Homotopy orbits of free loop spaces

Marcel Bökstedt & Iver Ottosen ^{∗}

March 9, 1999

Abstract Let X be a space with free loop space ΛX and mod two cohomology R = H^{∗}X. We construct functors Ω_{λ}(R) and ℓ(R) together with algebra homomorphisms e : Ω_{λ}(R) → H^{∗}(ΛX) and ψ : ℓ(R) → H^{∗}(ES^{1} ×_{S^{1}} ΛX). When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

1 Preface

The purpose of this paper is to present a new approach to the cohomology of the S^{1} homotopy orbits of the free loop space ΛX.

Let X be a space, and C_{n} the cyclic group considered as acting on X^{n} by permuting coordinates. The (ℤ/2-) cohomology of EC_{n} ×_{C_{n}} X^{n} can be computed from R = H^{∗}(X; ℤ/2) by a Serre spectral sequence. It is a fundamental fact, derived from the equivariance of the Eilenberg-Zilber map, that this spectral sequence collapses for all spaces X and all n. In particular this cohomology is a functor in R, and does not depend in any other way on the topology of X. The second author has investigated this situation closely in [Ottosen].

We begin the study of the cohomology of the orbits with some preliminaries and generalities about spaces with an S^{1} action in §3 and §4.

We continue in §6 by using the S^{1} transfer to construct maps connecting the cohomology of X respectively EC_{2}×_{C_{2}}X^{2} with ES^{1}×_{S^{1}}ΛX. These maps are not multiplicative, but they do satisfy a version of Frobenius reciprocity. This forces the images of the maps to satisfy universal relations, some of these involving multiplications.

∗We would like to thank the University of Oslo where the second author did part of the work. He was supported by the European Union network: K-theory and Algebraic Groups. 1991 Mathematics Subject Classification: 55N91, 55P35, 55R12.

In this way, taking sums of products of the cohomology classes and taking the universal relations into account, we obtain a group ℓ(R), which is again a functor of R. This group maps by its definition to H^{∗}(ES^{1} ×_{S^{1}} ΛX; ℤ/2), and we think of the map as an approximation.

Via relations to the Steenrod diagonal, the approximation is shown to be nontrivial in many cases in §7.

Surprisingly, in §8 it turns out that the approximating group has nice algebraic properties. Investigating it by purely algebraic methods we find that it is closely related to a version of de Rham cohomology of the ring R. There is a map from L(R) (a slight variation on ℓ(R)) to the de Rham cycles. The de Rham differential factors over this map. Using this, we see that in the (very) special case when R is a polynomial ring, ℓ is precisely given by de Rham cycles and de Rham cohomology.

Finally, in §9, §10 we discover that in the polynomial case, the approximation is precise, that is ℓ(R) → H^{∗}(ES^{1} ×_{S^{1}} ΛX; ℤ/2) is an isomorphism. This is the main result of the paper.

In the two appendices §11 and §12 we show that the approximation functor has other good algebraic properties. In §11 we extend the functor ℓ(R) to the case where R non graded. In §12, we show that L(R) is easily computable from R in terms of generators and relations.

The authors wonder if there is some variation on the functor ℓ that computes the cohomology of the S^{1}-homotopy orbits of the free loop space in greater generality.

They would also like to know about possible connections to the Lanne’s functor, which in certain situations computes the cohomology of spaces of homotopy fixed points.

2 Notation

Let us fix some notation and conventions. The coefficient ring in all cohomology groups is the field 𝔽_{2} = ℤ/2 when no other ring appears in the notation. A denotes the mod two Steenrod algebra. We only consider spaces X which are of finite 𝔽_{2}-type i.e. H_{i}(X; 𝔽_{2}) is finite dimensional for each i.

The cohomology of the circle and its classifying space is denoted

H^{∗}(S^{1}) = Λ(v)
H^{∗}(BS^{1}) = 𝔽_{2}[u]

where v has degree one and u has degree two.

For the cohomology of the classifying space of a cyclic group of two power order we use the notation

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

here t and v are one dimensional and u is two dimensional. The inclusion j : C_{2^{m}} ⊆ S^{1} satisfies (Bj)^{∗}(u) = u when m ≥ 2 and (Bj)^{∗}(u) = t^{2} when m = 1 as seen by the Leray-Serre spectral sequence associated to the fibration S^{1} → BC_{2^{m}} → BS^{1}. Also i^{∗}(v) = v when i : S^{1} → BC_{2^{m}} denotes the inclusion of the fiber.

For any unstable A-algebra we define the operation λx = Sq^{|x|−1}x. By the Cartan formula this is a derivation over Frobenius i.e.

λ(xy) = x^{2}λy + y^{2}λx

3 Connected S^{1}-spaces

Let Y be a connected S^{1}-space with action map η : S^{1} × Y → Y. The map η^{∗} is injective since η ∘ γ = id where γ is the map defined by γ(y) = (1, y). We write Y_{0} for the space Y with trivial S^{1}-action.

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

η^{∗}(y) = 1 ⊗ y + v ⊗ dy

Proposition 3.2. The map d satisfies the following equations.

d ∘ d = 0  (1)
d(x + y) = dx + dy  (2)
d(xy) = xdy + ydx  (3)
Sq^{i}(dx) = d(Sq^{i}x)  (4)
(dx)^{2} = d(λx)  (5)

Proof. (1) There is a commutative diagram

S^{1} × S^{1} × Y ⟶^{1×η} S^{1} × Y
μ×1↓  ^{η}↓
S^{1} × Y ⟶^{η} Y

where μ : S^{1} ×S^{1} → S^{1} is the multiplication map. In cohomology it satisfies μ^{∗}(v) = v ⊗ 1 + 1 ⊗ v. The result follows by pulling a class y ∈ H^{∗}Y back along the two different ways in the diagram.

(2) Follows from η^{∗}(x + y) = η^{∗}(x) + η^{∗}(y).

(3) Follows from η^{∗}(xy) = η^{∗}(x)η^{∗}(y).

(4) Follows by the Cartan formula and the fact that η^{∗} is A-linear.

(5) Follows from (4).

By our next result the differential d also appears in the Leray-Serre spectral sequence.

Proposition 3.3. 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^{∗}(ES^{1} ×_{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 differential from Definition 3.1

Proof. The action map η : S^{1} × Y_{0} → Y is an S^{1}-equivariant map hence it induces a map of fibrations.

S^{1} × Y_{0} ⟶ ES^{1} ×_{S^{1}} (S^{1} × Y_{0}) ⟶ BS^{1}
η↓  ^{1×η}↓  ^{id}↓
Y ⟶  ES^{1} ×_{S^{1}} Y  ⟶ BS^{1}

We get a map of the corresponding spectral sequences η^{∗} : E → Ê where Ê is the spectral sequence associated to the upper fibration. Since the upper fibration has total space homeomorphic to ES^{1} × Y it looks like

Ê_{2}^{∗∗} = H^{∗}(BS^{1}) ⊗ H^{∗}(S^{1}) ⊗ H^{∗}(Y) ⇒ H^{∗}(Y)

In Ê_{2} the differential on the fiber elements are given by

d_{2}(v ⊗ z) = u(1 ⊗ z) , d_{2}(1 ⊗ z) = 0  (6)

To see this, assume that Y is contractible. Then the E_{∞} term has a 𝔽_{2} placed at (0, 0) and zero at all other places. It follows that d_{2}(v) = u and d_{2}(1) = 0. Since the action on Y_{0} was trivial this case imply (6). Let y ∈ H^{∗}Y be a fiber class in E_{2}. By η^{∗} it maps to 1⊗y +v ⊗dy and applying the Ê_{2}- differential to this we get u(1 ⊗ dy). We see that d_{2}(y) = ux where x ∈ H^{∗}Y is a class with η^{∗}(x) = 1 ⊗ dy. Since x = dy satisfies this equation and η^{∗} is injective the result follows.

4 Homotopy orbits of connected S^{1}-spaces

Definition 4.1. Define the spaces E_{n}Y for n = 0, 1, 2, …, ∞ by

E_{n}Y = ES^{1} ×_{C_{2^{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.

Theorem 4.2. Let m ≥ 1 be an integer. Consider the following diagram

\begin{matrix}E_{m}Y & ⟶^{Q} & E_{∞}Y \\ ^{pr_{1}}↓ & & ^{pr_{1}}↓ \\ BC_{2^{m}} & ⟶^{Bj} & BS^{1} \end{matrix}  (7)

where Q denotes the quotient map. There is an isomorphism

θ : H^{∗}(BC_{2^{m}}) ⊗_{H^{∗}(BS^{1})} H^{∗}(E_{∞}Y) ≅ H^{∗}(E_{m}Y)

defined by 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) (here v = t when m = 1). We get an isomorphism

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

Proof. Filling in the fibers of diagram (7) we get

∗ ⟶ Y ⟶^{id} Y
↓  ↓  ↓
S^{1} ⟶ E_{m}Y ⟶^{Q} E_{∞}Y
↓  ^{pr_{1}}↓  ^{pr_{1}}↓
S^{1} ⟶ BC_{2^{m}} ⟶^{Bj} BS^{1}

Since the fundamental group of BS^{1} is zero, the Leray-Serre spectral sequence Ê of the lowest horizontal fibration has trivial coefficients, and since Q is the pull back of this, the Leray-Serre spectral sequence E associated to Q also has trivial coefficients. We get

E_{2}^{∗∗} = H^{∗}(E_{∞}Y) ⊗ H^{∗}(S^{1}) ⇒ H^{∗}(E_{m}Y)
Ê_{2}^{∗∗} = H^{∗}(BS^{1}) ⊗ H^{∗}(S^{1}) ⇒ H^{∗}(BC_{2^{m}})

and pr_{1} gives a map of these two spectral sequences. We know that d_{2}(v) = 0 in Ê. Since pr_{1}^{∗}(v) = v we get that d_{2}v = 0 in E hence E_{2} = E_{∞}. It is now obvious that the map θ is an isomorphism. Using the map pr_{1}^{∗} we get that τ_{m}^{m+1}(v) = v. By Frobenius reciprocity the description of the transfer map follows.

The above theorem is inspired by the following result of Tom Goodwillie, which can be found in an errata to [BHM].

Theorem 4.3. (Goodwillie) For any based S^{1}-space Z, there is a map

τ : Q̃(ΣES_{+}^{1} ∧_{S^{1}} Z) → holimQ̃(ES_{+}^{1} ∧_{C_{p^{n}}} Z)

which induces an isomorphism on homotopy groups with 𝔽_{p} coefficients.

We may now give our own definition of the S^{1}-transfer

Definition 4.4. 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 the degree of τ_{n}^{∞} is -1.

Definition 4.5. 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_{2^{n}} × Y → E_{m}Y , m < ∞
θ_{∞} : ∗ × Y → E_{∞}Y

Proposition 4.6. 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. The first three diagrams are obvious. The last diagram follows by Theorem 4.2 when passing to the direct limit over m in the diagram with the finite transfers.

Proposition 4.7. For any non negative integer n Frobenius reciprocity holds i.e.

τ_{n}^{∞}(q_{∞}^{n}(x)y) = xτ_{n}^{∞}(y)

Proof. When n ≥ 1 this is an easy consequence of the description of the transfer in Theorem 4.2. For n = 0 we use Frobenius reciprocity of τ_{0}^{2} as follows

τ_{0}^{∞}(q_{∞}(x)y) = τ_{2}^{∞} ∘ τ_{0}^{2}(q_{2}^{0}(q_{∞}^{2}(x))y) = τ_{2}^{∞}(q_{∞}^{2}(x)τ_{0}^{2}(y))

and use the n = 2 case.

Proposition 4.8.

τ_{0}^{∞} ∘ q_{∞}^{0} = 0  (8)
q_{∞}^{0} ∘ τ_{0}^{∞} = d  (9)

Proof. (8) We can factor the composite as τ_{0}^{∞} ∘ q_{∞}^{0} = τ_{1}^{∞} ∘ τ_{0}^{1} ∘ q_{1}^{0} ∘ q_{∞}^{1} and τ_{0}^{1} ∘ q_{1}^{0} is multiplication by 2 thus the zero map.

(9) We can factor θ_{0} as

θ : S^{1} × Y_{0} ⟶^{Δ×1} S^{1} × S^{1} × Y ⟶^{1×η} S^{1} × Y ⟶ ES^{1} × Y

and by this we find θ_{0}^{∗}(1 ⊗ y) = 1 ⊗ y + v ⊗ dy. The result follows from the last diagram of Proposition 4.6.

Definition 4.9. Define the map

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

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

f_{Y} : S^{1}/C_{2} × Y → ES^{1} ×_{C_{2}} Y^{2}

Note that this map is natural in Y with respect to C_{2}-equivariant maps. Recall that there is a natural isomorphism.

H^{∗}(ES^{1} ×_{C_{2}} Y^{2}) ≅ H^{∗}(C_{2}; H^{∗}(Y)^{⊗2})

For a homogeneous element y ∈ H^{∗}Y the C_{2} invariant y ⊗ y defines an element 1 ⊗ y^{⊗2} in the zeroth cohomology group of C_{2}. If x ∈ H^{∗}Y is another homogeneous element with x ≠ y then the invariant x ⊗ y + y ⊗ x also defines an element 1 ⊗ (1 + T)x ⊗ y in the zeroth cohomology group of C_{2}. Using this notation we have the following results.

Lemma 4.10.

f_{Y_{0}}^{∗}(1 ⊗ y^{⊗2}) = 1 ⊗ y^{2} + v ⊗ λy

Proof. We have a factorization

f_{Y_{0}} : S^{1}/C_{2} × Y_{0} ⟶^{i×1} ES^{1}/C_{2} × Y_{0} ⟶^{1×Δ} ES^{1} ×_{C_{2}} Y_{0}^{2}

This together with Steenrod’s definition of the squares gives the result.

Lemma 4.11.

f_{Y}^{∗} (1 ⊗ (1 + T)x ⊗ y) = v ⊗ d(xy)

Proof. There is a commutative diagram

H^{∗}(S^{1}/C_{2} × Y) ⟵^{f_{Y}^{∗}} H^{∗}(ES^{1} ×_{C_{2}} Y^{2})
τ_{0}^{1}⊗1↑  ^{τ_{0}^{1}}↑
H^{∗}(S^{1} × Y) ⟵^{f_{Y}^{′∗}} H^{∗}(ES^{1} × Y^{2})

The lower horizontal map is given by f_{Y}^{′∗}(1 ⊗ x ⊗ y) = 1 ⊗ xy + v ⊗ d(xy) as seen by the factorization

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

Since the norm class is hit by the transfer τ_{0}^{1}(1 ⊗ x ⊗ y) = 1 ⊗ (1 + T)x ⊗ y the result follows.

Theorem 4.12.

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

Proof. Because of the degrees there is a constant k ∈ 𝔽_{2} such that

f_{S^{1}}^{∗}(1 ⊗ v^{⊗2}) = kv ⊗ v

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 Lemma 4.10 and the above equation respectively to find the first two equations of the three equations below

f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (1 ⊗ y)^{⊗2}) = 1 ⊗ 1 ⊗ y^{2} + v ⊗ 1 ⊗ λy
f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (v ⊗ 1)^{⊗2}) = kv ⊗ v ⊗ 1
f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (v ⊗ dy)^{⊗2}) = kv ⊗ v ⊗ (dy)^{2}

The third equation follows from the other two.

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

We compute the pull back of the class 1 ⊗ y^{⊗2} to the cohomology of the upper left corner. First we find

(1 × η^{2})^{∗}(1 ⊗ y^{⊗2}) = 1 ⊗ (1 ⊗ y + v ⊗ dy)^{⊗2} =
1 ⊗ (1 ⊗ y)^{⊗2} + 1 ⊗ (v ⊗ dy)^{⊗2} + 1 ⊗ (1 + T)(1 ⊗ y) ⊗ (v ⊗ dy)

By Lemma 4.11 we can compute f_{S^{1}×Y_{0}}^{∗} applied to the last term

f_{S^{1}×Y_{0}}^{∗}(1 ⊗ (1 + T)(1 ⊗ y) ⊗ (v ⊗ dy)) = v ⊗ d_{S^{1}×Y_{0}}(v ⊗ ydy)
= v ⊗ (d_{S^{1}}(v) ⊗ ydy + v ⊗ d_{Y_{0}}(ydy))
= v ⊗ 1 ⊗ ydy

Altogether we have

(1 ⊗ η^{∗}) ∘ f_{Y}^{∗} (1 ⊗ y^{⊗2}) = f_{S^{1}×Y_{0}}^{∗} ∘ (1 × η^{2})^{∗}(1 ⊗ y^{⊗2})
= 1 ⊗ 1 ⊗ y^{2} + v ⊗ 1 ⊗ (ydy + λy) + kv ⊗ v ⊗ (dy)^{2}

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

f_{Y}^{∗} (1 ⊗ y^{⊗2}) = (1 ⊗ γ^{∗}) ∘ (1 ⊗ η^{∗}) ∘ f_{Y}^{∗} (1 ⊗ y^{⊗2})

= 1 ⊗ γ^{∗}(1 ⊗ 1 ⊗ y^{2} + v ⊗ 1 ⊗ (ydy + λy) + kv ⊗ v ⊗ (dy)^{2})

= 1 ⊗ y^{2} + v ⊗ (ydy + λy)

5 The approximation functors

Let A be a graded ring over 𝔽_{2}. Let λ : A → A be a linear map with |λx| = 2|x| − 1 which is a derivation over Frobenius i.e. λ(ab) = λ(a)b^{2} + a^{2}λ(b). We consider a version of the de Rham complex, relative to the derivation λ.

Definition 5.1. The algebra Ω_{λ}(A) is generated by

a, da for a ∈ A

where d(a) is given the degree |a| − 1. The relations are

d(a + b) = da + db  (10)
d(ab) = (da)b + a(db)  (11)
(da)^{2} = d(λ(a))  (12)

Of course there is also a differential on Ω_{λ}(A), defined as a derivation over A by the formula d(a) = da. This is compatible with the relations, since d((da)^{2}) = 0 = d(d(λa)). Note that this differential depends on λ. When λ is the trivial derivation λ(a) = 0 we get the ordinary de Rham complex.

Clearly Ω_{λ} is an functor from the category of graded rings with derivations over Frobenius to the category of differential graded algebras.

Definition 5.2. We define ℓ(A) to be the graded algebra over 𝔽_{2} generated by u of degree two and the following symbols, defined for each homogeneous element a ∈ A^{n}

ϕ(a),  deg(ϕ(a)) = 2n
δ(a),  deg(δ(a)) = n − 1
q(a),  deg(q(a)) = 2n − 1

Satisfying the following relations:

ϕ(a + b) = ϕ(a) + ϕ(b)  (13)
δ(a + b) = δ(a) + δ(b)  (14)
q(a + b) = q(a) + q(b) + δ(ab)  (15)
0 = δ(xy)δ(z) + δ(yz)δ(x) + δ(zx)δ(y)  (16)
ϕ(ab) = ϕ(a)ϕ(b) + uq(a)q(b)  (17)
q(ab) = q(a)ϕ(b) + ϕ(a)q(b)  (18)
δ(a)^{2} = δ(λ(a))  (19)
q(a)^{2} = ϕ(λ(a)) + δ(a^{2}λ(a))  (20)
δ(a)ϕ(b) = δ(ab^{2})  (21)
δ(a)q(b) = δ(aλ(b)) + δ(ab)δ(b)  (22)
uδ(a) = 0  (23)

We note some immediate consequences of these relations: By (13), (14) and (15), ϕ(0) = δ(0) = q(0) = 0. By (18), q(1) = q(1)ϕ(1) + ϕ(1)q(1) = 0. Using this and (15), we see that δ(1) = 0. Using (17), (18) and (21) we now see that ϕ(1) acts as a one element. So we can put 1 = ϕ(1), to make ℓ(A) into a unital algebra. Finally, q(a^{2}) = q(a)ϕ(a) + ϕ(a)q(a) = 0, and also δ(a^{2}) = δ(1)ϕ(a) = 0.

Definition 5.3. Define L(A) to be the quotient algebra ℓ(A)/(u).

Note that the map ϕ : A → L(A) is a unital ring homomorphism, and doubles the degree. ℓ and L are functors from graded rings with derivation over Frobenius to graded rings.

6 The free loop space.

Let X be a connected space and let ΛX denote its free loop space. We also assume that ΛX is connected (Otherwise we may apply the following theory on each component of ΛX). Then we may take Y = ΛX in §3 and §4.

Definition 6.1. Define the evaluation maps ev_{0} : ΛX → X

ev′_{1} : ΛX → X^{2}
ev_{1} : ES^{1} ×_{C_{2}} ΛX → ES^{1} ×_{C_{2}} X^{2}

by ev_{0}(ω) = ω(1), ev′_{1}(ω) = (ω(1), ω(−1)) and ev_{1} = 1 × ev′_{1}. Definition 6.2. Define an algebra homomorphism

e : Ω_{λ}^{∗}(H^{∗}X) → H^{∗}(ΛX)

by e(x) = ev_{0}^{∗}(x) and e(dx) = dev_{0}^{∗}(x).

Note that e is well defined by Proposition 3.2. Definition 6.3. For any homogeneous class x ∈ H^{∗}X define the classes δ(x), ϕ(x), q(x) in H^{∗}E_{∞}ΛX of degrees |x| − 1, 2|x|, 2|x| − 1 respectively by

δ(x) = τ_{0}^{∞} ∘ ev_{0}^{∗}(x)
ϕ(x) = τ_{1}^{∞} ∘ ev_{1}^{∗}(t ⊗ x^{⊗2})
q(x) = τ_{1}^{∞} ∘ ev_{1}^{∗}(1 ⊗ x^{⊗2})

Any constant loop is a S^{1}-fixpoint in the free loop space, which defines a section to the fibration pr_{1} : E_{∞}ΛX → BS^{1}. Thus pr_{1}^{∗} is injective and we define u in H^{∗}E_{∞}ΛX by u = pr_{1}^{∗}(u). Lemma 6.4. There is a commutative diagram

H^{∗}(ES^{1} ×_{C_{2}} X^{2}) ⟶^{ev_{1}^{∗}} H^{∗}(ES^{1} ×_{C_{2}} ΛX) ⟶^{τ_{1}^{∞}} H^{∗}(ES^{1} ×_{S^{1}} ΛX)
T r_{0}^{1}↑  ^{τ_{0}^{1}}↑
H^{∗}(X^{2})  ⟶^{ev_{1}^{′∗}}  H^{∗}(ΛX)
Δ^{∗}↓  ∥
H^{∗}(X)  ⟶^{ev_{0}^{∗}}  H^{∗}(ΛX)

and by this we have δ(xy) = τ_{1}^{∞} ∘ ev_{1}^{∗}(1 ⊗ (1 + T)x ⊗ y) and δ(x^{2}) = 0.

Proof. The upper square is obviously commutative. The lower square is induced by a homotopy commutative diagram. The homotopy

H : I × ΛX → X^{2} ; (s, ω) ↦ (ω(1), ω(e^{πis}))

has the evaluation map ev′_{1} in one end and Δ ∘ ev_{0} in the other end. The last statement follows since T r_{0}^{1}(x ⊗ y) = 1 ⊗ (1 + T)x ⊗ y.

Proposition 6.5.

ev_{1}^{∗}(1 ⊗ x^{⊗2}) = ϕ(x) + tq(x)

Proof. Follows directly from Theorem 4.2.

Theorem 6.6. The map q_{∞}^{0} : H^{∗}E_{∞}ΛX → H^{∗}ΛX maps the above classes as follows.

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

Proof. It follows directly from Proposition 4.8 that δ(x) is mapped as stated. For the other two classes we use Proposition 4.6

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

Let f = f_{ΛX} be the map from Definition 4.9. We see that ev_{1} ∘ θ_{1} equals the composite

S^{1}/C_{2} × ΛX ⟶^{f} ES^{1} ×_{C_{2}} (ΛX)^{2} ⟶^{1×ev_{0}^{2}} ES^{1} ×_{C_{2}} X^{2}

Thus we have the following equality.

q_{∞}^{0} ∘ τ_{1}^{∞} ∘ ev_{1}^{∗} = τ_{1}^{∞} ∘ f^{∗} ∘ (1 × ev_{0}^{2})^{∗}

Applying this and Theorem 4.12 we get

q_{∞}^{0}(q(a)) = τ_{1}^{∞} ∘ f^{∗}(1 ⊗ ev_{0}^{∗}(a)^{⊗2}) = e(ada + λa)
q_{∞}^{0}(ϕ(a)) = τ_{1}^{∞} ∘ f^{∗}(t ⊗ ev_{0}^{∗}(a)^{⊗2}) = e(a^{2})

Proposition 6.7. The sub algebra of H^{∗}E_{∞}ΛX generated by the classes δ(x), ϕ(x), q(x) for x ∈ H^{∗}X and u is closed under the A-action. The action is explicitly given by the formulas

Sq^{i}δ(x) = δ(Sq^{i}x)  (24)
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)) (25)

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))(26)

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)

Proof. The first formula (24) is obvious. For the other two we use the formulas for the A-action on H^{∗}(ES^{1} ×_{C_{2}} X^{2}) which can be found in eg. [Milgram].

Sq^{i}(t^{k} ⊗ x^{⊗2}) = ∑_{j≥0} \binom{k + |x| − j}{i − 2j}t^{k+i−2j} ⊗ (Sq^{j}x)^{⊗2}

+ δ_{k,0} ∑_{r=0}^{[ \frac{i−1}{2} ]} 1 ⊗ (1 + T)Sq^{r}x ⊗ Sq^{i−r}x

Here δ_{i,j} is the Kronecker delta. The second formula (25) follows directly and the last formula (26) follows when applying Lemma 6.4.

Theorem 6.8. Let X be a connected space with connected free loop space ΛX. Then there are natural algebra homomorphisms

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

The first is defined by e(x) = ev_{0}^{∗}(x) and e(dx) = dev_{0}^{∗}(x). The second is defined by ψ(u) = u and

ψ(δ(x)) = τ_{0}^{∞} ∘ ev_{0}^{∗}(x)
ψ(ϕ(x)) = τ_{1}^{∞} ∘ ev_{1}^{∗}(t ⊗ x^{⊗2})
ψ(q(x)) = τ_{1}^{∞} ∘ ev_{1}^{∗}(1 ⊗ x^{⊗2})

Proof. We have already defined the map e. To establish the map ψ we must show that the classes δ(x), ϕ(x), q(x) from Definition 6.3 satisfy the relations (13)-(23) from the definition of the ℓ-functor.

(13) We have the expansion

1 ⊗ (a + b)^{⊗2} = 1 ⊗ a^{⊗2} + 1 ⊗ b^{⊗2} + 1 ⊗ (1 + T)a ⊗ b

if we multiply this equation with t the norm element goes to zero and the relation follows.

(14) is obvious.

(15) We apply τ_{1}^{∞}∘ev_{1}^{∗} to the expansion and identify the last term by Lemma 6.4.

(16) By Frobenius reciprocity we get the first equality in the equation below, and by Proposition 4.8 we get the second equality.

τ_{0}^{∞} ∘ ev_{0}^{∗}(xy)τ_{0}^{∞} ∘ ev_{0}^{∗}(z) = τ_{0}^{∞}(q_{∞}^{0} ∘ τ_{0}^{∞} ∘ ev_{0}^{∗}(xy)ev_{0}^{∗}(z))
= τ_{0}^{∞}(d(ev_{0}^{∗}(xy))ev_{0}^{∗}(z))

From this we see that the right hand side of (16) equals τ_{0}^{∞} ∘ e applied to

d(xy)z + d(yz)x + d(zx)y

Since this element is zero the relation follows.

(17) & (18) The map ev_{1}^{∗} is a ring homomorphism. Thus we have

ev_{1}^{∗}(1 ⊗ (ab)^{⊗2}) = ev_{1}^{∗}(1 ⊗ a^{⊗2})ev_{1}^{∗}(1 ⊗ b^{⊗2})

This and Proposition 6.5 gives the result (like e^{iθ} = cos θ + i sin θ).

(19) & (20) These are special cases of Proposition 6.7.

(21) We first use Frobenius reciprocity and then Theorem 6.6.

ϕ(b)δ(a) = τ_{0}^{∞}(q_{∞}^{0}(ϕ(b))ev_{0}^{∗}(a)) = τ_{0}^{∞}(ev_{0}^{∗}(b^{2}a)) = δ(ab^{2})

(22) Again we first use Frobenius reciprocity and then Theorem 6.6.

q(a)δ(b) = τ_{0}^{∞}(q_{∞}^{0}(q(b))ev_{0}^{∗}(a)) = τ_{0}^{∞}(ev_{0}^{∗}(ab)dev_{0}^{∗}(b) + ev_{0}^{∗}(aλ(b)))

The second term gives δ(aλb) and if we use 4.8 to replace the d by q_{∞}^{0} ∘ τ_{0}^{∞} in the first term we see by Frobenius reciprocity that it gives δ(ab)δ(b).

(23) We use Lemma 6.4. When we multiply a norm element by t^{2} we get zero.

The functors Ω_{λ}(H^{∗}X) and ℓ(H^{∗}x) are viewed as approximations to H^{∗}(ΛX) and H^{∗}(ES^{1} ×_{S^{1}} ΛX) respectively via these maps.

We can motivate the definition of the functor L from Definition 5.3 as follows. Since the composite q_{0}^{∞} ∘ ψ vanish on the ideal (u) ⊆ ℓ(H^{∗}X), it induces a map

L(H^{∗}X) → Im(q_{0}^{∞} : H^{∗}(ES^{1} ×_{S^{1}} ΛX) → H^{∗}(ΛX))

Hence L(H^{∗}X) may be viewed as an approximation to the image of q_{0}^{∞} via this map.

7 Relations with the Steenrod diagonal

Assume that X is a connected space with connected free loop space. Let i : X → ΛX denote the map which sends a point x in X to the constant loop with value x. This is an S^{1}-equivariant map when X is given the trivial S^{1}-action. Thus 1 × i define maps

i_{n} : E_{n}ΛX → BC_{2^{n}} × X , n ≥ 0
i_{∞} : E_{∞}ΛX → BS^{1} × X

Proposition 7.1. For each integer m ≥ 1 there is a commutative diagram

H^{∗}(E_{∞}ΛX) ⟶^{i_{∞}^{∗}} H^{∗}(BS^{1}) ⊗ H^{∗}(X)
τ_{m}^{∞}↑  ^{τ_{m}^{∞}⊗1}↑
H^{∗}(E_{m}ΛX) ⟶^{i_{m}^{∗}} H^{∗}(BC_{2^{m}}) ⊗ H^{∗}(X)

Proof. There is a commutative square

BC_{2^{m}} × X ⟶^{Bj×1} BS^{1} × X
pr_{1}↓  ^{pr_{1}}↓
BC_{2^{m}}  ⟶^{Bj} BS^{1}

If we take Y = ΛX in Theorem 4.2, it maps into the square (7) by the maps i_{m} and i_{∞} which sits over the respective identity maps. The resulting cube is commutative. From the cube we get a commutative diagram

H^{∗}E_{m}ΛX  ⟶^{i_{m}^{∗}}  H^{∗}(BC_{2^{m}} × X)
θ↑  ↑
H^{∗}(BC_{2^{m}}) ⊗_{H^{∗}(BS^{1})} H^{∗}(E_{∞}ΛX) ⟶^{1⊗i_{∞}^{∗}} H^{∗}(BC_{2^{m}}) ⊗_{H^{∗}(BS^{1})} H^{∗}(BS^{1} × X)

Here θ is an isomorphism and it is easy to see that the vertical map to the right is also an isomorphism. The result follows by passing to the direct limit over transfer maps.

Note that ev_{1} ∘ i_{1} is the diagonal map. Hence we have the following equation

i_{1}^{∗} ∘ ev_{1}^{∗}(t^{k} ⊗ x^{⊗2}) = ∑_{j≥0} t^{k+j} ⊗ Sq^{|x|−j}x  (27)

according to the definition of the Steenrod squares.

Proposition 7.2. Let X be a connected space with connected free loop space. Then the composite map

ℓ(H^{∗}X) ⟶^{ψ} H^{∗}(ES^{1} ×_{S^{1}} ΛX) ⟶^{i_{∞}^{∗}} 𝔽_{2}[u] ⊗ H^{∗}X

is given by δ(x) ↦ 0 and

ϕ(x) ↦ ∑_{i≥0} u^{i} ⊗ Sq^{|x|−2i}x , q(x) ↦ ∑_{i≥0} u^{i} ⊗ Sq^{|x|−2i−1}x

Proof. The classes ϕ(x) and q(x) are mapped as stated by (27) and Proposition 7.1. By Lemma 6.4 we find

i_{∞}^{∗}(δ(x)) = i_{∞}^{∗} ∘ τ_{1}^{∞} ∘ ev_{1}^{∗}(1 ⊗ (1 + T)x ⊗ 1)
= (τ_{1}^{∞} ⊗ 1) ∘ i_{1}^{∗} ∘ ev_{1}^{∗}(1 ⊗ (1 + T)x ⊗ 1)

This is zero since the Steenrod diagonal maps norm elements to zero (see [St-Ep]).

As a consequence the approximation is nontrivial in many cases. The additive map ϕ : H^{∗}X → H^{∗}(ES^{1} ×_{S^{1}} ΛX) is eg. always injective when restricted to the subalgebra of H^{∗}X consisting all even dimensional classes.

8 de Rham cohomology

In this section we relate the L-functor to the de Rham cycles. This will be used later in the proof of our main result, Theorem 10.1.

There is a filtration on Ω_{λ}(A) induced by giving each element in A itself the filtration 0, and the symbol a_{0}(da_{1}) … (da_{n}) the filtration n. The multiplication decreases filtration, even if it does not necessarily preserve it.

The corresponding graded algebra can be identified with the de Rham complex corresponding to the trivial derivation λ, λ(a) = 0.

The differential increases the filtration above by 1, but it does not preserve the corresponding grading.

As usual, the composite of the de Rham differential with itself is zero, so we can consider the cohomology groups. Sometimes these can be computed explicitly.

We consider another version of the de Rham complex. Let Ω̃_{λ}(A) be the algebra generated by the same generators as A, satisfying (10) and (11) but instead of (12) the relation

(da)^{2} = λ(a)  (28)

Note that this is not a homogeneous relation!

Let us now define a Frobenius map. It is a multiplicative map, defined on generators by

Φ : Ω̃_{λ}(A) → H^{∗}(Ω_{λ}(A))
Φ(a) = a^{2}
Φ(da) = ada + λ(a)

Proposition 8.1. This formula defines a ring map. In case A is a polynomial algebra (no matter what λ is), Φ : Ω̃_{λ}(A) → H^{∗}(Ω_{λ}(A)) is an isomorphism of rings. This also holds when the polynomial algebra has infinitely many generators.

Proof. The map Φ is a linear map, compatible with the relations in Ω̃_{λ} modulo boundaries, since

Φ(d(a + b)) = (a + b)d(a + b) + λ(a + b)
= d(ab) + ada + bdb + λ(a) + λ(b)
= d(ab) + Φ(da) + Φ(db)
Φ(d(ab)) = abd(ab) + λ(ab) = a^{2}bdb + b^{2}ada + a^{2}λ(b) + b^{2}λ(a)
= Φ(a)Φ(db) + Φ(da)Φ(b)
Φ(d(a)^{2}) = (ada + λ(a))^{2} = a^{2}dλ(a) + λ(a)^{2}
= d(a^{2}λ(a)) + Φ(λ(a))

The map goes into the homology, since d is a derivation, dΦ(a) = 0 and dΦ(da) = d(ada+λ(a)) = (da)^{2}+dλ(a) = 0. Let A be a polynomial algebra on generators a_{1}, …, a_{n}. Assume first that λ = 0. Then Ω_{λ}^{∗}(A) and Ω̃_{λ}^{∗}(A) are both isomorphic to the tensor product of the differential graded algebras 𝔽_{2}[a_{i}, da_{i}]/(da_{i})^{2} = 0. By the Künneth formula it suffices to check the n = 1 case which is easily done by inspection.

Let A = 𝔽_{2}[a_{1}, …, a_{n}], and λ an arbitrary derivation over Frobenius. In this general case, we compare the chain complex Ω_{λ} with the corresponding graded complex.

… ⟶ F^{i}/F^{i−1} ⟶^{d} F^{i+1}/F^{i} ⟶ …

This graded complex is the complex of the case λ = 0. We can define a corresponding filtration on Ω̃_{λ} and the map Φ is filtration preserving. By the above argument, Φ induces an isomorphism between graded rings.

It follows that if z_{i} ∈ F^{i}(Ω_{λ}) is a cycle, we can find an a ∈ F^{i}Ω̃_{λ} and an x ∈ F^{i−1}Ω_{λ} so that z_{i−1} = z_{i} − Φ(a) + dx is contained in F^{i−1}Ω_{λ}. Besides, z_{i−1} has to be a cycle. So by induction, we conclude that any cycle in F^{i}Ω_{λ} is homologous to an element in the image of Φ. A similar argument proves injectivity.

Since both functors in A commute with direct limits, the statement also holds in the case of infinitely many polynomial generators.

Theorem 8.2. There is a well-defined map Ψ : Ω_{λ}(A) → LA defined by

Ψ(a_{0}da_{1} … da_{n}) = δ(a_{0})δ(a_{1}) … δ(a_{n})

There is a ring map R : LA → Ω_{λ}(A) given on generators by

R(ϕ(a)) = a^{2}
R(q(a)) = ada + λa
R(δ(a)) = da

Note that d ∘ R = 0.

Proof. We only have to check relations. For the map Ψ is suffices to check that

Ψ(a_{0}da_{1} … da_{i−1}d(a_{i}a_{i+1})da_{i+2} … da_{n}) =
Ψ(a_{0}a_{i}da_{1} … da_{i−1}da_{i+1} … da_{n} + a_{0}a_{i+1}da_{1} … da_{i}da_{i+2} … da_{n})

But this is a consequence of relation 16:

δ(a_{0})δ(a_{i}a_{i+1}) + δ(a_{0}a_{i})δ(a_{i+1}) + δ(a_{0}a_{i+1})δ(a_{i}) = 0

For the R map, there are more relations to check. Again, (13), (14), (17),

(21) and (23) are obvious from linearity. The rest needs small computations.

R(q(a + b) + q(a) + q(b) + δ(ab))

= (a + b)d(a + b) + λ(a + b) + ada + λ(a) + bdb + λ(b) + d(ab)

= adb + bda + d(ab) = 0

R(δ(ab)δ(c) + δ(bc)δ(a) + δ(ca)δ(b))

= d(ab)dc + d(bc)da + d(ca)db = 0

R(q(ab) + q(a)ϕ(b) + ϕ(a)q(b))

= abd(ab) + λ(ab) + (ada + λ(a))b^{2} + a^{2}(bdb + λ(b))

= a^{2}bdb + b^{2}ada + b^{2}λ(a) + a^{2}λ(b) + ab^{2}da + b^{2}λ(a) + a^{2}bdb + a^{2}λ(b) = 0

R(δ(a)^{2} + δ(λ(a))) = (dx)^{2} + d(λx) = 0
R(q(a)^{2} + ϕ(λ(a)) + δ(a^{2}λ(a)))

= (ada + λ(a))^{2} + (λ(a))^{2} + d(a^{2}λ(a))

= a^{2}(da)^{2} + λ(a)^{2} + λ(a)^{2} + a^{2}dλ(a) = 0

R(δ(a)ϕ(b) + δ(ab^{2})) = (da)b^{2} + d(ab^{2}) = 0
R(δ(a)q(b) + δ(aλ(b)) + δ(ab)δ(b))
= d(a)(bdb + λ(b)) + d(aλ(b)) + d(ab)d(b)
= bdadb + daλ(b) + daλ(b) + ad(λ(b)) + a(db)^{2} + bdadb = 0

The composition of these maps is not so hard to compute. Let z = a_{0}da_{1} … da_{n}.

R ∘ Ψ(z) = R ∘ Ψ(a_{0}da_{1} … da_{n})
= R(δ(a_{0}) … δ(a_{n}))
= da_{0} … da_{n}
= dz

Moreover

Theorem 8.3. Let Φ be as above. Then the image of Ψ is an ideal in L(A), and the quotient ring is isomorphic to Ω_{λ}(A). The isomorphism is realized by a surjective ring map P : L(A) → Ω̃_{λ}(A) satisfying

P (δ(a)) = 0
P (ϕ(a)) = a
P (q(a)) = da

In particular, the map R : L(A) → Ω_{λ}(A) discussed above factors after projection to Ω_{λ}(A)/dΩ_{λ}(A) as Φ ∘ P

Proof. The image of Ψ clearly consists of all sums of all products of elements of the form δ(a). By using relations (21) and (22), we see that these classes do indeed form an ideal, and the quotient L(A)/Im(Ψ) is generated by the classes ϕ(a) and q(a). The relations these symbols satisfy are given by setting δ(a) equal to 0 in the list of relations above. We obtain:

ϕ(a + b) = ϕ(a) + ϕ(b)
q(a + b) = q(a) + q(b)
ϕ(ab) = ϕ(a)ϕ(b)
q(ab) = ϕ(a)q(b) + q(a)ϕ(b)
q(a)^{2} = ϕ(λ(a))

It is immediately clear that the formulas above for P define a ring homomorphism and that ImΨ ⊆ ker P. On the other hand, it is as easy to define an inverse homomorphism P^{−1} : Ω̃_{λ}(A) → L(A)/Im(Ψ) by P^{−1}(a) = ϕ(a), P^{−1}(da) = q(a). The claimed factorization of R can be checked on generators:

Φ ∘ P (ϕ(a)) = Φ(a) = a^{2} = R(ϕ(a))
Φ ∘ P (δ(a)) = Φ(0) = 0 = da + da = R(δ(a)) + da
Φ ∘ P (q(a)) = Φ(da) = ada + λ(a) = R(q(a))

Theorem 8.4. The composite Ψ ∘ d : Ω_{λ}(A) → L(A) is trivial, so we can define Ψ as a map on Ω_{λ}(A)/Im(d). This allows us to consider the composite Ψ ∘ Φ : Ω̃_{λ}(A) → L(A). This composite is zero.

Proof. The first claim is easy, since the image of d are the classes of the form da_{1}da_{2} … da_{n}. And by definition, Ψ vanishes on those classes.

To show the second claim, we claim the following rules: Let b ∈ A, and z = a_{0}da_{1} … da_{n} ∈ Ω̃_{λ}

Ψ(Φ(b)z) = ϕ(b)Ψ(z)
Ψ(Φ(db)z) = q(b)Ψ(z)

The first follows from relation (21), the second is a calculation, using relation (22) at an essential point.

Ψ(Φ(db)a_{0}da_{1} … da_{n}) = Ψ((bdb + λ(b))a_{0}da_{1} … da_{n}))
= Ψ((ba_{0})dbda_{1} … da_{n} + λ(b)a_{0}da_{1} … da_{n})
= (δ(ba_{0})δ(b) + δ(λ(b)a_{0}))δ(a_{1}) … δ(a_{n})
= δ(a_{0})q(b)δ(a_{1}) … δ(a_{n})
= q(b)Φ(z)

Now the theorem follows from these two formulas and the observation that Ψ(1) = 0

We can collect the information we have gathered so far in a diagram:

\xymatrix{ & Ω̃_{λ}(A) \ar[d]_{Φ} \\ Ω_{λ}(A) \ar[r] \ar[drr]_{d} & Ω_{λ}(A)/dΩ_{λ}(A) \ar[r]^{Ψ} & L(A) \ar[r]^{P} \ar[d]^{R} & Ω̃_{λ}(A) \ar[d]^{Φ} \\  &  & Ω_{λ}(A) \ar[r] & Ω_{λ}(A)/dΩ_{λ}(A)}

The results above say that the right hand square commutes, the composite Ψ∘Φ vanishes, ker P = ImΨ and finally that the composite map from Ω_{λ}(A) to itself is the de Rham differential.

Theorem 8.5. Assume that the map Φ : Ω̃_{λ}(A) → H^{∗}(Ω_{λ}(A)) is an isomorphism. Then

R : L(A) → ker {d : Ω_{λ}(A) → Ω_{λ}(A)}

is an isomorphism.

Proof. We first prove that R is surjective. By assumption, every cycle in Ω_{λ}(A) is the sum of a boundary and an element in the image of Φ. But since P is surjective, this says that every element in the kernel of d is the sum of a boundary and an element of the image of R. On the other hand, d factors over R, so R is surjective by itself.

Now we prove injectivity. Assume that z ∈ L(A), and that R(z) = 0. Since Φ is injective, P (z) = 0, and z = Ψ(y) for some y ∈ Ω_{λ}(A). But then dy = 0, so that y is a cycle. Using again that Φ is surjective onto the de Rham cohomology, we can write y as a sum of a boundary and an element Φ(x), where x ∈ Ω̃_{λ}. But both summands map to zero in L(A), so z = 0.

9 Cohomology of free loop spaces

The Eilenberg-Moore spectral sequence can sometimes be used to compute the cohomology of free loop spaces. According to [Smith81] there is a fiber square F(X) of the following form

ΛX ⟶^{ev} X
ev↓  ^{Δ}↓
X ⟶^{Δ} X × X

with common fiber ΩX. Assume that X is a 1-connected space and let R^{∗} = H^{∗}X. The Eilenberg-Moore spectral sequence associated to F(X) is a second quadrant spectral sequence of cohomology type. It has the following form

E_{2}^{−p,q} = Tor_{p}^{R^{∗}⊗R^{∗}}(R^{∗}, R^{∗})^{q} ⇒ H^{∗}(ΛX)

where the action of R^{∗} ⊗ R^{∗} on R^{∗} is via the multiplication map

μ : R^{∗} ⊗ R^{∗} → R^{∗}

Since R^{∗} is a commutative ring the E_{2}-term can be expressed by Hochschild homology.

E_{2}^{−p,q} = HH_{p}(R^{∗})^{q}

If we assume that R^{∗} is a smooth 𝔽_{2}-algebra we get the following isomorphism by the Hochschild-Konstant-Rosenberg theorem ([Loday] p. 102)

HH_{∗}(R^{∗}) ≅ Ω^{∗}(R^{∗})

Here Ω^{∗}(R^{∗}) is the graded exterior differential module of the 𝔽_{2} algebra R^{∗} see [Loday] p. 26. Since all algebra generators sits in E_{2}^{0,∗} or E_{2}^{−1,∗} the spectral sequence collapses i.e. E_{2} = E_{∞}.

Since X is 1-connected we have zeros below the line q = −2p in the E_{2}-term. Thus the decreasing filtration F^{i}H^{∗} of H^{∗} := H^{∗}(ΛX) is bounded

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

Note also that F^{0}H^{n} = R^{n}.

10 Spaces with polynomial cohomology

Theorem 10.1. Let X be a 1-connected space. Assume that H^{∗}X = S(V) is a symmetric algebra on the graded vector space V, where V^{i} is finite dimensional for each i. Then the two ring homomorphisms

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

are isomorphisms of unstable A-algebras, where the A-action on Ω_{λ}(H^{∗}X) is given by Sq^{i}(dx) = d(Sq^{i}x) and the A-action on ℓ(H^{∗}X) is given by the formulas of Proposition 6.7.

Proof. We first show that e is an isomorphism. Since any polynomial algebra is smooth we have an isomorphism of graded 𝔽_{2} vector spaces

H^{∗}(ΛX) ≅ E_{∞}^{∗∗} ≅ Ω^{∗}(R^{∗})

Let x ∈ V. We first observe that e(x) represents x in E_{∞}^{0,∗}. This is a consequence of the construction of the Eilenberg Moore spectral sequence. The composite

R^{∗} ⊗_{R^{∗}⊗R^{∗}} R^{∗} = Tor_{0}^{R^{∗}⊗R^{∗}}(R^{∗}, R^{∗}) = E_{2}^{0,∗} → E_{∞}^{0,∗} ⊆ H^{∗}(ΛX)

is given by a ⊗ b ↦ ev_{0}^{∗}(a)ev_{0}^{∗}(b) since we consider the spectral sequence associated to the fiber square F(X). From this we see that ev_{0}^{∗}(x) represents x in E_{∞}^{0,∗}.

Next we show that e(dx) represents dx in E_{∞}^{−1,∗}. Put n = |x| and choose a map f : X → K(𝔽_{2}, n) such that f^{∗}(ι_{n}) = x. Clearly ev_{0}∘Λf = f ∘ev_{0} where Λf is the map between corresponding free loop spaces. By Proposition 4.8 we see that d commutes with the cohomology of such maps thus we have:

d ∘ ev_{0}^{∗}(x) = d ∘ ev_{0}^{∗} ∘ f^{∗}(ι_{n}) = d ∘ (Λf)^{∗} ∘ ev_{0}^{∗}(ι_{n}) = (Λf)^{∗} ∘ d ∘ ev_{0}^{∗}(ι_{n})

By naturality of the Eilenberg Moore spectral sequence it is enough to show that dev_{0}^{∗}(ι_{n}) represents dι_{n} in E_{∞}^{−1,n}. But H^{n−1}(ΛK(𝔽_{2}, n)) which additively equals (Ω^{∗}H^{∗}(K(𝔽_{2}, n)))^{n−1} is one dimensional. Thus it is enough to show that dev_{0}^{∗}(ι_{n}) ≠ 0. This is done in the Lemma below.

Note that Ω_{λ}^{∗}(R^{∗}) is generated by

{dx_{1} … dx_{r}|x_{s} ∈ S, x_{i} ≠ x_{j} for i ≠ j, r ≥ 0}

as an R^{∗} module. This follows since each time we replace a square (dx)^{2} in a monomial by d(λx) the number of d’s in any of the monomials in the resulting sum is lowered by one. Thus the replacement process terminates. Define F^{−p} ⊆ Ω_{λ}^{∗}(R^{∗}) as the subspace generated by

{dx_{1} … dx_{r}|x_{s} ∈ S, x_{i} ≠ x_{j} for i ≠ j, r ≤ p}

as an R^{∗} module. Then we have a decreasing filtration F^{i}, i ≤ 0 of Ω_{λ}^{∗}(R^{∗}) satisfying e(F^{i}) ⊆ F^{i}H^{∗}. The map induced by e between the corresponding graded objects is clearly an isomorphism hence e is an isomorphism. The statement about the A-action follows directly by Proposition 3.2.

We now prove that ψ is an isomorphism. By Proposition 3.3 we can write the E_{3}-term of the Leray-Serre spectral sequence of the fibration ΛX → E_{∞}X → BS^{1} as

E_{3}^{∗∗} = ker(d) ⊕ uH(d) ⊕ u^{2}H(d) ⊕ …

By Proposition 8.1 and Theorem 8.5 the map R is an isomorphism. This together with Theorem 6.6 makes the spectral sequence collapse i.e. E_{2} = E_{∞}. It is then easy to check that ψ is an isomorphism. The statement about the A-action is immediate by Proposition 6.7.

Lemma 10.2. Let n ≥ 2 and put K = K(𝔽_{2}, n) with fundamental cohomology class ι_{n}. Then the class dev_{0}^{∗}(ι_{n}) is non zero in H^{∗}(ΛK).

Proof. By Lemma 3 of [Smith84] we have K_{n} × K_{n−1} ≃ K_{n} × ΩK_{n} ≃ ΛK_{n}. We can write d(ι_{n}) = kι_{n−1} where k ∈ 𝔽_{2} and we must show k = 1. Consider the composite

g : S^{1} × ΛK ⟶^{η} ΛK ⟶^{ev_{0}} K

which is simply given by evaluation g(z, ω) = ω(z). The pull back of the fundamental class is g^{∗}(ι_{n}) = 1 ⊗ ι_{n} + kv ⊗ ι_{n−1}. Assume that k = 0. Since K classifies mod two cohomology in degree n we see that g is homotopic to

S^{1} × ΛK ⟶^{pr_{2}} ΛK ⟶^{ev_{0}} K

But then the adjoined of g, which is id_{ΛK}, is homotopic to the adjoined of ev_{0} ∘pr_{2} which is the map from ΛK to itself sending a loop ω to the constant loop ω(1). Since this is not the case we have k = 1.

11 Appendix: The ℓ functor for non homogeneous classes

Actually, the ℓ functor is not really limited to homogeneous elements. Let a be a not necessarily homogeneous element. Write it as sum of homogeneous elements a_{i} ∈ A_{i}. We define the following elements in ℓ(A).

ϕ(∑_{i} a_{i}) = ∑_{i} ϕ(a_{i})
δ(∑_{i} a_{i}) = ∑_{i} δ(a_{i})
q(∑_{i} a_{i}) = ∑_{i} q(a_{i}) + ∑_{i<j} δ(a_{i}a_{j})

Theorem 11.1. These classes satisfy all the relations (13)-(23).

Proof. The relations (13), (14), (16), (17), (19), (21), (23) only involve the additive operations λ, ϕ, uq(−) and δ, and follow immediately from linearity. The relations (15), (18), (20) and (22) involve the quadratic operation q, so they each require a computation.

(15) We compute:

q(∑_{i} a_{i} + ∑_{i} b_{i}) = q(∑_{i} (a_{i} + b_{i}))

= ∑_{i} q(a_{i} + b_{i}) + ∑_{i<j} δ((a_{i} + b_{i})(a_{j} + b_{j}))
= ∑_{i}(q(a_{i}) + q(b_{i}) + δ(a_{i}b_{i}))

+ ∑_{i<j}(δ(a_{i}a_{j}) + δ(b_{i}b_{j}) + δ(a_{i}b_{j}) + δ(b_{i}a_{j}))

= q(a) + q(b) + ∑_{i} δ(a_{i}b_{i}) + ∑_{i<j}(δ(a_{i}b_{j}) + δ(b_{i}a_{j}))
= q(a) + q(b) + δ((∑_{i} a_{i})(∑_{j} b_{j}))
= q(a) + q(b) + δ(ab)

(18) This is the most complicated computation. We do it in two steps, first assuming that b is a homogeneous element, but a = ∑_{i} a_{i}. Under this assumption:

q(ab) = q(∑_{i} a_{i}b)
= ∑_{i} q(a_{i}b) + ∑_{i<j} δ(a_{i}ba_{j}b)
= ∑_{i} (ϕ(a_{i})q(b) + q(a_{i})ϕ(b)) + ∑_{i<j} ϕ(b)δ(a_{i}a_{j})
= ( ∑_{i} ϕ(a_{i}))q(b) + ϕ(b)( ∑_{i} q(a_{i}) + ∑_{i<j} δ(a_{i}a_{j}))
= ϕ(a)q(b) + q(a)ϕ(b)

In the general case, we do a very similar calculation, and taking into account that in addition to the special case above, we have also proved relation (15) in the general case.

q(ab) = q(∑_{i} ab_{i})
= ∑_{i} q(ab_{i}) + ∑_{i<j} δ(ab_{i}ab_{j})
= ∑_{i}(q(a)ϕ(b_{i}) + ϕ(a)q(b_{i})) + ∑_{i<j} ϕ(a)δ(b_{i}b_{j})
= q(a)ϕ(b) + ϕ(a)q(b)

The first step in proving relation (20) is to notice that

δ(a^{2}λ(a)) = δ((∑_{i} a_{i})^{2} ∑_{j} λ(a_{j})) = δ(∑_{i,j} a_{i}^{2}λ(a_{j}))

The rest is calculation:

q(a)^{2} = ( ∑_{i} q(a_{i}) + ∑_{i<j} δ(a_{i}a_{j}))^{2}
= ∑_{i} q(a_{i})^{2} + ∑_{i<j} δ(a_{i}a_{j})^{2}
= ∑_{i} (ϕ(λ(a_{i})) + δ(a_{i}^{2}λ(a_{i}))) + ∑_{i<j} δ(λ(a_{i}a_{j}))
= ϕ(λ(a)) + δ( ∑_{i} a_{i}^{2}λ(a_{i}) + ∑_{i<j}(a_{i}^{2}λ(a_{j}) + a_{j}^{2}λ(a_{i})))
= ϕ(λ(a)) + δ( ∑_{i,j} a_{i}^{2}λ(a_{j}))
= ϕ(λ(a)) + δ(a^{2}λ(a))

To prove (22), we first note that the relation is additive in a, so we can as well assume that a is homogeneous.

δ(a)q(b) = δ(a)( ∑_{i} q(b_{i}) + ∑_{i<j} δ(b_{i}b_{j}))
= ∑_{i} δ(a)q(b_{i}) + ∑_{i<j} δ(a)δ(b_{i}b_{j})
= ∑_{i} δ(aλ(b_{i})) + ∑_{i} δ(ab_{i})δ(b_{i}) + ∑_{i<j} δ(a)δ(b_{i}b_{j})

Now we use relation (16) on the last term! We get

δ(a)q(b) = ∑_{i} δ(aλ(b_{i})) + ∑_{i} δ(ab_{i})δ(b_{i}) + ∑_{i<j}(δ(b_{i})δ(ab_{j}) + δ(b_{j})δ(ab_{i}))
= δ(aλ(b)) + ∑_{i,j} δ(b_{i})δ(ab_{j})
= δ(aλ(b)) + δ(ab)δ(b)

Given a not necessarily graded algebra A, we can define a new functor ℓ′(A) to be the ring generated by the symbols δ(a), ϕ(a), q(a), where we now allow any a ∈ A, and satisfying relations (13)-(23).

If A is indeed graded there is a forgetful map

F : ℓ(A) → ℓ′(A)

The previous computations can be interpreted as saying that there is a ring map G : ℓ′(A) → ℓ(A), mapping q(a) etc, to the sums defined above. It is easy to check that both F ∘ G and G ∘ F are the identity on generators, so F and G identify the two constructions.

So we see that in this way we can extend ℓ to a functor on all 𝔽_{2}-algebras with derivation over Frobenius.

12 Appendix: Generators and relations for the algebra L(A)

Suppose that we have somehow computed L(A) for a ring A. Let B = A/I. The purpose of this section is to give an description of L(B). In conjunction with the computation in the last paragraph of L(A) for a polynomial ring A this gives a method of computing L(B) in general.

Let B = A/I where I is an ideal, p : A → B the natural map sending an element to its equivalence class. Let {x_{i}} be a set of generators for I. Let {y_{i}} be a set of generators of I as a module over A with the Frobenius action, that is, so that any element m ∈ I can be written as a sum

m = ∑ b_{i}^{2}y_{i}

If A is a Noetherian ring, we can always choose the sets {x_{i}} and {y_{i}} to be finite.

Theorem 12.1. L(B) = L(A)/J where J is the ideal generated by the classes ϕ(x_{i}), q(x_{i}) and δ(y_{i}).

Proof. From the definition it follows that L(A) → L(B) is surjective. More specifically, L(B) ≅ L(A)/J′ where J′ is the ideal generated by ϕ(a) − ϕ(b), δ(a) − δ(b), q(a) − q(a) for any pair of classes a, b which satisfy that p(a) = p(b). It is clear that J ⊂ J′.

The idea of the proof is to show that the ideal J′ is also generated by the classes listed in the theorem, so that J = J′. That, is we have to show that any of the generators for J′ that we just listed are also elements of J

First we note that ϕ(x) ∈ J and δ(x) ∈ J for all x ∈ I. This follows from the formulas

ϕ(∑ a_{i}x_{i}) = ∑ ϕ(a_{i})ϕ(x_{i}) , δ(∑ b_{i}^{2}y_{i}) = ∑ ϕ(b_{i})δ(y_{i})

Assume that a − b ∈ I. By linearity and the above remark we see that ϕ(a) − ϕ(b) and δ(a) − δ(b) lies in J. Hence we only need to verify that q(a) − q(b) ∈ J. We have

q(a) − q(b) = q(a − b) + δ(ab)

We check that each of the two terms lies in J. Write a − b = ∑ a_{i}x_{i}. Then

q(a − b) = ∑_{i} q(a_{i}x_{i}) + ∑_{i<j} δ(a_{i}a_{j}x_{i}x_{j})

Since q(a_{i}x_{i}) = ϕ(a_{i})q(x_{i})+q(a_{i})ϕ(x_{i}) the first sum is in J. Since a_{i}a_{j}x_{i}x_{j} ∈ I the last sum also lies in J by the remark above. Finally we can use that δ(a^{2}) = 0 to get δ(ab) = −δ(a(a − b)). This lies in J since a(a − b) ∈ I.

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Department of Mathematics University of Aarhus Ny Munkegade DK-8000 Århus C Denmark Email: marcel@mi.aau.dk

Département de Mathématiques Université de Paris Nord, URA 742 du CNRS Avenue Jean-Baptiste Clément F-93430 Villetaneuse France Email: ottosen@math.univ-paris13.fr