Homology, Homotopy and Applications, vol.6(1), 2004, pp.137–152

REAL COBORDISM AND GREEK LETTER ELEMENTS IN THE GEOMETRIC CHROMATIC SPECTRAL SEQUENCE

PO HU and IGOR KRIZ

(communicated by Gunnar Carlsson)

Abstract In this paper, we give a basic application of our Adams-Novikov spectral sequence analogue based on Real cobordism. Concretely, using that technique, we prove restrictions on Greek letter elements which can be permanent cycles or targets of differentials in the geometric chromatic spectral sequence.

1. Introduction

In a previous paper [5], the authors investigated extensively the Landweber-Araki Real cobordism spectrum Mℝ ([1, 7]), and its stable summand at p = 2 called BP ℝ. An Adams-like spectral sequence based on BP ℝ was constructed, and homotopy groups of many spectra related to BP ℝ were calculated.

The goal of the present note is to begin investigating the question as to how BP ℝ-theory contributes to the known information about stable homotopy groups of spheres. This question is not easy. The dimension to which stable stems have been calculated to date is, (if somewhat hazy), certainly high enough to make calculating past that point from scratch a substantial challenge for any new method. At this point, the authors did not get far enough in calculating with BP ℝ to get any new information that way. On the other hand, any effort to compare the BP ℝ-based spectral sequence with other known spectral sequences is fraught with the usual difficulty: elements can get renamed.

Nevertheless, there is one basic case when a rigorous comparison can be made, namely on the edge of a spectral sequence. To illustrate this, assume we have a series of cofibrations

X_{n−1} → Y_{n−1} → X_{n}  (1.1)

where X_{−1} = S^{0}. Then we have a spectral sequence

E_{1} = π_{∗}Y_{n−1} ⇒ π_{∗}S^{0}.  (1.2)

An element ω ∈ π_{∗}Y_{n−1} is a permanent cycle if it lifts to π_{∗}X_{n−1}. Now however it may not even be easy to know π_{∗}Y_{n−1} explicitly, and suppose we use the Adams-

The first author was an NSF postdoctoral research fellow, both authors are currently supported by the NSF Received August 5, 2002, revised December 18, 2003; published on April 9, 2004. 2000 Mathematics Subject Classification: 55Q10, 55Q45, 55T15, 55T25. Key words and phrases: homotopy theory, homotopy groups, Adams spectral sequence, generalized cohomology. \begin{matrix}c \\ ◯ \end{matrix} 2004, Po Hu and Igor Kriz. Permission to copy for private use granted.

Novikov spectral sequence to calculate it. Suppose further that in this spectral sequence, ω has filtration degree 0, i.e. has non-trivial Hurewicz homomorphism image ω_{0} ∈ BP_{∗}Y_{n}. Suppose further that in BP -homology, (1.1) induce short exact sequences, so we cannot use BP to obtain a necessary condition for ω to lift. But now we have a factorization of the unit 1 ∈ π_{∗}BP of the form

\xymatrix{S^{0} \ar[r] & BP ℝ \ar[r]^{κ} & BP}

(actually, BP ℝ is a ℤ/2-equivariant spectrum, so we should really write BP ℝ^{ℤ/2}, but that is a detail for now). Suppose we can completely calculate the map

BP ℝ_{∗}X_{n} → BP ℝ_{∗}Y_{n}  (1.3)

and suppose we know that no element ω_{1} ∈ BP ℝ_{∗}Y_{n} with κ_{∗}ω_{1} = ω_{0} is in the image of (1.3). Then ω cannot be a permanent cycle in the spectral sequence (1.2).

On the other hand, suppose we know that there is a lift ω_{2} ∈ BP ℝ_{∗}X_{n}, and that further any such lift (of any choice of ω_{1}) has the property that

δ^{∗}ω_{2} ≠ 0

where δ^{∗} : BP ℝ_{∗}X_{n−1} → Σ^{n}BP ℝ_{∗}S^{0} is the connecting map. Then the in particular the same must hold for any ω_{2} which lifts to stable homotopy, and we conclude that ω cannot be a target of a differential in (1.2).

In this note, we shall apply this simple method to the geometric chromatic spectral sequence, where we shall see that strikingly, it does give new information on possible permanent cycles in all families of Greek-letter elements, thus generalizing, and in fact even somewhat explaining, the numerology of the known permanent cycles in α, β, γ. This signals that however difficult to extract it may be, BP ℝ does contain new useful information about the stable stems. Throughout this note, we work locally at the prime 2, as BP ℝ is just BP at odd primes. Nevertheless, algebraic cobordism (cf. [6]) gives tantalizing hints of some possible analogues of the present method at odd primes. Such extension, however, at present is unknown, as is the equally interesting question of possible connections of BP ℝ with the root invariant.

To state our results, we must recall the chromatic spectral sequence of Miller-Ravenel-Wilson [8]:

E_{1} = ⊕_{n} Ext_{BP_{∗}BP} (BP_{∗}, v_{n}^{−1}BP_{∗}/(v_{0}^{∞}, ..., v_{n−1}^{∞})) ⇒ Ext_{BP_{∗}BP} (BP_{∗}, BP_{∗}).

(1.4)

In the E_{1}-term (1.4), an element of Ext_{BP_{∗}BP}^{0} (BP_{∗}, v_{n}^{−1}BP_{∗}/(v_{0}, ..., v_{n−1})) represented by

\frac{v_{n}^{k}}{v_{n−1}^{i_{n−1}}v_{n−2}^{i_{n−2}} · ... · v_{0}^{i_{0}}} mod (v_{0}, ..., v_{n−1})  (1.5)

is denoted by

α_{k/i_{n−1},i_{n−2},...,i_{0}}^{(n)}  (1.6)

where α^{(n)} is the n-th Greek letter. If the numbers i_{0}, ...i_{ℓ−1} are equal to 1, they are omitted.

In [9], Ravenel constructed BP -local spectra Y_{n} such that

BP_{∗}Y_{n} = v_{n}^{−1}BP_{∗}/(v_{0}^{∞}, ..., v_{n−1}^{∞}),

and a geometric chromatic spectral sequence

E_{1} = ⊕_{n} π_{∗}Y_{n} ⇒ π_{∗}(S^{0})_{2}^{∧}.  (1.7)

At this point, relatively little is known about the spectral sequence (1.7). However, call the element (1.5) a geometric Greek letter element if it is the image of an element of π_{∗}Y_{n} via the Hurewicz map π_{∗}Y_{n} → BP_{∗}Y_{n}, i.e. an element of the E_{1}-term of the geometric chromatic spectral sequence (1.7). In that case, we also use the notation (1.6) for that element. Then the main results of this paper are:

Theorem 1.8. If α_{k/i_{n−1},i_{n−2},...,i_{ℓ}}^{(n)} is a geometric Greek letter element which is a permanent cycle in the geometric chromatic spectral sequence, then there exists a j, 0 ⩽ j ⩽ ℓ (we can have ℓ = n) such that

k(2^{n} − 1) ≡ ∑_{m=0}^{n−1} i_{m}(2^{m} − 1) + 2^{j} − 1 mod 2^{j+1}.

Theorem 1.9. Let

k_{n} = n + 2 if n is even 2^{n} + n + 2 if n is odd.

Then a geometric Greek letter element α_{k}^{(n)} is not a target of a differential in the geometric chromatic spectral sequence if k ≡ k_{n} mod 2^{n+1}, and supports a differential in the geometric chromatic spectral sequence if k ≡ k_{n} + 2^{n} mod 2^{n+1}.

For n = ℓ = 1, Theorem 1.8 says that if α_{k} is a permanent cycle then k is not congruent to 3 mod 4, which is well known.

For n = ℓ = 2, Theorems 1.8, 1.9 imply that β_{k} is not a permanent cycle when k is divisible by 8. For n = 2 and ℓ = 1, Theorem 1.8 says that if β_{k/i_{1}} is a permanent cycle, then k + i_{1} is not congruent to 1 mod 4. Doug Ravenel points out that the first examples of these elements are β_{8} and β_{8/5}, and conjectures that we have d_{3}(β_{8}) = η^{3}β_{8/3} and d_{3}(β_{8/5}) = η^{3}η_{5} in the Adams-Novikov spectral sequence.

For n = ℓ = 3, Theorem 1.8 implies that if γ_{k} is a permanent cycle, then k is not congruent to 5 mod 16. The first element excluded is γ_{5}, in dimension 59.

Theorems 1.8, 1.9 should not be thought of as results directly about the stable 2-stems (π_{∗}S^{0})_{(2)}, since they do not give any sufficient condition when the said elements are permanent cycles in the geometric chromatic spectral sequence. However, the nilpotence and periodicity theorems of Devinatz, Hopkins and Smith [2], [3] give a general existence theorem for geometric Greek letter elements as elements of stable 2-stems. There was, as far as we know, no detection theorem for such elements beyond the γ family, and this was a well known problem (although it is usually phrased in terms of the Adams-Novikov spectral sequence). The present paper does contribute to answering this question. The difficulty is that the existence theorems [2],[3] are phrased globally, and as far as we know, concrete names of elements which these theorems construct have not been worked out. Therefore we still do not know what, if any, is the intersection between the existence and uniqueness theorems.

The present paper is organized as follows: Preliminaries, including the construction of the spectra Y_{n}, are recalled in Section 2. The necessary BP ℝ-homology calculations are presented in Section 3, and proofs of Theorems 1.8, 1.9 following the method we outlined are deduced. The BP ℝ-calculations are proved in Section 4.

2. Preliminaries

First we shall recall the construction of the spectra Y_{n} in (1.7) ([9]). We refer the reader to [3] for the definition of v_{n−1}-spectra and v_{n}-self maps.

Lemma 2.1. Let f : V → V′ be any map of v_{n−1}-spectra (see [3]). Let v : Σ^{k}V → V, v′ : Σ^{k}V′ → V′ be v_{n}-self maps. Then there exists an N such that the following diagram commutes up to homotopy:

\xymatrix{Σ^{Nk}V \ar[r]^{f} \ar[d]_{v^{N}} & Σ^{Nk}V′ \ar[d]^{v^{′N}} \\ V \ar[r]^{f} & V′.}  (2.2)

Proof: Consider the two maps

Dv ∧ 1, 1 ∧ v′ : Σ^{k}DV ∧ V′ → DV ∧ V′.

They are both v_{n}-self maps, so by Hopkins-Smith [3], there is an N such that

(Dv ∧ 1)^{N} ≃ (1 ∧ v′)^{N}.

Therefore, considering f as an element of π_{∗}(DV ∧ V′), we have

(Dv ∧ 1)_{∗}^{N}(f) = (1 ∧ v′)_{∗}^{N}(f).  (2.3)

But the two sides of (2.3), considered as maps Σ^{Nk}V → V′, are the two ways around the diagram (2.2). □

Now Ravenel’s construction is essentially as follows: One constructs a spectrum X′_{n−1} as a telescope of v_{n−1}-spectra

\xymatrix{V_{n−1,1} \ar[r]^{f_{n−1,1}} & V_{n−1,2} \ar[r]^{f_{n−1,2}} & V_{n−1,3} \ar[r]^{f_{n−1,3}} & …}  (2.4)

For n = 0, let X′_{−1} = S^{0} = V_{−1,i} for all i, where the maps f_{−1,i} are equal to the identity. Provided (2.4) is constructed, we next construct a spectrum Y′_{n−1} and a cofibration sequence of the form

X′_{n−1} → Y′_{n−1} → X′_{n}.  (2.5)

Concretely, by Lemma 2.1, we can inductively find v_{n}-self maps

v_{n,i} : Σ^{k_{n−1,i}}V_{n−1,i} → V_{n−1,i}

and numbers N_{n−1,i} > 1 such that

k_{n−1,i+1} = N_{n−1,i}k_{n−1,i}

and the following diagrams commute:

\xymatrix{Σ^{k_{n−1,i+1}}V_{n−1,i} \ar[r]^{f_{n−1,i}} \ar[d]_{(v_{n,i})^{N_{n−1,i}}} & Σ^{k_{n−1,i+1}}V_{n−1,i+1} \ar[d]^{v_{n,i+1}} \\ V_{n−1,i} \ar[r]^{f_{n−1,i}} & V_{n−1,i+1}.}

Then consider the diagram

\xymatrix{V_{n−1,1} \ar[r]^{f_{n−1,1}} \ar[d]_{v_{n,1}} & V_{n−1,2} \ar[r]^{f_{n−1,2}} \ar[d]_{v_{n,2}} & V_{n−1,3} \ar[r]^{f_{n−1,3}} \ar[d]_{v_{n,3}} & … \\ Σ^{−k_{n−1,1}}V_{n−1,1} \ar[r]_{f′_{n−1,1}} & Σ^{−k_{n−1,2}}V_{n−1,2} \ar[r]_{f′_{n−1,2}} & Σ^{−k_{n−1,3}}V_{n−1,3} \ar[r]_{f′_{n−1,3}} & ….} (2.6)

where

f′_{n−1,i} = f_{n−1,i}(v_{n,i})^{N_{n−1,i}−1}.

The cofiber of the vertical rows of (2.6) is, by definition,

\xymatrix{V_{n,1} \ar[r]^{f_{n,1}} & V_{n,2} \ar[r]^{f_{n,2}} & V_{n,3} \ar[r] & ….}  (2.7)

Now let Y′_{n−1} be the telescope of the bottow row (2.6), and let X′_{n} be the telescope of (2.7). Thus, we have (2.5). One easily proves by induction that when smashing (2.5) with BP, one obtains the cofiber sequence of MU-modules

BP/(v_{0}^{∞}, ...v_{n−1}^{∞}) → v_{n}^{−1}BP/(v_{0}^{∞}, ..., v_{n−1}^{∞}) → BP/(v_{0}^{∞}, ..., v_{n}^{∞}).  (2.8)

Now absent a proof of the telescope conjecture, we do not know that the Adams-Novikov spectral sequence converges for X′_{n}, Y′_{n}, n > 1. However, Ravenel [9] proved that if we denote by X_{n}, Y_{n} the Bousfield localizations of X′_{n}, Y′_{n} at BP, then the Adams-Novikov spectral sequence

Ext_{BP_{∗}BP} (BP_{∗}, v_{n}^{−1}BP_{∗}Y_{n−1}/(v_{0}^{∞}, ..., v_{n−1}^{∞})) ⇒ π_{∗}(Y_{n−1})  (2.9)

converges. We call (2.9) the Chromatic Adams-Novikov spectral sequence). Also, since stable Bousfield localization preserves cofibration sequences, we have cofibrations

\xymatrix{X_{n−1} \ar[r]^{α_{n−1}} & Y_{n−1} \ar[r]^{γ_{n−1}} & X_{n}.}  (2.10)

Applying π_{∗} to (2.10), we obtain an exact couple, which leads to the spectral sequence (1.7). Ravenel [9] proved that this spectral sequence converges. Note that we now have the following diagram of spectral sequences

[figure: ⊕Ext_{BP_{∗}BP} (v_{n}^{−1}BP_{∗}/(v_{0}^{∞}, ...v_{n−1}^{∞})); CANSS,E_{2}; CSS,E_{1}; ⊕π_{∗}Y_{n−1}; GCSS,E_{1}; Ext_{BP_{∗}BP} (BP_{∗}); ANSS,E_{2}; (π_{∗}S^{0})_{2}^{∧}.]

(2.11)

The superscript of each arrow indicates the abbreviated name of the spectral sequence, and its initial term. The Greek letter elements (1.6) are native in the top corner of (2.11). Geometric Greek letter elements are those which are permanent cycles in the CANSS, and therefore live in the left corner of (2.11). Theorems 1.8, 1.9 concern aspects of the behaviour of these elements in the GCSS, caused by Real cobordism. This will be discussed in the next section. Note, however, that the statements of the Theorems would be stronger if we could phrase them in terms of Greek letter elements of the Adams-Novikov spectral sequence, i.e. those which are permanent cycles of the CSS, and therefore live in the right corner of (2.11). The reason this would be better is that the CSS is purely algebraic, and hence in principle completely computable. Unfortunately, we were unable to prove such stronger results using the present methods. Without referring to the methods, we can say that is possible for renaming to occur in the ANSS, so that elements labelled as Greek letters do not correspond to such elements in the GCSS.

3. Real cobordism and proof of the main results

We begin by recalling some facts about Real cobordism, proved in [5]. The Real cobordism spectrum Mℝ is a ℤ/2-equivariant spectrum obtained by considering the ℤ/2-action on MU by complex conjugation. This can be taken almost literally, if we consider the usual prespectrum defining MU, consisting of Thom spaces of ndimensional universal complex bundles. Then complex conjugation acts non-trivially on both the spaces and structure maps of the prespectrum: denoting by α the 1dimensional real sign representation of ℤ/2, then the 1-point compactification of ℂ with respect to complex conjugation is S^{1+α}. As a result, Mℝ is an RO(ℤ/2)- graded spectrum, or spectrum indexed by a complete ℤ/2-universe. The reader is referred to [5] for details and other relevant references.

Before proceeding further, we will make certain crucial conventions: First of all, all ℤ/2-equivariant spectra will be RO(ℤ/2)-graded, and we will use the subscript ⋆ to denote RO(ℤ/2)-graded coefficients; therefore, the possible dimensions represented by ⋆ are k + ℓα, k, ℓ ∈ ℤ. We will use the subscript ∗ if we are referring only to the “twist 0” dimensions, i.e. k + 0α.

Next, we will make notational conventions of certain RO(ℤ/2)-graded homotopy and homology elements. First, let a be the element of π_{−α}S_{ℤ/2}^{0} represented by the non-trivial unstable map S^{0} → S^{α}. Next, recall from [5] that the ℤ/2-equivariant Borel cohomology ring with coefficients in ℤ/2 is

ℤ/2[σ, σ^{−1}][a].

The element σ has dimension α − 1. This element does not lift to the coefficients of S_{ℤ/2}^{0} (or Mℝ), but as we shall see, its powers survive as multipliers of certain elements of Mℝ_{⋆}. Also, using the fact that Milnor manifolds are defined over ℝ, we get a map

BP_{⋆} → Mℝ_{⋆}

where by BP_{⋆} we mean ℤ[v_{1}, v_{2}, ...], but where v_{i} is in dimension (2^{i} − 1)(1 + α). We will also denote by I_{n} the ideal (v_{0}, v_{1}, ...v_{n−1}) in BP_{⋆}. We will rely on the ⋆ to indicate the fact that we are working in the RO(ℤ/2)-graded dimensions (this notation was also used in [4]).

Araki [1], [5] has developed a theory of Real-oriented ℤ/2-spectra very parallel with the classical theory of complex-oriented spectra. In particular, 2-locally, there is a Quillen-idempotent e : Mℝ → Mℝ. The spectrum eMℝ is denoted by BP ℝ, and is easier to work with. When forgetting ℤ/2-equivariant structure, BP ℝ becomes just BP, but when applying the geometric fixed point functor to BP ℝ, one obtains Hℤ/2. Recall that a G-spectrum E is called complete with respect to G-action if the canonical map E → F (EG_{+}, E) is an equivalence. A crucial result stated in [5] is

Theorem 3.1. The spectrum BP ℝ is complete with respect to ℤ/2-action and we have

BP ℝ_{⋆} =

⨁_{ℓ=(2s+1)2^{n}∈ℤ} Ker(BP_{⋆}[a]/(a^{2^{i+1}−1}v_{i}|i ⩾ 0) →

BP_{⋆}[a]/(v_{0}, ..., v_{n}, a^{2^{i+1}−1}v_{i}|i ⩾ n + 1)) · σ^{2ℓ}

In the summand for ℓ = 0, we count n as ∞. Moreover, the multiplicative structure is the obvious one, i.e. as a subring of

BP_{⋆}[σ, σ^{−1}, a]/(a^{2^{i+1}−1}v_{i}|i ⩾ 0).
□

We now recall from [5] that Mℝ is also a ℤ/2-equivariant E_{∞} ring spectrum, and therefore we can construct cofiber sequences of Mℝ-modules

BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) → v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) → BP ℝ/(v_{0}^{∞}, ..., v_{n}^{∞}). (3.2)

Here we consider v_{i} ∈ BP_{⋆}. The significance of these ℤ/2-equivariant spectra for our purposes is in the following

Proposition 3.3. When smashing (2.10) with BP ℝ, we obtain the cofibration sequence of ℤ/2-equivariant spectra (3.2).

Our main calculational result on (3.2) is contained in the following Proposition 3.4. The spectrum BP ℝ/(v_{0}^{∞}, v_{1}^{∞}, ..., v_{n−1}^{∞}) is complete with respect to ℤ/2-action and we have

\begin{matrix}(BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}))_{⋆} = \\ ⨁_{k=0}^{n−1} BP_{⋆}/(v_{0}, ..., v_{k−1}, v_{k}^{∞}, ...v_{n−1}^{∞})[σ^{±2^{k+1}}][a]/(a^{2^{k+1}−1}) \\ ·\{v_{0}^{−1} · ... · v_{k−1}^{−1}σ^{−2^{k}+1}\} \\ ⊕BP ℝ_{⋆}/(σ^{2ℓ_{0}}v_{0}, σ^{4ℓ_{1}}v_{1}, ..., σ^{2^{n}ℓ_{n−1}}v_{n−1})\{v_{0}^{−1} · ... · v_{n−1}^{−1}σ^{−2^{n}+1}\}. \end{matrix}  (3.5)

We should explain that in our notation, when we are writing algebra, elements enclosed in the braces {} indicate additive generators, while elements enclosed in the brackets [] indicate multiplicative (polynomial) generators. While the notation for the generators in (3.5) indicates their origin in the computation, at the moment the significance of introducing additive generators is just suspension by their dimension.

Corollary 3.6. The coimage of the forgetful map

\xymatrix{(BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}))_{⋆} \ar[r]^{λ} & (BP/(v_{0}^{∞}, ...v_{n−1}^{∞}))_{⋆}}  (3.7)

is spanned by the following elements:

\begin{matrix}⨁_{j=0}^{n−1} BP_{⋆}/(v_{0}, ..., v_{j−1}, v_{j}^{∞}, ..., v_{n−1}^{∞})\{v_{0}^{−1} · ... · v_{j−1}^{−1} · σ^{−2^{j}+1}\}[σ^{±2^{j+1}}] \\ ⨁_{j=n}^{∞} BP_{⋆}/(v_{0}, ..., v_{j−1})\{v_{0}^{−1} · ... · v_{n−1}^{−1} · v_{j} · σ^{−2^{n}+1}\}[σ^{±2^{n+1}}] \end{matrix}  (3.8)

where, as before, dim(v_{k}) = (2^{k} − 1)(1 + α) and dim(σ) = α − 1.

Proof: It will be obvious from the proof of Proposition 3.4 via the Borel cohomology spectral sequence (see next section) that the elements (3.8) map non-trivially, while all other elements are multiples of a. □

Finally, we shall need information on the connecting maps of the cofibration sequences (3.2). This is given by

Proposition 3.9. The connecting map

∂_{n} : (BP ℝ/(v_{0}^{∞}, ..., v_{n}^{∞}))_{⋆} → Σ(BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}))_{⋆}  (3.10)

associated with (3.2) is a map of BP ℝ_{⋆}-modules given by

∂_{n} : σ^{−2^{n}}v_{n}^{−1} ↦ a^{2^{n+1}−1}  (3.11)

on the last summand (3.5), and by 0 on the other summands.

Propositions 3.4, 3.9, 3.3 will be proved in the next section. We will now apply these propositions to prove our main results.

Proof of Theorem 1.8: Consider the diagram

\xymatrix{π_{∗}X_{n−1} \ar[r]^{ι} \ar[d]_{π_{∗}α_{n−1}} & BP_{∗}X_{n−1} = BP_{∗}/(v_{0}^{∞}, ..., v_{n−1}^{∞}) \ar[d]^{BP_{∗}α_{n−1}} \\ π_{∗}Y_{n−1} \ar[r]^{κ} & BP_{∗}Y_{n−1} = v_{n}^{−1}BP_{∗}/(v_{0}^{∞}, ..., v_{n−1}^{∞})}  (3.12)

where ι, κ are Hurewicz maps. The Greek letter element (1.5) is an element of BP_{∗}Y_{n−1}. Now assuming that (1.5) is a geometric Greek letter element is equivalent to the existence of an element y ∈ π_{∗}Y_{n−1} such that (1.5) is equal to κ(y). Assuming further that y is a permanent cycle in the GCSS is equivalent to the existence of an element x ∈ π_{∗}X_{n−1} such that

π_{∗}α_{n−1}(x) = y.

Since BP_{∗}α_{n−1} is injective, this implies that

ι(x) = \frac{v_{n}^{k}}{v_{n−1}^{i_{n−1}}v_{n−2}^{i_{n−2}} · ... · v_{ℓ}^{i_{ℓ}}v_{ℓ−1} · ... · v_{0}}  (3.13)

(provided we are considering the Greek letter element figuring in the statement of the Theorem).

But now the map ι factors as

\xymatrix{π_{∗}X_{n−1} \ar[r]^{ν} & BP ℝ_{∗}X_{n−1} \ar[r]^{λ} & BP_{∗}X_{n−1}.}  (3.14)

Therefore, (3.13) must be in the image of λ in twist 0, i.e. of the form

λ(z), z ∈ BP ℝ_{∗+0α}X_{n−1}.  (3.15)

Inspecting (3.8), we find that by Corollary 3.6, the leading term of z must be of the form

\frac{v_{n}^{k}}{v_{n−1}^{i_{n−1}}v_{n−2}^{i_{n−2}} · ... · v_{ℓ}^{i_{ℓ}}v_{ℓ−1} · ... · v_{0}} σ^{−2^{j}+1+m·2^{j+1}}  (3.16)

with dimensional conventions as in Corollary 3.6. The requirement that (3.16) be of twist 0 then gives the condition of the Theorem. □

Lemma 3.17. Let z ∈ BP ℝ_{∗}X_{n−1} be such that

λ(z) = \frac{v_{n}^{k}}{v_{n−1}v_{n−2} · ... · v_{0}}  (3.18)

where

2^{n+1} | k(2^{n} − 1) − ∑_{m=0}^{n}(2^{m} − 1).  (3.19)

Then the image of z under the connecting map

δ^{∗} : BP ℝ_{⋆}X_{n−1} → ΣBP_{⋆}X_{n−2} → ... → Σ^{n}BP ℝ_{⋆}S^{0}  (3.20)

is non-zero.

Proof: Corollary 3.6 implies that under the condition (3.19),

z = \frac{v_{n}^{k}}{v_{n−1}v_{n−2}...v_{0}} σ^{−2^{n}+1+m2^{n+1}} mod (a)  (3.21)

for some m ∈ ℤ. Note that the element on the right hand side of (3.21) is in the summand of (3.8) which has j = n.

But by Proposition 3.9,

δ^{∗} (\frac{v_{n}^{k}}{v_{n−1}v_{n−2}...v_{0}} σ^{−2^{n}+1+m2^{n+1}}) = v_{n}^{k}σ^{m2^{n+1}}a ^{∑_{m=0}^{n−1} (2^{m+1}−1)},  (3.22)

and, moreover,

Im(δ^{∗}) ⊂ (a) ^{∑_{m=0}^{n−1} (2^{m+1}−1)}.  (3.23)

We conclude that the a-multiples in (3.21) map to

(a)^{1+ ∑_{m=0}^{n−1} (2^{m+1}−1)}

by δ^{∗}, and hence cannot cancel the non-zero element (3.22). The Lemma follows. □

Proof of Theorem 1.9: We begin with the second statement, which is a consequence of Theorem 1.8. To this end, note that for i_{0} = ... = i_{m−1} = 1, the condition of Theorem 1.8 reads

2^{j+1} | k(2^{n} − 1) − ∑_{m=0}^{n−1}(2^{m} − 1) − 2^{j} + 1  (3.24)

for some 0 ⩽ j ⩽ n. Processing (3.24) further gives

2^{j+1} | (k − 1)(2^{n} − 1) + (n + 1) − 2^{j}.  (3.25)

Note that the subsets of ℤ/2^{n+1}ℤ satisfying (3.25) for different j = 0, ..., n are disjoint, and the class for j has 2^{n−j} elements, which form a congruence class mod 2^{j+1}. It follows that there is precisely one class q + 2^{n+1}ℤ which does not satisfy (3.25) for any j = 0, ..., n, and it is

q = 2^{n} + k  (3.26)

where k satisfies (3.25) with j = n. To determine k, note that

−(2^{n} + 1) = (2^{n} − 1)^{−1} ∈ (ℤ/2^{n+1}ℤ)^{×},

so (3.25) with j = n gives

k − 1 ≡ ((n + 1) − 2^{n})(2^{n} + 1) ≡ n + 1 mod 2^{n+1} for n even
≡ 2^{n} + n + 1 mod 2^{n+1} for n odd.

(3.27)

It follows from (3.26) that q ≡ k_{n} + 2^{n} mod 2^{n+1}, as claimed.

To prove the first statement of Theorem 1.9, we turn to Lemma 3.17. The condition 3.19 is clearly equivalent to 3.25 with j = n, and hence to k ≡ k_{n} mod 2^{n+1}. But under this condition, Lemma 3.17 implies

δ^{∗}ν(x) = δ^{∗}(z) ≠ 0 ∈ BP ℝ_{⋆}S^{0}.  (3.28)

Therefore, if we denote also by δ^{∗} the connecting map

δ^{∗} : π_{∗}X_{n−1} → Σπ_{∗}X_{n−2} → ... → Σ^{n}π_{∗}S^{0},

we conclude from (3.28) that

δ^{∗}(x) ≠ 0 ∈ π_{∗}S^{0}  (3.29)

(since δ^{∗} obviously commutes with Hurewicz maps). But (3.29) occuring for every lift xof y is equivalent to y not being hit by a differential in the GCSS. □

4. BP ℝ-homology calculations

The purpose of this section is to prove Propositions 3.4, 3.9, 3.3. We begin with Proposition 3.4.

Lemma 4.1. We have

\begin{matrix}BP ℝ/(v_{0}, ..., v_{n−1})_{⋆} = \\ BP ℝ_{⋆}/(σ^{2ℓ_{0}}v_{0}, σ^{4ℓ_{1}}v_{1}, ..., σ^{2^{n}ℓ_{n−1}}v_{n−1}) · \{1, σ^{−1}, ..., σ^{−2^{n}+1}\}. \end{matrix}  (4.2)

On the right hand side of (4.2), we quotient out over all values of ℓ_{i} ∈ ℤ. Moreover, BP ℝ/(v_{0}, ..., v_{n−1}) is a complete spectrum with respect to ℤ/2-action, and the differentials of its Borel cohomology spectral sequence have the form

d_{2^{m+1}−1}σ^{−2^{m}} · q · σ^{−i} = v_{m}a^{2^{m+1}−1}q · σ^{−i}  (4.3)

where 0 ⩽ i ⩽ 2^{n} − 1, m ⩾ n and q = σ^{ℓ2^{m+1}}v_{R} where ℓ ∈ ℤ and R = (0, 0, ..., 0, r_{m}, r_{m+1}, ...).

Here we denote

v_{(r_{0},r_{1},...)} = v_{0}^{r_{0}}v_{1}^{r_{1}}...

(of course, only finitely many of the r_{i}’s are allowed to be non-zero).

Proof: To establish completeness, recall from [5] that BP ℝ is complete with respect to ℤ/2-action, and that we have cofibrations of Mℝ-modules

\xymatrix{Σ^{(2^{n}−1)(1+α)}BP ℝ/(v_{0}, ..., v_{n−1}) \ar[d]^{v_{n}} \\ BP ℝ/(v_{0}, ..., v_{n−1}) \ar[d] \\ BP ℝ/(v_{0}, ..., v_{n}).}  (4.4)

The differentials (4.3) are established exactly in the same way as in the case n = 0, which is done in [5]. □

Proof of Proposition 3.4: First consider the cofibration sequence of Mℝ-modules (3.2). Recall from [5] that

Φ^{ℤ/2}BP ℝ = Hℤ/2.  (4.5)

Since multiplication by v_{i} is 0 on (4.5), we conclude inductively that

Φ^{ℤ/2}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) = Hℤ/2  (4.6)

and

Φ^{ℤ/2}v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) = 0.  (4.7)

Therefore, the completeness statement of the Proposition will follow if we can show that

(v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}))^{∧} = 0  (4.8)

where the hat indicates the Tate spectrum. We will show this by induction on n, jointly with the following

Claim 4.9. The differentials in the Borel cohomology spectral sequence of

BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞})

are as follows:

\begin{matrix}d_{2^{k+1}−1}(v_{0}^{−1}v_{1}^{−1} · ... · v_{k−1}^{−1}v_{k}^{−i_{k}} · .. · v_{n−1}^{−i_{n−1}}v_{R}σ^{−2^{k+1}+1+ℓ2^{k+1}}) = \\ v_{0}^{−1}v_{1}^{−1} · ... · v_{k−1}^{−1}v_{k}^{−i_{k}+1}v_{k+1}^{−i_{k+1}} · ... · v_{n−1}^{−i_{n−1}}v_{R}σ^{−2^{k}+1+ℓ2^{k+1}}a^{2^{k+1}−1} \end{matrix}  (4.10)

where 0 ⩽ k ⩽ n − 1, i_{k} > 1, i_{k+1}, ..., i_{n−1} ⩾ 1, ℓ ∈ ℤ, R = (0, ..., 0, r_{n}, r_{n+1}, ...) and

\begin{matrix}d_{2^{m+1}−1}(v_{0}^{−1} · ... · v_{n−1}^{−1}v_{R}σ^{−2^{n}+1−2^{m}+ℓ2^{m+1}}) = \\ v_{0}^{−1} · ... · v_{n−1}^{−1}v_{m}v_{R}σ^{−2^{n}+1+ℓ2^{m+1}}a^{2^{m+1}−1} \end{matrix}  (4.11)

where m ⩾ n, ℓ ∈ ℤ, R = (0, ..., 0, r_{m}, r_{m+1}, ...).

More precisely, we will show that the Claim implies (4.8), and also the Claim with n replaced by n + 1. Note that these statements jointly imply the Proposition, by computing the Borel cohomology spectral sequence E_{∞} term via (4.10), (4.11).

Thus, assume the Claim is valid for a fixed n. Then we can compute the Tate spectral sequence for v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) by inverting v_{n} and a (for v_{n}, use the Mℝ-module structure). Note, however, that then the differential (4.10) wipes out the k-th summand of (3.5), and (4.11) with m = n wipes out the last summand. Consequently, the Tate spectral sequence for v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) collapses to E_{2^{n}} = 0, and thus (4.8) follows.

To prove the Claim with n replaced by n + 1, we will construct a map of spectral sequences

E′ ⊕ E′′ → E  (4.12)

which will be onto each E_{r}-term. To this end, let E′ be the Borel cohomology spectral sequence for v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) where the map E′ → E is induced by the second map (3.2). Let, on the other hand, E′′ be the Borel cohomology spectral sequence for BP ℝ/(v_{0}, ..., v_{n}) with E′′ → E induced by the obvious map of Mℝ-modules

{v_{0}^{−1} · ... · v_{n}^{−1}}BP ℝ/(v_{0}, ..., v_{n}) → BP ℝ/(v_{0}^{∞}, ..., v_{n}^{∞}).

To examine the map (4.12), write, as usual ([4], [5]), the Borel cohomology spectral sequence for BP ℝ/(v_{0}^{∞}, ..., v_{n}^{∞})_{⋆} in the form

E_{1} = BP_{⋆}/(v_{0}^{∞}, ..., v_{n}^{∞})[a][σ^{±1}] ⇒ BP ℝ/(v_{0}^{∞}, ..., v_{n}^{∞})_{⋆}.  (4.13)

In this notation, let, for i = 0, ..., n, _{i}E_{1} be spanned by those monomials in E_{1} which involve σ^{s} where the exponents of v_{0}, ..., v_{i−1} are −1, and in addition either s ≡ −2^{i} + 1 mod 2^{i+1}, or s ≡ −2^{i} + 1 ≡ 1 mod 2^{i} and the exponent of v_{i} is < −1.

Let, further, _{∞}E_{1} be the summand of E_{1} spanned by the remaining monomials, i.e. those involving σ^{s} where s ≡ −2^{n+1} + 1 ≡ 1 mod 2^{n+1}, and the exponents of v_{0}, ..., v_{n} are −1.

Now we know inductively that the Borel cohomology spectral sequence for

BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞})

splits as a sum of n+1 summands corresponding to the n+1 summands on the right hand side of (3.5). Inverting v_{n}, v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) correspondingly splits into n + 1 summands, which we will denote by _{0}E′_{r}, ..., _{n}E′_{r}. Then we find that

[figure: _{i}E′_{1}; _{i}E_{1},]

where i = 0, ..., n, while

[figure: _{i}E′_{2^{i+1}}; H(_{i}E_{1}, d_{2^{i+1}−1}).]  (4.14)

On the other hand, recalling Lemma 4.1,

E′′ = _{0}E′′ ⊕ ... ⊕_{2^{n}−1}E′′

where _{i}E′′ is the sub-spectral sequence of E′′ involving the factor σ^{−i} in (4.3). We then see that

\xymatrix{_{2^{n}−1}E′′ \ar[r]^{≃} & _{∞}E.}  (4.15)

Now (4.14), (4.15) imply that (4.12) is onto every E_{r} term, completing the induction step. □

Proof of Proposition 3.9: As noted in the preceding proof, the last summand of (3.5) (with n replaced by n + 1) is in the image of the map

{v_{0}^{−1} · ... · v_{n}^{−1}}(BP ℝ/(v_{0}, ..., v_{n}))_{⋆} → (BP ℝ/(v_{0}^{∞}, ..., v_{n}^{∞}))_{⋆},  (4.16)

so the connecting map on these summands can be figured out from the connecting map of the cofibration sequence

\xymatrix{\{v_{n}\}BP ℝ/(v_{0}, ..., v_{n−1}) \ar[r]^{v_{n}} & BP ℝ/(v_{0}, ..., v_{n−1}) \ar[r] & BP ℝ/(v_{0}, ..., v_{n}).}

(4.17)

The Borel cohomology spectral sequence gives

\begin{matrix}(BP ℝ/(v_{0}, ..., v_{n−1}))_{⋆} = \\ BP ℝ_{⋆}/(σ^{2ℓ_{0}}v_{0}, σ^{4ℓ_{1}}v_{1}, ..., σ^{2^{n}ℓ_{n−1}}v_{n−1})\{1, σ^{−1}, ..., σ^{−2^{n−1}+1}\}. \end{matrix}  (4.18)

The target of the connecting map δ_{n} of (4.17) is the kernel of the self map v_{n} in (4.18), which clearly consists of multiples of a^{2^{n+1}−1}. Differentials in the Borel cohomology spectral sequence of BP ℝ/(v_{0}, ..., v_{n−1}) then give the formula

δ_{n} : σ^{−2^{n}} ↦ a^{2^{n+1}−1}(v_{n})

which remains valid when multiplied by σ^{−i} with 0 ⩽ i < 2^{n}, and hence implies (3.11).

To show that the target of ∂_{n} is in the last summand of (3.5), note that this target is the direct limit of the targets of connecting maps of the form

{v_{n}^{−k}}BP ℝ/(v_{0}^{∞}, v_{1}^{∞}, ..., v_{n−1}^{∞}, v_{n}^{k})_{⋆} → ΣBP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞})_{⋆}.  (4.19)

But elements in the image of (4.19) must be in the kernel of v_{n}^{k}, which is clearly injective on all but the last summand of (3.5). □

Proof of Proposition 3.3: Induction on n. Suppose the statement is true with n replaced by n − 1. Consider a Hopkins-Smith v_{n−1}-spectrum V such that

BP_{∗}V = BP_{∗}/(v_{0}^{k_{0}}, ..., v_{n−1}^{k_{n−1}}),

and the Hurewicz map

η : V → BP ℝ ∧ V.  (4.20)

We shall prove that for a v_{n}-self map v : V → V (ignoring suspensions in the notation), we have a commutative diagram of the form

\xymatrix{V \ar[r]^{η} \ar[d]_{v} & BP ℝ ∧ V \ar[d]^{v_{n}^{N} σ^{K}} \\ V \ar[r]_{η} & BP ℝ ∧ V,}  (4.21)

for some N and K, at least when v is replaced by its suitable power. Similarly as above, this can be done by considering η as an element of the BP ℝ cohomology of V ∧ DV. The question then becomes what map in BP ℝ-cohomology the map w = v ∧ Id : V ∧ DV → V ∧ DV induces. We can assume by Hopkins-Smith [3] that w induces a power of v_{n} in BP_{∗}. Thus, by the Borel cohomology spectral sequence,

BP ℝ^{⋆}w = v_{n}^{N}σ^{K} mod (a).

However, considering the structure of BP ℝ_{⋆} (Theorem 3.1), we see that for any finite fixed spectrum X, a is nilpotent on elements of BP ℝ_{k+ℓα}X with k >> 0 (use the Atiyah-Hirzebruch-type spectral sequence associated with a finite celldecomposition of X). Now assume

BP ℝ^{⋆}w = v_{n}^{N}σ^{K} + at.

Then

BP ℝ^{⋆}w^{2^{M}} = (v_{n}^{N}σ^{K})^{2^{M}} + a^{M}t^{2^{M}}

(since, in BP ℝ_{⋆}, 2a = 0). Now for M >> 0, the second term disappears, thus proving (4.21).

Now passing to homotopy colimit and BP -localization, we get a commutative diagram

\xymatrix{X_{n−1} \ar[r] \ar[d] & BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) \ar[d] \\ Y_{n−1} \ar[r] & v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}).}  (4.22)

(Note that

v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) ≃ (v_{n}^{N}σ^{K})^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}),

since each of the elements v_{n}, v_{n}^{N}σ^{K} divides a power of the other.) Using the ring structure of BP ℝ, we get a diagram

\xymatrix{BP ℝ ∧ X_{n−1} \ar[r]^{ϕ} \ar[d] & BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) \ar[d] \\ BP ℝ ∧ Y_{n−1} \ar[r]^{ψ} & v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞})}  (4.23)

where the top horizontal arrow ϕ is an equivalence, and the bottom horizontal arrow ψ is an equivalence non-equivariantly. But both BP ℝ∧Y_{n−1}, v_{n}^{−1}BP ℝ/(v_{0}^{∞}, ..., v_{n−1}^{∞}) are free spectra (v induces 0 in homology), so ψ is an equivalence. Passing to cofibers of the vertical arrows of (4.23) gives the induction step. □

References

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[4] P.Hu: The Ext^{0}-term of the Real-oriented Adams-Novikov Spectral Sequence, in Homotopy Methods in Algebraic Topology, J. Greenlees, ed. Contemporary Mathematics 271 (2001) 141-154

[5] P.Hu, I.Kriz: Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence. Topology 40 (2001) 317-399

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2004/n1a9/v6n1a9.(dvi,ps,pdf)

Po Hu pohu@math.uchicago.edu

Department of Mathematics University of Chicago Chicago, IL 60637

Igor Kriz ikriz@math.lsa.umich.edu

Department of Mathematics University of Michigan Ann Arbor, MI 48109