%! TEX root = AlgT.tex
% vim: tw=50
% 22/01/2024 11AM

\begin{flashcard}[homoteq-basics-prop]
\begin{proposition*}
  We have
  \begin{enumerate}[(i)]
    \item $X \homoteq X$,
    \item If $X \homoteq Y$ then $Y \homoteq X$,
    \item If $X \homoteq Y$ and $Y \homoteq Z$,
      then $X \homoteq Z$.
  \end{enumerate}
\end{proposition*}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(i)]
    \item \cloze{Take $f = g = \id_X$, and constant
      \glspl{homotopy}.}
    \item \cloze{Given $f : X \to Y$, $g : Y \to X$ with
      $f \circ g \homotopic_H \id_Y$, $g \circ f
      \homotopic_G \id_X$, this is the same data as
      a $Y \homoteq X$.}
    \item \cloze{Suppose we have \glspl{map} $f : X \to Y$, $f'
      : Y \to Z$, $g : Y \to X$, $g' : Z \to Y$,
      with $f \circ g \homotopic \id_X$, $f'
      \circ g' \homotopic \id_X$, $f' \circ g'
      \homotopic \id_Z$, $g' \circ f' \homotopic
      \id_Y$. Consider $f' \circ f : X \to Z$, $g
      \circ g' : Z \to X$. Then
      \[ (g \circ g') \circ (f' \circ f) = g \circ
      (g' \circ f') \circ f \homotopic g \circ
      \id_Y \circ f = g \circ f \homotopic \id_X ,\]
      and the other composition is similar.}
      \qedhere
  \end{enumerate}
\end{proof}
\end{flashcard}

\begin{flashcard}[retraction-defn]
\begin{definition*}[Retraction]
  \glsverbdefn{retract}{retract}{retract}
  \cloze{If $i : A \to X$ is the inclusion of a subspace,
  then
  \begin{enumerate}[(i)]
    \item A \emph{retraction} is a $r : X \to
      A$ such that $r \circ i = \id_A$.
    \item A \emph{deformation} retraction is a
      retraction such that also $i \circ r
      \homotopic \id_X$.
  \end{enumerate}
  }
\end{definition*}
\end{flashcard}

\subsubsection*{Paths}

\begin{flashcard}[path-defn]
\begin{definition*}[Path]
  \glsnoundefn{path}{path}{paths}
  \cloze{For a space $X$ and points $x_0, x_1 \in X$, a
  path from $x_0$ to $x_1$ is a \gls{map} $\gamma : I =
  [0, 1] \to X$ such that $\gamma(0) = x_0$,
  $\gamma(1) = x_1$.

  If $x_0 = x_1$, call $\gamma$
  a \emph{loop} based at $x_0$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[concatenation-notation]
\begin{notation*}[Concatenation]
  \glssymboldefn{concat}{$\cdot$}{$\cdot$}
  \cloze{If $\gamma$ is a \gls{path} from $x_0$ to $x_1$,
  and $\gamma'$ is a path from $x_1$ to $x_2$,
  then we can form the \emph{concatenation}
  $\gamma \cdot \gamma' : I \to X$ via
  \[ (\gamma \cdot \gamma')(t) = \begin{cases}
    \gamma(2t) & 0 \le t \le \half \\
    \gamma'(2t + 1) & \half \le t \le 1
  \end{cases} \]
  (continuous by the \nameref{gluing_lemma}). This
  is a \gls{path} from $x_0$ to $x_2$.}
\end{notation*}
\end{flashcard}

\begin{flashcard}[inverse-path-notation]
\begin{notation*}[Inverse path]
  \glssymboldefn{inverse_path}{$\gamma^{-1}$}{$\gamma^{-1}$}
  \cloze{Define the \emph{inverse} $\gamma^{-1} : I \to
  X$ via
  \[ \gamma^{-1}(t) = \gamma(1 - t) .\]
  This is a \gls{path} from $x_1$ to $x_0$.}
\end{notation*}
\end{flashcard}

\begin{flashcard}[constant-path-notation]
\begin{notation*}[Constant path]
  \glssymboldefn{constpath}{$c_{x_0}$}{$c_{x_0}$}
  \cloze{Define the \emph{constant path} $c_{x_0} : I \to
  X$ via $c_{x_0}(t) = x_0$.}
\end{notation*}
\end{flashcard}

\begin{flashcard}[path-components-defn]
\begin{definition*}[Path components]
  \glssymboldefn{piz}{$\pi_0$}{$\pi_0$}
  \glsadjdefn{pathconn}{path-connected}{space}
  \cloze{
  Using the above, we can define an equivalence
  relation $\sim$ on $X$ via
  \[ X_0 \sim X_1 \iff \text{there exists a
  \gls{path} $\gamma$ from $x_0$ to $x_1$} \]
  The equivalence classes of $\sim$ are called
  \emph{path components} of $X$. Say $X$ is
  \emph{path-connected} if there is only $1$
  equivalence class. Let $\pi_0(X) \defeq X /
  \sim$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[locally-path-connected-defn]
\begin{definition*}[Locally path-connected]
  \glsadjdefn{locpathconn}{locally
  path-connected}{space}
  \cloze{We say that a space $X$ is \emph{locally
  path-connected} if for every $x \in X$, and
  neighbourhood $U \ni x$, there
  exists a smaller neighbourhood $U \supset V \ni x$
  such that $V$ is \gls{pathconn}.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[pi0-function-quotient-prop]
\begin{proposition*}
  For a \gls{map} $f : X \to Y$, there is a
  well-defined function $\piz(f) :
  \cloze{\piz(X) \to \piz(Y)}$
  given by $\cloze{\piz(f)([x]) = [f(x)]}$.
  Furthermore:
  \begin{enumerate}[(i)]
    \item \cloze{If $f \homotopic g$ then $\piz(f)
      = \piz(g)$.}
    \item \cloze{If $A \stackrel{h}{\to} B
      \stackrel{k}{\to} C$ then $\piz(k \circ h) =
    \piz(k) \circ \piz(h)$.}
  \item \cloze{$\piz(\id_X) = \id_{\piz(X)}$.}
  \end{enumerate}
\end{proposition*}

\begin{proof}
  To see well-defined, let $[x] = [x']$. Then
  there is a \gls{path} $\gamma$ from $x$ to $x'$. Then
  $f \circ \gamma : I \stackrel{\gamma}{\to} X
  \stackrel{f}{\to} Y$ is a \gls{path} for $f(x)$
  to $f(x')$, so $[f(x)] = [f(x')]$. Properties
  (ii) and (iii) are immediate. For (i), let $H :
  X \times I \to Y$ be a \gls{homotopy} from $f$
  to $g$. Now $H|_{\{x\} \times I} : \{x\} \times
  I \to Y$ is a \gls{path} from $f(x)$ to $g(x)$,
  so
  \[ \piz(f)([x]) = [f(x)] = [g(x)] = \piz(g)([x])
  . \qedhere \]
\end{proof}
\end{flashcard}

\begin{flashcard}[homoteq-piz-bijection-coro]
\begin{corollary*}
  If $f : X \to Y$ is a \gls{homequivce}, then
  $\piz(f)$ \cloze{is a bijection.}
\end{corollary*}

\begin{proof}
  If $g : Y \to X$ is a \gls{homotopy} inverse,
  then
  \[ \piz(f) \circ \piz(g) = \piz(f \circ g) =
  \piz(\id_Y) = \id_{\piz(Y)} \]
  and $\piz(g) \circ \piz(f) = \id_{\piz(X)}$, so
  $\piz(f)$ is a bijection.
\end{proof}
\end{flashcard}

\begin{example*}
  The space $\{-1, +1\}$ with the discrete
topology is \emph{not} \gls{contractible}. This is
  because any \gls{path} in this space is constant, so
  \[ \piz(\{-1, +1\}) = \{-1, +1\} \]
  of cardinality $2$, so $\piz(\{*\}) = \{*\}$ has
  cardinality $1$.
\end{example*}

\begin{example*}
  The space $[-1, 1]$ does \emph{not}
  \gls{retract} onto $\{-1, +1\}$. Suppose it
  does. Then:
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/aa39bf94def84a65.png}
  \end{center}
\end{example*}

\begin{flashcard}[homotopic-paths-defn]
\begin{definition*}[Homotopic paths]
  \glsadjdefn{hompath}{homotopic as paths}{paths}
  \glssymboldefn{hompath}{$\simeq$}{$\simeq$}
  \cloze{Two \glspl{path} $\gamma, \gamma' : I \to X$
  both from $x_0$ to $x_1$ are called
  \emph{homotopic as paths} if they are
  \gls{homotopic} relative to $\{0, 1\} \subset I$
  as in the last lecture. So $\gamma
  \homotopic \gamma' \homrel{\{x_0, x_1\}}$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[hom-path-concat-lemma]
\begin{lemma*}
  If $\gamma_0 \hompath \gamma_1$ as \glspl{path}
  from $x_0$ to $x_1$, and $\gamma_0' \hompath
  \gamma_1'$ as \glspl{path} from $x_1$ to $x_2$,
  then $\gamma_0 \concat \gamma_0' \hompath
  \gamma_1 \concat \gamma_1'$ as \glspl{path} from
  $x_0$ to $x_2$.
\end{lemma*}

\begin{proof}
  \cloze{Let $H$ be the \gls{homotopy} from $\gamma_0$ to
  $\gamma_1$ relative to $\{x_0, x_1\}$, and $H'$
  the \gls{homotopy} from $\gamma_0'$ to
  $\gamma_1'$ relative to $x_1$ and $x_2$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/f055a5061cc54256.png}
  \end{center}
  \[ H''(s, t) = \begin{cases}
    H(2s, t) & 0 \le s \le \half \\
    H'(2s - 1, t) & \half \le s \le 1
  \end{cases} \]
  (continuous as usual by the
  \nameref{gluing_lemma}), is a \gls{homotopy}
  from $\gamma_0 \concat \gamma_0'$ to $\gamma_1
  \concat \gamma_1'$ relative to $x_0$ and $x_2$.
  }
\end{proof}
\end{flashcard}