k(n)-torsion-free H-spaces and P (n)-cohomology

J. Michael Boardman  W. Stephen Wilson

June 2004

REVISED EDITION

Now with p = 2!

Abstract

In [Wi75], for each k, the H-space that represents Brown–Peterson cohomology BP^{k}(−) was split into indecomposable factors, which have torsion-free homotopy and homology. Here, we do the same for the related spectrum P (n), by constructing idempotent operations in P (n)-cohomology P (n)^{k}(−) in the style of [BJW95]; this relies heavily on the Ravenel–Wilson determination [RW96] of the relevant Hopf ring. The resulting (i−1)-connected H-spaces Y_{i} have free connective Morava K-homology k(n)_{∗}(Y_{i}), and may be built from the spaces in the Ω-spectrum for k(n) using only v_{n}-torsion invariants.

We also extend Quillen’s theorem on complex cobordism to show that for any space X, the P (n)_{∗}-module P (n)^{∗}(X) is generated by elements of P (n)^{i}(X) for i ≥ 0. This result is essential for the work of Ravenel–Wilson–Yagita [RWY98], which in many cases allows one to compute BP -cohomology from Morava K-theory.

Introduction

We exploit the close relationship between the connective Morava K-theory spectrum k(n), whose coefficient ring is k(n)_{∗} = 𝔽_{p}[v_{n}], and the spectrum P (n) with P (n)_{∗} = 𝔽_{p}[v_{n}, v_{n+1}, v_{n+2}, …], where 𝔽_{p} denotes the field with p elements. These ring spectra are defined for each prime p (suppressed from almost all the notation) and integer n ≥ 0. Most of our work generalizes the case n = 0 (see [Wi75]), where k(0) = H(ℤ_{(p)}), the Eilenberg–Mac Lane spectrum for ℤ_{(p)} (the integers localized at p), and P (0) = BP, the Brown–Peterson spectrum, with BP_{∗} = ℤ_{(p)}[v_{1}, v_{2}, v_{3}, …].

§1 The main results 2

In §1, we present three groups of results. First, we give a structure theorem for a class of H-spaces that may be defined entirely in terms of k(n). Second, starting from P (n), we construct examples of such H-spaces which we use to prove our structure theorem. Third, there are consequences for the structure of P (n)-(co)homology: we find (i) a Quillen-type result, that P (n)^{∗}(X) is generated as a module by elements of P (n)^{i}(X) for i ≥ 0, (ii) a Landweber-type filtration theorem, and (iii) a bound on the homological dimension of P (n)-homology.

All these results depend on the Ravenel–Wilson calculation [RW96] of the Hopf ring for P (n), which encodes the unstable operations in P (n)-cohomology. All the machinery of [Bo95, BJW95] becomes available, making P (n)^{∗}(−) our sixth example of a cohomology theory whose operations we can handle in a uniform manner.

The authors acknowledge the influence of D. C. Johnson and D. C. Ravenel on this paper through their work with us on [BJW95, RW96].

Notation We fix throughout a prime p and an integer n > 0. Because it occurs so frequently, we find it convenient to write N = p^{n} − 1.

[For completeness, we include the results for p = 2. Modifications are required because: (i) our ring spectra are no longer commutative, and (ii) one of our test spaces, real projective space, has different cohomology. Shorter comments, like this one, are enclosed within square brackets. Longer comments form a subsection. A few proofs are substantial enough to be deferred to a separate paper [Bo].]

All spaces are assumed to be homotopy-equivalent to cw-complexes. Identity maps and homomorphisms are denoted by id.

We use much notation and terminology from [BJW95]. A ring spectrum E defines a homology theory E_{∗}(−) and a cohomology theory E^{∗}(−), both multiplicative with coefficient ring E_{∗} = π_{∗}^{S}(E). Then E^{i}(−) is represented (on the homotopy category Ho of unbased spaces) by the i-th space \underline{E}_{i} of the Ω-spectrum for E.

Because we deal mainly with homology and homotopy groups rather than cohomology, we use homology degrees throughout (unlike [BJW95]), assigning the degree i to elements of E_{i}(X) and π_{i}(X). This forces elements of E^{i}(X) to have degree −i. We thus write E_{∗} for the coefficient ring, even when working with cohomology; in particular, E^{i}(point) = E_{−i}. So the Hazewinkel generator v_{i} has degree 2(p^{i}−1).

The algebraic suspension ΣM of a graded group M is a copy of M with all degrees raised by one: an element x ∈ M_{i} gives rise to Σx ∈ (ΣM)_{i+1}.

As in [RW96], E(x, …) denotes the exterior algebra on generator(s) x, …, P (x, …) the polynomial algebra, and T P_{h}(x) the truncated polynomial algebra P (x)/(x^{p^{h}}).

1 The main results

Splittings of H-spaces We regard the standard generator u_{k} of P (n)^{∗}(S^{k}) as a map u_{k}: S^{k} → \underline{P (n)}_{k}. We consider spaces X that satisfy the axioms:

(i) X is a connected H-space of finite type (meaning that each homo- topy group π_{i}(X) is finitely generated); (ii) k(n)_{∗}(X) is a free k(n)_{∗}-module (equivalently, has no v_{n}-torsion); (iii) For any k > 0, any map S^{k} → X factors through the map u_{k} to give a map \underline{P (n)}_{k} → X. (1.1)

Our first theorem classifies these spaces.

Theorem 1.2 Given n > 0, the spaces X that satisfy the axioms (1.1) have the following properties:

(a) For each k > 0, there is (up to homotopy) a unique (k−1)-connected (but not k-connected) example Y_{k} that does not decompose as a product of spaces;

(b) Every X is homotopy equivalent to some product Y = ∏_{i} Y_{k_{i}}, where the number of copies of each Y_{k} is finite and is uniquely determined by X;

(c) Every retract of X is another example;

(d) Every product of examples is an example, provided it has finite type;

(e) The loop space ΩX is another example, provided X is simply connected.

Shortly, in Definition 1.10, we shall reveal the spaces Y_{k} explicitly.

Remark The above decompositions and equivalences are not as H-spaces. Nevertheless, no information is lost, because in (b) for example, the given multiplication on X corresponds to some multiplication on Y; as we (shall) have complete information on the possible maps Y × Y → Y, we can in principle detect which of them are H-space multiplications.

Part (c) is clear. So is (d), with the help of the Künneth formula for k(n)-homology (as in [Bo95, Thm. 4.2]). Part (e) will follow immediately from (b) and Theorem 1.15. We prove (a) and (b) in §3.

Towers built from k(n) Although axiom (1.1)(iii) is technically convenient, it lacks intuitive content. Here, we replace it by a more appealing axiom. This makes Theorem 1.2 analogous to the results of [Wi75], as we discuss later in this section.

We consider spaces that are built from the spaces \underline{k(n)}_{i} in a particularly nice way, using only v_{n}-torsion invariants. We recall that k(n)_{∗} = 𝔽_{p}[v_{n}], where v_{n} has degree 2N = 2(p^{n}−1).

Definition 1.3 Given a space Y, we call a map z: Y → \underline{k(n)}_{q+1} a v_{n}-torsion map if, considered as an element of k(n)^{∗}(Y), it satisfies v_{n}^{c}z = 0 for some c. (We assume q ≥ 0. Indeed, z must be zero unless q ≥ 2N + 1 = 2p^{n} − 1.)

We call a space X a k(n)-tower with v_{n}-free homotopy if it is the homotopy limit of a sequence of spaces and maps

… − ; − ; → X_{3} − ; − ; → X_{2} − ; − ; → X_{1} − ; − ; → X_{0} = point  (1.4)

in which each map X_{i} → X_{i−1} (for i > 0) is the homotopy fibre of some v_{n}-torsion map z_{i}: X_{i−1} → \underline{k(n)}_{q(i)+1}. (We allow the possibility of a finite tower, X = X_{m} for some m, or even a tower having only one stage, X = X_{1} = \underline{k(n)}_{q(1)}, as well as the degenerate case where X is contractible.)

A v_{n}-torsion map z: Y → \underline{k(n)}_{q+1} necessarily induces the zero homomorphism on homotopy. Then for each i > 0 (assuming X is connected, so that q(i) ≥ 1), the homotopy long exact sequence of z_{i} reduces to the short exact sequence of groups

0 − ; − ; → Σ^{q(i)}𝔽_{p}[v_{n}] − ; − ; → π_{∗}(X_{i}) − ; − ; → π_{∗}(X_{i−1}) − ; − ; → 0.  (1.5)

Thus π_{∗}(X) is an iterated extension of suspensions of 𝔽_{p}[v_{n}]. (Our terminology is abusive to the extent that we do not have a natural action of v_{n} on π_{∗}(X_{i}) for i > 1.)

We study such towers in more detail in §4 and prove the following equivalence.

Theorem 1.6 If we replace axiom (iii) in (1.1) by the axiom

(iii)′ X is a k(n)-tower with v_{n}-free homotopy, (1.7)

we obtain the same class of H-spaces. Thus Theorem 1.2 remains valid.

Examples based on P (n) The prime ideal

I_{n} = (p, v_{1}, v_{2}, …, v_{n−1}) ⊂ BP_{∗} = ℤ_{(p)}[v_{1}, v_{2}, v_{3}, …]

is invariant and therefore of particular interest. (We set v_{0} = p, and take I_{1} = (p) and I_{0} = (0).) The spectrum P (n) is constructed (see §2) to have the quotient ring

P (n)_{∗} = BP_{∗}/I_{n} = 𝔽_{p}[v_{n}, v_{n+1}, v_{n+2}, …]

as its homotopy. In particular, P (0) = BP and P (1) is just BP mod p.

Further, given m ≥ n, we kill off the ideal

J_{m} = (v_{m+1}, v_{m+2}, v_{m+3}, …) ⊂ P (n)_{∗}  (1.8)

to produce the spectrum we call P (n, m) (but known to Yosimura [Yo76] as BP [n, m+1) and to Yagita [Ya76] as BP (p, v_{1}, …, v_{n−1}, v_{m+1}, …)), with homotopy

P (n, m)_{∗} = P (n)_{∗}/J_{m} = 𝔽_{p}[v_{n}, v_{n+1}, …, v_{m}].

It comes equipped with a canonical map ρ(m): P (n) → P (n, m). These spectra are intimately connected with the spectra E(n, m) = v_{m}^{−1}P (n, m), which are essential in Ravenel–Wilson–Yagita [RWY98]. We recognize P (n, n) as k(n).

Remark Unlike I_{n}, the ideal J_{m} is not at all canonical, as it depends on the choice of the generators v_{i} of P (n)_{∗}. Nevertheless, our results are independent of this choice, as we are concerned only with the additive structure of P (n, m).

The behavior of these spectra depends on the numerical function

g(n, m) = 2(p^{n} + p^{n+1} + … + p^{m}),  (1.9)

where it is reasonable to define g(n, n−1) = 0.

Definition 1.10 Given k > 0, we define the H-space Y_{k} = \underline{P (n, m)}_{k}, where the integer m is defined in terms of equation (1.9) by

g(n, m−1) < k ≤ g(n, m).  (1.11)

For convenience, we also define Y_{0} = 𝔽_{p}, viewed as a discrete group.

These are the spaces Y_{k} that appear in Theorem 1.2. In particular, Y_{k} = \underline{k(n)}_{k} for 0 < k ≤ 2p^{n}. As the spaces \underline{P (n)}_{k} satisfy the axioms (1.1), they must decompose according to Theorem 1.2(b). We establish the following splittings in §3.

Theorem 1.12 Assume k ≥ 0. If k > 0, define m by equation (1.11); if k = 0, take m = n − 1. Then we have homotopy decompositions

\underline{P (n)}_{k} ≃ Y_{k} × ∏_{j>m} Y_{k+2(p^{j}−1)}  (1.13)

and, for any h > m,

\underline{P (n, h)}_{k} ≃ Y_{k} × ∏_{j=m+1}^{h} Y_{k+2(p^{j}−1)}.  (1.14)

These are equivalences of H-spaces [except in the extreme case when p = 2 and k = g(n, m)].

We showed in [BW01, Thm. 1.1] that such splittings exist, though without making them explicit as we do here in §3. They are patterned after the splittings of the spaces \underline{BP}_{k} in [Wi75], which were recovered explicitly in [BJW95] and are reviewed below.

We note that equation (1.14) reduces to Definition 1.10 when h = m.

Remark No such result holds for \underline{P (n, m)}_{k} when k > g(n, m), as axiom (1.1)(ii) definitely fails (otherwise this space would contradict Theorem 1.2(b)).

We use equation (1.14) to decompose ΩY_{k} = \underline{P (n, m)}_{k−1} explicitly.

Theorem 1.15 The loop space ΩY_{k} is given for all k > 0 as follows:

(a) If k does not have the form g(n, q) + 1 for any q, then ΩY_{k} ≃ Y_{k−1};

(b) If k = g(n, q) + 1, where q ≥ n − 1, then ΩY_{k} ≃ Y_{k−1} × Y_{k−1+2(p^{q+1}−1)}.

Since Ω is a right adjoint functor and so preserves products, this gives part (e) of Theorem 1.2. We leave it as an exercise to decompose the negative spaces \underline{P (n)}_{−k} for k > 0, by writing them as Ω^{k+1}\underline{P (n)}_{1}, and similarly \underline{P (n, m)}_{−k}.

Some history For n = 0, the results differ slightly. Recall that k(0) = H(ℤ_{(p)}), P (0) = BP, and (see [Wi75]) P (0, m) = BP ⟨m⟩ has BP ⟨m⟩_{∗} = ℤ_{(p)}[v_{1}, v_{2}, …, v_{m}]. Axioms (1.1) (with (iii) replaced by (1.7)) then yield connected H-spaces X whose homotopy groups π_{k}(X) and homology groups H_{k}(X) are all free ℤ_{(p)}-modules of finite rank. The Postnikov k-invariants of such spaces are necessarily torsion elements. Theorem 1.2 remains valid exactly as stated, with m still defined by equation (1.11). However, Theorem 1.12 gives H-space equivalences only for g(0, m−1) < k < g(0, m); for k = g(0, m), we have merely a homotopy equivalence. (Of course, Y_{0} = ℤ_{(p)} rather than 𝔽_{p}.) These are the main results of [Wi75] or Theorem 1.16 of [BJW95], and form the motivation for this work.

The structure of P (n)-cohomology We extend Quillen’s theorem on complex cobordism to P (n).

Theorem 1.16 For any space X, the cohomology P (n)^{∗}(X) is generated, as a P (n)_{∗}-module (topologically if X is infinite), by elements of P (n)^{i}(X) for i > 0, together with one element of P (n)^{0}(X) for each component of X.

This result is essential for the calculations in Ravenel–Wilson–Yagita [RWY98]. One version was stated as Theorem 1.11 of [Ya84], without proof (although the approach suggested is now known not to work). In §8, our machinery of additive unstable operations provides a very short direct proof in terms of an explicit formula.

We also refine Landweber’s filtration theorem. Yosimura [Yo76, Thm. 3.4] and Yagita [Ya76] both observed that Landweber’s theorem generalizes to stable P (n)-co-homology comodules M. The only finitely generated invariant prime ideals in P (n)_{∗} are I_{m} = (v_{n}, v_{n+1}, …, v_{m−1}) for n ≤ m < ∞ (where I_{n} is interpreted as (0)). We find in §8 that an unstable comodule structure on M (in the sense of [BJW95, Defn. 6.32]) restricts the possible Landweber factors as follows.

Lemma 1.17 Let M be a P (n)_{∗}-module with a single generator x ∈ M^{k} (in homology degree −k) and annihilator ideal Ann(x) = I_{m}, where n ≤ m < ∞, so that

M ≅ Σ^{−k}P (m)_{∗} ≅ Σ^{k}P (m)^{∗} ≅ Σ^{k}(P (n)^{∗}/I_{m}).

Then M admits an unstable P (n)-cohomology comodule structure if and only if k satisfies the appropriate condition (depending on m and p):

(i) k ≥ 0 if m = n;
(ii) k ≥ g(n, n) − 1 if m = n + 1;
(iii) k ≥ g(n, m−1) − 1 if m ≥ n + 2 and p is odd;
(iv) k ≥ g(n, m−1) − 2 if m ≥ n + 2 and p = 2;

and this comodule structure is unique.

This leads directly to the filtration theorem.

Theorem 1.18 Let M be an unstable P (n)-cohomology comodule of finite type (each M^{i} a finitely generated 𝔽_{p}-module) and bounded above (M^{i} = 0 for all i > i_{0}). Then M admits a finite filtration by subcomodules

0 = M_{0} ⊂ M_{1} ⊂ … ⊂ M_{h} = M

in which each quotient M_{i}/M_{i−1} is a monogenic comodule Σ^{−k_{i}}P (m_{i})_{∗} with generator x_{i}, as listed in Lemma 1.17. In particular, M is a finitely presented P (n)_{∗}-module.

If, in addition, M is a P (n)_{∗}-algebra of any of the forms:

(i) M = P (n)^{∗}(X), for a finite complex X; (ii) M = Im[f^{∗}: P (n)^{∗}(Y) → P (n)^{∗}(X)], for a map of spaces f: X → Y, where

X is a finite complex; (iii) A spacelike (see [BJW95, Defn. 7.14]) unstable P (n)_{∗}-cohomology algebra;

we may take each M_{i} to be an invariant ideal in M. At the last stage, we may take x_{h} = 1 and m_{h} = n.

Our proof in §8 quotes the method of proof of Theorem 20.11 in [BJW95]. However, here we prove that M is finitely presented, instead of assuming it. (Of course, it has long been known that for finite X, P (n)^{∗}(X) is a coherent P (n)_{∗}-module and hence finitely presented.) In [ibid.], we overlooked the fact that this modification applies equally well to BP = P (0), as follows. (Again, (i) is not new. However, (ii) is non-trivial and new when BP^{∗}(Y) has phantom classes.)

Theorem 1.19 Let M be an unstable BP -cohomology comodule of finite type (each M^{i} a finitely generated ℤ_{(p)}-module) and bounded above, for example:

(i) M = BP^{∗}(X), for a finite complex X; (ii) M = Im[f^{∗}: BP^{∗}(Y) → BP^{∗}(X)], for a map of spaces f: X → Y, where X

is a finite complex.

Then M is a finitely presented BP_{∗}-module.

Homological dimension Our starting point is the Conner–Floyd Theorem [CF66, Thm. 10.1], that the map of ring spectra from the unitary Thom spectrum MU to the K-theory spectrum K determined by the Todd genus induces for finite X an isomorphism of cohomology theories

K_{∗} ⊗_{MU_{∗}} MU^{∗}(X) − ; − ; →^{≅} K^{∗}(X).

A far-reaching analogue is the result

E(n, m)_{∗} ⊗_{P (n)_{∗}} P (n)^{∗}(X) ≅ E(n, m)^{∗}(X),  (1.20)

where E(n, m) = v_{m}^{−1}P (n, m). A key ingredient of such results is knowledge of the homological dimension of various (co)homology modules.

The case m = n of (1.20) is due to Morava [Mo85] as part of his structure theorem, and is quoted and reproved in [JW75], as well as by Yagita [Ya76]. The case n = 0, along with results on the homological dimension of BP_{∗}(X), was proved by Johnson– Wilson [JW73, Rk. 5.13] by means of the splitting theorem for BP in [Wi75]. Shortly afterwards, Landweber [La76] reproved this case by using cohomology operations instead of the splitting, establishing his exact functor theorem in the process; however, he was unable to recover Corollary 4.4 of [JW73], which gave an upper bound on the homological dimension of BP_{∗}(X). Later, Morava and Yagita [Ya77, Thm. 3.11] showed that P (n)^{∗}(X) is a BP^{∗}(BP)-module. Yagita and Yosimura [Yo76] both used this fact to generalize the exact functor theorem to P (n), which fully includes (1.20), and obtain homological dimension results for P (n)_{∗}(X).

We have now gone full circle, and with our splitting for P (n) in hand, can use the techniques of [JW73] to recover these results as well as (1.20), with the added benefit of the following estimate, which we establish in §5.

Theorem 1.21 Assume that X is a finite complex of dimension less than g(n, m)/2. Then the homological dimension of the P (n)_{∗}-module P (n)_{∗}(X) is at most m − n.

Although the exact functor theorem does not apply, ρ(m): P (n) → P (n, m) still induces a natural homomorphism of P (n, m)_{∗}-modules

\overline{ρ(m)}: P (n, m)_{∗} ⊗_{P (n)_{∗}} P (n)^{∗}(X) − ; − ; → P (n, m)^{∗}(X).

This is an isomorphism when X is a point, but not in general, as the left side is not a cohomology theory. Classically, as in [JW73], one then asks for which X it is an isomorphism. Instead, we show in §5 that it is always an isomorphism in a certain range of degrees [with no modification if p = 2]. Explicitly, its components are

\overline{ρ(m)}: P (n)^{h}(X)/ ∑_{j>m} v_{j}P (n)^{h+2(p^{j}−1)}(X) − ; − ; → P (n, m)^{h}(X).  (1.22)

§2 The ring spectrum P (n) 8

Theorem 1.23 Assume that X is finite-dimensional and that m ≥ n > 0. Then (1.22) is an isomorphism for all h ≤ g(n, m), and therefore a P (n, m)_{∗}-module isomorphism in this range.

In particular, for m = n we have the isomorphism

\overline{ρ(n)}: P (n)^{h}(X)/ ∑_{j>n} v_{j}P (n)^{h+2(p^{j}−1)}(X) ≅ k(n)^{h}(X)

for all h ≤ 2p^{n}, which preserves the v_{n}-action in this range.

2 The ring spectrum P (n)

As the literature is somewhat conflicted [especially when p = 2], we review the construction of P (n) in fair detail. In this section, we work entirely in the graded stable homotopy category Stab_{∗}.

The spectrum P (n), so named by Johnson–Wilson [JW75], was based on work of Morava. It may conveniently be constructed directly from the Thom spectrum MU by applying Sullivan–Baas theory [Ba73] to kill off the unwanted generators of MU_{∗}, as well as p (with no need for localization). (As stable P (n)-cohomology operations act faithfully on P (n)-homology, no information is lost by working in homology.)

It is automatically a BP -module spectrum, with an action map λ: BP ∧ P (n) → P (n) that satisfies the usual two module axioms, and the canonical map BP → P (n) is BP -linear. It comes equipped with an exterior algebra E(Q_{0}, Q_{1}, …, Q_{n−1}) of BP - linear operations, where Q_{i} has homology degree −(2p^{i}−1); we write the monomial basis elements as Q^{I} = Q_{0}^{i_{0}} ^{∘}Q_{1}^{i_{1}} ^{∘} …^{∘}Q_{n−1}^{i_{n−1}} for each multi-index I = (i_{0}, i_{1}, …, i_{n−1}), where each i_{r} is 0 or 1.

The multiplication The canonical map η: S^{0} → BP → P (n) serves as the unit map of P (n), where S^{0} denotes the sphere spectrum, but there is no obvious multiplication on P (n). It is known that for p ≠ 2, there is a unique multiplication ϕ: P (n) ∧ P (n) → P (n) having the following properties:

(i) ϕ is BP -bilinear; (ii) BP → P (n) is multiplicative; (iii) ϕ has η: S^{0} → P (n) as two-sided unit; (iv) ϕ is commutative; (v) ϕ is associative; (vi) Each Q_{i}: P (n) → P (n) is a derivation, in the sense that (2.1)

Q_{i}^{∘}ϕ = ϕ^{∘}(Q_{i} ∧ id) + ϕ^{∘}(id ∧ Q_{i}): P (n) ∧ P (n) − ; − ; → P (n).  (2.2)

Historically, three quite different approaches have been used. First, for p ≠ 2, Morava [Mo79] used averaging over the symmetric group Σ_{2} to produce idempotent operations in (co)bordism with repeated singularities. These operations yield a canonical multiplication ϕ on P (n) that is automatically commutative (cf. Mironov [Mi78, Thm. 4.2]). Associativity by this method involves averaging over Σ_{3} and requires p ≥ 5 [ibid., Thm. 4.1].

The second method is heavily geometric. Mironov [Mi75] and Shimada–Yagita [SY76] constructed (roughly equivalent) explicit multiplications on P (n) in the Baas bordism context for any prime p. These apparently depend on a sequence of choices of Morava manifolds. They automatically satisfy axioms (i), (ii) and (iii). Moreover, Shimada–Yagita [SY76, Thm. 5.25] and Mironov [Mi78, Thm. 2.4] both show that the obstructions to associativity lie in groups that vanish, and also obtain (vi). The disadvantage of this approach is that uniqueness is difficult to handle.

Third, Würgler [Wü77] developed an entirely algebraic cohomological approach in terms of comodules, which leads to the existence of ϕ and the following results.

Lemma 2.3 In the graded stable homotopy category Stab_{∗}:

(a) Any BP -linear map P (n) → P (n), of any degree, can be uniquely written in the form

∑_{I} c_{I}Q^{I}: P (n) − ; − ; → P (n),  (2.4)

with coefficients c_{I} ∈ P (n)_{∗} of the appropriate degrees;

(b) Any BP -bilinear map P (n) ∧ P (n) → P (n), of any degree, can be uniquely written in the form

∑_{I,J} c_{I,J} ϕ^{∘}(Q^{I} ∧ Q^{J}): P (n) ∧ P (n) − ; − ; → P (n),  (2.5)

with coefficients c_{I,J} ∈ P (n)_{∗} of the appropriate degrees;

(c) Any BP -trilinear map P (n) ∧ P (n) ∧ P (n) → P (n), of any degree, can be uniquely written in the form

∑_{I,J,K} c_{I,J,K} ϕ^{∘}(ϕ ∧ id)^{∘}(Q^{I} ∧ Q^{J} ∧ Q^{K}): P (n) ∧ P (n) ∧ P (n) − ; − ; → P (n),  (2.6)

with coefficients c_{I,J,K} ∈ P (n)_{∗} of the appropriate degrees.

Proof Part (a) is a strengthened form of Proposition 3.5 of Würgler [Wü77]. Part (b) is Proposition 4.12 of [ibid.], and (c) is entirely analogous.

Lemma 2.7 The canonical map ρ: P (n) → P (n+1) is a map of ring spectra.

Proof By a slight generalization of (2.5) (also proved by Würgler), any BP -bilinear map P (n) ∧ P (n) → P (n+1), in particular ϕ^{∘}(ρ ∧ ρ), can be written

∑_{I,J} c_{I,J} ρ^{∘}ϕ^{∘}(Q^{I} ∧ Q^{J}): P (n) ∧ P (n) − ; − ; → P (n+1),

with coefficients c_{I,J} ∈ P (n+1)_{∗}. Since ϕ^{∘}(ρ ∧ ρ)^{∘}(η ∧ η) = η, the sparseness of P (n+1)_{∗} leaves ρ^{∘}ϕ as the only candidate for ϕ^{∘}(ρ ∧ ρ). [This works even for p = 2, regardless of the choices of multiplication on P (n) and P (n+1).]

If we write ϕ^{∘}(η ∧ id) in the form (2.4), the sparseness of P (n)_{∗} yields axiom (iii) [even for p = 2], since we know ϕ^{∘}(η ∧ η) = η. Then (ii) is a formal consequence of (i), (iii), and the BP -linearity of the map BP → P (n).

Since any BP -bilinear multiplication can be written in the form (2.5), the sparseness of P (n)_{∗} ensures that ϕ is unique, as long as p ≥ 3. Further, (iv) holds, since ϕ^{∘}T also satisfies (i) and (iii), where T: P (n) ∧ P (n) → P (n) ∧ P (n) denotes the switch map.

We may similarly deduce the associativity of ϕ, provided p ≥ 5, by writing ϕ^{∘}(id∧ ϕ) in the form (2.6). We also obtain (vi), provided p ≥ 3, by writing Q_{i}^{∘}ϕ in the form (2.5); since (Q_{i}^{∘}ϕ)^{∘}(η ∧ id) = Q_{i} = (Q_{i}^{∘}ϕ)^{∘}(id ∧ η), the only candidate is (2.2).

Finally, we should mention that there is now a fourth approach, the brave new ring context of Elmendorf–Kriz–Mandell–May. See [EKMM96] for p odd [or Strickland [St99] for p = 2].

The case p = 2 It is well known that there is no commutative multiplication on P (n) when p = 2. Instead, we see in [Bo] that there are exactly two multiplications that satisfy all the axioms (2.1) except (iv). To make P (n) a ring spectrum, we arbitrarily choose one of the two good multiplications as ϕ; then the other is its opposite, \overline{ϕ} = ϕ^{∘}T, which defines the opposite ring spectrum \overline{P (n)}. Nassau [Na02, Thm. 3] shows that complex conjugation defines an isomorphism of ring spectra Ξ: P (n) ≅ \overline{P (n)}.

Mironov [Mi78, Thm. 4.7] computed \overline{ϕ} explicitly in the form (2.5) as

\overline{ϕ} = ϕ^{∘}T = ϕ + v_{n}ϕ^{∘}(Q_{n−1} ∧ Q_{n−1}): P (n) ∧ P (n) − ; − ; → P (n).  (2.8)

From now on, we write Q = Q_{n−1}, in view of its frequent occurrence.

Products in homology and cohomology We review briefly the various products in P (n)-(co)homology. Their properties are familiar enough [except when p = 2]. We remind the reader that the operations Q_{i} act on both homology and cohomology.

Given x ∈ P (n)^{∗}(X) and y ∈ P (n)^{∗}(Y), we have the cohomology cross product x × y ∈ P (n)^{∗}(X×Y); by taking Y = X and using the diagonal map of X, we deduce the cup product xy ∈ P (n)^{∗}(X), which makes P (n)^{∗}(X) a ring. Given a ∈ P (n)_{∗}(X) and b ∈ P (n)_{∗}(Y), we have the homology cross product a × b ∈ P (n)_{∗}(X ×Y). All three products are associative. For p ≠ 2, they are also commutative, in the sense that T^{∗}(y×x) = ± x × y, yx = ±xy, and T_{∗}(b×a) = ± a × b. By equation (2.2), each Q_{i} is a derivation for all three products.

By taking X as a one-point space, P (n)^{∗}(Y) and P (n)_{∗}(Y) become P (n)_{∗}-modules, and both cross products are P (n)_{∗}-bilinear [even for p = 2; see below].

There is also the scalar or Kronecker product ⟨x, a⟩ ∈ P (n)_{∗} of x ∈ P (n)^{∗}(X) and a ∈ P (n)_{∗}(X), which is P (n)_{∗}-bilinear [even for p = 2; see [Bo]].

The case p = 2 There are of course no signs, but the noncommutativity of ϕ forces us to watch carefully for any shuffling of copies of P (n). Nevertheless, we find [Bo] that the Künneth and duality formulae continue to hold, exactly as stated in [Bo95].

It is immediate from equation (2.8) that

T^{∗}(y×x) = x \overline{×} y = x×y + v_{n}Qx×Qy in P (n)^{∗}(X ×Y),  (2.9)

where x \overline{×} y denotes the twisted cross product formed using the opposite multiplication \overline{ϕ} on P (n). For cup products, this implies

yx = xy + v_{n}(Qx)(Qy) in P (n)^{∗}(X),  (2.10)

§3 Proofs of the main theorems 11

so that P (n)^{∗}(X) is not commutative in general in the ordinary sense. Alternatively, these products are T_{Q}-commutative if we replace the standard commutativity isomorphism T: A ⊗ B ≅ B ⊗ A everywhere by T_{Q}: A ⊗ B ≅ B ⊗ A, defined by

T_{Q}(a ⊗ b) = b ⊗ a + v_{n}Qb ⊗ Qa in B ⊗ A.  (2.11)

Similarly, homology is also T_{Q}-commutative, in the sense that

T_{∗}(b×a) = a \overline{×} b = a × b + v_{n}Qa × Qb in P (n)_{∗}(X ×Y).  (2.12)

Taking X to be a point shows that the P (n)_{∗}-actions on P (n)^{∗}(Y) and P (n)_{∗}(Y) are independent of the choice of ϕ. In [Bo], we find that ⟨x, a⟩ is also independent of this choice.

There is one surprise, on account of the hidden shuffling, proved in [Bo].

Proposition 2.13 Given x ∈ P (n)^{∗}(X), y ∈ P (n)^{∗}(Y), a ∈ P (n)_{∗}(X), and b ∈ P (n)_{∗}(Y), we have

⟨x×y, a×b⟩ = ⟨x, a⟩⟨y, b⟩ + v_{n}⟨x, Qa⟩⟨Qy, b⟩.  (2.14)

If instead we mix the products, we find

⟨x×y, a\overline{×}b⟩ = ⟨x, a⟩⟨y, b⟩.  (2.15)

3 Proofs of the main theorems

In this section, we establish Theorems 1.2 and 1.12. More precisely, we reduce them to two key lemmas: Lemma 3.1 provides our main splitting and Lemma 3.8 will imply that our splittings are best possible.

Splittings All our splittings are derived from the following splitting.

Lemma 3.1 For k ≤ g(n, m), where m ≥ n, there is a map

\overline{θ(m)}: \underline{P (n, m)}_{k} − ; − ; → \underline{P (n)}_{k}

that splits the canonical map ρ(m): \underline{P (n)}_{k} → \underline{P (n, m)}_{k}, i. e. ρ(m)^{∘}\overline{θ(m)} ≃ id. It is a map of H-spaces [except when p = 2 and k = g(n, m)].

We express this in terms of idempotent P (n)-cohomology operations in §5.

A short direct proof of Lemma 3.1 is presented in [BW01], based on the bar spectral sequence. For such k, we show that E_{∗}(\underline{P (n, m)}_{k}) is a quotient of E_{∗}(\underline{P (n)}_{k}), first for E = P (n), then for E = P (n, m), and that these are free E_{∗}-modules. It follows by duality that \overline{θ(m)} exists, but its status as an H-map is left unclear.

We deduce other useful splittings. The canonical map ρ(m−1, m): P (n, m) → P (n, m−1), which kills v_{m}, fits into the exact triangle of spectra

P (n, m) − ; − ; − ; →^{v_{m}} P (n, m) − ; − ; − ; − ; − ; − ; − ; →^{ρ(m−1,m)} P (n, m−1) − ; − ; →^{δ} P (n, m).  (3.2)

On homotopy groups, this induces the obvious short exact sequence

0 → 𝔽_{p}[v_{n}, v_{n+1}, …, v_{m}] − ; − ; − ; →^{v_{m}} 𝔽_{p}[v_{n}, v_{n+1}, …, v_{m}] → 𝔽_{p}[v_{n}, v_{n+1}, …, v_{m−1}] → 0.

Unstably, we have the H-space fibration

\underline{P (n, m)}_{k+2(p^{m}−1)} − ; − ; − ; →^{v_{m}} \underline{P (n, m)}_{k} − ; − ; − ; − ; − ; − ; − ; →^{ρ(m−1,m)} \underline{P (n, m−1)}_{k}.

For k ≤ g(n, m−1), the composite

\underline{P (n, m−1)}_{k} − ; − ; − ; − ; − ; →^{\overline{θ(m−1)}} \underline{P (n)}_{k} − ; − ; − ; − ; →^{ρ(m)} \underline{P (n, m)}_{k}  (3.3)

automatically splits ρ(m−1, m), to yield the decomposition

\underline{P (n, m)}_{k} ≃ \underline{P (n, m−1)}_{k} × \underline{P (n, m)}_{k+2(p^{m}−1)},  (3.4)

where the two injections are (3.3) and v_{m}. This is a decomposition of H-spaces [except when p = 2 and k = g(n, m−1)].

Proof of Theorem 1.12 This is completely analogous to the proof of Theorem 1.16 of [BJW95]. Everything we need is contained in the commutative diagram

[figure: \underline{P (n)}_{k+2(p^{j}−1)}; \underline{P (n)}_{k}; \underline{P (n, m)}_{k}; \underline{P (n, j)}_{k+2(p^{j}−1)}; \underline{P (n, j)}_{k}; v_{j}; ρ(j); ρ(m); ρ(j); v_{j}; ρ(m,j)]  (3.5)

of H-spaces and canonical H-maps, where j > m.

With m given by (1.11), we observe that the spaces Y_{k} and Y_{k+2(p^{j}−1)} appear in the diagram disguised as \underline{P (n, m)}_{k} and \underline{P (n, j)}_{k+2(p^{j}−1)}. We insert the splittings \overline{θ(m)} and \overline{θ(j)} from Lemma 3.1 to produce the desired decomposition of \underline{P (n)}_{k}, as suggested by the decomposition of abelian groups

𝔽_{p}[v_{n}, v_{n+1}, v_{n+2}, …] = 𝔽_{p}[v_{n}, v_{n+1}, …, v_{m}] ⊕ ⨁_{j>m} v_{j}𝔽_{p}[v_{n}, v_{n+1}, …, v_{j}].

(But we warn that our splittings cannot be expected to induce exactly this decomposition of the coefficient ring P (n)_{∗}, and it seems likely that they never do.)

In detail, we map Y_{k} into \underline{P (n)}_{k} by \overline{θ(m)}, which is an H-map [unless p = 2 and k = g(n, m)], and Y_{k+2(p^{j}−1)}, for each j > m, by the H-map (in all cases)

Y_{k+2(p^{j}−1)} − ; − ; − ; →^{\overline{θ(j)}} \underline{P (n)}_{k+2(p^{j}−1)} − ; − ; →^{v_{j}} \underline{P (n)}_{k}.

We multiply these together, using the H-space structure of \underline{P (n)}_{k}, to form a map f: W → \underline{P (n)}_{k} from the restricted direct product W (the union of all finite subproducts) of the based Y-spaces mentioned.

We filter P (n)_{∗} by the ideals J_{j}. We note that v_{j}^{∘}\overline{θ(j)} induces a homomorphism P (n, j)_{∗} → J_{j−1} on homotopy groups that induces the quotient isomorphism

P (n, j)_{∗} = 𝔽_{p}[v_{n}, v_{n+1}, …]/J_{j} − ; − ; →^{v_{j}} J_{j−1}/J_{j}.

This is enough to guarantee that f induces an isomorphism on homotopy groups and is thus a homotopy equivalence. Because the connectivities of the Y-spaces increase, W is homotopy-equivalent to the desired full product and we have (1.13).

The same method applies to \underline{P (n, h)}_{k}, with the simplification that the product W is now finite. (One can also produce decompositions like (1.14) directly from the splittings (3.4) by induction on h, though the resulting maps are different and far more complicated.)

For k = 0, the splitting \overline{θ(n−1)}: Y_{0} = 𝔽_{p} → \underline{P (n)}_{0} is obvious and unique up to homotopy. We can still use diagram (3.5).

Indecomposability On the other hand, we need to know that Y_{k} does not split.

Lemma 3.6 A map f: Y_{k} → Y_{k} is a homotopy equivalence if and only if it induces an isomorphism on the bottom homotopy group π_{k}(Y_{k}) ≅ 𝔽_{p}.

Corollary 3.7 The space Y_{k} does not decompose as a product.

In §12, we prove the following about P (n) and deduce Lemma 3.6 from it.

Lemma 3.8 Represent an unstable operation r: P (n)^{k}(−) → P (n)^{m}(−), where k > 0 and m > 0, by the map r: \underline{P (n)}_{k} → \underline{P (n)}_{m}. Then the induced homomorphism on homotopy groups

r_{∗}: Σ^{k}P (n)_{∗} ≅ π_{∗}(\underline{P (n)}_{k}) − ; − ; − ; − ; →^{π_{∗}(r)} π_{∗}(\underline{P (n)}_{m}) ≅ Σ^{m}P (n)_{∗}  (3.9)

has the properties, for any element v ∈ P (n)_{∗}:

(a) r_{∗}Σ^{k}(v_{n}v) = v_{n}r_{∗}Σ^{k}v;

(b) r_{∗}Σ^{k}(v_{q}v) ≡ v_{q}r_{∗}Σ^{k}v mod I_{q} = (v_{n}, v_{n+1}, …, v_{q−1}), provided k > g(n, q−1).

Construction of maps Our strategy for proving Theorem 1.2 is to construct enough maps to and from the spaces \underline{P (n)}_{k}.

Lemma 3.10 If X is a space for which k(n)_{∗}(X) is a free k(n)_{∗}-module, then P (n)_{∗}(X) is a free P (n)_{∗}-module.

Proof Lemmas 4.7 (with k = m = n) and 2.1 of Yosimura [Yo76] show that P (n)_{∗}(X) is a flat P (n)_{∗}-module. Such modules are free by [ibid., Prop. 1.5].

Lemma 3.11 Let X be a (k−1)-connected space with π_{k}(X) a nonzero finite abelian p-group and suppose k(n)_{∗}(X) is a free k(n)_{∗}-module. Then there exists a map f: X → \underline{P (n)}_{k} that induces a nonzero homomorphism f_{∗}: π_{k}(X) → π_{k}(\underline{P (n)}_{k}) ≅ 𝔽_{p} on the bottom homotopy groups.

Proof Since P (n)_{∗}(X) is a free P (n)_{∗}-module by Lemma 3.10, the universal coefficient theorem [Bo95, Thm. 4.14] gives

P (n)^{∗}(X) ≅ Hom_{P (n)_{∗}}^{∗} (P (n)_{∗}(X), P (n)_{∗}).

As X is (k−1)-connected, P (n)_{k}(X) ≅ H_{k}(X; 𝔽_{p}) ≅ π_{k}(X) ⊗ 𝔽_{p} ≠ 0, and it is clear that suitable cohomology classes f ∈ P (n)^{k}(X), i. e. maps f: X → \underline{P (n)}_{k}, exist.

Proof of parts (a) and (b) of Theorem 1.2 We first note that for k > 0, the space \underline{P (n)}_{k} satisfies the axioms (1.1). Axiom (i) is clear. Axiom (ii) holds by [RW96].

§4 k(n)-towers with v_{n}-free homotopy 14

Axiom (iii) is easy. Take any element Σ^{k}v ∈ Σ^{k}P (n)_{∗} ≅ π_{∗}(\underline{P (n)}_{k}), where v ∈ P (n)_{h}. Viewed as a cohomology class, it is vu_{k+h} ∈ P (n)^{∗}(S^{k+h}). Multiplication by v on P (n)^{∗}(−) is represented by the map we want, v: \underline{P (n)}_{k+h} → \underline{P (n)}_{k}.

Then Y_{k}, being a retract of \underline{P (n)}_{k}, also satisfies the axioms. By Corollary 3.7, it is indecomposable. Uniqueness of Y_{k} and our decompositions will follow from (b), under the assumption that all our H-spaces have finite type.

For the induction step in (b), given any (k−1)-connected space X that satisfies the axioms, define m by (1.11). Then Lemma 3.11 provides a map

h: X − ; − ; → \underline{P (n)}_{k} − ; − ; − ; − ; →^{ρ(m)} \underline{P (n, m)}_{k} = Y_{k}

that induces a nonzero homomorphism h_{∗}: π_{k}(X) → π_{k}(Y_{k}) ≅ 𝔽_{p}. Choose α ∈ π_{k}(X) such that h_{∗}α = 1 ∈ 𝔽_{p}; then axiom (iii) provides a map

f: Y_{k} = \underline{P (n, m)}_{k} − ; − ; − ; − ; →^{\overline{θ(m)}} \underline{P (n)}_{k} − ; − ; → X

that induces f_{∗}1 = α. By Lemma 3.6, h^{∘}f: Y_{k} → Y_{k} is a homotopy equivalence. We use the homotopy fibre j: F → X of h and the multiplication μ on X to construct a homotopy equivalence

Y_{k} × F − ; − ; − ; →^{f×j} X × X − ; − ; →^{μ} X.

Then F, being a retract of X, again satisfies the axioms.

We begin the induction with Z_{0} as the given space, and find a sequence of equivalences Z_{i} ≃ Y_{k_{i}} × Z_{i+1} for i ≥ 0. By finiteness, the spaces Z_{i} become more and more highly connected as i increases, and we deduce Z_{0} ≃ ∏_{i} Y_{k_{i}} as required.

4 k(n)-towers with v_{n}-free homotopy

In this section, we prove Theorem 1.6. We must show that the original axiom (iii) of (1.1) is equivalent (in the presence of the other axioms) to axiom (iii)′ stated in (1.7), which asserts that X is a k(n)-tower with v_{n}-free homotopy. Lemma 4.1 shows that (iii)′ implies (iii), while Lemma 4.3 gives the converse.

Lemma 4.1 Suppose the connected H-space X is a k(n)-tower of finite type with v_{n}-free homotopy. Then axiom (iii) holds: given k > 0, any map S^{k} → X factors through the standard map u_{k}: S^{k} → \underline{P (n)}_{k} to yield a map \underline{P (n)}_{k} → X.

We first show that it does not matter how far up the tower we can lift.

Lemma 4.2 In diagram (1.4), any map f: \underline{P (n)}_{k} → X_{i−1} lifts to a map \underline{P (n)}_{k} → X.

Proof With z_{i} as in Definition 1.3, we note that v_{n}^{c}(z_{i}^{∘}f) = f^{∗}(v_{n}^{c}z_{i}) = 0 in k(n)^{∗}(\underline{P (n)}_{k}). But by [RW96], k(n)_{∗}(\underline{P (n)}_{k}) and hence k(n)^{∗}(\underline{P (n)}_{k}) contain no v_{n}-torsion; therefore z_{i}^{∘}f ≃ 0 and f lifts to f′: \underline{P (n)}_{k} → X_{i}.

By induction and limits, f lifts all the way to X.

Proof of Lemma 4.1 For any connected space Y and k > 0, let us call an element α ∈ π_{k}(Y), or map α: S^{k} → Y, extendable if it extends over u_{k} to a map \underline{P (n)}_{k} → Y.

All elements of π_{∗}(\underline{k(n)}_{q}) ≅ Σ^{q}𝔽_{p}[v_{n}] are obviously extendable. It follows from diagram (1.5) that every element in Ker[π_{∗}(X_{i}) → π_{∗}(X_{i−1})] is extendable.

By Lemma 4.2, any extendable element of π_{∗}(X_{i−1}) lifts in diagram (1.4) to some extendable element of π_{∗}(X).

The sum of any two extendable elements of π_{k}(X) is again extendable: given f_{1}, f_{2}: \underline{P (n)}_{k} → X, we use the given multiplication μ on X to construct the map

\underline{P (n)}_{k} − ; − ; →^{Δ} \underline{P (n)}_{k} × \underline{P (n)}_{k} − ; − ; − ; − ; →^{f_{1}×f_{2}} X × X − ; − ; →^{μ} X.

Together, these facts imply that every element of π_{∗}(X) is extendable.

The space Y_{k} A countable product of k(n)-towers with v_{n}-free homotopy is another such tower (provided it has finite type). In view of Theorem 1.2(b), it suffices to prove the following.

Lemma 4.3 For each k > 0, the space Y_{k} is a k(n)-tower with v_{n}-free homotopy.

We first destabilize the Johnson–Wilson construction [JW75, §4] of a filtration of the spectrum P (n) whose subquotients are suspensions of k(n), and adapt it for P (n, m). The result will be a tower

… − ; − ; → W_{3} − ; − ; → W_{2} − ; − ; → W_{1} − ; − ; → W_{0} = \underline{P (n, m)}_{k}  (4.4)

with trivial homotopy limit, where each W_{i} is the homotopy fibre of a map W_{i−1} → \underline{k(n)}_{q(i)} that is epic on homotopy groups. This depends on the following lemma,

where we recall that π_{∗}(\underline{P (n, m)}_{k}) ≅ Σ^{k}P (n, m)_{∗} etc.

Lemma 4.5 Given v ∈ P (n, m)_{h} and k ≤ g(n, m), there exist an integer c and stable P (n)-operation r such that the composite

s: \underline{P (n, m)}_{k} − ; − ; − ; − ; →^{\overline{θ(m)}} \underline{P (n)}_{k} − ; − ; →^{r} \underline{P (n)}_{k+h−2cN} − ; − ; − ; →^{ρ(n)} \underline{k(n)}_{k+h−2cN}  (4.6)

induces s_{∗}Σ^{k}v = Σ^{k+h−2cN}v_{n}^{c} on homotopy groups.

Proof Lemma 1.12 of [JW75], viewed unstably, supplies c and r.

We construct the tower (4.4) by induction, starting from W_{0} = \underline{P (n, m)}_{k}. Given j: W_{i−1} → \underline{P (n, m)}_{k}, where W_{i−1} is (k+h−1)-connected and j_{∗}: π_{∗}(W_{i−1}) →

π_{∗}(\underline{P (n, m)}_{k}) is monic, we choose a bottom nonzero element u ∈ π_{k+h}(W_{i−1}) to kill. Lemma 4.5 provides a map s: \underline{P (n, m)}_{k} → \underline{k(n)}_{k+h−2cN} such that s_{∗}j_{∗}u = Σ^{k+h−2cN}v_{n}^{c}. For dimensional reasons, s^{∘}j factors through v_{n}^{c}: \underline{k(n)}_{k+h} → \underline{k(n)}_{k+h−2cN} to produce the desired map W_{i−1} → \underline{k(n)}_{k+h}, with fibre W_{i}.

This is the wrong kind of tower for Definition 1.3. To correct it, we could take the homotopy fibre X_{i} of each map W_{i} → \underline{P (n)}_{k}, to express Ω\underline{P (n, m)}_{k} = \underline{P (n, m)}_{k−1} as a k(n)-tower with v_{n}-free homotopy. This approach fails to produce a suitable tower for \underline{P (n, m)}_{k} when k = g(n, m). Our solution is to observe that it is inefficient to deloop and then take fibres; instead, we prove only what we actually need.

Lemma 4.7 Given a (k+h)-connected map q: \underline{P (n, m)}_{k} → X and any map s: \underline{P (n, m)}_{k} → \underline{k(n)}_{k+h−2cN}, there exists a v_{n}-torsion map z: X → \underline{k(n)}_{k+h+1} such that s is one value of the following Toda bracket,

s ∈ ⟨v_{n}^{c}, z, q⟩: \underline{P (n, m)}_{k} − ; − ; → \underline{k(n)}_{k+h−2cN}.

Proof We are using the adjoint (but equivalent) description of a Toda bracket in terms of loop spaces instead of suspensions. We build the commutative diagram Figure 4.1 in which the two rows are fibration sequences. We start with the obvious

[figure: \underline{P (n, m)}_{k}; \underline{k(n)}_{k+h}; X′; X; \underline{k(n)}_{k+h+1}; \underline{k(n)}_{k+h}; \underline{k(n)}_{k+h−2cN}; \underline{G}_{k+h−2cN}; \underline{k(n)}_{k+h+1}; \underline{k(n)}_{k+h−2cN+1}; q′; q; =; g; z; f; =; −v_{n}^{c}; π; δ; v_{n}^{c}]

Figure 4.1: Construction of the Toda bracket ⟨v_{n}^{c}, z, q⟩

fibration as the bottow row, where (stably) G denotes the cofibre of v_{n}^{c}: k(n) → k(n), with homotopy 𝔽_{p}[v_{n}]/(v_{n}^{c}). By the connectivity of q, q^{∗}: G^{j}(X) → G^{j}(\underline{P (n, m)}_{k}) is an isomorphism for j ≤ k + h − 2cN + 2N − 1, so that π^{∘}s factors uniquely through q to yield a map f such that f^{∘}q = π^{∘}s. We put z = δ^{∘}f, which automatically satisfies v_{n}^{c}z = 0. We define X′ as the homotopy fibre of z, and fill in g to form a morphism of fibrations.

Since X′ may be constructed as a pullback, we can fill in q′ to lift q and satisfy g^{∘}q′ = s. (Equivalently, v_{n}^{c}^{∘}[\underline{P (n, m)}_{k}, \underline{k(n)}_{k+h}] is part of the indeterminacy of the Toda bracket.) Then by definition, g^{∘}q′ = s is one value of the Toda bracket.

Proof of Lemma 4.3 We build the desired tower for \underline{P (n, m)}_{k} by induction, starting from a point as X_{0}. Suppose we have constructed a map q_{i−1}: \underline{P (n, m)}_{k} → X_{i−1} that induces a surjection q_{i−1∗}: Σ^{k}P (n, m)_{∗} → π_{∗}(X_{i−1}) on homotopy groups, with kernel K an 𝔽_{p}[v_{n}]-submodule. We choose a bottom nonzero element Σ^{k}v ∈ K_{k+h} to kill, where K_{i} = 0 for i < k + h. Then Lemma 4.5 provides a map s: \underline{P (n, m)}_{k} → \underline{k(n)}_{k+h−2cN}. We use Lemma 4.7 to build Figure 4.1, taking q_{i−i} as q and X_{i−1} as X.

We next take homotopy groups of Figure 4.1. By Lemma 3.8(a), applied to r^{∘}\overline{θ(m)}^{∘}ρ(m): \underline{P (n)}_{k} → \underline{P (n)}_{k+h−2cN}, s_{∗} is a homomorphism of 𝔽_{p}[v_{n}]-modules. By exactness and the hypothesis that q_{∗}Σ^{k}(v_{n}^{i}v) = 0, q′_{∗}(v_{n}^{i}v) must lift to −Σ^{k+h}v_{n}^{i} ∈ π_{∗}(\underline{k(n)}_{k+h}). It now follows that q′_{∗} is also epic, with kernel

K′ = Ker [s_{∗}|K: K − ; − ; → Σ^{k+h−2cN}𝔽_{p}[v_{n}]] ⊂ K,

a strictly smaller 𝔽_{p}[v_{n}]-submodule of Σ^{k}P (n, m)_{∗}. We take X′ as X_{i} and q′ as q_{i}.

§5 Splittings of P (n)-cohomology 17

The kernels K become more and more highly connected as i increases, hence \underline{P (n, m)}_{k} is the homotopy limit of the spaces X_{i}.

5 Splittings of P (n)-cohomology

In this section, we translate the H-space splittings in §3 into splittings of P (n)- cohomology. We also deduce Theorems 1.21 and 1.23.

We have yet to prove Lemmas 3.1, 3.6 and 3.8. Lemma 3.1 is equivalent to the following statement for the represented functors. (We do not mention Lemmas 3.6 and 3.8 again until §12.)

Lemma 5.1 Assume that k ≤ g(n, m), where m ≥ n. Then there is a splitting

\overline{θ(m)}: P (n, m)^{k}(X) − ; − ; → P (n)^{k}(X)

of ρ(m): P (n)^{k}(X) → P (n, m)^{k}(X) that satisfies ρ(m)^{∘}\overline{θ(m)} = id and is natural for spaces X. It is additive [except when p = 2 and k = g(n, m)].

This we actually prove in §9 [except the nonadditive case; see [Bo]], by constructing an idempotent cohomology operation θ(m) in P (n)^{k}(X). Unlike the case of BP, the use of nonadditive operations yields no further splittings [unless p = 2].

We next translate equation (3.4).

Corollary 5.2 For k ≤ g(n, m−1), where m > n, we have the natural short exact sequence of abelian groups

0 → P (n, m)^{k+2(p^{m}−1)}(X) − ; − ; − ; →^{v_{m}} P (n, m)^{k}(X) − ; − ; − ; − ; − ; − ; − ; →^{ρ(m−1,m)} P (n, m−1)^{k}(X) → 0.

This splits naturally [unless p = 2 and k = g(n, m−1)].

This implies our homological dimension bound, by the methods of [JW73].

Proof of Theorem 1.21 Following Yosimura [Yo76, Thm. 4.8], we need to show that

ρ(m−1, m): P (n, m)_{i}(X) − ; − ; → P (n, m−1)_{i}(X)  (5.3)

is epic for all i. For i ≤ 2(p^{m}−1), this is trivial, by the exact sequence

P (n, m)_{i}(X) − ; − ; − ; − ; − ; − ; − ; →^{ρ(m−1,m)} P (n, m−1)_{i}(X) − ; − ; →^{δ} P (n, m)_{i−2(p^{m}−1)−1}(X)

arising from the exact triangle (3.2).

For i > 2(p^{m}−1), we embed X in ℝ^{2q+1}, where q is the dimension of X, and take a regular neighborhood V of X. By Poincaré duality, (5.3) is equivalent to

ρ(m−1, m): P (n, m)^{2q+1−i}(V, ∂V) − ; − ; → P (n, m−1)^{2q+1−i}(V, ∂V).

This is epic by Corollary 5.2, because by hypothesis

2q + 1 − i ≤ (g(n, m) − 2) + 1 − (2(p^{m}−1) + 1) = g(n, m−1).

We also translate Theorem 1.12, using the splittings made explicit in §3, and finally deduce Theorem 1.23. (Decompositions like (5.6) also follow directly from Corollary 5.2 by induction on h, though the resulting homomorphisms are different.)

Theorem 5.4 Let X be any space and suppose that m ≥ n > 0.

(a) If k ≤ g(n, m) [replaced by k < g(n, m) if p = 2], we have the natural abelian group decomposition

P (n)^{k}(X) ≅ P (n, m)^{k}(X) ⊕ ∏_{j>m} P (n, j)^{k+2(p^{j}−1)}(X),  (5.5)

where the first factor on the right is injected by \overline{θ(m)}, and the others by

P (n, j)^{k+2(p^{j}−1)}(X) − ; − ; − ; →^{\overline{θ(j)}} P (n)^{k+2(p^{j}−1)}(X) − ; − ; →^{v_{j}} P (n)^{k}(X).

Hence, by composition with ρ(h): P (n)^{k}(X) → P (n, h)^{k}(X) for any h > m,

P (n, h)^{k}(X) ≅ P (n, m)^{k}(X) ⊕ ⨁_{j=m+1}^{h} P (n, j)^{k+2(p^{j}−1)}(X).  (5.6)

These decompositions are maximal if k > g(n, m−1) [also for k = g(n, m−1) if p = 2]. (They are in no sense decompositions as P (n)_{∗}-modules.)

(b) If p = 2 and k = g(n, m), we replace equations (5.5) and (5.6) by the natural short exact sequences

0 → ∏_{j>m} P (n, j)^{k+2^{j+1}−2}(X) → P (n)^{k}(X) − ; − ; − ; − ; →^{ρ(m)} P (n, m)^{k}(X) → 0

and

0 → ⨁_{j=m+1}^{h} P (n, j)^{k+2^{j+1}−2}(X) → P (n, h)^{k}(X) − ; − ; − ; − ; − ; →^{ρ(m,h)} P (n, m)^{k}(X) → 0.

Because P (n, n) = k(n) is so familiar, we break out the special case m = n. For h < 2(p^{n}−1), we can even replace k(n) by the periodic Morava K-theory K(n).

Corollary 5.7 For h ≤ 2p^{n}, where n > 0, we have, for all spaces X, the natural abelian group decomposition

P (n)^{h}(X) ≅ k(n)^{h}(X) ⊕ ∏_{j>n} P (n, j)^{h+2(p^{j}−1)}(X),

except that if p = 2 and h = 2^{n+1}, we have only the natural short exact sequence

0 → ∏_{j>n} P (n, j)^{h+2^{j+1}−2}(X) → P (n)^{h}(X) − ; − ; − ; →^{ρ(n)} k(n)^{h}(X) → 0.

Remark All the splittings exhibited above depend on the choice of θ(m), which is not canonical and does not respect multiplication by v_{j}.

Proof of Theorem 1.23 As X is finite-dimensional, the sum in equation (1.22) is essentially finite. Lemma 5.1 shows that \overline{ρ(m)} is epic. It is clear from Theorem 5.4 that Ker \overline{ρ(m)} is contained in the sum, and must therefore be the sum.

§6 Stable operations in P (n)-cohomology 19

6 Stable operations in P (n)-cohomology

In this section, we describe the stable operations in P (n)-cohomology P (n)^{∗}(−) in the style of [Bo95]. The results are old and well known [except for p = 2], but we include them for completeness and ease of reference; more importantly, they serve as a pattern for §§7, 10.

Monoidal structure (For the language of monoidal categories and functors, see e. g. Mac Lane [Ma71, Ch. VII].) Since P (n)_{∗} is a commutative ring [even if p = 2], the graded category (FMod_{∗}, ˆ⊗, P (n)_{∗}) of complete Hausdorff filtered P (n)_{∗}-modules is a symmetric monoidal category, with all (completed) tensor products taken over P (n)_{∗}. The cross product makes P (n)-cohomology a monoidal functor,

P (n)^{∗}(−): (Ho^{op}, ×, point) − ; − ; → (FMod, ˆ⊗, P (n)_{∗}).  (6.1)

(Conveniently, P (n)^{∗}(X) has no phantom classes and so is already complete Hausdorff.) For homology, we similarly have the monoidal functor

P (n)_{∗}(−): (Ho, ×, point) − ; − ; → (Mod, ⊗, P (n)_{∗}),  (6.2)

with values in the category Mod of discrete P (n)_{∗}-modules. Both functors are symmetric for p ≠ 2.

The cohomology version for spectra and graded maps is

P (n)^{∗}(−, o): (Stab_{∗}^{op}, ∧, S^{0}) − ; − ; → (FMod_{∗}, ˆ⊗, P (n)_{∗}),

and similarly for homology. (We include the basepoint subspectrum o in our notation as a reminder that all stable (co)homology is reduced, and to distinguish it from the (co)homology of a space, which here will generally be absolute.)

Operations Because Γ = P (n)_{∗}(P (n), o) is a free P (n)_{∗}-module, we may identify its dual P (n)_{∗}-module DΓ with A = P (n)^{∗}(P (n), o), the algebra of all stable operations in P (n)-cohomology, and have available all the stable machinery and results of [Bo95]. In particular, we have the monoidal functor

S: (FMod_{∗}, ˆ⊗, P (n)_{∗}) − ; − ; → (FMod_{∗}, ˆ⊗, P (n)_{∗})  (6.3)

defined by SM = FMod_{∗}(A, M). If M is filtered by submodules F^{a}M, we filter SM by the submodules F^{a}SM = SF^{a}M; as in [ibid.], SM is again complete Hausdorff. The ring spectrum structure of P (n) gives S its monoidal structure (see diagram (6.10) below), which is symmetric for p ≠ 2. (As in [Bo95], care is needed in keeping track of the many P (n)_{∗}-module actions, some of which are not obvious.)

The action of stable P (n)-cohomology operations is visibly encoded in the monoidal natural transformation

ρ_{X}: P (n)^{∗}(X) − ; − ; → S(P (n)^{∗}(X)) = FMod_{∗}(P (n)^{∗}(P (n), o), P (n)^{∗}(X))  (6.4)

defined by ρ_{X}x = x^{∗}, where we treat x ∈ P (n)^{∗}(X) as a map of spectra x: X_{+} → P (n) and X_{+} denotes the disjoint union of X and a (new) basepoint.

The coaction To convert the action of A into a coaction by Γ, we recall the natural isomorphism [Bo95, (11.4)]

θM: S′M = M ˆ⊗ Γ ≅ FMod_{∗}(DΓ, M) ≅ FMod_{∗}(A, M) = SM,

given on x ∈ M, c ∈ Γ, and r ∈ A ≅ DΓ by

((θM)(x ⊗ c))r = ±⟨r, c⟩x,  (6.5)

with the expected sign. We use it to transfer all the structure from the functor S to S′ and replace (6.4) by the equivalent natural transformation

ρ_{X}: P (n)^{∗}(X) − ; − ; → S′P (n)^{∗}(X) = P (n)^{∗}(X) ˆ⊗ Γ.  (6.6)

The monoid The resulting monoidal structure on S′ is necessarily induced by a monoid structure on the P (n)_{∗}-module Γ (as we see by naturality from the case M = N = P (n)_{∗} in diagram (6.10), below), and conversely. We simply need to compute it.

Lemma 6.7 The following monoid structure on Γ, which is inherited from the monoidal functor S, makes the natural transformation (6.6) monoidal:

(a) If p is odd, the multiplication on Γ is the obvious one,

\begin{matrix}Γ ⊗ Γ = P (n)_{∗}(P (n), o) ⊗ P (n)_{∗}(P (n), o) − ; − ; →^{×} P (n)_{∗}(P (n) ∧ P (n), o) \\ − ; − ; →^{ϕ_{∗}} P (n)_{∗}(P (n), o) = Γ, \end{matrix}  (6.8)

as inferred by writing Γ = P (n)_{∗}(P (n), o). The unit homomorphism of Γ is

P (n)_{∗} = P (n)_{∗}(S^{0}, o) − ; − ; →^{η_{∗}} P (n)_{∗}(P (n), o) = Γ.

(b) If p = 2, the multiplication is instead

\begin{matrix}Γ ⊗ Γ = P (n)_{∗}(P (n), o) ⊗ P (n)_{∗}(P (n), o) − ; − ; →^{\overline{×}} P (n)_{∗}(P (n) ∧ P (n), o) \\ − ; − ; →^{ϕ_{∗}} P (n)_{∗}(P (n), o) = Γ, \end{matrix}  (6.9)

which is better suggested by writing Γ = \overline{P (n)}_{∗}(P (n), o). The unit is unaffected.

Proof The multiplication ϕ on P (n) induces

ϕ^{∗}: DΓ ≅ P (n)^{∗}(P (n), o) − ; − ; → P (n)^{∗}(P (n) ∧ P (n), o) ≅ DΓ ˆ⊗ DΓ,

with the help of the Künneth formula [Bo95, Thm. 4.19]. The natural transformations ζ(M, N) for S′ and S form the left and right sides of the commutative diagram

[figure: (M ˆ⊗ Γ) ˆ⊗(N ˆ⊗ Γ); FMod_{∗}(DΓ, M) ˆ⊗ FMod_{∗}(DΓ, N); M ˆ⊗ N ˆ⊗(Γ ⊗ Γ); FMod_{∗}(DΓ ˆ⊗ DΓ, M ˆ⊗ N); M ˆ⊗ N ˆ⊗ Γ; FMod_{∗}(DΓ, M ˆ⊗ N); θM⊗θN; ≅; ≅; M⊗N⊗ϕ; FMod_{∗}(ϕ^{∗},id); θ(M _{⊗} N)]  (6.10)

which features the multiplication ϕ: Γ ⊗ Γ → Γ. We evaluate on x ⊗ c ⊗ y ⊗ d, where x ∈ M, y ∈ N, and c, d ∈ Γ. By (6.5), the lower route gives the element r ↦ ±⟨r, cd⟩x⊗y of FMod_{∗}(DΓ, M ˆ⊗ N). The upper route gives r⊗s ↦ ±⟨r, c⟩⟨s, d⟩x⊗y in FMod_{∗}(DΓ ˆ⊗ DΓ, M ˆ⊗ N). Assuming p ≠ 2, we can rewrite this as ±⟨r×s, c×d⟩x⊗ y; then in FMod_{∗}(DΓ, M ˆ⊗ N) we find

r ↦ ±⟨ϕ^{∗}r, c×d⟩x ⊗ y = ±⟨r, ϕ_{∗}(c×d)⟩x ⊗ y.

Thus cd = ϕ_{∗}(c×d) (with no sign) as expected, which is (a).

[If p = 2, this calculation is false; we must use equation (2.15) instead, which states that ⟨r, c⟩⟨s, d⟩ = ⟨r×s, c\overline{×}d⟩. Then cd = ϕ_{∗}(c\overline{×}d), for (b).]

The unit z: P (n)_{∗} → SP (n)_{∗} takes 1 = η ∈ P (n)_{∗} to the homomorphism

DΓ ≅ P (n)^{∗}(P (n), o) − ; − ; →^{η^{∗}} P (n)^{∗}(S^{0}, o) ≅ P (n)_{∗},

in other words, r ↦ ⟨η^{∗}r, 1⟩ = ⟨r, η_{∗}1⟩. Comparison with (6.5) shows that the corresponding element of S′P (n)_{∗} = P (n)_{∗} ⊗ Γ ≅ Γ is η_{∗}1.

If X is a point in (6.6), we find the right unit ring homomorphism

η_{R}: P (n)_{∗} − ; − ; → S′P (n)_{∗} = P (n)_{∗} ⊗ Γ ≅ Γ,  (6.11)

which is used to make Γ a right P (n)_{∗}-module (hence a bimodule). Since ρ is monoidal, this action makes (6.6) a homomorphism of P (n)_{∗}-modules.

The Hopf algebroid Now we add the algebra structure of A. Exactly as in [Bo95, §10], composition of operations and the identity operation induce natural transformations ψ: S → SS and ϵ: S → I. These make S a monoidal comonad in the category FMod, and (6.4) makes P (n)^{∗}(X) an S-coalgebra.

We transfer this structure too to S′. The resulting monoidal comonad structure on S′ is necessarily induced by a Hopf algebroid structure on Γ (as we see by taking M = P (n)_{∗}), and conversely. This structure consists of a coassociative comultiplication ψ_{S}: Γ → Γ ⊗ Γ with counit ϵ_{S}: Γ → P (n)_{∗}. These behave exactly as in Adams [Ad74] or [Bo95, Thm. 11.35]; in particular, ψ_{S} and ϵ_{S} are homomorphisms of P (n)_{∗}- bimodules and algebras. [This all works without change for p = 2; see [Bo].]

Proposition 6.12 The stable operations in P (n)-cohomology are encoded in the Hopf algebroid Γ = P (n)_{∗}(P (n), o) [replaced by Γ = \overline{P (n)}_{∗}(P (n), o) for p = 2].

The discussion of the structure of Γ carries over from the case K(n) in [Bo95] with little change [except that we allow p = 2]. We even use the same test spaces.

The one-point space We already discussed this in (6.11). The coaction ρ reduces to the ring homomorphism η_{R}, which is determined by the elements

w_{k} = η_{R}v_{k} ∈ Γ_{2(p^{k}−1)} for k ≥ n.  (6.13)

Complex orientation Our next test space is complex projective space ℂP^{∞}. As P (n) inherits a complex orientation from BP (or MU), we have P (n)^{∗}(ℂP^{∞}) = P (n)_{∗}[[x]], the formal power series ring generated by the Chern class x = x(ξ) of the Hopf line bundle ξ over ℂP^{∞}, filtered by powers of the ideal (x).

The coaction ρ for ℂP^{∞} defines elements b_{j} ∈ Γ_{2j−2} by the formula [Bo95, (13.2)]

ρx = b(x) = ∑_{j=1}^{∞} x^{j} ⊗ b_{j}  in P (n)^{∗}(ℂP^{∞}) ˆ⊗ Γ ≅ Γ[[x]].  (6.14)

Here, b(x) is a useful formal abbreviation for the right side. As always in the stable context [ibid., Prop. 13.4], b_{1} = 1 and b_{0} = 0.

Further, the comultiplication ψ_{S} is given on b_{i} as the coefficient of x^{i} in

ψ_{S}b(x) = ∑_{j=1}^{∞} b(x)^{j} ⊗ b_{j}  in (Γ ⊗ Γ)[[x]],  (6.15)

and ϵ_{S}b_{j} = 0 for all j > 1.

Since P (n) is p-local, we need only the accelerated elements b_{(j)} = b_{p^{j}} ∈ Γ_{2(p^{j}−1)} for j ≥ 0, where b_{(0)} = 1; the other b’s are expressible in terms of these and the v’s and w’s by [ibid., Lemma 13.7].

The p-th power map ζ: ℂP^{∞} → ℂP^{∞}, whose bundle interpretation is ζ^{∗}ξ = ξ^{⊗p}, induces in cohomology

ζ^{∗}x = [p](x) = ∑_{i=N}^{∞} g_{i}x^{i+1} in P (n)^{∗}(ℂP^{∞}) = P (n)_{∗}[[x]]

for certain coefficients g_{i} ∈ P (n)_{2i}. This formal power series is known as the p-series for P (n). There are no lower terms as g_{0} = p = 0 in P (n)_{0}. (The elements g_{i} are traditionally written a_{i}, but we rename them in order to avoid confusion with other elements, also named a_{i}, that appear shortly.)

We need only one standard fact [RW77, Thm. 3.11(b)] about the p-series:

[p](x) ≡ v_{k}x^{p^{k}}  mod (v_{n}, …, v̂_{k}, …)  (6.16)

for any k ≥ n, where the ideal is generated by all the v’s except v_{k}. In words, [p](x) contains terms v_{k}x^{p^{k}} but not λv_{k}^{i}x^{q} for any i > 1. In particular,

[p](x) = v_{n}x^{p^{n}} + v_{n+1}x^{p^{n+1}} + higher terms.  (6.17)

Hence as k varies, we have

[p](x) ≡ ∑_{k=n}^{∞} v_{k}x^{p^{k}}  mod V^{2},  (6.18)

where V denotes the maximal ideal (v_{n}, v_{n+1}, v_{n+2}, …) ⊂ P (n)_{∗}.

Naturality of ρ with respect to the map ζ yields the identity [Bo95, (13.11)]

b([p](x)) = [p]_{R}(b(x)) = ∑_{i=N}^{∞} b(x)^{i+1}η_{R}g_{i} in Γ[[x]].  (6.19)

The lowest power of x that occurs is x^{p^{n}}.

Definition 6.20 For each k ≥ n, we define the k-th main stable relation (R_{k}) as the coefficient of x^{p^{k}} in equation (6.19).

Since b_{(0)} = 1, the first relation (R_{n}) is simply v_{n}1 = w_{n}, which implies that every stable operation is v_{n}-linear. For k > n, equation (6.18) shows that (6.19) has a term w_{k}x^{p^{k}} on the right, and (R_{k}) becomes an inductive formula for w_{k} in terms of the v’s and b’s and lower w’s.

Cohomology of a lens space, for p odd Our final test space is the 2N-skeleton L of the lens space K(ℤ/p, 1). Geometrically, L is the orbit space of the standard ℤ/p -action on the unit sphere S^{2N+1} ⊂ ℂ^{N+1} given as complex multiplication by ℤ/p ⊂ S^{1} ⊂ ℂ, with the top cell omitted by requiring the last coordinate to be real non-negative, up to the action of ℤ/p. (Retaining the top cell, as in [Bo95], adds some extra complication but offers little benefit.)

Following [Bo95, §14], its cohomology is

P (n)^{∗}(L) = (E(u) ⊗ T P_{n}(x))/(ux^{N}),  (6.21)

because the Atiyah–Hirzebruch spectral sequence can support no differential. Here, x is induced from the Chern class of the Hopf line bundle on ℂP^{N}, which is a quotient space of L, and u is uniquely defined as restricting to the standard generator u_{1} ∈ P (n)^{∗}(S^{1}), where we recognize the 1-skeleton L^{1} of L as the circle S^{1}.

Since x is a Chern class, the coaction ρ_{L} is given on x by naturality as ρ_{L}x = b(x). Although L is not an H-space, there are, as in [ibid., (14.31)], partial multiplications L^{2k} × L^{2m} → L on the skeletons whenever k + m = N, which imply that

ρ_{L}u = u⊗1 + ∑_{i=0}^{n−1} x^{p^{i}} ⊗a_{(i)} in P (n)^{∗}(L) ⊗ Γ  (6.22)

for certain elements a_{(i)} ∈ Γ_{2p^{i}−1} that this equation defines. (We warn that these generators differ from Würgler’s [Wü77] and Yagita’s [Ya77] generators a_{i} by the conjugation in Γ; as a result, certain formulae become transposed. Our generators are chosen for compatibility with [Bo95] and [Wi84], because they destabilize properly in §§7, 10.) The element a_{(n)} does not exist because u fails to lift to the 2p^{n}-skeleton of the lens space. As in [Bo95, Thm. 14.32], the coalgebra structure is given by

ψ_{S}a_{(k)} = a_{(k)}⊗1 + ∑_{i=0}^{k−1} b_{(k−i)}^{p^{i}}⊗a_{(i)} + 1⊗a_{(k)}  (6.23)

and ϵ_{S}a_{(k)} = 0.

Cohomology of real projective space, for p = 2 Here, the same test space L is better known as real projective space ℝP^{2N}. It remains true that the Atiyah– Hirzebruch spectral sequence can support no differential, so that

P (n)^{∗}(ℝP^{2N}) = P (n)_{∗}[t]/(t^{2N+1}),  (6.24)

generated by the unique nonzero element t ∈ P (n)^{1}(ℝP^{2N}). As above, we find that

ρt = t⊗1 + ∑_{i=0}^{n−1} t^{2^{i+1}} ⊗a_{(i)},  (6.25)

which defines elements a_{(i)} ∈ Γ_{2^{i+1}−1}. Indeed, this formula is identical to equation (6.22), since x = t^{2} is the Chern class of the complexified real Hopf line bundle. Thus equation (6.23) remains valid for p = 2.

Summary Würgler [Wü77] and Yagita [Ya77] both proved that we now have enough elements of Γ to handle all stable operations.

§7 Additive operations in P (n)-cohomology 24

Theorem 6.26 The stable operations in P (n)-cohomology are dual to the Hopf algebroid Γ = P (n)_{∗}(P (n), o) [replaced by \overline{P (n)}_{∗}(P (n), o) if p = 2], which is generated as a P (n)_{∗}-algebra by the elements b_{(j)} and a_{(i)} defined by equations (6.14) and (6.22) [replaced by (6.25) if p = 2].

(a) For odd p, as a P (n)_{∗}-algebra,

Γ = P (n)_{∗}(P (n), o) = E(a_{(0)}, a_{(1)}, …, a_{(n−1)}) ⊗ P (b_{(1)}, b_{(2)}, b_{(3)}, …);

(b) For p = 2, as a P (n)_{∗}-algebra,

Γ = \overline{P (n)}_{∗}(P (n), o) = P (a_{(0)}, a_{(1)}, …, a_{(n−1)}, b_{(n+1)}, b_{(n+2)}, …),

and the elements b_{(j)} for j ≤ n are given by the relations

a_{(i)}^{2} = b_{(i+1)} for 0 ≤ i ≤ n − 1;  (6.27)

(c) As a left P (n)_{∗}-module, Γ is free with a basis consisting of all monomials

a^{I}b^{J} = a_{(0)}^{i_{0}}a_{(1)}^{i_{1}} … a_{(n−1)}^{i_{n−1}}b_{(1)}^{j_{1}}b_{(2)}^{j_{2}}b_{(3)}^{j_{3}} …,

with multi-indices I = (i_{0}, i_{1}, …, i_{n−1}) and J = (j_{1}, j_{2}, …) in which each i_{r} = 0 or 1;

(d) The right P (n)_{∗}-action on Γ is given by multiplication by the elements w_{k} = η_{R}v_{k}, where w_{n} = v_{n}1 and w_{k} is determined inductively for k > n by the main relation (R_{k}) (see Definition 6.20);

(e) The comultiplication ψ_{S}: Γ → Γ⊗Γ is the P (n)_{∗}-algebra homomorphism given on the generators by equations (6.15) and (6.23);

(f) The counit ϵ_{S}: Γ → P (n)_{∗} is the P (n)_{∗}-algebra homomorphism given on the generators by ϵ_{S}a_{(i)} = 0 for all i and ϵ_{S}b_{(j)} = 0 for j > 0.

Proof What survives intact from Würgler [Wü77, Thm. 2.13] and Yagita [Ya77, Lemma 3.5], even for p = 2, is (c) (using the conjugate generators to the a_{(i)}). Parts (a), (d), (e) and (f) need no further comment.

[In (b), commutativity is not trivial; see Nassau [Na02] or [Bo]. Since t^{2} is a Chern class, (ρt)^{2} = ρ(t^{2}) = b(t^{2}). Comparing the coefficients of t^{2^{i+2}} with the help of (6.14) and (6.25), we deduce (6.27) for i < n−1.

This argument fails for i = n−1, as t^{2^{n+1}} = 0; nevertheless, the result still holds by [Na02, Thm. 2], which corrects [KW87]. Alternatively, the map of ring spectra P (n) → P (n+1) in Lemma 2.7 sends each generator of Γ(n) = Γ to its namesake in Γ(n+1). As a_{(n−1)}^{2} = b_{(n)} in Γ(n+1), the only candidates for a_{(n−1)}^{2} in Γ(n) are b_{(n)} and b_{(n)} + v_{n}1. Since ϵ_{S}(a_{(n−1)}^{2}) = (ϵ_{S}a_{(n−1)})^{2} = 0, we must choose b_{(n)}.]

7 Additive operations in P (n)-cohomology

In this section, we describe the additive unstable operations in P (n)-cohomology in the style of [BJW95], in terms of a certain bigraded algebra Q_{∗}^{∗}, which, like Γ, is a P (n)_{∗}-bimodule equipped with a coalgebra structure (ψ_{A}, ϵ_{A}) (called (Q(ψ), Q(ϵ)) in [BJW95]) that encodes the composition of operations and the identity operation. Although the results bear a strong formal resemblance to the stable results in §6, the stable proofs do not carry over; instead, one has to compute the whole Hopf ring in §11 and then take the indecomposables.

For p odd, we define Q_{∗}^{∗} = QP (n)_{∗}(\underline{P (n)}_{∗}), the algebra of indecomposables in

the Hopf ring P (n)_{∗}(\underline{P (n)}_{∗}). Specifically, Q_{i}^{k} denotes the group of indecomposables

in degree i of the Hopf algebra P (n)_{∗}(\underline{P (n)}_{k}); its elements have total degree i − k in Q_{∗}^{∗} (and this is the degree that governs signs). The multiplication and unit in Q_{∗}^{∗} are induced from ^{∘}-multiplication and the element [1] in the Hopf ring by the homomorphisms (10.1). The left P (n)_{∗}-module action is induced from the Hopf ring: if v ∈ P (n)_{j} and c ∈ Q_{i}^{k}, we have vc ∈ Q_{j+i}^{k}. [When p = 2, it should be no surprise after Proposition 6.12 that the correct Hopf ring to consider is not P (n)_{∗}(\underline{P (n)}_{∗})

but \overline{P (n)}_{∗}(\underline{P (n)}_{∗}); in this case, we set Q_{∗}^{∗} = Q\overline{P (n)}_{∗}(\underline{P (n)}_{∗}). This is the same left

P (n)_{∗}-module as QP (n)_{∗}(\underline{P (n)}_{∗}), but with slightly different multiplication.]

By [RW96, Cor. 1.5], both Q_{∗}^{∗} and the Hopf ring are free P (n)_{∗}-modules. These conditions ensure [BJW95, Lemma 4.16(a)] that the dual module to Q_{∗}^{∗} is indeed the module of all additive unstable operations on P (n)-cohomology, and make available all the machinery and results on additive operations. We thus identify:

(i) The additive unstable operation r: P (n)^{k}(−) → P (n)^{m}(−);
(ii) The primitive cohomology class rι_{k} ∈ P (n)^{m}(\underline{P (n)}_{k});

(iii) The representing H-map of H-spaces r: \underline{P (n)}_{k} → \underline{P (n)}_{m}, up to homotopy;

(iv) The P (n)_{∗}-linear functional ⟨r, −⟩: Q_{∗}^{k} → P (n)_{∗}, of degree k − m.

The action of additive operations on P (n)^{∗}(X) is encoded in coactions

ρ_{X}: P (n)^{k}(X) − ; − ; → P (n)^{∗}(X) ˆ⊗ Q_{∗}^{k}  (7.1)

(one for each k), which are monoidal as k varies [even if p = 2].

To construct the generators of Q_{∗}^{∗}, we use the same test spaces as stably in §6, together with the circle. We record the values of ψ_{A} and ϵ_{A} on each generator.

Cohomology of a point The right unit ring homomorphism η_{R}: P (n)_{∗} → Q_{0}^{∗} is just the coaction ρ for the one-point space, and so is determined by the elements

w_{k} = η_{R}v_{k} ∈ Q_{0}^{−2(p^{k}−1)} for k ≥ n.  (7.2)

We use η_{R} to make Q_{∗}^{∗} a right P (n)_{∗}-module and the coactions ρ_{X} in (7.1) into a P (n)_{∗}-module homomorphism.

Cohomology of a circle The coaction for the circle S^{1} defines the suspension element e ∈ Q_{1}^{1} by

ρu_{1} = u_{1} ⊗ e in P (n)^{∗}(S^{1}) ⊗ Q_{∗}^{1} = E(u_{1}) ⊗ Q_{∗}^{1}.  (7.3)

As in [BJW95, Prop. 12.3(d)], ψ_{A}e = e ⊗ e and ϵ_{A}e = 1.

Then for any j > 0, the coaction for the j-sphere S^{j} is given by

ρu_{j} = u_{j} ⊗ e^{j}  in P (n)^{∗}(S^{j}) ⊗ Q_{∗}^{j} = E(u_{j}) ⊗ Q_{∗}^{j}.  (7.4)

Given any additive operation r: P (n)^{k}(−) → P (n)^{m}(−), represented by the map r: \underline{P (n)}_{k} → \underline{P (n)}_{m}, where k, m > 0, we use P (n)^{k}(S^{j}) ≅ π_{j}(\underline{P (n)}_{k}) ≅ Σ^{k}P (n)_{j−k} to rewrite the induced homomorphism on homotopy groups as

r_{∗}: Σ^{k}P (n)_{∗} ≅ π_{∗}(\underline{P (n)}_{k}) − ; − ; − ; − ; →^{π_{∗}(r)} π_{∗}(\underline{P (n)}_{m}) ≅ Σ^{m}P (n)_{∗}.  (7.5)

By [BJW95, Cor. 12.4], this is given on Σ^{k}v, where v ∈ P (n)_{i}, by the formula

r_{∗}(Σ^{k}v) = Σ^{m}⟨r, e^{k+i}(η_{R}v)⟩.  (7.6)

Complex orientation The coaction for ℂP^{∞} defines elements b_{j} ∈ Q_{2j}^{2} by

ρx = b(x) = ∑_{j=1}^{∞} x^{j} ⊗ b_{j}  in P (n)^{∗}(ℂP^{∞}) ˆ⊗ Q_{∗}^{2} ≅ Q_{∗}^{2}[[x]],  (7.7)

which is formally identical to equation (6.14), except that now b_{1} = e^{2} by [BJW95, Prop. 14.4(a)]. As in [ibid.], ψ_{A}b_{i} is the coefficient of x^{i} in

ψ_{A}b(x) = ∑_{j=1}^{∞} b(x)^{j} ⊗ b_{j}  in (Q_{∗}^{∗} ⊗ Q_{∗}^{2})[[x]],  (7.8)

and ϵ_{A}b_{j} = 0 for j > 1.

Again [ibid., Lemma 14.6], we need only the accelerated elements b_{(j)} = b_{p^{j}} for j ≥ 0, so b_{(0)} = e^{2}. The additive version of equation (6.19) also looks the same,

b([p](x)) = [p]_{R}(b(x)) = ∑_{i=N}^{∞} b(x)^{i+1}η_{R}g_{i} in Q_{∗}^{2}[[x]].  (7.9)

Definition 7.10 For each k ≥ n, we define the k-th main additive relation (R_{k}) as the coefficient of x^{p^{k}} in equation (7.9).

In view of equation (6.17), the first two main relations are simply

(R_{n}) b_{(0)}^{p^{n}}w_{n} = v_{n}b_{(0)} in Q_{∗}^{2}  (7.11)

and

(R_{n+1}) b_{(1)}^{p^{n}}w_{n} + b_{(0)}^{p^{n+1}}w_{n+1} = v_{n+1}b_{(0)} + v_{n}^{p}b_{(1)} in Q_{∗}^{2}.  (7.12)

We shall find in equation (10.10) that (R_{n}) desuspends once to

(R′_{n}) eb_{(0)}^{N}w_{n} = v_{n}e in Q_{∗}^{1}.  (7.13)

By equation (6.18), the general main relation for k ≥ n has the form

(R_{k})  ∑_{i=n}^{k} b_{(k−i)}^{p^{i}}w_{i} ≡ 0 in Q_{∗}^{2} mod 𝔙 + 𝔚^{2},  (7.14)

where 𝔙 = (v_{n}, v_{n+1}, v_{n+2}, …) and 𝔚 = (w_{n}, w_{n+1}, w_{n+2}, …) denote ideals in Q_{∗}^{∗}.

Cohomology of a lens space Our last test space is the lens space skeleton L, whose cohomology is given by equation (6.21), assuming p is odd. We already know ρ_{L}x = b(x) from equation (7.7). For u, we find, as in [BJW95, (16.21)], that

ρ_{L}u = u⊗e + ∑_{i=0}^{n−1} x^{p^{i}} ⊗a_{(i)} in P (n)^{∗}(L) ⊗ Q_{∗}^{1}  (7.15)

for certain elements a_{(i)} ∈ Q_{2p^{i}}^{1} that this equation defines. We deduce that

ψ_{A}a_{(k)} = a_{(k)} ⊗ e + ∑_{i=0}^{k} b_{(k−i)}^{p^{i}} ⊗ a_{(i)}  (7.16)

and ϵ_{A}a_{(k)} = 0.

[If p = 2, L = ℝP^{2N} has different cohomology (6.24), and we replace equation (7.15) by

ρt = t ⊗ e + ∑_{i=0}^{n−1} t^{2^{i+1}} ⊗ a_{(i)} in P (t)/(t^{2N+1}) ⊗ Q_{∗}^{1}.  (7.17)

Nevertheless, equation (7.16) and ϵ_{A}a_{(k)} = 0 remain valid for p = 2. By [Bo], equation (6.27) destabilizes in the obvious way, to

a_{(i)}^{2} = b_{(i+1)} for 0 ≤ i ≤ n − 1.]  (7.18)

More relations We shall find in equation (10.14) that one more suspension factor can be squeezed out of (7.13) if we first multiply by a_{(0)}, to give

(R′′_{n}) a_{(0)}b_{(0)}^{N}w_{n} = v_{n}a_{(0)} in Q_{∗}^{1}.  (7.19)

[When p = 2, we can multiply this by another a_{(0)} and use equation (7.18) to obtain the unexpected formula

b_{(0)}^{N}b_{(1)}w_{n} = v_{n}b_{(1)}.  (7.20)

This is not all; if we multiply (R_{n+1}) (7.12) by b_{(0)}^{N}, we obtain the reduction formula

b_{(0)}^{N+2^{n+1}}w_{n+1} = v_{n}^{2}b_{(0)}^{N}b_{(1)} + v_{n}b_{(1)}^{2^{n}} + v_{n+1}b_{(0)}^{2^{n}},  (7.21)

by using (7.20) to simplify one of the terms.]

Summary We have the additive version of the Hopf algebroid Γ.

Theorem 7.22 The additive unstable operations in P (n)-cohomology are dual to the P (n)_{∗}-algebra Q_{∗}^{∗} = QP (n)_{∗}(\underline{P (n)}_{∗}) [replaced by Q\overline{P (n)}_{∗}(\underline{P (n)}_{∗}) if p = 2], which has the properties:

(a) Q_{∗}^{∗} is the commutative bigraded P (n)_{∗}-algebra generated by the elements:

w_{k} ∈ Q_{0}^{−2(p^{k}−1)} for k ≥ n, defined by η_{R} in equation (7.2); e ∈ Q_{1}^{1}, the suspension element, defined by equation (7.3); b_{(j)} ∈ Q_{2p^{j}}^{2} for j ≥ 0, defined by equation (7.7); a_{(i)} ∈ Q_{2p^{i}}^{1} for 0 ≤ i < n, defined by (7.15) [replaced by (7.17) if p = 2];

subject to the relations e^{2} = b_{(0)}, the main relations (R_{k}) for k > n (see Definition 7.10), and the two variants (7.13) and (7.19) of (R_{n}) [also (7.18) if p = 2];

(b) Q_{∗}^{∗} is a free left P (n)_{∗}-module;

(c) Multiplication by the elements w_{k} makes Q_{∗}^{∗} a right P (n)_{∗}-module;

(d) The comultiplication ψ_{A}: Q_{∗}^{∗} → Q_{∗}^{∗} ⊗ Q_{∗}^{∗} is the homomorphism of algebras and of P (n)_{∗}-bimodules given on each generator as noted above;

(e) The counit ϵ_{A}: Q_{∗}^{∗} → P (n)_{∗} is the P (n)_{∗}-algebra homomorphism given on generators by ϵ_{A}e = 1, ϵ_{A}a_{(i)} = 0, ϵ_{A}b_{(j)} = 0 for j > 0, ϵ_{A}b_{(0)} = 1, and ϵ_{A}w_{k} = v_{k}.

§8 Relations for additive operations 28

Parts (c), (d) and (e) need no further comment. Part (b) is included in Theorem 8.4. Part (a) can be read off from Theorem 11.1. [For commutativity when p = 2, we refer to [Bo].]

We recall [BJW95, (6.3)] the stabilization homomorphism Q(σ): Q_{∗}^{k} → Γ, which has degree zero. We may use it to recover the structure on Γ in Theorem 6.26 from Q_{∗}^{∗} simply by setting e = 1. The coalgebra structure (ψ_{A}, ϵ_{A}) stabilizes to (ψ_{S}, ϵ_{S}).

8 Relations for additive operations

We noted in Theorem 7.22 that Q_{∗}^{∗} is a free P (n)_{∗}-module, which is not at all obvious from the generators and relations given. In this section, we exhibit a basis of Q_{∗}^{∗} and prove in Lemma 8.5 that it spans the module.

We also establish some direct applications of additive operations.

The Ravenel–Wilson basis Since e^{2} = b_{(0)} and a_{(i)}^{2} = 0 trivially if p is odd [replaced by a_{(i)}^{2} = b_{(i+1)} if p = 2, from equation (7.18)], any monomial in the listed generators of the P (n)_{∗}-algebra Q_{∗}^{∗} can be written in the abbreviated form

e^{ϵ}a^{I}b^{J}w^{K} = e^{ϵ}a_{(0)}^{i_{0}}a_{(1)}^{i_{1}} … a_{(n−1)}^{i_{n−1}}b_{(0)}^{j_{0}}b_{(1)}^{j_{1}}b_{(2)}^{j_{2}} … w_{n}^{k_{n}}w_{n+1}^{k_{n+1}}w_{n+2}^{k_{n+2}} …,  (8.1)

with multi-indices I = (i_{0}, i_{1}, …, i_{n−1}), J = (j_{0}, j_{1}, j_{2}, …), and K = (k_{n}, k_{n+1}, …), where each i_{r}, also ϵ, is 0 or 1. (We keep the w’s to the right, as a reminder that they define the right action of P (n)_{∗} on Q_{∗}^{∗}.) We introduce the following parameters:

The b-length is ∑_{r} j_{r}, the total number of factors of the form b_{(j)}; The w-length is ∑_{r} k_{r}, the total number of factors of the form w_{k}.

As with BP in [RW77], it is easier to specify which monomials are not wanted in forming the basis than those which are. [For p = 2, the basis is not written out in detail in [RW96], and contains some surprises.] There are two variants; we shall need the second in §§10, 11.

Definition 8.2 We call the monomial (8.1) Q-allowable if it does not have any of the following forms [note that (iv) and (v) apply only if p = 2]:

(i) b_{(d_{n})}^{p^{n}}b_{(d_{n+1})}^{p^{n+1}} … b_{(d_{q})}^{p^{q}}w_{q}c, with 0 ≤ d_{n} ≤ d_{n+1} ≤ … ≤ d_{q}, q ≥ n; (ii) eb_{(0)}^{N}b_{(d_{n+1})}^{p^{n+1}} … b_{(d_{q})}^{p^{q}}w_{q}c, with 0 ≤ d_{n+1} ≤ … ≤ d_{q}, q ≥ n; (iii) a_{(0)}b_{(0)}^{N}b_{(d_{n+1})}^{p^{n+1}} … b_{(d_{q})}^{p^{q}}w_{q}c, with 0 ≤ d_{n+1} ≤ … ≤ d_{q}, q ≥ n; (iv) b_{(0)}^{N}b_{(1)}b_{(d_{n+1})}^{2^{n+1}} … b_{(d_{q})}^{2^{q}}w_{q}c, where p = 2, with 0 ≤ d_{n+1} ≤ … ≤ d_{q}, q ≥ n; (v) b_{(0)}^{N+2^{n+1}}b_{(d_{n+2})}^{2^{n+2}} … b_{(d_{q})}^{2^{q}}w_{q}c, where p = 2, with 0 ≤ d_{n+2} ≤ … ≤ d_{q}, q ≥ n + 1; (8.3)

where c is any monomial (c = 1 is permitted) in the generators e, a_{(i)}, b_{(j)}, and w_{k}. More generally, we call the monomial allowable if it is not of the form (i) or (ii).

Remark In [RW96], a monomial is called n-allowable (lies in A_{n}) if it is not of the form (i). If it contains a factor e or a_{(0)}, it is called n-plus allowable (lies in A_{n}^{+}) if it is not of the form (i), (ii) or (iii).

From [RW96, Thm. 1.3], we have the Ravenel–Wilson basis of Q_{∗}^{∗}.

Theorem 8.4 (Ravenel–Wilson) The Q-allowable monomials (8.1) form a basis of the free P (n)_{∗}-module Q_{∗}^{∗} = QP (n)_{∗}(\underline{P (n)}_{∗}) [or Q\overline{P (n)}_{∗}(\underline{P (n)}_{∗}) if p = 2].

Later in this section, we shall reprove half the theorem.

Lemma 8.5 The relations e^{2} = b_{(0)}, the main relations (R_{k}) for k > n, [relation (7.18) if p = 2,] and the variants (7.13) and (7.19) of (R_{n}) imply that the Q-allowable monomials (8.1) span the P (n)_{∗}-module Q_{∗}^{∗} = QP (n)_{∗}(\underline{P (n)}_{∗}) [or Q\overline{P (n)}_{∗}(\underline{P (n)}_{∗}) if p = 2].

Generators of cohomology Just as in [BJW95, Thm. 20.2], Theorem 1.16 follows directly from the fact that the additive operations on P (n)^{−k}(−) form the P (n)_{∗}-dual of the free P (n)_{∗}-module Q_{∗}^{−k}, whose generators all lie in groups Q_{j}^{−k} with j ≥ 0.

We combine the following two lemmas, which correspond to Theorem 20.3 and Lemma 20.5 of [ibid.]. We study the linear functional ϵ_{A} = ⟨ι_{−k}, −⟩: Q_{∗}^{−k} → P (n)_{∗} defined by the identity operation ι_{−k} on P (n)^{−k}(−), which is plainly additive.

Lemma 8.6 Given any integer k > 0, there exist:

(i) a sequence of additive unstable operations r_{i}: P (n)^{−k}(−) → P (n)^{m(i)}(−) with m(i) ≥ 0;

(ii) a sequence of elements v(i) ∈ P (n)_{∗} with deg(v(i)) → ∞;

such that in any additively unstable P (n)-cohomology comodule M (e. g. P (n)^{∗}(X) for any space X), any x ∈ M^{−k} decomposes as the (topological infinite) sum

x = ∑_{i} v(i)r_{i}x.

Proof Let {c_{1}, c_{2}, c_{3}, …} be the Ravenel–Wilson (or any other) basis of the free P (n)_{∗}-module Q_{∗}^{−k}, with c_{i} ∈ Q_{m(i)}^{−k}. Trivially, m(i) ≥ 0. For fixed x ∈ M^{−k} and any additive operation r, the linearity of rx in r may be expressed, as in [BJW95, (6.39)], by the formula

rx = ∑_{i}⟨r, c_{i}⟩r_{i}x,  (8.7)

where r_{i} denotes the operation dual to c_{i}. We take r = ι_{−k}, put v(i) = ⟨ι_{−k}, c_{i}⟩, and note that deg(v(i)) = m(i) + k → ∞.

Remark The coefficients are readily computed: v(i) = ϵ_{A}c_{i} = v^{K} if the monomial c_{i} has the form e^{ϵ}b_{(0)}^{j}w^{K}, and v(i) = 0 otherwise. Thus many terms are zero.

To get the more precise information for Theorem 1.16, we write the space X as the disjoint union of its components and reduce to the case when X is connected.

Lemma 8.8 Let M be a connected (see [BJW95, Defn. 7.14]) additively unstable P (n)-cohomology algebra (e. g. P (n)^{∗}(X) for any connected space X). Then as a topological P (n)_{∗}-module, M is generated by 1_{M} ∈ M^{0} and elements of M^{i} for i > 0. The generator 1_{M} is never redundant.

Proof Let L be the submodule generated (topologically) by the elements of all the M^{i} for i > 0. By Lemma 8.6, we need only consider x ∈ M^{0}. We choose a basis {c_{1}, c_{2}, c_{3}, …} of Q_{∗}^{0} with c_{1} = 1.

We recall from [BJW95, Defn. 7.13] the collapse operation κ_{j} on P (n)^{j}(−) for any j; since M is connected, on any x ∈ M^{j} it satisfies κ_{j}x = v1_{M} for some v ∈ P (n)_{−j}. But (8.7) gives κ_{0}x ≡ r_{1}x mod L and also x = ι_{0}x ≡ r_{1}x mod L. Thus x ≡ κ_{0}x = λ1_{M} mod L for some λ ∈ 𝔽_{p}.

Since κL = 0 and κ_{0}1_{M} = 1_{M}, 1_{M} never lies in L.

Higher-order relations The proof of Lemma 8.5 resembles that of [BJW95, Thm. 18.16]. The Nakayama Lemma [Bo95, §15] (which is easier for P (n)_{∗} than for BP_{∗}, as p = 0) allows us to work throughout modulo the ideal 𝔙 ⊂ Q_{∗}^{∗}. We also work modulo powers of 𝔚. (These ideals were introduced in equation (7.14), which displays the w-linear terms in the relation (R_{k}).)

When q = n and c = 1, we observe that (8.3)(i) is the first term in (R_{d_{n}+n}), and is thus expressible by equation (7.14) in terms of Q-allowable monomials mod 𝔙 + 𝔚^{2}. Equation (7.13) shows that (R′_{n}) takes care of (ii), while (7.19) shows that (R′′_{n}) takes care of (iii). [If p = 2, we use (7.20) and (7.21) to handle (iv) and (v).]

Otherwise, the relations (R_{k}) are not at all transparent. We handle the general disallowed monomial (8.3)(i) by eliminating the q − n variables w_{n}, w_{n+1}, …, w_{q−1} from the q −n+1 relations (R_{d_{n}+n}), (R_{d_{n+1}+n+1}), …, (R_{d_{q}+q}), expressed in the form (7.14), to obtain the higher-order derived relation

Δ_{q}w_{q} + ∑_{r>q} Δ_{r}w_{r} ≡ 0 mod 𝔙 + 𝔚^{2},  (8.9)

for certain determinants Δ_{r}. Explicitly, for any r ≥ q,

Δ_{r} = ∑_{π} ϵ_{π}b_{(d_{n}+n−πn)}^{p^{πn}} … b_{(d_{q−1}+q−1−π(q−1))}^{p^{π(q−1)}}b_{(d_{q}+q−πr)}^{p^{πr}},  (8.10)

where we sum over all permutations π of {n, …, q−1, r}, write ϵ_{π} for the sign of π, and adopt the convention that meaningless factors b_{(j)} with j < 0 are taken as 0.

We order the b -monomials lexicographically (b^{J} < b^{K} if and only if there exists t ≥ 0 such that j_{r} = k_{r} for all r < t, and j_{t} < k_{t}).

Lemma 8.11 For any r ≥ q, the determinant Δ_{r} in (8.10) has the form

Δ_{r} = b_{(d_{n})}^{p^{n}}b_{(d_{n+1})}^{p^{n+1}} … b_{(d_{q−1})}^{p^{q−1}}b_{(d_{q}+q−r)}^{p^{r}} + higher terms.

Proof The displayed term is the diagonal term with π = id. For any other permutation π, there is a first index t such that πt > t, so that n ≤ t ≤ q − 1 and πk = k for all k < t. The corresponding term ϵ_{π}b_{(d_{n})}^{p^{n}} … b_{(d_{t−1})}^{p^{t−1}}b_{(d_{t}+t−πt)}^{p^{πt}} … in (8.10) is higher, because d_{t} + t − πt < d_{t}.

Proof of Lemma 8.5 We show that each Q-disallowed monomial in (8.3) is a linear combination mod 𝔙 of higher monomials with the same w-length, and monomials of greater w-length, where we partially order all monomials according to the factor b^{J} (and ignore e, a_{(i)}, and w_{k}). Since there are only finitely many monomials in each bidegree, the result follows.

For (8.3)(i), Lemma 8.11 shows that we can use (8.9) to express b_{(d_{n})}^{p^{n}} … b_{(d_{q})}^{p^{q}}w_{q} as a linear combination mod 𝔙 of higher monomials and monomials with w-length ≥ 2, since for r > q, the diagonal term of Δ_{r} is higher than the diagonal term of Δ_{q}. Multiplication by c preserves the ordering.

For (ii), (iii) [and (iv), if p = 2], we modify equation (8.9) by eliminating the variables w_{n+1}, …, w_{q−1} from the relations (R_{d_{n+1}+n+1}), …, (R_{d_{q}+q}) to obtain

Δ′_{n}w_{n} + Δ′_{q}w_{q} + ∑_{r>q} Δ′_{r}w_{r} ≡ 0 mod 𝔙 + 𝔚^{2}.

When we multiply by eb_{(0)}^{N}c, the first term drops out by (7.13). Lemma 8.11, slightly modified (or with n replaced by n + 1), shows that (ii) is the lowest of the remaining terms. If we multiply by a_{(0)}b_{(0)}^{N}c instead and use (7.19), we obtain (iii). [For (iv), we multiply by b_{(0)}^{N}b_{(1)}c and use (7.20).]

[For (v), we eliminate the variables w_{n+2}, …, w_{q−1} from the relations (R_{d_{n+2}+n+2}), …, (R_{d_{q}+q}) to obtain a higher-order relation

Δ′′_{n}w_{n} + Δ′′_{n+1}w_{n+1} + Δ′′_{q}w_{q} + ∑_{r>q} Δ′′_{r}w_{r} ≡ 0 mod 𝔙 + 𝔚^{2}.

When we multiply this by b_{(0)}^{N+2^{n+1}}c, the first two terms drop out by (7.11) and (7.21). The diagonal term in the determinant Δ′′_{q} gives (v).]

The first higher-order relation The first relation for a given q, where we eliminate w_{n}, w_{n+1}, …, w_{q−1} from (R′_{n}), (R_{n+1}), …, (R_{q}), is particularly important. The additive version for P (n) of Bendersky’s Lemma [Be86, Thm. 6.2] (or see [BJW95, Lemma 18.23]) gives more precise information than our proof of Lemma 8.5, and follows immediately from Lemma 12.2.

We recall the ideal I_{q} = (v_{n}, v_{n+1}, …, v_{q−1}) ⊂ P (n)_{∗} (where I_{n} = (0)).

Lemma 8.12 In Q_{∗}^{∗} = QP (n)_{∗}(\underline{P (n)}_{∗}) [replaced by Q\overline{P (n)}_{∗}(\underline{P (n)}_{∗}) if p = 2], we have the relation

e^{g(n,q)−1}w_{q} ≡ v_{q}e^{g(n,q−1)+1} mod I_{q}Q_{∗}^{∗} for q ≥ n.  (8.13)

[If p = 2, this is almost superseded by the relation

e^{g(n,q)−2}w_{q} ≡ v_{q}e^{g(n,q−1)} mod I_{q}Q_{∗}^{∗} for q ≥ n + 1.]  (8.14)

Primitive elements Let M be an unstable P (n)-cohomology comodule (in the sense of [BJW95, Defn. 6.32]). An element x ∈ M^{k} is called (additively unstably) primitive if the coaction ρ_{M} has the value ρ_{M}x = x⊗e^{k} on x. Then for any v ∈ P (n)_{∗},

ρ_{M}(vx) = x ⊗ e^{k}(η_{R}v).  (8.15)

Of course, all this requires k ≥ 0, but more is true, as in [ibid., Lemma 20.8].

§9 Idempotent operations 32

Lemma 8.16 Let x ∈ M^{k} be a nonzero primitive element of the unstable P (n)- cohomology comodule M, and assume q ≥ n.

(a) If I_{q}x = 0 and k satisfies the condition (depending on p and q):

(i) k ≥ g(n, q) − 1 if p is odd or q = n; (ii) k ≥ g(n, q) − 2 if p = 2 and q ≥ n + 1; (8.17)

then v_{q}x is primitive (possibly zero);

(b) If k does not satisfy the condition (8.17), then for all i > 0, v_{q}^{i}x is nonzero and is not primitive.

Proof For (a), (8.15) gives ρ(v_{q}x) = x ⊗ e^{k}w_{q}. By Lemma 8.12, this is the same as x ⊗ v_{q}e^{k−2(p^{q}−1)} = v_{q}x ⊗ e^{k−2(p^{q}−1)}, since I_{q}x = 0.

For (b), we have ρ(v_{q}^{i}x) = x ⊗ e^{k}w_{q}^{i}. Here, e^{k}w_{q}^{i} is Q-allowable by (8.3) and hence a basis element of Q_{∗}^{∗}, which shows that v_{q}^{i}x is not primitive.

Proof of Lemma 1.17 We must have ρx = x ⊗ e^{k}. If m > n, we have v_{m−1}x = 0, and case (b) of Lemma 8.16 with q = m − 1 does not apply; hence the lower bound on k.

Conversely, (8.15) specifies the coaction on all of M, and Lemma 8.12 shows it is well defined.

Proof of Theorem 1.18 We build an increasing sequence

0 = M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ … ⊂ M

of subcomodules of M. For each i > 0, just as in the proof of Theorem 20.11 in [BJW95], we construct a primitive element x_{i} ∈ M/M_{i−1} with Ann(x_{i}) = I_{m_{i}} for some m_{i}, using Lemma 8.16 in place of Lemma 20.8 of [ibid.]. We take M_{i}/M_{i−1} ⊂ M/M_{i−1} as the P (n)_{∗}-submodule generated by x_{i}. Lemma 1.17 describes M_{i}/M_{i−1}.

Because each k_{i} ≥ 0 in Theorem 1.18 and each M^{k} is a finitely generated 𝔽_{p}- module, this sequence must terminate after finitely many steps. We deduce that M is a finitely presented P (n)_{∗}-module.

9 Idempotent operations

Lemma 9.1 delivers the promised additive idempotent operations θ(m) in P (n)- cohomology that we need for Lemma 5.1, which is equivalent to Lemma 3.1. In fact, we find a large class of θ(m), among which none seems to be preferred. The rest of this section applies the work in §8 to prove Lemma 9.1.

Lemma 9.1 Assume that k ≤ g(n, m) [replaced by k ≤ g(n, m) − 1 if p = 2], where m ≥ n. Then there exists an additive idempotent operation θ(m) on P (n)^{k}(−) having the following properties:

(i) The image of the operation θ(m) is represented by the space \underline{P (n, m)}_{k}; (ii) The map θ(m): \underline{P (n)}_{k} → \underline{P (n)}_{k} factors to yield an H-space splitting \overline{θ(m)}: \underline{P (n, m)}_{k} → \underline{P (n)}_{k} of the canonical H-map ρ(m): \underline{P (n)}_{k} → \underline{P (n, m)}_{k}; (iii) For all spaces X, \overline{θ(m)} naturally embeds P (n, m)^{∗}(X) ⊂ P (n)^{∗}(X) as a summand, in the sense of abelian groups (but not as P (n)_{∗}-modules).

Remark Exactly as in [BJW95, end of §22], we can make the splittings θ(m) compatible as k and m vary if we wish. The decomposition factors of \underline{P (n)}_{k} resulting from this approach must of course be the same as in Theorem 1.12, according to Theorem 1.2(b), but the injection maps are different, in general.

However, we emphasize that the splitting theorems as stated in §§1, 5 do not require any compatibility.

The ideals 𝔍_{m} As in [BJW95], the ideal J_{m} = (v_{m+1}, v_{m+2}, …) ⊂ P (n)_{∗}, introduced in equation (1.8), gives rise to an analogous ideal for the right action of P (n)_{∗} on Q_{∗}^{∗}.

Definition 9.2 Given any m ≥ n, we define the ideal

𝔍_{m} = (w_{m+1}, w_{m+2}, w_{m+3}, …) ⊂ Q_{∗}^{∗}.

We need to know how 𝔍_{m} sits inside Q_{∗}^{∗}. As in [ibid.], the answer is remarkably clean, in a certain range.

Lemma 9.3 For k ≤ g(n, m) [replaced by k ≤ g(n, m) − 1 if p = 2], Q_{∗}^{k} ∩ 𝔍_{m} is the left P (n)_{∗}-submodule of Q_{∗}^{k} spanned by all the Q-allowable monomials (8.1) that lie in it and contain an explicit factor w_{q} for some q > m.

Remark By Lemma 8.12, v_{m+1}eb_{(0)}^{g(n,m)/2} − z lies in 𝔍_{m}, where z ∈ I_{m+1}Q_{∗}^{∗}, so the result definitely fails for k = g(n, m) + 1 [also for k = g(n, m) if p = 2].

Proof Any monomial that contains w_{h} with h > m visibly lies in 𝔍_{m}. To show the converse, we fix k and i_{0} and prove by downward induction on h that for all i ≤ i_{0}, all elements in Q_{i}^{k} of the form cw_{h} lie in the indicated P (n)_{∗}-submodule. This statement is trivial for sufficiently large h (depending on k and i_{0}).

We therefore choose q > m, assume the statement holds for all h > q, and prove it for h = q. Take cw_{q} ∈ Q_{i}^{k}, where i ≤ i_{0}, so that c ∈ Q_{i}^{k+2(p^{q}−1)}. By Lemma 8.5, we may reduce to the case where c is a Q-allowable monomial. We note that in (8.3), the Q-disallowed monomials (i) and (iv) have b -length \frac{1}{2}g(n, q), while (ii), (iii) and (v) have b -length \frac{1}{2}g(n, q) − 1.

Case 1: c has no factor e, a_{(0)}, or w_{j}. For odd p, the b -length of c is at most

\frac{1}{2}(k + 2(p^{q}−1)) ≤ \frac{1}{2}(g(n, m) + 2p^{q} − 2) < \frac{1}{2}g(n, q),

which makes cw_{q} also Q-allowable, as only rule (i) of (8.3) is relevant. [If p = 2, we need to assume k ≤ g(n, m) − 1 to get the stronger bound \frac{1}{2}g(n, q) − 1.]

Case 2: c = ey or c = a_{(0)}y, where y has no factor w_{j}. In this case, the b -length of c is at most

\frac{1}{2}(k − 1 + 2(p^{q}−1)) ≤ \frac{1}{2}(g(n, m) + 2p^{q} − 3) < \frac{1}{2}g(n, q) − 1,

which makes cw_{q} automatically Q-allowable.

Case 3: c = yw_{j}, where j ≤ q. Then cw_{q} remains Q-allowable, by the form of Definition 8.2.

Case 4: c = yw_{j}, where j > q. By induction, cw_{q} = (yw_{q})w_{j} lies in the indicated submodule.

Linear functionals To establish Lemma 9.1, we actually construct the associated P (n)_{∗}-linear functional ⟨θ(m), −⟩: Q_{∗}^{k} → P (n)_{∗}.

Lemma 9.4 Assume the linear functional ⟨θ(m), −⟩: Q_{∗}^{k} → P (n)_{∗} corresponding to the additive operation θ(m): P (n)^{k}(−) → P (n)^{k}(−) satisfies the conditions:

(i) ⟨θ(m), Q_{∗}^{k} ∩ 𝔍_{m}⟩ = 0;
(ii) ⟨θ(m), c⟩ ≡ ϵ_{A}c mod J_{m} for all c ∈ Q_{∗}^{k};

(9.5)

where ϵ_{A}: Q_{∗}^{k} → P (n)_{∗} is the augmentation. Then:

(a) The homology homomorphism Q(θ(m)_{∗}): Q_{∗}^{k} → Q_{∗}^{k} induced by the representing map θ(m): \underline{P (n)}_{k} → \underline{P (n)}_{k} satisfies

(i) Q(θ(m)_{∗})(Q_{∗}^{k} ∩ 𝔍_{m}) = 0;
(ii) Q(θ(m)_{∗}) ≡ id: Q_{∗}^{k} → Q_{∗}^{k} mod 𝔍_{m};

(b) Q(θ(m)_{∗}) induces a splitting of the short exact sequence

0 − ; − ; → Q_{∗}^{k} ∩ 𝔍_{m} − ; − ; → Q_{∗}^{k} − ; − ; → Q_{∗}^{k}/(Q_{∗}^{k} ∩ 𝔍_{m}) − ; − ; → 0

of left P (n)_{∗}-modules; (c) The operation θ(m) is idempotent and has the properties listed in Lemma 9.1.

Proof The proof is patterned after that of Lemma 22.2 in [BJW95]. We require the commutative diagram

[figure: Q_{∗}^{k}; Q_{∗}^{∗} ⊗ Q_{∗}^{k}; Q_{∗}^{∗} ⊗ P (n)_{∗}; Q_{∗}^{∗}; Q_{∗}^{k}/𝔍_{m}; Q_{∗}^{∗} ⊗ Q_{∗}^{k}/𝔍_{m}; Q_{∗}^{∗} ⊗ P (n)_{∗}/J_{m}; Q_{∗}^{∗}/𝔍_{m}; ψ_{A}; q′; id⊗⟨θ(m),−⟩; λ_{R}; q′; \overline{ψ_{A}}; id⊗\overline{ϵ_{A}}; \overline{λ_{R}}]

of P (n)_{∗}-module homomorphisms, where \overline{ψ_{A}}, \overline{ϵ_{A}} and \overline{λ_{R}} denote quotients of ψ_{A}, ϵ_{A}, and the right action λ_{R} of P (n)_{∗} on Q_{∗}^{∗}, Q_{∗}^{k}/𝔍_{m} is really Q_{∗}^{k}/(Q_{∗}^{k} ∩ 𝔍_{m}), and the vertical arrows are the obvious projections. The conditions (9.5) on ⟨θ(m), −⟩ are exactly what we need to fill in the diagonal.

By [BJW95, Lemma 6.51(c)], the top row gives the homology homomorphism Q(θ(m)_{∗}), while by [ibid., (6.31)], the bottom row reduces to the identity homomorphism of Q_{∗}^{k}/𝔍_{m}. Thus the diagonal provides a splitting we call j′: Q_{∗}^{k}/𝔍_{m} → Q_{∗}^{k} that satisfies j′^{∘}q′ = Q(θ(m)_{∗}) and q′^{∘}j′ = id and so yields (a). Part (b) is merely a restatement of (a).

It follows by faithfulness that θ(m) is an idempotent operation, so that the image h(−) = θ(m)P (n)^{k}(−) ⊂ P (n)^{k}(−) is an ungraded cohomology theory. By [Bo95, Thm. 3.6], h(−) is represented (on Ho) by some H-space Y, and the additive operations h(−) ⊂ P (n)^{k}(−) and θ(m): P (n)^{k}(−) → h(−) are represented by H-maps j: Y → \underline{P (n)}_{k} and q: \underline{P (n)}_{k} → Y respectively, that satisfy j^{∘}q = θ(m) and q^{∘}j = id.

To finish (c), we apply the homotopy group functor π_{∗}(−) to obtain homomorphisms q_{∗}: π_{∗}(\underline{P (n)}_{k}) → π_{∗}(Y) and j_{∗}: π_{∗}(Y) → π_{∗}(\underline{P (n)}_{k}) that satisfy q_{∗}^{∘}j_{∗} = id

§10 Unstable operations in P (n)-cohomology 35

and j_{∗}^{∘}q_{∗} = θ(m)_{∗}. Recall that π_{∗}(\underline{P (n)}_{k}) ≅ Σ^{k}P (n)_{∗}. Given v ∈ P (n)_{i}, (7.6) evaluates θ(m)_{∗}Σ^{k}v = Σ^{k}⟨θ(m), e^{k+i}(η_{R}v)⟩. Then (9.5)(i) yields θ(m)_{∗}Σ^{k}v = 0 if v ∈ J_{m}, while for any v, (ii) gives θ(m)_{∗}Σ^{k}v ≡ Σ^{k}v mod J_{m}. It follows that ρ(m)^{∘}j: Y → \underline{P (n, m)}_{k} induces an isomorphism of homotopy groups and is therefore a homotopy equivalence. To establish the properties listed in Lemma 9.1, we put \overline{θ(m)} = j^{∘}g, where g: \underline{P (n, m)}_{k} → Y is a homotopy inverse to ρ(m)^{∘}j.

Proof of Lemma 9.1 Lemma 9.3 makes it obvious that linear functionals ⟨θ(m), −⟩ exist that satisfy the conditions (9.5), so that Lemma 9.4 applies.

Remark As an explicit example, choose ⟨θ(m), −⟩ on the Ravenel–Wilson basis as ⟨θ(m), c⟩ = v^{K} if c has the form e^{ϵ}b_{(0)}^{j}w^{K} but contains no factor w_{k} with k > m, and ⟨θ(m), c⟩ = 0 otherwise. To determine ⟨θ(m), c⟩ for c not in the basis, we must first express c in terms of the basis.

10 Unstable operations in P (n)-cohomology

In this section, we use all unstable operations in P (n)-cohomology to obtain generators and relations for the Hopf ring P (n)_{∗}(\underline{P (n)}_{∗}), in the style of [BJW95]. The two multiplications are c∗d = μ_{∗}(c×d) and c^{∘}d = ϕ_{∗}(c×d), induced respectively by the maps μ: \underline{P (n)}_{k} × \underline{P (n)}_{k} → \underline{P (n)}_{k} and ϕ: \underline{P (n)}_{k} × \underline{P (n)}_{m} → \underline{P (n)}_{k+m} that represent addition and multiplication in P (n)-cohomology, and 1_{k} will denote the ∗-identity element of P (n)_{∗}(\underline{P (n)}_{k}). [If p = 2, we use the Hopf ring \overline{P (n)}_{∗}(\underline{P (n)}_{∗}) instead, replacing c × d by c \overline{×} d in both multiplications.] We still assume that 0 < n < ∞.

We deduce the results of §7 on additive operations by applying the homomorphism

q_{k}: P (n)_{∗}(\underline{P (n)}_{k}) − ; − ; → Q_{∗}^{k},  (10.1)

which neglects 1_{k} and decomposables, shifts degrees by −k, and (as k varies) takes ∘-products to products (with a sign, on account of the degree shift). However, the Hopf ring structure maps ψ and ϵ are unrelated to ψ_{A} and ϵ_{A}.

Since the Hopf ring is a free P (n)_{∗}-module by [RW96, Cor. 1.5], Theorem 4.14 of [BJW95] allows us to identify:

(i) The cohomology operation r: P (n)^{k}(−) → P (n)^{m}(−); (ii) The cohomology class r(ι_{k}) ∈ P (n)^{m}(\underline{P (n)}_{k}); (iii) The representing map of spaces r: \underline{P (n)}_{k} → \underline{P (n)}_{m}, up to homotopy;

(iv) The P (n)_{∗}-linear functional ⟨r, −⟩: P (n)_{∗}(\underline{P (n)}_{k}) → P (n)_{∗} of degree −m

[or ⟨r, −⟩: \overline{P (n)}_{∗}(\underline{P (n)}_{k}) → P (n)_{∗} if p = 2].

Hopf rings for p = 2 When p = 2, P (n)_{∗}(\underline{P (n)}_{∗}) and \overline{P (n)}_{∗}(\underline{P (n)}_{∗}) are not Hopf

rings in the ordinary sense (though H_{∗}(\underline{P (n)}_{∗}; 𝔽_{2}) is one, and is described in [BW01] and after Theorem 11.8). (A few things are simpler: there are no signs and χ is the identity.) Because P (n) is not commutative, all Hopf ring axioms that shuffle factors must be modified to use the commutativity isomorphism T_{Q} of equation (2.11), which results in extra terms; see §2 or [Bo] for details. Neither multiplication is commutative in the ordinary sense, nor is ψ cocommutative.

As a P (n)_{∗}-module, the Hopf ring \overline{P (n)}_{∗}(\underline{P (n)}_{∗}) is identical to P (n)_{∗}(\underline{P (n)}_{∗}). The choice of multiplication on P (n) does not affect the P (n)_{∗}-module structure on \overline{P (n)}_{∗}(\underline{P (n)}_{∗}), nor does it affect the ^{∘}-generators that we construct below. However, switching to the other good multiplication on P (n) replaces c^{∘}d = ϕ_{∗}(c\overline{×}d) by

\overline{ϕ}_{∗}(c×d) = ϕ_{∗}T_{∗}(c×d) = ϕ_{∗}(d\overline{×}c) = d^{∘}c,

which is different in general; and similarly for c ∗ d.

The Cartan formulae Assume first that p is odd. Given a cohomology class x ∈ P (n)^{k}(X), we encode the action of operations on x by a formula of the form

r(x) = ∑_{α}⟨r, c_{α}⟩x_{α} for all r,

for suitable choices c_{α} ∈ P (n)_{∗}(\underline{P (n)}_{k}) and x_{α} ∈ P (n)^{∗}(X). (Here and elsewhere, we mean all operations r that have the correct domain degree. The sum may be infinite if X is not finite-dimensional.) Similarly, given y ∈ P (n)^{m}(X), suppose

r(y) = ∑_{β}⟨r, d_{β}⟩y_{β} for all r.

Then the two Cartan formulae [BJW95, (10.23) and (10.36)] are:

r(x + y) = ∑_{α} ∑_{β}(−1)^{deg(x_{α}) deg(y_{β})}⟨r, c_{α} ∗ d_{β}⟩x_{α}y_{β}  (10.2)

and

r(xy) = ∑_{α} ∑_{β}(−1)^{deg(x_{α}) deg(y_{β})}⟨r, c_{α}^{∘}d_{β}⟩x_{α}y_{β}.  (10.3)

We use them repeatedly without further reference.

The case p = 2 Examination reveals that the proof of the Cartan formulae in [BJW95] relies on the identity ⟨x×y, a×b⟩ = ±⟨x, a⟩⟨y, b⟩, which is false for p = 2; we must replace a × b by a \overline{×} b and use equation (2.15) instead. When we use the Hopf ring \overline{P (n)}_{∗}(\underline{P (n)}_{∗}), both Cartan formulae remain valid as stated.

Cohomology of a point Our first test space is the one-point space. For each v ∈ P (n)_{q}, the Hopf ring element [v] ∈ P (n)_{0}(\underline{P (n)}_{−q}) [or \overline{P (n)}_{0}(\underline{P (n)}_{−q}) if p = 2] is defined by the identity

r(v) = ⟨r, [v]⟩ in P (n)^{∗}(point) = P (n)_{∗}, for all r.  (10.4)

The properties of these elements were listed in [BJW95, Prop. 11.2]. As [v + v′] = [v] ∗ [v′] and [vv′] = [v]^{∘}[v′], we are primarily interested in the elements

[v_{k}] ∈ P (n)_{0}(\underline{P (n)}_{−2(p^{k}−1)}) for k ≥ n

[or in \overline{P (n)}_{0}(\underline{P (n)}_{−2(2^{k}−1)}) if p = 2]. Then (10.1) maps [v_{k}] to w_{k}.

We have the important relation

[1]^{∗p} = [p] = [0_{0}] = 1_{0}.  (10.5)

Cohomology of a circle Our second test space is the circle S^{1}. The suspension element e = e_{1} ∈ P (n)_{1}(\underline{P (n)}_{1}) [or \overline{P (n)}_{1}(\underline{P (n)}_{1}) if p = 2] is defined by the action of operations r on the standard generator u_{1} ∈ P (n)^{1}(S^{1}),

r(u_{1}) = ⟨r, 1_{1}⟩1_{S} + ⟨r, e⟩u_{1} in P (n)^{∗}(S^{1}) = E(u_{1}), for all r.  (10.6)

The properties of e were listed in [BJW95, Prop. 13.7].

Complex orientation Our third test space is ℂP^{∞}. The Hopf ring elements b_{j} ∈ P (n)_{2j}(\underline{P (n)}_{2}) [or \overline{P (n)}_{2j}(\underline{P (n)}_{2}) if p = 2] for j ≥ 0 are defined by the identity

r(x) = ⟨r, b(x)⟩ = ∑_{j=0}^{∞}⟨r, b_{j}⟩x^{j}  in P (n)^{∗}(ℂP^{∞}) ≅ P (n)_{∗}[[x]], for all r, (10.7)

where b(x) is a convenient formal abbreviation for ∑_{j} b_{j}x^{j}. Their properties were listed in [BJW95, Prop. 15.3]. In particular, b_{0} = 1_{2} is now nonzero and b_{1} = −e^{∘}e. Again, the accelerated elements b_{(j)} = b_{p^{j}} ∈ P (n)_{2p^{j}} (\underline{P (n)}_{2}) [or in \overline{P (n)}_{2^{j+1}}(\underline{P (n)}_{2}) if p = 2] suffice, as [ibid., Lemma 15.9] shows how to express the other b’s inductively in terms of these and the v’s and [v]’s.

Naturality of equation (10.7) with respect to the p-th power map ζ: ℂP^{∞} → ℂP^{∞}, with massive use of the Cartan formulae, yields the identity

b([p](x)) = ∗_{i=N}^{∞}\{b(x)^{^{∘}i+1}^{∘}[g_{i}]\} in P (n)_{∗}(\underline{P (n)}_{2})[[x]]  (10.8)

[or in \overline{P (n)}_{∗}(\underline{P (n)}_{2})[[x]] if p = 2], as in [ibid., (15.14)]. The lowest power of x that occurs is still x^{p^{n}}, apart from the term 1_{2} on each side.

Definition 10.9 For each k ≥ n, we define the k-th main unstable relation (R_{k}) as the coefficient of x^{p^{k}} in equation (10.8).

The first relation is simply

(R_{n}) v_{n}b_{(0)} = b_{(0)}^{^{∘}p^{n}} ^{∘}[v_{n}] in P (n)_{∗}(\underline{P (n)}_{2})

[or in \overline{P (n)}_{∗}(\underline{P (n)}_{2}) if p = 2]. By [RW96, Prop. 2.1(j)], it desuspends once to

(R′_{n}) v_{n}e = e^{∘}b_{(0)}^{^{∘}N}^{∘}[v_{n}] in P (n)_{∗}(\underline{P (n)}_{1})  (10.10)

[or in \overline{P (n)}_{∗}(\underline{P (n)}_{1}) if p = 2]. The second relation is almost as easy, in view of equation (6.17):

(R_{n+1}) b_{(1)}^{^{∘}p^{n}} ^{∘}[v_{n}] + b_{(0)}^{^{∘}p^{n+1}} ^{∘}[v_{n+1}] = v_{n}^{p}b_{(1)} + v_{n+1}b_{(0)}.  (10.11)

Cohomology of a lens space, for p odd Our final test space is the lens space skeleton L, whose cohomology (6.21) has two generators u and x. As x is a Chern class, equation (10.7) gives r(x) by naturality. We define Hopf ring elements a_{i} and c_{i} by the identity

r(u) = ∑_{i=0}^{N}⟨r, a_{i}⟩x^{i} + ∑_{i=0}^{N−1}⟨r, c_{i}⟩ux^{i} in P (n)^{∗}(L), for all r.  (10.12)

Not by coincidence, their formal properties are exactly the same as in the case E = K(n) of [BJW95]. The formal abbreviation a(x) = ∑_{i} a_{i}x^{i} is convenient.

Proposition 10.13 For p odd, the Hopf ring elements a_{i} ∈ P (n)_{2i}(\underline{P (n)}_{1}) (for

0 ≤ i < p^{n}), a_{(i)} = a_{p^{i}} ∈ P (n)_{2p^{i}}(\underline{P (n)}_{1}) (for 0 ≤ i < n), and c_{i} ∈ P (n)_{2i+1}(\underline{P (n)}_{1}) (for 0 ≤ i ≤ p^{n} − 2) defined by equation (10.12) have the following properties:

(a) a_{0} = 1_{1} and c_{0} = e;

(b) ψa_{k} = ∑_{i+j=k} a_{i} ⊗ a_{j};

(c) ϵa_{i} = 0 for all i > 0, in particular, ϵa_{(i)} = 0 for all i;

(d) a_{i} ∗ a_{j} = \binom{i+j}{i}a_{i+j}, provided i + j < p^{n};

(e) a_{(i)}^{∗p} = 0 for 0 ≤ i < n − 1;

(f) χa_{i} = (−1)^{i}a_{i}, in particular, χa_{(i)} = −a_{(i)};

(g) c_{i} = e ∗ a_{i};

(h) a_{(i)}^{∘}a_{(j)} = −a_{(j)}^{∘}a_{(i)};

(i) a_{(i)}^{∘}a_{(i)} = 0;

(j) For all r, r_{∗}a_{k} is the coefficient of x^{k} in the formal identity

r_{∗}a(x) = ∗_{i=0}^{N}{b(x)^{^{∘}i}^{∘}[⟨r, a_{i}⟩]} ∗ ∗_{i=0}^{N−1}{a(x)^{∘}b(x)^{^{∘}i}^{∘}[⟨r, c_{i}⟩]}

in P (n)_{∗}(\underline{P (n)}_{∗})[x]/(x^{p^{n}}).

Proof The statement and proof are identical to [BJW95, Prop. 17.16], except that we offer a simpler proof of (f) (and could have also in [ibid.]; compare the divided power Hopf algebra Γ(a_{1})).

If m is odd, say m = 2k + 1, we can write the defining equation for χa_{m} as

χa_{m} + ∑_{i=1}^{k}(χa_{m−i} ∗ a_{i} + χa_{i} ∗ a_{m−i}) + a_{m} = 0.

By induction, the terms in the sum cancel in pairs, as m−i and i have opposite parity. If m is even, (d) decomposes a_{m} as a ∗-product, and we again use induction.

We emphasize that (e) is not valid for i = n − 1; instead, [RW96, Prop. 2.1(i)] shows that the unstable analogue of equation (7.19) is

(R′′_{n}) a_{(n−1)}^{∗p} = v_{n}a_{(0)} − a_{(0)}^{∘}b_{(0)}^{^{∘}N} ^{∘}[v_{n}] in P (n)_{∗}(\underline{P (n)}_{1}).  (10.14)

Cohomology of real projective space, for p = 2 In this case, L = ℝP^{2N}, with cohomology (6.24). We define Hopf ring elements f_{i} by the identity

r(t) = ∑_{i=0}^{2N}⟨r, f_{i}⟩t^{i} in P (n)^{∗}(ℝP^{2N}) = P (n)_{∗}[t]/(t^{2N+1}), for all r.  (10.15)

Again, we mimic equation (10.12) by writing a_{i} = f_{2i}, a_{(i)} = a_{2^{i}} = f_{2^{i+1}}, and c_{i} = f_{2i+1}. We make the obvious changes to Proposition 10.13 and write f(t) = ∑_{i} f_{i}t^{i}.

§11 Structure of the Hopf ring 39

We warn that the analogy is not perfect; ψa_{k} acquires many extra terms. Also, (d) now requires proof; see [Bo].

Proposition 10.16 For p = 2, the Hopf ring elements f_{i} ∈ \overline{P (n)}_{i}(\underline{P (n)}_{1}) (for 0 ≤ i ≤ 2N) and a_{(i)} = f_{2^{i+1}} ∈ \overline{P (n)}_{2^{i+1}}(\underline{P (n)}_{1}) (for 0 ≤ i ≤ n − 1) defined by equation (10.15) have the following properties:

(a) f_{0} = 1_{1} and f_{1} = e;

(b) ψf_{k} = ∑_{i+j=k} f_{i} ⊗ f_{j};

(c) ϵf_{i} = 0 for all i > 0, in particular, ϵa_{(i)} = 0 for all i;

(d) a_{(i)}^{∘}a_{(j)} = a_{(j)}^{∘}a_{(i)};

(e) f_{i} ∗ f_{j} = \binom{i+j}{i}f_{i+j}, provided i + j ≤ 2N;

(f) a_{(i)} ∗ a_{(i)} = 0 for 0 ≤ i < n − 1;

(g) For all r, r_{∗}f_{k} is the coefficient of t^{k} in the formal identity

r_{∗}f(t) = ∗_{i=0}^{2N}\{f(t)^{^{∘}i}^{∘}[⟨r, f_{i}⟩]\} in \overline{P (n)}_{∗}(\underline{P (n)}_{∗})[t]/(t^{2N+1}).

Again, for i = n − 1, (f) is replaced by (10.14), now taken in \overline{P (n)}_{∗}(\underline{P (n)}_{1}).

Finally, we prove in [Bo] that equation (7.18) lifts in the obvious way.

Lemma 10.17 In the Hopf ring \overline{P (n)}_{∗}(\underline{P (n)}_{∗}) for p = 2, we have

a_{(i)}^{∘}a_{(i)} = b_{(i+1)} for 0 ≤ i ≤ n − 1.  (10.18)

Remark There is a case for writing e here as a_{(−1)}, so that the identity e^{∘}e = b_{(0)} becomes a natural extension of (10.18).

11 Structure of the Hopf ring

In this section, we present two descriptions of the Hopf ring P (n)_{∗}(\underline{P (n)}_{∗}) [replaced by \overline{P (n)}_{∗}(\underline{P (n)}_{∗}) if p = 2]: a clean concise description in terms of the generators and relations developed in §10, and a concrete computational description that specifies exactly what the elements of the Hopf ring are. (This relies heavily on the technical work of Ravenel–Wilson [RW96], and in no way replaces it.)

Theorem 11.1 (Ravenel–Wilson) The Hopf ring P (n)_{∗}(\underline{P (n)}_{∗}) [which is replaced by \overline{P (n)}_{∗}(\underline{P (n)}_{∗}) if p = 2] over P (n)_{∗} has the ^{∘}-generators:

[v_{k}] ∈ P (n)_{0}(\underline{P (n)}_{−2(p^{k}−1)}) for k ≥ n, defined by equation (10.4);

e ∈ P (n)_{1}(\underline{P (n)}_{1}), defined by equation (10.6);

b_{(j)} = b_{p^{j}} ∈ P (n)_{2p^{j}} (\underline{P (n)}_{2}) for j ≥ 0, defined by equation (10.7); a_{(i)} = a_{p^{i}} ∈ P (n)_{2p^{i}}(\underline{P (n)}_{1}) for 0 ≤ i < n, defined by equation (10.12) [replaced by (10.15) if p = 2]; subject to the relations [1]^{∗p} = 1_{0}, e^{∘}e = −b_{(0)}, the main relations (R_{k}) for k > n (see Definition 10.9), and the two variants (10.10) and (10.14) of (R_{n}) [also (10.18) if p = 2].

Allowable monomials For our second description of the Hopf ring, we reinterpret the general monomial (8.1) as the ^{∘}-monomial

\begin{matrix}e^{^{∘}ϵ}^{∘}a^{^{∘}I}^{∘}b^{^{∘}J}^{∘}[v^{K}] \\ = e^{^{∘}ϵ}^{∘}a_{(0)}^{^{∘}i_{0}}^{∘}a_{(1)}^{^{∘}i_{1}}^{∘} …^{∘}a_{(n−1)}^{^{∘}i_{n−1}}^{∘}b_{(0)}^{^{∘}j_{0}} ^{∘}b_{(1)}^{^{∘}j_{1}} ^{∘}b_{(2)}^{^{∘}j_{2}} ^{∘} …^{∘}[v_{n}]^{^{∘}k_{n}} ^{∘}[v_{n+1}]^{^{∘}k_{n+1}} ^{∘} … \end{matrix}  (11.2)

(We adopt the usual convention (e. g. [RW77]) that for any element d with ϵd = 0, d^{^{∘}0} = [1] − 1_{0}, so that d^{^{∘}0}^{∘}d = d holds. We also set [v_{k}]^{^{∘}0} = [v_{k}^{0}] = [1].)

We define it to be allowable or Q-allowable exactly as in Definition 8.2.

A direct description of the allowable monomials is useful, to replace the indirectness of Definition 8.2. As in [RW96], Δ_{0} denotes the multi-index (1, 0, 0, …).

Proposition 11.3 Any allowable ^{∘}-monomial c in the Hopf ring can be written uniquely in one of the standard forms

(a) c = a^{^{∘}I}^{∘}b^{^{∘}G+L}^{∘}[v^{K}] if c does not involve e;
(b) c = e^{∘}a^{^{∘}I}^{∘}b^{^{∘}G+L−Δ_{0}} ^{∘}[v^{K}] if c involves e;

(11.4)

where the multi-index G is defined by

b^{^{∘}G} = b_{(d_{n})}^{^{∘}p^{n}}^{∘}b_{(d_{n+1})}^{^{∘}p^{n+1}}^{∘} …^{∘}b_{(d_{q−1})}^{^{∘}p^{q−1}},  (11.5)

L = (l_{0}, l_{1}, l_{2}, …), and the indices satisfy

(i) q ≥ n; (ii) 0 ≤ d_{n} ≤ d_{n+1} ≤ … ≤ d_{q−1}; (iii) 0 ≤ l_{t} < p^{r} for all t < d_{r}, for n ≤ r < q; (iv) 0 ≤ l_{t} < p^{q} for all t; (v) k_{r} = 0 (i. e. v^{K} contains no factor v_{r}) for all r < q; (vi) In Case (b), d_{n} = 0 or l_{0} > 0. (11.6)

Conversely, any such monomial is allowable.

Proof If the allowable monomial c does not involve e, we choose each d_{r} in turn as small as possible, so that (iii) holds; moreover, (iii) requires this choice of d_{r}. (If we cannot even start, q = n, c = a^{^{∘}I}^{∘}b^{^{∘}L}^{∘}[v^{K}], and (ii), (iii) and (v) become vacuous.) We continue as long as possible, until (iv) holds. In view of (8.3)(i), c does not contain [v_{r}] for any r < q, and (v) holds.

If c has the form e^{∘}c′, we note that e^{∘}c = b_{(0)}^{∘}c′ remains allowable, and apply case (a) to it. Here, we need (vi) so that Δ_{0} can be subtracted off.

Conversely, the monomials (11.4) are easily seen to be allowable.

The algebra structure We recall that a simple system of generators of a graded algebra A with multiplication ∗ over a graded ring R of characteristic p is a set of elements z_{1}, z_{2}, z_{3}, … such that the finite products

z^{∗M} = z_{1}^{∗m_{1}} ∗ z_{2}^{∗m_{2}} ∗ z_{3}^{∗m_{3}} ∗ …,  (11.7)

where 0 ≤ m_{r} < p for each z_{r} of even degree and m_{r} = 0 or 1 for each z_{r} of odd degree, form a set of free R-module generators of A.

The following description is also essentially included in [RW96, Thms. 1.3, 1.4] [except that for p = 2, (d) was not written out explicitly and contains the surprise (iii)]. For I ≠ (1, 1, …, 1), ρ(I) denotes the smallest t such that i_{n−t} = 0.

Theorem 11.8 (Ravenel–Wilson) Assume 0 < n < ∞, and let k be any integer. Then

(a) The Hopf algebra P (n)_{∗}(\underline{P (n)}_{k}) [or \overline{P (n)}_{∗}(\underline{P (n)}_{k}) if p = 2] has as a simple system of ∗-generators the set of all allowable ^{∘}-monomials (11.2) (that lie in it);

(b) The Q-allowable ^{∘}-monomials form a minimal set of algebra ∗-generators of P (n)_{∗}(\underline{P (n)}_{∗}) [or \overline{P (n)}_{∗}(\underline{P (n)}_{∗}) if p = 2];

(c) For p odd, P (n)_{∗}(\underline{P (n)}_{k}) is the tensor product of the following subalgebras, one for each Q-allowable ^{∘}-monomial (that lies in it):

(i) T P_{ρ(I)}(a^{^{∘}I}^{∘}b^{^{∘}J}^{∘}[v^{K}]) for I ≠ (1, 1, …, 1);
(ii) P (a^{^{∘}I}^{∘}b^{^{∘}J}^{∘}[v^{K}]) for I = (1, 1, …, 1);
(iii) E(e^{∘}a^{^{∘}I}^{∘}b^{^{∘}J}^{∘}[v^{K}]);

(d) For p = 2, \overline{P (n)}_{∗}(\underline{P (n)}_{k}) contains the following subalgebras, one for each Qallowable ^{∘}-monomial (that lies in it), and is additively (but not multiplicatively) isomorphic to their tensor product:

(i) T P_{ρ(I)}(e^{^{∘}ϵ}^{∘}a^{^{∘}I}^{∘}b^{^{∘}J}^{∘}[v^{K}]) for I ≠ (1, 1, …, 1);
(ii) P (a^{^{∘}I}^{∘}b^{^{∘}J}^{∘}[v^{K}]) for I = (1, 1, …, 1);
(iii) T P_{n+1}(e^{∘}a^{^{∘}I}^{∘}b^{^{∘}J}^{∘}[v^{K}]) for I = (1, 1, …, 1).

Remark For p = 2, the quotient algebra

H_{∗}(\underline{P (n)}_{k}; 𝔽_{2}) ≅ 𝔽_{2} ⊗_{P (n)_{∗}} \overline{P (n)}_{∗}(\underline{P (n)}_{k})

is the tensor product of the subalgebras listed in (d), interpreted as 𝔽_{2}-algebras.

To complete this description, we need the structure maps ∗, ^{∘}, ψ, ϵ, and χ, which are all (bi)linear. We know ψ, ϵ, and χ on each generator e, a_{(i)}, b_{(j)} and [v_{k}]; then the Hopf ring laws determine these operations in general.

Reduction to standard form We reprove part of Theorem 11.1 by showing that we have enough relations to reduce any Hopf ring expression to a P (n)_{∗}-linear combination of ∗-products (11.7) of allowable ^{∘}-monomials.

For ^{∘}, we need to know how to ^{∘}-multiply any two ^{∘}-monomials (11.2); then the distributive laws for (a∗b)^{∘}c and a^{∘}(b∗c) [modified if p = 2] take care of general ∗-monomials z^{∗M} as in (11.7). As the ^{∘}-generators ^{∘}-commute up to sign [even for p = 2], all we need is a reduction formula for each non-allowable ^{∘}-monomial (11.2).

The relation e^{∘}e = −b_{(0)} takes care of e^{^{∘}2}. If p is odd, a_{(i)}^{^{∘}2} = 0 is automatic, by Proposition 10.13(i). [If p = 2, we use a_{(i)}^{^{∘}2} = b_{(i+1)} instead, from equation (10.18).]

For the disallowed monomials (i) and (ii) of (8.3), we use the same relations as in Lemma 8.5, now working modulo ∗-decomposables as well. These use only the relations (R_{k}) for k > n and (10.10), which implies (R_{n}).

For the ∗-product of two ∗-monomials (11.7), we shuffle the ^{∘}-monomials into the desired order (with the appropriate sign), and deal with excess ∗-powers of any ∘-monomial. [If p = 2, shuffling introduces extra terms, but the process quickly terminates, because the ∗-commutator c ∗ d − d ∗ c of any two ^{∘}-monomials is ∗- central; see [Bo] for details.]

The Frobenius operator To finish the reduction to standard form, we need a formula for the Frobenius operator F c = c^{∗p} = c ∗ c ∗ … ∗ c on each allowable ∘-monomial c of even degree [or any degree if p = 2].

We start from the relation [1]^{∗p} = 1_{0}, which we rewrite as F ([1] − 1_{0}) = 0. We next reverse the identity [BJW95, (15.13)] as

F (c^{∘}b^{^{∘}J}) = (F c)^{∘}b^{^{∘}0,J},  (11.9)

where 0, J denotes the extended multi-index (0, j_{0}, j_{1}, j_{2}, …). The proof used only the property ψb_{k} = ∑_{i+j=k} b_{i} ⊗ b_{j}. Since a_{k} has the same property when p is odd, according to Proposition 10.13(b), we similarly have

F (a^{^{∘}I,0}^{∘}c) = a^{^{∘}0,I}^{∘}F c

for any multi-index I = (i_{0}, i_{1}, i_{2}, …, i_{n−2}). [For p = 2, Proposition 10.16 delivers the same result, and also

F (e^{∘}c) = a_{(0)}^{∘}F c.]

For a_{(n−1)}, we rewrite the relation (10.14) as

F a_{(n−1)} = v_{n}a_{(0)} − a_{(0)}^{∘}b_{(0)}^{^{∘}N} ^{∘}[v_{n}].

Since applying −^{∘}[v^{K}] preserves ∗-multiplication, we immediately have

F (c^{∘}[v^{K}]) = (F c)^{∘}[v^{K}].

Combining these, we find the general formulae

F (a^{^{∘}I,0}^{∘}b^{^{∘}J}^{∘}[v^{K}]) = 0  (11.10)

and (with attention to the shuffles needed and the resulting signs)

F (a^{^{∘}I,1}^{∘}b^{^{∘}J}^{∘}[v^{K}]) = (−1)^{|I|+1}a^{^{∘}1,I} ^{∘}b^{^{∘}N,J} ^{∘}[v_{n}v^{K}] + (−1)^{|I|}v_{n}a^{^{∘}1,I} ^{∘}b^{^{∘}0,J} ^{∘}[v^{K}], (11.11)

where |I| = ∑_{r} i_{r}.

[If p = 2, we need also the formulae involving e, which are

F (e^{∘}a^{^{∘}I,0}^{∘}b^{^{∘}J}^{∘}[v^{K}]) = 0  (11.12)

and

F (e^{∘}a^{^{∘}I,1}^{∘}b^{^{∘}J}^{∘}[v^{K}]) = a^{^{∘}0,I} ^{∘}b_{(1)}^{∘}b^{^{∘}N,J} ^{∘}[v_{n}v^{K}] + v_{n}a^{^{∘}0,I} ^{∘}b_{(1)}^{∘}b^{^{∘}0,J} ^{∘}[v^{K}], (11.13)

in which we make use of a_{(0)}^{∘}a_{(0)} = b_{(1)}. For example,

F (e^{∘}a_{(n−1)}^{∘}b_{(0)}^{^{∘}N}) = b_{(0)}^{^{∘}N+2^{n+1}} ^{∘}[v_{n+1}] + v_{n}b_{(1)}^{^{∘}2^{n}} + v_{n}^{2}b_{(0)}^{^{∘}N}^{∘}b_{(1)} + v_{n+1}b_{(0)}^{^{∘}2^{n}}

after reduction to standard form, which recovers equation (7.21).] A reduction formula There is a difficulty with equation (11.11) which obscures the algebraic structure of the Hopf ring. Even in the simple case

F (a_{(n−1)}^{∘}b^{^{∘}G}) = −a_{(0)}^{∘}b^{^{∘}N,G}^{∘}[v_{n}] + v_{n}a_{(0)}^{∘}b^{^{∘}0,G},

with G as in equation (11.5), the first term on the right is visibly not allowable (unless q = n, so that G = 0). What we need is a reduction formula for

b^{^{∘}0,G}^{∘}[v_{n}] = b_{(d_{n}+1)}^{^{∘}p^{n}}^{∘}b_{(d_{n+1}+1)}^{^{∘}p^{n+1}}^{∘} …^{∘}b_{(d_{q−1}+1)}^{^{∘}p^{q−1}}^{∘}[v_{n}],

which is essentially Lemma 3.8 of [RW96]. It involves the p-th ^{∘}-power of b^{^{∘}G},

(b^{^{∘}G})^{^{∘}p} = b^{^{∘}pG} = b_{(d_{n})}^{^{∘}p^{n+1}} ^{∘}b_{(d_{n+1})}^{^{∘}p^{n+2}}^{∘} …^{∘}b_{(d_{q−1})}^{^{∘}p^{q}}.

Lemma 11.14 Using only the main relations (R_{k}), the ^{∘}-monomial b^{^{∘}0,G}^{∘}[v_{n}], with G as in equation (11.5), reduces to an allowable monomial by a formula of the form

b^{^{∘}0,G}^{∘}[v_{n}] ≡ (−1)^{q−n}b^{^{∘}pG}^{∘}[v_{q}] + …,

where the omitted terms do not involve any a_{(i)} and either (i) have the form b^{^{∘}J}^{∘}[v_{k}] with b^{^{∘}J} lexicographically higher than b^{^{∘}pG} (see §8), (ii) lie in the ideal 𝔙 = (v_{n}, v_{n+1}, …), (iii) have [v]-length at least 2, or (iv) are ∗-decomposable.

We apply this to equation (11.11) [also (11.13) if p = 2].

Corollary 11.15 For the general allowable ^{∘}-monomial (11.4)(a) without e, we have

F (a^{^{∘}I,1}^{∘}b^{^{∘}G+L}^{∘}[v^{K}]) ≡ (−1)^{q−n+|I|+1}a^{^{∘}1,I}^{∘}{b^{^{∘}N,L}^{∘}b^{^{∘}pG}^{∘}[v_{q}v^{K}] + …}.  (11.16)

[If p = 2, we similarly obtain

F (e^{∘}a^{^{∘}I,1}^{∘}b^{^{∘}G+L−Δ_{0}} ^{∘}[v^{K}]) ≡ a^{^{∘}0,I}^{∘}{b^{^{∘}N,L}^{∘}b^{^{∘}2G}^{∘}[v_{q}v^{K}] + …}  (11.17)

from (11.4)(b).]

The leading term on the right in equation (11.16) is always allowable: written in standard form (11.4), it is a^{^{∘}1,I}^{∘}b^{^{∘}G+L′} ^{∘}[v^{K′}], with the same G, v^{K′} = v_{q}v^{K}, and b^{^{∘}L′} = b^{^{∘}N,L}^{∘}b^{^{∘}(p−1)G}. Careful bookkeeping shows that, as the indices vary, it runs through all the Q-disallowed ^{∘}-monomials of type (8.3)(iii) that are nevertheless allowable, once each. [Similarly, (11.17) accounts for types (iv) and (v).]

It follows that F never kills anything unexpected. Now we can read off parts (c) and (d) of Theorem 11.8.

Proof of Lemma For n ≤ r ≤ q, we set c_{r} = b_{(d_{n})}^{^{∘}p^{n+1}} ^{∘}b_{(d_{n+1})}^{^{∘}p^{n+2}}^{∘} …^{∘}b_{(d_{r−1})}^{^{∘}p^{r}}, so that (conventionally) c_{n} = b^{^{∘}0} and c_{q} = b^{^{∘}pG}.

We show first that c_{r}^{∘}[v_{s}] ≡ 0 whenever n ≤ s < r ≤ q, by induction on s. We ^{∘}-multiply (R_{d_{s}+s}) by c_{s}; by (7.14), the k-th term is c_{s}^{∘}b_{(d_{s}+s−k)}^{^{∘}p^{k}}^{∘}[v_{k}]. If k < s, this term is neglected by induction. (If s = n, there are no such terms.) If k > s, we have d_{s} + s − k < d_{s}, and this term is lexicographically higher. If k = s, we have c_{s}^{∘}b_{(d_{s})}^{^{∘}p^{s}}^{∘}[v_{s}], which gives c_{s+1}^{∘}[v_{s}] ≡ 0 when we ^{∘}-multiply by b_{(d_{s})}^{^{∘}p^{s+1}−p^{s}}; hence c_{r}^{∘}[v_{s}] ≡ 0 for any r > s, if we ^{∘}-multiply by further factors.

§12 Effect on homotopy groups 44

Then we show that c_{s}^{∘}b_{(d_{s}+1)}^{^{∘}p^{s}}^{∘}[v_{s}] ≡ −c_{s+1}^{∘}[v_{s+1}] for n ≤ s < q, from which the result follows by induction, starting from c_{n} = b^{^{∘}0}. We ^{∘}-multiply (R_{d_{s}+s+1}) by c_{s}. The k-th term is c_{s}^{∘}b_{(d_{s}+s+1−k)}^{^{∘}p^{k}}^{∘}[v_{k}], which we have just shown is negligible if k < s. If k > s + 1, we have d_{s} + s + 1 − k < d_{s}, and the term is lexicographically higher. The two remaining terms, with k = s and k = s + 1, are the desired terms.

12 Effect on homotopy groups

Given an unstable operation r: P (n)^{k}(−) → P (n)^{m}(−), where k, m > 0, consider the homomorphism of homotopy groups r_{∗}: Σ^{k}P (n)_{∗} → Σ^{m}P (n)_{∗} (see diagram (3.9)) induced by the representing map r: \underline{P (n)}_{k} → \underline{P (n)}_{m}. By [BJW95, Lemma 13.9], it is given on Σ^{k}v, where v ∈ P (n)_{i}, by the unstable analogue of equation (7.6), namely

r_{∗}Σ^{k}v = Σ^{m}⟨r, e_{k+i}^{∘}[v]⟩,  (12.1)

where e_{2j} = b_{(0)}^{^{∘}j} and e_{2j+1} = e^{∘}b_{(0)}^{^{∘}j}.

We therefore seek more information on the relations in the Hopf ring.

The first higher-order relation We need the Hopf ring version for P (n) of Bendersky’s Lemma [Be86, Thm. 6.2], which immediately implies Lemma 8.12.

Lemma 12.2 For q ≥ n, we have in P (n)_{∗}(\underline{P (n)}_{g(n,q−1)+1}) the reduction formula

e_{g(n,q)−1}^{∘}[v_{q}] ≡ v_{q}e_{g(n,q−1)+1} mod I_{q}P (n)_{∗}(\underline{P (n)}_{g(n,q−1)+1}).  (12.3)

[If p = 2, this is almost superseded by

e_{g(n,q)−2}^{∘}[v_{q}] ≡ v_{q}e_{g(n,q−1)} + F (e_{g(n,q−1)−1}^{∘}a_{(n−1)}) mod I_{q}, for q ≥ n + 1.] (12.4)

Proof We establish (12.3) by induction on q. For q = n, it follows immediately from (10.10). For q > n, we return to the definition of the relation (R_{q}). We expand [p](x) = ∑_{K} λ(K)v^{K}x^{d(K)}, summing over multi-indices K, with coefficients λ(K) ∈ 𝔽_{p} and exponents d(K); then if we write b(x) = 1_{2} + \overline{b}(x), (10.8) becomes

1_{2} + \overline{b}( ∑_{K} λ(K)v^{K}x^{d(K)}) = ∗_{K} \{1_{2} + \overline{b}(x)^{^{∘}d(K)}^{∘}[v^{K}]\}^{∗λ(K)}.  (12.5)

We apply the suspension e_{h}^{∘}−, where h = g(n, q−1) − 1, which kills 1_{2} and most ∗-products and thus drastically simplifies (12.5) to

e_{h}^{∘}\overline{b}( ∑_{K} λ(K)v^{K}x^{d(K)}) = ∑_{K} λ(K)e_{h}^{∘}\overline{b}(x)^{^{∘}d(K)}^{∘}[v^{K}].

We take the coefficients of x^{p^{q}} and work mod I_{q}. On the left, by (6.18), the only surviving term in [p](x) is v_{q}x^{p^{q}}, giving e_{h}^{∘}v_{q}b_{(0)}, the right side of (12.3). On the right, e_{h}^{∘}[v_{k}] ≡ 0 for all k < q, by induction on q, since h = g(n, q−1) − 1 ≥ g(n, k) − 1. This leaves only e_{h}^{∘}b_{(0)}^{^{∘}p^{q}} ^{∘}[v_{q}], as required.

[For (12.4), we take h = g(n, q−1) − 2 instead. We still have enough e’s to kill [v_{k}] for any k < q − 1, but not [v_{q−1}]. By (6.16), the only terms of interest in [2](x) are v_{q−1}x^{2^{q−1}} and v_{q}x^{2^{q}}. Instead of (12.3), we find

e_{h}^{∘}v_{q}b_{(0)} ≡ e_{h}^{∘}b_{(1)}^{^{∘}2^{q−1}} ^{∘}[v_{q−1}] + e_{h}^{∘}b_{(0)}^{^{∘}2^{q}} ^{∘}[v_{q}].

The first and third terms appear as the second and first terms in (12.4). The second term is not allowable; but if q = n + 1, we use (11.13) to write it as

b_{(0)}^{^{∘}N}^{∘}b_{(1)}^{^{∘}2^{n}} ^{∘}[v_{n}] ≡ F (e^{∘}a_{(n−1)}^{∘}b_{(0)}^{^{∘}N}).

If q > n + 1, we have e_{h}^{∘}[v_{q−1}] ≡ F (e_{g(n,q−2)−1}^{∘}a_{(n−1)}) by induction. Then

e_{h}^{∘}b_{(1)}^{^{∘}2^{q−1}} ^{∘}[v_{q−1}] ≡ b_{(1)}^{^{∘}2^{q−1}} ^{∘}F (e_{g(n,q−2)−1}^{∘}a_{(n−1)}) = F (b_{(0)}^{^{∘}2^{q−1}} ^{∘}e_{q(n,q−2)−1}^{∘}a_{(n−1)}),

as required, with the help of equation (11.9).]

Proofs for §3 Now we can finish the proofs of two lemmas.

Proof of Lemma 3.8 For (a), by (12.1),

r_{∗}Σ^{k}(v_{n}v) = Σ^{m}⟨r, e_{k+q+2N}^{∘}[v_{n}v]⟩ = Σ^{m}⟨r, e_{k+q+2N}^{∘}[v_{n}]^{∘}[v]⟩.

Since k + q > 0, we can use (10.10) to rewrite this as

r_{∗}Σ^{k}(v_{n}v) = Σ^{m}⟨r, v_{n}e_{k+q}^{∘}[v]⟩ = v_{n}r_{∗}Σ^{k}v.

Part (b) is similar, with (12.3) in place of (10.10).

Lemma 12.6 Let r: \underline{P (n)}_{k} → \underline{P (n)}_{k} be any map, where k > g(n, m−1), and suppose that on homotopy, r_{∗}: Σ^{k}P (n)_{∗} → Σ^{k}P (n)_{∗} is given on the bottom class by r_{∗}Σ^{k}1 = λΣ^{k}1, where λ ∈ 𝔽_{p}. Then on any monomial v^{K} in the elements v_{n}, v_{n+1}, …, v_{m}, r_{∗} has the form

r_{∗}Σ^{k}v^{K} = λΣ^{k}v^{K} + ∑_{L>K} c_{L}Σ^{k}v^{L}  (12.7)

with coefficients c_{L} ∈ 𝔽_{p}, where we order the multi-indices L = (l_{n}, l_{n+1}, …) lexicographically (as in §8).

Proof We use induction on the length of v^{K}, starting from K = 0. If (12.7) holds for Σ^{k}v^{K}, Lemma 3.8(b) gives

r_{∗}Σ^{k}(v_{q}v^{K}) ≡ λΣ^{k}(v_{q}v^{K}) + ∑_{L>K} c_{L}Σ^{k}(v_{q}v^{L}) mod I_{q}.

If we assume (as we may) that v^{K} contains no factors v_{t} with t < q, all monomials in I_{q} will be larger lexicographically than v_{q}v^{K}, and we have the result for v_{q}v^{K}.

Proof of Lemma 3.6 Take any map f: \underline{P (n, m)}_{k} → \underline{P (n, m)}_{k}, where g(n, m−1) < k ≤ g(n, m). Suppose f_{∗}Σ^{k}1 = λΣ^{k}1. We apply Lemma 12.6 to the composite

\underline{P (n)}_{k} − ; − ; − ; − ; →^{ρ(m)} \underline{P (n, m)}_{k} − ; − ; →^{f} \underline{P (n, m)}_{k} − ; − ; − ; − ; →^{\overline{θ(m)}} \underline{P (n)}_{k}

to deduce that

\overline{θ(m)}_{∗}f_{∗}Σ^{k}v^{K} = λΣ^{k}v^{K} + ∑_{L>K} c_{L}Σ^{k}v^{L}

for any monomial v^{K} in the generators v_{n}, v_{n+1}, …, v_{m}. We apply ρ(m)_{∗}, to see that f_{∗}Σ^{k}v^{K} has the same form (possibly with some terms v^{L} deleted). It is now clear that if λ ≠ 0, f_{∗} is an isomorphism and f is a homotopy equivalence.

References 46

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Department of Mathematics, Johns Hopkins University 3400 N. Charles St., Baltimore, MD 21218-2689, U.S.A.

                                      E-mail: boardman@math.jhu.edu, wsw@math.jhu.edu