The Thomified Eilenberg-Moore spectral sequence

Mark Mahowald, Douglas C. Ravenel and Paul Shick

November 21, 2000

1. Introduction

In this paper we will construct a generalization of the Eilenberg-Moore spectral sequence, which in some interesting cases turns out to be a form of the Adams spectral sequence. We recall the construction of both of these in general terms. Suppose we have a diagram of spectra of the form

X_{0}  X_{1}  X_{2}  ⋯ K_{0}  K_{1}  K_{2}_{_{g_{0}} u}  _{_{g_{1}} u}_{u}  _{_{g_{2}} u}_{u}  _{u} (1.1)

where X_{s+1} is the fiber of g_{s}. We get an exact couple of homotopy groups and a spectral sequence with

E_{1}^{s,t} = π_{t−s}(K_{s}) and d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t+r−1}.

This spectral sequence converges to π_{∗}(X) (where X = X_{0}) if the homotopy inverse limit lim_{←} X_{s} is contractible and certain lim^{1} groups vanish. When X is connective, it is a first quadrant spectral sequence. For more background, see [Rav86].

The second author acknowledges support from NSF grant DMS-9802516 and the Centre de Recerca Matematica in Barcelona.

In the case of the classical Adams spectral sequence, we have some additional conditions on on (1.1), namely

• Each spectrum K_{s} is a generalized mod p Eilenberg-Mac Lane spectrum, and

• each map g_{s} induces a monomorphism in mod p homology

These conditions enable us to identify the E_{2}-term as an Ext group over the Steenrod algebra, and to prove convergence when X is connective and p-adically complete.

For the Eilenberg-Moore spectral sequence, let

X ⟶^{i} E ⟶^{h} B (1.2)

be a fiber sequence with simply connected base space B. Then one uses this (in a manner to be described below) to produce a diagram of the form (1.1) where X_{0} is the suspension spectrum of X. This will yield a spectral sequence converging to the stable homotopy of X, but in practice it is not very useful. However if we smash everything in sight with the mod p Eilenberg-Mac Lane spectrum H/p, we get the Eilenberg-Moore spectral sequence converging to H_{∗}(X), where E_{2} is a certain Cotor group over H_{∗}(B).

In this paper we will explain a way to twist this construction using a p-local spherical fibration over the total space E. The entire construction can be Thomified to yield a spectral sequence converging to the homotopy of the Thom spectrum for the induced bundle over X. In §2 we recall a geometric construction of the Eilenberg-Moore spectral sequence, and in §3 we explain how it can be Thomified. In §4 we identify the E_{2}-term under certain circumstances as an Ext group over the Massey-Peterson algebra of the base space of the fibration in question, and in §5 we show that in some other cases we get a BP - theoretic analog of this result. In §6, we show that a special case of the Z/(p)-equivariant Adams spectral sequence of Greenlees can be constructed using the Thomified Eilenberg-Moore spectral sequence.

The authors wish to thank Bill Dwyer, John Greenlees and Brooke Shipley for helpful conversations and correspondence.

2. A geometric construction of the Eilenberg-Moore spectral sequence

We begin by recalling the stable cosimplicial construction associated with the Eilenberg-Moore spectral sequence, due to Larry Smith [Smi69] and Rector [Rec70]. Given the fibration (1.2), for s ≥ 0 let

G_{s} = E × \overbrace{B × ⋯ × B}^{s factors} .

Define maps h_{t} : G_{s−1} → G_{s} for 0 ≤ t ≤ s by

h_{t}(e, b_{1}, b_{2}, …, b_{s−1})

= \{ \begin{matrix}(e, b_{1}, b_{2}, …, b_{s−1}, ∗) & if t = 0 \\ (e, b_{1}, b_{2}, …, b_{t}, b_{t}, b_{t+1}, …, b_{s−1}) & if 1 ≤ t ≤ s − 1 \\ (e, h(e), b_{1}, b_{2}, …, b_{s−1}) & if t = s. \end{matrix}

Let E_{0} = E, X_{0} = X, X_{1} = E/Im i, and for s ≥ 1 we define spectra

Σ^{s}E_{s} = G_{s}/Im h_{0} ∪ ⋯ ∪ Im h_{s−1}
Σ^{s+1}X_{s+1} = G_{s}/Im h_{0} ∪ ⋯ ∪ Im h_{s}

i.e., the spectra X_{s} and E_{s} are desuspensions of suspension spectra of the indicated spaces. Then for s ≥ 0, h_{s} induces a map X_{s} → E_{s} giving a cofiber sequence

X_{s} ⟶^{h_{s}} E_{s} ⟶^{∂_{s}} ΣX_{s+1}, (2.1)

where ∂_{s} is projection from the topological quotient of G_{s} by one subspace to the quotient by a bigger subspace.

For s ≥ 0 there is a homology isomorphism

H_{∗}(E_{s}) = Σ^{−s}H_{∗}(E) ⊗ \overline{H}_{∗}(B^{(s)}),

where \overline{H} denotes reduced homology. Since B is simply connected, the connectivity of E_{s} increases without bound with s. Note also that

H_{∗}(X_{s}) = Σ^{−s}H_{∗}(X) ⊗ \overline{H}_{∗}(B^{(s)}),

for s > 0, so the homotopy inverse limit of the X_{s} is contractible. The homology exact couple associated with the cofiber sequences (2.1) leads to the Eilenberg-Moore spectral sequence for the fibration (1.2).

The Eilenberg-Moore spectral sequence also converges for non-simply-connected B, also. Dwyer has proved ([D74]) that the Eilenberg-Moore spectral sequence for the fibration

X → E → B

converges strongly to H_{∗}(X) if and only if π_{1}(B) acts nilpotently on H_{i}(E) for all i ≥ 0.

3. The Thomified Eilenberg-Moore spectral sequence

Now suppose that in addition to the fibration (1.2) we also have a p-local stable spherical fibration ξ over E which is oriented with respect to mod p homology. Projection onto the first coordinate gives compatible maps of the G_{s} to E, and hence a stable spherical fibration over each of them. This means that we can Thomify the entire construction. To each of the quotients X_{s} and E_{s} we associate a reduced Thom spectrum, which is defined as follows. Given a space A with a spherical fibration and a subspace B ⊂ A, the reduced Thom space for A/B is the space D_{A}/(S_{A} ∪ D_{B}) where D_{X} and S_{X} denote disk and sphere bundles over the space X. Thus we can associate reduced Thom spectra to the topological quotients E_{s} and X_{s+1} of G_{s}.

Let Y, K, Y_{s} and K_{s} be the Thomifications of X, E, X_{s} and E_{s}. Then the cofiber sequence of (2.1) Thomifies to

Y_{s} ⟶ K_{s} ⟶ ΣY_{s+1}. (3.1)

and we have

H_{∗}(K_{s}) = Σ^{−s}H_{∗}(K) ⊗ \overline{H}_{∗}(B^{(s)}).

The exact couple of homotopy groups for (3.1) leads to a spectral sequence converging to π_{∗}(Y). There is an associated diagram

Y  Y_{0}  Y_{1}  Y_{2}  ⋯ K_{0}  K_{1}  K_{2}_{_{g_{0}} u}  _{_{g_{1}} u}_{u}  _{_{g_{2}} u}_{u}  _{u} (3.2)

where Y_{s+1} is the fiber of g_{s}. This is similar to the Adams diagram of (1.1), but H_{∗}(g_{s}) need not be a monomorphism in general. We will call this the Thomified Eilenberg-Moore spectral sequence. We will use the indexing conventions of Adams rather than Eilenberg-Moore, namely

E_{1}^{s,t} = π_{t−s}(K_{s}) with E_{r}^{s,t} ⟶^{d_{r}} E_{r}^{s+r,t+r−1}.

This puts our spectral sequence in the first rather than the second quadrant.

We will see below (Theorem 4.4(ii) and Corollary 4.5) that under suitable hypotheses (including that the map i of (1.2) induces a monomorphism in homology), the Thomified Eilenberg-Moore spectral sequence coincides with the usual Adams spectral sequence for π_{∗}(Y).

The following lemma will be useful.

Lemma 3.3. For each prime p there is a p-local spherical fibration over Ω^{2}S^{3} whose Thom spectrum is the mod p Eilenberg-Mac Lane spectrum H/p.

Proof. For p = 2 we can use an ordinary vector bundle. We extend the nontrivial map S^{1} → BO to Ω^{2}S^{3} using the double loop space structure on BO. It was shown in [Mah79] that the resulting Thom spectrum is H/2.

The following argument for odd primes is due to Mike Hopkins. Let BF (n)_{(p)} denote the classifying space for the monoid of homotopy equivalences of the p-local n-sphere. Its fundamental group is Z_{(p)}^{×}. A p-local n-dimensional spherical fibration of a space X, i.e., a fibration with fiber S_{(p)}^{n}, is classified by a map X → BF (n)_{(p)}. Its Thom space is the cofiber of the projection map to X. Such fibrations and Thom spectra can be stabilized in the usual way. We denote the direct limit of the BF (n)_{(p)} by BF_{(p)}.

Now consider a p-local spherical fibration over S^{1} corresponding to an element u ∈ Z_{(p)}^{×}. It Thomifies to the Moore spectrum S^{0} ∪_{1−u} e^{1}. If we set u = 1 − p (which is a p-local unit) we get the mod p Moore spectrum V (0).

As in the case p = 2, we can extend this map S^{1} → BF_{(p)} to Ω^{2}S^{3} using the double loop space structure on BF_{(p)}, and similar arguments to those of [Mah79] identify the resulting Thom spectrum as H/p.

4. Identifying the E_{2}-term

Observe that H_{∗}(K) is simultaneously a comodule over A_{∗} and (via the Thom isomorphism and the map h_{∗}) H_{∗}(B), which is itself a comodule over A_{∗}. Following Massey-Peterson [MP67], we combine these two structures by defining the Massey-Peterson coalgebra (they called the dual object the semitensor product)

R_{∗} = H_{∗}(B) ⊗ A_{∗} (4.1)

in which the coproduct is the composite

H_{∗}(B) ⊗ A_{∗} H_{∗}(B) ⊗ H_{∗}(B) ⊗ A_{∗} ⊗ A_{∗} H_{∗}(B) ⊗ A_{∗} ⊗ H_{∗}(B) ⊗ A_{∗} ⊗ A_{∗} H_{∗}(B) ⊗ A_{∗} ⊗ A_{∗} ⊗ H_{∗}(B) ⊗ A_{∗} (H_{∗}(B) ⊗ A_{∗}) ⊗ (H_{∗}(B) ⊗ A_{∗}),_{_{Δ_{B}⊗Δ_{A}} u}_{_{H_{∗}(B)⊗ψ_{B}⊗A_{∗}⊗A_{∗}} u}_{_{H_{∗}(B)⊗A_{∗}⊗T ⊗A_{∗}} u}_{_{H_{∗}(B)⊗m_{A}⊗H_{∗}(B)⊗A_{∗}} u} (4.2)

where Δ_{A} and Δ_{B} are the coproducts on A_{∗} and H_{∗}(B), T is the switching map, ψ_{B} : H_{∗}(B) → A_{∗} ⊗ H_{∗}(B) is the comodule structure map, and m_{A} is the multiplication in A_{∗}.

Massey-Peterson gave this definition in cohomological terms. They denoted the semitensor algebra R by H^{∗}(B) ⊙ A, which is additively isomorphic to H^{∗}(B) ⊗ A with multiplication given by

(x_{1} ⊗ a_{1})(x_{2} ⊗ a_{2}) = x_{1}a′_{1}(x_{2}) ⊗ a′′_{1}a_{2},

where x_{i} ∈ H^{∗}(B), a_{i} ∈ A, and a′_{1} ⊗ a′′_{1} denotes the coproduct expansion of a_{1} given by the Cartan formula. Our definition is the homological reformulation of theirs.

Note that given a map f : V → B and a subspace U ⊂ V, H̄^{∗}(V/U) = H^{∗}(V, U) is an R-module since it is an H^{∗}(V)-module via relative cup products, even if the map f does not extend to the quotient V/U. In our case we have maps G_{s} → B for all s ≥ 0 given by

(e, b_{1}, …, b_{s}) ↦ h_{e}.

These are compatible with all of the maps h_{t}, so H_{∗}(Y_{s}) and H_{∗}(K_{s}) are R_{∗}-comodules, and the maps between them respect this structure.

We will see in the next theorem that under suitable hypotheses, the E_{2}-term of the Thomified Eilenberg-Moore spectral sequence is Ext_{R_{∗}}(Z/(p), H_{∗}(K)). When B is an H-space we have a Hopf algebra extension (see [Rav86, A1.1.15] for a definition)

A_{∗} ⟶ R_{∗} ⟶ H_{∗}(B).

This gives us a Cartan-Eilenberg spectral sequence ([CE56, page 349] or [Rav86, A1.3.14]) converging to this Ext group with

(4.3) E_{2} = Ext_{A_{∗}}(Z/(p), Ext_{H_{∗}(B)}(Z/(p), H_{∗}(K))).

Note that the inner Ext group above is the same as Ext_{H_{∗}(B)}(Z/(p), H_{∗}(E)), the E_{2}-term of the classical Eilenberg-Moore spectral sequence converging to H_{∗}(X). If the latter collapses from E_{2}, then the Ext group of (4.3) can be thought of as

Ext_{A_{∗}}(Z/(p), H_{∗}(Y)),

where H_{∗}(Y) is equipped with the Eilenberg-Moore bigrading. This is the usual Adams E_{2}-term for Y when H_{∗}(Y) is concentrated in Eilenberg-Moore degree 0, but the Ext group of (4.3) is graded differently in general.

Theorem 4.4. (i) Suppose that B is simply connected. Then the Thomified Eilenberg-Moore spectral sequence associated with the homotopy of (3.2) converges to π_{∗}(Y). If, in addition, H^{∗}(K) is a free A-module, then

E_{2} = Ext_{R_{∗}}(Z/(p), H_{∗}(K)),

where R_{∗} is the Massey-Peterson coalgebra of (4.1). (ii) If, in addition, the map i : X → E induces a monomorphism in

mod p homology, then the Thomified Eilenberg-Moore spectral sequence coincides with the classical Adams spectral sequence for Y.

The hypotheses on H_{∗}(K) may be unnecessary, but they are adequate for our purposes. The result may not be new, but we know of no published proof. Before proving the theorem we give a corollary that indicates that the hypotheses are not as restrictive as they may appear.

Corollary 4.5. Given a fibration

X ⟶ E ⟶ B

with X p-adically complete, a p-local spherical fibration over E, and B simply connected, there is a spectral sequence converging to π_{∗}(Y) (where Y is the Thomification of X) with

E_{2} = Ext_{H_{∗}(B)⊗A_{∗}}(Z/(p), H_{∗}(K)),

where K as usual is the Thomification of E.

Proof. We can apply 4.4 to the product of the given fibration with pt. → Ω^{2}S^{3} → Ω^{2}S^{3}, where Ω^{2}S^{3} is equipped with the p-local spherical fibration of Lemma 3.3. Then the Thomified total space is K∧H/p, so its cohomology is a free A-module. Thus the E_{2}-term is

Ext_{H_{∗}(B∧H/p)⊗A_{∗}}(Z/(p), H_{∗}(K ∧ H/p)) = Ext_{H_{∗}(B)⊗A_{∗}}(Z/(p), H_{∗}(K)).

Proof of Theorem 4.4 (i) The freeness of H_{∗}(K) over A_{∗} does not make (3.2) an Adams resolution because H_{∗}(g_{s}) need not be a monomorphism and the cofiber sequence

Σ^{s}Y_{s} ⟶^{g_{s}} Σ^{s}K_{s} ⟶ Σ^{s+1}Y_{s+1}

need not induce a short exact sequence in homology.

We will finesse this problem by producing a commutative diagram

(4.6)  Σ^{s}Y_{s}  Σ^{s}K_{s}  Σ^{s+1}Y_{s+1} Σ^{s}K_{s}  Σ^{s}W_{s}  Σ^{s+1}K_{s+1}^{g_{s}}  _{w}_{u_{g_{s}}}  _{w}_{u}  _{u_{h_{s+1}}}_{w}  _{w}  for s ≥ 0.

in which the cofiber sequence in the bottom row does induce a short exact sequence in homology with

H_{∗}(W_{s}) = H_{∗}(K_{s}) ⊗ H_{∗}(B). (4.7)

By the change-of-rings isomorphism of Milnor-Moore [MM65], this implies that

(4.8) Ext_{R_{∗}}(Z/(p), H_{∗}(W_{s})) = Ext_{A_{∗}}(Z/(p), H_{∗}(K_{s})).

Splicing the short exact sequences in homology from the bottom row of (4.6) gives a long exact sequence

0 ⟶ H_{∗}(K) ⟶ H_{∗}(W_{0}) ⟶ H_{∗}(ΣW_{1}) ⟶ ⋯,

which gives an algebraic spectral sequence (see [Rav86, A1.3.2]) converging to Ext_{R_{∗}}(Z/(p), H_{∗}(K)) with

E_{1} = Ext_{R_{∗}}(Z/(p), H_{∗}(W_{s})),

suitably indexed.

The freeness hypothesis on H_{∗}(K) implies (via (4.7)) that H_{∗}(W_{s}) is free over R_{∗}, so the algebraic spectral sequence collapses from E_{2}, i.e., Ext_{R_{∗}}(Z/(p), H_{∗}(K)) is the cohomology of the cochain complex

Ext_{R_{∗}}^{0}(Z/(p), H_{∗}(W_{0})) ⟶ Ext_{R_{∗}}^{0}(Z/(p), H_{∗}(ΣW_{1})) ⟶ ⋯

By (4.8) this is the same as

Ext_{A_{∗}}^{0}(Z/(p), H_{∗}(K_{0})) ⟶ Ext_{A_{∗}}^{0}(Z/(p), H_{∗}(ΣK_{1})) ⟶ ⋯

and our freeness hypothesis along with (4.6) allows us to identify this cochain complex with the E_{1}-term of the Thomified Eilenberg-Moore spectral sequence.

Thus the Thomified Eilenberg-Moore spectral sequence has the desired E_{2}-term if we can produce the diagram (4.6) satisfying (4.7). We shall do this now by geometric construction.

We define the following subspaces of G_{s} for s ≥ 1:

A_{s} = Im h_{0} ∪ Im h_{2} ∪ … ∪ Im h_{s−1},
B_{s} = A_{s} ∪ Im h_{1}
and C_{s} = B_{s} ∪ Im h_{s}.

Then it follows that h_{s} sends C_{s−1} to B_{s} and B_{s−1} to A_{s}, B_{s}/A_{s} = G_{s−1}/B_{s−1} and C_{s}/B_{s} = G_{s−1}/C_{s−1}. Thus for s ≥ 0 we get the following pointwise commutative diagram in which each row is a cofiber sequence.

X  E  E/i(X) E  G_{1}/A_{1}  G_{1}/B_{1}^{i}  _{w}_{u_{i}}  ^{∂_{0}}  _{w}_{u_{h_{1}}}  _{u_{h_{1}}}_{^{h_{1}} w}  _{^{∂_{1}} w}  for s = 0

and  G_{s−1}/C_{s−1}  G_{s}/B_{s}  G_{s}/C_{s} G_{s}/B_{s}  G_{s+1}/A_{s+1}  G_{s+1}/B_{s+1}^{h_{s}}  _{w}_{u_{h_{s}}}  ^{∂_{s}}  _{w}_{u_{h_{s+1}}}  _{u_{h_{s+1}}}_{^{h_{s+1}} w}  _{^{∂_{s+1}} w}  for s ≥ 1.

We define Σ^{s−1}W_{s−1} to be the Thomification of G_{s}/A_{s}, and we have previously defined Σ^{s}K_{s} and Σ^{s+1}X_{s+1} to be the Thomifications of G_{s}/B_{s} and G_{s}/C_{s}, so Thomification converts the diagrams above to (4.6).

Let p_{s} : G_{s+1} → G_{s} × B be the homeomorphism given by
p_{s}(e, b_{1}, …, b_{s+1}) = ((e, b_{2}, …, b_{s+1}), b_{1}).

Then we have

p_{s}h_{0} = (h_{0} × B)p_{s−1}
and p_{s}h_{t} = (h_{t−1} × B)p_{s−1} for 2 ≤ t ≤ s.

It follows that

G_{s+1}/A_{s+1} = (G_{s} × B)/(B_{s} × B) = (G_{s}/B_{s}) × B

and (4.7) follows.

(ii) If the H_{∗}(i) is monomorphic and H^{∗}(K_{s}) is a free A-module, then the diagram (3.2) is an Adams resolution for Y. Thus, the identity map on the resolution provides a comparison map from the Thomified Eilenberg-Moore spectral sequence to the Adams spectral sequence. We can identify the inner Ext group of (4.3) with H_{∗}(Y) concentrated in degree 0, the Cartan-Eilenberg spectral sequence collapses and our E_{2}-term is the usual

Ext_{A_{∗}}(Z/(p), H_{∗}(Y)).

So the comparison map induces an isomorphism on the E_{2} term of the spectral sequences, completing the proof of the theorem.

5. An Adams-Novikov analog

We now describe a case of the Thomified Eilenberg-Moore spectral sequence leading to variants of the Adams-Novikov spectral sequence. Suppose that in the fibration of (1.2), the spherical fibration over E is a complex vector bundle and that MU_{∗}(K) is free as a comodule over MU_{∗}(MU). If in addition MU_{∗}(i) is a monomorphism, then we get the usual Adams-Novikov spectral sequence converging to π_{∗}(Y).

We want an analog of 4.4 in the p-local case identifying the E_{2}- term for more general i. For this we need a BP-theoretic analog of the Massey-Peterson algebra R_{∗} of (4.1), additively isomorphic to

Γ(B) = BP_{∗}(B) ⊗_{BP_{∗}} Γ, (5.1)

where Γ = BP_{∗}(BP). In order to define a coproduct on this as in (4.2), we need a coalgebra structure on BP_{∗}(B). This does not exist in general, but it does when H_{∗}(B) is torsion free and BP_{∗}(B) is therefore a free BP_{∗}-module. If B is also an H-space, then BP_{∗}(B) is a Hopf algebra over BP_{∗} and (BP_{∗}, Γ(B)) is a Hopf algbebroid (defined in [Rav86, A1.1.1])

(BP_{∗}, Γ) ⟶ (BP_{∗}, Γ(B)) ⟶ (BP_{∗}, BP_{∗}(B))

is a Hopf algebroid extension as defined in [Rav86, A1.1.15]. This means there is a Cartan-Eilenberg spectral sequence (see [CE56, page 349] or [Rav86, A1.3.14]) converging to Ext_{Γ(B)}(BP_{∗}, BP_{∗}(K)) with

E_{2} = Ext_{Γ}(BP_{∗}, Ext_{BP_{∗}(B)}(BP_{∗}, BP_{∗}(K))). (5.2)

Then we get the following analog of Theorem 4.4, which can be proved in the same way.

Theorem 5.3. (i) Suppose that BP_{∗}(K) is free as a BP_{∗}(BP)- comodule and B is simply connected with torsion free homology. Then the Thomified Eilenberg-Moore spectral sequence associated with the homotopy of (3.2) converges to π_{∗}(Y) with

E_{2} = Ext_{Γ(B)}(BP_{∗}, BP_{∗}(K)),

where Γ(B) is the Massey-Peterson coalgebra of (5.1). (ii) If in addition the map i : X → E induces a monomorphism

in BP-homology, then the Thomified Eilenberg-Moore spectral sequence coincides with the Adams-Novikov spectral sequence for Y.

There is an analog of 4.5 in which we retain the hypothesis on B while dropping the one on K.

Corollary 5.4. Given a fibration

X ⟶ E ⟶ B

with X p-local, a a complex vector bundle over E, and B simply connected with torsion free homology, there is a spectral sequence converging to π_{∗}(Y) (where Y is the Thomification of X) with

E_{2} = Ext_{Γ(B)}(BP_{∗}, BP_{∗}(K)),

where K as usual is the Thomification of E.

This can be proved by applying 5.3 to the product of the given fibration with

pt. ⟶ BU ⟶ BU

with the universal complex vector bundle over BU.

6. A construction of the equivariant Adams spectral sequence

In this section we provide an alternative construction of a special case of the equivariant Adams spectral sequence, due to Greenlees ([G88] and [G90].) We first recall Greenlees’ approach.

Let G be a finite p-group. (Later, we will restrict our attention to the case where G is elementary abelian.) We work in the equivariant stable homotopy category of [LMS86], with all spaces pointed and all homology groups reduced. In this setting, G-free means that the action of G is free away from the base point. Greenlees’ version of the equivariant Adams spectral sequence is based on mod p Borel cohomology, defined for a based G-spectrum X as

b_{G}^{∗}(X) = H^{∗}(EG_{+} ∧_{G} X; Z/(p)),

where, as above, the Z/(p) coefficient groups will hereafter be suppressed. This is an RO(G)-graded cohomology theory, defined as follows for α any virtual real representation of G:

b_{G}^{α}(X) = H^{|α|}(EG_{+} ∧_{G} X).

Since G is a p-group, all representations are orientable, and the suspension isomorphisms in b_{G}^{∗} are given by the Thom maps, so the theory is really Z-graded in this case. This cohomology theory b_{G} is representable in the equivariant stable category. Adams and Greenlees identify the algebra b_{G}^{∗}(b_{G}) of natural cohomology operations as

b_{G}^{∗}(b_{G}) ≅ H^{∗}(BG_{+}) ˜⊗A,

where ˜⊗ denotes the Massey-Peterson semitensor product. Greenlees actually defines the spectral sequence in terms of a variant of Borel cohomology, namely f- or coBorel-cohomology, represented by

c_{G} = b_{G} ∧ EG_{+}.

Greenlees shows in [G88] that c_{G}^{∗}(c_{G}) ≅ b_{G}^{∗}(b_{G}).

Greenlees’ main result is the following cohomology version of the spectral sequence.

Theorem 6.1. ([G88]) For G a finite p-group, X and Y any G-spectra, with Y p-complete, bounded below, G-free and homologically locally finite, there is a convergent Adams spectral sequence

E_{2}^{s,t} = Ext_{c_{G}^{∗}(c_{G})}^{s,t}(c_{G}^{∗}Y, c_{G}^{∗}X) ⟹ [X, Y]_{∗}^{G},

natural in both variables.

One can define a similar spectral sequence based on b_{G}^{∗}(·), but this requires the additional hypothesis that X is G-free to guarantee proper convergence. A homology version of the spectral sequence can be written using the homology theory represented by the G-spectrum b_{G}, ([G90]) which does calculate [X, Y]_{∗}^{G} when X or Y is not G-free, provided we take G to be elementary abelian. The hypotheses on Y can just be checked nonequivariantly, if Y is G-free, by looking at the non-equivariant spectrum EG_{+} ∧_{G} Y.

Greenlees’ construction involves building a resolution of b_{G}^{∗}Y by free b_{G}^{∗}(b_{G})-modules,

0⟵b^{∗}Y ⟵^{ϵ}P_{0}⟵^{δ_{0}}P_{1}⟵^{δ_{1}}P_{1}⟵ ⋯

and realizing this resolution geometrically. Apply the functor [X, −]^{G} to this geometric resolution, obtaining a spectral sequence with

E_{1} = [Σ^{t−s}X, Q_{s}]^{G} ⟹ [Σ^{t−s}X, Y/ holim_{s} Y_{s}]^{G},

where Q_{s} is a locally finite wedge of copies of the spectrum representing b_{G} made free (i.e. a wedge of copies of c_{G} = b_{G} ∧ EG_{+},) with P_{s} = b_{G}^{∗}Σ^{s}Q_{s}. One identifies the E_{2} term in the usual manner, and proves convergence by comparing c_{G}^{∗}- (or b_{G}^{∗}-) connectivity with H^{∗}- connectivity to show that holim_{s} Y_{s} ≃ ∗.

We now show how to identify this equivariant Adams spectral sequence as a case of the Thomified Eilenberg-Moore spectral sequence, with certain restrictive hypotheses. From here onward we take G to be Z/(p), and we’ll work with the spectrum X G-fixed (so that we’ll use the c_{G}^{∗}(c_{G})-based spectral sequence, rather than it b_{G}^{∗}(b_{G})-based analog.) Let Z be a p-complete free G-spectrum with a spherical Gfibration F → E(ξ)⟶^{p}Z. Consider the Borel fibration

Z → EG_{+} ∧_{G} Z → BG.

The spherical G-fibration over Z induces a G-fibration

EG_{+} ∧_{G} F → EG_{+} ∧_{G} E(ξ) → EG_{+} ∧_{G} Z,

so we have the desired fibration over the total space of the Borel fibration.

We smash the Borel fibration with

pt. → Ω^{2}S^{3} → Ω^{2}S^{3}

(with the trivial G-action) and apply the Thomified Eilenberg-Moore spectral sequence construction to the resulting fibration. The resulting resolution is EG_{+} ∧_{G} HZ/(p)-free. Now for a G-fixed spectrum W (like HZ/(p) here,) the Borel construction is very simple: EG_{+}∧_{G}W ≃ BG_{+}∧W, So the Thomified Eilenberg-Moore spectral sequence resolution is free over BG_{+} ∧ HZ/(p). Let T (Z) denote the Thom spectrum of the bundle over Z. Then the resulting Thomified Eilenberg-Moore spectral sequence has

E_{2} = Ext_{H_{∗}(BG_{+}) ˜⊗A_{∗}}(H_{∗}BG_{+}, H_{∗}(T (EG_{+} ∧_{G} Z)))
= Ext_{H_{∗}(BG_{+}) ˜⊗A_{∗}}(H_{∗}BG_{+}, H_{∗}(EG_{+} ∧_{G} T (Z)))
= Ext_{b_{G}^{∗}(b_{G})}(b_{G}^{∗}(T (Z)), b_{G}^{∗}(S^{0})),

by F_{p}-duality, so that the Thomified Eilenberg-Moore spectral sequence E_{2} agrees with the equivariant Adams spectral sequence E_{2} term.

The Thomified Eilenberg-Moore spectral sequence E_{1} term here is given by applying the (nonequivariant) functor π_{∗}(−) to the Thomified Eilenberg-Moore spectral sequence diagram

Y  Y_{0}  Y_{1}  Y_{2}  ⋯ K_{0}  K_{1}  K_{2}_{_{g_{0}} u}  _{_{g_{1}} u} u  _{_{g_{2}} u} u  u

where Y is the Thom spectrum of EG_{+} ∧_{G} Z ∧ Ω^{2}S^{3}. The equivariant b_{G}^{∗}(b_{G})-Adams spectral sequence E_{1} term arises from applying [S^{0}, −]_{∗}^{G} to the geometric resolution

W  W_{0}  W_{1}  W_{2}  ⋯ Q_{0}  Q_{1}  Q_{2}_{_{g_{0}} u}  _{_{g_{1}} u} u  _{_{g_{2}} u} u  u

where W is the Thom spectrum of Z and Q_{s} is a wedge of copies of the coBorel spectrum c_{G}. But the Adams isomorphism ([Ad84], 5.3) shows that

[S^{0}, Q_{s}]^{G} = [S^{0}, K_{s}]^{1},

so that the isomorphism of E_{2} terms above is induced by one on the E_{1} level. This proves the following.

Theorem 6.2. Let Z be a p-complete based free Z/(p)-spectrum, with a spherical Z/(p)-fibration over Z. The Thomified Eilenberg-Moore spectral sequence for the smash product of the fibrations

pt. → Ω^{2}S^{3} → Ω^{2}S^{3}

and

Z → EZ/(p)_{+} ∧_{Z/(p)} Z → BZ/(p)

agrees with the b_{G}^{∗}(b_{G})-based equivariant Adams spectral sequence converging to π_{∗}(T (Z))^{Z/(p)} from E_{2} onward.

Unfortunately, the Thomified Eilenberg-Moore spectral sequence is known to converge only in the case where the base space in the fibration is simply-connected, from 4.4, which is not the case for the Borel fibration. Note that we would hope that the case of the Thomified Eilenberg-Moore spectral sequence above would converge to [EG_{+}, T (Z)]_{∗}^{G}, rather than [S^{0}, T (Z)]_{∗}^{G}, the target of the Z/p-equivariant Adams spectral sequence. Thus, despite the lack of simple-connectivity for the base space, this special case of the Thomified Eilenberg-Moore spectral sequence does converge if [S^{0}, T (Z)]_{∗}^{G} is isomorphic to [EG_{+}, T (Z)]_{∗}^{G} = [S^{0}, F (EG_{+}, T (Z))]_{∗}^{G} via the comparison map T (Z) → F (EG_{+}, T (Z)), which is indeed an equivalence when T (Z) is finite, by the (cinfirmed)

The Thomified Eilenberg-Moore spectral sequence 17

Segal Conjecture ([Car84].) Thus, when Z is finite and G-free, the Thomified Eilenberg-Moore spectral sequence converges to

π_{∗}(T (EG_{+} ∧_{G} Z)) ≅ π_{∗}(EG_{+} ∧_{G} T (Z)) ≅ π_{∗}(T (Z)/G) ≅ π_{∗}(T (Z)^{G}),

as we wish, where we think of π_{∗}(T (Z))^{G} as [S^{0}, T (Z)]^{G} with the sphere G-fixed. If Z is not finite, then the Thomified Eilenberg-Moore spectral sequence need not converge. For example, if T (Z) = EG_{+} ∧ HZ/p, then π_{∗}(T (Z)) = H_{∗}(G), which is bounded below, while F (EG_{+}, EG_{+}∧ HG) = F (EG_{+}, HG), which has homotopy H^{∗}(G), which is unbounded. This example was pointed out by the referee.

References

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18 Mark Mahowald, Douglas C. Ravenel and Paul Shick

[MP67] W. S. Massey and F. P. Peterson. The mod 2 cohomology structure of certain fibre spaces, volume 74 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence, Rhode Island, 1967.

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Northwestern University, Evanston IL 60208, University of Rochester, Rochester NY 14627, John Carroll University, University Hts OH 44118