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\topmatter	

\title On Non-separating Simple Closed Curves in a Compact Surface
\endtitle
\author Feng Luo \endauthor
\communicatedby{Walter Neumann}
\address Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 \endaddress
\email fluo\@math.rutgers.edu \endemail
\subjclass 57 \endsubjclass
\keywords Simple closed curve, surface \endkeywords
\date April 22, 1995\inRevisedForm March 22, 1995 \enddate


\abstract
We introduce a semi-algebraic structure on the
set $\Cal S$ of all isotopy classes of non-separating simple
closed curves in any
compact oriented surface and show that the structure is finitely generated.
As a consequence, we produce a natural finite dimensional linear 
representation of the mapping class group of the surface. 
Applications to 
the Teichm\"uller space,
Thurston's measured lamination space, the harmonic Beltrami differentials,
and the first cohomology group of the surface are discussed.
\endabstract


\cyear{1995}
\cvolyear{1995}
\cvolno{1}
\cvol{1}

\endtopmatter

\rm

%\vskip3em

\S 1. {\bf MAIN RESULTS}

\vskip1em

Given a compact oriented surface of positive genus $\Sigma$,
let $\Cal S$ = $\Cal S$($\Sigma$) be
the set of all isotopy classes of non-separating simple closed unoriented
curves in $\Sigma$.  We introduce two relations, \it  orthogonal \rm
and \it disjoint \rm in $\Cal S$ as follows.
Two classes $\alpha$ and $\beta$ in $\Cal S$ are 
said to be \it orthogonal, \rm denoted by $\alpha$ $\perp$ $\beta$,  if they
have representing simple closed curves $a$ and $b$ intersecting 
transversely at one point (in this case, 
we also say that $a$ is \it orthogonal \rm to $b$ and denote
it by $a$  $\perp$  $b$). Two classes $\alpha$ and $\beta$ in $\Cal S$ are
said to be \it disjoint \rm if they
have representing simple closed curves $a$ and $b$  so that $a \cap b = \phi$.
Our goal is to study $\Cal S$ under these two relations.

Given two orthogonal simple closed curves
$p$ and $q$,
define the product $pq$ of $p$ and $q$ to be
$D_p (q)$  where $D_c$ is the positive Dehn
twist about   the simple closed curve $c$.
Geometrically, $pq$ and $qp$ are obtained from $p \cup q$ by
by breaking the intersection into two embedded 
arcs as in figure 1.
 
Clearly the isotopy classes of $pq$ and $qp$ depend only on the isotopy
classes of $p$ and $q$. We use [$p$] to denote the isotopy class of
a simple closed curve $p$ and  define the product  [$p$][$q$] to be
[$pq$] when $[p] \perp [q]$. 
Our first result states that $\Cal S$ is finitely generated in the product.
Actually, a stronger form of the finiteness result holds. It is on  the
stronger form that we will focus. To this end, we introduce
the following definition.

\vskip1ex
{\bf Definition}. Given a  subset $\Cal X$ of $\Cal S$, the  \it derived set \rm
 $\Cal X' $ of $\Cal X$ is $\Cal X \cup \{ \alpha \beta | \alpha, \beta,$ and
$\beta \alpha$ are in $\Cal X $ \}. 
Inductively define $\Cal X^{n}$
to be the derived set of $ \Cal X^{n-1}$ for $n>1$. 
We define $\cup_{n=1}^{\infty} \Cal X^n$ to be the
set \it generated by \rm  $\Cal X$, and denoted it by $\Cal X^{\infty}$.
\vskip1ex 


Our  first theorem is the following.

\proclaim{Theorem 1}  If $\Sigma$ is a compact orientable surface of positive
 genus, then there is a finite subset $\Cal F$  of isotopy classes
of non-separating simple closed curves so that $\Cal F^{\infty} = $
 $\Cal S$($\Sigma$). 
\endproclaim


The next theorem is an analogue of Dehn-Nielsen's theorem that
the mapping class group of a closed surface is the outer automorphism
group of the fundamental group of the surface. To be more precise,
given any orientation preserving
self-homeomorphism $\phi$ of $\Sigma$,  $\phi$ induces a
bijective map of $\Cal S$ by the formula $\phi_*$([$a$]) = [$\phi (a)$]. 
Clearly, $\phi_*$ preserves disjointness, orthogonality and the
product.

\input psfig.tex
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\centerline{The orientation is the right handed orientation in the plane}
\centerline{Figure 1}
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\vskip2em

\proclaim{Theorem 2}  If $h$ is a bijective map of the set of
all isotopy classes of non-separating simple closed curves in a 
closed orientable surface so that $h$ preserves disjointness,
orthogonality and the product, then $h$ is induced by an orientation
preserving self-homeomorphism of the surface. 
\endproclaim

\vskip3em

\S 2. {\bf  MOTIVATIONS }

\vskip1em

Theorems  1 and 2  are well known for the torus $T^2$. As usual, choose an
oriented meridian $m$ and an oriented longitude  $l$  for $T^2$. 
Each element in $\Cal S$
is uniquely represented by $ \pm ( p [m] + q [l])$ in the first homology group
where $p$ and $q$ are two relatively prime integers. 
By  assigning the rational
number $p/q$ to the class,  we identify $\Cal S$ with
$\bold  Q \cup \{ \infty \}$ $\subset$ $\bold  R \cup$\{$\infty$\} which 
is considered to be the natural boundary of the hyperbolic upper half-plane.
Two classes $p/q$ and $r/s$ are orthogonal if and only if $ ps - qr = \pm$1.
Furthermore, if 
$\beta \perp \gamma$, then  $\beta \gamma$ and $\gamma \beta$ 
are symmetric 
with respect to 
the hyperbolic reflection about the geodesic ending at $\beta$ and $\gamma$.
Thus by the well known modular picture, one sees that
$\Cal S$ is generated by 0, 1, and $\infty$ as in figure 2.

To see theorem 2 for the torus, let us take a bijective map  $h$ of
$\Cal S$ = $\bold  Q \cup \{ \infty \}$ preserving the orthogonality.
Thus  $h(0)$, $h(1)$ and $h(\infty)$ are three pairwise orthogonal
rational numbers. Therefore, there is an element $\psi \in$ GL(2, $\bold Z$)
acting on $\bold Q \cup  \{ \infty \}$ as M\"obius transformations
so that $\psi (0) = h(0)$, $\psi (1) = h(1)$ and $\psi( \infty) = h(\infty)$.
Since  GL(2, $\bold Z$) is the mapping class group of the torus, we may
assume at that $h$ leaves the three curves $0$, $1$ and $\infty$ fixed. Thus
by the modular pictures above, the bijection $h$ leaves each rational
number fixed.

 
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\centerline{Figure 2}

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The other motivation of the theorem 1 comes from 
Thurston's compactification of the
Teichm\"uller space and the trace formula tr($XY$) + tr($X^{-1}Y$) =
tr($X$)tr($Y$) for $X$ and $Y$ in SL(2, $\bold C$) (see [FLP]). 
If two elements
$A$ and $B$ in $\pi_1(\Sigma)$ have representatives in the
free homotopy class  $[a]$, $[b]$ in $\Cal S$ so that $[a]$ $\perp$ $[b]$,
then the products $AB$ and $A^{-1}B$ are
represented by the classes $[ab]$ and $[ba]$.
As a consequence
of the trace formula,  the hyperbolic
lengths of the classes $A$, $B$, $AB$ and $A^{-1} B$ with the property above
satisfy the following nonlinear relation:
{ cosh$( l_{AB}/2) + $ cosh$( l_{A^{-1}B}/2)$ = 2 cosh$(l_A /2) $ cosh$(l_B/2)$
},
i.e., it satisfies the following,
$$ f(\alpha \beta) + f(\beta \alpha) = 2 f(\alpha) f(\beta),  \quad
  \alpha \perp \beta .
 \tag1$$ 
Now if we consider a degenerate
family of hyperbolic metrics which tends projectively
to a simple closed curve $c$ (or more generally  a measured lamination)
in the Thurston's compactification of the Teichm\"uller space, then
the relation (1) degenerates to
{$ I([a], [c]) + I([b], [c]) =  $  max$( I([ab], [c]), I([ba], [c]))$},
i.e., it satisfies the following,
$$ f(\alpha ) + f(\beta ) = \text{max} ( f(\alpha \beta), f(\beta \alpha)),
 \quad \alpha \perp \beta . \tag 2$$ where
I( $\cdot,\cdot$ ) is the geometric intersection number.
Since equation (2) is piecewise linear (max($x$,$y$) = 
$1/2( x+ y + | x - y|)$), this may be one reason that
 the action of the mapping class group on
the measured lamination space is piecewise linear.


Since both the Teichm\"uller space and the measured lamination
space are finite dimensional,  this prompts us to ask  about the existence of
finite generators for $\Cal S$.

\vskip3em

\S 3. {\bf  SOME  CONSEQUENCES }

\vskip1em

As a consequence of theorem 1, we produce  a finite dimensional linear
representation of the mapping class group $\Cal G$ = 
$\Cal G(\Sigma)$  of the surface $\Sigma$.
Recall that the mapping class group $\Cal G(\Sigma)$ is defined
to be Homeo$(\Sigma)/$isotopies and $\Cal G$
acts naturally on $\Cal S$
by permuting the isotopy classes. 
The finite dimensional linear representations of  $\Cal G$ are constructed 
as follows.
Take V to be the vector space $\bold C^{\Cal S}$
of all complex valued  functions on $\Cal S$, i.e., 
V = \{$ f : \Cal S \to \bold C$\}. 
The mapping class group $\Cal G$
acts naturally linearly on V by permuting $\Cal S$ coordinates.
Let $a,b,c$ and $d$ be four given  complex
numbers with $cd \neq 0$. Define a linear subspace $V_{a,b,c,d}$ of
 V by the equations  
$$ a f(\alpha) +  b f(\beta) = c f(\alpha \beta) +  d f(\beta \alpha ), \quad
\alpha \perp \beta .
 \tag3$$
By theorem 1, $V_{a,b,c,d}$ is finite dimensional and is clearly
invariant under the linear action of the mapping class group.

\proclaim{Corollary}. If the genus $g$ of the surface is bigger than one,
then the only space  $V_{a,b,c,d}$ which  contains a non-constant
function is $V_{2a, 2a, a, a} = V_{2,2,1,1}$.  Furthermore,
the  dimension of $V_{2,2,1,1}$ is at least
 $(2g+n) (2g+n -1)/2$ where $g$ is the genus and  $n$ is the
numbers of boundary components of
the surface. 
\endproclaim

The basic idea of the
proof is the following. If a simple closed curve $s$ is orthogonal to
another  simple closed curve $t$, then there is a universal relation 
that $[s (ts )] = [t]$ (and also $[(st)s] = [t]$). This relation  $[s(ts)] =[t] $  implies the
well known braid relation $D_s D_t D_s = D_t D_s D_t$ mod(isotopy) in 
the mapping class group.
By iterating  the relation  $[s (ts )] = [t]$ several times for three
pairwise orthogonal simple closed curves which are not in a torus with a hole,
we obtain   that  $2a$ = $2 b$ = $c$  = $d$.
To show that $V_{2,2,1,1}$ contains non-constant solutions, we take 
the square of algebraic intersection number $f(\alpha) = i^2(\alpha, \gamma)$
for a fixed class $\gamma$ in $\Cal S$.
One sees easily that $f$  satisfies the skein relation
$f(\alpha \beta) + f(\beta \alpha) = 2 f(\alpha) + 2 f(\beta)$ whenever
$\alpha \perp \beta$.
Since for all choice of $\gamma$ in $\Cal S$,
these functions  span a linear subspace of dimension
at least  $(2g+n) (2g+n -1)/2$ in $V_{2,2,1,1}$ (the actual dimensions
of the subspace is $(2g+n) (2g+n -1)/2$ if $n \neq 0$ and is
$g(2g+1)$ if $n=0$), the estimate on the dimension follows. 

Theorem 1 may give rise to a characterization of the
length spectrum of a hyperbolic metric in a closed surface. Given a
closed orientable surface $\Sigma$, let Teich( $\Sigma$) be the Teichm\"uler
space of  $\Sigma$. For each equivalence class $[d]$ of hyperbolic
metric in Teich( $\Sigma$), we produce a function $f_{[d]}$ from  $\Cal S$
to $\{  \bold R | t > 1 \}$ by setting $f_{[d]} (\alpha)$ = cosh($l_d (\alpha)/2$)
where $l_d (\alpha)$ is the length of the geodesic in the class $\alpha$
in the metric $d$. By the remark above, we have
$$ f_{[d]}(\alpha \beta) + f_{[d]}(\beta \alpha) = 2 f_{[d]}(\alpha) f_{[d]}(\beta),  \quad
 \alpha \perp \beta.  \tag4 $$  The set $\Cal T$ of all functions from
$\Cal S $  to $\{  \bold R | t > 1 \}$ satisfying (1) is finite dimensional by theorem 1. It
is  well known that the map from Teich($\Cal \Sigma$) to $\Cal T$ sending
$[d]$ to $f_{[d]}$ is injective and continuous. 
It is natural to ask whether the map is an onto map.
 
One possible approach to the above problem is to find the dimension of
$\Cal T$.
Since harmonic Beltrami differentials  are deformations of the
hyperbolic metrics in the Teichm\"uller space,  given a function $f$ satisfying
the relation $(1)$ above, it is  tempting to call a function
$g$ satisfying
$$
g(\alpha \beta) + g(\beta \alpha) = 2 f(\alpha) g(\beta) + 2f(\beta)g(\alpha), 
\quad \alpha \perp \beta.   \tag5 $$ 
a
\it ``harmonic Beltrami differential" \rm  in the  \it 
``conformal structure" \rm given by $f$.  Denote $T_{f}(\Cal T)$ the  
linear space (the tangent space of $\Cal T$ at $f$)
of all functions $g$ satisfying $(5)$.
One would expect to have a Riemann-Roch theorem which calculates the
dimension of $T_{f}(\Cal T)$. 

\vskip3em

\S 4. {\bf SKETCH OF THE PROOFS}

\vskip1em

The main step in the proof of theorem 1 is an induction on a semi-norm
$ ||\Sigma|| = 3g+n$ where $\Sigma$ is a compact orientable
surface of genus $g$ with $n$ boundary components. We show that there
are finitely many subsurfaces $\Sigma_i$ in $\Sigma$ of smaller norms so that
non-separating simple closed curves in $\Sigma_i$  
generate $\Cal S (\Sigma)$. The main technical 
difficult is due to the fact that separating simple closed
curves in $\Sigma_i$ may become non-separating in $\Sigma$.

%\midspace{0.1cm}
\vskip3em
\centerline{\psfig{file=3.ps,width=10cm}}
%caption{Figure 1.2}
%\midspace{0.1cm}
%\centerline{\epsfxsize=10cm \epsfbox{3.ps}}
\vskip-5em
\centerline{The orientation is the right handed orientation in the plane}
\centerline{Figure 3}
\vskip2em
 

We uses several induction steps to achieve the above result. In each step
of the induction process, we construct a 1-complex $G$ in $\Sigma$ and
induct on a semi-norm $|| \alpha ||_G$ for $\alpha \in \Cal S$ where
 $|| \alpha ||_G$ = $inf \{ | G \cap a | | a \in \alpha \}$,
i.e., the geometric intersection number.  Given a class
$\alpha \in \Cal S$, we would like to  find two
 classes $\beta$ and $\gamma$ in $\Cal S$ so that $\beta \perp \gamma$,
$\alpha = \beta \gamma$ and the semi-norms of $\beta$, $\gamma$ and $\gamma \beta$ are smaller than  $|| \alpha ||_G$.
To this end, let us take $a \in \alpha$ so that  $|| \alpha ||_G$
= $|G \cap a| $ and find
 an arc $c$ in $\Sigma$ so that $ c \cap a$ = $\partial c$ and 
$c$ approaches both end points of
$\partial c$ from different sides of $a$. Indeed, assuming $c$ has been 
constructed, we obtain three new non-separating simple closed curves
representing $\beta$, $\gamma$ and $\gamma \beta$ as in the following figure 3.
Furthermore, $\alpha = \beta \gamma$.
We call this an H-reduction on the curve $a$.


Now if the arc $c$ satisfies  $\partial c $ = $ c \cap G$,
then clearly $ || \beta||_G$, $|| \gamma ||_G$ and $|| \gamma \beta||_G$
are all less than $|| \alpha ||_G$.

Let us illustrate this by considering the case of a torus, i.e., the
modular picture is generated by hyperbolic reflections on the three
sides of the ideal triangle with vertices $0$ = $[l]$, $1$ = $[ml]$ and
$\infty$ = $[m]$. Given a class $\alpha$ = $\pm ( p[m] + q[l])$ where $p$ and
$q$ are relatively prime integers, define a norm 
$|| \alpha ||$ = $|p| + |q|$ = inf$\{ |a \cap m| + | a \cap l| | a \in \alpha \}$. Let $a \in \alpha$ be a
representative so that $||\alpha||$  $= |a \cap m| + |a \cap l|$ where
$|a \cap l| = |p|$ and $|a \cap m| = |q|$. Then all intersection
points in $a \cap m$ (and in $ a \cap l$ respectively) 
have the same intersection signs.
Thus, if one of the numbers $| a \cap m |$ or $| a \cap l|$ 
is bigger than 1, say $| a \cap l | \geq 2$, than there are 
two adjacent intersection
points $x$ and $y$ in 
the curve $l$ so that there is an arc $c$ 
in $l$ joining $x$ and $y$ with $c \cap a = \partial c$ and $c \cap m = \phi$. 
Then, the
H-reduction on the curve $a$ at $c$ produces two classes $\beta$ and $\gamma$
so that $\beta \perp \gamma$,
$\alpha = \beta \gamma$ and the norms of $\beta$, $\gamma$ and $\gamma 
\beta$ are  smaller than  $|| \alpha ||$.
Finally, if both $p$ and $q$ are at most 1,
then $\alpha$ is one of the four  classes $[m]$, $[l]$, $[ml]$ and $[lm]$.
Thus the result follows.

There are two steps in the proof of theorem 1. In the first step, we
show that there are finitely many non-separating simple closed
curves \{$c_1, ..., c_k $\} in $\Cal S$ so that $\Cal X$ =
$\{ \alpha \in \Cal S | \alpha $ is disjoint from one of $[c_i] \}$ satisfies
$ (\Cal X)^{\infty}$ = $\Cal S$. This step is relatively easy to achieve.
The next step is the major  
step in which we  replace these curves $c_i$ by  
subsurfaces  each of them
has only one boundary component. Knowing this, we apply 
the induction on 
the semi-norm $||\Sigma||$  of the subsurface  and end the proof.

To prove theorem 2,  we show that the Lickorish-Humphries basis \newline
$F_0$ = $\{ [a_1], ..., [a_{2g+1}] \}$ (as a subset of $\Cal S$ see figure 4) for the mapping class group of a closed surface of genus $g$
satisfies the following property. 

\vskip3em

\input psfig.tex
%\midspace{0.1cm}
\centerline{\psfig{file=4.ps,width=10cm}}
%caption{Figure 1.2}
%\midspace{0.7cm}
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\vskip-4em
\centerline{Figure 4 A Lickorish-Humphries basis}
\vskip2em

 Let $F_n$ be the set $\{ \alpha |
\alpha = \beta \gamma,$ where $ \beta$ and $\gamma$ are in $F_{n-1}$ \} $\cup F_{n-1}$.
Then $\Cal S$ = $\cup_{n=1} ^{\infty} F_n$. 
Now, any two Lickorish-Humphries
bases are related by a self homeomorphism of the surface and
a bijective map of $\Cal S$ preserving disjointness and orthogonality
sends a Lickorish-Humphries basis to a Lickorish-Humphries basis. Thus
the result follows. This is an analogue of  the modular group case where
we use two curves $0$ and $\infty$ as basis.  
The basic technique of the proof is the same as that used in theorem 1.
We do not have to consider the third curve $\gamma \beta$ here but
we do need to produce the minimal set of generators in the new sense.


\vskip3em

\S 5. {\bf SOME OBSERVATIONS AND QUESTIONS}

\vskip1em

There is an interesting similarity  between
the relations (1), (2) and (3) above  and  the
functional equations for the well known elementary functions from
the set of integers {\bf Z} to {\bf R}. 
Namely, the hyperbolic trigonometric functions
cosh$(\lambda t)$ satisfy $f(x+y)+f(x-y) = 2f(x)f(y)$ with $f(x) > 1$, 
the absolute
value functions $| \lambda t|$ satisfy $f(x) + f(y) = $ max($f(x+y), f(x-y))$,
and the square functions $\lambda t^2$ satisfy
$f(x+y)+f(x-y) = 2f(x) + 2 f(y)$.  
Here $\lambda$ is a parameter. It is easy to show that these are 
all non-constant solutions to the functional equations above.
Note that  cos$(\lambda t)$ also satisfies $f(x+y)+f(x-y) = 2f(x)f(y)$.

This suggests one to ask several questions.

\vskip1ex
{\bf Question 1.}  What is the dimension of the vector space $V_{2,2,1,1}$?
\vskip1ex

{\bf Question 2.}  Given a closed orientable surface $\Sigma$ and
a function  $f$ $: \Cal S \to \{ t \in \bold R | t > 1 \}$
satisfies the  relation
$ f(\alpha \beta) + f(\beta \alpha) = 2 f(\alpha) f(\beta)$   
whenever $\alpha \perp \beta$,
is it true that $f$ = $f_{[d]}$ for some hyperbolic metric $d$ in the surface?
\vskip1ex
	
It can be shown easily that there are non-constant solutions to
the relation
$ f(\alpha \beta) + f(\beta \alpha) = 2 f(\alpha) f(\beta)$ which
take some values equal to one.

In view of the linear skein relations for Jones
type knot invariants, one may call a function in $V_{2,2,1,1}$ a
two-dimensional ``Jones invariant". 
More generally, one may define a 2-dimensional ``Jones invariants" as
follows. Take a finite collection of classes $\alpha_1$,...,
$\alpha_k$ in $\Cal S$  (or more generally the set of all
isotopy classes of simple closed curves in the surface) 
and finite collection of non-zero 
numbers $d_1$, ..., $d_k$. 
A  2-dimensional  ``Jones invariant" is a function
$f: \Cal S \to \bold C$ satisfying 
a linear  skein relation:
$$ \sum _{i = 1} ^k d_i f(\phi _* ( \alpha _i)) = 0 \tag6$$
for all  $\phi$ in the mapping class group of the surface.
As an example, take
three pairwise orthogonal simple closed curves $a$, $b$ and $c$
so that $a \cap b \cap c \neq \phi$ and $a \cup b \cup c$ is not
in a torus with a hole. Let $\alpha =[a]$, $\beta =[b]$ and $\gamma =[c]$.
Clearly each function $f$ in $V_{2,2,1,1}$ satisfies the relation (7)
below.
$$ f(\alpha \beta) + f(\beta \alpha ) + f(\beta \gamma) + f(\gamma \beta) +
f(\gamma \alpha) + f(\alpha \gamma) = 4f(\alpha) + 4 f(\beta) + 4 f(\gamma) \tag
7 $$
Is it true that the set of all
solutions to the relation (7)
forms a finite dimensional  vector space?

Finally, suppose the surface $\Sigma$ is closed and $\Cal G$
is the mapping class group of the surface. 
If each function $f$ in $V_{2,2,1,1}$  comes from the square of an
algebraic intersection number, then the  canonical Miller-Morita-Mumford
class in $H^2(\Cal G)$ comes from the  finite dimensional linear 
representation $V_{2,2,1,1}$. 
One  may ask whether each  Miller-Morita-Mumford class 
(or more generally each class in the cohomology group of $\Cal G$) 
in $H^{2k}(\Cal G)$ comes from a finite dimensional linear
representation of the group $\Cal G$.

\vskip1em 


{\bf Acknowledgement}. I would like to thank X.-S. Lin for many
helpful discussions and Peter Landweber for careful reading of the
manuscript. This work is supported in part by the NSF.

\vskip3em

%\centerline{\bf  Bibiliography }

%\vskip1em

\Refs
\widestnumber\key{[FLP]}

\ref\key De \by Dehn, M.\book Papers on group theory and topology, J. Stillwell, ed.\publ Springer-Verlag\publaddr New York\yr 1987\endref

\ref\key FLP \by Fathi, A., Laudenbach, F., Po\`enaru, V.\paper Travaux de Thurston sur les
surfaces \jour Ast\'erisque  \pages 66-67 \yr 1979\endref

\ref\key Li \by Lickorish, W.\paper A representation of oriented combinatorial 3-manifolds
\jour Ann. Math. \vol 72 \pages 531-540 \yr 1962\endref

\ref\key Hu \by Humphries \paper Generators for the mapping class group \jour Lecture Notes in Math.
\vol 722 \publ Springer \publaddr Berlin \yr (1979) \pages 44-47 \endref

\ref\key Lu1 \by Luo, F. \paper On non-separating simple closed curves in a compact surface,
preprint \endref

\ref\key Lu2 \by Luo, F. \paper On the mapping class groups of compact surfaces, in preparation \endref

\endRefs


%\bigskip
%Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

%E-mail: fluo\@math.rutgers.edu

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