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\section*{Quotient smooth manifold.}


�� ���ǿ��� $M$�� smooth manifold�̸� $M$�� smooth structure��
covering $\widetilde{M}$�� pull back��ų �� �ִٴ� ����� ���Ҵ�.
�̹����� �ݴ�� $\widetilde{M}$�� smooth structure�� ������ ����
��, ���� $M$���� smooth structure�� ���� �� �ִ����� ���캸��.\\

covering $p:\widetilde{M}\rightarrow M$ �� ���� deck
transformation group $G$�� action�� �����غ���. (�������2��
Covering space-Deck transformation �κ��� ����.) $p^{-1}(x)$��
���� G�� action�� transitive �� �� regular covering�̶�� �Ѵ�. ��
��� $M$�� $\widetilde{M}/G$�� �� �� �ְ� ���� ������
�����Ѵ�.\\

{\it Let $p:\widetilde{M}\rightarrow M$ be a regular covering.
Suppose $\widetilde{M}$ has a smooth structure such that the deck
transformation group acts on $\widetilde{M}$ as diffeomorphisms.
Then $M$ inherits the smooth structure of $\widetilde{M}$ such
that $p$ becomes $\ccn$.}

(����) Exercise.\\

{\bf ��.} covering $p:\rb\rightarrow \sbb^1$, $p(x)=e^{2\pi ix}$.






$\rb$�󿡼� $\tau (x)=x+1$�� �ΰ� $G=<\tau>\cong \mathbb{Z}$��
�θ� �̴� deck transformation group�� �ȴ�. �� ��� $p$�� �翬��
regular covering�� �ȴ�. $\tau$�� $\ccn$�̹Ƿ�
$\sbb^1=\rb/G=\rb/\mathbb{Z}$ �� smooth structure�� �� �� �ִ�.
���� �Ϲ������� universal covering
$p:\rb^n\rightarrow\mathbb{T}^n=\sbb^1\times\cdot\cdot\cdot\times\sbb^1$
�� deck transformation group
$G=<\tau_1,\cdot\cdot\cdot,\tau_n>\cong\mathbb{Z}^n(\tau_i(x)=x+e_i)$��
�̿��ϸ� $\mathbb{T}^n$�� standard �� smooth structure�� ������
�� �ִ�. �� �ٸ� ���δ� covering $p:\sbb^n\rightarrow \rb P^n$ ��
�ִ�. �� ��� antipodal map $\alp(x)=-x$�� deck transformation
group�� generate�ϰ� $\alp^2=id$�̹Ƿ�
$G=<\alp>\cong\mathbb{Z}/2$�� �ȴ�. ���� $\sbb^n/G=\rb P^n$ ��
�ȴ�.(Exercise) ������ �� �� ���� $\rb P^n$�� structure�� ������
����� structure�� ����.



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