THE COBAR CONSTRUCTION AS A HOPF

ALGEBRA AND THE LIE DIFFERENTIAL

Hans-Joachim Baues

Abstract. We show that the integral cobar construction ΩC_{∗}X of a 1-reduced simplicial set X has the structure of a homotopy Hopf algebra. This leads to a rational differential Lie algebra L(X) given by a differential d_{L} on the free Lie algebra L(s^{−1}C̃_{∗}X). The universal enveloping algebra UL(X) is naturally isomorphic to ΩC_{∗}X ⊗ ℚ.

Let X be a 1-reduced simplicial set and let s^{−1}C̃_{∗}X be the desuspension of the reduced chain complex C̃_{∗}X of X. The cobar construction

ΩC_{∗}X = (T (s^{−1}C̃_{∗}X), d_{Ω})

is given by a differential d_{Ω} on the tensor algebra T (s^{−1}C̃_{∗}X) defined over ℤ. The homology of ΩC_{∗}X coincides with the homology of the loop space Ω|X|. A formula for d_{Ω} was given by Adams [Ad]. We show that the cobar construction ΩC_{∗}X has over ℤ the additional natural structure of a “homotopy Hopf algebra” consisting of a coassociative diagonal

ψ : Ω(C_{∗}X) → Ω(C_{∗}X) ⊗ Ω(C_{∗}X)

and a derivation homotopy E : T ψ ≃ ψ. Explicit formulas for ψ and E are given in (2.9). The result is based on the geometric cobar construction in [B]. Using the fundamental theory of Anick [A] we derive from the data (ΩC_{∗}X, ψ, E) a differential Lie algebra

L(X) = (L(s^{−1}C̃_{∗}X), d_{L})

which is defined by a differential d_{L} on the free rational Lie algebra L(s^{−1}C̃_{∗}X). The homology of L(X) coincides with the rational homotopy groups of Ω|X|.

Moreover we describe a cocommutative diagonal

ϕ : (ΩC_{∗}X) ⊗ ℚ → (ΩC_{∗}X) ⊗ (ΩC_{∗}X) ⊗ ℚ

together with a derivation homotopy H : ψ ⊗ ℚ ≃ ϕ and a natural isomorphism of differential Hopf algebras

1991 \underline{Mathematics Subject Classification} 55P62, 55U10, 55P35. \underline{Key words and phrases}. Cobar construction, Quillen’s differential Lie algebra of a space, Anick’s Hopf algebra up to homotopy, geometric cobar construction, cubical chain complex.

Typeset by AMS-TEX

β : UL(X) ≅ (ΩC_{∗}X ⊗ ℚ, ϕ)

Formulas for ϕ, H and β are obtained by an induction. A map f : X → Y between 1-reduced simplicial sets induces the map F_{∗} = L(s^{−1}C̃_{∗}f) : L(s^{−1}C̃_{∗}X) → L(s^{−1}C̃_{∗}Y) between free Lie algebras which is compatible with the differential d_{L}. Hence X ↦ L(X) yields a canonical functor from 1-reduced simplicial sets to free differential Lie algebras. This functor induces an isomorphism of rational homotopy categories. In fact, L is up to homotopy naturally equivalent to the functor λ = NP Q̂G of Quillen [Q] which is the composite of four intricate functors. We describe in section 5 inductively a sequence of elements β(2), β(3), … which determine the differential d_{L} and we compute the quadratic part of d_{L}. A complete formula for d_{L}, however, is highly intricate since it can be considered as a kind of “simplicial form” of the classical Hausdorff formula; see (4.9) (4).

We cannot expect a simple form of the differential d_{L} since the homology of L(X) yields the rational homotopy groups of X; that is

π_{∗+1}(X) ⊗ ℚ = H_{∗}L(X) = kernel d_{L}/image d_{L}

This supplements the result of Kan [K] who gave a combinatorial description of π_{∗}(X) in terms of the free simplicial group GX. Hence π_{∗}(X) requires computations in free groups while the formula for π_{∗}(X)⊗ℚ requires only the computation of the homology of a chain complex of rational vector spaces. We point out that L(X) is a chain complex of finite type if X has only finitely many non degenerate simplices. Hence in this case d_{L} yields a finite formula for π_{n}(∥ X ∥) ⊗ ℚ. Such a formula was not yet achieved in the literature.

For example using the Sullivan method one obtains π_{∗}(X) ⊗ ℚ only implicitly by computing a minimal model of the De Rham algebra A(X); see Bousfield-Gugenheim [BG]. Here A(X) is not of finite type if X has only finitely many non degenerate simplices so that A(X) does not give finite formula for π_{n}(∥ X ∥) ⊗ ℚ. In this case the minimal model M_{X} of A(X) is of finite type and hence also gives a finite formula for π_{n}(∥ X ∥) ⊗ ℚ. There is, however, no explicit formula for M_{X} in the literature and M_{X} is not natural in X.

On the other hand also the model λ(X) of Quillen is considerably larger than L(X) above and is only defined as the normalization of a simplicial object; see (4.9) (4).

§ 1 Algebras, coalgebras and Hopf algebras

We fix in this section the main algebraic definitions we shall use; compare for example [MM, HMS, A, BL]. Let R be a commutative ring with unity 1 which we use as ring of coefficients. A \underline{graded module} is a sequence V = \{V_{n}, n ∈ ℤ\} of R-modules V_{n}. The ring R is the graded module which is R in degree 0 and which is trivial otherwise. A map f : V → W of degree |f| = r between graded modules is a sequence of R-linear maps f_{n} : V_{n} → W_{n+r}, n ∈ ℤ. Let s^{r}V be the suspension of Y defined by (s^{r}V)_{n} = V_{n−r} and let s^{r} : V → s^{r}V be the map of degree r obtained by the identity V_{n} = (sV)_{n+r}. We define the tensor product V ⊗ W by (V ⊗ W)_{n} = ⊕{V_{i} ⊗ V_{j}, i + j = n}. The interchange isomorphism

T : V ⊗ W ≅ W ⊗ V

carries x ⊗ y to (−1)^{|x||y|}y ⊗ x. Here we write |x| = n if x ∈ V_{n} and |x| is termed the degree of x ∈ V. For maps f : V → V′ and g : W → W′ we obtain f ⊗ g : V ⊗ W → V′ ⊗ W′ by

(f ⊗ g)(x ⊗ y) = (−1)^{|g||x|}f(x) ⊗ g(y)

A \underline{differential module} V is a graded module together with a map d : V → V of degree −1 satisfying dd = 0. The tensor product of differential modules V, W is the graded module V ⊗ W with the differential d ⊗ 1 + 1 ⊗ d; that is d(x ⊗ y) = (dx) ⊗ y + (−1)^{|x|}x ⊗ dy. The maps f : V → W between differential modules are maps of degree 0 compatible with the differential, that is df = fd. We say that V is positive if V_{n} = 0 for n < 0. The positive differential modules and maps define a category DM^{+} which by the tensor product and the unit object R is a monoidal category termed (DM^{+}, ⊗, R). A \underline{homology isomorphism} is a map f : V → W in DM^{+} which induces an isomorphism in homology HV ≅ HW where HV = kernel (d)/image (d).

(1.1) Definition. A monoid in DM^{+} is a positive differential module A together with an associative multiplication μ : A⊗A → A, a⊗b ↦ a·b, and a unit 1 : R → A. In particular R is such a monoid. A \underline{differential algebra} A is a monoid (A, d, μ, 1) in DM^{+} together with a map ε : A → R of monoids termed the augmentation of A. Let Ā = ker(ε) ∈ DM^{+} be the augmentation ideal. Then μ defines the map μ̄ : Ā⊗Ā → Ā in DM^{+}. We say that A is \underline{connected} if 1 : R ≅ A_{0} is an isomorphism. A map f : A → B between differential algebras is a map DM^{+} compatible with the multiplication, unit and augmentation. Let DA be the category of differential algebras and such maps. For A, B ∈ DA we obtain A ⊗ B ∈ DA by the tensor product of the underlying differential modules. The multiplication of A ⊗ B is defined by (μ ⊗ μ)(1 ⊗ T ⊗ 1); that is

(a ⊗ b) · (c ⊗ d) = (−1)^{|b||c|}(a · c) ⊗ (b · d)

and 1 = 1 ⊗ 1 and ε = ε ⊗ ε are unit and augmentation of A ⊗ B. Therefore (DA, ⊗, R) is again a monoidal category.

The next definition of coalgebras is dual to the definition of algebras in (1.1).

(1.2) Definition. A comonoid in DM^{+} is a positive differential module C together with an associative comultiplication (diagonal) Δ : C → C ⊗ C and a counit 1 : C → R. In particular R is such a comonoid. A \underline{differential coalgebra} C is a comonoid (C, d, Δ, 1) in DM^{+} together with a map η : R → C of comonoids termed the coaugmentation of C. Let C̃ = coker (η) be the cokernel of η in DM^{+} and let Δ̃ : C̃ → C̃ ⊗ C̃ be the map in DM^{+} induced by Δ. The map Δ̃ is called the reduced diagonal of C. We say that C is \underline{connected} if 1 : C_{0} ≅ R is an isomorphism. A map f : C → D between differential coalgebras is a map in DM^{+} compatible with the comultiplication, counit and coaugmentation. Let DC be the category of differential colagebras and such maps.

Let V be a positive graded module. Then we obtain the \underline{tensor algebra} T (V) which as graded module is the direct sum in DM^{+}

(1.3)  T (V) = ⨁_{n≥0} V^{⊗n}

where V^{⊗0} = R and V^{⊗n} = V ⊗ … ⊗ V (n-fold). The canonical multiplication μ as T (V) is defined by

μ(a_{1} ⊗ … ⊗ a_{r}, a_{r+1} ⊗ … ⊗ a_{n}) = a_{1} ⊗ … ⊗ a_{n}.

Let i^{n} : V^{⊗n} → T (V) be the inclusion. Then the unit 1 : R → T (V) is given by i^{0} and the augmentation ε : T (V) → R is defined by εi^{0} = 1, εi^{n} = 0 for n ≥ 1. Let E be a graded set with E_{n} = ϕ (= empty set) for n < 0. Then E generates the free graded R-module V = span_{R}(E) and the tensor algebra

(1.4)  T (V) = T (span_{R}(E)) = R < E >

is termed the \underline{free graded R-algebra} generated by E. We call a differential algebra A \underline{free} if the underlying graded algebra of A admits an isomorphism A ≅ T (V) = R < E > of augmented algebras where V is a free graded R-module. Note that A need not be a free object in the category DA.

(1.5) Definition. Let C be a differential coalgebra which is positive and connected. Then the \underline{cobar construction} Ω(C) due to Adams [Ad] (see also [Mo]) is the differential algebra

Ω(C) = (T (s^{−1}C̃), d_{Ω})

with the differential d_{Ω} determined by the restriction

d_{Ω}i^{1} = −i^{1}(s^{−1}ds) + i^{2}(s^{−1} ⊗ s^{−1})Δ̃s.

If C_{1} = 0 then Ω(C) is connected. Moreover if C is free as a graded R-module then Ω(C) is free. Let DC_{0} be the full subcategory of DC consisting of connected differential coalgebras. The cobar construction is a functor Ω : DC_{0} → DA. Clearly a map f : C → D in DC_{0} is carried to Ω(f) = T (s^{−1}f̃) where f̃: C̃ → D̃ is induced by f.

(1.6) Definition. A \underline{differential Hopf algebra} is a differential algebra A together with an associative comultiplication (diagonal)

Δ : A → A ⊗ A

which is a map in DA such that (A, Δ) is a comonoid in the monoidal catgeory (DA, ⊗). Moreover the counit of (A, Δ) coincides with the augmentation ε : A → R and the unit 1 : R → A is a coaugmentation of (A, Δ) so that (A, Δ) is a coalgebra in the sense of (1.2). The Hopf algebra is \underline{cocommutative} if T Δ = Δ. Let HA be the category of differential Hopf algebras. Morphisms are maps in DA which are compatible with Δ; that is (f ⊗ f)Δ = Δf.

We shall use the following notion of homotopy for maps in DA.

(1.7) Definition. Let f, g : A → B be maps between differential algebras in DA. A map F : A → B of degree r between the underlying graded modules is an (f, g)- \underline{derivation} if it satisfies

F (xy) = F (x)g(y) + (−1)^{r|x|}f(x)F (y)

A \underline{derivation homotopy} from f to g is an (f, g) -derivation F of degree +1 satisfying

dF + F d = f − g

We write F : f ≃ g. The relation of derivation homotopy is an equivalence relation if A is free and connected.

Anick in § 2 [A] uses a more general notion of homotopy but in this paper we need only derivation homotopies. In particular such homotopies are used in the following definition of homotopy Hopf algebras.

(1.8) Definition. A \underline{homotopy Hopf algebra} (A, ψ, G_{1}, G_{2}) is a connected differential algebra A together with a map ψ : A → A⊗A in DA such that the augmentation ε : A → R is a counit for ψ (that is (ε ⊗ 1)ψ = 1 and (1 ⊗ ε)ψ = 1) and

G_{1} : T ψ ≃ ψ
G_{2} : (ψ ⊗ 1)ψ ≃ (1 ⊗ ψ)ψ

are derivation homotopies. A map f : (A, ψ, G_{1}, G_{2}) → (A′, ψ′, G′_{1}, G′_{2}) between homotopy Hopf algebras is a map f : A → A′ in DA which commutes with diagonals and homotopies; that is (f ⊗ f)ψ = ψ′f, (f ⊗ f)G_{1} = G′_{1}f, (f ⊗ f ⊗ f)G_{2} = G′_{2}f. Let hHA_{0} be the category of such homotopy Hopf algebras and maps.

One has the canonical functors

(1.9)  CHA_{0} ⊂ hHA_{0} ⟶^{π} DA_{0}

Here CHA_{0} ⊂ HA is the full subcategory of connected cocommuttaive objects and DA_{0} ⊂ DA is the full subcategory of connected objects. The inclusion carries (A, Δ) to (A, Δ, G_{1} = 0, G_{2} = 0) and the forgetful functor π carries (A, ψ, G_{1}, G_{2}) to A. We point out that any homotopy Hopf algebra above yields canonically a “Hopf algebra up to homotopy” in the sense of Anick; see 4.1 in [A].

§ 2 The cobar construction on a simplicial set

Let ℕ = {0, 1, 2, … } be the set of natural numbers and let Set be the category of sets. The simplicial category Δ is the subcategory of Set with objects the finite sets

(2.1)  {0 … n} = {t ∈ ℕ, 0 ≤ t ≤ n} with n ∈ ℕ.

Morphisms in Δ are the functions g : {0 … m} → {0 … n} which are order preserving; that is i ≤ j implies g(i) ≤ g(j). A \underline{simplicial set} X is a contravariant functor X : Δ → Set. A map X → Y between simplicial sets is a natural transformation of such functors. Let sSet be the category of simplicial sets and maps between them. We write for X ∈ sSet also X_{n} = X({0 … n}) so that X determines a graded set {X_{n}} also denoted by X. For a subset

a = {a_{0} < a_{1} … < a_{r}} ⊂ {0 … n}

we obtain the unique order preserving injective function i_{a} : {0 … r} → {0 … n} with image(i_{a}) = a. We define for σ ∈ X_{n} the element

(2.2)  σ(a) = σ(a_{0}, a_{1}, …, a_{r}) = i_{a}^{∗}(σ) ∈ X_{r}

The subset ∂_{i}a = a − \{a_{i}\} with i = 0, …, r is termed a \underline{face} of a. This yields the faces of σ ∈ X_{n} by

d_{i}(σ) = σ(∂_{i}{0 … n}) ∈ X_{n−1}

for i = 0, 1, …, n. An element σ ∈ X_{n} is \underline{degenerate} if there exists a surjective function g : {0 … n} → {0 … m} in Δ with m < n such that σ ∈ image (g^{∗} : X_{m} → X_{n}). Let (DX)_{n} be the subset of X_{n} consisting of degenerate elements. Hence DX ⊂ X is an inclusion of graded sets. We obtain the \underline{normalized chain complex} of X by the quotient of graded R-modules

(2.3)  C_{∗}X = span_{R}(X)/span_{R}(DX)

This is a positive differential module with differential

d(σ) = ∑_{i=0}^{n}(−1)^{i}d_{i}(σ)

We say that X is r-\underline{reduced} if X_{0} = X_{1} = … = X_{r} = \{∗\} consists of a single point ∗. If X is 0-reduced we obtain the \underline{Alexander-Whitney} diagonal

Δ : C_{∗}X → C_{∗}X ⊗ C_{∗}X

Δ(σ) = ∑_{i=0}^{n} σ(0, …, i) ⊗ σ(i, …, n) (2.4)

Here X_{0} = {∗} implies that C_{0}X = ℤ and one can check that C_{∗}X = (C_{∗}X, d, Δ) is a connected differential coalgebra. Moreover C_{∗}X as a graded R-module is freely generated by the graded set EX = X − DX of non-degenerate elements in X. The cobar construction Ω(C_{∗}X) on the coalgebra C_{∗}X is the free algebra

Ω(C_{∗}X) = T (s^{−1}C̃_{∗}(X)) = R⟨s^{−1}ẼX⟩

generated by the desuspension of the graded set ẼX = EX − {∗}. The differential d = d_{Ω} is defined for σ ∈ X_{n} by

d(s^{−1}σ) = − ∑_{i=0}^{n}(−1)^{i}s^{−1}(d_{i}σ)

+ ∑_{i=1}^{n−1}(−1)^{i}s^{−1}σ(0, …, i) ⊗ s^{−1}σ(i, …, n) (2.5)

This is the explicit formula derived from (1.4) and (2.4). Any degenerate element in the formula represents the zero element. The cobar construction yields the functor

(2.6)  ΩC_{∗} : sSet_{1} → DA_{0}

Here sSet_{1} is the category of 1-reduced simplicial sets. Adam’s result in [Ad] shows that for X ∈ sSet_{1} there is a natural isomorphism of algebras

(2.7)  H_{∗}(ΩC_{∗}X) ≅ H_{∗}(Ω|X|, R)

where Ω|X| is the loop space of the realization |X| and where H_{∗}(−, R) denotes the singular homology with coefficients in R.

We are going to describe the natural structure of a homotopy Hopf algebra for ΩC_{∗}X. For this we introduce the following notation. The number of elements of a subset a ⊂ ℕ is denoted by ♯(a). Let

(2.8)  N = {a; a ⊂ ℕ, 2 ≤ ♯(a) < ∞}

be the set of finite subsets of ℕ with at least 2 elements. We define the \underline{degree} of a ∈ N by |a| = ♯(a) − 2. Moreover we define for a, b ∈ N

(1) a ≤ b ⇔ a ⊂ b and Min(a) = Min(b) and Max(a) = Max(b)

where Min and Max denote the minimum and maximum respectively. Let N^{∗} be the \underline{free monoid} generated by the set N. The elements of N^{∗} are words

(2)  w = a^{1}a^{2} … a^{k} with a^{i} ∈ N for i = 1, …, k

Here k = l(w) is termed the \underline{length} of W and the \underline{degree} of w is the sum |w| = |a^{1}|+ …+|a^{k}| of degrees. The product wv of words v, w ∈ N^{∗} is given by \underline{concatenation}. For n, m ∈ ℕ with n < m we obtain the \underline{interval}

{n … m} = {t ∈ ℕ, n ≤ t ≤ m}

Moreover each b ∈ N determines by its total ordering b = {b_{0} < … < b_{r}} with r = |b| + 1 the \underline{word of intervals}

(3)  b̄ = {b_{0} … b_{1}}{b_{1} … b_{2}} … {b_{r−1} … b_{r}}

This gives us the function N → N^{∗} which carries b to b̄. We define for b ∈ N and w in (2) the words

(4)  w|_{b} = (a^{1} ∩ b)(a^{2} ∩ b) … (a^{k} ∩ b) and

(5)  \{ \begin{matrix}b ∪ w & = (b ∪ a^{1})a^{2}a^{3} … a^{k} \\ w ∪ b & = a^{1}a^{2} … a^{k−1}(a^{k} ∪ b). \end{matrix}

We shall use the word w |_{b} in (3.5) (3) where the letters of w |_{b} are again elements in N. Here a ∩ b and a ∪ b denote the intersection, resp. union of the sets a, b ⊂ ℕ. Moreover the \underline{carrier} of a word w = a^{1} … a^{k} ∈ N^{∗} is the union of letters of w; that is,

(6)  car(w) = a^{1} ∪ … ∪ a^{k} ∈ N.

The \underline{sign} function τ : N^{∗} → \{+1, −1\} is defined by τ(a) = 1 for a ∈ N and

(7)  τ(wv) = (−1)^{l(w)|v|}τ(w) · τ(v) for w, v ∈ N^{∗}

or equivalently by τ(w) = (−1)^{t} with

t = ∑_{i=1}^{k}(i − 1)|a^{i}|.

(2.9) Definition. Let X be a 1-reduced simplicial set and let σ ∈ X_{n}, n ≥ 2. For w ∈ N^{∗} with car(w) ⊂ {0 … n} let

(1)  (w)_{σ} = s^{−1}σ(a^{i_{1}}) ⊗ … ⊗ s^{−1}σ(a^{i_{s}}) ∈ T (s^{−1}C̃_{∗}X)

where i_{1} < … < i_{s} are exactly those indices of the letters a^{1}, …, a^{k} in w = a^{1} … a^{k} for which |a^{i_{1}}|, …, |a^{i_{s}}| > 0. If |a^{i}| = 0 for all i = 1, …, k or equivalently if |w| = 0 we set (w)_{σ} = 1 where 1 is the unit in the tensor algebra T (s^{−1}C̃_{∗}X). The degree of the element (w)_{σ} in the tensor algebra coincides with the degree of the word w in (2.8). For σ ∈ X_{n}, n ≥ 2, we have s^{−1}σ = ({0 … n})_{σ} and the differential d in the cobar construction ΩC_{∗}X is given by the formula (see (2.5))

d(s^{−1}σ) = − ∑_{i=0}^{n}(−1)^{i}(∂_{i}{0 … n})_{σ}

+ ∑_{i=2}^{n−2}(−1)^{i}({0 … i}{i … n})_{σ}

We define a diagonal

(2)  ψ : Ω(C_{∗}X) → Ω(C_{∗}X) ⊗ ΩC_{∗}X

for σ ∈ X_{n} by

(3)  ψ(s^{−1}σ) = ∑_{f≤{0...n}} τ(f̄) · (f̄)_{σ} ⊗ (f)_{σ}

We define a derivation homotopy E : T ψ ≃ ψ by

(4)  E(s^{−1}σ) = ∑_{i=1}^{n−1} ∑_{\begin{matrix}f≤\{0...i\} \\ g≤\{i...n\} \end{matrix}} τ · (f ∪ ḡ)_{σ} ⊗ (f̄∪ g)_{σ}

where the sign τ ∈ {+1, −1} is

(5)  τ = τ(f̄)τ(ḡ) · (−1)^{(|f|+i)(|g|+n)}

In (3) and (4) we use the notation in (2.8) (1) ... (7).

(2.10) Theorem. Let X be a 1-reduced simplicial set. Then the cobar construction ΩC_{∗}X is a well defined homotopy Hopf algebra with diagonal ψ defined in (2.9). This diagonal is coassociative and the augmentation ε : ΩC_{∗}X → R is a counit for ψ. Moreover E in (2.9) is a derivation homotopy E : T ψ ≃ ψ. Both ψ and E are natural in X ∈ sSet_{1} so that we get the commutative diagram of functors

sSet_{1}  ⟶^{ΩC_{∗}} DA_{0}
(ΩC_{∗})_{ψ,E} ↘  ↗ π
hHA_{0}

with (ΩC_{∗})_{ψ,E}(X) = (ΩC_{∗}(X), ψ, G_{1} = E, G_{2} = 0); see (1.8).

We prove this result together with theorem (3.5) below.

Remark. We obtained the diagonal ψ of the cobar construction ΩC_{∗}X in (2.9) already in [B] so that only the explicit formula for the derivation homotopy E is new. Majewski [M] independently showed that ΩC_{∗}X has the natural structure of a homotopy Hopf algebra; his diagonal however is not associative and he does not describe explicit formulas. The diagonal ψ in (2.9) was used for the double cobar construction and double bar construction in [B′]. Gerstenhaber-Voronov [GV] showed that the diagonal ψ corresponds to a “homotopy Gerstenhaber-algebra structure” on the cochain complex C^{∗}X of the simplicial set X; see corollary 6 in [GV]. Moreover Getzler-Jones [GJ] showed that this structure induced by ψ is a special “B_{∞} -algebra structure” on C^{∗}X. Therefore it is likely that theorem (2.10) and the explicit formula for the homotopy E has implication for the theory of homotopy Gerstenhaber algebras and B_{∞} -algebras respectively.

§ 3 Cubical chains

For the proof of theorem (2.10) we need the cubical chain algebra of a topological monoid and the geometric cobar construction introduced in [B]. We begin by introducing classical formalism concerning cubes and cubical chains; see [HW, AH, A].

Let CU_{∗}(Y) denote the \underline{cubical singular chain complex} of a topological space Y with coefficients in R. To construct CU_{∗}(Y), take the R-module generated freely by all the singular n-cubes (i.e. continuous maps σ : I^{n} → Y, where I^{n} is the unit cube), and divide out by the R-submodule generated by degenerate cubes. (Here σ is degenerate if it factors through any of the n standard projections I^{n} → I^{n−1}.) As is well known CU_{∗} is naturally chain equivalent with the functor CS_{∗} of simplicial singular chains and H_{n}(Y; R) = H_{n}CU_{∗}(Y) is the singular homology of Y. For a CW-complex Y let C_{∗}(Y) be the \underline{cellular chain complex} with C_{n}(Y) = H_{n}(Y^{n}, Y^{n−1}, R) where Y^{n} is the n-skeleton of Y. The set of n-cells of Y is a basis in the free module C_{n}(Y).

The unit interval I = I^{1} is a CW-complex with two 0-cells e′ = {0}, e′′ = {1} and one 1-cell e. Let {e, e′, e′′}^{∗} be the free graded monoid generated by e, e′, e′′ with degree |e| = 1, |e′| = |e′′| = 0 and let

(3.1)  I(∞) = R⟨e, e′, e′′⟩ = span_{R}{e, e′, e′′}^{∗}

be the corresponding free graded algebra. Moreover I(∞) is a free differential algebra with differential

(1)  d(e) = e′′ − e′

A second grading of I(∞) is given by \underline{word length} l with l(e) = l(e′) = l(e′′) = 1. We denote a word of length n in I(∞) by

(2)  (a, b)_{n} = e_{1} … e_{n}

where a, b ⊂ {1 … n} are disjoint subsets with e_{i} = e for i ∈ b, e_{i} = e′′ for i ∈ a and e_{i} = e′ for i ∈ {1 … n} − a ∪ b. Let I(n) be the span of all words of length n in I(∞). Then I(n) is a differential submodule of I(∞) and we get maps of differential modules

(3)  I(n) ≅ C_{∗}(I^{n}) ⊂ CU_{∗}(I^{n})

Here C_{∗}(I^{n}) is the cellular chain complex of the cube I^{n} and the isomorphism carries the generator (2) to the corresponding cell e_{1} × … × e_{n} of dimension ♯b. The inclusion in (3) carries this cell to the face I^{♯b} ⊂ I^{n} which is the closure of e_{1} × … × e_{n}. The differential d(σ) of the singular cube (σ : I^{n} → Y) ∈ CU_{∗}(Y) is

(4)  d(σ) = CU_{∗}(σ)(de^{n})

where e^{n} = (ϕ, {1 … n})_{n} is the n-dimensional cell in I^{n}. There is a unique map in DA

(5)  ψ_{I} : I(∞) → I(∞) ⊗ I(∞)

given by ψ_{I}(e′) = e′ ⊗ e′, ψ_{I}(e′′) = e′′ ⊗ e′′ and ψ_{I}(e) = e′ ⊗ e + e ⊗ e′′. The map ψ_{I} restricted to I(n) is induced by a cellular map I^{n} → I^{n} × I^{n} which is homotopic to the diagonal. In fact ψ_{I} is coassociative and we can define an explicit derivation homotopy E_{I} : T ψ_{I} ≃ ψ_{I} by setting

(6)  E_{I}(e′) = E_{I}(e′′) = 0 and E_{I}(e) = e ⊗ e.

One readily checks that dE_{I}(e) + E_{I}d(e) = T ψ_{I}(e) − ψ_{I}(e).

(3.2) Definition. Let CU′_{∗}(Y) be the first \underline{Eilenberg subcomplex}; i.e. the subcomplex of CU_{∗}(Y) generated by all singular cubes which send the 0-skeleton of I^{n} to the base point ∗ of Y. Assume that Y is a topological monoid with multiplication μ : Y × Y → Y and unit ∗ ∈ Y. We obtain an algebra structure of CU′_{∗}(Y) by

(1)  σ · σ′ = μ(σ × σ′) : I^{n+m} = I^{n} × I^{m} → Y × Y → Y

for the singular cubes σ, σ′ in CU′_{∗}(Y). We define a diagonal ψ on CU′_{∗}(Y) by

(2)  ψ(σ) = (CU′_{∗}(σ) ⊗ CU′_{∗}(σ))(ψ_{I}e^{n})

and we define a derivation homotopy E : T ψ ≃ ψ by

(3)  E(σ) = (CU′_{∗}(σ) ⊗ CH′_{∗}(σ))(E_{I}e^{n})

Here we use ψ_{I} and E_{I} defined in (3.1) (5), (6).

Let Mon_{0} be the category of path connected topological monoids.

(3.3) Proposition. For Y ∈ Mon_{0} the singular cubical chain complex CU′_{∗}(Y) is a well defined homotopy Hopf algebra with diagonal ψ defined in (3.2). This diagonal is associative and the augmentation ε : CU′_{∗}(Y) → R is a counit of ψ. Moreover E in (3.2) is a derivation homotopy E : T ψ ≃ ψ. Both ψ and E are natural in Y ∈ Mon_{0} so that we get the commutative diagram of functors

\begin{matrix}Mon_{0} & ⟶^{CU′_{∗}} & DA_{0} \\ (CU′_{∗})_{ψ,E} ↘ & & ↗ π \end{matrix}

hHA_{0}

with (CU′_{∗})_{ψ,E}(X) = (CU′_{∗}(X), ψ, G_{1} = E, G_{2} = 0); see (1.8).

This result is essentially well known; compare for example [HW] and section 7 [A]. In [B] we define the \underline{geometric cobar construction} which is a functor

(3.4)  \underline{Ω} : sSet_{1} → Mon_{0}

where \underline{Ω}X is a CW-complex with \underline{Ω}X ≃ Ω|X|. The cells of \underline{Ω}X have canonical characteristic maps I^{n} → \underline{Ω}X defined on a cube. This defines the inclusion

C_{∗}(\underline{Ω}X) ⊂ CU′_{∗}(\underline{Ω}X)

where C_{∗}(\underline{Ω}X) is the cellular chain complex of \underline{Ω}X. Moreover we have by [B] the natural isomorphism

Ω(C_{∗}X) ≅ C_{∗}(\underline{Ω}X)

of differential algebras. Now (3.3) and (2.10) fit together by the following result.

(3.5) Theorem. Let X ∈ sSet_{1}. Then the inclusion of differential algebras

I : Ω(C_{∗}X) = C_{∗}(\underline{Ω}X) ⊂ CU′_{∗}(\underline{Ω}X)

is compatible with the diagonals ψ and derivation homotopies E defined in (2.9) and (3.2) respectively. Hence one has an inclusion

i : (ΩC_{∗})_{ψ,E}(X) ⊂ (CU′_{∗})_{ψ,E}(\underline{Ω}X)

in the category hHA_{0} which induces an isomorphism in homology and which is natural in X. Here (ΩC_{∗})_{ψ,E} and (CU′_{∗})_{ψ,E} are the functors in (2.10) and (3.3).

Proof of (3.5) and (2.10). Let n ≥ 2. For each non-degenerate element σ ∈ X_{n} we have a cell e_{σ} in \underline{Ω}X together with a characteristic map, m = n − 1,

(1)  σ̄ : I^{m} → \underline{Ω}X

of e_{σ}. Moreover the following diagram of differential modules commutes:

(2)  \begin{matrix}C_{∗}I^{m} & ⟶^{C_{∗}(σ̄)} & C_{∗}\underline{Ω}X \\ ∥ & & ∥ \\ I(m) & ⟶^{σ_{♯}} & Ω(C_{∗}X) \end{matrix}

Here σ_{♯} carries the generator (a, b) = (a, b)_{m} ∈ I(m) to the element (see (2.9))

(3)  σ_{♯}(a, b) = (c̄|_{d})_{σ}

where c = {0 … n} − a ∪ b and d = {0 … n} − a. Recall that C̄ is the word of intervals defined by c as in (2.8) (3) and that c̄|_{d} is defined by (2.8) (4). The commutativity of (2) is a consequence of III.3.5 in [B]. One can check that σ_{♯} is compatible with differentials in (3.1) and (2.5); that is dσ_{♯} = σ_{♯}d. For the proof of (3.5) it now suffices to show that the following equations hold:

(4)  ψσ_{♯} = (σ_{♯} ⊗ σ_{♯})ψ_{I}

(5)  Eσ_{♯} = (σ_{♯} ⊗ σ_{♯})E_{I}

Moreover it is sufficient to show these equations only on e^{m} ∈ I(m). For this we have

ψ_{I}(e^{m}) = (e′ ⊗ e + e ⊗ e′′)^{m}

= ∑ τ_{a,b}(ϕ, a) ⊗ (a, b) (6)

Here the sum is taken over all partitions (a, b) of {1 … m} and τ_{a,b} is the sign of the permutation (a_{1} < … < a_{r}, b_{1} < … < b_{s}) with r = ♯a, s = ♯b. One can check that this sign coincides with

(7)  τ_{a,b} = τ(f̄), see (2.8) (7),

with f = b ∪ {0, n} ∈ N and f̄ = {0 … b_{1}}{b_{1} … b_{2}} … {b_{s} … n} ∈ N^{∗}. Moreover using the definition of σ_{♯} in (3) we get

(8)  σ_{♯}(ϕ, a) = (f̄)_{σ}

(9)  σ_{♯}(a, b) = (f)_{σ}

Thus we derive from (6) ... (9)

(10)  (σ_{♯} ⊗ σ_{♯})ψ_{I}(e^{m}) = ∑_{f≤{0...n}} τ(f̄)(f̄)_{σ} ⊗ (f)_{σ}

Hence by (10) and (2.9) (3) one gets

(11)  ψ(s^{−1}σ) = (σ_{♯} ⊗ σ_{♯})ψ_{I}(e^{m})

where s^{−1}σ = σ_{♯}(e^{m}). This completes the proof of (4). We proceed similarly for the proof of (5). We have by (1.7) and (3.1) (6)

(12)  E_{I}(e^{m}) = ∑_{i=1}^{m}(−1)^{i−1}T ψ_{I}(e^{i−1})E_{I}(e)ψ_{I}(e^{m−i})

(13)  = ∑_{i=1}^{m}(−1)^{i−1}(∑_{A} τ̄_{a,b}(a, b) ⊗ (ϕ, a)) · (e ⊗ e) · (∑_{B} τ_{c,d}(ϕ, c) ⊗ (c, d))

Here A is the set of all partitions (a, b) of {1 … (i − 1)} and B is the set of all partitions (c, d) of {(i + 1) … m}. Moreover

(14)  τ̄_{a,b} = (−1)^{♯a·♯b}τ_{a,b}

Using the multiplication law in (1.1) we see that the summands in (13) have the form:

((a, b) ⊗ (ϕ, a)) · (e ⊗ e) · ((ϕ, c) ⊗ (c, d))
= (−1)^{♯a}((a, b ∪ {i}) ⊗ (ϕ, a ∪ {i})) · ((ϕ, c) ⊗ (c, d))

= (−1)^{♯a+(1+♯a)♯c}(a, b ∪ {i} ∪ c) ⊗ (c, a ∪ {i} ∪ d) (15)

Hence by (12) ... (15) we get

(16)  E_{I}(e^{m}) = ∑_{i=1}^{m} ∑_{A} ∑_{B} τ · (a, b ∪ {i} ∪ c) ⊗ (c, a ∪ {i} ∪ d)

where τ is the sign

τ = (−1)^{t}τ_{a,b}τ_{c,d} with
(17)  t = i − 1 + ♯a + ♯c + ♯a♯c + ♯a♯b

We have by (3)

σ_{♯}(a, b ∪ {i} ∪ c) = (f ∪ ḡ)_{σ}

σ_{♯}(c, a ∪ {i} ∪ d) = (f̄∪ g)_{σ} (18)

with g = d∪{i, n} and f = b∪{0, 1}. As in (7) one gets τ(f̄) = τ_{a,b} and τ(ḡ) = τ_{c,d}. Since ♯a + ♯b = i − 1, ♯c + ♯d = m − i and |f| = ♯b, |g| = ♯d we see that τ in (17) coincides with τ in (2.9) (5). This completes the proof of (5) above since we get by (16), (18) and (2.9) (4) the following equation

(19)  E(s^{−1}σ) = E(σ_{♯}e^{m}) = (σ_{♯} ⊗ σ_{♯})E_{I}(e^{m})

q.e.d.

§ 4 The cobar construction as a cocommutative differential Hopf algebra

In this section we assume that the ring ℚ of rational numbers is a subring of the coefficient ring; that is ℚ ⊆ R. Let V be a positive graded free module with V_{0} = 0. The tensor algebra T (V) is a graded Lie algebra with the Lie bracket

(4.1)  [x, y] = x · y − (−1)^{|x||y|}y · x

for x, y ∈ T (V). Let L(V) be the Lie subalgebra of T (V) generated by V ⊂ T (V). The inclusion L(V) ⊂ T (V) induces the isomorphism UL(V) = T (V) where UL(V) is the universal enveloping algebra of L(V). Let S_{k} denote the symmetric group on k letters. Let σ ∈ S_{k} and let \underline{x} = (x_{1}, …, x_{k}) be a list of (homogeneous) elements in L(V). Define the graded sign [σ : \underline{x}] of σ relative to \underline{x} to be the sign (i.e., +1 or −1) that σ has when viewed as permuting k-blocks of sizes |x_{1}|, …, |x_{k}|. Put

(4.2)  C(\underline{x}) = C(x_{1}, …, x_{k}) = \frac{1}{k!} ∑_{σ∈S_{k}}[σ : \underline{x}]x_{σ(1)} · . . . · x_{σ(k)}

which defines an element in T (V). Let U_{0}(V) = R and U_{1}(V) = L(V) and

(1)  U_{n}(V) = span_{R}{C(x_{1}, …, x_{n})|x_{i} ∈ L(V)} ⊆ T (V)

Then the Poincaré-Birkhoff-Witt theorem implies that the inclusions (1) induce an isomorphism

(2)  ⨁_{n≥0} U_{n}(V) = T (V)

which is natural in V; compare 0.4.3 [R]. The isomorphism yields the \underline{canonical projections}

(3)  π_{n} = π_{n}^{V} : T (V) → U_{n}(V)

which are natural in V. Clearly π_{0} = ε is the augmentation of T (V). Moreover we have the map

(4)  C : U_{1}(V) ⊗ U_{n}(V) → U_{n+1}(V)

which carries x ⊗ C(x_{1}, …, x_{n}) to C(x, x_{1}, …, x_{n}) with x, x_{1}, …, x_{n} ∈ L(V). Again C is natural in V. Let

(5)  μ_{j} : T (V) ⊗ T (V) ⟶^{π_{1}⊗π_{j}} U_{1}(V) ⊗ U_{j}(V) ⟶^{C} U_{j+1}(V)

be the composite given by (3) and (4). The formula at the bottom of page 60 [R] yields an explicit description for π_{n} in (3) and μ_{j} in (5).

(4.3) Definition. Let I(V) be the augmentation ideal in the tensor algebra T (V); that is, I(V) = ⊕{V^{⊗n}, n ≥ 1}. Anick defines on page 429 of [A] the following maps between graded modules which are natural in V.

I(V)^{⊗2} ⟶^{ϕ} I(V)^{⊗2} ⊕ I(V)^{⊗3} ⟶^{λ} I(V)^{⊗2} ⟶^{μ} I(V)

We also write ϕ = ϕ^{V} , λ = λ^{V} and μ = μ^{V}. For a, b, c ∈ I(V) put

(1)  μ(a ⊗ b) = ∑_{j≥1} \frac{1}{1 + j} μ_{j}(a ⊗ b)

(2)  λ(a ⊗ b) = ∑_{i,j≥1} \frac{−i}{i + j} T (π_{i}(a) ⊗ π_{j}(b))

(3)  λ(a ⊗ b ⊗ c) = ∑_{j,k≥1} \frac{1}{1 + j + k} (1 + T )(μ_{j}(a ⊗ b) ⊗ π_{k}(c))

Here T is the interchange map and 1 + T is the sum of the identity and the interchange map. Moreover the coordinates of ϕ = (ϕ_{1}, ϕ_{2}) are

(4)  \{ \begin{matrix}ϕ_{1} = 1 − T \\ ϕ_{2} = Δ̄ ⊗ 1 − 1 ⊗ Δ̄ \end{matrix}

Here Δ̄ with Δ̄(a) = Δ(a) − 1 ⊗ a − a ⊗ 1 for a ∈ I(V) ⊂ T (V) is the reduced diagonal given by the algebra homomorphism

(5)  Δ : T (V) → T (V) ⊗ T (V)

with Δ(x) = x ⊗ 1 + 1 ⊗ x for x ∈ V.

(4.4) Definition. Let (A, ψ, G_{1}, G_{2}) be a homotopy Hopf algebra as in (1.8) for which A is a free differential algebra generated by a submodule V ⊂ A with V_{0} = 0. We use the inclusion V ⊂ A to obtain the identification T (V) = A and I(V) = A_{+} where A_{+} is the augmentation ideal. Let n ≥ 1. We call

(A, ψ, G_{1}, G_{2}, V)

n-\underline{good} if the following conditions (1), (2) and (3) hold. Let V_{<n} and V_{≤n} be the submodules of V consisting of elements x ∈ V with |x| < n and |x| ≤ n respectively.

(1)  ψ = T ψ and (ψ ⊗ 1)ψ = (1 ⊗ ψ)ψ on V_{≤n}

(2)  d(V_{≤n+1}) ⊂ L(V_{≤n}) ⊂ T (V) = A

(3)  V_{≤n} ⊂ ker(ψ̄)

Here ψ̄ : A_{+} → A_{+} ⊗ A_{+} is the map defined by ψ̄(a) = ψ(a) − 1 ⊗ a − a ⊗ 1 for a ∈ A_{+}.

(4.5) Anick theorem. Let n ≥ 1 and let X = (A, ψ, G_{1}, G_{2}, V) be n-good. Then α(X) = (A, χ, J_{1}, J_{2}, W) defined by the following conditions (1) ... (8) is (n + 1) -good. Moreover there is a derivation homotopy F : ψ ≃ χ with F (x) = 0 for |x| < n, x ∈ A.

Let q = 1 − ε : A → A_{+}. We also allow q to denote any of the homomorphisms q^{⊗2} : A^{⊗2} → A_{+}^{⊗2}, q^{⊗3} : A^{⊗3} → A_{+}^{⊗3} or q^{⊗2} ⊕ q^{⊗3}. Let v ∈ V. We define χ together with F by the following properties (1) ... (5). Via the identification I(V) = A_{+} we can use the maps μ = μ^{V} , λ = λ^{V} and ϕ = ϕ^{V} in (4.3).

(1)  χ(v) = ψ(v) for |v| ≤ n

(2)  χ(v) = ψ(v) − λϕψ̄(v) for |v| = n + 1

(3)  F (v) = λϕλq(G_{1}(v), G_{2}(v)) for |v| = n, n + 1

(4)  F (v) = 0 for |v| < n and |v| > n + 1

(5)  χ(v) = ψ(v) − F (dv) for |v| > n + 1

Here (4) and (5) are used to obtain F and χ inductively for |v| > n + 1. Moreover we define J_{1}, J_{2} and W by:

(6)  J_{1} = −F + G_{1} + T F

J_{2} = −(χ ⊗ 1 − 1 ⊗ χ)F − (F ⊗ 1 − 1 ⊗ F)ψ + G_{2}

(8)  W_{i} = \{ \begin{matrix}(1 − μχ̄)V_{i} for i = n + 1, n + 2 \\ V_{i} otherwise \end{matrix}

The Anick-theorem above is essentially a reformulation of proposition 5.5 in [A] which we combine with corollary 3.12 in [A].

The construction of α(X) in (4.5) is natural in the following sense.

(4.6) Lemma. Let X = (A, ψ, G_{1}, G_{2}, V) and X′ = (A′, ψ′, G′_{1}, G′_{2}, V′) be n-good and let f : X → X′ be a map in hHA with f(V) ⊂ V′. Then f : A → A′ is also a map f : α(X) → α(X′) in hHA_{0} with f(W) ⊂ W′. Moreover (f ⊗ f)F = F′f for the homotopies F : ψ ≃ χ and F′ : ψ′ ≃ χ′.

The lemma follows from the naturality properties of μ, λ, ϕ in (4.3) by inspection of (4.5) (1) ... (8).

As an application of the Anick-Theorem (4.5) and theorem (2.10) we get the following cocommutative diagonal for the cobar construction.

(4.7) Theorem. Assume ℚ ⊆ R and let X be a 1-reduced simplicial set. Then there is a diagonal ϕ on the cobar construction ΩC_{∗}X such that (ΩC_{∗}X, ϕ) is a cocommutative differential Hopf algebra. Moreover there is a derivation homotopy F : ψ ≃ ϕ where ψ is the diagonal in (2.10). Both ϕ and F are natural for X ∈ sSet_{1} so that one gets a commutative diagram of functors

sSet_{1}  ⟶^{ΩC_{∗}} DA_{0}
(ΩC_{∗})_{ϕ} ↘  ↗ π
CDA_{0}

where (ΩC_{∗})_{ϕ}(X) = (ΩC_{∗}X, ϕ); see (1.9).

Proof of (4.7). One readily checks that

Ω^{(1)} = (ΩC_{∗}X, ψ, G_{1} = E, G_{2} = 0, V = s^{−1}C̃_{∗}X)

is 1-good in the sense of (4.4). Hence by applying the Anick theorem inductively we obtain objects Ω^{(n)} with Ω^{(n)} = α(Ω^{(n−1)}) for n ≥ 2 such that

Ω^{(n)} = (ΩC_{∗}X, ψ^{(n)}, G_{1}^{(n)}, G_{2}^{(n)}, V^{(n)})

is n-good and such that F^{(n)} : ψ^{(n)} ≃ ψ^{(n+1)} is a derivation homotopy. Since ψ^{(n)}(x) = ψ^{(n+1)}(x) for |x| ≤ n, x ∈ ΩC_{∗}X, and since F^{(n)}(x) = 0 for |x| < n we see that ϕ, F in (4.7) are well defined by

ϕ(x) = ψ^{(n)}(x) for |x| = n

F (x) = ∑_{n≥1} F^{(n)}(x)

One readily checks that ϕ is cocommutative and coassociative; compare (4.4) (1). The naturality of ϕ and F follows from (4.6). q.e.d.

We start in the induction of theorem (4.7) with explicit data Ω^{(1)} given by (2.9). Therefore the diagonal ϕ in (4.7) is computable by applying inductively the Anickconstruction (4.5). This yields a formula for ϕ in terms of generators of the cobar construction. Moreover we obtain by the construction in the proof of (4.7) the following result (4.8).

Let DL_{0} be the category of differential Lie algebras L = (L, d) over ℚ with H_{0}L = 0; see [Q]. For a graded rational vector space V let L(V) be the free Lie algebra generated by V. The tensor algebra T (V) is a Lie algebra with the bracket [x, y] = xy − (−1)^{|x||y|}yx. Then L(V) is the sub Lie algebra of T (V) generated by V and T (V) = UL(V) is the universal enveloping algebra of L(V). Clearly a degree 0-map φ : V → V′ induces a Lie algebra map L(φ) : L(V) → L(V′).

(4.8) Theorem. Let X be a 1-reduced simplicial set. Then there is a differential d = d_{L} on the free Lie algebra L(s^{−1}C̃_{∗}X) such L(X) = (L(s^{−1}C̃_{∗}X), d_{L}) defines a functor

L : sSet_{1} → DL_{0}

Here L carries a map f : X → Y to the induced map L(f) = L(s^{−1}C̃_{∗}f) which is a differential map. Moreover there is an isomorphism of cocommutative differential Hopf algebras

β : (UL(X), Δ) ≅ (ΩC_{∗}X, ϕ)

Here UL(X) = (UL(s^{−1}C̃_{∗}X), d) is the universal enveloping for which the diagonal Δ is primitive on L(s^{−1}C̃_{∗}X) and ϕ is the diagonal on the cobar construction in (4.7). The isomorphism β is natural in X ∈ sSet_{1} and β induces the identity on s^{−1}C̃_{∗}X if one divides out all decomposable elements.

Proof of (4.8). Using Ω^{(n)} in the proof of (4.7) we define

(1)  V (X) ⊂ ΩC_{∗}X

by V (X)_{n} = (V^{(n)})_{n}. Then we see that X ↦ V (X) is a functor from sSet_{1} to the category of graded modules such that (1) is natural in X. Moreover by definition of ϕ we have the isomorphism of Hopf algebras

(2)  (UL(V (X)), Δ) = (T (V (X)), Δ) = (ΩC_{∗}X, ϕ)

which is induced by the inclusion (1). We now obtain a natural isomorphism

(3)  β̄ : V (X) ≅ s^{−1}C̃_{∗}X

by dividing out in (2) all decomposable elements. Hence (2) and (3) yield the isomorphism β in (4.8). We define the differential d_{L} by d_{L} = β^{−1}d_{Ω}β. q.e.d.

The functor L in (4.8) induces an equivalence of rational homotopy categories

(4.9)  Ho(L) : Ho_{ℚ}sSet_{1} ⟶^{∼} Ho_{ℚ}DL_{0}

Here Ho_{ℚ} denotes the localization with respect to rational homology isomorphisms. In fact, by (4.11) below and the main result of Majewski [M] we know that Ho(L) is naturally isomorphic to Quillen’s functor NPQ̂G which is the composite of four complicated functors; see [Q]. Hence theorem (4.8) replaces Quillen’s functor by the functor L which is given by a differential d_{L} on L(s^{−1}C̃_{∗}X). A formula for d_{L} is obtained by an inductive process; see § 5 below.

As pointed out by Majewski [M] the complete simplicial Lie algebra

(1)  L′(X) = P Q̂G(X), X ∈ \underline{\underline{sSet}}_{1},

(given by Quillen’s functor) can be described as follows. Let L̂V be the free complete Lie algebra over R generated by the free (filtered) graded module V. Then L′(X) is the simplicial free complete Lie algebra given by

(2)  \begin{aligned}(L′X)_{n−1} & = L̂(X_{n} − s_{0}X_{n−1}) \\ & = L̂(X_{n})//L̂(s_{0}X_{n−1}) \end{aligned}

with d_{j}, j ≠ 0 and s_{j} defined as in X and with d_{0}(x) given by the Hausdorff formula

(3)  d_{0}(x) = log(exp(d_{0}x) · exp(−d_{1}x))

Quillen’ s functor NP Q̂G is now obtained by the normalization (Moore chain complex) NL′(X) of L′(X). Using results of [M] we obtain a map

(4)  L(X) ⟶^{∼} NL′(X) = NP Q̂G(X)

in \underline{\underline{DL}}_{0} which is a homology isomorphism and natural in the homotopy category Ho(\underline{\underline{DL}}_{0}). Here the differential Lie algebra L(X) is considerably smaller than the differential Lie algebra NL′(X). For example we have in degree 1 the equation L(X)_{1} = s^{−1}C_{2}X while (NL′(X))_{1} = L̂(s^{−1}C_{2}X). Since L′ is defined by the Hausdorff formula (3) we consider the differential d_{L} of L(X) via the homology isomorphism (4) as a “simplicial form” of the classical \underline{Hausdorff formula}:

(5)  log(exp(a) · exp(b)) = a + b + \frac{1}{2}[a, b] + …

Here … denotes an intricate sum of Lie monomials of length ≥ 3; see [R]. For the differential d_{L} in (4.8) we have the canonical decomposition

(4.10)  d_{L} = d_{L}^{(1)} + d_{L}^{(2)} + … + d_{L}^{(t)} + …

where d_{L}^{(t)}(s^{−1}σ), σ ∈ X_{n}, is a sum of Lie monomials of length t; that is

d_{L}^{(t)}(s^{−1}σ) ∈ L(s^{−1}C̃_{∗}X) ∩ (s^{−1}C̃_{∗}X)^{⊗t}

For degree reasons we have d_{L}^{(t)}(s^{−1}σ) = 0 for t ≥ n − 1. Now let R be a subring of ℚ and let x ∈ span_{ℚ}(E) be an element in the ℚ -vector space with fixed basis E. Then we say that x \underline{is defined over} R if x ∈ span_{R}(E) ⊂ span_{ℚ}(E). Following Anik [A] let ρ(R) be the least non invertible prime in the ring R.

(4.11) Theorem. Let X be a 1-reduced simplicial set. Then the differential d_{L} of L(X) in (4.8) satisfies for σ ∈ X_{n} the following formulas

\begin{matrix}d_{L}^{(1)}(s^{−1}σ) = − ∑_{i=0}^{n}(−1)^{i}s^{−1}(d_{i}σ); see (2.5) \\ d_{L}^{(2)}(s^{−1}σ) = \frac{1}{2} ∑_{i=2}^{n−2}(−1)^{i}[s^{−1}σ(0 … i), s^{−1}σ(i … n)] \end{matrix}

Moreover d_{L}^{(t)}(s^{−1}σ) for t ≥ 1 is defined over any subring R with t < ρ(R), R ⊂ ℚ.

We prove theorem (4.11) in (5.15) below.

(4.12) Corollary. Let X be an r-reduced simplicial set (i.e. X_{i} = ∗ for i ≤ r) and let dim |X| ≤ n. Then L(X) is defined over R ⊂ ℚ with n − 2 < r · ρ(R).

The corollary says that the top attaching map S^{n−1} → X^{n−1} of X is “r-mild”; compare theorem 4.8 [A]. The corollary is the basis of Halperin’s result [Ha] on H_{∗}(ΩX, R) who however assumes n − 1 < r · ρ(R).

Theorem (4.8) is also related to the following result of Anick; see 7.5 in [A].

(4.13) Theorem. (Anick’s cubical Hopf algebra): Let ℚ ⊆ R and let Y be a path connected topological monoid. Then the cubical chain complex CU′_{∗}(Y) is a differential algebra which admits a diagonal ϕ such that (CU′_{∗}(Y), ϕ) is a cocommutative differential Hopf algebra. Moreover there is a derivation homotopy F : ψ ≃ ϕ where ψ is the diagonal in (3.2). Both ϕ and F are natural for Y ∈ Mon_{0} so that one has a commutative diagram of functors

\begin{matrix}Mon_{0} & ⟶^{CU′_{∗}} & DA_{0} \\ (CU′_{∗})_{ϕ} ↘ & & ↗ π \end{matrix}

CDA_{0}

where (CU′_{∗})_{ϕ}(Y) = (CU′_{∗}(Y), ϕ).

The functor E of Anick is given by the primitive elements of (CU′_{∗}(Y), ϕ); see (7.5) [A]. In theorem (3.5) we obtained the embedding of the cobar construction ΩC_{∗}X into CU′_{∗}(\underline{Ω}X) where \underline{Ω}X is the geometric cobar construction. This embedding was compatible with the structure as a homotopy Hopf algebra. We now show that it is as well an embedding of cocommutative differential Hopf algebras given by the cocommutative diagonal ϕ in (4.13) and (4.7) respectively.

(4.14) Theorem. Let ℚ ⊆ R and let X ∈ sSet_{1}. Then the inclusion of differential algebras in (3.5)

i : Ω(C_{∗}X) = C_{∗}(\underline{Ω}X) ⊂ CU′_{∗}(\underline{Ω}X)

is compatible with the diagonal ϕ on both sides; that is, i is an inclusion of differential Hopf algebras in CDA_{0}. Moreover the homotopy F in (4.7) is via the inclusion i the restriction of the homotopy F in (4.13).

Theorem (4.14) in connection with Majewski’s result (4.9) (4) solves a problem of Anick; see the remark following 7.7 in [A]. In fact by (4.14) and (4.9) (4) we see immediately that Anick’s functor defined by the primitives of the Hopf algebra (CU′_{∗}(\underline{Ω}X), ϕ) coincides up to natural equivalence in Ho(\underline{\underline{DL}}_{0}) with Quillen’s functor NP Q̂G.

Proof of (4.14). The algebra I(∞) in (3.1) is termed S in [A]. Let S^{1} be the submodule of elements x ∈ I(∞) with |x| ≥ 1 and let T S^{1} be the tensoralgebra generated by S^{1}. Anick describes in 7.2 [A] a homotopy Hopf algebra (T S^{1}, D, ψ, G_{1} = E, G_{2} = 0) and in 7.3 [A] he applies (4.5) to obtain a cocommutative Hopf algebra (T S^{1}, D, ϕ) together with a derivation homotopy F : ψ ≃ ϕ. Now let Δ(∞) be the simplicial set in sSet_{1} for which the realization |Δ(∞)| is the quotient Δ^{∞}/Δ^{∞,1}. Here Δ^{∞,1} is the 1-skeleton of the infinite dimensional simplex Δ^{∞} which is the direct limit of Δ^{1} ⊂ Δ^{2} ⊂ Δ^{3} ⊂ … where Δ^{n} is the standard simplex. Hence Δ^{n} correspond to an element σ_{n} ∈ Δ(∞)_{n} which is non degenerate for n ≥ 2. This yields the map of differential modules

σ_{♯} : I(∞) = ⨁_{n≥0} I(n) → ΩC_{∗}Δ(∞)

which is (σ_{n})_{♯} on I(n − 1), n ≥ 1; compare (2) in the proof of (3.5). Moreover σ_{♯} induces canonically a map in hHA_{0}

σ_{∗} : (T S^{1}, D, ψ, E) → (ΩC_{∗}Δ(∞), ψ, E)

We now apply the Anick theorem on both sides to obtain the diagonal ϕ and the homotopy F : ψ ≃ ϕ which both commute with σ_{∗}. This implies the proposition of (4.14) since (ΩC_{∗}Δ(∞), ϕ) is the universal example for the Hopf algebra (ΩC_{∗}X, ϕ) in (4.14) with X ∈ sSet_{1}. On the other hand ϕ on T S^{1} is used by Anick to define ϕ on CU′_{∗}(Y) for Y ∈ Mon_{0}; see 7.4 [A]. q.e.d.

§ 5 On the computation of the Lie differential d_{L}

In this section we describe an inductive construction of elements β(2), β(3), … which determine the differential d_{L} by the formula in (5.9) below. This leads to a proof of theorem (4.11); see (5.13).

The graded set N is defined as in (2.8) with |a| = ♯(a) − 2 for a ∈ N. Let

(5.1)  Ω = R⟨N⟩/(N_{0} ∼ 1)

be the free algebra R⟨N⟩ divided by the ideal generated by a − 1 with a ∈ N_{0}. We clearly have

(1)  Ω = R⟨N_{≥1}⟩ = T (U)

where U = span_{R}(N_{≥1}). For a ∈ N let N[a] = {b ∈ N, b ⊂ a}. If |a| = n − 1 we have a canonical order preserving bijection

(2)  j_{a} : \underline{n} = \{0 … n\} ≈ a

This defines the bijection j_{a} : N[\underline{n}] ≈ N[a] which carries b to j_{a}(b). Moreover let Ω[a] be the subalgebra of Ω generated by N[a]. Then j_{a} also induces the isomorphism of algebras

(3)  j_{a} : Ω[\underline{n}] ≈ Ω[a]

We use the following convention. Given an element w(\underline{n}) ∈ Ω[\underline{n}] we obtain for any a ∈ N with |a| = n − 1 the corresponding element

(5.2)  w(a) = j_{a}(w(\underline{n})) ∈ Ω[a] ⊂ Ω

22

Hence if elements d(\underline{n}), G(\underline{n}), Λ(\underline{n}), … are defined in Ω[\underline{n}] then also the elements d(a), G(a), Λ(a), … ∈ Ω are defined. We therefore can define a differential d_{Ω} on Ω by putting

d_{Ω}(\underline{n}) = − ∑_{i=0}^{n}(−1)^{i}∂_{i}(\underline{n}) + ∑_{i=2}^{n−2}\{0 … i\} · \{i … n\}

We also write

∂(\underline{n}) = ∑_{i=0}^{n}(−1)^{i}∂_{i}(\underline{n})

∂^{+}(\underline{n}) = ∑_{i=2}^{n−2}\{0 … i\} · \{1 … n\}

so that d_{Ω}(\underline{n}) = −∂(\underline{n}) + ∂^{+}(\underline{n}). This yields by (5.2) the formula for d_{Ω}(a) for all a ∈ N_{≥1} and one can check that (Ω, d_{Ω}) is a well defined differential algebra. In fact, let Δ(∞) be the simplicial set for which the realization is Δ^{∞}/Δ^{∞,1} as in the proof of (4.11). Then Ω = Ω(C_{∗}Δ(∞)) is the cobar construction defined in (2.5). Therefore we have by (2.9) the diagonal

ψ_{Ω} : Ω → Ω ⊗ Ω

and the derivation homotopy E : T ψ_{Ω} ≃ ψ_{Ω} with

(1)  ψ_{Ω}(\underline{n}) = ∑_{f≤\underline{n}} τ(f̄)f̄⊗ f

(2)  E(\underline{n}) = ∑_{i=1}^{n−1} ∑_{\begin{matrix}f≤\{0...i\} \\ g≤\{i...n\} \end{matrix}} τ · (f ∪ ḡ) ⊗ (f̄∪ g)

For the definition of τ see (2.9) (5). Using the convention (5.2) also ψ_{Ω}(a) and E(a) are defined for a ∈ N. We are going to define inductively elements, n ≥ 2,

(5.4)  \{ \begin{matrix}β(\underline{n}) ∈ Ω[\underline{n}],  β(\underline{2}) = \underline{2} \\ H(\underline{n − 1}) ∈ Ω[\underline{n − 1}] ⊗ Ω[\underline{n − 1}], H(\underline{1}) = 0 \\ β′(\underline{n + 1}) ∈ Ω[\underline{n + 1}],  β′(\underline{3}) = \underline{3} \\ H′(\underline{n}) ∈ Ω[\underline{n}] ⊗ Ω[\underline{n}],  H′(\underline{2}) = 0 \end{matrix}

If Q(n) = \{β(\underline{n}), H(\underline{n − 1})\} and Q′(n) = \{β′(\underline{n + 1}), H′(\underline{n}\} then we shall see that Q(2), Q(3), …, Q(n) and Q′(n) determine Q(n + 1) and Q′(n + 1). Given the elements (5.4) we have by the convention (5.2) the corresponding elements β(a), H(b), β′(c), H′(a) with a, b, c ∈ N and |a| = n − 1, |b| = n − 2, |c| = n. The elements β(a) define an isomorphism

(1)  β : Ω ⟶^{≅} Ω

which carries a to β(a). We define the differential d_{L} on Ω by

(2)  d_{L}(a) = β^{−1}d_{Ω}β(a)

so that β is an isomorphism of differential algebras. The choice of elements in (5.4) has the property that d_{L}(a) is a Lie element, that is

(3)  d_{L}(a) ∈ L(U) ⊂ T (U) = Ω

Here Δ : Ω → Ω ⊗ Ω is the diagonal which is primitive on L(U), that is Δ(a) = 1 ⊗ a + a ⊗ 1 for a ∈ N_{≥1}. Then the elements H(a) given by (5.4) determine a derivation homotopy for the diagram

(4)

Ω ⊗ Ω ⟶_{≅}^{β⊗β} Ω ⊗ Ω
Δ↑  ^{↖H}  ↑^{ψ_{Ω}}
Ω ⟶_{≅}^{β} Ω

That is, H : Ω → Ω ⊗ Ω is the derivation homotopy

(5)  H : ψ_{Ω}β ≃ (β ⊗ β)Δ

which is defined on generators a ∈ N by the elements H(a).

Let Ω(n) be the subalgebra of Ω generated by elements a ∈ N with |a| ≤ n. Hence \underline{n + 1} ∈ Ω(n). We use the data Q(2), Q(3), …, Q(n) and Q′(n) in (5.4) to define the following diagram which we denote by D(n),

(6)  D(n) = \begin{cases}Ω(n − 1) ⊗ Ω(n − 1) & ⟶_{≅}^{β⊗β} & Ω(n − 1) ⊗ Ω(n − 1) \\ ^{Δ}↑  ^{↖H′_{n}} & ↑^{ψ_{Ω}} \\ Ω(n − 1) & ⟶_{≅}^{β} & Ω(n − 1) \\ ∩ & ∩ \\ Ω(n) & ⟶_{β′_{n+1}}^{≅} & Ω(n) \end{cases}

Here β is the restriction of β in (1) and β′_{n+1} is an extension of β|Ω(n − 1) defined by

(7)  β′_{n+1}(a) = \{ \begin{matrix}β(a) & for |a| ≤ n − 1, \\ β′(a) & for |a| = n, a ∈ N \end{matrix}

Moreover the derivation homotopy H′_{n} is an extension of H|Ω(n − 2) defined by

(8)  H′_{n}(a) = \{ \begin{matrix}H(a) & for |a| ≤ n − 2, \\ H′(a) & for |a| = n − 1, a ∈ N. \end{matrix}

The isomorphism β′_{n+1} defines a differential d′_{L} for Ω(n) by d′_{L}(a) = d_{L}(a) for |a| ≤ n − 1 and

(9)  d′_{L}(\underline{n + 1}) = β^{−1}d_{Ω}β′(\underline{n + 1})

Again this is a Lie element in L(U) ∩ Ω(n − 1). Given a diagram D(n) in (6) with these properties we shall use the Anick theorem (4.5) to construct a diagram D(n + 1); see the proof of (5.11) below. Then D(n + 1) determines Q(n + 1) and Q′(n + 1) so that the inductive step is obtained this way. We start the induction with diagram D(2) with H′_{2} = 0 and β′_{3} = identity.

(5.5) Notation. Using diagram D(n) in (5.4) (6) we introduce the following types of elements. Let

\begin{matrix}β^{−1} : Ω(n − 1) ≅ Ω(n − 1), \\ (β′_{n+1})^{−1} : Ω(n) ≅ Ω(n) \end{matrix}

be the inverses of the isomorphisms β and β′_{n+1} respectively. Then the elements ν(\underline{n}), ν′(\underline{n + 1}), ▽(\underline{n}) and ▽′(\underline{n + 1}) defined by

β(\underline{n}) = \underline{n} + ν(\underline{n}),
β′(\underline{n + 1}) = \underline{n + 1} + ν′(\underline{n + 1})
β^{−1}(\underline{n}) = \underline{n} + ▽(\underline{n})
(β′_{n+1})^{−1}(\underline{n + 1}) = \underline{n + 1} + ▽′(\underline{n + 1})

are decomposable elements ∈ Ω_{+} · Ω_{+}. One readily checks the equations

(1)  \{ \begin{matrix}▽ (\underline{n}) = −β^{−1}ν(\underline{n}) \\ ▽′ (\underline{n + 1}) = −β^{−1}ν′(\underline{n + 1}) \end{matrix}

Moreover we define for j = 0, 1, …, n + 1 the elements v_{j}(\underline{n + 1}), u_{j}(\underline{n + 1}) ∈ Ω_{1} = U_{1} = span_{R}(N_{1}) by

(2)  ▽′(\underline{n + 1}) = ξ + ∑_{j=0}^{n+1}(∂_{j}(\underline{n + 1}) · v_{j}(\underline{n + 1}) + u_{j}(\underline{n + 1}) · ∂_{j}(\underline{n + 1})).

Here ξ ∈ Ω is an element which has no summands involving a ∈ N with |a| = n−1. Let q, Δ, λ = λ^{U}, ϕ = ϕ^{U}, μ = μ^{U} be defined for Ω = T (U) as in (4.3). Then we set

(3)  Λ(\underline{n}) = λϕλq(G_{1}(\underline{n}), G_{2}(\underline{n}))

where

G_{1}(\underline{n}) = (β ⊗ β)^{−1}((1 − T )H′(\underline{n}) + E β(\underline{n}))
G_{2}(\underline{n}) = (1 ⊗ Δ − Δ ⊗ 1)(β ⊗ β)^{−1}H′(\underline{n})
+ (1 ⊗ K − K ⊗ 1)(β ⊗ β)^{−1}ψ_{Ω}β(\underline{n})

with K = (β ⊗ β)^{−1}H′_{n}. Moreover we set

(4)  Λ′(\underline{n + 1}) = λϕλq(G′_{1}(\underline{n + 1}), G′_{2}(\underline{n + 1}))

where

\begin{aligned}G′_{1}(\underline{n + 1}) & = (β′_{n+1} ⊗ β′_{n+1})^{−1}E β′(\underline{n + 1}) \\ G′_{2}(\underline{n + 1}) & = (1 ⊗ (β ⊗ β)^{−1}H′_{n} − (β ⊗ β)^{−1}H′_{n} ⊗ 1)(β′_{n+1} ⊗ β′_{n+1})^{−1}ψ̄_{Ω}β′(\underline{n + 1}) \end{aligned}

Recall that here ψ̄_{Ω}(a) = ψ_{Ω}(a) − 1 ⊗ a − a ⊗ 1 is the reduced diagonal defined by ψ_{Ω} and that E is the homotopy in (5.3)(2). Using the elements (1) ... (4) we are ready to describe the explicit formulas which yield the induction.

(5.6) Formulas for the induction.

(1)  β(\underline{n + 1}) = β′(\underline{n + 1}) − βμ(β ⊗ β)^{−1}(A) with
A = A(\underline{n + 1}) = ψ̄_{Ω}β′(\underline{n + 1}) − H′_{n}β^{−1}d_{Ω}β′(\underline{n + 1})

(2)  H(\underline{n}) = H′(\underline{n}) + (β ⊗ β)Λ(\underline{n})

(3) β′(\underline{n + 2}) = \underline{n + 2} − β′_{n+1}μ(X + Y + Z_{1} + Z_{2}) with
X = X(\underline{n + 2}) = (β′_{n+1} ⊗ β′_{n+1})^{−1}ψ̄_{Ω}(\underline{n + 2})
Y = Y (\underline{n + 2}) = (β ⊗ β)^{−1}H′_{n} (∑_{i=0}^{n+2}(−1)^{i} ▽′ (∂_{i}(\underline{n + 2})) − β^{−1} ∑_{i=2}^{n}(−1)^{i}\{0 … i\} · \{1 … n + 2\})

Z_{1} = Z_{1}(\underline{n + 2}) = ∑_{i=0}^{n+2}(−1)^{i} ∑_{j=0}^{n+1} Z_{ji}
Z_{ji} = (Λ ∂_{j}∂_{i}(\underline{n + 2})) · Δ v_{j}(∂_{i}(\underline{n + 2})) − Δ u_{j}(∂_{i}(\underline{n + 2})) · (Λ ∂_{j}∂_{i}(\underline{n + 2}))
Z_{2} = Z_{2}(\underline{n + 2}) = Δ\{0, 1, 2\} · Λ(\{2 … n + 2\}) − (−1)^{n}Λ(\{0 … n\}) · Δ\{n, n + 1, n + 2\}

(4) H′(\underline{n + 1}) = (β′_{n+1} ⊗ β′_{n+1})Λ′(\underline{n + 1}) − R − \frac{1}{4}(β ⊗ β)(S)
R = R(\underline{n + 1}) = H′_{n}μ(β ⊗ β)^{−1}ψ̄_{Ω}β′(\underline{n + 1})
S = S(\underline{n + 1}) = (Λ\underline{n}) · Δ\{0, n, n + 1\} + (−1)^{n}Δ\{0, n, n + 1\} · (Λ \underline{n})

− (−1)^{n}(Λ{1 … n + 1}) · Δ{0, 1, n + 1} − Δ{0, 1, n + 1} · (Λ{1 … n + 1})

+ ∑_{i=1}^{n}(−1)^{i}(Δ\{i − 1, i, i + 1\} · (Λ∂_{i}\underline{n + 1}) + (−1)^{n}(Λ∂_{i}\underline{n + 1}) · Δ\{i − 1, i, i + 1\})

In (3) and (4) we have Δ(x) = x ⊗ 1 + 1 ⊗ x since the corresponding elements x in (3) and (4) are of degree 1. We point out that all equations (1) ... (4) are well defined in terms of the data Q(2), Q(3), …, Q(N), Q′(n) in (8.9) so that the induction is well defined. The formulas above imply the following results (5.7) and (5.8) which simplify the expressions for the Lie differential d_{L}.

(5.7) Proposition. d_{Ω}β(\underline{n + 1}) = d_{Ω}β′(\underline{n + 1}) and hence

d_{L}(\underline{n + 1}) = d′_{L}(\underline{n + 1}) = β^{−1}d_{Ω}β′(\underline{n + 1})

Proof. By (5.6) (1) we have:

(1)  d_{Ω}β(\underline{n + 1}) = d_{Ω}β′(\underline{n + 1}) − d_{Ω}βμ(β ⊗ β)^{−1}(A)

d_{Ω}βμ(β ⊗ β)^{−1} = βd_{L}μ(β ⊗ β)^{−1}
= βμd_{L}(β ⊗ β)^{−1}

= βμ(β ⊗ β)^{−1}d_{Ω} (2)

since μ is a Der(L) -homomorphism by 3.7 (a) [A] with d_{L} ∈ Der(L).

(3)  d_{Ω}A = d_{Ω}ψ̄_{Ω}β′(\underline{n + 1}) − d_{Ω}H′_{n}β^{−1}d_{Ω}β′(\underline{n + 1}) where

d_{Ω}H′_{n} = ψ_{Ω}β − (β ⊗ β)Δ − H′_{n}d_{L} on Ω(n − 1)
= ψ̄_{Ω}β − (β ⊗ β)Δ̄ − H′_{n}d_{L}

(4)

d_{Ω}A = d_{Ω}ψ̄_{Ω}β′(\underline{n + 1}) − ψ̄_{Ω}d_{Ω}β′(\underline{n + 1})
+ (β ⊗ β)Δ̄β^{−1}d_{Ω}β′(\underline{n + 1}) + H′_{n}d_{L}β^{−1}d_{Ω}β′(\underline{n + 1})

Here the last term vanishes since d_{L}β^{−1} = β^{−1}d_{Ω}. Moreover the first two terms cancel so that

(5)  d_{Ω}A = (β ⊗ β)Δ̄β^{−1}d_{Ω}β′(\underline{n + 1}) = (β ⊗ β)Δ̄d′_{L}(\underline{n + 1}) = 0

Here Δ̄d′_{L}(\underline{n + 1}) = 0 since d′_{L}(\underline{n + 1}) is a Lie element. Hence the lemma is a consequence of (1) and (2). q.e.d.

Recall that

π_{1} = 1 − μ Δ̄ : T (U) → L(U)

is the canonical retraction from the tensor algebra Ω = T (U) to the Lie algebra.

(5.8) Theorem. The Lie differential is given by

\begin{aligned}d_{L}(\underline{n + 2}) = & − ∑_{i=0}^{n+2}(−1)^{i}∂_{i}\underline{n + 2} \\ & + (1 − μΔ̄) (− ∑_{i=0}^{n+2}(−1)^{i} ▽′ (∂_{i}\underline{n + 2}) + β^{−1} ∑_{i=2}^{n}(−1)^{i}\{0 … i\} · \{1 … n + 2\}) \end{aligned}

The proof of (5.8) is given in (5.12) below. We point out that d_{L}(\underline{n + 2}) in (5.8) is determined by the data β(\underline{2}), β(\underline{3}), …, β(\underline{n}) and β′(\underline{n + 1}). Equivalently to (5.8) one has the following formulas

(5.9) Corollary.

\begin{aligned}d_{L}(\underline{n + 2}) & = (1 − μΔ̄)((β′_{n+1})^{−1}d_{Ω}\underline{n + 2}) \\ d_{L}(\underline{n + 2}) & = (1 − μΔ̄)(β^{−1}d_{Ω}\underline{n + 2}) \end{aligned}

Proof. The first formula follows from (5.8) since by (5.3) and (5.5)

\begin{aligned}(β′_{n+1})^{−1}d_{Ω}(\underline{n + 2}) & = −(β′_{n+1})^{−1}∂ \underline{n + 2} + β^{−1}(∂^{+}\underline{n + 2}) \\ & = −∂ \underline{n + 2} − ▽′(∂ \underline{n + 2}) + β^{−1}(∂^{+} \underline{n + 2}) \end{aligned}

where (1 − μΔ̄)∂ \underline{n + 2} = ∂ \underline{n + 2}. The second formula is obtained as follows. We have by (5.5) (1)

(1 − μΔ̄) ▽′ (\underline{n + 1}) = −(1 − μΔ̄)β^{−1}ν′(\underline{n + 1})

where by (5.6)(1)

ν′(\underline{n + 1}) = ν(\underline{n + 1}) + βμ(β ⊗ β)^{−1}(A)

Since (1 − μΔ̄)μ = 0 we thus get by (5.5) (1)

\begin{aligned}(1 − μΔ̄) ▽′ (\underline{n + 1}) & = −(1 − μΔ̄)β^{−1}ν(\underline{n + 1}) \\ & = (1 − μΔ̄) ▽ (\underline{n + 1}) \end{aligned}

Hence we can replace ▽′ in (5.8) by ▽ and therefore one obtains the second formula in a similar way as the first formula above. q.e.d.

The formulas in (5.6) yield a well defined sequence of elements as in (5.4) and by the proof of the theorem below all the properties in (5.4) are satisfied. We now use (Ω, d_{L}) as a universal example. For this we define for each 1-reduced simplicial set X and σ ∈ X_{n} the homomorphism of algebras

σ_{∗} : Ω[\underline{n}] → UL(s^{−1}C̃_{∗}X) = T (s^{−1}C̃_{∗}X)

which carries a ∈ N[\underline{n}] to σ_{∗}(a) = s^{−1}σ(a); see (2.2). Then we obtain a differential d_{L} on L(s^{−1}C̃_{∗}) by setting

d_{L}(s^{−1}σ) = σ_{∗}d_{L}(\underline{n})

This defines a differential Lie algebra

(5.10)  L(X) = (L(s^{−1}C̃_{∗}X), d_{L})

We obtain an isomorphism

(1)  β : UL(X) ≅ Ω(C_{∗}X)

by setting β(s^{−1}σ) = σ_{∗}β(\underline{n}) and we obtain a derivation homotopy

(2)  H : ψβ ≃ (β ⊗ β)Δ

by setting H(s^{−1}σ) = (σ_{∗} ⊗ σ_{∗})H(\underline{n}).

(5.11) Theorem. The definition in (5.10) yields a well defined functor

L : \underline{\underline{sSet}}_{1} → \underline{\underline{DL}}_{0}

with the properties in (4.8).

In fact, this functor coincides with the one in (4.8) up to natural isomorphism.

Proof. Assume we have a diagram D(n), n ≥ 2, with the properties described in (5.4). Using D(n) we define a homotopy Hopf algebra

(1)  X = (A, ψ, G_{1}, G_{2}, V)

which is (n − 1) -good as follows. Let V = U_{≤n+1} and let A = T (V). Then an isomorphism

(2)  γ : A = T (U_{≤n+1}) ≅ T (U_{≤n+1}) = Ω(n + 1)

is defined by

γ(a) = \{ \begin{matrix}β(a) & for |a| ≤ n − 1 \\ β′_{n+1}(a) & for |a| = n \\ a & for |a| = n + 1, a ∈ N. \end{matrix}

We define a differential d on A by the formula d(a) = γ^{−1}d_{Ω}γ(a) where d_{Ω} is the differential of Ω = T (U). Hence we have

(3)  d(a) = \{ \begin{matrix}d_{L}(a) & for |a| ≤ n − 1, see (5.4) (2) \\ d′_{L}(a) & for |a| = n; see (5.4) (9) \\ (β′_{n+1})^{−1}d_{Ω}(a) & for |a| = n + 1, a ∈ N \end{matrix}

We now have the following commutative diagram

(3a)  \begin{matrix}A ⊗ A & ⟶^{γ⊗γ} & Ω(n + 1_{⊗}Ω(n + 1) \\ ↑^{ψ′} & & ↑^{ψ_{Ω}} \\ A & ⟶_{γ} & Ω(n + 1) \end{matrix}

where ψ′ = (γ ⊗ γ)^{−1}ψ_{Ω}γ. Since γ extends β : Ω(n − 1) ≅ Ω(n − 1) in D(n) we see that (β ⊗ β)^{−1}H′_{n} is a derivation homotopy

ψ′|Ω(n − 1) = (β ⊗ β)^{−1}ψ_{Ω}β ≃ Δ|Ω(n − 1)

The homotopy extension of ψ′ via (β ⊗ β)^{−1}H′_{n} thus yields a diagonal

(4)  ψ : A → A ⊗ A

and a homotopy K : ψ′ ≃ ψ with the properties:

(5)  K(a) = \{ \begin{matrix}(β ⊗ β)^{−1}H′_{n}(a) & for |a| ≤ n − 1 \\ 0 & for |a| = n, n + 1 \end{matrix}

(6)  ψ(a) = \{ \begin{matrix}Δ(a) & for |a| ≤ n − 1 \\ ψ′(a) − K(da) & for |a| = n, n + 1 \end{matrix}

We now obtain the homotopies G_{1}, G_{2} in (1) as follows.

T ψ ⟵^{T K} T ψ′ ⟶^{γ⊗γ)^{−1}Eγ} ψ′ ⟶^{K} ψ

(7)  G_{1} = K + (γ ⊗ γ)^{−1}Eγ − T K

Here we use E in (5.3) (2). Moreover G_{2} is given by

(ψ ⊗ 1)ψ ⟵^{(ψ⊗1)K} (ψ ⊗ 1)ψ′ ⟵^{(K⊗1)ψ′} (ψ′ ⊗ 1)ψ′
∥
(1 ⊗ ψ)ψ ⟵^{(1⊗ψ)K} (1 ⊗ ψ)ψ′ ⟵^{(1⊗K)ψ′} (1 ⊗ ψ′)ψ′

Here we use the fact that ψ in (5.3) (1) is coassociative so that

(8)  G_{2} = (1 ⊗ ψ − ψ ⊗ 1)K + (1 ⊗ K − K ⊗ 1)ψ′

This completes the definition of X in (1). The properties of D(n) in (5.4) show that X is (n − 1) -good in the sense of (4.4). For this we need the assumption that d′_{L}(\underline{n + 1}) in (5.4) (9) is a Lie element. We now apply Anick’s theorem (4.5) and we obtain

(9)  α(X) = (A, χ, J_{1}, J_{2}, W) and F : ψ ≃ χ,

which is n-good. We point out that for this we have to \underline{replace} n in (4.5) (1) ... (8) \underline{by} n − 1. We only use (A, χ, W, F) for the description of diagram D(n + 1). We define the isomorphism

(10)  γ′ : Ω(n + 1) ≅ A

by the map which carries V to W, that is

γ′(a) = \{ \begin{matrix}a & for |a| ≤ n − 1 \\ (1 − μχ̄)(a) & for |a| = n, n + 1 \end{matrix}

Then diagram D(n + 1) is defined by

(11)  β′_{n+2} = γγ′ : Ω(n + 1) ≅ Ω(n + 1),
(12)  H′_{n+1} = (γ ⊗ γ)(K + F)γ′, restricted to Ω(n)

Accordingly we obtain the elements

(13)  \{ \begin{matrix}β(\underline{n + 1}) & = γγ′(\underline{n + 1}) \\ H(\underline{n}) & = (γ ⊗ γ)(K + F)γ′(\underline{n}) with γ′(\underline{n}) = \underline{n} \\ β′(\underline{n + 2}) & = γγ′(\underline{n + 2}) \\ H′(\underline{n + 1}) & = (γ ⊗ γ)(K + F)γ′(\underline{n + 1}) \end{matrix}

It remains to show that the elements in (13) coincide with the elements described in (5.6). For this we need the definition of χ and F in (4.5) where we replace n by n − 1. Recall that |\underline{n + 1}| = n.

(14)  \begin{aligned}β(\underline{n + 1}) & = β′_{n+1}(\underline{n + 1} − μχ̄(\underline{n + 1})); see (2) and (10) \\ & = β′(\underline{n + 1}) − βμ(ψ̄(\underline{n + 1}) − λϕψ̄(n + 1)) \end{aligned}

Here we have β′_{n+1}μ = βμ in degree n since image (μ) is decomposable and we use (4.5) (2). By (3.8) [A] we know that Δ̄μ + λϕ = 1 where μΔ̄μ = μ by 3.7 [A] since μ has only coordinates in U_{j+1}(V), j ≥ 1; see (4.2) (5). Hence we see that μλϕ = 0 and we get by (6)

β(\underline{n + 1}) = β′(\underline{n + 1}) − βμ(ψ̄′(\underline{n + 1}) − K(d(\underline{n + 1})))

ψ̄′(\underline{n + 1}) = (γ ⊗ γ)^{−1}ψ̄_{Ω}γ(\underline{n + 1})
= (β ⊗ β)^{−1}ψ̄_{Ω}β′(\underline{n + 1}); see (2),

K(d(\underline{n + 1})) = K(d′_{L}(\underline{n + 1})); see (3),
= (β ⊗ β)^{−1}H′_{n}(β^{−1}d_{Ω}β′(\underline{n + 1})); see (5) and (5.4) (9).

This completes the proof of (5.6) (1). Next we get the following formulas in degree n − 1.

(15)  \begin{aligned}H(\underline{n}) & = (γ ⊗ γ)(K(\underline{n}) + F (\underline{n})) \\ & = (β ⊗ β)((β ⊗ β)^{−1}H′(\underline{n}) + F (\underline{n})); see (5) and (5.8) (8) \\ & = H′(\underline{n}) + (β ⊗ β)F (\underline{n}) \end{aligned}

By (4.5) (3) we have

(15a)  F (\underline{n}) = λϕλq(G_{1}(\underline{n}), G_{2}(\underline{n}))

G_{1}(\underline{n}) = (1 − T )K(\underline{n}) + (γ ⊗ γ)^{−1}E γ(\underline{n}); see (7)
= (1 − T )(β ⊗ β)^{−1}H′(\underline{n}) + (β ⊗ β)^{−1}E β(\underline{n}); see (5) and (2)
= (β ⊗ β)^{−1}((1 − T )H′(\underline{n}) + E β(\underline{n}))

G_{2}(\underline{n}) = (1 ⊗ ψ − ψ ⊗ 1)K(\underline{n}) + (1 ⊗ K − K ⊗ 1)ψ′(\underline{n}), see (8)
= (1 ⊗ Δ − Δ ⊗ 1)(β ⊗ β)^{−1}H′(\underline{n}); see (5), (6)
+ (1 ⊗ K − K ⊗ 1)(β ⊗ β)^{−1}ψ_{Ω}β(\underline{n}); see (3a), (2)

Here K can be replaced by (β ⊗ β)^{−1}H′_{n} since we are in degree n − 1; see (5). By (15) we thus obtain a proof of (5.6) (2). Here we have Λ(\underline{n}) = F (\underline{n}) by (5.5) (3).

Next we obtain in degree n + 1 the following equations.

(16)  \begin{aligned}β′(\underline{n + 2}) & = γγ′(\underline{n + 2}) \\ & = γ(1 − μχ̄)(\underline{n + 2}); see (10), \\ & = γ(\underline{n + 2}) − γμχ̄(\underline{n + 2}) \\ & = \underline{n + 2} − β′_{n+1}μχ̄(\underline{n + 2}); see (2). \end{aligned}

By (4.5) (5) we have

χ̄(\underline{n + 2}) = ψ̄(\underline{n + 2}) − F (d(\underline{n + 2}))
= ψ̄′(\underline{n + 2}) − K(d(\underline{n + 2})) − F (d(\underline{n + 2})); see (6)
X = ψ̄′(\underline{n + 2}) = (β′_{n+1} ⊗ β′_{n+1})^{−1}ψ̄_{Ω}(\underline{n + 2}); see (2), (3a)
d(\underline{n + 2}) = (β′_{n+1})^{−1}d_{Ω}(\underline{n + 2}); see (3)

Hence we get by (16)

(17)  β′(\underline{n + 2}) = \underline{n + 2} − β′_{n+1}μ(X + Y + Z_{1} + Z_{2})

with X above and Y and Z_{1}, Z_{2} computed as follows:

\begin{aligned}Y & = −K(β′_{n+1})^{−1}d_{Ω}(\underline{n + 2}) \\ & = K(β′_{n+1})^{−1}(∑_{i=0}^{n+2}(−1)^{i}∂_{i}(\underline{n + 2}) − ∑_{i=2}^{n}(−1)^{i}\{0 … i\}\{i … n + 2\}) \end{aligned}

Here we have by definition of ▽′(\underline{n + 1}) in (5.5) (1)

(18)  (β′_{n+1})^{−1}(∂_{i}(\underline{n + 2})) = ∂_{i}(\underline{n + 2}) + ▽′(∂_{i}(\underline{n + 2}))

and we have K∂_{i}(\underline{n + 2}) = 0 by (5). Thus we get the formula for Y in (5.6) (3) since K coincides with (β ⊗ β)^{−1}H′_{n} and β′_{n+1} coincides with β on Ω(n − 1). Next we compute by (18)

−F (d(\underline{n + 2})) = −F ((β′_{n+1})^{−1}d_{Ω}(\underline{n + 2}))

= F (∑_{i=0}^{n+2}(−1)^{i}(∂_{i}(\underline{n + 2}) + ▽′(∂_{i}(\underline{n + 2}))) − β^{−1} ∑_{i=2}^{n}(−1)^{i}\{0 … i\}\{i … n + 2\}) (19)

By (4.5) (4) we know that F (a) = 0 for a ∈ N with |a| < n − 1 and |a| > n. Here we have by (4.5) (3) for v = ∂_{i}(\underline{n + 2})

μF (v) = μλϕλq(G_{1}v, G_{2}v) = 0

since μλϕ = 0; see the argument in (14). Hence we have

(20)  −μF (d(\underline{n + 2})) = μ(Z_{1} + Z_{2})

where

\begin{aligned}Z_{1} & = F (∑_{i=0}^{n+2}(−1)^{i} ▽′ (∂_{i}(\underline{n + 2}))) \\ Z_{2} & = −F β^{−1} ∑_{i∈\{2,n\}}(−1)^{i}\{0 … i\}\{1 … n + 2\} \end{aligned}

We know for ξ in (5.5) (2) that F (ξ) = 0 hence we get by (5.5) (2)

Z_{1} = ∑_{i=0}^{n+2}(−1)^{i} ∑_{j=0}^{n+1} F (∂_{j}∂_{i}(\underline{n + 2}) · v_{j}(∂_{i}(\underline{n + 2})) + u_{j}(∂_{i}(\underline{n + 2})) · ∂_{j}∂_{i}(\underline{n + 1}))

Now F : ψ ≃ χ is the derivation homotopy in (9) where ψ and χ coindide with Δ in degree 1. Hence we obtain the formula for Z_{1} in (5.6) (3) since F (∂_{j}∂_{i}(\underline{n + 2})) = Λ(∂_{j}∂_{i}(\underline{n + 2})) by (15a). Moreover since β is the identity in degree 1 we get

Z_{2} = −F ({0, 1, 2} · β^{−1}{2 … n + 2} + (−1)^{n}β^{−1}{0 … n} · {n, n + 1, n + 2})

Here we have β^{−1}\underline{n} = \underline{n} + η where η = ▽(\underline{n}) is decomposable so that F (η) = 0. Hence we get F β^{−1}\underline{n} = F (\underline{n}) = Λ(\underline{n}) as computed in (15a). This proves the formula for Z_{2} in (5.6) (3) and hence the proof of (5.6) (3) is complete.

Finally we compute by (10) and (5)

(21)  \begin{aligned}H′(\underline{n + 1}) & = (γ ⊗ γ)(K + F)γ′(\underline{n + 1}) \\ & = (γ ⊗ γ)(K + F)(\underline{n + 1} − μχ̄(\underline{n + 1})) \\ & = (γ ⊗ γ)(F (\underline{n + 1}) − (K + F)μχ̄(\underline{n + 1})) \end{aligned}

where we can replace Kμ by (β ⊗ β)^{−1}H′_{n}μ (see (5)) and where by (4.5) (2)

μχ̄(\underline{n + 1}) = μ(ψ̄(\underline{n + 1}) − λϕψ̄(\underline{n + 1}))

with μλϕ = 0 (see (14)) so that

μχ̄(\underline{n + 1}) = μψ̄(\underline{n + 1})
= μ(ψ̄′(\underline{n + 1}) − F (d \underline{n + 1})), see (6)
= μ((γ ⊗ γ)^{−1}ψ̄_{Ω}γ(\underline{n + 1}) − F (d′_{L}\underline{n + 1})), see (3) and (3a)

= μ((β ⊗ β)^{−1}ψ̄_{Ω}β′(n + 1) − F (β^{−1}d_{Ω}β′(\underline{n + 1}))), see (2) and (5.4) (9).

Here we have β′(\underline{n + 1}) = \underline{n + 1} + ξ where ξ is decomposable so that:

μF β^{−1}d_{Ω}β′(n + 1) = μF β^{−1}d_{Ω}(\underline{n + 1} + ξ)

= μF β^{−1}(− ∑_{i=0}^{n−1}(−1)^{i}∂_{i}\underline{n + 1}), since F (a) = 0 for |a| < n − 1,

= μF (− ∑_{i=0}^{n−1}(−1)^{i}∂_{i}\underline{n + 1}) = 0

since β^{−1}(\underline{n}) = \underline{n} + η with F η = 0 and since μF (\underline{n}) = 0 by (15a) and μλϕ = 0. Hence we have proved

μχ̄(\underline{n + 1}) = μ(β ⊗ β)^{−1}ψ̄_{Ω}β′(\underline{n + 1})

so that by (21):

H′(\underline{n + 1}) = (γ ⊗ γ)F (\underline{n + 1})
− (γ ⊗ γ)(β ⊗ β)^{−1}H′_{n}μ(β ⊗ β)^{−1}ψ̄_{Ω}β′(n + 1)
− (γ ⊗ γ)F μ(β ⊗ β)^{−1}ψ̄_{Ω}β′(\underline{n + 1})

and hence by (2)

(22)  \begin{aligned}H′(\underline{n + 1}) & = (β′_{n+1} ⊗ β′_{n+1})F (\underline{n + 1}) − R − \frac{1}{4}(β ⊗ β)(S) \\ R & = H′_{n}μ(β ⊗ β)^{−1}ψ̄_{Ω}β′(n + 1) \\ S & = 4 · F μ(β ⊗ β)^{−1}ψ̄_{Ω}β′(\underline{n + 1}) \end{aligned}

Taking into account that F (a) = 0 for |a| < n − 1 we obtain the following formulas

\frac{1}{4}S = F μ(β ⊗ β)^{−1}ψ̄_{Ω}β′(\underline{n + 1)}
= F μ(β ⊗ β)^{−1}ψ̄_{Ω}(\underline{n + 1})
= F μψ̄_{Ω}(\underline{n + 1}) = F μ(S′) with
S′ = ∑_{f∈M} τ(f̄)f̄⊗ f

where M is the set of all f, f ≤ \{0 … n+1\}, f ≠ \{0, n+1\} and f ≠ \underline{n + 1} for which |f| = n − 1 or for which f̄ contains a factor of degree n − 1. Hence M consists of the following subsets of \underline{n + 1},

\{0, 1, n + 1\}, \{0, n, n + 1\}, ∂_{i}(\underline{n + 1}) with i = 1, …, n

Hence we have

S′ = S′(\underline{n + 1}) = +\{0 … n\} ⊗ \{0, n, n + 1\}

− (−1)^{n}{1 … n + 1} ⊗ {0, 1, n + 1}

− ∑_{i=1}^{n}(−1)^{i}\{i − 1, i, i + 1\} ⊗ ∂_{i}(\underline{n + 1})

By definition of μ in (4.3) we thus get

μ(S′) = \frac{1}{2}C(S′) = \frac{1}{4}m(1 + T )(S′) ∈ Ω

Here m denotes the multiplication m : Ω ⊗ Ω → Ω. Thus we get

(23)  S = 4F μ(S′) = F (1 + T )(S′)

By (15a) this coincides with the formula for S in (5.6) (4) since ψ and χ coincide with Δ in degree 1. Finally we consider the first summand in (22). By (4.5) (3) we have

(24)  F (\underline{n + 1}) = λϕλq(G_{1}\underline{n + 1}, G_{2}\underline{n + 1})

Here (7) yields

\begin{aligned}G_{1}(\underline{n + 1}) & = (1 − T )K(\underline{n + 1}) + (γ ⊗ γ)^{−1}E γ(\underline{n + 1}) \\ & = (β′_{n+1} ⊗ β′_{n+1})^{−1}E β′(\underline{n + 1}), see (5) and (2). \end{aligned}

For this we have to check that Eβ′(\underline{n + 1}) ∈ Ω(n) ⊗ Ω(n); but this is a consequence of the fact that E(\underline{n + 1}) ∈ Ω(n) ⊗ Ω(n). Hence G_{1}(\underline{n + 1}) coincides with the element G′_{1}(\underline{n + 1}) in (5.5) (4). On the other hand we get by (8) the formula

(25)  \begin{aligned}G_{2}(\underline{n + 1}) & = (1 ⊗ ψ − ψ ⊗ 1)K(\underline{n + 1}) + (1 ⊗ K − K ⊗ 1)ψ′(\underline{n + 1}) \\ & = (1 ⊗ K − K ⊗ 1)(γ ⊗ γ)^{−1}ψ_{Ω}γ(\underline{n + 1}); see (5), (3a). \end{aligned}

Here γ can be replaced by β′_{n+1} and K can be replaced by (β ⊗ β)^{−1}H′_{n}. To see this we write ψ_{Ω}(x) = 1 ⊗ x + x ⊗ 1 + ψ̄_{Ω}(x) so that

\begin{aligned}(γ ⊗ γ)^{−1}ψ_{Ω}γ(\underline{n + 1}) & = (γ ⊗ γ)^{−1}(1 ⊗ γ(\underline{n + 1}) + γ(\underline{n + 1}) ⊗ 1 + ψ̄_{Ω}γ(\underline{n + 1})) \\ & = 1 ⊗ \underline{n + 1} + \underline{n + 1} ⊗ 1 + (γ ⊗ γ)^{−1}ψ̄_{Ω}β′(\underline{n + 1}); see (2). \end{aligned}

Here we have (γ ⊗ γ)^{−1}ψ̄_{Ω} = (β′_{n+1} ⊗ β′_{n+1})^{−1}ψ̄_{Ω} in degree n. Moreover we have K(\underline{n + 1}) = 0 by (5) so that we get

G_{2}(\underline{n + 1}) = G′_{2}(n + 1)

where the right hand side is the element in (5.4) (4). Hence F (\underline{n + 1}) = Λ′(\underline{n + 1}) coincides by (24) with the element in (5.4) (4) and therefore the proof of (5.6) (4) is complete. q.e.d.

(5.12) Proof of (5.8). We have by (5.6) (3)

(1)  \begin{aligned}d_{L}(\underline{n + 2}) & = β^{−1}d_{Ω}β′(\underline{n + 2}) \\ & = β^{−1}d_{Ω}(\underline{n + 2}) − β^{−1}d_{Ω}β′_{n+1}μ(X + Y + Z_{1} + Z_{2}) \end{aligned}

where d_{Ω}β′_{n+1} = βd_{L} by (5.7) and d_{L}μ = μd_{L} since μ is a Der(L) -homomorphism; see 3.7 (a) [A]. Hence we get

(2)  d_{L}(\underline{n + 2}) = β^{−1}(−∂ + ∂^{+}) − μd_{L}(X + Y + Z_{1} + Z_{2})

where we put ∂ = ∂(\underline{n + 2}), ∂^{+} = ∂^{+}(\underline{n + 2}) as in (5.3). We have

β^{−1}(∂) = ∂ + ▽(∂)

where ▽ is the linear map defined by ▽ in (5.5). Using (5.6) (A) and (5.5) we get

▽(∂) = −β^{−1}ν(∂)
= −β^{−1}(ν′(∂) − βμ(β ⊗ β)^{−1}(A(∂))
= ▽′(∂) + μ(β ⊗ β)^{−1}(A(∂))
A(∂) = ψ̄_{Ω}β′(∂) − H′_{n}d_{L}(∂)
= ψ̄_{Ω}(∂ + ν′(∂)) − H′_{n}d_{L}(∂)

where ν′(∂) = −β ▽′ (∂). Hence we get for the summand −β^{−1}(∂) of (2) the equation

(3)  \begin{matrix}− β^{−1}(∂) = −∂ − ▽′(∂) \\ − μ(β ⊗ β)^{−1}(ψ̄_{Ω}∂ − ψ̄_{Ω}β ▽′ (∂) − H′_{n}d_{L}(∂)) \end{matrix}

On the other hand the summand −μ d_{L}X of (2) has the form

(4)  \begin{aligned}−μ d_{L}X & = −μ d_{L}(β′_{n+1} ⊗ β′_{n+1})^{−1}ψ̄_{Ω}(\underline{n + 2}) \\ & = −μ(β ⊗ β)^{−1}ψ̄_{Ω}d_{Ω}\underline{n + 2} \\ & = μ(β ⊗ β)^{−1}ψ̄_{Ω}(∂ − ∂^{+}) \end{aligned}

Here the first summand of (4) and the corresponding summand of (3) cancel. Next we get

−μ d_{L}Y = μ d_{L}(β ⊗ β)^{−1}H′_{n}(− ▽′ (∂) + β^{−1}(∂^{+}))

with

\begin{aligned}d_{L}(β ⊗ β)^{−1}H′_{n} & = (β ⊗ β)^{−1}d_{Ω}H′_{n} \\ & = (β ⊗ β)^{−1}(ψ̄_{Ω}β − (β ⊗ β)Δ̄ − H′_{n}d_{L}). \end{aligned}

See (5.4) (6). Hence we get

−μ d_{L}Y = μ(β ⊗ β)^{−1}ψ̄_{Ω}(−β ▽′ (∂) + ∂^{+})

− μΔ̄(− ▽′ (∂) + β^{−1}(∂^{+})) (5)

− μ(β ⊗ β)^{−1}H′_{n}d_{L}(− ▽′ (∂) + β^{−1}∂^{+}) (6)

Here we have

− ▽′ (∂) + β^{−1}(∂^{+}) = β^{−1}ν′(∂) + β^{−1}(∂^{+})
= β^{−1}(β′_{n+1}(∂) − ∂ + ∂^{+})
= β^{−1}(β′_{n+1}(∂) + d_{Ω}\underline{n + 2}) = ξ_{1}

with

\begin{aligned}d_{L}ξ_{1} & = β^{−1}d_{Ω}β′_{n+1}(∂) \\ & = β^{−1}β d_{L}(∂) = d_{L}(∂) \end{aligned}

Hence we get

(7)  −μ(β ⊗ β)^{−1}H′_{n}d_{L}(∂) = (6)

It remains to check that

(8)  μ d_{L}(Z_{1} + Z_{2}) = 0.

Then putting all equations together the proof of (5.8) is complete. For the proof of (8) we use (20) in the proof of (5.11) so that we get

(9)  μ d_{L}(Z_{1} + Z_{2}) = μ d_{L}F (d(\underline{n + 2}))

Recall that F : A_{+} → A_{+} ⊗ A_{+} is a homotopy F : ψ̄ ≃ χ̄. Moreover in degree ≤ n the differential of A coincides with d_{L}, see (5.11) (3) and (5.7). Hence we have in (9)

(10)  \begin{aligned}μ d_{L}(Z_{1} + Z_{2}) & = μ(dF)(d \underline{n + 2}) \\ & = μ(ψ̄ − χ̄ − F d)(d \underline{n + 2}) \\ & = μ(ψ̄ − χ̄)d \underline{n + 2} \end{aligned}

Here we have ψ̄ = χ̄ in degree ≤ n − 1 by (4.5) (1) (where we have to replace n by n − 1). This implies by (5.11) (3)

(11)  \begin{aligned}(ψ̄ − χ̄)d \underline{n + 2} & = (ψ̄ − χ̄)(β′_{n+1})^{−1}d_{Ω}\underline{n + 2} \\ & = (ψ̄ − χ̄)(β′_{n+1})^{−1}(− ∑_{i=0}^{n+2}(−1)^{i}∂_{i}\underline{n + 2}) \\ & = (ψ̄ − χ̄) (− ∑_{i=0}^{n+2}(−1)^{i}∂_{i}\underline{n + 2}) \end{aligned}

Here the last equation uses the fact that ▽′(\underline{n + 1}) in (5.5) is decomposable. Finally we get by (4.5) (2) (where we replace n by n − 1) the equation

(12)  μ(ψ̄ − χ̄)(\underline{n + 1}) = μλϕψ̄(\underline{n + 1}) = 0

where μλϕ = 0; compare (5.11) (14). q.e.d.

Let m : Ω ⊗ Ω → Ω be the multiplication of the tensor algebra Ω = T (U). The following elements are needed in (5.8).

(5.13) Lemma. (1 − μΔ̄) ▽′ \underline{n + 1} = \frac{1}{4}(1 − μΔ̄)m(1 + T )A′(\underline{n + 1}) where

A′(\underline{n + 1}) = μ(β ⊗ β)^{−1}A\{0 … n\} ⊗ \{0, n, n + 1\}

− (−1)^{n}μ(β ⊗ β)^{−1}A{1 … n + 1} ⊗ {0, 1, n + 1}

− ∑_{i=1}^{n}(−1)^{i}\{i − 1, i, i + 1\} ⊗ μ(β ⊗ β)^{−1}A(∂_{i}\underline{n + 1})

Here A is defined as in (5.6) (1).

Proof. For π_{1} = 1 − μΔ̄ we have by (5.5) and (5.6) (3)

\begin{aligned}π_{1} ▽′ \underline{n + 1} & = −π_{1}β^{−1}ν′(\underline{n + 1}) \\ & = +π_{1}β^{−1}β′_{n}μ(X + Y + Z_{1} + Z_{2})(\underline{n + 1}) \end{aligned}

Here we have β′_{n} = β in degree < n−1 so that β^{−1}β′_{n}μ(Y +Z_{1}+Z_{2}) = μ(Y +Z_{1}+Z_{2}) since Y, Z_{1}, Z_{2} are decomposable. Using this type of argument we get the following equations since π_{1}μ = 0.

π_{1} ▽′ \underline{n + 1} = π_{1}β^{−1}β′_{n}μ(X(\underline{n + 1}))

= π_{1}β^{−1}β′_{n}μ(β′_{n} ⊗ β′_{n})ψ̄_{Ω}\underline{n + 1}

= π_{1}β^{−1}β′_{n}μ(β′_{n} ⊗ β′_{n})S′(\underline{n + 1})

= π_{1}β^{−1}β′_{n}μS′(\underline{n + 1})

= π_{1}β^{−1}β′_{n}(\frac{1}{4}m(1 + T ))S′(\underline{n + 1}), see (4.3) (1),

= \frac{1}{4}π_{1}m(1 + T )(β^{−1}β′_{n} ⊗ β^{−1}β′_{n})S′(\underline{n + 1})

Here we use S′(\underline{n + 1}) in (5.11) (23) so that S′(\underline{n + 1}) = \underline{n} ⊗ \{0, n, n + 1\} + . . .. Here we have

(β^{−1}β′_{n} ⊗ β^{−1}β′_{n})(\underline{n} ⊗ \{0, n, n + 1\}) =
β^{−1}β′_{n}(\underline{n}) ⊗ \{0, n, n + 1\} =
(\underline{n} + μ(β ⊗ β)^{−1}A(\underline{n})) ⊗ \{0, n, n + 1\}

In the last equation we use (5.6) (1). This yields the equation in the lemma since

\frac{1}{4}π_{1} m(1 + T )(\underline{n} ⊗ \{0, n, n + 1\}) = π_{1}μ(\underline{n} ⊗ \{0, n, n + 1\}) = 0

q.e.d.

(5.14) Corollary. (1 − μΔ̄) ▽′ (\underline{n + 1}) ∈ Ω_{+}^{3} = ⊗_{i≥3}U^{⊗i}.

The corollary is a consequence of the lemma since A(\underline{n}) is decomposable; i.e. A(\underline{n}) ∈ Ω_{+}^{2}. For this we need the fact that π_{1} carries a word w ∈ N^{∗} to a linear combination of words obtained by permuting the letters of w, see 3.2.3 [R].

(5.15) Proof of (4.11). By (5.8) we already know d_{L}^{(1)}. Next we see by (5.8) and (5.14) that d_{L}^{(2)}(\underline{n + 2}) is the quadratic part of

(1 − μΔ̄)(∑_{i=2}^{n}(−1)^{i}β^{−1}{0 … i} · β^{−1}{1 … n + 2})

where β^{−1}\underline{n} = \underline{n} + ▽(\underline{n}) where ▽(\underline{n}) is decomposable. Hence we get

d_{L}^{(2)}(\underline{n + 2}) = (1 − μΔ̄)(∑_{i=2}^{n}(−1)^{i}\{0 … i\} · \{2 … n + 2\})

and this yields the formula for d_{L}^{(2)} in (4.11) since

(1 − μΔ̄)\{0 … i\} · \{i … n + 2\} = \frac{1}{2}[\{0 … i\}, \{i … n + 2\}].

Finally we refer to the properties of μ and λ in (4.3) for the proof of the fact that d_{L}^{(t)} is defined over R with t < ρ(R); for this compare the corresponding argument of Anick in [A]. q.e.d.

(5.16) Example. Below we consider only subsets of {0, 1, …, 9}. Therefore we can simplify the notation by setting 135 = \{1, 3, 5\} or \underline{2} = 012, \underline{3} = 0123. Moreover the product in Ω is denoted by 012 · 234 = {0, 1, 2} · {2, 3, 4}. With this notation we get by (5.2) (1), (2):

(1)  ψ̄_{Ω}(\underline{2}) = 0 and
ψ̄_{Ω}(\underline{3}) = −123 ⊗ 013 + 012 ⊗ 023 and
ψ̄_{Ω}(\underline{4}) = 1234 ⊗ 014 + 0123 ⊗ 034
+ 012 ⊗ 0234 + 234 ⊗ 0124 − 123 ⊗ 0134
− (012 · 234) ⊗ 024

(2)  E(\underline{2}) = 012 ⊗ 012 and
E(\underline{3}) = −0123 ⊗ (013 + 123)
+ (023 + 012) ⊗ 0123

We now consider the first step in the induction (5.6) when we put n = 2. We get by (5.5) (3), (4) the formulas

(3)  \begin{aligned}Λ(\underline{2}) & = λϕλq(E(\underline{2})) = \frac{1}{4}(1 − T )(E(\underline{2})) \\ & = \frac{1}{4}[012, 012] and \\ Λ′(\underline{3}) & = λϕλq(E(\underline{3})) = \frac{1}{4}(1 − T )(E(\underline{3})) \\ & = −\frac{1}{4}[0123, 013 + 123] + \frac{1}{4}[023 + 012, 0123] \end{aligned}

Here we see by the definition of λ, ϕ that we can put λ = −\frac{1}{2}T and ϕ = (1 − T ). Moreover we get by definition of μ

(4)  \begin{aligned}μ ψ̄_{Ω}(\underline{3}) & = \frac{1}{4}m(1 + T )ψ̄_{Ω}\underline{3} \\ & = \frac{1}{4}(−123 · 013 + 013 · 123 + 012 · 023 − 023 · 012) \end{aligned}

Now we get by (5.6) and (5.4) the formulas (5)

β(\underline{3}) = \underline{3} − μ ψ̄_{Ω}(\underline{3}),
H(\underline{2}) = Λ(\underline{2}),
β′(\underline{4}) = \underline{4} − μ(ψ̄_{Ω}(\underline{4}) + Δ(012) · Λ(234) − Λ(012) · Δ(234)),
H′(\underline{3}) = Λ′(\underline{3}) − S(\underline{3})

Compare the definition of S in (5.6) (4). We are now reday to compute d_{L}\underline{5}. Using (5) we have

(6)  β^{−1}(\underline{3}) = \underline{3} + μ ψ̄_{Ω}(\underline{3})

Hence we get by (5.8) the following formula with π_{1} = 1 − μ Δ̄.

\begin{aligned}d_{L}\underline{5} & = − ∑_{i=0}^{5}(−1)^{i}(∂_{i}\underline{5} + π_{1} ▽′ (∂_{i}\underline{5})) \\ & + π_{1}((012) · β^{−1}(2345) − β^{−1}(0123) · (345)) \end{aligned}

Hence we obtain d_{L}(\underline{5}) = d_{L}^{(1)}(\underline{5}) + d_{L}^{(2)}(\underline{5}) + d_{L}^{(3)}(\underline{5}) with

d_{L}^{(1)}(\underline{5}) = − ∑_{i=0}^{5}(−1)^{i}∂_{i}\underline{5}
d_{L}^{(2)}(\underline{5}) = π_{1}(012 · 2345 − 0123 · 345)

= \frac{1}{2}[012, 2345] − \frac{1}{2}[0123, 345]

d_{L}^{(3)}(\underline{5}) = − ∑_{i=0}^{5}(−1)^{i}π_{1} ▽′ (∂_{i}\underline{5})

+ π_{1}((012) · μψ̄_{Ω}(2345) − μψ̄_{Ω}(0123) · (345))

Here we have by (5.13)):

π_{1} ▽′ (\underline{4}) = \frac{1}{4}π_{1}m(1 + T )A′(\underline{4}) where
A′(\underline{4}) = μ ψ̄_{Ω}(0123) ⊗ 034 + μ ψ̄_{Ω}(1234) ⊗ 014
+ 012 ⊗ μ ψ̄_{Ω}(0234) − 123 ⊗ μ ψ̄_{Ω}(0134)
+ 234 ⊗ μψ̄_{Ω}(0124)

Now d_{L}^{(3)}(5) is completely computed by use of (4).

References

[A] Anick, D.J., Hopf algebras up to homotopy, Journal of the AMS, 2 (1989), 417-443.

[Ad] Adams, J.F., On the cobar construction, Prof. Nat. Acad. Sci. (USA) 42 (1956), 409-412.

[AH] Adams, J.F. and Hilton, P.J., On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305-330.

[B] Baues, H.J., Geometry of loop spaces and the cobar construction, Memoirs of the AMS 25 (1980), 170 pages.

[B′] Baues, H.J., The double bar and cobar construction, Compositio Math. 43 (1981), 331-341.

[BG] Bousfield, A.K. and Gugenheim, V.K.A.M., On P L de Rham theory and rational homotopy type, Memoirs of the AMS 179 (1976).

[BL] Baues, H.J. and Lemaire, J.M., Minimal models in homotopy theory, Math. Ann. 225 (1977), 219-242.

[GJ] Getzler, E. and Jones, J.D.S., Operads, homotopy algebra, and iterated integrals for double loop spaces, preprint 1995.

[GV] Gerstenhaber, M. and Voronov, A.A., Homotopy G-algebras and moduli space operad, IMRN International Math. Research Notices No 3 (1995), 141-153.

[H] Hess, K., Mild and tame homotopy theory, Pure and Appl. Algebra 84 (1993), 277-310.

[Ha] Halperin, S., Universal enveloping algebras and loop space homology, Preprint.

[HMS] Husemöller, D. and Moore, J.C. and Stasheff, J., Differential homological algebra and homogeneous spaces, J. pure appl. Algebra 5 (1974), 113-185.

[HW] Hilton, P.J. and Wylie, S., Homology Theory, Cambridge University Press (1965).

[K] Kan, D.M., A combinatorial definition of homotopy groups, Ann. of Math. 67 (1958), 288-312.

[M] Majewski, M., A cellular Lie algebra model for spaces and its equivalence with the models of Quillen and Sullivan, Thesis Freie Universität Berlin, Fachbereich Math. (1966).

[MM] Milnor, J. and Moore, J.C., On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211-264.

[Mo] Moore, J.C., Differential homological algebra, Actes du Congr. Intern. des Math. (1970), 335-336.

[Q] Quillen, D.G., Rational homotopy theory, Ann of Math. (2) 90 (1969), 205-295.

[R] Reutenauer, C., Free Lie Algebras, London Math. Soc. Monographs. New Series 7. Oxford University Press (1993), 269 pages.