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\subsection{Simplicial complexes}

\begin{flashcard}[affinely-indep-defn]
\begin{definition*}[Affinely independent]
  \glsadjdefn{affindep}{affinely independent}{set
  of points}
  \cloze{A finite set of points $a_0, a_1, \ldots, a_r
  \in \RR^m$ is \emph{affinely independent} if
  \[ \begin{cases}
    \sum_{i = 1}^n t_i a_i = 0 \\
    \text{and } \sum_{i = 1}^n t_i = 0
  \end{cases} \iff (t_1, \ldots, t_n) = 0 .\]}
\end{definition*}
\end{flashcard}

\begin{flashcard}[affindep-lemma-iff]
\begin{lemma*}
  $a_0, \ldots, a_n \in \RR^m$ is \gls{affindep}
  if and only if \cloze{$a_1 - a_0, a_2 - a_0,
  \ldots, a_n - a_0$ is linearly independent.}
\end{lemma*}

\begin{proof}
  \cloze{Let $a_0, \ldots, a_n$ be \gls{affindep}, and
  suppose
  \[ \sum_{i = 1}^n s_i(a_i - a_0) = 0 .\]
  Then
  \[ \left( -\sum_{i = 1}^n s_i \right) a_0 + s_1
  a_1 + \cdots + s_n a_n = 0 \]
  and
  \[ \left( -\sum_{i = 1}^n s_i \right) + s_1 +
  \cdots + s_n = 0 \]
  hence $(s_1, \ldots, s_n) = 0$. So $a_1 - a_0,
  \ldots, a_n - a_0$ are linearly independent.

  Similarly for the converse.}
\end{proof}
\end{flashcard}

\begin{flashcard}[n-simplex-defn]
\begin{definition*}[$n$-simplex]
  \glsnoundefn{vert}{vertex}{vertices}%
  \glsnoundefn{nsimp}{$n$-simplex}{$n$-simplexes}%
  \glsnoundefn{simp}{simplex}{simplexes}
  \glsnoundefn{bcoord}{barycentric
  coordinate}{barycentric coordinates}
  \glssymboldefn{simplexspan}{$\langle a_0,
  \ldots, a_n \rangle$}{$\langle a_0, \ldots, a_n
  \rangle$}
  \cloze{If $a_0, \ldots, a_n \in \RR^m$ are
  \gls{affindep}, then they define an
  \emph{$n$-simplex}
  \[ \sigma = \langle a_0, \ldots, a_n \rangle =
  \left\{ \sum_{i = 0}^n t_i a_i ~\Bigg|~ \sum_{i =
  0}^n t_i = 1 \text{ and }t_i \ge 0 \right\}
  \subseteq \RR^m \]
  given by the convex hull of the points $a_0,
  \ldots, a_n$. These are called the
  \emph{vertices} of $\sigma$, and we say that
  they \emph{span} $\sigma$.

  If $x \in \langle a_0, \ldots, a_n \rangle$,
  then $x$ can be written \emph{uniquely} as $x =
  \sum_{i = 0}^n t_i a_i$ for real numbers $t_0,
  \ldots, t_n$ summing to $1$. Call the $t_i$'s
  the \emph{barycentric coordinates} of $x$.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[face-defn]
\begin{definition*}[Face]
  \glsnoundefn{face}{face}{faces}
  \glssymboldefn{faceof}{$\le$}{$\le$}
  \glssymboldefn{propface}{$<$}{$<$}
  \cloze{A \emph{face} of a \gls{nsimp} $\sigma =
  \simplex<a_0, \ldots, a_n>$ is a \gls{simp}
  $\tau$ spanned by a subset of $\{a_0, \ldots,
  a_n\}$. Write $\tau \le \sigma$. Write $\tau <
  \sigma$ if $\tau$ is a proper face.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[simplex-boundary-defn]
\begin{definition*}[Boundary]
  \glsnoundefn{bound}{boundary}{boundaries}
  \glssymboldefn{bound}{$\partial$}{$\partial$}
  \cloze{The \emph{boundary} of a \gls{simp} $\sigma$,
  written $\partial \sigma$ is the union of all
  its proper \glspl{face}.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[simplex-interior-defn]
\begin{definition*}[Interior]
  \glssymboldefn{interior}{interior}{interior}
  \glsnoundefn{int}{interior}{interiors}
  \cloze{The \emph{interior} of $\sigma$, written
  $\mathring{\sigma}$, is $\sigma - \bound
  \sigma$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[p-simplex-homeo-lemma]
\begin{lemma*}
  Let $\sigma$ be a \nsimp{p} in $\RR^m$ and
  $\tau$ be a \nsimp{p} in $\RR^n$. Then $\sigma$
  and $\tau$ are homeomorphic.
\end{lemma*}

\begin{proof}
  \cloze{Let $\sigma = \simplex<a_0, \ldots, a_p>$, $\tau
  = \simplex<b_0, \ldots, b_p>$. Define
  \begin{align*}
    h : \sigma
    &\to \tau \\
    \sum_{i = 0}^p t_i a_i
    &\mapsto \sum_{i = 0}^p t_i b_i
  \end{align*}
  This is well-defined and a bijection, by
  uniqueness of \glspl{bcoord}. As the $a_i - a_0$
  are linearly independent, $h$ extends to an
  affine map $\hat{h} : \RR^n \to \RR^m$, so $h$
  is continuous. So is its inverse.}
\end{proof}
\end{flashcard}

\begin{flashcard}[geometric-simplicial-complex-defn]
\begin{definition*}[Geometric simplicial complex]
  \glsnoundefn{gsimpcomp}{geometric simplicial
  complex}{geometric simplicial complexes}
  \glsnoundefn{simpcomp}{simplicial
  complex}{simplicial complexes}
  \cloze{A \emph{geometric (or Euclidean) simplicial
  complex} in $\RR^m$ is a finite set $K$ of
  \glspl{simp} in $\RR^m$ such that:
  \begin{enumerate}[(i)]
    \item If $\sigma \in K$ and $\tau \face
      \sigma$, then $\tau \in K$.
    \item If $\sigma, \tau \in K$, then $\sigma
      \cap \tau = \emptyset$ or $\sigma \cap \tau$
      is a \gls{face} of $\sigma$ and of $\tau$.
  \end{enumerate}}
\end{definition*}
\end{flashcard}

\begin{example*}
  \phantom{}
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/0536688fee704ce0.png}
  \end{center}
\end{example*}

\begin{flashcard}[dimension-defn]
\begin{definition*}[Dimension of a simplicial
  complex]
  \glsadjdefn{scdim}{dimension}{\gls{simpcomp}}
  \cloze{The \emph{dimension} of a \gls{simpcomp} $K$ is
  the largest $p$ such that $K$ contains a
  \nsimp{p}.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[polyhedron-of-simpcomp-defn]
\begin{definition*}[Polyhedron of a simplicial
  complex]
  \glsnoundefn{polyh}{polyhedron}{polyhedra}
  \glssymboldefn{polyh}{$|K|$}{$|K|$}
  \cloze{The \emph{polyhedron} of $K$ is the space
  \[ |K| = \bigcup_{\sigma \in K} \sigma \subseteq
  \RR^m .\]}
\end{definition*}
\end{flashcard}

\begin{remark*}
  $\polyh|K|$ is compact and Hausdorff.
\end{remark*}

\begin{flashcard}[d-skeleton-defn]
\begin{definition*}[$d$-skeleton of a simplicial
  complex]
  \glssymboldefn{dskel}{$K_{(d)}$}{$K_{(d)}$}
  \cloze{The \emph{$d$-skeleton} $K_{(d)}$ of $K$ is the
  sub-\gls{simpcomp} containing all \glspl{simp}
  of $K$ of \gls{scdim} $\le d$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[triangulation-defn]
\begin{definition*}[Triangulation]
  \glsnoundefn{triang}{triangulation}{triangulations}
  \cloze{A \emph{triangulation} of a space $X$ is a
  \gls{gsimpcomp} $K$ and a homeomorphism $h :
  \polyh|K| \homeoto X$.}
\end{definition*}
\end{flashcard}

\begin{example*}
  \glssymboldefn{ssimp}{$\Delta$}{$\Delta$}
  The \emph{standard \nsimp{n}} is $\Delta^n =
  \simplex<e_1, \ldots, e_{n + 1}> \subseteq
  \RR^{n + 1}$. It, along with its \glspl{face},
  defines a \gls{simpcomp}.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/94c2e8334ae546be.png}
  \end{center}
\end{example*}

\begin{example*}
  The \emph{simplicial $(n - 1)$-sphere} is the
  \gls{simpcomp} given by the proper \glspl{face}
  of $\ssimp^n$. Its \gls{polyh} is $\bound
  \ssimp^n \subseteq \RR^{n + 1}$.
\end{example*}

\begin{example*}
  In $\RR^{n + 1}$ consider the $2^{n + !}$
  \glspl{simp} given by $\simplex<\pm e_1, \pm
  e_2, \ldots, \pm e_{n + 1}>$ and let $K$ be
  given by these and all their faces.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/d5d6d19e12ae417e.png}
  \end{center}
  Define
  \begin{align*}
    h : \RR{n + 1} \setminus \{0\} \supseteq \polyh|K|
    &\to S^n \\
    x
    &\mapsto \frac{x}{|x|}
  \end{align*}
  this is continuous, and a bijection. As
  $\polyh|K|$ and $S^n$ are compact Hausdorff, it
  is a homeomorphism.
\end{example*}

\begin{flashcard}[simplicial-map-defn]
\begin{definition*}[Simplicial map]
  \glsnoundefn{simpmap}{simplicial map}{simplicial
  maps}
  \glssymboldefn{VK}{$V_K$}{$V_K$}
  \cloze{
  Write $V_K$ for the set of \glspl{vert} (i.e.
  the $0$-simplces) of $K$.

  A \emph{simplicial map} $f$ from $K$ to $L$ is a
  frunction $f : V_K \to V_L$ such that if $\sigma
  = \simplex<a_0, \ldots, a_n> \in K$ then
  $\{f(a_0), \ldots, f(a_n)\}$ spans a \gls{simp}
  of $L$, called $f(\sigma)$. Write $f : K \to
  L$.}
\end{definition*}
\end{flashcard}

\begin{example*}
  The map
  \begin{align*}
    f : \ssimp^1
    &\to \ssimp^2 \\
    (1, 0)
    &\mapsto (1, 0, 0) \\
    (0, 1)
    &\mapsto (0, 1, 0)
  \end{align*}
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/0c83c74143dc4f9b.png}
  \end{center}
\end{example*}

\begin{example*}
  The map
  \begin{align*}
    g : \ssimp^2
    &\to \ssimp^1 \\
    (1, 0, 0)
    &\mapsto (1, 0) \\
    (0, 1, 0)
    &\mapsto (1, 0) \\
    (0, 0, 1)
    &\mapsto (0, 1)
  \end{align*}
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/082ce9b3f7834661.png}
  \end{center}
\end{example*}
