\input amstex
\magnification=1200
\centerline{THE HOMOLOGY OF ITERATED LOOP SPACES}
\vskip .5cm
\centerline{V.A.Smirnov}
\centerline{\font\small=cmti8 \small
            with a computational appendix by F. Sergeraert}
\vskip 6pt
\document
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
$H_*(\Omega X)$ of the loop space $\Omega X$ over a topological space
$X$ he introduced the notion of the cobar construction $FK$ over a
coalgebra $K$, [1].

Recall that a chain complex $K$ is called a coalgebra if there is given
a chain mapping $\nabla\colon K\to K\otimes K$ satisfying the
associativity relation 
$$(\nabla\otimes 1)\nabla=(1\otimes\nabla)\nabla.$$ 

The cobar construction $FK$ over a coalgebra $K$ is by the definition
a differential algebra which as a graded algebra coincides with the
tensor algebra $TS^{-1}K$ over the desuspension $S^{-1}K$ over $K$. The
generator elements in $FK$ usually denoted as $[x_1,\dots,x_n]$ where
$x_i\in K$, $1\le i\le n$. The product is defined by the formula
$$\pi([x_1,\dots,x_n]\otimes
[x_{n+1},\dots,x_{n+m}])=[x_1,\dots,x_{n+m}].$$
The differential on the generators $[x]\in FK$ is defined by the
formula
$$d[x]=-[d(x)]+\sum(-1)^\epsilon [x',x''],$$
where $\nabla(x)=\sum x'\otimes x''$, $\epsilon =dim(x')$.

On the chain complex $C_*(X)$ of a
topological space $X$ there exist a natural coalgebra structure
determined on simplexes $x$ with vertixes $0,1,\dots,n$ by the formula
$$\nabla(x(0,1,\dots,n))=\sum_ix(0,\dots,i)\otimes x(i,\dots,n).$$
So there is defined the cobar construction $FC_*(X)$.
 
J.F.Adams proved that for a simply connected topological space $X$
there is a chain equivalence of differential algebras
$$C_*(\Omega X)\simeq FC_*(X).$$
 
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 $C_*(\Omega X)$ will be chain
equivalent to the tensor algebra $TC_*(X)$ 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 $FK$ 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 $FC_*(X)$. This structure was determined using the family
of operations $$\nabla_{n,m}\colon C_*(X)\to C_*(X)^{\otimes n}\otimes
C_*(X)^{\otimes m}$$ of dimensions $n+m-1$. So he defined the double
cobar construction $F^2C_*(X)$ and proved that for $2$-connected
topological space $X$ there is a chain equivalence $$C_*(\Omega^2X)
\simeq F^2C_*(X).$$
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 $H_*(\Omega^nS^nX)$, [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
$E_\infty$-coalgebra structure which determines the weak homotopy type
of $X$. Using this structure the chain complex $C_*(\Omega^nX)$ of the
$n$-fold loop space $\Omega^nX$ 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 $G^nX$. 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.

\vskip .5cm
\centerline{\S 1. Operads and algebras over operads}
\vskip 6pt
Consider the category of chain complexes over a ring $R$. By a
symmetric family $\Cal E$ in this category is meant a family $\Cal
E=\{\Cal E(j)\}_{j\ge 1}$ of chain complexes $\Cal E(j)$ operated on
by the symmetric groups $\Sigma_j$.

Given two symmetric families $\Cal E'$, $\Cal E''$ we define the
symmetric families $\Cal E'\otimes\Cal E''$, $\Cal E'\times\Cal E''$
by putting $$(\Cal E'\otimes\Cal E'')(j)=\Cal E'(j)\otimes\Cal
E''(j),$$ and $(\Cal E'\times\Cal E'')(j)$ equal to the quotiont module
of the free $\Sigma_j$-module generated by the module
$$\sum_{j_1+\dots+j_k=j}\Cal E'(k)\otimes\Cal E''(j_1)\otimes\dots
\otimes\Cal E''(j_k)$$
with respect to the equivalence generated by the relations
$$\gather
x'\sigma\otimes x''_1\otimes \dots\otimes x''_k\sim x'\otimes 
x''_{\sigma^{-1}(1)}\otimes\dots\otimes x''_{\sigma^{-1}(k)}\cdot 
\sigma(j_1,\dots,j_k),\\
x'\otimes x''_1\sigma_1\otimes\dots\otimes x''_k\sigma_k\sim
x'\otimes x''_1\otimes\dots\otimes x''_k\cdot\sigma_1\times\dots
\times\sigma_k,\endgather $$
where $\sigma(j_1,\dots,j_k)$ is the permutation of a set of $j$ 
elements obtained by partitioning the set into $k$ blocks of
$j_1,\dots,j_k$ elements, respectively, and carrying out on these
blocks the permutation $\sigma$, while $\sigma_1\times\dots\times
\sigma_k$ means the image of the element $(\sigma_1,\dots,\sigma_k)$
under the imbedding $\Sigma_{j_1}\times\dots\times\Sigma_{j_k}\to
\Sigma_j$.

It is easy to see that for the symmetric families $\Cal E',\Cal E'',
\Cal F',\Cal E''$ there is the interchanging mapping
$$T\colon (\Cal E'\otimes\Cal E'')\times(\Cal F'\otimes\Cal E'')\to
(\Cal E'\times\Cal F')\otimes(\Cal E''\times\Cal F'').$$

A symmetric family $\Cal E$ is called an operad if there is given
a symmetric-family mapping $\gamma\colon\Cal E\times\Cal E\to
\Cal E$ such that $\gamma(\gamma\times 1)=\gamma(1\times\gamma)$, or
that is the same, the following diagram is commutative
$$\CD
\Cal E\times\Cal E\times\Cal E@>\gamma\times 1>>\Cal E\times\Cal E\\
@V1\times\gamma VV @VV\gamma V\\
\Cal E\times\Cal E@>\gamma >>\Cal E\endCD $$
If there exist an element $1\in\Cal E(1)$ such that $\gamma(1\otimes
x)=\gamma(x\otimes 1^{\otimes k})=x$ for all $x\in\Cal E(k)$ then
we say that $\Cal E$ is an operad with identity.

An operad $\Cal E$ is called a Hopf operad if besides the operation
$\gamma\colon\Cal E\times\Cal E\to\Cal E$ there is given an
operation $\nabla\colon\Cal E\to\Cal E\otimes\Cal E$ which is
associative and satisfy the Hopf relation, i.e. the following
diagram is commutative
$$\CD
\Cal E\times\Cal E@>\nabla\times\nabla >> (\Cal E\otimes\Cal E)
\times(\Cal E\otimes\Cal E)@>T>>(\Cal E\times\Cal E)\otimes
(\Cal E\times \Cal E)\\
@V\gamma VV @. @VV\gamma\otimes\gamma V\\
\Cal E@>\nabla >>\Cal E\otimes\Cal E@>=>>\Cal E\otimes\Cal E
\endCD $$

A mapping of operads $f\colon\Cal E'\to\Cal E''$ is a mapping of 
symmetric families for which the folowing diagram is commutative
$$\CD \Cal E'\times\Cal E'@>\gamma'>>\Cal E'\\
@Vf\times fVV @VVfV\\
\Cal E''\times\Cal E''@>\gamma''>>\Cal E''\endCD $$
If $\Cal E'$ and $\Cal E''$ are operads with identities $1'$ and $1''$
respectively, then it is required that $f(1')=1''$.

We shall say that an operad $\Cal E$ acts on a symmetric family $\Cal
F$ on the left (right) if there is given a mapping $\mu'\colon
\Cal E\times\Cal F\to\Cal F$ ($\mu''\colon\Cal F\times\Cal E\to\Cal
F$) such that $$\mu'(\gamma\times 1)=\mu'(1\times\mu')\quad
(\mu''(1\times\gamma)=\mu''(\mu''\times 1)),$$
or that is the same, the following diagrams are commutative
$$\CD
\Cal E\times\Cal E\times\Cal F@>\gamma\times 1>>\Cal E\times\Cal F
\\@V1\times\mu'VV @VV\mu'V\\
\Cal E\times\Cal F@>\mu'>>\Cal F\endCD \qquad
\left(\CD
\Cal F\times\Cal E\times\Cal E@>1\times\gamma >>\Cal F\times\Cal E\\
@V\mu''\times 1VV @VV\mu''V\\
\Cal F\times\Cal E@>\mu''>>\Cal F\endCD\right) $$

For any chain complex $X$ and symmetric family $\Cal E$ we define
chain complexes $\Cal E(X)$, $\overline{\Cal E}(X)$ by putting
$$\Cal E(X)=\sum_k\Cal E(k)\otimes_{\Sigma_k}X^{\otimes k},
\quad\overline{\Cal E}(X)=\prod_kHom_{\Sigma_k}(\Cal E(k);X^{\otimes
k}).$$
If $\Cal E$ is an operad, then the operad structure in $\Cal E$
determines a mapping 
$$\gamma\colon\Cal E^2(X)=\Cal E(\Cal E(X))\to\Cal E(X)$$
such that the correspondence $X\longmapsto \Cal E(X)$ is a monad 
in the category of chain complexes.

Dually, the operad structure in $\Cal E$ determines a mapping
$$\overline\gamma\colon\overline{\Cal E}(X)\to\overline{\Cal E}^2
(X)=\overline{\Cal E}(\overline{\Cal E}(X))$$
such that the correspondence $X\longmapsto \overline{\Cal E}(X)$
is a comonad in the category of chain complexes.

A chain complex $X$ is called an algebra over an operad $\Cal E$,
or simply $\Cal E$-agebra, if there is given a mapping $\mu\colon
\Cal E(X)\to X$ satisfying the associativity relation:
$\mu\circ\gamma(X)=\mu\circ\Cal E(\mu)$, or that is the same,
the following diagram is commutative
$$\CD \Cal E^2(X)@>\gamma(X)>>\Cal E(X)\\
@V\Cal E(\mu)VV @VV\mu V\\
\Cal E(X)@>\mu >>X\endCD $$ 

Dually, a chain complex $X$ is called a coalgebra over an operad
$\Cal E$, or simply $\Cal E$-coalgebra, if there is given a mapping
$\tau\colon X\to\overline{\Cal E}(X)$ satisfying the associativity
relation: $\overline\gamma(X)\circ\tau=\overline{\Cal E}(\tau)
\circ\tau$, or that is the same, the following diagram is commutative
$$\CD X@>\tau >>\overline{\Cal E}(X)\\
@V\tau VV @VV\overline\gamma(X)V\\
\overline{\Cal E}(X)@>\overline{\Cal E}(\tau)>>\overline{\Cal E}^2(X)
\endCD $$

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 $\Sigma_j$. So $E_0(j)\cong R$. The operation 
$\gamma\colon E_0\times E_0\to E_0$ is defined by the formula
$$\gamma(e(k)\otimes e(j_1)\otimes\dots\otimes e(j_k))=
e(j_1+\dots+j_k).$$
The operation $\nabla\colon E_0\to E_0\otimes E_0$ is defined by the
formula
$$\nabla(e(j))=e(j)\otimes e(j).$$
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
$\Sigma_j$ - module with one zero dimensional generator $a(j)$.
So $A(j)\cong R(\Sigma_j)$.
The operation $\gamma\colon A\times A\to A$ is defined by the
formula
$$\gamma(a(k)\otimes a(j_1)\otimes\dots\otimes a(j_k))=
a(j_1+\dots+j_k).$$
The operation $\nabla\colon A\to A\otimes A$ is defined by the
formula $$\nabla(a(j))=a(j)\otimes a(j).$$
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 $\Cal E$ define the suspension $S\Cal E$
by putting $(S\Cal E)(j)=S^{j-1}\Cal E(j)$ - $(j-1)$-fold suspension
over $\Cal E(j)$. It is clear that if $\Cal E$ is an operad then the
suspension $S\Cal E$ will be also an operad, and if $X$ is an algebra
(coalgebra) over an operad $\Cal E$ then the suspension $SX$ will be
an algebra (coalgebra) over the operad $S\Cal E$.

4. For operads $\Cal E'$, $\Cal E''$ its tensor product $\Cal E'
\otimes \Cal E''$ evidently is an operad.

5. Let $L$ be a suboperad of the operad $A$ generated by the element
$$b(2)=a(2)-a(2)T,\quad T\in\Sigma_2.$$ Then algebras (coalgebras) over
the operad $L$ will be simply Lie algebras (coalgebras).

Similary, let $L_n$ be a suboperad of the operad $S^nA$ generated by
the element $$b_n(2)=s^na(2)+(-1)^ns^na(2)T.$$ Then algebras
(coalgebras) over the operad $L_n$ will be simply $n$-Lie algebras
(coalgebras).

6. Let $P_n=E_0\times L_n$ and the operad structure $\gamma$ is
determined by the corresponding structures in $E_0$, $L_n$ and
by the formulas
$$\gamma(b_n(2)\otimes 1\otimes e(2))=e(2)\otimes b_n(2)\otimes 1
+e(2)\otimes 1\otimes b_n(2)\cdot (213).$$
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 $\Cal E_X$, $\Cal E^X$
by putting 
$$\Cal E_X(j)=Hom(X^{\otimes j};X);\quad
\Cal E^X(j)=Hom(X;X^{\otimes j}).$$
The actions of the symmetric groups are determined by the permutations
of factors of $X^{\otimes j}$ and operad structures are defined by
the formulas
$$\gather \gamma_X(f\otimes g_1\otimes\dots\otimes g_k)=
f\circ(g_1\otimes\dots\otimes g_k),\quad f\in\Cal E_X(k),~
g_i\in\Cal E_X(j_i);\\
\gamma^X(f\otimes g_1\otimes\dots\otimes g_k)=
(g_1\otimes\dots\otimes g_k)\circ f,\quad f\in \Cal E^X(k),~
g_i\in\Cal E^X(j_i).\endgather$$
Directly from the definition it follows that a chain complex $X$
is an algebra (coalgebra) over an operad $\Cal E$ if and only if
there is given an operad mapping $\xi\colon\Cal E\to\Cal E_X$
($\xi\colon \Cal E\to\Cal E^X$).

Analogously for chain complexes $X$ and $Y$ there are defined 
symmetric families $\Cal F_{X,Y}$, $\Cal F^{X,Y}$:
$$\Cal F_{X,Y}(j)=Hom(X^{\otimes j};Y),\quad
\Cal F^{X,Y}(j)=Hom(X;Y^{\otimes j})$$
and actions
$$\gather \mu'\colon\Cal E_Y\times\Cal F_{X,Y}\to\Cal F_{X,Y},~
\mu''\colon\Cal F_{X,Y}\times\Cal E_X\to\Cal F_{X,Y};\\
\mu'\colon\Cal E^X\times\Cal F^{X,Y}\to\Cal F^{X,Y},~
\mu''\colon\Cal F^{X,Y}\times\Cal E^Y\to\Cal F^{X,Y}.\endgather $$
 
8. One of the most important topological operad is the Bordman and
Vogt's ``little $n$-cubes'' 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
$f\colon J^n\to J^n$ with parallel axes.
Then $E_n(j)$ is the set of ordered $j$-tuples $(f^1,\dots,f^j)$ of
$n$-dimensional little cubes $f^i\colon J^n\to J^n$ that images
don't intersect.
This operad acts on the $n$-fold loop space $\Omega^nX$ over a space $X$.

The direct limite of the operads $E_n$ over the inclusions $E_n
\subset E_{n+1}$ denotes as $E_\infty$. It is acyclic operad
with the free actions of the symmetric groups.

9. It is easy to see that if $\Cal E$ is a topological operad
then it's singular chain complex $C_*(\Cal E)$ is an operad in
the category of chain complexes, and if $\Cal E$ acts on a spaces
$X$ then $C_*(\Cal E)$  acts on $C_*(X)$.

Similary, the homology $H_*(\Cal E)$ of a topological operad $\Cal E$
is an operad in the category of graded modules, and if $\Cal E$ acts
on a space $X$ then $H_*(\Cal E)$ 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 $H_*(\Omega^nX)$ of the $n$-fold loop space $\Omega^nX$
is an algebra over the Poisson operad $P_n$.

The operad $C_*(E_\infty)$ 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 $\Sigma_j$-homotopy
equivalent chain complexes. We will call such operads as 
$E_\infty$-operads.

10. Another example of $E_\infty $-operad give us the simplicial
resolutions of the symmetric groups. Denote by $E\Sigma_*(j)$ the free
simplicial resolution of the symmetric group $\Sigma_j$, i.e.
$$E\Sigma_*(j):\Sigma_j@<<<\Sigma_j\times\Sigma_j@<<<\dots $$
The mappings $\gamma\colon\Sigma_k\times\Sigma_{j_1}\times\dots
\times\Sigma_{j_k}\to\Sigma_{j_1+\dots+j_k}$ induce the operad
structure 
$$\gamma_*\colon E\Sigma_*(k)\times E\Sigma_*(j_1)\times\dots\times 
E\Sigma_*(j_k)\to E\Sigma_*(j_1+\dots+j_k).$$
So $E\Sigma_*$ will be the acyclic operad with the free actions of the
symmetric groups in the category of simplicial sets.

Taking the chain complex $C_*(E\Sigma_*)$ we obtain the $E_\infty$-operad
in the category of chain complexes. Denote it simply as $E\Sigma$.

Note that for any chain operad $\Cal E$ the operad $\Cal E\otimes
E\Sigma$ will has the same homology and the free actions of the
symmetric groups. The projection $E\Sigma\to E_0$ induces the
projection $\Cal E\otimes E\Sigma\to \Cal E$. So $\Cal E\otimes
E\Sigma$ may be considered as $\Sigma$-free resolution of the operad
$\Cal E$. If $\Cal E$ is an acyclic operad then $\Cal E\otimes E\Sigma$
will be $E_\infty$-operad.
 
\vskip .5cm
\centerline{\S 2. On the chain complex of a topological space}
\vskip 6pt
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 $$\nabla\colon C_*(X)\to C_*(X)
\otimes C_*(X)$$ on the chain complex of a topological space there are
coproducts $$\nabla_i\colon C_*(X)\to C_*(X)\otimes C_*(X)$$
increasing dimensions by $i$ and such that $$d(\nabla_i)=\nabla_{i-1}
+(-1)^iT\nabla_{i-1},$$ where $T\colon C_*(X)\otimes C_*(X)\to
C_*(X)\otimes C_*(X)$ permutes factors.

Dually, on the cochain complex $C^*(X)$ besides an algebra structure
$$\cup\colon C^*(X)\otimes C^*(X)\to C^*(X)$$ there are products
$$\cup_i\colon C^*(X)\otimes C^*(X)\to C^*(X)$$ such that
$d(\cup_i)=\cup_{i-1}+(-1)^i\cup_{i-1}T$.

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 $n\ge 0$ denote by $\Delta^n$ the normalized chain complex of the
standard $n$-dimensional simplex. Then $\Delta^*=\{\Delta^n\}$ is the
cosimplicial object in the category of chain complexes. Consider also
the cosimplisial object $(\Delta^*)^{\otimes j}=\Delta^*\otimes\dots
\otimes\Delta^*$ and its realization 
$$E^\Delta(j)=Hom(\Delta^*;(\Delta^*)^{\otimes j}),$$
where $Hom$ is considered in the category of cosimplicial objects.

The family $E^\Delta=\{E^\Delta(j)\}$ 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 $\Cal E^X$, where instead of $X$ we take $\Delta^*$.

Note that since the complexes $\Delta^n$ are acyclic then the operad
$E^\Delta$ will be also acyclic.

{\bf Theorem 1.} {\sl On the chain complex $C_*(X)$ of a topological
space $X$ there exists a natural coalgebra structure $\tau\colon
C_*(X)\to\overline E^\Delta(C_*(X))$ with the following universal
property: if for any operad $\Cal E$ there is a natural 
$\Cal E$ - coalgebra structure $\widetilde\tau\colon C_*(X)\to
\overline{\Cal E}(C_*(X))$, then there exist a unique operad
mapping $\xi\colon\Cal E\to E^\Delta$ such that $\widetilde\tau =
\overline\xi(C_*(X))\circ\tau.$}

{\sl Proof.} Our aim is to define operations 
$$\tau\colon E^\Delta(j)\otimes C_*(X)\to C_*(X)^{\otimes j}.$$
Let $x_n\in C_n(X)$, $y\in E^\Delta(j)=Hom(\Delta^*;(\Delta^*)
^{\otimes j})$. The element $x_n\in C_n(X)$ determines the chain
mapping $\overline x_n\colon\Delta^n\to C_*(X)$ such that the generator
$u_n\in\Delta^n$ maps to $x_n$. 

Note that there is the operation $\tau^n\colon E^\Delta(j)\otimes
\Delta^n\to(\Delta^n)^{\otimes j}$. Define the required operation $\tau$
putting $$\tau(y\otimes x_n)=(\overline x_n)^{\otimes j}\circ\tau^n
(y\otimes u_n).$$
Then we will have the following commutative diagram

$$\CD E^\Delta(j)\otimes C_*(X)@>\tau >> C_*(X)^{\otimes j}\\
@A1\otimes \overline x_nAA @AA(\overline x_n)^{\otimes j}A\\
E^\Delta(j)\otimes\Delta^n @>\tau^n>>(\Delta^n)^{\otimes j}\endCD$$
It is easy to see that the required relations are satisfied.

The $E^\Delta$ - coalgebra structure on the chain complex $C_*(X)$
of a topological space $X$ induces  the $E^\Delta$ - algebra structure
on the cochain complex $C^*(X)=Hom(C_*(X);R)$. The corresponding
operations $\mu\colon E^\Delta(j)\otimes C^*(X)^{\otimes j}\to C^*(X)$
are defined by the formulas
$$\mu(y\otimes f_1\otimes\dots\otimes f_j)(x)=(f_1\otimes\dots\otimes
f_j)\circ\tau(y\otimes x),$$
where $y\in E^\Delta(j)$, $f_i\colon C_*(X)\to R$, $x\in C_*(X)$.
So we have

{\bf Theorem 1'.} {\sl On the cochain complex $C^*(X)$ of a topological
space $X$ there exists a natural algebra structure $\mu\colon
E^\Delta(C^*(X))\to C^*(X)$ with the following universal property:
if fo any operad $\Cal E$ there is a natural $\Cal E$ - algebra
structure $\widetilde\mu\colon\Cal E(C^*(X))\to C^*(X)$, then there
exists a unique operad mapping $\xi\colon\Cal E\to E^\Delta$ such
that $\widetilde\mu=\mu\circ\xi(C^*(X))$.}

Let $R(\Sigma_2)$ - the $\Sigma_2$-free resolution with generator
elements $e_i$ of dimensions $i$ and the differential defined by the
formula $$d(e_i)=e_{i-1}+(-1)^ie_{i-1}T,\quad T\in\Sigma_2.$$

Since $E^\Delta(2)$ is acyclic, there is the $\Sigma_2$-chain mapping
$R(\Sigma_2)\to E^\Delta(2)$ and hence the mapping $$R(\Sigma_2)
\otimes_{\Sigma_2}C^*(X)^{\otimes 2}\to C^*(X).$$ Its restriction on
the elements $e_i$ usually denoted as $$\cup_i\colon C^*(X)\otimes
C^*(X)\to C^*(X)$$ and called cup-$i$ product.

Let $p^n\colon\Delta^n\to S\Delta^{n-1}$ be the projection obtained
by contracting the ($n-1$)-dimensional face spanned by the vertixes
with numbers $0,1,\dots,n-1$. These projections induce a projection
of operads $E^\Delta\to SE^\Delta$. For a topological space
$X$ the suspension $SC_*(X)$ will be a coalgebra over the operad 
$SE^\Delta$ and the following diagram commutes
$$\CD SC_*(X)@>>>\overline{SE}^\Delta(SC_*(X))\\
@VVV @VVV\\
C_*(SX)@>>>\overline E^\Delta(C_*(SX))\endCD $$

Iterating this construction we obtain the projections 
$E^\Delta\to S^nE^\Delta$ and the commutative diagrams
$$\CD S^nC_*(X)@>>>\overline{S^nE}^\Delta(S^nC_*(X))\\
@VVV @VVV\\
C_*(S^nX)@>>>\overline E^\Delta(C_*(S^nX))\endCD $$

Note that the operad $E^\Delta$ may be not $\Sigma$-free and so it
is not an $E_\infty$-operad. To obtain the $E_\infty$-operad we
consider the operad $E^\Delta\otimes E\Sigma$ and denote it simply as 
$E$.

The projection $E\to E^\Delta$ induces the $E$ - coalgebra structure
on the chain complex $C_*(X)$ of a topological space $X$. The mapping
$E^\Delta\to SE^\Delta$ induces the operad mapping  $E\to SE$ and
for the chain complexes $C_*(S^nX)$ of the suspesions $S^nX$ over a
topological space $X$ there are the corresponding commutative
diagrams similar to the diagrams for the operad $E^\Delta$.

\vskip .5cm
\centerline{\S 3. Bar and cobar constructions over operads}
\vskip 6pt
Let $\Cal E$ be an operad with right action on a symmetric family
$\Cal F$, $\nu\colon\Cal F\times\Cal E\to\Cal F$, and let $X$ be
an algebra over the operad $\Cal E$, $\mu\colon\Cal E(X)\to X$.
Consider the simplicial object $B_*(\Cal F,\Cal E,X)=
\{B_n(\Cal F,\Cal E,X)\}$ for which $B_n(\Cal F,\Cal E,X)=
\Cal F\Cal E^n(X)$ with the face and degeneracy operators
given by the formulas
$$\gather d_0=\nu 1^n,~d_i=1^i\gamma 1^{n-i},~0<i<n;\\
d_n=1^n\mu,~s_j=1^{j+1}i1^{n-j+1},0\le j\le n.
\endgather $$
Its realization 
$$B(\Cal F,\Cal E,X)=|B_*(\Cal F,\Cal E,X)|=\sum_n\Delta^n
\otimes B_n(\Cal F,\Cal E,X)/\sim $$
is called the bar construction.

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

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.
$$B(A,X)\simeq S^{-1}BX,~ [8].$$ 

Dually, if $X$ is a coalgebra over the operad $\Cal E$, $\tau
\colon X\to\overline{\Cal E}(X)$, then we can consider the
cosimplicial object $F^*(\Cal F,\Cal E,X)=\{F^n(\Cal F,\Cal E,X)\}$
for which $F^n(\Cal F,\Cal E,X)=\overline{\Cal F}\overline{\Cal E}^n
(X)$ with the coface and codegeneracy operators given by the
formulas
$$\gather \delta^0=\overline\nu 1^n,~\delta^i=1^i\overline\gamma 
1^{n-i},~0<i<n;\\
\delta^n=1^n\tau,~\sigma^j=1^{j+1}p1^{n-j+1},0\le j\le n.
\endgather $$
Its realization 
$$F(\Cal F,\Cal E,X)=|F^*(\Cal F,\Cal E,X)|=Hom(\Delta^*;
F^*(\Cal F,\Cal E,X)),$$ where $Hom$ is considered in the 
category of cosimplicial objects, is called the cobar construction.

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

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.
$$F(A,X)\simeq SFX,~ [8].$$ 

Let now $X$ be an $n$-connected topological space. As it was shown
above on the chain complex $S^nC_*(\Omega^nX)$ there is $S^nE$ -
coalgebra structure 
$$\tau\colon S^nC_*(\Omega^nX)\to\overline{S^nE}(S^nC_*(\Omega^nX)).$$ 
This structure  and the mapping $j\colon
S^n\Omega^nX\to X$ induce the mapping
$$S^nC_*(\Omega^nX)\to\overline{S^nE}(S^nC_*(\Omega^nX))\to
\overline{S^nE}(C_*(X)).$$
This mapping is a coaugmentation of the cosimplicial object
$F^*(S^nE,E,C_*(X))$ and hence it induces the mapping of $S^nE$-
coalgebras
$$S^nC_*(\Omega^nX)\to F(S^nE,E,C_*(X)).$$

{\bf Theorem 2.} {\sl For each $n$-connected topological space $X$
the mapping 
$$S^nC_*(\Omega^nX)\to F(S^nE,E,C_*(X))$$
is a chain equivalence of $S^nE$-coalgebras.}

{\sl Proof.} For $n=1$ the chain equivalence 
$SC_*(\Omega X)\to F(SE,E,C_*(X))$ follows from the Adams chain
equivalence $$SC_*(\Omega X)\simeq SFC_*(X)$$
and the chain equivalence
$$SFC_*(X)\simeq F(SE,E,C_*(X)),~[8].$$
Suppose that for each ($n-1$)-connected topological space $X$ we have
a chain equivalence
$$S^{n-1}C_*(\Omega^{n-1}X)\to F(S^{n-1}E,E,C_*(X)).$$
Then for an $n$-connected topological space $X$ we will have the
following sequence of chain equivalences
$$\gather S^nC_*(\Omega^nX)\simeq SF(S^{n-1}E,E,C_*(\Omega X))
\simeq F(S^nE,SE,SC_*(\Omega X))\simeq \\
F(S^nE,SE,F(SE,E,C_*(X)))\simeq F(S^nE,E,C_*(X)).\endgather$$
Its composition will be the desirable chain equivalence
$$S^nC_*(\Omega^nX)\simeq F(S^nE,E,C_*(X)).$$

\vskip .5cm
\centerline{\S 4. The spectral sequence for the homology of iterated
loop spaces}
\vskip 6pt
Let $X$ be an $E$-coalgebra. Consider the spectral sequence of the
cobar construction $F(S^nE,E,X)$ over the filtration determined by
the operad grading of $E$. Namely, for any operad $\Cal E$ we define
the grading of mappings $f\colon \Cal E(j)\to X^{\otimes j}$
equal to $j$. So we will have the grading of the elements of
$\overline{S^nE}(X)$, $\overline{S^nE}\circ\overline E(X)$ and so on.
Thus we will have the grading of the elements of the cobar
construction $F(S^nE,E,X)$  which determines in it the decreasing 
filtration.

Note that the mappings
$\overline\mu\colon\overline{S^nE}\to\overline{S^nE}\circ\overline E$,
$\overline\gamma\colon\overline E\to\overline E\circ\overline E$
preserve gradings and the mapping $\tau\colon X\to \overline E(X)$
increases it. Therefore the first term of the corresponding spectral 
sequence will be isomorphic  to the homology of the cobar construction
$F(S^nE,E,X)$, 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 $\nabla
\colon FK\to FK\otimes FK$ defined on the generators $[x]\in FK$
by the formula
$$\nabla[x]=[x]\otimes 1+1\otimes [x].$$
 
Note that a commutative coalgebra $K$ may be considered as 
$E$-coalgebra. The required $E$-coalgebra structure is induced by the
projection $E\to E_0$. Moreover, the chain equivalence $SFK\simeq
F(SE,E,K)$ will be the chain equivalence of $SE$-coalgebras.

Thus in this case the Adams cobar construction may be iterated and
there are chain equivalences
$$S^nF^nK\simeq F(S^nE,E,K).$$

In particular, if a chain complex $X$ has the trivial $E$-coalgebra
structure then there are the chain equivalences
$$S^nF^nX\simeq F(S^nE,E,X).$$
So to calculate the homology of the cobar construction $F(S^nE,E,X)$
over a trivial $E$-coalgebra $X$ it is sufficient to calculate the 
homology of the iterated Adams cobar construction $F^nX$.
We will do it for chain complexes over a field.

Denote $X_*=H_*(X)$. The homology $H_*(FX)$ is isomorphic to the
tensor algebra $TS^{-1}X_*$ over the desuspension $S^{-1}X_*$ over
$X_*$, i.e.
$$H_*(FX)\cong TS^{-1}X_*.$$

The double cobar construction $F^2X$ will be chain equivalent to the 
cobar construction $FTS^{-1}X_*$, where $TS^{-1}X_*$ is considered 
as a Hopf algebra with the coproduct 
$$\nabla\colon TS^{-1}X_*\to TS^{-1}X_*\otimes TS^{-1}X_*$$ 
determined on the generators $[x]\in S^{-1}X_*$ by the formula
  $$\nabla[x]=[x]\otimes 1+1\otimes [x].$$

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 $$TS^{-1}X_*\cong T_sLS^{-1}X_*,$$ where 
$T_sLS^{-1}X_*$ denotes the free commutative algebra generated by the 
module $LS^{-1}X_*$. 

The homology of $FT_sLS^{-1}X_*$ is isomorphic to the free commutative
algebra generated by the module $S^{-1}PT_sLS^{-}X_*$, i.e.
$$H_*(F^2X)\cong T_sS^{-1}PT_sLS^{-1}X_*,$$
where $PT_sLS^{-1}X_*$ denotes the module of primitive elements of 
$TS^{-1}X_*$, i.e.
$$PT_sLS^{-1}X_*=\{y\in T_sLS^{-1}X_*|~\nabla(y)=y\otimes 1+1\otimes y\}.$$

So this construction may be iterated and by induction we obtain the 
isomorphisms
$$H_*(F^nX)\cong T_s(S^{-1}PT_s)^{n-1}LS^{-1}X_*,$$
and hence the isomorphisms
$$S^{-n}H_*(F(S^nE,E,X))\cong T_s(S^{-1}PT_s)^{n-1}LS^{-1}X_*.$$
Thus we have 

{\bf Theorem 3.} {\sl If $X$ is an $n$-connected topological space,
then there is the spectral sequence which converges to
$H_*(\Omega^nX)$ and for the first term $E^1$ of this spectral sequence 
there is the isomorphism
$$E^1\cong T_s(S^{-1}PT_s)^{n-1}LS^{-1}H_*(X).$$}

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 $[~,~]\colon L\otimes L\to L$ called
a Lie bracket and satisfying the relations
$$\gather
[x,x]=0;\\
[x,y]+[y,x]=0;\\
[x,[y,z]]=[[x,y],z]+[y,[x,z]].\endgather $$
 
Similary an $n$-Lie algebras are defined.

Note that the module $PT_sM$ of primitive elements of the Hopf algebra 
$T_sM$ is generated by the elements of the form $x^{2^k}$, $x\in M$, 
$k\ge 0$.

For a graded module $M$ denote by $\Cal E_n(M)$ the module generated
by the elements $e_{i_1}\dots e_{i_k}x$, where $x\in M$, $0\le i_1\le
\dots \le i_k\le n$, and the dimensions of these elements are
defined equal to $i_1+2i_2+\dots+2^{k-1}i_k+2^kdim(x)$.
So for an $n$-Lie algebra $L_n$ we will have the module $\Cal E_nL_n$.  

Denote also by $T_s\Cal E_nL_n$ the quotient algebra of the free
commutative algebra generated by the module $\Cal E_nL_n$ over the
relations $x\cdot x=e_0x$.

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
$$S^{-n}H_*(F(S^nE,E,X))\cong T_s\Cal E_{n-1}L_{n-1}S^{-n}X_*.$$
Hence we have 

{\bf Theorem 3'.} {\sl For the first term of the considered
spectral sequence of the homology $H_*(\Omega^nX)$ (over $Z/2$) there 
is the isomorphism
$$E^1\cong T_s\Cal E_{n-1}L_{n-1}S^{-n}H_*(X).$$}

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 $[~,~]\colon L\otimes L\to L$ called
a Lie bracket and satisfying the relations
$$\gather
[x,y]+(-1)^\epsilon [y,x]=0;\\
[x,[y,z]]=[[x,y],z]+(-1)^\epsilon [y,[x,z]];\endgather $$
where $\epsilon = dim(x)\cdot dim(y)$.

Similary a graded module $L$ called an $n$-Lie algebra
if there is given an operation $[~,~]\colon L\otimes L\to L$ of
dimension $n$, called a Lie bracket and satisfying the relations
$$\gather
[x,y]+(-1)^\epsilon [y,x]=0;\\
[x,[y,z]]=[[x,y],z]+(-1)^\epsilon [y,[x,z]];\endgather $$
where $\epsilon = (dim(x)+n)\cdot (dim(y)+n)$.

Note that the module $PT_sM$ of primitive elements of the Hopf algebra 
$T_sM$ is generated by the elements $x\in M$ and the elements 
$x^{p^k}$ for which $k>0$ and $dim(x)$ - even.

For a graded module $M$ denote by $\Cal E_n(M)$ the module generated
by the elements $e_{i_1}\dots e_{i_k}x$, where $x\in M$, $0\le i_1\le
\dots \le i_k\le n$, and besides that $i_1,\dots,i_k$ are odd if
$dim(x)$ is odd and $i_1,\dots,i_k$ are even if $dim(x)$ is even.
The dimensions of these elements are defined equal to 
$(p-1)(i_1+pi_2+\dots+p^{k-1}i_k)+p^kdim(x)$.
So for an $n$-Lie algebra $L_n$ we will have the module $\Cal E_nL_n$.  

Denote also by $T_s\Cal E_nL_n$ the quotient algebra of the free
commutative algebra generated by the module $\Cal E_nL_n$ over the
relations $e_0x=x^p$.

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
$$S^{-n}H_*(F(S^nE,E,X))\cong T_s\Cal E_{n-1}L_{n-1}S^{-n}X_*.$$
Hence we have 

{\bf Theorem 3''.} {\sl For the first term of the considered spectral 
sequence of $H_*(\Omega^nX)$ (over $Z/p$) there is the isomorphism
$$E^1\cong T_s\Cal E_{n-1}L_{n-1}S^{-n}H_*(X).$$}

Note that in the case of characteristic zero coefficients the module
$PT_sM$ of primitive elements of $T_sM$ is isomorphic to $M$. Hence
we have

{\bf Theorem 3'''.} {\sl The first term of the considered spectral 
sequence of $H_*(\Omega^nX)$ (over a field of characteristic zero) 
is the isomorphic to the free $(n-1)$-Poisson algebra generated by
$S^{-n}H_*(X)$, i.e.
$$E^1\cong T_sL_{n-1}S^{-n}H_*(X)=P_{n-1}S^{-n}H_*(X).$$}


\vskip .5cm
\centerline{\S 5. The second term of the spectral sequence of the
homology}
\centerline{of the iterated loop spaces}
\vskip 6pt
To determine the second term of the spectral sequence of the
homology of the iterated loop spaces firstly we define
the notion of a $\Cal P_n$-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 
$\Cal P_n$-algebra if

1. There is given a structure of a commutative algebra
$$x\otimes y\longmapsto x\cdot y,\quad x,y\in M.$$

2. There is given a structure of an $n$-Lie algebra
$$x\otimes y\longmapsto [x,y],\quad x,y\in M,$$
and the $n$-Lie algebra structure with the commutative algebra
structure form an $n$-Poisson algebra structure.

3. There are given operations
$$e_i\colon M\to M,\quad 0\le i\le n,~dim(e_i(x))=2\cdot dim(x)
+i,$$ and the following relations are satisfied
$$\gather e_0(x)=x\cdot x,\\
e_i(x\cdot y)=\sum_ke_k(x)\cdot e_{i-k}(y),\\
[e_i(x),y]=0,~i<n,\\
[e_n(x),y]=[x,[x,y]],\\
e_i(e_j(x))=\sum_k\binom{k-j-1}{i-k-1}e_{i+2(j-k)}(e_k(x)),
~i>j.\endgather $$

Note that instead of the operations $e_i\colon M\to M$ we can
consider the cup-$i$-products $$\cup_i\colon M\otimes M\to M$$
defined on the generators $x,y\in M$ by the formula
$$x\cup_iy=\cases 0,&x\ne y,\\e_i(x),&x=y.\endcases $$ 

Denote by $\Cal P_n$ the monad which corresponds to the graded
module $M$ the free $\Cal P_n$-algebra generated by $M$. It is
easy to see that there is the isomorphism $\Cal P_n(M)\cong
T_s\Cal E_nL_nM$ and so we have 

{\bf Theorem 3''''.} {\sl The first term of the considered spectral 
sequence of $H_*(\Omega^nX)$ (over $Z/2$) is isomorphic to the free 
$\Cal P_{n-1}$-algebra generated by $S^{-n}H_*(X)$, i.e. 
$$E^1\cong \Cal P_{n-1}S^{-n}H_*(X).$$}

A chain complex $M$ (over $Z/2$) will be called a differential
$\Cal P_n$-algebra if considered as graded module it is a 
$\Cal P_n$-algebra and the differential $d$ satisfies the
following relations
$$\gather 
d(x\cdot y)=d(x)\cdot y+x\cdot d(y)+d(x)\cup_1d(y),\\
d[x,y]=[d(x),y]+[x,d(y)],\\
d(e_i(x))=e_{i+1}(d(x)),~i<n,\\
d(e_n(x))=[d(x),x].\endgather $$

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
$$\nabla_*\colon H_*(X)\to H_*(X)\otimes H_*(X)$$ and of the
action of the Steenrod algebra
$$\Cal A\otimes H_*(X)\to H_*(X).$$

For an $n$-connected topological space $X$ these structures on
its homology $H_*(X)$ induce on $\Cal P_{n-1}S^{-n}H_*(X)$ the
differential $d_\varphi$ defined on the generators 
$s^{-n}x_{i+n}\in H_{i+n}(X)$ by the formula
$$d_\varphi(s^{-n}x_{i+n})=\sum_{x'<x''}[s^{-n}x',s^{-n}x'']+
\sum_{\scriptstyle k\atop\scriptstyle i-2k\le n}e_{i-2k-1}
(s^{-n}Sq^{i-k}(x_{i+n})),$$
where $\sum x'\otimes x''=\nabla_*(x_{i+n})$.

Denote the corresponding differential $\Cal P_{n-1}$-algebra as
$\Cal P_{n-1\varphi}S^{-n}H_*(X)$. Then we will have

{\bf Theorem 4.} {\sl For the second term of the considered spectral 
sequence of $H_*(\Omega^nX)$ (over $Z/2$) there is the isomorphism
$$E^2\cong H_*(\Cal P_{n-1\varphi}S^{-n}H_*(X)).$$}

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
$$\nabla_*\colon H_*(X)\to H_*(X)\otimes H_*(X).$$ 
Hence we have

{\bf Theorem 4'.} {\sl For the second term of the considered spectral 
sequence of $H_*(\Omega^nX)$ (over a field of characteristic zero) 
there is the isomorphism
$$E^2\cong H_*(P_{n-1\varphi}S^{-n}H_*(X)),$$
where the differential $d_\varphi$ defined on the generators
$s^{-n}x_{i+n}\in H_{i+n}(X)$ by the formula
$$d_\varphi(s^{-n}x_{i+n})=\sum_{x'<x''}[s^{-n}x',s^{-n}x''],$$
where $\sum x'\otimes x''=\nabla_*(x_{i+n})$.}

\vskip .5cm
\centerline{\S 6. The homology of iterated loop spaces over the
real projective spaces}
\vskip 6pt

Ones of the most important spaces in Algebraic Topology besides
spheres are the real projective spaces $RP^n$ and $RP^\infty$.
In some sense these spaces are opposite to spheres. The homology
$H_*(RP^\infty)$ (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 $RP^{\infty}$ 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
$$H_*(\Omega^m(RP^\infty/RP^n)),\quad m\le n,$$ of the iterated loop
spaces over the space $RP^\infty/RP^n$ with $Z/2$ coefficients.

Denote the $i$-dimensional generator of $H_*(RP^\infty)$ as $e_i$.
The coalgebra structure on this generators is determined by the
formula
$$\nabla(e_i)=\sum_ke_k\otimes e_{i-k}.$$

The action of the Steenrod algebra is determined by the
formula
$$Sq^j(e_i)=\binom{i-j}je_{i-j}.$$

Since $H_*(RP^\infty)$ is the free $\overline E_*$-coalgebra
there are no higher $\overline E_*$-operations. $H_*(RP^n)$ is
the $\overline E_*$-subcoalgebra of $H_*(RP^\infty)$ and hence
the $\overline E_*$-coalgebra structure on $H_*(RP^\infty)$
induces the $\overline E_*$-coalgebra structure on $H_*(RP^
\infty)/H_*(RP^n)\cong H_*(RP^\infty/RP^n)$. Therefore on the
homology $H_*(RP^\infty/RP^n)$ there are no higher 
$\overline E_*$-operations. So the spectral sequence for the 
homology of the iterated loop spaces over $RP^\infty/RP^n$ has
no higher differential and we have

{\bf Theorem 5.} {\sl The homology $H_*(\Omega^m(RP^\infty/RP^n))$,
$n\ge m$ (over $Z/2$) is isomorphic to the differential $\Cal
P_{m-1}$-algebra $\Cal P_{m-1\varphi}S^{-m}H_*(RP^\infty/RP^n)$,
where the differential $d_\varphi$ on the generators 
$u_i=s^{-m}e_{i+m}$ is defined by the formula
$$d_\varphi(u_i)=
\sum_{\scriptstyle k\atop \scriptstyle i-2k>m}
[u_k,u_{i-k-m}]+\sum_{\scriptstyle k\atop \scriptstyle i-2k\le m}
\binom{k+m}{i-k}e_{i-2k-1}(u_k).$$}

Note that in the case $m=1$ we have the isomorphisms
$$\Cal P_0S^{-1}H_*(RP^\infty/RP^n)\cong T_sLS^{-1}H_*(RP^\infty/
RP^n)\cong TS^{-1}H_*(RP^\infty/RP^n),$$
and the differential in the tensor algebra
$TS^{-1}H_*(RP^\infty/RP^n)$ has very simple form
$$d_\varphi(u_i)=\sum_{\scriptstyle k\ge n\atop\scriptstyle i-k-1
\ge n}u_k\otimes u_{i-k-1}.$$ From here it follows that
the homology $H_*(\Omega(RP^\infty/RP^n))$ (over $Z/2$) is isomorphic 
to the algebra generated by the elements $u_i$, $n\le i\le 2n$, 
of dimensions $i$ and relations
$$\gather u_n\cdot u_n=0;\\
u_n\cdot u_{n+1}+u_{n+1}\cdot u_n=0;\\
\dots \\
u_n\cdot u_{2n}+\dots+u_{2n}\cdot u_n=0.\endgather $$

So as graded module the homology $H_*(\Omega(RP^\infty/RP^n))$
is generated by the noncommutative products $u_{n_1}\cdot\dots
\cdot u_{n_k}$ with $n\le n_1\le 2n$, $n<n_2,\dots,n_k\le 2n$.

Consider the homology $H_*(\Omega^2(RP^\infty/RP^2))$. It
is isomorphic to the homology of the differential $\Cal P_1$-
algebra $\Cal P_{1\varphi}\{u_i|~i\ge 1\}$, where the
differential is determined by the formulas
$$\align
d_\varphi(u_{2i+1})&=\sum_{k<i}[u_k,u_{2i-k-1}]+\binom{i+2}{i+1}
e_0(u_i);\\
d_\varphi(u_{2i+2})&=\sum_{k<i}[u_k,u_{2i-k}]+e_1(u_i).
\endalign $$

In small dimensions we will have

$d(u_1)=0$;

$d(u_2)=0$;

$d(u_3)=u_1u_1$;

$d(u_4)=e_1(u_1)$;

$d(u_5)=[u_1,u_2]$;

$d(u_6)=[u_1,u_3]+e_1(u_2)$;

$d(u_7)=[u_1,u_4]+[u_2,u_3]+u_3u_3$;

$d(u_8)=[u_1,u_5]+[u_2,u_4]+e_1(u_3)$.
\vskip 6pt
From these formulas it follows that the homology $H_i=H_i(\Omega^2
(RP^\infty/RP^2))$ in small dimensions $i$ has the following
generators

$H_1:\quad u_1$;

$H_2:\quad u_2$;

$H_3:\quad u_1u_2$;

$H_4:\quad u_2^2$;

$H_5:\quad u_1u_2^2,~ e_1(u_2)$;

$H_6:\quad u_1e_1(u_2),~ u_2^3,~ [u_2,u_3],~ u_3^2$;

$H_7:\quad u_1u_2^3,~ u_1u_3^2,~ u_1[u_2,u_3],~ u_2e_1(u_2),~ 
e_1(u_3)$.
\vskip 6pt
Consider the homology $H_*(\Omega^3(RP^\infty/RP^3))$. It
is isomorphic to the homology of the differential $\Cal P_2$-
algebra $\Cal P_{2\varphi}\{u_i|~i\ge 1\}$, where the
differential is determined by the formulas
$$\align
d_\varphi(u_{2i+1})&=\sum_{k<i-1}[u_k,u_{2i-k-2}]+\binom{i+3}{i+1}
e_0(u_i)+e_2(u_{i-1});\\
d_\varphi(u_{2i+2})&=\sum_{k\le i-1}[u_k,u_{2i-k-1}]+
\binom{i+3}{i+2}e_1(u_i).\endalign $$

In small dimensions we will have

$d(u_1)=0$;

$d(u_2)=0$;

$d(u_3)=0$;

$d(u_4)=0$;

$d(u_5)=e_2(u_1)$;

$d(u_6)=[u_1,u_2]+e_1(u_2)$;

$d(u_7)=[u_1,u_3]+e_2(u_2)+u_3u_3$;

$d(u_8)=[u_1,u_4]+[u_2,u_3]$.
\vskip 6pt

From these formulas it follows that the homology $H_i=H_i(\Omega^3
(RP^\infty/RP^3))$ in small dimensions $i$ has the following
generators

$H_1:\quad u_1$;

$H_2:\quad u_1^2,~ u_2$;

$H_3:\quad u_1^3,~ u_1u_2,~ e_1(u_1),~ u_3$;

$H_4:\quad u_1^4,~ u_1^2u_2,~ u_1e_1(u_1),~ u_1u_3,~ u_2^2,~ u_4$;

$H_5:\quad u_1^5,~u_1^3u_2,~u_1^2u_3,~u_1^2e_1(u_1),~u_1u_2^2,~
u_1u_4,~u_2u_3,~u_2e_1(u_1),~e_1(u_2)$;

$H_6:\quad u_1^6,~u_1^4u_2,~u_1^3u_3,~u_1^3e_1(u_1),~u_1^2u_2^2,~
u_1^2u_4,~u_1u_2u_3,~u_1u_2e_1(u_1),~u_1e_1(u_2),~u_2u_4,$

$u_2^3,~u_3e_1(u_1),~u_3^2,~e_2(u_2),~e_1(u_1)^2$;

$H_7:\quad u_1^7,~u_1^5u_2,~u_1^4u_3,~u_1^4e_1(u_1),~u_1^3u_2^2,~
u_1^3u_4,~u_1^2u_2u_3,~u_1^2u_2e_1(u_1),~u_1^2e_1(u_2),$

$u_1u_2u_4,~u_1u_2^3,~u_1u_3e_1(u_1),~u_1u_3^2,~u_1e_2(u_2),
~u_1e_1(u_1)^2,~u_2^2u_3,~u_2^2e_1(u_1),~u_2e_1(u_2)$,

$u_3u_4,~[u_2,u_3],~e_1(u_3),~e_1e_1(u_1)$.
\vskip 6pt

Consider also the homology $H_*(\Omega^4(RP^\infty/RP^4))$. It
is isomorphic to the homology of the differential $\Cal P_3$-
algebra $\Cal P_{3\varphi}\{u_i|~i\ge 1\}$, where the
differential is determined by the formulas
$$\align
d_\varphi(u_{2i+1})&=\sum_{k<i-1}[u_k,u_{2i-k-3}]+\binom{i+4}{i+1}
e_0(u_i)+\binom{i+3}{i+2}e_2(u_{i-1});\\
d_\varphi(u_{2i+2})&=\sum_{k<i-1}[u_k,u_{2i-k-2}]+
\binom{i+4}{i+2}e_1(u_i)+e_3(u_{i-1}).\endalign $$

In small dimensions we will have

$d(u_1)=0$;

$d(u_2)=0$;

$d(u_3)=0$;

$d(u_4)=0$;

$d(u_5)=e_2(u_1)$;

$d(u_6)=e_1(u_2)+e_3(u_1)$;

$d(u_7)=[u_1,u_2]+u_3u_3$;

$d(u_8)=[u_1,u_3]+e_1(u_3)+e_3(u_2)$.
\vskip 6pt

From these formulas it follows that the homology $H_i=H_i(\Omega^4
(RP^\infty/RP^4))$ in small dimensions $i$ has the following
generators

$H_1:\quad u_1$;

$H_2:\quad u_1^2,~ u_2$;

$H_3:\quad u_1^3,~ u_1u_2,~ e_1(u_1),~ u_3$;

$H_4:\quad u_1^4,~ u_1^2u_2,~ u_1e_1(u_1),~ u_1u_3,~ u_2^2,~ u_4$;

$H_5:\quad u_1^5,~u_1^3u_2,~u_1^2u_3,~u_1^2e_1(u_1),~u_1u_2^2,~
u_1u_4,~u_2u_3,~u_2e_1(u_1),~e_1(u_2)$;

$H_6:\quad u_1^6,~u_1^4u_2,~u_1^3u_3,~u_1^3e_1(u_1),~u_1^2u_2^2,~
u_1^2u_4,~u_1u_2u_3,~u_1u_2e_1(u_1),~u_1e_1(u_2),~u_2^3$,

$u_2u_4,~u_3e_1(u_1),~u_3^2,~e_2(u_2),~e_1(u_1)^2$;

$H_7:\quad u_1^7,~u_1^5u_2,~u_1^4u_3,~u_1^4e_1(u_1),~u_1^3u_2^2,~
u_1^3u_4,~u_1^2u_2u_3,~u_1^2u_2e_1(u_1),~u_1^2e_1(u_2),~u_1u_2^3$,


$u_1u_2u_4,~u_1u_3e_1(u_1),~u_1u_3^2,~u_1e_2(u_2),~u_1e_1(u_1)^2,~
u_2^2u_3,~u_2^2e_1(u_1),~u_2e_1(u_2),~u_3u_4$,

$e_1(u_3),~e_3(u_2),~e_1e_1(u_1)$.

\vskip .5cm
\centerline{\S 7. Appendix (F. Sergeraert)}
\centerline{Computations with the Kenzo program}
\vskip 6pt

{\it Not yet written.}

\newpage
\centerline{REFERENCES}
\vskip 6pt

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2. H.J.Baues. The double bar and cobar constructions. Comp. Math.
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3. J.P.May. The Geometry of Iterated Loop Spaces. Lect. Notes in
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4. F.R.Cohen, T.J.Lada, J.P.May. The Homology of Iterated Loop
Spaces. Lect. Notes in Math. 533(1976).

5. E.Dyer, R.Lashoff. Homology of iterated loop spaces. Amer.
J. Math. 84(1962), 35-88.

6. R.J.Milgram. Iterated loop spaces. Ann. of Math. 84, N 3,
1966, 386-403.

7. V.A.Smirnov. On the cochain complex of topological spaces.
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8. V.A.Smirnov. Homotopy theory of coalgebras. Izv. Ac. Nauk
(Russia), 49(1985), 1302-1321.

9. V.A.Smirnov. On the chain complex of an iterated loop
space. Izv. Ac. Nauk (Russia), 53(1989), 1108-1119.

10. F.Sergeraert. The computability Problem in Algebraic
Topology. Advances in Math.104(1994), N 1, 1-29.

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