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\begin{document}

\parindent=0cm
\section*{VIII.4 Euler characteristic and Lefschetz fixed point theorem}

\begin{defn}
Let $K$ be a finite simplicial complex. The Euler characteristic
of $K$ is defined by\\
\begin{center}

$\chi(K) := \sum (-1)^{p} rk (C_{p}(K)) = \sum (-1)^{p}
 \sharp \{\textrm{p-simplices of $K$ }\}.$\\
$\textrm{And let} \bet_{p}=rk (H_{p}(K)) = \textrm{the $p$-th
Betti number} = rk (H_{p}(|K|)).$

\end{center}

\end{defn}

������ �տ��� Betti number�� ������ �� $H_{p}(K)$�� simplicial
homology�̰� $H_{p}(|K|)$�� singular homology�̴�. �� ���ǰ�
�ڿ������� ������ simplicial homology�� singular homology���̿�
natural equivalence�� �ֱ� �����̴�.\\

{\bf 1. $\chi(K) = \sum(-1)^{p} \bet_{p}$}\\

\begin{pf}

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & C_{p+1} \ar[r] & C_{p} \ar[r] & C_{p-1} \ar[r] & \cdots \hspace{2.0em} & (\ast)}
\]

�켱 ���� chain complex�� �����Ѵٴ� ���� �Ʒ��� ���� �ΰ��� short exact sequence�� �����Ѵٴ� �Ͱ� ��ġ�̴�.\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & Z_{p} \ar[r]^{i} & C_{p} \ar[r]^{\bd} & B_{p-1} \ar[r] & 0 \\
0 \ar[r] & B_{p} \ar[r]_{i} & Z_{p} \ar[r] & H_{p} \ar[r] & 0 }
\]

����,\\

$rk(C_{p}) = rk(Z_{p}) + rk(B_{p-1}) , \hspace{0.5em} rk(Z_{p}) = rk(B_{p}) + rk(H_{p})$\\

$\Rightarrow \chi = \sum (-1)^{p}rk(C_{p}) = \sum (-1)^{p} (
rk(H_{p}) + rk(B_{p}) + rk(B_{p-1})) \\
  \hspace*{1.5em}= \sum (-1)^{p} rk(H_{p})$\\

\end{pf}

{\bf Structure theorem of $R$-module}\\
Assume that $R$ is a P.I.D, $F$ is a free $R$-module of rank $s$
and $N$ is a submodule of $F$.
 Then there exists a basis $\{f_{1}, f_{2} , \cdots, f_{s} \}$ of $F$ and $d_{1}, d_{2}, \cdots d_{s} \in R$
 such that \\

 (1) nonzero elements of $\{d_{1}f_{1}, \cdots d_{s}f_{s}\}$ form a basis of $N$.\\
 (2) $d_{1} | d_{2} | \cdots | d_{s}$\\

\begin{pf}

Reference\\
Serge Lang, Algebra , pp. 521-522.\\
Jacobson, Basic Algebra 1, pp. 179-180.\\

\end{pf}

\begin{cor}

Let $G$ be an abelian group. Then $G$=free part $\bigoplus$
torsion part.
And define $rk (G) := rk (\textrm{free part of $G$})$\\

\end{cor}

\begin{cor}

$rk (F) = rk (N) + rk (F/N)$.\\ In general,$rk (M) = rk (N) + rk (M/N)$ if\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & N \ar[r] & M \ar[r] & M/N \ar[r] & 0}
\]

is exact.\\

\end{cor}

\begin{pf}

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
 & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & K \ar[d] \ar[r]^{=} & K \ar[d] \ar[r] & 0 \ar[d] \ar[r] & 0 \\
0 \ar[r] & p^{-1}(N) \ar[d] \ar[r] & F \ar[d]^{p} \ar[r] & \clubsuit \ar[d]^{f} \ar[r] & 0\\
0 \ar[r] & N \ar[d] \ar[r] & M \ar[d] \ar[r] & M/N \ar[d] \ar[r] & 0\\
 & 0 & 0 & 0 & }
\]

���� commutative diagram�� ���캸�� nine lemma�� ���ؼ� $f$��
isomorphism�� �ǰ� ����, $rk (M) = rk (F) - rk (K) = rk
(p^{-1}(N)) + rk (\clubsuit) - rk (K) = rk (N) + rk (M/N)$��
�����Ѵ�.\\

\end{pf}

\begin{cor}

If ($\ast$) is acyclic (i.e., exact), then $\sum(-1)^{p} rk (C_{p}) = 0$.

\end{cor}

{\bf Examples}\\
$\chi(S^{n}) = 1 +(-1)^{n} = \begin{cases} 0 & \textrm{n is odd} \\ 2 & \textrm{n is even}
\end{cases}$\\

$\hspace*{1.5em}(\because H_{0}(S^{n}) = H_{n}(S^{n}) = \zb)$\\

$\chi(\sum_{g}) = 1 - 2g + 1 = 2 - 2g\\
\hspace*{1.5em}(\because H_{0}(\sum_g) = \zb, H_{1}(\sum_g) = \zb^{2g}, H_{2}(\sum_{g}) = \zb)$\\

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{\bf 2.}More generally, given a diagram of homomorphisms of finitely generated modules over P.I.D,

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & A \ar[d]^{\phi' = \phi|} \ar[r]^{i} & B \ar[d]^{\phi} \ar[r]^{p} & C \ar[d]^{\bar{\phi}} \ar[r] & 0
& \textrm{: exact}\\
0 \ar[r] & A \ar[r]_{i} & B \ar[r]_{p} & C \ar[r] & 0 & }
\]

where $B$ is free, we have\\
\begin{center}

$tr (\phi) = tr (\phi') + tr (\bar{\phi})$

\end{center}

where $tr (\bar{\phi})$ is taken on free part $FC$ of $C$.\\
Ư��, ���⼭ $\phi$�� identity�� ��� 1�� ����� �����ϴ�.\\

\begin{pf}

Let $\{b_{1}, \cdots , b_{k}, b_{k+1}, \cdots , b_{n}\}$ be a basis for $B$ such that
$\{d_{1}b_{1}, \cdots , d_{k}b_{k}\}$ is a basis for $A$. And $\phi(b_{j}) := \sum r_{ij}b_{i}$. Then\\
Matrix of $\phi = (r_{ij}) = \begin{pmatrix} \ast_{1} & \ast \\ 0 & \ast_{2} \end{pmatrix}$.\\

where $\ast_{1}$ is the matrix of $\phi|$ and $\ast_{2}$ is the matrix of $\bar{\phi}$ on $FC = <\bar{b_{k+1}},
\cdots , \bar{b_{n}}>$.\\

{\bf Note}\\
$\phi|(d_{j}b_{j}) = \phi(d_{j}b_{j}) = \sum_{i} r_{ij}d_{j}b_{i}$�ε� ����, $\phi|(d_{j}b_{j}) = \sum a_{ij}d_{i}b_{i}$
�̶� �θ�, $a_{ij}d_{i} = r_{ij}d_{j}$�� �����Ѵ�. ����, $tr (\phi|) = \sum a_{ii} = \sum r_{ii}$�̴�.\\

Note���뿡 ���� ���� ������ �ϼ��ȴ�. \\

\end{pf}

{\bf 3.Hopf trace formula}\\
Let $C$ be a finitely generated chain complex (of $R$-module), i.e., each $C_{p}$ is finitely generated.
And $\phi : C \to C$ be a chain map. Then\\

\begin{center}

$\sum (-1)^{p}tr (\phi_{p},C_{p}) = \sum (-1)^{p}tr
(\phi_{p*},H_{p}(C)/T_{p}(C))$

\end{center}

���⼭ $T_{p}(C)$�� $H_{p}(C)$�� Torsion part�̴�.\\

\begin{pf}

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\cdots \ar[r] & C_{p+1} \ar[d]^{\phi_{p+1}} \ar[r] & C_{p} \ar[d]^{\phi_{p}} \ar[r] &
C_{p-1} \ar[d]^{\phi_{p-1}} \ar[r] & \cdots \\
\cdots \ar[r] & C_{p+1} \ar[r] & C_{p} \ar[r] & C_{p-1} \ar[r] & \cdots}
\]

���� commutative diagram���� 1�� ���� ���������� �Ʒ� �ΰ��� commutative diagram�� ��´�.

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & Z_{p} \ar[d]^{\phi_{p}^{'}} \ar[r] & C_{p}
\ar[d]^{\phi_{p}} \ar[r] & B_{p-1}
\ar[d]^{\phi_{p-1}^{''}} \ar[r] & 0\\
0 \ar[r] & Z_{p} \ar[r] & C_{p} \ar[r] & B_{p-1} \ar[r] & 0 }
\]


\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
0 \ar[r] & B_{p} \ar[d]^{\phi_{p}^{''}} \ar[r] & Z_{p}
\ar[d]^{\phi_{p}^{'}} \ar[r] & H_{p}
\ar[d]^{\phi_{p*}} \ar[r] & 0\\
0 \ar[r] & B_{p} \ar[r] & Z_{p} \ar[r] & H_{p} \ar[r] & 0 }
\]

�׸��� ������ diagram���� $tr (\phi_{p}) = tr (\phi_{p}^{'}) + tr
(\phi_{p-1}^{''})$
�� $tr (\phi_{p}^{'}) = tr (\phi_{p}^{''}) + tr (\phi_{p*})$���� �� �� �ְ�, ����\\

$tr (\phi_{p}) = tr (\phi_{p*}) + tr (\phi_{p}^{''}) + tr (\phi_{p-1}^{''})\\
\Rightarrow \sum (-1)^{p}tr (\phi_{p}) = \sum (-1)^{p}tr (\phi_{p*})$�� �����Ѵ�.\\

\end{pf}

{\bf Note} If $\phi$= id. $\Rightarrow tr (\phi_{p})= rk (C_{p})$ and 1 is a special case.\\

\begin{defn}

Let $K$ be a finite simplicial complex and $f :|K| \to |K|$. \\
$\lam(f) := \sum (-1)^{p}tr (f_{*},H_{p}(|K|)/T_{p}(|K|)) = \textrm{Lefschetz number of $f$}$\\

\end{defn}

{\bf 4.Lefschetz fixed point theorem}\\
Let $K$ be a finite simplicial complex and $f : |K| \to |K|$.\\
If $\lam(f) \neq 0$, then $f$ has a fixed point.\\

�켱 $S^{n}$�� ��츦 ���캸��. $H_{0}(S^{n}) \to H_{0}(S^{n})$�� identity�̰�,
$H_{n}(S^{n}) \to H_{n}(S^{n})$���� $tr (f_{*})=deg f$�̹Ƿ� $\lam(f)= 1+ (-1)^{n} deg f$��
�Ǿ� $\lam(f) \neq 0$�̶�� ���� $deg f \neq (-1)^{n+1}$�̶�� ���̴�.
�׸��� �� ���� �̹� �տ��� �ٷ����.\\

\begin{pf}
Assume $f$ has no fixed point and we will prove $\lam(f)=0$. Since
$|K|$ is compact, there exists $\eps > 0$ such that $d(x,f(x)) >
\eps, \forall x \in |K|$.

May assume $mesh(K) < \eps$ by passing to a subdivision.\\
Let $g : K^{'} \to K$ be a simplicial approximation of $f$. Recall
$\forall \sig \in K^{'}, g(\sig)$ and $f(\sig)$ lie in the same
simplex $\tau \in K$. Then, $g(\sig) \cap \sig = \emptyset$. If
not, $g(\sig) \cap \sig \neq \emptyset, g(\sig) \cap \tau \neq
\emptyset , f(\sig) \subset \tau$ and $ diam(\tau) <\eps.
\Rightarrow d(x,f(x)) < \eps$ for $x \in \sig \cap g(\sig)$. It is a contradiction.\\

Now consider $\phi_{p} : C_{p}(K^{'}) \overset{g_{\sharp}}{\to} C_{p}(K) \overset{sd}{\to}
C_{p}(K^{'})$. Then $tr (\phi_{p})=0$ , since the coefficient of $\sig$ in $\phi_{p}(\sig)$ is 0.\\

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
\phi_{*} : & H_{p}(K^{'}) \ar[d]^{\eta_{*}^{\cong}} \ar[r]^{g_{*}}
& H_{p}(K)
\ar[d]^{\eta_{*}^{\cong}} \ar[r]^{sd_{*}} & H_{p}(K^{'}) \ar[d]^{\eta_{*}^{\cong}} & \textrm{commutes}\\
& H_{p}(|K|) \ar[r]_{g_{*}=f_{*}} & H_{p}(|K|) \ar[r]_{sd_{*}=id}
& H_{p}(|K^{'}|) & }
\]

���� diagram���� ������ �κ��� commutativity�� ������ diagram���κ��� �翬�ϴ�.

\[
\xymatrix @M=1ex @C=1em @R=1em @*[c]{%
C_{p}(K) \ar[r]^{sd} & C_{p}(K^{'})&\\
\triangle_{p}(K) \ar[d] ^{\theta} \ar[u]^{\mu} \ar[r]^{sd} &
\triangle_{p}(K^{'})
\ar[d]^{\theta} \ar[u]^{\mu}& \textrm{commutes}\\
S_{p}(|K|) \ar[r]^{sd} & S_{p}(|K^{'}|) & }
\]

$\Rightarrow tr (\phi_{*}) = tr (f_{*})$ and $\lam(f) = \sum
(-1)^{p} tr(f_{*}) = \sum (-1)^{p}tr (\phi_{*}) = \sum (-1)^{p}tr
(\phi_{p}) = 0$

\end{pf}


\begin{cor}

(A generalization of Brouwer fixed point theorem)\\
Let $|K|$ be acyclic (i.e. $\widetilde{H}(|K|) = 0$) and $f:|K| \to |K|$.\\
Then, $f$ has a fixed point.\\

\end{cor}

$\because H_{0}(|K|)= \zb \Rightarrow \lam(f)=1 \neq 0$\\

{\bf Example} 1. $f: S^{n} \to S^{n} \Rightarrow
\lam(f)=1+(-1)^{n} deg f$.\\
2. $f \simeq id. : |K| \to |K| \Rightarrow \lam(f) = \lam(id.) = \chi(|K|)$\\
If $\chi(|K|) \neq 0 \Rightarrow f \simeq id. $has a fixed point.\\

$S^{2n}$�� nonvanishing vector field�� �ִٰ� ��������. �׷���, ��
vector field�� ���� flow�� �� �� �ְ�, flow�� ���� identity��
homotopic�� map�� ������ �� �ִ�. �׷��� ���� ���� fixed point��
���� �� map�� ���缺�� $S^{2n}$�� Euler characteristic�� 0�̾��
���� ����ϴ� ���̹Ƿ� ����̴�. ���� $S^{2n}$���� nonvanishing
vector field��
������ �� ����.\\
���������� compact smooth manifold $M$�� nonvanishing vector
field�� �����ϸ�, $\chi(M) = 0$�� �Ǿ�� �Ѵ�. �����δ� ����
�����ϰ� Euler characteristic�� nonvanishing vector field��
�����ϱ� ���� abstraction���ε� �ľǵ� �� �ִ�.\\

\newpage
{\bf ���� 4}\\

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\pscircle(0,-0.9){0.03}\pscircle(0.3,-0.9){0.03}\pscircle(0.6,-0.9){0.03}\pscircle(0.9,-0.9){0.03}
\pscircle(-0.3,-0.9){0.03}\pscircle(-0.6,-0.9){0.03}\pscircle(-0.9,-0.9){0.03}
\rput(1.1,1.1){$\rb^{2}$}}

%map F%
\psline{->}(2.0,3.6)(3.0,3.6) \rput(2.5,3.4){$F$}

%down%
\psline{->}(1.0,2.6)(1.0,1.9) \psline{->}(4.0,2.6)(4.0,1.9)

\end{pspicture}
\end{center}

�켱 $\rb^{2}$�� ����$\{(m,n)| m,n \in \zb\}$�� �־��� �ִٰ�
����. �׸��� ���������� �����ϴ� linear map $F (\in GL(2,\zb))$��
��������. ���� torus�� universal covering�� �� $\rb^{2}$��� ��
��, $\rb^{2}$������ map $F$�� $x+\{(m,n)\}$�� $F(x)+\{(m,n)\}$����
������ map�� �ǹǷ� torus ������ map
$f=\bar{F}$�� induce�Ѵ�. �̶�, $\lam(f)$�� ��� �Ǵ°�?\\

\end{document}
