\chapter{Classification of 2-Manifolds: Organizing Surfaces}

\begin{exercise} 
Show that the torus $T^2$ is homeomorphic to
$\Sph^1\times \Sph^1$.
\end{exercise}

\oneperpage

\begin{exercise}
For a given number of holes, demonstrate that the $n$-holed torus where the holes are lined up is homeomorphic to an $n$-holed torus where the holes are arranged in a circle.
\end{exercise}

\oneperpage

\begin{exercise}
\be \item Show that $\PP^2\cong\Sph^2/\langle x\sim -x\rangle$, that is,
the projective plane is homeomorphic to the $2$-sphere with
diametrically opposite points identified.
\item Show that $\PP^2$ is also homeomorphic to a disk with two
edges on its boundary (called a \textbf{bigon}),\index{bigon}
identified as indicated in Figure \ref{rptwo}.  
\begin{figure}[h]
\begin{center}
%\includegraphics[height = 2.3cm]{Figures/rp2}
\caption{\label{rptwo} $\PP^2$.}
\end{center}
\end{figure}
\item Show that the Klein bottle can be realized as a square with
certain edges identified. \ee
\end{exercise}

\oneperpage

\begin{thm} 
Suppose $M$ is a compact, connected 1-manifold. Then
$M$ is triangulable. That is, $M$ is homeomorphic to a subset $C$ of $\R^n$ consisting of a finite collection of straight line segments where any two segments of $C$ are either disjoint or meet at an endpoint of each.
\end{thm}

\oneperpage

\begin{exercise}  
Provide a complete classification of compact, connected 1-manifolds.  
That is, describe a collection of topological spaces such that every compact, 
connected $1$-manifold is homeomorphic to one member of the collection.
\end{exercise}

\oneperpage

\begin{exercise} 
Provide a complete classification of compact $1$-manifolds.
\end{exercise}

\oneperpage

\begin{thm}
Every compact $2$-manifold is triangulable, that is, it is homeomorphic to a subset $C$ of $\R^n$ consisting of a finite collection $T = \{\sigma_i\}_{i=1}^k$ of rectilinear triangles (a fancy word for a rectilinear triangle is a $2$-simplex) where each pair of triangles are disjoint or they meet in one vertex of each or they share a single edge. Since the space $C$ is homeomorphic to a $2$-manifold, each edge of each triangle making up $C$ is shared by exactly two triangles, and around each vertex is a circle of triangles whose union is a disk. 
\end{thm}

\oneperpage

\begin{exercise}  The boundary of a tetrahedron is naturally
triangulated with a triangulation $T$ consisting of four
$2$-simplexes, having six edges and four vertices. 
\begin{enumerate}
\item On the boundary of a tetrahedron draw the first and second barycentric
subdivisions of $T$. 
\item Locate the edges of the four triangles in $T$. 
\item Draw the regular neighborhood of the union of all the edges of $T$. 
\item Draw the regular neighborhood of a single edge of a triangle in $T$.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
In the second barycentric subdivisions of a triangulation of the torus (Figure \ref{torussecondbary}),
%and the Klein bottle (Figure \ref{kleinbary}), 
find regular neighborhoods of various
subsets of the edges.
\end{exercise}

\oneperpage

\begin{exercise}
Consider the triangulation of the torus in Figure \ref{torustriangles}.
%and Klein bottle that are drawn above. 
Describe those graphs created from edges in the $1$-skeleton of $T$ that have
regular neighborhoods homeomorphic to a disk.
\end{exercise}

\oneperpage

\begin{thm}  
Let $M^2$ be a compact, triangulated $2$-manifold with triangulation $T$. 
Let $S$ be a tree whose edges
are $1$-simplices in the $1$-skeleton of $T$. Then $N(S)$, the
regular neighborhood of $S$, is homeomorphic to $\D^2$.
\end{thm}

\oneperpage

\begin{thm}  
Let $M^2$ be a compact, triangulated
$2$-manifold with triangulation $T$. Let $S$ be a tree equal to a
union of 'edges' in the dual $1$-skeleton of $T$. Then $\cup \{
\sigma''_j \mid \sigma''_j \in T''$ and $\sigma''_j \cap S\ne
\emptyset\}$ is homeomorphic to $\D^2$.
\end{thm}

\oneperpage

\begin{thm}  Let $M^2$ be a connected, compact, triangulated
$2$-manifold with triangulation $T$. Let $S$ be a tree in the
$1$-skeleton of $T$. Let $S'$ be the subgraph of the dual
$1$-skeleton of $T$ whose 'edges' do not intersect $S$. Then $S'$ is
connected.
\end{thm}

\oneperpage

\begin{thm}  Let $M^2$ be a connected, compact, triangulated
$2$-manifold. Then $M^2 = D_0 \cup D_1 \cup \bigl( \bigcup_{i=1}^k
H_i \bigr)$ where $D_0$, $D_1$, and each $H_i$ is homeomorphic to
$\D^2$, $\hbox{\rm Int } D_0 \cap D_1 =\emptyset$, the $H_i$'s are
disjoint, $\bigcup_{i=1}^k \hbox{\rm Int }H_i \cap ( D_0 \cup D_1) =
\emptyset$, and for each $i$, $H_i \cap D_1$ equals $2$ disjoint arcs each
arc on the boundary of each of $H_i$ and $D_1$.
\end{thm}

\oneperpage

\begin{thm} Let $M^2$ be a connected, compact, triangulated
$2$-manifold. Then: \be
\item There is a disk $D_0$ in $M^2$ such that $M^2 -
(\interior D_0)$ is homeomorphic to the following subset of $\R^3$:
a disk $D_1$ with a finite number of disjoint strips, $H_i$ for
$i\in\{1,\ldots n\}$, attached to boundary of $D_1$ where each strip
has no twist or a $1/2$ twist. (See  Figure \ref{3handles1twist}.)
\item
Furthermore, the boundary of the disk with strips, $D_1 \cup \bigl(
\bigcup_{i=1}^k H_i \bigr)$, is connected. \ee

\begin{figure}[h]
\begin{center}
%\includegraphics[height = 1.7in]{Figures/3handles1twist}
\caption{\label{3handles1twist} A disk with four handles attached.}
\end{center}
\end{figure}

\end{thm}

\oneperpage

\begin{exercise}
In the conclusion of the previous theorem, any strip $H_i$ divides the
boundary of $D_1$ into two arcs, $e_i^1$ and $e_i^2$, where $H_i$ is
\emph{not} attached, that is, the two arcs that make up $(D_1 \bigcap H_i)$ are disjoint from the two arcs $e_i^1$ and $e_i^2$ except at their endpoints. Show that if a strip $H_j$ is attached to $D_1$
with no twists, then there must be a strip $H_k$ that is attached to
both $e_j^1$ \emph{and} $e_j^2$.
\end{exercise}

\oneperpage

\begin{thm} Let $M^2$ be a connected, compact, triangulated
$2$-manifold. Then there is a disk $D_0$ in $M^2$ such that $M^2 -
\hbox{\rm Int } D_0$ is homeomorphic to a disk $D_1$ with strips
attached as follows: first come a finite number of strips with $1/2$
twist each of whose attaching arcs are consecutive along $\bd D_1$, and
next come a finite number of pairs of untwisted strips, each pair
with attaching arcs entwined as pictured with the four arcs from
each pair consecutive along $\bd D_1$.
\end{thm}

\oneperpage

\begin{thm} Let $X$ be the union of a disk with three strips attached as follows: a disk $E_0$ with one strip attached with a
$1/2$ twist with its attaching arcs consecutive along $\bd E_0$ and
one pair of untwisted strips with attaching arcs entwined as
pictured with the four arcs consecutive along $\bd E_0$. Let $Y$ again be a union of a disk with three strips attached, but the three are attached differently. The set $Y$ consists of a disk $E_1$ with three strips with a $1/2$ twist each whose attaching arcs are consecutive along $\bd E_1$. Then $X$ is
homeomorphic to $Y$.

\begin{figure}[h]
\begin{center}
%\includegraphics[width =4in]{Figures/xandy}
\caption{\label{xandy} These spaces are homeomorphic.}
\end{center}
\end{figure}

\end{thm}

\oneperpage

\begin{thm}  Let $M^2$ be a connected, compact, triangulated
$2$-manifold. Then there is a disk $D_0$ in $M^2$ such that $M^2 -
\interior D_0$ is homeomorphic to one of the following:
\smallskip
\bi
\item[a)] a disk $D_1$,
\item[b)] a disk $D_1$ with $k$ $\frac{1}{2}$-twisted strips with
consecutive attaching arcs, or
\item[c)] a disk $D_1$ with $k$ pairs of untwisted strips, each pair
in entwining position with the four attaching arcs from each pair
consecutive. \ei
\end{thm}

\oneperpage

\begin{thm} 
Suppose $M_1$ and $M_2$ are compact, triangulated, connected
$2$-manifolds and $M$ is a connected sum of $M_1$ and $M_2$ (that is,
we can select triangulations of $M_1$ and $M_2$, apply the process
above and arrive at a space homeomorphic to $M$). Then $M$ is a
compact, connected, triangulable 2-manifold.
\end{thm}

\oneperpage

\begin{exercise} 
Suppose $M$ is a compact, connected, triangulated $2$-manifold.  What is
$\Sph^2\num M$?
\end{exercise}

\oneperpage

\begin{exercise} 
Sketch $\num_{i=1}^n\T^2$.
\end{exercise}

\oneperpage

\begin{thm}[Classification of compact, connected $2$-manifolds]
\index{$2$-manifold!Classification Theorem} Any connected, compact,
triangulated $2$-manifold is homeomorphic to the $2$-sphere
$\Sph^2$, a connected sum of tori, or a connected sum of projective
planes.
\end{thm}

\oneperpage

\begin{exercise} 
Identify the following spaces and give
justification.\label{basicpoly}

\begin{tabular}{ccc}
%\includegraphics[height = 3cm]{Figures/spherewithgluing} &
%\includegraphics[height = 3cm]{Figures/kleinsquarec} &
%\includegraphics[height = 3cm]{Figures/rp2square}\\
(a) & (b) & (c)\end{tabular}
\end{exercise}

\oneperpage

\begin{thm}  
Let $P$ be a polygonal presentation. Then $P$ is a $2$-manifold.
\end{thm}

\oneperpage

\begin{thm}
Suppose $M$ is a compact, connected, triangulable $2$-manifold. Then $M$ is homeomorphic to a polygonal presentation.
\end{thm}

\oneperpage

\begin{thm} 
Let $P$ be a polygonal presentation. Then $P$ is a
compact, connected, triangulable 2-manifold.
\end{thm}

\oneperpage

\begin{thm} 
Let $Abb^{-1}C$ be a string of $2n$ letters where each
letter occurs twice, neglecting superscripts (and there is at least
one pair other than $b$ and $b^{-1}$).  Then the $2$-manifold
obtained from the word $Abb^{-1}C$ is homeomorphic to that obtained
from $AC$.
\end{thm}

\oneperpage

\begin{thm} 
Suppose $P$ is a polygonal presentation not homeomorphic
to $\Sph^2$.  Then there is a homeomorphic polygonal presentation
where all the vertices are in the same equivalence class, that is,
all the vertices are identified to each other.
\end{thm}

\oneperpage

\begin{thm} 
Suppose $P$ is a polygonal presentation not homeomorphic
to $\Sph^2$. Then $P$ is homeomorphic to a polygonal presentation
where all the vertices are identified and for every pair of edges with
the same orientation, the two edges of that pair are consecutive.
\end{thm}

\oneperpage

\begin{thm} 
Suppose $P$ is a polygonal presentation not homeomorphic
to $\Sph^2$. Then $P$ is homeomorphic to a polygonal presentation
where all the vertices are identified, every pair of edges with the
same orientation are consecutive, and all other edges are grouped in
disjoint sets of two intertwined pairs following the pattern
$aba^{-1}b^{-1}$.
\end{thm}

\oneperpage

\begin{thm} 
\label{torus-plus-projplane}
If $A$ and $C$ are (possibly empty) words, then the
polygonal presentation $Aaba^{-1}b^{-1}ccC$ is homeomorphic to that
represented by $AddeeffC$.
\end{thm}

\oneperpage

\begin{thm} 
Any compact, connected, triangulated $2$-manifold $M$ is
homeomorphic to the polygonal presentation given by one of the
following words: $aa^{-1}$, $a_1a_1\ldots a_na_n$ (where $n\geq1$)
or $a_1a_2a_1^{-1}a_2^{-1}\ldots a_{n-1}a_na_{n-1}^{-1}a_{n}^{-1}$
(where $n\geq 2$ is even).  
\end{thm}

\oneperpage

\begin{exercise} 
Suppose that we have two compact, connected 2-manifolds
represented by the words $w_1$ and $w_2$, respectively.  Suppose in
addition that $w_1$ and $w_2$ have no letters in common.  What can
you say about the $2$-manifold corresponding to the concatenated word
$w_1w_2$ in terms of the connected sum?
\end{exercise}

\oneperpage

\begin{exercise}
Re-state Theorem \ref{torus-plus-projplane} above in the case that $A$ and $C$ are empty, in terms of connected sum.
\end{exercise}

\oneperpage

\begin{thm}[Classification of compact, connected $2$-manifolds]
\index{$2$-manifold!Classification Theorem} Any compact, connected, triangulated $2$-manifold is homeomorphic to exactly one of the
following:
\begin{enumerate} 
\item$\Sph^2$,
\item a connected sum of $n$ tori, and
\item a connected sum of $n$ projective planes.
\end{enumerate} 
\end{thm}

\oneperpage

\begin{exercise} 
Describe heuristically a strategy by which you
would define a consistent clockwise direction on the standard
embedding of the 2-sphere in $\R^3$.  What is the relevant
property?
\end{exercise}

\oneperpage

\begin{exercise}
Show that the induced orientation is well defined; in other words,
that it is independent of the choice of positive equivalence class
representative for the original $2$-simplex.
\end{exercise}

\oneperpage

\begin{thm}\label{orient}
Show that the following are equivalent for a 2-manifold $M$.
\begin{enumerate}
\item Every triangulation of $M$ is not orientable (that is if $K_{T}$
is a simplicial complex to which $M$ is homeomorphic, $K_{T}$ is not
orientable).
\item $M$ admits a triangulation that is not orientable.
\item $M$ admits a triangulation that contains a collection of
simplices whose union is homeomorphic to the M\"{o}bius band.
\item $M$ admits an embedding of a M\"{o}bius band.
\item There is a map $F:\Sph^1\times[0,1]\to M$ such that
$F(\cdot,t)$ is an embedding for each $t$ and such that
$F(\cdot,1)=F(r(\cdot),0)$, where $r$ is a reflection map of
$\Sph^1$ about some line through its center.
\end{enumerate}
\end{thm}

\oneperpage

\begin{thm}
Let $M_1,\ldots,M_n$ be connected, compact, triangulated $2$-manifolds.  Let $M$ be a connected sum of $M_1,\ldots,M_n$.  Then
$M$ is orientable if and only if $M_i$ is orientable for each
$i\in\{1,\ldots,n\}$.
\end{thm}

\oneperpage

\begin{exercise} State and prove which compact, connected, triangulated $2$-manifolds are orientable and which are not.
\end{exercise}

\oneperpage

\begin{exercise} 
Calculate the Euler characteristic of the following
spaces.\be
\item $\Sph^2$
\item $\T^2$
\item $\K^2$
\item $\PP^2$
%\item $\Sph^n$
%\item the solid torus
%\item the M\"{o}bius band
%\item a finite tree
%\item a finite graph
\ee\end{exercise}

\oneperpage

\begin{lem}  
Suppose $M_1$ and $M_2$ are compact 2-manifolds.  If $M_1\num M_2$ is \emph{any} choice for the connect sum of $M_1$
and $M_2$, then $\chi(M_1\num M_2)=\chi(M_1)+\chi(M_2)-2$.
\end{lem}

\oneperpage

\begin{exercise}  
\begin{enumerate}
\item Calculate the Euler characteristic of $\num_{i=1}^n\PP^2$.
\item Calculate the Euler characteristic of $\num_{i=1}^n\T^2$.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm} 
The combination of Euler characteristic and
orientability is a complete invariant of compact, connected
$2$-manifolds.
\end{thm}

\oneperpage

\begin{exercise}  
Identify the following $2$-manifolds as a sphere, a connected sum
of $n$ tori (specifying $n$), or a connected sum of $n$ projective
planes (specifying $n$). 
\be
\item[a.]   $\T\#\PP$
\item[b.]   $\K\#\PP$
\item[c.]   $\PP\#\T\#\K\#\PP$
\item[d.]   $\K\#\T\#\T\#\PP\#\K\#\T$
\ee
\end{exercise}

\oneperpage

\begin{exercise}
%Identify the following surfaces:
%\begin{enumerate}
%\item The surface obtained by identifying the edges of
%the octagon as indicated in Figure \ref{octogenustwothrice}.
%\begin{figure}[h]
%\begin{center}
%\includegraphics[height = 2 in]{Figures/octogenustwo}
%\caption{\label{octogenustwothrice} The genus two surface}
%\end{center}
%\end{figure}
Identify the surface obtained by identifying the edges of the
decagon as indicated in Figure \ref{tensides}.

\begin{figure}[h]
\begin{center}
%\includegraphics[height = 1.8 in]{Figures/tensides}
\caption{\label{tensides} The decagon with edges identified in
pairs.}
\end{center}
\end{figure}

%\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise} 
Notice that the edge (or boundary) of a M\"{o}bius
band is a simple close curve.  Construct a space by gluing a disk to
the M\"{o}bius band along their respective boundaries.  Show that
this space is homeomorphic to the projective plane.
\end{exercise}

\oneperpage

