v_{n}-Bockstein Operations on Iterated Loop Spaces

Dai Tamaki

Department of Mathematical Sciences Faculty of Science Shinshu University Matsumoto, 390-8621 Japan

Abstract

In this note, we determine the action of the first differential in the Atiyah-Hirzebruch spectral sequence for the n-th Morava K-theory of the p-adic construction D_{p}^{(ℓ)}(X) = C_{ℓ}(p) ∧_{Σ_{p}} X^{∧p} of a pointed space X for n ≥ 1. Our formula gives us a way to compute the first differentials in the Atiyah-Hirzebruch spectral sequence for the Morava K-theory of iterated loop spaces.

Analogous formulas are also proved for the Steenrod operations.

Key words: homology operation, Morava K-theory. 1991 MSC: 55S05, 55S12.

1 Introduction

For a prime p, let A be the mod p Steenrod algebra and A_{∗} be its dual. Milnor proved that, when p is odd, there exist elements ξ_{1}, ξ_{2}, ⋯ and τ_{0}, τ_{1}, ⋯ in A_{∗} with degrees

deg ξ_{i} = 2p^{i} − 2 and deg τ_{i} = 2p^{i} − 1

and

A_{∗} ≅ 𝔽_{p}[ξ_{1}, ξ_{2}, ⋯] ⊗ Λ(τ_{0}, τ_{1}, ⋯),

as algebras [Milnor58]. When p = 2, we have

A_{∗} ≅ 𝔽_{2}[ξ_{1}, ξ_{2}, ⋯]

  Email address: rivulus@math.shinshu-u.ac.jp (Dai Tamaki).

Preprint submitted to Elsevier Science 18 May 2001

with

deg ξ_{i} = 2^{i} − 1.

He denote the dual of τ_{n} (the dual of ξ_{n+1} when p = 2) in A by Q_{n}.

On the other hand, there are another kind of “Q”-operations, namely the homology operations on the mod p homology of infinite loop spaces. For an infinite loop space X and any nonnegative integer i, there is a linear map

Q_{i} : H_{∗}(X) ⟶ H_{p∗+i(p−1)}(X)

which is natural with respect to (i + 1)-fold loop maps. This Q_{i} can also be defined on the mod p homology of (i + 2)-fold loop spaces.

To avoid confusion, we denote the Milnor’s Q_{n} operation by β_{1}^{v_{n}} and, throughout the rest of this paper, Q_{i} always denotes the i-th homology operation.

The operation β_{1}^{v_{n}} is called the (first) v_{n}-Bockstein, for this is known to be the first nontrivial differential in the Atiyah-Hirzebruch spectral sequence for the n-th Morava K-theory whose coefficient ring is 𝔽_{p}[v_{n}, v_{n}^{−1}]. Thus knowing the relation between the v_{n}-Bockstein and the homology operations is the first step toward the understanding of the n-th Morava K-theory of iterated loop spaces, such as Ω^{ℓ}Σ^{ℓ}X.

The aim of the first three sections of this paper is to find explicit formulas for the action of β_{1}^{v_{n}} on homology operations.

In order to describe the formulas, let us recall several well-known facts on the mod p homology of iterated loop spaces. Proofs can be found in the chapter III of [Cohen-Lada-May].

For an ℓ-fold loop space X, the evaluation map induces a surjective map

H_{∗}(Ω^{ℓ}Σ^{ℓ}X) ⟶ H_{∗}(X).

Thus it is enough to determine the action of β_{v_{n}}^{1}, the dual operation of β_{1}^{v_{n}}, on H_{∗}(Ω^{ℓ}Σ^{ℓ}X). When ℓ = 1, it is easy to see that K(n)_{∗}(ΩΣX) is the tensor algebra on K(n)_{∗}(X). Thus, in the following, we assume that ℓ ≥ 2.

For ℓ ≥ 2, we have the following three kinds of operations on the mod p homology of ℓ-fold loop spaces.

Q_{i}: H_{k}(X) ⟶ H_{i(p−1)+kp}(X) for 0 ≤ i ≤ ℓ − 2
ξ_{ℓ−1} : H_{k}(X) ⟶ H_{(ℓ−1)(p−1)+kp}(X)
λ_{ℓ−1} : H_{k}(X) ⊗ H_{k′}(X) ⟶ H_{ℓ−1+k+k′}(X)

together with the mod p Bockstein operation β. It is proved by F. Cohen in [Cohen-Lada-May] that, for any pointed path connected space X, the mod p homology of Ω^{ℓ}Σ^{ℓ}X is generated as algebras by the elements produced from H̃_{∗}(X) by applying the above operations. Since β and β_{v_{n}}^{1} are commutative in the graded sense, it is enough to study the relation between β_{v_{n}}^{1} and Q_{i}, ξ_{ℓ−1} and λ_{ℓ−1} on H_{∗}(Ω^{ℓ}Σ^{ℓ}X). Theorem 1 For a path-connected space X and n > 0, the following formulas hold in H_{∗}(Ω^{ℓ}Σ^{ℓ}X).

(1) When p is odd, for 0 ≤ i ≤ ℓ − 2 and x ∈ H̃_{∗}(X)

β_{v_{n}}^{1}Q_{i}(x) = Q_{i−1}(β_{v_{n−1}}^{1}(x)) − Q_{i−2p^{n−1}−1}(β_{v_{n−2}}^{1}(x)) + ⋯
+(−1)^{n−1}Q_{i−2p^{n−1}−2p^{n−2}−···−2p−1}(β_{v_{0}}^{1}(x))
+(−1)^{n}βQ_{i−2p^{n−1}−···−2p−2}(x)
= ∑_{k=1}^{n}(−1)^{n−k}Q_{i−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)) + (−1)^{n}βQ_{i−2d_{n}}(x) (1)
β_{v_{n}}^{1}ξ_{ℓ−1}(x) = ∑_{k=1}^{n}(−1)^{n−k}Q_{ℓ−1−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)) + (−1)^{n}βQ_{ℓ−1−2d_{n}}(x)
+ \underbrace{λ_{ℓ−1}(β_{v_{n}}^{1}x, λ_{ℓ_{1}}(x, ⋯, λ_{ℓ−1}}_{p−1}(x, x) ⋯))  (2)

where d_{n,k} = \frac{p^{n}−p^{k}}{p−1} and d_{n} = d_{n,0} = \frac{p^{n}−1}{p−1}. By applying β, we obtain

β_{v_{n}}^{1}βQ_{i}(x) = ∑_{k=1}^{n}(−1)^{n−k+1}βQ_{i−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)).

(2) When p = 2, for 0 ≤ i ≤ ℓ − 2 and x ∈ H̃_{∗}(X)

β_{v_{n}}^{1}Q_{i}(x) = Q_{i−1}(β_{v_{n−1}}^{1}(x)) + Q_{i−2^{n}−1}(β_{v_{n−2}}^{1}(x)) + ⋯
+Q_{i−2^{n}−2^{n−1}−···−2^{2}−1}(β_{v_{0}}^{1}(x))
+Q_{i−2^{n}−···−2−1}(x)
= ∑_{k=1}^{n} Q_{i−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)) + Q_{i−2d_{n}−1}(x)
β_{v_{n}}^{1}ξ_{ℓ−1}(x) = ∑_{k=1}^{n} Q_{ℓ−1−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)) + Q_{ℓ−1−2d_{n}−1}(x)
+λ_{ℓ−1}(β_{v_{n}}^{1}x, x).

(3) For x, y ∈ H̃_{∗}(X),

β_{v_{n}}^{1}λ_{ℓ−1}(x, y) = λ_{ℓ−1}(β_{v_{n}}^{1}(x), y) + (−1)^{|x|}λ_{ℓ−1}(x, β_{v_{n}}^{1}(y))  (3)

Notice that, since we need only homological information, our result should depend only on the stable homotopy type of Ω^{ℓ}Σ^{ℓ}X. By the stable splitting theorem [Snaith74], if X is path-connected, we have

Ω^{ℓ}Σ^{ℓ}X ≃_{S} ⋁_{j} D_{j}^{(ℓ)}(X),

where

D_{j}^{(ℓ)}(X) = C_{ℓ}(j)_{+} ∧_{Σ_{j}} X^{∧j}

and C_{ℓ}(j) is the configuration space of j little ℓ-cubes. This is called the j-adic construction on X.

Under this splitting, the elements Q_{i}(x) and ξ_{ℓ−1}(x) for x ∈ H_{∗}(X) lie in H_{∗}(D_{p}^{(ℓ)}(X)), which means that it is sufficient to find formulas for the action of β_{v_{n}}^{1} on H_{∗}(D_{p}^{(ℓ)}(X)) or, dually, the action of β_{1}^{v_{n}} on H^{∗}(D_{p}^{(ℓ)}(X)), together with the formula for λ_{ℓ−1}.

The formula (1) says that β_{1}^{v_{n}} penetrates Q_{i} to reveal an “internal structure” of D_{p}^{(ℓ)}(X) concerning β_{1}^{v_{n−1}}, ⋯, β_{1}^{v_{0}} on X. This gives another example of the “prism”-like property of the functor D_{p}(ℓ)(−); i.e. the functor D_{p}^{(ℓ)}(−) decomposes information related to v_{n} into information related to v_{0}, ⋯, v_{n−1}. This property of D_{p}^{(ℓ)}(−) was first observed by Toda [Toda68] in the form of important relations in the stable homotopy groups of spheres.

The above formula also indicates that K(n)_{∗}(Ω^{ℓ}Σ^{ℓ}X) is not a functor of K(n)_{∗}(X) It might be a functor of K(n)_{∗}(X) together with the actions of (higher) v_{m}-Bocksteins for 0 ≤ m ≤ n − 1.

Since the Steenrod operations are also defined by using the cohomology of D_{p}(X) = D_{p}^{(∞)}(X), we can easily obtain a similar formula for the Steenrod operations.

Corollary 2 In the odd primary Steenrod algebra, we have the following relations

β_{1}^{v_{n}}P^{j} = ∑_{k=1}^{n}(−1)^{n−k}P^{j+d_{n,k−1}}β_{1}^{v_{k−1}} + (−1)^{n}βP^{j+d_{n}}.  (4)

In the mod 2 Steenrod algebra, we have

β_{1}^{v_{n}}Sq^{2j} = ∑_{k=1}^{n} Sq^{2(j+d_{n,k−1})}β_{1}^{v_{k−1}} + Sq^{2(j+d_{n})+1}

A special case of the 2-primary case

β_{1}^{v_{n}}Sq_{1} = Sq_{0}β_{1}^{v_{n−1}}

was used in [Zarati91], where Sq_{i}x = Sq^{i+deg x}x.

This paper is organized as follows.

In §2, (3) is proved and the proof of (2) is reduced to the proof of (1).

In §3, we recall the cohomological properties of D_{p}(X) used in later sections.

In §4, we show that

β_{v_{n}}^{1}Q_{i}(x) = ∑_{k=1}^{n} A_{k−1}Q_{i−d_{n,k}}(β_{v_{k−1}}^{1}(x)) + A_{n}βQ_{i−d_{n}}(x)

for some constants A_{0}, A_{1}, ⋯, A_{n} ∈ 𝔽_{p}.

In §5, these constants are determined by evaluating β_{v_{n}}^{1} on H_{∗}(D_{p}(BΣ_{p})) directly by the Nishida relations.

In §6, as a corollary, we prove the formula (4).

Acknowledgement and Comments: The stable formula (1) was a part of the author’s Ph.D. thesis at the University of Rochester under the supervision of Fred Cohen in 1993. The author would like to thank him for helpful conversations.

The author first planned to use these formulas togethor with the cobar spectral sequence constructed in [Tamaki94] to study the Morava K-theory of Ω^{ℓ}Σ^{ℓ}X. But it turns out that there exist lots of higher differentials in the cobar spectral sequnece which cannnot be detected by (1) even when X is a sphere.

Recently, the author finds another cobar-type spectral sequence [Tamaki01] with which the author succeeds in determining the Morava K-theory of Ω^{4}S^{2ℓ+1}.

(1) is useful and necessary in the calculation and the author has decided to publish this result.

2 Reduction to the Stable Case

Throughout the rest of this paper, let X denote a pointed space. In this section, we consider unstable operations ξ_{ℓ_{1}} and λ_{ℓ−1}.

The formula (3) follows from the next lemma.

Lemma 3 We have the following commutative diagram.

[figure: A_{∗} ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X) ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X); A_{∗} ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X); A_{∗} ⊗ A_{∗} ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X) ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X); H_{∗}(Ω^{ℓ}Σ^{ℓ}X); A_{∗} ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X) ⊗ A_{∗} ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X); H_{∗}(Ω^{ℓ}Σ^{ℓ}X) ⊗ H_{∗}(Ω^{ℓ}Σ^{ℓ}X); λ_{ℓ−1}; Δ⊗1⊗1; 1⊗T ⊗1; λ_{ℓ−1}]

where T is the switching map of graded modules. PROOF. This is immediate from the definition of λ_{ℓ_{1}}. Corollary 4 For x, y ∈ H_{∗}(X), we have

β_{v_{n}}^{1}(λ_{ℓ−1}(x, y)) = λ_{ℓ−1}(β_{v_{n}}^{1}(x), y) + (−1)^{|x|}λ_{ℓ−1}(x, β_{v_{n}}^{1}(y))

in H_{∗}(Ω^{ℓ}Σ^{ℓ}X). In order to determine β_{v_{n}}^{1}(ξ_{ℓ−1}(x)), consider the Freudenthal suspension map

E : Ω^{ℓ}Σ^{ℓ}X ⟶ Ω^{∞}Σ^{∞}X.

The induced map on homology

E_{∗} : H_{∗}(Ω^{ℓ}Σ^{ℓ}X) ⟶ H_{∗}(Ω^{∞}Σ^{∞}X)

maps Q_{i}(x) to Q_{i}(x) for 0 ≤ i ≤ ℓ − 2 and ξ_{ℓ−1} to Q_{ℓ−1}(x). Suppose we have proved (1). Then

β_{v_{n}}^{1}(ξ_{ℓ−1}(x)) ≡ ∑_{k=1}^{n}(−1)^{n−k}Q_{ℓ−1−d_{n,k}}(β_{v_{k−1}}^{1}(x))+(−1)^{n}βQ_{ℓ−1−d_{n}}(x) (mod Ker E_{∗}).

For simplicity, let us denote

S_{n,ℓ}(x) = ∑_{k=1}^{n}(−1)^{n−k}Q_{ℓ−1−d_{n,k}}(β_{v_{k−1}}^{1}(x)) + (−1)^{n}βQ_{ℓ−1−d_{n}}(x).

We follow the idea of the proof of the Nishida relation in [Cohen-Lada-May] to estimate the unstable error term

Γ = β_{v_{n}}^{1}(ξ_{ℓ−1}(x)) − S_{n,ℓ}(x).

We first note that Γ can be written as

Γ = ∑ λ_{I_{1}} ⋯ λ_{I_{j}}

where each λ_{I_{k}} is a basic λ_{ℓ−1}-product (see p.222 of [Cohen-Lada-May] for the definition of basic λ_{ℓ−1}-product) and if λ_{I_{i}} = ⋯ = λ_{I_{j}} then I_{i} = ⋯ = I_{j}.

Let M_{X} be the submodule of H_{∗}(Ω^{ℓ}Σ^{ℓ}X) spanned by the elements of the form αλ_{I}^{m} where λ_{I} is a basic λ_{ℓ−1}-product, m ≢ 0 (mod p) and α does not contain additional factors of λ_{I}. An important fact proved in [Cohen-Lada-May] is the restriction of

λ_{ℓ−1}(ι, −) : H_{∗}(Ω^{ℓ}Σ^{ℓ}X) ⟶ H_{∗}(Ω^{ℓ}Σ^{ℓ}(X ∨ S^{m}))

to M_{X} is a monomorphism, where ι ∈ H_{m}(S^{m}) is a generator.

Since our error term Γ belongs to M_{X}, it is enough to compute λ_{ℓ−1}(ι, Γ).

From the formulas

λ_{ℓ−1}(ι, ξ_{ℓ−1}(x)) = \underbrace{λ_{ℓ−1}(ι, λ_{ℓ−1}(x, ⋯, λ_{ℓ−1}}_{p−1}(x, x)) ⋯)
λ_{ℓ−1}(x, Q_{i}(y)) = 0

together with (3) and the Cartan formula for the Browder operation, we see that Γ has no decomposable summand. Thus Γ is a sum of (p−1)-fold iterated Browder operations. By the transgression theorem for the Browder operation, Γ is an image of the homology suspension and our formula follows from

β_{v_{n}}^{1}(ξ_{0}(x)) = β_{v_{n}}^{1}(x^{p}) = \underbrace{[β_{v_{n}}^{1}(x), [x, ⋯, [}_{p−1} x, x]] ⋯].

Thus the proof of (2) is reduced to the proof of (1) which will be proved in the next three sections.

3 (Co)homological Properties of D_{p}(X)

Let us denote

D_{p}(X) = D_{p}^{(∞)}(X) = EΣ_{p} ∧_{Σ_{p}} X^{∧p}.

In this section, we state important properties of H_{∗}(D_{p}(X)) and H^{∗}(D_{p}(X)) used in later calculations. There is nothing new in here. Proofs can be found, for example, in [Steenrod-Epstein] or [May70].

Let us consider a special case first. When X = S^{0}, we have D_{p}(S^{0}) = BΣ_{p} ∪ {∗}. We first need H^{∗}(BΣ_{p}) together with the action of cohomology operations. Recall in §1 we defined d_{n} = \frac{p^{n}−1}{p−1}.

Lemma 5 There are elements u ∈ H^{2p−3}(BΣ_{p}) and v ∈ H^{2p−2}(BΣ_{p}) with the following properties.

(1) As algebras,

H^{∗}(BΣ_{p}) ≅ \begin{cases}Λ(u) ⊗ 𝔽_{p}[v] & p > 2 \\ 𝔽_{2}[u] & p = 2. \end{cases}

(2) For any n ≥ 0,

β_{1}^{v_{n}}(u) = v^{1+d_{n}},
β_{1}^{v_{n}}(v) = 0

where v = u^{2} when p = 2.

(3) For any nonnegative integers j and ℓ,

P^{j}(v^{ℓ}) = \binom{ℓ(p − 1)}{j}v^{ℓ+j},
P^{j}(uv^{ℓ−1}) = \binom{ℓ(p − 1) − 1}{j}uv^{ℓ+j−1},

where P^{j} = Sq^{2j} when p = 2.

Let e_{2ℓ(p−1)−1} and e_{2ℓ(p−1)} be the duals of uv^{ℓ−1} and v^{ℓ} in H_{∗}(BΣ_{p}), respectively. By dualizing Lemma 5, we obtain the following.

Corollary 6 In H_{∗}(BΣ_{p}), the following formulas hold.

β_{v_{n}}^{1}(e_{2ℓ(p−1)}) = e_{2(ℓ−d_{n})(p−1)−1}.
β_{v_{n}}^{1}(a_{2ℓ}) = 0.
P_{∗}^{j}(e_{2ℓ(p−1)}) = \binom{(ℓ − j)(p − 1)}{j}e_{2(ℓ−j)(p−1)}.
P_{∗}^{j}(e_{2ℓ(p−1)−1}) = \binom{(ℓ − j)(p − 1) − 1}{j}e_{2(ℓ−j)(p−1)−1}.

In general, consider the spectral sequences defined by the skeletal filtration on EΣ_{p},

E_{∗,∗}^{2} ≅ H_{∗}(BΣ_{p}; H̃_{∗}(X)^{⊗p}) ⟹ H̃_{∗}(D_{p}(X)),
E_{2}^{∗,∗} ≅ H^{∗}(BΣ_{p}; H̃^{∗}(X)^{⊗p}) ⟹ H̃^{∗}(D_{p}(X)).

These spectral sequences are known to collapse at E^{2} and E_{2}, respectively and

H̃_{∗}(D_{p}(X)) ≅ H_{∗}(BΣ_{p}; H̃_{∗}(X)^{⊗p})
H̃^{∗}(D_{p}(X)) ≅ H^{∗}(BΣ_{p}; H̃^{∗}(X)^{⊗p})

as 𝔽_{p}-vector spaces.

In order to describe these (co)homologies more explicitly, let us fix an arbitrary basis B_{X} = {x_{α}|α ∈ A} for H̃_{∗}(X), and let B_{X}^{∗} = {y_{α}|α ∈ A} be the dual basis for H̃^{∗}(X) with respect to the Kronecker pairing. Let “<” be a fixed ordering on A.

Proposition 7 H_{∗}(BΣ_{p}; H̃_{∗}(X)^{⊗p}) has a basis

{e_{i(p−1)} ⊗ x_{α}^{p}|α ∈ A, i ≡ deg x_{α} mod 2, i ≥ 0}
∪ {e_{i(p−1)−1} ⊗ x_{α}^{p}|α ∈ A, i ≡ deg x_{α} mod 2, i > 0}
∪ {x_{α_{1}} ⊗ ⋯ ⊗ x_{α_{p}}|α_{1} ≤ ⋯ ≤ α_{p}, α_{i} < α_{i+1} for some i and x_{α_{k}} does not

appear twice if deg x_{α_{k}} is odd}

when p > 2 and

{e_{i} ⊗ x_{α}^{p}|α ∈ A, i ≥ 0} ∪ {x_{α_{1}} ⊗ x_{α_{2}}|α_{1} < α_{2}}

when p = 2, where

e_{i(p−1)} ⊗ x_{α}^{p} ∈ H_{i(p−1)}(BΣ_{p}; H̃_{∗}(X)^{⊗p})
e_{i(p−1)−1} ⊗ x_{α}^{p} ∈ H_{i(p−1)−1}(BΣ_{p}; H̃_{∗}(X)^{⊗p})
x_{α_{1}} ⊗ ⋯ ⊗ x_{α_{p}} ∈ H_{0}(BΣ_{p}; H̃_{∗}(X)^{⊗p}),

so that

e_{2i(p−1)−1} ⊗ ι_{0}^{p} = e_{2i(p−1)−1},
e_{2i(p−1)} ⊗ ι_{0}^{p} = e_{2i(p−1)},

if X = S^{0}, where ι_{0} ∈ H̃_{0}(S^{0}) is the canonical generator. We define another basis for H_{∗}(BΣ_{p}; H̃_{∗}(X)^{⊗p}) as follows. Definition 8 Let p be an odd prime. For an integer q, let

ν(q) = (−1)^{\frac{(p−1)q(q−1)}{4}} ((\frac{p − 1}{2})!)^{q} .

This is a unit in 𝔽_{p} for any q. Define

Q_{i}(x_{α}) = (−1)^{\frac{i+q}{2}} ν(q)e_{i(p−1)} ⊗ x_{α}^{p},
βQ_{i}(x_{α}) = (−1)^{\frac{i+q}{2}} ν(q)e_{i(p−1)−1} ⊗ x_{α}^{p},

where q = deg x_{α}. Definition 9 When p = 2, define

Q_{i}(x_{α}) = e_{i} ⊗ x_{α}^{2}.

The introduction of this basis is justified by the following Nishida relations and the Cartan formulas. Proposition 10 When p is an odd prime, we have

P_{∗}^{r}Q_{i}(x) = ∑_{j}(−1)^{r+j}\binom{(i + k − 2r)\frac{p−1}{2}}{r − pj}Q_{i−2r+2pj}P_{∗}^{j}(x),
P_{∗}^{r}βQ_{i}(x) = ∑_{j}(−1)^{r+j}\binom{(i + k − 2r)\frac{p−1}{2} − 1}{r − pj}βQ_{i−2r+2pj}P_{∗}^{j}(x)
+ ∑_{j}(−1)^{r+j}\binom{(i + k − 2r)\frac{p−1}{2}}{r − pj − 1}Q_{i−2r+2pj+1}P_{∗}^{j}β(x),

where k = deg x. Proposition 11 When p = 2, we have

Sq_{∗}^{r}Q_{i}(x) = ∑_{j} \binom{k + i − r}{r − 2j}Q_{i−r+2j}Sq_{∗}^{j}(x)

where k = deg x. Proposition 12 The diagonal map of EΣ_{p} induces a map

Δ : D_{p}(X ∧ Y) ⟶ D_{p}(X) ∧ D_{p}(Y),

under which we have the external Cartan formulas

Δ_{∗}(Q_{i}(x ⊗ y)) = ∑_{k} Q_{k}(x) ⊗ Q_{i−k}(y),
Δ_{∗}(βQ_{i}(x ⊗ y)) = ∑_{k} βQ_{k}(x) ⊗ Q_{i−k}(y) + (−1)^{deg x} ∑_{k} Q_{k}(x) ⊗ βQ_{i−k}(y).

For simplicity, we use cohomology rings in the rest of this section and §4. Definition 13 Define

u_{i(p−1)} ⊗ y_{α}^{p} to be the dual of Q_{i}(x_{α})
u_{i(p−1)−1} ⊗ y_{α}^{p} to be the dual of βQ_{i}(x_{α})
v_{i(p−1)} ⊗ y_{α}^{p} to be the dual of e_{i(p−1)} ⊗ x_{α}^{p}
v_{i(p−1)−1} ⊗ y_{α}^{p} to be the dual of e_{i(p−1)−1} ⊗ x_{α}^{p}

in the dual basis for H̃^{∗}(D_{p}(X)), respectively. Then the basis for H̃^{∗}(D_{p}(X)) dual to the basis in Definition 8 is given by

{u_{i(p−1)} ⊗ y_{α}^{p}|α ∈ A, i ≡ deg y_{α} mod 2, i ≥ 0}
∪ {u_{i(p−1)−1} ⊗ y_{α}^{p}|α ∈ A, i ≡ deg y_{α} mod 2, i > 0}
∪ { ∑_{σ∈Σ_{p}} y_{α_{σ(1)}} ⊗ ⋯ ⊗ y_{α_{σ(p)}}|α_{1} ≤ ⋯ ≤ α_{p}, α_{i} < α_{i+1} for some i and y_{α_{k}}

does not appear twice if deg y_{α_{k}} is odd}.

when p is odd and by

{u_{i(p−1)} ⊗ y_{α}^{p}|α ∈ A, i ≥ 0} ∪ {y_{α_{1}} ⊗ y_{α_{2}} + y_{α_{2}} ⊗ y_{α_{1}}|α_{1} < α_{2}} when p = 2. With this basis Proposition 12 can be translated as follows. Proposition 14 Under the map Δ : D_{p}(X ∧Y) ⟶ D_{p}(X)∧D_{p}(Y), we have

Δ^{∗}((u_{i(p−1)} ⊗ x^{p}) ⊗ (u_{j(p−1)} ⊗ y^{p})) = u_{(i+j)(p−1)} ⊗ (x ⊗ y)^{p}
Δ^{∗}((u_{i(p−1)−1} ⊗ x^{p}) ⊗ (u_{j(p−1)} ⊗ y^{p})) = u_{(i+j)(p−1)−1} ⊗ (x ⊗ y)^{p}
Δ^{∗}((u_{i(p−1)} ⊗ x^{p}) ⊗ (u_{j(p−1)−1} ⊗ y^{p})) = (−1)^{deg x}u_{(i+j)(p−1)−1} ⊗ (x ⊗ y)^{p}

for x ∈ H^{∗}(X) and y ∈ H^{∗}(Y). As a special case of this formula, we obtain Corollary 15 Under the map Δ : D_{p}(ΣX) ⟶ D_{p}(X) ∧ D_{p}(S^{1}), we have

Δ^{∗}((u_{i(p−1)} ⊗ x^{p}) ⊗ (u_{p−1} ⊗ ι_{1}^{p})) = u_{(i+1)(p−1)} ⊗ (σx)^{p}
Δ^{∗}((u_{i(p−1)−1} ⊗ x^{p}) ⊗ (u_{p−1} ⊗ ι_{1}^{p})) = u_{(i+1)(p−1)−1} ⊗ (σx)^{p}

where x ∈ H^{∗}(X), ι_{1} ∈ H^{1}(S^{1}) is the canonical generator, and σx is the suspension of x. Definition 16 For a pointed space X, consider the map

D_{p}(X) ⟶^{D_{p}(Δ_{X})} D_{p}(X ∧ X) ⟶^{Δ} D_{p}(X) ∧ D_{p}(X).

For u_{i} ⊗ x^{p} and u_{j} ⊗ y^{p} ∈ H^{∗}(D_{p}(X)), the image of (u_{i} ⊗ x^{p}) ⊗ (u_{j} ⊗ y^{p}) under the map D_{p}(Δ_{X})^{∗} ∘ Δ^{∗} is denoted by (u_{i} ⊗ x^{p}) · (u_{j} ⊗ y^{p}). We also need the action of β_{1}^{v_{n}} on H^{∗}(D_{p}(S^{1})). Lemma 17 In H̃^{∗}(D_{p}(S^{1})),

β_{1}^{v_{n}}(u_{(2i+1)(p−1)} ⊗ ι_{1}^{p}) = 0

for any n and i.

PROOF. H̃^{∗}(D_{p}(S^{1})) has a basis {u_{(2i+1)(p−1)} ⊗ι_{1}^{p}, u_{(2i+1)(p−1)−1} ⊗ι_{1}^{p}} and we have

β(u_{(2i+1)(p−1)−1} ⊗ ι_{1}^{p}) = u_{(2i+1)(p−1)} ⊗ ι_{1}^{p}.

Thus

β_{1}^{v_{0}}(u_{(2i+1)(p−1)} ⊗ ι_{1}^{p}) = β(u_{(2i+1)(p−1)} ⊗ ι_{1}^{p}) = 0.

Suppose, inductively, that we have proved β_{1}^{v_{n−1}}(u_{(2i+1)(p−1)} ⊗ ι_{1}^{p}) = 0.

Recall that

β_{1}^{v_{n}} = \begin{cases}[P^{p^{n−1}}, β_{1}^{v_{n−1}}] & p > 2 \\ [Sq^{2^{n}}, β_{1}^{v_{n−1}}] & p = 2. \end{cases}

By the inductive assumption, when p > 2,

P^{p^{n−1}}β_{1}^{v_{n−1}}(u_{(2i+1)(p−1)} ⊗ ι_{1}^{p}) = 0.

Since P^{p^{n−1}}(u_{(2i+1)(p−1)}⊗ι_{1}^{p}) is of odd degree, it is a scalar multiple of u_{(2k+1)(p−1)}⊗ ι_{1}^{p} for some k. Thus

β_{1}^{v_{n−1}}P^{p^{n−1}}(u_{(2i+1)(p−1)} ⊗ ι_{1}^{p}) = 0.

The case p = 2 follows from the same argument.

4 Proof of the Formula up to Scalar Multiples

In order to prove our Main Theorem 1, we use cohomology. By the anticommutativity of β_{1}^{v_{n}} and β, it is enough to consider β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ x^{p}).

Lemma 18 There are cohomology operations ϕ_{i,j} and ψ_{i,j}, with which we have

β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ x^{p}) = ∑_{j} u_{(i+j)(p−1)−1} ⊗ ϕ_{i,j}(x)^{p} + ∑_{j} u_{(i+j)(p−1)} ⊗ ψ_{i,j}(x)^{p}.
β_{1}^{v_{n}}(u_{i(p−1)} ⊗ x^{p}) = − ∑_{j} u_{(i+j)(p−1)} ⊗ ϕ_{i,j}(x)^{p}.

PROOF. Since β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ x^{p}) is an element of H^{∗}(D_{p}(X)), we can write this in the following form.

β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ x^{p}) = ∑_{j} u_{(i+j)(p−1)−1} ⊗ ϕ_{i,j}(x)^{p}

+ ∑_{j} u_{(i+j)(p−1)} ⊗ ψ_{i,j}(x)^{p}

+ ∑ y_{1} ⊗ ⋯ ⊗ y_{p}.

But in homology, no elements of the form x_{1} ⊗ ⋯ ⊗ x_{p} can hit an element of the form βQ_{i}(x) by β_{v_{n}}^{1}. Thus

β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ x^{p}) = ∑_{j} u_{(i+j)(p−1)−1} ⊗ ϕ_{i,j}(x)^{p}

+ ∑_{j} u_{(i+j)(p−1)} ⊗ ψ_{i,j}(x)^{p},

for some elements ϕ_{i,j}(x), ψ_{i,j}(x) ∈ H^{∗}(X). But obviously the correspondences

x ⟼ ϕ_{i,j}(x), x ⟼ ψ_{i,j}(x)

are 𝔽_{p}-linear and natural, hence ϕ_{i,j} and ψ_{i,j} are cohomology operations.

Apply the Bockstein β and obtain the second formula.

Lemma 19 The operations ϕ_{i,j} and ψ_{i,j} are independent of i, and for this reason, we write these operations as ϕ_{j} and ψ_{j}, respectively.

PROOF. Since we are dealing with cohomological information, our result should only depend on the stable homotopy type of D_{p}(X). It is well-known that the p-adic construction D_{p}(X) is a stable construction [Lewis-May-Steinberger], namely the functor D_{p} : Spaces ⟶ Spaces can be extended to the category of spectra

D_{p} : Spectra ⟶ Spectra .

Therefore we can assume that X in D_{p}(X) is desuspendable and we obtain a map in the stable category

Δ : D_{p}(X) ⟶ D_{p}(Σ^{−1}X) ∧ D_{p}(S^{1}).

For x ∈ H̃^{∗}(X), by Corollary 15, we have

u_{(i+1)(p−1)−1} ⊗ x^{p} = Δ^{∗}((u_{i(p−1)−1} ⊗ (σ^{−1}x)^{p}) ⊗ (u_{p−1} ⊗ ι_{1}^{p})).

Apply β_{1}^{v_{n}} to the both sides to obtain

β_{1}^{v_{n}}(u_{(i+1)(p−1)−1} ⊗ x^{p}) = Δ^{∗}(β_{1}^{v_{n}}((u_{i(p−1)−1} ⊗ σ^{−1}(x)^{p}) ⊗ (u_{p−1} ⊗ ι_{1}^{p})))
= Δ^{∗}((β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ σ^{−1}(x)^{p})) ⊗ (u_{p−1} ⊗ ι_{1}^{p}))

+(−1)^{deg x}Δ^{∗}((u_{i(p−1)−1} ⊗ σ^{−1}(x)^{p}) ⊗ β_{1}^{v_{n}}(u_{p−1} ⊗ ι_{1}^{p}))

= ∑_{j} Δ^{∗}((u_{(i+j)(p−1)−1} ⊗ ϕ_{i,j}(σ^{−1}(x))^{p}) ⊗ (u_{p−1} ⊗ ι_{1}^{p}))

+ ∑_{j} Δ^{∗}((u_{(i+j)(p−1)} ⊗ ψ_{i,j}(σ^{−1}(x))^{p}) ⊗ (u_{p−1} ⊗ ι_{1}^{p}))

= ∑_{j} u_{(i+j+1)(p−1)−1} ⊗ ϕ_{i,j}(x)^{p}

+ ∑_{j} u_{(i+j+1)(p−1)} ⊗ ψ_{i,j}(x)^{p}.

Here we used the fact that β_{1}^{v_{n}}(u_{p−1} ⊗ ι_{1}^{p}) = 0 from Lemma 17.

On the other hand,

β_{1}^{v_{n}}(u_{(i+1)(p−1)−1} ⊗ x^{p}) = ∑_{j} u_{(i+j+1)(p−1)−1} ⊗ ϕ_{i+1,j}(x)^{p}

+ ∑_{j} u_{(i+j+1)(p−1)} ⊗ ψ_{i+1,j}(x)^{p}.

Therefore

ϕ_{i+1,j}(x) = ϕ_{i,j}(x)
ψ_{i+1,j}(x) = ψ_{i,j}(x)

for all i, j and x ∈ H^{∗}(X) for any X.

We see more restrictions on these operations ϕ_{j} and ψ_{j} arising from Proposition 14.

Lemma 20 The operation ϕ_{j} is a derivation for each j, hence determines a primitive element in the Steenrod algebra A. Since ϕ_{j}’s are of odd degree, each ϕ_{j} is a scalar multiple of β_{1}^{v_{m}} for some m.

PROOF. Take x, y ∈ H^{∗}(X) and consider β_{1}^{v_{n}}((u_{k(p−1)} ⊗ x^{p}) · (u_{l(p−1)} ⊗ y^{p})). Since β_{1}^{v_{n}} is a derivation

β_{1}^{v_{n}}((u_{k(p−1)} ⊗ x^{p}) · (u_{l(p−1)} ⊗ y^{p}))

= (β_{1}^{v_{n}}(u_{k(p−1)} ⊗ x^{p})) · (u_{l(p−1)} ⊗ y^{p}))

+(−1)^{deg x}(u_{k(p−1)} ⊗ x^{p}) · β_{1}^{v_{n}}(u_{l(p−1)} ⊗ y^{p})

= (− ∑_{j} u_{(k+j)(p−1)} ⊗ ϕ_{j}(x)^{p}) · (u_{l(p−1)} ⊗ y^{p})

+(−1)^{deg x}(u_{k(p−1)} ⊗ x^{p}) · (− ∑_{j} u_{(l+j)(p−1)} ⊗ ϕ_{j}(y)^{p})

= − ∑_{j} u_{(k+j+l)(p−1)} ⊗ (ϕ_{j}(x) · y)^{p}

−(−1)^{deg x} ∑_{j} u_{(k+j+l)(p−1)} ⊗ (x · ϕ_{j}(y))^{p}.

On the other hand, by Proposition 14,

β_{1}^{v_{n}}((u_{k(p−1)} ⊗ x^{p}) · (u_{l(p−1)} ⊗ y^{p})) = β_{1}^{v_{n}}(u_{(k+l)(p−1)} ⊗ (xy)^{p})
= − ∑_{j} u_{(k+l+j)(p−1)} ⊗ ϕ_{j}(xy)^{p}.

Hence

ϕ_{j}(xy) = ϕ_{j}(x) · y + (−1)^{deg x}x · ϕ_{j}(y)

and ϕ_{j} is a derivation for any j.

It is easy to see that ϕ_{j}’s are of odd degree. Milnor proved that the only odd degree primitive elements in A are β_{1}^{v_{m}}’s [Milnor58].

Lemma 21 ψ_{j}(xy) = ψ_{j}(x)y for any x, y ∈ H^{∗}(X), hence Δ(ψ_{j}) = ψ_{j} ⊗ 1 in A. This means that ψ_{j} is a scalar multiple of the identity map.

PROOF. By calculating β_{1}^{v_{n}}((u_{k(p−1)−1} ⊗ x^{p}) · (u_{l(p−1)} ⊗ y^{p})) similarly to the above lemma, we obtain ψ_{j}(xy) = ψ_{j}(x)y. Hence Δ(ψ_{j}) = ψ_{j} ⊗ 1. The only elements which have this kind of coproduct in the Steenrodalgebra are scalar multiples of the unit.

Summarizing the above arguments, we have the following.

Proposition 22 There exist A_{0}, A_{1}, ⋯, A_{n} ∈ 𝔽_{p}, with which we have the following formulas;

β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ x^{p}) = ∑_{k=1}^{n} A_{k−1}u_{(i+1+2d_{n,k})(p−1)−1} ⊗ β_{1}^{v_{k−1}}(x)^{p}
+A_{n}u_{(i+2d_{n})(p−1)} ⊗ x^{p},
β_{1}^{v_{n}}(u_{i(p−1)} ⊗ x^{p}) = − ∑_{k=1}^{n} A_{k−1}u_{(i+1+2d_{n,k})(p−1)} ⊗ β_{1}^{v_{k−1}}(x)^{p},

where d_{n,k} = \frac{p^{n}−p^{k}}{p−1} and d_{n} = d_{n,0} = \frac{p^{n}−1}{p−1}.

PROOF. It follows from easy degree counting that, if ϕ_{j} = A_{k−1}β_{1}^{v_{k−1}} for some k and some A_{k−1} ∈ 𝔽_{p}, then j = 2d_{n,k} + 1. Similarly, if ψ_{j} = A_{n} · 1 for some A_{n} ∈ 𝔽_{p}, then j = 2d_{n}.

Dually,

Corollary 23 When p is odd, we have

β_{v_{n}}^{1}Q_{i}(x) = − ∑_{k=1}^{n} A_{k−1}Q_{i−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)) + A_{n}βQ_{i−2d_{n}}(x)
β_{v_{n}}^{1}βQ_{i}(x) = ∑_{k=1}^{n} A_{k−1}βQ_{i−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)),

for some A_{0}, ⋯, A_{n} ∈ 𝔽_{p}.

When p = 2, we have

β_{v_{n}}^{1}Q_{i}(x) = ∑_{k=1}^{n} A_{k−1}Q_{i−2d_{n,k}−1}(β_{v_{k−1}}^{1}(x)) + A_{n}Q_{i−2d_{n}−1}(x)

for some A_{0}, ⋯, A_{n} ∈ 𝔽_{2}.

5 Evaluation on D_{p}(BΣ_{p})

In order to determine the coefficients in Corollary 23, we wish to find a space whose homology has nontrivial actions of β_{v_{0}}^{1}, β_{v_{1}}^{1}, ⋯, β_{v_{n−1}}^{1}. The smallest of such spaces is the finite complex called “V (n)”, but unfortunately the existence of V (n) is known only for small n and large primes. BΣ_{p} also has nontrivial actions of β_{v_{k}}^{1}’s as stated in Lemma 5 but the homology is much bigger than that of V (n). Corollary 23, however, simplifies the calculations and enables the evaluation of the coefficients.

In this section, we evaluate the action of β_{v_{n}}^{1} on H_{∗}(D_{p}(BΣ_{p})) directly via Nishida relation and finish the proof of Main Theorem 1.

The method is a simple induction on n based on the following facts;

β_{1}^{v_{0}} = \begin{cases}β & p > 2, \\ Sq^{1} & p = 2 \end{cases}

β_{1}^{v_{n}} = \begin{cases}[P^{p^{n−1}}, β_{1}^{v_{n−1}}] & p > 2 \\ [Sq^{2^{n}}, β_{1}^{v_{n−1}}] & p = 2. \end{cases}

PROOF OF Main Theorem 1.

Consider the case when p is odd. The 2-primary case is easier and the proof is omitted.

When n = 0, there is nothing to prove.

Suppose we have proved the formula (1) for β_{v_{n−1}}^{1}.

β_{v_{n−1}}^{1}Q_{i}(x) = ∑_{k=1}^{n−1}(−1)^{n−k−1}Q_{i−2d_{n−1,k}−1}(β_{v_{k−1}}^{1}(x)) + (−1)^{n−1}βQ_{i−2d_{n−1}}(x).

For any x ∈ H_{ℓ}(X), by the Nishida relation,

β_{v_{n}}^{1}Q_{i}(x) = (β_{v_{n−1}}^{1}P_{∗}^{p^{n−1}} − P_{∗}^{p^{n−1}}β_{v_{n−1}}^{1})Q_{i}(x)
= β_{v_{n−1}}^{1} (∑_{j}(−1)^{p^{n−1}+j}\binom{(i + ℓ − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − pj}Q_{i−2p^{n−1}+2pj}P_{∗}^{j}(x))
− P_{∗}^{p^{n−1}} (∑_{k=1}^{n−1}(−1)^{n−k−1}Q_{i−2d_{n−1,k}−1}β_{v_{k−1}}^{1}(x))
− P_{∗}^{p^{n−1}}((−1)^{n−1}βQ_{i−2d_{n−1}}(x))
= ∑_{k=1}^{n−1} ∑_{j}(−1)^{p^{n−1}+j+n−k−1}\binom{(i + ℓ − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − pj}Q_{i+2pj−2d_{n,k}−1}β_{v_{k−1}}^{1}P_{∗}^{j}(x)
+ ∑_{j}(−1)^{p^{n−1}+j+n−1}\binom{(i + ℓ − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − pj}βQ_{i+2pj−2d_{n}}P_{∗}^{j}(x)
+ ∑_{k=1}^{n−1} ∑_{j}(−1)^{p^{n−1}+j+n−k}\binom{(i + ℓ − 2d_{n,k−1})\frac{p−1}{2}}{p^{n−1} − pj}Q_{i+2pj−2d_{n,k}−1}P_{∗}^{j}β_{v_{k−1}}^{1}(x)
+ ∑_{j}(−1)^{p^{n−1}+j+n}\binom{(i − 2d_{n−1} + ℓ − 2p^{n−1})\frac{p−1}{2} − 1}{p^{n−1} − pj}βQ_{i−2d_{n}+2pj}P_{∗}^{j}(x)
+ ∑_{j}(−1)^{p^{n−1}+j+n}\binom{(i − 2d_{n} + ℓ)\frac{p−1}{2}}{p^{n−1} − pj − 1}Q_{i−2d_{n}+2pj}P_{∗}^{j}β(x).

Since

\binom{(i + ℓ − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − pj} − \binom{(i − 2d_{n−1} + ℓ − 2p^{n−1})\frac{p−1}{2} − 1}{p^{n−1} − pj}

= \binom{(i + ℓ − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − pj} − \binom{(i + ℓ − 2p^{n−1})\frac{p−1}{2} − p^{n−1}}{p^{n−1} − pj}
= \{ \begin{matrix}0 & if j ≠ 0 \\ 1 & if j = 0, \end{matrix}

the second and the fourth term together give (−1)^{n}βQ_{i−d_{n}}(x). Thus

β_{v_{n}}^{1}Q_{i}(x) = ∑_{k=1}^{n−1} ∑_{j}(−1)^{j+n−k}\binom{(i + ℓ − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − pj}Q_{i+2pj−2d_{n,k}−1}β_{v_{k−1}}^{1}P_{∗}^{j}(x)

+ ∑_{k=1}^{n−1} ∑_{j}(−1)^{j+n−k+1}\binom{(i + ℓ − 2d_{n,k−1})\frac{p−1}{2}}{p^{n−1} − pj}Q_{i+2pj−2d_{n,k}−1}P_{∗}^{j}β_{v_{k−1}}^{1}(x)

+ ∑_{j}(−1)^{j+n+1}\binom{(i − 2d_{n} + ℓ)\frac{p−1}{2}}{p^{n−1} − pj − 1}Q_{i−2d_{n}+2pj}P_{∗}^{j}β(x)
+ (−1)^{n}βQ_{i−2d_{n}}(x).

By Corollary 6, in H_{∗}(D_{p}(BΣ_{p})), we obtain

β_{v_{n}}^{1}Q_{i}(e_{2ℓ(p−1)})
= ∑_{k,j}(−1)^{j+n−k}\binom{\frac{(i+2ℓ(p−1)−2p^{n−1})(p−1)}{2}}{p^{n−1}−pj}\binom{(ℓ−j)(p−1)}{j}Q_{i+2pj−2d_{n,k}−1}(e_{2(ℓ−j−d_{k−1})(p−1)−1})
+ ∑_{k,j}(−1)^{j+n−k+1}\binom{\frac{(i+2ℓ(p−1)−2d_{n,k−1})(p−1)}{2}}{p^{n−1}−pj}\binom{(ℓ−j)(p−1)−p^{k−1}}{j}Q_{i+2pj−2d_{n,k}−1}(e_{2(ℓ−j−d_{k−1})(p−1)−1})
+ ∑_{j}(−1)^{j+n+1}\binom{\frac{(i−2dn+2ℓ(p−1))(p−1)}{2}}{p^{n−1}−pj−1}\binom{(ℓ−j)(p−1)−1}{j}Q_{i−2d_{n}+2pj}(e_{2(ℓ−j)(p−1)−1})
+ (−1)^{n}βQ_{i−2d_{n}}(e_{2ℓ(p−1)}).

Consider the coefficient of Q_{i−2d_{n,s}−1}(e_{2(ℓ−d_{s−1})(p−1)−1}).

∑_{k=1}^{n−1}(−1)^{n−s}\binom{(i + 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − d_{s,k}}\binom{(ℓ − d_{s−1,k−1})(p − 1)}{d_{s−1,k−1}}

+ ∑_{k=1}^{n−1}(−1)^{n−s+1}\binom{(i + 2ℓ(p − 1))\frac{p−1}{2} − p^{n} + p^{k−1}}{p^{n−1} − d_{s,k}}\binom{(ℓ − d_{s−1,k−1})(p − 1) − p^{k−1}}{d_{s−1,k−1}}
+ (−1)^{n−s}\binom{(i + 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2} − p^{n−1} + 1}{p^{n−1} − d_{s,1} − 1}\binom{(ℓ − \frac{1}{2}d_{s−1})(p − 1) − 1}{d_{s−1}}
= ∑_{k=1}^{n−1}(−1)^{n−s}\binom{(i + 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − (p^{s−1} + ⋯ + p^{k})}\binom{ℓ(p − 1) − p^{s−1} + p^{k−1}}{p^{s−2} + ⋯ + p^{k−1}}

− ∑_{k=1}^{n−1}(−1)^{n−s}\binom{(i + 2ℓ(p − 1))\frac{p−1}{2} − p^{n} + p^{k−1}}{p^{n−1} − (p^{s−1} + ⋯ + p^{k})}\binom{ℓ(p − 1) − p^{s−1}}{p^{s−2} + ⋯ + p^{k−1}}
+ (−1)^{n−s}\binom{(i − 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2} − p^{n−1} + 1}{p^{n−1} − (p^{s−1} + ⋯ + 1)}\binom{ℓ(p − 1) − p^{s−1}}{p^{s−2} + ⋯ + 1}
= ∑_{k=1}^{n−1}(−1)^{n−s}\binom{(i + 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2}}{p^{n−1} − (p^{s−1} + ⋯ + p^{k})}\binom{ℓ(p − 1) + p^{k−1}}{p^{s−2} + ⋯ + p^{k−1}}

− ∑_{k=1}^{n−1}(−1)^{n−s}\binom{(i + 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2} − p^{n−1} + p^{k−1}}{p^{n−1} − (p^{s−1} + ⋯ + p^{k})}\binom{ℓ(p − 1)}{p^{s−2} + ⋯ + p^{k−1}}
+ (−1)^{n−s}\binom{(i − 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2} + 1}{p^{n−1} − (p^{s−1} + ⋯ + 1)}\binom{ℓ(p − 1)}{p^{s−2} + ⋯ + 1}.

We want to prove that this is equal to (−1)^{n−s}. At first glance, it looks very complicated. Fortunately, however, our formula should be independent of i and ℓ, thanks to Lemma 19, Corollary 6, and Corollary 23. Therefore it is enough to compute this summation for some special i and ℓ. We choose ℓ ≡ 0 mod p^{s−1} so that

\binom{ℓ(p − 1)}{p^{s−2} + ⋯ + 1} = 0,

\binom{ℓ(p − 1) + p^{k−1}}{p^{s−2} + ⋯ + p^{k−1}} = \begin{cases}0 & if k ≠ s − 1 \\ 1 & if k = s − 1, \end{cases}

\binom{ℓ(p − 1)}{p^{s−2} + ⋯ + p^{k−1}} = \begin{cases}0 & if k ≠ s − 1 \\ 1 & if k = s − 1. \end{cases}

Let C = (i + 2ℓ(p − 1) − 2p^{n−1})\frac{p−1}{2} and take i so that C ≡ 0 mod p^{n}. Then the coefficient of Q_{i−2d_{n,s}−1}(e_{2(ℓ−d_{s−1})(p−1)−1}) is

(−1)^{n−s} (\binom{C}{p^{n−1}} − \binom{C − p^{n−1} + p^{s−2}}{p^{n−1}}) = (−1)^{n−s}(0 − (p − 1)) = (−1)^{n−s},

for n − 1 > s − 2.

This completes the proof of (1).

Corollary 24

β_{1}^{v_{n}}(u_{i(p−1)−1} ⊗ x^{p}) = ∑_{k=1}^{n}(−1)^{n−k+1}u_{(i+2d_{n,k}+1)(p−1)−1} ⊗ β_{1}^{v_{k−1}}(x)^{p}
+(−1)^{n}u_{(i+2d_{n})(p−1)} ⊗ x^{p},
β_{1}^{v_{n}}(u_{i(p−1)} ⊗ x^{p}) = ∑_{k=1}^{n}(−1)^{n−k}u_{(i+2d_{n,k}+1)(p−1)} ⊗ β_{1}^{v_{k−1}}(x)^{p}.  (5)

6 β_{1}^{v_{n}} and P^{j}

In order to prove the formula (4) for the Steenrod operations, let us recall the definition of the Steenrod operations. See [Steenrod-Epstein] for details.

We concentrate on the odd primary cases, in this section. The 2-primary case is easier and left to the reader.

Definition 25 Define the extended diagonal map

d : (BΣ_{p} ∪ {∗}) ∧ X ⟶ D_{p}(X)

by smashing EΣ_{p} ∪ {∗} with the diagonal map X ⟶ X^{p} and taking quotient by the action of Σ_{p}.

In H^{∗}(BΣ_{p}), write v_{2i(p−1)−1} = uv^{i−1} and v_{2i(p−1)} = v^{i}, for consistency. Write the image of v_{2k(p−1)} ⊗ x^{p} under d^{∗} as

d^{∗}(v_{i(p−1)} ⊗ x^{p}) = ∑_{i+ℓ≡0 mod 2} v_{(i+ℓ)(p−1)−1} ⊗ D_{ℓ(p−1)−1}(x)
+ ∑_{i+ℓ≡0 mod 2} v_{(i+ℓ)(p−1)} ⊗ D_{ℓ(p−1)}(x).  (6)

The j-th reduced power operation is defined by

P^{j}(x) = (−1)^{j}ν(q)D_{(q−2j)(p−1)}(x)

where q = deg x. We also use the lower-indexed operation

P_{i}(x) = P^{\frac{q−i}{2}} (x) = (−1)^{\frac{q−i}{2}} ν(q)D_{i(p−1)}(x)

The following fact is well-known.

Lemma 26 For any i,

βD_{i(p−1)} = −D_{i(p−1)−1}.

PROOF OF Formula (4) Recall from Definition 8 and Definition 13 that, for any x ∈ H^{q}(X),

u_{i(p−1)} ⊗ x^{p} = (−1)^{\frac{i+q}{2}} ν(q)v_{i(p−1)} ⊗ x^{p}.

Multiply (−1)^{\frac{i+q}{2}} ν(q) to (6) to obtain

d^{∗}(u_{i(p−1)} ⊗ x^{p}) = ∑_{i+ℓ≡0 mod 2}(−1)^{\frac{i+q}{2}} ν(q)v_{(i+ℓ)(p−1)−1} ⊗ D_{ℓ(p−1)−1}(x)
+ ∑_{i+ℓ≡0 mod 2}(−1)^{\frac{i+q}{2}} ν(q)v_{(i+ℓ)(p−1)} ⊗ D_{ℓ(p−1)}(x).

Apply β_{1}^{v_{n}} and use Lemma 5

d^{∗}β_{1}^{v_{n}}(u_{i(p−1)} ⊗ x^{p}) = ∑_{i+ℓ≡0 mod 2}(−1)^{\frac{i+q}{2}} ν(q)v_{(i+ℓ+2d_{n})(p−1)} ⊗ D_{ℓ(p−1)−1}(x)
+ ∑_{i+ℓ≡0 mod 2}(−1)^{\frac{i+q}{2} +1}ν(q)v_{(i+ℓ)(p−1)−1} ⊗ β_{1}^{v_{n}}D_{ℓ(p−1)−1}(x)
+ ∑_{i+ℓ≡0 mod 2}(−1)^{\frac{i+q}{2}} ν(q)v_{(i+ℓ)(p−1)} ⊗ β_{1}^{v_{n}}D_{ℓ(p−1)}(x).

By Corollary 24, the left hand side is equal to

d^{∗}β_{1}^{v_{n}}(u_{i(p−1)} ⊗ x^{p})
= d^{∗} (∑_{k=1}^{n}(−1)^{n−k}u_{(i+2d_{n,k}+1)(p−1)} ⊗ β_{1}^{v_{k−1}}(x)^{p})
= ∑_{k=1}^{n} ∑_{m}(−1)^{n−k+ \frac{q+i}{2} +d_{n,k−1}}ν(2p^{k−1} − 1 + q)v_{(i+2d_{n,k}+1+m)(p−1)−1} ⊗ D_{m(p−1)−1}β_{1}^{v_{k−1}}(x)
+ ∑_{k=1}^{n} ∑_{m}(−1)^{n−k+ \frac{q+i}{2} +d_{n,k−1}}ν(2p^{k−1} − 1 + q)v_{(i+2d_{n,k}+1+m)(p−1)} ⊗ D_{m(p−1)}β_{1}^{v_{k−1}}(x).

Thus

ν(q)β_{1}^{v_{n}}D_{ℓ(p−1)}(x) = −ν(q)D_{(ℓ−2d_{n})(p−1)−1}(x)

+ ∑_{k=1}^{n}(−1)^{n−k+d_{n,k−1}}ν(2p^{k−1} − 1 + q)D_{(ℓ−2d_{n,k}−1)(p−1)}β_{1}^{v^{k−1}}(x).

= ν(q)βD_{(ℓ−2d_{n})(p−1)}(x)

+ ∑_{k=1}^{n}(−1)^{n−k+d_{n,k−1}}ν(2p^{k−1} − 1 + q)D_{(ℓ−2d_{n,k}−1)(p−1)}β_{1}^{v^{k−1}}(x).

= (−1)^{n}(−1)^{d_{n}}ν(q)βD_{(ℓ−2d_{n})(p−1)}(x)

+ ∑_{k=1}^{n}(−1)^{n−k+d_{n,k−1}}ν(2p^{k−1} − 1 + q)D_{(ℓ−2d_{n,k}−1)(p−1)}β_{1}^{v^{k−1}}(x).

Multiply (−1)^{\frac{q−ℓ}{2}} to the both sides to get

(−1)^{\frac{q−ℓ}{2}} ν(q)β_{1}^{v_{n}}D_{ℓ(p−1)}(x)

= (−1)^{n}(−1)^{\frac{q−ℓ+2dn}{2}} ν(q)βD_{(ℓ−2d_{n})(p−1)}(x)

+ ∑_{k=1}^{n}(−1)^{n−k}(−1)^{\frac{q−ℓ}{2} +d_{n,k−1}}ν(2p^{k−1} − 1 + q)D_{(ℓ−2d_{n,k}−1)(p−1)}β_{1}^{v^{k−1}}(x).

Hence by the definition of the Steenrod operations,

β_{1}^{v_{n}}P_{i} = ∑_{k=1}^{n}(−1)^{n−k}P_{i−2d_{n,k}−1}β_{1}^{v_{k−1}} + (−1)^{n}βP_{i−2d_{n}}.

In the usual upper-indexed form,

β_{1}^{v_{n}}P^{j} = ∑_{k=1}^{n}(−1)^{n−k}P^{j+d_{n,k−1}}β_{1}^{v_{k−1}} + (−1)^{n}βP^{j+jd_{n}}.

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