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Martin Thoma 2014-01-30 18:38:37 +01:00
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4 changed files with 9 additions and 18 deletions
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documents/GeoTopo/GeoTopo.pdf
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documents/GeoTopo/Kapitel3.tex
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@ -54,7 +54,7 @@
H''(t, 2s-1) &\text{falls } \frac{1}{2} \leq s \leq 1\end{cases}$
$\Rightarrow$ $H$ ist stetig und Homotopie von $\gamma_1$ nach
$\gamma_3$
$\gamma_3$.
\end{itemize}
$\qed$
\end{beweis}
@ -177,7 +177,6 @@
\begin{figure}[htp]
\centering
%\includegraphics[width=0.5\linewidth, keepaspectratio]{figures/todo/skizze-bemerkung-10-6.jpg}
\input{figures/topology-homotop-paths-2.tex}
\caption{Situation aus \cref{kor:bemerkung-10-6}}.
\label{fig:situation-bemerkung-10-6}
@ -193,8 +192,8 @@
H_2(2t-1,s) &\text{falls } \frac{1}{2} \leq t \leq 1
\end{cases}\]
Homotopie zwischen $\gamma_1 * \gamma_2$ und $\gamma_1' * \gamma_2 '$ (!)
\todo[inline]{Hier fehlt noch was}
eine Homotopie zwischen
$\gamma_1 * \gamma_2$ und $\gamma_1' * \gamma_2 '$.
\end{beweis}
\section{Fundamentalgruppe}
@ -327,7 +326,7 @@ Wenn $\pi_1(X,x) = \Set{e}$ für ein $x \in X$ gilt, dann wegen
\begin{beispiel}
\begin{bspenum}
\item $f:S^1 \hookrightarrow \mdr^2$ ist injektiv, aber
$f_*:\pi_1(S^1, 1) \cong \mdz \rightarrow \pi_1(\mdr^2, 1) = 0 \Set{e}$
$f_*:\pi_1(S^1, 1) \cong \mdz \rightarrow \pi_1(\mdr^2, 1) = \Set{e}$
ist nicht injektiv
\item $f: \mdr \rightarrow S^1, t \mapsto (\cos 2 \pi t, \sin 2 \pi t)$
ist surjektiv, aber $f_*: \pi_1(\mdr, 0) = \Set{e} \rightarrow \pi_1(S^2, 1) \cong \mdz$
@ -658,16 +657,6 @@ Haben wir Häufungspunkt definiert?}
mit $\tilde{\gamma}(0)=y$ und $p \circ \tilde{\gamma} = \gamma$.
\end{satz}
\begin{beweis}
Existenz: Siehe \Cref{fig:satz-12.6}.
\begin{figure}[htp]
\centering
\includegraphics[width=0.6\linewidth, keepaspectratio]{figures/todo/skizze-1.jpg}
\caption{Skizze für den Beweis von \cref{thm:ueberlagerung-weg-satz-12.6}}
\label{fig:satz-12.6}
\end{figure}
\end{beweis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Sebastians Mitschrieb vom 17.12.2013 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -744,12 +733,14 @@ $p|V_j: V_j \rightarrow U$ Homöomorphismus.
nur von $[\gamma] \in \pi_1(X,x_0)$ ab.
Für geschlossene Wege $\gamma_0, \gamma_1$ um $x$ gilt:
\begin{align*}
\tilde{\gamma_0}(1) &= \tilde{\gamma_1}(1)\\
\Leftrightarrow [\tilde{\gamma_0} * \tilde{\gamma_1}^{-1}] &\in \pi_1(Y, y_0)\\
\Leftrightarrow [\gamma_0 * \gamma_1^{-1}] &\in p_* (\pi_1(Y,y_0))\\
\Leftrightarrow [\gamma_0] \text{ und } [\gamma_1] \text{liegen in der selben Nebenklasse bzgl.} p_*(\pi_1(Y, y_0))
\Leftrightarrow [\gamma_0] \text{ und } [\gamma_1] &\text{liegen in der selben Nebenklasse bzgl.} p_*(\pi_1(Y, y_0))
\end{align*}
Zu $i \in \Set{0, \dots, d-1}$ gibt es Weg $\delta_i$ in
$Y$ mit $\delta_i(0) = y_0$ und $\delta_i(1) = y_i$\\
$\Rightarrow p \cup \delta_i$ ist geschlossener Weg in

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documents/GeoTopo/figures/topology-homotop-paths-2.tex
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@ -22,5 +22,5 @@
\draw [rounded corners, dashed] (b) edge[bend right=-20] (c);
\draw [rounded corners, dashed] (b) edge[bend right=10] (c);
\draw [rounded corners, dashed] (b) edge[bend right=-10] (c);
\draw [rounded corners, thick, orange] (b) edge[bend left] node[above] {$\gamma_1'$} (c);
\draw [rounded corners, thick, orange] (b) edge[bend left] node[above] {$\gamma_2$} (c);
\end{tikzpicture}

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tikz/topology-homotop-paths-2/topology-homotop-paths-2.tex
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@ -28,6 +28,6 @@
\draw [rounded corners, dashed] (b) edge[bend right=-20] (c);
\draw [rounded corners, dashed] (b) edge[bend right=10] (c);
\draw [rounded corners, dashed] (b) edge[bend right=-10] (c);
\draw [rounded corners, thick, orange] (b) edge[bend left] node[above] {$\gamma_1'$} (c);
\draw [rounded corners, thick, orange] (b) edge[bend left] node[above] {$\gamma_2$} (c);
\end{tikzpicture}
\end{document}