
\documentclass[12pt ]{article}
\setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true
cm}

\usepackage{hangul}
\usepackage{amscd,amsmath}
\usepackage{amsfonts}
\usepackage{amssymb,theorem}
\usepackage{longtable}
\usepackage{floatflt}
\usepackage{texdraw, epic, pstcol ,pstricks ,pst-3d, pst-poly,pst-grad,pst-node, pst-text}
\newcommand{\Aff}{\mbox{\it Aff}}
\newcommand{\aff}{\mbox{\it aff}}


\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\del}{\delta}
\newcommand{\gam}{\gamma}
\newcommand{\vep}{\varepsilon}
\newcommand{\eps}{\epsilon}
\newcommand{\lam}{\lambda}
\newcommand{\kap}{\kappa}
\newcommand{\sig}{\sigma}
\newcommand{\ome}{\omega}
\newcommand{\Gam}{\Gamma}
\newcommand{\Ome}{\Omega}
\newcommand{\Sig}{\Sigma}
\newcommand{\Del}{\Delta}
\newcommand{\Lam}{\Lambda}
\newcommand{\dis}{\displaystyle}
\newcommand{\disi}{\displaystyle{\sum_{i=1}^n}}
\newcommand{\disj}{\displaystyle{\sum_{j=1}^m}}
\newcommand{\pa}{\partial}
\newcommand{\cb}{\mathbb{C}}
\newcommand{\cc}{\mathcal{C}}
\newcommand{\ccn}{\mathcal{C}^{\infty}}
\newcommand{\sbb}{\mathbb{S}}
\newcommand{\rb}{\mathbb{R}}
\newcommand{\zb}{\mathbb{Z}}
\newcommand{\tx}{(\widetilde{X}, \widetilde{x}_0)}
\newcommand{\x}{(X,x_0)}
\newcommand{\pnx}{\pi_n(X,x_0)}
\newcommand{\xt}{\widetilde{x}}
\newcommand{\yt}{\widetilde{y}}
\newcommand{\Xt}{\widetilde{X}}

\newtheorem{thm}{����}
\newtheorem{cor}[thm]{��������}
\newtheorem{lem}[thm]{��������}
\newtheorem{prop}[thm]{����}
\newtheorem{cl}{Claim}


{\theorembodyfont{\rm}
\newtheorem{ex}{��}
\newtheorem{que}{����}
\newtheorem{notation}{Notation}[section]
\newtheorem{defn}{����}
\newtheorem{rem}{��}
\newtheorem{note}{Note}
}
\renewcommand{\thenote}{}
\renewcommand{\therem}{}

\newenvironment{proof}{{\bf ����}}{\hfill\framebox[2mm]{}}
\newenvironment{proof1}{{\bf ��������}}{\hfill\framebox[2mm]{}}

\begin{document}
\parindent=0cm
\section*{V.3 Simplicial approximation theorem}

% baricenter %

\begin{floatingfigure}[r]{4.5cm}
\psset{unit=1.5cm}
\begin{pspicture}(-0.2,-0.2)(2,2)%
\pspolygon(0,0)(2,0)(1,1.732)
\psline(0,0)(1.5,0.866)\psline(2,0)(.5,0.866)\psline(1,0)(1,1.732)
\rput(-0.2,-0.2){$a_1$} \rput(1,1.9){$a_0$} \rput(2.2,-0.2){$a_2$}
\rput(1.95,1.6){$\widehat{\sig}$} \psline{->}(1.8,1.4)(1.02,0.62)
\psdots[dotscale=1](0,0)(2,0)(1,1.732)
\end{pspicture}
\end{floatingfigure}
{\bf Barycentric subdivision}\\
Given $\sig =<a_0, \cdots, a_n>\subset \rb^N$, the
\emph{\textbf{barycenter}} $\widehat{\sig}$ of $\sig$ is defined
by
\[
\widehat{\sig}= \displaystyle{\frac{1}{n+1} \sum^n_{i=0}} a_i.
\]
\vspace{1em}
\begin{defn}
Let $K$ be a simplicial complex in $\rb^N$, the
\emph{\textbf{baricentric subdivision}} of $K$, denoted by
$ sd K$, is defined inductively as follows\\
1. $L_0:=V(K)$.\\
2. Given $L_p$, $L_{p+1}$ is the simplicial complex determined by
$ \displaystyle{\bigcup_{\sig\in
K^{p+1}}}\{\widehat{\sig}*\partial\sig \}$\\
\hspace*{1em} where $\partial\sig$ is viewed as a subcomplex of
$L_p$. \\
3. $sdK=\bigcup L_p$.
\end{defn}

\begin{center}
\psset{unit=1.3cm}
\begin{pspicture}(0,-0.4)(2.5,2)%
\pspolygon[fillstyle=solid,fillcolor=lightgray](0,0)(2,0)(1,1.732)
\psline(2,0)(2.5,1)
\psdots[dotscale=1.5](0,0)(2,0)(1,1.732)(2.5,1)
\rput(1.3,-0.4){$K$}
\end{pspicture}

\vspace{2em}
\begin{pspicture}(0,-0.4)(2.5,2)%
\psdots[dotscale=1.5](0,0)(2,0)(1,1.732)(2.5,1)
\rput(1.3,-0.4){$L_0$}
\end{pspicture}
\hspace{1.5em}
\begin{pspicture}(0,-0.4)(2.5,2)%
\pspolygon(0,0)(2,0)(1,1.732)
\psdots[dotscale=1.5](0,0)(2,0)(1,1.732)(1,0)(.5,0.866)(1.5,0.866)(2.5,1)(2.25,0.5)
\psline(2,0)(2.5,1) \rput(1.3,-0.4){$L_1$}
\end{pspicture}
\hspace{1.5em}
\begin{pspicture}(0,-0.4)(2.5,2)%
\pspolygon[fillstyle=solid,fillcolor=lightgray](0,0)(2,0)(1,1.732)
\psdots[dotscale=1.5](0,0)(2,0)(1,1.732)(1,0)(.5,0.866)(1.5,0.866)(2.5,1)(2.25,0.5)(1,0.577)
\psline(2,0)(2.5,1)
\psline(0,0)(1.5,0.866)\psline(2,0)(.5,0.866)\psline(1,0)(1,1.732)
\rput(1.3,-0.4){$L_2$}
\end{pspicture}
\end{center}

\vspace{2em}
{\bf Note.} Abstractly,\\
1. $V(sdK)= \displaystyle{\bigcup_{\sig\in K}}\{\widehat{\sig}\}$\\
2. $sd K= \{<\widehat{\sig}_1,\cdots,\widehat{\sig}_p>  |
~\sig_1 <\cdots <\sig_p, p=1,2,\cdots, \sig_i\in K \}$,\\
\hspace*{1em} where $<$ denotes proper face. \\
3. $|sdK|=|K|$ as a topological space.

\begin{defn}
$K$ : a finite simplicial complex in $\rb^N$. \\
\hspace*{3.2em}$mesh(K) = max\{diam(\sig)~|~ \sig\in K\}$.
\end{defn}

\begin{thm}
\hfill\\
(1) $\sig$ : $n$-simplex in $\rb^N$ $\Rightarrow
mesh(sd\sig)\leq \frac{n}{n+1}mesh(\sig)$.\\
(2) $K$ : $n$-dimensional finite simplicial complex in $\rb^N$\\
\hspace*{2em} $\Rightarrow mesh(sdK)\leq \frac{n}{n+1}mesh(K)$
\end{thm}
\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_0|=|~v_0-\displaystyle{\frac{1}{n+1}
\sum_{i=0}^n} v_i| = |~\displaystyle{\frac{1}{n+1} \sum_{i=1}^n
(v_i-v_0)} | \leq \displaystyle{\frac{1}{n+1} \sum_{i=1}^n
|~v_i-v_0|} $\\
\hspace*{3.8em}$ \leq \displaystyle{ \frac{n}{n+1}max |~v_i-v_0|}
$ $ \displaystyle{\leq\frac{n}{n+1}mesh(\sigma)
} $\\

���� (1)�� ������ ����. $\forall\,\tau \in sd(\sigma)$�� ����
$\tau$�� ��� 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 converge to $0$ as
$N\rightarrow \infty$.\\
\end{cor}


{\bf Note.} $g:K \rightarrow L$, a simplicial map \\
\hspace*{3em}$\Rightarrow g(st(v))\subset st(g(v))$, $\forall v
\in V(K)$.\\
\begin{proof} $x\in st(v)\Leftrightarrow t_v(x)>0$.\\
$g$�� simplicial map�̹Ƿ� $t_{g(v)}(g(x))\geq t_v(x)$�̰� ����
$x\in st(v)$�̸� $ t_{g(v)}(g(x))>0$�̰� $g(x)\in st(g(v))$.
\end{proof}\\

\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) $v\in V(K),f(st(v))\subset st(g(v))$.
\end{prop}
\begin{defn}
Such simplicial map $g$ is called a \emph{\textbf{Simplicial
approximation}} of $f$.
\end{defn}

\begin{proof}
(2)$\Rightarrow $(1)�� �翬�ϴ�. \\
(1)$\Rightarrow$(3) $x\in st(v)$,$f(x)\in\overset{\circ}\tau$ ��
����. �������κ��� $t_v(x)>0,\,\,g(x)\in\tau$�̴�. �̶�,
$t_{g(v)}(g(x))\geq t_v(x)>0$�̹Ƿ�, $g(x)\in st(g(v))$�̴�.
�׷��� $g(x)\in \tau$���� �˰� �����Ƿ�, $g(v)$�� $\tau$��
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�. ������
$v\in V(\sigma)$�� ���Ͽ� �����ϹǷ�, $g(\sigma)\subset \tau$ �̰�
$g(x)\in \tau$�̴�.
\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 there 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
$\sigma=<v_0,\cdot\cdot\cdot,v_k>$�� ������
$x\in\overset{\circ}\sigma$�� �����ϰ�,
$x\in\displaystyle{\bigcap_{i=0}^k}st(v_i)$�̴�. \\�� ��, $f(x)\in
\displaystyle{f(\bigcap st(v_i))\subset \bigcap f(st(v_i))\subset
\bigcap st(w_i)}\,,\,w_i=g(v_i)$�� �Ǿ� ��� $i$�� ����
$t_{w_i}(f(x))>0$�̴�. ����
$<w_0,\cdot\cdot\cdot,w_k>=<g(v_0),\cdot\cdot\cdot,g(v_k))>$��
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 $g$�� ����
�� �ִ�.

\begin{thm}[Simplicial approximation theorem]
\hfill\\
$K,L$ : finite simplicial complexes in $\rb^N$. \\
(1) Given a map $f:|K|\rightarrow |L|$, $\exists N>0$ such that
$f$ has a simplicial approximation $g:sd^N K\rightarrow L$.\\
(2) If $g$ is a simplicial approximation of $f$, $f\simeq g$.
\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�� ���� simplicial approximation $g$�� �����Ѵ�.

(2) $f$�� $g$������  homotopy�� $F:|K|\times I \rightarrow |L|$,
$F(x,t)=tf(x)+(1-t)g(x)$�� �����Ѵ�. ���� ���� 3�� ���� ��
$f(x)$�� $g(x)$�� ���� simplex �ȿ� �����Ƿ� $F$�� �� ���ǵȴ�.
���� �� simplex $\sig$���� $f$�� $g$�� �����̹Ƿ� $F$��
$|~\sig|\times I$���� �����̴�. $K$�� finite�̹Ƿ� $F$�� �����̴�.
���� $F$�� ���ϴ� homotopy�̴�.
\end{proof}\\


{\bf Remark.} \\
1. Theorem 5 holds for arbitrary simplicial complexes $K$ and $L$,
and for a suitable subdivision (satisfying mesh condition in the
proof) $K'$ of $K$.\\
(Reference: Munkres, 16.5, 19.4, 20.5)\\
2. In general, The topology of $|K|\times I$ is coherent with the
subspaces\\ $\{|\sigma|\times I \,\,|\,\,\sigma \in K\}$. (see
Munkres, Elements of algebraic topology.) ({\bf ���� 16.})



\end{document}
