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\begin{document}
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\section*{Simplicial approximation theorem}

{\bf Barycentric subdivision, $sd K$}

$\sigma=<a_0,\cdot\cdot\cdot,a_n>\subset \mathbf{R}^n$ ��
�־����ٰ� ����. �� ��, $\sigma$�� barycenter ��

$\hspace{7em}\hat{\sigma}=\displaystyle{\frac{1}{n+1}}\sum_{i=0}^{n}a_i$

�� ��������. �׷���, $\mathbf{R}^n$�� simplicial complex $K$��
���� barycentric subdivision $sd K$�� ������ ���� ���ǵȴ�.\\

$V(sdK)= \{{\hat{\sigma}}\,|\,\sigma\in K\}$
�̰�,
$sd(K)=\{\{\hat{\sigma_1},\cdot\cdot\cdot, \hat{\sigma_p}
\}\,\,|\,\,\sigma_1\lneq\cdot\cdot\cdot\lneq
\sigma_p\,,\,\,\sigma_i\in K,p=1,2,\cdot\cdot\cdot \}$.\\

{\bf Note.} $|sdK|=|K|$.


\begin{defn}\textit{(mesh(K))}
$K$�� $\mathbf{R}^N$ �� finite simplicial complex �� ��

$mesh(K):=max\{diam(\sigma)\,\,|\,\,\sigma\in K\}$.


\end{defn}

\begin{prop}

$\hspace{20em}\\$ (1) $\sigma$ �� $\mathbf{R}^N$�� n���� simplex��
��, $mesh(sd(\sigma))\leq\frac{n}{n+1} mesh(\sigma)\,\,$ �̴�.

(2) $K\,$�� $\mathbf{R}^N$�� n���� finite simplicial complex�� ��,

$mesh(sd(K))\leq\frac{n}{n+1} mesh(K)\,\,$ �̴�.

\end{prop}

\begin{proof}
(2)�� (1)�� ����� $\frac{x}{x+1}$�� �����Լ���� ��Ƿκ��� �ٷ�
�����Ƿ�, (1)�� ��������. �̸� �����ϱ⿡ �ռ� ���� Note��
���캸��.

{\bf Note.} 1.$\forall\,\, \sigma,\,\, \exists\,$ an edge $
e<\sigma\,$ such that $\,diam(\sigma)=length(e).$

2.$\forall \,\,x\in\sigma,\,\,|\hat{\sigma}-x|\leq
|\hat{\sigma}-v|$ for some vertex $v$ of $\sigma$, and
$|\hat{\sigma}-v|\leq \frac{n}{n+1}mesh(\sigma)$.

(����)$|\hat{\sigma}-v_0|=|\displaystyle{\frac{1}{n+1}
\sum_{i=0}^n} v_i - v_0|\\=
| \frac{1}{n+1}\displaystyle{\sum_{i=0}^n (v_i-v_0)} |$\\
=$ |\displaystyle{\frac{1}{n+1} \sum_{i=1}^n (v_i-v_0)} | $

$\displaystyle{ \leq \frac{1}{n+1} \sum_{i=1}^n |v_i-v_0|} $

$\displaystyle{\leq \frac{n}{n+1}max_i|v_i-v_0|} $ $
\displaystyle{\leq\frac{n}{n+1}mesh(\sigma)
} $ �� �ȴ�.\\

���� (1)�� ������ �ϸ�, $\forall\,\tau \in sd(\sigma)$�� ����
$\tau$�� ��� edge $e$�� barycenter���� Ȥ�� face�� barycenter����
�����Ƿ� Note 2�� ���� $length(e)\leq \frac{n}{n+1}mesh(\sigma)$
�̴�. ���� Note 1�� ����
$mesh(\tau)\leq\frac{n}{n+1}mesh(\sigma)$ �̴�.
\end{proof}

\begin{cor}

$mesh(sd^N K)\leq C(\frac{n}{n+1})^N$ and converges to 0 if
$N\rightarrow \infty$.
\end{cor}


{\bf Note.} $g:|K|\rightarrow |L|$�� simplicial map �̸�,
$\forall\,\,v\in V(K)$ �� ����


$\hspace{3em}g(st(v))\subset st(g(v))$ �̴�.

(����)$x\in st(v)\Leftrightarrow t_v(x)>0$.

$g$�� simplicial map�̹Ƿ� $t_v(x) \leq t_{g(v)}(g(x))$ �̰� ����

$ t_{g(v)}(g(x))>0 \Rightarrow g(x)\in st(g(v)).$

\begin{prop}
Let $f:|K|\rightarrow |L|$ be a map and $g:|K|\rightarrow |L|$ be
a simplicial map. Then the followings are equivalent.

(1) $\forall\,x\in|K|$, $f(x)\in \overset{\circ}\tau \Rightarrow
g(x) \in \tau$.

(2) $\forall\,x\in|K|$, $f(x)\in \tau \Rightarrow g(x) \in \tau$.

(3) $\forall\,v\in V(K),f(st(v))\subset st(g(v))$.

�� ��� $g$�� $f$�� simplicial approximation �̶�� �Ѵ�.
\end{prop}

\begin{proof}
(2)$\Rightarrow $(1) �� �翬�ϴ�.

(1)$\Rightarrow$(3) $x\in st(v)$,$f(x)\in\overset{\circ}\tau$ ��
����. �׷���

$t_v(x)>0,\,\,g(x)\in\tau$

$\Rightarrow t_{g(v)}(g(x))>0,\,\,g(x)\in\tau$

$\Rightarrow g(x)\in st(g(v))\,\,$ and $\,\,g(x)\in \tau$

�� $g(v)$�� $\tau$�� vertex�̰�, $\overset{\circ}\tau\subset
st(g(v))$�̴�. ���� $f(x)\in\overset{\circ}\tau\subset
st(g(v))$.

(3)$\Rightarrow $(2) $x\in\overset{\circ}\sigma$ �̰�,
$f(x)\in\tau$��� ��������. �� ��, ������ $v\in V(\sigma)$ �� ����

$x\in st(v)$�̴�. $f(st(v))\subset st(g(v))$ �κ���$ f(x)\in
st(g(v))$ �̰� ���� $g(v)$�� $\tau$�� vertex�� �ȴ�. ����
$g(\sigma)\subset \tau$ �̰� $g(x)\in \tau$�̴�.

\end{proof}

\begin{thm}
Let $f:|K|\rightarrow |L|$ be a map which satisfies "star
condition", i.e.,

$\forall\,\,v\in V(K),\,\,\exists\,\,w\in V(L)$ such that
$f(st(v))\subset st(w)$. Then

$\,\,\exists\,\,g:K\rightarrow L$ which is a simplicial
approximation of $f$.
\end{thm}

\begin{proof}
��� $v\in V(K)$�� ����  $f(st(v))\subset st(w)$�� �����ϴ� �ƹ���
$w$ �� �����Ͽ�  $g(v)=w$�� ��������. ���� �� $g$�� simplicial
map���� ���̱� ����

$<v_0,\cdot\cdot\cdot,v_k>$ �� simplex �̸�,
$<g(v_0),\cdot\cdot\cdot,g(v_k)>$�� simplex ���� ������. simplex
$<v_0,\cdot\cdot\cdot,v_k>$�� $\sigma$�� ������
$\overset{\circ}\sigma$�� ���� $x\in\overset{\circ}\sigma$ ��
�����ϰ�, �� $x$�� ����
$x\in\displaystyle{\bigcap_{i=0}^n}st(v_i)$ �̴�. �� ��,

$f(x)\in \displaystyle{f(\bigcap st(v_i))\subset \bigcap
f(st(v_i))\subset \bigcap st(w_i)}\,,\,w_i=g(v_i)$ �� �Ǿ� ���
$i$�� ���� $f(x)\subset st(w_i)$ �̹Ƿ�
$t_{w_i}(f(x))>0,\,\forall\,\,i$ �̴�. ����

$<w_0,\cdot\cdot\cdot,w_k>=<g(v_0),\cdot\cdot\cdot,g(v_k))>$ ��
simplex�� �����Ѵ�.(interior point $f(x)$�� �����ϹǷ�). ����
$g$�� simplicial map�� �ǰ� $f(st(v))\subset st(g(v))$ ��
�����ϹǷ� ���� ���� 3�� (3)�� �����Ͽ� $g$�� $f$�� simplicial
approximation�� �ȴ�.
\end{proof}\\

{\bf Remark.} $f:|K|\rightarrow |L|$�� $K$�� subcomplex $M$����
�̹� simplicial map�̶�� ����. �׷��� �� ������ ������������
$g$�� ���� ��, $M$������ ���� �״�� �ְ� (�� Note���� simplicial
map�� star condition�� �����ϹǷ� ) ������ �κи� ������ ����ó��
�ϸ� �ǹǷ� $g|_{|M|}=f|_{|M|}$ �� �ǵ��� approximation
��ų �� �ִ�.\\

\begin{thm}(Simplicial approximation theorem)

K,L �� finite simplicial complex ���� ��,

(1) �־��� map $f:|K|\rightarrow |L|$ �� ���� � $N$ �� �����ؼ�
$f$�� simplicial approximation $g:sd^N \,K\rightarrow L$ �� ����
�� �ִ�.

(2) $g$�� $f$�� simplicial approximation�̸�, $f\simeq g$ �̴�.
\end{thm}

\begin{proof}

(1) $\mathcal{U}=\{f^{-1}(st(w))\,|\,w\in V(L)\}$ �� $|K|$�� open
covering �� �ȴ�. $K$�� finite�̹Ƿ� $|K|$�� compact�̰� ����
open covering $\mathcal{U}$�� ���� Lebesgue number $\eps$��
�����Ѵ�. �� ��, $mesh(sd^N K)<\frac{\eps}{2}$ �� �����ϵ��� $N$��
����� ũ�� ������ $sd^N K$ ���� �� star���� diam�� $\eps$����
�۰� ���� � $U \in \mathcal{U} $�� ���Եȴ�. �׷��� $f:|sd^N
K|\rightarrow |L|$ �� star condition�� �����ϰ� $|sd^N K|=|K|$
�̹Ƿ� ���� 4�� ���� �츮�� ���ϴ� $g$ �� �����Ѵ�.

(2) $g$�� $f$�� simplicial approximation�̶�� ����. ���� $L$��
finite�̹Ƿ� $\mathbf{R}^N$�� embedded�Ǿ� �ִٰ� �����ص� ����.
�� �� $F:|K|\times I \rightarrow |L|$, $F(x,t)=tf(x)+(1-t)g(x)$ ��
$f$�� $g$������  homotopy�� ���� ������. ���� �� $f(x)$�� $g(x)$��
���� simplex�� ���� �����Ƿ� $F$�� �� ���ǵȴ�. ���� $f$�� $g$��
�����̹Ƿ� $F$ ���� �����̴�. ���� $F$�� ���ϴ� homotopy�̴�.\\

{\bf Remark.} ���� 5���� �� $sd^N K$ ���� �� �ʿ�� ����. ��������
$mesh$ ������ �����ϴ� � subdivision�� ���ؼ���
�����Ѵ�.

���� K�� L�� finite�� �ƴ϶� �����Ѵ�.(See Munkres
16.5,19.4,20.5.)
�̸� ���̱� ���� ���� ����� �ʿ��ϴ�.\\

{\bf Fact.} $F:|K| \times I \rightarrow |L|$ is continuous

$\Leftrightarrow \,\,F:|\sigma|\times I\rightarrow |L|$ is
continuous.

(�̰��� ������ ���� �����κ��� �ڸ��ϴ�.)\\

{\bf ���� 16.} \\The topology of $|K|\times I$ is coherent with
the subspaces $\{|\sigma|\times I \,\,|\,\,\sigma \in K\}$.



\end{proof}


\end{document}
