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\titrecourant{\hfill\eightbf On the combinatorics of the graph-complex}

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{\parindent=0pt\eightbf
C. R. Acad. Sci. Paris, t.~334, S\'erie~I,
p.~1--6, 2002

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Alg\`ebres de Lie/\hskip -.5mm{\eightbfti Lie Algebras}

(Combinatoire/\hskip -.5mm{\eightbfti Combinatorics})


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\noindent
{\fourteenbf
On the combinatorics of the graph-complex
}

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\parindent=-1mm\footnote{}{
Note pr\'esent\'ee par Michel D{\sixbf UFLO.}
}

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\noindent{\bf Bodo LASS}

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{\parindent=0mm
D\'epartement de math\'ematique, Universit\'e Louis-Pasteur, 67084 Strasbourg, France

Courriel~: lass@math.u-strasbg.fr
\par}

\medskip


{\parindent=0mm
Lehrstuhl II f\"ur Mathematik, RWTH Aachen, 52062 Aachen, Allemagne

Courriel~: lass@math2.rwth-aachen.de
\par}

\medskip

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(Re\c{c}u le 26 octobre 2001, accept\'e le 13 novembre 2001)
}

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%%%%%%%%%%%%%%%%%%
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{\parindent=2cm\rightskip 1.5cm\baselineskip=9pt
\item{\bf Abstract.~}{\eightrm In their recent preprint~[3] Kontsevich and Shoikhet have introduced two graph-com\-plexes: 
the complex on the even (resp.~odd) space in order to study the cohomology of the Lie algebra Ham$\scs _0$ 
(resp.~Ham$\scs _0^{odd}$) of Hamiltonian vector fields vanishing at the origin on the 
infinite-dimensional even (resp.~odd) space.
We construct an isomorphism between those two graph-complexes, proving in particular that their cohomologies coincide. 
This solves a problem posed by Shoikhet. \copyright~Acad\'emie des Sciences/Elsevier,
Paris}
\par}

\vskip 10pt

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{\parindent=2cm\rightskip 1.5cm
{\bfti Sur la combinatoire du complexe de graphes}
\par}

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{\parindent=2cm\rightskip 1.5cm\baselineskip=9pt
\item{\bf R\'esum\'e.~}{\eightit Dans leur pr\'epublication r\'ecente~[3], Kontsevich et Shoikhet ont introduit deux
complexes de graphes~: 
le complexe sur l'espace pair (resp.~impair) pour \'etudier la cohomologie de l'alg\`ebre de Lie Ham$\scs _0$
(resp.~Ham$\scs _0^{odd}$) des champs vectoriels hamiltoniens sans terme constant sur l'espace pair (resp.~impair)
de dimension infinie.
Nous construisons un isomorphisme entre ces deux complexes de graphes, d\'emontrant notamment que leur cohomologies 
co\"\i ncident. Ceci r\'esoud un probl\`eme pos\'e par Shoikhet.~}{\eightrm\copyright~Acad\'emie des
Sciences/Elsevier, Paris}
\par}

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\noindent
{\bf Version fran\c caise abr\'eg\'ee}

\bigskip

Le probl\`eme de calculer les groupes de cohomologie de beaucoup d'alg\`ebres de Lie peut \^etre ramen\'e au calcul
des groupes de cohomologie de diff\'erents complexes de graphes (voir~[1],[2],[3]). Un {\it complexe (cohomologique) de graphes}
est un complexe
\vskip-10pt
$$\diagram{
0 & \rarr{}{} & {\bb C} & \rarr{0}{} & C^1 & \rarr{\delta}{} & C^2 & \rarr{\delta}{} & C^3 & \rarr{\delta}{} & \dots \cr
}$$
\vskip-5pt
\noindent
o\`u $C^n$ est l'espace vectoriel engendr\'e par les {\it classes d'isomorphie} de paires $(G,\hbox{or}_G)$ telles que $G$ est un
(multi)graphe de $n$~sommets et $\hbox{or}_G$ est une {\it orientation} d'un espace vectoriel r\'eel (ou plut\^ot d'un ensemble
fini) associ\'e \`a $G$. Un {\it (multi)graphe} $G = (H;V,E)$ est un ensemble de {\it demi-ar\^etes} $H$ ($|H| = 2m$) muni de 
deux partitions $V$ et $E$. Les $n$ blocs $v \in V$ sont appel\'es {\it sommets} tandis que les $m$ blocs $e \in E$ de la 
deuxi\`eme partition sont des sous-ensembles de $H$ de cardinalit\'e deux, appel\'es {\it ar\^etes}. Une ar\^ete est une 
{\it boucle} si et seulement si ses deux demi-ar\^etes font partie du m\^eme sommet. Le {\it degr\'e} d'un sommet 
$v \in V$ est sa cardinalit\'e, c'est-\`a-dire le nombre de ses demi-ar\^etes. On dit que $v$ est {\it pair} 
(resp.~{\it impair}) si et seulement si son degr\'e est pair (resp.~impair). Notons $Q$ (resp.~$O$) 
l'ensemble des sommets pairs (resp.~impairs) de~$G$. 

On {\it oriente} un ensemble fini (ou bien l'espace r\'eel engendr\'e par cet ensemble) en le munissant d'une {\it forme alternante}
(multilin\'eaire) $[ \; \cdot \; ]$ prenant ses valeurs dans l'ensemble~$\{1,-1\}$. 
\'Evidemment, le nombre d'orientations diff\'erentes vaut deux. 
L'union disjointe de plusieurs ensembles orient\'es est naturellement {\it pseudo-orient\'ee} (i.e.~orient\'ee par rapport \`a la 
partition) par le produit ordinaire des formes alternantes des blocs de la partition. Une {\it pseudo-orientation} d'un ensemble 
partitionn\'e est alors une forme alternante par rapport \`a toutes les permutations respectant la partition. \'Evidemment,
le nombre de pseudo-orientations est aussi \'egal \`a deux. 
%
L'orientation d'un graphe dite {\it paire} est la pseudo-orientation de l'union disjointe $V \uplus (\biguplus_{e \in E} e)$;
autrement dit, on \hbox{oriente} l'ensemble~$V$ et, pour chaque ar\^ete $e \in E$, l'ensemble des deux demi-ar\^etes de~$e$
(voir~[2],[3],[4]). 
L'orientation d'un graphe dite {\it impaire}, cependant, est la pseudo-orientation de l'union disjointe 
$Q \uplus (\biguplus_{v \in V} v)$, ce qui veut dire que l'on oriente l'ensemble~$Q$ des sommets pairs et, pour 
chaque sommet $v \in V$, l'ensemble des demi-ar\^etes de~$v$
(voir~[3]).

Un {\it automorphisme}~$P$ du graphe~$G$ est une permutation~$P_H$ de l'ensemble des demi-ar\^etes~$H$ pr\'eservant les structures
de~$V$ et de~$E$. Voil\`a pourquoi $P$ agit sur l'orientation paire (resp.~impaire) par une multiplication avec un nombre not\'e
$\Theta_{even}(P) \in \{1,-1\}$ (resp.~$\Theta_{odd}(P)$). Pour la permutation~$T$ des deux demi-ar\^etes d'une boucle, on a, par
exemple, $\Theta_{even}(T) = \Theta_{odd}(T) = -1$, puisque l'orientation d'une seule ar\^ete (pour l'orientation paire) et
d'un seul sommet (pour l'orientation impaire) est renvers\'ee. De fa\c con plus g\'en\'erale, nous d\'emontrerons pour 
chaque automorphisme~$P$ que $\Theta_{even}(P) = \Theta_{odd}(P) = \hbox{sign}(P_H) \cdot \hbox{sign}(P_V)$, 
si $P_V$ d\'esigne la permutation induite sur l'ensemble des sommets de~$G$. 
%
On impose la relation $(G,-\hbox{or}_G) = -(G,\hbox{or}_G)$, de sorte que $(G,\hbox{or}_G) = 0$ si et seulement si $G$ a
un automorphisme tel que $\Theta_{even}(P) = \Theta_{odd}(P) = -1$, comme dans le cas des boucles (d'o\`u $C^1 = 0$). 
Par cons\'equent, les g\'en\'erateurs du complexe de graphes sont (\`a une multiplication par $\pm 1$ pr\`es) les m\^emes pour
l'orientation paire et impaire. 
En fait, nous montrerons que ces deux orientations correspondent de fa\c con naturelle \`a une orientation ordinaire 
de l'ensemble $H \uplus V$, que nous appelons {\it universelle}. 

Cette correspondance permet \'egalement de comparer les diff\'erentielles, d\'efinies par Kontsevich et Shoikhet~[3]  
de la mani\`ere suivante~:
%
Pour $\delta_{even}(G)$ et $\delta_{odd}(G)$ on prend toujours la somme sur {\it tous} les {\lfq gonflages\rfq} de {\it tous} 
les sommets de~$G$, o\`u un gonflage d'un sommet veut dire que l'ensemble de ses demi-ar\^etes est partitionn\'e (de fa\c con 
non-ordonn\'ee) en deux blocs (ce sont des sommets nouveaux), qui sont reli\'es par une ar\^ete nouvelle (c'est-\`a-dire 
chaque bloc nouveau obtient une demi-ar\^ete nouvelle, et ces deux demi-ar\^etes nouvelles forment une ar\^ete).
\'Evidemment, chaque gonflage augmente le nombre de sommets d'une unit\'e. 
Soit $G =  (H;V,E)$ avec $V = \{v_1,v_2,\dots,v_n\}$; $E = \{e_1,\dots,e_m\}$;
$v_1 = \{h_{a_1},\dots,h_{x_1},h_{y_1},\dots,h_{b_1}\}$, $v_2 = \{h_{a_2},\dots,h_{b_2}\}$, \dots, $v_n = \{h_{a_n},\dots,h_{b_n}\}$;
$e_1 = \{h_{i_1},h_{t_1}\}$, \dots, $e_m = \{h_{i_m},h_{t_m}\}$; et $H = \biguplus_{k=1}^n v_k = \biguplus_{k=1}^m e_k$.
Consid\'erons un gonflage de $v_1$ avec $v'_1 = \{h_{a_1},\dots,h_{x_1}\}$, $v''_1 = \{h_{y_1},\dots,h_{b_1}\}$ et avec la nouvelle
ar\^ete $\{h'_1,h''_1\}$. Pour l'orientation, Kontsevich et Shoikhet~[3] d\'efinissent alors
$$
[v'_1,v''_1,v_2,\dots,v_n] \cdot [h'_1,h''_1] \prod_{k=1}^m [h_{i_k},h_{t_k}] 
\; := \; [v_1,v_2,\dots,v_n] \cdot \prod_{k=1}^m [h_{i_k},h_{t_k}] \quad \hbox{sur l'espace pair et}
$$
$$
[v'_1,v''_1,v_2,\dots,v_n]_Q \cdot [h'_1,h_{a_1},\dots,h_{x_1}] [h''_1,h_{y_1},\dots,h_{b_1}] \prod_{k=2}^n [h_{a_k},\dots,h_{b_k}] 
$$
$$
:= \; [v_1,v_2,\dots,v_n]_Q \cdot \prod_{k=1}^n [h_{a_k},\dots,h_{b_k}] \quad \hbox{sur l'espace impair,} 
$$
o\`u $[ \; \cdot \; ]_Q$ signifie qu'il faut supprimer tous les sommets impairs entre les crochets.
%
Nous d\'emontrons le th\'eor\`eme suivant ($o := |O|$, $n := |V|$).

\medskip

{\sc Th\'eor\`eme.~-} {\it Du point de vue de l'orientation universelle, il suffit de multiplier l'orientation de chaque graphe~$G$
par $(-1)^{n-o/2}$ pour obtenir un isomorphisme entre l'orientation paire et impaire qui est compatible avec les diff\'erentielles.
C'est pourquoi le complexe sur l'espace pair et le complexe sur l'espace impair sont isomorphes.}


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\centerline{\hbox to 2cm{\hrulefill}}
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\bigskip


%%%%%%%%%%%%%%%%%%%%%%%
%%  TEXTE PRINCIPAL  %%
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\noindent
{\bf 1. Introduction}

\medskip

The underlying vector space of the Lie algebra Ham$_0(N)$ of Hamiltonian vector fields vanishing at the origin {\it on the 
even space} is equal to $\bigoplus_{d \ge 2} S^d V$, where $V$ is the defining $2N$-dimensional representation of sp$(2N)$, 
$S^2 V = \hbox{sp}(2N)$, etc. It is classical (see [1] and [2]) that the cochain complex 
$$
C_{Lie}^\bullet(\hbox{Ham}_0(N);\bb C) = \bigoplus_{d_2 \ge 0, \, d_3 \ge 0, \, \dots} \textstyle
                                         \bigwedge^{d_2}(S^2 V) \wedge \bigwedge^{d_3}(S^3 V) \wedge \cdots
$$
is quasi-isomorphic to its sp-invariant part $[C_{Lie}^\bullet(\hbox{Ham}_0(N);\bb C)]^{sp}$. Let us consider 
$\bigwedge^{d_2}(S^2 V) \wedge \bigwedge^{d_3}(S^3 V) \wedge \cdots$ as a quotient space of 
$(V^{\otimes 2})^{\otimes d_2} \otimes (V^{\otimes 3})^{\otimes d_3} \otimes \cdots$.
If $2d_2 + 3d_3 + \cdots =: 2m$ is even, then any perfect matching on $H := \{1,2,\dots,2m\}$ defines a (multi)graph $G = (H;V,E)$
with $2m$ half-edges in~$H$, $m$ edges in~$E$ and $n := d_2 + d_3 + \cdots$ vertices in~$V$, $d_2$ of which have degree~$2$, $d_3$ 
of which have degree~$3$, etc. There exists a canonical symplectic invariant in $\bigwedge^2 V \hookrightarrow V^{\otimes 2}$,
that can be chosen in $V_i \otimes V_j$ for every edge $\{i,j\} \in E$ ($i,j \in H$) to obtain an invariant in
$V^{\otimes 2m}$.
By the main theorem of invariant theory those tensors form a basis of $[V^{\otimes 2m}]^{sp}$ and there are no invariants for
$(2d_2 + 3d_3 + \cdots)$ odd, if $2N = \hbox{dim}(V)$ is sufficiently large.
The (multi)graphs $G = (H;V,E)$ carry the {\it even orientation} corresponding to the exterior product over the set of vertices
(in the definition of the cochain complex) and to the exterior products over the set of half-edges of any edge (in the definition
of the canonical symplectic invariants). In fact, the consideration of those oriented graphs modulo the action of symmetric groups
provides a bijection with the elements of the complex $[C_{Lie}^\bullet(\hbox{Ham}_0(N);\bb C)]^{sp}$. The differentials of this
complex can be easily expressed in graph-theoretical terms (see Section 3). Therefore the cohomology of the resulting 
{\it graph-complex on the even space} is equal to the cohomology of Hamiltonian vector fields vanishing at the origin on the 
infinite-dimensional even space.

The underlying vector space of the Lie algebra Ham$_0^{odd}(N)$ of Hamiltonian vector fields vanishing at the origin 
{\it on the odd space} is equal to $\bigoplus_{d \ge 2} \bigwedge^d V$, where $V$ is the defining $N$-dimensional 
representation of $\hbox{sp}^{odd}(N) = \hbox{\tengoth o}(N)$, $\bigwedge^2 V = \hbox{sp}^{odd}(N)$, etc.
The main difference from the even case is that the canonical invariant lies in 
$S^2 V \hookrightarrow V^{\otimes 2}$ and that the cochain complex is given by
$$
C_{Lie}^\bullet(\hbox{Ham}_0^{odd}(N);\bb C) = \bigoplus_{d_2 \ge 0, \, d_3 \ge 0, \, \dots} \textstyle
\bigwedge_{super}^{d_2}(\bigwedge^2 V) \wedge_{super} \bigwedge_{super}^{d_3}(\bigwedge^3 V) \wedge_{super} \cdots ,
$$
where $\wedge_{super}$ denotes the graded exterior product (which is anticommutative if the degrees of both factors are even
and commutative otherwise). Therefore the cohomology of Hamiltonian vector fields vanishing at the origin on the infinite-dimensional 
odd space is equal to the cohomology of the {\it graph-complex on the odd space}, which differs from the graph-complex on the even 
space just by its orientation. In fact, the (multi)graphs $G = (H;V,E)$ carry the {\it odd orientation} corresponding to the graded 
exterior product over the set of vertices and to the exterior products over the set of half-edges of any vertex, and the 
differentials of the graph-complex on the odd space can be expressed in graph-theoretical terms with the help of the odd 
orientation. 

With respect to this situation Shoikhet suggested that 
{\it
``you obtain at first look a different graph-complex, but our idea with Kontsevich was that the two cohomologies coincide - 
and to prove it by something like an isomorphism of orientations. {\dots} 
So, up to me, a really interesting problem in this direction is to prove that the even and the odd graph-complexes give the same 
orientation.''
}

We solve this problem in the next two sections using still another orientation, which we call {\it universal} and which is more
classical from the point of view of algebraic topology.

\bigskip

\noindent
{\bf 2. The generators of the graph complexes on the even space and on the odd space}

\medskip

A (multi)graph $G = (H;V,E)$ is a set of {\it half-edges}~$H$ consisting of $2m$ elements, together with two partitions of this 
set into disjoint unions of subsets. The $n$ blocks of the first partition $v \in V$ are called {\it vertices} whereas the 
$m$ blocks of the second partition $e \in E$ are two-element subsets of~$H$ called {\it edges}. The {\it degree} of a 
vertex $v \in V$ is the number of its half-edges and $v$ is called {\it even} (resp.~{\it odd}) iff its degree is 
even (resp.~odd). 
%
Let $[ \; \cdot \; ]$ be an alternating (multilinear) form on (the real space generated by) the disjoint union of all half-edges 
$h_1,h_2,\dots,h_{2m}$ and all vertices $v_1,v_2,\dots,v_n$. The form $[ \; \cdot \; ]$ is uniquely defined by determining the
value $[h_1,h_2,\dots,h_{2m};v_1,v_2,\dots,v_n] = [v_1,v_2,\dots,v_n;h_1,h_2,\dots,h_{2m}] \in \{1,-1\}$. A (multi)graph with 
such a form will be called {\it universally oriented}. 
In order to determine $[h_1,h_2,\dots,h_{2m};v_1,v_2,\dots,v_n]$ we can write the elements $h_1,h_2,\dots,h_{2m};v_1,v_2,\dots,v_n$
in different orders, all of which will start with the half-edges.

One natural order makes use of the block structure determined on the set of half-edges~$H$ by the set of edges~$E$, i.e.~we consider
$[h_{i_1},h_{t_1};h_{i_2},h_{t_2};\dots;h_{i_m},h_{t_m};v_{j_1},v_{j_2},\dots,v_{j_n}]$, where $\{h_{i_1},h_{t_1}\}$, 
$\{h_{i_2},h_{t_2}\}$, \dots, $\{h_{i_m},h_{t_m}\}$ are the $m$ edges of the graph $G$.
Permuting whole edges does not change the value of                  
$[h_{i_1},h_{t_1};h_{i_2},h_{t_2};\dots;h_{i_m},h_{t_m};v_{j_1},v_{j_2},\dots,v_{j_n}]$
whereas permuting two half-edges within an edge does change this value.
Therefore, orienting a graph universally is canonically the same as orienting the set $V$ of vertices and orienting, for every
edge $e \in E$, the set of half-egdes of~$e$. This corresponds to the orientation in the case of the graph-complex 
on the even space (see [2],[3],[4]).

Another natural order makes use of the block structure determined on the set of half-edges~$H$ by the set~$Q$ ($|Q|=q$) of even 
vertices and by the set~$O$ ($|O|=o$) of odd vertices. In other words, we consider 
$$
w \; := \; [h_{a_1},\dots,h_{b_1}; \dots; h_{a_q},\dots,h_{b_q}; h_{c_1},\dots,h_{d_1}; \dots; h_{c_o},\dots,h_{d_o};
           v_{f_1},\dots,v_{f_o}; v_{g_1},\dots,v_{g_q}], \leqno (*)
$$
where $\{h_{a_1},\dots,h_{b_1}\}$, \dots, $\{h_{a_q},\dots,h_{b_q}\}$ as well as $v_{g_1},\dots,v_{g_q}$ are the $q$ even vertices 
of $G$ and where $v_{f_1} = \{h_{c_1},\dots,h_{d_1}\}$, \dots, $v_{f_o} = \{h_{c_o},\dots,h_{d_o}\}$ are its $o$ odd vertices.
Permuting whole even vertices (i.e.~their sets of half-edges) does not change the value of $w$. But permuting whole odd vertices
(i.e.~their sets of half-edges) does not change $w$ either, because in that case the vertices $v_{f_1},\dots,v_{f_o}$ have to
be permuted in the same way by our convention. On the other hand, permutations within whole vertices (i.e.~within their sets 
of half-edges) as well as permutations of $Q = \{v_{g_1},\dots,v_{g_q}\}$ do change $w$. 
Therefore, orienting a graph universally is equivalent to orienting the set $Q$ of even vertices and for every vertex $v \in V$
orienting the set of half-egdes of~$v$. This corresponds to the orientation in the case of the graph-complex on the odd space
(see~[3]).

An {\it automorphism} $P$ of the graph~$G$ is an arbitrary permutation $P_H$ of the set~$H$ which preserves the structures of~$V$ 
and~$E$. In particular, it induces permutations $P_Q$ of $Q$, $P_O$ of $O$ and $P_V$ of the set~$V$ of vertices.  
The automorphism $P$ acts on the even (resp.~odd) orientation by a multiplication by 
$$
\Theta_{even}(P) \; = \; \hbox{sign}(P_V) \, \cdot \, \prod_{e \in E}\epsilon_P(e), \quad
\Theta_{odd}(P) \; = \; \hbox{sign}(P_Q) \, \cdot \, \prod_{v \in V}\epsilon_P(v),  \quad \hbox{respectively},
$$
where $\epsilon_P(e)$ equals $1$ (resp.~$-1$) if and only if $P$ is sign-preserving (resp.~sign-reversing) on 
(the set of half-edges of) $e$, and 
where $\epsilon_P(v)$ equals $1$ (resp.~$-1$) if and only if $P$ is sign-preserving (resp.~sign-reversing) on 
(the set of half-edges of) $v$.
Finally, $P$ acts on the universal orientation by a multiplication by $\hbox{sign}(P_H) \cdot \hbox{sign}(P_V)$.
Now our established correspondence between those three orientations implies:
$$
\Theta_{even}(P) \; = \; \Theta_{odd}(P) \; = \; \hbox{sign}(P_H) \cdot \hbox{sign}(P_V).
$$

The $n$-dimensional generators of the different graph-complexes are always {\it equivalence classes} of pairs 
$(G,\hbox{or}_G)$, where $G$ is a graph with $n$ vertices (whose degrees are all greater than one), 
and where $\hbox{or}_G$ is one of our orientations.
One imposes the relation $(G,-\hbox{or}_G) = -(G,\hbox{or}_G)$, so that $(G,\hbox{or}_G) = 0$ if and only if $G$ has an 
automorphism whose sign is $-1$. 
We have established the following theorem.

\medskip

{\sc Theorem 1.~-} 
{\it There is a canonical correspondence between the generators of the graph-complex on the even space, 
     the generators of the graph-complex on the odd space and the generators of the graph-complex 
     using the universal orientation.}\qed

\bigskip


\noindent
{\bf 3. The differentials of the graph complexes on the even space and on the odd space}

\medskip

In order to get an isomorphism of complexes we have to verify whether our canonical correspondences between the generators
are compatible with the differentials. Let us first define a differential for the universal orientation. It is essentially
given by the dual of the operation of contracting each edge. More precisely, some vertex $v_s \in V$ has to be split up into
two vertices $v'_s$ and $v''_s$, i.e.~the set of its half-edges has to be partitioned. Moreover, a new edge $\{h'_s,h''_s\}$
will be introduced, where $h'_s \in v'_s$ and $h''_s \in v''_s$. Finally, the orientation can be defined by
$$\openup2pt\eqalign{
&[h_1,h_2,\dots,h_{2m},h'_s,h''_s;v_1,v_2,\dots,v_{s-1},v'_s,v''_s,v_{s+1},\dots,v_n] \cr
&:= \; (-1)^{s-1} [h_1,h_2,\dots,h_{2m};v_1,v_2,\dots,v_{s-1},v_s,v_{s+1},\dots,v_n]. \cr
}$$
It is evident that this definition is well defined and leads in fact to a differential $\delta$ (for the differential, the sum over 
all splittings is taken; if we split twice in the same way but in a different order, then the corresponding terms cancel, 
i.e.~$\delta^2 = 0$).

The expression of the definition in terms of the block structure determined by the set of edges of the graph~$G$ is immediate.
It leads precisely to the definition of~[3] (and~[2]) in the case of the graph-complex on the even space
(corresponding to the cochain complex $C_{Lie}^\bullet(\hbox{Ham}_0;\bb C)$).

In order to find the expression for the definition in terms of the block structure determined by the set of vertices of~$G$,
it is sufficient to consider what happens to $w$ (see equation $(*)$ in the preceding section) if we partition $v_{f_1}$ or 
$v_{g_1}$ (these vertices are both on odd places because the number~$o$ of odd vertices of $G$ is even). 
First suppose that $v_{f_1} = \{ h_{c_1},\dots,h_{x_1},h_{y_1},\dots,h_{d_1} \}$ and that the half-edges of $v_{f_1}$ are 
partitioned into the blocks $v'_{f_1} = \{ h'_{f_1},h_{c_1},\dots,h_{x_1} \}$ and $v''_{f_1} = \{ h''_{f_1},h_{y_1},\dots,h_{d_1} \}$,
where $\{h'_{f_1},h''_{f_1}\}$ is the new edge.
If $|v'_{f_1}|$ is odd and $|v''_{f_1}|$ is even, then, according to our universal orientation, $w$ is transformed into 
$$\openup2pt\eqalign{
[&h_{a_1},\dots,h_{b_1}; \dots; h_{a_q},\dots,h_{b_q};
  h'_{f_1},h''_{f_1};
  h_{c_1},\dots,h_{x_1}; h_{y_1},\dots,h_{d_1}; h_{c_2},\dots,h_{d_2}; \dots; h_{c_o},\dots,h_{d_o}; \cr
 &v'_{f_1},v''_{f_1};v_{f_2},\dots,v_{f_o}; v_{g_1},\dots,v_{g_q}] \; =  \cr
[&h_{a_1},\dots,h_{b_1}; \dots; h_{a_q},\dots,h_{b_q}; 
  h''_{f_1},h_{y_1},\dots,h_{d_1}; h'_{f_1},h_{c_1},\dots,h_{x_1}; h_{c_2},\dots,h_{d_2}; \dots; h_{c_o},\dots,h_{d_o}; \cr
 &v'_{f_1},v_{f_2},\dots,v_{f_o}; v''_{f_1},v_{g_1},\dots,v_{g_q}] \cdot (-1). \cr  
}$$
If $|v'_{f_1}|$ is even and $|v''_{f_1}|$ is odd, we get
$$\openup2pt\eqalign{
[&h_{a_1},\dots,h_{b_1}; \dots; h_{a_q},\dots,h_{b_q}; 
  h'_{f_1},h_{c_1},\dots,h_{x_1}; h''_{f_1},h_{y_1},\dots,h_{d_1}; h_{c_2},\dots,h_{d_2}; \dots; h_{c_o},\dots,h_{d_o}; \cr
 &v''_{f_1},v_{f_2},\dots,v_{f_o}; v'_{f_1},v_{g_1},\dots,v_{g_q}] \cdot (-1). \cr  
}$$
Now suppose that $v_{g_1} = \{ h_{a_z},\dots,h_{x_z},h_{y_z},\dots,h_{b_z}\}$ and that the half-edges of $v_{g_1}$ are 
partitioned into the blocks $v'_{g_1} = \{ h'_{g_1},h_{a_z},\dots,h_{x_z} \}$ and $v''_{g_1} = \{ h''_{g_1},h_{y_z},\dots,h_{b_z} \}$.
If $|v'_{g_1}|$ and $|v''_{g_1}|$ are odd, then 
$$\openup2pt\eqalign{
[&h_{a_1},\dots,h_{b_1}; \dots; h'_{g_1}, h''_{g_1}; h_{a_z},\dots,h_{x_z}; h_{y_z},\dots,h_{b_z}; \dots; h_{a_q},\dots,h_{b_q};
  h_{c_1},\dots,h_{d_1}; \dots; h_{c_o},\dots,h_{d_o}; \cr
 &v_{f_1},\dots,v_{f_o}; v'_{g_1}, v''_{g_1}; v_{g_2},\dots,v_{g_q}] \; =  \cr
[&h_{a_1},\dots,h_{b_1}; \dots; h_{a_{z-1}},\dots,h_{b_{z-1}}; h_{a_{z+1}},\dots,h_{b_{z+1}}; \dots; h_{a_q},\dots,h_{b_q}; \cr   
 &h'_{g_1},h_{a_z},\dots,h_{x_z}; h''_{g_1},h_{y_z},\dots,h_{b_z}; h_{c_1},\dots,h_{d_1}; \dots; h_{c_o},\dots,h_{d_o}; \cr
 &v'_{g_1},v''_{g_1},v_{f_1},\dots,v_{f_o}; v_{g_2},\dots,v_{g_q}]. \cr  
}$$
Finally, if $|v'_{g_1}|$ and $|v''_{g_1}|$ are even, we get
$$\openup2pt\eqalign{
[&h_{a_1},\dots,h_{b_1}; \dots; h'_{g_1},h_{a_z},\dots,h_{x_z}; h''_{g_1},h_{y_z},\dots,h_{b_z}; \dots; h_{a_q},\dots,h_{b_q};
  h_{c_1},\dots,h_{d_1}; \dots; h_{c_o},\dots,h_{d_o}; \cr
 &v_{f_1},\dots,v_{f_o}; v'_{g_1},v''_{g_1},v_{g_2},\dots,v_{g_q}] \cdot (-1). \cr
}$$
If there were no factors $-1$, the preceding relations would be precisely those used by Kontsevich and Shoikhet~[3] 
for defining their differential of the graph-complex on the odd space (corresponding to the cochain complex 
$C_{Lie}^\bullet(\hbox{Ham}_0^{odd};\bb C)$). 
In order to get rid of those factors $-1$, however, 
{\it it suffices to use the isomorphism which multiplies every graph~$G$ (i.e.~its orientation) by $(-1)^{n-o/2}$, 
if $G$ has $n$ vertices, $o$ of which are odd}. In fact, the multiplication by $(-1)^n$ provides a minus for every differential,
whereas the multiplication by $(-1)^{o/2}$ (note that $o$ is always even) kills this minus if two new odd vertices are created.
Therefore we have proved the following theorem.

\medskip

{\sc Theorem 2.~-} {\it The cohomological graph-complex on the even space, the cohomological graph-complex on the odd space and 
the universally oriented cohomological graph-complex are isomorphic. In particular, their cohomologies coincide.}\qed

\medskip

By the classical arguments reproduced in the introduction this implies our final corollary, a surprising duality relation
which (according to Feigin) seems to be difficult to derive by different means.

\medskip

{\sc Corollary.~-} {\it The cohomologies of the Lie algebras $\hbox{Ham}_0$ and $\hbox{Ham}_0^{odd}$ of Hamiltonian vector fields
vanishing at the origin on the infinite-dimensional even and odd spaces coincide.}\qed


\bigskip

{\it Acknowledgements.} I should like to thank B.~Shoikhet for kindly sending me the interesting preprint~[3],
for asking me the question that led to this Note and for very useful discussions; 
I am also grateful to P.~Cartier and D.~Foata for all their help and encouragement. 


\bigskip

%%%%%%%%%%%%%%%%%%%%%%%
%%   BIBLIOGRAPHIE   %%
%%%%%%%%%%%%%%%%%%%%%%%

{\bf References}

\medskip

{\baselineskip=9pt\eightrm
\item{[1]} Fuchs D.B., Kogomologii beskonechnomernykh algebr Lie, Nauka, Moskva, 1984.
\smallskip 

\item{[2]} Kontsevich M., Formal (non)-commutative symplectic geometry,
           in~: Corwin L., Gelfand I., Lepowsky J.~(Eds.),
           The Gelfand Mathema\-tical Seminars 1990-1992, Birkh\"auser, 1993, pp.~173-187.
\smallskip 

\item{[3]} Kontsevich M., Shoikhet B., Formality conjecture, geometry of complex manifolds and 
           combinatorics of the graph-complex, preprint.
\smallskip

\item{[4]} Lass B., Une conjecture de Kontsevich et Shoikhet et la caract\'eristique d'Euler,
           Rev.~Math.~Pures Appl.~(\`a para\^\i tre).
\smallskip 
}

\end
