From 5d76d15a6b4d8f3cf11f9bf07189a9b68cc18de5 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Tue, 17 Dec 2013 16:33:14 +0100 Subject: [PATCH] Vorlesung vom 17.12.2013 digitalisiert --- .../torus-three-paths/torus-three-paths.tex | 2 +- documents/GeoTopo/Arbeitszeit.txt | 2 + documents/GeoTopo/Bildquellen.tex | 1 + documents/GeoTopo/GeoTopo.pdf | Bin 5397901 -> 5677525 bytes documents/GeoTopo/Kapitel1.tex | 6 +- documents/GeoTopo/Kapitel3.tex | 192 ++++++++++++++++-- documents/GeoTopo/Readme.md | 9 + .../figures/todo/torus-three-paths.jpg | Bin 494639 -> 0 bytes documents/GeoTopo/shortcuts.sty | 1 + 9 files changed, 196 insertions(+), 17 deletions(-) delete mode 100644 documents/GeoTopo/figures/todo/torus-three-paths.jpg diff --git a/asymptote/torus-three-paths/torus-three-paths.tex b/asymptote/torus-three-paths/torus-three-paths.tex index 61355c1..b99895e 100644 --- a/asymptote/torus-three-paths/torus-three-paths.tex +++ b/asymptote/torus-three-paths/torus-three-paths.tex @@ -2,7 +2,7 @@ \usepackage{asymptote} \begin{document} \begin{asy} -settings.render = 8; +settings.render = 0; settings.prc = false; import graph3; diff --git a/documents/GeoTopo/Arbeitszeit.txt b/documents/GeoTopo/Arbeitszeit.txt index 160a738..1a41d44 100644 --- a/documents/GeoTopo/Arbeitszeit.txt +++ b/documents/GeoTopo/Arbeitszeit.txt @@ -9,3 +9,5 @@ Datum | Uhrzeit 13.12.2013 | 13:10 - 13:47 14.12.2013 | 13:00 - 14:45 15.12.2013 | 20:30 - 21:20 +16.12.2013 | 15:00 - 15:30 +17.12.2013 | 07:30 - 07:45, 14:30 - 15:40 diff --git a/documents/GeoTopo/Bildquellen.tex b/documents/GeoTopo/Bildquellen.tex index 80b2f90..11b05c6 100644 --- a/documents/GeoTopo/Bildquellen.tex +++ b/documents/GeoTopo/Bildquellen.tex @@ -21,5 +21,6 @@ modifiziert. \item[Abb. \ref{fig:reidemeister-zuege}] Reidemeister-Züge: YAMASHITA Makoto (\href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{1}, \href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{2}, \href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{3}) \item[Abb. \ref{fig:treefoil-knot-three-colors}] Kleeblattknoten, 3-Färbung: Jim.belk, \href{https://commons.wikimedia.org/wiki/File:Tricoloring.png}{commons.wikimedia.org/wiki/File:Tricoloring.png} \item[Abb. \ref{fig:double-torus}] Doppeltorus: Oleg Alexandrov, \href{https://commons.wikimedia.org/wiki/File:Double_torus_illustration.png}{commons.wikimedia.org/wiki/File:Double\_torus\_illustration.png} + \item[Abb. \ref{fig:torus-three-paths}] 3 Pfade auf Torus: \href{http://tex.stackexchange.com/users/484/charles-staats}{Charles Staats}, \href{http://tex.stackexchange.com/a/149991/5645}{tex.stackexchange.com/a/149991} \item[Abb. \ref{fig:ueberlappung-r1-spirale-s1}] Überlappung vom $S^1$ mit $\mdr$: \href{http://tex.stackexchange.com/users/22467/alex}{Alex}, \href{http://tex.stackexchange.com/a/149706/5645}{tex.stackexchange.com/a/149706} \end{itemize} diff --git a/documents/GeoTopo/GeoTopo.pdf 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\input{figures/stereographic-projection} + \resizebox{0.9\linewidth}{!}{\input{figures/stereographic-projection}} \caption{Visualisierung der stereographischen Projektion} \label{fig:stereographic-projection} \end{figure} @@ -865,11 +865,11 @@ $\qed$ \begin{figure}[htp] \centering \subfloat[Spirale $S$ mit Kreis $C$]{ - \input{figures/topology-spiral} + \resizebox{0.25\linewidth}{!}{\input{figures/topology-spiral}} \label{fig:topology-spiral} }% \subfloat[Sinus]{ - \input{figures/topology-sinx.tex} + \resizebox{0.65\linewidth}{!}{\input{figures/topology-sinx.tex}} \label{fig:sinx} }% diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index a65ad6a..d92a425 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -63,21 +63,9 @@ \begin{enumerate}[label=\arabic*)] \item Sei $X = S^1$. $\gamma_1$ und $\gamma_2$ aus Abb.~\ref{fig:circle-two-paths} nicht homöotop. - \begin{figure} - \centering - \input{figures/topology-circle-two-paths.tex} - \caption{Kreis mit zwei Wegen} - \label{fig:circle-two-paths} - \end{figure} \item Sei $X = T^2$. $\gamma_1, \gamma_2$ und $\gamma_3$ aus Abb.~\ref{fig:torus-three-paths} sind paarweise nicht homöotop. - \begin{figure} - \centering - \includegraphics[width=0.5\linewidth, keepaspectratio]{figures/todo/torus-three-paths.jpg} - \caption{Torus mit drei Wegen} - \label{fig:torus-three-paths} - \end{figure} \item Sei $X = \mdr^2$ und $a=b=(0,0)$. Je zwei Wege im $\mdr^2$ mit Anfangs- und Enpunkt $(0,0)$ @@ -98,6 +86,20 @@ $H(t,0) = \gamma(t)\; \forall t \in I$ und $H(t,1) = 0 \; \forall t \in I$ \end{enumerate} + + \begin{figure}[ht] + \centering + \subfloat[Kreis mit zwei Wegen]{ + \input{figures/topology-circle-two-paths.tex} + \label{fig:circle-two-paths} + }% + \subfloat[Torus mit drei Wegen]{ + \includegraphics[width=0.45\linewidth, keepaspectratio]{figures/torus-three-paths.pdf} + \label{fig:torus-three-paths} + }% + \label{Formen} + \caption{Beispiele für (nicht)-homotopie von Wegen} + \end{figure} \end{beispiel} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -315,7 +317,7 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \label{fig:kor-bem-11.5} \end{figure} -\begin{beweis} +\begin{beweis}\leavevmode \begin{enumerate}[label=\alph*)] \item $f_*$ ist wohldefiniert: Seien $\gamma_1, \gamma_2$ homotope Wege von $x$. z.Z.: $f \circ \gamma_1 \sim f \circ \gamma_2$: @@ -660,5 +662,169 @@ Existenz: Siehe Skizze (Abbildung~\ref{fig:satz-12.6}). \end{figure} \end{beweis} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Sebastians Mitschrieb vom 17.12.2013 % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +$p:Y \rightarrow X$ Überlagerung, $X,Y$ wegzusammenhängend. +$p$ stetig und surjektiv, zu $x \in X \exists$ Umgebung $U$, so dass +$p^{-1}(U) = \bigcup V_j$ + +$p|V_j: V_j \rightarrow U$ Homöomorphismus. + +\begin{bemerkung}%Bemerkung 12.6 der Vorlesung + Wege in $X$ lassen sich zu Wegen in $Y$ liften. + + Zu jedem $y \in p^{-1}(\gamma(0))$ gibt es genau einen Lift von + $\gamma$. +\end{bemerkung} + +\begin{proposition}\label{proposition:12.7}%Proposition 12.7 der Vorlesung + Seien $p: Y \rightarrow X$ eine Überlagerung, $a,b \in X$, + $\gamma_0, \gamma_1: I \rightarrow X$ homotope Wege von $a$ nach + $b$, $\tilde{a} \in p^{-1}(a), \tilde{\gamma_0}, \tilde{\gamma_1}$ + Liftungen von $\gamma_0$ bzw. $\gamma_1$ mit + $\tilde{\gamma_i}(0) = \tilde{0}$. + + Dann ist $\tilde{\gamma_0}(1) = \tilde{\gamma_1}(1)$ und + $\tilde{\gamma_0} \sim \tilde{\gamma_1}$. +\end{proposition} + +\begin{beweis} + Sei $H: I \times I \rightarrow X$ Homotopie zwischen $\gamma_1$ + und $\gamma_2$. + + Für $s \in [0,1]$ sei $\gamma_s: I \rightarrow X$, $t \mapsto H(t,s)$. + + Sei $\tilde{\gamma_s}$ Lift von $\gamma_s$ mit $\tilde{\gamma_s}(0) = \tilde{a}$ + + Sei $\tilde{H}: I \times I \rightarrow Y,\;\;\; \tilde{H}(t,s) := (\tilde{\gamma_s}(t), s)$ + + Dann gilt: + \begin{enumerate}[label=(\roman*)] + \item $\tilde{H}$ ist stetig (Beweis wie für Korollar~\ref{kor:12.5}) + \item $\tilde{H}(t,0) = \tilde{\gamma_s}(t) = \tilde{H}(t,1) = \tilde{\gamma_1}(t)$ + \item $\tilde{H}(0,s) = \tilde{\gamma_s}(0) = \tilde{0}$ + \item $\tilde{H}(1,s) \in p^{-1}(b)$ + \end{enumerate} + + Da $p^{-1}(b)$ diskrete Teilmenge von $Y$ ist\\ + $\Rightarrow \tilde{H}(1,s) = \tilde{H}(1,0) = \tilde{b_s} \forall s \in I$\\ + $\Rightarrow \tilde{b_0} = \tilde{b_1}$ und $H$ ist Homotopie + zwischen $\tilde{\gamma_0}$ und $\tilde{\gamma_1}$. $\qed$ +\end{beweis} + +\begin{korollar}%In Vorlesung: "Folgerung 12.8" + Sei $p: Y \rightarrow X$ eine Überlagerung, $x_0 \in X, y_0 \in p^{-1}(x_0)$ + \begin{enumerate}[label=\alph*)] + \item $p_1: \pi_1(Y, y_0) \rightarrow \pi_1(X, x_0)$ ist injektiv\label{kor:12.8a} + \item $[\pi_1(X, x_0): p_* (\pi_1(Y, y_0))] = \deg(p)$\label{kor:12.8b} + \end{enumerate} +\end{korollar} + +\begin{beweis}\leavevmode + \begin{enumerate}[label=\alph*)] + \item Sei $\tilde{\gamma}$ ein Weg in $Y$ um $y_0$ und + $p_* ([\tilde{\gamma}]) = e$, also $p \circ \tilde{\gamma} \sim \gamma_{x_0}$ + + Nach Proposition~\ref{proposition:12.7} ist dann + $\tilde{\gamma}$ homotop zum Lift des konstanten Wegs + $\gamma_{x_0}$ mit Anfangspunkt $y_0$, also zu + $\gamma_{y_0} \Rightarrow [\tilde{\gamma}] = e$ + \item Sei $d = \deg{p}, p^{-1}(x_0) = \Set{y_0, y_1, \dots, y_{d-1}}$. + Für einen geschlossenen Weg $\gamma$ in $X$ um $x_0$ + sei $\tilde{\gamma}$ die Liftung mit $\tilde{\gamma}(0) = y_0$. + + $\tilde{\gamma}(1) \in \Set{y_0, \dots, y_{d-1}}$ hängt + nur von $[\gamma] \in \pi_1(X,x_0)$ ab. + + Es 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)) + \end{align} + Zu $i \in \Set{0, \dots, d-1}$ gibt es Weg $\delta_i$ in + $Y$ mit $\delta_i(0) = y_0$ und $\sigma_i(1) = y_i$\\ + $\Rightarrow p * \delta_i$ ist geschlossener Weg in + $X$ um $x_0$.\\ + $\Rightarrow \sigma_i = \widetilde{p*\delta_i}$\\ + $\Rightarrow$ Jedes $y_i$ mit $i=0, \dots, d-1$ ist + $\tilde{\gamma}(1)$ für ein $[\gamma] \in \pi_1(X,x_0)$ + \end{enumerate} +\end{beweis} + +\begin{korollar}%In Vorlesung: "Folgerung 12.9" + Sei $p: Y \rightarrow X$ Überlagerung und $X$ einfach zusammenhängend. + + Dann ist $p$ ein Homöomorphismus. +\end{korollar} + +\begin{beweis} + Wegen Korollar~\ref{kor:12.8a} ist auch $Y$ einfach zusammenhängend + und wegen Korollar~\ref{kor:12.8b} ist $\deg(p)=1$, $p$ ist also + bijektiv. + + Nach \todo{Was ist das?}{12.2} ist $p$ offen $\Rightarrow p^{-1}$ + ist stetig. $\qed$ +\end{beweis} + +\begin{definition}%In Vorlesung: "Definition 12.10" + Eine Überlagerung $p: \tilde{X} \rightarrow X$ heißt + \textbf{universell}\xindex{Überlagerung!universelle}, wenn + $\tilde{X}$ einfach zusammenhängend ist. +\end{definition} + +\begin{beispiel} + $\mdr \rightarrow S^1, \;\;\; t \mapsto (\cos 2 \pi t, \sin 2 \pi t)$ + + $\mdr^2 \rightarrow T^2 = \mdr^2 / \mdz^2$ + + $S^n \rightarrow \praum^n(\mdr)$ für $n \geq 2$ +\end{beispiel} + +\begin{satz}%In Vorlesung: Satz 12.11 + Sei $p: \tilde{X} \rightarrow X$ universelle Überlagerung, + $q:Y \rightarrow X$ weitere Überlagerung. + + Sei $x_0 \in X, \tilde{x_0} \in \tilde{X}, y_0 \in Y$ mit + $q(y_1) = x_0, p(\tilde{x_0}) = x_0$. + + Dann gibt es genau eine Überlagerung $\tilde{p}: \tilde{X} \rightarrow Y$ + mit $\tilde{p}(\tilde{x_0}) = y_0$. +\end{satz} + +\begin{beweis} + Sei $z \in \tilde{X}, y_z: I \rightarrow \tilde{X}$ ein Weg von + $\tilde{x_0}$ nach $z$. + + Sei $\delta_Z$ \underline{die} Liftung von $p \circ \gamma_z$ + nach $y$ mit $\delta_2(0) = y_0$. + + Setze $\tilde{p}(z) = \delta_Z(1)$. + + Da $\tilde{X}$ einfach zusammenhängend ist, hängt $\tilde{p}(z)$ + nicht vom gewählten $y_z$ ab. + + Offensichtlich ist $q(\tilde{p}(z)) = p(z)$. + + $\tilde{p}$ ist stetig (in $z \in \tilde{X}$). Sei $W \subseteq Y$ + offene Umgebung von $\tilde{p}(z)$. + + $\xRightarrow{q \text{ offen}} q(W)$ ist offene Umgebung von $p(z) \cdot d(\tilde{p}(z))$. + + Sei $U \subseteq q(W)$ offen wie in Defintion~\ref{def:12.1} und + $V \subseteq q^{-1}(U)$ die \todo{Was?}{Komp.} die $\tilde{p}(z)$ + enthält. + + \Obda sei $V \subseteq W$. + + Sei $Z := p^{-1}(U)$. Für $u \in Z$ sei $\delta$ ein Weg in $Z$ + von $z$ nach $u$. + + $\Rightarrow \gamma_Z * \delta$ ist Weg von $x_0$ nach $u$\\ + $\Rightarrow \tilde{p}(u) \in V$\\ + $\Rightarrow Z \subseteq \tilde{p^{-1}}(W)$\\ + $\Rightarrow \tilde{p}$ ist stetig +\end{beweis} % Die Übungsaufgaben sollen ganz am Ende des Kapitels sein. \input{Kapitel3-UB} diff --git a/documents/GeoTopo/Readme.md b/documents/GeoTopo/Readme.md index 3bed5fb..3e73716 100644 --- a/documents/GeoTopo/Readme.md +++ b/documents/GeoTopo/Readme.md @@ -44,3 +44,12 @@ Commit-Nachricht von Git zu sehen. Bilder habe ich entweder selbst erstellt oder von tex.stackexchange.com. Bei Bildern von tex.stackexchange.com steht der Link auf die Quelle im Quelltext des Bildes (siehe Ordner `figures`). + +Was noch kommen soll +==================== + +1. Alle `TODOS` auflösen +2. Reviews (Mathematik, LaTeX und Bilder) +3. A5-Version drucken + * Momentan sind es ca. 60 Seiten in A4. In A5 sind es ca. +4. 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zefa+X@?3u(_Rsn+y#D~Z`G5Hz{SVeSC;gAyf7yOB{)_9J{-5?|f4BPM{{ZAq`yZxp z{{ZCa-e359a>tL@{{Vmf3jYAH`_KE|lRwJm`k(2Y{{Z?TwSTnF`tyIw{-6HI=lF~~ z{{Y`y=l;ECANuS60Ky;iKEM70{*UiJ&tsSA{+a&(fAP=ypN#(iu=}6oKifaRM_$U3x%>Mwk`@{VB{{UzE^RN2% Z{{ZR+{{XZ0&-s7bIsN|t-yinJ|Jjq$ns@*J diff --git a/documents/GeoTopo/shortcuts.sty b/documents/GeoTopo/shortcuts.sty index 5f2652d..c2d5ccd 100644 --- a/documents/GeoTopo/shortcuts.sty +++ b/documents/GeoTopo/shortcuts.sty @@ -39,6 +39,7 @@ \newframedtheorem{satz}{Satz}[chapter] \newframedtheorem{lemma}[satz]{Lemma} +\newframedtheorem{proposition}[satz]{Proposition} \newtheorem{korollar}[satz]{Korollar} \newtheorem{plaindefinition}{Definition} \newenvironment{definition}{\begin{plaindefinition}}{\end{plaindefinition}}

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