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\begin{flashcard}[ps-covmap-inj-hom-lemma]
\begin{lemma*}
  Let $p : \tilde{X} \to X$ be a \gls{covmap},
  $x_0 \in X$, $\tilde{x}_0 \in p^{-1}(x_0)
  \subset \tilde{X}$. Then
  \[ \ps p : \pio\bs(\tilde{X}, \tilde{x}_0) \to
  \pio\bs(X, x_0) \]
  is an injective homomorphism.
\end{lemma*}

\begin{proof}
  \cloze{Let $\gamma : I \to \tilde{X}$ be a loop
  based at $\tilde{x}_0$, and suppose $\ps
  p[\gamma] = [p \circ \gamma] = [\constpath{x_0}]$ so $p \circ
  \gamma \homotopic_H \constpath{x_0}$ as
  loops. Now \gls{lift} $H$ to a
  \gls{homotopy} $\tilde{H}$ starting at $\gamma$
  (by \nameref{path_lifting}): then $\tilde{H}$ is a
  \gls{homotopy} of \glspl{path} from $\gamma$ to
  a \gls{lift} of $\constpath{x_0}$, which must be
  $\constpath{\tilde{x}_0}$ (by uniqueness). So
  $[\gamma] = [\constpath{\tilde{x}_0}] = e \in
  \pio\bs(\tilde{X}, \tilde{x}_0)$.}
\end{proof}
\end{flashcard}

\glsnoundefn{covaction}{action}{actions}
\glssymboldefn{covact}{$\cdot$}{$\cdot$}
In the proof of the previous Corollary we
constructed for a path $\gamma : I \to X$ from $x_0$
to $x_1$ a bijection $\gamma_* : p^{-1}(x_0) \to
p^{-1}(x_1)$. It only depended on the
\gls{homotopy} class of the path $\gamma$. This
defines a (right) action of $\pio\bs(X, x_0)$ on
$p^{-1}(x_0)$, via
\[ y_0 \covact [\gamma] \defeq \tilde{\gamma}(1) \]
for $\tilde{\gamma}$ \emph{the} \gls{lift} of
$\gamma$ starting at $y_0$.

\begin{flashcard}[pathconn-pio-lemma]
\begin{lemma*}
  Let $p : \tilde{X} \to X$ be a \gls{covmap}, $X$
  \gls{pathconn} and $x_0 \in X$. Then:
  \begin{enumerate}[(i)]
    \item \cloze{$\pio\bs(X, x_0)$
      \glsref[covaction]{acts} transitively on
      $p^{-1}(x_0)$ if and only if $\tilde{X}$ is
      \gls{pathconn}.}
    \item \cloze{The stabiliser of $y_0 \in p^{-1}(x_0)$
      is $\Im(\pio\bs(\tilde{X}, y_0)
      \stackrel{\ps p}{\to} \pio\bs(X, x_0)) \le
      \pio\bs(X, x_0)$.}
    \item \cloze{If $\tilde{X}$ is \gls{pathconn} then
      there is a bijection
      \[ \frac{\pio\bs(X, x_0)}{\ps p
      \pio\bs(\tilde{X}, y_0)} \to p^{-1}(x_0) \]
      induced by \glsref[covaction]{acting} on $y_0 \in p^{-1}(x_0)$.}
  \end{enumerate}
\end{lemma*}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(i)]
    \item \cloze{Let $\tilde{X}$ be \gls{pathconn}, and $y_0, z_0
      \in p^{-1}(x_0)$. Let $\tilde{\gamma} : I \to
      \tilde{X}$ be a \gls{path} from $y_0$ to $z_0$, so
      $\gamma = p \circ \tilde{\gamma} : I \to X$ is a
      loop based at $x_0$. \emph{The} lift of
      $\gamma$ starting at $y_0$ is
      $\tilde{\gamma}$, and ends at $z_0$, so $y_0
      \covact [\gamma] = z_0$, so transitive.
      \begin{center}
        \includegraphics[width=0.6\linewidth]{images/ac2a0c58d9604873.png}
      \end{center}
      
      Conversely, suppose the \gls{covaction} is
      transitive, let $y_0, z_0 \in \tilde{X}$.
      Choose a \gls{path} $\gamma$ from $p(y_0)$
      to $p(z_0)$, \gls{lift} it starting at
      $y_0$, then it ends at $z_1$, lying in the
      same fibre as $z_0$.
      \begin{center}
        \includegraphics[width=0.6\linewidth]{images/9afa0bfd418b42ce.png}
      \end{center}
      Suppose $y_0$ and $z_0$ are \emph{not} in
      the same path-component. Then $z_1$ and
      $z_0$ are also not. But taking $x_0' =
      p(z_0)$, $\pio\bs(X, x_0')$
      \glsref[covaction]{acts} transitively on
      $p^{-1}(x_0') \ni z_0, z_1$. So there exiss a
      loop $\gamma$ in $X$ whose \gls{lift}
      starting at $z_0$ ends at $z_1$, i.e. $z_0$
      and $z_1$ are in the same path-component.}
    \item \cloze{Suppose $y_0 \covact [\gamma] = y_0$,
      then the lift $\tilde{\gamma}$ of $\gamma$
      starting at $y_0$ also ends at $y_0$, so is
      a loop. Then $[\gamma] = \ps p
      [\tilde{\gamma}]$, so
      \[ \Stab_{\pio\bs(X, x_0)}(p^{-1}(x_0)) \le
      \Im(\ps p) .\]
      If $[\gamma] = \ps p[\gamma']$ then $\gamma'$
      is the \gls{lift} of $\gamma$ starting at
      $y_0$, but also ends at $y_0$ so $y_0
      \covact [\gamma] = y_0$.}
    \item \cloze{Orbit-stabiliser. \qedhere}
  \end{enumerate}
\end{proof}
\end{flashcard}

\begin{flashcard}[n-sheeted-defn]
\begin{definition*}[$n$-sheeted]
  \glsadjdefn{nshtd}{$n$-sheeted}{\gls{covmap}}
  \cloze{If $p : \tilde{X} \to X$ is a \gls{covmap} we
  say that it is \emph{$n$-sheeted} if
  $p^{-1}(x_0)$ all have the same cardinality $n
  \in \NN \cup \{\infty\}$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[unicov-defn]
\begin{definition*}[Universal cover]
  \glsadjdefn{unicov}{universal
  cover}{\gls{covmap}}
  If $p : \tilde{X} \to X$ is a \gls{covmap} we
  say that it is a \emph{universal cover} if
  $\tilde{X}$ is \gls{simpconn}.
\end{definition*}
\end{flashcard}

\begin{flashcard}[unicov-coro-bij-coro]
\begin{corollary*}
  If $p : \tilde{X} \to X$ is a \gls{unicov}, then
  \cloze{each $\tilde{x}_0 \in p^{-1}(x_0)$ determines a
  bijection
  \begin{align*}
    l : \pio\bs(X, x_0)
    &\stackrel{\sim}{\to} p^{-1}(x_0) \\
    [\gamma]
    &\mapsto \tilde{\gamma}(1)
  \end{align*}
  for $\tilde{\gamma}$ the \gls{lift} of $\gamma$
  starting at $\tilde{x}_0$.}
\end{corollary*}
\end{flashcard}

\vspace{-1em}
This induces a group-law on $p^{-1}(x_0)$ via
\[ y_0 * z_0 \defeq l(l^{-1}(y_0) \concat
l^{-1}(z_0)) \]
Spelling out, this is given by:
\begin{itemize}
  \item Choose a \gls{path} $\tilde{\gamma} : I
    \to \tilde{X}$ from $\tilde{x}_0$ to $z_0$,
  \item Let $\gamma$ be the \gls{lift} of $p \circ
    \tilde{\gamma}$ starting at $y_0$,
  \item $y_1 * z_1 = \gamma(1)$
\end{itemize}
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/5d222df5af4744ff.png}
\end{center}

\setcounter{section}{3}
\setcounter{subsection}{0}

\subsection{Fundamental group of $S^1$}

\begin{flashcard}[fundgp-s1-iso-Z]
\begin{theorem*}
  Let $u : I \to S^1$ be $u(s) = e^{2\pi i s}$,
  which is based at $1 \in S^1 \subset \CC$. Then
  there is an isomorphism $\pio\bs(S^1, 1) \cong
  (\ZZ, +, 0)$ sending $u$ to $1 \in \ZZ$.
\end{theorem*}

\begin{proof}
  We know $p : \RR \to S^1$, $t \mapsto e^{2\pi i
  t}$ is a \gls{covmap}. $\RR$ is
  \gls{contractible} so \gls{simpconn}. So this is
  a \gls{unicov}. Hence
  \[ l : \pio\bs(S^1, 1) \stackrel{\sim}{\to}
  p^{-1}(1) = \ZZ \subset \RR \]
  is a bijection. To compute $l^{-1}(m)$ we can
  take $\tilde{u}_m : I \to \RR$, $t \mapsto mt$,
  so $u_m = p \circ \tilde{u}_m$ is a loop in
  $S^1$. Take $\tilde{x}_0 = 0 \in \ZZ$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/4ca1deeccceb4a0b.png}
  \end{center}
  So
  \begin{align*}
    n * m
    &= \text{the end point of the \gls{lift} of
    $u_m$ starting at $n$} \\
    &= (t \mapsto n + mt)|_{t = 1} \\
    n + m
  \end{align*}
  So $* = +$.
\end{proof}
\end{flashcard}

\begin{flashcard}[no-retraction-mapping-thm]
\begin{theorem*}
  The disc $D^2$ does not \gls{retract} to its
  boundary $S^1$.
\end{theorem*}

\begin{proof}
  \cloze{Suppose $r : D^2 \to S^1$ is a
  \glsref[retract]{retraction}, $i : S^1
  \hookrightarrow D^2$, so $r \circ i =
  \id_{S^1}$:
  \[ \id : \ub{\pio\bs(S^1, 1)}_{\cong \ZZ}
  \stackrel{\ps i}{\to} \ub{\pio\bs(D^2,
  1)}_{\cong \{e\}} \stackrel{\ps r}{\to}
  \ub{\pio\bs(S^1, 1)}_{\cong \ZZ} \]
  This is clearly not possible, so $r$ does not
  exist.}
\end{proof}
\end{flashcard}

\begin{flashcard}[brouwer-fp-thm]
\begin{corollary*}[Brouwer fixed point theorem]
  \cloze{Any map $f : D^2 \to D^2$ has a fixed
  point.}
\end{corollary*}

\begin{proof}
  \cloze{Suppose not. Define $r : D^2 \to S^1$ as shown:
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/b1ff72c61c1d4702.png}
  \end{center}
  This would be a retraction onto the boundary,
  contradictin the previous theorem.}
\end{proof}
\end{flashcard}