THE HOMOLOGY OF ITERATED LOOP SPACES

V.A.Smirnov

with a computational appendix by F. Sergeraert

Last years to solve different probems in Agebraic Topology it needs to consider more and more complicate structures on the chain complex C_{∗}(X) of a topological space X and its homology H_{∗}(X).

One of the most difficult problem is the problem of calculation of the homology groups of the iterated loop spaces. First steps to solve this problem were made by J.F.Adams. To calculate the homology H_{∗}(ΩX) of the loop space ΩX over a topological space X he introduced the notion of the cobar construction F K over a coalgebra K, [1].

Recall that a chain complex K is called a coalgebra if there is given a chain mapping ∇: K → K ⊗ K satisfying the associativity relation

(∇ ⊗ 1)∇ = (1 ⊗ ∇)∇.

The cobar construction F K over a coalgebra K is by the definition a differential algebra which as a graded algebra coincides with the tensor algebra T S^{−1}K over the desuspension S^{−1}K over K. The generator elements in F K usually denoted as [x_{1}, …, x_{n}] where x_{i} ∈ K, 1 ≤ i ≤ n. The product is defined by the formula

π([x_{1}, …, x_{n}] ⊗ [x_{n+1}, …, x_{n+m}]) = [x_{1}, …, x_{n+m}].

The differential on the generators [x] ∈ F K is defined by the formula

d[x] = −[d(x)] + ∑(−1)^{ϵ}[x′, x′′],

where ∇(x) = ∑ x′ ⊗ x′′, ϵ = dim(x′).

On the chain complex C_{∗}(X) of a topological space X there exist a natural coalgebra structure determined on simplexes x with vertixes 0, 1, …, n by the formula

∇(x(0, 1, …, n)) = ∑_{i} x(0, …, i) ⊗ x(i, …, n).

So there is defined the cobar construction F C_{∗}(X).

J.F.Adams proved that for a simply connected topological space X there is a chain equivalence of differential algebras

C_{∗}(ΩX) ≃ F C_{∗}(X).

In particular, if on the C_{∗}(X) there is the trivial coalgebra structure, it is so for example if X is the suspension SY over a space Y, then the chain complex C_{∗}(ΩX) will be chain equivalent to the tensor algebra T C_{∗}(X) over the chain complex C_{∗}(X).

But passing to the cobar construction we lose the coalgebra structure. There is no natural coalgebra structure on the Adams cobar construction F K over a coalgebra K. So the Adams cobar construction can’t be iterated.

However in [2] H.J.Baues introduced a coalgebra structure in the Adams cobar construction F C_{∗}(X). This structure was determined using the family of operations

∇_{n,m}: C_{∗}(X) → C_{∗}(X)^{⊗n} ⊗ C_{∗}(X)^{⊗m}

of dimensions n+m−1. So he defined the double cobar construction F^{2}C_{∗}(X) and proved that for 2-connected topological space X there is a chain equivalence

C_{∗}(Ω^{2}X) ≃ F^{2}C_{∗}(X).

But there is no appropriate coalgebra structure on the double cobar construction and therefore further iterations is not possible.

In [3] J.P.May introduced the notion of an operad and investigated structure on the iterated loop spaces. This structure used to calculate the homology H_{∗}(Ω^{n}S^{n}X), [4]. Also this homology investigated in [5], [6] and others.

In [7], [8], [9] operad methods were transfered from the category of topological spaces to the category of chain complexes. It was shown that on the chain complex C_{∗}(X) of a topological space X there is E_{∞}-coalgebra structure which determines the weak homotopy type of X. Using this structure the chain complex C_{∗}(Ω^{n}X) of the n-fold loop space Ω^{n}X over an n-connected topological space X was expressed through the chain complex C_{∗}(X) of a space X.

Note that a general method to calculate the homology of the iterated loop space gives us simplicial theory. There is the simplicial construction GX of the loop space over a simplicial set X and its iteration G^{n}X. This construction is very complicate and real calculations may be produced using computer methods [10].

Our aim here is to construct the spectral sequence for the homology of the iterated loop spaces, to determine its first and second terms, and to produce some calculations for the real projective spaces.

§1. Operads and algebras over operads

Consider the category of chain complexes over a ring R. By a symmetric family E in this category is meant a family E = {E(j)}_{j≥1} of chain complexes E(j) operated on by the symmetric groups Σ_{j}.

Given two symmetric families E′, E′′ we define the symmetric families E′ ⊗ E′′, E′ × E′′ by putting

(E′ ⊗ E′′)(j) = E′(j) ⊗ E′′(j),

and (E′ × E′′)(j) equal to the quotiont module of the free Σ_{j}-module generated by the module

∑_{j_{1}+···+j_{k}=j} E′(k) ⊗ E′′(j_{1}) ⊗ ⋯ ⊗ E′′(j_{k})

with respect to the equivalence generated by the relations

x′σ ⊗ x′′_{1} ⊗ ⋯ ⊗ x′′_{k} ∼ x′ ⊗ x′′_{σ^{−1}(1)} ⊗ ⋯ ⊗ x′′_{σ^{−1}(k)} · σ(j_{1}, …, j_{k}),
x′ ⊗ x′′_{1}σ_{1} ⊗ ⋯ ⊗ x′′_{k}σ_{k} ∼ x′ ⊗ x′′_{1} ⊗ ⋯ ⊗ x′′_{k} · σ_{1} × ⋯ × σ_{k},

where σ(j_{1}, …, j_{k}) is the permutation of a set of j elements obtained by partitioning the set into k blocks of j_{1}, …, j_{k} elements, respectively, and carrying out on these blocks the permutation σ, while σ_{1} × ⋯ × σ_{k} means the image of the element (σ_{1}, …, σ_{k}) under the imbedding Σ_{j_{1}} × ⋯ × Σ_{j_{k}} → Σ_{j}.

It is easy to see that for the symmetric families E′, E′′, F′, E′′ there is the interchanging mapping

T: (E′ ⊗ E′′) × (F′ ⊗ E′′) → (E′ × F′) ⊗ (E′′ × F′′).

A symmetric family E is called an operad if there is given a symmetric-family mapping γ: E × E → E such that γ(γ × 1) = γ(1 × γ), or that is the same, the following diagram is commutative

E × E × E ⟶^{γ×1} E × E
1×γ↓  ↓^{γ}
E × E ⟶^{γ} E

If there exist an element 1 ∈ E(1) such that γ(1 ⊗ x) = γ(x ⊗ 1^{⊗k}) = x for all x ∈ E(k) then we say that E is an operad with identity.

An operad E is called a Hopf operad if besides the operation γ: E × E → E there is given an operation ∇: E → E ⊗ E which is associative and satisfy the Hopf relation, i.e. the following diagram is commutative

E × E ⟶^{∇×∇} (E ⊗ E) × (E ⊗ E) ⟶^{T} (E × E) ⊗ (E × E)
γ↓  ↓^{γ⊗γ}
E ⟶^{∇}  E ⊗ E  ⟶^{=}  E ⊗ E

A mapping of operads f: E′ → E′′ is a mapping of symmetric families for which the folowing diagram is commutative

E′ × E′ ⟶^{γ′} E′
f×f↓  ↓^{f}
E′′ × E′′ ⟶^{γ′′} E′′

If E′ and E′′ are operads with identities 1′ and 1′′ respectively, then it is required that f(1′) = 1′′.

We shall say that an operad E acts on a symmetric family F on the left (right) if there is given a mapping μ′: E × F → F (μ′′: F × E → F) such that

μ′(γ × 1) = μ′(1 × μ′) (μ′′(1 × γ) = μ′′(μ′′ × 1)),

or that is the same, the following diagrams are commutative

\begin{matrix}E × E × F & ⟶^{γ×1} & E × F \\ ^{1×μ′}↓ & & ↓^{μ′} \\ E × F & ⟶^{μ′} & F \end{matrix}  (\begin{matrix}F × E × E & ⟶^{1×γ} & F × E \\ ^{μ′′×1}↓ & & ↓^{μ′′} \\ F × E & ⟶^{μ′′} & F \end{matrix})

For any chain complex X and symmetric family E we define chain complexes E(X), \overline{E}(X) by putting

E(X) = ∑_{k} E(k) ⊗_{Σ_{k}} X^{⊗k}, \overline{E}(X) = ∏_{k} Hom_{Σ_{k}}(E(k); X^{⊗k}).

If E is an operad, then the operad structure in E determines a mapping

γ: E^{2}(X) = E(E(X)) → E(X)

such that the correspondence X ⟼ E(X) is a monad in the category of chain complexes. Dually, the operad structure in E determines a mapping

\overline{γ}: \overline{E}(X) → \overline{E}^{2}(X) = \overline{E}(\overline{E}(X))

such that the correspondence X ⟼ \overline{E}(X) is a comonad in the category of chain complexes.

A chain complex X is called an algebra over an operad E, or simply E-agebra, if there is given a mapping μ: E(X) → X satisfying the associativity relation: μ ∘ γ(X) = μ ∘ E(μ), or that is the same, the following diagram is commutative

E^{2}(X) ⟶^{γ(X)} E(X)
E(μ)↓  ↓^{μ}
E(X) ⟶^{μ} X

Dually, a chain complex X is called a coalgebra over an operad E, or simply Ecoalgebra, if there is given a mapping τ: X → \overline{E}(X) satisfying the associativity relation: \overline{γ}(X) ∘ τ = \overline{E}(τ) ∘ τ, or that is the same, the following diagram is commutative

X ⟶^{τ} \overline{E}(X)
τ↓  ↓^{\overline{γ}(X)}
\overline{E}(X) ⟶^{\overline{E}(τ)} \overline{E}^{2}(X)

Consider some examples of operads and algebras over operads.

1. A Hopf operad E_{0} = {E_{0}(j)}, where E_{0}(j) - the free module with one zero dimensional generator e(j) and trivial action of the symmetric group Σ_{j}. So E_{0}(j) ≅ R. The operation γ: E_{0} × E_{0} → E_{0} is defined by the formula

γ(e(k) ⊗ e(j_{1}) ⊗ ⋯ ⊗ e(j_{k})) = e(j_{1} + ⋯ + j_{k}).

The operation ∇: E_{0} → E_{0} ⊗ E_{0} is defined by the formula

∇(e(j)) = e(j) ⊗ e(j).

It is easy to see that the required relations are satisfied and algebras (coalgebras) over the operad E_{0} are simply commutative and associative algebras.

2. A Hopf operad A = {A(j)}, where A(j) - the free Σ_{j} - module with one zero dimensional generator a(j). So A(j) ≅ R(Σ_{j}). The operation γ: A × A → A is defined by the formula

γ(a(k) ⊗ a(j_{1}) ⊗ ⋯ ⊗ a(j_{k})) = a(j_{1} + ⋯ + j_{k}).

The operation ∇: A → A ⊗ A is defined by the formula

∇(a(j)) = a(j) ⊗ a(j).

It is easy to see that the required relations are satisfied and that algebras (coalgebras) over the operad A are simply associative algebras.

3. For a symmetric family E define the suspension SE by putting (SE)(j) = S^{j−1}E(j)

- (j − 1)-fold suspension over E(j). It is clear that if E is an operad then the suspension SE will be also an operad, and if X is an algebra (coalgebra) over an operad E then the suspension SX will be an algebra (coalgebra) over the operad SE.

4. For operads E′, E′′ its tensor product E′ ⊗ E′′ evidently is an operad.

5. Let L be a suboperad of the operad A generated by the element

b(2) = a(2) − a(2)T, T ∈ Σ_{2}.

Then algebras (coalgebras) over the operad L will be simply Lie algebras (coalgebras). Similary, let L_{n} be a suboperad of the operad S^{n}A generated by the element

b_{n}(2) = s^{n}a(2) + (−1)^{n}s^{n}a(2)T.

Then algebras (coalgebras) over the operad L_{n} will be simply n-Lie algebras (coalgebras).

6. Let P_{n} = E_{0} × L_{n} and the operad structure γ is determined by the corresponding structures in E_{0}, L_{n} and by the formulas

γ(b_{n}(2) ⊗ 1 ⊗ e(2)) = e(2) ⊗ b_{n}(2) ⊗ 1 + e(2) ⊗ 1 ⊗ b_{n}(2) · (213).

The operad P_{n} is called the n-Poisson operad. Algebras (coalgebras) over this operad are called n-Poisson algebras (coalgebras).

7. For any chain complex X define operads E_{X}, E^{X} by putting

E_{X}(j) = Hom(X^{⊗j}; X); E^{X}(j) = Hom(X; X^{⊗j}).

The actions of the symmetric groups are determined by the permutations of factors of X^{⊗j} and operad structures are defined by the formulas

γ_{X}(f ⊗ g_{1} ⊗ ⋯ ⊗ g_{k}) = f ∘ (g_{1} ⊗ ⋯ ⊗ g_{k}), f ∈ E_{X}(k), g_{i} ∈ E_{X}(j_{i});
γ^{X}(f ⊗ g_{1} ⊗ ⋯ ⊗ g_{k}) = (g_{1} ⊗ ⋯ ⊗ g_{k}) ∘ f, f ∈ E^{X}(k), g_{i} ∈ E^{X}(j_{i}).

Directly from the definition it follows that a chain complex X is an algebra (coalgebra) over an operad E if and only if there is given an operad mapping ξ: E → E_{X} (ξ: E → E^{X}).

Analogously for chain complexes X and Y there are defined symmetric families F_{X,Y}, F^{X,Y}:

F_{X,Y} (j) = Hom(X^{⊗j}; Y), F^{X,Y} (j) = Hom(X; Y^{⊗j})

and actions

μ′: E_{Y} × F_{X,Y} → F_{X,Y} , μ′′: F_{X,Y} × E_{X} → F_{X,Y};
μ′: E^{X} × F^{X,Y} → F^{X,Y} , μ′′: F^{X,Y} × E^{Y} → F^{X,Y} .

8. One of the most important topological operad is the Bordman and Vogt’s “little ncubes” operad E_{n}, [2]. Let J denote the open interval (0, 1) and J^{n} the open n-dimensional cube. By an n-dimensional little cube it calles a liner embedding f: J^{n} → J^{n} with parallel axes. Then E_{n}(j) is the set of ordered j-tuples (f^{1}, …, f^{j}) of n-dimensional little cubes f^{i}: J^{n} → J^{n} that images don’t intersect. This operad acts on the n-fold loop space Ω^{n}X over a space X.

The direct limite of the operads E_{n} over the inclusions E_{n} ⊂ E_{n+1} denotes as E_{∞}. It is acyclic operad with the free actions of the symmetric groups.

9. It is easy to see that if E is a topological operad then it’s singular chain complex C_{∗}(E) is an operad in the category of chain complexes, and if E acts on a spaces X then C_{∗}(E) acts on C_{∗}(X).

Similary, the homology H_{∗}(E) of a topological operad E is an operad in the category of graded modules, and if E acts on a space X then H_{∗}(E) acts on the homology H_{∗}(X). In particular, the homology H_{∗}(E_{n}) of the topological n-cubes operad E_{n} is isomorphic to the n-Poisson operad P_{n}. So the homology H_{∗}(Ω^{n}X) of the n-fold loop space Ω^{n}X is an algebra over the Poisson operad P_{n}.

The operad C_{∗}(E_{∞}) gives us an example of the acyclic operad in the category of chain complexes with the free actions of the symmetric groups. Note that all acyclic operads with the free actions of the symmetric groups consist of the Σ_{j}-homotopy equivalent chain complexes. We will call such operads as E_{∞}-operads.

10. Another example of E_{∞}-operad give us the simplicial resolutions of the symmetric groups. Denote by EΣ_{∗}(j) the free simplicial resolution of the symmetric group Σ_{j}, i.e.

EΣ_{∗}(j) : Σ_{j} ⟵ Σ_{j} × Σ_{j} ⟵ …

The mappings γ: Σ_{k} × Σ_{j_{1}} × ⋯ × Σ_{j_{k}} → Σ_{j_{1}+···+j_{k}} induce the operad structure

γ_{∗}: EΣ_{∗}(k) × EΣ_{∗}(j_{1}) × ⋯ × EΣ_{∗}(j_{k}) → EΣ_{∗}(j_{1} + ⋯ + j_{k}).

So EΣ_{∗} will be the acyclic operad with the free actions of the symmetric groups in the category of simplicial sets.

Taking the chain complex C_{∗}(EΣ_{∗}) we obtain the E_{∞}-operad in the category of chain complexes. Denote it simply as EΣ.

Note that for any chain operad E the operad E ⊗ EΣ will has the same homology and the free actions of the symmetric groups. The projection EΣ → E_{0} induces the projection E ⊗ EΣ → E. So E ⊗ EΣ may be considered as Σ-free resolution of the operad E. If E is an acyclic operad then E ⊗ EΣ will be E_{∞}-operad.

§2. On the chain complex of a topological space

Here we consider a structure on the singular chain complex C_{∗}(X) of a topological space X, and dually on the singular cochain complex C^{∗}(X).

Besides a coalgebra structure

∇: C_{∗}(X) → C_{∗}(X) ⊗ C_{∗}(X)

on the chain complex of a topological space there are coproducts

∇_{i}: C_{∗}(X) → C_{∗}(X) ⊗ C_{∗}(X)

increasing dimensions by i and such that

d(∇_{i}) = ∇_{i−1} + (−1)^{i}T ∇_{i−1},

where T: C_{∗}(X) ⊗ C_{∗}(X) → C_{∗}(X) ⊗ C_{∗}(X) permutes factors.

Dually, on the cochain complex C^{∗}(X) besides an algebra structure

∪: C^{∗}(X) ⊗ C^{∗}(X) → C^{∗}(X)

there are products

∪_{i}: C^{∗}(X) ⊗ C^{∗}(X) → C^{∗}(X)

such that d(∪_{i}) = ∪_{i−1} + (−1)^{i} ∪_{i−1} T.

To describe all operations on the singular chain complex C_{∗}(X) of a toplogical space X and its dual cochain complex C^{∗}(X) we consider the corresponding operad.

For n ≥ 0 denote by Δ^{n} the normalized chain complex of the standard n-dimensional simplex. Then Δ^{∗} = {Δ^{n}} is the cosimplicial object in the category of chain complexes. Consider also the cosimplisial object (Δ^{∗})^{⊗j} = Δ^{∗} ⊗ ⋯ ⊗ Δ^{∗} and its realization

E^{Δ}(j) = Hom(Δ^{∗}; (Δ^{∗})^{⊗j}),

where Hom is considered in the category of cosimplicial objects.

The family E^{Δ} = {E^{Δ}(j)} will be the operad for which the actions of the symmetric groups and an operad structure are defined similary to the corresponding structures for the above defined operad E^{X}, where instead of X we take Δ^{∗}.

Note that since the complexes Δ^{n} are acyclic then the operad E^{Δ} will be also acyclic. Theorem 1. On the chain complex C_{∗}(X) of a topological space X there exists a

natural coalgebra structure τ: C_{∗}(X) → \overline{E}^{Δ}(C_{∗}(X)) with the following universal property: if for any operad E there is a natural E - coalgebra structure τ̃: C_{∗}(X) → \overline{E}(C_{∗}(X)), then there exist a unique operad mapping ξ: E → E^{Δ} such that τ̃ = \overline{ξ}(C_{∗}(X)) ∘ τ.

Proof. Our aim is to define operations

τ: E^{Δ}(j) ⊗ C_{∗}(X) → C_{∗}(X)^{⊗j}.

Let x_{n} ∈ C_{n}(X), y ∈ E^{Δ}(j) = Hom(Δ^{∗}; (Δ^{∗})^{⊗j}). The element x_{n} ∈ C_{n}(X) determines the chain mapping \overline{x}_{n}: Δ^{n} → C_{∗}(X) such that the generator u_{n} ∈ Δ^{n} maps to x_{n}.

Note that there is the operation τ^{n}: E^{Δ}(j) ⊗ Δ^{n} → (Δ^{n})^{⊗j}. Define the required operation τ putting

τ(y ⊗ x_{n}) = (\overline{x}_{n})^{⊗j} ∘ τ^{n}(y ⊗ u_{n}).

Then we will have the following commutative diagram

E^{Δ}(j) ⊗ C_{∗}(X) ⟶^{τ} C_{∗}(X)^{⊗j}
1⊗\overline{x}_{n}↑  ↑^{(\overline{x}_{n})^{⊗j}}
E^{Δ}(j) ⊗ Δ^{n} ⟶^{τ^{n}} (Δ^{n})^{⊗j}

It is easy to see that the required relations are satisfied.

The E^{Δ} - coalgebra structure on the chain complex C_{∗}(X) of a topological space X induces the E^{Δ} - algebra structure on the cochain complex C^{∗}(X) = Hom(C_{∗}(X); R). The corresponding operations μ: E^{Δ}(j) ⊗ C^{∗}(X)^{⊗j} → C^{∗}(X) are defined by the formulas

μ(y ⊗ f_{1} ⊗ ⋯ ⊗ f_{j})(x) = (f_{1} ⊗ ⋯ ⊗ f_{j}) ∘ τ(y ⊗ x),

where y ∈ E^{Δ}(j), f_{i}: C_{∗}(X) → R, x ∈ C_{∗}(X). So we have

Theorem 1’. On the cochain complex C^{∗}(X) of a topological space X there exists a natural algebra structure μ: E^{Δ}(C^{∗}(X)) → C^{∗}(X) with the following universal property: if fo any operad E there is a natural E - algebra structure μ̃: E(C^{∗}(X)) → C^{∗}(X), then there exists a unique operad mapping ξ: E → E^{Δ} such that μ̃ = μ ∘ ξ(C^{∗}(X)).

Let R(Σ_{2}) - the Σ_{2}-free resolution with generator elements e_{i} of dimensions i and the differential defined by the formula

d(e_{i}) = e_{i−1} + (−1)^{i}e_{i−1}T, T ∈ Σ_{2}.

Since E^{Δ}(2) is acyclic, there is the Σ_{2}-chain mapping R(Σ_{2}) → E^{Δ}(2) and hence the mapping

R(Σ_{2}) ⊗_{Σ_{2}} C^{∗}(X)^{⊗2} → C^{∗}(X).

Its restriction on the elements e_{i} usually denoted as

∪_{i}: C^{∗}(X) ⊗ C^{∗}(X) → C^{∗}(X)

and called cup-i product.

Let p^{n}: Δ^{n} → SΔ^{n−1} be the projection obtained by contracting the (n−1)-dimensional face spanned by the vertixes with numbers 0, 1, …, n − 1. These projections induce a projection of operads E^{Δ} → SE^{Δ}. For a topological space X the suspension SC_{∗}(X) will be a coalgebra over the operad SE^{Δ} and the following diagram commutes

SC_{∗}(X) ⟶ \overline{SE}^{Δ}(SC_{∗}(X))
↓  ↓
C_{∗}(SX) ⟶ \overline{E}^{Δ}(C_{∗}(SX))

Iterating this construction we obtain the projections E^{Δ} → S^{n}E^{Δ} and the commutative diagrams

S^{n}C_{∗}(X) ⟶ \overline{S^{n}E}^{Δ}(S^{n}C_{∗}(X))
↓  ↓
C_{∗}(S^{n}X) ⟶ \overline{E}^{Δ}(C_{∗}(S^{n}X))

Note that the operad E^{Δ} may be not Σ-free and so it is not an E_{∞}-operad. To obtain the E_{∞}-operad we consider the operad E^{Δ} ⊗ EΣ and denote it simply as E.

The projection E → E^{Δ} induces the E - coalgebra structure on the chain complex C_{∗}(X) of a topological space X. The mapping E^{Δ} → SE^{Δ} induces the operad mapping E → SE and for the chain complexes C_{∗}(S^{n}X) of the suspesions S^{n}X over a topological space X there are the corresponding commutative diagrams similar to the diagrams for the operad E^{Δ}.

§3. Bar and cobar constructions over operads

Let E be an operad with right action on a symmetric family F, ν: F × E → F, and let X be an algebra over the operad E, μ: E(X) → X. Consider the simplicial object B_{∗}(F, E, X) = {B_{n}(F, E, X)} for which B_{n}(F, E, X) = FE^{n}(X) with the face and degeneracy operators given by the formulas

d_{0} = ν1^{n}, d_{i} = 1^{i}γ1^{n−i}, 0 < i < n;
d_{n} = 1^{n}μ, s_{j} = 1^{j+1}i1^{n−j+1}, 0 ≤ j ≤ n.

Its realization

B(F, E, X) = |B_{∗}(F, E, X)| = ∑_{n} Δ^{n} ⊗ B_{n}(F, E, X)/ ∼

is called the bar construction.

In the case of trivial F, i.e. F(1) = R and F(j) = 0 if j ≥ 2, the corresponding bar construction denotes B(E, X).

If X is an A-algebra, i.e. simply an algebra then the bar construction B(A, X) will be chain equivalent to the desuspension over the usual Adams bar construction B(X), i.e.

B(A, X) ≃ S^{−1}BX, [8].

Dually, if X is a coalgebra over the operad E, τ: X → \overline{E}(X), then we can consider the cosimplicial object F^{∗}(F, E, X) = \{F^{n}(F, E, X)\} for which F^{n}(F, E, X) = \overline{F}\overline{E}^{n}(X) with the coface and codegeneracy operators given by the formulas

δ^{0} = \overline{ν}1^{n}, δ^{i} = 1^{i}\overline{γ}1^{n−i}, 0 < i < n;
δ^{n} = 1^{n}τ, σ^{j} = 1^{j+1}p1^{n−j+1}, 0 ≤ j ≤ n.

Its realization

F (F, E, X) = |F^{∗}(F, E, X)| = Hom(Δ^{∗}; F^{∗}(F, E, X)),

where Hom is considered in the category of cosimplicial objects, is called the cobar construction.

In the case of trivial F the corresponding bar construction denotes F (E, X).

If X is an A-coalgebra, i.e. simply a coalgebra then the cobar construction F (A, X) will be chain equivalent to the suspension over the usual Adams cobar construction F (X), i.e.

F (A, X) ≃ SF X, [8].

Let now X be an n-connected topological space. As it was shown above on the chain complex S^{n}C_{∗}(Ω^{n}X) there is S^{n}E - coalgebra structure

τ: S^{n}C_{∗}(Ω^{n}X) → \overline{S^{n}E}(S^{n}C_{∗}(Ω^{n}X)).

This structure and the mapping j: S^{n}Ω^{n}X → X induce the mapping

S^{n}C_{∗}(Ω^{n}X) → \overline{S^{n}E}(S^{n}C_{∗}(Ω^{n}X)) → \overline{S^{n}E}(C_{∗}(X)).

This mapping is a coaugmentation of the cosimplicial object F^{∗}(S^{n}E, E, C_{∗}(X)) and hence it induces the mapping of S^{n}E- coalgebras

S^{n}C_{∗}(Ω^{n}X) → F (S^{n}E, E, C_{∗}(X)).

Theorem 2. For each n-connected topological space X the mapping

S^{n}C_{∗}(Ω^{n}X) → F (S^{n}E, E, C_{∗}(X))

is a chain equivalence of S^{n}E-coalgebras.

Proof. For n = 1 the chain equivalence SC_{∗}(ΩX) → F (SE, E, C_{∗}(X)) follows from the Adams chain equivalence

SC_{∗}(ΩX) ≃ SF C_{∗}(X)

and the chain equivalence

SF C_{∗}(X) ≃ F (SE, E, C_{∗}(X)), [8].

Suppose that for each (n − 1)-connected topological space X we have a chain equivalence

S^{n−1}C_{∗}(Ω^{n−1}X) → F (S^{n−1}E, E, C_{∗}(X)).

Then for an n-connected topological space X we will have the following sequence of chain equivalences

S^{n}C_{∗}(Ω^{n}X) ≃ SF (S^{n−1}E, E, C_{∗}(ΩX)) ≃ F (S^{n}E, SE, SC_{∗}(ΩX)) ≃
F (S^{n}E, SE, F (SE, E, C_{∗}(X))) ≃ F (S^{n}E, E, C_{∗}(X)).

Its composition will be the desirable chain equivalence

S^{n}C_{∗}(Ω^{n}X) ≃ F (S^{n}E, E, C_{∗}(X)).

§4. The spectral sequence for the homology of iterated loop spaces

Let X be an E-coalgebra. Consider the spectral sequence of the cobar construction F (S^{n}E, E, X) over the filtration determined by the operad grading of E. Namely, for any operad E we define the grading of mappings f: E(j) → X^{⊗j} equal to j. So we will have the grading of the elements of \overline{S^{n}E}(X), \overline{S^{n}E} ∘ \overline{E}(X) and so on. Thus we will have the grading of the elements of the cobar construction F (S^{n}E, E, X) which determines in it the decreasing filtration.

Note that the mappings \overline{μ}: \overline{S^{n}E} → \overline{S^{n}E} ∘ \overline{E}, \overline{γ}: \overline{E} → \overline{E} ∘ \overline{E} preserve gradings and the mapping τ: X → \overline{E}(X) increases it. Therefore the first term of the corresponding spectral sequence will be isomorphic to the homology of the cobar construction F (S^{n}E, E, X), where X is considered as a trivial E-coalgebra. To calculate its homology recall the notion of the iterated Adams cobar construction.

Let K be a cocommutative coalgebra. Then the Adams cobar construction will be a cocommutative Hopf algebra with the coproduct ∇: F K → F K ⊗ F K defined on the generators [x] ∈ F K by the formula

∇[x] = [x] ⊗ 1 + 1 ⊗ [x].

Note that a commutative coalgebra K may be considered as E-coalgebra. The required E-coalgebra structure is induced by the projection E → E_{0}. Moreover, the chain equivalence SF K ≃ F (SE, E, K) will be the chain equivalence of SE-coalgebras.

Thus in this case the Adams cobar construction may be iterated and there are chain equivalences

S^{n}F^{n}K ≃ F (S^{n}E, E, K).

In particular, if a chain complex X has the trivial E-coalgebra structure then there are the chain equivalences

S^{n}F^{n}X ≃ F (S^{n}E, E, X).

So to calculate the homology of the cobar construction F (S^{n}E, E, X) over a trivial Ecoalgebra X it is sufficient to calculate the homology of the iterated Adams cobar construction F^{n}X. We will do it for chain complexes over a field.

Denote X_{∗} = H_{∗}(X). The homology H_{∗}(F X) is isomorphic to the tensor algebra T S^{−1}X_{∗} over the desuspension S^{−1}X_{∗} over X_{∗}, i.e.

H_{∗}(F X) ≅ T S^{−1}X_{∗}.

The double cobar construction F^{2}X will be chain equivalent to the cobar construction F T S^{−1}X_{∗}, where T S^{−1}X_{∗} is considered as a Hopf algebra with the coproduct

∇: T S^{−1}X_{∗} → T S^{−1}X_{∗} ⊗ T S^{−1}X_{∗}

determined on the generators [x] ∈ S^{−1}X_{∗} by the formula

∇[x] = [x] ⊗ 1 + 1 ⊗ [x].

Denote by LS^{−1}X_{∗} the free Lie algebra generated by the module S^{−1}X_{∗}. Then by the Poincare-Birkhoff-Witt theorem there is the isomorphism of coalgebras

T S^{−1}X_{∗} ≅ T_{s}LS^{−1}X_{∗},

where T_{s}LS^{−1}X_{∗} denotes the free commutative algebra generated by the module LS^{−1}X_{∗}.

The homology of F T_{s}LS^{−1}X_{∗} is isomorphic to the free commutative algebra generated by the module S^{−1}P T_{s}LS^{−}X_{∗}, i.e.

H_{∗}(F^{2}X) ≅ T_{s}S^{−1}P T_{s}LS^{−1}X_{∗},

where P T_{s}LS^{−1}X_{∗} denotes the module of primitive elements of T S^{−1}X_{∗}, i.e.

P T_{s}LS^{−1}X_{∗} = {y ∈ T_{s}LS^{−1}X_{∗}| ∇(y) = y ⊗ 1 + 1 ⊗ y}.

So this construction may be iterated and by induction we obtain the isomorphisms

H_{∗}(F^{n}X) ≅ T_{s}(S^{−1}P T_{s})^{n−1}LS^{−1}X_{∗},

and hence the isomorphisms

S^{−n}H_{∗}(F (S^{n}E, E, X)) ≅ T_{s}(S^{−1}P T_{s})^{n−1}LS^{−1}X_{∗}.

Thus we have

Theorem 3. If X is an n-connected topological space, then there is the spectral sequence which converges to H_{∗}(Ω^{n}X) and for the first term E^{1} of this spectral sequence there is the isomorphism

E^{1} ≅ T_{s}(S^{−1}P T_{s})^{n−1}LS^{−1}H_{∗}(X).

Rewrite the first term of the spectral sequence using the notion of a Lie algebra and a Poisson algebra. Firstly we will consider Z/2-coefficients.

Recall that a graded module L (over Z/2) is called a Lie algebra if there is given an operation [ , ]: L ⊗ L → L called a Lie bracket and satisfying the relations

[x, x] = 0;
[x, y] + [y, x] = 0;
[x, [y, z]] = [[x, y], z] + [y, [x, z]].

Similary an n-Lie algebras are defined.

Note that the module P T_{s}M of primitive elements of the Hopf algebra T_{s}M is generated by the elements of the form x^{2^{k}}, x ∈ M, k ≥ 0.

For a graded module M denote by E_{n}(M) the module generated by the elements e_{i_{1}} … e_{i_{k}}x, where x ∈ M, 0 ≤ i_{1} ≤ ⋯ ≤ i_{k} ≤ n, and the dimensions of these elements are defined equal to i_{1} + 2i_{2} + ⋯ + 2^{k−1}i_{k} + 2^{k}dim(x). So for an n-Lie algebra L_{n} we will have the module E_{n}L_{n}.

Denote also by T_{s}E_{n}L_{n} the quotient algebra of the free commutative algebra generated by the module E_{n}L_{n} over the relations x · x = e_{0}x.

From the above considerations it follows that if X is a chain complex (over Z/2) considered as the trivial E-coalgebra, then there are the isomorphisms

S^{−n}H_{∗}(F (S^{n}E, E, X)) ≅ T_{s}E_{n−1}L_{n−1}S^{−n}X_{∗}.

Hence we have

Theorem 3’. For the first term of the considered spectral sequence of the homology H_{∗}(Ω^{n}X) (over Z/2) there is the isomorphism

E^{1} ≅ T_{s}E_{n−1}L_{n−1}S^{−n}H_{∗}(X).

Consider now Z/p-coefficients, p > 2. Recall that a graded module L (over Z/p) is called a Lie algebra if there is given an operation [ , ]: L ⊗ L → L called a Lie bracket and satisfying the relations

[x, y] + (−1)^{ϵ}[y, x] = 0;
[x, [y, z]] = [[x, y], z] + (−1)^{ϵ}[y, [x, z]];

where ϵ = dim(x) · dim(y).

Similary a graded module L called an n-Lie algebra if there is given an operation [ , ]: L ⊗ L → L of dimension n, called a Lie bracket and satisfying the relations

[x, y] + (−1)^{ϵ}[y, x] = 0;
[x, [y, z]] = [[x, y], z] + (−1)^{ϵ}[y, [x, z]];

where ϵ = (dim(x) + n) · (dim(y) + n).

Note that the module P T_{s}M of primitive elements of the Hopf algebra T_{s}M is generated by the elements x ∈ M and the elements x^{p^{k}} for which k > 0 and dim(x) - even.

For a graded module M denote by E_{n}(M) the module generated by the elements e_{i_{1}} … e_{i_{k}}x, where x ∈ M, 0 ≤ i_{1} ≤ ⋯ ≤ i_{k} ≤ n, and besides that i_{1}, …, i_{k} are odd if dim(x) is odd and i_{1}, …, i_{k} are even if dim(x) is even. The dimensions of these elements are defined equal to (p − 1)(i_{1} + pi_{2} + ⋯ + p^{k−1}i_{k}) + p^{k}dim(x). So for an n-Lie algebra L_{n} we will have the module E_{n}L_{n}.

Denote also by T_{s}E_{n}L_{n} the quotient algebra of the free commutative algebra generated by the module E_{n}L_{n} over the relations e_{0}x = x^{p}.

From the above considerations it follows that if X is a chain complex (over Z/p) considered as the trivial E-coalgebra, then there are the isomorphisms

S^{−n}H_{∗}(F (S^{n}E, E, X)) ≅ T_{s}E_{n−1}L_{n−1}S^{−n}X_{∗}.

Hence we have

Theorem 3”. For the first term of the considered spectral sequence of H_{∗}(Ω^{n}X) (over Z/p) there is the isomorphism

E^{1} ≅ T_{s}E_{n−1}L_{n−1}S^{−n}H_{∗}(X).

Note that in the case of characteristic zero coefficients the module P T_{s}M of primitive elements of T_{s}M is isomorphic to M. Hence we have

Theorem 3”’. The first term of the considered spectral sequence of H_{∗}(Ω^{n}X) (over a field of characteristic zero) is the isomorphic to the free (n−1)-Poisson algebra generated by S^{−n}H_{∗}(X), i.e.

E^{1} ≅ T_{s}L_{n−1}S^{−n}H_{∗}(X) = P_{n−1}S^{−n}H_{∗}(X).

§5. The second term of the spectral sequence of the homology of the iterated loop spaces

To determine the second term of the spectral sequence of the homology of the iterated loop spaces firstly we define the notion of a P_{n}-algebra generalizing the notion of an n-Poisson algebra. To simplify constructions we will consider Z/2-coefficients.

A graded module M (over Z/2) will be called a P_{n}-algebra if

1. There is given a structure of a commutative algebra

x ⊗ y ⟼ x · y, x, y ∈ M.

2. There is given a structure of an n-Lie algebra

x ⊗ y ⟼ [x, y], x, y ∈ M,

and the n-Lie algebra structure with the commutative algebra structure form an n-Poisson algebra structure.

3. There are given operations

e_{i}: M → M, 0 ≤ i ≤ n, dim(e_{i}(x)) = 2 · dim(x) + i,

and the following relations are satisfied

e_{0}(x) = x · x,

e_{i}(x · y) = ∑_{k} e_{k}(x) · e_{i−k}(y),

[e_{i}(x), y] = 0, i < n,
[e_{n}(x), y] = [x, [x, y]],

e_{i}(e_{j}(x)) = ∑_{k} \binom{k − j − 1}{i − k − 1}e_{i+2(j−k)}(e_{k}(x)), i > j.

Note that instead of the operations e_{i}: M → M we can consider the cup-i-products

∪_{i}: M ⊗ M → M

defined on the generators x, y ∈ M by the formula

x ∪_{i} y = \{ \begin{matrix}0, & x ≠ y, \\ e_{i}(x), & x = y. \end{matrix}

Denote by P_{n} the monad which corresponds to the graded module M the free P_{n}- algebra generated by M. It is easy to see that there is the isomorphism P_{n}(M) ≅ T_{s}E_{n}L_{n}M and so we have

Theorem 3””. The first term of the considered spectral sequence of H_{∗}(Ω^{n}X) (over Z/2) is isomorphic to the free P_{n−1}-algebra generated by S^{−n}H_{∗}(X), i.e.

E^{1} ≅ P_{n−1}S^{−n}H_{∗}(X).

A chain complex M (over Z/2) will be called a differential P_{n}-algebra if considered as graded module it is a P_{n}-algebra and the differential d satisfies the following relations

d(x · y) = d(x) · y + x · d(y) + d(x) ∪_{1} d(y),
d[x, y] = [d(x), y] + [x, d(y)],
d(e_{i}(x)) = e_{i+1}(d(x)), i < n,
d(e_{n}(x)) = [d(x), x].

Denote by \overline{E}_{∗} the comonad in the category of graded modules which corresponds to a graded module M the graded module H_{∗}(\overline{E}(M)). The E-coalgebra structure on the chain complex C_{∗}(X) of a topological space X induces on its homology H_{∗}(X) the \overline{E}_{∗}-coalgebra structure. This structure consists of the commutative coalgebra structure

∇_{∗}: H_{∗}(X) → H_{∗}(X) ⊗ H_{∗}(X)

and of the action of the Steenrod algebra

A ⊗ H_{∗}(X) → H_{∗}(X).

For an n-connected topological space X these structures on its homology H_{∗}(X) induce on P_{n−1}S^{−n}H_{∗}(X) the differential d_{φ} defined on the generators s^{−n}x_{i+n} ∈ H_{i+n}(X) by the formula

d_{φ}(s^{−n}x_{i+n}) = ∑_{x′<x′′}[s^{−n}x′, s^{−n}x′′] + ∑_{{k \atop i−2k≤n}} e_{i−2k−1}(s^{−n}Sq^{i−k}(x_{i+n})),

where ∑ x′ ⊗ x′′ = ∇_{∗}(x_{i+n}).

Denote the corresponding differential P_{n−1}-algebra as P_{n−1φ}S^{−n}H_{∗}(X). Then we will have

Theorem 4. For the second term of the considered spectral sequence of H_{∗}(Ω^{n}X) (over Z/2) there is the isomorphism

E^{2} ≅ H_{∗}(P_{n−1φ}S^{−n}H_{∗}(X)).

Similar theorem may be obtained for Z/p-coefficients.

Note that in the case of characteristic zero coefficients the \overline{E}_{∗}-coalgebra structure on the homology H_{∗}(X) of a topological space X consists of only the commutative coalgebra structure

∇_{∗}: H_{∗}(X) → H_{∗}(X) ⊗ H_{∗}(X).

Hence we have

Theorem 4’. For the second term of the considered spectral sequence of H_{∗}(Ω^{n}X) (over a field of characteristic zero) there is the isomorphism

E^{2} ≅ H_{∗}(P_{n−1φ}S^{−n}H_{∗}(X)),

where the differential d_{φ} defined on the generators s^{−n}x_{i+n} ∈ H_{i+n}(X) by the formula

d_{φ}(s^{−n}x_{i+n}) = ∑_{x′<x′′}[s^{−n}x′, s^{−n}x′′],

where ∑ x′ ⊗ x′′ = ∇_{∗}(x_{i+n}).

§6. The homology of iterated loop spaces over the real projective spaces

Ones of the most important spaces in Algebraic Topology besides spheres are the real projective spaces RP^{n} and RP^{∞}. In some sense these spaces are opposite to spheres. The homology H_{∗}(RP^{∞}) (over Z/2) is the free \overline{E}_{∗}-coalgebra with one 1-dimensional generator, inspite of spheres that homology are trivial \overline{E}_{∗}-coalgebras.

From the other side the Adams spectral sequence of stable homotopy groups of RP^{∞} is very similar to the corresponding spectral sequence of spheres, and it is the problem to find the relations between them.

Here we consider the problem of calculation the homology

H_{∗}(Ω^{m}(RP^{∞}/RP^{n})), m ≤ n,

of the iterated loop spaces over the space RP^{∞}/RP^{n} with Z/2 coefficients.

Denote the i-dimensional generator of H_{∗}(RP^{∞}) as e_{i}. The coalgebra structure on this generators is determined by the formula

∇(e_{i}) = ∑_{k} e_{k} ⊗ e_{i−k}.

The action of the Steenrod algebra is determined by the formula

Sq^{j}(e_{i}) = \binom{i − j}{j}e_{i−j}.

Since H_{∗}(RP^{∞}) is the free \overline{E}_{∗}-coalgebra there are no higher \overline{E}_{∗}-operations. H_{∗}(RP^{n}) is the \overline{E}_{∗}-subcoalgebra of H_{∗}(RP^{∞}) and hence the \overline{E}_{∗}-coalgebra structure on H_{∗}(RP^{∞}) induces the \overline{E}_{∗}-coalgebra structure on H_{∗}(RP^{∞})/H_{∗}(RP^{n}) ≅ H_{∗}(RP^{∞}/RP^{n}). Therefore on the homology H_{∗}(RP^{∞}/RP^{n}) there are no higher \overline{E}_{∗}-operations. So the spectral sequence for the homology of the iterated loop spaces over RP^{∞}/RP^{n} has no higher differential and we have

Theorem 5. The homology H_{∗}(Ω^{m}(RP^{∞}/RP^{n})), n ≥ m (over Z/2) is isomorphic to the differential P_{m−1}-algebra P_{m−1φ}S^{−m}H_{∗}(RP^{∞}/RP^{n}), where the differential d_{φ} on the generators u_{i} = s^{−m}e_{i+m} is defined by the formula

d_{φ}(u_{i}) = ∑_{{k \atop i−2k>m}}[u_{k}, u_{i−k−m}] + ∑_{{k \atop i−2k≤m}} \binom{k + m}{i − k}e_{i−2k−1}(u_{k}).

Note that in the case m = 1 we have the isomorphisms

P_{0}S^{−1}H_{∗}(RP^{∞}/RP^{n}) ≅ T_{s}LS^{−1}H_{∗}(RP^{∞}/RP^{n}) ≅ T S^{−1}H_{∗}(RP^{∞}/RP^{n}),

and the differential in the tensor algebra T S^{−1}H_{∗}(RP^{∞}/RP^{n}) has very simple form

d_{φ}(u_{i}) = ∑_{{k≥n \atop i−k−1≥n}} u_{k} ⊗ u_{i−k−1}.

From here it follows that the homology H_{∗}(Ω(RP^{∞}/RP^{n})) (over Z/2) is isomorphic to the algebra generated by the elements u_{i}, n ≤ i ≤ 2n, of dimensions i and relations

u_{n} · u_{n} = 0;
u_{n} · u_{n+1} + u_{n+1} · u_{n} = 0;
…
u_{n} · u_{2n} + ⋯ + u_{2n} · u_{n} = 0.

So as graded module the homology H_{∗}(Ω(RP^{∞}/RP^{n})) is generated by the noncommutative products u_{n_{1}} · . . . · u_{n_{k}} with n ≤ n_{1} ≤ 2n, n < n_{2}, …, n_{k} ≤ 2n.

Consider the homology H_{∗}(Ω^{2}(RP^{∞}/RP^{2})). It is isomorphic to the homology of the differential P_{1}- algebra P_{1φ}{u_{i}| i ≥ 1}, where the differential is determined by the formulas

d_{φ}(u_{2i+1}) = ∑_{k<i}[u_{k}, u_{2i−k−1}] + \binom{i + 2}{i + 1}e_{0}(u_{i});
d_{φ}(u_{2i+2}) = ∑_{k<i}[u_{k}, u_{2i−k}] + e_{1}(u_{i}).

In small dimensions we will have d(u_{1}) = 0; d(u_{2}) = 0; d(u_{3}) = u_{1}u_{1}; d(u_{4}) = e_{1}(u_{1}); d(u_{5}) = [u_{1}, u_{2}]; d(u_{6}) = [u_{1}, u_{3}] + e_{1}(u_{2}); d(u_{7}) = [u_{1}, u_{4}] + [u_{2}, u_{3}] + u_{3}u_{3}; d(u_{8}) = [u_{1}, u_{5}] + [u_{2}, u_{4}] + e_{1}(u_{3}).

From these formulas it follows that the homology H_{i} = H_{i}(Ω^{2}(RP^{∞}/RP^{2})) in small dimensions i has the following generators

H_{1} : u_{1}; H_{2} : u_{2}; H_{3} : u_{1}u_{2}; H_{4} : u_{2}^{2}; H_{5} : u_{1}u_{2}^{2}, e_{1}(u_{2}); H_{6} : u_{1}e_{1}(u_{2}), u_{2}^{3}, [u_{2}, u_{3}], u_{3}^{2}; H_{7} : u_{1}u_{2}^{3}, u_{1}u_{3}^{2}, u_{1}[u_{2}, u_{3}], u_{2}e_{1}(u_{2}), e_{1}(u_{3}).

Consider the homology H_{∗}(Ω^{3}(RP^{∞}/RP^{3})). It is isomorphic to the homology of the differential P_{2}- algebra P_{2φ}{u_{i}| i ≥ 1}, where the differential is determined by the formulas

d_{φ}(u_{2i+1}) = ∑_{k<i−1}[u_{k}, u_{2i−k−2}] + \binom{i + 3}{i + 1}e_{0}(u_{i}) + e_{2}(u_{i−1});

d_{φ}(u_{2i+2}) = ∑_{k≤i−1}[u_{k}, u_{2i−k−1}] + \binom{i + 3}{i + 2}e_{1}(u_{i}).

In small dimensions we will have d(u_{1}) = 0; d(u_{2}) = 0; d(u_{3}) = 0; d(u_{4}) = 0; d(u_{5}) = e_{2}(u_{1}); d(u_{6}) = [u_{1}, u_{2}] + e_{1}(u_{2}); d(u_{7}) = [u_{1}, u_{3}] + e_{2}(u_{2}) + u_{3}u_{3}; d(u_{8}) = [u_{1}, u_{4}] + [u_{2}, u_{3}].

From these formulas it follows that the homology H_{i} = H_{i}(Ω^{3}(RP^{∞}/RP^{3})) in small dimensions i has the following generators

H_{1} : u_{1}; H_{2} : u_{1}^{2}, u_{2}; H_{3} : u_{1}^{3}, u_{1}u_{2}, e_{1}(u_{1}), u_{3}; H_{4} : u_{1}^{4}, u_{1}^{2}u_{2}, u_{1}e_{1}(u_{1}), u_{1}u_{3}, u_{2}^{2}, u_{4}; H_{5} : u_{1}^{5}, u_{1}^{3}u_{2}, u_{1}^{2}u_{3}, u_{1}^{2}e_{1}(u_{1}), u_{1}u_{2}^{2}, u_{1}u_{4}, u_{2}u_{3}, u_{2}e_{1}(u_{1}), e_{1}(u_{2}); H_{6} : u_{1}^{6}, u_{1}^{4}u_{2}, u_{1}^{3}u_{3}, u_{1}^{3}e_{1}(u_{1}), u_{1}^{2}u_{2}^{2}, u_{1}^{2}u_{4}, u_{1}u_{2}u_{3}, u_{1}u_{2}e_{1}(u_{1}), u_{1}e_{1}(u_{2}), u_{2}u_{4}, u_{2}^{3}, u_{3}e_{1}(u_{1}), u_{3}^{2}, e_{2}(u_{2}), e_{1}(u_{1})^{2}; H_{7} : u_{1}^{7}, u_{1}^{5}u_{2}, u_{1}^{4}u_{3}, u_{1}^{4}e_{1}(u_{1}), u_{1}^{3}u_{2}^{2}, u_{1}^{3}u_{4}, u_{1}^{2}u_{2}u_{3}, u_{1}^{2}u_{2}e_{1}(u_{1}), u_{1}^{2}e_{1}(u_{2}), u_{1}u_{2}u_{4}, u_{1}u_{2}^{3}, u_{1}u_{3}e_{1}(u_{1}), u_{1}u_{3}^{2}, u_{1}e_{2}(u_{2}), u_{1}e_{1}(u_{1})^{2}, u_{2}^{2}u_{3}, u_{2}^{2}e_{1}(u_{1}), u_{2}e_{1}(u_{2}), u_{3}u_{4}, [u_{2}, u_{3}], e_{1}(u_{3}), e_{1}e_{1}(u_{1}).

Consider also the homology H_{∗}(Ω^{4}(RP^{∞}/RP^{4})). It is isomorphic to the homology of the differential P_{3}- algebra P_{3φ}{u_{i}| i ≥ 1}, where the differential is determined by the formulas

d_{φ}(u_{2i+1}) = ∑_{k<i−1}[u_{k}, u_{2i−k−3}] + \binom{i + 4}{i + 1}e_{0}(u_{i}) + \binom{i + 3}{i + 2}e_{2}(u_{i−1});

d_{φ}(u_{2i+2}) = ∑_{k<i−1}[u_{k}, u_{2i−k−2}] + \binom{i + 4}{i + 2}e_{1}(u_{i}) + e_{3}(u_{i−1}).

In small dimensions we will have d(u_{1}) = 0; d(u_{2}) = 0; d(u_{3}) = 0; d(u_{4}) = 0; d(u_{5}) = e_{2}(u_{1}); d(u_{6}) = e_{1}(u_{2}) + e_{3}(u_{1}); d(u_{7}) = [u_{1}, u_{2}] + u_{3}u_{3}; d(u_{8}) = [u_{1}, u_{3}] + e_{1}(u_{3}) + e_{3}(u_{2}).

From these formulas it follows that the homology H_{i} = H_{i}(Ω^{4}(RP^{∞}/RP^{4})) in small dimensions i has the following generators

H_{1} : u_{1}; H_{2} : u_{1}^{2}, u_{2}; H_{3} : u_{1}^{3}, u_{1}u_{2}, e_{1}(u_{1}), u_{3}; H_{4} : u_{1}^{4}, u_{1}^{2}u_{2}, u_{1}e_{1}(u_{1}), u_{1}u_{3}, u_{2}^{2}, u_{4}; H_{5} : u_{1}^{5}, u_{1}^{3}u_{2}, u_{1}^{2}u_{3}, u_{1}^{2}e_{1}(u_{1}), u_{1}u_{2}^{2}, u_{1}u_{4}, u_{2}u_{3}, u_{2}e_{1}(u_{1}), e_{1}(u_{2}); H_{6} : u_{1}^{6}, u_{1}^{4}u_{2}, u_{1}^{3}u_{3}, u_{1}^{3}e_{1}(u_{1}), u_{1}^{2}u_{2}^{2}, u_{1}^{2}u_{4}, u_{1}u_{2}u_{3}, u_{1}u_{2}e_{1}(u_{1}), u_{1}e_{1}(u_{2}), u_{2}^{3}, u_{2}u_{4}, u_{3}e_{1}(u_{1}), u_{3}^{2}, e_{2}(u_{2}), e_{1}(u_{1})^{2}; H_{7} : u_{1}^{7}, u_{1}^{5}u_{2}, u_{1}^{4}u_{3}, u_{1}^{4}e_{1}(u_{1}), u_{1}^{3}u_{2}^{2}, u_{1}^{3}u_{4}, u_{1}^{2}u_{2}u_{3}, u_{1}^{2}u_{2}e_{1}(u_{1}), u_{1}^{2}e_{1}(u_{2}), u_{1}u_{2}^{3}, u_{1}u_{2}u_{4}, u_{1}u_{3}e_{1}(u_{1}), u_{1}u_{3}^{2}, u_{1}e_{2}(u_{2}), u_{1}e_{1}(u_{1})^{2}, u_{2}^{2}u_{3}, u_{2}^{2}e_{1}(u_{1}), u_{2}e_{1}(u_{2}), u_{3}u_{4}, e_{1}(u_{3}), e_{3}(u_{2}), e_{1}e_{1}(u_{1}).

§7. Appendix (F. Sergeraert) Computations with the Kenzo program

Not yet written.

REFERENCES

1. J.F.Adams. On the cobar construction. Proc. Nat. Acad. Sci. 42(1956), 409-412.

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