Cohomology operations and the Deligne conjecture
Martin MARKL
The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
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\begin{document}
\bibliographystyle{plain}
\baselineskip 1.3em

\title{Cohomology operations and the Deligne conjecture}
\author[M. Markl]{Martin MARKL}
\thanks{The author was supported by the grant GA \v CR 201/05/2117 and by
   the Academy of Sciences of the Czech Republic, 
   Institutional Research Plan No.~AV0Z10190503.}
\date{June~10, 2005}
\email{markl@math.cas.cz}

\begin{abstract}
The aim of this note, which raises more questions than it answers,  
is to study natural operations acting on the cohomology of various types of
algebras. It contains a lot of very surprising partial results and examples. 
\end{abstract}


\address{Mathematical Institute of the Academy, {\v Z}itn{\'a} 25, 
         115 67 Prague 1, The Czech Republic}

\maketitle

\section*{Introduction}

In this note, 
all algebraic objects will be defined over a fixed field $\bfk$ of
characteristic zero. 
An~{\em algebra\/} means an algebra over a quadratic Koszul
operad~$\calP$~\cite[II.3.3]{markl-shnider-stasheff:book}. 
This generality covers all ``reasonable'' algebras -- associative, Lie, 
commutative associative, Poisson, Gerstenhaber, Leibniz,~\&c.

By the {\em cohomology\/} of a $\calP$-algebra $A$ we mean the
operadic cohomology $H^*_\calP(A;A)$ of $A$ with coefficients in
itself~\cite[II.3.100]{markl-shnider-stasheff:book}, defined as the
cohomology of the cochain complex $\CAP* = (\CAP*,d_\calP)$ recalled
in A.\ref{Ferda} of the appendix to this note. The complex $\CAP*$
generalizes the ``standard constructions'' and $H^*_\calP(A;A)$ the
``classical'' cohomology (Hochschild for associative algebras,
Chevalley-Eilenberg for Lie algebras, Harrison for associative
commutative algebras,~\&c.)  In general, $H^*_\calP(A;A)$ agrees with
the triple cohomology~\cite[Proposition~8.6]{fox-markl:ContM97} and governs
deformations of $A$ in the category of $\calP$-algebras.

By a {\em natural operation\/} we mean a multilinear operation on
$\HAP *$ induced by a natural multilinear cochain operation on
$\CAP*$. Naturality means being defined using data that do not
depend on a concrete algebra $A$ only. An example is the classical cup
product $f,g \mapsto f \cup g$ of Hochschild cochains, resp.~the
induced graded commutative associative multiplication on the
Hochschild cohomology of associative algebras~\cite{gerstenhaber:AM63}.  Our
definition excludes some operations that are also ``natural'' in some
sense, such as the degree zero unary operation defined as the
projection $\pi_n : \HAP * \to \HAP n$, $n \geq 0$, because this
operation is not induced by any natural cochain map. A precise
definition of natural operations is given in
Section~\ref{natural}. Our aim is to {\em describe the homotopy type\/} of the
dg operad $\calB_\calP = \{\calB_\calP(n)\}_{n \geq 0}$ of all these
natural operations, see Problem~\ref{prob1} and its baby version
Problem~\ref{prob3}. The reward would be an ultimate understanding of
the structure of the cohomology of a given type of algebras.

Our original hope was that the homotopy type of $\calB_\calP$ would be
that of another Koszul quadratic operad $\calQ_\calP$ determined by
$\calP$ in an explicit and simple manner. Examples we had the in mind
were $\calP = \As$ for which probably $\calQ_\calP = \Gerst$, the
operad for Gerstenhaber algebras, and $\calP = \Lie$ for which
probably $\calQ_\calP = \Lie$, the operad for Lie algebras.
Calculations presented in this note however show that the homotopy
type of $\calB_\calP$ is in general more complicated, therefore the
property that makes the homotopy type of $\calB_\calP$ for $\calP =
\As$ or $\calP = \Lie$ so nice must be finer than just the Koszulity
of $\calP$. We have no idea what this property is.


We feel that our formulations are somehow unsatisfactory -- 
we would certainly prefer a concept that would not depend on a
``representation'' of the cohomology by a concrete cochain complex.
In an ideal world, we should be working with 
natural operations in an appropriate ``derived'' category in which 
the cohomology is the hom-functor. The
possibility of such a more conceptual approach for associative algebras and 
their Hochschild cohomology was indicated
by~\cite{kock-toen}, see also~\cite{hu,hu-kriz}. 

Another possibility could be to consider $H_\calP^*(A;A)$ as the
cohomology of the cotangent complex of a suitable suitably derived
stack of the variety of structure constants of $\calP$-algebras,
see~\cite{fontanine-kapranov:JAMS,fontanine-kapranov,markl:JPAA96},
and study automorphisms of the point of this stack representing the
algebra $A$. Our feeling is, however, that these fancier approaches,
despite their beauty and generality, are still not developed enough to
give {\em concrete\/} answers to {\em concrete\/} questions.


Let us explain the title of this note. In his famous
letter~\cite{deligne:letter}, P.~Deligne asked whether the
Gerstenhaber algebra structure on the Hochschild cohomology of an
associative algebra given by the cup product and the intrinsic bracket
is induced by a natural action of singular chains on the little discs
operad. There are several proofs of this so-called {\em Deligne
conjecture\/}
today~\cite{kaufmann,kontsevich-soibelman,mcclure-smith,mcclure-smith:JAMS03,%
tamarkin:deligne,hinich:FM03}. Assume one can prove that the
operad of all natural operations on the Hochschild complex (that is,
$\calB_\As$ in our notation) has the homotopy type of the operad for
Gerstenhaber algebras. The formality of the little discs
operad~\cite{tamarkin:formality} would then immediately imply the
Deligne conjecture by simple homological considerations.

In fact, most of the proofs of the Deligne conjecture we are aware
of~\cite{kaufmann,kontsevich-soibelman,mcclure-smith,mcclure-smith:JAMS03},
involve a conveniently chosen suboperad of $\calB_\Ass$ whose homotopy
type is detected by Fiedorowicz' recognition principle for
$E_2$-operads~\cite{fiedorowicz}. We will discuss these proofs in
Section~\ref{pisi_na_novem_notebooku}. Other proofs based on the
Etingof-Kazhdan (de)quantization were given
in~\cite{tamarkin:deligne,hinich:FM03}. Several attempts have
also been made to find a suitable filtration of the Fulton-MacPherson
compactification of the configuration space of points in the plane to
prove the conjecture~\cite{getzler-jones:preprint,voronov:99}. The
Deligne conjecture has surprising implications for the existence of
the deformation quantization of Poisson
manifolds~\cite{hinich:FM03,tamarkin:deligne}.


{\bf Acknowledgments.} I would like to express my thanks to
F.~Chapoton, E.~Getzler, V.~Hinich M.~Livernet,
P.~Somberg and A.A.~Voronov for many useful comments and remarks.  
My special thanks are due to D.~Tamarkin for inspiring discussions
during my stay at the Northwestern University in April~2004.


\section{Formulation of the problem}


In this section we state the problems sketched out in the
introduction more concretely and formulate also some conjectures.  
Let $\calB_\calP =
(\calB_\calP,\delta_\calP)$ be the dg-operad of all natural
multilinear operations on the cochain complex $C^*_\calP(A;A) =
(C^*_\calP(A;A),d_\calP)$.  The $n$-th component
$\calB_\calP(n)$ of $\calB_\calP$ is the space of all $n$-linear
natural operations $C^*_\calP(A;A)^{\ot n} \to \CAP*$ with the grading
induced by the grading of $C^*_\calP(A;A)$: $U \in \calB_\calP(n)$ has
degree $d$ if
\[
U(f_1,\ldots,f_n) \in \CPAA{m_1+\cdots +m_n + d},
\]
whenever $f_i \in \CPAA {m_i}$ for $1 \leq i \leq n$. In this case we
write $U \in \calB^d_\calP(n)$. Each $\calB_\calP(n)$ is equipped with
the degree $+1$ differential $\delta_\calP$ induced by the
differential $d_\calP$ of $C^*_\calP(A;A)$ in the usual way.

A precise definition of the operad $\calB_\calP$ is given in
Section~\ref{natural}.  Here we emphasize only that $\calB^d_\calP(n)
= 0$ for $d < 0$ and that $\calB_\calP(0) \not = 0$ for any nontrivial
$\calP$. The central problem of the paper reads:

\begin{problem}
\label{prob1}
Describe the homotopy type (in the non-abelian derived category) of
the dg operad $\calB_\calP$. In particular, calculate the cohomology
of $\calB_\calP$.
\end{problem}

A baby-version of this problem is Problem~\ref{prob3} of
Section~\ref{spaci_medvidek}. Closely related is:  


\begin{problem}
\label{zase_prace_nad_hlavu}
Find a property characterizing operads $\calP$ for which $\calB_\calP$
is formal and has the homotopy type of some Koszul quadratic operad.
\end{problem}

We will see, in Example~\ref{ve_stredu_poletim_do_Minneapolis}, a
simple quadratic Koszul operad $\calD$ such that
\hbox{$H^*(\calB_\calD(0),\delta_\calD) \not= 0$}. This clearly means
that $\calB_\calD$ does not have the homotopy type of a quadratic Koszul
operad, therefore the property answering
Problem~\ref{zase_prace_nad_hlavu} must be stronger than Koszulness of
$\calP$. 


Suppose that $\calP$ is the symmetrization of a non-$\Sigma$ operad
$\ucalP$~\cite[Remark~II.1.15]{markl-shnider-stasheff:book}. In this
case there exists a dg-suboperad $\calB_\ucalP$ of $\calB_\calP$
consisting of natural operations that preserve the
order of inputs of $\calP$-cochains.  For example, the classical cup product
$f\cup g \in C^1_\Ass(A;A) \cong \Lin(\otexp A2,A)$ of Hochschild
cochains $f,g \in C^0_\Ass(A;A) \cong \Lin(A,A)$ defined as 
\[
(f \cup g)(a \ot b) := f(a)\cdot g(b)\ 
\mbox { for } a \ot b \in A \ot A,
\]
with $\cdot$ denoting the associative multiplication of $A$, belongs to
$\calB_\uAss$, while the operation 
\[
U(f,g):= f(b)\cdot g(a)\ \mbox { for } a \ot b \in A \ot A,
\]
does not, see Definition~\ref{letel_jsem_do_Nymburka!!}
of Section~\ref{natural} for details.

Since $\calB_\ucalP(n)$ is a $\Sigma_n$-closed
subspace of $\calB_\calP(n)$, $n \geq 0$, $\calB_\ucalP$
is a usual, not only a non-$\Sigma$, operad.  We will see in
Example~\ref{zase_nebude_pocasi} that, surprisingly, the homotopy type of
$\calB_\ucalP$ in general {\em differs\/} from the homotopy type of
$\calB_\calP$. We therefore formulate:

\begin{problem}
\label{Pastviny}
Let $\ucalP$ a non-$\Sigma$ quadratic Koszul operad.  Describe the
homotopy type of the dg operad $\calB_\ucalP$. In particular,
calculate the cohomology of $\calB_\ucalP$.
\end{problem}

In Section~\ref{s6}(i) we give some indications that the operad
$\calB_\uAss$ has the homotopy type of the operad $\Gerst$ for
Gerstenhaber algebras, see~A.\ref{ma_prejit_fronta_v_sobotu} for a
definition of $\Gerst$.

One may consider also {\em strongly homotopy versions\/} of the above
problems.  Recall that a strongly homotopy $\calP$-algebra is,
by~\cite{markl:zebrulka}, an algebra over the minimal model $sh\calP$
of the operad $\calP$. Let us denote by $sh\calB_\calP =
\calB_{sh\calP}$ the dg-operad of natural operations on the cochain
complex $C^*_{sh\calP}(A,A)$ for the cohomology of a strongly homotopy
algebra $A$ with coefficients in itself. An example of this type of
operad is the ``minimal operad'' $M$ considered
in~\cite{kontsevich-soibelman}, which is a certain suboperad of
$\calB_{sh\Ass}$, see Section~\ref{pisi_na_novem_notebooku}(iii).


It is clear that there exists a canonical map $\calB_{\calP} \to
\calB_{sh\calP}$, but simple examples show that, again rather
surprisingly, this map is in general not a homotopy equivalence. Let
us formulate:


\begin{problem}
Describe the homotopy type of the dg-operad $\calB_{sh\calP}$ of
natural operations on the cohomology of strongly homotopy $\calP$-algebras.
\end{problem}

 Other problems formulated in
this paper are Problem~\ref{-} of Section~\ref{s2} and
Problems~\ref{prob3},\ref{+} of Section~\ref{s3}.

Let us finally formulate also some conjectures.
Although the operads $\calB_\calP$ and $\calB_{\calP^!}$ are
not isomorphic (see Section~\ref{natural}), computational evidences
together with an equivalence between the derived category of $\calP$
algebras and the derived category of $\calP^!$-algebras lead us to
believe in:

\begin{conjecture}
\label{zase}
The homotopy type of the operad $\calB_\calP$ is the same as the
homotopy type of $\calB_{\calP^!}$.
\end{conjecture}


The following two conjectures concern the homotopy type of
$\calB_\calP$ for $\calP = \Ass$ and $\calP = \Lie$.


\begin{conjecture}
The operad $\calB_\Ass$ has the homotopy type of the operad $\Gerst$ for
Gerstenhaber algebras.
\end{conjecture}

Some results which may be helpful in the proof of the above
conjecture are recalled in Section~\ref{pisi_na_novem_notebooku}.

\begin{conjecture}
\label{Kuratko}
The operad $\calB_\Lie$ has the homotopy type of the operad $\Lie$.
\end{conjecture}

According to a formality theorem~\cite[Proposition~3.4]{markl:zebrulka}, it
is enough to prove that 
\[
H^*(\calB_\Lie,\delta_\Lie) \cong \Lie.
\]
Since $H^0(\calB_\Lie,\delta_\Lie) \cong \Lie$ (see
Section~\ref{jsem_otrok}), Conjecture~\ref{Kuratko} is equivalent to
the acyclicity of $\calB_\Lie$ in positive degrees. Another conjecture,
Conjecture~\ref{ib}, is given in Section~\ref{s4}.

 

Let us finish this section with one exceptional example.
The trivial operad $\triv$ is a Koszul quadratic
self-dual operad. A $\triv$-algebra is a vector space $A$ with no
operations. Clearly $C^*_\triv(A;A)$ is just the space $\Lin(A,A)$ of
linear maps $f : A \to A$
concentrated in degree zero with trivial differential, thus
$H^*_\triv(A;A) = \Lin(A,A)$. It is also clear that
all natural operations on $\Lin(A,A)$ are the identity $\id_A \in
\Lin(A,A)$ considered as a degree zero constant, and 
iterated compositions 
\[
\Lin(A,A) \ni f_1,f_2,\ldots,f_n \mapsto f_1 \circ f_2 \circ
\cdots \circ f_n \in \Lin(A,A),\ n \geq 1.
\]
Therefore
\[
\calB_\triv \cong U\hskip -.2em  \As,
\] 
the operad for unital associative algebras. This example is
pathological in that the canonical element introduced in
Definition~\ref{canon} equals zero. Therefore, from now on all
quadratic Koszul operads in this note will be {\em nontrivial\/} in
the sense that $\calP \not= \triv$.


\section{The constants $\calB_\calP(0)$ -- soul without body}
\label{duneni_subwooferu}
\label{s2}

This section, as well as the rest of the paper, relies on terminology
and notation recalled in the Appendix.  The main result of this part
is Proposition~\ref{kozulka} which describes the dg-vector space
$\calB_\calP(0) = (\calB_\calP(0),\delta_\calP)$ of ``constants.''  It
is not difficult to see (compare also Example~\ref{Kocicka_zrcatko} of
Section~\ref{natural}) that
\[
\calB^{m-1}_\calP(0)  \cong \ss (\calP(m) \ot \calP^!(m))^{\Sigma_m}, \ m
\geq 1,
\]
with the action $\calB_\calP^{m-1}(0) \to \CAP{m-1}$ given as the composition
\begin{eqnarray}
\label{napsala_mi_Jituska!}
\lefteqn{ 
\ss(\calP(m) \ot \calP^!(m))^{\Sigma_m}
\stackrel{\cong}{\longrightarrow}
(\ss\calP(m) \ot \calP^!(m))^{\Sigma_m}
\stackrel{\ss \alpha \ot \id}{\longrightarrow}
(\ss\End_A(m) \ot \calP^!(m))^{\Sigma_m}} \hskip 1em
\\
\nonumber 
&&
\hskip 1em
\cong 
(\End_{\downarrow   A}(m) \ot \calP^!(m))^{\Sigma_m}
=
\left[
\Lin ((\downarrow \hskip -.2em A)^{\ot m},
\downarrow \hskip -.2em A) \ot \calP^!(m)\right]^{\Sigma_m}
=
\CAP{m-1}.
\end{eqnarray}

Since composition~(\ref{napsala_mi_Jituska!}) is monic for all
``generic'' $\calP$-algebras $A$, $(\calB^*_\calP(0),\delta_\calP)$ is
``morally'' the subcomplex of natural elements in $(\CAP *,d_\calP)$.
Before going further, we must recall the following general
construction.  Let $\calT$ be an operad. It is well-known that the
formula
\[
[f,g] := f \circ g - (-1)^{(m-1)(n-1)} g \circ f,
\]
where $f \circ g$ is, for $f \in \calT(m)$ and $g \in \calT(n)$,
defined by
\[
f \circ g := \sum_{1 \leq i \leq m} (-1)^{(n-1)(i-1)} f \circ_i g,
\]
makes the direct sum $\calT_* = \bigoplus_{m \geq
0}\calT_*$, with 
\[
\calT_{m-1} := \susp^{m-1} \calT(m) = \ss \calT(m),
\] 
a graded Lie algebra. Another standard fact is that each
element $\omega \in \calT_1 = \ss{}\calT(2)$ satisfying
$[\omega,\omega] = 0$ defines a degree $+1$ differential
$\delta_\omega : \calT_* \to \calT_{*+1}$ by
\[
\delta_\omega(t) := [t,\omega], \ \mbox { for } t \in \calT_*.
\]
It is helpful to observe that the condition $[\omega,\omega] =
0$ means the associativity:
\begin{equation}
\label{vcera_jsme_se_koupali_na_Hradistku}
\omega \circ_1 \omega = \omega \circ_2 \omega
\end{equation}
and that the differential $\delta_\omega$ in terms of
$\circ_i$-operations equals
\[
\delta_{\omega}(t) =  t \circ_1 \omega - t \circ_2 \omega
+ \cdots -(-1)^m  t \circ_m \omega + (-1)^{m}\omega \circ_1 t  - 
\omega \circ_2 t,
\ \mbox { for } t \in \calT(m).
\]

As proved in~\cite{laan-defo}, the graded Lie algebra structure
$(\calT_*,[-,-])$ descents to the space of coinvariants therefore it
induces, via the canonical isomorphism between invariants and
coinvariants, a Lie bracket, denoted again $[-,-]$, on the graded
vector space $\calT^\Sigma_* = \bigoplus_{m \geq 0}\calT^\Sigma_m$
with pieces
\[
\calT^\Sigma_{m-1} := \hskip .2em \uparrow^{m-1} 
(\calT(m) \ot \sgn_m)^{\Sigma_{m}} = \ss \calT(m)^{\Sigma_{m}}.
\]
As usual, an element $\phi \in \calT^\Sigma_1 
= \ss \calT(2)^{\Sigma_2}$ satisfying
$[\phi,\phi]=0$ induces a degree $+1$ differential $\delta^\Sigma_\phi:
\calT^\Sigma_* \to \calT^\Sigma_{*+1}$ by
\begin{equation}
\label{dnes_poslan_ib_do_MA}
\delta^\Sigma_\phi t := [\phi,t],\ \mbox { for } t \in \calT^\Sigma_*.
\end{equation}


In Proposition~\ref{kozulka} below we put $\calT := (\calP \ot
\calP^!)$ and define the differential~(\ref{dnes_poslan_ib_do_MA}) by
taking as $\phi$ the canonical element $\chi$ introduced in the
following definition in which $\#$ denotes the linear dual.

\begin{definition}
\label{canon}
Let $\calP$ be a quadratic Koszul operad. The {\em canonical
element\/} $\chi$ is the element of $\ss(\calP \ot
\calP^!)(2)^{\Sigma_2}$ corresponding, under the standard
identification 
\[
\ss(\calP \ot \calP^!)(2) 
\cong \hskip .2em \uparrow \hskip -.2em (\calP \ot \calP^\#)(2) 
\cong \hskip .2em \uparrow \hskip -.2em (\calP(2) \ot \calP(2)^\#) 
\cong \hskip .2em \uparrow \hskip -.2em\Lin(\calP(2), \calP(2)),
\]
to the suspension of the identity map $\uparrow \hskip -.2em
\id_{\calP(2)} \in \hskip .2em\uparrow \hskip -.2em\Lin(\calP(2),\calP(2))$.
\end{definition}
%
Observe that $\chi$ is symmetric,
\begin{equation}
\label{porad_to_knika}
\chi \tau = \chi\ \mbox { for
$\tau \in \Sigma_2$,}
\end{equation}
therefore indeed $\chi \in \ss(\calP \ot \calP^!)(2)^{\Sigma_2}$. The
condition $[\chi, \chi]=0$ is equivalent to the Jacobi
identity~(\ref{zase_jsem_se_opil}) for $\chi$ which follows
from~\cite[Corollary~2.2.9(b)]{ginzburg-kapranov:DMJ94}, see also the proof of
Proposition~\ref{lisasek}.


\begin{proposition}
\label{kozulka}
There is a natural isomorphism of cochain complexes
\[
(\calB^*_\calP(0),\delta_\calP) \cong ( (\opt)^\Sigma_*,\delta^\Sigma_\chi).
\]
\end{proposition}

If $\calP$ is the symmetrization of a \ns\ operad
$\ucalP$~\cite[Remark~II.1.15]{markl-shnider-stasheff:book}, then
there is a similar description of the chain complex
$(\calB^*_\ucalP(0),\delta_\ucalP)$ obtained as follows. 
The definition of the graded Lie algebra $(\calT_*, [-,-])$ given
above clearly makes sense also when $\calT$ is a non-$\Sigma$ operad. Observe
also that there exists the \ns\ quadratic dual $\ucalP^!$ of $\ucalP$
and that one may introduce the {\em \ns\ canonical element\/} $\uchi
\in \ss (\ucalP \ot \ucalP^!)(2)$ in exactly the same manner as its
symmetric version. The element $\uchi$ obviously satisfies the associativity
condition~(\ref{vcera_jsme_se_koupali_na_Hradistku}):
\[
\uchi \circ_1 \uchi = \uchi \circ_2 \uchi.
\]
Our non-$\Sigma$ version of Proposition~\ref{kozulka} reads:

\begin{proposition}
\label{Jituska}
Let $\calP$ be the symmetrization of a quadratic Koszul non-$\Sigma$
operad $\ucalP$.  Then
\[
(\calB^*_\ucalP(0),\delta_\ucalP) \cong ((\ucalP \ot \ucalP^!)_*,\delta_\uchi).
\]
\end{proposition}


Let us make a comment on the meaning of the cohomology
$H^*(\calB_\calP(0),\delta_\calP)$.  The natural morphism
\[
M: H^*(\calB_\calP(0),\delta_\calP) \to \HAP*
\]
induced by action~(\ref{napsala_mi_Jituska!}) is monic for any
``generic'' $\calP$-algebra $A$, therefore elements
$H^*(\calB_\calP(0),\delta_\calP)$ represent {\em natural generically
nontrivial homology classes\/} in the cohomology of
$\calP$-algebras. This leads one to believe that
$H^*(\calB_\calP(0),\delta_\calP)=0$ for all well-behaved operads,
since otherwise people would stumble upon nontrivial natural classes -- compare
the Casimir element in the cohomology of simple Lie
algebras. Example~\ref{ve_stredu_poletim_do_Minneapolis} however
contradict this reasonable assumption. We believe that
$H^*(\calB_\calP(0),\delta_\calP)$ is an important invariant of the
operad $\calP$ that deserves its own name:

\begin{definition}
We call the graded vector space $H^*(\calB_\calP(0),\delta_\calP)$ described
in Proposition~\ref{kozulka} the
{\em soul of the cohomology\/} of $\calP$-algebras.
\end{definition}

 
It is easy to prove that $H^0(\calB_\calP(0),\delta_\calP)$
is always trivial.

\begin{example}
\label{ctvrek_a_zase_je_lat}
{\rm\
Let us describe the complex calculating the soul
$H^*(\calB_\As(0),\delta_\As)$ of the Hochschild cohomology. Clearly
\[
(\calP \ot \calP^!)^\Sigma_{m-1} = (\As \ot \As)^\Sigma_{m-1} \cong
\ss(\As \ot \As)(m)^{\Sigma_m} \cong \ss \As(m),
\]
therefore the complex $((\As \ot \As)^\Sigma,\delta^\Sigma_\chi)$ has the
form
\begin{equation}
\label{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu}
\bfk \stackrel{\delta^\Sigma_\chi}{\longrightarrow} \bfk[\Sigma_2] 
\stackrel{\delta^\Sigma_\chi}{\longrightarrow}  \bfk[\Sigma_3] 
\stackrel{\delta^\Sigma_\chi}{\longrightarrow} \bfk[\Sigma_4]
\stackrel{\delta^\Sigma_\chi}{\longrightarrow} \cdots
\end{equation}
Is also easy to describe the differential $\delta^\Sigma_\chi$; on a permutation
$\sigma \in \Sigma_m$ it acts as
\[
\delta^\Sigma_\chi(\sigma) := d_0(\sigma) -  d_1(\sigma) +  d_2(\sigma) -
\cdots + (-1)^{m+1}  d_{m+1}(\sigma) \in \bfk[\Sigma_{m+1}],   
\]
where $ d_0(\sigma) := \id \times \sigma$, $d_{m+1}(\sigma) := \sigma
\times \id$ and $d_i(\sigma) \in \Sigma_{m+1}$ is the permutation
obtained by doubling the $i$th input of $\sigma$. In
Theorem~\ref{krystal_mame} below we prove
that~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu})
is acyclic.

Since $\Ass$ is the symmetrization of the non-$\Sigma$ operad $\uAss$,
it makes sense to consider also the subcomplex
$(\calB^*_\uAss(0),\delta_\uAss)$ of $(\calB^*_\Ass(0),\delta_\Ass)$
described in Proposition~\ref{Jituska}.  This subcomplex is obviously
isomorphic to the acyclic complex
\begin{equation}
\label{Andulka}
\bfk \stackrel{d_0}{\longrightarrow} \bfk
\stackrel{d_1}{\longrightarrow}  \bfk
\stackrel{d_2}{\longrightarrow} \bfk
\stackrel{d_3}{\longrightarrow} \cdots
\end{equation}
in which $d_{2i} = \id_\bfk$ and $d_{2i+1} = 0$, $i \geq 0$. The
inclusion $(\calB^*_\uAss(0),\delta_\uAss) \hookrightarrow
(\calB^*_\Ass(0),\delta_\Ass)$ sends the generator $1 \in \bfk$ of the
$n$th piece of~(\ref{Andulka}) into the identity permutation
$\id_{n-1} \in \bfk[\Sigma_{n-1}]$ in
complex~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu}).
}
\end{example}


\begin{theorem}
\label{krystal_mame}
The soul $H^*(\calB_\As(0),\delta_\As)$ of the Hochschild cohomology
is trivial.
\end{theorem}

\noindent
{\bf Proof.}  We must prove
that~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu})
is an acyclic complex. The idea will be to show that it
decomposes into a direct sum of acyclic subcomplexes indexed by
primitive, in the sense introduced below, permutations.

We define first, for each $\sigma \in \Sigma_n$, a natural number
$g(\sigma)$, $0 \leq g(\sigma) \leq n$, which we call the {\em
grade\/} of $\sigma$, as follows.  The grade of the unit $\id_n \in
\Sigma_n$ is $n-1$, $g(\id_n) := n-1$.  For $\sigma \not= \id_n$, let
\begin{eqnarray*}
a(\sigma) &:=& \max\{i;\ \sigma = \id_i \times \tau \mbox { for some }
\tau \in \Sigma_{n-i}\}, \mbox { and }
\\
c(\sigma) &:=& \max\{j;\ \sigma = \lambda \times \id_j \mbox { for some }
\lambda \in \Sigma_{n-i}\}.
\end{eqnarray*}
There clearly exists a unique $\omega = \omega(\sigma) \in \Sigma_{n
  - a(\sigma) - c(\sigma)}$ such that
$\sigma = \id_{a(\sigma)} \times \omega(\sigma) \times
\id_{c(\sigma)}$.
Let, finally, $b(\sigma)$ be the number of ``doubled strings'' in
$\omega(\sigma)$, 
\[
b(\sigma) := \{1 \leq s < k;\ \omega(s+1) = \omega(s) +1\}.
\]
The grade of $\sigma$ is then defined by 
\[
g(\sigma) := a(\sigma) + b(\sigma) + c(\sigma),
\]
see Figure~\ref{Jitka_prisla} for examples. Observe that the
differential $\delta^\Sigma_\chi$
of~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu})
raises the grade by $+1$.
\begin{figure}
\begin{center}
\unitlength 2mm 
\thicklines
\begin{picture}(65,35)(4,5)
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\put(63,35){\vector(0,1){3}}
\put(21,26){\vector(0,1){3}}
\put(23,26){\vector(4,3){4}}
\put(27,26){\vector(-4,3){4}}
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\put(36,30){\makebox(0,0)[bc]{$1$}}
\put(36,25){\makebox(0,0)[tc]{$1$}}
\put(36,21){\makebox(0,0)[bc]{$1$}}
\put(36,16){\makebox(0,0)[tc]{$1$}}
\put(50,36){\makebox(0,0)[cb]{$\longmapsto$}}
\put(50,27){\makebox(0,0)[cb]{$\longmapsto$}}
\put(50,18){\makebox(0,0)[cb]{$\longmapsto$}}
\put(50,9){\makebox(0,0)[cb]{$\longmapsto$}}
\put(10,36){\makebox(0,0)[cb]{grade $0$:}}
\put(10,27){\makebox(0,0)[cc]{grade $1$:}}
\put(10,18){\makebox(0,0)[cc]{grade $2$:}}
\put(10,8){\makebox(0,0)[cc]{grade $3$:}}
\end{picture}
\end{center}
\caption{\label{Jitka_prisla}%
Left: Examples of elements of $\Sigma_4$ of grade $0$ (first line),
grade $1$ (second line), grade $2$ (third line) and grade $3$ (fourth
line). Right: the corresponding primitive elements.}
\end{figure}


Let us call $\chi \in \Sigma_k$, $k \geq 1$, {\em primitive\/} if
$g(\sigma) = 0$. Observe that, according to our definitions, $\id_n
\in \Sigma_n$ is primitive if and only if $n=1$.  For each $\sigma \in
\Sigma_n$, $\sigma \not= \id_n$, we define a unique primitive $\kappa
= \kappa(\sigma) \in \Sigma_k$, $k = n - g(\sigma)$, by contracting all
``multiple strings'' of $\omega(\sigma)$ into ``simple'' ones, see
Figure~\ref{Jitka_prisla}. We put $\chi(\id_n):= \id_1$.

For a primitive $\chi$, let $P^*(\chi)$ be the graded subspace
of~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu})
spanned by all permutations $\sigma$ with $\chi = \kappa(\sigma)$. The
following statements can be easily verified:

(i) Each $P^*(\chi)$ is a subcomplex
of~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu}).

(ii)
Complex~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu})
decomposes as the summation $\bigoplus_\chi P^*(\chi)$
over all primitive permutations~$\chi$.

(iii) For each primitive $\chi \in \Sigma_n$,
\[
P^*(\chi) \cong P^*(\id_1) \ot \cdots \ot P^*(\id_1)\ \mbox { ($n+2$ times).}
\]
The proof is finished by observing that $P^*(\id_1)$ is isomorphic
to the acyclic complex~(\ref{Andulka}) and applying the K\"unneth formula.
\qed


\begin{example}
{\rm\ 
In this example we describe the soul of the Chevalley-Eilenberg
cohomology of Lie algebras which is, due to the obvious self-duality
of Proposition~\ref{kozulka}, the same as the soul of the Harrison
cohomology of commutative associative algebras. In both cases
\[
(\calP \ot \calP^!)^\Sigma_{m-1} = (\Com \ot \Lie)^\Sigma_{m-1} \cong
\ss \Lie(m)^{\Sigma_m} = \hskip .4em \uparrow^{m-1} \hskip -.2em
({\Lie(m) \otimes \sgn_m})^{\Sigma_m} \cong \bfk
\] 
(see~\cite{klya}) and one may identify $((\Com \ot
\Lie)^\Sigma,\delta^\Sigma_\chi)$ with
the acyclic complex~(\ref{Andulka}). Therefore the
souls of both the Chevalley-Eilenberg cohomology and the
Harrison cohomology are trivial.
}
\end{example}


\begin{example}
\label{ve_stredu_poletim_do_Minneapolis}
{\rm\ 
This example presents a Koszul quadratic operad with a nontrivial soul.
Let $\calD := \Ass * \Ass$ be the free product of two copies of
the associative operad. Operad $\calD$ is a
Koszul quadratic operad, 
whose quadratic dual ${\calD^!}$ equals the coproduct
$\Ass\vee\Ass$ defined by
\[
(\Ass\vee\Ass{})(m) := \cases{\bfk}{if $m=1$
                                  and}{\Ass(m)\oplus\Ass(m)}{if $m \geq 2$.}
\]
%
\begin{figure}[t]
\begin{center}
{% Picture saved by xtexcad 2.4
\unitlength=1.000000pt
\begin{picture}(230.00,175.00)(73.00,-10.00)
\put(0.00,120.00){\makebox(0.00,0.00){$d_2$}}
\put(0.00,70.00){\makebox(0.00,0.00){$d_2$}}
\put(0.00,20.00){\makebox(0.00,0.00){$d_2$}}
\put(270.00,10.00){\makebox(0.00,0.00)[l]{$d_1$}}
\put(160.00,10.00){\makebox(0.00,0.00)[l]{$d_1$}}
\put(50.00,10.00){\makebox(0.00,0.00)[l]{$d_1$}}
\put(380.00,0.00){\makebox(0.00,0.00){$\cdots$}}
\put(10.00,170.00){\makebox(0.00,0.00){$\vdots$}}
\put(330.00,0.00){\makebox(0.00,0.00){$\Lin(A^{\otimes 4},A)$}}
\put(10.00,150.00){\makebox(0.00,0.00){$\Lin(A^{\otimes 4},A)$}}
\put(220.00,0.00){\makebox(0.00,0.00){$\Lin(A^{\otimes 3},A)$}}
\put(10.00,100.00){\makebox(0.00,0.00){$\Lin(A^{\otimes 2},A)$}}
\put(110.00,0.00){\makebox(0.00,0.00){$\Lin(A^{\otimes 2},A)$}}
\put(10.00,50.00){\makebox(0.00,0.00){$\Lin(A^{\otimes 2},A)$}}
\put(10.00,0.00){\makebox(0.00,0.00){$\Lin(A,A)$}}
\put(10.00,110.00){\vector(0,1){30.00}}
\put(10.00,60.00){\vector(0,1){30.00}}
\put(10.00,10.00){\vector(0,1){30.00}}
\put(260.00,0.00){\vector(1,0){30.00}}
\put(150.00,0.00){\vector(1,0){30.00}}
\put(40.00,0.00){\vector(1,0){30.00}}
\end{picture}}
\end{center}
\caption{\label{jsem_nachlazeny}%
The ``meager'' bicomplex 
describing the cohomology of $\calD$-algebras.}
\end{figure} 

\noindent 
Obviously,
$\calD$-algebras are triples $A =
(A,\mu_1,\mu_2)$ consisting of two independent associative
multiplications 
$\mu_1,\mu_2 : A \ot A \to A$.
The cohomology of these algebras is the cohomology of the total
complex $(\CAD *,d_\calD)$ of the ``meager'' bicomplex
in Figure~\ref{jsem_nachlazeny}. The
horizontal line is the Hochschild cochain complex of the associative algebra
$A_1 := (A,\mu_1)$, the
vertical line the Hochschild complex of $A_2 := (A,\mu_2)$.

Let $e$ denote the identity $\id \in \Lin(A,A)$ considered as a natural
element of $\CAD 0$. Clearly
\[
d_\calD(d_1 e) = d_\calD(d_2 e) = 0
\]
therefore both $d_1 e$ and $d_2 e$ are natural cochains in $\CAD 1$
thus representing $\delta_\calD$-cochains in
$\calB_\calD^1(0)$. The equality
\[
d_\calD e = d_1 e + d_2 e
\]
implies that $d_1 e + d_2 e$ is $\delta_\calD$-homologous to zero in
$\calB^1_\calD(0)$. We conclude that
\[
H^1(\calB_\calD(0),\delta_\calD) \cong \Span([d_1 e]) \cong \Span([d_2 e]) \cong \bfk.
\] 
}
\end{example}

We saw that the souls of the Hochschild ($\calP = \As)$,
Chevalley-Eilenberg ($\calP = \Lie$) and Harrison ($\calP = \As$)
cohomologies were trivial, while the soul of the cohomology for
$\calD$-algebras analyzed in
Example~\ref{ve_stredu_poletim_do_Minneapolis} was not.  This leads us
to formulate:

\begin{problem}
\label{-}
Which property of a quadratic Koszul operad $\calP$ implies the
triviality of the soul $H^*((\calP \ot \calP^!)^\Sigma,\delta^\Sigma_\chi)$
of the $\calP$-cohomology?
\end{problem}


\begin{example}
\label{zase_nebude_pocasi}
{\rm\ In this example we describe a non-$\Sigma$ quadratic Koszul
operad with the property that $(\calB^*_\ucalP(0),\delta_\ucalP)$ is
acyclic but the soul $(\calB^*_\calP(0),\delta_\calP)$ is not. This
shows that the homotopy type of $\calB_\calP$ is in general different
from the homotopy type of $\calB_\ucalP$.
 
Let $\uMag$ be the free non-$\Sigma$ operad generated by one bilinear
operation, $\uMag : = \underline{\Gamma}( \underline{\mu})$, and $\Mag$ its
symmetrization. The corresponding cochain complex $C^*_\Mag (A;A)$ is
the truncation
\[
\Lin(A,A) \stackrel{d}{\longrightarrow} \Lin (\otexp A2,A)
\]  
of the Hochschild complex. The complex $(\calB^*_\Mag(0),\delta_\Mag)$
defining the soul of $\Mag$ is the truncation
\[
\bfk \stackrel{\delta^\Sigma_\chi}{\longrightarrow} \bfk[\Sigma_2]
\]
of~(\ref{to_jsem_zvedav_jestli_budu_moct_ten_notebook_zaplatit_z_grantu}),
and is manifestly not acyclic. On the other hand,
$(\calB^*_\uMag(0),\delta_\uMag)$ is acyclic, isomorphic to the truncation
$
\bfk \stackrel{d_0}{\longrightarrow} \bfk
$
of~(\ref{Andulka}). We conclude that
$H^*(\calB^*_\uMag(0),\delta_\uMag) = 0$ while
\[
H^*(\calB^*_\Mag(0),\delta_\Mag) = H^1(\calB^*_\Mag(0),\delta_\Mag)
\cong \bfk.
\]
}
\end{example}

\section{Homotopy type  of $\calB(1)$ -- surprises continue}
\label{spaci_medvidek}
\label{s3}

In this section we study, as a next step toward the understanding of
$\calB_\calP$, the homotopy type of the associative dg-algebra
$\calB_\calP(1) = (\calB^*_\calP(1),\delta_\calP)$.  Since the operad
$\calP^!$ is a module, in the sense of~\cite{markl:zebrulka}, over itself,
it makes sense to consider the space $\Endom_{\calP^!}({\calP^!})$ of
all ${\calP^!}$-module endomorphisms $\alpha : {\calP^!} \to
{\calP^!}$. Very crucially,
\begin{equation}
\label{volal_jsem_Martinovi}
\Endom_{\calP^!}({\calP^!}) \cong \bfk,
\end{equation}
because each $\alpha \in \Endom_{\calP^!}({\calP^!})$ is uniquely
determined by the value $\alpha_1(1) \in {\calP^!}(1)\cong \bfk$ and,
conversely, for each $\varphi \in \bfk$ the homomorphism $\alpha :=
\varphi \cdot \id_{\calP^!}$ is such that $\alpha_1(1) = \varphi$.


\begin{proposition}
\label{modules}
There is a {\em canonical\/} identification of associative unital algebras
\begin{equation}
\label{8}
H^0(\calB^*_\calP(1),\delta_\calP) 
\cong  \Endom_{\calP^!}({\calP^!}) \cong \bfk.
\end{equation}
\end{proposition}


\noindent 
{\bf Proof.}
Since there are no elements in negative degrees, 
\[
H^0(\calB_\calP(1)) = \Ker\left( \delta : \calB^0_\calP(1) \to
\calB^1_\calP(1)\right).
\]
By definition, elements of the kernel  $\Ker(\delta)$ are natural chain
maps
\[
\{\varphi_m : C^m_\calP(A;A) \to  C^m_\calP(A;A)\}_{m \geq 0}.
\]
As explained in Example~\ref{Deda_Mraz}, the naturality of $\varphi_m$
means that it is induced by a $\Sigma_{m+1}$-equivariant map
$\alpha_{m+1} : {\calP^!}(m+1) \to {\calP^!}(m+1)$ .  It is easy to
verify that the collection $\{\alpha_m\}$ determines a chain map if
and only if it assembles into a ${\calP^!}$-module endomorphism
$\alpha : {\calP^!} \to {\calP^!}$.
\qed


The following example shows that the dg-associative algebra
$\calB_\calP(1)$ might in general have nontrivial cohomology in
positive degrees.


\begin{example}
\label{sym}
{\rm Let $\Sym$ be the operad describing algebras with one commutative
bilinear multiplication and no axioms. Explicitly, $\Sym$ is the free
operad generated by the trivial representation of $\Sigma_2$ placed in
arity two.  It is a Koszul quadratic operad whose quadratic dual
$\Sym^!$ is given by $\Sym^!(1) = \bfk$, $\Sym^!(2) = \sgn_2$ (the
signum representation of $\Sigma_2$), and $\Sym^!(m) = 0$ for $m \geq
3$.

The cohomology of a $\Sym$-algebra $A = (A,\ \cdot\ )$ is the
cohomology of the two-term complex (which should be interpreted as a
truncation of the Harrison complex)
\[
\Lin(A,A) \stackrel{d}{\longrightarrow} \Lin(S^2 A,A),
\]
where $S^2 A$ is the second symmetric power of $A$.
The differential $d$ is given by the usual formula
\[
(d \phi)(a,b) := a \cdot \phi(b) - \phi(a\cdot b) + \phi(b) \cdot a,
\]
for $\phi \in \Lin(A,A)$ and $a,b \in A$.

We are going to describe the dg-algebra $\calB^*_\Sym(1)$.
Let $\alpha$ be the projection of $\Lin(A,A) \oplus \Lin(S^2 A,A)$
onto $\Lin(A,A)$ and $\beta$ the
projection onto $\Lin(S^2 A,A)$. Let $u$ and $v$ be degree $+1$
operations given by 
\[
u(\phi)(a,b) := a \cdot \phi(b) + \phi(a) \cdot b
\mbox { and }
v(\phi)(a,b) := \phi(a\cdot b),
\] 
for $\phi \in \Lin(A,A)$ and $a,b \in A$. Then clearly 
$\calB^0_\Sym(1)$ is the semisimple algebra $\bfk \oplus \bfk$ spanned by
$\alpha$ and $\beta$, 
and the space $\calB^1_\Sym(1)$ is two-dimensional, spanned by
$u$ and $v$. The higher $\calB^i_\Sym(1)$ are, for $i \geq 2$, trivial. To
describe the multiplication in $\calB^*_\Sym(1)$, it is enough to specify
how $\calB^0_\Sym(1)$ acts on $\calB^1_\Sym(1)$. This action is given by
\[
\alpha b = 0 = b\beta\ \mbox { and } b \alpha = b = \beta b, \ \mbox {
for $b \in \calB^1_\Sym(1)$.} 
\]
The differential $\delta_\Sym$ of $\calB^*_\Sym(1)$ acts by
\[
\delta \alpha = -\delta \beta = u - v,\
\delta u = \delta v = 0.
\]
The cohomology of $(\calB^*_\Sym(1),\delta_\Sym)$ can be easily calculated,
\[
H^*(\calB^*_\Sym(1)) \cong \bfk \oplus W,
\]
where $W$ is the vector space spanned by the class
$[u]$. We leave as a simple exercise to prove that there
exist a quasi-isomorphism $H^*(\calB^*_\Sym(1)) \to \calB^*_\Sym(1)$. 
The dg-associative algebra $\calB_\Sym(1)$ is therefore {\em formal\/}.
}
\end{example}

Here is a baby version of Problem~\ref{prob1}:

\begin{problem}
\label{prob3}
Describe the homotopy type of the unital differential graded
associative algebra $\calB_\calP(1) =
(\calB^*_\calP(1),\delta_\calP)$.  In particular, calculate the
cohomology of $\calB_\calP(1)$.
\end{problem}

We expect that the homotopy type of $\calB_\calP(1)$ is that of $\bfk$
for all ``reasonable'' operads, though we do not know what
``reasonable'' means -- the operad $\Sym$ of Example~\ref{sym} seems
reasonable enough, yet the homotopy type of $\calB_\Sym(1)$ is
nontrivial.
Let us close this section by formulating:

\begin{problem}
\label{+}
Which property of the operad $\calP$ implies that the
dg associative algebra $(\calB^*_\calP(1),\delta_\calP)$ has the
homotopy type of $\bfk$?
\end{problem}

\section{The operad $H^0(\calB_\calP)$ and the intrinsic bracket}
\label{jsem_otrok}
\label{s4}

It is well-known~\cite{markl:ib,schlessinger-stasheff:JPAA85} that the
chain complex $C^*_\calP(A;A)$ always carries a natural dg Lie algebra
structure given by the {\em intrinsic bracket\/}.  The easiest way to
construct such a bracket is to identify $C^*_\calP(A;A)$ with the dg
Lie algebra ${\it Coder\/}^*({\mathcal F}^c_{\calP^!}(\downarrow \hskip
-.2em A))$ of coderivations of the cofree nilpotent
$\calP^!$-coalgebra cogenerated by the desuspension $\downarrow \hskip
-.2em A$ as it was done
in~\cite[Definition~II.3.99]{markl-shnider-stasheff:book}.  In this
way we obtain a natural homomorphism
\begin{equation}
\label{lepidlo}
I: (\Lie,0) \to (\calB_\calP,\delta_\calP)
\end{equation}
of dg operads. If $\calP$ is the symmetrization of a non-$\Sigma$ operad
$\ucalP$, then $\Im(I) \subset \calB_\ucalP$, therefore the map $I$
of~(\ref{lepidlo}) factorizes through the natural inclusion
$(\calB_\ucalP,\delta_\ucalP) \hookrightarrow
(\calB_\calP,\delta_\calP)$. Computational evidences lead us to:

\begin{conjecture}
\label{ib}
The natural homomorphism $I: (\Lie,0) \to (\calB_\calP,\delta_\calP)$ 
induces an isomorphism of operads
\[
 H^0(\calB_\calP) \cong \Lie, 
\]
for an arbitrary nontrivial quadratic Koszul $\calP$.
\end{conjecture}

We were able to verify Conjecture~\ref{ib} for $\calP = \Lie$, that is,
to prove
\begin{equation}
\label{naspet_od_MF}
H^0(\calB_\Lie) \cong \Lie.
\end{equation}
This isomorphism turned out to be related to a certain
characterization of free Lie algebras inside free pre-Lie
algebras. More precisely, let $\freepreLie(X)$ denote the free pre-Lie
algebra generated by a set $X$. The commutator of the pre-Lie product
makes $\freepreLie(X)$ a Lie algebra. Let $\freeLie(X) \subset
\freepreLie(X)$ be the Lie algebra generated by $X$ in
$\freepreLie(X)$. It is not hard to see that $\freeLie(X)$ is in fact
isomorphic to the free Lie algebra on $X$, see also~\cite{dzhu-lof:HHA02}.
Then~(\ref{naspet_od_MF}) is implied by a very explicit
characterization of the subspace $\freeLie(X)$ of $\freepreLie(X)$.

Similarly, the conjectured isomorphism 
$H^0(\calB_\uAss) \cong \Lie$ can be
translated into a certain characterization of free Lie algebras inside free
brace algebras.
We were also able to prove that, for an arbitrary quadratic Koszul operad
\begin{equation}
\label{co_to_ten_Jim_ji?}
H^0(\calB_\calP(2)) \cong \sgn_2,
\end{equation}
the signum representation of $\Sigma_2$, by describing
$H^0(\calB_\calP(2))$ in terms of suitably defined {\em pairings\/}
$\calP^! \ot \calP^! \to \calP^!$.

Let us close this section by a couple of remarks which will be useful
in the proof of Proposition~\ref{lisasek}. As we recalled at the
beginning of this section, there is a canonical isomorphism
$C^*_\calP(A;A) \cong {\it Coder\/}^*({\mathcal F}^c_{\calP^!}(\downarrow 
\hskip -.2em A))$. It is well-known that coderivations of a cofree
nilpotent algebra form a natural pre-Lie
algebra~\cite[II.3.9]{markl-shnider-stasheff:book}, therefore one has
a natural homomorphism of non-dg operads
\begin{equation}
\label{nasadil_jsem_retez}
{\it pre}I : \preLie \to \calB_\calP.
\end{equation}
The map~(\ref{lepidlo}) is then the composition
\[
\Lie \longrightarrow \preLie \stackrel{{\it pre}I}{\longrightarrow}
\calB_\calP 
\]
of ${\it pre}I$ with the anti-symmetrization map $\Lie \to \preLie$. 

\section{The cup products}
\label{cup}
\label{s5}


The central statement of this section is Theorem~\ref{zitra_prednaska}
that claims that the suspension $\ss(\calP \ot \calP^!)$
(see~A.\ref{odpalil_jsem_notebook}) of the operad $(\calP \ot
\calP^!)$ acts on $\CAP*$, and Theorem~\ref{poslouchal_jsem_Kryla}
that characterizes which elements of $\ss(\calP \ot \calP^!)$ act via
chain maps.  Observe that the operad $\ss(\calP \otimes \calP^!)$ need
not be quadratic even when $\calP$ is.


\begin{theorem}
\label{zitra_prednaska}
There is a canonical action of the operad $\ss(\opt)$
on  the graded vector space $\CAP*$, via natural operations.
This action can be interpreted as an inclusion of non-differential
graded operads
\begin{equation}
\label{dnes_jsem_mel_prednasku_ve_Filadelfii}
\ccc: \ss(\calP \otimes \calP^!) \hookrightarrow \calB_\calP.
\end{equation}
\end{theorem}


\noindent 
{\bf Proof.}  The proof relies on the notation introduced/recalled
in~A.\ref{Ferda} and~A.\ref{zase_do_Prahy}.  
The ``tautological'' action of the endomorphism
operad $\End_A$ on $A$ tensored with the action of $\calP^!$ on itself
makes the graded vector space $\tildeCAP* = \bigoplus_{m \geq
0}\tildeCAP m$ an $\ss(\End_A \ot \calP^!)$-algebra. It is
straightforward to prove that this action induces, via
\begin{equation}
\label{zitra_letim_do_Raleigh}
t(f_1,\ldots,f_n) := {\it Aver\/}\left(\rule{0em}{1em}
t(\iota(f_1),\ldots,\iota(f_n))\right),
\end{equation}
for $t \in \ss(\End_A \ot\calP^!)(n)$ and $f_1,\ldots,f_n \in \CAP*$,
an action of $\ss(\End_A \ot \calP^!)$ on the graded vector space
$\CAP*$.  Suppose that $A$ is a $\calP$-algebra, with the structure
given by $\alpha : \calP \to \End_A$.
Action~(\ref{dnes_jsem_mel_prednasku_ve_Filadelfii}) is obtained by
composing action~(\ref{zitra_letim_do_Raleigh}) with the homomorphism
$\ss(\alpha \ot \id) : \ss(\calP \ot \calP^!) \to \ss(\End_A \ot
\calP^!)$.  An alternative description
of~(\ref{dnes_jsem_mel_prednasku_ve_Filadelfii}) is given in
Example~\ref{fit} of Section~\ref{natural}.% 
\qed


We use inclusion~(\ref{dnes_jsem_mel_prednasku_ve_Filadelfii})
to view $\ss(\calP \ot \calP^!)$ 
as a suboperad of $\calB_\calP$.
Elements of $\ss(\calP \ot \calP^!)$ need not be $\delta_\calP$-closed 
in $\calB_\calP$;
let $\calZ_\calP \subset \ss(\calP \otimes \calP^!)$ 
denote the suboperad of $\delta_\calP$-cocycles. 
In Proposition~\ref{poslouchal_jsem_Kryla}, which describes
$\calZ_\calP$ explicitly, we use
the canonical element $\chi$ introduced in Definition~\ref{canon}.

\begin{theorem}
\label{poslouchal_jsem_Kryla}
The suboperad $\calZ_\calP$ of $\delta_\calP$-closed elements in
$\ss(\calP \ot \calP^!)$ is characterized as follows: $t \in \ss(\calP
\otimes \calP^!)(n)$ belongs to $\calZ_\calP(n)$ if and only if
\begin{equation}
\label{s_ni_jsem_se_natrapil}
\chi \circ_2 t + t \circ_1 \chi +   (t \circ_2 \chi) (12) +
 (t \circ_3 \chi) (123)  + \cdots +  (t\circ_n \chi) (123 \cdots n) = 0,
\end{equation}
where $(123\cdots k) \in \Sigma_{n+1}$ is the cycle
\[
\left(
\begin{array}{cccccccc}
1&2&3&\cdots&k&k+1&\cdots&n+1
\\
2&3&4&\cdots&1&k+1&\cdots&n+1
\end{array}
\right).
\]
\end{theorem}

The {\bf proof} is a completely straightforward calculation. We
recommend as an exercise to verify that solutions
of~(\ref{s_ni_jsem_se_natrapil}) are indeed closed under operadic
composition.  The meaning of equation~(\ref{s_ni_jsem_se_natrapil})
should be clear from Figure~\ref{jj}. The importance of the operad
$\calZ_\calP$ is explained by

\begin{figure}[t]
\begin{center}
{% Picture saved by xtexcad 2.4
\unitlength=1.000000pt
\begin{picture}(330.00,55.00)(0.00,10.00)
\thicklines
\put(330.00,40.00){\makebox(0.00,0.00){$=\ 0$}}
\put(240.00,40.00){\makebox(0.00,0.00){$+$}}
\put(160.00,40.00){\makebox(0.00,0.00){$+$}}
\put(70.00,40.00){\makebox(0.00,0.00){$+$}}
%CTVRTY OBRAZEK
%prvni dve zalomene
\put(280.00,30.00){\line(0,-1){10.00}}
\put(270.00,30.00){\line(0,-1){10.00}}
\put(280.00,20.00){\line(1,-2){10.00}}
\put(270.00,20.00){\line(1,-2){10.00}}
%
\put(-5,0){
\put(310.00,20.00){\makebox(0.00,0.00){$\chi$}}
\put(300.00,20.00){\makebox(0.00,0.00){$\bullet$}}
\put(300.00,20.00){\line(-3,-1){29.5}}
\put(310.00,10.00){\line(0,-1){10.00}}
\put(270.00,10.00){\line(0,-1){10.00}}
\put(300.00,20.00){\line(1,-1){10.00}}
}

\put(280.00,40.00){\makebox(0.00,0.00){$t$}}
\put(290.00,30.00){\line(1,-2){5.00}}
\put(280.00,50.00){\line(0,1){20.00}}
\put(280.00,50.00){\line(-1,-1){20.00}}
\put(300.00,30.00){\line(-1,1){20.00}}
\put(260.00,30.00){\line(1,0){40.00}}
%
%TRETI OBRAZEK
\put(203.00,20.00){\makebox(0.00,0.00)[lb]{$\chi$}}
\put(200.00,40.00){\makebox(0.00,0.00){$t$}}
\put(200.00,20.00){\makebox(0.00,0.00){$\bullet$}}
%cara co jsem zlomil
\put(190.00,30.00){\line(0,-1){10.00}}
\put(190.00,20.00){\line(1,-2){10.00}}
\put(190.00,10.00){\line(-1,-3){3.33333}}
\put(210.00,10.00){\line(0,-1){10.00}}
\put(200.00,20.00){\line(1,-1){10.00}}
\put(200.00,20.00){\line(-1,-1){10.00}}
\put(200.00,30.00){\line(0,-1){10.00}}
\put(220.00,20.00){\line(0,-1){20.00}}
\put(210.00,30.00){\line(1,-1){10.00}}
\put(200.00,70.00){\line(0,-1){20.00}}
\put(200.00,50.00){\line(-1,-1){20.00}}
\put(220.00,30.00){\line(-1,1){20.00}}
\put(180.00,30.00){\line(1,0){40.00}}
%
%DRUHY OBRAZEK
\put(100.00,20.00){\makebox(0.00,0.00){$\bullet$}}
\put(120.00,40.00){\makebox(0.00,0.00){$t$}}
\put(90.00,20.00){\makebox(0.00,0.00){$\chi$}}
\put(130.00,30.00){\line(0,-1){30.00}}
\put(120.00,30.00){\line(0,-1){30.00}}
\put(107.00,10.00){\line(0,-1){10.00}}
\put(100.00,20.00){\line(2,-3){6.7}}
\put(100.00,20.00){\line(-2,-3){6.7}}
\put(93.00,10.00){\line(0,-1){10.00}}
\put(110.00,30.00){\line(-1,-1){10.00}}
\put(120.00,50.00){\line(0,1){20.00}}
\put(120.00,50.00){\line(-1,-1){20.00}}
\put(140.00,30.00){\line(-1,1){20.00}}
\put(100.00,30.00){\line(1,0){40.00}}
%
%PRVNI OBRAZEK
\put(20.00,70.00){\line(0,-1){20.00}}
\put(20.00,50.00){\makebox(0.00,0.00){$\bullet$}}

\put(10.00,60.00){\makebox(0.00,0.00){$\chi$}}
\put(6.8,30.00){\line(0,-1){30.00}}
\put(20.00,50.00){\line(-2,-3){13}}
\put(20.00,50.00){\line(2,-3){13}}
%pravy trojuhelnik
\put(-7,0){
\put(40.00,20.00){\makebox(0.00,0.00){$t$}}
\put(50.00,10.00){\line(0,-1){10.00}}
\put(40.00,10.00){\line(0,-1){10.00}}
\put(30.00,10.00){\line(0,-1){10.00}}
\put(40.00,30.00){\line(-1,-1){20.00}}
\put(60.00,10.00){\line(-1,1){20.00}}
\put(20.00,10.00){\line(1,0){40.00}}
}
\end{picture}}
\end{center}
\caption{\label{jj}%
Equation~(\protect\ref{s_ni_jsem_se_natrapil}) for $n=3$.}
\end{figure}

\begin{corollary}
The map $\ccc$ of~(\ref{dnes_jsem_mel_prednasku_ve_Filadelfii}) 
induces a canonical map (denoted again $\ccc$)
\begin{equation}
\label{kralicek}
\ccc : \calZ_\calP \to H^*(\calB_\calP,\delta_\calP),
\end{equation}
therefore $\HAP*$ is a natural $\calZ_\calP$-algebra.
\end{corollary}

From reasons which become clear later we call operations induced by
elements of $\calZ_\calP$ the {\em cup products\/}.  The following
proposition in which $\Lie$ is the operad for Lie algebras
(see~A.\ref{posloucham_Boroveho}) shows that the operad $\calZ_\calP$
is {\em always\/} nontrivial (provided $\calP \not = {\bf 1}$) while the
map~(\ref{kralicek}) is {\em never\/} monic.


\begin{proposition}
\label{lisasek}
The operad $\calZ_\calP$ contains the canonical element $\chi$.  There
exists a unique map $L: \ss \Lie \to \calZ_\calP$ that sends the
generator $\ss \lambda \in \ss \Lie(2)$ into $\chi \in
\calZ_\calP(2)$.  All elements in the image of this map are
$\delta_\calP$-cohomologous to zero in $\calB_\calP$.
\end{proposition}


\noindent
{\bf Proof.}
Recall~\cite[Corollary~2.2.9(b)]{ginzburg-kapranov:DMJ94} that, for
each quadratic operad $\calP$, there exits a morphism of operads $\Lie
\to \calP \ot \calP^!$ that takes the generator $\lambda \in \Lie(2)$
into the identity operator in $\calP(2) \ot \calP(2)^\# \cong \calP(2)
\ot \calP^!(2)$. Let $L : \ss{} \Lie \to \ss{}(\calP \ot \calP^!)$ be
the suspension of this morphism.  Let us prove, using
Theorem~\ref{poslouchal_jsem_Kryla}, that $\chi \in \calZ_\calP(2)$.
Equation~(\ref{s_ni_jsem_se_natrapil}) for $t = \chi$ reads
\[
\chi \circ_2 \chi + \chi \circ_1 \chi + (\chi \circ_2 \chi)(12) =0,
\]
which can be written, due to the symmetry~(\ref{porad_to_knika}) of
$\chi$, as the Jacobi identity for a degree $1$ ``multiplication''
$\chi$:
\begin{equation}
\label{zase_jsem_se_opil}
\chi \circ_1 \chi +  (\chi \circ_1 \chi)(123) + (\chi \circ_1 \chi)(132) =0,
\end{equation}
or, pictorially,
\[
{% Picture saved by xtexcad 2.4
\unitlength=1.00000pt
\thicklines
\begin{picture}(170.00,55.00)(-160.00,0)
\put(-110.00,30.00){\makebox(0.00,0.00){$+$}}
\put(-50.00,30.00){\makebox(0.00,0.00){$+$}}
\put(15.00,30.00){\makebox(0.00,0.00){$=\ 0$}}
\put(-120.00,0.00){\makebox(0.00,0.00){$3$}}
\put(-80.00,0.00){\makebox(0.00,0.00){$1$}}
\put(-40.00,0.00){\makebox(0.00,0.00){$2$}}
\put(-160.00,0.00){\makebox(0.00,0.00){$1$}}
\put(-60.00,0.00){\makebox(0.00,0.00){$2$}}
\put(-20.00,0.00){\makebox(0.00,0.00){$3$}}
\put(-140.00,0.00){\makebox(0.00,0.00){$2$}}
\put(-100.00,0.00){\makebox(0.00,0.00){$3$}}
\put(-0.00,0.00){\makebox(0.00,0.00){$1$}}
\multiput(-35.00,20.00)(-60,0){3}{\makebox(0.00,0.00)[br]{$\chi$}}
\multiput(-25.00,30.00)(-60,0){3}{\makebox(0.00,0.00)[br]{$\chi$}}
\put(-30.00,20.00){\makebox(0.00,0.00){$\bullet$}}
\put(-90.00,20.00){\makebox(0.00,0.00){$\bullet$}}
\put(-150.00,20.00){\makebox(0.00,0.00){$\bullet$}}
\put(-140.00,30.00){\makebox(0.00,0.00){$\bullet$}}
\put(-80.00,30.00){\makebox(0.00,0.00){$\bullet$}}
\put(-20.00,30.00){\makebox(0.00,0.00){$\bullet$}}
\put(-150.00,20.00){\line(0,1){0.00}}
\put(-140.00,10.00){\line(-1,1){10.00}}
\put(-140.00,50.00){\line(0,-1){20.00}}
\put(-140.00,30.00){\line(-1,-1){20.00}}
\put(-120.00,10.00){\line(-1,1){20.00}}
\put(-80.00,50.00){\line(0,-1){20.00}}
\put(-90.00,20.00){\line(1,-1){10.00}}
\put(-80.00,30.00){\line(-1,-1){20.00}}
\put(-60.00,10.00){\line(-1,1){20.00}}
\put(-20.00,10.00){\line(-1,1){10.00}}
\put(-20.00,30.00){\line(-1,-1){20.00}}
\put(-0.00,10.00){\line(0,1){0.00}}
\put(-20.00,30.00){\line(1,-1){20.00}}
\put(-20.00,50.00){\line(0,-1){20.00}}
\end{picture}}
\]

\noindent 
But~(\ref{zase_jsem_se_opil}) is satisfied, because $\chi = L(\ss{}
\lambda)$ by definition, and $\ss{} \lambda \in \ss{}\Lie (2)$
satisfies the same condition in $\ss \Lie$. The inclusion $\Im(L)
\subset \calZ_\calP$ follows from the fact that $\Im(L)$ is generated
by $\chi$ and that $\calZ_\calP$ is a suboperad of $\ss{}(\calP \ot
\calP^!)$.

Let us prove that all elements in the image of $L$ are
$\delta_\calP$-cohomologous to zero. Let $\ell \in \preLie(2)$ be the
generator of the quadratic operad $\preLie$ for pre-Lie algebras and
let $\circ := {\it pre}I(\ell) \in \calB^0_\calP(2)$, where ${\it
preI} : \preLie \to \calB_\calP$ is the map considered
in~(\ref{nasadil_jsem_retez}) at the end of
Section~\ref{jsem_otrok}. It is easy to verify that then
$\chi = \delta_\calP (\circ)$. This finishes the proof of
Proposition~\ref{lisasek}, because $\Im(L)$ is generated by $\chi$.%
\qed

Suppose that $\calP$ is the symmetrization of a non-$\Sigma$ operad
$\ucalP$.  Given $t \in \ss(\calP \ot \calP^!)(n)$ as in
Theorem~\ref{poslouchal_jsem_Kryla}, $\ccc(t) \in \calB_\ucalP(n)$ if
and only if $t$ belongs to the $\Sigma_n$-closure of $\ss(\ucalP \ot
\ucalP^!)(n)$ in $\ss(\calP \ot \calP^!)(n)$, that is, if $t =
\underline{t}\sigma$ for some $\underline{t} \in \ss(\ucalP \ot
\ucalP^!)(n)$ and $\sigma \in \Sigma_n$. In the following non-$\Sigma$
version of Theorem~\ref{poslouchal_jsem_Kryla}, $\uchi \in \ss
(\ucalP \ot \ucalP^!)(2)$ is the non-$\Sigma$ canonical element
introduced in Section~\ref{s2}.

\begin{theorem}
\label{tuzka}
An element $\ut \in \ss(\ucalP \ot \ucalP^!)(n)\subset \ss(\calP \ot
\calP^!)(n)$ belongs to
$\calZ_\calP(n)$ if and only if
\begin{equation}
\label{kocicka}
\uchi \circ_2 \ut = \ut \circ_1 \uchi =  \ut \circ_2 \uchi
= \cdots =  \ut \circ_n \uchi= \uchi \circ_1 \ut,
\end{equation}
see Figure~\ref{iii}.
\end{theorem}

\begin{figure}[t]
\begin{center}
{% Picture saved by xtexcad 2.4
\unitlength=1.000000pt
\begin{picture}(400.00,60.00)(0.00,10.00)
\thicklines
%PRVNI OBRAZEK
\put(20.00,70.00){\line(0,-1){20.00}}
\put(20.00,50.00){\makebox(0.00,0.00){$\bullet$}}
\put(10.00,60.00){\makebox(0.00,0.00){$\underline{\chi}$}}
\put(6.8,30.00){\line(0,-1){30.00}}
\put(20.00,50.00){\line(-2,-3){13}}
\put(20.00,50.00){\line(2,-3){13}}
%pravy trojuhelnik
\put(-7,0){
\put(40.00,20.00){\makebox(0.00,0.00){$\underline{t}$}}
\put(50.00,10.00){\line(0,-1){10.00}}
\put(40.00,10.00){\line(0,-1){10.00}}
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\put(60.00,10.00){\line(-1,1){20.00}}
\put(20.00,10.00){\line(1,0){40.00}}
}
\put(280.00,40.00){\makebox(0.00,0.00){$\underline{t}$}}
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\put(330.00,40.00){\makebox(0.00,0.00){$=$}}
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\put(300.00,20.00){\line(-1,-1){10.00}}
\put(310.00,10.00){\line(0,-1){10.00}}
\put(300.00,20.00){\line(1,-1){10.00}}
\put(290.00,30.00){\line(1,-1){10.00}}
\put(280.00,30.00){\line(0,-1){30.00}}
\put(270.00,30.00){\line(0,-1){30.00}}
\put(280.00,50.00){\line(0,1){20.00}}
\put(280.00,50.00){\line(-1,-1){20.00}}
\put(300.00,30.00){\line(-1,1){20.00}}
\put(260.00,30.00){\line(1,0){40.00}}
\put(240.00,40.00){\makebox(0.00,0.00){$=$}}
\put(160.00,40.00){\makebox(0.00,0.00){$=$}}
\put(70.00,40.00){\makebox(0.00,0.00){$=$}}
\put(190.00,20.00){\makebox(0.00,0.00){$\underline{\chi}$}}
\put(90.00,20.00){\makebox(0.00,0.00){$\underline{\chi}$}}
\put(200.00,40.00){\makebox(0.00,0.00){$\underline{t}$}}
\put(120.00,40.00){\makebox(0.00,0.00){$\underline{t}$}}
\put(200.00,20.00){\makebox(0.00,0.00){$\bullet$}}
\put(100.00,20.00){\makebox(0.00,0.00){$\bullet$}}
\put(210.00,10.00){\line(0,-1){10.00}}
\put(200.00,20.00){\line(1,-1){10.00}}
\put(190.00,10.00){\line(0,-1){10.00}}
\put(200.00,20.00){\line(-1,-1){10.00}}
\put(200.00,30.00){\line(0,-1){10.00}}
\put(220.00,20.00){\line(0,-1){20.00}}
\put(210.00,30.00){\line(1,-1){10.00}}
\put(180.00,20.00){\line(0,-1){20.00}}
\put(190.00,30.00){\line(-1,-1){10.00}}
\put(200.00,70.00){\line(0,-1){20.00}}
\put(200.00,50.00){\line(-1,-1){20.00}}
\put(220.00,30.00){\line(-1,1){20.00}}
\put(180.00,30.00){\line(1,0){40.00}}
\put(130.00,30.00){\line(0,-1){30.00}}
\put(120.00,30.00){\line(0,-1){30.00}}
\put(110.00,10.00){\line(0,-1){10.00}}
\put(100.00,20.00){\line(1,-1){10.00}}
\put(90.00,10.00){\line(0,-1){10.00}}
\put(110.00,30.00){\line(-1,-1){20.00}}
\put(120.00,50.00){\line(0,1){20.00}}
\put(120.00,50.00){\line(-1,-1){20.00}}
\put(140.00,30.00){\line(-1,1){20.00}}
\put(100.00,30.00){\line(1,0){40.00}}
%POSLEDNI OBRAZEK
\put(400,0){
\put(-20.00,70.00){\line(0,-1){20.00}}
\put(-20.00,50.00){\makebox(0.00,0.00){$\bullet$}}
\put(-10.00,60.00){\makebox(0.00,0.00){$\underline{\chi}$}}
\put(-6.8,30.00){\line(0,-1){30.00}}
\put(-20.00,50.00){\line(2,-3){13}}
\put(-20.00,50.00){\line(-2,-3){13}}
%pravy trojuhelnik
\put(7,0){
\put(-40.00,20.00){\makebox(0.00,0.00){$\underline{t}$}}
\put(-50.00,10.00){\line(0,-1){10.00}}
\put(-40.00,10.00){\line(0,-1){10.00}}
\put(-30.00,10.00){\line(0,-1){10.00}}
\put(-40.00,30.00){\line(1,-1){20.00}}
\put(-60.00,10.00){\line(1,1){20.00}}
\put(-20.00,10.00){\line(-1,0){40.00}}
}
}
\end{picture}}
\end{center}
\caption{\label{iii}
Equation~(\protect\ref{kocicka}) for $n=3$.}
\end{figure}

{\bf Proof} of Theorem~\ref{tuzka} 
is a straightforward verification. The {\bf proof}
of the following proposition is similar to that of
Proposition~\ref{lisasek}. 

 
\begin{proposition}
\label{posledni_den_v_Phily}
Let $\calP$ be the symmetrization of a non-$\Sigma$ operad
$\ucalP$. Then $\uchi \in \calZ_\calP$ and there exists a unique map
$A : \ss\As \to \calZ_\calP$ defined by $A(\ss \mu) := \uchi$, where
$\ss\mu \in \ss\As(2)$ is the suspension of the generator $\mu$
(see~A.\ref{posloucham_Boroveho}). Moreover, the diagram
\[
\Dtriangle {\ss\Lie}{\calZ_\calP}{\ss\As}{L}{}{A}
\]
where $L$ is as in Proposition~\ref{lisasek},
with the vertical map given by the anti-commutator of the
associative product, commutes.
\end{proposition}


\begin{example}
{\em -- Hochschild cohomology.
Let $\calP = \Ass{}$ be the operad for associative algebras. 
Then $\ss(\calP \ot \calP^!) = \ss(\As \ot \As)$
and a simple calculation reveals that the map $A$ of
Proposition~\ref{posledni_den_v_Phily} is the suspended diagonal $\ss\Delta:
\ss\As \to \ss(\As \ot \As)$ and that $\calZ_\Ass = \Im(A)$. Therefore 
\[
\calZ_\As \cong \ss \As.
\] 
The generator $\ss\mu\in \ss \As(2)$ is mapped to the ``classical'' cup product
$f,g \mapsto f\cup g$ of Hochschild cochains~\cite{gerstenhaber:AM63}, 
and the generator $\ss\lambda \in \ss\Lie(2)$ to the anti-commutator
of this cup product:
\[
f,g \mapsto f\cup g +(-1)^{|f||g|} g \cup f
\]
which cohomologous to zero, because the cup product of Hochschild
cochains is homotopy commutative~\cite[Theorem~3]{gerstenhaber:AM63}.
}  
\end{example}

\begin{example}
\label{oslicek1} 
{\em -- Chevalley-Eilenberg cohomology. If
$\calP = \Lie$ is the operad for Lie algebras, 
then $\ss(\calP \ot \calP^!) = \ss(\Lie \ot \Com) \cong
\ss\Lie$
and we see immediately that
\begin{equation}
\label{hhh}
\calZ_\Lie \cong \ss\Lie = \Im(L).
\end{equation}
The generator $\ss\lambda \in \Lie(2)$ is mapped to the product $f,g
\mapsto \{f,g\}$, which is cohomologous to zero,
see~\cite[Exercise~7]{lm:sb}. 
}  
\end{example}


\begin{example}
\label{oslicek2}
{\em -- Harrison cohomology. Here
$\calP = \Com$ is the operad for commutative associative algebras 
and  $\ss(\calP \ot \calP^!) = \ss(\Com \ot \Lie)  \cong \ss\Lie$,
therefore, as in Example~\ref{oslicek1},
\begin{equation}
\label{1hhh}
\calZ_\Com \cong \ss \Lie= \Im(L).
\end{equation}
Equations~(\ref{hhh}) and~(\ref{1hhh}) illustrate the obvious
self-duality of the space of cup products:
\[
\calZ_{\calP^!} \cong \calZ_{\calP},
\]
compare Conjecture~\ref{zase}.
}  
\end{example}


\begin{example}
{\em
If $\calD = \As * \As$ is as in 
Example~\ref{ve_stredu_poletim_do_Minneapolis}, then
\[
\calZ_\calD = \ss \As \vee \ss \As.
\]
Let us describe products corresponding to the
generators of the $4$-dimensional vector space
\[
(\ss \As \vee \ss \As)(2) =
\ss\As(2) \oplus \ss\As(2) 
\cong \hskip .2em \uparrow \bfk[\Sigma_2] \oplus  \uparrow
\bfk[\Sigma_2].
\]

Recall that $\CAD*$ is the total complex of the meager bicomplex
in Figure~\ref{jsem_nachlazeny}.
Let $\cup^1$ (resp.~$\cup^2$) be the cup product
in the horizontal (resp.~vertical) subcomplex in
Figure~\ref{jsem_nachlazeny}.  
Let $\pi_1$ (resp.~$\pi_2$) be the projection of $\CAD*$ onto the horizontal
(vertical) subcomplex. Likewise, let $\iota_1$
(resp.~$\iota_2$) be the inclusion. Although neither
$\pi_i$, $\iota_i$ nor $\cup^i$ are chain maps  ($i =
1,2$), the compositions
\[
f \cup_1 g : = \iota_1(\pi_1 f \cup^1 \pi_1 g)\
\mbox{ and } 
f \cup_2 g : = \iota_2(\pi_2 f \cup^2 \pi_2 g)
\]
are chain operations. The generators of $\ss\As(2) \oplus \ss\As(2)$
then correspond to the four operations 
\[
f,g \mapsto f\cup_1 g,\  f,g \mapsto f\cup_2 g,\
f,g \mapsto g\cup_1 f\ \mbox { and } f,g \mapsto g\cup_2 f.
\] 
The combination
\[
(f\cup_1 g + f\cup_2 g) + (-1)^{|f||g|} (g\cup_1 f + g\cup_2 f)
\]
is cohomologous to zero and the image $T(\calZ_\calD(2))$ of
$\calZ_\calD$ in $H^1(\calB_\calD(2))$ is easily seen to be
$3$-dimensional.  }
\end{example}


\section{Operad $\calB_\Ass$ and the Deligne conjecture}
\label{pisi_na_novem_notebooku}
\label{s6}

In this section we recall some results related to $\calB_\Ass$ and the
Deligne conjecture. Let us make a necessary comment about our degree
convention. We use the grading such that the intrinsic bracket
of Section~\ref{s4} has degree $0$ in $\calB^*_\calP(2)$, while the
$n$-fold cup products of Section~\ref{s5} are of degree $n-1$ in
$\calB^*_\calP(n)$. In the literature related to the Deligne
conjecture, the convention under which the intrinsic bracket has degree
$1$ and the $n$-fold cup products are of degree
$0$ is often used. These two conventions are
tied by the following regrading operator:
\[
\Reg(\calB^*_\calP (n)) := \calB^{n-1-*}_\calP(n).
\]

In what follows we identify operads that differ only by the above
regrading. In particular, the operad $\Gerst$ for Gerstenhaber algebras
becomes identified with the operad $\Braid$ for braid algebras (also
called 1-algebras), see~A.\ref{ma_prejit_fronta_v_sobotu}. 

Let us recall that a topological operad $\calA$ is an {\em
$E_2$-operad\/} if it has the homotopy type of the little discs operad
${\disc}_2$~\cite{may:1972}. According to the Formality
Theorem~\cite{tamarkin:formality}, the operad $S_*(\calA)$ of singular
chains on such an operad has the homotopy type of the operad 
$\Braid$ for braid algebras.

(i) {\em D.~Tamarkin\/} and {\em B.~Tsygan\/} studied
in~\cite[Section~3]{tamarkin-tsygan:LMP01} a certain operad $F =
\{F(n)\}_{n \geq 1}$ of natural operations on the cosimplicial
Hochschild complex $C^\bullet(X,X)$ of a topological unital
monoid~$X$. The $n$-th space of this cosimplicial set is the space
${\it Cont\/}(X^{\times n},X)$ of continuous maps from the $n$-th
cartesian power of $X$ to $X$. For each $n \geq 1$, $F(n)$ is
a functor $(\Delta^{\it op})^n \times \Delta \to
{\sf Sets}$. They then considered a topological operad $E =
\{E(n)\}_{n \geq 1}$ whose pieces are the topological realizations of
these functors and claimed that $E$ is an $E_2$-operad. 

It is not difficult to see that the operad ${\it CN\/}_*(F)$ of
normalized chains of $F$ coincides with the operad
$\{\calB_\uAss(n)\}_{n \geq 1}$ (our $\calB_\uAss$ without
constants). Since $(\calB^*_\uAss(0),\delta_\uAss)$ is acyclic (see
Example~\ref{ctvrek_a_zase_je_lat}), we could conclude that
$\calB_\uAss$ has the homotopy type of $\Gerst$, but we must bear in
mind that the arguments in~\cite{tamarkin-tsygan:LMP01} were merely
sketched.

(ii) {\em J.E.~McClure\/} and {\em J.H.~Smith\/} considered
in~\cite{mcclure-smith:JAMS03} a dg-suboperad $\calS_2$ of their
``sequence'' operad $\calS$ and proved that $\calS_2$ naturally acts
on the Hochschild cochain complex of an associative algebra. In our
terminology this means that they constructed a canonical map $\calS_2
\to \calB_\Ass{}$. They then verified the Deligne conjecture by showing,
using a result of~\cite{berger:CM202}, that $\calS_2$ has the homotopy
type of the singular chain complex $S_* (\disc_2)$ of the little discs
operad.  Their proof is a very reliable one.

(iii) {\em M.~Kontsevich\/} and {\em
Y.~Soibelman\/}~\cite{kontsevich-soibelman} introduced a ``minimal
operad'' $M$ naturally acting on the Hochschild cochain complex of an
$A_\infty$-algebra. In our terminology, $M$ was a suboperad of
$\calB_{sh\Ass}$ generated by braces and cup products.  They then
argued that $M$ has the homotopy type of the operad of suitably
defined piecewise algebraic chains on the operad
$FM_2$ of the Fulton-MacPherson compactification of the configuration space of
points in ${\mathbb R}^2$. Since $FM_2$ is,
by~\cite[Proposition~3.9]{salvatore}, an $E_2$-operad,
they concluded that $M$ has the homotopy type of $\Gerst$. 

(iv) {\em R.M.~Kaufmann\/} realized in~\cite{kaufmann}
that the cellular chains ${\it CC\/}_*(\Cact^1)$ on his operad-up-to-homotopy
$\Cact^1$ of spineless normalized cacti is a honest operad which
naturally acts on the Hochschild cochain complex,
via braces and cup products. By comparing $\Cact^1$ to the operad
$\Cact$ of spineless (non-normalized) cacti, he concluded that
$CC_*(\Cact^1)$ is a model for chains on the little discs operad
$\disc_2$.


All the proofs of the Deligne conjecture mentioned above use
some special features of associative algebras and
$E_2$-operads, such as the cosimplicial structure of the Hochschild
cochain complex, Fiedorowicz' detection principle, or a relation to the
Fulton-MacPherson and cacti operads. None of these features are
available for a general operad $\calP$, we therefore think that the
analysis of the homotopy type of $\calB_\calP$ for a general Koszul
quadratic $\calP$ is substantially more difficult than the analysis of
$\calB_\As$. 

Let us mention that there are other approaches to the Deligne
conjecture, as D.~Tamarkin's proofs that use the Etingof-Kazhdan
quantization~\cite{hinich:FM03,tamarkin:deligne}, or those
based on a suitable filtration of the Fulton-MacPherson
compactification $FM_2$, see E.~Getzler and
J.D.S.~Jones~\cite{getzler-jones:preprint} or
A.A.~Voronov~\cite{voronov:99}.


\section{Natural operations}
\label{natural}

Let use recall the following definitions which can be found for
example in~\cite[Section~II.1.5]{markl-shnider-stasheff:book}.  By a
{\em tree\/} we mean a connected graph $T$ without loops. A {\em
valence\/} of a vertex $v$ of $T$ is the number of edges adjacent to
$v$. A {\em leg\/} or {\em leaf\/} of $T$ is an edge adjacent to a
vertex of valence one, other edges of $T$ are {\em interior\/}.  We in
fact discard vertices of valence one at the endpoints of the legs,
therefore the legs become ``half-edges'' having only one vertex. By a
{\em rooted\/} or {\em directed\/} tree we mean a tree with a
distinguished {\em output\/} leg called the {\em root\/}.  The
remaining legs are called the {\em input legs\/} of the tree. A tree
with $a$ input legs labelled by elements of the set $\{1,2,\ldots,a\}$
is called an {\em $a$-tree\/}.  A rooted tree is automatically {\em
oriented\/}, each edge pointing towards the root.  The edges pointing
towards a given vertex $v$ are called the {\em input edges\/} of $v$,
the number of these input edges is then the {\em arity\/} of $v$
denoted ${\it ar\/}(v)$. Vertices of arity one are called unary,
vertices of arity two binary, vertices of arity three ternary,~etc.


\noindent 
{\bf Notation.}
{\it\
Let $n,m$ and $m_1,\ldots,m_n$ be non-negative integers. 
In the rest of this section, $i$ will always denote an integer between
$0$ and $n$,  $a:= m+1$ and $a_i := m_i + 1$. We will also assume the
notation introduced in~A.\ref{Ferda}.\/}

An {\em $n$-linear natural operation\/} 
\[
U : \CAP{m_1} \ot \cdots \ot \CAP{m_n} \to \CAP m
\]
is given by the following data.

(i) A rooted $a$-tree $T$ with $n$ white vertices
$w_1,\ldots,w_n$ of  arities $a_1,\ldots,a_n$,
and $k$ at least binary black vertices, $k \geq 0$. 

(ii) A linear order on the set of input edges of each white vertex of $T$.

(iii) A decoration of black vertices of $T$ by elements of $\calP$.

(iv) A linear map $\Phi : \ss \calP^!(a_1) \ot \cdots \ot
\ss \calP^!(a_n)  \to \ss \calP^!(a)$.

Given the above data and $f_i \in \CAP{m_i}$, the value
$U(f_1,\ldots,f_n) \in \CAP m$ is defined as follows. Let us decompose
\[
f_i = \sum_{\kappa_i} \phi_i^{\kappa_i} \ot q^i_{\kappa_i} \in [\Lin
  (A^{\otimes a_i},A) \ot \ss \calP^!(a_i)]^{\Sigma_{a_i}} \cong
\CAP{m_i},
\]
where $\phi_i^{\kappa_i} \in \Lin(A^{\ot a_i},A)$, $q^i_{\kappa_i} \in
\ss\calP^!(a_i)$ and $\kappa_i$ is a summation index.  Since the
inputs of white vertices are linearly ordered, each
$\phi_i^{\kappa_i}$ determines a decoration of the white vertex $w_i$
by an element of $\Lin( A ^{\ot a_i}, A) = \End_A(a_i)$. Recall that
$A$ is a $\calP$-algebra with the structure homomorphism $\alpha :
\calP \to \End_A$. Applying $\alpha$ to the decorations of the black
vertices we decorate also black vertices with elements of $\End_A$. So
$T$ is now a tree with all vertices decorated by $\End_A$.  The
composition in the operad $\End_V$ along
$T$~\cite{ginzburg-kapranov:DMJ94} determines, for each
$k_1,\ldots,k_n$, the element
\[
T(\phi_1^{\kappa_1},\ldots,\phi_n^{\kappa_n}) \in \Lin(A^{\ot a}, A).
\]
Let 
\[
\widetilde U(f_1,\ldots,f_n) := \sum_{\kappa_1,\ldots,\kappa_n} 
T(\phi_1^{\kappa_1},\ldots,\phi_n^{\kappa_n}) \ot
\Phi(q^1_{\kappa_1},\ldots,q^n_{\kappa_n}) \in \Lin
 (A^{\ot a},A) \ot \ss\calP^!(a) \cong \tildeCAP m.
\]
Finally, let $U(f_1,\ldots,f_n) := {\it Aver\/}(\widetilde
U(f_1,\ldots,f_n)) \in \CAP m$.
It follows from an elementary combinatorics of trees that
\[
\deg ( U) = {\it ar\/}(b_1) + \cdots + {\it ar\/}(b_k) - k,
\]
therefore $\deg(U)$ is {\em always non-negative\/} and
$\deg(U) = 0$ if and only if $T$ has no black vertex.


\begin{definition}
\label{sam_s_detickami}
Let $\calB_\calP := \{\calB_\calP(n)\}_{n \geq 0}$ be the operad
spanned by all natural operations $U = U_{(T,\Phi)}$ in the above sense.
Since the differential $d_\calP$ of $\CAP*$ is itself a natural
operation living in $\calB_\calP^1(1)$, it induces a differential
$\delta_\calP$ on $\calB_\calP$ by the standard formula
\[
\delta_\calP(U)(f_1,\ldots,f_n) := d_\calP U(f_1,\ldots,f_n) -
(-1)^{|U|} \sum_{1 \leq i \leq n} 
(-1)^{|f_1| + \cdots + |f_{i-1}|}
U(f_1,\ldots,d_\calP f_i,\ldots, f_n),
\]
making $\calB_\calP = (\calB^*_\calP,\delta_\calP)$ a dg-operad.
\end{definition}


Heuristically, the value $U_{(T,\Phi)}(f_1,\ldots,f_n)$ is given by
inserting $f_i$ at the vertex $w_i$ of $T$, $1\leq i \leq n$, and then
performing the composition along $\Phi$. The operadic composition of
$\calB_\calP$ is the vertex insertion similar to that
of~\cite{chapoton-livernet:pre-lie} and the symmetric group permutes
the labels of white vertices. In the following definition we introduce
a non-$\Sigma$ version of $\calB_\calP$.

\begin{definition}
\label{letel_jsem_do_Nymburka!!}
Suppose $\calP$ is the symmetrization of a non-$\Sigma$ operad
$\ucalP$. Let $\calB_\ucalP$ be the dg-suboperad of $\calB_\calP$
spanned by natural operations $U_{(T,\Phi)}$ as in
Definition~\ref{sam_s_detickami} such that the tree $T$ is
planar, with black vertices decorated by elements of \hskip .1em $\ucalP$,
and the map $\Phi$ such that 
\[
\Phi(\ss \ucalP^!(a_1)
\ot \cdots \ot \ss \ucalP^!(a_n)) \subset \ss \ucalP^!(a).
\]
\end{definition}

\begin{example}
{\rm
--\label{Kocicka_zrcatko}
Constants. Let us see what happens if $T$ is the
$a$-corolla with one black vertex decorated by $p \in \calP(a)$ and no
white vertices as in Figure~\ref{rozbil_jsem_notebook}. 
\begin{figure}[t]
\begin{center}
\unitlength=.95pt
\begin{picture}(60.00,80.00)(20.00,0.00)
\thicklines
%LEVY OBRAZEK
\put(-70,0){
\put(0.00,0.00){\makebox(0.00,0.00){$1$}}
\put(10.00,0.00){\makebox(0.00,0.00){$2$}}
\put(60.00,0.00){\makebox(0.00,0.00){$a$}}
\put(30.00,10.00){\makebox(0.00,0.00)[b]{$\cdots$}}
\put(40.00,50.00){\makebox(0.00,0.00){$p$}}
\put(30.00,40.00){\makebox(0.00,0.00){$\bullet$}}
\put(60.00,10.00){\line(-1,1){30.00}}
\put(10.00,10.00){\line(2,3){20.00}}
\put(0.00,10.00){\line(1,1){30.00}}
\put(30.00,80.00){\line(0,-1){40.00}}
}
%PRAVY OBRAZEK
\put(70,0){
\put(42.00,50.00){\makebox(0.00,0.00){$w_1$}}
\put(-4.00,0.00){\makebox(0.00,0.00){$\sigma(1)$}}
\put(22.00,0.00){\makebox(0.00,0.00){$\sigma(2)$}}
\put(64.00,0.00){\makebox(0.00,0.00){$\sigma(a)$}}
\put(36.00,10.00){\makebox(0.00,0.00)[b]{$\cdots$}}
\put(30.00,40.00){\makebox(0.00,0.00){$\circ$}}
\put(60.00,10.00){\line(-1,1){28.00}}
\put(18.00,10.00){\line(2,5){11.00}}
\put(0.00,10.00){\line(1,1){28.00}}
\put(30.00,80.00){\line(0,-1){37.00}}
}
\end{picture}
\end{center}
\caption{\label{rozbil_jsem_notebook}
The tree defining a constant in $\calB^m_\calP(0)$ 
(left) and a unary operation in $\calB^0_\calP(1)$ (right), where $m =
a+1$ as always.}
\end{figure}
The map
$\Phi: \bfk \to \ss \calP^!(a)$ is given by specifying an element
$\varphi := \Phi(1) \in \ss\calP^!(a)$ and  $\widetilde U$ determined
by this $\Phi$ equals $\alpha
(p) \ot \varphi \in \tildeCAP m$. Since $\alpha$ is equivariant,
\[
{\it Aver\/}(\alpha(p) \ot \varphi) = (\alpha \ot \id)({\it Aver\/}(p \otimes
\varphi)) 
\]
therefore $U := {\it Aver\/}(\alpha(p) \ot \varphi) \in \CAP m$ 
is parametrized by an element in the image of the averaging map 
\[
{\it  Aver\/}: \calP(a) \ot \ss \calP^!(a) \to (\calP(a) \ot \ss
\calP^!(a))^{\Sigma_a},
\] 
in other words, 
\[
\calB_\calP^{m}(0) \cong \ss (\calP \ot \calP^!)(a)^{\Sigma_a},\ m \geq 0.
\]
It is equally easy to see that, for a quadratic Koszul non-$\Sigma$
operad $\ucalP$,
\[
\calB^m_\ucalP(0) \cong \ss (\ucalP \ot\ucalP^!)(a),\ m \geq 0.
\]
}
\end{example}


\begin{example}
-- \label{Deda_Mraz} {\rm\ Unary operations of degree $0$. Now $T$ is
  an $a$-corolla with one white planar vertex and no black vertices,
  with input legs labelled $\sigma(1),\ldots,\sigma(a)$, $\sigma \in
  \Sigma_a$, as shown in Figure~\ref{rozbil_jsem_notebook}, and $\Phi :
  \ss\calP^!(a) \to \ss\calP^!(a)$ is a linear map. If
\[
f = \sum_\kappa \phi^\kappa \ot q_\kappa \in [\Lin( \otexp Aa ,A) \ot
  \ss \calP^!(a)]^{\Sigma_a} \cong \CAP m,
\]
then 
$U(f) = {\it Aver\/}(\sum_\kappa\phi^\kappa \sigma^{-1} \ot \Phi(q_\kappa))$.
Since $f = \sum \phi^\kappa \ot q_\kappa$ is $\Sigma_a$-stable,
\[
U(f) =
\sum \phi^\kappa \ot {\it Aver\/}(\Phi_\kappa)(q_\kappa).
\]
Therefore $U(f)$ is given  by a $\Sigma_a$-equivariant map 
$\Psi := \desusp^{a-1}{\it Aver\/}(\Phi) : \calP^!(a) \to \calP^!(a)$,
thus
\begin{equation}
\label{ll}
\calB_\calP^0(1) \cong \Lin_\Sigma(\calP^!,\calP^!),
\end{equation}
the space of all collections $\{\psi_n: \calP^!(n) \to \calP^!(n)\}_{n
\geq 0}$ of equivariant maps. We leave as an exercise to verify that,
for a non-$\Sigma$ quadratic Koszul operad $\ucalP$,
\[
\calB_\ucalP^0(1) \cong \Lin(\ucalP^!,\ucalP^!).
\]
}
\end{example}

\begin{example}
-- {\rm 
Projections. 
Let $p_m \in \calB_\calP^0(1)$ be given, in identification~(\ref{ll}), by
$\Psi \in \Lin_\Sigma(\calP^!,\calP^!)$ defined as
\[
\Psi|_{\calP^!(a)} =
\cases{\id_{\calP^!(a)}}{for $a=m+1$ and}0{othewise.}
\] 
Clearly, $p_m$ is the projection $\CAP* \epi \CAP
m$. The system of all these
projections makes $\calB_\calP$ an $\BbbZ^{\geq 0}$-colored operad, where
$\BbbZ^{\geq 0}$ is the set of non-negative integers. Since 
these projections do not commute with $d_\calP$ (that is
$\delta_\calP(p_m) \not= 0$ for a generic $\calP$),
$(\calB_\calP,\delta_\calP)$ {\em is not\/} a {\em dg\/}
$\BbbZ^{\geq 0}$-colored operad.
}
\end{example}

\begin{example}
\label{fit}
-- {\rm Cup products.  In this example we explain how an element 
\[
t = p
  \ot \ss q \in \calP(n) \ot \ss\calP^!(n) \cong \ss(\calP \ot
  \calP^!)(n)
\] 
determines a natural operation in $\calB_\calP^{n-1}(n)$.  Let $T$ be
as in Figure~\ref{dnes_novy_notebook}, with the black vertex
decorated by $p \in \calP(n)$,
\begin{figure}[t]
\begin{center}
{% Picture saved by xtexcad 2.4
\thicklines
\unitlength=.8pt
\begin{picture}(240.00,100.00)(0.00,10.00)
\put(220.00,0.00){\makebox(0.00,0.00){$\underbrace{\rule{33pt}{0pt}}_{a_n}$}}
\put(70.00,0.00){\makebox(0.00,0.00){$\underbrace{\rule{33pt}{0pt}}_{a_2}$}}
\put(20.00,0.00){\makebox(0.00,0.00){$\underbrace{\rule{33pt}{0pt}}_{a_1}$}}
\put(130.00,30.00){\makebox(0.00,0.00){$\cdots$}}
\put(224.00,10.00){\makebox(0.00,0.00)[b]{\scriptsize $\cdots$}}
\put(74.00,10.00){\makebox(0.00,0.00)[b]{\scriptsize $\cdots$}}
\put(24.00,10.00){\makebox(0.00,0.00)[b]{\scriptsize $\cdots$}}
\put(130.00,100.00){\makebox(0.00,0.00){$p$}}
\put(220.00,30.00){\makebox(0.00,0.00){$\circ$}}
\put(70.00,30.00){\makebox(0.00,0.00){$\circ$}}
\put(20.00,30.00){\makebox(0.00,0.00){$\circ$}}
\put(10.00,35.00){\makebox(0.00,0.00)[rb]{$w_1$}}
\put(50.00,35.00){\makebox(0.00,0.00)[lb]{$w_2$}}
\put(230.00,35.00){\makebox(0.00,0.00)[lb]{$w_n$}}
\put(120.00,90.00){\makebox(0.00,0.00){$\bullet$}}
\put(120.00,120.00){\line(0,-1){30.00}}
%
\put(222,28){\line(1,-1){18}}
\put(218.8,27.5){\line(-1,-2){8.5}}
\put(218,28){\line(-1,-1){18}}
%
\put(-200,0){
\put(222,28){\line(1,-1){18}}
\put(218.8,27.5){\line(-1,-2){8.5}}
\put(218,28){\line(-1,-1){18}}
}
\put(-150,0){
\put(222,28){\line(1,-1){18}}
\put(218.8,27.5){\line(-1,-2){8.5}}
\put(218,28){\line(-1,-1){18}}
}
\put(120.00,90.00){\line(5,-3){97}}
\put(120.00,90.00){\line(-5,-6){48}}
\put(120.00,90.00){\line(-5,-3){97}}
\end{picture}}
\end{center}
\caption{\label{dnes_novy_notebook}
The tree defining the cup product.}
\end{figure}
and let the linear map 
$\Phi : \ss\calP^!(a_1) \ot \cdots \ot
\ss\calP^!(a_n) \to \ss \calP^!(a)$ be given by the operadic composition in
$\ss{}\calP^!$:
\[
\Phi(\ss q_1,\ldots,\ss  q_n) :=
\ss q(\ss q_1,\ldots,\ss q_n),\ q_i \in \calP^!(a_i),\ 1 \leq i \leq n.
\]
It is more or less clear that the natural operation $U_{(T,\Phi)}$ 
determined by the above data
agrees with the cup product $\ccc(t)$ of
Theorem~\ref{zitra_prednaska}. We recommend as another exercise to verify
that also the intrinsic bracket described in~(\ref{lepidlo}) is
given by natural operations in the sense of this section.  
}
\end{example}


\begin{example}
{\rm
Let us describe all natural operations $\CAP 1 \ot \CAP 1 \to \CAP 2$
for some particular choices of $\calP$.

(i) Hochschild cohomology.  For $\calP = \As$, $\CAP 1 =
\Lin (\otexp A2,A)$, $\CAP 2 =  \Lin (\otexp A3,A)$, and the
only natural operations $\CAP 1 \ot \CAP 1 \to \CAP 2$ are linear
combinations of
\[
f,g \to (f \circ_1 g)\sigma,\ f,g \to (f \circ_2 g)\sigma,\ f,g \to 
(g \circ_1 f)\sigma,\ f,g \to (g \circ_2 f)\sigma,\ \sigma \in \Sigma_3,
\]
where $\circ_1$, $\circ_2$ are Gerstenhaber-type
products~\cite{gerstenhaber:AM63} given by
\begin{equation}
\label{pisu_v_susarne}
(u \circ_1 v) (a,b,c) := u(v(a,b),c)),\
(u \circ_2 v) (a,b,c) := u(a,v(b,c)),\ 
\end{equation}
for $u,v \in \CAP 2$, $a,b,c \in A$, and $\sigma \in \Sigma_3$
permutes the factors of $\otexp A3$ in the usual way. Operations
belonging to $\calB^{0}_\uAss(2)$ are linear combinations of the
operations~(\ref{pisu_v_susarne}) with $\sigma
= \id_3$, the unit of $\Sigma_3$.

(ii) Chevalley-Eilenberg cohomology.
If $\calP = \Lie$, then 
 $\CAP 1 =  \Lin (\land^2,A)$, $\CAP 2 =
\Lin (\land^3 A,A)$, where $\land^n A$ denotes the $n$-th
exterior power. The only natural operations  $\CAP 1
\ot \CAP 1 \to \CAP 2$ are linear combinations of
\[
f,g \to f \circ g\ \mbox { and } \ f,g \to g \circ f,
\]
where
\[
(u\circ v)(a,b,c) := u(v(a,b),c) +  u(v(b,c),a) + u(v(c,a),b)
\] 
for $u,v \in \CAP 2$ and $a,b,c \in A$.

(iii) Harrison cohomology.
If $\calP = \Com$, then 
\[
\CAP 1 =  \{u \in   \Lin (\otexp A2,A);\
u(a,b) - u(b,a) = 0\}
\]  
and $\CAP 2$ consists of all 
$w \in \Lin (\otexp A3,A)$ such that 
\[
w(a,b,c) - w(b,a,c) + w(b,c,a) =
w(a,b,c) - w(a,c,b) + w(c,a,b) = 0,
\]
for $a,b,c \in A$.
Natural operations $\CAP 1 \ot \CAP 1 \to \CAP 2$ 
are linear combinations of
\[
f,g \to f \circ g\ \mbox { and } \ f,g \to g \circ f,
\]
where
\[
u \circ v := u(v(a,b)c) - u(v(b,c),a),
\] 
for $u,v \in \CAP 2$ and $a,b,c \in A$.
}
\end{example}


\section*{Appendix: Notations, conventions and background material}


\begin{Appendix}
\label{jitka_neodpovida}
{\rm
In this note, an {\em operad\/} means an operad in the category of
differential graded (dg) vector spaces, that is, a sequence 
$\calP = \{\calP(n)\}_{n \geq 0}$ of right $\Sigma_n$-modules with structure
operations 
\[
\gamma : \calP(n) \ot \calP(k_1) \ot \cdots \ot \calP(k_n) 
\to
\calP(k_1 + \cdots + k_n),
\]
for $n \geq 1$ and $k_1,\ldots,k_n \geq 0$, and a unit map $\eta :
\bfk \to \calP(1)$ that satisfy the usual
axioms~\cite{may:1972,kriz-may}. Instead of $\gamma(p \ot p_1 \ot
\ldots \ot p_n)$ we will often write $\gamma(p, p_1, \ldots , p_n)$ or
$p(p_1,\ldots,p_n)$. Recall~\cite{markl:zebrulka} that operads can be
equivalently defined using the $\circ_i$-operations
\[
\circ_i : \calP(m) \ot \calP(n) \to \calP(m+n-1)
\] 
defined, for $m,n \geq 0$, $1 \leq i \leq m$, by
\[
p\circ_i q := \gamma(p \ot e^{\ot (i-1)}\ot q \ot \otexp e{m-i}), 
\]
where $e : = \eta(1)$.

If we remove from the above definition all references to the symmetric group
actions, we get a definition of a {\em non-$\Sigma$ operad\/}. Each
non-$\Sigma$ operad $\ucalP$ generates a unique (usual) operad $\calP$
such that $\calP(n) \cong \ucalP(n) \ot \bfk[\Sigma_n]$, $n \geq 0$.
}
\end{Appendix}


\begin{Appendix}
\label{posloucham_Boroveho}
{\rm For each set of operations $E$, there exists the {\em free
operad\/} $\Gamma(E)$ generated by
$E$~\cite[Proposition~II.1.92]{markl-shnider-stasheff:book}.  Let
$\mu$ denote a bilinear operation placed in degree $0$. The operad
$\Ass{}$ for {\em associative algebras\/} is the quotient
\[
\Ass{} := \Gamma(\mu)/(\mu \circ_1 \mu - \mu \circ_2 \mu),
\]
where $(\mu \circ_1 \mu - \mu \circ_2 \mu)$ denotes the operadic ideal
generated by the associativity axiom for $\mu$.

If $\lambda$ is a skew-symmetric bilinear operation,
then the operad for {\em Lie algebras\/} is the quotient
\[
\Lie := \Gamma(\lambda)/({\it Jacobi}(\lambda)),
\]
where 
\[
{\it Jacobi}(\lambda) := \sum_{\sigma\in C_3} (\lambda \circ_1 \lambda) \sigma
\]
with the summation taken over the order $3$ cyclic subgroup $C_3$ of
$\Sigma_3$, denotes the Jacobi identity for $\lambda$.

Finally, for an arbitrary differential graded vector space $V$, there
is the {\em endomorphism operad\/} $\End_V = \{Lin(V^{\ot n},V)\}_{n
\geq 0}$, with structure operations given as the usual composition of
multilinear maps. A {\em $\calP$-algebra\/} is then a homomorphism
$\alpha : \calP \to \End_V$. We sometimes call $\alpha$ also an {\em
action\/} of $\calP$ on $V$.  }
\end{Appendix}


\begin{Appendix}{\rm
\label{odpalil_jsem_notebook}
The {\em suspension\/} $\ss A = 
\{\ss A(n)\}_{n \geq 0}$ of a $\Sigma$-module $A = \{A(n)\}_{n \geq 0}$
is defined by
\[
\ss A(n) := \hskip .2em \uparrow^{n-1} \hskip -.2em A(n) \otimes
\sgn_n,
\] 
where $\sgn_n$ denotes the signum representation of
$\Sigma_n$, see~\cite[Definition~II.3.15]{markl-shnider-stasheff:book}. 
If $\calP$ is an operad, then the collection 
$\ss \calP$ carries a canonical induced
operad structure and the operad $\ss \calP$ is called the {\em
operadic suspension\/} of $\calP$.
For any two operads $\calP$ and $\calQ$, 
\[
\ss (\calP \otimes \calQ) \cong \ss \calP \ot \calQ \cong \calP \ot
\ss \calQ. 
\] 
}
\end{Appendix}



\begin{Appendix}
\label{ma_prejit_fronta_v_sobotu}
{\rm 
An $(m,n)$-{\em algebra\/}
is~\cite[Example~9.4]{fox-markl:ContM97} a graded vector space $A$
together with two bilinear maps, $-\cup-:A \ot A \to A$ of degree $m$,
and $[-,-]:A\ot A\to A$ of degree $n$ ($m$ and $n$ are natural
numbers), such that, for any homogeneous $a,b,c\in A$,
\begin{itemize}
\item[(i)]
$a\cup b = (-1)^{|a|\cdot |b|+m}\cdot b\cup a$,
\item[(ii)]
$[a,b] = -(-1)^{|a|\cdot |b|+n}\cdot [b,a]$,
\item[(iii)]
$-\cup-$ is associative in the sense that
$$
a\cup (b\cup c) = (-1)^{m\cdot(|a|+1)}\cdot(a\cup b)\cup c,
$$
\item[(iv)]
$[-,-]$ satisfies the following form of the Jacobi identity:
$$
(-1)^{|a|\cdot(|c|+n)}\cdot [a,[b,c]]+(-1)^{|b|\cdot(|a|+n)}\cdot
[b,[c,a]]
+(-1)^{|c|\cdot(|b|+n)}\cdot [c,[a,b]] = 0,
$$
\item[(v)]
the operations $-\cup-$ and $[-,-]$ are compatible in the sense that
$$
(-1)^{m\cdot|a|}[a,b\cup c] =
[a,b]\cup c + (-1)^{(|b|\cdot |c|+m)}[a,c]\cup b.
$$
\end{itemize}
$(0,1)$-algebras were considered in~\cite{getzler-jones:preprint}
under the name {\em $2$-algebras\/} or {\em braid algebras\/}. The
corresponding operad $\Braid$ is isomorphic to the homology of the
little discs operad $\disc_2$, $\Braid \cong H_*(\disc_2)$.
Following~\cite[Section~10]{gerstenhaber-voronov:FAP95}, we call
$(1,0)$-algebras {\em Gerstenhaber algebras\/}, though the terminology
is not unique, compare for instance~\cite[Subsection~10.2]{ADT88}
where a Gerstenhaber algebra means a $(0,-1)$-algebra.
}
\end{Appendix}


\begin{Appendix}
\label{zase_do_Prahy}
{\rm
Let $M$ be a right module over a finite group $G$. We denote, as usual
\[
M^G := \{ m \in M;\ m g =g \mbox { for all } g \in G\}
\ \mbox { and }\
M_G := \frac M{(m-mg;\ m \in M,\ g \in G)}.
\]
Let ${\it Aver\/}: M \to M^G$ be the ``averaging'' map given by
\[
{\it Aver\/}(m):= \frac 1{|G|}\sum_{g \in G} mg.
\]
It is a standard fact that the composition $\pi \iota$ of the
projection $\pi : M \epi M_G$ with the inclusion $\iota : M^G
\hookrightarrow M$ is the identity and that ${\it Aver\/}$ is a right
inverse to $\iota$.
}
\end{Appendix}


\begin{Appendix}
\label{Ferda}
{\rm
Let us recall the operadic cochain complex
and introduce some useful notations.
As a graded vector space, the operadic cochain complex is defined
by~\cite[Definition~II.3.99]{markl-shnider-stasheff:book}:
\begin{equation}
\label{neco_tady_hrozne_kvika}
\CAP{n-1} = \left[ 
\Lin(\otexp {(\downarrow \hskip -.2em A)}n,\downarrow\hskip -.2em A) 
\ot \calP^!(n)
\right]^{\Sigma_n} \hskip -.9em, \ n \geq 1,
\end{equation}
where $\downarrow A$ denotes the desuspension of the graded vector
space $A$. 
It will be convenient to denote 
\[
\tildeCAP{n-1} := \Lin(\otexp {(\downarrow \hskip -.2em
  A)}n,\downarrow  \hskip -.2em A) \ot \calP^!(n),
\]
so that $\CAP {n-1} \cong \tildeCAP{n-1}^{\Sigma_n}$.
The averaging over the $\Sigma_n$-action defines an epimorphism
\[
{\it Aver\/} : \tildeCAP{n-1} \epi \CAP{n-1}
\]
of graded modules which is a left inverse to the inclusion
\[
\iota : \CAP{n-1} \hookrightarrow \tildeCAP{n-1}.
\] 

We will often use the following canonical isomorphisms of graded
$\Sigma_n$-modules:
\begin{eqnarray*}
\tildeCAP {n-1} &=&
\Lin(\otexp {(\downarrow \hskip -.2em A)}n,\downarrow\hskip -.2em A) 
\ot \calP^!(n) \cong
\hskip .2em \uparrow^{\ot {n-1}} \hskip -.2em 
(\Lin(\otexp An,A) \ot
\calP^!(n) \ot \sgn_n) 
\\
&\cong&
\ss(\Lin(\otexp An,A) \ot \calP^!(n))
\cong  \ss\End_A(n) \ot \calP^!(n) 
\\
&\cong& \End_A(n) \ot \ss\calP^!(n) \cong \End_{\desusp A}(n) \ot \calP^!(n).
\end{eqnarray*}
}
\end{Appendix}


\baselineskip 1.1em
%\bibliography{b}
\def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{berger:CM202}
C.~Berger.
\newblock Combinatorial models for real configuration spaces and
  {$E_n$}-operads.
\newblock In J.L. Loday, J.D Stasheff, and A.A. Voronov, editors, {\em Operads:
  Proceedings of Renaissance Conferences}, volume 202 of {\em Contemporary
  Math.}, pages 37--52. Amer. Math. Soc., 1997.

\bibitem{chapoton-livernet:pre-lie}
F.~Chapoton and M.~Livernet.
\newblock Pre-{Lie} algebras and the rooted trees operad.
\newblock {\em Internat. Math. Res. Notices}, 8:395--408, 2001.

\bibitem{fontanine-kapranov:JAMS}
I.~{Ciocan}-{Fontanine} and M.~Kapranov.
\newblock Derived {Hilbert} schemes.
\newblock {\em J. Amer. Math. Soc.}, 15(4):787--815, June 2002.

\bibitem{fontanine-kapranov}
I.~{Ciocan}-{Fontanine} and M.M. Kapranov.
\newblock Derived {Quot} schemes.
\newblock {\em Ann. Sci. Ecole Norm. Sup.}, 34(3):403--440, May-June 2001.

\bibitem{deligne:letter}
P.~Deligne.
\newblock A letter to {Stasheff}, {Gerstenhaber}, {May}, {Schechtman} and
  {Drinfeld}.
\newblock Unpublished, 1993.

\bibitem{dzhu-lof:HHA02}
A.~Dzhumadil'daev and C.~L\"ofwall.
\newblock Trees, free right-symmetric algebras, free {Novikov} algebras and
  identities.
\newblock {\em Homotopy, Homology and Applications}, 4(2):165--190, 2002.

\bibitem{fiedorowicz}
Z.~Fiedorowicz.
\newblock The symmetric bar construction.
\newblock Preprint, available from the author's home page.

\bibitem{fox-markl:ContM97}
T.F. Fox and M.~Markl.
\newblock Distributive laws, bialgebras, and cohomology.
\newblock In J.L. Loday, J.D. Stasheff, and A.A. Voronov, editors, {\em
  Operads: Proceedings of Renaissance Conferences}, volume 202 of {\em
  Contemporary Math.}, pages 167--205. Amer. Math. Soc., 1997.

\bibitem{gerstenhaber:AM63}
M.~Gerstenhaber.
\newblock The cohomology structure of an associative ring.
\newblock {\em Ann. of Math.}, 78(2):267--288, 1963.

\bibitem{ADT88}
M.~Gerstenhaber and S.D. Schack.
\newblock Algebraic cohomology and deformation theory.
\newblock In {\em Deformation Theory of Algebras and Structures and
  Applications}, pages 11--264. Kluwer, Dordrecht, 1988.

\bibitem{gerstenhaber-voronov:FAP95}
M.~Gerstenhaber and A.A. Voronov.
\newblock Higher operations on the {Hochschild} complex.
\newblock {\em Functional Anal. Appl.}, 29(1):1--6, 1995.
\newblock In Russian.

\bibitem{getzler-jones:preprint}
E.~Getzler and J.D.S. Jones.
\newblock Operads, homotopy algebra, and iterated integrals for double loop
  spaces.
\newblock Preprint {\tt hep-th/9403055}, March 1994.

\bibitem{ginzburg-kapranov:DMJ94}
V.~Ginzburg and M.M. Kapranov.
\newblock Koszul duality for operads.
\newblock {\em Duke Math. J.}, 76(1):203--272, 1994.

\bibitem{hinich:FM03}
V.~Hinich.
\newblock Tamarkin's proof of {Kontsevich} formality theorem.
\newblock {\em Forum Mathematicum}, 15:591--614, 2003.

\bibitem{hu}
P.~Hu.
\newblock Higher string topology on general spaces.
\newblock Preprint {\tt math.AT/0401081}, January 2004.

\bibitem{hu-kriz}
P.~Hu, I.~{K\v r\'\i \v z}, and A.A. Voronov.
\newblock On {Kontsevich}'s {Hochschild} cohomology conjecture.
\newblock Preprint {\tt math.AT/0309369}, September 2003.

\bibitem{kaufmann}
R.M. Kaufmann.
\newblock On spineless cacti, {Deligne}'s conjecture and {Connes--Kreimer}'s
  {Hopf} algebra.
\newblock Preprint {\tt math.QA/0308005}, August 2003.

\bibitem{klya}
A.A. Klyatchko.
\newblock Lie elements in the tensor algebra.
\newblock {\em Siberian Math. J.}, 15:914--920, 1974.

\bibitem{kock-toen}
J.~Kock and B.~{To\"en}.
\newblock Simplicial localization of monoidal structures, and a non-linear
  version of {Deligne's} conjecture.
\newblock Preprint {\tt math.AT/0304442}, April 2003.

\bibitem{kontsevich-soibelman}
M.~Kontsevich and Y.~Soibelman.
\newblock Deformations of algebras over operads and {Deligne's} conjecture.
\newblock Preprint {\tt math.QA/0001151}, January 2000.

\bibitem{kriz-may}
I.~{K\v r\'\i\v z} and J.P. May.
\newblock {\em Operads, Algebras, Modules and Motives}, volume 233 of {\em
  Ast\'erisque}.
\newblock Soci\'et\'e Math\'ematique de France, 1995.

\bibitem{lm:sb}
T.~Lada and M.~Markl.
\newblock Symmetric brace algebras with applications to particles of hight
  spin.
\newblock Preprint {\tt math.QA/0307054}, July 2003.

\bibitem{markl:JPAA96}
M.~Markl.
\newblock Cotangent cohomology of a category and deformations.
\newblock {\em J. Pure Appl. Algebra}, 113:195--218, 1996.

\bibitem{markl:zebrulka}
M.~Markl.
\newblock Models for operads.
\newblock {\em Comm. Algebra}, 24(4):1471--1500, 1996.

\bibitem{markl:ib}
M.~Markl.
\newblock The existence of intrinsic brackets on the cohomology of bialgebras.
\newblock Preprint {\tt math.AT/0411456}, November 2004.

\bibitem{markl-shnider-stasheff:book}
M.~Markl, S.~Shnider, and J.~D. Stasheff.
\newblock {\em Operads in Algebra, Topology and Physics}, volume~96 of {\em
  Mathematical Surveys and Monographs}.
\newblock American Mathematical Society, Providence, Rhode Island, 2002.

\bibitem{may:1972}
J.P. May.
\newblock {\em The Geometry of Iterated Loop Spaces}, volume 271 of {\em
  Lecture Notes in Mathematics}.
\newblock Springer-Verlag, New York, 1972.

\bibitem{mcclure-smith}
J.E. McClure and J.H. Smith.
\newblock A solution of {Deligne's} {Hochschild} cohomology conjecture.
\newblock In {\em Proceedings of the JAMI conference on Homotopy Theory},
  volume 293 of {\em Contemp. Math.}, pages 153--193, 2002.

\bibitem{mcclure-smith:JAMS03}
J.E. McClure and J.H. Smith.
\newblock Multivariable cochain operations and little $n$-cubes.
\newblock {\em J. Amer. Math. Soc.}, 16(3):681--704, January 2003.

\bibitem{salvatore}
P.~Salvatore.
\newblock Configuration spaces with summable labels.
\newblock In J.~Aguad\'e, editor, {\em Cohomological methods in homotopy
  theory. Proceedings of the Barcelona conference on algebraic topology (BCAT),
  Bellaterra, Spain, June 4-10, 1998}, volume 196 of {\em Prog. Math.}, pages
  375--395, Basel, 2001. Birkh\"auser Publ.

\bibitem{schlessinger-stasheff:JPAA85}
M.~Schlessinger and J.D. Stasheff.
\newblock The {Lie} algebra structure of tangent cohomology and deformation
  theory.
\newblock {\em J. Pure Appl. Algebra}, 38:313--322, 1985.

\bibitem{tamarkin-tsygan:LMP01}
D.~Tamarkin and B.~Tsygan.
\newblock Cyclic formality and index theorems.
\newblock {\em Lett. Math. Phys.}, 56:85--97, 2001.

\bibitem{tamarkin:deligne}
D.E. Tamarkin.
\newblock Another proof of {M}. {K}ontsevich formality theorem.
\newblock Preprint {\tt math.QA/9803025}, March 1998.

\bibitem{tamarkin:formality}
D.E. Tamarkin.
\newblock Formality of chain operad of little discs.
\newblock {\em Lett. Math. Phys.}, 66(1-2):65--72, 2003.

\bibitem{laan-defo}
P.~van~der Laan.
\newblock Operads up to homotopy and deformations of operad maps.
\newblock Preprint {\tt math.QA/0208041}, August 2002.

\bibitem{voronov:99}
A.A. Voronov.
\newblock Homotopy {Gerstenhaber} algebras.
\newblock In G.~Dito, editor, {\em Conference Moshe Flato 1999: Quantization,
  deformations, and symmetries, {II}.}, volume~22 of {\em Math. Phys. Stud.},
  pages 307--331. Kluwer Academic Publishers, 2000.

\end{thebibliography}


\end{document}

math0506170.extracted_bib
\begin{thebibliography}{10}

\bibitem{berger:CM202}
C.~Berger.
\newblock Combinatorial models for real configuration spaces and
  {$E_n$}-operads.
\newblock In J.L. Loday, J.D Stasheff, and A.A. Voronov, editors, {\em Operads:
  Proceedings of Renaissance Conferences}, volume 202 of {\em Contemporary
  Math.}, pages 37--52. Amer. Math. Soc., 1997.

\bibitem{chapoton-livernet:pre-lie}
F.~Chapoton and M.~Livernet.
\newblock Pre-{Lie} algebras and the rooted trees operad.
\newblock {\em Internat. Math. Res. Notices}, 8:395--408, 2001.

\bibitem{fontanine-kapranov:JAMS}
I.~{Ciocan}-{Fontanine} and M.~Kapranov.
\newblock Derived {Hilbert} schemes.
\newblock {\em J. Amer. Math. Soc.}, 15(4):787--815, June 2002.

\bibitem{fontanine-kapranov}
I.~{Ciocan}-{Fontanine} and M.M. Kapranov.
\newblock Derived {Quot} schemes.
\newblock {\em Ann. Sci. Ecole Norm. Sup.}, 34(3):403--440, May-June 2001.

\bibitem{deligne:letter}
P.~Deligne.
\newblock A letter to {Stasheff}, {Gerstenhaber}, {May}, {Schechtman} and
  {Drinfeld}.
\newblock Unpublished, 1993.

\bibitem{dzhu-lof:HHA02}
A.~Dzhumadil'daev and C.~L\"ofwall.
\newblock Trees, free right-symmetric algebras, free {Novikov} algebras and
  identities.
\newblock {\em Homotopy, Homology and Applications}, 4(2):165--190, 2002.

\bibitem{fiedorowicz}
Z.~Fiedorowicz.
\newblock The symmetric bar construction.
\newblock Preprint, available from the author's home page.

\bibitem{fox-markl:ContM97}
T.F. Fox and M.~Markl.
\newblock Distributive laws, bialgebras, and cohomology.
\newblock In J.L. Loday, J.D. Stasheff, and A.A. Voronov, editors, {\em
  Operads: Proceedings of Renaissance Conferences}, volume 202 of {\em
  Contemporary Math.}, pages 167--205. Amer. Math. Soc., 1997.

\bibitem{gerstenhaber:AM63}
M.~Gerstenhaber.
\newblock The cohomology structure of an associative ring.
\newblock {\em Ann. of Math.}, 78(2):267--288, 1963.

\bibitem{ADT88}
M.~Gerstenhaber and S.D. Schack.
\newblock Algebraic cohomology and deformation theory.
\newblock In {\em Deformation Theory of Algebras and Structures and
  Applications}, pages 11--264. Kluwer, Dordrecht, 1988.

\bibitem{gerstenhaber-voronov:FAP95}
M.~Gerstenhaber and A.A. Voronov.
\newblock Higher operations on the {Hochschild} complex.
\newblock {\em Functional Anal. Appl.}, 29(1):1--6, 1995.
\newblock In Russian.

\bibitem{getzler-jones:preprint}
E.~Getzler and J.D.S. Jones.
\newblock Operads, homotopy algebra, and iterated integrals for double loop
  spaces.
\newblock Preprint {\tt hep-th/9403055}, March 1994.

\bibitem{ginzburg-kapranov:DMJ94}
V.~Ginzburg and M.M. Kapranov.
\newblock Koszul duality for operads.
\newblock {\em Duke Math. J.}, 76(1):203--272, 1994.

\bibitem{hinich:FM03}
V.~Hinich.
\newblock Tamarkin's proof of {Kontsevich} formality theorem.
\newblock {\em Forum Mathematicum}, 15:591--614, 2003.

\bibitem{hu}
P.~Hu.
\newblock Higher string topology on general spaces.
\newblock Preprint {\tt math.AT/0401081}, January 2004.

\bibitem{hu-kriz}
P.~Hu, I.~{K\v r\'\i \v z}, and A.A. Voronov.
\newblock On {Kontsevich}'s {Hochschild} cohomology conjecture.
\newblock Preprint {\tt math.AT/0309369}, September 2003.

\bibitem{kaufmann}
R.M. Kaufmann.
\newblock On spineless cacti, {Deligne}'s conjecture and {Connes--Kreimer}'s
  {Hopf} algebra.
\newblock Preprint {\tt math.QA/0308005}, August 2003.

\bibitem{klya}
A.A. Klyatchko.
\newblock Lie elements in the tensor algebra.
\newblock {\em Siberian Math. J.}, 15:914--920, 1974.

\bibitem{kock-toen}
J.~Kock and B.~{To\"en}.
\newblock Simplicial localization of monoidal structures, and a non-linear
  version of {Deligne's} conjecture.
\newblock Preprint {\tt math.AT/0304442}, April 2003.

\bibitem{kontsevich-soibelman}
M.~Kontsevich and Y.~Soibelman.
\newblock Deformations of algebras over operads and {Deligne's} conjecture.
\newblock Preprint {\tt math.QA/0001151}, January 2000.

\bibitem{kriz-may}
I.~{K\v r\'\i\v z} and J.P. May.
\newblock {\em Operads, Algebras, Modules and Motives}, volume 233 of {\em
  Ast\'erisque}.
\newblock Soci\'et\'e Math\'ematique de France, 1995.

\bibitem{lm:sb}
T.~Lada and M.~Markl.
\newblock Symmetric brace algebras with applications to particles of hight
  spin.
\newblock Preprint {\tt math.QA/0307054}, July 2003.

\bibitem{markl:JPAA96}
M.~Markl.
\newblock Cotangent cohomology of a category and deformations.
\newblock {\em J. Pure Appl. Algebra}, 113:195--218, 1996.

\bibitem{markl:zebrulka}
M.~Markl.
\newblock Models for operads.
\newblock {\em Comm. Algebra}, 24(4):1471--1500, 1996.

\bibitem{markl:ib}
M.~Markl.
\newblock The existence of intrinsic brackets on the cohomology of bialgebras.
\newblock Preprint {\tt math.AT/0411456}, November 2004.

\bibitem{markl-shnider-stasheff:book}
M.~Markl, S.~Shnider, and J.~D. Stasheff.
\newblock {\em Operads in Algebra, Topology and Physics}, volume~96 of {\em
  Mathematical Surveys and Monographs}.
\newblock American Mathematical Society, Providence, Rhode Island, 2002.

\bibitem{may:1972}
J.P. May.
\newblock {\em The Geometry of Iterated Loop Spaces}, volume 271 of {\em
  Lecture Notes in Mathematics}.
\newblock Springer-Verlag, New York, 1972.

\bibitem{mcclure-smith}
J.E. McClure and J.H. Smith.
\newblock A solution of {Deligne's} {Hochschild} cohomology conjecture.
\newblock In {\em Proceedings of the JAMI conference on Homotopy Theory},
  volume 293 of {\em Contemp. Math.}, pages 153--193, 2002.

\bibitem{mcclure-smith:JAMS03}
J.E. McClure and J.H. Smith.
\newblock Multivariable cochain operations and little $n$-cubes.
\newblock {\em J. Amer. Math. Soc.}, 16(3):681--704, January 2003.

\bibitem{salvatore}
P.~Salvatore.
\newblock Configuration spaces with summable labels.
\newblock In J.~Aguad\'e, editor, {\em Cohomological methods in homotopy
  theory. Proceedings of the Barcelona conference on algebraic topology (BCAT),
  Bellaterra, Spain, June 4-10, 1998}, volume 196 of {\em Prog. Math.}, pages
  375--395, Basel, 2001. Birkh\"auser Publ.

\bibitem{schlessinger-stasheff:JPAA85}
M.~Schlessinger and J.D. Stasheff.
\newblock The {Lie} algebra structure of tangent cohomology and deformation
  theory.
\newblock {\em J. Pure Appl. Algebra}, 38:313--322, 1985.

\bibitem{tamarkin-tsygan:LMP01}
D.~Tamarkin and B.~Tsygan.
\newblock Cyclic formality and index theorems.
\newblock {\em Lett. Math. Phys.}, 56:85--97, 2001.

\bibitem{tamarkin:deligne}
D.E. Tamarkin.
\newblock Another proof of {M}. {K}ontsevich formality theorem.
\newblock Preprint {\tt math.QA/9803025}, March 1998.

\bibitem{tamarkin:formality}
D.E. Tamarkin.
\newblock Formality of chain operad of little discs.
\newblock {\em Lett. Math. Phys.}, 66(1-2):65--72, 2003.

\bibitem{laan-defo}
P.~van~der Laan.
\newblock Operads up to homotopy and deformations of operad maps.
\newblock Preprint {\tt math.QA/0208041}, August 2002.

\bibitem{voronov:99}
A.A. Voronov.
\newblock Homotopy {Gerstenhaber} algebras.
\newblock In G.~Dito, editor, {\em Conference Moshe Flato 1999: Quantization,
  deformations, and symmetries, {II}.}, volume~22 of {\em Math. Phys. Stud.},
  pages 307--331. Kluwer Academic Publishers, 2000.

\end{thebibliography}