%! TEX root = AlgT.tex
% vim: tw=50
% 04/03/2024 11AM

\begin{flashcard}[Mayer-Vietoris-thm]
\begin{theorem*}[Mayer-Vietoris Theorem]
  Let $K$ be a \gls{simpcomp}, $M$, $N$ be
  subcomplexes, and $L = M \cap N$. Suppose also
  that $M$ and $N$ cover $K$, i.e. every
  \gls{simp} of $K$ is in $M$ or $N$. Write
  \[ \begin{tikzcd}[ampersand replacement=\&]
    L \ar[r, "j", hook] \ar[d, "i", hook]
    \& N \ar[d, "l", hook] \\
    M \ar[r, "k", hook]
    \& K
  \end{tikzcd} \]
  for the inclusion maps. There are natural
  homomorphisms $\partial_* : \Hcc_n(K) \to
  \Hcc_{n - 1}(L)$ such that
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/ad35272b1a474ddd.png}
  \end{center}
  is a long \gls{exs}.
\end{theorem*}

\begin{proof}
  \cloze{Have a \gls{sescc}
  \[ 0 \to \Cgp_\bullet(L) \stackrel{(\cmapstar i,
  \cmapstar j)}{\longrightarrow} \Cgp_\bullet(M)
  \oplus \Cgp_\bullet(N) \stackrel{\cmapstar k -
  \cmapstar l}{\longrightarrow} \Cgp_\bullet(K) \to
  0 \]
  because:
  \begin{enumerate}[(i)]
    \item $i_\bullet$ and $j_\bullet$ are both
      injective, so \gls{exact} at left-hand term.
    \item $k_\bullet$ and $l_\bullet$ are jointly
      surjective, as $M$ and $N$ cover $K$.
    \item Suppose $(x, y) \in \Cgp_n(M) \oplus
      \Cgp_n(N)$ is in $\Ker(k_n - l_n)$, i.e.
      $k_n(x) = l_n(y)$. So both sides are a
      linear combination of simplices in $M \cap N
      = L$, i.e. $\exists z \in \Cgp_n(L)$ such
      that $x = i_n(z)$, $y = j_n(z)$. So
      $\Ker(k_n - l_n) \subseteq \Im(i_n \oplus j_n)$.
      Other constraint is clear.
  \end{enumerate}
  Now apply the above algebraic theorem.}
\end{proof}
\end{flashcard}

\subsection{Continuous maps and homotopy inverse}

\begin{flashcard}[contiguous-simp-maps-defn]
\begin{definition*}
  \glsadjdefn{contig}{contiguous}{\gls{simpmap}}
  \cloze{\glsref[simpmap]{Simplicial maps} $f, g : K \to
  L$ are \emph{contiguous} if for each $\sigma \in
  K$, $f(\sigma)$ and $g(\sigma)$ are \glspl{face}
  of some \gls{simp} $\tau$ of $L$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[simp-approx-contig-lemma]
\begin{lemma*}
  If $f$ and $g$ are both \glspl{simpapprox} to $F
  : \polyh|K| \to \polyh|L|$, then $f$ and $g$ are
  \gls{contig}.
\end{lemma*}

\begin{proof}
  \cloze{If $x \in \interior{\sigma}\subset \polyh|K|$
  and $F(x) \in \interior \tau \subset \polyh|L|$
  then as in the proof of
  ``\nameref{simpapprox_are_simpmap}'',
  $f(\sigma)$ and $g(\sigma)$
  are \glspl{face} of $\tau$.}
\end{proof}
\end{flashcard}

\begin{flashcard}[contig-implies-chomic-lemma]
\begin{lemma*}
  If $f, g : K \to L$ are \gls{contig}, then
  $f_\bullet \chomic g_\bullet : \Cgp_\bullet(K)
  \to \Cgp_\bullet(L)$, and so $\cmapstar f =
  \cmapstar g$.
\end{lemma*}

\begin{proof}
  Choose an ordering $\prec$ of $\vset K$, and
  represent a basis of $\Cgp_n(K)$ by $[a_0,
  \ldots, a_n]$ with $a_0 \prec a_1 \prec \cdots
  \prec a_n$. Define a homomorphism
  \begin{align*}
    h_n : \Cgp_{n + 1}(K)
    &\to \Cgp_{n + 1}(L) \\
    [a_0, \ldots, a_n]
    &\mapsto \sum_{i = 0}^n (-1)^i [f(a_0),
    \ldots,
    f(a_i), g(a_i), \ldots, g(a_k)]
  \end{align*}
  Direct calculation shows that
  \[ d_{n + 1} \circ h_n + h_{n - 1} \circ d_n =
  g_n - f_n .\]
  This is the \gls{chom}.
\end{proof}
\end{flashcard}


\begin{flashcard}[simpapprox-to-id-lemma]
\begin{lemma*}
  Let $K\subdiv$ be the \gls{bsubdiv}, choose $a : K \cong
  \vset{K\subdiv} \to \vset K$ any function sending
  $\bcentre{\sigma}$ to some \gls{vert} of $\sigma$.
  Such an $a : \vset{K\subdiv} \to \vset K$ is a
  \gls{simpapprox} to the identity map on
  $\polyh|K| = \polyh|K\subdiv|$. Furthermore, all
  \gls{simpapprox} to $\id$ are of this form.
\end{lemma*}

\begin{proof}
  \cloze{Seen the first part (just before the definition
  of \gls{mesh}). If $g : \vset{K\subdiv} \to
  \vset K$ is a \gls{simpapprox} to $\id$, then
  \[ \interior{\sigma} \subseteq
  \id(\St_{K\subdiv}(\bcentre{\sigma}))
  \subseteq \St_K(g(\bcentre{\sigma})) .\]
  So $g(\bcentre{\sigma})$ must be a \gls{vert} of
  $\sigma$.}
\end{proof}
\end{flashcard}

\begin{flashcard}[subdiv-simpapprox-id-iso-prop]
\begin{proposition*}
  If $a : K\subdiv \to K$ is a \gls{simpapprox} to
  $\id$, then $\cmapstar a : \Hgp_n(K\subdiv) \to
  \Hgp_n(K)$ is an isomorphism for all $n$.
  Furthermore, this isomorphism is independent of
  the choice of $a$.
\end{proposition*}

\begin{proof}
  \cloze{First suppose $K$ is a \gls{simp} $\sigma =
  \ssimp^n \subset \RR^{n + 1}$. Then $K\subdiv$
  is a \gls{cone} with \gls{conep}
  $\bcentre{\sigma}$. Also $K$ is a \gls{cone}
  with any \gls{vert} as \gls{conep}. So
  $\cmapstar a : \Hgp_n(K\subdiv) \to \Hgp_n(K)$
  is an isomorphism for $n > 0$ (both sides are
  $0$). Also, $\cmapstar \Hgp_0(K\subdiv) \to
  \Hgp_0(K)$ is an isomorphism too, with
  $\cmapstar a[\bcentre{\sigma}] =
  [a(\bcentre{\sigma})]$. So done if $K$ is a
  \gls{simp}.

  \textbf{General case}: double induction on:
  \begin{enumerate}[(i)]
    \item $\dim K$
    \item number of \glspl{simp} of $K$
  \end{enumerate}
  Let $\sigma \in K$ be a \gls{simp} of maximal
  dimansion. Then $L = K - \{\sigma\}$ is a
  \gls{simpcomp}. Let
  \[ S = \{\text{$\sigma$ and all its faces}\},
  \qquad T = S \cap L = \{\text{all proper faces
  of $\sigma$}\} .\]
  Any \gls{simpapprox} $a : K\subdiv \to K$ to
  $\id$ sends $L\subdiv$ into $L$, $S\subdiv$ into
  $S$, $T\subdiv$ into $T$. So we get a map of
  Mayer-Vietoris sequences:
  \[ \begin{tikzcd}[ampersand replacement=\&]
    \Hgp_n(T\subdiv) \ar[r] \ar[d, "\simeq"
    description, "\cmapstar{(a|_{T\subdiv})}"]
    \& \Hgp_n(S\subdiv) \oplus \Hgp_n(L\subdiv)
    \ar[r] \ar[d, "\simeq" description,
    "\cmapstar{(a_{K\subdiv})} \oplus
    \cmapstar{(a|_{L\subdiv})}"]
    \& \Hgp_n(K\subdiv) \ar[r, "\partial_*"]
    \ar[d, "\cmapstar a"]
    \& \Hgp_{n - 1}(T\subdiv) \ar[r] \ar[d,
    "\simeq" description, "\cmapstar{(a_{T\subdiv})}"]
    \& \Hgp_{n - 1}(S\subdiv) \oplus \Hgp_{n -
    1}(L\subdiv) \ar[d, "\simeq" description] \\
    \Hgp_n(T) \ar[r]
    \& \Hgp_n(S) \oplus \Hgp_n(L) \ar[r]
    \& \Hgp_n(K) \ar[r, "\partial_*"]
    \& \Hgp_{n - 1}(T) \ar[r]
    \& \Hgp_{n - 1}(S) \oplus \Hgp_{n - 1}(L)
  \end{tikzcd} \]
  $T\subdiv$ has dimension strictly smaller than
  $K$, so by induction hypothesis, we may assume
  that $\cmapstar{(a|_{T\subdiv})}$ and
  $\cmapstar{(a|_{T\subdiv})}$ are isomorphisms.
  We can also use the induction hypothesis to
  assume that the second and fifth downward maps
  are isomorphisms: we have an isomorphism on the
  left since $S$ is a \gls{simp} (and we did the
  \gls{simp} case earlier), and on the right
  factor we have strictly fewer \glspl{simp}, so
  we use the induction hypothesis.

  Now by the ``Five lemma'' (see \es[3]{4}), we
  deduce that $\cmapstar a$ is an isomorphism.}
\end{proof}
\end{flashcard}
