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a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index e0257d3..7cecc66 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -701,7 +701,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \input{figures/topology-triangle-to-line.tex} \item Tori können simplizial auf Sphären abgebildet werden: - + \resizebox{0.9\linewidth}{!}{\input{figures/topology-2.tex}} \end{bspenum} \end{beispiel} @@ -792,7 +792,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \item Jeder zusammenhängende Graph $\Gamma$ enthält einen Teilbaum $T$, der alle Ecken von $\Gamma$ enthält.% \footnote{$T$ wird \enquote{Spannbaum} genannt.} - \item Ist $n = a_1(\Gamma) = a_1(T)$, so ist $\chi(\Gamma) = 1 - n$. + \item Ist $n = a_1(\Gamma) - a_1(T)$, so ist $\chi(\Gamma) = 1 - n$. \end{bemenum} \end{bemerkung} @@ -829,7 +829,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \end{figure} \begin{beweis} - $\chi(K) = \chi(\Delta) - \underbrace{\underbrace{(-1)^n}_{n-\text{Simplex}} + \sum_{k=0}^n (-1)^k}_{(1+(-1))^{n+1}} = \chi(\Delta) \qed$ + $\chi(K) = \chi(\Delta) - \underbrace{\underbrace{(-1)^n}_{n\text{-Simplex}} + \sum_{k=0}^n (-1)^k \binom{n+1}{k}}_{(1+(-1))^{n+1}} = \chi(\Delta) \qed$ \end{beweis} \begin{satz}[Eulersche Polyederformel]\xindex{Eulersche Polyederformel}% @@ -844,7 +844,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \begin{enumerate}[label=\arabic*)] \item Die Aussage ist richtig für den Tetraeder. \item \Obda{} sei $0 \in P$ und $P \subseteq \fB_1(0)$. Projeziere - $0P$ von $0$ aus auf $\partial \fB_1(0) = S^2$. + $\partial P$ von $0$ aus auf $\partial \fB_1(0) = S^2$. Erhalte Triangulierung von $S^2$. \item Sind $P_1$ und $P_2$ konvexe Polygone und $T_1, T_2$ die zugehörigen Triangulierungen von $S^2$, so gibt es @@ -863,7 +863,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \end{beweis} \begin{bemerkung}[Der Rand vom Rand ist 0]\label{kor:9.11} - Sei $K$ ein \todo{Warum in Klammern?}{(endlicher)} Simplizialkomplex mit Knotenmenge $V$ + Sei $K$ ein endlicher Simplizialkomplex mit Knotenmenge $V$ und $<$ eine Totalordnung auf $V$. Sei $A_n$ die Menge der $n$-Simplizes in $K$, d.~h. @@ -890,11 +890,23 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \caption{Simplizialkomplex mit Totalordnung} \end{figure} - $a < b < c$ + Sei $a < b < c$. Dann gilt: - $d_2 \sigma = e_1 - e_2 + e_3 = (c - b) - (c-a) + (b - a) = 0$ + \begin{align*} + d_2 \sigma &= e_1 - e_2 + e_3\\ + d_1(e_1- e_2 + e_3) &= (c - b) - (c-a) + (b - a)\\ + &= 0 + \end{align*} - \todo[inline]{Beispiel auf Tetraeder übertragen} + Sei $a, very thick] (a) edge node[label=below:$e_3$] {} (b); \draw[->, very thick] (b) edge node[label=right:$e_1$] {} (c); - \draw[->, very thick] (c) edge node[label=left:$e_2$] {} (a); + \draw[->, very thick] (a) edge node[label=left:$e_2$] {} (c); \end{tikzpicture} diff --git a/tikz/topology-oriented-triangle/topology-oriented-triangle.png b/tikz/topology-oriented-triangle/topology-oriented-triangle.png index 7e535752f70eadb2515165adc490b5fcd22f8001..171e720671ddac9c0ed8afe977880455388cf7bb 100644 GIT binary patch literal 19191 zcmd432Uk_B1j9pt3=&OkRVlQ zQBhQyMhP7PiWmXuz2wfg_c`a@U+{g7;Rq|qnrpsuzWsS-?wqzU=iha37X$*~$5^1x zLLe~kC-fT+7kD{;{IL~y;Rrd2ImZM3#qf9&z%_5Mg<}W=V%Ew2551fO-vVzQ2sL#G zwF|Ka{dKtzT1Sr 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