GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 5, 1998, 441-452

ITERATING THE BAR CONSTRUCTION

T. KADEISHVILI AND S. SANEBLIDZE

Abstract. For a 1-connected space X Adams’s bar construction B(C^{∗}(X)) describes H^{∗}(ΩX) only as a graded module and gives no information about the multiplicative structure. Thus it is not possible to iterate the bar construction in order to determine the cohomology of iterated loop spaces Ω^{i}X. In this paper for an nconnected pointed space X a sequence of A(∞)-algebra structures {m_{i}^{(k)}}, k = 1, 2, …, n, is constructed, such that for each k ≤ n there exists an isomorphism of graded algebras

H^{∗}(Ω^{k}X) ≅

≅ (H(B(⋯ (B(B(C^{∗}(X); {m_{i}^{(1)}}); {m_{i}^{(2)}}); ⋯); {m_{i}^{(k−1)}})); m_{2}^{(k)∗}).

Introduction

For a 1-connected pointed space X Adams [1] found a natural isomorphism of graded modules

H(B(C^{∗}(X)) ≅ H^{∗}(ΩX),

where B(C^{∗}(X)) is the bar construction of DG-algebra C^{∗}(X). The method cannot be extended directly for iterated loop spaces Ω^{k}X for k ≥ 2, since the bar construction B(A) of a DG-algebra A is just a DG-coalgebra, and it does not carry the structure of a DG-algebra in order to produce a double bar construction B(B(A)).

However, for A = C^{∗}(X) Baues [2] has constructed an associative product

μ : B(C^{∗}(X)) ⊗ B(C^{∗}(X)) → B(C^{∗}(X)),

which turns B(C^{∗}(X)) into a DG-algebra and which is geometric: for 1connected X there exists an isomorphism of graded algebras

H(B(C^{∗}(X)) ≅ H^{∗}(ΩX)

1991 Mathematics Subject Classification. 55P35. Key words and phrases. Bar construction, A(∞)-algebra, perturbation lemma.

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and for a 2-connceted X there exists an isomorphism of graded modules

H(B(B(C^{∗}(X))) ≅ H^{∗}(Ω^{2}X).

But again, as mentioned in [2], this method cannot be extended for Ω^{3}X either, since “it is impossible to construct a ‘nice’ product on B(B(C^{∗}(X)).”

We remark here that in order to produce the bar construction B(M) it is not necessary to have, on a DG-module M, a strict associative product

μ : M ⊗ M → M;

it suffices to have a strong homotopy associative product, or, which is the same, to have an A(∞)-algebra structure on M. This notion was introduced by Stasheff in [3]. An A(∞)-algebra (M, {m_{i}}) is a graded module M equipped with a sequence of operations

{m_{i} : M ⊗ ⋯ (i-times) ⋯ ⊗ M → M, i = 1, 2, 3, …; deg m_{i} = 2 − i},

which satisfies the suitable associativity conditions (see below). Such a sequence defines on B(M) the correct differential

d_{m} : B(M) → B(M),

which is a coderivation with respect to the standard coproduct. This DGcoalgebra (B(M); d_{m}) is denoted by B(M, {m_{i}}) and called the bar construction of A(∞)-algebra (M, {m_{i}}).

In particular, an A(∞)-algebra of the type

(M, {m_{1}, m_{2}, 0, 0, … })

is just a DG-algebra with a differential m_{1} and a strict associative product m_{2} (up to signs). For such an A(∞)-algebra, B(M, {m_{i}}) coincides with the usual bar construction. For a general A(∞)-algebra (M, {m_{i}}) the first operation m_{1} : M → M is a differential, which is a derivation with respect to the second operation m_{2} : M ⊗M → M; this operation is not neccesarily associative, but is homotopy associative (the operation m_{3} is a suitable homotopy). Thus we can consider homology of DG-module (M, m_{1}). Then the product m_{2} induces, on H(M, m_{1}), the strict associative product m_{2}^{∗}.

Now we can formulate the main result of this paper.

Theorem A. Let X be an n-connected pointed space. Then there exists a sequence of A(∞)-algebra structures {m_{i}^{(k)}}, k = 1, 2, …, n, such that for each k ≤ n there exists an isomorphism of graded algebras

H^{∗}(Ω^{k}X) ≅
≅ (H(B(⋯ (B(B(C^{∗}(X); {m_{i}^{(1)}}); {m_{i}^{(2)}}); ⋯); {m_{i}^{(k−1)}})); m_{2}^{(k)∗}).

Remark 0.1. The latter A(∞)-algebra structure {m_{i}^{(n)}} allows one to produce the next bar construction

(B(⋯ (B(B(C^{∗}(X); {m_{i}^{(1)}); {m_{i}^{(2)}); ⋯); {m_{i}^{(n)}})),

but it is not clear whether it is geometric, i.e., whether homology of this bar construction is isomorphic to H^{∗}(Ω^{n+1}X) if X is not (n + 1)-connected.

The proof is based on

Theorem B. Let (M, {m_{i}}) be an A(∞)-algebra, (C, d) be a DG-module, and

f : (M, m_{1}) → (C, d)

be a weak equivalence of DG-modules. Assume further that M and C are connected and free as graded modules. Then there exist

(1) an A(∞)-algebra structure {m′_{i}} on C with m′_{1} = d;

(2) a morphism of A(∞)-algebras

{f_{i}} : (M, {m_{i}}) → (C, {m′_{i}})

with f_{1} = f.

Let us mention some results from the literature dedicated to the problem of iterating the bar construction.

In [4] Khelaia has constructed, on C^{∗}(X), an aditional structure which, in particular, contains Steenrod’s ∪_{1} product, and which is used to introduce, in the bar construction BC^{∗}(X), a homotopy associative product which is geometric: there exists an isomorphism of graded algebras

H^{∗}(B(C^{∗}(X)) ≅ H^{∗}(ΩX).

One can show that this product is strong homotopy associative. Thus there is the possibility to produce the next bar construction B(B(C^{∗}(X))), but the aditional structure itself is lost in B(C^{∗}(X)) and hence this structure is not enough to determine the product in B(B(C^{∗}(X))).

Later Smirnov [5], using the technique of operands, introduced a more powerful aditional structure – the E_{∞}-structure which can be transferred on B(C^{∗}(X)). Hence there is the possibility of an iteration.

The structure of an m-algebra introduced by Jastin Smith [6] is also transferable on the bar constuction and, as mentioned in [6], is smaller and has computational advantages against Smirnov’s one.

Seemingly, the structure introduced in this paper, i.e., the sequence of A(∞)-algebra structures {m_{i}^{(k)}}, should be the smallest one because the A(∞)-algebra structure is a minimal structure required to produce the bar construction.

The disadvantage of our structure is that it cannot be considered as a sequence of operations from Hom(⊗^{k}C^{∗}(X), C^{∗}(X)) as in [5] and [6]. In the forthcoming publication we are going to make up for this disadvantage.

The proof of Theorem B is based on the perturbation lemma (see [7,8]), extended for chain equivalences in [9].

In Section 1 Stasheff’s notion of an A(∞)-algebra is given. Section 2 is dedicated to the perturbation lemma and in Section 3 Theorem A is proved.

1. A(∞)-Algebras

The notion of an A(∞)-algebra was introduced by J. Stasheff in [3].

An A(∞)-algebra is a graded module M = ∑_{i=0}^{∞} M^{i} with a given sequence

of operations

{m_{i} : ⊗^{i}M → M; i = 1, 2, 3, … }

which satisfies the following conditions:

(1) deg m_{i} = 2 − i;

(2) ∑_{k=1}^{n−1} ∑_{j=1}^{n−k+1} m_{n−j+1}(a_{1} ⊗⋯⊗a_{k} ⊗m_{j}(a_{k+1} ⊗⋯⊗a_{k+j})⊗⋯⊗a_{n})=0

for each a_{i} ∈ M and n > 0.

The sequence of operations {m_{i}} defines on the tensor coalgebra

T′(s^{−1}M) = R + s^{−1}M + s^{−1}M ⊗ s^{−1}M + ⋯ = ∑_{i=0}^{∞} ⊗^{i}s^{−1}M

(here s^{−1}M is the desuspension of M) a differential d_{m} : T (s^{−1}M) → T (s^{−1}M), given by

d_{m}(s^{−1}a_{1} ⊗ ⋯ ⊗ s^{−1}a_{n}) =

=∑_{k=1}^{n−1} ∑_{j=1}^{n−k+1}s^{−1}a_{1} ⊗ ⋯ ⊗ s^{−1}a_{k} ⊗ s^{−1}m_{j}(a_{k+1} ⊗ ⋯ ⊗ a_{k+j}) ⊗ ⋯ ⊗ s^{−1}a_{n}.

This differential turns (T′(s^{−1}M), d_{m}) into a differential graded coalgebra called the bar-construction of an A(∞)-algebra (M, {m_{i}}) and denoted by B(M, {m_{i}}). Conversely, using the cofreeness of the tensor coalgebra one can show that any differential d : T′(s^{−1}M) → T′(s^{−1}M) which is a coderivation at the same time coincides with d_{m} for some A(∞)-algebra structure {m_{i}} (see [10] for details).

For an A(∞)-algebra of the type (M; {m_{1}, m_{2}, 0, 0, … }), i.e., for a differential algebra with the differential m_{1} and the multiplication m_{2} the bar-construction coincides with the usual one.

A morphism of A(∞)-algebras

{f_{i}} : (M; {m_{i}}) → (N; {n_{i}})

is defined as a sequence of homomorphisms

{f_{i} : ⊗^{i}M → N; i = 1, 2, 3, … }

which satisfies the following conditions:

(1) deg f_{i} = 1 − i;

(2) ∑_{k=1}^{n−1} ∑_{j=1}^{n−k+1}f_{n−j+1}(a_{1}⊗⋯⊗a_{k}⊗m_{j}(a_{k+1}⊗⋯⊗a_{k+j})⊗a_{k+j+1}⊗···⊗a_{n})=

= ∑_{t=1}^{n} ∑_{k_{1}+···+k_{t}=n} n_{t}(f_{k_{1}}(a_{1} ⊗ ⋯ ⊗ a_{k_{1}}) ⊗ ⋯ ⊗ f_{k_{t}}(a_{n−k_{t}+1} ⊗ ⋯ ⊗ a_{n}))

for each a_{i} ∈ M and n > 0.

Each A(∞)-algebra morphism induces a DG-coalgebra morphism

B({f_{i}}) : B(M, {m_{i}}) → B(N, {n_{i}})

by

B({f_{i}})(s^{−1}a_{1} ⊗ ⋯ ⊗ s^{−1}a_{n}) = ∑_{t=1}^{n} ∑_{k_{1}+···+k_{t}=n} s^{−1}f_{k_{1}}(a_{1} ⊗ ⋯ ⊗ a_{k_{1}}) ⊗

⊗ ⋯ ⊗ s^{−1}f_{k_{t}}(a_{n−k_{t}+1} ⊗ ⋯ ⊗ a_{n}).

Conversely, as above, because of the cofreeness of a tensor coalgebra, each DG-colagebra morphism

F : B(M, {m_{i}}) → B(N, {n_{i}})

coincides with B({f_{i}}) for a suitable A(∞)-algebra morphism {f_{i}}.

Remark 1.1. The first component f_{1} : (M; m_{1}) → (N; n_{1}), which is a chain map, is homotopy multiplicative with respect to the homotopy associative products m_{2} and n_{2}. Therefore it induces a map of graded algebras

f_{1}^{∗} : (H(M, m_{1}), m_{2}∗) → (H(M, m_{1}), m_{2}∗).

We call an A(∞)-algebra morphism {f_{i}} a weak equivalence if f_{1}^{∗} is an isomorphism.

2. Perturbation Lemma

The general perturbation lemma [7,8] deals with the following problem. Let (M, d_{M}) and (N, d_{N}) be cochain complexes, and suppose they are

equivalent in some sense, which will be discussed. A perturbation of the differential d_{N} is a homomorphism t : N → N which satisfies

d_{N}t + td_{N} + tt = 0

so that d_{t} = d_{N} +t is a new differential on N. Then, according to the perturbation lemma, in some suitable circumstances, there exists a perturbation t′ : M → M so that the complexes (M, d_{t′}) and (N, d_{t}) remain “equivalent.” There are (co)algebraic versions of the perturbation lemma (see [11,9]): if M and N are DG-(co)algebras and t is a (co)derivation, then, in suitable circumstances, t′ is a (co)derivation too.

Using the so-called “tensor trick” (see [11]), the perturbation lemma allows one to transfer not only differentials, but certain algebraic structures too. Suppose, for example, that N is equipped with a DG-algebra structure. Applying the functor T′s^{−1} we get new complexes T′(s^{−1}M) and T′(s^{−1}N) with differentials induced by d_{M} and d_{N}, respectively. The product operation of N determines the perturbation t so that

(T′(s^{−1}N), d_{t}) = B(N).

Then, according to the coalgebra perturbation lemma, there appears a new differential on T′(s^{−1}N), which is a coderivation too and therefore can be interpreted as an A(∞)-algebra structure on M.

Let us specify what an “equivalence” means.

The basic perturbation lemma [7,8] requires of M and N to form a filtered SDR (strong defomation retraction)

((M, d_{M}) ⇄^{α}_{β} (N, d_{N}), ν)

which consists of the following data:

(1) the cochain complexes (M, d_{M}) and (N, d_{N}), both filtered with complete filtrations;

(2) the filtration preserving chain maps α and β such that βα = id_{M};

(3) the filtration preserving chain homotopy which is a homomorphism ν : N → N of degree −1 such that

αβ − id_{N} = d_{N}ν + νd_{N}.

We can now formulate the following Perturbation Lemma. Let

((M, d_{M}) ⇄^{α}_{β} (N, d_{N}), ν)

be a filtered SDR and t : N → N be a perturbation of d_{N} which increases filtration. Then there are formulas for t′, α′, β′, ν′ such that

((M, d_{t′}) ⇄^{α′}_{β′} (N, d_{t}), ν′)

is a filtered SDR and t′, α − α′, β − β′, ν − ν′ increase filtrations.

Remark 2.1. In [11,9] the (co)algebra version of this lemma is shown: if the initial SDR is (co)algebraic, i.e., M and N are filtered DG-(co)algebras, α and β are multiplicative, ν is a (co)derivation homotopy, and if t is a (co)derivation, then the resulting SDR is (co)algebraic too.

In [9] the perturbation lemma is extended for chain equivalences in the following sense: a filtered chain equivalence

(μ, (M, d_{M}) ⇄^{α}_{β} (N, d_{N}), ν)

consists of

– the filtered cochain complexes M and N;

– the filtration preserving chain maps α and β;

– the filtration preserving chain homotopies μ : M → and ν : N → N such that

βα − id_{M} = d_{M}μ + μd_{M}, αβ − id_{N} = d_{N}ν + νd_{N}.

Extended Perturbation Lemma. Let

(μ, (M, d_{M}) ⇄^{α}_{β} (N, d_{N}), ν)

be a filtered chain equivalence and t : N → N be a perturbation of d_{N}, which increases filtration. Then there are formulas for t′, α′, β′, μ′, ν′ such that

(μ′, (M, d_{t′}) ⇄^{α′}_{β′} (N, d_{t}), ν′)

is a filtered chain equivalence and t′, α − α′, β − β′, μ − μ′, ν − ν′ increase filtrations.

Although there is no (co)algebraic version of this extended perturbation lemma in a general setting, we are going to use in this paper the following result, dual to Theorem (2.3^{∗}) from [9].

Proposition 1. Let

(μ, X ⇄^{α}_{β} Y, ν)

be a chain equivalence, let

(T′μ, T′(X) ⇄^{T′α}_{T′β} (Y), T′ν)

be the corresponding filtered chain equivalence of DG-coalgebras, and let t be a comultiplicative perturbation of the differential on T′(Y) (i.e., it increases the augmentation filtration of T′(Y) and is a coderivation). Then there exsist a comultiplicative perturbation t′ of the differential on T′(X) and a filtered chain equivalence of DG-coalgebras

(T′_{t}μ, T′_{t}(X) ⇄^{T′_{t}α}_{T′_{t}β} T′_{t}(Y), T′_{t}ν),

where T′_{t}(X) and T′_{t}(Y) refer to new chain complexes.

This proposition (a weak form of the extended coalgebra perturbation lemma) will be used to prove

Theorem B. Let (M, {m_{i}}) be an A(∞)-algebra, (C, d) be a DG-module, and

f : (M, m_{1}) → (C, d)

be a weak equivalence of DG-modules. Assume further that M and C are connected and free as graded modules. Then there exist

(1) an A(∞)-algebra structure {m′_{i}} on C with m′_{1} = d;

(2) a morphism of A(∞)-algebras

{f_{i}} : (M, {m_{i}}) → (C, {m′_{i}})

with f_{1} = f.

Proof. Since M and C are free, it is possible to construct a chain equivalence

(μ, (C, d) ⇄^{g}_{f} (M, m_{1}), ν).

Using the desuspension functor s^{−1}we get the chain equivalence

(s^{−1}μs, (s^{−1}C, s^{−1}ds) ⇄^{s^{−1}gs}_{s^{−1}fs} (s^{−1}M, s^{−1}m_{1}s), s^{−1}νs).

Applying the functor T′, we get the filtered chain equivalence

(T′s^{−1}μs, (T′(s^{−1}C), T′(s^{−1}ds)))

⇄^{T′(s^{−1}gs)}_{T′(s^{−1}fs)} ((T′(s^{−1}M), T′(s^{−1}m_{1}s)), T′(s^{−1}νs)).

Let t be the perturbation of T′(s^{−1}m_{1}s)) given by

t = d_{B} − T′(s^{−1}m_{1}s)),

where d_{B} is the differential of the bar construction B(M, {m_{i}}). Clearly, t is a coderivation and increases the standard filtration of T′(s^{−1}M). Then, by virtue of the proposition there exist a perturbation t′ and a filtered chain equivalence of DG-algebras

(T′_{t}s^{−1}μs, (T′_{t}(s^{−1}C), d′ = T′(s^{−1}ds) + t′)

⇄^{T′_{t}(s^{−1}gs)}_{T′_{t}(s^{−1}fs)} (T′_{t}((s^{−1}M), d_{B} = T′(s^{−1}m_{1}s) + t), T′_{t}s^{−1}νs),

The differential d′ can be interpreted as the A(∞)-algebra structure {m′_{i}} and the map T′_{t}(s^{−1}fs) as the morphism of A(∞)-algebras

{f_{i}} : (M, {m_{i}}) → (C, {m′_{i}}),

which completes the proof.

3. Proof of Theorem A

For an n-connected space X and 0 ≤ k ≤ n we are going to construct a sequences of A(∞)-algebra structures {m_{i}^{(k)}(X)} and weak equivalences of A(∞)-algebras

{f_{i}^{(k)}(X)} : C^{∗}(Ω^{k}X) →
((B(⋯ (B(B(C^{∗}(X); {m_{i}^{(1)}(X)}); {m_{i}^{(2)}(X)}); ⋯); {m_{i}^{(k)}(X)}).

Then

f_{1}^{(k)∗}(X) : H^{∗}(Ω^{k}X) →
(H(B(⋯ (B(B(C^{∗}(X); {m_{i}^{(1)}(X)}); ⋯); {m_{i}^{(k−1)}(X)})); m∗_{2}^{(k)(X)})

will be the required isomorphism from Theorem A.

According to Adams and Hilton [1] (see also Brown [12]), for a 1-connec-ted space X there exists a weak equivalence of DG-coalgebras

f : C^{∗}(ΩX) → BC^{∗}(X).

Furthermore, the cup product

μ : C^{∗}(ΩX) ⊗ C^{∗}(ΩX) → C^{∗}(ΩX)

turns (C^{∗}(ΩX), d, μ) into a DG-algebra, i.e., it can be considered as an A(∞)-algebra

(C^{∗}(ΩX), {m_{1} = d, m_{2} = μ, m_{>2} = 0}).

Now by Theorem B there exists, on the bar construction BC^{∗}(X), a structure of A(∞)-algebra (BC^{∗}(X), {m_{i}^{(1)}(X)}) and a weak equivalence of A(∞)-algebras

{f_{i}^{(1)}(X)}:(C^{∗}(ΩX), {m_{1} =d, m_{2} =μ, m_{>2} =0})→(BC^{∗}(X), {m_{i}^{(1)}(X)}).

For the next inductive step, assume that X is 2-connected. Then, if instead of X we consider the loop space ΩX (which is 1-connected), by virtue of the preceding we have a weak equivalence of A(∞)-algebras

{f_{i}^{(1)}(ΩX)} : C^{∗}(Ω^{2}X) → (BC^{∗}(ΩX), {m_{i}^{(1)}(ΩX)}).

Moreover, the A(∞)-morphism {f_{i}^{(1)}(X)} induces a weak equivalence of DG-coalgebras

B({f_{i}^{(1)}(X)}) : B(C^{∗}(ΩX) → B(BC^{∗}(X), {m_{i}^{(1)}(X)}).

By Theorem B this weak equivalence transfers the A(∞)-algebra structure {m_{i}^{(1)}(ΩX)} of B(C^{∗}(ΩX) to B(BC^{∗}(ΩX), {m_{i}^{(1)}(ΩX)}) and we get the A(∞)-algebra structure {m_{i}^{(2)}(X)} and the weak equivalence of A(∞)- algebras

{f̄_{i}^{(2)}(X)} : (BC^{∗}(ΩX), {m_{i}^{(1)}(ΩX)}) →
(B(BC^{∗}(X), {m_{i}^{(1)}(X)}), {m_{i}^{(2)}(X)}).

Now we can define the A(∞)-morphism {f_{i}^{(2)}(X)} as the composition

{f̄_{i}^{(2)}(X)} ∘ {f_{i}^{(1)}(ΩX)} : C^{∗}(Ω^{2}X) → (BC^{∗}(ΩX), m_{i}^{(1)}(ΩX)) →

(B(BC^{∗}(X), {m_{i}^{(1)}(X)}), {m_{1}^{(2)}(X)}).

Suppose now that {m_{i}^{(k−1)}(X)} and {f_{i}^{(k−1)}(X)} have already been constructed for k ≤ n. Then, since Ω^{k−1}X is at least 1-connected, there exists a weak equivalence of A(∞)-algebras

{f_{i}^{(1)}(Ω^{k−1}X)} : C^{∗}(Ω^{k}X) → (BC^{∗}(Ω^{k−1}X), {m_{i}^{(1)}(Ω^{k−1}X)}).

Moreover, we also have a weak equivalence of DG-coalgebras

B({f_{i}^{(k−1)}(X)}) : B(C^{∗}(Ω^{k−1}X) →
B(⋯ (BC^{∗}(X), {m_{i}^{(1)}(X)}), ⋯), {m_{i}^{(k−1)}(X)}).

By Theorem B this weak equivalence transfers the A(∞)-algebra structure ({m_{i}^{(1)}(Ω^{k−1}X)} of B(C^{∗}(Ω^{k−1}X)) to

B(⋯ (BC^{∗}(X), {m_{i}^{(1)}(X)}), ⋯), {m_{i}^{(k−1)}(X)})

and we get the A(∞)-algebra structure {m_{i}^{(k)}(X)} and the morphism of A(∞)-algebras

{f̄_{i}^{(k)}(X)} : (B(C^{∗}(Ω^{k−1}X), {m_{i}^{(1)}(Ω^{k−1}X)}) →

(B(⋯ (BC^{∗}(X), {m_{i}^{(1)}(X)}), ⋯), {m_{i}^{(k−1)}(X)}), {m_{i}^{(k−1)}(X)}).

Now we can define the A(∞)-morphism {f_{i}^{(k)}(X)} as the composition

{f̄_{i}^{(k)}(X)} ∘ {f_{i}^{(1)}(Ω^{k−1}X)}:C^{∗}(Ω^{k}X)→(B(C^{∗}(Ω^{k−1}X), {m_{i}^{(1)}(Ω^{k−1}X)})

→ (B(⋯ (BC^{∗}(X), {m_{i}^{(1)}(X)}), ⋯), {m_{i}^{(k−1)}(X)}), {m_{i}^{(k−1)}(X)}).

This completes the proof.

Acknowledgment

The research described in this publication was made possible in part by Grant No. RVC2OO from the International Science Foundation.

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(Received 27.05.1996)

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