\documentclass[11pt]{article}

\usepackage{fullpage,amssymb,amsmath}

\setlength{\parskip}{1 ex}

\newtheorem{theorem}{Theorem}[section]      
\newtheorem{definition}[theorem]{Definition}

\begin{document}

\begin{center}A quick intro to \LaTeX\end{center}
\medskip

\LaTeX\ (LAH-tek) is an enhanced version of the mathematical typesetting
software \TeX, developed by Donald Knuth during the late 1070's and
1980's. Today, it is freely available in many forms, with enhancements for
almost any kind of document preparation. It requires a certain initial
investment of time and effort to learn, but the time invested is very well
repaid. Nothing else comes close to the freedom and power that even the
occasional user of \LaTeX\ can draw upon.

The best way to get started is looking at some examples, as we do in the
first section below. In the second section, we will give a bigger picture
of what \LaTeX\ is and does, sufficient, we hope, to get the reader going on
any of the many fine introductory books.
\bigskip

\noindent\textsl{1. Some examples}

In this section we give a sequence of examples. Each is given twice.
The first version, in \texttt{this font}, is what appears in the file that
you write and feed to the \LaTeX\ software. The second time, in the regular
font we are using now, shows how it will appear in the final document.

\noindent The first example is just ordinary text. Note how \LaTeX\ ignores
much of the spacing and line breaking in the input. Note also the way that
it ignores any comments that appear on a line after a percent sign \%. If
you actually want a percent sign to appear in the final document, it is
entered with a preceding backslash as ``\texttt{$\backslash$\%}''.

\begin{verbatim}
The authors are grateful to many sources of support during the lengthy
preparation of this work. These 
include the Australian % putting a percent sign % tells 
     % LaTeX to ignore everything that comes after it on the same line
Research Council, the
Korea Research     Foundation, the Basic Science 
                         Research Center of Korea
University, Saint Louis University, the U. S. National Science Foundation,
the Mathematical Sciences  Research Institute, the University of Oklahoma
Vice President for Research, and the University of Oklahoma College of Arts
and Sciences. We also thank the referees of versions of this work for
occasional corrections and 
many helpful suggestions.
\end{verbatim}

The authors are grateful to many sources of support during the lengthy
preparation of this work. These 
include the Australian Research Council, the
Korea Research     Foundation, the Basic Science 
                         Research Center of Korea
University, Saint Louis University, the U. S. National Science Foundation,
the Mathematical Sciences  Research Institute, the University of Oklahoma
Vice President for Research, and the University of Oklahoma College of Arts
and Sciences. We also thank the referees of versions of this work for
occasional corrections and 
many helpful suggestions.

\newpage 

Text given between two dollar signs is treated in ``math mode''. In math
mode, fonts are selected differently, spacing is different and adapted to
formulas, and certain commands become available, such as
\texttt{$\backslash$pi} to produce a letter $\pi$ (commands usuallly start
with a backslash $\backslash$). Note that to use ordinary Roman font in math
mode, the \texttt{$\backslash$text} command is used.

\begin{verbatim}
The elliptic 3-manifolds were completely classified long ago. They are
exactly the 3-manifolds whose universal cover can be uniformized as the
unit sphere $S^3$ in $\mathbb{R}^4$ so that $\pi_1(M)$ acts freely as 
a subgroup
of $\text{Isom}_+(S^3)=\text{SO}(4)$. The subgroups of $\text{SO}(4)$ that
act freely were first determined by Hopf and Seifert-Threlfall, and
reformulated using quaternions by Hattori.
\end{verbatim}

The elliptic 3-manifolds were completely classified long ago. They are
exactly the 3-manifolds whose universal cover can be uniformized as the
unit sphere $S^3$ in $\mathbb{R}^4$ so that $\pi_1(M)$ acts freely as a
subgroup of $\text{Isom}_+(S^3)=\text{SO}(4)$. The subgroups of
$\text{SO}(4)$ that act freely were first determined by Hopf and
Seifert-Threlfall, and reformulated using quaternions by Hattori.

The next example shows a little more math mode. Note also the way that
quotes work in ordinary text, with opening and closing quotes both available.

\begin{verbatim}
We interpret $K_U$ as the ``upper'' knot, contained in $T\times (3/4,1]$,
and $K_L$ as the ``lower'' knot, contained in $T\times [0,1/4)$.  The
vertical $1$-handle with cocore $\sigma$ is assumed to run between $T\times
\{1/4\}$ and $T\times \{3/4\}$, with the separating disk $\sigma$ being its
intersection with~$T\times \{1/2\}$.
\end{verbatim}

We interpret $K_U$ as the ``upper'' knot, contained in $T\times (3/4,1]$,
and $K_L$ as the ``lower'' knot, contained in $T\times [0,1/4)$.  The
vertical $1$-handle with cocore $\sigma$ is assumed to run between $T\times
\{1/4\}$ and $T\times \{3/4\}$, with the separating disk $\sigma$ being its
intersection with~$T\times \{1/2\}$.

\newpage

More complicated commands often use the begin-end structure, as in the
\texttt{pmatrix} command in the next example. The
\texttt{$\backslash$[}-\texttt{$\backslash$]} pair is like the dollar
signs, but escapes to mathematics ``display'' mode. The main difference
from ordinary math mode is there is that the text is centered on the page,
but there are a number of other differences as well. You can also use
double dollar signs for displyed text, as in \$\$\texttt{$\backslash$alpha}\$\$,
and can use a
\texttt{$\backslash$(}-\texttt{$\backslash$)} enclosure equivalently to
single dollar signs, as in \texttt{$\backslash$($\backslash$alpha$\backslash$)}

\begin{verbatim}The notations here indicate the underlying algebra. 
Assume that $T_{p,q}$ is normalized, with $p+r\geq q+s>0$; as mentioned
above, all other cases can be reduced to this one. Write $(p+r)/(q+s)$ as
a continued fraction $[n_1,\ldots,n_k]$ with all $n_i$ positive.  Write
$U=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ and
$L=\begin{pmatrix}1&0\\1&1\end{pmatrix}$. We call the matrix
\[M_{p+r,q+s}=(U\text{ or }L)^{n_k-1}\cdots U^{n_2}L^{n_1} = 
\begin{pmatrix}p&q\\r&s\end{pmatrix}\]
the matrix \textit{associated to $T_{p+r,q+s}$.} 
\end{verbatim}

The notations here indicate the underlying algebra. Assume that $T_{p,q}$
is normalized, with $p+r\geq q+s>0$; as mentioned above, all other cases
can be reduced to this one. Write $(p+r)/(q+s)$ as a continued fraction
$[n_1,\ldots,n_k]$ with all $n_i$ positive.  Write
$U=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ and
$L=\begin{pmatrix}1&0\\1&1\end{pmatrix}$. We call the matrix
\[M_{p+r,q+s}=(U\text{ or }L)^{n_k-1}\cdots U^{n_2}L^{n_1} = 
\begin{pmatrix}p&q\\r&s\end{pmatrix}\]
the matrix \textit{associated to $T_{p+r,q+s}$.} 

\newpage

The next example features definitions using the
\texttt{$\backslash$newcommand} command, that allows you to define your own
\LaTeX\ commands. Note also the cross-referencing commands 
\texttt{$\backslash$label}, which associates its argument 
(the ``label'' you give it) to the
local environment (in this case, a \texttt{definition} environment). Later,
the \texttt{$\backslash$ref} command is given, using as its argument the given
label. \LaTeX\ inserts the correct reference number in the final document,
allowing you to add and take out references without worrying about
cross-referencing.

\begin{verbatim}
\newcommand{\Diff}{\text{Diff}} 
\newcommand{\Imb}{\text{Emb}}
\newcommand{\I}{\text{I}}
\newcommand{\rel}{\text{rel}}

Throughout this chapter, all Riemannian metrics are assumed to be products
near the boundary, or near the horizontal boundary for total spaces of
bundles, such that any submanifolds under consideration meet the collars
in $\I$-fibers. Let $V$ be a submanifold of $M$. The notation
$\mathcal{X}(V,TM)$ means the Fr\'echet space of sections from $V$ to the
restriction of the tangent bundle of $M$ to $V$ that are tangent to
$\partial M$ at all points of $V\cap \partial M$. We also utilize various
kinds of control, as indicated in the following definitions.
\begin{definition}\label{def:diffcontrol}
The notations $\Diff(M\rel X)$ and $\Diff^{M-X}(M)$ mean the space of
diffeomorphisms which restrict to the identity map on each point of the
subset $X$ of $M$.  These notations may be combined, for example
$\Diff^L(M\rel X)$ is the space of diffeomorphisms that are the identity
on $X\cup (M-L)$. \end{definition}$\cdots$

For a neighborhood $K$ of $V$, the superscript notation of
Definition~\ref{def:diffcontrol} may be used, as in $\Imb^K(V,M\rel S)$.
\end{verbatim}

\newcommand{\Diff}{\operatorname{Diff}} 
\newcommand{\Imb}{\operatorname{Emb}}
\newcommand{\I}{\operatorname{I}}
\newcommand{\rel}{\operatorname{rel}}
Throughout this chapter, all
Riemannian metrics are assumed to be products near the boundary, or near
the horizontal boundary for total spaces of bundles, such that any
submanifolds under consideration meet the collars in $\I$-fibers. Let $V$
be a submanifold of $M$. The notation $\mathcal{X}(V,TM)$ means the
Fr\'echet space of sections from $V$ to the restriction of the tangent
bundle of $M$ to $V$ that are tangent to $\partial M$ at all points of
$V\cap \partial M$. We also utilize various kinds of control, as indicated
in the following definitions.
\begin{definition}\label{def:diffcontrol}
The notations 
$\Diff(M\rel X)$ and 
$\Diff^{M-X}(M)$ mean the space of
diffeomorphisms which restrict to the identity map on each point of the
subset $X$ of $M$.  These notations may be combined, for example
$\Diff^L(M\rel X)$ is the 
space of diffeomorphisms that are the identity on $X\cup (M-L)$.
\end{definition}

$\cdots$

For a neighborhood $K$ of $V$, the superscript notation of
Definition~\ref{def:diffcontrol} may be used, as in 
$\Imb^K(V,M\rel S)$.

\newpage

A very useful \LaTeX\ environment is the ``enumerate''
environment:

\begin{verbatim}
\begin{enumerate}
\item For $m$ odd, $\Diff(L(m,1))\approx P_2 \times S^1 \times S^3 \times
\mathbb{R}^\infty$.
\item For $m$ even, $\Diff(L(m,1))\approx P_2 \times S^1 \times 
\text{SO}(3) \times \mathbb{R}^\infty$. 
\end{enumerate}
or this
\begin{enumerate}
\item[(a)] For $m$ odd, $\Diff(L(m,1))\approx P_2 
\times S^1 \times S^3 \times \mathbb{R}^\infty$.
\item[(b)] For $m$ even, $\Diff(L(m,1))\approx P_2 \times S^1 \times 
\text{SO}(3) \times \mathbb{R}^\infty$. 
\end{enumerate}
Notice that the \texttt{item} command takes an ``optional'' parameter,
allowing you to specify the numbering of the items, which defaults to 1, 2,
and so on if you do not specify anything.

The enumerate environment can be customized to your stylistic needs. It
also has many cousins that are environments suited to different kinds of
listings.
\end{verbatim}

\begin{enumerate}
\item For $m$ odd, $\Diff(L(m,1))\approx P_2 \times S^1 \times S^3 \times
\mathbb{R}^\infty$.
\item For $m$ even, $\Diff(L(m,1))\approx P_2 \times S^1 \times 
\text{SO}(3) \times \mathbb{R}^\infty$. 
\end{enumerate}
or this
\begin{enumerate}
\item[(a)] For $m$ odd, $\Diff(L(m,1))\approx P_2 \times S^1 \times S^3 \times
\mathbb{R}^\infty$.
\item[(b)] For $m$ even, $\Diff(L(m,1))\approx P_2 \times S^1 \times 
\text{SO}(3) \times \mathbb{R}^\infty$. 
\end{enumerate}

Notice that the \texttt{$\backslash$item} command takes an ``optional'' parameter,
allowing you to specify the numbering of the items, which defaults to 1, 2,
and so on if you do not specify anything.

The enumerate environment can be customized to many stylistic needs. It
also has many cousins that are environments suited to different kinds of
listings.

\newpage

The \texttt{$\backslash$tabular} command allows you to define the format of a
table, then enter the lines of the table with each item separated by a \&
symbol.

\begin{verbatim}
Here is a simple table, to illustrate the ``tabular'' environment:
\begin{center}
\begin{tabular}{c|c}
Quantified statement&Negation\\
\hline %\noalign{\hrule height 0.25 pt}
$\exists x\in S, P(x)$ & $\forall x \in S, \neg P(x)$ \\
$\forall x\in S, P(x)$ & $\exists x \in S, \neg P(x)$ \\
\end{tabular}
\end{center}

\bigskip

\noindent and here is a more elaborate one:
\begin{small}
\setlength{\fboxsep}{0pt}
\setlength{\fboxrule}{0.5pt}
\begin{center}
\fbox{%
\begin{tabular}{c|c|c|c|c}
$C_i$&$L_1$&$R_1$&$L_2$&$R_2$\\
\hline
$M_i$&$\begin{pmatrix}1 & 0\\ 1 & 1 \end{pmatrix}$
&$\begin{pmatrix}1 & 1\\ 0 & 1 \end{pmatrix}$
&$\begin{pmatrix}1 & 0\\ 1 & 0 \end{pmatrix}$
&$\begin{pmatrix}0 & 1\\ 0 & 1 \end{pmatrix}$\\
\end{tabular}}
\end{center}
\end{small}
\end{verbatim}

Here is a simple table, to illustrate the ``tabular'' environment:
\begin{center}
\begin{tabular}{c|c}
Quantified statement&Negation\\
\hline 
$\exists x\in S, P(x)$ & $\forall x \in S, \neg P(x)$ \\
$\forall x\in S, P(x)$ & $\exists x \in S, \neg P(x)$ \\
\end{tabular}
\end{center}

\bigskip

\noindent and here is a more elaborate one:
\begin{small}
\setlength{\fboxsep}{0pt}
\setlength{\fboxrule}{0.5pt}
\begin{center}
\fbox{%
\begin{tabular}{c|c|c|c|c}
$C_i$&$L_1$&$R_1$&$L_2$&$R_2$\\
\hline
$M_i$&$\begin{pmatrix}1 & 0\\ 1 & 1 \end{pmatrix}$
&$\begin{pmatrix}1 & 1\\ 0 & 1 \end{pmatrix}$
&$\begin{pmatrix}1 & 0\\ 1 & 0 \end{pmatrix}$
&$\begin{pmatrix}0 & 1\\ 0 & 1 \end{pmatrix}$\\
\end{tabular}}
\end{center}
\end{small}

\newpage

\noindent\textsl{2. The bigger picture}

\noindent \textsl{Question:} What is \LaTeX?

\noindent \textsl{Answer No.\ 1:} \LaTeX\ is a programming
language. 

In this way of looking at it, what you write is 
actually a sequence of commands telling
\LaTeX\ what to put in your final document and how to format it.
For example, putting
\begin{verbatim}f(x)-g(x)\end{verbatim}
in your file is a command telling \LaTeX\ to
typeset the letters f(x)-g(x), using the ordinary Roman font, while putting 
\begin{verbatim}\textit{f(x)-g(x)}\end{verbatim}
is a command telling \LaTeX\ to
typeset the letters \textit{f(x)-g(x)}, using italic font, and putting
\begin{verbatim}$f(x)-g(x)$\end{verbatim}
tells it to enter math mode, typeset the math mode letters $f(x)-g(x)$, 
and exit math mode. Note how in math mode, the letters are in italic font,
but the hyphen is interpreted as a subtraction symbol and the spacing is
much better for formulas.

\bigskip

\noindent \textsl{Answer No.\ 2:} \LaTeX\ is a computer program that
accepts your input file as data, and outputs certain other
files. 

The input file that you prepare should carry a \texttt{.tex} suffix,
as in \texttt{root.tex}. When you enter
\begin{verbatim}latex root.tex\end{verbatim}
on the command line, \LaTeX\ will read root.tex and check it for errors. If
there are errors, it will describe the to you. If not (or if there are few
enough errors that it can still process the document), it
will write out several files:
\medskip

\begin{enumerate}
\item root.dvi -- A ``device-independent'' file that encodes the
 information necessary to produce the typset output. The dvi file can be
 viewed on an appropriate reader, or changed to a postscript file
 root.ps (printable on a standard printer) using another program whose
 name is dvips). There are also versions of \LaTeX\ that produce pdf files
as output.
\item root.log -- A copy of the output that \LaTeX\ feeds to the
command line as it runds.
\item root.aux -- Some technical stuff that you don't need
to worry about until you are fairly expert in \LaTeX.
\end{enumerate}

There are many enhancements of \LaTeX\ available. For example, there are
ways to draw pictures or to import graphics files into your document. Many
people have written ``packages'' that you can import into your document just
by putting
\begin{verbatim}\usepackage{packagename1,packagename2,etc.}\end{verbatim}
in the ``preamble'' of your document (more on this coming up shortly). 

\newpage

\noindent \textsl{Question: What appears in the \LaTeX\ ``source'' file?}
\medskip

Here is a short example:
\begin{verbatim}
\documentclass[11pt]{article}

\usepackage{fullpage} 

\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\renewcommand{\P}{{\mathbb P}}

\begin{document}

The stuff that is supposed to be typset goes here. Only this part will be
seen directly in the final document.

\end{document}
\end{verbatim}

The part before the \texttt{$\backslash$begin{document}} line is the ``preamble'',
 where you can define commands to be used anywhere in your document, and
 give \LaTeX\ many other kinds of special instructions about how to format
 your document, There are ways to define sophisticated commands that take
 parameters, and so on. Note the use of \texttt{$\backslash$renewcommand}. It is
 needed because \texttt{$\backslash$P} is already a built-in command in \LaTeX, and
 this definition will override that. It's usually not a good idea to
 redefine a built-in command, but this one is OK.

The file root.dvi produced from root.tex by \LaTeX would print out as:

The stuff that is supposed to be typset goes here. Only this part will be
seen directly in the final document.

There are also many user-friendly interfaces and implementations of
\LaTeX\ that help  you prepare the source file,
so that you don't have to learn much about \LaTeX\ to use
it for many purposes. If you start using it more, though, it's worth your
time to read a book about it and learn how to control it much more
precisely than is possible using the point and click setups. You can also
find many tutorials online, and can get
help and references on most any aspect of
\LaTeX, say by googling \texttt{latex math symbols} or whatever  you want
to look up.

\end{document}


\begin{verbatim}
In the ``verbatim'' environment, \LaTeX\ just puts exactly what you
typed. So
\begin{verbatim}
Let $G = \pi_1^{orb}(\mathcal{O})$ and let $H$ be any subgroup of $G$.
\end{verbatim}
produces
\begin{verbatim}
Let $G = \pi_1^{orb}(\mathcal{O})$ and let $H$ be any subgroup of $G$.
\end{verbatim}
rather than
\medskip

\noindent Let $G = \pi_1^{orb}(\mathcal{O})$ and let $H$ be any subgroup of $G$.
\end{verbatim}

In the ``verbatim'' environment, \LaTeX\ just puts exactly what you
typed. So
\begin{verbatim}
Let $G = \pi_1^{orb}(\mathcal{O})$ and let $H$ be any subgroup of $G$.
\end{verbatim}
produces
\begin{verbatim}
Let $G = \pi_1^{orb}(\mathcal{O})$ and let $H$ be any subgroup of $G$.
\end{verbatim}
rather than
\medskip

\noindent Let $G = \pi_1^{orb}(\mathcal{O})$ and let $H$ be any subgroup of $G$.


\bigskip




\begin{verbatim}
\begin{table}
\begin{scriptsize}
\renewcommand{\arraystretch}{1.5}
\setlength{\fboxsep}{0pt}
\fbox{%
\begin{tabular}{c|c|c}
$m,n$ values&$M$&$\isom(M)$\\
\hline
\hline
$m= n= 1$&$L(4,1)$&%
\makebox[13 ex][c]{$\SO(3)\times S^1$}$=$%
\makebox[27 ex][l]{\ $\set{f(q,x)\vbar (q,x)\in S^3\times J}$}\\
\hline
$m= 1,\,n>1$&$L(4n,2n-1)$&%
\makebox[13 ex][c]{$S^1\times S^1$}$=$%
\makebox[27 ex][l]{\ $\set{f(x,y)\vbar (x,y)\in S^1\times J}$}\\
\hline
$m>1,\,n=
1$&\begin{minipage}[m]{3.24cm}\begin{center}\rule[2.4ex]{0mm}{0mm}quaternionic 
($m=2$)\\
or \rule[-1.3ex]{0mm}{0mm}prism ($m>2$)\end{center}\end{minipage}&%
\makebox[13 ex][c]{$\SO(3)$}$=$%
\makebox[27 ex][l]{\ $\set{f(1,q)\vbar q\in S^3}$}\\
\hline
$m>1,\,n> 1$&\begin{minipage}[m]{3.24cm}\begin{center}\rule[2.4ex]{0mm}{0mm}quaternionic 
($m=2$)\\
or \rule[-1.3ex]{0mm}{0mm}prism ($m>2$)\end{center}\end{minipage}&%
\makebox[13 ex][c]{$S^1$}$=$%
\makebox[27 ex][l]{\ $\set{f(x,1)\vbar x\in S^1}$}\\
\end{tabular}}
\end{scriptsize}
\bigskip\medskip
\caption{Isometry groups of the $M(m,n)$}
\label{Kleintable1}
\end{table}
\end{verbatim}

\begin{table}
\begin{scriptsize}
\renewcommand{\arraystretch}{1.5}
\setlength{\fboxsep}{0pt}
\fbox{%
\begin{tabular}{c|c|c}
$m,n$ values&$M$&$\isom(M)$\\
\hline
\hline
$m= n= 1$&$L(4,1)$&%
\makebox[13 ex][c]{$\SO(3)\times S^1$}$=$%
\makebox[27 ex][l]{\ $\set{f(q,x)\vbar (q,x)\in S^3\times J}$}\\
\hline
$m= 1,\,n>1$&$L(4n,2n-1)$&%
\makebox[13 ex][c]{$S^1\times S^1$}$=$%
\makebox[27 ex][l]{\ $\set{f(x,y)\vbar (x,y)\in S^1\times J}$}\\
\hline
$m>1,\,n=
1$&\begin{minipage}[m]{3.24cm}\begin{center}\rule[2.4ex]{0mm}{0mm}quaternionic 
($m=2$)\\
or \rule[-1.3ex]{0mm}{0mm}prism ($m>2$)\end{center}\end{minipage}&%
\makebox[13 ex][c]{$\SO(3)$}$=$%
\makebox[27 ex][l]{\ $\set{f(1,q)\vbar q\in S^3}$}\\
\hline
$m>1,\,n> 1$&\begin{minipage}[m]{3.24cm}\begin{center}\rule[2.4ex]{0mm}{0mm}quaternionic 
($m=2$)\\
or \rule[-1.3ex]{0mm}{0mm}prism ($m>2$)\end{center}\end{minipage}&%
\makebox[13 ex][c]{$S^1$}$=$%
\makebox[27 ex][l]{\ $\set{f(x,1)\vbar x\in S^1}$}\\
\end{tabular}}
\end{scriptsize}
\bigskip\medskip
\caption{Isometry groups of the $M(m,n)$}
\label{Kleintable1}
\end{table}

\end{document}

