to 0. Moreover, it is shown there that the morphism v_{1}^{−1}E_{2}(S^{2n+1})⟶^{H_{2}}v_{1}^{−1}Ext_{U}(W (n)) of [4, (2.5)] factors through i_{∗}, yielding the morphism, also called H_{2}, of our (3.18).

The analogue of Theorem 3.17 before inverting v_{1} is claimed in the metastable range in [5, 5.7]. The metastable condition was required in order to guarantee that the image of H_{2} lies on the bottom cell, which is true without the metastability condition after localization.

Implicit in Theorem 3.17 is the statement that the image of the James-Hopf invariant double desuspends. This is not always true in periodic homotopy or in the unlocalized UNSS, but is true in the localized E_{2}-term.

Now we begin the proof of Theorem 3.17. Throughout this proof, let M = M_{2n} denote the free BP_{∗}-module on a generator of degree 2n. We consider the cosimplicial resolution (of coalgebras, by [7, 6.10])

BP_{∗}(ΩS^{2n+1})⟶^{ξ}G(M) \overrightarrow{→} G(V (M)) \begin{matrix}→ \\ → \\ → \end{matrix} G(V^{2}(M)) ⋯,  (3.19)

analogous to (3.10). Taking primitives in (3.19), we obtain, as in [5, §4], a chain complex

\overline{U}(M)⟶^{d_{0}}\overline{U}(V (M))⟶^{d_{1}}\overline{U}(V^{2}(M)) → ⋯  (3.20)

where d_{i} is the alternating sum of the coface operators. Here \overline{U}(M) denotes the primitives in G(M). As noted earlier, this \overline{U}(M) maps isomorphically to U(M) ⊂ Γ⊗ M. We use \overline{U}(−) when we are interested in the specific representations of elements in G(−) rather than their image in Γ⊗−. Since G(M) is a free coalgebra, the homology of (3.20) in homological degree i is the ith derived functor of P (BP_{∗}(ΩS^{2n+1})). In particular, it has no homology in homological degree greater than 1, by [5, 4.1].

We now embed (3.20) into a double complex whose homology is σ^{−1}E_{2}(S^{2n+1}), but from which we can identify the James-Hopf map. This double complex has, for p, q ≥ 0, (p, q)th group U^{p}(\overline{U}(V^{q}(M)), with vertical differential that of (3.20). The homology of the qth row is Ext_{U}^{p}(UV^{q}M) = 0 if p > 0, while if p = 0 it is V^{q}M ≈ σ^{−1}U^{q}(σM). (This is true since, for any N, U(N) is injective in U, satisfying Ext_{U}^{0}(U(N)) ≈ N.) Hence the homology of the total complex is the homology of the complex whose qth term is V^{q}(M) ≈ σ^{−1}U^{q}(σM), and this homology can be interpreted as either E_{2}(ΩS^{2n+1}) or σ^{−1}E_{2}(S^{2n+1}), by (3.4).

We let T (ΩS^{2n+1}) denote this double complex. A double complex analogous to this can be constructed for ΩSp(n) using N = BP_{∗}{g_{1}, …, g_{n}} as M. Thus T^{s}(ΩSp(n)) = ⨁ U^{s−q}(\overline{U}(V^{q}(N))). It is the \overline{U} here which causes ∗-products of g_{i}-classes to come into play here.

Returning to the case M = M_{2n}, we now take homology in the opposite direction. The homology of the pth column is obtained by applying U^{p} to the homology of (3.20). This is possibly nonzero only in q = 0 and q = 1, where it is U^{p}(R^{q}P), since U commutes with homology. Here and throughout the remainder of this proof, we let P = P (BP_{∗}(ΩS^{2n+1})), and R^{q} denotes the qth right derived functor. Taking horizontal homology of this yields a 2-line spectral sequence which reduces to the double suspension exact sequences as in [8, §8].

The homology of the total complex in homological dimension s is carried by

U^{s}(\overline{U}(M)) ⊕ U^{s−1}(\overline{U}(V (M))).  (3.21)

Recall that \overline{U}(M) = P (BP_{∗}(BP_{2n})). By [29, 5.3], the first two indecomposables in BP_{∗}(BP_{2n}) are b_{0}^{∘n} and b_{1}^{∘n}, and, as discussed earlier, the images of these in Γ⊗M are 1⊗ι_{2n} and h_{1}^{n} ⊗ι_{2n}. By [5, 3.6], 2h_{1}^{n} ⊗ι_{2n} is the image of a primitive of \overline{U}(M), and by 3.6 this primitive must be 2h_{1}^{n} ⊗ ι_{2n} − 1 ⊗ ι_{2n}^{∗2}. (Here we are using the stable/unstable notation introduced earlier.) Thus an element in the first term of (3.21) can be written as α_{1}(1⊗ι_{2n})+α_{2}(2h_{1}^{n}⊗ι_{2n}−1⊗ι_{2n}^{∗2}) mod later primitives, with α_{i} ∈ BP_{∗}BP^{⊗s}. As for the second term of (3.21): By [5, 4.3] and the paragraph which precedes it, the vertical homology of our total complex at \overline{U}(V (M)) is R^{1}P, and the bottom class of this equals 1 ⊗ h_{1}^{n} ⊗ ι_{2n}, mod terms that desuspend.^{3} Under the isomorphism R^{1}P ≈ W (n), this bottom class equals the class called x_{4n−1} in the paragraph following the statement of Theorem 3.17.

3There is a minor misprint in [5, 4.3], which should say P (BP_{∗}(ΩS^{2n+1})) instead of P (A(2n)).

Now we note, by [4] or, more explicitly, [5, 5.3ii], that any class y ∈ v_{1}^{−1}E_{2}^{s}(S^{2n+1}) may be represented, mod terms that desuspend, as y = α ⊗h_{1}^{n} ⊗ι_{2n} with α ∈ Γ^{⊗(s−1)}. The projection of this y to v_{1}^{−1}Ext_{U}^{s−1}(R^{1}P) = v_{1}^{−1}Ext_{U}^{s−1}(W (n)) is just H_{2}(y) = α ⊗ x_{4n−1}. In terms of the double complex, y is represented by

\overline{y} = [α_{1} ⊗ (1 ⊗ ι_{2n}) + α_{2} ⊗ (2h_{1}^{n} ⊗ ι_{2n} − 1 ⊗ ι_{2n}^{∗2})] + [α ⊗ 1 ⊗ h_{1}^{n} ⊗ ι_{2n}]
∈ [U^{s}\overline{U}(M)] + [U^{s−1}R^{1}P],  (3.22)

mod terms of higher vertical filtration in the double complex and terms with higher powers of ι_{2n}.

Since \overline{y} is a cycle, and α is a mod 2 cycle, evaluating the coefficient of 1⊗2h_{1}^{n} ⊗ι_{2n} in d(\overline{y}) yields

0 = α_{2}(1 ⊗ 2h_{1}^{n} ⊗ ι_{2n}) + \frac{d(α)}{2}1 ⊗ 2h_{1}^{n} ⊗ ι_{2n}.

Thus α_{2} = \frac{d(α)}{2} = δ(α), where δ is the Bockstein. Now the James-Hopf map yields a map of double complexes T (ΩS^{2n+1})⟶^{H}T (ΩS^{4n+1}) which sends ι_{2n}^{∗2} to ι_{4n}. Hence H(\overline{y}) = α_{2} = δ(α), establishing the following proposition, since α = H_{2}(y).

Proposition 3.23 An element x ∈ v_{1}^{−1}E_{2}^{s}(ΩS^{2n+1}) can be represented in T (ΩS^{2n+1}) as α ⊗ h_{1}^{n} ⊗ ι_{2n} + β ⊗ 1 ⊗ ι_{2n}^{∗2} mod higher powers of ι_{2n} and terms which desuspend. Then α = H_{2}(x) and β = δ(α). The James-Hopf map H sends this to β ⊗ ι_{4n} ∈ v_{1}^{−1}E_{2}^{s}(ΩS^{4n+1}).

Theorem 3.17 is just a restatement of this proposition. Note that the thing that makes Theorem 3.17 work in the v_{1}-periodic setting but not for unlocalized E_{2} is the fact that v_{1}^{−1}Ext_{U}(W (n)) is built entirely on the bottom class of W (n). If we had to worry about elements of Ext_{U}(W (n)) built on x_{8n−1}, then the next Husemoller primitive,

4h_{2}^{n} ⊗ ι_{2n} − 2h_{1}^{2n} ⊗ ι_{2n}^{∗2} + 1 ⊗ ι_{2n}^{∗4},

would have to be taken into account.

Proposition 3.23 links h_{1}^{n} ⊗ ι_{2n} with ι^{∗2}, while the following result links it with ι^{2}.

Proposition 3.24 If the mod 2 reduction of ψ(g_{n}) ∈ BP_{∗}(ΩSp(n)) contains the term h_{1} ⊗ g_{n/2}^{2}, then it also contains the term v_{1}h_{1}^{n−1} ⊗ g_{n/2}.

Proof. We first note that if Δ(g_{n}) contains the term ϵv_{1}g_{n/2} ⊗g_{n/2}, then, assuming the hypothesis of the proposition, (1 ⊗ Δ)(ψ(g_{n})) contains the terms 2h_{1} ⊗ g_{n/2} ⊗ g_{n/2} and 1 ⊗ ϵv_{1}g_{n/2} ⊗ g_{n/2}, while m_{Γ}(ψ ⊗ ψ)Δ(g_{n}) contains ϵv_{1} ⊗ g_{n/2} ⊗ g_{n/2}. Using η_{R}(v_{1}) = v_{1} − 2h_{1}, the equating of these which is implied by diagram (3.9) yields 2h_{1} + ϵ(v_{1} − 2h_{1}) = ϵv_{1}, and hence ϵ = 1. Thus Δ(g_{n}) contains the term v_{1}g_{n/2} ⊗ g_{n/2}.

In the proof of Proposition 3.13, it was shown that the map Δ_{G} of (3.8) sends only v_{1}h_{1}^{n−1} ⊗ g_{n/2} to v_{1}g_{n/2} ⊗ g_{n/2}. Thus the composite (3.8) implies that η_{ΩSp(n)}(g_{n}) goes to v_{1}h_{1}^{n−1} ⊗ g_{n/2} plus other monomials. Now the composite (3.7) implies the proposition.

We will need to make frequent reference to the chart of the periodic UNSS for S^{2n+1} given on page 58 of [4]. We reprint it in Figure 3.25 for the convenience of the reader. We have included a few classes which are involved in d_{3}-differentials which were not included in [4]. The notation is explained on page 53 of [4]. For us, the most important part is that an element y denoted {Ah^{i}} means that y pulls back to S^{2i+1}, where it satisfies H_{2}(y) = A. The horizontal component of the chart is the mod 8 value of the stem, not the homotopy group.

We provide one other bit of background before proceeding with the proof of Proposition 1.8: the explicit manner in which the remaining differentials are computed. This is similar to the method in Section 2 for the stable NSS. It involves the chain complex T (X) described above, whose homology equals the E_{2}-term of the UNSS for X = ΩS^{2n+1} or ΩSp(n). If \overline{U}(−) is identified with U(−), then T^{s}(X) becomes a subset of Γ ⊗ ⋯ ⊗ Γ ⊗ QBP_{∗}(X), with s + 1 factors of Γ. However, the only terms that we will ever have need to consider are the third and fourth terms in (3.22), both of which involve a 1⊗ which can be omitted, so that we actually view the elements as involving the s-fold tensor product of Γ.

As in the stable situation,

∂ : E_{2}^{s}(ΩS^{4n−1}) → E_{2}^{s+1}(ΩSp(n − 1))

is calculated on a class z by finding a representative cycle c for z for which the reduced coaction \overline{ψ}(c) pulls back to a class in T^{s+1}(ΩSp(i)) for i as small as possible. The projection of \overline{ψ}(c) into T^{s+1}(ΩS^{4i−1}) yields ∂(z). This can be done for the complex

Figure 3.25: E_{2} of periodic UNSS for spheres

[figure: 1; 1; 2; 2; 3; 3; 4; 4; 0; 1; 2; 3; 4; 5; 6; 7; ηα̃_{4k}; {ηh^{4k}}_{h^{4k+1}}; η^{2}{uv^{ν(k)}h^{4k−4−ν(k)}}; {ηh^{4k+2}}_{h^{4k+3}}; η^{2}{uv^{−1}h^{4k−1}}; η^{3}α_{4k+1}; n≡1,2; n≡1,2; n≡0,3; n≡0,3; α_{4k+1}; η^{2}α̃_{4k}; η{ηh^{4k}}; n≡1,2; η{ηh^{4k+2}}; n≡0,3; ηα_{4k+1}; {uv^{−1}h^{4k−1}}_{h^{4k+2}}; η^{2}{ηh^{4k}}; (η^{3}α̃_{4k})_{α_{4k+2}}; {uv^{ν(k+1)}h^{4k−ν(k+1)}}_{h^{4k+4}}; η^{2}{ηh^{4k+2}}; n≡1,2; n≡0,3; n≡1,2; n≡0,3; α_{(4k+2)/2}; a_{4k+2}; η{uv^{−1}h^{4k−1}}; η^{2}α_{4k+1}; n≡0,3; α̃_{4(k+1)}; η{uv^{ν(k+1)}h^{4k−ν(k+1)}}; α_{4k+3}; ηα_{4k+3}; η^{2}α_{4k+3}; n≡1,2; n≡1,2]

T (−) just as well as for the complex C(−) used in Section 2, because the fibration

ΩSp(n − 1) → ΩSp(n) → ΩS^{4n−1}

induces a short exact sequence of T (−), and the boundary morphism in the associated long exact homology sequence is as described above.

Explicit calculation of the coaction in BP_{∗}(ΩSp(n)) can be performed by a method similar to that of [10], but the formulas are extremely complicated. Instead we opt for calculating a homomorphic image of the coaction which contains the information relevant to our application, exactly as we did for Q^{n} in Section 2.

This formula is obtained similarly to that given earlier in Q^{n}. We use the description of bu_{∗}(ΩSp(n)) given in 3.2. Since Ψ^{3} is a ring homomorphism, it suffices to determine Ψ^{3} on the generators g_{i}. Although we do not have complete information about the action of Ψ^{3} in bu_{∗}(ΩSp(n)), the following will suffice for our applications.

Proposition 3.26 In bu_{∗}(ΩSp(n)/Sp([n/2])),

Ψ^{3}(g_{n}) = ∑_{i=0}^{[(n−1)/2]}(−3)^{−i}\binom{2n+i−1}{i}v^{2i}g_{n−i}.

Since the inclusion maps ΩSp(i) → ΩSp(n) send g_{j} to g_{j}, this proposition can be used to read off Ψ^{3}(g_{i}) in bu_{∗}(ΩSp(n)/Sp([n/2])) for all i ≤ n.

Proof. Let Q^{n} denote the quaternionic quasi-projective space which embeds naturally into Sp(n), and let Q_{k}^{n} = Q^{n}/Q^{k−1}. Then Q_{[n/2]+1}^{n} embeds in Sp(n)/Sp([n/2]), and is a suspension for dimensional reasons. Thus we shall work with Q_{[n/2]+1}^{n}, and let Y satisfy ΣY = Q_{[n/2]+1}^{n}. By adjointing, we obtain a map Y ⟶^{f}ΩSp(n)/Sp([n/2]). Letting g_{i} denote the image under f_{∗} of the generators of bu_{∗}(Y), the result follows from Proposition 2.7.

Using the \binom{−2n}{i}-form for the binomial coefficient immediately yields the following corollary.

Corollary 3.27 In bu/(2)_{∗}(ΩSp(n)/Sp([n/2])),

Ψ^{3}(g_{n}) = g_{n} + v^{2^{ν(n)+2}}g_{n−2^{ν(n)+1}} + T,

where T ∈ ⟨v^{2i}g_{n−i} : i > 2^{ν(n)+1}⟩.

Of course, if n − 2^{ν(n)+1} ≤ [(n − 1)/2], then this corollary must be interpreted as saying Ψ^{3}(g_{n}) ≡ g_{n} (mod 2). This will be the case iff n = 2^{e} or 3 · 2^{e} for some e.

Exactly analogous to Proposition 2.11, we have the following result.

Proposition 3.28 Let ϕ : BP_{∗}BP → Z_{2}[v] be the homomorphism defined in Proposition 2.11. Let g_{i} ∈ BP_{4i−2}(ΩSp(n)/Sp([n/2])) be the generator defined, similarly to the proof of 2.7, using S-duality and the standard generators of BP^{∗}(HP^{m}). Suppose the coaction

BP_{∗}(ΩSp(n)/Sp([n/2]))⟶^{ψ}Γ ⊗ BP_{∗}(ΩSp(n)/Sp([n/2]))

satisfies

ψ(g_{n}) = ∑ z_{i} ⊗ g_{i}.

Then

ϕ(z_{i}) = \{ \begin{matrix}0 & if n − 2^{ν(n)+1} < i < n \\ v^{2^{ν(n)+2}} & if i = n − 2^{ν(n)+1}. \end{matrix}

Note that ψ(g_{n}) cannot contain any g_{i}^{2}-terms for dimensional reasons.

Now we are ready to calculate the differentials in Proposition 1.8. We begin with the stable differentials (1.11). These follow by the same argument that was used to prove Proposition 2.22. Or they can be deduced from it, using the map from the UNSS of Sp(n) to the stable NSS of Q^{n} discussed in Proposition 4.6. The reason that the differentials of (1.11) are not given on classes 2^{e} or 3 · 2^{e} with e > 0 is that differentials (1.9) are seen first on these classes.

Next we will prove the differentials (1.9) and (1.10). This proof will be by induction on i. We assume that it has been proved for all values of i less than n. This, along with (1.11), implies that there are no stable classes (i or i′) for the η-tower 2n to hit. Thus we want to determine the boundary homomorphism

v_{1}^{−1}E_{2}^{1,4k}(ΩS^{8n−1})⟶^{∂}v_{1}^{−1}E_{2}^{2,4k}(ΩSp(2n − 1))  (3.29)

in (3.5), and the target group consists only of unstable classes. By Figure 3.25, the source of the morphism (3.29) is the element α_{2k−4n+1}g_{2n}, where, as above, α_{j} is the element of order 2 on the 1-line in the (2j − 1)-stem for spheres, as described, for example, in [4, p. 52]. By [7, p. 246], α_{2k−4n+1}g_{2n} is represented by v_{1}^{2k−4n}h_{1} ⊗ g_{2n}.

We work now in the complex T^{∗}(ΩSp(n)), which involves \overline{U}(N) ⊂ G(N). Recall that N = QBP_{∗}(ΩSp(n)). This involves classes g_{i}^{∗2} but no classes g_{i}^{2}. Note that \overline{ψ}(g_{2n}) ∈ Γ ⊗ G(N) cannot contain any terms B ⊗ g_{i}^{∗2} with i > n for dimensional reasons. It does contain the term h_{1} ⊗ g_{n}^{∗2} since Proposition 3.2 implies that (z_{4n−1})Sq^{2} = z_{2n−1}^{2}. Here we use that h_{1} is detected by Sq^{2}, and that the ∗-product reduces to the homology product. This observation about the effect of the nonzero Sq^{2} is the key to this family of differentials. This sort of information about ψ was not seen in Proposition 3.28, which was only capable of seeing the stable range.

Thus there is a representative γ_{2n−1} for ∂(α_{2k−4n+1}⊗g_{2n}) which lies in T^{2}(ΩSp(2n− 1)) and has v_{1}^{2k−4n}h_{1} ⊗ h_{1} ⊗ g_{n}^{∗2} as its leading g_{i}^{∗2}-term. That is, it contains no terms of the form B ⊗ g_{i}^{∗2} with i > n.

We will inductively pull our class γ_{2n−1} back to a class γ_{n} ∈ T^{2}(ΩSp(n)). Assume that for some j satisfying n < j < 2n, there is a representative γ_{j} for ∂(α_{2k−4n+1}g_{2n}) which lies in T^{2}(ΩSp(j)) and has v_{1}^{2k−4n}h_{1} ⊗ h_{1} ⊗ g_{n}^{∗2} as its leading g_{i}^{∗2}-term. We project this to \overline{γ}_{j} ∈ T^{2}(ΩS^{4j−1}). As observed earlier, this cannot equal the nonzero stable class here, because that already supported a differential. If \overline{γ}_{j} is to be the nonzero unstable class, then it must contain the following terms in T^{2}(ΩS^{4j−1}):

v_{1}^{2k−4j+1}h_{1} ⊗ h_{1}^{2j−1} ⊗ g_{j} + v_{1}^{2k−4j}h_{1} ⊗ h_{1} ⊗ g_{j}^{∗2}.  (3.30)

To see this, we use Figure 3.25 to see that the unstable element w at height 2 and stem ≡ 0 mod 4 satisfies H_{2}(w) = v_{1}^{2k−4j+1}h_{1}. Then we use Proposition 3.23 to see that this must be accompanied by δ(H_{2}(w))g_{j}^{∗2}, and use [27, 5.3.13] to evaluate the Bockstein δ(v_{1}^{2k−4j+1}h_{1}) = v_{1}^{2k−4j}h_{1} ⊗ h_{1}.

But our induction hypothesis is that \overline{γ}_{j} does not contain this g_{j}^{∗2}-term. Thus \overline{γ}_{j} must be a boundary in T^{2}(ΩS^{4j−1}). We can vary γ_{j} by the boundary of an element of T^{∗}(ΩSp(j)) which corresponds to the bounding element in T^{∗}(ΩS^{4j−1}), obtaining a class γ_{j−1} which extends the induction hypothesis. The boundary can be chosen to be of the form d(A ⊗ vg_{j}) with ϕ(v) = 0 as in the stable situation considered in proving 2.25, and hence cannot produce detecting terms. Consequently by induction we obtain a representative γ_{n} which pushes to an element of T^{2}(ΩS^{4n−1}) containing the term v_{1}^{2k−4n}h_{1} ⊗ h_{1} ⊗ ι_{4n−2}^{∗2}. By Proposition 3.23 and Figure 3.25, this must be the nonzero unstable class n_{u}.

Next we will use the 2n → n_{u} differential just established to prove the differential (2n)′ → n′_{u} by precomposing the first differential with the E_{2}-element corresponding to the Hopf map σ in the 7-stem. We will use the notation of [27, p. 199], which denotes this element of Ext(BP_{∗}) by y_{4}, and its reduction to Ext(BP_{∗}/(2)) by u. Since we have shown that (3.29), which represents the (2n → n_{u})-differential, sends the generator of v_{1}^{−1}E_{2}^{1,4k}(ΩS^{8n−1}) into an element in the image of

v_{1}^{−1}E_{2}^{2,4k}(ΩSp(n))⟶^{i_{∗}}v_{1}^{−1}E_{2}^{2,4k}(ΩSp(2n − 1)),

the same will be true when this generator is precomposed by y_{4}. The element of v_{1}^{−1}E_{2}^{2,4k+8}(ΩS^{8n−1}) obtained as this composite is y_{4} ⊗ v_{1}^{2k−4n}h_{1} ⊗ g_{2n}. When this is reduced mod 2 (which we may do since we are looking at mod 2 classes), one obtains as the coefficient the stable class u ⊗ v_{1}^{2k−4n}h_{1} = h_{1} ⊗ v_{1}^{2k−4n}u, since stable Ext is commutative. By [27, 5.3.13c], this equals the mod 2 reduction of what is called ηα̃_{4k−4n+4} in Figure 3.25, which is the coefficient of the element (2n)′. Thus the source of our proposed differential is the desired σ-composition.

The element in E_{2}^{3,4k+8}(ΩS^{4n−1}) which is n_{u} precomposed with σ has leading term y_{4} ⊗ h_{1} ⊗ v_{1}^{2k−4n+1}h_{1}^{2n−1}g_{n}. By [5, 5.3], this unstable class x satisfies

H_{2}(x) = y_{4} ⊗ h_{1}v_{1}^{2k−4n+1} = h_{1} ⊗ uv_{1}^{2k−4n+1}.

Here H_{2} denotes the double suspension Hopf invariant, which yields a stable mod 2 class. But this is exactly what Figure 3.25 tells us is H_{2} of our proposed target class n′_{u}, namely the element at height 3 in the chart with horizontal coordinate 3 or 7 depending upon the parity of n. This establishes the differential (2n)′ → n′_{u}.

Now we are ready to derive (1.12). By naturality, it suffices to prove it on the top class n_{u} in the spectral sequence for v_{1}^{−1}π_{∗}(Sp(n)/Sp([n/2])). By 3.23, the unstable classes involved in the proposed differential are detected by their g_{i}^{∗2}-terms, and so we begin by studying the reduced coaction \overline{ψ}(g_{n}^{∗2}). As we are looking at elements of order 2, and d ∘ d = 0, this will consist of terms of the form z_{i} ⊗ g_{i}^{∗2}, where z_{i} is a mod 2 cycle. Here we are also using the multiplicativity of ψ with respect to the ∗-product. It is shown in [27, 5.3.13a] that any such cycle can be written, mod boundaries, as

z_{i} = ϵv_{1}^{k+3}h_{1} + ϵ′v_{1}^{k}u,  (3.31)

where ϵ and ϵ′ = 0 or 1, and u = t_{1}^{4} + v_{1}t_{1}^{3} + v_{1}^{2}t_{1}^{2} + v_{1}t_{2} + v_{2}t_{1}. The homomorphism ϕ defined in (2.12) sends the class in (3.31) to (ϵ + ϵ′)v^{4}. Proposition 2.21 shows that it sends any boundary to 0. Combining these remarks with Proposition 3.28 and the multiplicativity of ψ on ∗-products shows that

\overline{ψ}(g_{n}^{∗2}) ≡ ∑_{i<n}(ϵ_{i}v_{1}^{4(n−i)−1}h_{1} + ϵ′_{i}v_{1}^{4(n−i)−4}u + d(a_{i})) ⊗ g_{i}^{∗2} mod 2,  (3.32)

with a_{i} ∈ BP_{∗} and ϵ_{i} + ϵ′_{i} = 0 if i > n − 2^{ν(n)+1}, and ϵ_{i} + ϵ′_{i} = 1 if i = n − 2^{ν(n)+1}.

The source x of our proposed differential, also denoted n_{u}, is the element of E_{2}^{2,4k+8n}(ΩS^{4n−1}) corresponding to the unstable element at height 2 and horizontal component 0 or 4 in Figure 3.25. The notation of that chart says that it satisfies H_{2}(x) = v_{1}^{2k+1}h_{1}, and thus by Proposition 3.23

x ≡ v_{1}^{2k+1}h_{1} ⊗ h_{1}^{2n−1} ⊗ g_{n} + h_{1} ⊗ v_{1}^{2k}h_{1} ⊗ g_{n}^{∗2}  (3.33)

mod terms that desuspend or contain higher powers of g_{n}. Here we have used [27, 5.3.13d] to evaluate δ(H_{2}(x)). In the same chart, a target element y_{i}, labeled i′_{u} in (1.12), would be the element at height 3 and horizontal component 3 or 7, and so would satisfy

H_{2}(y_{i}) = h_{1} ⊗ uv_{1}^{2m+1}  (3.34)

for appropriate m. The differential, or boundary in the exact sequence, is calculated by applying \overline{ψ} to g_{n} and g_{n}^{∗2} in (3.33), and then modifying by boundaries. By Proposition 2.21, these boundaries cannot produce detecting terms. Thus the differential on n_{u} hits i′_{u} for largest possible i such that h_{1} ⊗ v_{1}^{2k}h_{1} ⊗ (ϵ_{i}v_{1}^{4l−1}h_{1} + ϵ′_{i}v_{1}^{4l−4}u) ⊗ g_{i}^{∗2} equals h_{1} ⊗ u ⊗ v_{1}^{2m}h_{1} ⊗ g_{i}^{∗2}. Here we have applied (3.32) to the second term of (3.33) to get the first expression, and have applied 3.23 and [27, 5.3.13d] to (3.34) to get the second expression. Since these coefficients are stable mod 2 classes, their factors commute. Thus the differential hits i′_{u} if and only if ϵ_{i} = 0 and ϵ′_{i} = 1. By (3.32), this can first happen when i = n − 2^{ν(n)+1}. Moreover, it must happen for this value of i, since we cannot have ϵ_{i} = 1 and ϵ′_{i} = 0 since that would imply the differential hits a class z whose James-Hopf invariant H(z) reduces mod 2 to v_{1}^{pwr}h_{1} ⊗ h_{1} ⊗ h_{1}. By Theorem 3.17 such a class would have δ(H_{2}(z)) = v_{1}^{pwr}h_{1} ⊗ h_{1} ⊗ h_{1}. But the only possible value of H_{2}(z) is given by (3.34), and so this possibility cannot occur since δ(h_{1} ⊗ uv_{1}^{2m+1}) = h_{1} ⊗ u ⊗ v_{1}^{2m}h_{1}.

4 Differentials and extensions

In this section we establish the differentials in the UNSS converging to v_{1}^{−1}π_{∗}(Sp(n)), whose E_{2}-term was depicted in Figure 1.16. We will also establish the exotic group extensions, and from this deduce Theorem 1.5.

The following theorem gives most of the higher differentials among the elements of Figure 1.16. Positions are the [x, y] = [t − s, s] coordinates of that chart. We use the phrase “log classes” for the elements of the Z_{2}-vector space of dimension [log_{2}(4n/3)] which is written “log” in Figure 1.16.

Theorem 4.1 In the UNSS for v_{1}^{−1}π_{∗}(Sp(n)), there are the following families of d_{3}- differentials, referring to Figure 1.16 for the description of v_{1}^{−1}E_{2}(Sp(n)).

1. d_{3} sends the η-tower of log classes starting in [8h + 3, 2] isomorphically to the η-tower of log classes starting in [8h + 2, 5].

2. d_{3} sends the η-tower of log classes starting in [8h + 2, 3] isomorphically to the η-tower of log classes starting in [8h + 1, 6]. (These latter elements are not actually represented in Figure 1.16, except as a shifted version of the η-tower going up from [8h + 6, 3].) It also sends the ηtower of •-classes beginning in [8h, 1] to the one beginning in [8h − 1, 4]

3. d_{3} sends the group in [8h + 1, 2] onto the group in [8h, 5] (which can be envisioned as a shifted version of the group in [8h + 8, 5]). Thus the group G(2^{e_{1}}) has [log_{2}(4n/3)] summands which support differentials (and perhaps also some which do not).

4. The η-tower on the • in [8h + 7, 2] is mapped isomorphically by d_{3} to the η-tower on the • in [8h + 6, 5].

Proof. Parts 1 and 2 follow immediately from the corresponding d_{3}-differentials in S^{4i−1}. Indeed, part 1 is the differential in Figure 3.25 emanating from filtration 2 in horizontal position 0 or 4, while the log-part of part 2 is the differential in Figure 3.25 going from filtration 3 and horizontal position 3 or 7. Remember that in Figure 3.25 horizontal position is stem, while in Figure 1.16 it is the number of the homotopy group. The classes involved in the •-differential in part 2 come from S^{3}. They appear in 3.25 as the d_{3}-differential on the element α_{4k+3}.

The differential in part 3 is immediate from the η-connection from G to log, which was already established, and the d_{3}’s on the log-part established in part 2. The classes involved in part 4 are stable classes coming from S^{4n−1} if n is odd, and S^{4n−5} if n is even. For example, if n = 10, they correspond to the 9′ in Figure 1.13. They should appear in Figure 3.25 as ηα̃_{4k+2} in [4, 2] with a d_{3} going to an element which is η times the top element listed in column 2 there. These differentials (with elements unlabeled) can be seen in [9, p. 488].

What remains from Figure 1.16 after the differentials of Theorem 4.1 are taken into account is pictured in Figure 4.2. Here H(2^{e_{1}}) is a group which fits into a short exact sequence

0 → H(2^{e_{1}}) → G(2^{e_{1}})⟶^{d_{3}}[log_{2}(4n/3)]Z_{2} → 0.

Thus the order of H(2^{e_{1}}) is 2^{e_{1}−[log_{2}(4n/3)]}.

Figure 4.2: A stage of the periodic UNSS for Sp(n)

[figure: s = 1; 2; 3; 4; t − s = 8h+; 1; 2; 3; 4; 5; 6; 7; 8; •; •; •; • log; log; log; log; H(2^{e_{1}}); • G(2^{e_{2}}); Z/2^{e_{1}}; Z/2^{e_{2}}; •; log]

It was proved in [10, 1.3] that the groups E_{2}^{1,4k−1} are cyclic of order 2^{e_{Sp}(k,n)}, where e_{Sp}(k, n) is defined as in (1.15). It is similar to the number e(k, n) which appears in Theorem 1.5 except that e_{Sp}(k, n) does not involve the terms ϵ_{k−1} and ϵ_{k−j} which appear in the definition, (1.3), of e(k, n). Recall that in our figures e_{j} = e_{Sp}(2h+j, n). In order to pass from Figure 4.2 to Theorem 1.5, we must consider extensions in the SS and additional d_{3}-differentials from the 1-line.

First, the extensions: There is an extension from the cyclic group v_{1}^{−1}E_{2}^{1,8h+7}(Sp(n)) to the element in v_{1}^{−1}E_{2}^{3,8h+9}(Sp(n)) which is in the image from E_{2}^{3,8h+9}(Sp(1)). This element is the one indicated by a • in [8h + 6, 3] in Figure 4.2. By “extension,” we mean a nontrivial extension in the short exact sequence

0 → v_{1}^{−1}E_{∞}^{3,8h+9}(Sp(n)) → v_{1}^{−1}π_{8h+6}(Sp(n))⟶^{∂}v_{1}^{−1}E_{∞}^{1,8h+7}(Sp(n)) → 0.

This accounts for the ϵ_{k−1} in e(k, n) in (1.3). To see this extension, one needs merely to observe that the element of order 2 in v_{1}^{−1}E_{∞}^{1,8h+7}(Sp(n)) is in the image from Sp(1), and the extension is present in the UNSS for v_{1}^{−1}π_{∗}(Sp(1)).

As for other extension questions: In 8h + 1, there could very well be nontrivial extensions. The two groups together, namely H(2^{e_{1}}) at height 2 and log at height 4, yield a group of order 2^{e_{1}} or 2^{e_{1}−1}, the latter being the case if there is a nonzero d_{3}- differential into v_{1}^{−1}E_{2}^{4,8h+5}(Sp(n)). Theorem 1.5 makes no claim about the structure of the group v_{1}^{−1}π_{8h+1}(Sp(n)). In 8h + 4, the extension is ruled out by the fact that the class in filtration 1 pulls back to Sp(1), while the class in filtration 3 does not. In 8h+5, there is a possible extension from G(2^{e_{2}}) to the •. This is incorporated into the G(2^{1+e(2h+2,n)}) in Theorem 1.5. If there is no d_{3}-differential into v_{1}^{−1}E_{2}^{4,8h+9}(Sp(n)), then e(2h + 2, n) = e_{2} + 1. In 8h + 7, we cannot solve the extension question.

The remainder of the work in this section is devoted toward proving the following result, which will complete the proof of Theorem 1.5.

Theorem 4.3 The differential

d_{3} : v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)) → v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n))

is nonzero if and only if e(k, n) < ϵ_{k−1} + e_{Sp}(k, n).

Note that the hypothesis on this proposition says exactly that the ϵ_{k−j} in Definition 1.3 affects the value of e(k, n).

Before embarking upon the lengthy proof of Theorem 4.3, we recapitulate the proof of Theorem 1.5.

• There is a spectral sequence converging to v_{1}^{−1}E_{2}(Sp(n)) with initial term given by summing Figure 1.7 over i ≤ n. The differentials among the η-towers are given by Proposition 1.8, while the S_{i}-groups build a cyclic 2-group of exponent e_{Sp}(k, n), and the U_{i}-groups build a group of the same order. The number of unstable Z_{2}’s in each bigrading surviving these differentials is [log_{2}(4n/3)], and so v_{1}^{−1}E_{2}(Sp(n)) is as in Figure 1.16.

• All the η-towers in v_{1}^{−1}E_{2}(Sp(n)) are involved in d_{3}-differentials implied by differentials in the spheres. What remains after these are accounted for is pictured in Figure 4.2; it includes the 1-line cyclic groups, part of the 2-line groups, and a few initial elements of target η-towers.

• There is a nonzero d_{3}-differential on the group in v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)) if and only if e(k, n) < ϵ_{k−1} + e_{Sp}(k, n). There is an extension in v_{1}^{−1}π_{4k−2}(Sp(n)) if k is even; this accounts for the ϵ_{k−1}. In Figure 4.4, we show how Figure 4.2 is modified by these considerations. The dotted differentials are present only when e(k, n) < ϵ_{k−1} + e_{Sp}(k, n).

The proof of Theorem 4.3 will proceed in a sequence of propositions. The first one handles the nonzero differentials.

Proposition 4.5 If e(k, n) < ϵ_{k−1} + e_{Sp}(k, n), then

d_{3} ≠ 0 : v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)) → v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n)).

Proof. In the exact sequence

E_{2}^{0,4k−1}(Sp(k))⟶^{p_{∗}}E_{2}^{0,4k−1}(Sp(k)/Sp(n)) → E_{2}^{1,4k−1}(Sp(n)) → 0,

the first two groups are infinite cyclic, and the third is Z/2^{e_{Sp}(k,n)}. In [32, 0.2] or [26, p. 868], it is shown that the mod torsion index of p_{∗}(π_{4k−1}(Sp(k))) in π_{4k−1}(Sp(k)/Sp(n)) has 2-exponent equal to or less than e(k, n). Assume, for starters, that ϵ_{k−1} = 0,

Figure 4.4: The final stage of the periodic UNSS for Sp(n)

[figure: s = 1; 2; 3; 4; t − s = 8h+; 1; 2; 3; 4; 5; 6; 7; 8; •; •; •; • log; log; log; log; H(2^{e_{1}}); • G(2^{e_{2}}); Z/2^{e_{1}}; Z/2^{e_{2}}; •; log]

or, equivalently, that k is odd. Then the hypothesis implies that the mod torsion index of the homotopy homomorphism p_{∗} is less than that of the E_{2}^{0,4k−1}- morphism p_{∗}. This implies that there must be a nonzero differential from the group E_{2}^{0,4k−1}(Sp(k)/Sp(n)).

Next we claim that this implies that E_{2}^{1,4k−1}(Sp(n)) must also support a nonzero differential. If not, then the element of E_{2}^{s,s+4k−2}(Sp(k)/Sp(n)) (s ≥ 2) which is the target of the differential in Sp(k)/Sp(n) must be in the image from E_{2}^{s,s+4k−2}(Sp(k)). This group is isomorphic to E_{2}^{s,s+4k−2}(Sp), which was shown in [10] to be 0 unless s = 3, in which case it has a single nonzero element which supports a nonzero d_{3}-differential in the UNSS of Sp. This differential must also take place in Sp(k). The target in E_{2}^{6,4k+3}(Sp(k)) of this differential must pull back to an element in E_{2}^{6,4k+3}(Sp(n)) which is hit by a d_{5} from the generator of E_{2}^{1,4k−1}(Sp(n)), contrary to our supposition that this group did not support a nonzero differential.

The proof of [12, 3.12] shows that there will be a periodic family of values of k for which e(k, n) and e_{Sp}(k, n) have the same values as they do for the value of k specified in the hypothesis of the proposition. The same argument as above applies to each of them, and so we conclude that the differential must be v_{1}-periodic. Hence it must be a d_{3}, since, after taking into account the differentials of Proposition 4.1, v_{1}^{−1}E_{3}^{s,s+4k−3}(Sp(n)) is nonzero only for s = 2 and 4.

A similar argument works when ϵ_{k−1} = 1, so that k is even. The hypothesis is that the mod torsion index of π_{4k−1}(Sp(k)) → π_{4k−1}(Sp(k)/Sp(n)) is equal to or less than that of E_{2}^{0,4k−1}(Sp(k)) → E_{2}^{0,4k−1}(Sp(k)/Sp(n)). This time there is a d_{3}- differential on E_{2}^{0,4k−1}(Sp(k)) depicted in [10, p. 74]. This would make impossible the inequality relating the mod torsion indexes unless there is a nonzero differential on E_{2}^{0,4k−1}(Sp(k)/Sp(n)). The target of that differential (and hence the differential itself) must map across to E_{2}^{s+1,s+4k−2}(Sp(n)), since there is nothing else in E_{2}^{s,s+4k−2}(Sp(k)) which can map to it, by [10, p. 74]. Thus we have deduced the asserted differential in Sp(n), and, as in the case when k is odd, it must be a v_{1}-periodic d_{3}.

In order to show that differentials are 0, we use the stable splitting map Σ^{∞}Sp(n) → Σ^{∞}(Q^{n}). See, for example, [23, p. 50].

Proposition 4.6 The stable splitting map induces a morphism from the UNSS of Sp(n) to the NSS of Q^{n} which sends E_{2}^{1,4k−1}(Sp(n)) isomorphically to

ker(E_{2}^{1,4k−1}(Q^{n}) → E_{2}^{1,4k−1}(Q^{∞})).

Proof. The morphism can be thought of as the composite of the stabilization morphism from the UNSS of Sp(n) to the stable NSS of Sp(n) followed by the map of NSS’s induced by the stable map from Sp(n) to Q^{n}. On E_{2}-terms, this is

Ext_{U}(P (BP_{∗}(Sp(n)))) → Ext_{G}(BP_{∗}(Sp(n))) → Ext_{G}(BP_{∗}(Q^{n})),

and P (BP_{∗}(Sp(n))) and BP_{∗}(Q^{n}) are isomorphic comodules.

Proposition 3.9 and Theorem 3.10 of [10] were proved for the UNSS based on MU; however, they also apply to spectral sequences based on BP, and to the stable NSS. The generators and coaction of BP_{∗}(Q^{n}) correspond precisely to those of P (BP_{∗}(Sp(n))). Hence Proposition 4.6 follows from the results of [10], which say that the groups are cyclic with order determined by \overline{e} applied to the coaction. For Sp(n), we did not need to write explicitly that we are computing the kernel into E_{2}^{1,4k−1}(Sp), since this group is 0.

The stable splitting map is useful because of the following result.

Proposition 4.7 The differential

d_{3} : v_{1}^{−1}E_{3}^{1,4k−1}(Q^{n}) → v_{1}^{−1}E_{3}^{4,4k+1}(Q^{n})

is nonzero if and only if e(k, n) < ϵ_{k−1} + e_{Sp}(k, n).

Proof. The mod torsion index of

π_{4k−1}^{s}(Q^{k}) → π_{4k−1}^{s}(Q^{k}/Q^{n})

is 2^{e(k,n)} by [26, p. 872], and it equals the order of the cokernel of

E_{∞}^{0,4k−1}(Q^{k}) → E_{∞}^{0,4k−1}(Q^{k}/Q^{n}).  (4.8)

By [10, 1.1], d_{3} is nonzero on E_{3}^{0,4k−1}(Q^{k}) if and only if ϵ_{k−1} = 1. Also from [10] is the result that the order of the cokernel of

E_{3}^{0,4k−1}(Q^{k}) → E_{3}^{0,4k−1}(Q^{k}/Q^{n})

is 2^{e_{Sp}(k,n)}. Thus the order of the cokernel of (4.8) has 2-exponent equal to e_{Sp}(k, n)+ ϵ_{k−1} − m, where m is the number of nontrivial differentials on E_{3}^{0,4k−1}(Q^{k}/Q^{n}). The proposition now follows by comparing with the earlier description of this exponent as e(k, n).

We can now easily obtain our desired deduction regarding d_{3} = 0 when k is even. Throughout the remainder of this section, E_{r} refers to the UNSS for Sp(m) and to the stable NSS for Q^{m}.

Proposition 4.9 If k is even and e(k, n) = 1 + e_{Sp}(k, n), then

d_{3} = 0 : v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)) → v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n)).  (4.10)

Proof. We prove the contrapositive. Suppose that d_{3} in (4.10) is nonzero. By Figure 4.2, there is only one nonzero element in v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n)) after the d_{3}-differentials of Proposition 4.1 are taken into account. If k is sufficiently large, this stable element pulls back to an element of E_{2}^{4,4k+1}(Sp(n)) which maps nontrivially to E_{2}^{4,4k+1}(Q^{n}). Thus d_{3} : E_{2}^{1,4k−1}(Q^{n}) → E_{2}^{4,4k+1}(Q^{n}) must also be nonzero, and hence by Proposition 4.7 e(k, n) < 1 + e_{Sp}(k, n), counter to the hypothesis of this proposition.

The argument when k is odd is a little bit more delicate because v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n)) has more than one nonzero element.

Proposition 4.11 Assume k is odd and e(k, n) = e_{Sp}(k, n). Then

i. If d_{3} is nonzero on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)), then d_{3} is nonzero on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n+ 1)) and e(k, n + 1) = e_{Sp}(k, n + 1).

ii. d_{3} = 0 on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)).

Proof. Using induction on n, part i implies that if d_{3} is nonzero on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)), then d_{3} is nonzero on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(N)) for all N ≥ n. This is certainly false for N > k, and hence d_{3} must have been zero on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)). Thus part i implies part ii, and so we concentrate on proving part i.

We begin with the case when n is even. The target of the differential in Sp(n) must map nontrivially to v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n + 1)). To see this, we note that if it mapped trivially, then it must be in the image from v_{1}^{−1}E_{2}^{3,4k+1}(S^{4n+3}), which is the group pictured in position [7, 3] in Figure 3.25. Each nonzero element in this group supports a d_{3}-differential in the UNSS of S^{4n+3}. The exact sequences in v_{1}^{−1}E_{2}(−) and v_{1}^{−1}π_{∗}(−) would then imply, similarly to the proof of Proposition 4.5, a nonzero d_{5}-differential in Sp(n + 1), contradicting [4, 5.5].

The map Sp(n) → Sp(n + 1) implies that d_{3} must be nonzero between the image classes in Sp(n + 1). Thus v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)) must map isomorphically to v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n+1)) since v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n+1)) is a Z_{2}-vector space. Thus e_{Sp}(k, n) = e_{Sp}(k, n + 1). Since k + n is odd, Proposition 5.2c says that e(k, n) = e(k, n + 1). Hence e(k, n + 1) = e_{Sp}(k, n + 1), completing the argument when n is even.

If n is odd, the above argument fails because v_{1}^{−1}E_{3}^{3,4k+1}(S^{4n+3}) has a class on which d_{3} is zero. Instead, we argue as follows.

Suppose that there is a nonzero differential on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n)) but not on v_{1}^{−1}E_{2}^{1,4k−1}(Sp(n + 1)). Then the target of the differential must be in the image of the morphism

v_{1}^{−1}E_{2}^{3,4k+1}(S^{4n+3})⟶^{∂}v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n)).

Now v_{1}^{−1}E_{2}^{3,4k+1}(S^{4n+3}) is spanned by η(n+1)′_{u} and η^{2}(n+1). Of these, only η^{2}(n+1) is sent nontrivially by ∂. We will show that under the stable splitting map

v_{1}^{−1}E_{2}^{4,4k+1}(Sp(n))⟶^{σ}v_{1}^{−1}E_{2}^{4,4k+1}(Q^{n})

∂(η^{2}(n + 1)) is sent nontrivially. This implies that there is a nonzero differential into v_{1}^{−1}E_{2}^{4,4k+1}(Q^{n}) and hence by Proposition 4.7 e(k, n) < e_{Sp}(k, n), contrary to our hypothesis.

To show that σ(∂(η^{2}(n + 1))) ≠ 0, we argue as follows. Consider n to be a fixed odd integer. It follows from Propositions 4.13 and 5.2a that there exist odd values of k such that e(k, n) < ϵ_{k−1} + e_{Sp}(k, n), and hence by Proposition 4.7 d_{3} ≠ 0 on E_{3}^{1,4k−1}(Q^{n}). For such a value of k, the morphism

v_{1}^{−1}E_{3}^{4,4k+1}(Sp(n))⟶^{σ}v_{1}^{−1}E_{3}^{4,4k+1}(Q^{n})  (4.12)

must satisfy σ(∂(η^{2}(n + 1))) ≠ 0. But the graded Z_{2}-vector spaces E_{3}^{4,4∗+1}(X) are acted on by period 8 Adams periodicity, which induces isomorphisms

v_{1}^{−1}E_{3}^{4,4k+1}(X)⟶^{A}v_{1}^{−1}E_{3}^{4,4k+9}(X)

for X = Sp(n) or Q^{n}, and A commutes with σ. Thus the morphism σ of (4.12) is nonzero on ∂(η^{2}(n + 1)) for all integers k, and so in particular it is nonzero for the value of k involved in the proposition.

Now we have shown that d_{3} is nonzero on v_{1}^{−1}E_{3}^{1,4k−1}(Sp(n)). We must also prove that e(k, n+1) = e_{Sp}(k, n+1). If this were not true, then by Proposition 4.7, d_{3} would be nonzero on v_{1}^{−1}E_{3}^{1,4k−1}(Q^{n+1}). Since, as we just showed, the nonzero d_{3}-differential mapped nontrivially from Sp(n) to Sp(n + 1), we must have e_{Sp}(k, n) = e_{Sp}(k, n + 1) and hence the nonzero differential on Q^{n+1} is the image of a nonzero differential on Q^{n}, which, with Proposition 4.7, contradicts the hypothesis that e(k, n) = e_{Sp}(k, n).

The following result was used in the above proof. It implies by Propositions 5.2a and 4.7 that for each odd n, there are some odd values of k for which there are nonzero d_{3}-differentials on v_{1}^{−1}E_{3}^{1,4k−1}(Q^{n}).

Proposition 4.13 If k ≡ 1 mod 2^{e−1}, and n is odd and satisfies n ≤ 2^{e−1}, then e_{Sp}(k, n) = 2n − 1 and e_{Sp}(k, n + 1) = 2n + 1.

Proof. Let v_{U}(K, J) = ν(∑_{i=1}^{J}(−1)^{i}\binom{J}{i}i^{K}). We will prove that if K ≡ 1 mod 2^{e}, and J ≤ 2^{e}+2, then v_{U}(K, J) = J −2 if J is odd, while if J is even, then v_{U}(K, J) ≥ J −1. This implies that the numbers e_{U}(K, N), defined as min{v_{U}(K, J) : J ≥ N}, satisfy e_{U}(K, N) = N − 2 (resp. N − 1) if N is odd (resp. even). Letting K = 2k − 1 and N = 2n + 1 and 2n + 2, and using the result of [10, 1.4] that

e_{U}(2k − 1, 2n) = e_{Sp}(k, n) = e_{U}(2k − 1, 2n + 1)  (4.14)

yields the desired result.

Now we prove the claim about v_{U}(K, J). Mod 2^{e+1}, the terms in the sum with i even are 0, while if i is odd, we have i^{k} ≡ i. Thus the result follows from

∑_{i odd} \binom{J}{i}i = J ∑_{i even} \binom{J−1}{i} = J ∑_{i=0}^{J−2} \binom{J−2}{i} = J · 2^{J−2}.

5 Applications to James numbers and exponents

In this section we discuss specific computations of the numbers e(k, n) and e_{Sp}(k, n), and give applications of our results to exponents of actual homotopy groups of Sp(n) and quaternionic James numbers.

The numbers e(k, n) and e_{Sp}(k, n) share many of the interesting properties of the slightly more basic numbers e_{U}(k, n) studied in [13] and [11]. Computer algebra packages allow extensive computation of these numbers, uncovering patterns that cry out for generalization and proof. We have used Mathematica to perform many calculations. The program consists of just three statements.

   nutwo[x ]:=If[OddQ[x],0,1+nutwo[x/2]]

   v[k ,j ,e ]:=nutwo[Sum[(-1)^i Binomial[2j,i] PowerMod[j-i,2k,2^e],
               {i,0,j-1}]]-nutwo[j]
   vt[k ,jo ,je ,e ]:=Table[v[k,j,e],{j,jo,je}]

The first line defines a function that calculates the exponent of 2. The second line calculates the basic numbers v(k, j) which are the exponents of 2 of the coefficients involved in (1.15), working mod 2^{e} so as to take advantage of the PowerMod function. Answers which it gives that are equal to or greater than e imply only that the correct number here is ≥ e. Here we have used the alternate form for the coefficients of (1.15) given in (1.4). The third line tabulates numbers for a range of values of j. In practice, a fourth line which performs vt[-] over an arithmetic progression of values of k is useful.

For example, if we run vt[20,3,12,20], the output

   {4, 4, 10, 15, 13, 11, 15, 16, 19, 19}

is almost immediate. Recalling that e_{Sp}(20, n) is the minimum of these numbers v(20, j) for j > n, this yields e_{Sp}(20, 2) = e_{Sp}(20, 3) = 4, e_{Sp}(20, 4) = 10, e_{Sp}(20, 5) = e_{Sp}(20, 6) = e_{Sp}(20, 7) = 11, etc. One can be assured that there will not be a smaller value of v(20, j) for some j > 12 since v(k, j) ≥ ν_{2}((2j − 1)!) by [26, 1.2,1.3]. As for the numbers e(20, n), we subtract 1 from each of the numbers v(20, j) with j odd, do a similar minimizing, and then add 1 (since ϵ_{19} = 1) to all numbers, obtaining e(20, 2) = 4, e(20, 3) = 5, e(20, 4) = 10, e(20, 5) = e(20, 6) = e(20, 7) = 12, etc.

The numbers e_{Sp}(k, n) and e(k, n) are periodic in k by [26, 1.7]. This makes it feasible to give complete formulas for these numbers for fixed n. The following proposition was obtained in a few hours of work using the above Mathematica program.

Proposition 5.1 a. If k is even, then

e_{Sp}(k, n) = \{ \begin{matrix}1 & n = 1 \\ 4 & n = 2 \\ 4 & n = 3 \\ min(11, 6 + ν(k − 4)) & n = 4 \\ min(11, 7 + ν(k − 4)) & n = 5 \\ min(15, 10 + ν(k − 6)) & n = 6 \\ min(18, 10 + ν(k − 230)) & n = 7 \end{matrix}

and

e(k, n) = \{ \begin{matrix}2 & n = 1 \\ min(5, 3 + ν(k − 2)) & n = 2 \\ 5 & n = 3 \\ min(12, 6 + ν(k − 36)) & n = 4 \\ min(12, 8 + ν(k − 4)) & n = 5 \\ 12 & n = 6, k ≡ 0 mod 4 \\ min(16, 10 + ν(k − 6)) & n = 6, k ≡ 2 mod 4 \\ min(18, 11 + ν(k − 102)) & n = 7 \end{matrix}

b. If k is odd, then

e_{Sp}(k, n) = \{ \begin{matrix}1 & n = 1 \\ 3 & n = 2 \\ min(8, 4 + ν(k − 7)) & n = 3 \\ min(8, 6 + ν(k − 3)) & n = 4 \\ min(14, 7 + ν(k − 37)) & n = 5 \\ min(14, 9 + ν(k − 37)) & n = 6 \\ min(16, 11 + ν(k − 13)) & n = 7, k ≡ 1 mod 4 \\ min(21, 11 + ν(k − 7)) & n = 7, k ≡ 3 mod 4 \end{matrix}

and

e(k, n) = \{ \begin{matrix}0 & n = 1 \\ 3 & n = 2 \\ min(7, 3 + ν(k − 7)) & n = 3 \\ 7 & n = 4 \\ min(13, 6 + ν(k − 37)) & n = 5 \\ min(13, 9 + ν(k − 5)) & n = 6 \\ min(15, 10 + ν(k − 13)) & n = 7, k ≡ 1 mod 4 \\ min(20, 10 + ν(k − 7)) & n = 7, k ≡ 3 mod 4 \end{matrix}

In the range of this proposition, we have e(k, n) = e_{Sp}(k, n) − 1 if k and n are odd. For awhile, it seemed important to know whether this was true for all odd values of k and n. We extended Proposition 5.1 through n = 18 for odd values of k, and found that the only case in this range in which e(k, n) = e_{Sp}(k, n) (with n odd) is when n = 13 and k ≡ 13 + 5 · 2^{15} mod 2^{19}. For such values of k, we have v(k, 13) = 22, v(k, 14) = 41, v(k, 15) = 40, and v(k, j) ≥ 41 for j ≥ 16. Thus e(k, 13) = e_{Sp}(k, 13) = 40.

To prove things about the numbers e_{Sp}(k, n), it might be useful to use (4.14), which relates them directly to the simpler numbers e_{U}(k′, n′). However, the numbers e(k, n) contain information not derivable from the e_{U}-numbers. We can easily derive some simple relationships between e_{Sp}(k, n), e_{Sp}(k, n + 1), and e(k, n).

Proposition 5.2 a. If k + n is even and e(k, n) = ϵ_{k−1} + e_{Sp}(k, n), then e_{Sp}(k, n) = e_{Sp}(k, n + 1).

b. If k + n is odd and e_{Sp}(k, n) < ϵ_{k−1} + e_{Sp}(k, n), then e_{Sp}(k, n) = e_{Sp}(k, n + 1). c. If k + n is odd and e_{Sp}(k, n) = e_{Sp}(k, n + 1), then e(k, n) = e(k, n + 1).

Proof. As in the computer program above, let v(k, j) equal the exponent of 2 in the coefficient relevant to the e_{Sp}-numbers.

a. The hypothesis implies that

min(v(k, n + 1) − 1, v(k, n + 2), v(k, n + 3) − 1, …)
= min(v(k, n + 1), v(k, n + 2), v(k, n + 3), …).

Thus the second minimum must equal min(v(k, n+2), v(k, n+3), …), which implies the desired result.

b. The hypothesis implies that

min(v(k, n + 1), v(k, n + 2) − 1, v(k, n + 3), …)
= min(v(k, n + 1), v(k, n + 2), v(k, n + 3), …) − 1,

and hence v(k, n + 1) cannot be less than min(v(k, n + 2), v(k, n + 3), …). c. The hypothesis implies that

min(v(k, n + 1), v(k, n + 2), v(k, n + 3), …)
= min(v(k, n + 2), v(k, n + 3), …).

Hence

min(v(k, n + 1), v(k, n + 2) − 1, v(k, n + 3), …)
= min(v(k, n + 2) − 1, v(k, n + 3), …).

We can apply our results to obtain a lower bound for the 2-exponent of Sp(n), similar to the applications made for SU(n) in [13] and [9]. This is new information about actual homotopy groups. Recall that the 2-exponent of a space X, denoted exp_{2}(X), is the minimal e such that 2^{e} annihilates the 2-torsion of π_{∗}(X). Since the v_{1}-periodic homotopy groups occur as direct summands of actual homotopy groups, the following results are immediate from Theorem 1.5 and Proposition 5.1.

Proposition 5.3 Let e_{2}^{Sp}(n) = max{e(k, n) : k ≥ n}. Then e_{2}^{Sp}(n) ≤ exp_{2}(Sp(n)).

Corollary 5.4 For n = 1, …, 7, the values of exp_{2}(Sp(n)) are equal to or greater than 2, 5, 7, 12, 13, 16, and 20, respectively.

This says, for example, that some homotopy group of Sp(7) contains an element of order 2^{20}. Corollary 5.4 is sharp when n = 1. An optimistic conjecture would be that exp_{2}(Sp(n)) = e_{2}^{Sp}(n), although the situation of SU(3) noted in [15] dampens one’s confidence.

The following theorem, similar to the result which was felt to be important enough to be highlighted as Theorem 1.1 in [13], gives a simple and fairly sharp lower bound for exp_{2}(Sp(n)). It says that, for every n, some homotopy group of Sp(n) has an element of order 2^{2n−1}.

Theorem 5.5 2n − 1 ≤ e_{2}^{Sp}(n) ≤ exp_{2}(Sp(n)).

Proof. The proof of (5.6) of [13] given on page 539 of that paper shows that max{e_{U}(k, m) : k ≢ m mod 2} ≥ m − 1. In other words, the lower bound for e_{2}^{U}(m) given there is obtained using values opposite in parity to m. Since, by (4.14), e_{Sp}(k, n) = e_{U}(2k − 1, 2n), we obtain e_{2}^{Sp}(n) ≥ 2n − 1. The result now follows from Proposition 5.3.

Finally we can deduce results about quaternionic James numbers similar to those proved for complex James numbers in [13] and [9]. Letting X_{n,k} denote the quaternionic Stiefel manifold, the quaternionic James number c_{n,k} is defined as the index of p_{∗}π_{4n−1}(X_{n,k}) in π_{4n−1}(S^{4n−1}). See [32] or [26] for earlier work on these numbers. The stable James numbers c_{n,k}^{s} are defined similarly using stable homotopy groups instead of ordinary homotopy groups. Our main result about quaternionic James numbers is that, if the numbers n and k are sufficiently large compared to their difference, then the stable and unstable James numbers are equal, and they equal their conjectured values. Here ϵ and e(−, −) are as in 1.5.

Theorem 5.6 If, for fixed n, k is sufficiently large, then

ν(c_{k,k−n}) = ν(c_{k,k−n}^{s}) = ν((2k − 1)!) + ϵ_{k−1} − e(k, n).

Proof. We prove

ν((2k − 1)!) + ϵ_{k−1} − e(k, n) ≤ ν(c_{k,k−n}^{s}) ≤ ν(c_{k,k−n}) ≤ ν((2k − 1)!) + ϵ_{k−1} − e(k, n).

The first inequality is from [32, 0.2] or [26, pp. 867-8], and the second is immediate from the definitions. The third follows similarly to [13, 4.3]. Indeed, we deduce from Theorem 1.5 that, if k is sufficiently large, then there is a summand of π_{4k−2}(Sp(n)) of order 2^{e(k,n)} which injects into π_{4k−2}(Sp(k −1)) ≈ Z/(2^{ϵ_{k−1}}(2k −1)!). The following commutative diagram then implies the inequality.

π_{4k−1}(Sp(k)) → π_{4k−1}(Sp(k)/Sp(n)) → π_{4k−2}(Sp(n)) → 0
↓ ↓ ↓
π_{4k−1}(Sp(k)) → π_{4k−1}(Sp(k)/Sp(k − 1)) → π_{4k−2}(Sp(k − 1)) → 0

6 Interpretation involving J-homology

When we began this project in [16], we intended to use charts of J-homology groups to determine v_{1}^{−1}π_{∗}(Sp(n)). For various reasons, this turned out to be infeasible. Nevertheless, it was important in lending insight toward the answer, and might lend insight to others. Therefore, we present a sketch of this point of view. In this section, we also use J-methods to prove Lemma 2.28, and introduce a new spectrum J_{1}, which has certain advantages over J.

Let J_{∗}(−) denote (stable) connective J-homology, as used, for example, in [25], where the crucial result

v_{1}^{−1}π_{∗}(S^{2n+1}) ≈ v_{1}^{−1}J_{∗}(Σ^{2n+1}P^{2n})  (6.1)

is proved.^{4} The determination of v_{1}^{−1}π_{∗}(Sp(n)) from the exact sequences in v_{1}^{−1}π_{∗}(−) associated to the fibrations

Sp(i − 1) → Sp(i) → S^{4i−1}, 2 ≤ i ≤ n,

4It was not until [19] that it was stated in this form.

may be viewed as the problem of inserting differentials and extensions in the chart for

⨁_{i=1}^{n} v_{1}^{−1}π_{∗}(S^{4i−1}) ≈ ⨁_{i=1}^{n} v_{1}^{−1}J_{∗}(Σ^{4i−1}P^{4i−2}).  (6.2)

We sketch how v_{1}^{−1}J_{∗}(P^{2m}) is determined, and refer the reader to [25] or [17] for more details. It is formed from

v_{1}^{−1}bo_{∗}(P^{2m}) ⊕ ϕ^{−2}σ^{−1}v_{1}^{−1}bo_{∗}(P^{2m}),  (6.3)

where ϕ^{i} increases filtration by i, and σ^{j} increases stem by j. The second summand is actually v_{1}^{−1}(Σ^{4}bsp)_{∗}(P^{2m}), suitably positioned to reflect its appearance in the exact sequence associated to the fibration

J → bo ⟶^{ψ^{3}−1} Σ^{4}bsp.  (6.4)

All these spectra are localized at 2. To form v_{1}^{−1}J_{∗}(P^{2m}) from (6.3), one inserts d_{r}- differentials (i.e., back 1, up r) from a tower in horizontal position 4j − 1 of the first summand to the corresponding tower from the second summand in position 4j − 2 if r = ν_{2}(j) + 1.

We illustrate with the computation of v_{1}^{−1}J_{∗}(P^{14}). In the left half of Figure 6.5 we depict the standard calculation of bo_{∗}(P^{14}). See, for example, [17] or [25]. In the right half is v_{1}^{−1}J_{∗}(P^{14}).

To form v_{1}^{−1}bo_{∗}(P^{14}), the part of Figure 6.5 in the parallelogram is continued in both directions with period [8, 4]. This involves extending the initial parts of the chart into negative filtration. The rectangle on the right side of Figure 6.5 displays a portion of v_{1}^{−1}J_{∗}(P^{14}) which appears for every integer h. There is a d_{2+ν(h)}-differential from the tower in 8h − 1 provided ν(h) ≤ 2. If ν(h) > 2, then the differential from 8h − 1 to 8h − 2 is zero.

These charts are summed as in (6.2) to obtain the initial term of a cellular SS converging to v_{1}^{−1}π_{∗}(Sp(n)). As we shall discuss later, the problem here is that we cannot prove that the s-filtrations are meaningful. In the left half of Figure 6.6, we have listed ⨁ v_{1}^{−1}π_{j}(Σ^{4i−1}P^{4i−2}) in the range 8k + 3 ≤ j ≤ 8k + 10, filtration s ≥ 4k − 13. Elements of the first type in (6.3) are indicated by •, and those of the second type by ∘. The label on a connected component of this chart is the

Figure 6.5: bo_{∗}(P^{14}) and v_{1}^{−1}J_{∗}(P^{14})

[figure: 4h; 8h; 1; 3; 7; 11; 15; 19]

integer i such that it comes from Σ^{4i−1}P^{4i−2}. We have omitted elements involved in d_{1}-differentials.

Every v_{1}^{−1}π_{∗}(S^{4i−1}) with 1 ≤ i ≤ n contributes four pairs of Z_{2}’s to the chart for v_{1}^{−1}π_{∗}(Sp(n)). The elements in each pair are connected by η ∈ π_{j+1}(S^{j}), which is represented in the charts by a line of length 1 and slope 1. On the right side of Figure 6.6, we indicate each of these pairs of Z_{2}’s by a single number, the i such that the pair comes from S^{4i−1}. The four pairs are characterized by whether they come from the bo-part or bsp-part, and whether they are stable or unstable. The bsp-parts, which always lie 1 unit to the left and 2 down from their bo-counterpart, were ∘’s in the left part of Figure 6.6, and are indicated by primes (′’s) in the right part of the chart. The unstable parts lie below their stable counterparts (except on S^{3}); they are indicated by a subscript u in the right side of Figure 6.6.

The pattern of differentials in the right side of Figure 6.6 is the way in which the Z_{2}’s cancel out in v_{1}^{−1}π_{∗}(Sp) if the s-filtrations are meaningful. These differentials are not quite the same as those given by Proposition 1.8. The differentials of that proposition involved minimal change of the number of the sphere, while those in Figure 6.6 involve minimal increase of s. On Sp(n) only the Z_{2}-pairs with number ≤ n will exist. This opens up other classes to a more complicated pattern of differentials,

Figure 6.6: J-type chart for v_{1}^{−1}π_{∗}(Sp(n))

[figure: 1; 1; 3; 3; 5; 5; 7; 7; 4; 2; 3; 6; 2; 2; 2; 4; 6; 8; 3; 3; 5; 7; 4; 8; 6; 6; 4; 5; 8; 5; 3; 4; 4k − 13; 4k + 3; 8k + 3; 8k + 10; 1′; 3′; 5′; 7′; 1; 3; 5; 7; 1′_{u}; 2′_{u}; 3′_{u}; 4′_{u}; 2′; 4′; 6′; 8′; 1_{u}; 2_{u}; 3_{u}; 4_{u}; 5_{u}; 2; 4; 6; 8; 8k + 3; 8k + 8; 4k − 9; 4k − 1; 4k − 5]

similar to that of Proposition 1.8.

When the Z_{2}’s are removed from consideration, what remains are the pairs of towers which would like to contribute to v_{1}^{−1}π_{4j+1} and v_{1}^{−1}π_{4j+2}. These have a pattern of differentials and extensions which is very complicated from this point of view. The Novikov point of view is much more convenient for handling these elements.

The biggest problem with using Figure 6.6 is that it is not clear that its s-filtrations are meaningful. The chart certainly depicts the initial term of a cellular SS converging to v_{1}^{−1}π_{∗}(Sp(n)); however, if it is to be useful, we need to know that the differentials in the SS increase s, and that they may be filtered by the amount by which they increase s.

In [18], it was proved that there is a finite spectrum X_{n} such that v_{1}^{−1}π_{∗}(X_{n}) ≈ v_{1}^{−1}π_{∗}(Sp(n)). The nice thing about this is that for a spectrum X there is an isomorphism v_{1}^{−1}π_{∗}(X) ≈ v_{1}^{−1}J_{∗}(X), and there is a SS converging to v_{1}^{−1}J_{∗}(X) with initial term

v_{1}^{−1}bo_{∗}(X) ⊕ ϕ^{−2}σ^{−1}v_{1}^{−1}bo_{∗}(X),

similarly to (6.3), and with meaningful s-filtrations. The spectrum X_{n} will be built by cofibrations from suspensions Σ^{k_{i}} of Σ^{4i−1}P^{4i−2}. It is the suspensions that are the problem. They cause bo_{∗}(X_{n}) to be spread out compared to Figure 6.6. That is, if k_{i} >> k_{i−1}, the contribution from Σ^{4i−1+k_{i}}P^{4i−2} will appear far to the right and below the contribution from Σ^{4i−5+k_{i−1}}P^{4i−6}. The chart for v_{1}^{−1}J_{∗}(X_{n}) formed in this way would have possible differentials that we would like to rule out. There is a possible solution to this problem. It involves controlling the Adams filtration of the attaching maps. If we know that the attaching map from Σ^{4i−2+k_{i}}P^{4i−2} to X_{i−1} has filtration s, then we can form a chart for J_{∗}(X_{i}) from J_{∗}(X_{i−1}) and ϕ^{s−1}J_{∗}(Σ^{4i−1+k}P^{4i−2}). (Recall that ϕ increases filtration.) The hope was (is) that these filtrations can be controlled so that the desired chart is obtained, but some difficulties in doing this have not been overcome.

This seems not to be terribly important anyway, since the Novikov methods seem to offer a cleaner way of handling the differentials among the Z_{2}’s as well as the larger summands. Figure 6.6 can be used for insight even if we don’t know that it represents anything real.

One of the unpleasant features of the spectrum J is that the usual J-charts are not actual ASS charts. They do not depict Ext_{A}(H^{∗}(X ∧ J), Z_{2}), as was discussed in [14]. However, in filtration ≥ 2, the usual J-chart is an Ext chart, as we can see by utilizing the spectrum J_{1}, defined to be the fibre of the nontrivial map J → HZ_{2}. The principal properties of J_{1} are established in the following result, in which A_{2} denotes the subalgebra of the mod 2 Steenrod algebra A generated by Sq^{1}, Sq^{2}, and Sq^{4}.

Proposition 6.7 As an A-module,

H^{∗}J_{1} ≈ ⟨g_{0}, g_{1}, g_{3} : Sq^{1}g_{0}, Sq^{2}g_{1} + Sq^{3}g_{0}, Sq^{1}g_{3} + Sq^{2}Sq^{1}g_{1} + Sq^{4}g_{0}, Sq^{5}g_{3} + Sq^{7}g_{1}⟩,

(6.8)

where |g_{i}| = i. For any connected spectrum X, the ASS converging to π_{∗}(X ∧ J_{1}) has

E_{2}^{s,t}(X ∧ J_{1}) = Ext_{A_{2}}^{s,t}(H^{∗}(X) ⊗ T_{2}, Z_{2}),  (6.9)

where T_{2} is the A_{2}-module with generators and relations as in (6.8). The usual chart for J_{∗}(X), formed, for example, as in [25, 7.1], has its E_{1}^{s,t} isomorphic to E_{2}^{s−1,t−1}(X∧ J_{1}) for s ≥ 2, with differentials agreeing under this isomorphism.

Proof. Let bo_{1} denote the fibre of the nontrivial map bo → HZ_{2}. Then there is a fibration

J_{1} → bo_{1}⟶^{θ′}Σ^{4}bsp,

and θ′ induces the 0-homomorphism in mod 2 cohomology. This is the advantage of J_{1} over J, for it implies a short exact sequence in cohomology and hence a long exact sequence

→ Ext_{A}^{s,t}(H^{∗}(Σ^{3}bsp)) → Ext_{A}^{s,t}(H^{∗}(J_{1})) → Ext_{A}^{s,t}(H^{∗}(bo_{1})) → .  (6.10)

This sequence holds with any H^{∗}Y in the second variable, although we shall always have Z_{2} in the second variable. The exact sequence also holds after smashing the spectra with a spectrum X. In a turnaround from the usual order of business, we can use the Ext-sequence to tell us the A-extensions in the short exact sequence

0 → H^{∗}(bo_{1}) → H^{∗}(J_{1}) → H^{∗}(Σ^{3}bsp) → 0.

The chart for Ext_{A}(H^{∗}(J_{1})) begins as in Figure 6.11. Here •’s represent classes from bo_{1}, and ∘’s from Σ^{3}bsp. The d_{1}-differentials represent the boundary homomorphism in (6.10), which is known by comparison with the exact sequence of E_{1}-terms of (6.4). The h_{1}-extension from 2 to 3 also follows from this comparison.

Figure 6.11: Chart for Ext_{A}(H^{∗}(J_{1}))

[figure: 0; 3; 7]

From Figure 6.11, we see that H^{∗}(J_{1}) has generators g_{0}, g_{1}, and g_{3}, with relations Sq^{1}g_{0}, Sq^{2}g_{1} + Sq^{3}g_{0}, Sq^{1}g_{3} + a_{1}g_{1} + a_{2}g_{0}, and Sq^{5}g_{3} + a_{3}g_{1} + a_{4}g_{0}, with a_{i} ∈ A. The relations correspond to the elements at height 1 in Figure 6.11, and the presence of Sq^{1} and Sq^{2} in these relations can be read off from the action of h_{0} (vertical lines) and h_{1} (diagonal lines) in the chart. The presence of Sq^{5}g_{3} is a consequence of the known structure of H^{∗}(bsp) = A/A(Sq^{1}, Sq^{5}). From the element in position [3,2], we deduce that there must be an element b in A such that

Sq^{1}(Sq^{1}g_{3} + a_{1}g_{1} + a_{2}g_{0}) + Sq^{2}(Sq^{2}g_{1} + Sq^{3}g_{0}) + b(Sq^{1}g_{0})

is identically 0. This implies that a_{1} = Sq^{2}Sq^{1}, a_{2} = Sq^{4}, and b = Sq^{4}. The term Sq^{3}Sq^{1} could also be included as a summand of a_{2}, but since we already know that Sq^{1}g_{0} = 0, there is no need to include it. Similarly, we omit an optional Sq^{3} summand from a_{1}.

The Ext chart tells us that Sq^{1}(Sq^{5}g_{3} + a_{3}g_{1} + a_{4}g_{0}) is expandable to an identity. This implies that Sq^{6}Sq^{1} cannot be a summand of a_{3}, for there is no way to cancel the Sq^{7}Sq^{1}g_{1} using the earlier relations. Moreover, it implies that this fourth relation in H^{∗}J_{1} must be of the form

Sq^{5}g_{3} + ϵSq^{7}g_{1} + ϵ′(Sq^{4}Sq^{2}Sq^{1}g_{1} + (Sq^{8} + Sq^{6}Sq^{2})g_{0}),  (6.12)

as Sq^{1} times this equals

ϵ′((Sq^{8} + Sq^{6}Sq^{2})(Sq^{1}g_{0}) + Sq^{4}Sq^{2}(Sq^{2}g_{1} + Sq^{3}g_{0})).

We need not consider Sq^{5}Sq^{2} as a summand of a_{3}, or Sq^{7}Sq^{1} or Sq^{5}Sq^{2}Sq^{1} as summands of a_{4}, because the resulting elements could be obtained from the first two relations. Finally the fact that Sq^{2}(Sq^{5}g_{3} + a_{3}g_{1} + a_{4}g_{0}) is expandable to an identity implies that ϵ′ = 0 in (6.12) since there would be no way to cancel the Sq^{10} which it would create, while ϵ = 1, since

Sq^{2}(Sq^{5}g_{3}+Sq^{7}g_{1}) = Sq^{6}(Sq^{1}g_{3}+Sq^{2}Sq^{1}g_{1}+Sq^{4}g_{0})+(Sq^{7}+Sq^{4}Sq^{2}Sq^{1})(Sq^{2}g_{1}+Sq^{3}g_{0}).

Since the coefficients in the relations for H^{∗}(J_{1}) are all in A_{2}, we deduce that H^{∗}(J_{1}) can be written as A ⊗_{A_{2}} T_{2}, where T_{2} is an A_{2}-module with generators and relations corresponding exactly to those of the A-module presentation of H^{∗}(J_{1}). The standard change-of-rings theorem now implies the Ext result.

It is standard that, for any A-module M,

Ext_{A}^{s,t}(M ⊗ H^{∗}(bo_{1}), Z_{2}) ≈ Ext_{A}^{s+1,t+1}(M ⊗ H^{∗}(bo), Z_{2})

if s > 0. The third part of the proposition then follows from a comparison of the “X∧”-version of (6.10) with the long exact sequence

→ Ext_{A}^{s−1,t}(H^{∗}(X ∧ Σ^{4}bsp)) → E_{1}^{s,t}(J ∧ X) → Ext_{A}^{s,t}(H^{∗}(X ∧ bo)) → .

Now we prove Lemma 2.28. We use the isomorphism v_{1}^{−1}π_{∗}(Q^{n}) ≈ v_{1}^{−1}J_{∗}(Q^{n}), and we pull the classes back under the map J_{1}_{∗}(Q^{n}) → v_{1}^{−1}J_{∗}(Q^{n}). We calculate J_{1}_{∗}(Q^{n}) by the ASS, for which an E_{1}-term is formed by summing charts like Figure 6.11, suspended 4i − 1 times, for 1 ≤ i ≤ n. The boundary homomorphism of Lemma 2.28 has source classes which appear as ∘’s in horizontal component congruent to 3 mod 4 in Figure 6.11, and target class of the type of the right-most • in Figure 6.11. Differentials (boundary homomorphisms) cannot go from ∘’s to •’s in that chart; this is implied by the exact sequence (6.10).

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