\chapter{Manifolds, Simplexes, Complexes, and Triangulability: Building Blocks}

\begin{exercise} 
Show that the standard $n$-ball
and the standard $n$-cube are homeomorphic spaces and each is
compact and connected.
\end{exercise}

\oneperpage

\begin{exercise} 
Show that for $n\geq1$, the $n$-sphere is compact and connected.
\end{exercise}

\oneperpage

\begin{exercise} 
Consider $\Sph^0$, $\Sph^1$, and $\Sph^2$. Is any
pair of them homeomorphic? If not, are there properties that allow you to
distinguish them?
\end{exercise}

\oneperpage

\begin{thm}  
For a separable, metric space $M^n$, the following are equivalent:
\begin{enumerate}
\item $M^n$ is an $n$-manifold;
\item for each point $p\in M^n$, $p$ has a neighborhood base of open sets each homeomorphic to the interior of an $n$-ball;
\item for every point $p \in M^n$, $p \in U$ where $U$ is an open set homeomorphic to $\R^n$.
\end{enumerate}
\end{thm}

\oneperpage

\begin{exercise} 
If you are comfortable with ordinal numbers, construct a topological space where every point has an open set containing it that is homeomorphic to $\R^1$, and yet the space is not metrizable. You might call your space the \textbf{long line}\index{long line}.
\end{exercise}

\oneperpage

\begin{exercise}
Show that a locally Euclidean space is Hausdorff and second countable if and only if it is separable and metrizable.
\end{exercise}

\oneperpage

\begin{exercise}  
Show that $\Sph^n$ is an $n$-manifold. 
\end{exercise}

\oneperpage

\begin{thm} 
If $M$ is an $n$-manifold and $U$ is an open subset of $M$, then $U$ is also an $n$-manifold.
\end{thm}

\oneperpage

\begin{thm} 
If $M$ is an $m$-manifold and $N$ is an $n$-manifold, then $M\times N$ is an $(m+n)$-manifold.
\end{thm}

\oneperpage

\begin{thm}
Let $M^n$ be an $n$-dimensional manifold with boundary. Then $\partial M^n$ is an $(n-1)$-manifold. 
\end{thm}

\oneperpage

\begin{exercise} 
Show that if $\sigma$ is a simplex and $\tau$ is one of its
faces, then $\tau \subset\sigma$.
\end{exercise}

\oneperpage

\begin{exercise} 
Show that an $n$-simplex is homeomorphic to a closed $n$-dimensional ball.
\end{exercise}

\oneperpage

\begin{exercise}
Exhibit a collection of simplices that satisfies condition (1) but not (2) in the definition of a simplicial complex.
\end{exercise}

\oneperpage

\begin{exercise}
\label{simple-complex}
Let $K$ be the simplicial complex in $\R^{2}$:
$$
K = \{
\sigma, 
e_1,e_2,e_3,e_4,e_5,
v_1,v_2,v_3,v_4
\}
$$
where
$\sigma=\{(0, 0)(0, 1)(1, 0)\}$, 
$e_1=\{(0, 0)(0, -1)\}$, 
$e_2=\{(0, -1)(1, 0)\}$,
$e_3=\{(0, 0)(0, 1)\}$,
$e_4=\{(0, 1)(1, 0)\}$,
$e_5=\{(1, 0)(0, 0)\}$,
$v_1=\{(0, 0)\}$,
$v_2=\{(0, 1)\}$,
$v_3=\{(1, 0)\}$, and
$v_4=\{(0,-1)\}$.
%$$ T = \{ \sigma_{1}, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, v_{1}, v_{2}, v_{3}, v_{4} \}$$
%where $\sigma_{1} = \{(0, 0)(0, 1)(1, 0)\}$, $e_{1} = \{(0, 0)(0, 1)\}$, $e_{2}=\{(0, 1)(1, 0)\}$, $e_{3}=\{(1, 0)(0, 0)\}$,
%$e_{4}=\{(0, 0)(0, -1)\}$, $e_{5}=\{(0, -1)(1, 0)\}$,
%$v_{1}= \{(0, 0)\}$ , $v_{2}=\{(0, 1)\}$, $v_{3}=\{(1, 0)\}$, $v_{4}=\{(0, -1)\}$.
%
%$$T=\left\{ \begin{tabular}{c}$\{(0, 0)(0, 1)(1, 0)\},
%\{(0, 0)(0, -1)\}, \{(0, -1)(1, 0)\},$\\
%$\{(0, 0)(0, 1)\}, \{(0, 1)(1, 0)\}, \{(1, 0)(0, 0)\},$\\
%$\{(0, 0)\} , \{(0, 1)\}, \{(1, 0)\}, \{(0, -1)\}$\end{tabular}
%\right\}.$$  
Draw $K$ and its underlying space.
\end{exercise}

\oneperpage

\begin{exercise}
Show that the space shown in Figure \ref{circledisk} is triangulable
by exhibiting a simplicial complex whose underlying space it is homeomorphic to.
\end{exercise}

\oneperpage

\begin{exercise} 
For each $n$, $\Sph^n$ is triangulable.
\end{exercise}

\oneperpage

\begin{thm} 
A simplicial map from $K$ to $L$ is determined by the images of the vertices of $K$.
\end{thm}

\oneperpage

\begin{thm} 
A composition of simplicial maps is a simplicial map.
\end{thm}

\oneperpage

\begin{thm}
\label{simpcor} 
If two complexes are simplicially homeomorphic, then there
are 1-1 correspondences between their $k$-simplices for each
$k\geq0$.
\end{thm}

\oneperpage

\begin{thm} A simplicial map $f:K \to L$ is continuous as a map on the underlying spaces.
In particular, simplicially homeomorphic complexes have homeomorphic underlying spaces.
\end{thm}

\oneperpage

\begin{thm} 
A composition of PL maps is PL.  A PL homeomorphism is an equivalence relation.
\end{thm}

\oneperpage

\begin{thm}
\label{PLequiv} PL homeomorphic complexes are homeomorphic as
topological spaces.
\end{thm}

\oneperpage

\begin{exercise}
\label{triangle-twice-around}
Let $K$ is a complex consisting of the boundary of a triangle (three vertices and three edges) and $L$ be an isomorphic complex.  Both $|K|$ are $|L|$ are topologically circles.  There is a continuous map that takes the circle $|K|$ and winds it twice around the circle $|L|$; however, show that there is no simplicial map from $K$ to $L$ that winds the circle $|K|$ twice around the circle $|L|$.
\end{exercise}

\oneperpage

\begin{exercise}
How many $n$-simplices are there in the first barycentric subdivision of an $n$-simplex?
\end{exercise}

\oneperpage

\begin{exercise}
Convince yourself that the barycentric subdivision of a complex $K$ is, in fact, a subdivision of $K$.
\end{exercise}

\oneperpage

\begin{thm}
\label{diameter}
Let $K$ be a finite simplicial complex and let $a_n$  be the maximum among the diameters of simplices in $\sd^n K$.  Then
$\lim_{n\to\infty}a_n=0$.
\end{thm}

\oneperpage

\begin{exercise}
The star of a vertex $v$ in a complex $K$ is an open set of $|K|$, and the collection of all vertex stars covers $|K|$.
\end{exercise}

\oneperpage

\begin{exercise}
If the simplex $\sigma=\{v_{0},\ldots,v_{k}\}$ in $K$ is the minimal face of a point $x \in |K|$, 
then $x \in \st(v_{0}) \cap \cdots \cap \st(v_{k})$.
\end{exercise}

\oneperpage

\begin{thm}
Suppose $K$ and $L$ are simplicial complexes.
Then a continuous function $f: |K| \to |L|$ satisfies the star condition with respect to $K$ and $L$ if and only if $f$ has a simplicial approximation $g:K \to L$.  
\end{thm}

\oneperpage

\begin{thm}
If $g,g':K \to L$ are both simplicial approximations to a continuous function $f: |K| \to |L|$, 
then for any point $x \in |K|$, if $\sigma$ is the minimal face of $x$ in $K$, 
the point $f(x)$ and the simplices $g(\sigma)$ and $g'(\sigma)$ all lie in a single simplex of $L$.
\end{thm}

\oneperpage

\begin{thm}
Let $K$ and $L$ be simplicial complexes. If $f:|K| \to |L| $ has a simplicial approximation $g:K \to L$, then $f$ is homotopic to $g:|K|\to|L|$.
\end{thm}

\oneperpage

\begin{thm}
Suppose $K$ and $L$ are finite simplicial complexes and $f: |K| \to |L|$ is a continuous function between their underlying spaces.  Then there exists $m\geq 1$ such that the function 
$f: |\sd^{m} K| \to |L|$ satisfies the star condition with respect to $\sd^{m} K$ and $L$.
\end{thm}

\oneperpage

\begin{thm}
Suppose $K$ and $L$ are simplicial complexes and $f: |K| \to |L|$ is a continuous function between their underlying spaces.  Then there exists $m\geq 1$ such that $f$ has a simplicial approximation $g: \sd^{m} K \to L$.
\end{thm}

\oneperpage

\begin{thm}[Simplicial Approximation Theorem]
\label{simplicial-approx}
Let $K$ and $L$ be simplicial complexes, let $f:|K| \to |L|$ be a continuous function between their underlying spaces, and let $\epsilon>0$. Then there exist $m,n\geq 1$ and a simplicial map $g: \sd^n K \to  \sd^m L$ such that $f$ is homotopic to $g$ and for every $x \in |K|$, $d(f(x),g(x))<\epsilon$. 
\end{thm}

\oneperpage

\begin{thm}
Let $K$ be a subdivision of a $1$-simplex $\sigma$. 
Label every vertex of $K$ with a $0$ or a $1$ such that one of the two vertices of $\sigma$ is labeled with a $0$ and the other is labeled with a $1$. Then there is a $1$-simplex $\tau$ in $K$ such that one vertex of $\tau$ is labeled $0$ and the other vertex of $\tau$ is labeled $1$.
\end{thm}

\oneperpage

\begin{thm}
Let $K$ be a subdivision of a $2$-simplex $\sigma$. Label every vertex of $K$ with $0$, $1$, or $2$ such that the three vertices of $\sigma$ are labeled with different numbers. Then there is a $2$-simplex $\tau$ in $K$ such that its vertices are labeled with all different numbers.
\end{thm}

\oneperpage

\begin{thm}[Sperner's Lemma]
Let $K$ be a subdivision of a $n$-simplex $\sigma$. 
Label every vertex of $K$ with one of $\{0, 1, \ldots, n\}$ such that the $(n+1)$ vertices of $\sigma$ are labeled with different numbers. Then there is an $n$-simplex $\tau$ in $K$ such that its vertices are labeled with all different numbers.
\end{thm}

\oneperpage

\begin{thm}
Let $\sigma$ be an $n$-simplex with boundary $\partial \sigma$. There does not exist a continuous function $r:\sigma \to \partial \sigma$ such that for every $x \in \partial \sigma$, $r(x)=x$.
\end{thm}

\oneperpage

\begin{thm}[$n$-dimensional Brouwer Fixed Point Theorem]
\index{Brouwer Fixed Point Theorem!$n$-dimensional} Let $\sigma^n$ be an $n$-simplex. For every
continuous function $f:\sigma^n\to\sigma^n$ there exists a point $x\in\sigma^n$
such that $f(x)=x$.
\end{thm}

\oneperpage

\begin{thm}
Let $h:\Sph^1 \to \R^2$ be an embedding that is a polygon, that is, $h(\Sph^1)$ consists of a finite number of straight line intervals. Then $h(\Sph^1)$ separates $\R^2$ into two components and each point of $h(\Sph^1)$ is a limit point of each component. 
\end{thm}

\oneperpage

\begin{thm}
Let $h:\Sph^1 \to \R^2$ be an embedding that is a polygon, that is, $h(\Sph^1)$ consists of a finite number of straight line intervals. Then there is a homeomorphism $H: \R^2 \to \R^2$ such that $H(h(S^1))$ is the unit circle. 
\end{thm}

\oneperpage

\begin{thm}
Let $A$ and $B$ be disjoint closed subsets of $[0,1] \times [0,1]$ such that $A \cap ([0,1] \times \{0,1\} \cup \{1\} \times [0,1]) = \emptyset$ and $B \cap ([0,1] \times \{0,1\} \cup \{0\} \times [0,1]) = \emptyset$. Then there exists a path in $[0,1] \times [0,1]$ from $(1/2, 0)$ to $(1/2,1)$ that misses $A \cup B$. 
\end{thm}

\oneperpage

\begin{thm}
Suppose $h:[0,1] \to \R^2$ is an embedding and suppose $p$ and $q$ are points in $\R^2$ not contained in $h([0,1])$. Then there exists a path $f:[0,1] \to \R^2$ such that $f(0)=p$, $f(1)=q$, and $f([0,1]) \cap h([0,1]) = \emptyset$.
\end{thm}

\oneperpage

\begin{thm}
Suppose $g,h:[0,1] \to \R^2$ are embeddings such that $g([0,1])$ is a straight line segment, $g(0)=h(0)$, $g(1)=h(1)$, and $g((0,1)) \cap h((0,1)) = \emptyset$. Then $g([0,1]) \cup h([0,1])$ separates $\R^2$ into two components and each point of $g([0,1]) \cup h([0,1])$ is a limit point of each component. 
\end{thm}

\oneperpage

\begin{thm}
Let $h:\Sph^1 \to \R^2$ be an embedding. Then $h(\Sph^1)$ separates $\R^2$ into two components and each point of $h(\Sph^1)$ is a limit point of each component. 
\end{thm}

\oneperpage

\begin{exercise}
\begin{enumerate}
\item Let $C_0$ be a disk with two holes. Construct a subset $C_1$ of $C_0$ such that $C_1$ is also homeomorphic to a disk with two holes, and for which each point $x \in C_1$ is within distance $1$ of points in each of the three components of $\R^2 - C_1$. 
\item Construct a continuum $C \subset \R^2$ such that $\R^2 -C$ has three components and each point $x \in C$ is a limit point of each component of $\R^2 -C$. 
\item Construct a continuum $C \subset \R^2$ such that $\R^2 -C$ has infinitely many components and each point $x \in C$ is a limit point of each component of $\R^2 -C$. 
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
Let $h:\Sph^1 \to \R^2$ be an embedding. Let $p$ be a point in the bounded component of $\R^2 - h(\Sph^1)$ and let $\epsilon > 0$. Then there exists an embedding $g:\Sph^1 \to \R^2$ such that $g(\Sph^1)$ is a polygonal simple closed curve in the bounded component of $\R^2 - h(\Sph^1)$, $g(\Sph^1)$ lies in the $\epsilon$-neighborhood of $h(\Sph^1)$, and $p$ is in the bounded component of $\R^2 - g(\Sph^1)$.
\end{thm}

\oneperpage

\begin{thm}
Let $h:\Sph^1 \to \R^2$ be an embedding. Let $\epsilon > 0$ and let $A$ be an arc on $\Sph^1$ with endpoints $a$ and $b$ such that the diameter of $h(A)$ is less than $\epsilon$ and let $p$ be a point in the bounded component of $\R^2 - h(\Sph^1)$. Then there exists an embedding $g:\Sph^1 \to \R^2$ such that $g(\Sph^1)$ is a polygonal simple closed curve in the bounded component of $\R^2 - h(\Sph^1)$, $p$ is in the bounded component of $\R^2 - g(\Sph^1)$, and for every $x \in A$, $d(g(x),h(x))<\epsilon$.
\end{thm}

\oneperpage

\begin{thm}
Let $h:\Sph^1 \to \R^2$ be an embedding, let $U$ be the bounded component of $\R^2 - h(\Sph^1)$, and let $D$ be the closed unit ball in $\R^2$. Then there is a homeomorphism $H: (U \cup h(\Sph^1) \to D$.
\end{thm}

\oneperpage

\begin{thm}
Let $h:\Sph^1 \to \R^2$ be an embedding. Then there is a homeomorphism $H: \R^2 \to \R^2$ such that $H(h(\Sph^1))$ is the unit circle.
\end{thm}

\oneperpage

\begin{thm}
Let $h:[0,1] \to \R^2$ be an embedding of $[0,1]$ in the plane and let $\epsilon > 0$. Then there exists an embedding $g:[0,1] \to \R^2$ such that $h(0)=g(0)$, $h(1)=g(1)$, and for every $x \in [0,1]$, $d(h(x),g(x))<\epsilon$. 
\end{thm}

\oneperpage

\begin{thm}
Let $f,g:[0,1] \to \R^2$ be two embeddings of $[0,1]$ in the plane such that $f(0)=g(0)$ and $f(1)=g(1)$. Let $\epsilon > 0$. Suppose for every $x \in [0,1]$, $d(f(x),g(x))<\epsilon$. Then there exists a homeomorphism $h:\R^2 \to \R^2$ such that for every $t \in [0,1]$, $h(f(t))=g(t)$ and for every $x \in \R^2$, $d(x, h(x))<\epsilon$. 
\end{thm}

\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) $2$-simplices where each pair of $2$-simplices 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 $2$-simplex 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{thm}
Every $2$-manifold is triangulable. 
\end{thm}

\oneperpage

\begin{lem}
Let $M_1$ and $M_2$ be two rectilinearly triangulated $2$-manifolds in $\R^n$. Let $h:M_1 \to M_2$ be a topological homeomorphism. Then there exists a homeomorphism $g:M_1 \to M_2$ such that the image of every edge in the triangulation of $M_1$ is a polyhedral arc in $M_2$.
\end{lem}

\oneperpage

\begin{thm}
Let $M_1$ and $M_2$ be two rectilinearly triangulated $2$-manifolds in $\R^n$. Let $h:M_1 \to M_2$ be a topological homeomorphism. Then there exists a homeomorphism $g:M_1 \to M_2$ such that the image of every triangle in a triangulation of $M_1$ is a rectilinear triangle in $M_2$.
\end{thm}

\oneperpage

\begin{thm}
Any two triangulations of a compact $2$-manifold are equivalent. 
\end{thm}

\oneperpage

\begin{thm}
The Euler characteristic is well-defined for compact $2$-manifolds.
\end{thm}

\oneperpage

\begin{thm}
Orientability is well-defined for compact $2$-manifolds.
\end{thm}

\oneperpage

