On the unstable Adams spectral sequence for SO and U, and splittings of unstable Ext

groups ^{∗}

A. K. Bousfield University of Illinois at Chicago Chicago, IL 60680 Donald M. Davis Lehigh University Bethlehem, PA 18015 ^{†}

This paper is dedicated to the memory of José Adem.

1 Introduction

Let U denote the category of unstable modules over the mod 2 Steenrod algebra A, and let Ext^{s,t}(−) = Ext_{U}^{s}(−, Σ^{t}Z_{2}). Let M_{∞} = H̃^{∗}(ΣCP_{+}^{∞}) denote the unstable A-module with nonzero classes x_{i} such that i is odd and positive, and

Sq^{2j}x_{2k+1} = \binom{k}{j}x_{2(j+k)+1}.

Then H^{∗}(U) ≈ U(M_{∞}), where the left side is the mod 2 cohomology of the infinite unitary group, and the right side the free unstable A-algebra generated by M_{∞}. Thus there is an unstable Adams spectral sequence (UASS), defined as in [5], converging to π_{∗}(U) with E_{2}^{s,t} ≈ Ext^{s,t}(M_{∞}). We shall construct an algebraic spectral sequence (SS) which we conjecture agrees with this UASS. In Section 3, we perform the minor modifications required to yield the analogous results for the infinite special orthogonal group SO.

Part of our conjecture is a splitting result for Ext(M_{∞}). Let M_{n} denote the subspace of M_{∞} spanned by those x_{i} such that α(i) ≤ n, where α(i) denotes the

1 number of 1’s in the binary expansion of i. This provides a filtration of M_{∞} by A-submodules, associated to which is a trigraded SS with

∗AMS Subject Classification 55T15, 55R45. Key words: unstable Adams spectral sequence, splittings of Ext groups, looping resolutions

†Both authors were supported by National Science Foundation research grants.

E_{1}^{n,s,t} = Ext^{s,t}(M_{n}/M_{n−1}),

d_{r} : E_{r}^{n,s,t} → E_{r}^{n+r,s+1,t}, and Ext^{s,t}(M_{∞}) filtered with nth subquotient E_{∞}^{n,s,t}.

Conjecture 1.1 This SS collapses to an isomorphism

Ext^{s,t}(M_{∞}) ≈ ⨁_{n≥1} Ext^{s,t}(M_{n}/M_{n−1}).

Conjecture 1.1, which should be of much interest in its own right, is implied by our main conjecture, because the algebraic SS which we construct has ⨁ Ext(M_{n}/M_{n−1}) as its E_{2}-term. The following analogue for SO may be of even more interest.

Conjecture 1.2 Let Q_{n} denote the subquotient of H̃^{∗}RP^{∞} spanned by classes x_{i} with α(i) = n. Then

Ext^{s,t}(H̃^{∗}RP^{∞}) ≈ ⨁_{n≥1} Ext^{s,t}(Q_{n}).

The UASS on which we focus is somewhat unusual, since we know the homotopy groups which it is computing, by Bott periodicity. We do not, however, know the Adams filtrations of their classes. One thing that is known in this direction is the complete Adams spectral sequence (ASS) converging to the homotopy groups of the connective unitary spectrum u localized at 2. The 2nth space of this spectrum is U[2n+1, ∞], the space obtained from U by killing π_{i}(−) for i < 2n+1. Using results of [6], we have H^{∗}(u) ≈ ΣA//AΛ_{1}, where Λ_{1} is the exterior subalgebra of A generated by the Milnor primitives of degree 1 and 3. Thus this SS has

E_{∞}^{s,t} ≈ E_{2}^{s,t} = Ext_{A}(H^{∗}u) ≈ Ext_{Λ_{1}}(ΣZ_{2}).  (1.3)

In the usual (t − s, s) depiction of ASS, this SS consists of, for each i ≥ 0, an infinite h_{0}-tower in stem t − s = 2i + 1 beginning in filtration s = i. The filtrations of the generators of the towers in E_{∞} of the UASS for U are certainly not this nice. But we use an algebraic SS converging to the nice E_{∞}-term of (1.3) as an aid to constructing our conjectural UASS(U).

Let F (n) denote the free unstable A-module on a generator ι_{n} of degree n, F′(n) = F (n)/ASq^{1}, and J_{n} ⊂ F′(2n − 1) the A-submodule generated by Sq^{3}ι_{2n−1}. Define

D^{s,t} = \{ \begin{matrix}Z_{2} & if t − s = 2i + 1 and 0 ≤ i ≤ s \\ 0 & otherwise \end{matrix}

and, for any integer n,

D_{n}^{s,t} = \{ \begin{matrix}D^{s,t} & if t − s ≤ 2n + 1 \\ Ext^{s−n−1,t−n}(J_{n+1}) & otherwise. \end{matrix}

If n < 0, then D_{n}^{s,t} = 0. Note that D^{s,t} agrees with the ASS for u described in (1.3). We will prove the following key result in Section 2.

Proposition 1.4 For n ≥ 0, there are exact sequences

⟶^{j_{n}^{s−1,t}} Ext^{s−n−1,t−n}(M_{n+1}/M_{n}) ⟶^{δ_{n}^{s−1,t}} D_{n−1}^{s,t} ⟶^{i_{n}^{s,t}} D_{n}^{s,t} ⟶^{j_{n}^{s,t}} Ext^{s−n,t−n}(M_{n+1}/M_{n}) ⟶^{δ_{n}^{s,t}} .

These can be spliced together to give an exact couple, and hence a SS. ([4]) Written in tableau form, the SS begins as follows.

→^{j} Ext^{s−1,t}(M_{1}) →^{δ}  0
↓^{i}
→^{j} Ext^{s−2,t−1}(M_{2}/M_{1}) →^{δ} D_{0}^{s,t} →^{j} Ext^{s,t}(M_{1}) →^{δ}  0
↓^{i}  ↓^{i}
→^{j} Ext^{s−3,t−2}(M_{3}/M_{2}) →^{δ} D_{1}^{s,t} →^{j} Ext^{s−1,t−1}(M_{2}/M_{1}) →^{δ} D_{0}^{s+1,t} →^{j}
↓^{i}  ↓^{i}
→^{j} Ext^{s−4,t−3}(M_{4}/M_{3}) →^{δ} D_{2}^{s,t} →^{j} Ext^{s−2,t−2}(M_{3}/M_{2}) →^{δ} D_{1}^{s+1,t} →^{j}
↓^{i}  ↓^{i}

If t − s < 2^{n+1} − 2, then i_{n}^{s,t} is bijective, since both groups surrounding it are 0. Thus limits are attained in this SS, and so the following corollary is immediate.

Corollary 1.5 There is a SS with

E_{2}^{n,s,t} = Ext^{s−n,t−n}(M_{n+1}/M_{n}),

with d_{r}^{n,s,t} : E_{r}^{n,s,t} → E_{r}^{n−r+1,s+1,t} and E_{∞}^{n,s,t} the nth subquotient of a filtration of D^{s,t}.

Thus this algebraic SS, which has a very complicated E_{2}-term, converges to the nice ASS for u. But its E_{2}-term is not the E_{2}-term of the UASS converging to π_{∗}(U), for it effectively pushes the chart of Ext^{∗,∗}(M_{n+1}/M_{n}) up by n units, by letting Ext^{s−n,t−n}(M_{n+1}/M_{n}) contribute to the limit (s, t)-group. In order to have a chance of obtaining the correct E_{2}-term, we regrade.

Define a new SS with

E_{2}^{s,t} = ⨁_{n} E_{2}^{n,s+n,t+n} = ⨁_{n} Ext^{s,t}(M_{n+1}/M_{n}),  (1.6)

with d_{r} on the nth summand of E_{r}^{s,t} equal to d_{r}^{n,s+n,t+n} of the SS of 1.5. The E_{∞}-term of this SS is a regraded version of E_{∞} of the SS of 1.5. When an element of D^{s,t} is pulled back to D_{n}^{s,t} for smallest possible n, it will now be seen in filtration s−n rather than s, as it was in E. These n’s are a nonincreasing function of s as we move up a tower (fixed t − s) of D^{s,t}, and will eventually stabilize. But changes in this n cause what look like filtration jumps in the SS of (1.6).

We illustrate with the situation through t − s = 9, where this jump first happens. The left SS is that of 1.5, and the right that of (1.6). Classes with n = 0 are indicated by x, with n=1 by •, and with n = 2 by ∘. As usual, coordinates are (t − s, s).

[figure: 1; 1; 3; 3; 5; 5; 7; 7; 9; 9; x; x; x; x; x; x; x; x; x; x; x; x]

Note that when the SS of 1.5 is pictured as above, all differentials look like d_{1}’s, for they all go from a group contributing to D^{s,t} to a group contributing to D^{s+1,t}. But the subscripts of differentials in the SS of 1.5 are related to changes of summand.

Our second main conjecture for U is as follows.

Conjecture 1.7 The SS of (1.6) agrees with the UASS of U.

As evidence, we observe that if Conjecture 1.1 is true, then the SS of (1.6) has the correct E_{2}-term, and its E_{∞} could be correct, for it has 0 in even stems and Z_{2}’s of strictly increasing filtrations in odd stems. Further evidence is given by the fact that our proof of Proposition 1.4 will involve the cohomology of the spaces in the Postnikov system for U.

As pointed out by Mark Mahowald, this algebraic SS for π_{∗}(U) can be demystified somewhat by thinking of it as arising from the destabilization of an Adams resolution of u. This is discussed in Section 4, where we also explain how the unstable A-modules M_{n+1}/M_{n} can be obtained as derived functors of the destabilization functor applied to a shifted version of H^{∗}u. In Section 5, we present a generalization of our results and conjectures. It is not clear whether the situation for SO and U contains essential ingredients not present in the much more general context of Section 5.

The reader attempting to prove Conjectures 1.1 and 1.2 should keep in mind that they are not true if an arbitrary module is allowed in the second variable. For example, Ext^{0,0}(H̃^{∗}RP^{∞}, Q_{1}) = 0, while Ext^{0,0}(Q_{1}, Q_{1}) ≠ 0.

2 Construction of the spectral sequence

In this section, we prove Proposition 1.4, which we have already seen implies the existence of the SS’s of 1.5 and (1.6).

Define F_{n} = F′(2n), and define K_{n} so that there is a short exact sequence (SES) in U

0 → K_{n} → F_{n} ⟶^{·Sq^{3}} J_{n−1} → 0.  (2.1)

Since J_{m} = 0 for m < 2, and

Ext^{s,t}(F_{n}) = \{ \begin{matrix}Z_{2} & t − s = 2n, s ≥ 0 \\ 0 & otherwise, \end{matrix}

the Ext sequence of (2.1) yields

Proposition 2.2 i) if n = 1 or 2, then Ext^{s,t}(K_{n}) = \{ \begin{matrix}Z_{2} & t − s = 2n, s ≥ 0 \\ 0 & otherwise; \end{matrix}

ii) if n ≥ 3, then Ext^{s,t}(J_{n−1}) ≈ Ext^{s−1,t}(K_{n}) unless t − s = 2n and s ≥ 0, in which case Ext^{s,t}(J_{n−1}) = Z_{2} and Ext^{s−1,t}(K_{n}) = 0.

We will need the following elementary result, where Ω is the loop functor in U as defined in [5].

Proposition 2.3 Ext^{s,t}(ΩK_{n}) ≈ Ext^{s,t+1}(K_{n}).

Proof. Letting Ω_{1} denote the first derived functor of Ω, [5, pp 43-44] and (2.1) imply that Ω_{1}K_{n} = 0, since it injects to Ω_{1}F_{n} = 0. Then the exact sequence

→ Ext^{s,t}(ΩK_{n}) → Ext^{s,t+1}(K_{n}) → Ext^{s−1,t}(Ω_{1}K_{n}) → Ext^{s+1,t}(ΩK_{n}) →

(see, e.g., [2, 3.7]) implies the desired result.

The Postnikov tower for U is a tower

U
↓
⋮  (2.2)
↓
K(Z, 5)⟶^{i} U[1, 5] ⟶^{k} K(Z, 8)
↓^{p}
K(Z, 3)⟶^{i} U[1, 3] ⟶^{k} K(Z, 6)
↓^{p}
K(Z, 1) ⟶^{k} K(Z, 4),

in which each ⟶^{i} ⟶^{p} ⟶^{k} is a fiber sequence, and the map U → U[1, 2n+1] induces an isomorphism in π(−) for i ≤ 2n + 1, while π_{i}(U[1, 2n + 1]) = 0 for i > 2n + 1.

There are isomorphisms of A-algebras H^{∗}(K(Z, 2n − 1)) ≈ U(ΩF_{n}) and, by [6] or [3], H^{∗}(U[1, 2n − 1])) ≈ U(X_{n}) for a certain unstable A-module X_{n}, which can be thought of as Σ^{−1}QH^{∗}(BU[2, 2n]), where Q denotes the indecomposable quotient. The morphisms of H^{∗}(−) of (2.2) are induced by morphisms of these unstable Amodules as below.

(2.4)

ΩF_{n+1} ⟵^{i_{n+1}} X_{n+1} ⟵^{f_{n+1}} F_{n+2}
↑
ΩF_{n} ⟵^{i_{n}} X_{n} ⟵^{f_{n}} F_{n+1}
↑
X_{n−1} ⟵^{f_{n−1}} F_{n}

The following result is culled from [6] (see esp. [6, 8.6] for (2.7) and [6, 7.1] for (2.8)), with K_{n}, J_{n}, and M_{n} as above.

Theorem 2.5 In the above diagram, i_{n}f_{n} = ·Sq^{3},

ker(f_{n}) = K_{n+1} and im(f_{n+1}) ≈ J_{n+1},  (2.6)

and there are SES’s in U

0 → M_{n} → X_{n+1} → Ωker(f_{n}) → 0  (2.7)

and

0 → im(f_{n+1}) → X_{n+1} → M_{n+1} → 0.  (2.8)

Note that (2.7) and (2.8) imply that

M_{n} ≈ im(X_{n} → X_{n+1}).  (2.9)

Taking quotients by M_{n} of the latter modules in (2.8), then using (2.7) to rewrite the middle group, and finally using (2.6) yields a SES

0 → J_{n+1} → ΩK_{n+1} → M_{n+1}/M_{n} → 0.  (2.10)

Remark 2.11 Note that this says that when the exact sequence

Σ^{3}A/ASq^{1}⟶^{·Sq^{3}} A/ASq^{1}⟶^{·Sq^{3}} Σ^{−3}A/ASq^{1}

is destabilized so that the generator of its middle group has dimension 2n + 1, then the homology is M_{n+1}/M_{n}. We will elaborate on this remark in Section 4.

From (2.10) and (2.3), we obtain a LES

→ Ext^{s−n−1,t−n+1}(K_{n+1}) → Ext^{s−n−1,t−n}(J_{n+1}) → Ext^{s−n,t−n}(M_{n+1}/M_{n})
→ Ext^{s−n,t−n+1}(K_{n+1}) → .  (2.11)

We will deduce the exact sequence of Proposition 1.4 from this.

When n = 0, the sequence of 1.4 reduces to the fact that

D_{0}^{s,t} ≈ Ext^{s,t}(M_{1}) ≈ \{ \begin{matrix}Z_{2} & t − s = 1 and s ≥ 0 \\ 0 & otherwise. \end{matrix}

When n = 1, both (2.11) and 1.4 reduce to the same isomorphism for t − s > 3, using 2.2 and the definition of D_{n}^{s,t}. The initial towers (t − s = 1) in D_{0}^{s,t} and D_{1}^{s,t} correspond under the morphism of 1.4. The tower in D_{1}^{s,t} in t − s = 3, s ≥ 1, maps in 1.4 to a tower in Ext(M_{2}/M_{1}) which in (2.11) maps to the tower in Ext(K_{2}).

If n ≥ 2, exactness of (2.11) is maintained if D^{s,t} for t − s ≤ 2n + 1 is added to both Ext^{s−n−1,t−n+1}(K_{n+1}) and Ext^{s−n−1,t−n}(J_{n+1}). The latter clearly becomes D_{n}^{s,t}, while 2.2 shows that the former becomes D_{n−1}^{s,t}, as desired.

Two remarks are in order. First, the reader may object that this is not an algebraic SS, as advertised, because it involves the cohomology of the spaces U[1, 2n − 1]. To this, we counter that the SS comes completely from (2.10), which is just algebraic. Moreover, a completely algebraic derivation of (2.10) can be given, but we felt the one presented here is more likely to lead to a proof of our conjectures.

Second, as more evidence for the relationship between our SS and the UASS, we point out that the Postnikov tower of U resembles an Adams-Postnikov tower, as defined in [5, p 82], because

ker(H^{∗}(U[1, 2n − 1]) → H^{∗}(U)) = ker(H^{∗}(U[1, 2n − 1]) → H^{∗}(U[1, 2n + 1])).

3 Results for SO

The entire argument can be directly adapted to SO, using results of [7] and [3] instead of [6]. We just list the replacements for the various symbols.

• M_{∞} = H̃^{∗}(RP^{∞}). Thus M_{∞} has classes x_{i} for all positive integers i, and M_{n} is spanned by those with α(i) ≤ n.

• For 0 ≤ b ≤ 3, let ρ(4a+b) = 8a+2^{b}. Thus ρ(n) is the grading of the nth nonzero homotopy group of BSO. Then X_{n} = Σ^{−1}QH^{∗}(BSO[2, ρ(n)]), and H^{∗}(SO[1, ρ(n) − 1]) ≈ U(X_{n}).

• F_{n} = \{ \begin{matrix}F (ρ(n)) & if n ≡ 0, 1 mod 4 \\ F′(ρ(n)) & if n ≡ 2, 3 mod 4 \end{matrix}

• i_{n}f_{n} : F_{n+1} → ΩF_{n} is ·Sq^{d}, where

d = ρ(n + 1) − ρ(n) + 1 = \{ \begin{matrix}2 & n ≡ 0, 3 mod 4 \\ 3 & n ≡ 1 mod 4 \\ 5 & n ≡ 2 mod 4 \end{matrix}

and, as before, K_{n+1} = ker(i_{n}f_{n}) = ker(f_{n}), J_{n} = im(i_{n}f_{n}) ≈ im(f_{n}), and M_{n} = im(X_{n} → X_{n+1}).

• Let h(i) = \{ \begin{matrix}1 & i ≡ 0, 1 mod 4 \\ ∞ & i ≡ 2, 3 mod 4. \end{matrix} Then

D^{s,t} = \{ \begin{matrix}Z_{2} & if t − s = ρ(i) − 1 and 0 ≤ i − 1 ≤ s < i − 1 + h(i) \\ 0 & otherwise \end{matrix}

and

D_{n}^{s,t} = \{ \begin{matrix}D^{s,t} & if t − s ≤ ρ(n + 1) − 1 \\ Ext^{s−n−1,t−n}(J_{n+1}) & otherwise \end{matrix}

• Proposition 2.2 is replaced by

Proposition 3.1 i) if n ≤ 3, then Ext^{s,t}(K_{n}) = \{ \begin{matrix}Z_{2} & t − s = ρ(n), 0 ≤ s < h(n) \\ 0 & otherwise; \end{matrix}

ii) if n ≥ 4, then Ext^{s,t}(J_{n−1}) ≈ Ext^{s−1,t}(K_{n}) unless t − s = ρ(n) and 0 ≤ s < h(n), in which case Ext^{s,t}(J_{n−1}) = Z_{2} and Ext^{s−1,t}(K_{n}) = 0.

With the new meaning of terms, Theorem 2.5, Proposition 1.4, and Corollary 1.5 are valid. Analogous to Remark 2.11 is that the following sequences on the left have homology as indicated on the right.

sequence  homology

F (8t + 1) → F′(8t − 1) → F′(8t − 6)  M_{4t−1}/M_{4t−2}
F (8t + 2) → F (8t) → F′(8t − 2) M_{4t}/M_{4t−1} (3.12)
F′(8t + 4) → F (8t + 1) → F (8t − 1)  M_{4t+1}/M_{4t}
F′(8t + 8) → F′(8t + 3) → F (8t)  M_{4t+2}/M_{4t+1}

Also, there is a SS (1.6), which we conjecture agrees with the UASS of SO. Conjecture 1.2 would be a corollary of this agreement, since it relates two ways of expressing the E_{2}-term.

We remark that a chart for D^{s,t} with coordinates (t − s, s) is the chart for the connective so spectrum, which begins as follows.

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

This can be seen since (see e.g., [3])

H^{∗}(so) ≈ ΣA/ASq^{3} = ΣA ⊗_{A_{1}} A_{1}/A_{1}Sq^{3},

where A_{1} is the subalgebra of A generated by Sq^{1} and Sq^{2}, and hence in the ASS for so

E_{∞}^{s,t} = E_{2}^{s,t} = Ext_{A_{1}}(ΣA_{1}/A_{1}Sq^{3}).

4 Topological and algebraic destabilizations

Our construction of the conjectured UASS for U (resp. SO) can be viewed as a consequence of destabilizing the ASS for u (resp. so). Let H_{n} = Σ^{n}HZ_{(2)} denote the Eilenberg-MacLane spectrum. An Adams-Postnikov tower for u has the simple form

u
↓
⋮
↓
H_{5} → u[1, 5] → H_{8}
↓
H_{3} → u[1, 3] → H_{6}
↓
H_{1} → H_{4}.

Diagram (2.2) is just the destabilization of this. Whereas the sequence

H^{∗}(H_{2n+4}) → H^{∗}(H_{2n+1}) → H^{∗}(H_{2n−2})

is exact, its destabilization

F′(2n + 4) → F′(2n + 1) → F′(2n − 2)

has homology M_{n+1}/M_{n}. Our SS deals with the way in which the groups Ext(M_{n+1}/M_{n}) build π_{∗}(U) ≈ π_{∗}(u).

As pointed out by Paul Goerss, there is an algebraic analogue of this. Let Ω^{∞} : Mod_{A} → U be the left adjoint to the inclusion functor. Thus, if M is an A-module, then Ω^{∞}M is the quotient of M mod relations Sq^{i}x = 0 if i > |x|. Let Ω_{s}^{∞} denote the sth derived functor of Ω^{∞}. Then we have

Proposition 4.1 (Goerss) If M_{i} is as in Section 1, then

Ω_{n}^{∞}(Σ^{−n+1}A//Λ_{1}) ≈ M_{n+1}/M_{n} ⊕ ΣZ/2,

while if M_{i} is as in Section 3, then

Ω_{n+1}^{∞}(Σ^{−n−2}A//A_{1}) ≈ M_{n}/M_{n−1}.

Proof. Define an acyclic chain complex C by

⋯ → C_{1} → C_{0} → A//Λ_{1} → 0

with C_{i} = Σ^{3i}A/ASq^{1} and boundary morphisms ·Sq^{3}. Apply Ω^{∞} to the modules in an A-resolution of each Σ^{−n+1}C_{i} to obtain a spectral sequence

E_{p,q}^{1} = Ω_{p}^{∞}Σ^{−n+1}C_{q} ⟹ Ω_{p+q}^{∞}Σ^{−n+1}A//Λ_{1},  (4.2)

with d^{r} : E_{p,q}^{r} → E_{p+r−1,q−r}^{r}. Noting that the only homology of the sequence

⋯ → F (3) → F (2) → F (1) → F (0) → 0,

with morphisms ·Sq^{1}, is F (0) = Z_{2} and ΣZ_{2} in F (1), we deduce

Ω_{p}^{∞}Σ^{m}A/ASq^{1} = \{ \begin{matrix}Σ^{m+p}Z_{2} & if p > 0 and m + p = 0 or 1 \\ F′(m) & if p = 0 \\ 0 & otherwise. \end{matrix}

Thus the only nonzero groups E_{p,q}^{1} in (4.2) satisfying p + q = n + ϵ with −1 ≤ ϵ ≤ 1 are E_{0,n+ϵ}^{1} = F′(2n + 1 + 3ϵ), E_{n,0}^{1} = ΣZ_{2}, and E_{n−1,0}^{1} = Z_{2}. By (2.10) and Remark (2.11), E_{0,n}^{2} = M_{n+1}/M_{n}. The only possible remaining differential involving groups with p + q = n is d^{n} : E_{0,n}^{n} → E_{n−1,0}^{n}, but this must be zero since these differentials preserve internal grading.

The SO-case is proved similarly, using the acyclic complex over Σ^{−n−2}A//A_{1} with C_{4t} = Σ^{12t−n−2}A/ASq^{1}, C_{4t+1} = Σ^{12t−n}A, C_{4t+2} = Σ^{12t−n+2}A, and C_{4t+3} = Σ^{12t−n+5}A/ASq^{1}. For ϵ = 0, 1, 2, and 3, let δ(ϵ) = 0, 2, 4, and 7, respectively. Then in the SS analogous to (4.2),

E_{p,4t+ϵ}^{1} = \{ \begin{matrix}Ω_{p}^{∞}Σ^{12t−n−2+δ(ϵ)}A/ASq^{1} & ϵ = 0, 3 \\ Ω_{p}^{∞}Σ^{12t−n−2+δ(ϵ)}A & ϵ = 1, 2 \end{matrix}

= \{ \begin{matrix}F′(12t − n − 2 + δ(ϵ)) & ϵ = 0, 3 and p = 0 \\ F (12t − n − 2 + δ(ϵ)) & ϵ = 1, 2 and p = 0 \\ Z_{2} & ϵ = 0, 3 and 12t − n − 2 + δ(ϵ) + p = 0 \\ ΣZ_{2} & ϵ = 0, 3 and 12t − n − 2 + δ(ϵ) + p = 1 \\ 0 & otherwise \end{matrix}

E_{0,n+1}^{2} is the homology of E_{0,n+2}^{1} → E_{0,n+1}^{1} → E_{0,n}^{1}, which by (3.12) is M_{n}/M_{n+1}. Higher differentials must be 0 for dimensional reasons as in the previous case. Since for ϵ = 0, 3 we have δ(ϵ) ≡ ϵ mod 4, the only Z_{2}’s or ΣZ_{2}’s occur in E_{p,q}^{1} with

p + q ≡ p + 4t + ϵ ≡ p + 12t + δ(ϵ) ≡ n + 2 or 3 mod 4,

and so there are none when p + q = n + 1.

5 A generalization

In this section, we propose conditions on an algebraic resolution which are satisfied by (2.4) and its SO analogue. We show that the analogue of Theorem 1.5 is true in this generality, and conjecture that the analogue of Conjecture 1.1 is true.

Suppose the diagram of unstable A-modules

F_{1}  F_{2}  F_{3}
↓^{f_{0}}  ↓^{f_{1}}  ↓^{f_{2}}
X_{0} ⟶^{p_{0}} X_{1}⟶^{p_{1}} X_{2} ⟶^{p_{2}} ⋯ → X
↓^{i_{1}}  ↓^{i_{2}}
ΩF_{1}  ΩF_{2}

satisfies • F_{n} → X_{n−1} → X_{n} → ΩF_{n} → ΩX_{n−1} is exact.

• F_{n} is a direct sum of F (m)’s and/or F′(k)’s

• (i_{n}f_{n})^{∗} = 0 : Ext(ΩF_{n}) → Ext(F_{n+1})

• ker(X_{n} → X) = ker(X_{n} → X_{n+1})

• X =dirlim(X_{n}).

Let M_{n} = im(X_{n} → X). Conjecture 5.1 Ext(X) = ⨁ Ext(M_{n+1}/M_{n})

The following proposition generalizes Theorem 1.5. Proposition 5.2 Let

D^{s,t} = ⨁_{n} Ext^{s−n,t−n}(ΩF_{n}).

There is a spectral sequence with

E_{2}^{n,s,t} = Ext^{s−n,t−n}(M_{n+1}/M_{n})

and E_{∞}^{n,s,t} the nth subquotient of a filtration of D^{s,t}.

The proof is identical to the proof for X = H̃^{∗}(ΣCP_{+}^{∞}) in Sections 1 and 2. One derives a SES

0 → im(f_{n+1}) → Ωker(f_{n}) → M_{n+1}/M_{n} → 0,

and modifies its exact Ext sequence using that Ext^{s,t}(imf_{n}) and Ext^{s−1,t}(kerf_{n}) differ only by Ext(F_{n+1}). This allows one to splice the exact sequences, yielding the desired spectral sequence.

One could modify the spectral sequence of the proposition to obtain a SS with E_{2}^{s,t} = ⨁ Ext^{s,t}(M_{n+1}/M_{n}). This E_{2}-term is the same as the E_{2}-term of the SS converging to Ext(X) whose collapsing we would like to prove. Somehow the fact that the SS of the proposition allowed us to consider the filtrations as being much higher is supposed to yield a proof of Conjecture 5.1. If the conjecture isn’t true in this generality, then what extra conditions are required to make it true?

References

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[2] A. K. Bousfield and E. B. Curtis, A spectral sequence for the homotopy of nice spaces, Trans Amer Math Soc 151 (1970) 457-479.

[3] J. Long, Two contributions to the homotopy theory of H-spaces, Princeton Univ thesis (1979).

[4] W. S. Massey, Exact couples in algebraic topology, Annals of Math, 56 (1952) 364-396.

[5] W. S. Massey and F. P. Peterson, The mod 2 cohomology of certain fibre spaces, Memoirs Amer Math Soc, 74 (1967).

[6] W. M. Singer, Connective fiberings over BU and U, Topology 7 (1968) 271-303.

[7] R. Stong, Determination of H^{∗}(BO(k, ∞)) and H^{∗}(BU(k, ∞)), Trans Amer Math Soc 107 (1963) 526-544.