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\begin{flashcard}[attaching-n-cell-fundamental-group-thm]
\begin{theorem*}
  \phantom{}
  \begin{enumerate}[(i)]
    \item If $n \ge 3$, then $\ps{\inc} :
      \pio\bs(X, x_0) \to \pio\bs(Y, [x_0])$ is \cloze{an
      isomorphism.}
    \item If $n = 2$, then $\ps{\inc} : \pio\bs(X,
      x_0) \to \pio\bs(Y, [x_0])$ is \cloze{the quotient
      by the normal subgroup generated by $[f] \in
      \pio\bs(X, x_0)$.}
  \end{enumerate}
\end{theorem*}

\begin{proof}
  \cloze{Let $U = \int(D^n)$, $V = X \cup_f (D^n -
  \{0\})$. These give an open cover of $Y$. Choose
  a \gls{path} $u$ in $U \cap V$ from $y_0$ to
  some
  $y_1 \in \int(D^n) = U$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/af5fa1d36c5f41dd.png}
  \end{center}
  If $n \ge 3$ then:
  \begin{enumerate}[(i)]
    \item $U \homoteq \{*\}$.
    \item $U \cap V = S^{n - 1} \times (0, 1)$ is
      \gls{simpconn}.
  \end{enumerate}
  So \nameref{SVK} gives us:
  \[ \pio\bs(U, y_1) \freepH{\ub{\pio\bs(U \cap V,
  y_1)}_{=\pio\bs(V, y_1)}} \pio\bs(V, y_1) \simto
  \pio\bs(Y, y_1) .\]
  So also, by change of basepoint isomorphism, we
  get
  \[ \pio\bs(V, y_0) \simto \pio\bs(Y, y_0) .\]
  But $V$ \glsref[retract]{strongly deformation
  retracts} to $X$, so:
  \[ \pio\bs(X, y_0) \simto \pio\bs(V, y_0)
  \simto \pio\bs(Y, y_0) .\]
  If $n = 2$ then:
  \begin{enumerate}[(i)]
    \item $U \homoteq \{*\}$.
    \item $U \cap V \cong S^1 \times (0, 1)$.
  \end{enumerate}
  Now \nameref{SVK} gives us:
  \[ \ub{\pio\bs(U, y_1)}_{\{e\}} \freepH{\ub{\pio\bs(U \cap V,
  y_1)}_{= \ZZ \ni 1}} \pio\bs(V, y_1) \simto
  \pio\bs(Y, y_1) .\]
  The $1 \in \ZZ$ corresponds to $e$ in
  $\pio\bs(U, y_1)$, and corresponds to
  $\pathisom{\invpath u} ([f])$ in $\pio\bs(V,
  y_1)$. So
  \[ \frac{\pio\bs(V,
  y_1)}{\relgen{\pathisom{\invpath u}[f]}} \simto
  \pio\bs(Y, y_0) \]
  Then change of basepoint and using the fact that
  $V$ strongly deformation retracts to $X$, we get
  \[ \frac{\pio\bs(X, y_0)}{\relgen{[f]}} \simto
  \frac{\pio\bs(V, y_0)}{\relgen{[f]}} \simto
  \pio\bs(Y, y_0) . \qedhere \]}
\end{proof}
\end{flashcard}

\begin{example*}
  The torus $T$:
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/f7c6b5636eaa41ee.png}
  \end{center}
  has a \gls{cellstruc} with:
  \begin{enumerate}
    \item A $0$-cell $x_0$,
    \item $1$-cells $a, b$,
    \item A $2$-cell.
  \end{enumerate}
  The $1$-skeleton is:
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/9a121798dce1423d.png}
  \end{center}
  So
  \[ \pio\bs(T^1, x_0) = \gpgen<a, b|> ,\]
  and so
  \[ \pio\bs(T, x_0) = \gpgen<a, b|aba^{-1}b^{-1}>
  \cong \ZZ \times \ZZ .\]
\end{example*}

\begin{flashcard}[group-as-dimensional-based-cell-complex-coro]
\begin{corollary*}
  For $G = \gpgen<S|R>$ with $S, R$ finite, there
  is a $2$-dimensional based
  \glsref[cellstruc]{cell complex} $\bs(X, x_0)$
  with $\pio\bs(X, x_0) \cong G$.
\end{corollary*}

\begin{proof}
  \cloze{Let $Y$ be the wedge of $|S|$-many circles.
  Sending $s \in S$ to the $s$-th circle $x_s$
  gives an isomorphism
  \[ \gpgen<S|> \simto \pio\bs(Y, y_0) .\]
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/6128904363d5450d.png}
  \end{center}
  Each $r \in R$ is an element of $\gpgen<S|>$, so
  gives a $[\gamma_r] \in \pio\bs(Y, y_0)$.
  Attaching $2$-cells to $Y$ along
  $\{\gamma_r\}_{r \in R}$ gives an $\bs(X, x_0)$
  with
  \[ \pio\bs(X, x_0) = \frac{\gpgen<S|>}{\relgen{r
  \in R}} = \gpgen<S|R> . \qedhere \]}
\end{proof}
\end{flashcard}

\subsection{A refinement of Seifert-van Kampen}

\begin{flashcard}[nbd-deformation-retract-defn]
\begin{definition*}[Neighbourhood deformation
  retract]
  \glsnoundefn{ndr}{neighbourhood deformation
  retract}{neighbourhood deformation retracts}
  \cloze{A subset $A \subset X$ is called a
  \emph{neighbourhood deformation retract} (NDR)
  if there is an open neighbourhood $A \subset U
  \subset X$ and $U$ \glsref[retract]{\emph{strongly} deformation
  retracts} to $A$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[SVK-refinement]
\begin{theorem*}[Refinement of Seifert-van Kampen]
  \cloze{Let $X$ be a space, $A, B \subset X$ closed
  subsets which cover $X$ and such that $A cap B$
  is \gls{pathconn} and is a \gls{ndr} in both $A$
  and $B$. Then
  \[ \pio\bs(A, x_0) \freepH{\pio\bs(A \cap B,
  x_0)} \pio\bs(B, x_0) \simto \pio\bs(X, x_0)
  .\]}
\end{theorem*}

\begin{proof}
  Let $A \cap B \subset U \subset A$, $A \cap B
  \subset V \subset B$, $U, V$ open, which
  \glsref[retract]{strongly deformation retract}
  to $A \cap B$. Observe
  \[ (A \cup V)^c = B - V \qquad (B \cup U)^c = A
  - U \]
  are closed, so $A \cup V$, $B \cup U$ is an open
  cover of $X$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/d1169e7335e84e90.png}
  \end{center}
  The \glsref[retract]{strong deformation retracts} of
  $U$ and $V$ to $A \cap B$ glue to give a
  deformation of $(A \cup V) \cap (B \cup U) = U
  \cap V$ to $A \cap B$. Then we get deformations
  of $A \cup V$ to $A$ and $B \cup U$ to $B$. Now
  we use \nameref{SVK} for the open cover and we
  get:
  \[ \begin{tikzcd}[ampersand replacement=\&]
    \pio(B) \ar[d, "\sim"]
    \& \pio(A \cap B) \ar[l] \ar[r], \ar[d, "\sim"]
    \& \pio(A) \ar[d, "\sim"] \\
    \pio(B \cup U)
    \& \pio((A \cup V) \cap (B \cup U)) \ar[l]
    \ar[r]
    \& \pio(A \cup V)
  \end{tikzcd} \qedhere \]
\end{proof}
\end{flashcard}

\subsection{Surfaces}

\begin{example*}
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/ce42214e24b440e0.png}
  \end{center}
  LHS \glsref[retract]{strongly deformation
  retracts} to
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/698c611bd0b14edc.png}
  \end{center}
  So
  \[ \pio\bs(X, x_0) = \gpgen<a, b|> ,\]
  with $[r] = aba^{-1}b^{-1}$.

  Applying \nameref{SVK} several times gives:
  \[ \pio\bs(F_g, x_0) \cong \gpgen<a_1, b_1, a_2,
  b_2, \ldots, a_g, b_g|> \]
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/691c7126206a4828.png}
  \end{center}
  The boundary is $r_1 r_2 \cdots r_g$, so
  attaching a $2$-cell along it to get $\Sigma_g$,
  \[ \pio\bs(\Sigma_g, x_0) = \gpgen<a_1 b_2,
  \ldots, a_g b_g| a_1 b_1 a_1^{-1} b_1^{-1} a_2
  b_2 a_2^{-1} b_2^{-1} \cdots a_g b_g a_g^{-1}
  b_g^{-1}> \]
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/5848469900084d9c.png}
  \end{center}
\end{example*}

\begin{example*}
  $\RR\PP^2$:
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/a2ad28aa20234967.png}
  \end{center}
  Has $1$-skeleton
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/6aeeede4c9704db9.png}
  \end{center}
  $2$-cell is attached along $aa$. So
  \[ \pio\bs(\RR\PP^2, *) \cong \gpgen<a|a^2> =
  \ZZ / 2\ZZ .\]
\end{example*}

\begin{example*}
  \phantom{}
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/2017ce3e51d34070.png}
  \end{center}
  $Y \homoteq S^1$, so
  \[ \pio\bs(Y, y_0) = \gpgen<a|> ,\]
  and $[r] = a^2 \in \pio\bs(Y, y_0)$.
  \nameref{SVK} again:
  \[ \pio\bs(E_n, y_0) \cong \gpgen<a_1, \ldots,
  a_n|> \]
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/74f7c13c4ca34896.png}
  \end{center}
  The boundary is $r_1 r_2 \cdots r_n$, so
  attaching a $2$-cell along it to get a closed
  surface $S_n$, we get
  \[ \pio\bs(S_n, y_0) = \gpgen<a_1, \ldots,
  a_n|a_1^2 a_2^2 \cdots a_n^2> .\]
\end{example*}
