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\hyphenation{gla-v-no-go gra-du-i-ro-va-n-no gra-du-i-ro-va-n-noe ig-ra-et 
ime-et kva-d-ra-tov li-e-vy ma-te-ma-tiche-s-kih  mno-go-ob-ra-zii od-no-ro-d-ny ope-rad
pe-re-s-ta-no-v-ke slu-zhit stru-k-tur stru-k-tu-ru stru-k-tu-ro
to-zh-destv to-zh-de-st-vo voz-ro-zh-de-niyu}

\newcommand{\h}{Hoh\-shilp1\-da}
\newcommand{\hc}{kom\-pleks Hoh\-shilp1\-da}
\newcommand{\hg}{go\-mo\-to\-pi\-che\-s\-kaya algebra Gerstenhabera\ }
\newcommand{\hl}{go\-mot\-o\-pi\-che\-s\-kaya algebra Lie\ }
\newcommand{\coh}{ko\-go\-mo\-lo\-gii}

\newcommand{\nz}{\Bbb{Z}}

\newcommand{\ca}{\cal{C}^\bullet (A)}
\newcommand{\CA}[1]{\cal{C}^{#1} (A)}
\newcommand{\hha}{\operatorname{HH}^\bullet (A)}

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\newtheorem{crl}[th]{Sledstvie}
\newtheorem{conj}[th]{Gipoteza}
\newtheorem{probl}[th]{Vopros}

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\begin{document}
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\title[\cyr
Vysshie operacii na komplekse Hohshilp1da]
{\cyr
Vysshie operacii na komplekse Hohshilp1da}

\author[\cyr A. A. Voronov, M. Gerstenhaber]
{\copyright  \ 1994. {\cyr A. A. Voronov, M. Gerstenhaber}}
\address
{Universitet Pensilp1vanii i Prinstonski\ui\ Universitet}

%\author
%{Aleksandr A. Voronov}
\address
%[A. A.~Voronov]
{%Department of Mathematics\\
%Princeton University%\\Princeton, NJ 08544-1000\\USA and Research Institute
%%for
%Mathematical Sciences\\Kyoto University\\Kyoto 606, Japan}
%curraddr{Department of Mathematics\\
Universitet Pensilp1vanii
%\\Philadelphia, PA 19104-6395\\USA
%and Department of Mathematics\\Princeton
%University\\Princeton, NJ 08544-1000\\USA
}
%\email{voronov@@math.princeton.edu}
\thanks{\cyr Issledovaniya pervogo avtora chastichno finansirovalisp1
Nacionalp1nym Nauchnym Fondom SSHA
}
\thanks{\cyr Issledovaniya vtorogo avtora chastichno finansirovalisp1
Nacionalp1nym Upravleniem Bezopasnosti SSHA}
\date{}

\maketitle

%\input{../amslatex/cyracc.def}

Nedavnee otkrytie ryada algebraicheskih struktur v kvantovo{\ui} teorii polya
\cite{WZ,Z} i teorii algebr vershinnyh operatorov \cite{lz} privelo k
vozrozhdeniyu operad i algebr s neskolp1kimi operaciyami. Videnie mira, prisuwee fizikam, obnaruzhilo naliqie novyh matematicheskih struktur, intensivno izuqaemyh teperp1 matematikami,
sm.\ \cite{grav,gj,gk,h,ksv,lz}. Sravnivaya dva rezulp1tata:
\begin{enumerate}
\item (Gerstenhaber, \cite{Gerst}) \coh\ \h\ $\hha := H^\bullet(A,A)$
associativno\ui\ algebry $A$ imeyut estestvennuyu strukturu graduirovanno\ui\
 puassonovo\ui\ algebry, nazyvaemo\ui\ teperp1 {\it algebro\ui\ Gerstenhabera},
\item (F. Koe1n,  \cite{C1}) algebra nad operado\ui\ gomologi\ui\ operady 
malenp1kih kvadratov estp1 ne chto inoe kak algebra Gerstenhabera --
\end{enumerate}
i, vozmozhno, novye idei \cite{gk,kon:sympl} otnositelp1no gomotopicheskih algebr,
P.~Delinp1 sformuliroval sleduyuwuyu gipotezu v pisp1me \cite{d:lett}.
\begin{conj}[Delinp1]
\label{deligne}
Kompleks Hohshilp1da $\ca := C^\bullet(A,A) $ imeet estestvennuyu 
strukturu 
algebry nad operado\ui\ cepe\ui\ operady malenp1kih kvadratov.
\end{conj}

Pod ''operado\ui\ cepe\ui'' zdesp1 mozhno podrazumevatp1 lyubuyu
 operadu kompleksov ({\it {\rm DG}-operadu\/}), operada gomologi\ui\ kotoro\ui\ izomorfna 
operade ma\-lenp1\-kih kvadratov. V duhe raboty Huanga \cite{h} e1tu 
gipotezu mozhno pereformulirovatp1 tak: {\it kompleks $\ca$ obladaet strukturo\ui, podobno\ui\ topologichesko\ui\
algebre 
vershinnyh operatorov}.

Poskolp1ku {\rm DG}-operada, operada gomologi\ui\ kotoro\ui\ opisyvaet 
associativnye (kommutativnye, lievy i t.\ d.) algebry, dolzhna opisyvatp1 
gomotopicheski associativnye (sootvet\(\)stvenno, kommutativnye, lievy
 i t.\ d.) algebry,
gipotezu \ref{deligne} mozhno ponimatp1 i tak.

\begin{conj}
\label{hg}
Kompleks \h\ $\ca$ imeet estestvennuyu strukturu 
gomotopichesko\ui\ algebry Gerstenhabera.
\end{conj}

V e1to\ui\ statp1e predstavleno dokazatelp1stvo gipotezy~\ref{hg} --
sm.\ teoremu~\ref{mainth}.  ''Go\-mo\-topichesko\ui\ algebro\ui\
Gerstenhabera'' v e1to\ui\ gipoteze mozhet slu\-zhitp1 lyu\-baya
gomotopicheskaya algebra, operacii kotoro\ui\ induciruyut strukturu
algebry Gerstenhabera v kogomologiyah. Priqina udivitelp1no\ui\ svyazi
kogomologi\ui\ \h\ s topologie\ui\ malenp1kih kvadratov do sih por 
ostaet\(\)sya zagadko\ui. Vozmozhnoe obp2yasnenie predlagaet\(\)sya v
punkte~\ref{conclusion}.

\subsection{{\protect\cyr Kompleks \h }}

Pustp1 $A$ -- associativnaya algebra nad polem i
$$V = \ca = C^\bullet(A,A)\; - $$
ee \hc, dayuwi\ui\ \coh\ \h\ algebry $A$ s koe1fficientami v sebe samo\ui.
Budem oboznaqatp1 e1lementy $V$ qerez $x$, $x_1$ i t.\ d., a e1lementy
$A$ -- qerez $a_1$, $a_2$
i t.\ d. Pustp1 takzhe $|x|$ -- stepenp1 e1lementa $x$ v sdvige $V[1]$
prostranstva
$V$:
\[
|x| = k-1 , \text{ esli } x \in \CA{k}.
\]
V qastnosti, my polagaem $|a|=-1$ dlya $a \in A = \CA{0}$. My
budem rassmatrivatp1 sleduyuwie operacii na komplekse \h.

\subsection{\protect\cyr Umnozhenie}
%\label{dot}

Opredelim dlya kocepe\ui\ $x \in \CA{k}$ i $y \in \CA{l}$ umnozhenie
\begin{equation}
\label{dot-eq}
(x \cdot y) (a_1, \dots , a_{k+l}) := (-1)^{(|x|+1)(|y|+1)} x(a_1, \dots, a_k)
\cdot y (a_{k+1}, \dots , a_{k+l}).
\end{equation}
E1to obyqnoe $\cup$-umnozhenie $x \cup y$ kocepe\ui\ s tochnostp1yu do znaka.
My polp1\-zu\-emsya sleduyuwim pravilom znakov: {\it pronesenie obp2ekta $x$
qerez obp2ekt $y$ vleqet znak $(-1)^{|x||y|}$}, gde my polagaem, qto 
stepenp1 ''$\cdot$'' ravna edinice,
stepenp1 $d$ (sm.\ nizhe) ravna edinice i t.\ d.
Legko proveritp1, qto $xy =
x\cdot y$ zadaet strukturu associativno\ui\ differencialp1no\ui\ graduirovanno\ui\ 
({\rm DG}-\nolinebreak[4]) \linebreak[0]
algebry na $V$, gde differencial
\begin{equation*}
(d x) (a_1, \dots, a_{n+1})
\begin{split}
\; := \; &  (-1)^{|x|} a_1 x(a_2, \dots, a_{n+1}) \\
& + (-1)^{|x|}  \sum_{i=1}^n (-1)^i x (a_1, \dots , a_{i-1}, a_i a_{i+1},
a_{i+2}, \dots , a_{n+1})\\
& +   x(a_1, \dots, a_n) a_{n+1}
\end{split}
\end{equation*}
v komplekse \h\ igraet rolp1 differencirovaniya:
\begin{equation}
\label{d-dot}
d (x \cdot y) =dx \cdot y +   (-1)^{|x|+1 } x \cdot dy.
\end{equation}
E1ta algebra yavlyaet\(\)sya differencialno\ui\ graduirovanno\ui\ v smysle 
oby\-q\-no\ui\ graduirovki $|x|+1$.

\subsection{\protect\cyr Skobki}
%\label{braces}

Polozhim
\begin{multline}
\label{brace}
x\{ x_1, \dots, x_m\} (a_1, \dots, a_n) \\
%\begin{split}
 :=  \sum_{0 \le i_1 \le \dots \le i_m \le n}
(-1)^\varepsilon x (%&
 a_1, \dots , a_{i_1},  x_1(a_{i_1+1}, \dots ), \dots,%\\
 %&
a_{i_m},  x_m (a_{i_m+1}, \dots), \dots, a_n),
%\end{split}
\end{multline}
gde $\varepsilon := \sum_{p=1}^{m} |x_{p}| i_{p} $, to estp1,
$(-1)^\varepsilon$ estp1 znak, poyavlyayuwi\ui sya pri perestanovke
po\-s\-le\-do\-va\-telp1\-no\-s\-ti bukv v levo\ui\ qasti formuly v
po\-s\-le\-do\-va\-telp1\-nostp1 bukv v pravo\ui\ soglasno nashemu pravilu znakov. Skobki
\linebreak[0]
$x\{ x_1, \dots, x_m\}$ odnorodny stepeni nolp1 v smysle graduirovki $|x|$. 
My budem takzhe obo\-z\-na\-qatp1 $x\{y\}$ qerez $x \circ y$ i ponimatp1 $x\{\}$
prosto kak $x$.

Pustp1 $m \in \CA{2}$, $m(a\otimes b ) = ab$, -- kocikl, opredelenny\ui\ umnozheniem v $A$. Neposredstvenno proveryayut\(\)sya sleduyuwie sootnosheniya:
\begin{gather}
\label{dot-m}
x \cdot y = (-1)^{|x|+1} m\{x,y\},\\
\label{d-m}
d x =  m \circ x -  (-1)^{|x|} x \circ m.
\end{gather}

\subsection{\protect\cyr Tozhdestva}
\label{identities}

Operacii, vvedennye vyshe, udovletvoryayut naboru tozhdestv, kotorye my sobiraemsya pereqislitp1. Vo-pervyh, $V$ yavlyaet\(\)sya associativno\ui\ 
{\rm DG}-algebro\ui\ po otnosheniyu k umnozheniyu i graduirovke $|x|+1$. Krome togo, ieyut\(\)sya menee trivialp1nye tozhdestva.

\subsection{\protect\cyr Vysshie predp2yakobievy tozhdestva}
\begin{multline}
\label{higher}
x\{x_1, \dots, x_m\} \{ y_{1}, \dots , y_{n}\} \\
%\begin{split}
= \sum_{0 \le i_1 \le \dots \le i_m \le n}  (-1)^\varepsilon  x\{ y_1, \dots,
y_{i_1}, x_1  \{ y_{i_1+1}, \dots \},  \dots, y_{i_{m}}, x_m \{  y_{i_{m}+1} ,
\dots \}, \dots, y_n \},
%\end{split}
\end{multline}
gde $\varepsilon := \sum_{p=1}^{m}   |x_{p}| \sum_{q=1}^{i_p} |y_{q}|$, to estp1,
$(-1)^\varepsilon$ -- znak, poyavlyayuwi\ui sya pri pronesenii $x_i$ qerez
 $y_j$ pri tasovke v formule. E1to tozhdestvo dokazyvaet\(\)sya pryamym vyqisleniem.

\subsection{\protect\cyr Distributivnostp1}

\begin{equation}
\label{dist}
(x_1 \cdot x_2) \{ y_1, \dots, y_n\} = \sum_{k=0}^n (-1)^\varepsilon x_1 \{
y_1, \dots, y_k\} \cdot x_2 \{ y_{k+1}, \dots, y_n\},
\end{equation}
gde $\varepsilon = (|x_2|+1) \sum_{p=1}^k |y_p|$. Uqityvaya
\eqref{dot-m}, e1to qastny\ui\ sluqa\ui\ tozhdestva \eqref{higher}.

\subsection{\protect\cyr Vysshie gomotopii}
\label{hhom}

\begin{multline}
\label{comm}
\begin{split}
&d ( x \{ x_1, \dots, x_{n+1}\} )
- (dx)\{x_1, \dots, x_{n+1}\} \\
&- (-1)^{|x|} \sum_{i=1}^{n+1} (-1)^{|x_1| + \dots + |x_{i-1}| }   x \{ x_1,
\dots, dx_i, \dots , x_{n+1}\}
\end{split}
\\
\begin{split}
\; = \; &  (-1)^{(|x|+1) |x_1| } x_1  \cdot x \{ x_2, \dots, x_{n+1} \} \\
& - (-1)^{|x|}  \sum_{i=1}^n (-1)^{|x_1| + \dots + |x_{i}| } x \{ x_1, \dots ,
x_i \cdot x_{i+1}, \dots , x_{n+1} \}\\
& +    (-1)^{|x| + |x_1| + \dots + |x_{n}|} x \{ x_1, \dots, x_n \} \cdot
x_{n+1}
\end{split}
\end{multline}
E1to tozhdestvo tozhe qastny\ui\ sluqa\ui\ formuly \eqref{higher} v silu
\eqref{dot-m} i \eqref{d-m}.

\subsection{\protect\cyr Gomotopiqeskie algebry Gerstenhabera}

{\it Gomotopiqeskaya algebra Gerstenhabera\/} -- e1to kompleks $V = \bigoplus_n V^n$ vektornyh
prostranstv s differencialom $d$ stepeni odin, umnozheniem $x y$,
prevrawayuwim nadstro\ui ku $V[-1]$, $V[-1]^n : = V^{n-1}$ -- funktor sdviga, v
associativnuyu {\rm DG}-algebru, i naborom skobok $ x
\{x_1, \dots, x_n\} $, $n \ge 0$, udovletvoryayuwih tozhdestvam
\eqref{higher}, \eqref{dist} i \eqref{comm}. Podrazumevaet\(\)sya, qto umnozhenie imeet stepenp1 odin v smysle izna\-qalp1\-no\ui\ graduirovki kompleksa $V$
qto skobki imeyut stepenp1 nolp1: $|x y | = |x|+ |y| +1$ i $|x
\{x_1, \dots, x_n\}| = |x| +  |x_1| + \dots + | x_n|$, esli stepenp1
$n$ e1lementa $x \in V^n$ oboznaqena qerez $|x|$.
Punkty \ref{identities}--\ref{hhom} v ramkah e1togo opredeleniya pereformuliruyut\(\)sya kak sleduyuwi\ui\ rezulp1tat.

\begin{th}
\label{mainth}
Sdvig $V = \ca [1]$ kompleksa \h\ as\-so\-ci\-a\-ti\-v\-no\ui\
algebry $A$ obrazuet gomotopiqeskuyu algebru Gerstenhabera otnositelp1no umnozheniya \eqref{dot-eq} i skobok \eqref{brace}.
\end{th}

\subsection{\protect\cyr Tozhdestva dlya pervyh skobok}

Pri $m=n=1$ ravenstvo \eqref{higher} vleqet {\it predp2yakobievo tozhdestvo}:
\begin{equation}
\label{prelie}
(x \circ y) \circ z - x \circ ( y \circ z)   =  (-1)^{|y||z|} ((x \circ z)
\circ y - x \circ ( z \circ y)).
\end{equation}
Pri $n=1$ formula \eqref{dist} prevrawaet\(\)sya v {\it pravoe pravilo Le\ui bnica}:
\begin{equation}
\label{right}
(xy ) \circ z  = (-1)^{(|y|+1)|z|} (x \circ z) y + x ( y \circ z)  .
\end{equation}
Polagaya $n=0$ v \eqref{comm}, my poluqim {\it gomotopiqeskuyu kommutativnostp1}:
\begin{equation}
\label{hcomm}
xy - (-1)^{(|x|+1)(|y|+1)}  yx  = (-1)^{|x|} (d(x \circ y) - dx \circ y -
(-1)^{|x|}x \circ dy) .
\end{equation}
Nakonec, priravnivaya $n=1$ v tom zhe tozhdestve \eqref{comm}, my poluqim
 {\it gomotopiqeskoe levoe pravilo Le\ui bnica}:
\begin{multline}
\label{hleft}
x \circ (yz) - (x \circ y) z - (-1)^{|x| (|y|+1)} y (x \circ z)
\\
 =  (-1)^{|x|+|y|+1}  (d ( x\{ y,z\} ) - (dx) \{  y, z\}
- (-1)^{|x|} x \{dy,  z\}   - (-1)^{|x|+|y|} x \{y, d z\}) .
\end{multline}

\subsection{\protect\cyr Algebry Gerstenhabera}

{\it Algebro\ui\ Gerstenhabera\/} nazyvaet\(\)sya graduirovannoe vektornoe prostranstvo $H$ s zadannym umnozheniem $xy$,
opredelyauwim strukturu graduirovanno-kommutativno\ui\ algebry na
$H[-1]$ i so skobko\ui\ $[x,y]$, opredelyauwe\ui\ strukturu graduirovanno\ui\
algebry Li na $H$, tak qto skobka s e1lementom iz $H$ yavlyaet\(\)sya differencirovaniem umnozheniya:
\[
[x , yz] = [x,y] z + (-1)^{|x| (|y|+1)} y [x,z] .
\]
Drugimi slovami, algebra Gerstenhabera -- e1to graduirovannuya algebra Puassona specialp1nogo vida.

\begin{crl}
Umnozhenie \eqref{dot-eq} i skobka $[x,y] := x \circ y - (-1)^{|x| |y|} y \circ x$
zadayut strukturu algebry Gerstenhabera na kogomologiyah \h\ $\hha$ associativno\ui\
algebry $A$.
\end{crl}

\begin{pf*}{Dokazatelp1stvo}
Prostoe vyqislenie pokazyvaet, qto predp2yakobievo tozhdestvo \eqref{prelie}
vleqet tozhdestvo Yakobi dlya skobki. Iz tozhdestva \eqref{hcomm} sleduet, qto
differencial yavlyaet\(\)sya differencirovaniem skobki:
\begin{equation}
\label{d-br}
d[x, y] - [dx ,  y ] - (-1)^{|x|} [x , dy ] = 0.
\end{equation}
Sledovatelp1no, ewe do perehoda k kogomologiyam kompleks \h\ obrazuet
{\rm DG}-algebru Li po otnosheniyu k skobke i associativnuyu {\rm DG}-algebru po otnosheniyu k umnozheniyu. 

Perehodya k kogomologiyam, ostaet\(\)sya pokazatp1, qto
\begin{enumerate}
\item obe operacii perevodyat kocikly v kocikly i ne zavisyat ot vybora predstavitele\ui\ klassov kogomologi\ui,
\item umnozhenie graduirovanno-kommutativno, i
\item skobka yavlyaet\(\)sya differencirovaniem umnozheniya.
\end{enumerate}
Spravedlivostp1 tozhdestva (1) legko uvidetp1 iz \eqref{d-dot} i \eqref{d-br},
(2) -- iz gomotopiqesko\ui\ kommutativnosti \eqref{hcomm} na urovne kocepe\ui\
i (3) -- iz pravogo i gomotopiqeskogo levogo pravil Le\ui bnica \eqref{right} 
i \eqref{hleft}.
\end{pf*}

\begin{rem}
Struktura algebry Gerstenhabera na kogomologiyah \h\ $\hha$ byla vvedena Gerstenhaberom v rabote
\cite{Gerst}. Tozhdestva \eqref{prelie}--\eqref{hleft} takzhe mogut bytp1 na\ui deny v e1to\ui\ rabote. Skobka $[z,z]$ igraet rolp1 glavnogo 
prepyat\(\)stviya v teorii deformaci\ui. Skobki \eqref{brace} byli vvedeny v 
rabote Geclera \cite{g:cartan}, gde oni ispolp1zovalisp1 dlya opredeleniya
 kogomologi\ui\ \h\ gomotopiqeski
associativnyh algebr.
\end{rem}

\subsection{\protect\cyr Otkrytye voprosy}
\label{conclusion}

Suwestvuet kanoniqeskoe opredelenie gomotopiqesko\ui\ algebry
Gerstenhabera kak algebry nad operado\ui, kobar-dvo\ui stvenno\ui\
operade gomologi\ui\ operady malenp1kih kvadratov, sm.\
\cite{gk}. Odnako nashe ponyatie gomotopiqesko\ui\ algebry Gerstenhabera 
otliqaet\(\)sya ot nego -- ono soderzhit menp1she gomotopi\ui\ dlya
umnozheniya i bolp1she gomotopi\ui\ dlya skobki. Na samom dele, nashe
opredelenie ne ukladyvet\(\)sya v obwuyu s\(\)hemu teorii kvadratiqnyh
operad
\cite{gk}. Vo-pervyh, my ispolp1zovali umnozhenie $x \circ y$, kotoroe 
yavlyaet\(\)sya biline\ui no\ui\ gomotopie\ui\ -- sm.\ \eqref{hcomm},
a v teorii kvadratichnyh operad biline\ui nye gomotopii nedopustimy.
Priqina v tom, qto tozhdestvo kommutativnosti v teorii operad
yavlyaet\(\)sya line\ui nym sootnosheniem (Ono de\ui stvitelp1no
line\ui no po strukturnym kon\-s\-tantam.), a v teorii kvadratiqnyh operad
vozmozhny tolp1ko kvadratiqnye sootnosheniya. Bolee togo, sootnosheniya
\eqref{higher} i
\eqref{comm} soderzhat ne tolp1ko kvadratiqnye (sootvet\(\)stvuyuwie kompoziciyam 
dvuh operaci\ui ), no i line\ui nye qleny. Takim obrazom, dvo\ui
stvennostp1 Koxulya dlya kvadratiqnyh operad \cite{gk} dolzhna bytp1
obobwena tak, qtoby vklyuqatp1 v sebya line\ui nye sootnoxeniya
naryadu s kvadratiqnymi. Vo-vtoryh, sootnosheniya
\eqref{dist} soderzhat kak kvadratiqnye, tak i kubiqeskie qleny. Vproqem, e1to ewe ne znaqit, qto \hc\ ne mozhet imetp1 struktury gomotopiqesko\ui\ algebry Gerstenhabera v smysle, vozmozhno, obobwenno\ui, teorii kvadratiqnyh operad:

\begin{probl}
Suwestvuet li na komplekse \h\ proizvolp1no\ui\ associativno\ui\ algebry estestvennaya struktura algebry nad operado\ui, kobar-dvo\ui \-s\-t\-ven\-no\ui\ operade, opisyvayuwe\ui\ algebry Gerstenhabera$?$
\end{probl}

My nadeemsya, qto nashe dokazatelp1stvo gipotezy~\ref{hg} sluzhit
podtverzhdeniem gipotezy~\ref{deligne}.
Vozmozhno, drugim podtverzhdeniem budet ee sleduyuwi\ui\ qastny\ui\ sluqa\ui.

 \begin{probl}
\label{mfld}
Pustp1 $A= \Omega^\bullet(X)$ -- algebra differencialp1nyh form na
mnogoobrazii $X$. Yavlyaet\(\)sya li \hc\ algebry $A$ algebro\ui\ nad
ope\-ra\-do\ui\ cepe\ui\ operady rimanovyh sfer s neperekryvayuwimisya
golomorfnymi diskami$?$
\end{probl}

E1tot vopros osnovan na sleduyuwem rassuzhdenii. Soglasno re\-zulp1\-ta\-tam Chena
\cite{chen}
i Dzhonsa
\cite{jones}, \hc\ $\ca$ kvaziizomorfen kompleksu $C_\bullet (\cal{L}X)$
singulyarnyh cepe\ui\ prostranstva svobodnyh petelp1 $\cal{L}X$.
Takim obrazom, gipotetiqeskaya struktura voprosa~\ref{mfld} dolzhna
in\-du\-ci\-ro\-vatp1 takuyu zhe strukturu na $C_\bullet (\cal{L}X)$.
No na prostranstve $\Omega^{\infty/2 + \bullet} (\cal{L} X)$
polubeskoneqnyh form, zanimayuwem promezhutoqnoe polozhenie mezhdu
$\Omega^\bullet (\cal{L}X)$ i $C_\bullet (\cal{L}X)$, e1ta struktura
imeet\(\)sya blagodarya sigma-modeli, v kotoro\ui\ $\Omega^{\infty/2 +
\bullet} (\cal{L} X)$ igraet rolp1 prostranstva sostoyani\ui.
Po-vidimomu, v voprose vyshe \hc\ algebry $A$ sleduet zamenitp1 na
kakuyu-libo ego po\-lu\-be\-s\-ko\-ne\-q\-nuyu versiyu. Togda
poluqayuweesya na $\Omega^{\infty/2 +
\bullet} (\cal{L} X)$ associativnoe gomoto\-pi\-qe\-s\-ki kommutativnoe
umnozhenie budet kvantovym umnozheniem zerkalp1no\ui\ simmetrii.
Otmetim takzhe, qto, kak sleduet iz rezulp1tatov Liana i Cukermana \cite{lz}, na po\-lu\-be\-s\-ko\-ne\-q\-nyh kogomologiyah prostranstva sostoyani\ui\ imeet\(\)sya struktura
algebry Gerstenhabera.

\begin{rem}
Posle togo, kak vyshel preprint e1to\ui\ raboty, Gecler soobwil nam,
qto v novom variante preprinta \cite{gj} on i Dzhons
privodyat re\-she\-nie gipotezy~\ref{deligne}. Oni ispolp1zuyut
konstrukciyu Foksa-Ne\ui virta kletoqnogo razbieniya prostranstva
konfiguraci\ui\ toqek na ploskosti, qtoby postroitp1 kletoqnuyu
modelp1 $\cal{E}_2$ operady prostranstv konfiguraci\ui, kotoraya
gomotopicheski e1kvivalentna operade malenp1kih kvadratov, i
po\-ka\-zy\-va\-yut, qto umnozhenie \eqref{dot-eq} i skobki
\eqref{brace} zadayut na $\ca$ strukturu algebry nad {\rm DG}-operado\ui\
$\cal{E}_2$.
\end{rem}

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\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother
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