New York Journal of Mathematics

New York J. Math. 14 (2008) 495–515.

The Morava K-Theory Eilenberg–Moore spectral sequence

John Carter

Abstract. In this article I consider the convergence of the Eilenberg– Moore spectral sequence for Morava K-theory. This spectral sequence can be constructed by applying Morava K-theory to D. L. Rector’s geometric cobar construction of the Eilenberg–Moore spectral sequence. I have shown that the Eilenberg–Moore spectral sequence for Morava Ktheory converges if the Eilenberg–Moore spectral sequence for ordinary homology collapses at E^{2} and the homology satisfies certain finiteness conditions.

Contents

1. Introduction 496 1.1. A computation 499 1.2. Rector’s construction 500 1.3. K(m)_{∗}-based Eilenberg–Moore spectral sequence 503

2. Analysis of the spectral sequence 504 3. Inverse limits of spectral sequences 504 4. The E^{∞} term of the inverse limit spectral sequence 506 5. The E^{∞} term of the Eilenberg–Moore spectral sequence 507 6. Results 509

6.1. Main theorem 509 6.2. Secondary result 512

References 513

Received December 12, 2007. Mathematics Subject Classification. 57T35. Key words and phrases. Morava K-theory, Eilenberg–Moore spectral sequence, Rector’s

construction.

495^{ISSN 1076-9803/08}

1. Introduction

This article concerns the convergence of the Eilenberg–Moore spectral sequence for computing the Morava K-theory of a homotopy pullback. We consider the special case associated to the path-loop fibration

ΩB → P B → B.

The calculation of the homology of loop spaces for generalized homology theories such as Morava K-theory has been a long-standing problem. There are two good reasons to suspect that the Eilenberg–Moore spectral sequence could be a useful tool toward this end. First it can be constructed from a tower of cofibrations and therefor can be used for any generalized homology theory. (We will make particular use of D. L. Rector’s construction of the Eilenberg–Moore spectral sequence from a tower of cofibrations.) And second for a generalized homology theory with Künneth isomorphisms the Eilenberg–Moore spectral sequence has a has a tractable (in theory anyway) E^{2}-page, namely E^{2} = Cotor^{K_{∗}(B)}(K_{∗}, K_{∗}).

Historically, for the Eilenberg–Moore spectral sequence, good convergence results are available in the case K_{∗} = H_{∗}(−, F_{p}) [6, 7, 8, 25]. For other cohomology theories with Künneth isomorphisms, namely where K_{∗} is one of Morava’s K-theories, useful general convergence criterion have proven elusive. Since the early 1980’s there have been a number of convergence results for the Eilenberg–Moore spectral sequence for Morava K-theory. For example in [29], Tamaki utilizes the Snaith splitting of Ω^{n}Σ^{n}X to show convergence for spaces of the form Ω^{n−1}Σ^{n}X. In [12], Jeanneret and Osse consider the pullback diagram

\xymatrix{X ×_{B} Y \ar[r] \ar[d] & X \ar[d]^{p} \\ Y \ar[r] & B}

in which B is connected. They prove that the spectral sequence converges to E^{∗}(X ×_{B} Y) when p is a fibration and E^{∗}(ΩB) is an exterior algebra on finitely many odd-degree generators. In [17], Mahowald, Ravenel, and Shick construct what they call the Thomified Eilenberg–Moore spectral sequence which is based on Rector’s construction and L. Smith’s related construction. In [24], Shipley considers the convergence of spectral sequence constructed from cosimplicial spaces which includes Rector’s construction as a particular example. More general criteria on the base space would go a long way toward making the Eilenberg–Moore spectral sequence for Morava K-theory more useful.

We will define the Eilenberg–Moore spectral sequence for Morava K-Theory as the spectral sequence that arises from applying Morava K-theory to the tower of cofibrations due to Rector.

Theorem 1.1. The Eilenberg–Moore spectral for Morava K-Theory comes from applying Morava K-Theory to the tower of cofibrations

(1.1)  \xymatrix{ & P (B) ≃ Σ^{∞}ΩB \ar[d] \\  & ⋮ \ar[d] \\ Σ^{−n}B^{∧n} \ar[r] & P_{n}(B) \ar[d] \\  & ⋮ \ar[d] \\ Σ^{−3}B^{∧3} \ar[r] & P_{3}(B) \ar[d] \\ Σ^{−2}B^{∧2} \ar[r] & P_{2}(B) \ar[d] \\  & P_{1}(B) ≃ Σ^{−1}B.}

and has

E^{2} = Cotor^{K(n)_{∗}(B)}(K(n)_{∗}, K(n)_{∗})

Under the assumption that the spectral sequence converges for ordinary homology with field coefficients we are justified in writing P (B) ≃ Σ^{∞}ΩB. Upon applying Morava K-theory to the tower (1.1), lim_{n} K(m)_{∗}(P_{n}(B)) is the obvious target of the spectral sequence, however convergence is by no means guaranteed.

To see that we cannot expect the K(m)_{∗}-based Eilenberg–Moore spectral sequence to converge in general note that for the n^{th} Eilenberg–Maclane space, K(n, ℤ/p).

K(m)_{∗}(K(n, ℤ/p)) = \begin{cases}≠ 0 & n ≤ m \\ 0 & n > m. \end{cases}

So, consider the K(m)_{∗}-based Eilenberg–Moore spectral sequence for

B = K(m + 1, ℤ/p).

If one applies K(m)_{∗}(−) to the tower (1.1), the target of the resulting spectral sequence is the associated graded, G(−), of the inverse limit, i.e.,

G(lim K(m)_{∗}(P_{n}(B))).

If this spectral sequence converged as hoped the target should be an associated graded group for

K(m)_{∗}(ΩK(m + 1, ℤ/p)) = K(m)_{∗}(K(m, ℤ/p)).

But this is not the case since K(m)_{∗}(B) = K(m)_{∗} and K(m)_{∗}(B^{∧n}) = K(m)_{∗} but K(m)_{∗}(ΩB) ≠ K(m)_{∗}. That is the m^{th} Morava K-theory does not ‘see’ K(m + 1, ℤ/p) but it does ‘see’ K(m, ℤ/p) so all of the groups in the Eilenberg–Moore spectral sequence are trivial but the hoped for target is not. In this case the K(m)_{∗}-based Eilenberg–Moore spectral sequence clearly cannot converge to K(m)_{∗}(ΩK(m + 1, ℤ/p)).

In this article I show that under some reasonable hypotheses on the base space B, that this spectral sequence calculates the Morava K-theory of ΩB. Specifically:

Theorem 1.2. Let B be a 1-connected CW complex with finitely many cells in each dimension such that the Eilenberg–Moore spectral sequence for mod p homology collapses at the E^{2} page and

Rank(⨁_{i} H_{j−i}(ΩB, K(m)_{i}))

is not infinite for adjacent j. Then the K(m)_{∗}-based Eilenberg–Moore spectral sequence for B converges to K(m)_{∗}(ΩB).

Here convergence is convergence in the sense of Boardman [4, Definition 5.2]. That is:

(1) The filtration exhausts G and there are isomorphisms E_{s}^{∞} ≅ F_{s}/F_{s+1} for all s. (This alone is weak convergence.)

(2) The filtration of G is Hausdorff, i.e., ⋂_{s} F_{s} = {0}.

The criteria for 1.2 come quite naturally from the method of proof which might be of interest in its own right.

To get at the convergence question we apply the Atiyah–Hirzebruch spectral sequence for K(n)_{∗} to all of the spaces in Rector’s tower. We are motivated here by the fact that the Atiyah–Hirzebruch spectral sequence for K(n)_{∗} has E^{∞} term K(n)_{∗}(B), so the inverse limit of the E^{∞} terms has the same form as the Eilenberg–Moore spectral sequence for Morava K-theory. This process gives rise to a tower of spectral sequences which gives rise to some obvious questions:

(1) Let E be the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P (B)), and E(n) the Atiyah–Hirzebruch sequence for K(m)_{∗}(P_{n}(B)). Is the spectral sequence E equal to lim E(n), the inverse limit of the spectral sequences E(n)?

(2) If the answer to question one is yes, what relationship holds between lim(K(m)_{∗}(P_{n}(B))) and K(m)_{∗}(P (B))?

(3) If we understand the answers to questions one and two, what does this say about the convergence of the K(m)_{∗}-based Eilenberg–Moore spectral sequence for ΩB?

Notice that if the answer to the first of these questions is yes then, assuming the Atiyah–Hirzebruch spectral sequence for all of the spaces in the tower converge, we get a close relationship between, lim E(n)^{∞}, an associated graded group for lim(K(m)_{∗}(P_{n}(B))) and, E^{∞}, an associated graded group for K(m)_{∗}(P (B)) = K(m)_{∗}(Σ^{∞}ΩB). The criteria of Theorem 1.2 are precisely the criteria that guarantee that lim E(n)^{∞} = E^{∞} as graded groups, and they arise from a careful analysis of how differentials can lift in a tower of spectral sequences. Once we have this result we can compare the Eilenberg–Moore spectral sequence for Morava K-theory with the spectral sequence that results from filtering lim E(n)^{∞} (an associated graded group for K(m)_{∗}(P (B)) = K(m)_{∗}(Σ^{∞}ΩB)) by the groups E(n)^{∞} which are associated graded groups for K(m)_{∗}(P_{n}(B)). This leads to our result and practical criteria for the computation of the Morava K-theory of at least some spaces.

1.1. A computation. It is important to note that the assumptions of 1.2 apply to some interesting spaces. For example this spectral sequence will converge for any space where the loopspace has finite mod-p, homology or has only even or only odd-dimensional homology in mod-p homology. Examples include SU(m)/ SU(n), BU, a product of odd spheres, any Hspace with homology or cohomology an exterior algebra etc. In all of these cases the ordinary Eilenberg–Moore spectral sequence will collapse at E^{2} since the differential lowers degree by 1 and ⨁_{i} H_{j−i}(P (B), K(m)_{i}) will be concentrated in all even dimensions.

Example 1.3. As an example of a space whose Morava K-theory can be calculated with the K(m)_{∗}-based Eilenberg–Moore spectral sequence consider B_{m,n} = SU(m)/ SU(n). To see that Theorem 1.2 applies recall that

H_{∗}(B_{m,n}) = E(b_{2n+1}, b_{2n+3}, …, b_{2m+1})

where |b_{2n+1}| = 2n + 1. That is the homology of B_{m,n} is an exterior algebra on generators in odd dimension. This implies that the Eilenberg–Moore spectral sequence for ordinary homology is concentrated in even degrees and hence collapses. Finally notice that H_{p,q}(ΩB_{m,n}) is concentrated in even dimensions and so is not infinite in adjacent dimensions, so that Theorem 1.2 applies.

It is known that for any multiplicative homology theory

h_{∗}(B_{m,n}) = E(b_{2n+1}, b_{2n+3}, …, b_{2m+1}),

so it is a primitively generated exterior algebra on odd-dimensional generators (see [2]). We show in this case Cotor is a polynomial algebra. To see this first notice that although Cotor is defined as the dual of Tor, it can be identified with Ext in this case. To simplify notation here let k = K(m)_{∗} i.e., k is a graded field.

Lemma 1.4. If P_{∗} is a free resolution of M such that A and M are of finite type, then

Cotor_{A}(M, k) ≅ Ext_{A^{∗}}(M^{∗}, k).

Proof. Since A and M are of finite type, we can choose a free A-resolution P_{∗} of finite type. Since linear duality of finite-dimensional vector spaces is an exact functor, P^{∗} is a free A^{∗}-resolution of M^{∗}. Then

Ext_{A^{∗}}(M^{∗}, k) = H_{∗}(hom_{A^{∗}}(P^{∗}, k)) = H_{∗}(hom_{A^{∗}}(P^{∗}, hom_{k}(k, k)))

≅ H_{∗}(hom_{k}(P^{∗} ⊗_{A^{∗}} k, k)) ≅ H_{∗}((P □_{A}k)^{∗∗})

≅ H_{∗}(P □_{A}k) = Cotor_{A}(M, k). □

A standard Ext calculation then gives the following [13, 14]:

Lemma 1.5. If E is a finitely generated primitive exterior algebra with generators x_{i} then

Cotor_{E}(k, k) = k(y_{i})

where deg(y_{i}) = (1, deg(x_{i})).

The E^{2} page of the K(m)_{∗}-based spectral sequence for computing ΩB_{m,n} is

Cotor_{K(m)_{∗}(B_{m,n})}(K(m)_{∗}, K(m)_{∗}) = k(y_{i})
deg(y_{i}) = (1, b_{i}).

Since K(m)_{∗}(B_{m,n}) is concentrated in even degrees E^{2} = E^{∞}. So

G(K(m)_{∗}(B_{m,n})) = k(y_{i}).

Since it is not well-known, we begin by recalling Rector’s construction of the Eilenberg–Moore spectral sequence.

1.2. Rector’s construction. The original motivation for Rector’s cosimplicial construction for the Eilenberg–Moore spectral sequence ([22]) was to show how the Steenrod operations interact with the spectral sequence structure. But his construction also allows one to construct the Eilenberg–Moore spectral sequence for generalized homology theories. Rector began with the cosimplicial space construction associated to a pullback. His construction proceeded as follows.

Assume B is a simply connected pointed space and start with a pullback diagram,

\xymatrix{E_{f} \ar[r] \ar[d] & E \ar[d]^{p} \\ X \ar[r]^{f} & B.}

Let

Δ : B → B × B

be the diagonal map. This is cocommutative, coassociative, and has a counit. We can thus create a cosimplicial space G_{∗}(X, B, E).

G_{0}(X, B, E)\begin{matrix}⟶ \\ ⟵ \\ ⟶ \end{matrix}G_{1}(X, B, E)\begin{matrix}⟶ \\ ⟵ \\ ⟶ \\ ⟵ \\ ⟶ \end{matrix}G_{2}(X, B, E) ⋯

where

G_{n}(E, B, X) = E × B^{n} × X,

δ^{t} : G_{s−1} → G_{s} for 0 ≦ t ≦ s is given by

δ^{t}(e, b_{1}, b_{2}, …, b_{s−1}, x)

= \begin{cases}(e, f(e), b_{1}, b_{2}, …, b_{s−1}, x) & if t = 0 \\ (e, b_{1}, b_{2}, …, b_{t}, b_{t}, b_{t+1}, …, b_{s−1}, x) & if 1 ≦ t ≦ s − 1 \\ (e, b_{1}, b_{2}, …, b_{s−1}, p(x), x) & if t = s, \end{cases}

and

σ^{t}(e, b_{1}, …, b_{n}, x) = (e, b_{1}, …, b_{t−1}, b_{t+1}, …, b_{n}, x).

For the pullback diagram above we have a natural weak homotopy equivalence [8, 9.2]

E_{f} ⋍_{w} Tot(G_{∗}(X, B, E))

From this he defined a sequence of cofibrations,

L_{0}(B) → G_{1}^{N}(E, B, X) → L_{1}(B)
L_{1}(B) → G_{2}^{N}(E, B, X) → L_{2}(B)
L_{2}(B) → G_{3}^{N}(E, B, X) → L_{3}(B)
⋮
L_{n−1}(B) → G_{n}^{N}(E, B, X) → L_{n}(B).

Here G_{∗}^{N}(E, B, X) is the normalized cosimplicial space (see [32, Definition 8.3.6] example) and L_{n−1}(B) is the cofiber of the induced map. This sequence of cofibrations induces an exact couple

D_{−p,q}^{1} = H_{q}(L_{p}(B))
E_{−p,q}^{1} = H_{q}(G_{p}^{N}(E, B, X)).

Rector then showed that the spectral sequence associated to this exact couple is isomorphic to the Eilenberg–Moore spectral sequence. (A good source for this is [30].)

In the language of spectra let G_{n}^{N}(E, B, X) and L_{n}(B) be the suspension spectra associated to these spaces. Define the spectra

E_{s} := Σ^{−s}G_{s}^{N}(E, B, X)
P_{s}(B) := Σ^{−s}L_{s}(B).

At the level of spectra for s ≧ 0, δ^{0} induces a map P_{s}(B) → E_{s} giving a cofiber sequence P_{s}(B) ⟶^{h_{s}} E_{s} ⟶^{∂_{s}} ΣP_{s+1}(B), where ∂_{s} is the projection from the topological quotient of G_{s} by one subspace to the quotient by a bigger subspace. One can obtain an exact couple and hence a spectral sequence by applying any homology theory to this sequence of cofibrations.

For the constructions in this paper we will specialize to the path-loop fibration, i.e.,

\xymatrix{ΩB \ar[r] \ar[d] & P B \ar[d] \\ ∗ \ar[r] & B.}

For the path-loop fibration, E^{s} and P_{s}(B) have a particularly nice form. For s ≥ 0 there is a homology isomorphism [17]

H_{∗}(E_{s}) = Σ^{−s}H_{∗}(∗) ⊗ H_{∗}(B^{∧s}) = Σ^{−s}H_{∗}(B^{∧s})

where H denotes reduced homology. Since B is simply connected, the connectivity of E^{s} is at least s − 1. The theorem below is a translation of Rector’s theorem [22].

Theorem 1.6. The Eilenberg–Moore spectral sequence comes from applying homology to the tower of cofibrations

(1.2)  \xymatrix{ & P (B) ≃ Σ^{∞}ΩB \ar[d] \\  & ⋮ \ar[d] \\ Σ^{−n}B^{∧n} \ar[r] & P_{n}(B) \ar[d] \\  & ⋮ \ar[d] \\ Σ^{−3}B^{∧3} \ar[r] & P_{3}(B) \ar[d] \\ Σ^{−2}B^{∧2} \ar[r] & P_{2}(B) \ar[d] \\  & P_{1}(B) ≃ Σ^{−1}B}

and has

E^{2} = Cotor^{H_{∗}(B)}(k, k) ⇒ H_{∗}(ΩB).

Remark 1.7. In this formulation of the Eilenberg–Moore spectral sequence, convergence is transparent when π_{1}(B) = 0. Formally we know that a spectral sequence of this form converges to lim_{n} H_{∗}(P_{n}(B)). In this case since the maps P_{n}(B) → P_{n−1}(B) are increasingly connective, the maps H_{∗}(P_{n}(B)) → H_{∗}(P_{n−1}(B)) are isomorphisms over a greater range and so the tower of homology groups is Mittag-Leffler. Therefore lim_{n} H_{∗}(P_{n}(B)) = 0, so lim_{n} H_{∗}(P_{n}(B)) ≅ H_{∗}(P (B)). Though convergence is transparent, one still has to compare it to Eilenberg and Moore’s original approach to prove that it abuts to H_{∗}(ΩX).

For any generalized homology theory we can use the tower (1.1) to define a spectral sequence.

Definition 1.8. The h_{∗}-based Eilenberg–Moore spectral sequence is the spectral sequence that results from applying h_{∗}(−) to Rector’s tower of cofibrations, (1.1). Furthermore, if h_{∗}(−) is a multiplicative homology theory with a perfect Künneth theorem for B, then

E^{2} = Cotor^{h_{∗}(B)}(h_{∗}, h_{∗}).

Clearly this process will give rise to a spectral sequence defined by the exact couple h_{q}(P_{l}(B)) = D_{l,q} and h_{q}(Σ^{−l}B^{∧l}) = E_{l,q}. This defines a right half-plane spectral sequence with exiting differentials. Since colim D_{l,∗} = 0 if the spectral sequence converges, it converges strongly to lim D_{l,∗} [4, 6.1b].

1.3. K(m)_{∗}-based Eilenberg–Moore spectral sequence. We review the properties of Morava K-theory, K(m)_{∗}, which are used in this paper. Fix a prime p. For each m there is a spectrum K(m) with

π_{∗}(K(0)) = K(0)_{∗} = ℚ

concentrated in degree 0, and

π_{∗}(K(m)) = K(m)_{∗} = ℤ/p[v_{m}, v_{m}^{−1}]

with |v_{m}| = 2(p^{m} − 1) for m > 0.

In addition K(m)_{∗}(−) has a perfect Künneth theorem; that is

K(m)_{∗}(X × Y) = K(m)_{∗}(X) ⊗ K(m)_{∗}(Y).

The spectrum K(m) is derived from complex cobordism and is characterized by the formal group of height m. For a definition and more complete discussion of the properties of K(m)_{∗} see [34].

Lemma 1.9. If the K(m)_{∗}-based Eilenberg–Moore spectral sequence converges for a CW spectrum B, it converges strongly to lim K(m)_{q}(P_{l}(B)).

Because the Morava K-theories have a perfect Künneth theorem

E^{2} = Cotor^{K(m)_{∗}(B)}(K(m)_{∗}, K(m)_{∗}),

which in some cases will be amenable to calculation.

We would like this spectral sequence to converge to K(m)_{∗}(ΩB) but it is unclear when it does so. The main subject of this paper is to establish some criteria which guarantee convergence to K(m)_{∗}(ΩB).

Acknowledgements. I would like to thank my advisor Hal Sadofsky for all of his help and Peter May who looked at and commented on an early version of this paper.

2. Analysis of the spectral sequence

Our goal is to give convergence conditions for the K(m)_{∗}-based Eilenberg– Moore spectral sequence. One major tool I will use for the analysis of the Eilenberg–Moore spectral sequence is the Atiyah–Hirzebruch spectral sequence. The Atiyah–Hirzebruch spectral sequence is a spectral sequence used to calculate generalized homology theories in terms of ordinary homology. For a CW complex X, and a generalized homology theory h_{∗}(−), this spectral sequence has E_{p,q}^{2} = H_{p}(X, h_{q}(∗)) ⇒ h_{p+q}(X). Similarly if X is a spectrum and h_{∗}(−) a generalized homology theory, the Atiyah–Hirzebruch spectral sequence has E_{p,q}^{2} = H_{p}(X, h_{q}(∗)) ⇒ h_{p+q}(X). See, for example, [28, Chapter 15].

The Eilenberg–Moore spectral sequence for K(m)_{∗} comes from applying K(m)_{∗}(−) to a tower of cofibrations of spectra (from (1.1)). To analyze this spectral sequence we consider the Atiyah–Hirzebruch spectral sequence for each P_{n}(B) with

E_{p,q}^{2} = H_{p}(P_{n}(B), K(m)_{q}) ⇒ K(m)_{∗}(P_{n}(B)).

This yields a tower of spectral sequences. We will use this tower to understand K(m)_{∗}(ΩB). To get information from this tower of spectral sequences there are several things we would like to know.

(1) Let E be the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P (B)), and E(n) the Atiyah–Hirzebruch sequence for K(m)_{∗}(P_{n}(B)). Is the spectral sequence E equal to lim E(n), the inverse limit of the spectral sequences E(n)?

(2) If the answer to question one is yes, what relationship holds between lim(K(m)_{∗}(P_{n}(B))) and K(m)_{∗}(P (B))?

(3) If we understand the answers to questions one and two, what does this say about the convergence of the K(m)_{∗}-based Eilenberg–Moore spectral sequence for ΩB?

3. Inverse limits of spectral sequences

In this section we will show that the inverse limit (over n) of the Atiyah– Hirzebruch spectral sequences for K(m)_{∗}(P_{n}(B)) is the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P (B)). First I will describe more specifically what I mean by the inverse limit of the Atiyah–Hirzebruch spectral sequences for K(m)_{∗}(P_{n}(B)).

Let {E(n)} be a sequence of spectral sequences with maps of spectral sequences E(n) → E(n − 1). We form the tri-graded abelian groups E_{p,q}^{r} = lim E_{p,q}^{r}(P_{n}(B)), and define differentials by taking the inverse limit of the differentials in the E(n). That is if d^{r}(n) is the r^{th} differential in E(n) and (x^{n}) ∈ E_{p,q}^{r} then d^{r}((x^{n})) = (d^{r}(n)(x^{n})). The resulting object is a spectral sequence provided that H(E_{p,q}^{r}, d^{r}) = E_{p,q}^{r+1}. This is equivalent to showing that H(lim_{n} E(n)_{p,q}^{r}, d^{r}) = lim_{n} H(E(n)_{p,q}^{r}, d^{r}). To see when H(lim_{n} E(n)_{p,q}^{r}, d^{r}) = lim_{n} H(E(n)_{p,q}^{r}, d^{r}), notice that (E_{p,q}^{r}, d^{r}) is a chain complex. Thus we can apply the following theorem due to Milnor. See [32, 3.5.8] for details.

Theorem 3.1. Let ⋯ C_{2} → C_{1} → C_{0} be a tower of chain complexes of abelian groups satisfying the Mittag-Leffler condition and let C = lim_{i} C_{i}. Then there is an exact sequence for each q:

0 → lim^{1} H_{q+1}(C_{i}) → H_{q}(C) → lim H_{q}(C_{i}) → 0.

So if lim^{1} H_{q+1}(C_{i}) = 0, the inverse limit of spectral sequences forms a spectral sequence. If, for example, the E_{p,q}^{r} are all finite-dimensional vector spaces, then the hypothesis of Theorem 3.1. will hold for all (E_{p,q}^{r}, d^{r}) and all lim^{1} terms of the homology will be 0.

Let E(n) be the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P_{n}(B)). It is not clear a priori that the inverse limit of Atiyah–Hirzebruch spectral sequence is the Atiyah–Hirzebruch spectral sequence of the inverse limit.

Lemma 3.2. Let B be a 1-connected CW complex with finitely many cells in each dimension. Take E(n) to be the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P_{n}(B)) Then:

(1) lim E(n) is a spectral sequence.

(2) The canonical map from the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(ΩB) to lim E(n) is an isomorphism.

Proof. The idea is to use the fact that the Eilenberg–Moore spectral sequence converges for ordinary homology to identify the E^{2} terms of these spectral sequences. First, since B has only finitely many cells in each dimension, the E_{p,q}^{r} are all finite-dimensional vector spaces, and lim E(n) is a spectral sequence. By construction we have a map from P (B) to each of the P_{n}(B) that is compatible with the maps in the tower. This gives us a map from the Atiyah–Hirzebruch spectral sequence for P (B) to the Atiyah– Hirzebruch spectral sequence for each P_{n}(B) that is compatible with the maps in the tower. By the universal property of the inverse limit we get a map Φ from the Atiyah–Hirzebruch spectral sequence for P (B) to the inverse limit of the Atiyah–Hirzebruch spectral sequences for the P_{n}(B).

Since B is 1-connected, the Eilenberg–Moore spectral sequence for ΩB converges. This implies that lim H_{∗}(P_{n}(B)) = H_{∗}(P (B)). Recall that E(n)_{p,q}^{2} = H_{p}(P_{n}(B), K(m)_{q}) and that P (B) = Σ^{∞}ΩB. So

lim E(n)_{p,q}^{2} = lim H_{p}(P_{n}(B), K(m)_{q})
= H_{p}(P (B), K(m)_{q})
= H_{p}(ΩB, K(m)_{q}).

Since the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(ΩB) has

E_{p,q}^{2} = H_{p}(ΩB, K(m)_{q})

this implies that Φ is an isomorphism at the E^{2} page. The isomorphism follows from the standard comparison theorem for spectral sequences. See [4, 5.3] for a precise statement. □

4. The E^{∞} term of the inverse limit spectral sequence

We would like to prove that lim K(m)_{∗}(P_{n}(B)) = K(m)_{∗}(P (B)). This would give strong convergence on the K(m)_{∗}-based Eilenberg–Moore spectral sequence. However lim K(m)_{∗}(P_{n}(B)) ≠ K(m)_{∗}(P (B)) even under some very restrictive conditions. So instead we will show that

lim G(K(m)_{∗}(P_{n}(B))) = G(K(m)_{∗}(P (B))).

First we must show that lim(E(n)^{∞}) is the same as (lim E(n))^{∞}. Recall that for each P_{n}(B) there is an Atiyah–Hirzebruch spectral sequence that converges strongly to K(m)_{∗}(P_{n}(B)). In other words E(n)^{∞} is the associated graded to K(m)_{∗}(P_{n}(B)). It seems natural to guess that the E^{∞} term of a spectral sequence defined as the inverse limit of spectral sequences would have as its E^{∞} term the inverse limit of the E^{∞} terms. We shall see that this is not necessarily the case. First we must analyze lim(E(n)^{∞}) and (lim E(n))^{∞} and determine how they are related.

Proposition 4.1. Let B be a CW complex with finitely many cells in each dimension. The canonical map Φ : (lim E(n))^{∞} → lim (E(n)^{∞}) is onto.

This map is not one-to-one in general (see below).

Proof. Let (x_{n}) be a class in lim (E(n)^{∞}) with x_{n} ∈ E(n)^{∞}. Define M_{n} ∈ E(n)^{2} as M_{n} = {x : x = x_{n} in E(n)^{∞}}. Since B has only finitely many cells in each dimension, this is a finite-dimensional vector space. We now have a tower

⋮

M_{k+2}
M_{k+1}
M_{k}.

Since this is a tower of finite-dimensional vector spaces, lim M_{n} ≠ 0. This gives an element (x_{n}) ∈ (lim E(n))_{2} = (lim E(n)_{2}) such that

Φ(\overline{(x_{n})}) = (x_{n}).  □

We now show that Φ : (lim E(n))^{∞} → lim (E(n)^{∞}) may not be one-to-one. Let x_{n} ∈ E^{2}(n) with x_{n} ↦ x_{n−1}. Suppose that x_{n} ∈ im(d^{r_{n}}) and lim_{n→∞} r_{n} = ∞. Then (x_{n}) is not a boundary in lim (E(n)) so (x_{n}) ≠ 0 in (lim E(n))^{∞} even though x_{n} = 0 in E(n)^{∞} for each n (see Figure 1). Thus \overline{(x_{n})} ≠ 0 in (lim E(n))^{∞} but Φ(x_{n}) = (x_{n}) = 0.

(4.1) [figure: •; •; •; •; •; (x_{n}); •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; q; •; •; •; •; •; •; •; •]

Figure 1. This diagram is meant to represent the superimposed pages of the Atiyah–Hirzebruch spectral sequence for three of the P_{n}(B). The arrows shown are the differential hitting x_{n} in P_{n}(B). This illustrates that as n increases so may the length of the differential hitting x_{n}. If this trend holds for all n then in the inverse limit spectral sequence there will be no differential hitting (x_{n}). This phenomenon is central in both our convergence results and nonconvergence examples.

5. The E^{∞} term of the Eilenberg–Moore spectral sequence for Morava K-theory

Recall the K(m)_{∗}-based Eilenberg–Moore spectral sequence results from applying K(m)_{∗}(−) to Rector’s tower of cofibrations (1.1). Recall also that if the spectral sequence converges, it has E^{∞} term [4, 6.1b]

G(lim D^{l,∗}) = G(lim K(m)_{∗}(P_{n}(B))).

Under certain conditions we will show that G(lim D^{l,∗}) is an associated graded of K(m)_{∗}(P (B)) with an exhaustive Hausdorff filtration.

Lemma 5.1. If B has only finitely many cells in each dimension and

lim(E(n)^{∞}) = (lim E(n))^{∞}

then:

(1) The canonical map Φ : K(m)_{∗}(P (B)) → lim K(m)_{∗}(P_{n}(B)) induces a morphism of filtered groups which is an isomorphism of their respective associated graded groups. That is

G(lim K(m)_{∗}(P_{n}(B))) ≅ G(K(m)_{∗}(P (B))).

The filtration on K(m)_{∗}(P (B)) is exhaustive and Hausdorff.

(2) The canonical map K(m)_{∗}(P (B)) → lim K(m)_{∗}(P_{n}(B)) is one-to-one.

Proof. First recall the E^{∞} term of the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P_{n}(B)) is G(K(m)_{∗}(P_{n}(B))). The filtration for lim(E(n)^{∞}) is the inverse limit of the filtrations for E(n)^{∞}, that is

F_{k} = lim_{n} (Im (K(m)_{∗}((P_{n}(B))^{(k)}) ⟶^{i} K(m)_{∗}(P_{n}(B)))) .

This is an increasing filtration on lim_{n} K(m)_{∗}(P_{n}(B)). In Lemma 3.2 we proved that for B a simply connected CW complex with finitely many cells in each dimension, the inverse limit spectral sequence is isomorphic to the Atiyah–Hirzebruch spectral sequence for P (B). The latter spectral sequence has E^{∞} term G(K(m)_{∗}(P (B))), so we have a homomorphism of filtered groups that is an isomorphism at the level of associated graded groups. Finally the filtration on G(K(m)_{∗}(P (B))) is the filtration from the Atiyah–Hirzebruch spectral sequence for P_{n}(B) which is both exhaustive and Hausdorff [4, 12.6]. □

Corollary 5.2. If B has only finitely many cells in each dimension and

lim(E(n)^{∞}) = (lim E(n))^{∞},

then the K(m)_{∗}-based Eilenberg–Moore spectral sequence for B converges.

Proof. Recall that convergence of a spectral sequence to a group G requires that [4, Definition 5.2]:

(1) The filtration exhausts G and there are isomorphisms E_{s}^{∞} ≅ F_{s}/F_{s+1} for all s. (This alone is weak convergence.)

(2) The filtration of G is Hausdorff. Let F_{n} be the filtration for the K(m)_{∗}-based Eilenberg–Moore spectral sequence. This spectral sequence formally converges to lim K(m)_{∗}(P_{n}(X)) and has a canonical filtration

F_{n} = Ker (K(m)_{∗}(lim P_{n}(X)) → K(m)_{∗}(P_{n}(X)) .

F_{n} is a complete, exhaustive filtration for (lim K(m)_{∗}(P_{n}(X)) [4, 5.4]. The map Φ induces a filtration, F′, on K(m)_{∗}(ΩB) with

F′_{n} = F_{n} ∩ K(m)_{∗}(ΩB).

Since Φ is one-to-one, ∩_{n}F′_{n} = ∩_{n}F_{n} = 0, so F′_{n} is a Hausdorff filtration. Also K(m)_{∗}(ΩB) = F′_{−1} so F′_{n} is exhaustive as well. Finally to see that F_{n}/F_{n+1} = F′_{n}/F′_{n+1} note that we have a commutative diagram,

(5.1)  \xymatrix{K(m)_{∗}(ΩB) \ar[r] \ar[d]^{Φ} & K(m)_{∗}(P_{k}B) \ar[d]^{≅} \\ lim_{n} K(m)_{∗}(P_{n}B) \ar[r]^{i_{k}} & K(m)_{∗}(P_{k}B)}

where Φ is one-to-one. And thus, in Boardman’s sense, the spectral sequence converges to K(m)_{∗}(P (X)). □

6. Results

6.1. Main theorem.

Theorem 6.1. Let B be a 1-connected space defined by a CW complex with finitely many cells in each degree such that the Eilenberg–Moore spectral sequence for ordinary homology collapses at the E^{2} page and

Rank(⨁_{i} H_{j−i}(ΩB, K(m)_{i}))

is not infinite for consecutive values of j. Then the Morava K-theory Eilenberg–Moore spectral sequence for B converges to K(m)_{∗}(ΩB).

Recall that ⨁_{i} H_{j−i}(ΩB, K(m)_{i}) consists of all elements of total degree j on the E^{2} page of the Atiyah–Hirzebruch spectral sequence for ΩB.

Proof. We consider the unraveled exact couple that forms the Eilenberg– Moore spectral sequence for ordinary homology. (6.1)

\xymatrix{ & ⋮ \ar[d]_{i} \\ H_{∗}(Σ^{−n}B^{∧n}) \ar[r]_{j_{n}} & H_{∗}(P_{n}(B)) \ar[r]_{k_{n}} \ar[d]_{i_{n}} & H_{∗}(Σ^{−n}B^{∧n+1}) \\ H_{∗}(Σ^{−n+1}B^{∧n−1}) \ar[r]_{j_{n−1}} & H_{∗}(P_{n−1}(B)) \ar[r]_{k_{n−1}} \ar[d]_{i_{n−1}} & H_{∗}(Σ^{−n+1}B^{∧n}) \\ H_{∗}(Σ^{−n+2}B^{∧n−2}) \ar[r]_{j_{n−2}} & H_{∗}(P_{n−2}(B)) \ar[r]_{k_{n−2}} \ar[d]_{i_{n−2}} & H_{∗}(Σ^{−n+2}B^{∧n−1}) \\  & ⋮}

Recall that d^{1} = k ∘ j and d^{n} is constructed by applying j, lifting through i_{n−1}, and applying k.

Represent a permanent cycle in the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(ΩB) by (x_{n}) where x_{n} ∈ E(n)^{2}. We are using Lemma 3.2 which identifies the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(ΩB) with the inverse limit of Atiyah–Hirzebruch spectral sequences for K(m)_{∗}(P_{n}B).

We wish to construct our proof by showing that under our hypothesis, the hypotheses of Corollary 5.2 hold. We need to show that under the above assumptions if for all n, x_{n} = 0 ∈ E(n)^{∞} then (x_{n}) = 0 ∈ (lim E(n))^{∞}. By abuse x_{n} represents both an element of the E^{2} page which is a permanent cycle, and its class on the E^{∞} page. In short we need to rule out something like the situation described by Diagram (4.1). The key fact here is that, by assumption, the only nontrivial differential in the Eilenberg–Moore spectral sequence is d^{1}.

Suppose d^{i_{n}}(y_{n}) = x_{n} in the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P_{n}B) where the i_{n} are increasing monotonically. If we can show that lim_{n→∞} i_{n} is finite, then we will have shown that (x_{n}) = 0 in (lim E(n))^{∞}. We investigate what circumstances can lead to i_{n+1} > i_{n}. Since we are really worried about i_{n+1} > i_{n}, for infinitely many n, we may begin by assuming n is large. In particular, we take advantage of the fact that our hypotheses guarantee that the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P_{n}B) stabilizes in any fixed dimension for n sufficiently large. So with out loss of generality, assume n is large enough so that H_{∗}(P_{n+1}) → H_{∗}(P_{n}) is an isomorphism in the dimension of x_{n}.

There are two possible ways to have i_{n+1} > i_{n}: either y_{n} does not lift to y_{n+1}, and x_{n+1} is in the image of a longer differential, or there is a lift of y_{n}, say ỹ_{n}, but ỹ_{n} supports a shorter differential than d^{i_{n}}. These two situations are summarized graphically in Diagram (6.2) and Diagram (6.3) in Figures 2 and 3, respectively.

The situation described under (6.2) does not happen under our hypotheses. To see why, assume y_{n} ∈ E(n)^{2} does not lift to a y_{n+1} ∈ E(n + 1)^{2}. This implies k_{n}(y_{n}) = b ≠ 0 ∈ H_{∗}(Σ^{−n}B^{∧n+1}). Let y_{n−s} be the image of y_{n} in H_{∗}(P (n − s)). Choose s so that y_{n−s} ≠ 0 in H_{∗}(P (n − s)) but y_{n−s−1} = 0 in H_{∗}(P (n − s − 1)). Then y_{n−s} = j_{n−s}(α). This shows d^{s+1}(α) = b in the Eilenberg–Moore spectral sequence for ΩB. But since d^{1} is the only nontrivial differential in that Eilenberg–Moore spectral sequence, d^{s+1} = 0 if s > 0. So b must be 0 on the E^{s+1} page and so it must be 0 on the E^{2} page of the Eilenberg–Moore spectral sequence. Choose a so that d^{1}(a) = b, and let y′_{n} = j_{n}(a). Let λ_{n} = y_{n} − y′_{n}, and notice that k(λ_{n}) = 0 so λ_{n} lifts to λ_{n+1} ∈ P (n + 1) with i(λ_{n+1}) = λ_{n}. Finally

d^{i_{n}}(λ_{n}) = d^{i_{n}}(y_{n} − y′_{n}) = d^{i_{n}}(y_{n}) − d^{i_{n}}(j_{n}(a)) = d^{i_{n}}(y_{n}) = x_{n},

and so d^{i_{n}}(λ_{n+1}) = x_{n+1} Here we are using the assumption that n is large enough so that x_{n+1} is determined by x_{n}. So the situation described in Diagram (6.2) cannot happen.

(6.2)  [figure: x_{n+1}; i; q; y_{n+1}; d^{i_{n+1}}; x_{n}; y_{n}; d^{in}; q]

Figure 2. This is an illustration of elements on the E^{2} pages of the Atiyah–Hirzebruch spectral sequences for K(m)_{∗}(P_{n+1}(B)) and K(m)_{∗}(P_{n}(B)) with higher differentials superimposed. In this diagram y_{n} does not lift and y_{n}, x_{n} ∈ E(n)^{2} while x_{n+1}, y_{n+1} ∈ E(n + 1)^{2}.

The situation described in Diagram (6.3) can happen. For (x_{n}) to be nonzero in (lim E(n))^{∞} it would have to happen for infinitely many n. Suppose this is the case: let (x_{n}) ∈ (lim E(n))^{2} such that x_{n} = 0 ∈ E(n)^{∞} for all n with (x_{n}) ≠ 0 ∈ (lim E(n))^{∞}. Also let x_{n} ∈ H_{j−i}(P_{n}(B), K(m)_{i}). Recall that in the Atiyah–Hirzebruch spectral sequence for homology the differential lowers total degree by one, so that all elements that can support a differential with image x_{n} are in ⨁_{p+q=j+1} H_{p}(P_{n}(B), K(m)_{q}). In the above proof for the impossibility of the scenario in Diagram (6.2) we show that if an element y_{n} on the E^{2} page of the Atiyah–Hirzebruch spectral sequence for P_{n}(B) supports a differential, there is an element on the E^{2} page of the Atiyah–Hirzebruch spectral sequence for P_{n+1}(B) that supports a differential with image x_{n+1}. As a consequence each y_{n} ∈ E(n) gives rise to an element (y_{n}) ∈ (lim E(n))^{2}. This means that under the assumption that the situation described in Diagram (6.3) happens an infinite number of times the rank of ⨁_{p+q=j+1} H_{p}(P (B), K(m)_{q}) is infinite. Furthermore since each y_{n} supports a differential, and the i_{n} are increasingly connective,

(6.3)  [figure: x_{n+1}; z_{n+1}; ỹ_{n}; d^{a}; i; q; y_{n+1}; d^{i_{n+1}}; x_{n}; y_{n}; d^{in}; q]

Figure 3. This is an illustration similar to Diagram 6.2, but in this case y_{n} does lift but y_{n+1} is 0 on the E(n)^{i_{n}} page of the Atiyah–Hirzebruch spectral sequence for K(m)_{∗}(P_{n+1}B). In this diagram y_{n}, x_{n} ∈ E(n)^{2} while x_{n+1}, y_{n+1}, ỹ_{n}, z_{n+1} ∈ E(n + 1)^{2}.

(y_{n}) supports a differential in lim (E(n)). One consequence of this is that for each y_{n} ∈ (lim E(n))^{2} there is an x_{n} ∈ (lim E(n))^{2}. Since the differential lowers dimension by one, if y_{n} is in total degree j + 1, x_{n} is in total degree j. As a result the rank of ⨁_{p+q=j} H_{p}(P (B), K(m)_{q}) would also be infinite. This contradicts the assumptions of Theorem 6.1 so the situation described in Diagram (6.3) can happen at most a finite number of times.

We have shown that under the assumptions of Theorem 6.1,

G(lim D^{l,∗}) = G(lim K(m)_{∗}(P_{n}(B)))

so by Lemma 5.1 the K(m)_{∗}-based Eilenberg–Moore spectral sequence converges. □

6.2. Secondary result.

Theorem 6.2. Let B be a space defined by a finite CW complex such that the Eilenberg–Moore spectral sequence for ordinary homology collapses at the E^{2} page and the Atiyah–Hirzebruch spectral sequence for Morava K-theory for P_{n}(B) collapses at the E_{k} page for all n and some fixed k. Then the Morava K-theory Eilenberg–Moore spectral sequence for B converges to K(m)_{∗}(ΩB).

We use the following lemma. (The proof is not given but hopefully obvious.)

Lemma 6.3. For any space B, if the K(m)_{∗} Atiyah–Hirzebruch spectral sequence for B collapses then the Atiyah–Hirzebruch spectral sequence for B^{∧n} also collapses.

Proof. As we have seen, the K(m)_{∗}-based Eilenberg–Moore spectral sequence converges under any conditions which rule out the scenarios described by Diagrams (6.2) and (6.3) occurring an infinite number of times. Since the Eilenberg–Moore spectral sequence for ordinary homology collapses at E^{2}, We know that Diagram (6.2) cannot occur by our previous argument. It suffices to prove that the situation described by Diagram (6.3) can occur at most a finite number of times, but this is clear since we know that the longest differential in the Atiyah–Hirzebruch spectral sequence for any P_{n}(B) is of length k − 1. So we have shown that under the assumptions of Theorem 6.1,

G(lim D^{l,∗}) = G(lim K(m)_{∗}(P_{n}(B)))

so by Lemma 5.1 the K(m)_{∗}-based Eilenberg–Moore spectral sequence converges. □

Remark 6.4. In practice it would be very hard to verify that the Atiyah– Hirzebruch spectral sequence for all P_{n}(B) collapses at E^{k} for a given k. As we will show in a sequel to this paper even if the Atiyah–Hirzebruch spectral sequence for B has only 1 differential the Atiyah–Hirzebruch spectral sequence for P_{n}(B) can have many nontrivial differentials.

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Franklin and Marshall College, Department of Mathematics, P.O. Box 3003, Lancaster, PA 17604-3003, USA jcarter@fandm.edu

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