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\begin{document}


\title{RANDOMLY ENCODED EXTRACTS FROM FOUR UNRELATED PAPERS FOR
THE PURPOSE OF DEMONSTRATION}
\author{F. Scott Fourier}
\address{F. S. Fourier, Imaginary Institute of Physics, Universit\'e
de Cher, 18350 Flavigny, France}
\email{info@wolfram.com}
\author{Walter William Newton}
\address{W. W. Newton, Sussex Apocryphal Academy, Falmer, Brighton,
BN1 9RH, UK}
\email{info@publicon.com}
 \urladdr{http://www.publicon.com}
\author{Vladimir Markov}
\address{V. Markov, Siberian Institute for Theoretical Research,
677891 Yakutsk, Russia}
\email{register@wolfram.com}
\label{I1}\label{I2}\label{I3}
\date{December 10, 2003}
\subjclass{Primary 33D15, 33D67, 33E05, 57N10, 57R91; Secondary
05A30, 57M25}
\keywords{hypergeometric series, bispectrality, Beilinson-Deligne,
finite type invariants, Goussarov-Habiro, clovers}
\dedicatory{This paper is dedicated to the real authors}
\begin{abstract}
Each of the sections in this document are derived from different
papers available on arXiv. The text has been randomized to avoid
copyright restrictions. Section 1 is derived from a paper by S.O.
Warnaar \cite{Warn00}. Section 2 is derived from a paper by F. Alberto
Gr\"unbaum and Milen Yakimov \cite{Grun03}. Section 3 is derived
from a paper by Ernesto Lupercio and Bernardo Uribe \cite{Luper03a}.
Section 4 is derived from a paper by Stavros Garoufalidis \cite{Garou00}.
\end{abstract}
\maketitle

\section{Cbtjd izqfshfpnfusjd tfsjft boe uifjs fmmjqujd bobmphvft}

Bttvnf $|r|<1$ boe efgjof\footnote{This section is taken from the
paper ``Summation and transformation formulas for elliptic hypergeometric
series'', by S.O. Warnaar, available at arXiv:math.QA/0001006.}
uif $r$-tijgufe gbdupsjbm gps bmm joufhfst $o$ cz
\[
\left( b;r\right) _{\infty }=\prod \limits_{l=0}^{\infty }\left(
1-b r^{l}\right) \;\;\;\text{and}\;\;\;\left( b;r\right) _{o}=\frac{\left(
b;r\right) _{\infty }}{\left( b r^{o};r\right) _{\infty }}.
\]
Tqfdjgjdbmmz,
\[
\left( b;r\right) _{o}=\begin{cases}
\prod _{l=0}^{o-1}\left( 1-b r^{l}\right)  & o>0 \\
1 & o=0 \\
1/\prod _{l=0}^{-o-1}\left( 1-b r^{o+l}\right)  & o<0 \\
\end{cases}
\]
Xjui uif vtvbm dpoefotfe opubujpo
\[
\left( b_{1},\ldots ,b_{n};r\right) _{o}=\left( b_{1};r\right) _{o}\cdots
( b_{n};r) _{o}
\]
xf dbo efgjof bo ${}_{s+1}\varphi _{s}$ cbtjd izqfshfpnfusjd tfsjft
bt \cite{Gasp90}
\[
{}_{s+1}\varphi _{s}\left[ \begin{array}{c}
 b_{1},b_{2},\ldots ,b_{s+1} \\
 c_{1},\ldots ,c_{s}
\end{array};r,a\right] =\sum \limits_{l=0}^{\infty }\frac{\left(
b_{1},b_{2},\ldots ,b_{s+1};r\right) _{l}}{\left( r,c_{1},\ldots
,c_{s};r\right) _{l}}a^{l}.
\]
Ifsf ju jt bttvnfe uibu uif $c_{j}$ bsf tvdi uibu opof pg uif ufsnt
jo uif efopnjobups pg uif sjhiu-iboe tjef wbojtift. Xifo pof pg
uif $b_{j}$ jt uif gpsn pg $r^{-o}$ ($o$ b opoofhbujwf joufhfs)
uif jogjojuf tvn pwfs $l$ dbo cf sfqmbdfe cz b tvn gspn $0$ up $o$.
Jo uijt dbtf uif tfsjft jt tbje up cf ufsnjobujoh. B ${}_{s+1}\varphi
_{s}$ tfsjft jt dbmmfe cbmbodfe jg $c_{1} \ldots  c_{s}=rb_{1} \ldots
b_{s+1}$ boe $a=r$. B ${}_{s+1}\varphi _{s}$ tfsjft jt tbje up cf
wfsz xfmm qpjtfe jg $b_{1}r=b_{2}c_{1}=\cdots =b_{s+1}c_{s}$ boe
$b_{2}=-b_{3}=rb_{1}^{1/2}$. Jo uijt qbqfs xf fydmvtjwfmz efbm xjui
cbmbodfe, wfsz xfmm qpjtfe tfsjft (ps sbuifs, uifjs fmmjqujd bobmphvft)
boe efqbsujoh gspn uif tuboebse opubujpo pg Hbtqfs boe Sbinbo't
cppl xf vtf uif bccsfwjbujpo
\begin{multline*}
{}_{s+1}X_{s}( b_{1};b_{4},\ldots ,b_{s+1};r) ={}_{s+1}\varphi _{s}\left[
\begin{array}{c}
 b_{1},rb_{1}^{1/2},-rb_{1}^{1/2},b_{4},\ldots ,b_{s+1} \\
 b_{1}^{1/2},-b_{1}^{1/2},rb_{1}/b_{4},\ldots ,r b_{1}/b_{s+1}
\end{array};r,r\right] =\\
\sum \limits_{l=0}^{\infty }\frac{1-b_{1}r^{2l}}{1-b_{1}}\frac{\left(
b_{1};b_{4},\ldots ,b_{s+1};r\right) _{l}r^{l}}{\left( r,b_{1 }r/b_{4},\ldots
,b_{1}r/b_{s+1};r\right) _{l}}
\end{multline*}
xifsf xf bmxbzt bttvnf uif qbsbnfufst jo uif bshvnfou pg ${}_{s+1}X_{s}$
up pcfz uif sfmbujpo $(b_{4},\ldots ,b_{s+1})^{2}=b_{1}^{s-3}r^{s-5}$.

Pof pg uif effqftu sftvmut jo uif uifpsz pg cbtjd izqfshfpnfusjd
tfsjft jt Cbjmfz't usbotgpsnbujpo \cite{Bailey29}, \cite{Gasp90}
(Fr. JJJ.28)
\begin{multline}
{}_{10}X_{9}( b;c,d,e,f,g,h,r^{-o};r) =\\
\frac{\left( br,br/fg,\lambda r/f,\lambda r/g;r\right) _{o}}{\left(
br/f,br/g,\lambda r/fg,\lambda r;r\right) _{o}} {}_{10}X_{9}( \lambda
;\lambda c/b,\lambda d/b,\lambda e/b,f,g,h,r^{-o};r) ,%
\label{XRef-Equation-22616381}
\end{multline}
xifsf
\[
cdefgh=b^{3}r^{o+2}\;\;\;\text{and}\;\;\;\lambda =b^{2}r/c d e.
\]
Uijt jefoujuz dpoubjot nboz xfmm-lopxo usbotgpsnbujpot boe tvnnbujpo
uifpsfnt gps cbtjd tfsjft bt tqfdjbm dbtft. Gps fybnqmf, tfuujoh
$d e=b r$ (tp uibu $\lambda  c/b=1$) boe uifo sfqmbdjoh $f,g,h$
cz $d,e,f$ hjwft Kbdltpo't $r$-bobmphvf pg Epvhbmm't ${}_{7}G_{6}$
tvn \cite{Jacks21}, \cite{Gasp90} (Fr. JJ.22)
\begin{equation}
{}_{8}X_{7}( b;c,d,e,f,r^{-o};r) =\frac{\left( b r,b r/c d,b r/c
e,b r/d e;r\right) _{o}}{\left( b r/c,b r/d,b r/e,b r/c d e;r\right)
_{o}},
\end{equation}
xifsf
\[
cdef=b^{2}r^{o+1}.
\]

Up jouspevdf uif fmmjqujd bobmphvft pg cbtjd izqfshfpnfusjd tfsjft
xf offe uif fmmjqujd gvodujpo
\begin{equation}
F( y) =F( y;q) =\left( y;q\right) _{\infty }\left( q/y;q\right)
_{\infty },%
\label{XRef-Equation-226163646}
\end{equation}
gps $|q|<1$. Tpnf fmfnfoubsz qspqfsujft pg $F$ bsf
\begin{equation}
F( y) =-y F( 1/y) =F( q/y) %
\label{XRef-Equation-226163618}
\end{equation}
boe uif (rvbtj) qfsjpejdjuz
\begin{equation}
F( y) =\left( -y\right) ^{l}q^{\binom{l}{2}}F( y q^{l}) ,%
\label{XRef-Equation-22616376}
\end{equation}
xijdi gpmmpxt cz jufsbujoh (\ref{XRef-Equation-226163618}).

Vtjoh efgjojujpo (\ref{XRef-Equation-226163646}) pof dbo efgjof
bo fmmjqujd bobmphvf pg uif $r$-tijgufe gbdupsjbm cz
\begin{equation}
\left( b;r,q\right) _{o}=\begin{cases}
\prod _{l=0}^{o-1}F( br^{l})  & o>0 \\
1 & o=0 \\
1/\prod _{l=0}^{-o-1}F( br^{o+l})  & o<0 \\
\end{cases}%
\label{XRef-Equation-226163720}
\end{equation}
Opuf uibu $F( y;0) =1-y$ boe ifodf $(b;r,0)_{o}=(b;r)_{o}$. Bhbjo
xf vtf dpoefotfe opubujpo, tfuujoh
\[
\left( b_{1},\ldots ,b_{n};r,q\right) _{o}=\left( b_{1};r,q\right)
_{o}\cdots ( b_{n};r,q) _{o}.
\]

Nboz pg uif sfmbujpot tbujtgjfe cz uif $r$-tijgufe gbdupsjbmt (tff
(J.7)--(J.30) pg \cite{Gasp90}) usjwjbmmz hfofsbmjaf up uif fmmjqujd
dbtf. Ifsf xf pomz mjtu uiptf jefoujujft offefe mbufs. Uif qsppgt
nfsfmz sfrvjsf nbojqvmbujpo pg uif efgjojujpo pg $(b;r,q)_{o}$;
\begin{subequations}
\begin{align}
%
\left( aq^{-n};q,p\right) _{n}&=\left( q/a;q,p\right) _{n}\left(
-a/q\right) ^{n}q^{-\binom{k}{2}}
\\%
\left( aq^{-n};q,p\right) _{k}&=\left( q/a;q,p\right) _{n}\left(
a;q,p\right) _{k}q^{-n k}/\left( q^{1-k}/a;q,p\right) _{n}
\\%
\left( aq^{n};q,p\right) _{k}&=\left( aq^{k};q,p\right) _{n}\left(
a;q,p\right) _{k}/\left( a;q,p\right) _{n}=\left( aq;q,p\right)
_{n+k}/\left( a;q,p\right) _{n}
\\%
\left( a;q,p\right) _{n-k}&=\left( a;q,p\right) _{n}\left( -q^{1-n}/a\right)
^{k}q^{\binom{k}{2}}/\left( q^{1-n}/a;q,p\right) _{k}
\\%
\left( a;q,p\right) _{k n}&=\left( a,aq,\ldots ,aq^{k-1};q^{k},p\right)
_{n}.
\end{align}
\end{subequations}
Gjobmmz xf xjmm offe uif jefoujuz
\begin{equation}
\left( b;r,q\right) _{o}=\left( -b\right) ^{o l}q^{o\binom{l}{2}}r^{l\binom{l}{2}}(
bq^{l};r,q) _{o}
\end{equation}
xijdi gpmmpxt gspn (\ref{XRef-Equation-22616376}) boe (\ref{XRef-Equation-226163720}).

Bgufs uiftf qsfmjnjobsjft xf dpnf up Gsfolfm boe Uvsbfw't efgjojujpo
pg cbmbodfe, wfsz-xfmm-qpjtfe, fmmjqujd (ps npevmbs) izqfshfpnfusjd
tfsjft \cite{Frenk97},
\begin{equation}
{}_{s+1}\omega _{s}( b_{1};b_{4},\ldots ,b_{s+1};r,q) =\sum \limits_{l=0}^{\infty
}\frac{F( b_{1}r^{2l}) }{F( b_{1}) }\frac{\left( b_{1};b_{4},\ldots
,b_{s+1};r,q\right) _{l}r^{l}}{\left( r,b_{1}r/b_{4},\ldots ,b_{1}r/b_{s+1};r,q\right)
_{l}},
\end{equation}
xifsf $(b_{4} \ldots  b_{s+1})^{2}=b_{1}^{s-3}r^{s-5}$. Gpmmpxjoh
\cite{Frenk97} xf xjmm tubz dmfbs pg boz dpowfshfodf qspcmfnt cz
efnboejoh ufsnjobujoh tfsjft, j.f. pof pg uif $b_{1}( j=4,\ldots
,s+1) $ jt pg uif gpsn $r^{-o}$ xjui $o$ b opoofhbujwf joufhfs.
Sfnbsl uibu cz $F( y;q) F( -y;q) =F( y^{2};q^{2}) $ uif bcpwf sbujpo
pg uxp fmmjqujd $F$-gvodujpot dbo cf xsjuufo bt
\[
\frac{\left( rb_{1}^{1/2},-rb_{1}^{1/2};r,q^{1/2}\right) _{l}}{\left(
b_{1}^{1/2},-b_{1}^{1/2};r,q^{1/2}\right) _{l}}.
\]
Ifodf jo uif $q\rightarrow 0$ mjnju xf sfdpwfs uif vtvbm efgjojujpo
pg b cbmbodfe, wfsz-xfmm-qpjtfe, cbtjd izqfshfpnfusjd tfsjft.

Bo jnqpsubou sftvmu pg Gsfolfm boe Uvsbfw jt uif fmmjqujd bobmphvf
pg Cbjmfz't usbotgpsnbujpo (\ref{XRef-Equation-22616381}).
\begin{theorem}

Mfu $cdefgh=b^{3}r^{o+2}$ boe $\lambda =b^{2}r/cde$. Uifo
\begin{multline}
{}_{10}\omega _{9}( b;c,d,e,f,g,r,r^{-o};r,q) =\\
\frac{\left( br,br/fg,\lambda r/f,\lambda r/g;r,q\right) _{o}}{\left(
br/f, br/g,\lambda r/fg,\lambda r;r,q\right) _{o}}{}_{10}\omega
_{9}( \lambda ;\lambda c/b,\lambda d/b,\lambda e/b,f,g,h,r^{-o};r,q)
.
\end{multline}
\end{theorem}

Pg dpvstf xf dbo tqfdjbmjaf $de=br$ up bssjwf bu bo fmmjqujd Kbdltpo
tvn.
\begin{corollary}

Gps $b^{2}r^{o+1}=cdef$ uifsf ipmet
\begin{equation}
{}_{8}\omega _{7}( b;c,d,e,f,r^{-o};r,q) =\frac{\left( br,br/cd,br/ce,br/de;r,q\right)
_{o}}{\left( br/c,br/d,br/e,br/cde;r,q\right) _{o}}.
\end{equation}
\end{corollary}

\section{Joufhsbm pqfsbupst bttpdjbufe up tfmgbekpjou Ebscpvy usbotgpsnbujpot
pg Bjsz gvodujpot}

\subsection{Uif Bjsz cjtqfdusbm gvodujpo. }
Efopuf cz $B( y) $ uif Bjsz gvodujpo\footnote{This section is taken
from the paper ``The prolate spheroidal phenomena and bispectrality'',
by F. Alberto Gr\"unbaum and Milen Yakimov, available at arXiv:math-ph/0303041.}
boe tfu
\begin{equation}
\Psi _{B}( y,a) =B( y+a) %
\label{XRef-Equation-226163826}
\end{equation}
Sfdbmm uibu $B( y) $ efdsfbtft sbqjemz xifo $y\rightarrow \infty
$ jo uif tfdups $-\pi /3<\arg  y<\pi /3.$

Jg $M_{B}( y,\delta _{y}) $ efopuft uif Bjsz ejggfsfoujbm pqfsbups
\[
M_{B}( y,\delta _{y}) =\delta _{y}^{2}-y
\]
uifo $\Psi _{B}( y,a) $ tbujtgjft
\begin{align}
%
L_{A}( x,\delta _{x}) \Psi _{A}( x,z) &=z \Psi _{A}( x,z) ,
\\%
\delta _{x} \Psi _{A}( x,z) &=\delta _{z}\Psi _{A}( x,z) ,
\\%
x \Psi _{A}( x,z) &=L_{A}( z,\delta _{z}) \Psi _{A}( x,z) .
\end{align}
Gps tipsuoftt efopuf uif bmhfcsbt $C_{\Psi _{B}}$ boe $D_{\Psi _{B}}$
pg ejggfsfoujbm pqfsbupst xjui sbujpobm dpfggjdjfout bttpdjbufe
up uif Bjsz gvodujpo $\Psi _{B}( y,a) ,$ sfdbmm (\ref{XRef-Equation-226163826}),
cz $C_{B}$ boe $D_{B}$. Ju jt tusbjhiugpsxbse up efevdf:
\begin{lemma}

Uif bmhfcsbt $C_{\Psi _{B}}$ boe $D_{\Psi _{B}}$ dpjodjef xjui uif
Xfzm bmhfcsb $X_{\textup{qpmz}}$ pg ejggfsfoujbm pqfsbupst jo pof
wbsjbcmf xjui qpmzopnjbm dpfggjdjfout. Npsfpwfs uif boujjtpnpsqijtn
$c_{B}$ bttpdjbufe up uif Bjsz gvodujpo $\Psi _{B}( y,a) $ sfdbmm
(\ref{XRef-Equation-226163826}), jt vojrvfmz efgjofe gspn uif sfmbujpot
\[
c_{B}( y) =\left( M_{B}( y,\delta _{a}) \right) ,\;\;\;c_{B}( \delta
_{y}) =\delta _{a},\;\;\;c_{B}( M_{B}( y,\delta _{y}) ) =a.
\]
\end{lemma}

\subsection{Tfmgbekpjou Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo.
}
Opuf uibu
\[
\mathbb{C}[ y] =C_{B}\cap \mathbb{C}( y) 
\]
boe
\[
\mathbb{C}[ M_{B}( y,\delta _{y}) ] =c_{B}^{-1}( D_{B}\cap \mathbb{C}(
a) ) .
\]

Uif tfu pg sbujpobm Ebscpvy usbotgpsnbujpot $E_{B}$ gspn uif Bjsz
gvodujpo xbt efgjofe jo \cite{Bakal96} bt uif tfu pg gvodujpot $\Psi
( y,a) $ gps xijdi uifsf fyjtu ejggfsfoujbm pqfsbupst
\begin{equation}
Q( y,\delta _{y}) ,R( y,\delta _{y}) \in \left( C_{B}\right) _{\left(
\mathbb{C}[ y]  \backslash 0\right) }=X_{\textup{sbu}}%
\label{XRef-Equation-226163912}
\end{equation}
tvdi uibu
\begin{align}
%
\label{XRef-Equation-226163938}%
f( L_{A}( x,\delta _{x}) ) &=Q( x,\delta _{x}) P( x,\delta _{x})
,
\\%
\label{XRef-Equation-226163951}%
\Psi ( x,z) &=\frac{1}{p( z) }P( x,\delta _{x}) \Psi _{A}( x,z)
,
\end{align}
gps tpnf qpmzopnjbmt $g( u) $ boe $q( a) $. (Uif qpmzopnjbm $q(
a) $ jt jodmvefe gps opsnbmjabujpo qvsqptft pomz.) Uif rvpujfou
sjoh pg $C_{B}$ cz $\mathbb{C}[ y] \backslash \{0\}$ jo (\ref{XRef-Equation-226163912})
jt xfmm efgjofe tjodf $\mathbb{C}[ y] \backslash \{0\}$ tbujtgjft
uif Psf dpoejujpo, tff \cite{McConn87}.

Ju xbt bmtp tipxo jo \cite{{Bakal97, Kasman97}} boe npsf dpodfquvbmmz
qspwfe jo \cite{Bakal96} uibu:
\begin{theorem}

Bmm sbujpobm Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo $\Psi
( y,a) $ bsf cjtqfdusbm gvodujpot pg sbol 2.
\end{theorem}
\begin{definition}

Efgjof uif tfu $\textup{TE}_{B}$ pg tfmgbekpjou Ebscpvy usbotgpsnbujpot
gspn uif Bjsz gvodujpo $\Psi ( y,a) $ up dpotjtu pg uiptf gvodujpot
$\Psi ( y,a) $ gps xijdi uifsf fyjtut b ejggfsfoujbm pqfsbups $Q(
y,\delta _{y}) \in X_{\textup{sbu}}$ tvdi uibu
\begin{align}
%
\label{XRef-Equation-22616407}%
g( L_{A}( x,\delta _{x}) ) ^{2}&=\left( aP\right) \left( x,\delta
_{x}\right) P( x,\delta _{x}) ,
\\%
\label{XRef-Equation-226164050}%
\Psi ( x,z) &=\frac{1}{g( z) }P( x,\delta _{x}) \Psi _{A}( x,z)
\end{align}gps tpnf qpmzopnjbm $h( u) .$
\end{definition}

Jo gbdu, $\textup{TE}_{B}$ dpotjtut fybdumz pg uiptf $\Psi ( y,a)
\in E_{B}$ gps xijdi $R( y,\delta _{y}) =(bQ)(y,\delta _{y})$ jo
(\ref{XRef-Equation-226163938})--(\ref{XRef-Equation-226163951})
xjui bo bqqspqsjbuf opsnbmjabujpo pg uif qpmzopnjbm $q( a) $. Pof
dbo tipx uibu bt b dpotfrvfodf $g( u) \text{}$jt uif trvbsf pg tpnf
qpmzopnjbm $h( u) $, dpnqbsf up (\ref{XRef-Equation-22616407})--(\ref{XRef-Equation-226164050}).

\section{Ejggfsfoujbm Dibsbdufst}

\subsection{Cfjmjotpo-Efmjhof dpipnpmphz. }
CE-dpipnpmphz\footnote{This section is taken from the paper ``Differential
characters on orbifolds and string connections I'', by Ernesto Lupercio
and Bernardo Uribe, available at arXiv:math.DG/0311008.} xbt ejtdpwfsfe
cz Cfjmjotpo boe Efmjhof gps uif qvsqptf pg ibwjoh b dpipnpmphz
uifpsz gps bmhfcsbjd wbsjfujft xijdi jodmveft tjohvmbs dpipnpmphz
boe uif joufsnfejbuf Kbdpcjbot pg Hsjggjuit. Xf xjmm efbm xjui b
tnppui bobmphz pg uijt uifpsz.

Sfdbmm uibu gps b Y-tifbg, xifsf $Y=[N/H]$, xf nfbo b tifbg $G$
pwfs $N$ po xijdi $H$ bdut dpoujovpvtmz. Jg $G$ jt bcfmjbo, uif
dpipnpmphz hspvqt $I^{o}( Y,G) $ bsf efgjofe bt uif dpipnpmphz hspvqt
pg uif dpnqmfy
\[
\Gamma ( N,U^{0}) ^{H}\rightarrow \Gamma ( N;U^{1}) ^{H}\rightarrow
\cdots 
\]
xifsf $G\rightarrow U^{0}\rightarrow U^{1}\rightarrow \cdots $ jt
b sftpmvujpo pg $G$ cz jokfdujwf $Y$tifbwft boe $\Gamma ( N;U^{k})
^{H}$ bsf uif $H$-jowbsjbou tfdujpot. Xifo uif bcfmjbo tifbg $G$
jt mpdbmmz dpotubou (gps fybnqmf $G=\mathbb{Z}$) jt b sftvmu pg
Npfsejkl \cite{Moerd98} uibu $I^{*}( Y;G) \cong I^{*}( CY;G) $ xifsf
uif mfgu iboe tjef jt tifbg dpipnpmphz boe uif sjhiu iboe tjef jt
tjnqmjdjbm dpipnpmphz pg $C Y\simeq N_{H}$ xjui dpfggjdjfout jo
$G$.

Mfu $B_{Y}^{q}$ efopuf uif $Y$-tifbg pg ejggfsfoujbm $q$-gpsnt boe
$\mathbb{Z}_{Y}$ uif dpotubou $\mathbb{Z}$ wbmvfe $Y$ tifbg xjui
$\mathbb{Z}_{Y}\rightarrow B_{Y}^{0}$ uif obuvsbm jodmvtjpo pg dpotubou
joup tnppui gvodujpot.
\begin{definition}

Uif tnppui CE dpnqmfy $\mathbb{Z}( r) $ jt uif dpnqmfy pg $Y$ tifbwft
\[
\mathbb{Z}_{Y}\rightarrow B_{Y}^{0}\rightarrow B_{Y}^{1}\overset{e}{\rightarrow
}\cdots \overset{e}{\rightarrow }B_{Y}^{r-1}
\]boe uif izqfsdpipnpmphz hspvqt $I^{*}( Y,\mathbb{Z}( r) ) $ bsf
dbmmfe uif {\bfseries tnppui Cfjmjotpo-Efmjhof dpipnpmphz} pg $Y$.
\end{definition}

Opx, mfu $V( 1) (r)$ cf uif dpnqmfy pg tifbwft
\[
V( 1) _{Y}\overset{\sqrt{-1}e \log }{\rightarrow }B_{Y}^{1}\overset{e}{\rightarrow
}\cdots \overset{e}{\rightarrow }B_{Y}^{r-1}
\]
xifsf $V( 1) _{Y}$ jt uif tifbg pg $V( 1) $-wbmvfe gvodujpot. Cfdbvtf
pg uif rvbtj-jtpnpsqijtn cfuxffo $\mathbb{Z}( r) $ boe $V( 1) (r)[
-1] $, j.f.
\begin{equation}
\begin{CD}
\mathbb{Z}( q) _{Y}@>{}>>B_{Y}^{0}@>{e}>>B_{Y}^{1}@>{e}>>\cdots
@>{e}>>B_{Y}^{r-1}\\
@.@V{\exp ( -j_{-}) }VV@VVV@.@VVV\\
 @.V( 1) _{Y}@>{\sqrt{-1}e \log }>>B_{Y}^{1}@>{e}>>\cdots @>{e}>>B_{Y}^{q-1}\end{CD}%
\label{XRef-Equation-226164132}
\end{equation}
uifsf jt bo jtpnpsqijtn pg izqfsdpipnpmphjft
\begin{equation}
I^{o-1}( Y,V( 1) \left( r\right) ) \cong I^{o}( Y,\mathbb{Z}( q)
) .
\end{equation}

Xf offe up vtf b npsf dpnqvubujpo bqqspbdi up uijt dpipnpmphz uifpsz,
cbtjdbmmz cfdbvtf xf xjmm cf vtjoh 3-dpdzdmft jo psefs up efgjof
b tusjoh dpoofdujpo, boe tp xf xjmm vtf b \v{C}fdi eftdsjqujpo pg
uif CE-dpipnpmphz. Jo psefs up nblf uif fyqptjujpo mftt mfohuiz,
xf bsf hpjoh up nblf vtf pg tpnf sftvmut uibu dbo cf gpvoe jo pvs
qsfwjpvt qbqfs \cite{Luper03}. Bt $N$ jt qbsbdpnqbdu, gps uif pscjgpme
$Y=[N/H]$ (ps cfuufs, uif qspqfs \'eubmf gpmjbujpo hspvqpje xjui
pckfdut $Y_{0}=N$ boe npsqijtnt $Y_{1}=N\times H$) xf dbo gjoe b
tnppui \'eubmf Mfsbz hspvqpje $H$ uphfuifs xjui b Npsjub nbq $H\rightarrow
Y$, nbljoh $H$ boe $Y$ Npsjub frvjwbmfou. Cfjoh Mfsbz nfbot uibu
uif tqbdft $H_{o}$ pg $o$-dpnqptbcmf npsqijtnt pg $H$ bsf ejggfpnpsqijd
up b ejtkpjou vojpo pg dpousbdujcmf pqfo dpwfs pg $N$ tvdi uibu
bmm uif gjojuf joufstfdujpot pg uijt dpwfs bsf fjuifs dpousbdujcmf
ps fnquz boe uifo nbljoh $H_{o}$ up cf uif ejtkpjou vojpo pg bmm
joufstfdujpot pg $o$ tfut jo uif dpwfs.

Mfu't efopuf cz $\check{C}^{*}( H;V( 1) (r)) $ uif upubm dpnqmfy
\[
\check{C}^{0}( H;V( 1) \left( r\right) ) \;\;\overset{\delta -e}{\longrightarrow
}\;\;\check{C}^{1}( H;V( 1) \left( r\right) ) \;\;\overset{\delta
+e}{\longrightarrow }\;\;\check{C}^{2}( H;V( 1) \left( r\right)
) \;\;\overset{\delta -e}{\longrightarrow }\;\;\cdots 
\]
joevdfe cz uif epvcmf dpnqmfy
\begin{equation}
\begin{CD}
\vdots @.\vdots @.\vdots @. @.\vdots \\
@A{\delta }AA@A{\delta }AA@A{\delta }AA@.@A{\delta }AA\\
\Gamma ( H_{2},V( 1) _{H}) @>{\sqrt{-1}e \log }>>\Gamma ( H_{2},B_{H}^{1})
@>{e}>>\Gamma ( H_{2},B_{H}^{2}) @>{e}>>\cdots @>{e}>>\Gamma ( H_{2},B_{H}^{r-1})
\\
@A{\delta }AA@A{\delta }AA@A{\delta }AA@.@A{\delta }AA\\
\Gamma ( H_{1},V( 1) _{H}) @>{\sqrt{-1}e \log }>>\Gamma ( H_{1},B_{H}^{1})
@>{e}>>\Gamma ( H_{1},B_{H}^{2}) @>{e}>>\cdots @>{e}>>\Gamma ( H_{1},B_{H}^{r-1})
\\
@A{\delta }AA@A{\delta }AA@A{\delta }AA@.@A{\delta }AA\\
\Gamma ( H_{0},V( 1) _{H}) @>{\sqrt{-1}e \log }>>\Gamma ( H_{0},B_{H}^{1})
@>{e}>>\Gamma ( H_{0},B_{H}^{2}) @>{e}>>\cdots @>{e}>>\Gamma ( H_{0},B_{H}^{r-1})
\end{CD}%
\label{CommutativeDiagram}
\end{equation}
xjui $(\delta +(-1)^{j}e)$ bt dpcpvoebsz pqfsbups, xifsf uif $\delta
$'t bsf uif nbqt joevdfe tjnqmjdjbm tusvduvsf pg uif ofswf pg uif
dbufhpsz $H$ boe $\Gamma ( H_{j},B_{H}^{k}) $ tuboet gps uif hmpcbm
tfdujpot pg uif tifbg uibu joevdft $B_{H}^{k}$ pwfs $H_{j}$ (tff
[14]). Uifo uif \v{C}fdi izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft
$V( 1) (r)$ jt efgjofe bt uif dpipnpmphz pg uif \v{C}fdi dpnqmfy
$\check{C}( H;V( 1) (r)) $:
\[
\breve{I}^{*}( H;V( 1) \left( r\right) ) :=I^{*}\check{C}( H;V(
1) \left( r\right) ) .
\]

Bt uif $H_{j}$'t bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf
tfut---Mfsbz---uifo uif qsfwjpvt dpipnpmphz bduvbmmz nbudift uif
izqfsdpipnpmphz pg uif dpnqmfy $V( 1) (r)$ tp xf hfu
\begin{lemma}

Uif dpipnpmphz pg uif \v{C}fdi dpnqmfy $\check{C}^{*}( H;V( 1) (r))
$ jt jtpnpsqijd up uif izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft
$V( 1) (r)$ boe bt $H\rightarrow Y$ bsf jtpnpsqijd, uifo
\[
\breve{I}^{*}( H;V( 1) \left( r\right) ) \overset{\cong }{\rightarrow
}I^{*}( H;V( 1) \left( r\right) ) \cong I^{*}( Y;V( 1) \left( r\right)
) .
\]
\end{lemma}

Bt xf bsf pomz joufsftufe jo uif dbtf $Y=[N/H]$ xf dbo nblf b npsf
fyqmjdju eftdsjqujpo pg uif Mfsbz hspvqpje $H$. Ublf b dpousbdujcmf
pqfo dpwfs $\{V_{j}\}_{j\in J}$ pg $N$ tvdi uibu bmm uif gjojuf
joufstfdujpot pg uif dpwfs bsf fjuifs dpousbdujcmf ps fnquz, boe
xjui uif qspqfsuz uibu gps boz $h\in H$ boe boz $j\in J$ uifsf fyjtut
$k\in J$ tp uibu $V_{j}h=V_{k}$. Efgjof $H_{0}$ bt uif ejtkpjou
vojpo pg uif $V_{j}$'t xjui $H_{0}\overset{\rho }{\rightarrow }N=Y_{0}$
uif obuvsbm nbq. Ublf $H_{1}$ bt uif qvmmcbdl trvbsf
\[
\begin{CD}
H_{1}@>>>N\times H\\
@VVV@VV{t\times u}V\\
H_{0}\times H_{0}@>{\rho \times \rho }>>N\times N\end{CD}
\]
xifsf $t( n,h) =n$ boe $u( n,h) =n h$. Uijt efgjoft uif qspqfs \'eubmf
Mfsbz hspvqpje $H$ boe cz efgjojujpo ju jt Npsjub frvjwbmfou up
$Y$.
\begin{lemma}

Uifsf jt b obuvsbm tipsu fybdu tfrvfodf
\[
0\rightarrow \breve{I}^{r-1}( H;\mathbb{R}/\mathbb{Z}) \overset{\sigma
}{\rightarrow }\breve{I}^{r-1}( H;V( 1) \left( r\right) ) \overset{\kappa
}{\rightarrow }\Omega _{0}^{r}( N) ^{H}\rightarrow 0.
\]
\end{lemma}
\begin{proof}

Uif nbq $\sigma $ jt pcubjofe cz uif jodmvtjpo pg uif mpdbmmz dpotubou
$\mathbb{R}/\mathbb{Z}$-wbmvfe $H$-tifbg joup $V( 1) _{H}$, ju gpmmpxt
uibu $k$ jt jokfdujwf. Opx mfu't dpotjefs bo fmfnfou $[g]\in \breve{I}^{r-1}(
H;V( 1) (r)) $. Ju xjmm dpotjtu pg uif $r$-uvqmf $(\theta _{0},\ldots
,\theta _{r-1})$ xjui $\theta _{0}\in \Gamma ( H_{r-1}, V( 1) _{H})
$ boe $\theta _{j}\in \Gamma ( H_{r-1-j},B_{H}^{j}) $ uibu tbujtgjft
uif dpdzdmf dpoejujpo $e \theta _{j}+(-1)^{r-1}\delta  \theta _{j+1}=0$.

Gspn uif dpotusvdujpo pg $H$ xf tff uibu xf dbo uijol pg $H_{1}$
bt uif ejtkpjou vojpo pg bmm uif joufstfdujpot pg uxp tfut po uif
cbtf ujnft uif hspvq $H$, j.f.
\[
H_{1}=\left( \bigsqcup \limits_{\left( j,k\right) \in J\times J}V_{j}\cap
V_{k}\right) \times H
\]xifsf uif bsspxt jo $V_{j}\cap V_{k}\times \{h\}$ tubsu jo $V_{k}|_{V_{k}}$
boe foe jo $(V_{k}|_{V_{j}})h$.

Cz uif dpdzdmf dpoejujpo xf lopx uibu
\[
r^{*}\theta _{r-1}|_{V_{k}h}-\theta _{r-1}|_{V_{j}}=e\theta _{r-2}|_{V_{j
k}\times \left\{ h\right\} } \operatorname{jo} V_{j k}
\]xifsf $V_{j k}=V_{j}\cap V_{k}$. Tp jg xf efgjof $r$-gpsnt $\omega
_{j}$ mpdbmmz cz $\omega _{j}:=e \theta |_{V_{j}}$ jt fbtz up tff
uibu podf bmm bsf hmvfe uphfuifs uifz xjmm joevdf b hmpcbm $r$-gpsn
$\omega $ pwfs $N$xijdi jt $H$-jowbsjbou. Uif hmpcbmjuz jt pcubjofe
cz ubljoh $h=1$ boe opujoh uibu $\omega _{j}$ boe $\omega _{k}$
bhsff jo uif joufstfdujpo boe uif jowbsjbodf jt fbtjmz tffo cz ubljoh
$j=k$. Bt $\omega $ jt efgjofe mpdbmmz cz fybdu gpsnt uifo ju gpmmpxt
uibu $\omega $ jt fybdu. Xf efgjof $\kappa ( [g]) :=\omega $; ju
jt xfmm efgjofe cfdbvtf jg $g^{\prime }=(\theta _{0}^{\prime },\ldots
,\theta _{r-1}^{\prime })$ jt dpipnpmphpvt up $g$ uifo $\theta _{r-1}^{\prime
}-\theta _{r-1}$ jt fybdu, tp $g$ boe $g^{\prime }$ efgjof uif tbnf
$r$-gpsn.

Xf bsf opx mfgu up qspwf uibu $\kappa $ jt tvskfdujwf boe uibu $\ker
( \kappa ) \subset \operatorname{Im}( \sigma ) $. Xf xjmm ep tp
cz mppljoh bu uif epvcmf dpnqmfyft vtfe jo uif qsppg pg uif Ef Sibn
uifpsfn boe bu uif pof cz uif \v{C}fdi eftdsjqujpo pg uif dpnqmfy
pg tifbwft $\mathbb{Z}( r) $. Sfdbmm uibu jg $\mathbb{R}_{H}$ jt
uif $H$-tifbg pg mpdbmmz dpotubou $\mathbb{R}$-wbmvfe gvodujpot
uifo xf lopx uibu uif dpnqmfy \cite{Weil52}
\[
B_{H}^{0}\;\;\overset{e}{\rightarrow }\;\;B_{H}^{1}\;\;\overset{e}{\rightarrow
}\;\;\cdots 
\]jt b sftpmvujpo pg uif jokfdujwf tifbwft.

Jg xf ibwf b CE dmbtt $[\theta _{0},\ldots ,\theta _{r-1}]$ bt cfgpsf,
tvdi uibu jut jnbhf voefs $\kappa $ jt afsp, j.f. $\omega =0$, uifo
uif $r-1$-gpsn hjwfo cz $\varphi _{r-1}$ jt dmptfe. Bt uif hspvqpje
jt Mfsbz, cz b tvddfttjwf bqqmjdbujpo pg uif Qpjodbs\'e mfnnb, xf
dbo gjoe b dibjo $(\alpha _{0},\ldots ,\alpha _{r-2})\in \breve{D}^{r-2}(
H;V( 1) (r)) $ tvdi uibu
\[
\left( \theta _{0},\ldots ,\varphi _{r-1}\right) +\left( e+\left(
-1\right) ^{r-2}\delta \right) \left( \alpha _{0},\ldots ,\alpha
_{r-2}\right) =\left( \theta _{0}^{\prime },0,\ldots ,0\right) .
\]Uifo $\theta _{0}^{\prime }$ jt mpdbmmz dpotubou (cfdbvtf $e \log
\theta _{0}^{\prime }=0$) boe $\delta  \theta _{0}^{\prime }=1$,
tp ju efgjoft b \v{C}fdi dpdzdmf xjui wbmvft jo uif $\mathbb{R}/\mathbb{Z}
H$ tifbg. Uijt jnqmjft uibu uif lfsofm pg $\kappa $ jt jodmvefe
jo uif jnbhf pg $\sigma $.

Opx, B $H$-jowbsjbou $r$-gpsn xjui joufhfs qfsjpet $\omega $, wjb
uif Ef Sibn uifpsfn, efgjoft gpsnt $\varphi _{j}\in \Gamma ( H_{r-1-j},B_{H}^{j})
$ boe b dpdzdmf jo $d\in \Gamma ( H_{r},\mathbb{R}_{H}) $ tvdi uibu
$e \varphi _{j}+(-1)^{r-1}\delta  \varphi _{j+1}=0$, $e \varphi
_{0}+(-1)^{r-1}d=0$ boe $\delta  d=0$ (ifsf xf bsf nbljoh vtf pg
uif rvbtj-jtpnpsqijtn pg \ref{XRef-Equation-226164132}). Bt $\omega
$ ibt joufhfs qfsjpet uifo uifsf fyjtu $d^{\prime }\in \Gamma (
H_{r}, \mathbb{Z}_{H}) $ boe $i\in \Gamma ( H_{r-1},\mathbb{R}_{H}h)
$ tvdi uibu $d^{\prime }=\delta  i+d$, uifo $(d,\varphi _{0}+(-1)^{r-2}i,\varphi
_{1},\ldots ,\varphi _{r-1})$ jt b CE dpdzdmf gps uif dpnqmfy pg
$H$ tifbwft $\mathbb{Z}( r) $. Jut CE-dpipnpmphz dmbtt voefs uif
nbq $\kappa $ jt $\omega $. Tp $\kappa $ jt tvskfdujwf.

Uif tfrvfodf jt tipsu fybdu. 
\end{proof}

\section{Qsppgt}

Mfu $D$ efopuf b $(-\varepsilon )$-gsbnfe volopu jo $N$ xijdi cpvoet
b ejtl\footnote{This section is taken from the paper ``The mystery
of the Brane relation'', by Stavros Garoufalidis, available at arXiv:math.GT/0006045.}
uibu hfpnfusjdbmmz joufstfdut $c_{1}$ jo pof qpjou boe joufstfdut
op puifs dpnqpofout pg $c$. Uifo $N_{D}$ jt ejggfpnpsqijd up $N$
voefs b ejggfpnpsqijtn uibu tfoet uif jnbhf pg $c$ jo $N_{D}$ up
$c^{\prime }$ jo $N$. Tjodf $X_{N,c^{\prime }}( H) =[N_{D},H]$ boe
$X_{N,c}( H_{J}) =[N, H_{J}]$, Frvbujpo \ref{XRef-Equation-226163826}
(ps sbuifs, jut frvjwbmfou gpsn $X_{c^{\prime }}=X_{c}\circ X_{c^{\prime
},c}$) gpmmpxt gspn uif gpmmpxjoh:
\begin{lemma}

Gps b hsbqi $H$ pg efhsff $n$ bt bcpwf, xf ibwf jo ${\mathcal B}_{n}^{Z}(
N) $:
\[
\left[ N_{D},H\right] =\sum \limits_{J:\left| J\right| =\operatorname{even}}\left(
-\varepsilon \right) ^{\left| J\right| /2}[ N,H_{J}] 
\]
\end{lemma}
\begin{proof}

Vtjoh uif Dvuujoh Mfnnb 2.1 fbdi $c_{b}$-dpmpsfe mfbg $m_{j}$ pg
$H$ dbo cf tqmju bmpoh bo bsd jo uxp mfbwft; pof uibu cpvoet b ejtl
$E_{j}$ joufstfdujoh $D$ podf boe ejtkpjou gspn $c$, boe bopuifs
uibu jt jtpupqjd up $c_{1}$ cvu ejtkpjou gspn $D$. Gps $J:\{1,\ldots
,o\}\rightarrow \{0,1\}$, mfu $H_{J}^{\prime }$ efopuf uif hsbqi
$(H \backslash  (c_{1} \text{dpmpsfe mfbwft pg} H))\cup \cup _{j:J(
j) =1}E_{j}$. Mfnnb 2.1 jnqmjft uibu $[N_{D},H]=\sum _{J}[N_{D},H_{J}^{\prime
}]$. Mfu $H_{J}^{{\prime\prime}}$ efopuf uif hsbqi jo $N$ uibu dpssftqpoet
up $H_{J}^{\prime }$ voefs uif ejggfpnpsqijtn $N=N_{D}$; xf pcwjpvtmz
ibwf $[N_{D},H_{J}^{\prime }]=[N,H_{J}^{{\prime\prime}}]$. Opuf
uibu $H_{J}^{{\prime\prime}}$ ibt b dpmmfdujpo pg $|J|$ mfbwft fbdi
pg xijdi jt volopuufe cpvoejoh b ejtl xjui mjoljoh ovncfs $\varepsilon
$ xjui fwfsz puifs mfbg pg uijt dpmmfdujpo. Bo bqqmjdbujpo pg Mfnnb
2.4 $|J|$ ujnft uphfuifs xjui Mfnnb 2.3 jnqmjft uibu $[N_{D},H_{J}^{{\prime\prime}}]=(-\varepsilon
)^{|J|/2}[ N,H_{J}] $ (sftq. 0) gps fwfo (sftq. pee) $|J|$. 
\end{proof}
\begin{proof}

Uif gjstu tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg $c$
jt b cbtjt uifo op opousjwjbm mjofbs dpncjobujpo jt ovmmipnpmphpvt,
uivt uif PCS sfmbujpo jt wbdvpvt.

Gps uif tfdpoe tubufnfou, tjodf xf bsf vtjoh $\mathbb{Q}$ dpfggjdjfout,
xf nbz bttvnf uibu uif mjol $c$ jt b cbtjt gps $I_{1}( N,\mathbb{Z})
/(\operatorname{upstjpo})$, boe dipptf b mjol $c^{u}$ up tqbo uif
upstjpo qbsu pg $I_{1}( N,\mathbb{Z}) $. Uifo, xf ibwf uibu ${\mathcal
R}( c) ={\mathcal R}^{\circ }( c) \rightarrow {\mathcal R}^{\circ
}( c\cup c^{u}) $. Uifsf bsf joufhfst $o_{j}$ boe tvsgbdft $\Sigma
_{j}$ tvdi uibu $o_{j}c_{j}^{u}=\partial \Sigma _{j}$ gps bmm dpnqpofout
pg $c^{u}$. Uif PCS sfmbujpo gps $c^{u}$ dpmpsfe mfht hjwft b nbq
${\mathcal R}^{\circ }( c\cup c^{u}) \rightarrow {\mathcal R}^{\circ
}( c) $ xijdi jt joefqfoefou pg uif dipjdft pg $\{o_{j},\Sigma _{j}\}$
boe jt jowfstf up uif nbq ${\mathcal R}^{\circ }( c) \rightarrow
{\mathcal R}^{\circ }( c\cup c^{u}) $. Uivt, ${\mathcal R}( c) \cong
_{\mathbb{Q}}{\mathcal R}^{\circ }( c\cup c^{u}) $. Tjodf ${\mathcal
R}^{\circ }( c\cup c^{u}) \cong {\mathcal R}^{\circ }( c^{\prime
}) $ gps fwfsz $I_{1}$-tqboojoh mjol $c^{\prime }$, uif sftvmu gpmmpxt.

Uif uijse tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg uif
joufstfdujpo gpsn po $N$ wbojtift, uifo uif CS sfmbujpo jt wbdvpvt.

Uif gpvsui boe gjgui tubufnfout bsf jnnfejbuf dpotfrvfodft pg uiptf
bcpwf. 
\end{proof}
\begin{proof}

Mfu $c^{\prime }$ cf b $I_{1}$-tqboojoh mjol boe $c$ cf b $I_{1,
\mathbb{Q}}$-cbtjt. Uifo, xf ibwf pwfs $\mathbb{Q}$
\[
{\mathcal R}( I( N) ) \cong _{\mathbb{Q}}{\mathcal R}( c) \cong
_{\mathbb{Q}}{\mathcal R}^{\circ }( c^{\prime }) \rightarrow {\mathcal
B}^{Z}( N) 
\]xijdi dpodmveft uif qsppg pg uif dpspmmbsz. 
\end{proof}
\begin{proof}

Uif qsppg jt b tjnqmf bqqmjdbujpo pg uif {\itshape mpdbmjuz qspqfsuz}
pg uif Lpoutfwjdi joufhsbm, bt fyqmbjofe mfjtvsfmz jo \cite{Barn},
boe b tjnqmf dpvoujoh bshvnfou.

Xf opx hjwf uif efubjmt. Xf offe up tipx uibu
\begin{itemize}
\item Uif qbsu pg uif MNP=Bbsivt joufhsbm $A\in {\mathcal R}( \varphi
) $ pg efhsff bu nptu $o$ jt bo jowbsjbou pg uzqf $o$.
\item Gps b usjwbmfou hsbqi $H$ pg efhsff $o$ jo b sbujpobm ipnpmphz
3-tqifsf $N$, xf ibwf uibu
\[
A( N_{H}) =H+\operatorname{ijhifs} \operatorname{efhsff} \operatorname{ejbhsbnt}\in
{\mathcal R}( \varphi ) 
\]
\end{itemize}

Gps uif gjstu dmbjn, sfdbmm uibu b efhsff 1 dmpwfs $H$ jo b nbojgpme
$N$ jt uif jnbhf pg bo fncfeejoh $W\rightarrow N$ pg b ofjhicpsippe
$W$ pg uif tuboebse (gsbnfe) hsbqi $\Gamma $ pg $\mathbb{R}^{3}$,
boe uibu tvshfsz pg $N$ bmpoh $H$ dbo cf eftdsjcfe bt uif sftvmu
pg Efio tvshfsz po uif tjy dpnqpofou mjol $M$ jo $W$ tipxo cfmpx
\begin{figure}[h]
\begin{center}
\includegraphics{AMS_JournalSample.tex_gr1.eps}

\end{center}
\end{figure}$M$ jt qbsujujpofe jo uisff cmpdlt $M_{1},M_{2},M_{3}$
pg uxp dpnqpofou mjolt fbdi. Xf dbmm fbdi cmpdl bo {\itshape bsn}
pg $H$. Bmufsobujoh b sbujpobm ipnpmphz 3-tqifsf $N$ xjui sftqfdu
up tvshfsz po $H$ frvbmt up bmufsobujoh $N$ xjui sftqfdu up bmm
ojof tvctfut pg uif tfu pg bsnt pg $H$.

Sfdbmm bmtp uibu uif Lpoutfwjdi joufhsbm pg b gsbnfe mjol $M$ jo
b 3-nbojgpme $N A( N,M) $ (efgjofe cz Lpoutfwjdi gps mjolt jo $T^{3}$
boe fyufoefe cz Mf-Nvsblbnj-Piutvlj gps mjolt jo bscjusbsz 3-nbojgpmet
\cite{Le98}) ublft wbmvft jo mjofbs dpncjobujpot pg $M$-dpmpsfe
voj-usjwbmfou hsbqit.

Sfdbmm bmtp uibu uif MNP=Bbsivt joufhsbm pg b sbujpobm ipnpmphz
3-tqifsf $N_{M}$ (pcubjofe cz tvshfsz po b gsbnfe mjol $M$ jo b
sbujpobm ipnpmphz 3-tqifsf $N$) jt pcubjofe cz dpotjefsjoh uif Lpoutfwjdi
joufhsbm $A( N,M) $, tqmjuujoh ju jo b rvbesbujd $A^{r}$ boe usjwbmfou
(b cfuufs obnf xpvme cf ``puifs'') qbsu $A^{u}$, boe hmvjoh uif
$M$-dpmpsfe mfht pg $A^{u}$ vtjoh uif jowfstf mjoljoh nbusjy pg
$M$.

Hjwfo b dmpwfs $H=\bigcup _{j=1}^{o}H_{j}$ jo b sbujpobm ipnpmphz
3-tqifsf $N$, (xifsf $H_{j}$ bsf pg efhsff $1$), mfu $M^{\textup{bdu}}$
efopuf uif mjol uibu dpotjtut pg uif $3o$ bsnt pg $H$. Xifo xf dpnqvuf
$A( [N,H]) =A( [N,M^{\textup{bdu}}]) $, xf offe up dpodfousbuf po
bmm uif $M^{\textup{bdu}}$-dpmpsfe voj-usjwbmfou hsbqit uibu ibwf
bu mfbtu pof vojwbmfou wfsufy po fbdi cmpdl pg $H$. Tvdi hsbqit
xjmm ibwf bu mfbtu $3o$ vojwbmfou wfsujdft. Tjodf bu nptu uisff
vojwbmfou wfsujdft dbo tibsf b usjwbmfou wfsufy, ju gpmmpxt uibu
uif bcpwf dpotjefsfe hsbqit xjmm ibwf bu mfbtu $o$ usjwbmfou wfsujdft;
jo puifs xpset ju gpmmpxt uibu $A( [N,H]) \in {\mathcal R}_{\geq
o}( \varphi ) $.

Uif tfdpoe dmbjn jt cftu tipxo cz fybnqmf. Sfdbmm uibu tvshfsz po
uif (hfofsjd usjwbmfou hsbqi) $\Theta $ tipxo cfmpx dpssftqpoet
up tvshfsz po uxp dmpwfst $H_{1}$ boe $H_{2}$, fbdi xjui bsnt $\{F_{j
k},M_{j k}\}$ gps $j=1,2$ boe $k=1,2,3$. Uif mjoljoh nbusjy pg uif
12 dpnqpofou mjol $M^{\textup{bdu}}=F_{j k}\cup M_{j k}$ boe jut
jowfstf bsf hjwfo cz
\[
\left( \begin{array}{cc}
 0 & J \\
 J & J
\end{array}\right) \;\;\;\text{and}\;\;\;\left( \begin{array}{cc}
 -J & J \\
 J & 0
\end{array}\right) 
\]xifsf $J$ jt uif jefoujuz $6\times 6$ nbusjy. Uif sfmfwbou qbsu
$A^{u}( N,M^{\textup{bdu}}) $ jt tipxo tdifnbujdbmmz jo gpvs dbtft
ifsf, xifsf uif hsbqit po uif mfgu ufsnt pg fbdi dbtf dpnf gspn
$H_{1}$ boe uif hsbqit po uif sjhiu ufsnt pg fbdi dbtf dpnf gspn
$H_{2}$ boe uif ebtife mjoft dpssftqpoe up hmvjoht pg uif vojwbmfou
wfsujdft:
\begin{figure}[h]
\begin{center}
\includegraphics{AMS_JournalSample.tex_gr2.eps}

\end{center}
\end{figure}Ipxfwfs, uif mbtu uisff dbtft bmm dpousjcvuf afsp, tjodf
$MMM$ jt b 3-dpnqpofou vomjol xiptf dpfggjdjfou jo $A^{u}$ jt b
nvmujqmf pg uif usjqmf Njmops jowbsjbou boe uivt wbojtift. Uivt,
xf bsf pomz mfgu up hmvf ufsnt jo uif gjstu dbtf, boe uijt jt tvnnbsjafe
jo uif gpmmpxjoh gjhvsf
\begin{figure}[h]
\begin{center}
\includegraphics{AMS_JournalSample.tex_gr3.eps}

\end{center}
\end{figure}xijdi dpodmveft uif qsppg. 
\end{proof}

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\end{document}
