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\subsubsection*{Example (non-orientable surfaces)}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/4e88f59494cf41c9.png}
\end{center}
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/0ba89b82208c425b.png}
\end{center}
Homology of $K$: the subcomplex $\polyh|W| \subset
\polyh|K|$ with
\begin{center}
  \includegraphics[width=0.2\linewidth]{images/b5401080a4f94210.png}
\end{center}
is such that $\polyh|K|$ deformation retracts to
it, so $\Hgp_0(K) = \ZZ\{\text{vertex}\}$,
$\Hgp_1(K) = \ZZ\{u\}$, $u = [1, 2] + [2, 3] + [3,
1]$ and $\Hgp_i(K) = 0$ for $i > 1$.

Note $r = [4, 5] + [5, 6] + [6, 4]$ is homologous
to $2u$.

Homology of $K_n$: $K_n = K_{n - 1} \cup K$, $K_{n
- 1} \cap K = \ssimp^1$. So Mayer-Vietoris gives:
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/e35fd3c4a1e14a74.png}
\end{center}
So inductively $\Hgp_0(K_n) = \ZZ\{\text{a
vertex}\}$, $\Hgp_1(K_n) = \ZZ^n = \ZZ\{u_1,
\ldots, u_n\}$, $\Hgp_2(K_n) = 0$ for $i \ge 2$.

The boundary $L$ of $K_n$ is the cycle $r_1 + r_2
+ \cdots + r_n = 2(u_1 + \cdots + u_n)$.

Homology of $K_n \cup CL$: Mayer-Vietoris again:
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/197b4ea7992a48c9.png}
\end{center}
Note ($*$) sends the generator to $r_1 + r_2 +
\cdots + r_n = 2(u_1 + \cdots + u_n)$. So ($*$) is
injective, which tells us that the previous map
$\partial_*$ is $0$.

So $\Hgp_0(K_n \cup CL) = \ZZ$, $\Hgp_1(K_n \cup
CL) = \frac{\ZZ\{u_1, \ldots, u_n\}}{\langle 2(u_1
+ \cdots + u_n) \rangle} \cong \ZZ / 2\ZZ \oplus
\ZZ^{n - 1}$, and finally $\Hgp_i(K_n \cup CL) =
0$ for $i \ge 2$. This is $H_*(S_n)$.

The surfaces:
\begin{center}
  \begin{tabular}{c|c|c|c}
    & $S^2$ & $\Sigma_g$ & $S_n$ \\
    \hline
    $H_1$ & $0$ & $\ZZ^{2g}$ & $\ZZ / 2\ZZ \oplus
    \ZZ^{n - 1}$ \\
    $H_2$ & $\ZZ$ & $\ZZ$ & $0$
  \end{tabular}
\end{center}
are all not homotopically equivalent to each
other.

\begin{theorem*}
  Every triangulable surface is homeomorphic to
  one of these.
\end{theorem*}

\subsection{Rational homology, Euler and Lefschetz
numbers}

\begin{flashcard}[On-with-Q]
\begin{definition*}[$O_n(K; \QQ)$]
  \glssymboldefn{Hqq}{$H(K; Q)$}{$H(K; Q)$}
  \cloze{For a \gls{simpcomp} $K$, let $O_n(K; \QQ)$ be
  the $\QQ$-vector space with basis the ordered
  \glspl{simp} of $K$. Define $T_n(K; \QQ)$ as
  usual, and so get $C_n(K; \QQ)$. Define $d_n$ by
  the same formula, and
  \[ H_n(K; \QQ) = \frac{\Ker(d_n : C_n(K; \QQ)
  \to C_{n - 1}(K, \QQ))}{\Im(d_{n + 1} : C_{n +
  1}(K; \QQ) \to C_n(K; \QQ)} .\]
  This is a $\QQ$-vector space.}
\end{definition*}
\end{flashcard}

\begin{lemma*}
  If $\Hgp_n(K) \cong \ZZ^r \oplus \text{(finite
  abelian group)}$, then $\Hqq_n(K; \QQ) \cong
  \QQ^r$.
\end{lemma*}

\begin{proof}
  See \es[5]{4}.
\end{proof}

\begin{example*}
  \[ \Hqq_i(S6n; \QQ) = \begin{cases}
    \QQ & i = 0, n \\
    0 & \text{otherwise}
  \end{cases} \]
  \[ \Hqq_i(\Sigma_g; \QQ) = \begin{cases}
    \QQ & i = 0, 2 \\
    \QQ^{2g} & i = 1 \\
    0 & \text{otherwise}
  \end{cases} \]
  \[ \Hqq_i(S_n; \QQ) = \begin{cases}
    \QQ & i = 0 \\
    \QQ^{n - 1} & i = 1 \\
    0 & \text{otherwise}
  \end{cases} \]
\end{example*}

\begin{flashcard}[lefschetz-number-defn]
\begin{definition*}[Lefschetz number]
  \glssymboldefn{lef}{$L(f)$}{$L(f)$}
  \glsnoundefn{lefnum}{Lefschetz number}{Lefschetz
  numbers}
  \cloze{Let $X$ be a polyhedron ($\cong \polyh|K|$) and
  $f : X \to X$ be a continuous map. The
  \emph{Lefschetz number} of $f$ is
  \[ L(f) = \sum_{i = 0}^\infty (-1)^i \Trace(f_* :
  \Hqq_i(X; \QQ) \to \Hqq_i(X; \QQ)) .\]}
\end{definition*}
\end{flashcard}

\begin{flashcard}[euler-char-defn]
\begin{definition*}[Euler characteristic]
  \glssymboldefn{eul}{$\chi$}{$\chi$}
  \glsnoundefn{eulnum}{Euler characteristic}{Euler
  characteristics}
  \cloze{The \emph{Euler characteristic} of $X$ is
  \[ \chi(X) = \lef(\id) = \sum_{i = 0}^\infty (1)^i
  \dim_{\QQ} \Hqq_i(X; \QQ) .\]}
\end{definition*}
\end{flashcard}

\begin{example*}
  \[ \eul(S^n) = 1 + (-1)^n = \begin{cases}
    2 & \text{$n$ even} \\
    0 & \text{$n$ odd}
  \end{cases} \]
  \[ \eul(\Sigma_g) = 2 - 2g, \qquad \eul(S_n) = 2
  - n .\]
\end{example*}

\begin{example*}
  The antipodal map $a : S^n \to S^n$ induces the
  identity map on $\Hqq_0(S^n; \QQ)$, and
  multiplication by $(-1)^{n + 1}$ on $\Hqq_n(S^n;
  \QQ)$. So $\lef(a) = 1 + (-1)^n(-1)^{n + 1} =
  0$.
\end{example*}

\begin{flashcard}[linalg-trace-fact-lemma]
\begin{lemma*}
  Let $V$ be a finite-dimensional vector space, $W
  \le V$ a subspace, and $A : V \to V$ a linear
  map such that $A(W) \le W$. Let $B = A|_W : W
  \to W$ and $C : V / W \to V / W$ the map induced
  by $A$. Then
  \[ \Trace(A) = \Trace(B) + \Trace(C) .\]
\end{lemma*}

\begin{proof}
  \cloze{Let $e_1, \ldots, e_r$ be a basis for $W$, and
  extend it to a basis of $V$. In this basis,
  \[ A = \begin{pmatrix}
    B & * \\
    0 & C
  \end{pmatrix} \]
  so $\Trace(A) = \Trace(B) + \Trace(C)$.}
\end{proof}
\end{flashcard}

\begin{flashcard}[lefschetz-equality-coro]
\begin{corollary*}
  For a \gls{cmap} $f_\bullet : C_\bullet(K; \QQ)
  \to C_\bullet(K; \QQ)$, we have
  \[ \cloze{\sum_{i = 0}^\infty (-1)^i \Trace(f_* :
  \Hqq_i(K; \QQ) \to \Hqq_i(K; \QQ)) = \sum_{i =
  0}^\infty (-1)^i \Trace(f_i : C_i(K; \QQ) \to
  C_i(K; \QQ))} \]
\end{corollary*}

\begin{proof}
  \cloze{Consider
  \[ 0 \to B_i(K; \QQ) \to Z_i(K; \QQ) \to H_i(K;
  \QQ) \to 0 \]
  \[ 0 \to Z_i(K; \QQ) \to C_i(K; \QQ) \stackrel{d_i}{\to}
  B_{i - 1}(K; \QQ) \to 0 \]
  Let $f^C, f^Z, f^B, f^H$ be the induced maps on
  chains / cycles / boundary / homology.
  \begin{align*}
    \sum (-1)^i \Trace(f_i^H)
    &= \sum (-1)^i (\Trace(f_i^Z) - \Trace(f_i^B))
    \\
    &= \sum (-1)^i (\Trace(f_i^C) - \Trace(f_{i -
    1}^B) - \Trace(f_i^B)) \\
    &= \sum (-1)^i \Trace(f_i^C)
  \end{align*}}
\end{proof}
\end{flashcard}

\begin{corollary*}
  \[ \eul(\polyh|K|) = \sum_{i = 0}^\infty (-1)^i
  \#\{\text{$i$-simplices of $K$}\} \]
\end{corollary*}