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\begin{document}

\title{Curves and surfaces}
\author{Moira Chas }
\date{}




\maketitle





              
\begin{ques} How would you define a
\emph{surface}? Can you  give examples of surfaces?
\end{ques}





\begin{proclamaRom}{Remark}  You probably know what the word
\emph{surface} means in everyday English. Mathematicians
use the word surface with a different (but related)
meaning.

\end{proclamaRom}



\begin{proclamaRom}{Examples of surfaces}
\begin{alphlist}
\item A \emph{very, very} thin sheet of paper.
\item A \emph{very, very} thin sheet of paper glued according the recipe
below.
\item The skin of a solid object (but not the solid object itself!).
\end{alphlist}
\end{proclamaRom}


\begin{proclamaRom}{A recipe to make surfaces}
\begin{numlist}
\item Cut polygons out of paper.
\item Choose pairs of straight edges of the same length
 and \textit{sew} them together (with tape.) 
\item The pieces of paper should not overlap (see Figure \ref{glu}).
\item Once a pair of edges has been sewed, it becomes ``just one edge"
and CANNOT be used again.
Hence, at the end of the process, any edge can belong to
one or two polygons, but not to three or more.
\end{numlist}
\end{proclamaRom}

\begin{figure}[ht]
\begin{pspicture}(16,3.5)  %\psgrid(16,4)
\rput(2,3){Polygons}
\pspolygon[fillcolor=lightgray,fillstyle=solid](1,1)(1.6,1.4)(1.9,2)(1.8,2.4)(1,3)(1.4,2.2)(1,2)
\pspolygon[fillcolor=lightgray,fillstyle=solid](2,1.8)(2.4,1)(1.7,1.3)

%middle
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fillstyle=solid](1,1)(1.6,1.4)(1.9,2)(1.8,2.4)(1,3)(1.4,2.2)(1,2)
} \rput{14}(3.34,0.99){\pspolygon[fillcolor=lightgray,
fillstyle=solid](2,1.8)(2.4,1)(1.7,1.3)
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%right
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fillstyle=solid](1,1)(1.6,1.4)(1.9,2)(1.8,2.4)(1,3)(1.4,2.2)(1,2)
\rput(-0.4,0.3){\pspolygon[fillcolor=lightgray,
fillstyle=solid](2,1.8)(2.4,1)(1.7,1.3)}
  }
%middle
%tape
\rput(-0.88,1){ \pspolygon[fillcolor=black,
fillstyle=hlines, hatchsep=2pt
](5.45,1.8)(5.7,1.75)(5.8,2.185)(5.6,2.384)}

\rput(8,2.5){tape}

\psline{->}(7.5,2.5)(5,3) \rput(6.9,3.1){Allowed sewing}


\rput(4,-1){ \pspolygon[fillcolor=lightgray,
fillstyle=solid](1,1)(1.6,1.4)(1.9,2)(1.8,2.4)(1,3)(1.4,2.2)(1,2)
} %\rput{18}(4.55,-1)
{\pspolygon[fillcolor=lightgray,
fillstyle=solid](5.54,0.4)(5.9,1)(7,0)
    }


\rput(11.6,3.3){Disallowed gluing}

\pspolygon[fillcolor=lightgray,fillstyle=solid](13,2)(12,1.2)(12,0)(13,0.4)

\pspolygon[fillcolor=lightgray,fillstyle=solid](13,2)(13,0.4)(14,0)(14,1)

\pspolygon[fillcolor=lightgray,
fillstyle=solid](13,2)(13.6,3)(13.6,1.4)

\end{pspicture}
\caption{Recipe for constructing surfaces}\label{glu}
\end{figure}

\begin{ques} Which of the rules of the recipe
for making surfaces are broken by the disallowed gluing of
Figure \ref{glu}?
\end{ques}


\begin{exper} Using the above recipe,
construct two surfaces:
\begin{numlist}
\item one made of at least four polygons and gluing at least seven pairs of edges.
\item one using only one polygon and gluing at least four pairs of edges.
\end{numlist}

\end{exper}



\begin{exper} Using the above recipe,
construct two surfaces:
\begin{numlist}
\item one which is not the skin of any solid.
\item one which is the skin of a solid.
\end{numlist}
What can you say about these two surfaces?
\end{exper}


\begin{exe} Determine whether each of the objects of
\figc{objects} is a surface.

\begin{figure}[ht]
\begin{pspicture}(16,8) %\psgrid[gridcolor=gray](16,8)


\rput(1,5){$a)$} 

\pspolygon[fillcolor=lightgray,
fillstyle=solid](1,6)(0.5,6.2)(0.6,7)(1.2,7)(2,6)(3,5)


\rput(7,5){$b)$}


\pscurve(6,6)(7,6)(8,7)(5,7)(6,5.5)(7,7)(6,7)(4,5)


\rput(13,5){$c)$}

\pspolygon[fillcolor=lightgray,
fillstyle=solid](13,7)(12,6)(12,5)(13,6)
\pspolygon[fillcolor=lightgray,
fillstyle=solid](13,7)(14,6)(14,5)(13,6)
\pspolygon[fillcolor=lightgray,
fillstyle=solid](13,7)(13.6,7.5)(13.5,6.5)


\rput(1,2.5){$d)$} 

\psccurve(1,3)(1,4)(1.2,3.5)
\psline(1,3)(4,3)
\psline(1,4)(4,4)
\pscurve(4,3)(4.16,3.5)(4,4)





\rput(7,2.5){$e)$}


\pspolygon[fillcolor=gray, fillstyle=solid](6,3)(8,3)(7,4.7)

\pspolygon[fillcolor=gray, fillstyle=solid](8,3.3)(8,3)(7,4.7)

\rput(13,2.5){$f)$}

\pspolygon[fillcolor=gray, fillstyle=solid](11,2)(11,4)(15,4)(15,2)(13.5,3)
\pspolygon[fillcolor=white, fillstyle=solid](12,3)(12,3.7)(11.5,3.4)

\rput(2,0){\pspolygon[fillcolor=white, fillstyle=solid](12,3)(12,3.7)(11.5,3.4)}


\rput(1,0){$g)$} 

\pspolygon[fillcolor=gray, fillstyle=solid](2,0)(4,0)(4,2)(2,2)

\rput(2,-2){\pspolygon[fillcolor=white, fillstyle=solid](0.3,2.2)(0.7,2.2)(0.7,2.6)(0.3,2.6)}

\rput(3,-2){\pspolygon[fillcolor=white, fillstyle=solid](0.3,2.2)(0.7,2.2)(0.7,2.6)(0.3,2.6)}


\rput(7,-0.0){$h)$}

\pspolygon[fillcolor=gray, fillstyle=solid](8,0)(9,0)(9,1)(8,1)
\pspolygon[fillcolor=gray, fillstyle=solid](8,1)(9,2)(10,2)(9,1)
\pspolygon[fillcolor=gray, fillstyle=solid](9,0)(9,1)(10,2)(10,1)

\rput(13,0){$i)$}

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\psline(1,4)(4,4)
\pscurve(4,3)(4.16,3.5)(4,4)


\
}

\end{pspicture}
\caption{Which are surfaces?}\label{objects}
\end{figure}



\end{exe}




\begin{proclamaRom}{Deforming objects} In
deforming an object, we are allowed to do the following
\begin{alphlist}\item Stretching \item Shrinking \item
Sliding one edge along another.
\item Pass an object through itself (this move is a complicated one and we'll think about it later)
\end{alphlist}
In deforming an object, we are NOT  allowed to
\begin{romlist}\item Cut. \item Sew or paste.\end{romlist}
\end{proclamaRom}

\begin{exe} \begin{numlist}
\item Using paper, scissors and tape, make two surfaces which can be deformed into one another.
\item Draw two surfaces which can be deformed into one another.
\item Using paper, scissors and tape, make two surfaces which can be transformed into one another by cutting and pasting.
\item Draw two surfaces which can be transformed into one another by cutting and pasting.

%\item Construct two surfaces on paper with  the following property: If our paper were rubber sheet, then one could stretch the first surface and get one that looks as the second one. \emph{(Short question: If we start with the second one, how can we get a surface that looks like the first?)}
%\item  Construct two surfaces on paper with  the following property: By cutting and pasting, one can make the first surface look like the second one.
\end{numlist}
\end{exe}


\begin{proclamaRom}{What does topology mean?} We are going to think of topology as
rubber sheet geometry. That is, we must think of the
surfaces we are going to consider as made of a very special
kind of rubber sheets, extremely thin (almost without
thickness!) and so flexible that we can stretch or shrink
them as much as we want, without tearing or losing the
``thinness."

For instance, in topology, a very small sheet of paper is
the ``same" as a huge one, because we can stretch the small
one to make it look exactly like the big one.

More generally, two objects are going to be considered
``equal" or the ``same" if it is possible to deform one into
the other  (see \figc{same}).

(Recall that in the geometry you know, a figure is called a
triangle if it has exactly three straight sides. ``Having three
straight sides"is the main property of the triangle from
the \emph{geometric} point of view. We are going to study some of
the properties that define objects from the \emph{topological}
point of view).


\end{proclamaRom}

\begin{exe}\label{list}  \begin{numlist}
\item Sketch five surfaces  which  are all the same topologically and
all different geometrically. 
\item Sketch three  surfaces  which  are all the same topologically,
all different geometrically  and different topologically from the first five objects you've sketched.
\item  Name  thee objects  from everyday  life which are the same topologically but different geometrically.
\end{numlist}
\end{exe}



\begin{figure}[ht]
\begin{pspicture}(16,3)
\end{pspicture}

\rput{20}(0.9,1.4){\PstPolygon[fillcolor=gray, fillstyle=solid, unit=1.3,PolyNbSides=3] }

\rput{20}(3.9,1.4){\PstPolygon[fillcolor=gray, fillstyle=solid,unit=1.3,PolyNbSides=4] }

\rput{20}(6.9,1.6){\PstPolygon[fillcolor=gray, fillstyle=solid,unit=1.3,PolyNbSides=5] }

\rput{20}(9.9,1.8){\PstPolygon[fillcolor=gray, fillstyle=solid,unit=1.3,PolyNbSides=7] }


\rput{20}(12.9,1.6){\PstPolygon[fillcolor=gray, fillstyle=solid,unit=1.3,PolyNbSides=15] }


\rput{20}(12.9,5.6){\PstPolygon[fillcolor=gray, fillstyle=solid,unit=1.3,PolyNbSides=35] }
\caption{In topology, all these objects are the
same!!}\label{same}
\end{figure}


\begin{proclamaRom}{Remark:}There is even a more general way
of deform surfaces, imagining our surface in a four
dimensional space (but we do not have time to study it,
maybe next time...).
\end{proclamaRom}




\begin{exe} Consider all the objects of
\figc{objects} which are surfaces. Which of them are
equal from the topological point of view? Group them so that two surfaces 
are in the same group only if they are topologically the same.
\end{exe}


\newpage

\subsection{Cylinders and Moebius bands}
\start{exper}{cylinder1} Take a sheet of paper and sew two
opposite edges to get a cylinder. Is the cylinder the same
as the original sheet of paper from the topological point
of view?
\end{exper}



\start{exper}{Moebius} Start with a long strip of paper.
Can you glue the two  short  edges one to each other and
obtain a surface which is NOT a cylinder?
\end{exper}



\start{exper}{twisting} Start with a long strip of paper.
Sew the two short edges to each other. Now take another strip of
paper and sew the two shorter edges together to obtain a surface which is
different from the first one from the topological point of
view. Is it possible to obtain more than two different surfaces in this way?
\end{exper}

\begin{proclama}{Harder question} How many different surfaces
can we obtain by sewing the two short edges of a strip of
paper?
\end{proclama}


\begin{ques} Suppose that we perform the previous experiments
sewing  the long edges instead of the short ones.   Will we
obtain the same results, from the topological point of
view?
\end{ques}



The surface obtained by sewing opposite edges of a strip of
papers with one twist is called a \emph{Moebius band}.

\start{exper}{curve on Moebius} Start drawing a curve on a
cylinder, parallel to the edges and halfway between them.
 Continue drawing it
until you arrive where you started.
You obtained a \emph{closed curve}.


Do the same with a Moebius band. Can you close the curve?
What happens when you come back to the place where you
started? How many times must you go around the band before you come
back where you started \emph{on the same side of the
surface}?
\end{exper}



\begin{proclamaRom}{Remark:} Observe that, in the \experc{curve on
Moebius}, in order to come back to the same place where you
started, on the same side of the surface, one has to go
once along the middle curve of the cylinder. On the other hand,
one has to go 
\emph{twice} along the middle curve of the Moebius band to come back to the same place where you started, on the same side of the surface.
\end{proclamaRom}

\start{exper}{cut Moebius} Cut the cylinder and  Moebius
band along the curve you drew in \experc{curve on Moebius}.
What do you get? And if you cut what you got in its middle
curve what do you get now?
\end{exper}


\emph{Topology} is a branch of mathematics which explains
phenomena such as the ones we observe in the previous
experiments. It is more general than our ``rubber sheet
geometry" (but it certainly includes it).

\newpage

\subsection{More about Cylinders and Moebius bands}




We have seen that if we start with two rectangles and sew an opposite pair of edges without a twist (so the arrows in Figure 4A) point in the same direction) we get a cylinder. And if we sew the edges with a twist (so the arrows in Figure 4B) point in the same direction) we get a Moebius band.



 \begin{figure}[ht]
\begin{pspicture}(16,3)%\psgrid[gridcolor=gray](16,3)

\pspolygon(1,1)(4,1)(4,2)(1,2)
\rput(0.8,1.4){$a$}\rput(4.2,1.4){$\bar{a}$}\psline[arrowsize=6pt]{->}(1,1)(1,1.5)

\rput(3,0){\psline[arrowsize=6pt]{->}(1,1)(1,1.5)}

\rput(1,0){$A)$} 

\rput(6,0){\pspolygon(1,1)(4,1)(4,2)(1,2)\rput(0.8,1.4){$a$}\rput(4.2,1.4){$\bar{a}$}
\psline[arrowsize=6pt]{->}(1,1)(1,1.5)

\rput(3,0){\psline[arrowsize=6pt]{->}(1,2)(1,1.5)}
}
\rput(7,-0.0){$B)$}
\end{pspicture}
\caption{How to obtain a cylinder and a Moebius band}\label{cylinder}
\end{figure}

\start{exper}{paint} Start painting a Moebius band with a marker and paint as much as you can without raising the marker from the surface. Do the same with a cylinder and observe the difference. 
\end{exper}


\textbf{Can you list some of the differences between  the cylinder and the Moebius band from the topological point of view?}

\vspace{0.5cm}

\begin{tabular}{|l|l|}
\hline
Cylinder  \hspace{5cm} & Moebius band \hspace{4cm} \\
\hline
&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\&\\
\hline
\end{tabular}


 

\begin{proclamaRom}{Remark} Notice what big difference can be made by the turn of a little arrow.

\end{proclamaRom}



\subsection{The sphere, the torus and the Klein bottle}

\start{exper}{sphere and torus} Start with three sheets of
paper. Sew the edges of each  of the sheets  as indicated
in \figc{first} a) b) and c). (For each of the figures,
first glue the  pair of edges labeled with $a, \bar{a}$, then the
pair labeled with $b, \bar{b}$, and so on until all the labeled
edges are sewn. The arrows should point in the same direction
where the edges are sewn.)


 \begin{figure}[ht]
\begin{pspicture}(16,3)%\psgrid[gridcolor=gray](16,3)

\pspolygon(1,1)(4,1)(4,2)(1,2)
\rput(0.8,1.4){$a$}\rput(4.2,1.4){$\bar{a}$}\psline[arrowsize=6pt]{->}(1,1)(1,1.5)
\psline[arrowsize=6pt]{->}(2,1)(3,1)
\psline[arrowsize=6pt]{->}(2,2)(3,2)
\rput(3,0){\psline[arrowsize=6pt]{->}(1,1)(1,1.5)}
\rput(2.5,0.5){$b$}\rput(2.5,2.4){$\bar{b}$}
\rput(1,0){$A)$} 

\rput(5,0){\pspolygon(1,1)(4,1)(4,2)(1,2)\rput(0.8,1.4){$a$}\rput(4.2,1.4){$\bar{b}$}
\psline[arrowsize=6pt]{->}(1,1)(1,1.5)

\rput(3,0){\psline[arrowsize=6pt]{->}(1,2)(1,1.5)}
\psline[arrowsize=6pt]{->}(2,1)(3,1)
\psline[arrowsize=6pt]{<-}(2,2)(3,2)

\rput(2.5,0.5){$b$}\rput(2.5,2.4){$\bar{a}$}


}
\rput(7,-0.0){$B)$}


\rput(10,0){\pspolygon(1,1)(4,1)(4,2)(1,2)
\rput(0.8,1.4){$a$}\rput(4.2,1.4){$\bar{a}$}\psline[arrowsize=6pt]{->}(1,1)(1,1.5)
\psline[arrowsize=6pt]{<-}(2,1)(3,1)
\psline[arrowsize=6pt]{->}(2,2)(3,2)
\rput(3,0){\psline[arrowsize=6pt]{->}(1,1)(1,1.5)}
\rput(2.5,0.5){$b$}\rput(2.5,2.4){$\bar{b}$}
}
\rput(12,-0.0){$C$}


\end{pspicture}

\caption{More sewing}\label{first}
\end{figure}
What do you get? Can you describe each of these surfaces as the skin
of a solid you know?\end{exper}



\begin{proclamaRom}{Remark} The surfaces obtained in the
previous experiment by the sewing as in  \figc{first}A) is
called a \emph{torus} and the one from \figc{first}B) is a
\emph{(topological) sphere}.  As you may have realized, the
one from \figc{first}C) cannot be constructed in our three
dimensional space,  but it exists in a "mathematical way",
and it is called a \emph{Klein bottle}.
\end{proclamaRom}

\begin{proclamaRom}{Remark} We could have used regular
letters without a bar to indicate the sewing of
\experc{sphere and torus} (and sew the pair of edges
labeled by exactly the same letter).  The reason why we do it
with pairs of the form $a, \bar{a}$ and $b, \bar{b}$, etc will appear
latter.
\end{proclamaRom}

\begin{proclama}{Exercise.} Name an object whose  skin is a
torus.
\end{proclama}

The torus we made of paper looks a little collapsed. If we
had used flexible rubber band  sheets the torus could look
as in \figc{tor}
\begin{figure}[ht]
\begin{pspicture}(7,4)
 \rput(-1,-3){%torus
\psccurve(8,6.4)(11,5)(8,3.2)(5,5)
% hole
\pscurve(7.2,4.8)(8,5)(8.6,4.8)
\pscurve(7,5.2)(7.2,4.8)(8,4.5)(8.6,4.8)(8.9,5.1) }

\end{pspicture}
\caption{The torus}\label{tor}
\end{figure}




\start{exper}{torus with boundary} Start with two sheets of
paper. Cut them and  sew the edges of one of the sheets  as
indicated in \figc{second} a) and the other, as in
\figc{second}b).
\begin{figure}[ht]
\begin{pspicture}(16,4)%\psgrid[gridcolor=gray](16,4)

\rput(2,2){\PstPolygon[unit=2,PolyNbSides=8] }
\rput(1,4){$a$}
\rput(4,3){$b$}
\rput(3,0){$\bar{a}$}
\rput(0,1){$\bar{b}$}


\rput(3,-0.6){$A)$}


\pspolygon(8,2)(8,3)(10,3)(10,4)(11,4)(11,3)(14,3)(14,2)(11,2)(11,0)(10,0)(10,2)
\rput(7.5,2.5){$a$}
\rput(10.5,4.4){$b$}
\rput(14.5,2.5){$\bar{a}$}
\rput(10.5,-0.3){$\bar{b}$}



 \rput(12,-0.2){$B)$}


\end{pspicture}
\caption{}\label{second}
\end{figure}


\end{exper}

Again, if we had used rubber sheets, the surfaces in \figc{second} would look like the surfaces in \figc{tor with hole}.


\begin{figure}[ht]
\begin{pspicture}(16,4)%\psgrid(16,4)
\psset{xunit=0.9cm}
 \rput(-6,-3){%torus
\psccurve(8,6.4)(11,5)(8,3.2)(5,5)
% hole
\pscurve(7.2,4.8)(8,5)(8.6,4.8)
\pscurve(7,5.2)(7.2,4.8)(8,4.5)(8.6,4.8)(8.9,5.1)}

\rput(4,2){\PstPolygon[unit=0.5,PolyNbSides=4] }

\rput(1,-3){%torus
\psccurve[linestyle=dashed](8,6.4)(11,5)(8,3.2)(5,5)
% hole
\pscurve[linestyle=dashed](7.2,4.8)(8,5)(8.6,4.8)
\pscurve[linestyle=dashed](7,5.2)(7.2,4.8)(8,4.5)(8.6,4.8)(8.9,5.1)}
\rput(3,-0.2){$A)$} \rput(10,-0.2){$B)$}

\psline(8.8,0.2)(8.8,0.6)
\psline(9.2,0.2)(9.2,0.6)

\psline(8.8,1.55)(8.8,1.2)
\psline(9.2,1.57)(9.2,1.2)


\pscurve(8.8,0.6)(7,2)(9,3)(11,2)(9.2,0.6)
\pscurve(8.8,1.2)(7.6,2)(9,2.55)(10.5,2)(9.2,1.2)

\rput(8,0){\pscurve[](9,3)(5,3)(5,0.6)(9,0.6)
\psccurve(9,3)(8.8,2.6)(8.8,1)(9,0.6)(9.2,1)(9.2,2.6)
\pscurve(6,2)(7,1.8)(8,2)
\pscurve(7.8,1.95)(7,2.1)(6.7,2.1)(6.2,1.96)}
\rput(15,-0.5){$C)$}

\end{pspicture}
 \caption{}\label{tor with hole}
\end{figure}

\begin{ques}
\figc{tor with hole}A) is a torus with one hole. What is
\figc{tor with hole}B)?   Can  \figc{tor with hole}B) be
deformed into \figc{tor with hole}C)? 
\end{ques}


\newpage

\subsection{Words and surfaces}


Let us consider the following set of letters, $T$. $T$ is an example of something called \emph{alphabet}.


$$T=\{a, b, c, \bar{a}, \bar{b}, \bar{c}\}$$

A \emph{``normal" word in the alphabet $T$} is  an array of letters of $T$, for instance, \emph{$\bar{b}cbb\bar{a}aa\bar{c}$} . 

A \emph{circular word in the alphabet $T$} is  an array of
a ``normal" word written in the letters of $T$ in a ring,
as, for instance, in  \figc{ring} with an arrow telling us
in which direction one should read it.



\begin{figure}[ht]
\begin{pspicture}(9,2)
\rput(7,1.2){$\curvearrowright$}
 \rput(7,0.1){$a$}
\rput(7,1.9){$b$}\rput(7.9,1){$\bar{a}$}\rput(6.1,1){$\bar{b}$}\rput(7.67,1.57){$b$}
\rput(6.45,1.57){$\bar{c}$} \rput(6.45,0.37){$\bar{c}$}
\rput(7.67,0.37){$a$}
\end{pspicture}
\caption{A ring in the letters of
$\ensuremath{\mathbb{A}}_7$}\label{ring}
\end{figure}

Observe that if we start with the ``normal" word
\emph{$\bar{b}cbb\bar{a}aa\bar{c}$} and we ``glue" the beginning to the end we
obtain the circular word of \figc{ring}.

\begin{exe} Give all the possible normal
words in the alphabet $T$, that arranged in a circle yield
the word of \figc{ring}.
\end{exe}


A \emph{surface word in the alphabet $T$} is a circular
word in which all the letters of $T$ appear exactly once.

\begin{exe} Give five surface words in the
alphabet $T$.
\end{exe}


\begin{proclamaRom}{Constructing surfaces using surface words}
\begin{numlist}
\item Choose a surface word.
\item Cut a polygon of twelve edges.
\item Choose one letter of the surface word and one side of
the polygon.
\item Label the chosen side of the polygon with the chosen
letter. 
\item Going clockwise, skip the edge following the
 one you started with and label the next one  with the next
letter in the surface word.
\item Continue like this, labeling alternate edges, until
six of the edges are labeled and one can ``read" the surface
word from the polygon.
\item Get a strip of paper, and sew one of its short edges
to the edge labeled with $a$ and the other, to the edge
labeled with $\bar{a}$, with no twisting.

\item Get a strip of paper, and sew one of its short edges
to the edge labeled with $b$ and the other, to the edge
labeled with $\bar{b}$, with no twisting.

\item Get a strip of
paper, and sew one of its short edges to the edge labeled
with $c$ and the other, to the edge labeled with $\bar{c}$, with
no twisting.
\end{numlist}

\end{proclamaRom}

\begin{exe} What surface word will produce a cylinder? What surface words will produce a torus with one hole?
\end{exe}

\start{exper}{two} For each pair of normal words, make the
corresponding circular words and  construct the associated
surface and decide if they are topologically the same.
\begin{numlist}
\item $a\bar{a}b\bar{b}c\bar{c}$, $a\bar{c}c\bar{a}b\bar{b}$
\item $abc\bar{a}\bar{b}\bar{c}$,  $ab\bar{a}\bar{b}c\bar{c}$
\end{numlist}

What is the name of each of the constructed surfaces?
\end{exper}


\begin{exe} Give a pair of different surface words
that yield the same surface from the topological point of
view.
\end{exe}


\subsection{Constructing new surfaces from old surfaces}


\begin{exe} Complete \tabc{adding}
\end{exe}

\begin{table}[ht]

\begin{tabular}{|l|l|l|l|}
\hline
Surface & Number of holes \hspace{1.5cm} & Number of sides\hspace{1.5cm} & Other description\hspace{1.5cm}\\
\hline
Sphere &&&\\&&&\\
Disk&&&\\&&&\\
Cylinder&&&\\&&&\\
Moebius band&&&\\&&&\\
Klein bottle&&&\\&&&\\
Torus&&&\\&&&\\
$ab\bar{a}\bar{b}$&&&\\&&&\\
$abc\bar{a}\bar{b}\bar{c}$&&&\\&&&\\
\hline
\end{tabular}
\caption{``Adding" surfaces}\label{adding}
\end{table}




\begin{ques}
We know two properties of the surfaces: number of holes and number of sides. Are those two properties enough to \emph{characterize} the surfaces? In other words, if two surfaces have the same number of holes and the same number of sides, are they topologically equivalent?
\end{ques}


\begin{proclama}{A recipe for constructing surfaces from other surfaces}(see Figure \ref{new recipe})

\begin{numlist}
\item Start with two surfaces. 
\item Cut a hole on each of them
\item Sew the two surfaces along the hole.
\end{numlist}
\end{proclama}





\begin{figure}[ht]
\begin{pspicture}(16,6)%\psgrid[gridcolor=gray](16,4)

\rput(-2,1){\psset{xunit=0.5cm, yunit=0.5cm}\torus}

\rput(0,-1){{\sphere}}
\rput(1,0){$1)$} 

\rput(6,0){
\rput(-2,1){\psset{xunit=0.5cm, yunit=0.5cm}\torus}

\rput(0,-1){\sphere}}

\pscircle[fillcolor=gray, fillstyle=solid](6.8,3.5){0.3}
\pscircle[fillcolor=gray, fillstyle=solid](8.68,0.94){0.3}
\rput(7,-0.0){$2)$}


\rput(11,0){\psset{xunit=0.5cm, yunit=0.5cm}\torus}
\rput(10.6,0.6){\sphere}

\pscircle[fillcolor=white, fillstyle=solid, linecolor=white](13.7,2.5){0.3}

\rput(13,-0.0){$3)$}


\end{pspicture}




\begin{pspicture}(16,5)%\psgrid[gridcolor=gray](16,4)

\rput(-2,1){\psset{xunit=0.5cm, yunit=0.5cm}\torus}

\rput(-1,-1){{\psset{xunit=0.5cm, yunit=0.5cm}\torus}}
\rput(1,0){$1)$} 
\rput(6,0){
\rput(-2,1){\psset{xunit=0.5cm, yunit=0.5cm}\torus}

\rput(-2.5,-1.5){\psset{xunit=0.5cm, yunit=0.5cm}\torus}}
\pscircle[fillcolor=gray, fillstyle=solid](6.8,3.5){0.3}
\pscircle[fillcolor=gray, fillstyle=solid](8.68,0.94){0.3}
\rput(7,-0.0){$2)$}
\rput(11,0){\psset{xunit=0.5cm, yunit=0.5cm}\torus}
\rput(8.1,-0){\psset{xunit=0.5cm, yunit=0.5cm}\torus}
\pscircle[fillcolor=white, fillstyle=solid, linecolor=white](13.7,2.5){0.3}

\rput(13,-0.0){$3$}


\end{pspicture}

\end{figure}



 \begin{figure}[ht]




\caption{Two examples of the new recipe }\label{new recipe}
\end{figure}



\begin{exe}  Complete \tabc{pairs}.
\end{exe}



\begin{table}[ht]
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Surface & Sphere  & Disk  & Cylinder & Torus & Torus with one hole \\
&&&&&\\&&&&&\\
\hline
Sphere &&&&&\\&&&&&\\&&&&&\\
\hline
Disk &&&&&\\&&&&&\\&&&&&\\
\hline
Cyllinder &&&&&\\&&&&&\\&&&&&\\
\hline
Torus &&&&&\\&&&&&\\&&&&&\\
\hline
Torus with hole &&&&&\\&&&&&\\&&&&&\\

\hline
\end{tabular}
\caption{Which surface does one obtain by sewing  pairs of surfaces?}\label{pairs}
\end{table}


The number of holes of a surface is a feature that does not change when we deform the surface. Another number that does not change is the number of \emph{handles} or tori (plural of torus) one has to sew together to obtain the surface.



\newpage

\subsection{Deformation classes of surfaces}

In our first class, we said that two surfaces are ``equal"
if they can deformed into one another. To be more
precise, one should say that two surfaces that can be
deformed into one another are \emph{equivalent}. Hence,
``can be deformed into one another" defines an equivalence
relation on the set of all surfaces. Two surfaces that are equivalent are said to be in the same \emph{equivalence class}.



\begin{proclama}{Very Important Theorem}  All the surfaces in one of these equivalence classes  have
the same number of sides (they all have two sides or one side),  the same number of handles and the same number of
holes. Reciprocally, if two surfaces both have two sides or one side, they have the same
 number of handles and the same number of boundary components, then they
are in the same equivalence class.


Therefore, number of sides, number of handles and number of holes \emph{characterize} the equivalence classes of
surfaces.
\end{proclama}

\begin{proclama}{Translation of the Very Important Theorem to technical math languange} All the surfaces in one of these equivalence classes  have
the same orientability property (they are all orientable or
all non-orientable),  the same genus and the same number of
boundary components. Reciprocally, if two surfaces are both
orientable or both non-orientable, and they have the same
genus and the same number of boundary components, then they
are in the same equivalence class.

Therefore, orientability, genus and number of boundary
components \emph{characterize} the equivalence classes of
surfaces.
\end{proclama}

\begin{exe} Complete the English-math dictionary

\begin{numlist}
\item  Number of sides:
\item  Having one side:
\item Having two sides:
\item \hspace{3cm}: boundary component
\item \hspace{3cm }: genus
\end{numlist}

\end{exe}

\begin{rem} We have not given a rigourous proof of the above theorem in the way that actual mathematicians do
(we would  need many more classes for that). Nevertheless,
we gave some of the ``first steps" of a proof, which
convinces our intuition that the result is true.
\end{rem}


From now on, we are going to study only surfaces with two sides.

\begin{exe} Complete the \tabc{figure}  with figures of two-sided surfaces.
\end{exe}



\begin{table}[ht]
\begin{tabular}{|l||l|l|l|l|l|}
\hline
handles & 0 \hspace{1.9cm} & 1 \hspace{1.9cm} & 2 \hspace{1.9cm} &3\hspace{1.9cm} & 4 \hspace{1.9cm} \\
holes&&&&&\\
\hline\hline
0&&&&&\\&&&&&\\&&&&&\\&&&&&\\&&&&&\\
\hline
1&&&&&\\&&&&&\\&&&&&\\&&&&&\\&&&&&\\
\hline
2 &&&&&\\&&&&&\\&&&&&\\&&&&&\\&&&&&\\
\hline
3 &&&&&\\&&&&&\\&&&&&\\&&&&&\\&&&&&\\

\hline
\end{tabular}
\caption{All surfaces with at most 4 handles and at most 3 holes}\label{figure}
\end{table}


\begin{rem} \tabc{figure} shows the beginning of a complete classification of two-sided surfaces with at most four handles and at most three holes. By continuing the table, we obtain a complete classification of all two sided surfaces. That is, any two-sided surface we can think of, can be deformed like one of the squares of the table.   
\end{rem}


\newpage

\begin{ques} In \tabc{figure} we list some of the two-sided surfaces. 
Which of the surfaces of \tabc{figure} can be the outer layer of a solid? 
 
 Can a one-sided surface be the outer layer of a solid?

More generally, which surfaces can be the outer layer of a solid?
\end{ques}
\section{Curves - a preview}

\begin{proclama}{From the Clay Institute webpage}
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is ``simply connected," but that the surface of the doughnut is not. Poincar\'e, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

\end{proclama}

\begin{proclama}{An old recipe to get a million dollars} Find a proof of the Poincar\'e conjecture.\emph{This recipe is now outdated.Perelman posted two short papers in 2002 and 2003, in which he proved a result that implies this conjecture}\end{proclama}


\section{Deformation classes of curves in a surface}



A \emph{curve on a surface} is a (topological) circle  on
that surface with an \emph{orientation}. In other words, it
is a closed path with an arrow which tell us in which
direction to walk along it. (By \emph{closed} we mean the
beginning and end of the path are glued together; the
\emph{orientation} is given by the arrow).

Our next goal is to understand better the set of all curves
on a (fixed) surface. As we did with surfaces, we are going
\emph{classify} them, that is, we are going to divide the
set of all curves on a surface into equivalence classes.



Again as  we did with surfaces, we are going to think that
the curves are made of extremely flexible rubber bands. We
can shrink, expand, slide curves along the surface. We are
not allowed to break them or separate them from the
surface.

Two curves on the same surface are \emph{equivalent} (or,
the ``same" from the topological point of view) if one can
be  deformed  into the other.

\begin{exe} Make a cylinder with paper and
draw five different curves on it.
\end{exe}


\begin{ques} How many deformation classes of curves do the following surfaces have?
\begin{numlist}
\item The sphere
\item A rectangle (or sphere with one boundary component)
\item The cylinder (or sphere with two boundary components)
\item The torus
\item A surface of genus five with four boundary components
\end{numlist}
Choose different deformation classes of curves on the
cylinder and draw a curve for each of these classes. Can
you give a label for each deformation class of curves in
the cylinder?
\end{ques}




From now on, we are going to choose a surface and study the
deformation classes of curves on a fixed surface.  The surface
we are going to choose is the one constructed by the rules
given in the previous notes, using the circular word
obtained by gluing both ends of the normal word $abABcC$.


\begin{rem} Instead of choosing the surface constructed
with the word $abABcC$, we  could have chosen any surface
and made this study. In most cases, using a different
surface, one obtains a different set of deformation
classes.

On the other hand, two surfaces that can be deformed one
into the other, have the ``same" set of deformation classes
of curves. By ``the same" we do not mean \textit{exactly} the same
(after all, we are considering curves on different
surfaces). We mean that there is a one to one
correspondence between these two sets of equivalence
classes. This correspondence is ``deep": for instance, if a
pair of classes of one surface ``behaves" in a certain way,
then the corresponding pair of classes of the other
surfaces will ``behave" in the same way.
\end{rem}



\begin{rem}  Note that \textbf{a surface does not intersect itself} (even if we can make it pass through itself to deform it, the final deformed surface cannot intersect itself). The glass model of the Klein bottle shown in
class is not a real Klein bottle. A real Klein bottle cannot be constructed in our three dimensional space. The model is just a \textit{trick} to help us visualize it.

On the other hand, \textbf{a curve on a surface may
intersect itself}.

\end{rem}




\begin{figure}[ht]
\begin{pspicture}(16,6)  %\psgrid(16,6)
\rput(2,-0.5){$i)$} \rput(12,-0.5){$ii)$}


\psline[linecolor=gray]{->}(2.2,5.8)(5.7,2)

\pscurve[linecolor=lightgray]{->}(2,5.7)(2,5)(0.3,4.2)

\pscurve[linecolor=lightgray]{->}(0.9,1)(1.6,1.6)(0.6,1.3)

\pscurve[linecolor=lightgray]{->}(0.2,3.6)(3,2)(3,1)(2,1)(3,2)(5.7,1.8)

\surface


\rput(7,3){\surface}
\end{pspicture}
\caption{Curves on the surface $abABcC$}\label{curves}
\end{figure}

\begin{exe}  Look at the path in dark gray in \figc{curves} $i)$. After gluing
together the edges of the polygon as indicated by the
labels, the path becomes a closed oriented curve (observe
that it has an arrow)

Analogously, the two pieces of light gray paths on \figc{curves}
$i)$ make a curve after gluing together the labeled edges.
Hence, we have two curves. Are both curves in the same
deformation class?

\end{exe}

The two curves of \figc{curves} $i)$  are equivalent. The
darker curve is shorter and simpler, and so, easier to deal
with. Hence, when we try to analyze a deformation class of
curves, we are going to consider the ``simplest" curves in
the deformation class.


The shortest curve in \figc{curves} $i)$ ``enters" through
the edge labeled with $a$ and ``leaves" through the edge
labeled by $A$. We will denote the deformation class of
this curve by the circular word $a$. 




\begin{figure}[ht]
\begin{pspicture}(0,-15)(16,6) % \psgrid(16,6)
\rput(2,-0.5){$i)$} \rput(12,-0.5){$ii)$}


%\psline[linecolor=gray]{->}(2.2,5.8)(5.7,2)

\pscurve[linecolor=gray]{->}(2,5.7)(1,3)(4,0.3)

\pscurve[linecolor=gray]{->}(5,5)(1.6,1.6)(5.8,2.7)

%\pscurve[linecolor=lightgray]{->}(0.2,3.6)(3,2)(3,1)(2,1)(3,2)(5.7,1.8)





\surface


\rput(7,3){\surface}

\rput(-3,-4){\rput(2,-0.5){$iii)$} \rput(14,-0.5){$iv)$}
\surface \rput(7,3){\surface}}


\rput(-3,-12){\rput(2,-0.5){$v)$} \rput(14,-0.5){$vi)$}
\surface \rput(7,3){\surface}}
\end{pspicture}

\caption{More curves on the surface $abABcC$}\label{second
curves}
\end{figure}


\begin{exe} In \figc{curves} $ii)$ draw \begin{numlist}
\item the shortest curve of the deformation class $b$.
\item the shortest curve of the class deformation class $B$.
\item a curve in the deformation class $b$ which is not the
shortest.\end{numlist}
\end{exe}



\begin{exe} Observe that
the two paths in \figc{second curves} $i)$ form a curve.
Draw one of the simplest curves \figc{second curves} $i)$
which can be deformed into the curve we have there. The new
curve has two pieces. For each of the pieces, say where
they enter and where they come out.

We will denote  the deformation class of this curve by the
circular word $BA$.
\end{exe}






\begin{rem} We are using circular words for two different
purposes that should not be confused: to construct surfaces
and to label deformation classes of curves on a fixed
surface.
\end{rem}


\begin{exe} \begin{numlist}\item Draw in  \figc{second curves} $iii)$ one of the
simplest curve of the class labeled by the circular word
$Bc$.
\item  Draw in  \figc{second curves} $iv)$ one of the
simplest curve of the class  $aBc$.

\item Write two cyclic words of at least four letters and draw
the corresponding shortest curves in \figc{second curves}
$v)$ and $vi)$.



\end{numlist}
\end{exe}

\newpage





\begin{exe} Draw in  \figc{third curves} $i)$ one of the
simplest curves of the class  $bBc$. Study your drawing. Can
you find a curve in the same deformation class which is shorter than the one you drew? 
\end{exe}

A circular word in which none of the pairs $aA$, $Aa$, $bB$, $Bb$,
$cC$ or $Cc$ appear is called a \emph{reduced word.}

\begin{exe} Write two reduced words of seven letters.
\end{exe}


\begin{exe} Draw in \figc{third curves} ii) one of the simplest curves of the class  $ABBc$. Does the  curve you draw intersect  itself? Is it possible to avoid those intersections?
\end{exe}

\begin{exe} Draw in \figc{third curves} iii) one of the simplest curves of the class  $ABCBc$. Does the  curve you draw intersect  itself? Is it possible to avoid those intersections?
\end{exe}

\begin{exe} Draw in \figc{third curves} iv) one of the simplest curves of the class  $AABaBc$. Does the  curve you draw intersect  itself? Is it possible to avoid those intersections?
\end{exe}


\begin{ques} What do the words of the three previous exercises have in common? (Hint: The words $ABcccdcdcDcccDBc$, $aaabABaabcadaBcccc$ and  $ABccBc$ share the same characteristic)  
\end{ques}

\begin{figure}[ht]
\begin{pspicture}(0,-15)(16,6) % \psgrid(16,6)
\rput(2,-0.5){$i)$} \rput(12,-0.5){$ii)$}
\surface
\rput(7,3){\surface}
\rput(-3,-4){\rput(2,-0.5){$iii)$} \rput(14,-0.5){$iv)$}
\surface \rput(7,3){\surface}}
\rput(-3,-12){\rput(2,-0.5){$v)$} \rput(14,-0.5){$vi)$}
\surface \rput(7,3){\surface}}
\end{pspicture}
\caption{Even more curves on the surface $abABcC$}\label{third curves}
\end{figure}




\begin{exe} Complete the following sentence:\newline \emph{ If a cyclic reduced word $\dots$\hspace{7cm}  then any curve labeled by the word $\dots$ \hspace{4cm}} 
\end{exe}


\enddocument





\enddocument

