Equivariant evaluation on free loop spaces

Iver Ottosen ^{∗}

January 30, 2000

Abstract

There is a natural evaluation map on the free loop space ΛX → X^{k} which sends a loop to its values at the kth roots of unity. This map is equivariant with respect to the action of the cyclic group on k elements C_{k}. We study the induced map in C_{k}-equivariant cohomology with mod two coefficients in the cases where k = 2^{m} for m ≥ 1.

1 Introduction

For each positive integer k there is an evaluation map on the free loop space ΛX → X^{k} which is equivariant with respect to the action of the cyclic group C_{k}. The evaluation map sends a loop to its values at the kth roots of unity. As k approaches infinity this evaluation distinguishes more loops. One hopes that the induced map in cohomology becomes more and more informative as k increases and hence provides new input to the ongoing problem of computing the cohomology of free loop spaces. Since evaluating at two different points of the circle gives homotopic maps, one should use C_{k}-equivariant cohomology in order to get interesting results.

In this paper we use cohomology with ℤ/2-coefficients. We consider the cases k = 2^{m} with m ≥ 1. Under these assumptions we compute the map induced by evaluation in C_{k}-equivariant cohomology in terms of the approximation functor ℓ introduced in [BO].

The functor ℓ comes equipped with a ring homomorphism as follows

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

∗1991 Mathematics Subject Classification: 55N91, 55P35. I would like to thank Université de Paris Nord and Universität Bielefeld where this work was developed, supported by the European Union TMR network ERB FMRX CT-97-0107: Algebraic K-Theory, Linear Algebraic Groups and Related Structures.

It is an isomorphism for 1-connected spaces X with polynomial cohomology. It is not hard to see that H^{∗}(BC_{k}) ⊗_{H^{∗}(BS^{1})} ℓ(H^{∗}X) approximates H^{∗}(ES^{1} ×_{C_{k}} ΛX) similarly.

In §2 we introduce some notation and conventions which are fixed through the paper. In §3 we review the definition and some basic properties of the functor ℓ and the related twisted de Rham functor Ω_{λ}.

In §4 we show that ℓ and Ω_{λ} are functors from the category of connected unstable A-algebras to the category of unstable A-algebras. Here A denotes the mod two Steenrod algebra. We give a useful pullback description of ℓ(K) when K is the cohomology of a product of Eilenberg-MacLane spaces of type K(𝔽_{2}, n) with n ≥ 1. The description is similar to the one for the quadratic construction C_{2}(K) as described in [HLS].

In §5 we generalize the exactness result for the approximation ψ to 0connected spaces with polynomial cohomology and a finite 2-group as fundamental group. The sections §4 and §5 give a convenient setup for working with the functor ℓ.

In §7 we set up the computation. There is a commutative diagram

\xymatrix{S^{1} × ΛX \ar[d]^{η} \ar[dr]^{f} \\ ES^{1} ×_{C_{k}} ΛX \ar[r]^{ev} & ES^{1} ×_{C_{k}} X^{k} \\ BC_{k} × X \ar[u]_{i} \ar[ur]_{Δ}}

where i comes from the constant loop inclusion X ↪ ΛX and η comes from the S^{1}-action on ΛX. The pair (i, η) induces an injective map in cohomology by the pullback in §5. The map induced by the diagonal Δ was computed in [O]. The induced map of f is computed in §7. From this data ev^{∗} is determined in §8. Finally in §9 we give an extension of the evaluation map ev^{∗} to an algebraic setting.

2 Notation and conventions

We always use 𝔽_{2} = ℤ/2 coefficients unless otherwise is specified. We say that a space X is of finite type if H_{i}X is finitely generated for each i. The mod two Steenrod algebra is denoted A, the category of unstable A-modules is denoted U and the category of unstable A-algebras is denoted K. The category K_{0} is the full subcategory of K consisting of connected unstable A-algebras. For any A-module M we define λ : M → M by λ(x) = Sq^{|x|−1}x.

The cyclic group of order k is denoted C_{k}. We view it as a subgroup of the circle group S^{1} generated by ζ_{k} = exp(2πi/k). Assume that k = 2^{m} with m ≥ 0. For an S^{1}-space Y and a C_{k}-space Z we use the following short hand notation for Borel constructions:

E_{∞}Y := ES^{1} ×_{S^{1}} Y , E_{m}Z := ES^{1} ×_{C_{k}} Z

For integers n and m with 0 ≤ n < m ≤ ∞ there is a map q_{m}^{n} induced by the quotient and a transfer map τ_{n}^{m} as follows

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

Here τ_{n}^{∞} is the S^{1}-transfer from Definition 4.4 in [BO].

The free loop space is the mapping space ΛX = X^{S^{1}}. Precomposition with the inclusion map j_{k} : C_{k} ↪ S^{1} defines a map j_{k}^{∗} : X^{S^{1}} → X^{C_{k}} which is C_{k}-equivariant. We get a map of Borel constructions.

1 ×_{C_{k}} j_{k}^{∗} : ES^{1} ×_{C_{k}} X^{S^{1}} → ES^{1} ×_{C_{k}} X^{C_{k}}

When k = 2^{m} we put C_{m}X = ES^{1}×_{C_{k}} X^{k} and we write ev_{m} : E_{m}ΛX → C_{m}X for the evaluation map 1 ×_{C_{k}} j_{k}^{∗}.

Finally, we fix the names on the algebra generators of the following cohomology groups. H^{∗}(S^{1}) = Λ(v) where |v| = 1, H^{∗}(BS^{1}) = 𝔽_{2}[u] where |u| = 2 and

H^{∗}(BC_{2^{m}}) = \begin{cases}Λ(v) ⊗ 𝔽_{2}[u] & when m ≥ 2 \\ 𝔽_{2}[t] & when m = 1 \end{cases}

where |t| = |v| = 1 and |u| = 1.

3 The functors Ω_{λ} and ℓ

In this section we review the definitions of the approximation functors Ω_{λ} and ℓ from [BO] and provide some extra material on them.

Definition 3.1. Let R be a unital, connected 𝔽_{2}-algebra. A linear map λ : R → R is called a derivation over Frobenius if |λx| = 2|x| − 1 and λ(xy) = x^{2}λy + y^{2}λx. We refer to (R, λ) as an algebra with derivation over Frobenius, and use the abbreviation FGA for such gadgets. They form a category F GA where the morphisms are algebra morphisms respecting the structure given by the derivations over Frobenius.

Definition 3.2. Let (R, λ) be a FGA. The algebra Ω_{λ}(R) is the quotient of the free commutative algebra on

x, dx for x ∈ R

where |dx| = |x| − 1, by the ideal generated by the elements

d(x + y) + dx + dy  (1)
d(xy) + xdy + ydx  (2)
(dx)^{2} + d(λx)  (3)

There is a differential on Ω_{λ}(R) defined by the formula d(x) = dx as a derivation over R.

Clearly Ω_{−} is a functor from the category of FGA’s to the category of DGA’s. The following result shows that the category of FGA’s is equipped with a tensor product. We omit the proof which is an easy direct computation.

Proposition 3.3. Let (A, λ) and (B, γ) be FGA’s and define an 𝔽_{2}-linear map λ ∗ γ : A ⊗ B → A ⊗ B by

λ ∗ γ(a ⊗ b) = λ(a) ⊗ b^{2} + a^{2} ⊗ γ(b)

Then (A ⊗ B, λ ∗ γ) is an FGA.

The functor Ω_{−} preserves tensor products in the following sense:

Proposition 3.4. The canonical inclusion maps i : A → A ⊗ B; a ↦ a ⊗ 1 and j : B → A ⊗ B; b ↦ 1 ⊗ b are morphisms of FGA’s and the composite map

κ : Ω_{λ}(A) ⊗ Ω_{γ}(B) ⟶^{i_{∗}⊗j_{∗}} Ω_{λ∗γ}(A ⊗ B) ⊗ Ω_{λ∗γ}(A ⊗ B) ⟶^{μ} Ω_{λ∗γ}(A ⊗ B)

is an isomorphism of DGA’s.

Proof. i and j are morphisms of FGA’s since an FGA is non negatively graded. Thus i_{∗} and j_{∗} are morphisms of DGA’s hence κ is a morphism of DGA’s. We check that there is a well defined map Ω_{λ∗γ}(A ⊗ B) → Ω_{λ}(A) ⊗ Ω_{γ}(B) such that y ↦ y and dy ↦ d_{⊗}y for y ∈ A ⊗ B Such a map is clearly an inverse to κ. As for the usual de Rham complex one sees that elements of the form (1) and (2) are mapped to zero. For a ∈ A and b ∈ B we have

(d(a ⊗ b))^{2} + d(λ ∗ γ(a ⊗ b)) =
(d(a ⊗ b))^{2} + d(λ(a) ⊗ b^{2} + a^{2} ⊗ γ(b)) ↦
(d(a) ⊗ b + a ⊗ d(b))^{2} + d(λa) ⊗ b^{2} + a^{2} ⊗ d(γb) = 0

and since (3) is additive in x we are done.

The forgetful functor U : K → F GA provide the main examples of FGA’s. For an object K in K we let λx = Sq^{|x|−1}x. It is a derivation over Frobenius by the Cartan formula.

Let Y be a S^{1}-space with action map η : S^{1} × Y → Y. Then there is a map d : H^{∗}Y → H^{∗−1}Y defined by η^{∗}(y) = 1 ⊗ y + v ⊗ dy. Proposition 3.2 in [BO] shows that (H^{∗}Y, d) is a DGA and Sq^{i}(dy) = d(Sq^{i}y) such that (dy)^{2} = d(λy). It is not necessary to assume that Y is connected for these results even though we did so in [BO]. When Y is a free loop space we get the following.

Proposition 3.5. For any connected space X there is a morphism of DGA’s

e : (Ω_{λ}(H^{∗}X), d) → (H^{∗}(ΛX), d) ; e(x) = ev_{0}^{∗}(x) ; e(dx) = dev_{0}^{∗}(x)

which is natural in X. For any pair of connected spaces X and Y there is a commutative diagram

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

where the lower horizontal map is the Künneth isomorphism.

Definition 3.6. Let (R, λ) be an FGA. The algebra ℓ(R) is the quotient of the free commutative algebra on generators

ϕ(x), q(x), δ(x) for x ∈ R and u

of degree |ϕ(x)| = 2|x|, |q(x)| = 2|x| − 1, |δ(x)| = |x| − 1 and |u| = 2, by the ideal generated by the elements

ϕ(a + b) + ϕ(a) + ϕ(b)  (4)
δ(a + b) + δ(a) + δ(b)  (5)
q(a + b) + q(a) + q(b) + δ(ab)  (6)
δ(xy)δ(z) + δ(yz)δ(x) + δ(zx)δ(y)  (7)
ϕ(xy) + ϕ(x)ϕ(y) + uq(x)q(y)  (8)
q(xy) + q(x)ϕ(y) + ϕ(x)q(y)  (9)
δ(x)^{2} + δ(λx)  (10)
q(x)^{2} + ϕ(λx) + δ(x^{2}λx)  (11)
δ(x)ϕ(y) + δ(xy^{2})  (12)
δ(x)q(y) + δ(xλy) + δ(xy)δ(y)  (13)
uδ(x)  (14)

where a, b, x, y, z are homogeneous elements in R and |a| = |b|.

Theorem 3.7. Let (R, λ) be an FGA. There is a natural algebra homomorphism DR : ℓ(R) → Ω_{λ}(R) defined by

ϕ(x) ↦ x^{2} , q(x) ↦ xdx + λx , δ(x) ↦ dx , u ↦ 0

If R is a symmetric algebra R = S(V) where V is a non negatively graded 𝔽_{2}-vector space with V^{n} finitely generated for each n, there is a short exact sequence

0 → (u) ⟶ ℓ(R) ⟶^{DR} ker(d) → 0

Proof. This was shown in [BO] Proposition 8.1, Theorem 8.2 and Theorem 8.5.

We can apply the results from chapter 6 of [BO] component vise and get the following.

Theorem 3.8. For any connected space X there is an algebra homomorphism ψ : ℓ(H^{∗}X) → H^{∗}(ES^{1} ×_{S^{1}} ΛX) defined by

\begin{matrix}ϕ(x) & ↦ & τ_{1}^{∞} ∘ ev_{1}^{∗}(t ⊗ x^{⊗2}) & ; & q(x) & ↦ & τ_{1}^{∞} ∘ ev_{1}^{∗}(1 ⊗ x^{⊗2}) \\ δ(x) & ↦ & τ_{0}^{∞} ∘ ev_{0}^{∗}(x) & ; & u & ↦ & τ_{1}^{∞} ∘ ev_{1}^{∗}(t^{3} ⊗ 1^{⊗2}) \end{matrix}

It is natural in X and the following diagram commutes

ℓ(H^{∗}X) ⟶^{ψ} H^{∗}(ES^{1} ×_{S^{1}} ΛX)
DR↓  ^{q_{∞}^{0}}↓
Ω_{λ}(H^{∗}X) ⟶^{e}  H^{∗}(ΛX)

4 Unstable algebras.

In this section we show that Ω_{λ} and ℓ restricts to functors K_{0} → K. We do this algebraically instead of using the result in the next chapter, since it gives a better understanding of the functors. For the functor ℓ we follow the treatment of the quadratic construction as close as possible. A certain pull back diagram appears, which will become useful later.

We use the notation Sq_{i}(x) := Sq^{|x|−i}x. For a sequence of positive integers n_{1}, …, n_{r} we define the following object in K_{0}:

A(n_{1}, …, n_{r}) = H^{∗}(K(𝔽_{2}, n_{1}) × ⋯ × K(𝔽_{2}, n_{r}))

Lemma 4.1. The following relations are valid in any unstable A-module

Sq^{2i}(λx) = λ(Sq^{i}x)  (15)
Sq^{2i+1}(λx) = (i + |x|)Sq^{i+|x|}Sq^{i}x  (16)

Proof. Follows by the Adem relation. See the proof of Proposition 5.2 in [O].

Proposition 4.2. For any K ∈ ob(K_{0}) we can define an A-action on Ω_{λ}(K) by Sq^{i}(dx) = d(Sq^{i}x) and the Cartan formula. With this action Ω_{λ} becomes a functor Ω_{λ} : K_{0} → K. The differential d : Ω_{λ}(K) → Ω_{λ}(K) is A-linear.

Proof. Let dK be the A-module given by (dK)^{n} = K^{n+1}. The symmetric algebra S(K ⊕ dK) is an A-module by the Cartan formula. Ω_{λ}(K) is by definition the quotient of this module by the ideal generated by (1),(2) and (3). We verify that this ideal is closed under the A-action. We have

Sq^{i}(d(x + y) + dx + dy) = d(Sq^{i}x + Sq^{i}y) + d(Sq^{i}x) + d(Sq^{i}y)

which is in the ideal (in fact (1) is already zero in dK). Further

Sq^{i}(d(xy) + xdy + ydx) =

∑_{j=0}^{i} (d(Sq^{j}(x)Sq^{i−j}(y)) + (Sq^{j}x)d(Sq^{i−j}y) + (Sq^{j}y)d(Sq^{i−j}x))

which is in the ideal by (2). We use Lemma 4.1 for the last type of elements:

Sq^{2i}((dx)^{2} + d(λx)) = (Sq^{i}(dx))^{2} + d(Sq^{2i}(λx)) = (d(Sq^{i}x))^{2} + d(λ(Sq^{i}x)) Sq^{2i+1}((dx)^{2} + d(λx)) = d(Sq^{2i+1}(λx)) = d((i + |x|)(Sq^{i}x)^{2}) = 0

We conclude that Ω_{λ}(K) has a natural A-module structure. It is a non negatively graded module since d1 = 0 by (2). We have Sq^{|x|}(dx) = d(x^{2}) = 0 and clearly Sq^{i}(dx) = 0 for i > |x| so Ω_{λ}(K) is an unstable A-module. The Cartan formula holds by definition and Sq^{|x|−1}(dx) = d(λx) = (dx)^{2} so Ω_{λ}(K) is an unstable A-algebra. The differential on Ω_{λ}(K) is A-linear by definition.

Definition 4.3. For K ∈ K_{0} we define the modified de Rham cohomology by H_{λ}^{∗}(K) = H^{∗}(Ω_{λ}(K), d).

Note that this cohomology theory has the nice property of being a functor H_{λ}^{∗} : K_{0} → K as well as it respects tensor products: H_{λ}^{∗}(A ⊗ B) ≅ H_{λ}^{∗}(A) ⊗ H_{λ}^{∗}(B).

Definition 4.4. Let K be an unstable A-algebra. For x ∈ K we define the elements St_{0}(x) = ∑ u^{i} ⊗ Sq_{2i}x and St_{1}(x) = ∑ u^{i} ⊗ Sq_{2i+1}x in 𝔽_{2}[u] ⊗ K where the sums are taken over all integers i ≥ 0. Further we let R(K) denote the sub 𝔽_{2}-algebra of 𝔽_{2}[u]⊗K generated by u and St_{0}(x), St_{1}(x) for x ∈ K.

Proposition 4.5. For i ≥ 0 one has Sq^{i}R(K) ⊆ R(K) hence we have defined a functor R : K → K. Explicitly, the following formula holds for ν = 0, 1:

Sq^{i}St_{ν}(x) = ∑_{j=0}^{∞} \binom{|x| − j + 1 − ν}{i − 2j}u^{[ \frac{i+1−ν}{2} ]−j}St_{i+ν}(Sq^{j}x)  (17)

where the lower index on the right should be taken modulo 2.

Proof. Let St(x) = ∑ t^{i} ⊗ Sq_{i}(x) in 𝔽_{2}[t] ⊗ K where |t| = 1. The following formula holds

Sq^{i}(St(x)) = ∑_{j=0}^{∞} \binom{|x| − j}{i − 2j}t^{i−2j}St(Sq^{j}x)  (18)

It expresses that the diagonal map Δ : C_{1}(K) → 𝔽_{2}[t] ⊗ K which is given by Δ(1 ⊗ (x ⊗ y + y ⊗ x)) = 0 and Δ(1 ⊗ x^{⊗2}) = St(x) is A-linear. Here C_{1} is the quadratic functor [HLS] page 58-59. The formulas for the A-module structure on C_{1}K can be found in [M]. There is an A-linear transfer map of degree -1

τ : 𝔽_{2}[t] → 𝔽_{2}[u] ; τ(t^{2i}) = 0 , τ(t^{2i+1}) = u^{i}

Note that the following formula holds for ν = 0, 1:

St_{ν}(x) = (τ ⊗ 1)(t^{1−ν}St(x))  (19)

It implies that Sq^{i}St_{ν}(x) = (τ ⊗ 1) ∘ Sq^{i}(t^{1−ν}St(x)). By this and formula (18) we see that equation (17) holds.

Proposition 4.6. For K ∈ ob(K_{0}) there is an 𝔽_{2}-algebra homomorphism

St : ℓ(K) → 𝔽_{2}[u] ⊗ K ;
ϕ(x) ↦ St_{0}(x) , q(x) ↦ St_{1}(x) , δ(x) ↦ 0 , u ↦ u

and R(K) is the image of this map.

Proof. We must check that St is well defined. The elements (5), (7), (10), (12), (13), (14) are clearly mapped to zero since St ∘ δ = 0. The elements (4), (6) are mapped to zero since St∘ϕ and St∘q are additive and St∘δ = 0. For the elements (8) and (9) we proceed as follows. Let again St(x) = ∑ t^{i} ⊗ Sq_{i}(x) instead of the above map. By the Cartan formula we have St(xy) = St(x)St(y) and applying (19) to this we find

St_{1}(ab) = St_{1}(a)St_{0}(b) + St_{0}(a)St_{1}(b)
St_{0}(ab) = St_{0}(a)St_{0}(b) + uSt_{1}(a)St_{1}(b)

Finally (11) is mapped to zero since Lemma 4.1 implies St_{0}(λx) = (St_{1}x)^{2}. By its definition R(K) is the image of the map St.

Lemma 4.7. The Eilenberg-MacLane space K(𝔽_{2}, n) with n ≥ 1 has the following mod two cohomology algebra.

𝔽_{2}[λ^{k}Sq^{I}ι_{n} | I admissible , e(I) ≤ n − 2 , k ≥ 0]

Proof. The usual set of algebra generators is {Sq^{J}ι_{n}} where J runs through the admissible sequences with excess e(J) ≤ n − 1. We verify that this set is in fact the same as the set of generators in the lemma.

Assume that I = (i_{1}, i_{2}, …, i_{s}) is admissible with excess e(I) ≤ n − 1. We have λSq^{I}ι_{n} = Sq^{(n−1+|I|,I)}ι_{n}. Since e(I) = i_{1} − i_{2} − ⋯ − i_{s} ≤ n − 1 we have n − 1 + |I| ≥ 2i_{1} such that (n − 1 + |I|, I) is admissible and clearly e(n − 1 + |I|, I) = n − 1. Thus the set of generators in the lemma is a subset of the usual set of algebra generators.

Conversely, if J = (j_{1}, …, j_{r}) is admissible and e(J) ≤ n − 2 then Sq^{J}ι_{n} is one of the generators with k = 0 in the lemma. Assume that e(J) = n − 1 and put J(t) = (j_{t}, j_{t+1}, …, j_{r}). The map t ↦ e(J(t)) is weakly decreasing since e(J(t)) − e(J(t + 1)) = j_{t} − 2j_{t+1} ≥ 0 and eventually it becomes zero. Let k = max{t|e(J(t)) = n − 1}. Then we have Sq^{J}ι_{n} = λ^{k}Sq^{J(k+1)}ι_{n} which is one of the generators in the lemma.

Definition 4.8. When R is an FGA we let J(R) denote the subring of ℓ(R) generated by the elements δ(x) for x ∈ R. By (12), (13) and (14) we see that J(R) is an ideal in ℓ(R).

Proposition 4.9. When K = A(n_{1}, …, n_{r}) where n_{i} ≥ 1 for each i we have ker(St) = J(K).

Proof. By definition J(K) ⊆ ker(St) so it suffices to show that the induced map St : ℓ(K)/J(K) → 𝔽_{2}[u] ⊗ K is injective. The algebra ℓ(K)/J(K) has generators u, ϕ(x), q(x) for x ∈ K. The relations are that ϕ and q are additive and further

ϕ(xy) = ϕ(x)ϕ(y) + uq(x)q(y)
q(xy) = q(x)ϕ(y) + ϕ(x)q(y)
q(x)^{2} = ϕ(λx)

Put x_{i} = ι_{n_{i}} for each i. We see that the following elements are algebra generators: ϕ(Sq^{I_{i}}x_{i}), q(λ^{k_{i}}Sq^{I_{i}}x_{i}), u where I_{i} is admissible, e(I_{i}) ≤ n_{i}−2 and 0 ≤ k_{i} for 1 ≤ i ≤ r. It suffices to show that these are mapped to algebraic independent elements. The coefficient to u^{0} in St_{0}(Sq^{I_{i}}x_{i}) is (Sq^{I_{i}}x_{i})^{2} and in St_{1}(λ^{k_{i}}Sq^{I_{i}}x_{i}) it is λ^{k_{i}+1}Sq^{I_{i}}x_{i}. Using the projection homomorphisms p_{i} : 𝔽_{2}[u] ⊗ K → K given by p_{i}(u^{j} ⊗ x) = δ_{i,j}x and induction the algebraic independence follows by Lemma 4.7.

Theorem 4.10. For K ∈ ob(K_{0}) there is a commutative diagram in the category of 𝔽_{2}-algebras

ℓ(K) ⟶^{St} R(K)
DR↓  ^{p}↓
ker(d) ⟶^{q} K

where p : 𝔽_{2}[u] ⊗ K → K and q : Ω_{λ}(K) → K are the projections defined by q(x) = x, q(dx) = 0 and p(x) = x and p(u) = 0 for x ∈ K. If K = A(n_{1}, …, n_{r}) where n_{i} ≥ 1 for each i the diagram is a pull back square.

Proof. By evaluating on generators it is easy to check that the diagram is commutative. Assume K is the cohomology of a product of Eilenberg-MacLane spaces as above. The diagram is a pull back if and only if the restriction St : ker(DR) → ker(p) is an isomorphism. By Proposition 4.9 and Theorem 3.7 we have

ker(DR) ∩ ker(St) = (u) ∩ J(K) = 0

so the restriction of St is injective. The kernel of p : 𝔽_{2}[u] ⊗ K → K is (u) ⊆ 𝔽_{2}[u] ⊗ K so (u) ∩ R(K) is the kernel of p : R(K) → K. Since St(ϕ(x)) = St_{0}(x), St(q(x)) = St_{1}(x) and St(u) = u it is obvious that the restriction of St is surjective.

Theorem 4.11. The functor ℓ restricts to a functor ℓ : K_{0} → K For K ∈ ob(K_{0}) the A-module structure on ℓ(K) is given by the following formulas:

Sq^{i}δ(x) = δ(Sq^{i}x)  (20)
Sq^{i}ϕ(x) = ∑_{j=0}^{∞} \binom{|x| + 1 − j}{i − 2j}u^{[ \frac{i+1}{2} ]−j}((i + 1)ϕ(Sq^{j}x) + iq(Sq^{j}x))  (21)

Sq^{i}q(x) = ∑_{j=0}^{∞} \binom{|x| − j}{i − 2j}u^{[ \frac{i}{2} ]−j}(iϕ(Sq^{j}x) + (i + 1)q(Sq^{j}x)) + δ(Q^{i}(x)) (22)

Sq^{i}u = \binom{2}{i}u^{[ \frac{i}{2} ]+1}  (23)

Here by convention a binomial coefficient is zero when its lower parameter is negative. The Q^{i} operation in the last formula is defined by

Q^{i}(x) = ∑_{r=0}^{[ \frac{i}{2} ]} Sq^{r}(x)Sq^{i−r}(x)

There is a commutative diagram in the category K

ℓ(K) ⟶^{St} R(K)
DR↓  ^{p}↓
ker(d) ⟶^{q} K

It is natural in K and when K = A(n_{1}, …, n_{r}) for a sequence of positive integers n_{1}, …, n_{r} it is a pull back square.

Proof. When K = A(n_{1}, …, n_{r}) we see that ℓ(K) is an unstable A-algebra by the pull back diagram in Theorem 4.10. We verify that (20)-(23) are formulas for this A-module structure by using the injectivity of the pair (St,DR). The verification of equation (20) and (23) is trivial. When we apply DR to the right hand side of (21) we only need to consider the j = [\frac{i+1}{2}] term since DR(u) = 0. This term equals ϕ(Sq^{i/2}x) for i even and zero for i odd. hence we get (Sq^{i/2}x)^{2} for i even and zero for i odd. This equals Sq^{i}DR(ϕ(x)) = Sq^{i}(x^{2}) as it should. When we apply DR to the right hand side of (22) only the j = [\frac{i}{2}] term contribute. This term equals q(Sq^{r}x) + δ(Q^{2r}(x)) for i = 2r and (|x| − r)ϕ(Sq^{r}x) + δ(Q^{2r+1}(x)) for i = 2r + 1. Thus we must check that

(Sq^{r}x)d(Sq^{r}x) + λ(Sq^{r}x) + dQ^{2r}(x) = Sq^{2r}(xdx + λx)
(|x| − r)(Sq^{r}x)^{2} + dQ^{2r+1}(x) = Sq^{2r+1}(xdx + λx)

We have dQ^{i}(x) = Sq^{i}(xdx) + (i + 1)Sq^{i/2}(x)Sq^{i/2}(dx) by the definition of Q^{i}(x) and this together with Lemma 4.1 gives the desired result. For the map St the verification follows directly by formula (17).

For a general connected unstable A-algebra K we proceed as follows. Let T denote the free, graded, unital, non commutative algebra on Sq^{i} for i ≥ 0 where Sq^{0} = 1. Let F (K) be the free, unital, graded, commutative algebra on u, ϕ(x), q(x), δ(x) for x ∈ K. We let T act on F (K) by the formulas (20)- (23) and the Cartan formula. Let I(K) ⊆ F (K) denote the ideal generated by the elements (4)-(14). First we check that T · I(K) ⊆ I(K) such that we have a well defined action T × ℓ(K) → ℓ(K). Let g(y_{1}, …, y_{r}) be one of the elements (4)-(14). Put n_{j} = |y_{j}|, x_{j} = ι_{n_{j}} and A = A(n_{1}, …, n_{r}). Let f : A(n_{1}, …, n_{r}) → K be the morphism in K defined by f(x_{j}) = y_{j}. By the first part of this proof we have T · F (A) ⊆ F (A) so naturality gives Sq^{i}g(y_{1}, …, y_{r}) ∈ I(K).

We have A = T/J where J ⊆ T is the two sided ideal generated by the Adem elements. We must check that J · ℓ(K) = 0 such that we have a well defined action A × K → K. But this follows by a similar naturality argument as above. Hence we have shown that the formulas (20)-(23) define a natural A-module structure on ℓ(K), which satisfy the Cartan formula. It is immediate from the formulas that the instability condition holds for each generator of ℓ(K) so the action is unstable. Using the relations (4)-(14) one verifies that Sq^{|y|}y = y^{2} when y is either of the generators ϕ(x), q(x), δ(x), u and so this relation holds in general. Thus the action makes ℓ(K) an unstable A-algebra. It is obvious that p and q are A-linear.

5 A more general condition for exactness of the approximation functors

In this section we prove a stronger version of the main theorem in [BO], Theorem 10.1, which states that e and ψ are isomorphisms for 1-connected spaces with polynomial cohomology. We start by an examination of the case X = Bℤ/2 = ℝP^{∞}.

For a connected space X one has π_{0}ΛX ≅ [S_{+}^{1}, X] ≅< π_{1}X > where < · > denotes the set of conjugacy classes. Especially π_{0}ΛBG ≅< G > for any group G. By Lemma 7.11 of [BHM] we have the following result.

Lemma 5.1. For a discrete group G

ΛBG = ∐_{[g]∈<G>} Λ_{[g]}BG and Λ_{[g]}BG ≅ BC_{G}[g]

where C_{G}[g] denotes the centralizer of g.

As a special case ΛBℤ/2 ≅ Λ_{0}Bℤ/2 ⊔ Λ_{1}Bℤ/2 where Λ_{ν}Bℤ/2 ≃ Bℤ/2 for ν = 0, 1. Define a group action a_{ν} : ℤ × ℤ/2 → ℤ/2 by (r, s) ↦ νr + s. Then Ba_{ν} : Bℤ × Bℤ/2 → Bℤ/2 is an S^{1}-action which is the trivial one for ν = 0. Let B_{ν}ℤ/2 denote the space Bℤ/2 with S^{1}-action Ba_{ν}.

Lemma 5.2. (Ba_{ν})^{∗}(t) = 1 ⊗ t + νt ⊗ 1 for ν = 0, 1.

Proof. Since π_{1}(Ba_{ν}) = a_{ν} we have H_{1}(Ba_{ν}; ℤ) = a_{ν}. The result follows by tensoring with 𝔽_{2} and dualizing.

Proposition 5.3. For ν = 0, 1 the following map is S^{1}-equivariant and also a homotopy equivalence

j_{ν} : B_{ν}ℤ/2 → Λ_{ν}Bℤ/2 ; j_{ν}(x)(z) = Ba_{ν}(z, x)

Proof. The conjugacy class of the neutral element corresponds to the component containing constant loops so the index of the target of j_{0} is 0 as stated.

Let i be the index of the target of j_{1}. Since Ba_{ν} is an action map we see that j_{μ} is equivariant and that ev_{0} ∘ j_{ν} = id. Thus π_{q}(j_{ν}) is injective. But the homotopy groups of its domain and target space are the same finite groups so it is injective as well. By the Whitehead theorem j_{ν} is a homotopy equivalence. H^{∗}(B_{ν}ℤ/2) is a DGA and dt = ν by Lemma 5.2. Since j_{ν} induces an isomorphism of DGA’s we have i = 1.

Proposition 5.4. The map e : (Ω_{λ}(H^{∗}Bℤ/2), d) → (H^{∗}(ΛBℤ/2), d) is an isomorphism.

Proof. Let R = H^{∗}Bℤ/2 = 𝔽_{2}[t]. It suffices to show that (j_{0} ⊔ j_{1})^{∗} ∘ e : Ω_{λ}(R) → R⊕R is an isomorphism. Since ev_{0}∘j_{ν} = id it follows that t ↦ (t, t) and by Lemma 5.2 we see that dt ↦ (0, 1). In Ω_{λ}(R) we have (dt)^{2} = dλt = dt such that (1 + dt)^{2} = 1 + dt, (dt)(1 + dt) = 0 and clearly dt + (1 + dt) = 1. This gives an algebra splitting: Ω_{λ}(R) = (dt)Ω_{λ}(R) ⊕ (1 + dt)Ω_{λ}(R). The components are simply (dt)Ω_{λ}(R) = (dt)R and (1 + dt)Ω_{λ}(R) = (1 + dt)R thus the above map is an isomorphism.

Recall that an action of a group π on an Abelian group M is said to be nilpotent if there is a finite π-filtration of M such that π acts trivially on the filtration quotients.

Theorem 5.5. Let X be a connected space of finite type. Then the two maps

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

are morphisms in K which are natural in X. Assume further that the group π_{1}(X)×π_{1}(X) acts nilpotently on H_{i}(ΩX) for each i and that the cohomology of X is a symmetric algebra H^{∗}X = S(V) where V is a graded 𝔽_{2}-vector space with V^{n} finitely generated for each n. Then both e and ψ are isomorphisms of unstable A-algebras.

Remark 5.6. Any action of a finite 2-group on a mod 2 vector space is nilpotent. Hence the nilpotent action condition in the theorem always holds when π_{1}X is a finite 2-group.

Proof. We must check that the 𝔽_{2}-algebra maps e and ψ are A-linear. The A-action on Ω_{λ}(H^{∗}X) was described in Proposition 4.2. Since the differential d on H^{∗}ΛX is A-linear we see that e is A-linear. The A-action on ℓ(H^{∗}X) is given in Theorem 4.11. By Proposition 6.7 of [BO] we see that the 𝔽_{2}-algebra map ψ is also A-linear.

Making the further assumptions of nilpotent action and polynomial cohomology we first show that e is an isomorphism. According to [S] there is a fiber square with common fiber ΩX as follows.

ΛX ⟶^{ev} X
ev↓  ^{Δ}↓
X ⟶^{Δ} X × X

The associated cohomology Eilenberg-Moore spectral sequence lies in the second quadrant and has the following E_{2}-page

E_{2}^{−p,q} = Tor_{p}^{K⊗K}(K, K)^{q}

where K = H^{∗}X = S(V). It converges strongly to H^{∗}(ΛX) by [B] 4.1. The E_{2}-page can be interpreted by Hochschild Homology of K and then via the Hochschild Konstant Rosenberg theorem by differential forms

E_{2}^{−p,∗} = HH_{p}(K) ≅ Ω^{p}(K)

Here the bidegree of x_{0}dx_{1} … dx_{p} is (−p, |x_{0}| + ⋯ + |x_{p}|). Since all algebra generators for E_{2} sits in E_{2}^{0,∗} or E_{2}^{−1,∗} we have E_{2} = E_{∞}. The assumption that V^{n} is finitely generated for each n combined with the formula for the bidegree shows that for m fixed E_{∞}^{−p,m+p} = 0 for p sufficiently large. Hence the filtration of H^{m} := H^{m}(ΛX) is finite for any m. As in the proof of Theorem 10.1 in [BO] one easily sees that e(x) ∈ H^{∗}(ΛX) represents x ∈ E_{2}^{0,∗} for x ∈ K. We now show that e(dx) ∈ H^{∗}(ΛX) represents dx ∈ E_{2}^{−1,∗}. By the naturality argument given in the proof of Theorem 10.1 in [BO] it suffices to verify this when X = K(𝔽_{2}, n) where n ≥ 1 and x = ι_{n} is the fundamental class. For n ≥ 2 this is done in [BO] and for n = 1 we proceed as follows. Since E_{∞} = E_{2} = Ω(𝔽_{2}[ι_{1}]) we have E_{∞}^{−p,∗} = 0 for p ≥ 2. Thus the filtration has length two:

H^{n} = F^{−1}H^{n} ⊇ F^{0}H^{n} ⊇ 0 , E_{∞}^{−p,n+p} ≅ F^{−p}H^{n}/F^{−p+1}H^{n}

From the E_{2} page we read that F^{−1}H^{0} ≅ 𝔽_{2} ⊕ 𝔽_{2} and F^{0}H^{0} ≅ 𝔽_{2} so it suffices to show that e(dι_{1}) ≠ 0 and e(dι_{1}) ≠ 1 = (1, 1) ∈ H^{∗}(ΛK(𝔽_{2}, 1)). But this is a consequence of Proposition 5.4.

Defining a compatible filtration of Ω_{λ}(K) one sees that e is an isomorphism as in [BO]. The same kind of Serre spectral sequence argument as given in [BO] shows that ψ is an isomorphism.

6 Detection maps

Let X be a connected space of finite type. In this section we shall see how it is possible to describe the cohomology of the evaluation map ev_{m} : E_{m}ΛX → C_{m}X via the functor ℓ. The cohomology of its target space has a classical interpretation. For a positive integer k and a positively graded 𝔽_{2}-vector space V we let C_{k}(V) denote the group cohomology C_{k}(V) := H^{∗}(C_{k}; V^{⊗k}) where the action on the coefficient module is by cyclic permutation. We put C_{m}(V) := C_{2^{m}}(V) for m ≥ 0. Our reason for choosing the same notation as for cyclic constructions is the following. By classical work of Dold and Steenrod there is an isomorphism of 𝔽_{2}-algebras

Φ : C_{m}(H^{∗}X) → H^{∗}(C_{m}X)  (24)

In [O] we found formulas for an A-action on C_{m}(M) when M is an unstable A-module. This gave a functor C_{m} : U → U such that (24) is an isomorphism of unstable A-modules.

The cohomology of the domain space of ev_{m} is of course unknown in general but the ℓ functor gives an approximation for it. It is easy to see that Theorem 4.2 in [BO] also holds for non connected spaces Y. Hence we have an isomorphism

θ : H^{∗}(BC_{2^{m}}) ⊗_{𝔽_{2}[u]} H^{∗}(E_{∞}ΛX) → H^{∗}(E_{m}ΛX)

for m ≥ 1. Via this we get a morphism of unstable A-algebras

Ψ = θ ∘ (1 ⊗ ψ) : H^{∗}(BC_{2^{m}}) ⊗_{𝔽_{2}[u]} ℓ(H^{∗}X) → H^{∗}(E_{m}ΛX)  (25)

We know that this is an isomorphism when X = K(𝔽_{2}, n) with n ≥ 1 so we can attempt to compute the map ev_{m}^{∗} in these cases. By naturality this would give information on the general case.

We shall use two detection maps, i_{m} and η_{m}, which we now describe. In the following a lower index 0 on a space means the space equipped with the trivial S^{1}-action. Firstly, the constant loop inclusion i : X_{0} → ΛX gives maps

i_{m} := E_{m}i : BC_{2^{m}} × X → E_{m}ΛX

Secondly, the action map η : S^{1} × (ΛX)_{0} → ΛX gives maps

E_{m}η : E_{m}(S^{1} × (ΛX)_{0}) → E_{m}ΛX

The domain space can be rewritten E_{m}(S^{1} × (ΛX)_{0}) ≅ E_{m}(S^{1}) × ΛX and the projection on the second factor pr_{2} : E_{m}(S^{1}) → S^{1}/C_{2^{m}} is a homotopy equivalence. For our purpose it is better with a map in the opposite direction of pr_{2}. We choose a point e ∈ ES^{1} and define s_{0} : S^{1} → ES^{1} × S^{1} by s_{0}(z) = (ze, z). It is an S^{1}-map so we can define

s_{m} = s_{0}/C_{2^{m}} : S^{1}/C_{2}^{m} → E_{m}(S^{1})
s_{0} = s_{0}/S^{1} : ∗ → E_{∞}(S^{1})

Since pr_{2} ∘ s_{m} = id we see that s_{m} induces an isomorphism in cohomology. Our second detection map is the composite

\begin{aligned}η_{m} : S^{1}/C_{2^{m}} × ΛX & ⟶^{s_{m}×1} & E_{m}(S^{1}) × ΛX & ⟶^{≅} & E_{m}(S^{1} × (ΛX)_{0}) \\ & ⟶^{E_{m}η} & E_{m}ΛX \end{aligned} (26)

The following result explains why we call i_{m} and η_{m} detection maps.

Theorem 6.1. Let X be a connected space of finite type and let m ≥ 2 be an integer. There is a commutative digram

\begin{matrix}Λ(v) ⊗ ℓ(H^{∗}X) & ⟶^{1⊗St} & Λ(v) ⊗ 𝔽_{2}[u] ⊗ H^{∗}X \\ ^{1⊗ψ}↓ & & ≅↓ \\ Λ(v) ⊗ H^{∗}(E_{∞}ΛX) & ⟶^{1⊗i_{∞}^{∗}} & Λ(v) ⊗ H^{∗}(BS^{1} × X) \\ ≅↓ & & ≅↓ \\ H^{∗}(E_{m}ΛX) & ⟶^{i_{m}^{∗}} & H^{∗}(BC_{2^{m}} × X) \end{matrix}  (27)

and a commutative diagram

Λ(v) ⊗ ℓ(H^{∗}X) ⟶^{1⊗DR} Λ(v) ⊗ Ω_{λ}(H^{∗}X)
1⊗ψ↓  ^{1⊗e}↓
Λ(v) ⊗ H^{∗}(E_{∞}ΛX) ⟶^{1⊗q_{∞}^{0}} Λ(v) ⊗ H^{∗}(ΛX)
≅↓  ≅↓
H^{∗}(E_{m}ΛX)  ⟶^{η_{m}^{∗}} H^{∗}(S^{1}/C_{2^{m}} × ΛX)

(28)

Thus the map

(i_{m}^{∗}, η_{m}^{∗}) : H^{∗}(E_{m}ΛX) → H^{∗}(BC_{2^{m}} × X) ⊕ H^{∗}(S^{1}/C_{2^{m}} × ΛX)

is injective when X = K(𝔽_{2}, n) with n ≥ 1.

Proof. By the argument given in the proof of Proposition 7.1 of [BO] we see that the lower square in (27) commutes. By Proposition 7.2 of [BO] the upper square in this diagram also commutes (in [BO] we assumed that ΛX was connected, but it is easy to see that this assumption is not necessary).

The composite (26) defining η_{m} sits over the following composite via quotient maps:

∗ × ΛX ⟶^{s_{∞}×1} E_{∞}(S^{1}) × ΛX ⟶^{≅} E_{∞}(S^{1} × (ΛX)_{0}) ⟶^{E_{∞}η} E_{∞}ΛX

which is given by (∗, ω) ↦ [e, ω]. So the top square in the following diagram commutes. Mapping down by projections pr_{1} we get the whole commutative cube:

[figure: S^{1}/C_{2^{m}} × ΛX; η_{m}; E_{m}ΛX; Q_{m}^{∞}; ∗ × ΛX; Q_{0}^{∞}; E_{∞}ΛX; S^{1}/C_{2^{m}}; j; BC_{2^{m}}; ∗; BS^{1}]

Here Q_{m}^{∞} and Q_{0}^{∞} are quotient maps and j is the fiber inclusion of the fibration following it in the cube. From this cube and the definition of θ in Theorem 4.2 of [BO] we see that the lower square of (28) commutes. The top square commutes by Theorem 3.8.

When X = K(𝔽_{2}, n) for n ≥ 1 the vertical maps in (27) and (28) are all isomorphisms so (i_{m}^{∗}, η_{m}^{∗}) is injective by Theorem 4.11.

7 The twisted diagonal map

Let Y be an S^{1}-space with action map η : S^{1} × Y → Y and associated differential d on H^{∗}Y.

Definition 7.1. For m ≥ 0 we define the map

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

where ξ_{m} = exp(2πi/2^{m}) and e is a point in ES^{1}. We let C_{2^{m}} act on the domain space by ξ_{m} · (z, y) = (ξ_{m}z, y) and on the target space by

ξ_{m} · (e, y_{1}, …, y_{2^{m}}) = (ξ_{m}e, y_{2}, …, y_{2^{m}}, y_{1})

Then the above map is C_{2^{m}}-equivariant. Passing to the quotients we get a map

f_{Y,m} : S^{1}/C_{2^{m}} × Y → C_{m}Y

which we call the twisted diagonal of order 2^{m}.

Remark 7.2. f_{Y,m} is natural in Y with respect to C_{2^{m}}-equivariant maps. Remark 7.3. Our reason for introducing the twisted diagonal map is that it can be used to compute η_{m}^{∗} ∘ ev_{m}^{∗} since there is a commutative diagram

S^{1}/C_{2^{m}} × ΛX ⟶^{η_{m}} E_{m}ΛX
f_{ΛX,m}↓  ^{ev_{m}}↓
E_{m}ΛX  ⟶^{E_{m}(ev_{0})} C_{m}X

Lemma 7.4. Let X be a space with trivial S^{1}-action and m ≥ 1 an integer. The map f_{X,m}^{∗} : H^{∗}(C_{m}X) → H^{∗}(S^{1} × X) satisfies

f_{X,m}(1 ⊗ x^{⊗2^{m}}) = 1 ⊗ x^{2^{m}} + v ⊗ x^{2^{m}−2}λx

Proof. There is a factorization

f_{X,m} : S^{1}/C_{2^{m}} × X ⟶^{i×1} ES^{1}/C_{2^{m}} × X ⟶^{1×Δ_{m}} C_{m}X

where i : S^{1} → ES^{1} is the inclusion associated to e and Δ_{m} is the diagonal Δ_{m}(y) = (y, …, y). When m = 1 the result follows by Steenrod’s formula for the diagonal. For m ≥ 2 we use Theorem 6.7 of [O] where the map (1×Δ_{m})^{∗} was determined. We find

f_{X,m}^{∗}(1 ⊗ x^{⊗2^{m}}) = 1 ⊗ x^{2^{m}} + v ⊗ Q_{m−1}^{2^{m−1}|x|−1}(x)

The operation Q_{n}^{i} is defined in Definition 5.5 of [O]. We have Q_{1}^{2|x|−1}(x) = x^{2}λx giving Q_{n}^{2^{n}|x|−1}(x) = x^{2^{n+1}−2}λx in general.

Proposition 7.5. There is a commutative diagram

H^{∗}(C_{m}Y)  ⟶^{f_{Y,m}^{∗}}  H^{∗}(S^{1}/C_{2^{m}} × Y)
τ_{m−1}^{m}↑  ^{τ_{m−1}^{m}⊗1}↑
H^{∗}(C_{m−1}(Y^{2})) ⟶^{(f′_{Y,m}/C_{2^{m−1}})^{∗}} H^{∗}(S^{1}/C_{2^{m−1}} × Y)
C_{m−1}(Δ_{1})^{∗}↓  ∥
H^{∗}(C_{m−1}Y)  ⟶^{f_{Y,m−1}^{∗}} H^{∗}(S^{1}/C_{2^{m−1}} × Y)

where Δ_{1} : Y → Y × Y denotes the diagonal Δ_{1}(y) = (y, y). Especially we have f_{Y,m}^{∗}(1 ⊗ Na_{1} ⊗ ⋯ ⊗ a_{2^{m}}) = v ⊗ d(a_{1} … a_{2^{m}}).

Proof. It is obvious that the upper square commutes. The lower square is induced by a homotopy commutative diagram of spaces. The norm class equals τ_{0}^{m}(1 ⊗ a_{1} ⊗ ⋯ ⊗ a_{2^{m}}) so we can find its image under f_{Y,m}^{∗} by use of the diagram. From the factorization

f_{Y,0} : S^{1} × Y ⟶^{Δ_{1}⊗1} S^{1} × S^{1} × Y ⟶^{i×η} ES^{1} × Y

we see that f_{Y,0}^{∗}(1 ⊗ y) = 1 ⊗ y + v ⊗ dy and the result follows.

Lemma 7.6. For m ≥ 1 the map f_{S^{1},m}^{∗} : H^{∗}(C_{m}S^{1}) → H^{∗}(S^{1}/C_{2^{m}} × S^{1}) is zero on all classes of positive degrees except for 1 ⊗ Nv ⊗ 1 ⊗ ⋯ ⊗ 1 which is mapped to v ⊗ 1.

Proof. When m = 1 this is a special case of Lemma 4.11 and Theorem 4.12 of [BO]. The differential on H^{∗}(S^{1}) is given by dv = 1 since η^{∗}(v) = 1⊗v+v⊗1.

Assume that m ≥ 2. Since the target space is Λ(v) ⊗ Λ(v) all classes of higher degree than two is mapped to zero. The degree two class

1 ⊗ (1 + T + ⋯ + T^{2^{m−1}−1})(v ⊗ 1^{⊗2^{m−1}−1})^{⊗2} = τ_{1}^{m}(1 ⊗ (v ⊗ 1^{⊗2^{m−1}−1})^{⊗2})

is mapped to zero by Proposition 7.5 and the m = 1 case. All other elements of degree at most two are norm elements and Proposition 7.5 takes care of these.

We now introduce an expansion formula which will be used in the proof of the next theorem and again later on. Let M be an 𝔽_{2}-vector space and k ≥ 1 an integer. The cyclic group C_{k} act on M^{⊗k} by cyclic permutation of the factors. For x, y ∈ M we let T_{k}(x, y) denote the subset of M^{⊗k} consisting of all strings a_{1} ⊗ ⋯ ⊗ a_{k} with a_{i} = x or a_{i} = y for each i. Note that T_{k}(x, y) is stable under the C_{k} action and that (x + y)^{⊗k} is the sum of the elements in T_{k}(x, y). For an orbit β ∈ T_{k}(x, y)/C_{k} we let |β| denote the length of β and sβ the sum of the elements in β. Thus we have the following expansion formula among C_{k}-invariants in M^{⊗k}:

(x + y)^{⊗k} = ∑ sβ  (29)

where the summation is taken over β ∈ T_{k}(x, y)/C_{k}.

Theorem 7.7. For m ≥ 1 we have

f_{Y,m}^{∗}(1 ⊗ y^{⊗2^{m}}) = 1 ⊗ y^{2^{m}} + v ⊗ y^{2^{m}−2}(ydy + λy)

Proof. Let Y_{0} denote the space Y with trivial S^{1}-action. The action map η : S^{1} × Y_{0} → Y is S^{1}-equivariant so by naturality there is a commutative diagram

S^{1}/C_{2^{m}} × S^{1} × Y_{0} ⟶^{f_{S^{1}×Y_{0},m}} C_{m}(S^{1} × Y_{0})
1×η↓  ^{C_{m}(η)}↓
S^{1}/C_{2^{m}} × Y  ⟶^{f_{Y,m}}  C_{m}(Y)

The map γ : Y → S^{1} ×Y defined by γ(y) = (1, y) has the property η ∘γ = id thus

f_{Y,m} = C_{m}(η) ∘ f_{S^{1}×Y_{0},m} ∘ (1 × γ)

We pull back step by step according to this factorization.

C_{m}(η)^{∗}(1 ⊗ y^{⊗2^{m}}) = 1 ⊗ (1 ⊗ y + v ⊗ dy)^{⊗2^{m}} = ∑ 1 ⊗ sβ

where the summation is taken over β ∈ T_{2^{m}}(1 ⊗ y, v ⊗ dy). Consider an orbit β with |β| = 2^{i} where 1 ≤ i ≤ m. A representative for this has the form \overline{β} = (a_{1} ⊗ ⋯ ⊗ a_{2^{i}})^{⊗2^{m−i}}. By Proposition 7.5 we have

f_{S^{1}×Y_{0},m}^{∗}(1 ⊗ sβ) = (τ_{m−i}^{m} ⊗ 1) ∘ f_{S^{1}×Y_{0},m−i}^{∗}(1 ⊗ (a_{1} … a_{2^{i}})^{⊗2^{m−i}}) (30)

Since v is an exterior class we see that (30) is zero when two or more of the elements a_{1}, …, a_{2^{i}} equals v ⊗ dy. If there is exactly one of these we have a_{1} … a_{2^{i}} = v ⊗ y^{2^{i}−1}dy and (30) equals zero unless i = m by Lemma 7.6. When i = m we have a norm element so Proposition 7.5 gives

f_{S^{1}×Y_{0},m}^{∗}(1 ⊗ sβ) = v ⊗ d_{S^{1}×Y_{0}}(v ⊗ y^{2^{m}−1}dy)
= v ⊗ d_{S^{1}}(v) ⊗ y^{2^{m}−1}dy + v ⊗ v ⊗ d_{Y_{0}}(y^{2^{m}−1}dy)
= v ⊗ 1 ⊗ y^{2^{m}−1}dy

We have now taken care of all the terms except of 1 ⊗ (1 ⊗ y)^{⊗2^{m}} and 1 ⊗ (v ⊗ dy)^{⊗2^{m}} which corresponds to orbits of length one. By Lemma 7.4 the first is mapped as follows

f_{S^{1}×Y_{0},m}^{∗}(1 ⊗ (1 ⊗ y)^{⊗2^{m}}) = 1 ⊗ 1 ⊗ y^{⊗2^{m}} + v ⊗ 1 ⊗ y^{2^{m}−2}λy

and Lemma 7.6 gives that the second is mapped to zero. We conclude that

f_{S^{1}×Y_{0},m}^{∗} ∘ C_{m}(η)^{∗}(1 ⊗ y^{⊗2^{m}}) = 1 ⊗ 1 ⊗ y^{⊗2^{m}} + v ⊗ 1 ⊗ y^{2^{m}−2}(ydy + λy)

The result follows by applying 1 ⊗ γ^{∗}.

Corollary 7.8. For m ≥ 2 one has

η_{m}^{∗} ∘ ev_{m}^{∗}(1 ⊗ x^{⊗2^{m}}) = 1 ⊗ e(x^{2^{m}}) + v ⊗ e(x^{2^{m}−2}(xdx + λx))

Proof. Follows directly from Theorem 7.7 and Remark 7.3.

8 Formulas for the evaluation maps

In this section we find a formula for ev_{m}^{∗}(1 ⊗ x^{⊗2^{m}}) in terms of the map (25). This is done by the two detection maps η_{m} and i_{m}. Note that i_{m} ∘ ev_{m} = Δ_{m} where Δ_{m} = E_{m}(D_{m}) with D_{m} : X → X^{2^{m}} the diagonal map. We computed Δ_{m}^{∗} in [O]. The result involved some notation which we now introduce. In this section K always denotes an unstable A-algebra.

Definition 8.1. For integers k ≥ 1 and n ≥ 0 we define λ_{k} : K → K and Q_{k}^{n} : K → K by

λ_{k}(x) = x^{2^{k}−2}λx  (31)
Q_{1}^{n}(x) = ∑_{i=0}^{[ \frac{n}{2} ]} Sq^{i}(x)Sq^{n−i}(x)  (32)

Q_{k}^{n}(x) = ∑_{i=0}^{n} Sq^{i}(x^{2^{k}−2})Q_{1}^{n−i}(x) , k ≥ 2  (33)

We put Q_{k}^{n} = 0 when n ≤ −1.

Definition 8.2. For integers k ≥ 1 we define three maps K → 𝔽_{2}[u] ⊗ K as follows

F_{k+1}(x) = ∑_{i=0}^{∞} u^{2^{k−1}(|x|−i)} ⊗ (Sq^{i}x)^{2^{k}}  (34)

G′_{k+1}(x) = ∑_{j=0}^{∞} u^{2^{k−1}|x|−j} ⊗ Sq^{2j}(λ_{k}x)  (35)

G′′_{k+1}(x) = ∑_{j=0}^{∞} u^{2^{k−1}|x|−j} ⊗ Q_{k}^{2j−1}(x)  (36)

Note that the sums are finite. F_{m} from Definition 6.5 of [O] is the same as above and G_{m} from Definition 6.5 of [O] equals the sum G′_{m} + G′′_{m}. Thus by Theorem 6.7 of [O] we have

Δ_{m}^{∗}(1 ⊗ x^{⊗2^{m}}) = F_{m}(x) + v(G′_{m}(x) + G′′_{m}(x))

for m ≥ 2.

Proposition 8.3. For x ∈ K and k ≥ 1 the following equations hold.

St(ϕ(x^{2^{k}})) = F_{k+1}(x)  (37)
St(uq(λ_{k}x)) = G′_{k+1}(x)  (38)
St(ϕ(x^{2^{k}−1})q(x)) = G′′_{k+1}(x)  (39)

Proof. Equation (37) and (38) follows easily from the definition of St in Proposition 4.6. We first prove (39) for k = 1. Since St is a ring homomorphism we have that

St(ϕ(x)q(x)) = ∑_{j=0}^{∞} u^{j} ⊗ ∑_{r+s=j} Sq_{2r}(x)Sq_{2s+1}(x)

So we must show that

Q_{1}^{2|x|−2j−1}(x) = ∑_{r+s=j} Sq_{2r}(x)Sq_{2s+1}(x)  (40)

We can rewrite the left hand side using the definition followed by the substitution t = |x| − i.

Q_{1}^{2|x|−2j−1}(x) = ∑_{t=0}^{|x|−j−1} Sq^{t}(x)Sq^{2|x|−2j−1−t}(x)

= ∑_{i=j+1}^{|x|} Sq_{i}(x)Sq_{2j+1−i}(x)

= ∑_{i=j+1}^{2j+1} Sq_{i}(x)Sq_{2j+1−i}(x)  (41)

The map, f_{j} : {0, 1, …, j} → {j + 1, j + 2, …, 2j + 1} defined by f_{j}(r) = 2(j − r) + 1 for r ≤ [j/2] and f_{j}(r) = 2r otherwise, is a bijection from the set of summation indexes of (40) to the set of summation indexes of (41). Since the r’th term of the right hand side of (40) equals the i = f_{j}(r)’th term of (41) the result follows.

Finally we prove (39) for k ≥ 2. By (9) we have q(x^{2^{k}−2}) = 0 such that ϕ(x^{2^{k}−1})q(x) = ϕ(x^{2^{k}−2})ϕ(x)q(x) by (8). Thus we can apply the k = 1 case as follows:

St(ϕ(x^{2^{k}−1})q(x)) = St(ϕ(x^{2^{k}−2}))St(ϕ(x)q(x))

= ∑_{j=0}^{∞} u^{j} ⊗ ∑_{r+s=j} Sq_{2r}(x^{2^{k}−2})Q_{1}^{2|x|−2s−1}(x)

and we must prove the following equation for each j:

Q_{k}^{2^{k}|x|−2j−1}(x) = ∑_{r+s=j} Sq_{2r}(x^{2^{k}−2})Q_{1}^{2|x|−2s−1}(x)  (42)

By its definition and the fact that Sq^{i}(y^{2}) = 0 we can rewrite the left hand side as

Q_{k}^{2^{k}|x|−2j−1}(x) = ∑_{t=0}^{∞} Sq^{2t}(x^{2^{k}−2})Q_{1}^{2^{k}|x|−2(j+t)−1}(x)

We substitute t = (2^{k−1} − 1)|x| − r and get that

Q_{k}^{2^{k}|x|−2j−1}(x) = ∑_{r=−∞}^{(2^{k−1}−1)|x|} Sq_{2r}(x^{2^{k}−2})Q_{1}^{2|x|−2(j−r)−1}(x)

If r < 0 the first factor in each summand is zero so we can start the summation at r = 0. If r > j the second factor in each summand is zero and if r > (2^{k−1} − 1)|x| the first factor in each summand is zero. So we can end the summation at j obtaining the formula (42).

Theorem 8.4. Let X be a connected space of finite type and let x, y ∈ H^{∗}X. For integers m ≥ 2 the following equation holds

ev_{m}^{∗}(1 ⊗ x^{⊗2^{m}}) = Ψ(ϕ(x^{2^{m−1}}) + vϕ(x^{2^{m−1}−1})q(x) + vuq(x^{2^{m−1}−2}λx))

For m = 1 we have

ev_{1}^{∗}(1 ⊗ x^{⊗2}) = Ψ(ϕ(x) + tq(x)) and ev_{1}^{∗}(1 ⊗ (1 + T)x ⊗ y) = Ψ(tδ(xy))

Further the square with transfer maps, the square with quotient maps and the triangle in the following diagram commutes.

\xymatrix{H^{∗}(C_{m}X) \ar[r]^{ev_{m}^{∗}} \ar[d]^{q_{m}^{m−1}} & H^{∗}(E_{m}ΛX) \ar[d]^{q_{m}^{m−1}} \\ H^{∗}(C_{m−1}(X^{2})) \ar[u]^{τ_{m−1}^{m}} \ar[d]_{Δ_{1}^{∗}} \ar[r] & H^{∗}(E_{m−1}ΛX) \ar[u]^{τ_{m−1}^{m}} \\ H^{∗}(C_{m−1}X) \ar@{-->}[ur]_{ev_{m−1}^{∗}}}

where Δ_{1} : X → X × X denotes the diagonal.

Proof. By naturality it is enough to prove the formula for ev_{m}^{∗} when X = K(𝔽_{2}, n) with n ≥ 1 and x = ι_{n} is the fundamental class. By Theorem 6.1 it suffices to check that the right hand side maps correctly by η_{m}^{∗} and i_{m}^{∗}. By Theorem 6.7 of [O] and Proposition 8.3 it does by i_{m}^{∗}. For η_{m}^{∗} we find

1 ⊗ DR(ϕ(x^{2^{m−1}}) + vϕ(x^{2^{m−1}−1})q(x) + vuq(x^{2^{m−1}−2}λx)) =
x^{2^{m}} + vx^{2^{m}−2}(xdx + λx)

which is the correct result by Corollary 7.8. The formulas for m = 1 follows from Lemma 6.4 and Proposition 6.5 of [BO].

The squares in the diagram commute since there is a corresponding commutative diagram of spaces with evaluation and quotient maps. The triangle comes from a homotopy commutative diagram of spaces.

Remark 8.5. The map ev_{m}^{∗} is H^{∗}(BC_{2^{m}})-linear and elements in its domain which are not of highest symmetry are hit by the transfer. So we have given a complete description of ev_{m}^{∗} in terms of the approximation Ψ.

9 Algebraic evaluation maps

In this section we show that the formulas for the map ev_{m}^{∗} also define a ring map when we replace H^{∗}X by a general FGA. This allows us to state our main theorem in a nice form.

Recall that the deviation from linearity of a set map F : A → B between 𝔽_{2}-algebras is defined as ΔF (x, y) = F (x + y) + F (x) + F (y). If L : A → B is linear and we let G(x) = L(x)F (x) we have

ΔG(x, y) = L(x)F (y) + L(y)F (x) + (L(x) + L(y))ΔF (x, y) (43)

Definition 9.1. Let K be an FGA. The maps f_{i}, g_{i}, h_{i} : K → ℓ(K) for integer i ≥ 0 are defined by f_{0}(x) = f_{1}(x) = 0, g_{0}(x) = δ(x) and

f_{i}(x) = uq(x^{2^{i−1}−2}λx) for i ≥ 2
g_{i}(x) = ϕ(x^{2^{i−1}−1})q(x) for i ≥ 1
h_{i}(x) = f_{i}(x) + g_{i}(x) for i ≥ 0

Lemma 9.2. For w = f, g, h and m ≥ 1 the following formulas are valid.

w_{m}(ab) = ϕ(a^{2^{m−1}})w_{m}(b) + ϕ(b^{2^{m−1}})w_{m}(a)  (44)
Δw_{m}(x, y) = ∑_{i=0}^{m−1} ∑_{r+s=2^{m−i}} w_{i}(x^{r}y^{s})  (45)

Proof. It is of course enough to prove this for w = f and w = g. We first prove (44). For m = 1 and w = f it is trivial. For m = 1 and w = g it holds by (9). The m = 2 case requires a little computation. For w = f we have

f_{2}(ab) = uq(λ(ab)) = uq(a^{2}λb + b^{2}λa)
= u(ϕ(a^{2})q(λb) + ϕ(b^{2})q(λa)) = ϕ(a^{2})f_{2}(b) + ϕ(b^{2})f_{2}(a)

and for w = g we find

g_{2}(xy) = (ϕ(x)ϕ(y) + uq(x)q(y))(ϕ(x)q(y) + q(x)ϕ(y))
= ϕ(x)^{2}g_{2}(y) + ϕ(y)^{2}g_{2}(x) + uq(y)^{2}g_{2}(x) + uq(x)^{2}g_{2}(y)
= (ϕ(x)^{2} + uq(x)^{2})g_{2}(y) + (ϕ(y)^{2} + uq(y)^{2})g_{2}(x)
= ϕ(x^{2})g_{2}(y) + ϕ(y^{2})g_{2}(x)

Note that when m ≥ 3 the following formula holds for both w = f and w = g.

w_{m}(x) = ϕ(x^{2^{m−1}−2})w_{2}(x)  (46)

So for these m we have

w_{m}(ab) = ϕ((ab)^{2^{m−1}−2})w_{2}(ab)
= ϕ(a^{2^{m−1}−2})ϕ(b^{2^{m−1}−2})(ϕ(a^{2})w_{2}(b) + ϕ(b^{2})w_{2}(a))
= ϕ(a^{2^{m−1}})w_{m}(b) + ϕ(b^{2^{m−1}})w_{m}(a)

Next we prove (45). From (44) and δ(a^{2}) = 0 we see that w_{i}(a^{2}) = 0 for any i ≥ 0. Thus the terms with both r and s even are zero. For m = 1 we have f_{1} = 0 and g_{1} = q such that Δg_{1}(x, y) = δ(xy) = g_{0}(xy). So here (45) holds. For m = 2 we have f_{2}(x) = uq(λx) which is linear. g_{2}(x) = ϕ(x)q(x) so by (43) and (44) we have

Δg_{2}(x, y) = g_{1}(xy) + (ϕ(x) + ϕ(y))δ(xy) = g_{1}(xy) + δ(xy^{3}) + δ(x^{3}y)

Thus (45) also holds for m = 2.

For m ≥ 3 we have w_{m}(x) = ϕ(x^{2^{m−2}})w_{m−1}(x) by (46) so (43) and (44) gives

Δw_{m}(x, y) = w_{m−1}(xy) + (ϕ(x^{2^{m−2}}) + ϕ(y^{2^{m−2}}))Δw_{m−1}(x, y)

Assume that Δw_{m−1}(x, y) is given by (45). By (44) we have ϕ(a^{2^{m−2}})w_{i}(b) = w_{i}(a^{2^{m−1−i}}b) for 1 ≤ i ≤ m − 2. This formula also holds for i = 0 trivially for w = f and by (12) for w = g. So we have

Δw_{m}(x, y) = w_{m−1}(xy) + ∑_{i=0}^{m−2} ∑_{r+s=2^{m−1−i}} (w_{i}(x^{2^{m−1−i}+r}y^{s}) + w_{i}(x^{r}y^{2^{m−1−i}+s}))

Since w_{i}(a^{2}) = 0 for any i ≥ 0 this sum equals the right hand side of (45).

The cyclic functor C_{2^{m}} : U → U was introduced in Theorem 4.8 of [O]. We denote it C_{m} = C_{2^{m}}. Recall that there are transfer and restriction maps as follows

τ_{m−1}^{m} : C_{m−1}(M ⊗ M) → C_{m}(M) and q_{m}^{m−1} : C_{m}(M) → C_{m−1}(M ⊗ M)

where M ∈ ob(U). Formulas for these were given in section 3 of [O].

Proposition 9.3. For each m ≥ 1 the functor C_{m} : U → U restricts to a functor C_{m} : K → K.

Proof. Let K be an unstable A-algebra. We must verify that the Cartan formula and the relation Sq^{|x|}x = x^{2} hold in C_{m}(K). This is done by induction on m. For m = 1 the result is well known. Assume that it holds for m−1. Let τ = τ_{m−1}^{m} and q = q_{m}^{m−1}. We know that these are A-linear. Using Frobenius reciprocity we can then check the Cartan formula when one of the factors in the product is hit by the transfer:

Sq^{n}(aτ(b)) = Sq^{n}(τ(q(a)b)) = τ(Sq^{n}(q(a)b))

= τ( ∑_{i+j=n} Sq^{i}(q(a))Sq^{j}(b)) = ∑_{i+j=n} τ(q(Sq^{i}(a))Sq^{j}(b))

= ∑_{i+j=n} Sq^{i}(a)τ(Sq^{j}(b)) = ∑_{i+j=n} Sq^{i}(a)Sq^{j}(τ(b))

Hence it suffices to verify the Cartan formula for a product of two highest symmetry classes. For x, y ∈ K with |x| = r and |y| = s there is a morphism of unstable A-algebras f : H^{∗}(K(𝔽_{2}, r)×K(𝔽_{2}, s)) → K given by f(ι_{r} ⊗1) = x and f(1 ⊗ ι_{s}) = y. If we apply C_{2^{m}} to the cohomology of a space we know that this is isomorphic to the cohomology of the cyclic construction of order 2^{m} on the space. So here the Cartan formula holds. So by the map f we see that it also holds for the product of v^{i}u^{j} ⊗x^{⊗2^{m}} and v^{k}u^{n}⊗y^{⊗2^{m}}. The relation Sq^{|x|}x = x^{2} is easy to verify directly from the formulas for the A-action.

We can now give the final description of the maps induced by the equivariant evaluations in mod two cohomology.

Theorem 9.4. Let K be an FGA with multiplication μ : K⊗K → K. There is a natural H^{∗}(C_{2^{m}})-linear ring homomorphism

Ev_{m} : C_{m}(K) → H^{∗}(C_{2^{m}}) ⊗_{𝔽_{2}[u]} ℓ(K)

for each m ≥ 1. When m = 1 it is defined by 1 ⊗ x^{⊗2} ↦ ϕ(x) + tq(x) and 1 ⊗ (1 + T)a ⊗ b ↦ tδ(ab) where t^{2} = u. For m ≥ 2 it is defined by the following two conditions

Ev_{m}(1 ⊗ x^{⊗2^{m}}) = ϕ(x^{2^{m−1}}) + vϕ(x^{2^{m−1}−1})q(x) + vuq(x^{2^{m−1}−2}λx) (47)
Ev_{m} ∘ τ_{m−1}^{m} = (τ_{m−1}^{m} ⊗ 1) ∘ Ev_{m−1} ∘ μ_{∗}  (48)

where the second condition relates maps in the following diagram.

\xymatrix{C_{m}(K) \ar[r]^{Ev_{m}} \ar[d]^{q_{m}^{m−1}} & H^{∗}(C_{2^{m}}) ⊗_{𝔽_{2}[u]} ℓ(K) \ar[d]^{q_{m}^{m−1}⊗1} \\ C_{m−1}(K^{⊗2}) \ar[u]^{τ_{m−1}^{m}} \ar[d]_{μ_{∗}} \ar[r] & H^{∗}(C_{2^{m−1}}) ⊗_{𝔽_{2}[u]} ℓ(K) \ar[u]^{τ_{m−1}^{m}⊗1} \\ C_{m−1}(K) \ar@{-->}[ur]_{Ev_{m−1}}}

The following equation holds for m ≥ 2:

(q_{m}^{m−1} ⊗ 1) ∘ Ev_{m} = Ev_{m−1} ∘ μ_{∗} ∘ q_{m}^{m−1}  (49)

If K is a connected unstable A-algebra then Ev_{m} is a morphisms in K for each m ≥ 1. Finally if K = H^{∗}X for a connected space X of finite type then we have a commutative diagram in K as follows.

H^{∗}(C_{m}X) ⟶^{ev_{m}^{∗}}  H^{∗}(E_{m}ΛX)
≅↑  ^{Ψ}↑
C_{m}(H^{∗}X) ⟶^{Ev_{m}} H^{∗}(C_{2^{m}}) ⊗_{𝔽_{2}[u]} ℓ(H^{∗}X)

Proof. We first check that Ev_{1} is well defined. ϕ(x) and uq(x) are linear in x and δ(ab) is bilinear in (a, b). From this it follows directly that most relations in H^{∗}(C_{2}; K^{⊗2}) are respected. In fact it suffices to verify that the following type is respected.

1 ⊗ (x + y)^{⊗2} = 1 ⊗ x^{⊗2} + 1 ⊗ y^{⊗2} + 1 ⊗ (1 + T)x ⊗ y

But since ϕ is linear this means that tq(x + y) = tq(x) + tq(y) + tδ(xy) which is true by (6).

Next we check that Ev_{1} is a ring homomorphism. The following relations must be respected:

(1 ⊗ x^{⊗2})(1 ⊗ y^{⊗2}) = 1 ⊗ (xy)^{⊗2}
(1 ⊗ x^{⊗2})(1 ⊗ (1 + T)a ⊗ b) = 1 ⊗ (1 + T)xa ⊗ xb
(1 ⊗ (1 + T)a ⊗ b)(1 ⊗ (1 + T)c ⊗ d) = 1 ⊗ (1 + T)ac ⊗ bd
+ 1 ⊗ (1 + T)ad ⊗ bc

The third is respected since both sides maps to zero. For the first and the second we must check that

(ϕ(x) + tq(x))(ϕ(y) + tq(y)) = ϕ(xy) + tq(xy)
(ϕ(x) + tq(x))tδ(ab) = tδ(x^{2}ab)

which follows by (8), (9) and (12), (14).

For the higher evaluation maps we use induction on m. Assume that the maps Ev_{1}, …, Ev_{m−1} have the stated properties. We first verify that Ev_{m} is well defined. All classes in its domain are hit by the transfer except for those of highest symmetry. So by induction it suffices to check that the defining conditions for Ev_{m} respect the relation coming from the expansion (29)

1 ⊗ (x + y)^{⊗2^{m}} = ∑ 1 ⊗ sβ

where the summation is taken over β ∈ T_{2^{m}}(x, y)/C_{2^{m}}. We rewrite this formula as

1 ⊗ (x + y)^{⊗2^{m}} + 1 ⊗ x^{⊗2^{m}} + 1 ⊗ y^{⊗2^{m}} = ∑_{k=1}^{m} σ_{k} where σ_{k} = ∑_{|β|=2^{k}} 1 ⊗ sβ (50)

When we apply Ev_{m} to the left hand side we get vΔh_{m}(x, y) by (47). We must check that this equals Ev_{m} applied to the right hand side where Ev_{m} was defined by (48). Thus we must find Ev_{m}(σ_{k}) where 1 ≤ k ≤ m. Let β be an orbit of length 2^{k}. It has a representative of the form (a_{1} ⊗⋯⊗a_{2^{k}})^{⊗2^{m−k}}. Let t(β) denote the order of the set {a_{i} | 1 ≤ i ≤ 2^{k} and a_{i} = x}. By (48) we have

Ev_{m}(1 ⊗ sβ) = vh_{m−k}(a_{1} … a_{2^{k}}) = vh_{m−k}(x^{t(β)}y^{2^{k}−t(β)})  (51)

If t(β) is even Ev_{m}(1 ⊗ sβ) = 0 since h_{i}(a^{2}) = 0 for all i ≥ 0. Let N(2^{k}, n) denote order of the set {β | |β| = 2^{k} and t(β) = n}. A string a_{1} ⊗ ⋯ ⊗ a_{2^{k}} with an odd number of a_{i}’s equal to x cannot be broken into two equal halfs so we have

N(2^{k}, 2j + 1) = \frac{1}{2^{k}} \binom{2^{k}}{2j + 1} = \frac{(2^{k} − 1)(2^{k} − 2) … (2^{k} − 2j)}{(2j + 1)!}

The 2-adic valuation of this number is

v_{2}(N(2^{k}, 2j + 1)) =v_{2}(2^{k} − 2) + v_{2}(2^{k} − 4) + ⋯ + v_{2}(2^{k} − 2j)
− v_{2}(2) − v_{2}(4) − ⋯ − v_{2}(2j) = 0

so N(2^{k}, 2j + 1) is odd and we conclude that

Ev_{m}(σ_{k}) = ∑_{j=0}^{2^{k−1}−1} vh_{m−k}(x^{2j+1}y^{2^{k}−2j−1}) = ∑_{r+s=2^{k}} vh_{m−k}(x^{r}y^{s})

Taking the sum over 1 ≤ k ≤ m and substituting i = m − k we obtain

Ev_{m}(∑_{k=1}^{m} σ_{k}) = ∑_{i=0}^{m−1} ∑_{r+s=2^{m−i}} vh_{i}(x^{r}y^{s})

By (45) this equals vΔh_{m}(x, y) so the relation (50) is respected and Ev_{m} is well defined.

Next we verify (49). For highest symmetry element we have

Ev_{m−1} ∘ μ_{∗} ∘ q_{m}^{m−1}(1 ⊗ x^{⊗2^{m}}) = Ev_{m−1} ∘ μ_{∗}(1 ⊗ (x ⊗ x)^{⊗2^{m−1}})
= Ev_{m−1}(1 ⊗ (x^{2})^{⊗2^{m−1}}) = ϕ(x^{2^{m−1}})
= (q_{m}^{m−1} ⊗ 1) ∘ Ev_{m}(1 ⊗ x^{⊗2^{m}})

For elements hit by the transfer the result follows since μ_{∗} ∘ q_{m}^{m−1} ∘ τ_{m−1}^{m} = 0 and (q_{m}^{m−1} ⊗ 1) ∘ (τ_{m}^{m−1} ⊗ 1) = 0.

We can now use Frobenius reciprocity and induction to show that Ev_{m} is a ring map. Assume that Ev_{m−1} is a ring map. We have

Ev_{m}(a · τ_{m−1}^{m}(b)) = Ev_{m}(τ_{m−1}^{m}(q_{m}^{m−1}(a) · b))
= (τ_{m−1}^{m} ⊗ 1) ∘ Ev_{m−1} ∘ μ_{∗}(q_{m}^{m−1}(a) · b)

= (τ_{m−1}^{m} ⊗ 1)(Ev_{m−1} ∘ μ_{∗} ∘ q_{m}^{m−1}(a) · Ev_{m−1} ∘ μ_{∗}(b))

= (τ_{m−1}^{m} ⊗ 1)((q_{m}^{m−1} ⊗ 1) ∘ Ev_{m}(a) · Ev_{m−1} ∘ μ_{∗}(b))
= Ev_{m}(a) · (τ_{m−1}^{m} ⊗ 1) ∘ Ev_{m−1} ∘ μ_{∗}(b)
= Ev_{m}(a) · Ev_{m}(τ_{m−1}^{m}(b))

Hence it suffices to check the following relation

Ev_{m}(1 ⊗ x^{⊗2^{m}}) · Ev_{m}(1 ⊗ y^{⊗2^{m}}) = Ev_{m}(1 ⊗ (xy)^{⊗2^{m}})

which is valid by (45).

The statement for K = H^{∗}X follows from Theorem 8.4 and Proposition 2.3 of [O].

References

[B] A. K. Bousfield, On the homology spectral sequence of a cosimplicial space, Amer. J. Math. 109 (1987), 361-394.

[BHM] M. Bökstedt, W. C. Hsiang, I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), no. 3, 465-539.

[BO] M. Bökstedt & I. Ottosen, Homotopy orbits of free loop spaces, Fund. Math. 162 (1999), 251-275.

[HLS] H.-W. Henn, J. Lannes, L. Schwartz, Localization of unstable A-modules and equivariant mod-p cohomology, Math. Ann. 301 (1995), 23-68.

[M] R. J. Milgram Unstable Homotopy from the Stable Point of View Springer Lecture Notes in Mathematics, Vol 368.

[O] I. Ottosen, Higher cyclic reduced powers, to appear in Journal of Pure and Applied Algebra.

[S] L. Smith, On the characteristic zero cohomology of the free loop space, Amer. J. Math. 103 (1981), 887-910

Iver Ottosen Fakultät für Mathematik Universität Bielefeld Postfach 100131 33501 Bielefeld, Germany Email: ottosen@Mathematik.Uni-Bielefeld.DE