Odd primary string cohomology

Iver Ottosen ^{∗}

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld

Fax number: +49 521 106 4743 E-mail address: ottosen@mathematik.uni-bielefeld.de 26 September 2000

Abstract

Let X be a connected space and let K = H^{∗}(X; 𝔽_{p}) where p is an odd prime. We construct functors ω and ℓ which approximate the cohomology of the free loop space ΛX as follows: There are morphisms ω(K) → H^{∗}(ΛX; 𝔽_{p}) and ℓ(K) → H_{S^{1}}^{∗}(ΛX; 𝔽_{p}). These are isomorphisms when X is a product of Eilenberg-MacLane spaces of type K(𝔽_{p}, n) for n ≥ 1.

MSC: 55N91; 55P35; 55R12

1 Introduction

The string cohomology of a topological space X with coefficients in a ring R is defined as follows:

H_{st}^{∗}(X; R) := H_{S^{1}}^{∗}(ΛX; R) = H^{∗}(ES^{1} ×_{S^{1}} ΛX; R)

where ΛX denotes the free loop space of X. These cohomology groups together with the cohomology of the free loop space itself H^{∗}(ΛX; R) plays a central role in geometry and topology. It is however not know how to compute these in general.

When R = 𝔽_{2} = ℤ/2, M. Bökstedt and I found computable functors of H^{∗}(X; 𝔽_{2}) which approximate these cohomology groups [2]. The purpose of this paper is to generalize these functors to the case R = 𝔽_{p} = ℤ/p where p is any of the odd primes. Certain algebra generators in string cohomology are more difficult to construct in the odd primary case. Hence method and strategy differs from [2] at various places.

2 Notation

Fix an odd prime p. We use 𝔽_{p}-coefficients everywhere unless otherwise is specified. A denotes the mod p Steenrod algebra, U the category of unstable A-modules and K the category of unstable A-algebras. We write K_{0} for the full subcategory of K with the connected unstable A-algebras as its objects. The category of differential graded algebras is denoted DGA.

∗The author was supported by the European Union TMR network ERB FMRX CT-97-0107: Algebraic K-theory, Linear Algebraic Groups and Related Structures.

For K ∈ K we define λ : K → K as follows

λx = \begin{cases}P^{\frac{|x|−1}{2}} x & , |x| odd \\ 0 & , |x| even \end{cases}

Note that |λx| = p(|x| − 1) + 1. Note also that λ is a derivation over Frobenius by the Cartan formula

λ(xy) = λ(x)y^{p} + x^{p}λ(y).

In any graded 𝔽_{p}-algebra K we define σ : K → 𝔽_{p} by σ(x) = 1 for |x| odd and σ(x) = 0 for |x| even. We also define σ̂ : K → 𝔽_{p} by σ̂(x) = 1 − σ(x).

3 The approximation functor ω

Let Y be a connected S^{1}-space with action map η : S^{1} × Y → Y. We have η ∘ γ = id where γ(y) = (1, y). Recall that H^{∗}(S^{1}) = Λ(v) where |v| = 1.

Definition 3.1. Define the map d : H^{∗}(Y) → H^{∗−1}(Y) by

η^{∗}(y) = 1 ⊗ y + v ⊗ dy.

Proposition 3.2. The map d satisfies the following:

d ∘ d = 0,  (1)
d(x + y) = dx + dy,  (2)
d(xy) = d(x)y + (−1)^{|x|}xd(y),  (3)
P^{i}(dx) = d(P^{i}x) for i ≥ 0,  (4)
β(dx) = −d(βx),  (5)
d(λx) = (dx)^{p},  (6)
d(βλx) = 0.  (7)

Proof. Similar to the p = 2 case proved in [2] Proposition 3.2.

Definition 3.3. For K ∈ K_{0} we define ω(K) as the symmetric product of K with the free graded commutative algebra on generators dx of degree |x| − 1 for x ∈ K divided by the ideal generated by the elements

d(x + y) − dx − dy,  (8)
d(xy) − d(x)y − (−1)^{|x|}xd(y),  (9)
d(λx) − (dx)^{p},  (10)
d(βλx).  (11)

There is a degree -1 differential d on ω(K) defined by d(x) = dx for x ∈ K as a derivation over K. Hence (ω(K), d) ∈ DGA.

Proposition 3.4. For K ∈ K we can define an A-action on ω(K) by θ(x) = θx and θ(dx) = (−1)^{|θ|}d(θx) for x ∈ K and θ ∈ A. In this way ω becomes a functor ω : K_{0} → K. Note that the differential d on ω(K) is graded A-linear.

Proof. See the appendix.

Definition 3.5. For a connected space X we define the map

e : ω(H^{∗}X) → H^{∗}(ΛX)

by x ↦ ev_{0}^{∗}(x) and dx ↦ dev_{0}^{∗}(x) where ev_{0} : ΛX → X; f ↦ f(1).

Remark 3.6. The map e is a morphism in K as well as a morphism in DGA. It is also natural in X. We view ω(H^{∗}X) as an approximation to H^{∗}(ΛX) via the morphism e.

Proposition 3.7. Let K, L ∈ K_{0} and let i : K → K ⊗ L and j : L → K ⊗ L be the inclusions given by i(x) = x ⊗ 1 and j(y) = 1 ⊗ y. The composite

κ : ω(K) ⊗ ω(L) ⟶^{ω(i)⊗ω(j)} ω(K ⊗ L) ⊗ ω(K ⊗ L) ⟶^{μ} ω(K ⊗ L),

where μ denotes the product, is an isomorphism in both DGA and K. For connected spaces X and Y with homology of finite type the following diagram commutes.

ω(H^{∗}X) ⊗ ω(H^{∗}Y) ⟶^{κ} ω(H^{∗}X ⊗ H^{∗}Y)
e⊗e↓  ^{e}↓
H^{∗}(ΛX) ⊗ H^{∗}(ΛY) ⟶^{≅} H^{∗}(Λ(X × Y))

Proof. The map κ is a morphism in both DGA and K since it is a composite of maps which are morphisms in both categories. For a ∈ K and b ∈ L we have that κ(a ⊗ b) = a ⊗ b and κ(d_{⊗}(a ⊗ b)) = d(a ⊗ b). We verify that there is a well defined 𝔽_{p}-algebra map γ in the opposite direction of κ with γ(a ⊗ b) = a ⊗ b and γ(d(a ⊗ b)) = d_{⊗}(a ⊗ b) such that κ is an isomorphism.

Elements of the form (8) are mapped to zero by definition. By direct computation one sees that the elements of the form (9) are also mapped to zero. Assuming that σ(a) = 1 and σ(b) = 0 we find

d(λ(a ⊗ b)) − (d(a ⊗ b))^{p} =d(λ(a) ⊗ b^{p}) − d(a)^{p} ⊗ b^{p} ↦
d(λa) ⊗ b^{p} − d(a)^{p} ⊗ b^{p} = 0,
d(β(λ(a ⊗ b))) =d(βλ(a) ⊗ b^{p}) ↦
d(βλ(a)) ⊗ b^{p} = 0

and if σ(a) = σ(b) the elements are already zero in ω(K ⊗ L). The above description of κ on generators shows that the diagram commutes.

4 The ω approximation for certain Eilenberg-MacLane spaces

For an Abelian group A and positive integer n we let B^{n}A denote the Eilenberg-MacLane space K(A, n). In this section we prove the following result:

Theorem 4.1. The map e : ω(H^{∗}B^{n}𝔽_{p}) → H^{∗}(ΛB^{n}𝔽_{p}) is an isomorphism in K and DGA for each n ≥ 1.

We first consider the case n ≥ 2. The Whitehead theorem together with the long exact sequence of homotopy groups and the five lemma proves the following:

Proposition 4.2. Let G be a connected topological group. Then there is a commutative diagram as follows

ΩG ⟶^{i_{2}} G × ΩG ⟶^{pr_{1}} G
id↓  ^{m}↓  ^{id}↓
ΩG ⟶ ΛG ⟶^{ev_{0}} G

where m(f, x) = (z ↦ f(z) · x). The map m is a weak homotopy equivalence.

Proposition 4.3. For n ≥ 2 there is an isomorphism

m^{∗} : H^{∗}(ΛB^{n}𝔽_{p}) → H^{∗}(B^{n}𝔽_{p}) ⊗ H^{∗}(B^{n−1}𝔽_{p})

with the property m^{∗} ∘ e(ι_{n}) = ι_{n} ⊗ 1 and m^{∗} ∘ e(dι_{n}) = c_{n}1 ⊗ ι_{n−1} for some nonzero constant c_{n} ∈ 𝔽_{p}.

Proof. The space B^{n}𝔽_{p} is a topological Abelian group since 𝔽_{p} is an Abelian group. Proposition 4.2 with G = B^{n}𝔽_{p} gives that m^{∗} is an isomorphism and that m^{∗} ∘ev_{0}^{∗}(ι_{n}) = 1⊗ι_{n} as stated. We have m^{∗}(dev_{0}^{∗}ι_{n}) = c_{n}1⊗ι_{n−1} for some c_{n} ∈ 𝔽_{p} since 1 ⊗ ι_{n−1} is the only class in degree n − 1 of the right hand side. Let h denote the composite

h : S^{1} × B^{n}𝔽_{p} × B^{n−1}𝔽_{p} ⟶^{m} S^{1} × ΛB^{n}𝔽_{p} ⟶^{η} ΛB^{n}𝔽_{p} ⟶^{ev_{0}} B^{n}𝔽_{p}

where η is the action map. Then h^{∗}(ι_{n}) = 1 ⊗ ι_{n} ⊗ 1 + c_{n}v ⊗ 1 ⊗ ι_{n−1}. If c_{n} = 0 then ev_{0} ∘ η is homotopic to the composite

k : S^{1} × ΛB^{n}𝔽_{p} ⟶^{pr_{2}} ΛB^{n}𝔽_{p} ⟶^{ev_{0}} B^{n}𝔽_{p}

since B^{n}𝔽_{p} classifies mod p cohomology in degree n. But we have ev_{0} ∘η(z, f) = f(z) so its adjoint is the identity on ΛB^{n}𝔽_{p}. The adjoint of k is the map ΛB^{n}𝔽_{p} → ΛB^{n}𝔽_{p} which sends a loop f to the constant loop with value f(1). The maps k and ev_{0} ∘ η cannot be homotopic since their adjoints are not.

Proposition 4.4. The map e : ω(H^{∗}B^{n}𝔽_{p}) → H^{∗}(ΛB^{n}𝔽_{p}) is an isomorphism for each n ≥ 2.

Proof. Since the cohomology of the spaces B^{m}𝔽_{p} are free objects in K we can define a morphism in K as follows:

f : H^{∗}(B^{n}𝔽_{p}) ⊗ H^{∗}(B^{n−1}𝔽_{p}) → ω(H^{∗}B^{n}𝔽_{p}) , ι_{n} ⊗ 1 ↦ ι_{n}
, 1 ⊗ ι_{n−1} ↦ c_{n}^{−1}dι_{n}.

We have m^{∗} ∘ e ∘ f = id and f ∘ m^{∗} ∘ e = id since these equalities hold on generators.

The case n = 1 is interesting since here ΛB𝔽_{p} splits in p components. We will use the following result to see that ω(H^{∗}B𝔽_{p}) split accordingly.

Lemma 4.5. There is an isomorphism of rings as follows

α : 𝔽_{p}[x]/(x^{p} − x) → (𝔽_{p})^{p} ; x ↦ (0, 1, 2, …, p − 1)

where 𝔽_{p}[x] is the polynomial ring in one variable x of degree zero and (𝔽_{p})^{p} is the p-fold Cartesian product of 𝔽_{p} by itself.

Proof. Use the factorization x^{p} − x = ∏_{n∈𝔽_{p}}(x − n) and the Chinese remainder theorem.

Remark 4.6. Let e_{n} = α^{−1}(0, …, 0, 1, 0, …, 0) with the 1 on the nth place for n ∈ 𝔽_{p}. Clearly e_{n}e_{m} = 0 for n ≠ m, e_{n}^{2} = e_{n} and ∑ e_{n} = 1. Also xe_{n} = ne_{n}. Finding eigenvectors for xf(x) = nf(x) and normalizing one gets

e_{0} = 1 − x^{p−1}

e_{m} = − ∑_{i=1}^{p−1}(\frac{x}{m})^{i} , m ≠ 0

Definition 4.7. For n ∈ 𝔽_{p} define the following action map

f_{n} : ℤ × 𝔽_{p} → 𝔽_{p} ; (r, [s]) ↦ [nr + s].

We let B𝔽_{p}(n) denote B𝔽_{p} equipped with S^{1}-action Bf_{n} and write d_{n} for the corresponding action differential on H^{∗}B𝔽_{p}(n).

Proposition 4.8. We have H^{∗}B𝔽_{p}(n) = Λ(v_{n}) ⊗ 𝔽_{p}[βv_{n}] where |v_{n}| = 1. The differential d_{n} on this algebra is given by d_{n}v_{n} = n and d_{n}βv_{n} = 0 for each n ∈ 𝔽_{p}.

Proof. We must show that (Bf_{n})^{∗}(v_{n}) = 1 ⊗ v_{n} + nv ⊗ 1. This follows from H_{1}(Bf_{n}) = π_{1}(Bf_{n}) = f_{n} by taking duals. Since λv_{n} = ι_{n} the class βv_{n} is mapped to zero.

Proposition 4.9. The map e : ω(H^{∗}B𝔽_{p}) → H^{∗}(ΛB𝔽_{p}) is an isomorphism.

Proof. From [1] Lemma 7.11 we have ΛB𝔽_{p} ≃ ⊔_{n∈𝔽_{p}}B𝔽_{p}. Define maps j_{n} : B𝔽_{p}(n) → ΛB𝔽_{p} by x ↦ (z ↦ Bf_{n}(z, x)) for n ∈ 𝔽_{p}. These are S^{1}-maps. Let (ΛB𝔽_{p})(n) denote the component of ΛB𝔽_{p} containing the image of j_{n}. Then the restriction j_{n}| : B𝔽_{p}(n) → (ΛB𝔽_{p})(n) is an S^{1}-map and a homotopy equivalence. Especially the induced in cohomology (j_{n}|)^{∗} is an isomorphism of differential graded algebras. By Proposition 4.8 we see that (ΛB𝔽_{p})(n) ≠ (ΛB𝔽_{p})(m) for n ≠ m such that ⊔j_{n} : ⊔B𝔽_{p}(n) → ΛB𝔽_{p} is an S^{1}-map and a homotopy equivalence. Especially (⊔j_{n})^{∗} = (j_{0}^{∗}, …, j_{p−1}^{∗}) is an isomorphism. So it suffices to show that g = (j_{0}^{∗}, …, j_{p−1}^{∗}) ∘ e is an isomorphism. We have

(j_{n}^{∗} ∘ e)(x) = j_{n}^{∗} ∘ ev_{0}^{∗}(x) = (ev_{0} ∘ j_{n})^{∗}(x) = x,  (12)
(j_{n}^{∗} ∘ e)(dx) = j_{n}^{∗}(dev_{0}^{∗}(x)) = d_{n} ∘ j_{n}^{∗} ∘ ev_{0}^{∗}(x) = d_{n}x  (13)

for x ∈ H^{∗}B𝔽_{p} which describes the map g on generators. By definition we have

ω(H^{∗}B𝔽_{p}) = Λ(ι_{1}) ⊗ 𝔽_{p}[βι_{1}] ⊗ (𝔽_{p}[dι_{1}]/((dι_{1})^{p} − dι_{1})).

From Lemma 4.5 and (13) we see that g is an isomorphism in degree zero. By (12) we conclude that it is an isomorphism in all degrees.

5 Steenrod diagonal elements

In this and the following four sections we describe algebra related to certain classes in string cohomology. The motivation comes later in Theorem 11.3. In the following K denotes an unstable A-algebra. The polynomial algebra 𝔽_{p}[u] where |u| = 2 is an object in K by the isomorphism 𝔽_{p}[u] ≅ H^{∗}(BS^{1}).

Definition 5.1. For x ∈ K and ϵ = 0, 1 we define St_{ϵ}(x) ∈ 𝔽_{p}[u] ⊗ K by

St_{ϵ}(x) = u^{−ϵσ̂(x)} ∑_{i≥0}(−u^{p−1})^{[|x|/2]−i} ⊗ β^{ϵ}P^{i}x.

Note that the terms where the total exponent of u is negative has β^{ϵ}P^{i}x = 0. Let R(K) ⊆ 𝔽_{p}[u] ⊗ K be the sub-𝔽_{p}-algebra generated by u ⊗ 1 and St_{ϵ}(x) for all x ∈ K and ϵ = 0, 1.

Theorem 5.2. For each θ ∈ A one has θR(K) ⊆ R(K). Thus R is a functor R : K → K. The explicit formulas are as follows where n = [|x|/2] and ϵ = 0, 1:

P^{i}St_{ϵ}(x) = ∑_{t} \binom{(p − 1)(n − t) + ϵσ(x)}{i − pt}u^{(p−1)(i−pt)}St_{ϵ}(P^{t}x)

−ϵ(−1)^{σ(x)} ∑_{t} \binom{(p − 1)(n − t) − 1 + σ(x)}{i − pt − 1}u^{(p−1)(i−pt)−1+(2−p)σ(x)}St_{0}(βP^{t}x),
βSt_{ϵ}(x) =(1 − ϵ)u^{σ̂(x)}St_{1}(x).

Proof. The formula for the Bockstein operation follows directly by the definition of St_{ϵ}(x). We use results from [3] to prove the other formula. By [8] we have that 𝔽_{p}[u, u^{−1}] is an A-algebra with β = 0 and

P^{i}u^{j} = \binom{j}{i}u^{j+i(p−1)} ; i, j ∈ ℤ ; i ≥ 0.

Here the following extended definition of binomial coefficients is used where r ∈ ℝ and k ∈ ℤ.

\binom{r}{k} = \begin{cases}\frac{r(r−1)...(r−k+1)}{k!} & , k > 0 \\ 1 & , k = 0 \\ 0 & , k < 0 \end{cases}

Let Δ = Λ(a) ⊗ 𝔽_{p}[b, b^{−1}] with |a| = 2p − 3, |b| = 2p − 2 be the A-algebra introduced in [3] (2.6). That is βa = b and

P^{i}(b^{j}) = (−1)^{i}\binom{(p − 1)j}{i}b^{i+j},
P^{i}(ab^{j−1}) = (−1)^{i}\binom{(p − 1)j − 1}{i}ab^{i+j−1}.

Note that we have changed the names of the generators. In [3] they were named u and v instead of a and b. We define an additive transfer map as follows:

τ : Δ → 𝔽_{p}[u, u^{−1}] ; b^{j} ↦ 0 ; ab^{j−1} ↦ (−u^{p−1})^{j}u^{−1}.

Note that |τ| = −1. A direct verification shows that τ is A-linear.

A functor R_{+} from the category of graded A-modules to itself is constructed in [3]. In the case of an unstable A-algebra K it comes with an A-linear map f : R_{+}K → ΣΔ ⊗ K defined by [3] (3.1), (3.2). The composite

R_{+}K ⟶^{f} σΔ ⊗ K ⟶^{Στ⊗1} Σ𝔽_{p}[u, u^{−1}] ⊗ K

is given by

sb^{k} ⊗ x ↦ −s ∑_{j}(−u^{p−1})^{k−j}u^{−1} ⊗ βP^{j}x,

sab^{k−1} ⊗ x ↦ s ∑_{j}(−u^{p−1})^{k−j}u^{−1} ⊗ P^{j}x.

Especially sb^{n} ⊗ x ↦ −su^{σ(x)}St_{1}(x) and sab^{n−1} ⊗ x ↦ su^{−1}St_{0}(x) where n = [|x|/2]. The formulas [3] (3.4), (3.5) for the A-action on R_{+}M gives the following formulas for the A-action on u^{σ(x)}St_{1}(x) and u^{−1}St_{0}(x):

P^{i}(u^{σ(x)}St_{1}(x)) = ∑_{t}\binom{(p − 1)(n − t)}{i − pt}u^{(p−1)(i−pt)−σ(x)}St_{1}(P^{t}x)

− ∑_{t}(−1)^{σ(x)}\binom{(p − 1)(n − t) − 1}{i − pt − 1}u^{(p−1)(i−pt−σ(x))−1}St_{0}(βP^{t}x),

P^{i}(u^{−1}St_{0}(x)) = ∑_{t}\binom{(p − 1)(n − t) − 1}{i − pt}u^{(p−1)(i−pt)−1}St_{0}(P^{t}x).

This proves the result directly for σ(x) = 0 and ϵ = 1. By the Cartan formula applied to uu^{−1}St_{ϵ}(x) we have that P^{i}St_{ϵ}(x) = uP^{i}(u^{−1}St_{ϵ}(x)) + u^{p}P^{i−1}(u^{−1}St_{ϵ}(x)). By combining this with the formulas above we get the result in the other cases.

6 The functor ℓ

In this section we describe the functor which approximates string cohomology. We also define maps which relate this functor to the functors R and ω.

Definition 6.1. For K ∈ K_{0} we define ℓ(K) as the graded commutative 𝔽_{p}- algebra generated by the classes

ϕ(x) of degree p|x| − σ(x)(p − 1),
δ(x) of degree |x| − 1,
q(x) of degree p|x| − 1 − σ(x)(p − 3)

for all homogeneous x ∈ K and a class u of degree 2; modulo the ideal generated by

ϕ(x + y) − ϕ(x) − ϕ(y) + σ(x) ∑_{i=0}^{p−2}(−1)^{i}δ(x)^{i}δ(y)^{p−2−i}δ(xy),  (14)
δ(x + y) − δ(x) − δ(y),  (15)
q(x + y) − q(x) − q(y) + σ̂(x) ∑_{i=1}^{p−1}(−1)^{i} \frac{1}{i}δ(x^{i}y^{p−i}),  (16)

(−1)^{σ(a)σ̂(c)}δ(a)δ(bc) + (−1)^{σ(b)σ̂(a)}δ(b)δ(ca) + (−1)^{σ(c)σ̂(b)}δ(c)δ(ab), (17)
ϕ(ab) − (−u^{p−1})^{σ(a)σ(b)}ϕ(a)ϕ(b),  (18)
q(ab) − (−u^{p−1})^{σ(a)σ(b)}(u^{σ(b)}q(a)ϕ(b) + (−u)^{σ(a)}ϕ(a)q(b)),  (19)
q(x)^{p} − u^{p−1}q(λx) − ϕ(βλx),  (20)
δ(a)ϕ(b) − δ(ab^{p}) − δ(aλb) + δ(ab)δ(b)^{p−1},  (21)
δ(a)q(b) − δ(ab^{p−1})δ(b) − δ(aβλb),  (22)
δ(x)u,  (23)
q(βλx),  (24)
δ(βλx)  (25)

where a, b, c, x, y ∈ K and |x| = |y|. It is understood that δ(1) = q(1) = 0. Remark 6.2. We have some immediate consequences of these relations:

• By (14)-(16) we have ϕ(0) = q(0) = δ(0) = 0.

• By (18), (19) and (21) the algebra ℓ(K) is unital with unit ϕ(1).

• By (18) and (19) we have q(a^{n}) = nϕ(a)^{n−1}q(a) such that q(a^{p}) = 0.

• By (22) we have δ(b^{p}) = 0 and also the important relation δ(λb) = δ(b)^{p}. Lemma 6.3. For any K ∈ K_{0} the following relations hold in ω(K):

∑_{i=1}^{p−1}(−1)^{i+1} \frac{1}{i}d(x^{i}y^{p−i}) = (x + y)^{p−1}d(x + y) − x^{p−1}dx − y^{p−1}dy,  (26)

∑_{j=0}^{p−2}(−1)^{j+1}(dx)^{j}(dy)^{p−2−j}d(xy) =

(d(x + y))^{p−1}(x + y) − (dx)^{p−1}x − (dy)^{p−1}y.  (27)

Here |x| = |y| is assumed to be even in (26) and odd in (27). Proof. We verify (26) and omit the proof of (27) which is similar. Since d is a derivation we have

∑_{i=1}^{p−1}(−1)^{i+1} \frac{1}{i}d(x^{i}y^{p−i}) = ∑_{i=1}^{p−1}(−1)^{i+1}(x^{i−1}y^{p−1}dx − x^{i}y^{p−i−1}dy).

By splitting the sum in two at the minus sign and substituting j = i − 1 in the first of the resulting sums we see that the above equals the following:

∑_{j=0}^{p−2}(−1)^{j}x^{j}y^{p−j−1}dx + ∑_{i=1}^{p−1}(−1)^{i}x^{i}y^{p−i−1}dy =

∑_{t=0}^{p−1}(−1)^{t}x^{t}y^{p−t−1}(dx + dy) − x^{p−1}dx − y^{p−1}dy.

For 0 ≤ t ≤ p − 1 we have that t! is invertible in 𝔽_{p} and also

\binom{p − 1}{t}t! = (p − 1)(p − 2) … (p − t) = (−1)^{t}t! mod p.

Thus we have \binom{p−1}{t} = (−1)^{t}. Substituting this in the above and using the binomial formula the result follows.

Proposition 6.4. For any K ∈ K_{0} there is a natural homomorphism of 𝔽_{p}- algebras which we call the de Rham map

DR : ℓ(K) → ω(K); ϕ(x) ↦ x^{p} + λx − x(dx)^{p−1},
q(x) ↦ x^{p−1}dx + βλx,
δ(x) ↦ dx, u ↦ 0.

and we have Im(DR) ⊆ ker(d : ω(K) → ω(K)). There is also a natural homomorphism of 𝔽_{p}-algebras which we call the Steenrod map

St : ℓ(K) → 𝔽_{p}[u] ⊗ K; ϕ(x) ↦ St_{0}(x), q(x) ↦ St_{1}(x),
δ(x) ↦ 0, u ↦ u ⊗ 1.

The image of this map is Im(St) = R(K). There is a commutative diagram of 𝔽_{p}-algebras as follows.

ℓ(K) ⟶^{St} 𝔽_{p}[u] ⊗ K
DR↓  ^{p_{1}}↓
ω(K) ⟶^{p_{2}}  K

where the algebra maps p_{1} and p_{2} are given by p_{1}(u) = 0 and p_{2}(dx) = 0, p_{2}(x) = x for each x ∈ K.

Proof. We check that the formulas for DR maps the relations (14)-(25) to zero. Formula (27) and the additivity of x ↦ x^{p} shows that (14) is mapped to zero. It is trivial that (15) is mapped to zero. By (26) and the additivity of x ↦ βλx it follows that (16) is mapped to zero.

Taking the derivative of products and permuting factors we find the following equations:

d(a)d(bc) = d(a)d(b)c + (−1)^{σ(b)}d(a)bd(c),

d(b)d(ca) = (−1)^{σ(a)(σ̂(b)+σ̂(c))}ad(b)d(c) + (−1)^{σ(c)+σ̂(a)(σ̂(b)+σ(c))}d(a)d(b)c,

d(c)d(ab) = (−1)^{σ̂(c)(σ̂(a)+σ(b))}d(a)bd(c) + (−1)^{σ(a)+σ̂(c)(σ(a)+σ̂(b))}ad(c)d(b).

After some reductions (17) follows from these.

One easily checks that (18) and (19) are mapped to zero in each of the cases σ(a) = σ(b) = 0, σ(a) = σ(b) = 1 and σ(a) = σ̂(b) = 1. It also follows by small direct computations that (20)-(25) are mapped to zero.

We check that (14)-(25) are mapped to zero by the formulas defining St. Since δ(x) is mapped to zero this is trivial for all elements except (18), (19),

(20) and (24).

By the Cartan formula and [\frac{|ab|}{2}] = [\frac{|a|}{2}] + [\frac{|b|}{2}] + σ(a)σ(b) one verifies that

St_{0}(ab) = (−u^{p−1})^{σ(a)σ(b)}St_{0}(a)St_{0}(b),

St_{1}(ab) = (−u^{p−1})^{σ(a)σ(b)}(u^{σ(b)}St_{1}(a)St_{0}(b) + (−u)^{σ(a)}St_{0}(a)St_{1}(b))

such that (18) and (19) are mapped to zero. Lemma 13.1 implies that (20) and (24) are mapped to zero. The diagram commutes by direct verification.

7 The de Rham map and cohomology of ω(K)

In this section K denotes a connected unstable A-algebra. The de Rham map gives a map from ℓ(K)/(u) to the cycles in ω(K). We give an important criterion which ensures that this map is an isomorphism. The material corresponds to section 8 in [2].

Definition 7.1. Let I_{δ} ⊆ ℓ(K) denote the ideal I_{δ} = (δ(x)|x ∈ K).

Proposition 7.2. There is an 𝔽_{p}-linear map as follows

Ψ : ω(K) → ℓ(K) ; a_{0}da_{1} … da_{n} ↦ δ(a_{0})δ(a_{1}) … δ(a_{n})

where a_{0}, …, a_{n} ∈ K. Its image is the ideal Im(Ψ) = I_{δ}.

Proof. We must show that Ψ is well defined. The relations arising from (8), (9) and (11) are respected since we have the same relations in ℓ(K) with d replaced by δ. We must verify that the following relation is respected:

a_{0}da_{1} … da_{i−1}d(a_{i}a_{i+1})da_{i+2} … da_{n} =
(−1)^{(k+σ̂(a_{i}))σ(a_{i+1})}a_{0}a_{i+1}da_{1} … da_{i}da_{i+2} … da_{n}
+(−1)^{(k+1)σ(a_{i})}a_{0}a_{i}da_{1} … da_{i−1}da_{i+1} … da_{n}

where k = |da_{1} … da_{i−1}|. This follows if the relation

xd(yz) = (−1)^{σ̂(y)σ(z)}xzd(y) + (−1)^{σ(y)}zyd(z)

is respected. By relation (17) one sees that it is after some work with the signs.

Definition 7.3. Define the 𝔽_{p}-algebra ω̃(K) as the quotient of ℓ(K) by the ideal I_{δ} + (u). Since DR(I_{δ}) ⊆ dω(K) we may define an 𝔽_{p}-algebra map Φ by the following diagram where P denotes the canonical projection:

ℓ(K)/(u) ⟶^{P}  ω̃(K)
DR↓  ^{Φ}↓
ω(K) ⟶ ω(K)/dω(K)

Since d ∘ DR = 0 we have in fact defined a morphism Φ : ω̃(K) → H^{∗}(ω(K)).

Remark 7.4. We have a filtration of ℓ(K) by ideals as follows:

ℓ(K) ⊇ (u) ⊇ (u^{2}) ⊇ ⋯ ⊇ (u^{i}) ⊇ …

and (u^{i})/(u^{i+1}) = u^{i}ω̃(K) for i ≥ 1.

Proposition 7.5. The composite Ψ ∘ d : ω(K) → ℓ(K) is trivial, so we can define Ψ as a map on ω(K)/dω(K). This allows us to consider the composite Ψ ∘ Φ : ω̃(K) → ℓ(K). This composite is zero.

Proof. By definition of Ψ we have Ψ ∘ d = 0. The following rules hold for b ∈ K and z = a_{0}da_{1} … da_{n} ∈ ω̃(K):

Ψ(Φ(ϕ(b))z) = (−1)^{|ϕ(b)|}ϕ(b)Ψ(z) , Ψ(Φ(q(b))z) = (−1)^{|q(b)|}q(b)Ψ(z).

In fact the first rule follows from (21) and the second from (22) as one sees by direct verification. By these rules and the observation Ψ(1) = 0 the result follows.

Remark 7.6. We can collect the information we have gathered so far in a commutative diagram:

ω̃(K) ⟶^{Φ} ω(K)/dω(K) ⟶^{Ψ} ℓ(K)/(u) ⟶^{P}  ω̃(K)
↑  ^{DR}↓  ^{Φ}↓
ω(K)  ⟶^{d} ω(K) ⟶ ω(K)/dω(K)

where the composite Ψ ∘ Φ vanishes and ker(P) = Im(Ψ).

Theorem 7.7. Assume that the map Φ : ω̃(K) → H^{∗}(ω(K)) is an isomorphism. Then so is

DR : ℓ(K)/(u) → ker(d : ω(K) → ω(K)).

Proof. The diagram is formally the same as the one above Theorem 8.5 of [2]. So the same diagram chase as in the proof of Theorem 8.5 in [2] gives the result.

Proposition 7.8. For any pair K, L ∈ K_{0} the following composite is an isomorphism

κ̃ : ω̃(K) ⊗ ω̃(L) ⟶^{ω̃(i)⊗ω̃(j)} ω̃(K ⊗ L) ⊗ ω̃(K ⊗ L) ⟶^{μ} ω̃(K ⊗ L)

where i : K → K ⊗ L; i(a) = a ⊗ 1 and j : L → K ⊗ L; j(b) = 1 ⊗ b are the canonical inclusions and μ denotes the product homomorphism. The following diagram commutes.

ω̃(K ⊗ L) ⟶^{Φ_{K⊗L}}  H^{∗}(ω(K ⊗ L))
κ̃↑  _{κ^{∗}}↑
ω̃(K) ⊗ ω̃(L) ⟶^{Φ_{K}⊗Φ_{L}} H^{∗}(ω(K)) ⊗ H^{∗}(ω(L))

Proof. We verify that κ̃ is an isomorphism. For a ∈ K and b ∈ L we have that a ⊗ b = (a ⊗ 1)(1 ⊗ b) giving the following relations in ω̃(K ⊗ L):

ϕ(a ⊗ b) = (1 − σ(a)σ(b))ϕ(a ⊗ 1)ϕ(1 ⊗ b),
q(a ⊗ b) = σ̂(b)q(a ⊗ 1)ϕ(1 ⊗ b) + σ̂(a)ϕ(a ⊗ 1)q(1 ⊗ b).

Since ϕ(a) ⊗ 1 ↦ ϕ(a ⊗ 1), q(a) ⊗ 1 ↦ q(a ⊗ 1), 1 ⊗ ϕ(b) ↦ ϕ(1 ⊗ b) and 1 ⊗ q(b) ↦ q(1 ⊗ b) we see that κ̃ is surjective. One checks that the following gives a well defined map γ̃ in the opposite direction of κ̃:

ϕ(a ⊗ b) ↦ (1 − σ(a)σ(b))ϕ(a) ⊗ ϕ(b),
q(a ⊗ b) ↦ σ̂(b)q(a) ⊗ ϕ(b) + σ̂(a)ϕ(a) ⊗ q(b).

By checking on generators we see that γ̃ ∘ κ̃ = id so κ̃ is injective as well and hence an isomorphism.

The isomorphisms κ and κ̃ have corresponding factorizations. The diagram splits in two commuting squares accordingly.

8 Frobenius algebras

Definition 8.1. A Frobenius algebra is a graded commutative 𝔽_{p}-algebra K equipped with two 𝔽_{p}-linear maps β, λ : K → K satisfying the following conditions:

• K is connected (K^{q} = 0 for q < 0 and K^{0} = 𝔽_{p}) and finite dimensional in each degree.

• |β| = 1, β ∘ β = 0 and β(xy) = β(x)y + (−1)^{|x|}xβ(y) for all x, y ∈ K.

• |λx| = p(|x| − 1) + 1, λx = 0 when |x| is even, λx = x when |x| = 1 and λ(xy) = λ(x)y^{p} + x^{p}λ(y) for all x, y ∈ K.

A morphism of Frobenius algebras f : (K, β, λ) → (K′, β′, λ′) is an 𝔽_{p}-algebra map f : K → K′ of degree zero such that f ∘ λ = λ′ ∘ f and f ∘ β = β′ ∘ f. The category of Frobenius algebras is denoted F.

Remark 8.2. There is a forgetful functor K_{0} → F.

Definition 8.3. Let v𝔽_{p} denote the category of positively graded 𝔽_{p}-vector spaces which are finite dimensional in each degree. Let I : F → v𝔽_{p} denote the functor which takes a Frobenius algebra to its augmentation ideal (that is I(K) = K^{∗>0}). Define the functor S_{F} : v𝔽_{p} → F as the left adjoint of I. For V ∈ v𝔽_{p} we call S_{F}(V) the free Frobenius algebra on V.

Remark 8.4. We have S_{F}(V ⊕ W) = S_{F}(V) ⊗ S_{F}(W). Furthermore there is an explicit description as follows

S_{F}(V) = S_{CA}(V ⊕ βV ⊕ ⨁_{i≥1,ν∈{0,1}} β^{ν}λ^{i}(βV^{even} ⊕ V^{odd,∗>1}))

where S_{CA} denotes the free graded commutative algebra functor.

Recall that a sequence of integers I = (ϵ_{1}, s_{1}, ϵ_{2}, s_{2}, …, ϵ_{k}, s_{k}, ϵ_{k+1}) with s_{i} ≥ 0 and ϵ_{i} ∈ {0, 1} is called admissible if s_{i} ≥ ps_{i+1} + ϵ_{i+1} and s_{k} ≥ 1 or if k = 0 when I = (ϵ). The degree of I is defined as |I| = ∑ ϵ_{j} + ∑ 2s_{j}(p − 1) and the excess is defined recursively by e((ϵ, s), J) = 2s + ϵ − |J|. We use the following notation P^{I} = β^{ϵ_{1}}P^{s_{1}}β^{ϵ_{2}}P^{s_{2}} … β^{ϵ_{k}}P^{s_{k}}β^{ϵ_{k+1}}.

Lemma 8.5. If n ≥ 2 then H^{∗}B^{n}𝔽_{p} is the free Frobenius algebra on the vector space generated by the following set:

{P^{I}ι_{n}|I is admissible , e(I) ≤ n − 2, ϵ_{1} = 0}.

Furthermore H^{∗}B𝔽_{p} is the free Frobenius algebra on the vector space generated by the class ι_{1}.

Proof. The case n = 1 is trivial. Assume that n ≥ 2 and define the set

A(n) = {I|I is admisseble , e(I) ≤ n − 1, |I| + n is odd }.

Remark that if I ∈ A(n) then (0, (|I| + n − 1)/2, I) ∈ A(n). To see this write I ∈ A(n) as I = (ϵ, s, I′). Then e(I) = 2s + ϵ − |I′| ≤ n − 1 or equivalently 2sp + 2ϵ − |I| ≤ n − 1 such that the sequence (0, (|I| + n − 1)/2, I) is admissible. Its excess is n − 1 and its degree plus n is odd since p − 1 is even.

By Cartan’s computation (a special case of [5], Theorem 10.3) we have that H^{∗}B^{n}𝔽_{p} is the free graded commutative algebra on the set

B = {P^{J}ι_{n}|J is admissible , e(J) < n or (e(J) = n and ϵ_{1} = 1)}.

Assume that P^{I}ι_{n} belongs to the set in the statement of the lemma. Then P^{I}ι_{n} and βP^{I}ι_{n} belongs to B. By the remark we see that if |I| + n is even then β^{ϵ}λ^{i}βP^{I}ι_{n} ∈ B and if |I| + n is odd then β^{ϵ}λ^{i}P^{I}ι_{n} ∈ B for ϵ = 0, 1 and i ≥ 1.

Conversely, assume that P^{J}ι_{n} ∈ B. If e(J) ≤ n − 2 or e(J) = n − 1 and ϵ_{1} = 1 it is clearly one of the generators described in the lemma. It suffices to handle the case e(J) = n−1, ϵ_{1} = 0 since the case e(J) = n, ϵ_{1} = 1 then follows. Write J as J = (0, s, J′) where e(J) = 2s − |J|′ = n − 1. Then 2s = n + |J′| − 1 such that P^{J}ι_{n} = λP^{J′}ι_{n} and e(J) ≤ e(J′). We can continue this process until the next ϵ equals one or the excess drops below n − 1.

Theorem 8.6. The map Φ : ω̃(K) → H^{∗}(ω(K)) is an isomorphism when K is a free Frobenius algebra.

Proof. It suffices to show this when K is a free Frobenius algebra on a one dimensional vector space. Let v be the generator of this vector space. We first check the case |v| = 1 where K = Λ(v)⊗𝔽_{p}[βv]. The idempotents from Remark 4.6 (with x = dv) gives the following splitting:

ω(K) = ⨁_{i∈𝔽_{p}} e_{i}ω(K).

For each i we have de_{i} = 0 and (dv)e_{i} = ie_{i}. Also λv = v such that dβv = 0. From this we see that d(v^{ϵ}(βv)^{r}e_{i}) = ϵi(βv)^{r}e_{i}. It follows that H^{∗}(e_{i}ω(K)) = 0 for i ≠ 0 and H^{∗}(e_{0}ω(K)) = K. So H^{∗}(ω(K)) = K and since Φ(ϕ(v)) = ve_{0} and Φ(q(v)) = βv we see that Φ is surjective. The relations ϕ(βv) = q(v)^{p} and q(βv) = 0 shows that ϕ(v) and q(v) generate ω̃(K) so Φ is also injective.

Assume that |v| is even. In the following we write [−] for the functor which takes a set to the vector space it generates. We have

K = S_{CA}[v, βv, λ^{i}βv, βλ^{i}βv|i ≥ 1]

and we find that ω(K) = K ⊗ S_{CA}[dv, dβv]. We change basis such that the differential becomes easier to describe:

ω(K) =S_{CA}[v, dv] ⊗ S_{CA}[βv, dβv] ⊗
S_{CA}[λ^{i}βv − (dλ^{i−1}βv)^{p−1}λ^{i−1}βv, βλ^{i}βv|i ≥ 1].

By the Künneth formula we find that H^{∗}(ω(K)) equals

S_{CA}[v^{p}, v^{p−1}dv] ⊗ S_{CA}[λ^{i}βv − (dλ^{i−1}βv)^{p−1}λ^{i−1}βv, βλ^{i}βv|i ≥ 1].

The algebra ω̃(K) is generated by the classes ϕ(v), ϕ(λ^{i}βv), q(v) and q(λ^{i}βv) where i ≥ 0. We see that Φ maps these generators to the free generators for the cohomology of ω(K). Hence Φ is an isomorphism. The case where |v| is odd is similar.

9 A pullback description of the functor ℓ

Proposition 9.1. Let (n_{i}) be a sequence of positive integers such that the set {i|n_{i} = N} is finite for each N. In particular (n_{i}) may be a finite sequence. If we let K = H^{∗}(∏ B^{n_{i}}𝔽_{p}) then ker(St) = I_{δ}.

Proof. We must show that St : ℓ(K)/I_{δ} → 𝔽_{p}[u] ⊗ K is injective. The algebra ℓ(K)/I_{δ} has generators ϕ(x), q(x) for x ∈ K together with u and the relations are that ϕ and q are additive and that (18), (19), (20) and (24) equals zero.

For a free Frobenius algebra K = S_{F}(V) we have listed a set of algebra generators for ℓ(K)/I_{δ} below and we have written how they are mapped by St modulo elements in the ideal (u^{p−1}). Here z ∈ V^{1}, v ∈ V^{odd,∗>1} and w ∈ V^{even} runs through a basis for V and i ≥ 0:

ϕ(z) ↦ 1 ⊗ z,  q(z) ↦ 1 ⊗ βz,
ϕ(βv) ↦ 1 ⊗ (βv)^{p},  q(βv) ↦ −u^{p−2} ⊗ βP^{(|v|−1)/2}βv,
ϕ(λ^{i}v) ↦ 1 ⊗ λ^{i+1}v,  q(λ^{i}v) ↦ 1 ⊗ βλ^{i+1}v,
ϕ(w) ↦ 1 ⊗ w^{p},  q(w) ↦ −u^{p−2} ⊗ βP^{|w|/2−1}w,
ϕ(λ^{i}βw) ↦ 1 ⊗ λ^{i+1}βw,  q(λ^{i}βw) ↦ 1 ⊗ βλ^{i+1}βw,
u ↦ u.

We claim that these generators are mapped to algebraically independent elements in 𝔽_{p}[u] ⊗ K when K is the cohomology of a product of Eilenberg-MacLane spaces as stated. It suffices to show this in the case of one single Eilenberg-MacLane space.

The claim is trivial for K = H^{∗}B𝔽_{p}. Let K = H^{∗}B^{n}𝔽_{p} where n ≥ 2. We can then cancel the first line in the list above. By Lemma 8.5 we have that K is the free Frobenius algebra on the vector space V with basis P^{I}ι_{n} where I is admissible, e(I) ≤ n − 2 and ϵ_{1} = 0.

If |I| + n is odd we must look closer at βP^{(|I|+n−1)/2}βP^{I}ι_{n}. Write I as I = (0, s, I′). We have e(I) = 2s − |I′| ≤ n − 2 which implies that (0, (|I| + n − 1)/2, 1, s, I′) is admissible. Its excess equals n − 2 and we see that P^{(|I|+n−1)/2}βP^{I}ι_{n} is a basis element in V^{even}.

If |I| + n is even we must look at βP^{(|I|+n−2)/2}P^{I}ι_{n}. As in the odd case we see that P^{(|I|+n−2)/2}P^{I}ι_{n} is a basis element in V^{even}. However there is no β between the first two P -operations from the left.

By the above list of the lowest terms and the description of K as a free Frobenius algebra we now see that the generators are mapped to algebraically independent elements. Especially they are free generators.

Lemma 9.2. Let a_{1}, …, a_{p} ∈ K be elements of odd degree and define the following element in I_{δ}:

Δ(a_{1}, …, a_{p}) = ∑_{i=2}^{p} δ(a_{1}a_{i})δ(a_{2}) … { \atop δ(a_{i})} … δ(a_{p}).

where the hat means that the factor is left out. Then for any permutation τ ∈ Σ_{p} one has Δ(a_{1}, …, a_{p}) = Δ(a_{τ(1)}, …, a_{τ(p)}). Further the element is mapped as follows by the de Rham map:

DR(Δ(a_{1}, …, a_{p})) = ∑_{i=1}^{p} a_{i}da_{1} … d̂a_{i} … da_{p}.

Proof. We first show the invariance under permutation. Since the degree of δ(a_{i}) is even it suffices to show that Δ(a_{1}, a_{2}, …, a_{p}) = Δ(a_{2}, …, a_{p}, a_{1}). We prove the following more general statement for n ≥ 3:

∑_{i=2}^{n} δ(a_{1}a_{i})δ(a_{2}) … { \atop δ(a_{i})} … δ(a_{n}) =

∑_{j=3}^{n} δ(a_{2}a_{j})δ(a_{1})δ(a_{3}) … { \atop δ(a_{j})} … δ(a_{n}) − (n − 1)δ(a_{2}a_{1})δ(a_{3}) … δ(a_{n}).

The proof is by induction on n. For n = 3 we have

δ(a_{1}a_{2})δ(a_{3}) + δ(a_{1}a_{3})δ(a_{2}) = δ(a_{1}a_{2})δ(a_{3}) − δ(a_{3}a_{1})δ(a_{2})
= 2δ(a_{1}a_{2})δ(a_{3}) + δ(a_{2}a_{3})δ(a_{1})
= −2δ(a_{2}a_{1})δ(a_{3}) + δ(a_{2}a_{3})δ(a_{1})

where we used (17) at the second equality sign. Assume that the formula holds for n − 1. Then we have

∑_{i=2}^{n} δ(a_{1}a_{i})δ(a_{2}) … { \atop δ(a_{i})} … δ(a_{n}) =

( ∑_{i=2}^{n−1} δ(a_{1}a_{i})δ(a_{2}) … { \atop δ(a_{i})} … δ(a_{n−1}))δ(a_{n}) + δ(a_{1}a_{n})δ(a_{2}) … δ(a_{n−1}) =

( ∑_{j=3}^{n−1} δ(a_{2}a_{j})δ(a_{1})δ(a_{3}) … { \atop δ(a_{j})} … δ(a_{n−1}))δ(a_{n})

− (n − 2)δ(a_{2}a_{1})δ(a_{3}) … δ(a_{n−1})δ(a_{n}) + δ(a_{1}a_{n})δ(a_{2}) … δ(a_{n−1}).

We have that δ(a_{1}a_{n})δ(a_{2}) + δ(a_{2}a_{1})δ(a_{n}) = δ(a_{2}a_{n})δ(a_{1}) by relation (17) such that the sum of the last two terms above equals

−(n − 1)δ(a_{2}a_{1})δ(a_{3}) … δ(a_{n}) + δ(a_{2}a_{n})δ(a_{1}) … δ(a_{n−1})

so we recover the formula for n.

We use that d(a_{1}a_{i}) = a_{i}da_{1} − a_{1}da_{i} to compute the image under the de Rham map:

DR(Δ(a_{1}, …, a_{p})) = ∑_{i=2}^{p} d(a_{1}a_{i})da_{2} … d̂a_{i} … da_{p}

= ∑_{i=2}^{p} a_{i}da_{1} … d̂a_{i} … da_{p} − (p − 1)a_{1}da_{2} … da_{p}

which modulo p gives the stated result.

Definition 9.3. For any non negative integer n we let B(n) denote the following set:

B(n) = {(β_{1}, …, β_{p}) ∈ ℤ^{p}|∀i : β_{i} ≥ 0, β_{1} + ⋯ + β_{p} = n, ∃i, j : β_{i} ≠ β_{j}}.

The cyclic group on p elements C_{p} act on B(n) by cyclic permutation of coordinates. For x ∈ K we define the following elements in I_{δ}:

D_{0}^{n}(x) = − σ(x) ∑ Δ(P^{β_{1}}(x), P^{β_{2}}(x), …, P^{β_{p}}(x)),
D_{1}^{n}(x) =σ̂(x) ∑ δ(P^{β_{1}}(x)P^{β_{2}}(x) … P^{β_{p}}(x))

where both sums are taken over β ∈ B(n)/C_{p}. Note that D_{0}(x) is well defined by Lemma 9.2

Lemma 9.4. For any x ∈ K the following formulas hold in ω(K):

P^{i} ∘ DR(ϕ(x)) = DR(ϕ(P^{i/p}x) + D_{0}^{i}(x)),  (28)
P^{i} ∘ DR(q(x)) = DR(q(P^{i/p}x) + D_{1}^{i}(x))  (29)

where by convention P^{t} = 0 when t is a rational number which is not a non negative integer.

Proof. We first prove (28). Recall that DR(ϕ(x)) = x^{p} + λx − x(dx)^{p−1}. We have P^{i}λx = λP^{i/p}x by Lemma 13.1 and also P^{i}(x^{p}) = (P^{i/p}x)^{p} so it suffices to prove the following for |x| odd:

P^{i}(x(dx)^{p−1}) = (P^{i/p}x)(dP^{i/p}x)^{p−1} − DR(D_{0}^{i}(x)).

By the Cartan formula we have

P^{i}(x(dx)^{p−1}) = ∑ P^{β_{1}}(x)dP^{β_{2}}(x) … dP^{β_{p}}(x)

where we sum over the tuples (β_{1}, …, β_{p}) with ∑ β_{j} = i. The cyclic group C_{p} acts on the set of such tuples and an orbit has length 1 or p. Arranging the terms according to this the result follows by the definition of D_{0}^{i}(x) and Lemma 9.2.

For the proof of (29) recall that DR(q(x)) = x^{p−1}dx + βλx. We have P^{i}(βλx) = βλ(P^{i/p}x) + (βP^{(i−1)/p}x)^{p} by Lemma 13.1 so when |x| is odd we are done. For |x| even we must show that

P^{i}(x^{p−1}dx) = (P^{i/p}x)^{p−1}dP^{i/p}x + DR(D_{1}^{i}(x)).

This follows by the Cartan formula and a similar argument on orbits as the above.

Theorem 9.5. For any K ∈ K_{0} there is an A-module structure on ℓ(K) such that ℓ becomes a functor ℓ : K_{0} → K. The explicit formulas for the action are as follows where x ∈ K, n = [|x|/2] and i ≥ 0. Firstly, the action on ϕ(x) is given by:

P^{i}ϕ(x) =D_{0}^{i}(x) + ∑_{t} \binom{(p − 1)(n − t)}{i − pt}u^{(p−1)(i−pt)}ϕ(P^{t}x),
βϕ(x) =u^{σ̂(x)}(q(x) − δ(x)^{p−2}δ(xβx)).

Secondly, the action on q(x) is given by:

P^{i}q(x) =D_{1}^{i}(x) + ∑_{t} \binom{(p − 1)(n − t) + σ(x)}{i − pt}u^{(p−1)(i−pt)}q(P^{t}x)

−(−1)^{σ(x)} ∑_{t} \binom{(p − 1)(n − t) − 1 + σ(x)}{i − pt − 1}u^{(p−1)(i−pt)−1+(2−p)σ(x)}ϕ(βP^{t}x),
βq(x) = − δ(x^{p−1}βx).

Thirdly, the actions on δ(x) and u are as follows:

P^{i}δ(x) = δ(P^{i}x), βδ(x) = −δ(βx), P^{1}u = u^{p}, βu = 0.

The maps DR and St becomes A-linear such that we have a commutative diagram in K:

ℓ(K) ⟶^{St} R(K)
DR↓  ^{p_{1}}↓
ker(d) ⟶^{p_{2}} K

If K = H^{∗}(∏ B^{n_{i}}𝔽_{p}) where (n_{i}) is a (possible finite) sequence of positive integers satisfying that the set {i|n_{i} = N} is finite for each N, then the diagram is a pullback square.

Proof. We first prove that we have a pullback when K is a product of Eilenberg-MacLane spaces as stated. By Proposition 7.8 and Theorem 8.6 the map Φ is an isomorphism. So by Theorem 7.7 the kernel of DR is the ideal (u) ⊆ ℓ(K). The kernel of p_{1} is the ideal (u⊗1) ⊆ R(K) and it suffices to show that the restriction of the Steenrod map to these kernels St| : (u) → (u ⊗ 1) is an isomorphism. It is surjective since St is surjective and St(u) = u ⊗ 1. By Proposition 9.1 we have ker(St) = I_{δ} such that ker(St|) = (u) ∩ I_{δ} which is trivial because of the relation δ(x)u = 0 in ℓ(K). Hence St| is also injective.

In the Eilenberg-MacLane case the pullback defines an A-module structure on ℓ(K). By Theorem 5.2 and Lemma 9.4 we see that the stated formulas describe this action. A standard argument using the fact that B^{n}𝔽_{p} classifies degree n cohomology together with naturality takes care of the statements for general K ∈ K.

10 Homotopy orbits of S^{1}-spaces

In this section we list some results on homotopy orbits of S^{1}-spaces. They are all similar to results for p = 2 considered in [2] and we often refer to the proofs given there. In the entire section Y denotes an S^{1}-space. We write C_{n} for the cyclic group of order n. We let u of degree |u| = 2 and v of degree |v| = 1 denote algebra generators as follows: H^{∗}S^{1} = Λ(v), H^{∗}BS^{1} = 𝔽_{p}[u] and H^{∗}BC_{p^{n}} = Λ(v) ⊗ 𝔽_{p}[u].

Proposition 10.1. The fibration Y → ES^{1} ×_{S^{1}} Y → BS^{1} has the following Leray-Serre spectral sequence:

E_{2}^{∗∗} = H^{∗}(BS^{1}) ⊗ H^{∗}(Y) ⇒ H_{S^{1}}^{∗}(Y).

The differential in the E_{2}-term is given by

d_{2} : H^{∗}(Y) → uH^{∗}(Y) ; d_{2}(y) = ud(y)

where d is the action differential.

Proof. Similar to the proof of [2] Proposition 3.3.

Definition 10.2. Define the spaces E_{n}Y for n = 0, 1, 2, …, ∞ by

E_{n}Y = ES^{1} ×_{C_{p^{n}}} Y , n < ∞
E_{∞}Y = ES^{1} ×_{S^{1}} Y.

For nonnegative integers n and m with m > n define the maps

q_{m}^{n} : H^{∗}E_{m}Y → H^{∗}E_{n}Y , τ_{n}^{m} : H^{∗}E_{n}Y → H^{∗}E_{m}Y

by letting q_{m}^{n} be the map induced by the quotient map and τ_{n}^{m} be the transfer map. Also define q_{∞}^{n} : H^{∗}E_{∞}Y → H^{∗}E_{n}Y as the map induced by the quotient.

The following theorem is inspired by a result of Tom Goodwillie which can be found in [4] p. 279. We use it to give a convenient definition of the S^{1}-transfer.

Theorem 10.3. There is a commutative diagram as follows for any m ≥ 1.

\begin{matrix}E_{m}Y & ⟶^{Q} & E_{∞}Y \\ ^{pr_{1}}↓ & & ^{pr_{1}}↓ \\ BC_{p^{m}} & ⟶^{Bj} & BS^{1} \end{matrix}  (30)

Here Q denotes the quotient map and j : C_{p^{m}} ↪ S^{1} the inclusion. The diagram gives rise to an isomorphism.

Θ : H^{∗}(BC_{p^{m}}) ⊗_{H^{∗}(BS^{1})} H^{∗}(E_{∞}Y) ≅ H^{∗}(E_{m}Y) ; x ⊗ y ↦ pr_{1}^{∗}(x)q_{∞}^{m}(y)

The transfer map τ_{m}^{m+1} : H^{∗}E_{m}Y → H^{∗}E_{m+1}Y is zero on elements of the form Θ(1 ⊗ y) and the identity on elements of the form Θ(v ⊗ y). We get an isomorphism

colimH^{∗}E_{m}Y = vH^{∗}E_{∞}Y ≅ H̃^{∗}(Σ(E_{∞}Y)_{+}).

Proof. Similar to the proof of [2] Theorem 4.2.

Definition 10.4. For any non negative integer n the S^{1}-transfer

τ_{n}^{∞} : H^{∗}E_{n}Y → H^{∗}E_{∞}Y

is defined as the composite

H^{∗}E_{n}Y ⟶ colimH^{∗}E_{m}Y ⟶^{v^{−1}} H^{∗}E_{∞}Y

where the direct limit is over the transfer maps τ_{n}^{m}. Note that |τ_{n}^{∞}| = −1.

Definition 10.5. Let θ_{0} denote the S^{1}-equivariant map

θ_{0} : S^{1} × Y_{0} → ES^{1} × Y ; (z, y) ↦ (ze, zy)

and let θ_{n} for n = 1, 2, …, ∞ be the maps one obtains by passing to the quotients

θ_{n} : S^{1}/C_{p^{n}} × Y → E_{m}Y , m < ∞
θ_{∞} : ∗ × Y → E_{∞}Y.

Proposition 10.6. For non negative integers m and n with n < m the following squares commutes.

\begin{matrix}H^{∗}E_{n}Y & ⟶^{θ_{n}^{∗}} & H^{∗}(S^{1} × Y) \\ ^{q_{m}^{n}}↑ & & ^{q_{m}^{n}⊗1}↑ \\ H^{∗}E_{m}Y & ⟶^{θ_{m}^{∗}} & H^{∗}(S^{1} × Y) \end{matrix}  \begin{matrix}H^{∗}E_{n}Y & ⟶^{θ_{n}^{∗}} & H^{∗}(S^{1} × Y) \\ ^{τ_{n}^{m}}↓ & & ^{τ_{n}^{m}⊗1}↓ \\ H^{∗}E_{m}Y & ⟶^{θ_{m}^{∗}} & H^{∗}(S^{1} × Y) \end{matrix}

There are also commutative squares

\begin{matrix}H^{∗}E_{n}Y & ⟶^{θ_{n}^{∗}} & H^{∗}(S^{1} × Y) \\ ^{q_{∞}^{n}}↑ & & ^{pr_{2}^{∗}}↑ \\ H^{∗}E_{∞}Y & ⟶^{θ_{∞}^{∗}} & H^{∗}Y \end{matrix}  \begin{matrix}H^{∗}E_{n}Y & ⟶^{θ_{n}^{∗}} & H^{∗}(S^{1} × Y) \\ ^{τ_{n}^{∞}}↓ & & ^{τ_{n}^{∞}}↓ \\ H^{∗}E_{∞}Y & ⟶^{θ_{∞}^{∗}} & H^{∗}Y \end{matrix}

where θ_{∞}^{∗} = q_{∞}^{0} is the map induced by the inclusion of the fiber, and the transfer on the right hand side is given by 1 ⊗ y ↦ 0 and v ⊗ y ↦ y.

Proof. Similar to the proof of [2] Proposition 4.6.

Proposition 10.7. Frobenius reciprocity holds for any n ≥ 0:

τ_{n}^{∞}(q_{∞}^{n}(x)y) = (−1)^{|x|}xτ_{n}^{∞}(y).

Furthermore the following composition formulas hold.

τ_{0}^{∞} ∘ q_{∞}^{0} = 0 , q_{∞}^{0} ∘ τ_{0}^{∞} = d.

Proof. Similar to the proof of [2] Proposition 4.7, Proposition 4.8.

Proposition 10.8. There is always an inclusion Im(q_{∞}^{0}) ⊆ ker(d). If we have equality Im(q_{∞}^{0}) = ker(d) then the Leray-Serre spectral sequence of the fibration Y → ES^{1} ×_{S^{1}} Y → BS^{1} collapses at the E_{3}-term.

Proof. By Proposition 10.7 we have d ∘ q_{∞}^{0} = q_{∞}^{0} ∘ τ_{0}^{∞} ∘ q_{∞}^{0} = 0. The collapse statement follows by Proposition 10.1.

Definition 10.9. Put ζ_{p} = exp(2πi/p) and define the map

f′_{Y} : S^{1} × Y → ES^{1} × Y^{p}
(z, y) ↦ (ze, zy, ζ_{p}zy, ζ_{p}^{2}zy, …, ζ_{p}^{p−1}zy).

We let C_{p} act on the space to the left by ζ_{p} · (z, y) = (ζ_{p}z, y) and on the space to the right by ζ_{p} · (e, y_{1}, …, y_{p}) = (ζ_{p}e, y_{2}, …, y_{p}, y_{1}). Then the above map is C_{p}-equivariant. Passing to the quotients we get a map

f_{Y} : S^{1}/C_{p} × Y → ES^{1} ×_{C_{p}} Y^{p}.

Note that this map is natural in Y with respect to C_{p}-equivariant maps.

Recall the followings facts on the order p cyclic construction [6], [5] and [7]. For any space X with homology of finite type there is a natural isomorphism

H^{∗}(ES^{1} ×_{C_{p}} X^{p}) ≅ H^{∗}(C_{p}; H^{∗}(X)^{⊗p})

where C_{p} acts on H^{∗}(X)^{⊗p} by cyclic permutation with the usual sign convention. For a homogeneous element y ∈ H^{∗}X the C_{p} invariant y^{⊗p} defines an element 1 ⊗ y^{⊗p} in the zeroth cohomology group of C_{p}. Let N = 1 + ζ_{p} + ζ_{p}^{2} + ⋯ + ζ_{p}^{p−1} be the norm element in the group ring 𝔽_{p}[C_{p}]. If x_{1}, …, x_{p} ∈ H^{∗}X are homogeneous elements, which are not all equal, then the invariant Nx_{1}⊗⋯⊗x_{p} also defines an element 1 ⊗ Nx_{1} ⊗ ⋯ ⊗ x_{p} in the zeroth cohomology group of C_{p}.

Theorem 10.10. The following formula holds where δ_{i,j} denotes the Kronecker delta:

f_{Y}^{∗} (1 ⊗ y^{⊗p}) = 1 ⊗ y^{p} + v ⊗ y^{p−1}dy + δ_{p,3}v ⊗ βλy.

Proof. We write Y_{0} for the space Y with trivial S^{1}-action. We first prove the theorem in the special case Y = Y_{0}. Here the differential is zero. There is a factorization

f_{Y_{0}} : S^{1}/C_{p} × Y_{0} ⟶^{i×1} ES^{1}/C_{p} × Y_{0} ⟶^{1×Δ} ES^{1} ×_{C_{p}} Y_{0}^{p}.

By this and the formula for the Steenrod diagonal, [7] p. 119 & Errata, the result follows.

Next we prove the following formula for a general S^{1}-space:

f_{Y}^{∗} (1 ⊗ Nx_{1} ⊗ ⋯ ⊗ x_{p}) = v ⊗ d(x_{1} … x_{p}).  (31)

There is a commutative diagram as follows:

H^{∗}(S^{1}/C_{p} × Y) ⟵^{f_{Y}^{∗}} H^{∗}(ES^{1} ×_{C_{p}} Y^{p})
τ_{0}^{1}⊗1↑  ^{τ_{0}^{1}}↑
H^{∗}(S^{1} × Y) ⟵^{f_{Y}^{′∗}} H^{∗}(ES^{1} × Y^{p})

The lower horizontal map is given by

f_{Y}^{′∗}(1 ⊗ x_{1} ⊗ ⋯ ⊗ x_{p}) = ∏_{i=1}^{p}(1 ⊗ x_{i} + v ⊗ dx_{i})

as seen by the factorization

\begin{aligned}f′_{Y} : S^{1} × Y & ⟶^{Δ_{2}} & (S^{1} × Y)^{2} & ⟶^{pr_{1}×Δ_{p}} & S^{1} × (S^{1} × Y)^{p} \\ & ⟶^{i×η^{p}} & ES^{1} × Y^{p} & ⟶^{1×1×ζ_{p}×···×ζ_{p}^{p−1}} & ES^{1} × Y^{p}. \end{aligned}

The norm class is hit by the transfer and by finding the coefficient to v in the above formula (31) follows.

Finally we prove the Theorem in the general case. Because of the degrees f_{S^{1}}^{∗}(1 ⊗ v^{⊗p}) = 0. The two projection maps pr_{1} : S^{1} × Y_{0} → S^{1} and pr_{2} : S^{1} × Y_{0} → Y_{0} are S^{1}-equivariant. Thus we can use naturality together with the case Y = Y_{0} and the above equation to find the equations below

f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (1 ⊗ y)^{⊗p}) = 1 ⊗ 1 ⊗ y^{p} + δ_{p,3}v ⊗ 1 ⊗ βλy,
f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (v ⊗ 1)^{⊗p}) = f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (v ⊗ dy)^{⊗p}) = 0.

The action map η : S^{1} × Y_{0} → Y is also an S^{1}-equivariant map, hence by naturality we have a commutative diagram

S^{1}/C_{p} × (S^{1} × Y_{0}) ⟶^{f_{S^{1}×Y_{0}}} ES^{1} ×_{C_{p}} (S^{1} × Y_{0})^{p}
1×η↓  ^{1×η^{p}}↓
S^{1}/C_{p} × Y  ⟶^{f_{Y}}  ES^{1} ×_{C_{p}} Y^{p}

We compute the pull back of the class 1 ⊗ y^{⊗p} to the cohomology of the upper left corner. First we find

(1 × η^{p})^{∗}(1 ⊗ y^{⊗p}) = 1 ⊗ (1 ⊗ y + v ⊗ dy)^{⊗p} =

1 ⊗ (1 ⊗ y)^{⊗p} + 1 ⊗ (v ⊗ dy)^{⊗p} + ∑_{i=1}^{p−1} 1 ⊗ N(1 ⊗ y)^{⊗i} ⊗ (v ⊗ dy)^{⊗(p−i)}.

By (31) we can compute f_{S^{1}×Y_{0}}^{∗} applied to the norm element terms. Only the i = p − 1 term contributes.

f_{S^{1}×Y_{0}}^{∗}(1 ⊗ N(1 ⊗ y)^{⊗(p−1)} ⊗ (v ⊗ dy)) = v ⊗ d_{S^{1}×Y_{0}}(v ⊗ y^{p−1}dy)
= v ⊗ (d_{S^{1}}(v) ⊗ y^{p−1}dy + v ⊗ d_{Y_{0}}(y^{p−1}dy))
= v ⊗ 1 ⊗ y^{p−1}dy

Altogether we have

(1 ⊗ η^{∗}) ∘ f_{Y}^{∗} (1 ⊗ y^{⊗p}) = f_{S^{1}×Y_{0}}^{∗} ∘ (1 × η^{p})^{∗}(1 ⊗ y^{⊗p})
= f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (1 ⊗ y)^{⊗p}) + v ⊗ 1 ⊗ y^{p−1}dy.

We now apply 1 ⊗ γ^{∗} on both sides (for the map γ, see the beginning of §3) and the result follows.

11 Construction of classes in string cohomology from classes in ordinary cohomology

In this section X denotes a connected space.

Definition 11.1. Put ζ_{p} = exp(2πi/p) and define evaluation maps as follows:

ev_{0} : ΛX → X ; γ ↦ γ(1) ,

ev_{1} : ES^{1} ×_{C_{p}} ΛX → ES^{1} ×_{C_{p}} X^{p} ; [e, γ] ↦ [e, γ(1), γ(ζ_{p}), …, γ(ζ_{p}^{p−1})].

Definition 11.2. The classes u, f(x), g(x), δ(x) ∈ H_{S^{1}}^{∗}(ΛX) for x ∈ H^{∗}X are defined by

f(x) = τ_{1}^{∞} ∘ ev_{1}^{∗}(v ⊗ x^{⊗p}),  g(x) = τ_{1}^{∞} ∘ ev_{1}^{∗}(1 ⊗ x^{⊗p}),
δ(x) = τ_{0}^{∞} ∘ ev_{0}^{∗}(x),  u = τ_{1}^{∞} ∘ ev_{1}^{∗}(vu ⊗ 1^{⊗p}).

Theorem 11.3. Let i_{0} : X ↪ ΛX denote the constant loop inclusion and let i_{∞} be the corresponding map of S^{1}-homotopy orbits. There is a commutative diagram as follows

\begin{matrix}H^{∗}(ES^{1} ×_{S^{1}} ΛX) & ⟶^{i_{∞}^{∗}} & H^{∗}(BS^{1} × X) \\ ^{q_{∞}^{0}}↓ & & ↓ \\ H^{∗}(ΛX) & ⟶^{i_{0}^{∗}} & H^{∗}(X) \end{matrix}  (32)

and an inclusion Im(q_{∞}^{0}) ⊆ ker(d : H^{∗}(ΛX) → H^{∗}(ΛX)). The constructed classes are mapped as follows under i_{∞}^{∗}.

i_{∞}^{∗}(f(x)) = σ̂(x)St_{0}(x) + σ(x)(−1)^{m}m!u^{m}St_{0}(x),
i_{∞}^{∗}(g(x)) = σ̂(x)St_{1}(x) + σ(x)(−1)^{m}m!u^{m−1}St_{1}(x),
i_{∞}^{∗}(δ(x)) = 0 and i_{∞}^{∗}(u) = u ⊗ 1.

Here m = (p − 1)/2. Under q_{∞}^{0} the images of the classes are as follows.

q_{∞}^{0}(f(x)) = σ̂(x)e(x^{p}),
q_{∞}^{0}(g(x)) = σ̂(x)e(x^{p−1}dx) + σ(x)δ_{p,3}e(βλx),
q_{∞}^{0}(δ(x)) = e(dx) and q_{∞}^{0}(u) = 0.

Here δ_{p,3} = 1 for p = 3 and zero otherwise.

Proof. A commutative diagram of spaces gives the diagram (32) and Proposition 10.7 gives the stated inclusion.

We check the formulas involving i_{∞}^{∗}. There is a commutative diagram as follows where Δ : X → X^{p} is the diagonal and i_{1} is the map of C_{p}-homotopy orbits induced by i_{0}.

H^{∗}(X)  ⟶^{ev_{0}^{∗}}  H^{∗}(ΛX)
Δ^{∗}↑  ^{id}↑
H^{∗}(X^{p})  ⟶  H^{∗}(ΛX)
T r_{0}^{1}↓  ^{τ_{0}^{1}}↓
H^{∗}(ES^{1} ×_{C_{p}} X^{p}) ⟶^{ev_{1}^{∗}} H^{∗}(ES^{1} ×_{C_{p}} ΛX) ⟶^{i_{1}^{∗}} H^{∗}(BC_{p} × X)
τ_{1}^{∞}↓  ^{τ_{1}^{∞}⊗1}↓
H^{∗}(ES^{1} ×_{S^{1}} ΛX) ⟶^{i_{∞}^{∗}} H^{∗}(BS^{1} × X)

The horizontal map with no label is the induced in cohomology of the map γ ↦ (γ(1), γ(ζ_{p}), …, γ(ζ_{p}^{p−1})). A homotopy commutative square of spaces shows that the upper square commutes and it is obvious that the other two are commutative. We see that i_{∞}^{∗}(u) = u ⊗ 1 as stated.

The composite ev_{1} ∘ i_{1} is the diagonal Δ_{1}. Its induced in cohomology is the Steenrod diagonal Δ_{1}^{∗} given by the following ([7] p. 119 & Errata):

ν(q)Δ_{1}^{∗}(1 ⊗ x^{⊗p}) = ∑_{i}(−1)^{i}u^{m(q−2i)} ⊗ P^{i}x + ∑_{i}(−1)^{i}vu^{m(q−2i)−1} ⊗ βP^{i}x

where q = |x| and ν(q) = (m!)^{q}(−1)^{m(q^{2}+q)/2}. From this formula and the lower part of the diagram we see that

ν(q)i_{∞}^{∗}(f(x)) = ∑_{i}(−1)^{i}u^{m(q−2i)} ⊗ P^{i}x = (−1)^{[q/2]}u^{σ(x)m}St_{0}(x),

ν(q)i_{∞}^{∗}(g(x)) = ∑_{i}(−1)^{i}u^{m(q−2i)−1} ⊗ βP^{i}x = (−1)^{[q/2]}u^{σ(x)(m−1)}St_{1}(x).

By [7] Lemma 6.3 one has (m!)^{2} = (−1)^{m+1} mod p and from this one sees that ν(q)^{−1}(−1)^{[q/2]} = 1 for q even and ν(q)^{−1}(−1)^{[q/2]} = (−1)^{m}m! for q odd. Hence we have verified the formulas for i_{∞}^{∗}(f(x)) and i_{∞}^{∗}(g(x)).

By the left part of the diagram we see that

δ(x) = τ_{1}^{∞} ∘ ev_{1}^{∗} ∘ T r_{0}^{1}(x ⊗ 1 ⊗ ⋯ ⊗ 1).

The composite Δ_{1}^{∗} ∘ T r_{0}^{1} is zero by [7] Lemma 4.1 so i_{∞}^{∗}(δ(x)) = 0.

Finally we check the formulas for q_{∞}^{0}. It follows directly from Proposition 10.7 that δ(x) is mapped as stated and clearly u is mapped to zero. For the classes f(x) and g(x) we use Proposition 10.6:

q_{∞}^{0} ∘ τ_{1}^{∞} ∘ ev_{1}^{∗} = τ_{1}^{∞} ∘ θ_{1}^{∗} ∘ ev_{1}^{∗} = τ_{1}^{∞} ∘ (ev_{1} ∘ θ_{1})^{∗}.

Note that ev_{1} ∘ θ_{1} equals the composite

S^{1}/C_{p} × ΛX ⟶^{f_{ΛX}} ES^{1} ×_{C_{p}} (ΛX)^{p} ⟶^{1×ev_{0}^{p}} ES^{1} ×_{C_{p}} X^{p}

where f_{ΛX} is the map from Definition 10.9. Thus we have

q_{∞}^{0} ∘ τ_{1}^{∞} ∘ ev_{1}^{∗} = τ_{1}^{∞} ∘ f_{ΛX}^{∗} ∘ (1 × ev_{0}^{p})^{∗}.

From this and Theorem 10.10 we get the stated results.

Proposition 11.4. The following diagram is a pullback square.

H^{∗}(ES^{1} ×_{S^{1}} ΛB𝔽_{p}) ⟶^{i_{∞}^{∗}} 𝔽_{p}[u] ⊗ H^{∗}B𝔽_{p}
q_{∞}^{0}↓  ↓
ker(d)  ⟶^{i_{0}^{∗}}  H^{∗}BF_{p}

Proof. In the proof of Proposition 4.9 we saw that ⊔j_{n} : ⊔BF_{p}(n) → ΛB𝔽_{p} was both an S^{1}-map and a homotopy equivalence. So the induced map of S^{1}-homotopy orbits (⊔j_{n})_{∞} is a weak homotopy equivalence. The maps in the diagram have nice descriptions in terms of this equivalence since there are commutative diagrams as follows where Q denotes quotient maps.

⊔ES^{1} ×_{S^{1}} B𝔽_{p}(n) ⟶^{(⊔j_{n})_{∞}} ES^{1} ×_{S^{1}} ΛB𝔽_{p}
⊔Q↑  ^{Q}↑
⊔ES^{1} × B𝔽_{p}(n) ⟶^{⊔j_{n}} ES^{1} × ΛB𝔽_{p}
⊔ES^{1} ×_{S^{1}} B𝔽_{p}(n) ⟶^{(⊔j_{n})_{∞}} ES^{1} ×_{S^{1}} ΛB𝔽_{p}
↑  ^{i_{∞}}↑
ES^{1} × B𝔽_{p}(0) ⟶^{≅} BS^{1} × B𝔽_{p}

Hence it suffices to show that the following diagram is a pullback where d_{(n)} denotes the differential on H^{∗}B𝔽_{p}(n).

⊕H^{∗}(ES^{1} ×_{S^{1}} B𝔽_{p}(n)) ⟶^{pr_{0}} H^{∗}(BS^{1} × B𝔽_{p})
⊕Q^{∗}↓  ↓
⊕ ker(d_{(n)})  ⟶  H^{∗}(B𝔽_{p})

We have H^{∗}(ES^{1} ×_{S^{1}} B𝔽_{p}(n)) ≅ ker(d_{(n)}) for n ≠ 0 since here the Leray-Serre spectral sequence has E_{3}^{i,∗} = 0 for i ≥ 1. The result follows.

As indicated by Theorem 11.3 above it turns out that when |x| is odd then both f(x) and g(x) can be written as a product of some power of u with another class. This was not the case for p = 2 as described in [2]. We construct new classes to get around this difficulty.

Theorem 11.5. Let x ∈ H^{∗}X be a cohomology class of odd degree. Then there exists classes ϕ(x), q(x) ∈ H^{∗}(ES^{1} ×_{S^{1}} ΛX) with |ϕ(x)| = p(|x| − 1) + 1 and |q(x)| = p(|x| − 1) + 2 such that

i_{∞}^{∗}(ϕ(x)) = St_{0}(x), q_{∞}^{0}(ϕ(x)) = λx − x(dx)^{p−1},
i_{∞}^{∗}(q(x)) = St_{1}(x), q_{∞}^{0}(q(x)) = βλx.

Proof. It suffices to prove the theorem when X = B^{n}𝔽_{p} for odd n ≥ 1. The general case then follows by defining ϕ(x) = (1 ×_{S^{1}} Λh)^{∗}ϕ(ι_{n}) and q(x) = (1 ×_{S^{1}} Λh)^{∗}q(ι_{n}) where |x| = n and h : X → B^{n}𝔽_{p} has h^{∗}(ι_{n}) = x.

For n = 1 we have St_{0}(ι_{1}) = 1 ⊗ ι_{1} and St_{1}(ι_{1}) = 1 ⊗ βι_{1} so here the result follows from Proposition 11.4.

Assume that n = 2r + 1 where r ≥ 1. By Proposition 10.1, Theorem 4.1 and Theorem 8.6 the E_{3}-term of the Leray-Serre spectral sequence for the fibration ΛB^{n}𝔽_{p} → ES^{1} ×_{S^{1}} ΛB^{n}𝔽_{p} → BS^{1} has the following form:

E_{3} ≅ Im(d) ⊕ (𝔽_{p}[u] ⊗ ω̃(K))

where K = H^{∗}B^{n}𝔽_{p}. Here u has bidegree (2, 0) and an element y in Im(d) or ω̃(K) has bidegree (0, |y|). Define s : BS^{1} → ES^{1} ×_{S^{1}} ΛB^{n}𝔽_{p} such that pr_{1} ∘ s = id by choosing a constant loop. By s^{∗} we see that the vertical line (∗, 0) survives to E_{∞}.

Up to dimension 2rp + 2p − 1 the only nonzero vertical lines are (∗, 0), (∗, 2rp + 1), (∗, 2rp + 2) and (∗, 2rp + 2p − 1) corresponding to the classes u, ϕ(ι_{n}), q(ι_{n}) and q(βι_{n}) respectively. Hence we can define ϕ(ι_{n}) and q(ι_{n}) by

q_{0}^{∞}(ϕ(ι_{n})) = λι_{n} − ι_{n}(dι_{n})^{p−1},
q_{0}^{∞}(q(ι_{n})) = βλι_{n} and s^{∗}(q(ι_{n})) = 0.

Since |f(ι_{n})| = 2rp + p and |g(ι_{n})| = 2rp + p − 1 we see that f(ι_{n}) = C_{1}u^{m}ϕ(ι_{n}) and g(ι_{n}) = C_{2}u^{m−1}q(ι_{n}) where C_{1}, C_{2} ∈ 𝔽_{p} and m = (p − 1)/2 as before. By Theorem 11.3 we conclude that

C_{1}u^{m}i_{∞}^{∗}(ϕ(ι_{n})) = (−1)^{m}m!u^{m}St_{0}(ι_{n}),
C_{2}u^{m−1}i_{∞}^{∗}(q(ι_{n})) = (−1)^{m}m!u^{m−1}St_{1}(ι_{n})

and the result follows.

Definition 11.6. For x ∈ H^{∗}X of even degree we define ϕ(x) = f(x) and q(x) = g(x).

12 String cohomology and the functor ℓ

In this section we prove the main result of this paper.

Theorem 12.1. Let X be a connected space which has finite dimensional mod p homology in each degree. Then there is a morphism of unstable A-algebras

ψ : ℓ(H^{∗}X) → H^{∗}(ES^{1} ×_{S^{1}} ΛX)

which sends ϕ(x), q(x), δ(x) for x ∈ H^{∗}X and u to the constructed classes with the same names. The morphism is natural in X. If both of the maps

e : ω(H^{∗}X) → H^{∗}(ΛX) , Φ : ω̃(H^{∗}X) → H^{∗}(ω(H^{∗}X))

are isomorphisms then so is ψ. In particular, when (n_{i}) is a sequence (possibly finite) of positive integers such that the set {i|n_{i} = N} is finite for each N and X = ∏ B^{n_{i}}𝔽_{p}, then ψ is an isomorphism.

Proof. Assume that both e and Φ are isomorphisms and put K = H^{∗}X. By Theorem 7.7 we have that DR surjects ker(d) and from the results in Section 11 we see that Im(DR) ⊆ Im(q_{∞}^{0}). Hence Im(q_{∞}^{0}) = ker(d). It then follows from Proposition 10.8 that the Leray-Serre spectral sequence associated to the fibration ΛX → ES^{1} ×_{S^{1}} ΛX → BS^{1} collapses at the E_{3}-term:

E_{∞} = E_{3} ≅ ker(d) ⊕ uω̃(K) ⊕ u^{2}ω̃(K) ⊕ …  (33)

By Remark 7.4 there is a filtration of ℓ(K) which associated graded object is also (33). If we fix a dimension the filtration is finite and we conclude that ℓ(K) and H^{∗}(ES^{1} ×_{S^{1}} ΛX) have the same dimension in each degree. Hence it suffices to show that the map ψ in the statement is a well defined morphism which is surjective.

The constructed classes are algebra generators for H^{∗}(ES^{1} ×_{S^{1}} ΛX) by the collapse, and the formulas for their images under i_{∞}^{∗} given in Section 11 shows that Im(i_{∞}^{∗}) = R(K). Hence we have a commutative diagram as follows:

H^{∗}(ES^{1} ×_{S^{1}} ΛX) ⟶^{i_{∞}^{∗}} R(K)
q_{∞}^{0}↓  ^{p_{1}}↓
ker(d)  ⟶^{p_{2}} K

The kernel of p_{1} is the ideal (u ⊗ 1) and i_{∞}^{∗}(u) = u ⊗ 1. Since u ∈ ker(q_{∞}^{0}) and i_{∞}^{∗} is surjective we conclude that the restriction i_{∞}^{∗}| : ker(q_{∞}^{0}) → ker(p_{1}) is surjective. Hence we have a surjection into the pullback.

We now restrict to the case where X is a product of Eilenberg-MacLane spaces as in the last part of the statement. Here e is an isomorphism by Proposition 3.7 and Theorem 4.1 and Φ is an isomorphism by Proposition 7.8 and Theorem 8.6.

The above surjection into the pullback together with Theorem 9.5 gives us a surjective morphism ψ′ : H^{∗}(ES^{1} ×_{S^{1}} ΛX) → ℓ(K) which is then an isomorphism. By definition it has inverse ψ.

By the fact that B^{n}𝔽_{p} classifies degree n cohomology and naturality of the constructed classes, we can now conclude that the defining relations for ℓ(K) are universal for the constructed classes. Hence ψ is a well defined morphism in general.

In the case where e and Φ are isomorphisms, the collapse ensures that ψ is surjective and hence an isomorphism.

Corollary 12.2. Let X be a connected space and assume that H^{∗}X is a polynomial algebra on a set of even dimensional generators. Assume also that H_{i}X is finite for each i. Then ψ is an isomorphism.

Proof. If K is zero in odd degrees then ω(K) is the ordinary de Rham complex Ω(K|𝔽_{p}). Furthermore, ω̃(K) is the de Rham complex Ω(K̄|𝔽_{p}) where K̄ is the algebra defined by K̄^{np} = K^{n} and K̄^{m} = 0 for m ≠ 0 mod p. The map Φ is the Cartier map.

The Eilenberg-Moore spectral sequence for H^{∗}(ΛX) has Hochschild homology of H^{∗}X as its E_{2}-term and it collapses since the algebra generators sit in E_{2}^{0,∗} and E_{2}^{−1,∗}. By the Hochschild-Konstant-Rosenberger theorem Hochschild homology is isomorphic to the de Rham complex and one concludes that e is an isomorphism. The Cartier map Φ is also an isomorphism.

Remark 12.3. Theorem 12.1 respects products in the following sense. Assume that X and Y are connected spaces with mod p homology of finite type and that e_{X}, Φ_{X}, e_{Y}, Φ_{Y} are isomorphisms. Then e_{X×Y} and Φ_{X×Y} are isomorphisms and ψ_{X×Y} is an isomorphism.

13 Appendix: Unstable A-algebras

In this appendix we prove Proposition 3.4. We shall need the following result.

Lemma 13.1. For any unstable A-algebra K and x ∈ K the following equations hold.

P^{i}λx = \begin{cases}λ(P^{\frac{i}{p}} x) & , i = 0 mod p \\ 0 & , otherwise \end{cases}  (34)

P^{i}βλx = \begin{cases}βλ(P^{\frac{i}{p}} x) & , i = 0 mod p \\ (βP^{\frac{i−1}{p}} x)^{p} & , i = 1 mod p \\ 0 & , otherwise \end{cases}  (35)

Proof. We just prove (34) since the proof of (35) is similar. When |x| is even both sides in the equation are zero. Assume that |x| is odd. By the instability condition P^{i}λx = 0 when 2i > p(|x| − 1) + 1. When i is divisible by p this inequality implies 2i ≥ p(|x| − 1) + p or \frac{2i}{p} ≥ |x| and since |x| is odd \frac{2i}{p} > |x|. So P^{i/p}x = 0 and the equation holds in this case. If 2i = p(|x| − 1) then P^{i}λx = λ^{2}x = λ(P^{i/p}x).

Finally assume that 2i < p(|x| − 1). Then we can apply the Adem relation:

P^{i}P^{\frac{|x|−1}{2}} x = ∑_{t=0}^{[ \frac{i}{p}]}(−1)^{i+t}\binom{(p − 1)(\frac{|x|−1}{2} − t) − 1}{i − pt}P^{i+ \frac{|x|−1}{2} −t}P^{t}x.

The instability condition shows that P^{i+ \frac{|x|−1}{2} −t}P^{t}x = 0 unless i ≤ pt. But the binomial coefficient is zero when i < pt. So we get zero when i ≠ 0 mod p and the term corresponding to t = i/p when i = 0 mod p.

We now prove Proposition 3.4.

Proof. Let dK denote the graded 𝔽_{p}-vector space given by (dK)^{n} = K^{n+1} and (dK)^{−1} = 0. We write dx for the element in dK corresponding to x in K hence d(x + y) = dx + dy. We define an A-algebra structure on dK by P^{i}dx = dP^{i}x and βdx = −dβx. Let F (dK) denote the free graded commutative algebra on the 𝔽_{p}-vector space dK. By the Cartan formula F (dK) is an A-algebra and the graded symmetric product K ⊙ F (dK) is an A-algebra. By definition ω(K) = K ⊙ F (dK)/I where I is the ideal generated by

1 ⊙ d(xy) − d(x) ⊙ y − (−1)^{|x|}x ⊙ d(y),  (36)
1 ⊙ (d(λx) − (dx)^{p}),  (37)
1 ⊙ d(βλx).  (38)

We verify that A · I ⊆ I such that ω(K) is an A-algebra. We have

P^{n}(1 ⊙ d(xy) − dx ⊙ y − (−1)^{|x|}x ⊙ dy) =

∑_{i+j=n}(1 ⊙ d(P^{i}(x)P^{j}(y)) − dP^{i}x ⊙ P^{j}y − (−1)^{|x|}P^{i}x ⊙ dP^{j}y)

which is in I by (36) since the degree of P^{i} is even. Further

β(1 ⊙ d(xy) − dx ⊙ y − (−1)^{|x|}x ⊙ dy) =
− (1 ⊙ d(β(x)y) − dβx ⊙ y − (−1)^{|βx|}βx ⊙ dy)
− (−1)^{|x|}(1 ⊙ d(xβy) − dx ⊙ βy − (−1)^{|x|}x ⊙ dβy)

which is also in I by (36).

In any A-algebra one has P^{i}(a^{p}) = (P^{i/p}a)^{p} when i = 0 mod p and zero otherwise, since this fact is a consequence of the Cartan formula alone. So by Lemma 13.1 we have the following relation in F (dK) when i = 0 mod p:

P^{i}(d(λx) − (dx)^{p}) = d(P^{i}λx) − (P^{\frac{i}{p}} dx)^{p} = d(λP^{\frac{i}{p}} x) − (dP^{\frac{i}{p}} x)^{p}.

For i ≠ 0 mod p we get zero. So P^{i} applied to an element of the form (37) lies in I. If we apply β to such an element we also land in I by (38). Finally Lemma 13.1 shows that P^{i}(1 ⊙ d(βλx)) ∈ I and trivially β(1 ⊙ d(βλx)) ∈ I.

We verify that ω(K) ∈ U. We must show that P^{i}dx = 0 if 2i > |x| − 1. This holds if 2i > |x| since K ∈ U. If 2i = |x| we have P^{i}dx = dP^{i}x = d(x^{p}) = 0. We must also show that βP^{i}dx = 0 when 2i + 1 > |x| − 1. This holds if 2i + 1 > |x| since K ∈ U and if 2i + 1 = |x| we have βP^{i}dx = −dβP^{i}x = −dβλx = 0. Since the action on products are by the Cartan formula we have shown that ω(K) ∈ U.

Finally we check that ω(K) ∈ K. The Cartan formula holds by definition. For |x| odd we have P^{\frac{|dx|}{2}} (dx) = dλx = (dx)^{p} and the result follows.

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