diff --git a/images/singular_chaincomplex_small.pdf b/images/singular_chaincomplex_small.pdf new file mode 100644 index 0000000..9bfd5b5 Binary files /dev/null and b/images/singular_chaincomplex_small.pdf differ diff --git a/images/singular_chaincomplex_small.svg b/images/singular_chaincomplex_small.svg new file mode 100644 index 0000000..a48797a --- /dev/null +++ b/images/singular_chaincomplex_small.svg @@ -0,0 +1,951 @@ + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/presentation/presentation.tex b/presentation/presentation.tex index 49c34be..77f660b 100644 --- a/presentation/presentation.tex +++ b/presentation/presentation.tex @@ -40,7 +40,7 @@ \begin{frame} \frametitle{Voorbeeld} \centering \vspace{-0.5cm} - Bekijk $\Delta^n \tot{f} X$,\, dwz...\, \raisebox{-.2\height}{\includegraphics{simplex_in_X}} + Bekijk $\Delta^n \tot{f} X$,\, dwz.\, \raisebox{-.2\height}{\includegraphics{simplex_in_X}} \bigskip \bigskip @@ -52,31 +52,30 @@ \begin{frame} \frametitle{Interessant?} - Gegeven een ketencomplex $C$: - $$ \cdots \to C_4 \tot{\del_3} C_3 \tot{\del_2} C_2 \tot{\del_1} C_1 \tot{\del_0} C_0 $$ - met $\del_n \circ \del_{n+1} = 0$ - \bigskip + Gegeven een ketencomplex $C$: \\ + $ \cdots \tot{\del_2} C_2 \tot{\del_1} C_1 \tot{\del_0} C_0 $ met $\del_n \circ \del_{n+1} = 0$ + \bigskip\bigskip + Dan geldt $im(\del_{n+1}) \trianglelefteq ker(\del_n)$ Definieer: $H_n(C) = ker(\del_{n-1}) / im(\del_n)$ - (met $ker(\del_0) = C_0$ per conventie) + met $ker(\del_{-1}) = C_0$ \end{frame} \begin{frame} \frametitle{Voorbeeld} - $ \cdots \to C_1 \tot{\del_0} C_0 $, wat is $ H_1 = \frac{ker(\del_0)}{im(\del_1)} $? + \raisebox{-.2\height}{\includegraphics[width=0.7\textwidth]{singular_chaincomplex_small}}, $ H_1 = \frac{ker(\del_0)}{im(\del_1)} $? \bigskip \begin{tabular}{m{0.3\textwidth} m{0.7\textwidth}} \includegraphics<1>{singular_homology1} \includegraphics<2->{singular_homology2} & - $ \sigma_1 - \sigma_2 + \sigma_3 \in ker (\del_0) $ \newline - \visible<2->{ - $ \del_1(\tau) = \sigma_1 - \sigma_2 + \sigma_3 $ \newline + $\sigma_1 - \sigma_2 + \sigma_3 \in ker (\del_0) $ \newline + \visible<2->{$\del_1(\tau) = \sigma_1 - \sigma_2 + \sigma_3 $ \newline Dus $ \sigma_1 + \sigma_2 - \sigma_3 \in im (\del_1) $ \newline Dus $ 0 = [\sigma_1 - \sigma_2 + \sigma_3] \in H_1 $} \end{tabular}