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\begin{document}
\parindent=0cm
\section*{deRham cohomology}

�̹����� �츮�� ��� exterior differential operator $d$�� ������
cohomology�� ������ ���ڴ�.\\

���� $M^n$�� ���Ͽ� ������ chain complex�� ���� �� �ִ�.\\

$0 \; \rightarrow \; \mathcal{E}^0 (M) \; \overset{d}{\rightarrow}
\; \mathcal{E}^1 (M) \; \overset{d}{\rightarrow} \; \mathcal{E}^2
(M) \; \overset{d}{\rightarrow} \; \cdots \;
\overset{d}{\rightarrow} \;
\mathcal{E}^n (M) \; \rightarrow \; 0$\\

( vector space�� $V_i$(�� �Ϲ������� module)�� sequence���� $d^2 =
0$�� �����ϴ� vector space homomorphism $d: V_i \rightarrow
V_{i+1}$�� �����ϴ� ��� chain complex��� �Ѵ�.)\\

�̰�� ������ stage $\mathcal{E}^p$���� $\mbox{im}\; d $��
$\mbox{ker} \;d$�� ���캸�� $d^2=0$�� ���� ������ $\mbox{im}\; d
\subset \mbox{ker}\; d$�� �������� �� �� �ִ�.

$B_p := \mbox{im}\; d = \{\mbox{exact p - forms}\}$ �� $Z_p :=
\mbox{ker}\; d =\{\mbox{closed p - forms}\}$�� ǥ������.

���� p-th deRham cohomology $H^p (M)$�� ���� �� �� �ִ�.\\

$H^p (M) = Z_p / B_p$\\

�� deRham cohomology�� exact�� �ƴ� closed form���� �󸶳� ����
�����ϴ����� ����(measure)�ϴ� ô���� �ȴ�.\\

ex) closed form �̸鼭 exact form�� �ƴ� ��\\

���� $M = \rb^2 \setminus O$ ���� ���� �Լ� $\theta$�� global�ϰ�
���ǵ��� ���Ѵ�. (branch cut�� ����) �׷��� local�ϰԴ� �׻�
$\theta$�� �� ���� �ǹǷ� �̰��� �̿���

$\alp = d \theta$ �� �����ϸ� $d$ �� local�� �����̹Ƿ� $\alp$��
��(global�ϰ�)���ǵȴ�.

�׷��� $\alp$�� ��ġ exactó�� �������� $\theta$�� global�ϰ� ����
���� �ʱ� ������ $\alp$�� exact�� �ƴϴ�.

�ݸ鿡 $d \alp$�� �����ϸ� local�ϰ� $d^2 \theta$�� �ǹǷ� $0$��
�ǰ� ���� $\alp$�� closed�̴�.

$H^1(\rb^2 \setminus O) = <\alp>$\\

���� 0��° cohomology�� ���� �����غ��� ker �� trivial�̹Ƿ�

$H^0 (M) = Z_0 (M)$ �� �Ǵµ� $Z_0$�� ���Ҷ� �ٸ��ƴ� locally
constant function���̴�. �׷��Ƿ� Ư���� $M$�� connected��
�����

locally constant $\Rightarrow$ constant �̹Ƿ� $H^0(M) = \{
\mbox{constant functions on M} \}$�̰�

����$\{ \mbox{constant functions on M} \} \cong \rb $�̹Ƿ�\\

$H^0(M) = \rb$�� ��´�.\\

���� ������ �ſ� �߿��ѵ� �츮�� ���������� �ϴ� Fact�� �޾Ƶ��̰�
������ �����Ѵ�.\\

\begin{thm}
(de Rham Theorem)

$H_{deRham}^p (M) \cong H_{sing}^p (M, \rb) $

(where $H_{sing}^p$ is the singular cohomology).
\end{thm}

���������� �̷��� ������ $H^p$���� manifold���� category���� group
category�� ���� functor(��Ȯ�ϰԴ� contravariant functor)�� ����
���캸�ڴ�.\\

���� manifold������ map  $\varphi : M \rightarrow N ; \ccn$��
���Ͽ�

$\varphi^* : \mathcal{E}(N) \rightarrow \mathcal{E}(M)$�� algebra
homorphism�� �ǰ� $\varphi^* \circ d = d \circ \varphi^*$��
�����ϴ� ���� �̹� ���캸�Ҵ�. �̰��� �״�� $H^p(N)$,
$H^p(M)$������ map���� Ȯ���� �� �ִµ� �װ��� ������ ���
�����̴�.\\

$\varphi^* Z_p (N) \subset Z_p (M)$

$\varphi^* B_p (N) \subset B_p (M)$\\

�� $\varphi^*$�� closedness�� exactness�� �����Ѵٴ� ���̴�. (��
��ǿ� ���� ������ diagram chasing���� �ٷ� �� �� �ִ�.)\\


$0 \; \rightarrow \; \mathcal{E}^0(M) \; \rightarrow \; \cdots \; \rightarrow \; \mathcal{E}^{i-1}(M) \; \rightarrow \;
\mathcal{E}^i(M)\; \rightarrow \;\mathcal{E}^{i+1}(M)\; \rightarrow \; \cdots$\\

$\hhhh \uparrow^{\varphi^*}\hhhh \hhh \uparrow^{\varphi^*} \hhhh \uparrow^{\varphi^*} \hhhh \uparrow^{\varphi^*}$\\

$0 \; \rightarrow \; \mathcal{E}^0(N) \; \rightarrow \; \cdots \;
\rightarrow \; \mathcal{E}^{i-1}(N) \; \rightarrow \;
\mathcal{E}^i(N)\; \rightarrow \;\mathcal{E}^{i+1}(N)\; \rightarrow \; \cdots$\\

�׷��Ƿ� $\varphi^* : H^p (N) = Z_p (N) / B_p (N) \rightarrow
Z_p(M) / B_p (M) = H^p (M)$�� ���� �� �� �ִ�.

(���� formal�ϰԴ� $\varphi$��ſ� $H^p (\varphi)$ )

�׹��� ���ǵ�(identity �� identity�� ������ ��, composition��
�����ϴ� �� ��)�� �ڸ��ϰ� �����ϹǷ� $H^p$�� contravariant
functor�� ���� �� �� �ִ�.


\end{document}
