
\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}
\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}


\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*{Morphisms of Covering Space}

  \begin{defn}
  \textit{ Let $p_i:\widetilde{X}_i\rightarrow X, i=1,2 $ be  covering
  maps.\\A morphism of $\widetilde{X}_1$ to $\widetilde{X}_2$ is a
  map $\phi:\widetilde{X}_1\rightarrow \widetilde{X}_2$ such that
  the diagram }

  $\hspace{5.5em}\phi$

  $\hspace{3em}\widetilde{X}_1\hspace{1em}\rightarrow \hspace{1em}\widetilde{X}_2$

  $\hspace{3em}p_1\searrow \hspace{2em} \swarrow p_2 \hspace{5em}$ \textit{commutes.}

  $\hspace{5.5em}X$
  \end{defn}

��, �� diagram �� commute�Ѵٴ� ���� fiber�� fiber�� �����ٴ�
���� �ȴ�. ���� ���� morphism�� �������� �� 1-1,onto��  morphism
�� isomorphism�̶�� �ϰ�, Ư�� �ڱ��ڽ����� ���� isomorphism��
deck transformation (or covering transformation) �̶�� �Ѵ�.

deck transformation�� ���� covering $p:\textbf{R}^1\rightarrow S^1$ ����\\ $\tau:\textbf{R}\rightarrow \textbf{R}
\,\,$,$\,\,\tau(x)=x+1$�� ������ ����.

$\hspace{5.5em}\tau$

  $\hspace{3em}\textbf{R}\hspace{1em}\rightarrow \hspace{1em}\textbf{R}$

  $\hspace{3em}p\searrow \hspace{1.5em} \swarrow p \hspace{5em}$

  $\hspace{5.3em}S^1$


�׷���, $\tau$�� ���ϱ� ���� $x$�� $p$�� ���� ���̳� $\tau$��
���� �� $p$�� ���� ���̳� ������ �� �� �ִ�. ��, $p\circ \tau=p$
�̹Ƿ� $\tau$�� deck transformation�� �ȴ�.

�� �ٸ� ���� $p:S^n\rightarrow P^n$ ���� �� $[x]$��
$p^{-1}$�̹����� $x,-x$ �� �ٲٴ� map, �� antipodal map $A$��
�����غ���. ���� ����������
 $ p(A(x))=p(-x)=p(x)$ �̹Ƿ�  $A$�� deck transformation�� �ȴ�.

\begin{thm}
Let $p_i:\widetilde{X}_i\rightarrow X, i=1,2 $ be  covering
  maps. Then

(1) $\exists$ a morphism
$\phi:(\widetilde{X}_1,\widetilde{x}_1)\rightarrow
(\widetilde{X}_2,\widetilde{x}_2)\Leftrightarrow$
$p_{1\sharp}\pi_1(\widetilde{X}_1,\widetilde{x}_1)\subset
p_{2\sharp}\pi_1(\widetilde{X}_2,\widetilde{x}_2)$. \\Such a
morphism
    is unique if it exists.

(2) A morphism $\phi$ is a covering map.
\end{thm}

\begin{proof}\\
(1)�� ������ ������ �ߴ� general lifting theorem ������ ����, uniqueness ���� ������ �ߴ� �Ͱ� ���������̴�.

(2)�� ������ $\phi$�� onto�̰�  evenly cover�Ǵ� �ٹ��� �����ٴ� ���� ���̸� �ȴ�.

���� $\phi$�� onto���� ���̱� ���� ������ $y\in \widetilde{X}_2$��
���� $\widetilde{x}_2$���� $y$ �� ���� path $\rho$�� ����. ��
$\rho$ �� $p_2$ �� ���� $X$�� ���� �� $\widetilde{X}_1$����
lifting�� ��Ų ���� $\widetilde{p_2\circ \rho}$�� ����.�׷���
$\phi$�� morphism�̶�� ��Ƿκ��� $\phi(\widetilde{p_2\circ
\rho})$�� $\widetilde{X}_2$������ $p_2\circ \rho$ �� lifting��
�ȴ�. �׷���, $\rho$ ���� $\widetilde{X}_2$������ $p_2\circ \rho$
�� lifting�̹Ƿ� uniqueness�� ���� ���� ����. ��,
$\phi(\widetilde{p_2\circ \rho})=\rho$ �̹Ƿ�, ������ ����.
���� $\phi$�� onto �̴�.

��������, $\phi$�� evenly cover�Ǵ� �ٹ��� ������ ������.

������ $y \in \widetilde{X}_2$�� ���� $p_2(y)=z\in X$ ��
�����غ���. �� $z$���� covering $p_1$�� ����  evenly cover�Ǵ�
$U_1$�� $p_2$�� ���� evenly cover�Ǵ� $U_2$�� �����Ѵ�. �� ����
�����տ� ���ԵǴ� path connected�� neighborhood�� $U$�� ����. �׷���
$p_1^{-1}(U)=\displaystyle{\coprod_{a\in p_1^{-1}(z)}}V_a$,
$\,\,p_2^{-1}(U)=\displaystyle{\coprod_{b\in p_2^{-1}(z)}}W_b$ ��
�� �� �ִ�.  Ư�� $y$�� �����ϴ� $\widetilde{X_2}$������ copy��
$W_y$�� ����. �� $V_a$������($a\in p_1^{-1}(z)$) $p_1$��
homeomorphism�̰�, �� $U$�� $W_y$������ $p_2$��  homeomorphism
�̹Ƿ�  $\phi|_{V_a}=p_2|_{U}^{-1}\circ p_1|_{V_a}$ �� �Ǿ��
�Ѵ�. ($p_1|_{V_a}$�� lifting�� uniqueness�� ����)�׷��� ��  $W_y$�� ������ ���� evenly covered �ȴ�.

$\displaystyle{\coprod_{a\in \phi^{-1}(y)}}V_a=\phi^{-1}(W_y)$.

\end{proof}




\begin{cor}
Let $\,\,p:\widetilde{X}\rightarrow X$ be a universal covering
and $p_1:\widetilde{X}_1\rightarrow X$ be a covering. Then
$\exists$ a covering $\,\,\phi:
\widetilde{X}\hspace{1em}\rightarrow
\hspace{1em}\widetilde{X}_1\,\,\,\,$ such that $p_1\circ \phi = p$

$\hspace{14em}p\searrow \hspace{2em}\swarrow p_1$

$\hspace{16em}X$\\
\end{cor}

�� ���������� ������, $X$�� universal covering $\widetilde{X}$ ��
�ֱ⸸ �ϸ� ��� �ٸ�  covering�� $\widetilde{X}$�� ���� covered
�ȴٴ� ���� �� �� �ִ�.\\

{\bf ��.} ������ ��� universal covering ���̴�.

1 . $p:\mathbf{R}\rightarrow S^1$.

2 . $p:S^n\rightarrow p^n\,\,\,\,,\,\,\,\,\,\,(n\geq 2)$.

3 . $p: \mathbf{R}^n\rightarrow T^n$.\\

\begin{thm}(Existence and Uniqueness.)

Let $p_i:\widetilde{X}_i\rightarrow X, i=1,2 $ be  covering
  maps.
 Then

$\exists$ an isomorphism
$\phi:(\widetilde{X}_1,\widetilde{x_1})\rightarrow(\widetilde{X}_2,\widetilde{x_2})$
$\Longleftrightarrow
p_{1{\sharp}}\pi_1(\widetilde{X}_1,\widetilde{x_1})=p_{2{\sharp}}\pi_1(\widetilde{X}_2,\widetilde{x_2})$
\end{thm}

\begin{proof} �� Section�� ���� 1���� ���ʹ����� �������Ƿ� ���ʿ�
���� �����ϸ� �ȴ�.
\end{proof}\\

�� �������� base point �� �����ϰ� ���� ������ �������� �� ��
�ִ�.

\begin{cor}

$\hspace{4.5em}\exists \phi$

$\hspace{7em}\widetilde{X}_1\hspace{1em}\rightarrow\hspace{1em}\widetilde{X}_2\hspace{3em},\phi:isomorphism.$

$\hspace{7em}p_1\searrow\hspace{2em}\swarrow p_2$

$\hspace{9.5em}X$

$\Leftrightarrow$ For some
$\,\,x=p_1(\widetilde{x}_1)=p_2(\widetilde{x}_2)$,
$\,\,p_{1\sharp}\pi_1(\widetilde{X}_1,\widetilde{x}_1)$ and
$p_{2\sharp}\pi_2(\widetilde{X}_2,\widetilde{x}_2)$ belong to the
same conjugacy class in $\pi_1(X,x)$.
\end{cor}

\begin{proof}
������ Application�κп���
$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}_0)$ ��
$p_{\sharp}\pi_1(\widetilde{X},\widetilde{x}_0')$  ���̿���
conjugation ���谡 ������ �������Ƿ� �̷κ��� ���� �� �ִ�.
\end{proof}\\

{\bf ���� 10.} Massey, "Algebraic topology : An introduction", p.
161, exc 6.4,\\

{\bf ���� 11.} Massey, "Algebraic topology : An introduction", p.
161, exc 6.5.









  \end{document}
