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a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 160d842..7cee2e1 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -171,15 +171,14 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{beispiel} Sei $X = \mdr^2$ und $(x_1, y_1) \sim (x_2, y_2) \gdw x_1 - x_2 \in \mdz$ - und $y_1 - y_2 \in \mdz$. - - $X /_\sim$ ist ein Torus. + und $y_1 - y_2 \in \mdz$. Dann ist $X /_\sim$ ein Torus. \end{beispiel} \begin{beispiel}\xindex{Raum!projektiver} \begin{align*} X= \mdr^{n-1} \setminus \Set{0},\;\;\; x \sim y &\gdw \exists \lambda \in \mdr^\times \text{ mit } y = \lambda x\\ - &\gdw x \text{ und } y \text{ liegen auf der gleichen Ursprungsgerade} + &\gdw x \text{ und } y \text{ liegen auf der gleichen}\\ + &\hphantom{\gdw} \text{Ursprungsgerade} \end{align*} \[\overline{X} = \praum^n(\mdr)\] Also für $n=1$:\nopagebreak\\ diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 974a1c4..0f553ef 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -634,11 +634,11 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{figure}[ht] \centering \subfloat[1D Simplizialkomplex]{ - \parbox[c][4cm]{4cm}{\centering\input{figures/topology-1-d-simplizialkomplex}} + \parbox[c][4cm]{3.5cm}{\centering\input{figures/topology-1-d-simplizialkomplex}} \label{fig:simplizialkomplex-1-d} }% \subfloat[2D Simplizialkomplex (ohne untere Fläche!)]{ - \parbox[c][4cm]{4cm}{\centering\input{figures/topology-pyramid.tex}} + \parbox[c][4cm]{3.5cm}{\centering\input{figures/topology-pyramid.tex}} \label{fig:simplizialkomplex-2-d} }% \subfloat[2D Simplizialkomplex]{ diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index 1ba6b69..6a1f696 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -464,7 +464,7 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \label{fig:ueberlappung-kaestchen-torus} }% \subfloat[$t \mapsto (\cos 4 \pi t, \sin 4 \pi t)$]{ - \resizebox{0.3\linewidth}{!}{\input{figures/topology-ueberlagerung.tex}} + \input{figures/topology-ueberlagerung.tex} \label{fig:liftung-s1-s1} }% \label{fig:ueberlagerungen} @@ -597,7 +597,7 @@ Haben wir Häufungspunkt definiert?} \begin{figure} \centering - \input{figures/liftung-torus-r.tex} + \resizebox{0.95\linewidth}{!}{\input{figures/liftung-torus-r.tex}} \caption{Beim Liften eines Weges bleiben geschlossene Wege im allgemeinen nicht geschlossen} \label{fig:satz-seifert-van-kampen} \end{figure} @@ -1170,7 +1170,9 @@ und der Fundamentalgruppe herstellen: Die Konstruktion aus \cref{kor:13.3} induziert zu der Gruppenoperation $\pi_1(X, x_0)$ aus \cref{bsp:13.4} einen Gruppenhomomorphismus $\varrho: \pi_1(X, x_0) \rightarrow \Homoo(X)$. Nach Satz~\ref{thm:12.15} -ist $\varrho(\pi_1(X, x_0)) = \Deck(\tilde{X} / X) = \Set{f: \tilde{X} \rightarrow \tilde{X} \text{ Homöomorphismus} | p \circ f = p}$ +ist \begin{align*}\varrho(\pi_1(X, x_0)) &= \Deck(\tilde{X} / X)\\ + &= \Set{f: \tilde{X} \rightarrow \tilde{X} \text{ Homöomorphismus} | p \circ f = p} + \end{align*} \begin{beispiel}% In Vorlesung: Beispiel 13.6 Sei $X := S^2 \subseteq \mdr^3$ und $\tau$ die Drehung um die $z$-Achse diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index 6a4f3e5..f638b94 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -707,11 +707,11 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \begin{figure}[ht] \centering \subfloat[Schritt 1]{ - \input{figures/coordinate-system-1.tex} + \resizebox{0.45\linewidth}{!}{\input{figures/coordinate-system-1.tex}} \label{fig:14.13.1} }% \subfloat[Schritt 2]{ - \input{figures/coordinate-system-2.tex} + \resizebox{0.45\linewidth}{!}{\input{figures/coordinate-system-2.tex}} \label{fig:14.13.2} }% \label{fig:14.13.0.1} @@ -737,11 +737,11 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \begin{figure}[ht] \centering \subfloat[Schritt 1]{ - \input{figures/coordinate-system-2.tex} + \resizebox{0.45\linewidth}{!}{\input{figures/coordinate-system-2.tex}} \label{fig:14.13.3} }% \subfloat[Schritt 2 (Bild 13)]{ - \input{figures/todo.tex} + \resizebox{0.45\linewidth}{!}{\input{figures/todo.tex}} \label{fig:14.13.4} }% \label{fig:14.13.0.2} @@ -793,11 +793,11 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \begin{figure}[ht] \centering \subfloat[Fall 1]{ - \input{figures/hyperbolische-geometrie-axiom-1-1.tex} + \resizebox{0.45\linewidth}{!}{\input{figures/hyperbolische-geometrie-axiom-1-1.tex}} \label{fig:hyperbolische-geometrie-axiom-1-1} }% - \subfloat[Fall1]{ - \input{figures/hyperbolische-geometrie-axiom-1-2.tex} + \subfloat[Fall 2]{ + \resizebox{0.45\linewidth}{!}{\input{figures/hyperbolische-geometrie-axiom-1-2.tex}} \label{fig:hyperbolische-geometrie-axiom-1-2} }% \label{fig:hyperbolische-geometrie-axiom-1-0} @@ -866,11 +866,11 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \begin{figure}[ht] \centering \subfloat[Fall 1]{ - \input{figures/hyberbolische-geometrie-1.tex} + \resizebox{0.45\linewidth}{!}{\input{figures/hyberbolische-geometrie-1.tex}} \label{fig:prop15.2.e.fall1.1} }% \subfloat[Fall 2 (Strahlensatz)]{ - \input{figures/hyberbolische-geometrie-2.tex} + \resizebox{0.45\linewidth}{!}{\input{figures/hyberbolische-geometrie-2.tex}} \label{fig:prop15.2.e.fall1.1} }% \label{fig:prop15.2.e.fall1.0} diff --git a/documents/GeoTopo/figures/topology-ueberlagerung.tex b/documents/GeoTopo/figures/topology-ueberlagerung.tex index 45dd9af..2426dd7 100644 --- a/documents/GeoTopo/figures/topology-ueberlagerung.tex +++ 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