euler char, traces, telescoping sums

3 files changed, 147 insertions, 1 deletions

diff --git a/buch/chapters/95-homologie/images/Makefile b/buch/chapters/95-homologie/images/Makefile
index d14a3a2..50c2b0d 100644
--- a/buch/chapters/95-homologie/images/Makefile
+++ b/buch/chapters/95-homologie/images/Makefile
@@ -3,7 +3,8 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	homocycles.pdf homoboundaries.pdf homoclasses.pdf gausshomoex.pdf gausshomobasis.pdf dreieck.pdf polyeder.pdf
+all:	complexbasis.pdf homocycles.pdf homoboundaries.pdf homoclasses.pdf \
+	gausshomoex.pdf gausshomobasis.pdf dreieck.pdf polyeder.pdf
 
 dreieck.pdf:	dreieck.tex
 	pdflatex dreieck.tex
@@ -26,3 +27,6 @@ homoboundaries.pdf:	homoboundaries.tex
 homoclasses.pdf:	homoclasses.tex
 	pdflatex homoclasses.tex
 
+complexbasis.pdf:	complexbasis.tex
+	pdflatex complexbasis.tex
+
diff --git a/buch/chapters/95-homologie/images/complexbasis.pdf b/buch/chapters/95-homologie/images/complexbasis.pdf
new file mode 100644
index 0000000..536f04f
--- /dev/null
+++ b/buch/chapters/95-homologie/images/complexbasis.pdf
diff --git a/buch/chapters/95-homologie/images/complexbasis.tex b/buch/chapters/95-homologie/images/complexbasis.tex
new file mode 100644
index 0000000..365789c
--- /dev/null
+++ b/buch/chapters/95-homologie/images/complexbasis.tex
@@ -0,0 +1,142 @@
+%
+% complexbasis.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\usetikzlibrary{decorations.pathreplacing}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\s{0.5}
+\def\h{0.02}
+
+\def\rechteck#1#2#3{
+	\fill[color=#3!20,rounded corners=2pt]
+		({-\s+\h},{(2*(#1)-1)*\s+\h})
+		rectangle
+		({\s-\h},{(2*(#2)+1)*\s-\h});
+	\draw[color=#3,rounded corners=2pt]
+		({-\s+\h},{(2*(#1)-1)*\s+\h})
+		rectangle
+		({\s-\h},{(2*(#2)+1)*\s-\h});
+	\foreach \y in {{#1},...,{#2}}{
+		\fill[color=#3] (0,{2*\y*\s}) circle[radius=0.05];
+	}
+}
+\def\Rechteck#1#2{
+	\draw[rounded corners=3pt]
+		({-\s-\h},{(2*(#1)-1)*\s-\h})
+		rectangle
+		({\s+\h},{(2*(#2)+1)*\s+\h});
+}
+
+\def\abbildung#1#2#3#4{
+	\fill[color=gray!20]
+		({\s},{(2*(#1)+1)*\s})
+		--
+		({3-\s},{-\s})
+		--
+		({3-\s},{(2*(#3)+1)*\s})
+		--
+		({\s},{(2*(#1)+1)*\s});
+	\fill[color=gray!40]
+		({\s},{(2*(#1+1)-1)*\s})
+		--
+		({3-\s},{(2*(#3+1)-1)*\s})
+		--
+		({3-\s},{(2*(#4)+1)*\s})
+		--
+		({\s},{(2*(#2)+1)*\s});
+}
+
+\clip (-4.5,-1.2) rectangle (7.5,11.7);
+
+\begin{scope}[xshift=-6cm]
+	\abbildung{6}{7}{10}{11}
+\end{scope}
+
+\begin{scope}[xshift=-3cm]
+	\abbildung{6}{10}{7}{11}
+	\rechteck{0}{6}{red}
+	\rechteck{7}{10}{darkgreen}
+	\rechteck{11}{11}{blue}
+	\Rechteck{0}{11}
+	\node[color=darkgreen] at ({0},{(9*2-1)*\s}) {$B_{k-2\mathstrut}$};
+	\node at (1.5,{9*2*\s}) {$\Delta_{k-1}$};
+	\node at (1.5,{-\s}) [above] {$\partial_{k-1\mathstrut}$};
+	\draw[decorate,decoration={brace,amplitude=4pt}]
+		({-\s-0.1},{-\s}) -- ({-\s-0.1},{(2*10+1)*\s});
+	\node at ({-\s-0.17},{10*\s}) [left] {$Z_{k-2\mathstrut}$};
+	\node at (0,{-\s}) [below] {$C_{k-2\mathstrut}$};
+\end{scope}
+
+\begin{scope}
+	\abbildung{2}{7}{5}{10}
+	\rechteck{8}{11}{blue}
+	\rechteck{3}{7}{darkgreen}
+	\rechteck{0}{2}{red}
+	\Rechteck{0}{11}
+	\node at (0,{-\s}) [below] {$C_{k-1\mathstrut}$};
+	\node[color=darkgreen] at ({0},{(5*2-1)*\s}) {$B_{k-1\mathstrut}$};
+	\node at (1.5,{6.5*2*\s}) {$\Delta_k$};
+	\node at (1.5,{-\s}) [above] {$\partial_{k\mathstrut}$};
+	\draw[decorate,decoration={brace,amplitude=4pt}]
+		({-\s-0.1},{-\s}) -- ({-\s-0.1},{(2*7+1)*\s});
+	\node at ({-\s-0.17},{7*\s}) [left] {$Z_{k-1\mathstrut}$};
+\end{scope}
+
+\begin{scope}[xshift=3cm]
+	\abbildung{3}{5}{5}{7}
+	\rechteck{6}{10}{blue}
+	\rechteck{4}{5}{darkgreen}
+	\rechteck{0}{3}{red}
+	\Rechteck{0}{10}
+	\node at (0,{-\s}) [below] {$C_{k\mathstrut}$};
+	\node[color=darkgreen] at ({-0.25},{9*\s})
+		{$B_{k\mathstrut}$};
+	\node[color=darkgreen] at (0.24,{2*4*\s}) {$b_1$};
+	\node[color=darkgreen] at (0.24,{2*4.5*\s+0.1}) {$\vdots$};
+	\node[color=darkgreen] at (0.24,{2*5*\s}) {$b_r$};
+	\node[color=red] at (0.24,{2*0*\s}) {$z_1$};
+	\node[color=red] at (0.24,{2*1*\s}) {$z_2$};
+	\node[color=red] at (0.24,{2*2*\s+0.1}) {$\vdots$};
+	\node[color=red] at (0.24,{2*3*\s}) {$z_l$};
+	\node[color=blue] at (0.24,{2*6*\s}) {$c_1$};
+	\node[color=blue] at (0.24,{2*7*\s}) {$c_2$};
+	\node[color=blue] at (0.24,{2*8*\s}) {$c_3$};
+	\node[color=blue] at (0.24,{2*9*\s}) {$\vdots$};
+	\node[color=blue] at (0.24,{2*10*\s}) {$c_s$};
+	\node at (1.5,{5.5*2*\s}) {$\Delta_{k+1}$};
+	\node at (1.5,{-\s}) [above] {$\partial_{k+1\mathstrut}$};
+	\draw[decorate,decoration={brace,amplitude=4pt}]
+		({-\s-0.1},{-\s}) -- ({-\s-0.1},{(2*5+1)*\s});
+	\node at ({-\s-0.17},{5*\s}) [left] {$Z_{k\mathstrut}$};
+\end{scope}
+
+\begin{scope}[xshift=6cm]
+	\abbildung{0}{5}{4}{8}
+	\rechteck{5}{7}{blue}
+	\rechteck{1}{5}{darkgreen}
+	\rechteck{0}{0}{red}
+	\Rechteck{0}{7}
+	\node at (0,{-\s}) [below] {$C_{k+1\mathstrut}$};
+	\node[color=darkgreen] at ({0},{(2.5*2+1)*\s})
+		{$B_{k+1\mathstrut}$};
+	\draw[decorate,decoration={brace,amplitude=4pt}]
+		({-\s-0.1},{-\s}) -- ({-\s-0.1},{(2*5+1)*\s});
+	\node at ({-\s-0.17},{5*\s}) [left] {$Z_{k+1\mathstrut}$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+