Georgian Mathematical Journal Volume 9 (2002), Number 3, 549–566

ON A MULTIPLICATIVITY UP TO HOMOTOPY OF THE GUGENHEIM MAP

Z. KHAREBAVA

Abstract. In the category of differential algebras with strong homotopy there is a Gugenheim’s map {ρ_{i}} : A^{∗} → C^{∗} from Sullivan’s commutative cochain complex to the singular cochain complex of a space, which induces a differential graded coalgebra map of appropriate Bar constructions. Both (BA^{∗}, d_{BA^{∗}}, △,) and (BC^{∗}, d_{BC^{∗}}, △,) carry multiplications. We show that the Gugenheim’s map

B{ρ_{i}} : (BA^{∗}, d_{BA^{∗}}, △) → (BC^{∗}, d_{BC^{∗}}, △)

is multiplicative up to homotopy with respect to these structures.

2000 Mathematics Subject Classification: 55N10, 57T05, 57T30, 58A10. Key words and phrases: De Rham theorem, twisting cochain, deformation functor, Bar-construction.

Let J denote either the category of C^{∞} manifolds (possibly with boundaries and corners) and C^{∞}-maps, or that of simplicial complexes and simplicial maps. Let V denote the category of graded vector spaces over the reals or rationals, and let A^{∗} : J → V denote either the classical de Rham functor of differential forms, or Sullivan’s functor of rational differential forms; see [11]. By C^{∗} : J → V we denote the functor of either normalized singular differentiable cochains or normalized simplicial cochains. The differentiable cochain (chain) functor is homology equivalent to usual (continuous) singular functors. This is a wellknown fact; for the proof see, for example, [23].

The transformation of functors ρ : A^{∗} → C^{∗} is defined by

⟨ρω, c⟩ = ∫_{c} ω,

where c is any singular (or simplicial) chain. This transformation was first constructed in the 1930’s in the context of smooth manifolds and ordinary differential forms. The de Rham theorem says, in either case, that ρ induces a homology isomorphism. Additionally, although ρ itself is not a map of algebras, the theorem asserts that the isomorphism

ρ^{∗} : H(A^{∗}) → H(C^{∗})

is a map of algebras [8].

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Gugenheim in [12] proved that ρ = ρ_{1} : A^{∗} → C^{∗} can be extended to a map in DASH (the category of differential algebras with strong homotopy) [13], [21], [22]. This means that there exists a whole family of “higher homotopies”

ρ_{i} : A^{∗⊗ i} → C^{∗} (i ≥ 1),

where A^{∗⊗ i} = A^{∗} ⊗ ⋯ ⊗ A^{∗} (i times), such that

Dρ_{i} = ∑_{j=1}^{i−1}(−1)^{j}{ϕ(ρ_{j} ⊗ ρ_{i−j}) − ρ_{i−1}(1^{(i−j)} ⊗ ϕ ⊗ 1^{(i−j−1)})}.

Here Dρ_{i} = d_{C^{∗}} · ρ_{i} + (−1)^{i} ρ_{i} · d_{A^{∗}}^{⊗}, where d_{A^{∗}}^{⊗} is a usual differential on A^{∗⊗ i}, ρ_{i} has degree −i + 1 and ϕ stands for multiplication. Thus

ϕ(α ⊗ β) = α ∧ β (exterior product) if α, β ∈ A^{∗}(X)
= α ⌣ β (cup product) if α, β ∈ C^{∗}(X).

For i = 1, 2 we get

Dρ_{1} = 0, Dρ_{2} = ρ_{1}ϕ − ϕ(ρ_{1} ⊗ ρ_{1}).

The latter statement contains, of course, the classical result that H(ρ) is multiplicative.

It is well known that any DASH map {ρ_{i}} : A^{∗} → C^{∗} induces a coalgebra map of appropriate Bar constructions

B{ρ_{i}} : (BA^{∗}, d_{BA^{∗}}, △) → (BC^{∗}, d_{BC^{∗}}, △);

both (BA^{∗}, d_{BA^{∗}}, △) equipped with shuffle μ_{sh} and (BC^{∗}, d_{BC^{∗}}, △) equipped with product μ_{E} introduced by Baues in [3], [4] are Hopf algebras. In this paper we shall prove the following

Theorem 1. Gugenheim’s map

B{ρ_{i}} : (BA^{∗}, d_{BA^{∗}}, △, μ_{sh}) → (BC^{∗}, d_{BC^{∗}}, △, μ_{E})

is multiplicative up to homotopy.

1. Review of the Notation and Known Results

In this section we recollect the definitions and facts we need; most can be found in [14], [15], [21]. Since these results are not new, we omit their proofs.

1.1. DG module. Let R be a commutative ring with identity 1_{R}. Let {M_{i} }_{i∈ℤ} be a sequence of R-modules; M = ∑ M_{i} is called a graded R-module (throughout this paper we assume M_{i} = 0 for i < 0). An element x ∈ M_{i} is said to be homogeneous of degree i, in which case we write |x| = i. A differential graded module (DGM) is a graded module M together with an endomorphism d ∈ End (M) of degree +1 and square zero. Given DGMs M, N and a homomorphism f : (M, d_{M}) → (N, d_{N}) of degree k, define the differential of f by Df = d_{N}f − (−1)^{k}fd_{M}. The category of DGMs is denoted by DM; its morphisms are chain maps, i.e., maps f of degree 0 with Df = 0.

1.2. DG algebra. DA denotes the category of augmented DGAs with unit. If (A, d, μ, η, ε) ∈ DA, then the multiplication μ : A⊗A → A, the unit η : R → A, the augmentation ε : A → R and the differential d : A → A satisfy the usual requirements. We say that A is connected if η : R → A_{0} is an isomorphism. A morphism in DA is a chain map preserving all structure. Let IA = ker(ε) be the augmentation ideal. The exact sequence 0 → IA →^{i} A →^{ε} R → 0 defines a functor I : DA → DM; the multiplication μ induces a map Iμ : IA ⊗ IA → IA given by i(Iμ) = μ(i ⊗ i).

1.3. DG coalgebra. DC denotes the category of coaugmented differential graded coalgebras (DGC’s) with counit. If (C, Δ, d, η, ε) ∈ DC, then the comultiplication Δ : C → C ⊗ C, the counit ε : C → R, the coaugmentation η : R → C and the differential d : C → C satisfy the usual requirements. We say that C is connected if ε : C_{0} → R is an isomorphism. A morphism in DC is a chain map preserving all structure. Let JC = co ker(η) be the coaugmentation ideal. The exact sequence 0 → R →^{η} C →^{p} JC → 0 defines a functor J : DC → DM; the comultiplication Δ induces a map JΔ : JC → JC ⊗ JC given by (JΔ)p = (p ⊗ p)Δ.

1.4. DG Hopf algebra. A differential graded Hopf algebra (DGHA) is a connected DGA (A, d, μ, η, ε) together with a coassociative comultiplication Δ : A → A ⊗ A such that (A, Δ, d, η, ε) is a DGC and Δ is a map in DA. The DGHA A is cocommutative if T Δ = Δ and commutative if μT = μ, where T : A ⊗ A → A ⊗ A is the twisting involution given by T (a ⊗ b) = (−1)^{|a||b|}b ⊗ a. Let HA denote the category of DGHA’s; morphisms in HA are the coalgebra maps f in DA, i.e., (f ⊗ f)Δ = Δf.

1.5. Suspension. The suspension (resp. desuspension) functor s : DM → DM (resp. s^{−1} : DM → DM) is given by (sM)_{n} = M_{n−1} and d_{sM} = −d_{M}. On an object M, the map s : M → sM has degree +1, is the identity in each dimension and Ds = 0. If f : M → N has degree k, so does sf : sM → sN and the following diagram commutes:

M ⟶^{f} N
s↓  ^{s}↓
sM ⟶^{sf} sN.

1.6. Tensor algebra. The free tensor algebra T (M) on a graded module M is the direct sum T (M) = ∑_{i≥0} M^{⊗i}, where M^{⊗0} = R, with multiplication given by

μ(a_{1} ⋯ a_{i}) ⊗ (a_{i+1} ⋯ a_{n}) = (a_{1} ⋯ a_{n}).

Let i_{1} : M → T (M) be the injection. T (M) has the following universal property: If A is any DGA and α : M → IA is any homomorphism, then there is a unique algebra map f_{α} : T (M) → A commuting the following diagram:

M ⟶^{i_{1}} T (M)
α↓  ^{f_{α}}↓
IA ⟶^{i} A

An explicit formula for f_{α} is given by

f_{α} = μ_{A}^{n}(α^{⊗n} ⊗ i^{⊗n}),

where μ_{A}^{1} = Id, μ_{A}^{2} = μ_{A} and μ_{A}^{n} = μ_{A}(μ_{A}^{n−1} ⊗ Id).

Let (C, d, △, η, ε) ∈ DC. The Cobar construction Ω(C, d, △, η, ε) due to Adams [1] is defined as a free tensor algebra T (s^{−1}(JC)) whose differential d_{Ω} is defined on generators s^{−1}JC by

d_{Ω} · i^{1} = −i^{1}(s^{−1}ds) + i^{2}(s^{−1} ⊗ s^{−1})(JΔ)s,

where i^{k} : (s^{−1}JC)^{⊗k} → T (s^{−1}JC) is an obvious inclusion.

Thus the Cobar construction of a DGC is a DGA with 1 : R → R as unit and augmentation.

1.7. Tensor coalgebra. The cofree tensor coalgebra T^{c} (M) on a graded module M is the direct sum T^{c} (M) = ∑_{i≥0} M^{⊗i}, where M^{⊗0} = R, with the diagonal given by

Δ(a_{1} ⋯ a_{n}) = ∑_{i=0}^{n}(a_{1} ⋯ a_{i}) ⊗ (a_{i+1} ⋯ a_{n}).

Let P_{1} : T^{c}(M) → M be the projection. T^{c} (M) has the following universal property: If C is any DGC and α : JC → M is any homomorphism, then there is a unique coalgebra map f_{α} : C → T^{c}(M) commuting the following diagram:

T^{c}(M) ⟶^{P_{1}} M
f_{α}↑  ^{α}↑
C ⟶^{p} JC

An explicit formula for f_{α} is given by

f_{α} = ∑_{i=1}^{∞}(p^{⊗i} ⊗ α^{⊗i})Δ_{C}^{i},

where Δ_{C}^{1} = Id, Δ_{C}^{2} = Δ_{C} and Δ_{C}^{i} = (Δ_{C}^{i−1} ⊗ Id)Δ_{C}.

Let (A, d, μ, η, ε) ∈ DA. The Bar construction on DGA (A, d, μ, η, ε) is a DGC T^{c} (s(IA)) with differential (coderivation)

d_{BA}[a_{1}| ⋯ |a_{n}] = ∑_{i=0}^{n−1}[\overline{a}_{1}| ⋯ |\overline{a}_{i}|da_{i+1}|a_{i+2}| ⋯ |a_{n}]

+ ∑_{i=0}^{n−2} [\overline{a}_{1}| ⋯ |\overline{a}_{i}|μ(\overline{a}_{i+1} ⊗ a_{i+2})|a_{i+3}| ⋯ |a_{n}],  (1.7.1)

where \overline{a} = (−1)^{|a|+1}a, and degree of [a_{1}| ⋯ |a_{n}] ∈ BA is ∑_{i=1}^{n} |a_{i}| + n.

Thus the Bar construction of DGA is a DGC with 1 : R → R as counit and coaugmentation.

Let S_{p,q} denote shuffle permutations in the set of all permutations ∑_{p+q}, i.e., σ ∈ S_{p,q} implies σ(1) < ⋯ < σ(p), σ(p + 1) < ⋯ < σ(p + q). The shuffle product μ_{sh} : BA ⊗ BA → BA is given by

μ_{sh} ([a_{1}| ⋯ |a_{p}] ⊗ [a_{p+1}| ⋯ |a_{p+q}]) = ∑_{σ∈S_{p,q}} ε(σ)[a_{σ^{−1}(1)}| ⋯ |a_{σ^{−1}(p+q)}],

where ε(σ) = +1 if the number of transpositions is even, and ε(σ) = −1 if the number of transpositions is odd. If DGA (A, d, μ, η, ε) is a commutative algebra, then differential d_{BA} : BA → BA on the Bar construction is a derivation of μ_{sh}, in which case (BA, Δ, d_{m}, μ_{sh}) is a DGHA.

1.8. Twisting cochain. In this section we recall briefly the definition of twisting cochains of Brown [7] and some notions and known facts connected with it.

Definition 2. Let (C, d_{C}, △, ε_{C}, η_{C}) be a DGC and (A, d_{A}, μ, ε_{A}, η_{A}) DGA. The complex Hom(C, A) is defined as a DGA with differential

Df = d · f − (−1)^{| f |}f · d_{C}

and multiplication

f ⌣ g = μ(f ⊗ g)△, f, g ∈ Hom(C, A).

A neutral element in Hom(C, A) is a composition e : C →^{ε_{C}} R →^{η_{A}} A.

It is easy to check that

D(f ⌣ g) = Df ⌣ g + (−1)^{| f |}f ⌣ Dg.

From now on, throughout the paper, we shall work only with the augmentation and coaugmentation ideals of DGA and DGC. Thus we consider the complex Hom(JC, IA).

Definition 3. An element t ∈ Hom(JC, IA) of degree -1 is a twisting cochain in the sense of Brown [7] if it satisfies

Dt = −t ⌣ t.

It is easy to show that for any DGA (A, d_{A}, μ, ε_{A}, η_{A}) (resp. DGC (C, d_{C}, △, ε_{C}, η_{C})) the natural map

p_{1} : BA → A (resp. i_{1} : C → ΩC)

is a twisting cochain.

Let α : JC → IA be a map of degree -1. By the universal properties, mentioned in Sections 1.6. and 1.7, any such homomorphism uniquely defines an algebra map f_{α} : ΩC → A and a coalgebra map g_{α} : C → BA. Homomorphisms f_{α} : ΩC → A and g_{α} : C → BA are chain maps, i.e., maps in DA and DC respectively, if and only if α : JC → IA is a twisting cochain; see [15], for example. Thus there are bijections

Hom_{DC}(C, BA) ↔ T (JC, IA) ↔ Hom_{DA}(ΩC, A),  (1.8.3)

where by T (JC, IA) we denote the set of all twisting cochains from JC to IA.

Berikashvili in [5] introduced the following equivalence relation on the set T (JC, IA).

Definition 4. Two twisting cochains α, α′ : JC → IA are equivalent (we write α ∼ α′) if there exists a homomorphism

β : JC → IA

of degree 0 satisfying the identity

α − α′ = βd_{C} − d_{A}β + β ⌣ α′ − α ⌣ β.

The set T (C, A) factored by this equivalence relation is denoted by D(C, A). Let f : A → B be a map in DA. It is trivial to check that for any α ∈

T (JC, IA) the composition Ifα, where If : IA → IB, is a twisting cochain in T (JC, IB). Moreover, if α ∼ α′, then Ifα ∼ Ifα′. Thus for any DGC C a DGA map f : A → B induces a set map

f_{∗} : D(JC, IA) → D(JC, IB)

defined by [α] ⟼ [Ifα]. Moreover, Berikashvili in [6] proved the following result.

Theorem 5. Let f : A → B be a map in DA inducing an isomorphism in homology. Then

f_{∗} : D(JC, IA) → D(JC, IB)

is a bijection.

Let f = {f_{i}} : A → B be a DASH map. Kadeishvili in [16] showed that if α ∈ T (JC, IA), then

ϕ = ∑_{i} Jf_{i}(α ⊗ ⋯ ⊗ α)△^{i}

belongs to T (JC, IB). Thus for any DGC C a DASH map f = {f_{i}} : A → B induces a set map

T (f) : T (JC, IA) → T (JC, IB)

(α ⟼ ∑_{i} Jf_{i}(α ⊗ ⋯ ⊗ α)J△^{i}), which itself induces a set map

f_{∗} : DT (JC, IA) → D(JC, IB)

defined by [α] ⟼ [∑_{i} Jf_{i}(α ⊗ ⋯ ⊗ α)J△^{i}]. Kadeishvili in [16] generalized Theorem 5 in the following way.

Theorem 6. Let f = {f_{i}} : A → B be a DASH map, such that its first component f_{1} : A → B induces an isomorphism in homology. Then

f_{∗} : D(JC, IA) → DT (JC, IB)

is a bijection.

Remark 7. Note that the surjectivity of f_{∗} : D(JC, IA) → DT (JC, IB) implies that for any twisting cochain β : JC → IB there exists a twisting cochain α : JC → IA making the diagram

JC ⟶^{α}  IA
β↘ {If_{i}}↓
IB

homotopy commutative; i.e., β ∼ ∑_{i} Jf_{i}(α ⊗ ⋯ ⊗ α)J△^{i} in T (JC, IB).

It is known that if α, α′ : JC → IA are equivalent twisting cochains, then f_{α}, f_{α′} : ΩC → A (resp. g_{α}, g_{α′} : C → BA) are homotopic in DA (resp. DC); see [17], for example.

We complete this section with

Theorem 8 ([18]). Let K be a simplicial set.

a) Then in DGA A = Hom(C_{∗}(K), ΩC_{∗}(K)⊗ΩC_{∗}(K)) there exists an element E of degree −1, satisfying the following conditions

I) DE = −E ⌣ E (i.e., E is a twisting cochain)

II) The components E^{1,0} : C → C ⊗ R and E^{0,1} : C → R ⊗ C are given by

E^{1,0}(x) = −x ⊗ 1,
E^{0,1}(x) = −1 ⊗ x.

III) E^{n,0} = E^{0,n} = 0 when n ≠ 1.

b) If E and E′ are elements of A of degree −1, satisfying I) and having E^{1,0} = E′^{1,0} and E^{0,1} = E′^{0,1}, then there exists P ∈ A of degree 0 having P^{0,0} = 0 and satisfying

E′ = E + P ⌣ E − E′ ⌣ P − DP.

In the next section we shall prove the dual theorem for Sullivan’s functor A^{∗} [10], [11], [19].

2. Sullivan Functor and Some Key Lemmas

We begin this section with a short sketch of the Sullivan functor A(K) mainly following [10], [19]. For more details see [11].

A simplicial object K with values in a category C is a sequence {K_{n}}_{n≥0} of objects in C, together with C-morphisms

d_{i} : K_{n+1} → K_{n}, 0 ≤ i ≤ n + 1 and s_{j} : K_{n} → K_{n+1}, 0 ≤ j ≤ n,

called face and degeneracy operators, satisfying certain identities, see [20] for example.

A simplicial morphism f : K → L between two such simplicial objects is a sequence of C-morphisms φ : K → L commuting with d_{i} and s_{j}.

A simplicial set K is a simplicial object in the category of sets. Thus it consists of a sequence of sets {K_{n}}_{n≥0} together with set maps.

A simplicial cochain algebra A is a simplicial object in the category of cochain algebras: it consists of a sequence of cochain algebras {A_{n}}_{n≥0} with an appropriate face and degeneracy operators. Similarly, simplicial cochain complexes, simplicial vector spaces … are simplicial objects in the category of cochain complexes, vector spaces ….

2.1. The construction of A(K). Let K be a simplicial set, and let A be a simplicial cochain complex or a simplicial cochain algebra. Then

A(K) = {A^{p}(K)}_{p≥0}

is an “ordinary” cochain complex (or cochain algebra) defined as follows: • A^{p}(K) is the set of simplicial set morphisms from K to A^{p}. Thus an element Φ ∈ A^{p}(K) is a mapping that to each n-simplex σ ∈ K_{n}

(n ≥ 0) assigns an element Φ_{σ} ∈ (A^{p})_{n} such that Φ_{d_{i}σ} = d_{i}Φ_{σ} and Φ_{s_{i}σ} = s_{i}Φ_{σ}. • Addition, scalar multiplication and differential are given by

(Φ + Ψ)_{σ} = Φ_{σ} + Ψ_{σ}, (λ · Φ)_{σ} = λ · Φ_{σ} and (dΦ)_{σ} = dΦ_{σ}.

• If A is a simplicial cochain algebra, multiplication in A(K) is given by

(Φ · Ψ)_{σ} = Φ_{σ} · Ψ_{σ}.

• If φ : K → L is a morphism of simplicial sets, then A(φ) : A(L) → A(K) is a morphism of cochain complexes (or cochain algebras) defined by

(A(φ)Φ)_{σ} = Φ_{φσ}.

• If Θ : A → B is a morphism of simplicial cochain complexes (or simplicial cochain algebras), then Θ(K) : A(K) → B(K) is a morphism defined by

(Θ(K)Φ)_{σ} = Θ(Φ_{σ}).

• When X is a topological space, we write A(X) for A(S_{∗}(X)) where S_{⋆}(X) = {S_{n}(X)}_{n≥0} is a simplicial set of singular simplices σ : △^{n} → X on the topological space X.

Remark 9. Note that the construction A(K) is covariant in A and contravariant in K.

Let us denote by △^{n} the Euclidean n-simplex ( n ; ∑_{i=0}t_{i} = 1, t_{i} ≥ 0) in R^{n+1}and

by △^{n} ⇄_{σ_{i}}^{𝔡_{i}} △^{n+1} the ordinary coface and codegeneracy operators given by

𝔡_{i}(t_{0}, …, t_{n}) = (t_{0}, …, t_{i−1}, 0, t_{i}, …, t_{n}) for 0 ≤ i ≤ n + 1,

and

σ_{i}(t_{0}, …, t_{n+1}) = (t_{0}, …, t_{i−1}, t_{i} + t_{i+1}, t_{i+2}, …, t_{n+1}) for 0 ≤ i ≤ n.

Let △ be a category whose objects are sets △^{n}, while morphisms are compositions of 𝔡_{i} and σ_{i}. A simplicial object in a category B we call any contravariant functor K : △ → B

Let us denote by A_{DR}(△^{n}) the classical cochain algebra of the C^{∞} real differential forms on △^{n}. The (differentiable) maps 𝔡_{i} and σ_{i} induce face and degeneracy operators

A_{DR}(△^{n−1}) ←^{d_{i}} A_{DR}(△^{n}) →^{s_{i}} A_{DR}(△^{n+1})

such that the family (A_{DR}(△^{n}), d_{i}, s_{i}) is a commutative simplicial DGA, i.e., the simplicial object in the category of commutative differential algebras over the reals. For the sake of simplicity we remove △^{n} from the notation by writing A_{DR} for A_{DR}(△^{n}). The Poincaré lemma allows us to prove that each A_{DR}(△^{n}) = (A_{DR})_{n} is acyclic (see [19], for example).

Let us denote by A_{Q} a commutative simplicial DGA (over rationals), where (A_{Q})_{n} is a subspace in (A_{DR})_{n} consisting of those differential forms

∑ ω_{i_{1}···i_{p}}dt^{i_{1}} ∧ ⋯ ∧ dt^{i_{p}}

coefficients ω_{i_{1}···i_{p}} of which relative in the barycentric coordinates (t_{0}, t_{1}, … t_{n}) are rational polynomials.

For a classical homotopy operator we have

h : A_{DR}^{p}(△^{n}) → A_{DR}^{p−1}(△^{n}),

with 1_{(A_{DR})_{n}^{p}} = dh + hd, h(A_{Q})_{n}^{p} ⊂ (A_{Q})_{n}^{p−1}. Thus (A_{Q})_{n} is acyclic as well (see [19], for example).

Definition 10. The Sullivan functor on a topological space X is defined as a DGA A^{∗}(X, Q) with

A^{p}(X, Q) = Hom(S_{∗}(X), (A_{Q})^{p}),

i.e., an element Φ ∈ A^{p}(X, Q) is a function assigning to each singular n-simplex of X a polynomial p-form on △^{n}, n ≥ 0, compatible with the face and degeneracy maps (see the construction of A(K) at the beginning of this subsection). The Sullivan functor A^{∗}(X, Q) is referred to as the cochain algebra of polynomial differential forms on X over rationals.

By analogy we have A^{p}(X, R) = Hom(S_{∗}(X), (A_{DR})^{p}) over the reals.

2.2. Acyclicity of the complex Hom(A(X)^{⊗i},A(X)). In this section we’ll prove two lemmas which will be used to prove the main theorem of the paper announced in Introduction.

Lemma 11. The complex Hom(A(X)^{⊗ i}, A(X)) is acyclic.

We will use the machinery of contravariant acyclic models used by Gugenheim in [12]; the machinery of acyclic models was introduced (covariantly) by Eilenberg and MacLane in [9].

For any contravariant functor K : J →{vector spaces} a new contravariant functor K̂ : J →{vector spaces} is defined by

K̂(X) = ∏_{u:M→X}{K(M), u},

where the product is over all maps u : M → X of J, and M is a “model”, i.e., one of the standard simplices △^{n}. If f : X → Y is a map, then K̂(f) : K̂(Y) → K̂(X) is defined by

K̂(f){m_{v}, v} = {m_{f·u}, u},

where u : M → X, v : M_{v} → Y, and m_{v} ∈ K(M_{v}).

If K, L : J → V are contravariant functors, and Θ : K → L is a transformation of functors, then there is a transformation of functors Θ̂ : K̂ → L̂ defined by

Θ̂(X){m, u} = {Θ(M)m, u}.

For any contravariant functor K : J → V, the transformation Φ : K → K̂ is defined by

Φ(X)h = {K(u)h, u} (h ∈ K(X)).

We easily verify that

Θ̂Φ = ΦΘ.

In particular, for the functors A^{∗} and C^{∗} the functors Â^{∗} and Ĉ^{∗} are defined by

(Â)^{n} = { \atop (A^{n})}, (Ĉ)^{n} = { \atop (C^{n})},

and we get differentials

d̂: Â^{∗} → Â^{∗}, d̂: Ĉ → Ĉ^{∗}.

To prove Lemma 11 we need the following

Proposition 12. The Functor A^{∗} is corepresentable, i.e., there is a transformation of functors Ψ : Â^{∗} → A^{∗} such that ΨΦ = the identity.

Proof. We define Ψ(X) : Â^{∗}(X) → A^{∗}(X) by

⟨Ψ(X){m_{u}, u}, v⟩ = ⟨m_{v}, 1_{M_{v}}⟩

where m_{u} ∈ A^{∗}(M_{u}), m_{v} ∈ A^{∗}(M_{v}), v : M_{v} → X, and 1_{M_{v}} is an identity map of M_{v} regarded as a singular chain. Let h ∈ A^{∗}(X), then

⟨ΨΦ(h), v⟩ = ⟨Ψ{A^{∗}(u)h, u}, v⟩ = ⟨A^{∗}(v)h, 1_{M_{v}}⟩ = ⟨h, v(1_{M_{v}})⟩ = ⟨h, v⟩,

i.e., ΨΦ = the identity.

We have mentioned in Subsection 2.1 that the functor A^{∗} is acyclic on models M. Moreover there is a homotopy operator (contraction)

h_{M} : A^{∗}(M) → A^{∗}(M)

with 1_{A^{∗}(M)} = dh_{M} + h_{M}d. It follows from the definition of tensor product that the functors

A^{∗}^{⊗i} : A^{∗}^{⊗i}(M) → A^{∗}^{⊗i}(M)

are acyclic on models as well. Let fix for each i ≥ 1 and a model M , once and for all, a contraction

h_{M}^{i} : A^{∗}^{⊗i}(M) → A^{∗}^{⊗i}(M).

For such cases Gugenheim [12] defined the transformation of functors

ĥ^{i} : Â^{∗}^{⊗i}(X) → ⊗^{i}Â^{∗}^{⊗i}(X)

by ĥ _{M}^{i}{ω_{u}, u} = {h_{M}^{i}ω_{u}, u}, where u : M → X. Clearly, Dĥ _{M}^{i} = 1_{⊗^{i}Â^{∗}(M)} (see [12], for instance). Proof of Lemma 11. Thus we have to show that if Df = 0 for any f ∈ Hom(A^{⊗i}(X), A(X)), deg(f) = k, then there exists g ∈ Hom(A^{⊗i}(X), A(X)) of degree k − 1 satisfying Dg = f.

Let us assume that g is defined in dimension < n, i.e., on (A^{⊗i}(X))_{< n}, and satisfies dg_{j−1}−(−1)^{k−1}g_{j}d = f_{j−1} for any j < n. We write the defining equation for g_{n} as

g_{n}d = (−1)^{k−1}(dg_{n−1} − f_{n−1}).

For simplicity, we denote Θ_{i} = (−1)^{k−1}(dg_{n−1} − f_{n−1}). Thus Θ_{i} is defined in dimension n − 1 and we have the following diagram:

(A^{∗})_{n−1+k}
Θ_{i}↗
(A^{∗⊗i})_{n−2} ⟶_{d} (A^{∗⊗i})_{n−1}  ⟶_{d}  (A^{∗⊗i})_{n} .

By the inductive hypothesis, Θ_{i}d = 0; indeed,

(−1)^{k−1}{dg_{n−1} − f_{n−1}}d = (−1)^{k−1}dg_{n−1}d + df_{n−2}
= d{(−1)^{k−1}g_{n−1}d + f_{n−2}} = ddg_{n−2} = 0.

We have used here Df = 0. Since in the above diagram is functorial, we get

(Â^{∗})_{n−1+k}
Θ̂_{i}↗  ↖^{k_{i}}
(Â^{∗⊗i})_{n−2} ⟶_{d̂} (Â^{∗⊗i})_{n−1}  \overleftarrow{⟶}_{d̂}^{ĥ^{i}}  (Â^{∗⊗i})_{n} ,

where Θ̂_{i}d̂= 0. Now we define k_{i} = Θ̂_{i}ĥ^{i} and obtain

k_{i}d̂= Θ̂_{i}ĥ^{i}d̂= Θ̂_{i}(1 − d̂ĥ^{i}) = Θ̂_{i}.

Next we define g_{n} on (A^{∗⊗i}(X))_{n} by

g_{n} = Ψk_{i}Φ,

and verify

g_{n}d = Ψk_{i}Φd = Ψk_{i}d̂Φ = ΨΘ̂_{i}Φ = ΨΦΘ_{i} = Θ_{i},

as required.

2.3. Twisting cochains on Hom(BA^{∗}⊗BA^{∗},A^{∗}). Sometimes it is be useful to represent the homomorphism E : BA^{∗} ⊗ BA^{∗} → A^{∗} by the collection of its components

{E^{p,q} = E|_{s^{−1}A^{⊗p}⊗s^{−1}A^{⊗q}}((s^{−1})^{⊗p} ⊗ (s^{−1})^{⊗q}) : (A^{∗})^{⊗p} ⊗ (A^{∗})^{⊗q} → A^{∗}}.

Note that if deg E = k, then deg E^{p,q} = k − (p + q).

In the set of all twisting cochains from BA^{∗} ⊗ BA^{∗} to A^{∗} there is one remarkable element. This is the “generator” E_{sh} = p_{1}μ_{sh} of the shuffle product μ_{sh} : BA^{∗} ⊗ BA^{∗} → BA^{∗}. Indeed, since the multiplication on A^{∗} is commutative, the Bar construction (BA^{∗}, d_{BA^{∗}}, μ_{sh}, △) of A^{∗} is a DGHA (see Subsection 1.7), i.e., μ_{sh} ∈ DC, by bijection (1.8.3)

E_{sh} = p_{1}μ_{sh} : BA^{∗} ⊗ BA^{∗} → A^{∗}

is a twisting cochain. As we easily see E_{sh} consists only of two nontrivial components: E_{sh}^{0,1} = 1 : R ⊗ A → A and E_{sh}^{0,1} = 1 : A ⊗ R → A to which we will return in the next section.

Let F ∈ Hom(BA^{∗} ⊗ BA^{∗}, A^{∗}). Let F^{n} denote the sum of components F^{p,q} of F with p + q = n:

F^{n} = ∑_{p+q=n} F^{p,q}.

We call a number p + q filtration of F.

Remark 13. It is easy to see that the product of two homogenous elements F^{p_{1},q_{1}} and F^{p_{2},q_{2}} in Hom(BA^{∗} ⊗ BA^{∗}, A^{∗}) is a homogenous element of type F^{p_{1}+p_{2},q_{1}+q_{2}}. Thus we see that the multiplication in Hom(BA^{∗} ⊗ BA^{∗}, A^{∗}) is compatible with filtration.

For the sake of simplicity we write \overline{A} for Hom(BA^{∗} ⊗ BA^{∗}, A^{∗}); if F ∈ \overline{A} has degree m, we write F ∈ \overline{A}^{m}.

Definition 14. Let n be a given number. We say the elements F_{1}, F_{2} ∈ \overline{A}^{m} = Hom^{m}(BA^{∗} ⊗ BA^{∗}, A^{∗}) coincide up to n (write F_{1} =^{n} F_{2}) if

F_{1}^{k} = F_{2}^{k}

for each k ≤ n.

Let X be a simplicial set. Then the differential D on Hom(BA^{∗}(X)⊗BA^{∗}(X), A^{∗}(X)) can be expressed as a sum

D = D_{1} + D_{2},

where

D_{1}F = F (d_{∧} ⊗ 1 + 1 ⊗ d_{∧}) D_{2}F = F (d_{⊗} ⊗ 1 + 1 ⊗ d_{⊗}) − (−1)^{|F |}d_{A^{∗}(X)}F,

d_{∧} and d_{⊗} are given by the formula

d_{∧}[Φ_{1}| ⋯ |Φ_{n}] = ∑_{i=0}^{n−2}[\overline{Φ}_{1}| ⋯ |\overline{Φ}_{i}|(\overline{Φ}_{i+1} · Φ_{i+2})|Φ_{i+3}| ⋯ |Φ_{n}],

d_{⊗}[Φ_{1}| ⋯ |Φ_{n}] = ∑_{i=0}^{n−1}[\overline{Φ}_{1}| ⋯ |\overline{Φ}_{i}|dΦ_{i+1}|Φ_{i+2}| ⋯ |Φ_{n}],

where Φ_{i} ∈ A^{∗}(X), \overline{Φ}_{i} = (−1)^{|Φ_{i}|+1}Φ_{i} (see Subsection 1.7).

Proposition 15. Let F be an element in \overline{A} having F^{n−1} = 0. Then the equality DF =^{n} 0 guarantees that D_{2}F^{n} = 0, i.e., F^{n} is a chain map of (chain) complexes

∑_{p+q=n}(A^{∗}(X) ⊗ ⋯^{p} ⊗ A^{∗}(X)) ⊗ (A^{∗}(X) ⊗ ⋯^{q} ⊗ A^{∗}(X)) → A^{∗}(X).

Proof. Clear.

Now we are ready to prove

Lemma 16. Let the elements E, E′ ∈ \overline{A}^{1} satisfy the following conditions I) DE = −E ⌣ E and DE′ = −E′ ⌣ E′ (i.e., E and E′ are twisting

cochains); II) E^{1,0} = E′^{1,0} and E^{0,1} = E′^{0,1}. Then there exists P ∈ \overline{A}^{0}, P^{0,0} = 0, satisfying

E′ = E + P ⌣ E − E′ ⌣ P − DP

(i.e., E and E′ are equivalent twisting cochains).

Proof. Note that it suffices to construct the sequence of elements in \overline{A}^{0}

P^{(0)}, P^{(1)}, P^{(2)}, …

satisfying the following conditions I) P^{(0)} = P^{(1)} = 0,

II) P^{(i)} =^{i} P^{(i+1)}, i = 0, 1, …,

III) (P^{(i)})^{k} = 0 when k = 0, 1 or k > i,

IV) DP^{(i)} =^{i} E − E′ + P^{(i−1)} ⌣ E − E′ ⌣ P^{(i−1)}.

After the construction of such a sequence we define P by

P =^{i} P^{(i)}, i = 0, 1, ….

Clearly, P thus defined satisfies the required identity E′ = E + P ⌣ E − E′ ⌣ P − DP.

Assume that P^{(0)}, P^{(1)}, …, P^{(n−1)} have already been constructed. Now we want to construct P^{(n)} with required conditions I–IV. Consider the following element in \overline{A}^{1}:

β(n − 1) = E − E′ + P^{(n−1)} ⌣ E − E′ ⌣ P^{(n−1)} − DP^{(n−1)}.

According to conditions I–IV and Remark 13 we get

Dβ(n − 1) = DE − DE′ + DP^{(n−1)} ⌣ E + P^{(n−1)} ⌣ DE − DE′ ⌣ P^{(n−1)}

+ E′ ⌣ DP^{(n−1)} =^{n} −E ⌣ E + E′ ⌣ E′ + (E − E′

+ P^{(n−2)} ⌣ E − E′ ⌣ P^{(n−2)}) ⌣ E − P^{(n−1)} ⌣ E ⌣ E

+ E′ ⌣ E′ ⌣ P^{(n−1)} + E′ ⌣ (E − E′ + P^{(n−2)} ⌣ E

− E′ ⌣ P^{(n−2)}) = P^{(n−2)} ⌣ E ⌣ E − P^{(n−1)} ⌣ E ⌣ E

+ E′ ⌣ E′ ⌣ P^{(n−1)} − E′ ⌣ E′ ⌣ P^{(n−2)} =^{n} 0.

(We have used here equalities like P^{(n−2)} ⌣ E ⌣ E =^{n} P^{(n−1)} ⌣ E ⌣ E; indeed, since P^{(n−2)}, P^{(n−1)} and E have no nontrivial components with filtration of 0, multiplication is compatible with filtration (Remark 13) and P^{(n−2)} =^{n−2} P^{(n−1)} (condition II), elements P^{(n−2)} ⌣ E ⌣ E and P^{(n−1)} ⌣ E ⌣ E coincide up to n.) Since β(n − 1) has no nontrivial components with filtration ≤ n − 1, by Proposition 15

D_{2}β(n − 1)^{n} = 0,

or, equivalently, β(n − 1)^{n} is a cycle in the chain complex

Hom ( ∑_{p+q=n}(A^{∗}(X) ⊗ ⋯^{p} ⊗ A^{∗}(X)) ⊗ (A^{∗}(X) ⊗ ⋯^{q} ⊗ A^{∗}(X)), A^{∗}(X)).

Since the complex above is acyclic (see Lemma 11) there exists an element P^{n} such that

D_{2}P^{n} = β(n − 1)^{n}.

Now we define P^{(n)} by

P^{(n)} = P^{(n−1)} + P^{n}.

Thus it remains to show that P^{(n)} together with P^{(0)}, P^{(1)}, …, P^{(n−1)} satisfies condition I–IV. The first three conditions are checked trivially. Let us show the validity of IV. Since D_{2}P^{n} = β(n−1)^{n} and β(n−1) has no nontrivial components with filtration ≤ n − 1, we have

DP^{n} =^{n} β(n − 1).

Then

DP^{(n)} = DP^{(n−1)} + DP^{n} =^{n} DP^{(n−1)} + β(n − 1)
= DP^{(n−1)} − DP^{(n−1)} + E − E′ + P^{(n−1)} ⌣ E − E′ ⌣ P^{(n−1)}

= E − E′ + P^{(n−1)} ⌣ E − E′ ⌣ P^{(n−1)}.

Thus Lemma 16 is completely proved.

3. Multiplicativity up to Homotopy

In this section we will prove the main theorem of this paper. We use all the lemmas and theorems which we have proved or mentioned in the preceding sections.

Theorem 17. Gugenheim’s map

B{ρ_{i}} : (BA^{∗}(X), d_{BA^{∗}}, △, μ_{sh}) → (BC^{∗}(X), d_{BC^{∗}}, △, μ_{E})

is multiplicative up to homotopy.

Proof. As we have mentioned in Introduction, Baues introduced a multiplication

μ_{E} : BC^{∗}(X) ⊗ BC^{∗}(X) → BC^{∗}(X)

on BC^{∗}(X) so that (BC^{∗}(X), d_{BC^{∗}(X)}, △, μ_{E}) is a DGHA, see [4]. Moreover, it is shown that this product is strict associative and homotopy commutative. Let consider the twisting cochain

BA^{∗}(X) ⊗ BA^{∗}(X) ⟶^{B{ρ_{i}}⊗B{ρ_{i}}} BC^{∗}(X) ⊗ BC^{∗}(X) →^{μ_{E}} BC^{∗}(X) →^{P_{1}} C^{∗}(X)

which is the composition of coalgebra maps and the twisting cochain P_{1}, see Subsection 1.8.

Since the first component ρ = ρ_{1} : A^{∗}(X) → C^{∗}(X) (⟨ρω, c⟩ = ∫_{c} ω) of Gugenheim’s (DASH) map

ρ_{i} : ⊗^{i}A^{∗}(X) → C^{∗}(X) ; (i ≥ 1)

induces an isomorphism in homology, Theorem 6 says that there exists a twisting cochain (see Remark 7)

\overline{E} : BA^{∗}(X) ⊗ BA^{∗}(X) → A^{∗}(X)

satisfying P_{1}μ_{E}(B\{ρ_{i}\} ⊗ B\{ρ_{i}\}) ∼ ∑_{i}ρ_{i}(\overline{E} ⊗ ⋯ ⊗ \overline{E})△^{i} = P_{1}B\{ρ_{i}\}μ\overline{E}, where

μ\overline{E} : BA^{∗}(X) ⊗ BA^{∗}(X) → BA^{∗}(X)

is the coalgebra map induced by \overline{E} (see the universal property of the tensor coalgebra, Subsection 1.7). As we know, equivalent twisting cochains on T (C, A) induce homotopic DG coalgebra maps on Hom(C, BA) (see Subsection 1.8); thus μ_{E}(B\{ρ_{i}\} ⊗ B\{ρ_{i}\}) ∼ B\{ρ_{i}\}μ\overline{E} in DC, i.e.,

B\{ρ_{i}\} : (BA^{∗}(X), d_{BA^{∗}}, △, μ\overline{E}) → (BC^{∗}(X), d_{BC^{∗}}, △, μ_{E})

is multiplicative up to homotopy, but there is no guarantee that such randomly constructed μ\overline{E} coincides with μ_{sh}.

Let us assume that the components \overline{E}^{0,1} and \overline{E}^{1,0} of \overline{E} are identity maps (we show below that this is always achievable for our \overline{E}). Then we have two twisting cochains E_{sh} (E_{sh} is the “generator” of the shuffle product μ_{sh}, see Subsection 2.3) and \overline{E} on Hom(BA^{∗}(X) ⊗ BA^{∗}(X), A^{∗}(X) having \overline{E}^{0,1} = E_{sh}^{1,0} = 1 and \overline{E}^{1,0} = E_{sh}^{1,0} = 1; according to Lemma 16 there exists P ∈ Hom(BA^{∗}(X) ⊗ BA^{∗}(X), A^{∗}(X)) such that E_{sh} ∼^{P} \overline{E}, hence μ_{sh} ∼ μ\overline{E} or, equivalently,

1 : (BA^{∗}(X), d_{BA^{∗}}, △, μ_{sh}) → (BA^{∗}(X), d_{BA^{∗}}, △, μ\overline{E})

is multiplicative up to homotopy. Taking the composition

B{ρ_{i}} · 1 : (BA^{∗}(X), d_{BA^{∗}}, △, μ_{sh})
→ (BA^{∗}(X), d_{BA^{∗}}, △, μ\overline{E}) → (BC^{∗}(X), d_{BC^{∗}}, △, μ_{E}),

we have that Gugenheim’s map

B{ρ_{i}} : (BA^{∗}(X), d_{BA^{∗}}, △, μ_{sh}) → (BC^{∗}(X), d_{BC^{∗}}, △, μ_{E})

is multiplicative up to homotopy.

Thus, it remains to show that the components \overline{E}^{0,1} and \overline{E}^{1,0} are the identity maps. First of all, note that μ\overline{E} is a homotopy commutative, homotopy associative and homotopy compatible with unit. Indeed,

\{ρ_{i}\}_{∗}(\overline{E}) = P_{1}B\{ρ_{i}\}μ\overline{E} ∼ P_{1}μ_{E}(B\{ρ_{i}\} ⊗ B\{ρ_{i}\}) ∼ P_{1}μ_{E}T (B\{ρ_{i}\} ⊗ B\{ρ_{i}\})

= P_{1}μ_{E}(B\{ρ_{i}\} ⊗ B\{ρ_{i}\})T ∼ P_{1}B\{ρ_{i}\}μ\overline{E}T = P_{1}B\{ρ_{i}\}μ_{\overline{E}T} = \{ρ_{i}\}_{∗}(\overline{E}T),

where we have used the fact that μ_{E} ∼ μ_{E}T on BC^{∗}(X) [4]. Since \{ρ_{i}\}_{∗}(\overline{E}) ∼ \{ρ_{i}\}_{∗}(\overline{E}T), according to the bijection of Theorem 6

{ρ_{i}}_{∗} : D(BA^{∗}(X) ⊗ BA^{∗}(X), A^{∗}(X)) → D(BA^{∗}(X) ⊗ BA^{∗}(X), C^{∗}(X)),

\overline{E} ∼ \overline{E}T, hence μ\overline{E} ∼ μ\overline{E}T; i.e., μ\overline{E} is homotopy commutative. By analogy, we can easily show that it is homotopy associative and has a homotopy unit. Then, according to Anick’s fundamental theory [2], μ\overline{E} can be replaced by the new product

μ_{F} ∼ μ\overline{E} : BA^{∗}(X) ⊗ BA^{∗}(X) → BA^{∗}(X)

which is strictly commutative, associative and compatible with unit (see Definition 5.3, Lemma 5.4 and Proposition 5.5 in [2]). The latter means that the twisting cochain

F = P_{1}μ_{F} : BA^{∗}(X) ⊗ BA^{∗}(X) → BA^{∗}(X) → A^{∗}(X)

has F^{0,1} = 1, F^{1,0} = 1 and, moreover, F^{0,k} = F^{k,0} = 0 when k ≠ 1. This completes the proof.

Acknowledgement

The research described in this publication was made possible in part by Award No. GM1–2083 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF).

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(Received 7.11.2001)

Authors address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia

Current address: Department of Mathematics North Carolina State University Harrelson Hall Box 8205, Raleigh NC 27695 U.S.A. E-mail: zkhareb@unity.ncsu.edu