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Just as we did for \gls{homotopy} of \glspl{map}
between spaces, one checks:
\begin{enumerate}[(1)]
  \item being \gls{chomic} defines an equivalence
    relation on the set of \glspl{cmap} from
    $\Ccc_\bullet$ to $D_\bullet$. Write
    $f_\bullet \chomic g_\bullet$.
  \item if $a_\bullet : A_\bullet \to
    \Ccc_\bullet$ is a \gls{cmap}, and $f_\bullet
    \chomic g_\bullet : \Ccc_\bullet : D_\bullet$,
    then $f_\bullet \circ a_\bullet \chomic
    g_\bullet \circ a_\bullet$ and similarly with
    post composition.
\end{enumerate}

\begin{flashcard}[chom-equivalence-defn]
\begin{definition*}[Chain homotopy equivalence]
  \glsnoundefn{chomequiv}{chain homotopy
  equivalence}{chain homotopy equivalences}
  \cloze{A \gls{cmap} $f_\bullet : \Ccc_\bullet \to
  D_\bullet$ is a \emph{chain homotopy
  equivalence} if there is a $g_\bullet :
  D_\bullet \to \Ccc_\bullet$, $g_\bullet \circ
  f_\bullet \chomic \id_{\Ccc_{\bullet}}$,
  $f_\bullet \circ g_\bullet \chomic
  \id_{D_\bullet}$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[chomy-equiv-iso-lemma]
\begin{lemma*}
  If $f_\bullet : \Ccc_\bullet \to D_\bullet$ is a
  \gls{chomequiv}, then \cloze{$\cmapstar f :
  \Hcc_n(\Ccc_\bullet) \to \Hcc_n(D_\bullet)$ is an
  isomorphism.}
\end{lemma*}

\begin{proof}
  \cloze{Using a homotopy inverse $g_\bullet$, have
  $\cmapstar f \circ \cmapstar f = \cmapstar{(f_\bullet
  \circ g_\bullet)} = \cmapstar{(\id_{D_\bullet})}
  = \id_{\Hcc_n(D_\bullet)}$ and simlarly for
  $\cmapstar g \circ \cmapstar f =
  \id_{\Hcc_n(\Ccc_\bullet)}$.}
\end{proof}
\end{flashcard}

\textbf{Exercise:} Let
\[ \ZZ[n] = (\to 0 \stackrel{d_{n + 1}}{\to} \ZZ
\stackrel{d_n}{\to} 0 \to \cdots) .\]
Describe
\[ \{\ZZ[n] \to \Ccc_\bullet\} / \text{\gls{chom}}
.\]

\subsection{Elementary calculations}

Return to the \glspl{cc} $\Ccc_\bullet(K)$
associated to a \gls{simpcomp} $K$.

\begin{flashcard}[simp-map-well-def-hom-lemma]
\begin{lemma*}
  Let $f : K \to L$ be a \gls{simpmap}. Then the
  formula
  \begin{align*}
    f_\bullet : \Cgp_n(K)
    &\to \Cgp_n(L) \\
    [a_0, \ldots, a_n]
    &\mapsto [f(a_0), \ldots, f(a_n)]
  \end{align*}
  is a \cloze{well-defined homomorphism, and defines a
  \gls{cmap} $f_\bullet : \Cgp_\bullet(K) \to
  \Cgp_\bullet(L)$, and hence gives a $\cmapstar f
  : \Hgp_n(K) \to \Hgp_n(L)$.}
\end{lemma*}

\begin{proof}
  \cloze{To be well-defined, need the given formula to
  send $\Tgp_n(K)$ into $\Tgp_n(L)$. It does. To
  be a \gls{cmap}, need:
  \begin{align*}
    f_{n - 1} d_n[a_0, \ldots, a_n]
    &= f_{n - 1} \left( \sum_{i = 0}^n (-1)^i
    [a_0, \ldots, \skipel{a_i}, \ldots, a_n]
  \right) \\
    &= \sum_{i = 0}^n (-1)^i [f(a_0), \ldots,
    \skipel{f(a_i)}, \ldots, f(a_n)] \\
    &= d_n f_n[a_0, \ldots, a_n] \qedhere
  \end{align*}}
\end{proof}
\end{flashcard}

\begin{flashcard}[simp-comp-cone-defn]
\begin{definition*}[Cone]
  \glsnoundefn{cone}{cone}{cones}
  \glsnoundefn{conep}{cone point}{cone points}
  \cloze{Say a \gls{simpcomp} $K$ is a \emph{cone} with
  \emph{cone point} $v_0 \in \vset K$ if every
  \gls{simp} of $K$ is a face of a \gls{simp}
  which has $v_0$ as a \gls{vert}.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[cone-inclusion-point-prop]
\begin{proposition*}
  If $K$ is a \gls{cone} with \gls{conep} $v_0$,
  then the inclusion $i : \{v_0\} \injto K$
  induces a \gls{chomequiv} $i_\bullet :
  \Cgp_\bullet(\{v_0\}) \to \Cgp_\bullet(K)$ and
  so
  \[ \Hgp_n(K) \cong \begin{cases}
    \ZZ\{[v_0]\} & n = 0 \\
    0 & \text{otherwise}
  \end{cases} \]
\end{proposition*}

\begin{proof}
  \cloze{The only map $r : \vset K \to \{v_0\}$ is a
  \gls{simpmap} $r : K \to \{v_0\}$, and $r \circ
  i \chomic \id_{\{v_0\}}$. I claim that $i \circ r
  \chomic \id_{\Cgp_\bullet(K)}$. Define
  \begin{align*}
    h_n : \Ogp_n(K)
    &\to \Ogp_{n + 1}(K) \\
    [a_0, \ldots, a_n]
    &\mapsto [v_0, a_0, \ldots, a_n]
  \end{align*}
  Note that this sends $\Tgp_n(K)$ into $\Tgp_{n +
  1}(K)$, so descends to $h_n : \Cgp_n(K) \to
  \Cgp_{n + 1}(K)$. For $n > 0$, then
  \begin{align*}
    (h_{n - 1} \circ d_n + d_{n + 1} \circ
    h_n)[a_0, \ldots, a_n]
    &= \left( \sum_{i = 0}^n (-1)^i [v_0, a_0,
    \ldots, \skipel{a_i}, \ldots, a_n] \right)
    \\
    &~~~~+ \left( [a_0, \ldots, a_n] + \sum_{i = 0}^n
    (-1)^{i + 1} [v_0, a_0, \ldots, \skipel{a_i},
    \ldots, a_n] \right) \\
    &= [a_0, \ldots, a_n] \\
    &= (\id - i_n \circ r_n) [a_0, \ldots, a_n]
  \end{align*}
  since $i_n \circ r_n [a_0, \ldots, a_n] = [v_0,
  \ldots, v_0] = 0$ as $n > 0$.

  For $n = 0$,
  \begin{align*}
    (\ub{h_{-1}}_{=0} \circ d_0 + d_1 \circ h_0)
    [a_0]
    &= d_1[v_0, a_0] - [a_0] - [v_0] \\
    &= (\id - i_0 r_0)[a_0]
  \end{align*}
  So $h$ provides a \gls{chom} from
  $\id_{\Cgp_\bullet(K)}$ to $i_\bullet \circ
  r_\bullet$.}
\end{proof}
\end{flashcard}

\begin{flashcard}[standard-n-simplex-homology-coro]
\begin{corollary*}
  The standard \nsimp{n} $\ssimp^n \subset \RR^{n
  + 1}$ and all its \glspl{face}, $L$, \cloze{is a
  \gls{cone} with any \gls{vert} as \gls{conep}. So}
  \[ \Hgp_i(L) \cong \cloze{\begin{cases}
      \ZZ & i = 0 \\
      0 & \text{otherwise}
  \end{cases}} \]
\end{corollary*}
\end{flashcard}

\begin{flashcard}[n-sphere-homology-coro]
\begin{corollary*}
  Let $K$ be the union of all the proper
  \glspl{face} of $\ssimp^n \subset \RR^{n + 1}$
  (i.e. the simplicial $(n - 1)$-sphere). Then for
  $n \ge 2$, we have
  \[ \Hgp_i(K) \cong \cloze{\begin{cases}
      \ZZ & i = 0 \\
      \ZZ & i = n - 1 \\
      0 & \text{otherwise}
  \end{cases}} \]
\end{corollary*}

\begin{proof}
  \cloze{Note that $K$ is the $(n - 1)$-skeleton of $L$,
  so $i : K \to L$ gives an isomorphism $\Cgp_i(K)
  \stackrel{\cong}{\to} \Cgp_i(L)$ for $i \le n -
  1$. These \glspl{cc} are
  \[ \begin{tikzcd}[ampersand replacement=\&]
    \Cgp_0(L) \ar[equal]{d}
    \& \Cgp_1(L) \ar[l, "d_1^L", swap] \ar[equal]{d}
    % \& \Cgp_2(L) \ar[l, "d_2^L"]
    \& \cdots \ar[l, "d_2^L", swap] \ar[equal]{d}
    % \& \Cgp_{n - 2}(L) \ar[l, "d_{n - 2}^L"]
    \& \Cgp_{n - 1}(L) \ar[l, "d_{n - 1}^L", swap] \ar[equal]{d}
    \& \Cgp_n(L) \ar[l, "d_n^L", swap]
    \& 0 \ar[l, "d_{n + 1}^L", swap] \\
    \Cgp_0(K)
    \& \Cgp_1(K) \ar[l, "d_1^K", swap]
    % \& \Cgp_2(K) \ar[l, "d_2^K"]
    \& \cdots \ar[l, "d_2^K", swap]
    % \& \Cgp_{n - 2}(K) \ar[l, "d_{n - 2}^K"]
    \& \Cgp_{n - 1}(K) \ar[l, "d_{n - 1}^K", swap]
    \& 0 \ar[l, "d_n^K", swap] \ar[u]
    \& 0 \ar[l]
  \end{tikzcd} \]
  For $i \le n - 2$,
  \[ \Hgp_i(K) = \Hgp_i(L) = \begin{cases}
    \ZZ & i = 0 \\
    0 & 1 \le i \le n - 2
  \end{cases} \]
  For $i > n - 1$, $\Hgp_i(K) = 0$ as there are no
  $i$-\glspl{simp}.
  \[ \Hgp_{n - 1}(K) = \frac{\Ker(d_{n -
  1}^K)}{\Im(d_n^K)} = \Ker(d_{n - 1}^K) =
  \Ker(d_{n -1}^L) .\]
  As $\Hgp_{n - 1}(L) = 0$, $\frac{\Ker(d_{n -
  1}^L)}{\Im(d_n^L)} = 0$, so $\Ker(d_{n - 1}^L) =
  \Im(d_n^L)$. As $\Hgp_n(L) = 0$, we see that
  $d_n^L : \ZZ \cong \Cgp_n(L) \to \Cgp_{n -
  1}(L)$ is injective. So $\Hgp_{n - 1}(K) =
  \Ker(d_{n - 1}^L) = \Im(d_n^L) \cong \ZZ$. It is
  generated by $d_n^L[e_1, \ldots, e_n] \in \Cgp_{n
  - 1}(K)$.}
\end{proof}
\end{flashcard}
