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The \emph{\glspl{vert}} of $K \subdiv$ are in
bijection with the \glspl{simp} of $K$. Choose a
function $K \to \vset K$ which assigns to $\sigma$
\emph{some} \gls{vert} of $\sigma$. So
\begin{align*}
  g : \vset{K \subdiv} \cong K
  &\to \vset K \\
  \bcentre{\sigma}
  (\leftrightarrow \sigma)
  &\mapsto v_\sigma
\end{align*}
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/90577b2a531749b6.png}
\end{center}
If $\simplex<\bcentre{\sigma_0}, \ldots,
\bcentre{\sigma_p}>$ is a \gls{simp} of $K
\subdiv$ then $\sigma_0 \face \sigma_1 \face
\cdots \face \sigma_p$ and $g(\bcentre{\sigma_i})$
is some \gls{vert} of $\sigma$, so a \gls{vert} of
$\sigma_p$. The $\{g(\bcentre{\sigma_i})\}$ thus
spans a \gls{face} of $\sigma$, so is a \gls{simp}
of $K$ if and only if $g$ is a \gls{simpmap}.

Also, if $\bcentre{\sigma} \in \tau' =
\simplex<\bcentre{\tau_0}, \ldots,
\bcentre{\tau_p}> \in K \subdiv$, then
$\bcentre{\sigma} \in \tau_p$, so $\sigma$ is a
\gls{face} of $\tau_p$. So $v_\sigma \in \sigma
\subseteq \tau_p$. So $\interior{\tau\subdiv}
\subset \interior{\tau_p} \subseteq
\St_K(v_\sigma)$. Thus
\[ \St_{K\subdiv}(\bcentre{\sigma}) \subset
\St_K(v_\sigma = g(\sigma)) \]
so $g$ is a \gls{simpapprox} to $\id :
\polyh|K\subdiv| \to \polyh|K|$. So $\smapc|g|
\homotopic \id$.

\begin{flashcard}[mesh-of-simpcomp-defn]
\begin{definition*}[Mesh]
  \glsnoundefn{mesh}{mesh}{meshes}
  \glssymboldefn{mesh}{$\mu(K)$}{$\mu(K)$}
  \cloze{The \emph{mesh} of $K$ is
  \[ \mu(K) = \max\{|v_0 - v_1| \st \simplex<v_0,
  v_1> \in K\} .\]}
\end{definition*}
\end{flashcard}

\begin{flashcard}[mesh-subdiv-ineq-lemma]
\begin{lemma*}
  Suppose $K$ has \gls{scdim} $\le n$, then
  $\mesh(K\rsubdiv{r}) \le \cloze{\left( \frac{n}{n + 1}
  \right)^r \mesh(K)}$, so $\mesh(K\rsubdiv{r}) \to
  0$ as $r \to \infty$.
\end{lemma*}

\begin{proof}
  \cloze{Enough to treat the case $r = 1$.

  Let $\simplex<\bcentre{\tau}, \bcentre{\sigma}>
  \in K \subdiv$, so $\tau \face \sigma$ in $K$.
  \[ |\bcentre{\tau} - \bcentre{\sigma}| \le
  \max\{|v - \bcentre{\sigma}| \st \text{$v$ is a
  \gls{vert} of $\sigma$}\} .\]
  \begin{center}
    \includegraphics[width=0.4\linewidth]{images/060bbe1d378049d1.png}
  \end{center}
  Let $\sigma = \simplex<v_0, v_1, \ldots, v_m>$
  with $m \le n$, and reorder so that the maximum
  is attained at $v = v_0$.
  \begin{align*}
    |v_0 - \bcentre{\sigma}|
    &= \left| v_0 - \frac{1}{m + 1} \sum_{i = 0}^m
    v_i \right| \\
    &= \left| \frac{m + 1}{m + 1} v_0 - \frac{1}{m
    + 1} \sum_{i = 0}^m v_i \right| \\
    &= \frac{1}{m + 1} \left| \sum_{i = 0}^m v_0 -
    v_i\right| \\
    &\le \frac{1}{m + 1} \sum_{i = 1}^m |v_0 -
    v_i| \\
    &\le \frac{m}{m + 1} \mesh(K) \\
    &\le \frac{n}{n + 1} \mesh(K) \qedhere
  \end{align*}}
\end{proof}
\end{flashcard}

\begin{flashcard}[simpapprox-thm]
\begin{theorem*}[Simplicial Approximation Theorem]
  \cloze{Let $f : \polyh|K| \to \polyh|L|$ be a
  continuous map. Then there is a $r \gg 0$ and a
  \gls{simpmap} $g : K\rsubdiv{r} \to L$ such that
  $g$ is a \gls{simpapprox} to $f$.

  If $f$ is \glsref[simpmap]{simplicial} on some
  $\polyh|N| \subset \polyh|K|$, can take
  $g|_{\vset{N}} = f|_{\vset{N}}$.}
\end{theorem*}

\begin{proof}
  \cloze{The $\St_L(\omega)$, $\omega \in \vset L$ is an
  open cover of $\polyh|L|$, so
  \[ \{f^{-1} \St_K(\omega)\}_{\omega \in \vset L} \]
  is an open cover of $\polyh|K|$; pick $\delta >
  0$ using \nameref{lebesgue_number_lemma} for
  this cover. Choose $r \gg 0$ such that
  $\mesh(K\rsubdiv{r}) < \delta$. For each $v \in
  \vset{K\rsubdiv{r}}$ have
  \[ \St_{K\rsubdiv{r}}(v) \subseteq
  B_{\mesh(K\rsubdiv{r})}(v) \subseteq
  f^{-1}(\St_L(\omega)) \]
  for some $w \in \vset L$. Define $g :
  \vset{K\rsubdiv{r}} \to \vset L$ by $g(v) = w$.
  Then
  \[ f(\St_{K\rsubdiv{r}}(v)) \subseteq
  \St_L(g(v)) .\]
  So $g$ is a \gls{simpapprox} to $f$. So $g$ is a
  \gls{simpmap}.

  The final step is by choosing $w$
  carefully when $v \in \vset N$.}
\end{proof}
\end{flashcard}

\begin{flashcard}[Sn-Sm-homotopic-constant-coro]
\begin{corollary*}
  If $\cloze{n < m}$, then any \gls{map} $f : S^n \to S^m$
  is \cloze{\gls{homotopic} to a constant map.}
\end{corollary*}

\begin{proof}
  \cloze{Spheres are polyhedra: $S^n = \polyh|K|$, $S^n
  = \polyh|L|$, then $f$ is \gls{homotopic} to
  $\smapc|g|$, for some $g : K\rsubdiv{r} \to L$.
  This can not hit any \nsimp{m} of $L$, as $K$
  has $\dim \le n$. So $\smapc|g|$ must miss every
  point on the \gls{int} of some \nsimp{m}: it is
  not onto. Thus it factors through $(S^n -
  \{*\}) \homoteq *$, so is \gls{homotopic} to a
  constant map.}
\end{proof}
\end{flashcard}
