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\begin{flashcard}[nuk-notation]
\begin{notation*}
  So: 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, and does not
  actually depend on $a$.

  \glssymboldefn{nuk}{$\nu_K$}{$\nu_K$}
  Call it $\nu_K : \Hgp_n(K\subdiv) \simto
  \Hgp_n(K)$. Write
  \[ \nuk_{K, r} \defeq \cloze{\nuk_{K\rsubdiv{r - 1}} \circ
  \cdots \circ \nuk_{K\subdiv} \circ \nuk_K :
  \Hgp_n(K\rsubdiv r) \simto \Hgp_n(K)} \]
  and
  \[ \nuk_{K, r, s} \defeq \cloze{\nu_{K\rsubdiv{r - 1}}
  \circ \cdots \circ \nuk_{K\rsubdiv s} :
  \Hgp_n(K\rsubdiv r) \simto \Hgp_n(K\rsubdiv s)} .\]
\end{notation*}
\end{flashcard}

\begin{flashcard}[cont-assoc-hom-prop]
\begin{proposition*}
  To each continuous $f : \polyh|K| \to \polyh|L|$
  there is an associated homomorphism $f_* :
  \Hgp_n(K) \to \Hgp_n(L)$ given by $f_* = s_*
  \circ \nuk_{K, r}^{-1}$, where $s : K\rsubdiv r
  \to L$ is a \gls{simpapprox} to $f$ (which
  exsists for some $r \gg 0$).
  \begin{enumerate}[(i)]
    \item This does not depend on the choices of
      $s$ and $r$.
    \item If $g : \polyh|M| \to \polyh|K|$ is
      another map, then $(f \circ g)_* = f_* \circ
      g_*$.
  \end{enumerate}
\end{proposition*}

\begin{proof}
  \cloze{For (i) let $s : K\rsubdiv r \to L$ and $t :
  K\rsubdiv q \to L$ be \gls{simpapprox} to $f$,
  with $r \ge q$. Let $a : K\rsubdiv r \to
  K\rsubdiv q$ be a \gls{simpapprox} to $\id$. Now
  $t \circ a, s : K\rsubdiv K \to L$ are both
  \gls{simpapprox} to $f$, so they are
  \gls{contig}, so $\cmapstar s = \cmapstar{(t
  \circ a)} = \cmapstar t \circ \cmapstar a$. But
  $\cmapstar a = \nuk_{K, r, q}$, so
  \[ \cmapstar s \circ \nuk_{K, r}^{-1} =
  \cmapstar t \circ \nuk_{K, r, q} \circ \nuk_{K,
  r}^{-1} = \cmapstar t \circ \nuk_{K, q}^{-1} .\]
  For (ii), let $s : K\rsubdiv r \to L$ approximate
  $f$, and let $t : M\rsubdiv q \to K\rsubdiv r$
  approximate $g$. Then $s \circ t$ approximates
  $f \circ g$, and
  \[ (f \circ g)_* = (s \circ t)_* \circ \nuk_{m,
  q}^{-1} = (s_* \circ \nuk_{K, r}^{-1}) \circ
  (\nuk_{K, r} \circ t_* \circ \nuk_{m, q}^{-1}) =
  f_* \circ g_* \qedhere \]}
\end{proof}
\end{flashcard}

\begin{flashcard}[homeo-hom-simplicial-complexes-coro]
\begin{corollary*}
  If $f : \polyh|K| \to \polyh|L|$ is a
  homeomorphism, then $f_* : \Hgp_n(K) \to
  \Hgp_n(L)$ is an isomorphism.
\end{corollary*}

\begin{proof}
  \cloze{$\id = (f \circ f^{-1})_* = f_* \circ (f^{-1})_*$
  and $\id = (f^{-1} \circ f)_* = (f^{-1})_* \circ
  f_*$, so $f_*$ is invertible.}
\end{proof}
\end{flashcard}

\begin{flashcard}[simpcomp-eps-lemma]
\begin{lemma*}
  For a \gls{simpcomp} $L$ in $\RR^m$, there is a
  $\eps = \eps(L) > 0$ such that \cloze{if $f, g :
  \polyh|K| \to \polyh|L|$ satisfy $|f(x) - g(x)|
  < \eps ~\forall x \in \polyh|K|$ then $f_* = g_*
  : \Hgp_n(K) \to \Hgp_n(L)$.}
\end{lemma*}

\begin{proof}
  \cloze{The $\{\St_L(w)\}_{w \in \vset L}$ give an open
  cover of $\polyh|L|$, so by the
  \nameref{lebesgue_number_lemma}, there exists
  $\eps > 0$ such that each $B_{2\eps}(x)$ lies in
  some $\St_L(w)$. Let $f, g : \polyh|K| \to
  \polyh|L|$ be as in the statement, using this
  $\eps$. Then $\{f^{-1}(B_\eps(y))\}_{y \in
  \polyh|L|}$ is an open cover of $\polyh|K|$, so
  there is a $\delta > 0$ such that each
  $f(B_\delta(x)) \subseteq B_\eps(y)$. Then also
  $g(B_\delta(x)) \subseteq B_{2\eps}(y)$. Let $r
  \gg 0$ such that $\mesh(K\rsubdiv r) < \half
  \delta$. Then for each $v \in \vset {K\rsubdiv
  r}$, the diameter of $\St_{K\rsubdiv r}(v)$ is
  $<\delta$. So $f(\St_{K\rsubdiv r}(v))$,
  $g(\St_{K\rsubdiv r}(v))$ lie in the common
  $B_{2\eps}(w) \subseteq \St_L(w)$.
  Let $s(v) = w$. Then $s$ is a
  \gls{simpapprox} to both $f$ and $g$. So $f_* =
  s_* \circ \nuk{K, r}^{-1} = g_*$.}
\end{proof}
\end{flashcard}

\begin{flashcard}[homotopic-same-homology-thm]
\begin{theorem*}
  If $f \homotopic g : \polyh|K| \to \polyh|L|$,
  then $f_* = g_* : \Hgp_n(K) \to \Hgp_n(L)$.
\end{theorem*}

\begin{proof}
  \cloze{Let $H : \polyh|K| \times I \to \polyh|L|$ be a
  \gls{homotopy} between them. As $\polyh|K|
  \times I$ is compact, $H$ is uniformly
  continuous. For the $\eps = \eps(L) > 0$ from
  the lemma, there is a $\delta > 0$ such that $|s
  - t| < \delta$ $\implies$ $|H(x, s) - H(x, t) <
  \eps$ for all $x \in \polyh|K|$. Choose
  \[ 0 = t_0 < t_1 < \cdots < t_k = 1 \]
  such that $|t_{i + 1} - t_i| < \delta$. Then let
  $f_i(x) \defeq H(x, t_i)$. Note $f_0 = f$, $f_k
  = g$. Also, $f_i$ and $f_{i + 1}$ are
  $\eps$-close. So $(f_i)_* = (f_{i + 1})_*$, and
  so $f_* = g_*$.}
\end{proof}
\end{flashcard}

\begin{flashcard}[h-triangulation-defn]
\begin{definition*}[$h$-triangulation]
  \glsnoundefn{htri}{$h$-triangulation}{$h$-triangulations}
  \glsadjdefn{htrid}{$h$-triangulated}{space}
  \glsadjdefn{htrible}{$h$-triangulable}{space}
  \cloze{A \emph{$h$-triangulation} of a space $X$ is a
  \gls{simpcomp} $K$ and a \gls{homequivce} $g :
  \polyh|K| \to X$. We define $\Hgp_n(X) \defeq
  \Hgp_n(K)$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[htri-choice-indep-lemma]
\begin{lemma*}
  The homology of a \gls{htrid} space does not
  depend on the choice of \gls{htri}.
\end{lemma*}

\begin{proof}
  \cloze{Let $\ol{g} : \polyh|\ol{K}| \to X$ be another
  \gls{htri}. Then consider
  \[ \polyh|\ol{K}| \stackrel{\ol{g}}{\to}
  \to \polyh|K| ,\]
  where the second arrow is a homotopy inverse of
  $g$. The composition is a \gls{homequivce}, so
  induces an isomorphism $\Hgp_n(\ol{K}) \simto
  \Hgp_n(K)$. If $g : \polyh|K| \to X$, $\ol{G} :
  \polyh|\ol{K}| \to \ol{X}$ are
  \glspl{htri} with homotopy inverses $f$ and
  $\ol{f}$, and $h : X \to \ol{X}$ is a map, then
  \[ \begin{tikzcd}[ampersand replacement=\&]
    \Hgp_n(X) \ar[r] \ar[equal]{d}
    \& \Hgp_n(\ol{X}) \ar[equal]{d} \\
    \Hgp_n(K) \ar[r, "(\ol{f} \circ h \circ g)_*"]
    \& \Hgp_n(\ol{K})
  \end{tikzcd} \]
  defines induced maps on homology for any map of
  \gls{htrible} spaces.}
\end{proof}
\end{flashcard}