\input amstex
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\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
$$\gather d_0=\nu 1^n,~d_i=1^i\gamma 1^{n-i},~0*2$.
Recall that a graded module $L$ (over $Z/p$) is called a Lie algebra
if there is given an operation $[~,~]\colon L\otimes L\to L$ called
a Lie bracket and satisfying the relations
$$\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,~ij.\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)),~im}
[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*