Merge branch 'AndreasFMueller:master' into master

13 files changed, 782 insertions, 1 deletions

diff --git a/buch/chapters/95-homologie/images/Makefile b/buch/chapters/95-homologie/images/Makefile
index ac964ff..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:	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
@@ -11,3 +12,21 @@ dreieck.pdf:	dreieck.tex
 polyeder.pdf:	polyeder.tex
 	pdflatex polyeder.tex
 
+gausshomobasis.pdf:	gausshomobasis.tex
+	pdflatex gausshomobasis.tex
+
+gausshomoex.pdf:	gausshomoex.tex
+	pdflatex gausshomoex.tex
+
+homocycles.pdf:	homocycles.tex
+	pdflatex homocycles.tex
+
+homoboundaries.pdf:	homoboundaries.tex
+	pdflatex 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..3a627d4
--- /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..6accb2a
--- /dev/null
+++ b/buch/chapters/95-homologie/images/complexbasis.tex
@@ -0,0 +1,158 @@
+%
+% 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})
+		--
+		({3.5-\s},{-\s})
+		--
+		({3.5-\s},{(2*(#3)+1)*\s})
+		--
+		({\s},{(2*(#1)+1)*\s})
+		--
+		cycle;
+	\fill[color=gray!40]
+		({\s},{(2*(#1+1)-1)*\s})
+		--
+		({3.5-\s},{(2*(#3+1)-1)*\s})
+		--
+		({3.5-\s},{(2*(#4)+1)*\s})
+		--
+		({\s},{(2*(#2)+1)*\s})
+		--
+		cycle;
+	\draw[<-,color=gray]
+		({\s},{(2*(#1+1)-1)*\s})
+		--
+		({3.5-\s},{(2*(#3+1)-1)*\s});
+	\draw[->,color=gray]
+		({3.5-\s},{(2*(#4)+1)*\s})
+		--
+		({\s},{(2*(#2)+1)*\s});
+	\draw[<-,color=gray!40]
+		({\s},{(2*(#1)+1)*\s})
+		--
+		({3.5-\s},{-\s});
+}
+
+\clip (-5.5,-1.2) rectangle (8.5,11.7);
+
+\begin{scope}[xshift=-7cm]
+	\abbildung{6}{7}{10}{11}
+\end{scope}
+
+\begin{scope}[xshift=-3.5cm]
+	\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 (1.75,{9*2*\s}) {$\Delta_{k-1}$};
+	\node at (1.75,{-\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 (1.75,{6.5*2*\s}) {$\Delta_k$};
+	\node at (1.75,{-\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=3.5cm]
+	\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 (1.75,{5.5*2*\s}) {$\Delta_{k+1}$};
+	\node at (1.75,{-\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=7cm]
+	\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},{(2.0*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}
+
diff --git a/buch/chapters/95-homologie/images/gausshomobasis.pdf b/buch/chapters/95-homologie/images/gausshomobasis.pdf
new file mode 100644
index 0000000..07414bb
--- /dev/null
+++ b/buch/chapters/95-homologie/images/gausshomobasis.pdf
diff --git a/buch/chapters/95-homologie/images/gausshomobasis.tex b/buch/chapters/95-homologie/images/gausshomobasis.tex
new file mode 100644
index 0000000..ba21f54
--- /dev/null
+++ b/buch/chapters/95-homologie/images/gausshomobasis.tex
@@ -0,0 +1,109 @@
+%
+% gaushomobasis.tex -- Bestimmung einer Basis der Homologiegruppen
+%
+% (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}
+\begin{document}
+\def\skala{1}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\s{0.5}
+\def\inset{0.05}
+\def\w{8}
+
+\def\zeile#1#2{
+	\fill[color=#2] ({0+\inset},{(12-#1)*\s+\inset})
+		rectangle ({\w*\s-\inset},{(13-#1)*\s-\inset});
+}
+\def\marke#1#2{
+\node at ({0.5*\w*\s},{12.5-#1)*\s}) {$#2\mathstrut$};
+}
+
+\def\gauss{
+\draw (0,0) rectangle ({\w*\s},{12*\s});
+\draw (0,{7*\s}) -- ({\w*\s},{7*\s});
+}
+
+\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-1)*\s})
+	to[out=180,in=90] (-3.6,-2);
+\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-2)*\s})
+	to[out=180,in=90] (-2.2,-2);
+\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-4)*\s})
+	to[out=180,in=90] (-0.7,-2);
+
+\draw[->,color=darkgreen,line width=1pt] ({0.1*\s},{(12.5-7)*\s})
+	to[out=180,in=90] (0.9,-2);
+\draw[->,color=darkgreen,line width=1pt] ({0.1*\s},{(12.5-8)*\s})
+	to[out=180,in=90] (1.6,-2);
+\draw[->,color=darkgreen,line width=1pt] ({(\w-0.1)*\s},{(12.5-12)*\s})
+	to[out=0,in=90] (2.6,-2);
+
+\draw[->,line width=2pt] ({\w*\s+0.1},{6*\s}) -- (5.4,{6*\s});
+\node at ({0.5*(\w*\s+5.5)},{6*\s}) [above] {Gauss};
+
+\begin{scope}
+\zeile{1}{red!30}
+\zeile{2}{red!30}
+\zeile{4}{red!30}
+\zeile{7}{darkgreen!30}
+\zeile{8}{darkgreen!30}
+%\zeile{10}{darkgreen!30}
+\zeile{12}{darkgreen!30}
+\marke{1}{\scriptstyle\partial_{k+1}e_1^{(k+1)}}
+\marke{2}{\scriptstyle\partial_{k+1}e_2^{(k+1)}}
+\marke{3}{\scriptstyle\partial_{k+1}e_3^{(k+1)}}
+\marke{4}{\vdots}
+\marke{5}{\scriptstyle\partial_{k+1}e_{n_{k+1}}^{(k+1)}}
+\marke{6}{\scriptstyle z_1^{(k)}}
+\marke{7}{\scriptstyle z_2^{(k)}}
+\marke{8}{\scriptstyle z_3^{(k)}}
+\marke{9}{\scriptstyle z_4^{(k)}}
+\marke{10}{\vdots}
+\marke{11}{\scriptstyle z_{l-1}^{(k)}}
+\marke{12}{\scriptstyle z_{l}^{(k)}}
+\gauss
+\end{scope}
+
+\begin{scope}[xshift=5.5cm]
+\zeile{1}{black!20}
+\zeile{2}{black!20}
+\zeile{3}{black}
+\marke{3}{\color{white}0}
+\zeile{4}{black!20}
+\zeile{5}{black}
+\marke{5}{\color{white}0}
+\zeile{6}{black}
+\marke{6}{\color{white}0}
+\zeile{7}{black!20}
+\zeile{8}{black!20}
+\zeile{9}{black}
+\marke{9}{\color{white}0}
+\zeile{10}{black}
+\marke{10}{\color{white}0}
+\zeile{11}{black}
+\marke{11}{\color{white}0}
+\zeile{12}{black!20}
+\gauss
+\end{scope}
+
+\node at (-4.4,-2) [below right] {$\{
+{\color{red}\partial_{k+1}e_1^{(k+1)}},
+{\color{red}\partial_{k+1}e_2^{(k+1)}},
+{\color{red}\partial_{k+1}e_{i_3}^{(k+1)}},\dots,
+{\color{darkgreen}z_2^{(k)}},
+{\color{darkgreen}z_3^{(k)}},
+\dots
+{\color{darkgreen}z_l^{(k)}}
+\} = {\color{red}\mathcal{B}_k} \cup {\color{darkgreen}\mathcal{Z}_k'}$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/95-homologie/images/gausshomoex.pdf b/buch/chapters/95-homologie/images/gausshomoex.pdf
new file mode 100644
index 0000000..bc0b766
--- /dev/null
+++ b/buch/chapters/95-homologie/images/gausshomoex.pdf
diff --git a/buch/chapters/95-homologie/images/gausshomoex.tex b/buch/chapters/95-homologie/images/gausshomoex.tex
new file mode 100644
index 0000000..df53f70
--- /dev/null
+++ b/buch/chapters/95-homologie/images/gausshomoex.tex
@@ -0,0 +1,120 @@
+%
+% gausshomoex.tex -- Beispiel für die Bestimmung einer Basis von H_1
+%
+% (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}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\s{2.0}
+
+\def\punkt#1#2{({((#1)+0.5*(#2))*\s},{(#2)*\s*sqrt(3)/2})}
+
+\def\knoten#1#2#3{
+	\fill[color=white] \punkt{#1}{#2} circle[radius=0.3];
+	\node at \punkt{#1}{#2} {$#3$\strut};
+	\draw \punkt{#1}{#2} circle[radius=0.3];
+}
+\def\dreieck#1#2#3{
+	\fill[color=gray] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+		-- \punkt{#1}{(#2)+1} -- cycle;
+	\node at \punkt{#1+0.3333}{#2+0.3333} {$#3$\strut};
+	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+		\punkt{#1}{#2} -- \punkt{#1+1}{#2};
+	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+		\punkt{#1+1}{#2} -- \punkt{#1}{#2+1};
+	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+		\punkt{#1}{#2+1} -- \punkt{#1}{#2};
+}
+
+\def\Dreieck#1#2#3{
+	\fill[color=gray!50] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+		-- \punkt{#1+1}{(#2)-1} -- cycle;
+	\node at \punkt{#1+0.3333}{#2+0.3333} {$#3$\strut};
+}
+
+\def\kante#1#2#3{
+	\fill[color=white,opacity=0.8] \punkt{#1}{#2} circle[radius=0.15];
+	\node at \punkt{#1}{#2} {$\scriptstyle #3$};
+}
+
+\dreieck{0}{0}{1}
+\dreieck{1}{0}{2}
+\dreieck{2}{0}{3}
+\dreieck{3}{0}{4}
+
+\dreieck{0}{1}{5}
+\dreieck{2}{1}{6}
+
+\dreieck{0}{2}{7}
+\dreieck{1}{2}{8}
+
+\dreieck{0}{3}{9}
+
+
+\knoten{0}{0}{1}
+\knoten{1}{0}{2}
+\knoten{2}{0}{3}
+\knoten{3}{0}{4}
+\knoten{4}{0}{5}
+
+\knoten{0}{1}{6}
+\knoten{1}{1}{7}
+\knoten{2}{1}{8}
+\knoten{3}{1}{9}
+
+\knoten{0}{2}{10}
+\knoten{1}{2}{11}
+\knoten{2}{2}{12}
+
+\knoten{0}{3}{13}
+\knoten{1}{3}{14}
+
+\knoten{0}{4}{15}
+
+\kante{0.5}{0}{1}
+\kante{1.5}{0}{2}
+\kante{2.5}{0}{3}
+\kante{3.5}{0}{4}
+
+\kante{0}{0.5}{5}
+\kante{0.5}{0.5}{6}
+\kante{1}{0.5}{7}
+\kante{1.5}{0.5}{8}
+\kante{2}{0.5}{9}
+\kante{2.5}{0.5}{10}
+\kante{3}{0.5}{11}
+\kante{3.5}{0.5}{12}
+
+\kante{0.5}{1}{13}
+\kante{2.5}{1}{14}
+
+\kante{0}{1.5}{15}
+\kante{0.5}{1.5}{16}
+\kante{2}{1.5}{17}
+\kante{2.5}{1.5}{18}
+
+\kante{0.5}{2}{19}
+\kante{1.5}{2}{20}
+
+\kante{0}{2.5}{21}
+\kante{0.5}{2.5}{22}
+\kante{1}{2.5}{23}
+\kante{1.5}{2.5}{24}
+
+\kante{0.5}{3}{25}
+
+\kante{0}{3.5}{26}
+\kante{0.5}{3.5}{27}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/95-homologie/images/homoboundaries.pdf b/buch/chapters/95-homologie/images/homoboundaries.pdf
new file mode 100644
index 0000000..644f334
--- /dev/null
+++ b/buch/chapters/95-homologie/images/homoboundaries.pdf
diff --git a/buch/chapters/95-homologie/images/homoboundaries.tex b/buch/chapters/95-homologie/images/homoboundaries.tex
new file mode 100644
index 0000000..ef8fd1a
--- /dev/null
+++ b/buch/chapters/95-homologie/images/homoboundaries.tex
@@ -0,0 +1,114 @@
+%
+% tikztemplate.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}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\s{0.4}
+
+\def\punkt#1#2{({((#1)+0.5*(#2))*\s},{(#2)*\s*sqrt(3)/2})}
+\def\A{\punkt{0}{0}}
+\def\B{\punkt{1}{0}}
+\def\C{\punkt{2}{0}}
+\def\D{\punkt{3}{0}}
+\def\E{\punkt{4}{0}}
+\def\F{\punkt{0}{1}}
+\def\G{\punkt{1}{1}}
+\def\H{\punkt{2}{1}}
+\def\I{\punkt{3}{1}}
+\def\J{\punkt{0}{2}}
+\def\K{\punkt{1}{2}}
+\def\L{\punkt{2}{2}}
+\def\M{\punkt{0}{3}}
+\def\N{\punkt{1}{3}}
+\def\O{\punkt{0}{4}}
+
+\def\dreieck#1#2#3{
+	\fill[color=gray] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+		-- \punkt{#1}{(#2)+1} -- cycle;
+}
+
+\def\blau#1#2{
+	\draw[color=blue] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+		-- \punkt{#1}{(#2)+1} -- cycle;
+}
+
+\def\gebiet{
+	\dreieck{0}{0}{1}
+	\dreieck{1}{0}{2}
+	\dreieck{2}{0}{3}
+	\dreieck{3}{0}{4}
+	\dreieck{0}{1}{5}
+	\dreieck{2}{1}{6}
+	\dreieck{0}{2}{7}
+	\dreieck{1}{2}{8}
+	\dreieck{0}{3}{9}
+}
+
+\begin{scope}
+\gebiet
+\blau{0}{0}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_1^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=3cm]
+\gebiet
+\blau{1}{0}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_2^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=6cm]
+\gebiet
+\blau{2}{0}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_3^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=9cm]
+\gebiet
+\blau{3}{0}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_4^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=1.5cm,yshift=2.59cm]
+\gebiet
+\blau{0}{1}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_5^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=7.5cm,yshift=2.59cm]
+\gebiet
+\blau{2}{1}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_6^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=3cm,yshift=5.19cm]
+\gebiet
+\blau{0}{2}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_7^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=6cm,yshift=5.19cm]
+\gebiet
+\blau{1}{2}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_8^{(2)}$};
+\end{scope}
+
+\begin{scope}[xshift=4.5cm,yshift=7.79cm]
+\gebiet
+\blau{0}{3}
+\node[color=blue] at ({2*\s},-0.5) {$\partial_2e_9^{(2)}$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/95-homologie/images/homoclasses.pdf b/buch/chapters/95-homologie/images/homoclasses.pdf
new file mode 100644
index 0000000..217ae75
--- /dev/null
+++ b/buch/chapters/95-homologie/images/homoclasses.pdf
diff --git a/buch/chapters/95-homologie/images/homoclasses.tex b/buch/chapters/95-homologie/images/homoclasses.tex
new file mode 100644
index 0000000..e325d9b
--- /dev/null
+++ b/buch/chapters/95-homologie/images/homoclasses.tex
@@ -0,0 +1,104 @@
+%
+% tikztemplate.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}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\s{0.4}
+
+\def\punkt#1#2{({((#1)+0.5*(#2))*\s},{(#2)*\s*sqrt(3)/2})}
+\def\A{\punkt{0}{0}}
+\def\B{\punkt{1}{0}}
+\def\C{\punkt{2}{0}}
+\def\D{\punkt{3}{0}}
+\def\E{\punkt{4}{0}}
+\def\F{\punkt{0}{1}}
+\def\G{\punkt{1}{1}}
+\def\H{\punkt{2}{1}}
+\def\I{\punkt{3}{1}}
+\def\J{\punkt{0}{2}}
+\def\K{\punkt{1}{2}}
+\def\L{\punkt{2}{2}}
+\def\M{\punkt{0}{3}}
+\def\N{\punkt{1}{3}}
+\def\O{\punkt{0}{4}}
+
+%\def\knoten#1#2#3{
+%	\fill[color=white] \punkt{#1}{#2} circle[radius=0.3];
+%	\node at \punkt{#1}{#2} {$#3$\strut};
+%	\draw \punkt{#1}{#2} circle[radius=0.3];
+%}
+\def\dreieck#1#2#3{
+	\fill[color=gray] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+		-- \punkt{#1}{(#2)+1} -- cycle;
+%	\node at \punkt{#1+0.3333}{#2+0.3333} {$#3$\strut};
+%	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+%		\punkt{#1}{#2} -- \punkt{#1+1}{#2};
+%	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+%		\punkt{#1+1}{#2} -- \punkt{#1}{#2+1};
+%	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+%		\punkt{#1}{#2+1} -- \punkt{#1}{#2};
+}
+
+%\def\Dreieck#1#2#3{
+%	\fill[color=gray!50] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+%		-- \punkt{#1+1}{(#2)-1} -- cycle;
+%	\node at \punkt{#1+0.3333}{#2+0.3333} {$#3$\strut};
+%}
+
+%\def\kante#1#2#3{
+%	\fill[color=white,opacity=0.8] \punkt{#1}{#2} circle[radius=0.15];
+%	\node at \punkt{#1}{#2} {$\scriptstyle #3$};
+%}
+
+\def\gebiet{
+	\dreieck{0}{0}{1}
+	\dreieck{1}{0}{2}
+	\dreieck{2}{0}{3}
+	\dreieck{3}{0}{4}
+	\dreieck{0}{1}{5}
+	\dreieck{2}{1}{6}
+	\dreieck{0}{2}{7}
+	\dreieck{1}{2}{8}
+	\dreieck{0}{3}{9}
+}
+
+\begin{scope}
+\gebiet
+\draw[color=darkgreen] \B -- \G -- \J -- \F -- cycle;
+\node[color=darkgreen] at ({2*\s},-0.5) {$z_5'$};
+\end{scope}
+
+\begin{scope}[xshift=2cm]
+\gebiet
+\draw[color=darkgreen] \D -- \I -- \L -- \H -- cycle;
+\node[color=darkgreen] at ({2*\s},-0.5) {$z_6'$};
+\end{scope}
+
+\begin{scope}[xshift=4cm]
+\gebiet
+\draw[color=darkgreen] \C -- \L -- \N -- \K -- \M -- \J -- cycle;
+\node[color=darkgreen] at ({2*\s},-0.5) {$z_9'$};
+\end{scope}
+
+\begin{scope}[xshift=6cm]
+\gebiet
+\draw[color=darkgreen] \K -- \N -- \O -- \M -- cycle;
+\node[color=darkgreen] at ({2*\s},-0.5) {$z_{12}'$};
+\end{scope}
+
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/95-homologie/images/homocycles.pdf b/buch/chapters/95-homologie/images/homocycles.pdf
new file mode 100644
index 0000000..075bb65
--- /dev/null
+++ b/buch/chapters/95-homologie/images/homocycles.pdf
diff --git a/buch/chapters/95-homologie/images/homocycles.tex b/buch/chapters/95-homologie/images/homocycles.tex
new file mode 100644
index 0000000..898cac6
--- /dev/null
+++ b/buch/chapters/95-homologie/images/homocycles.tex
@@ -0,0 +1,157 @@
+%
+% tikztemplate.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}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\s{0.4}
+
+\def\punkt#1#2{({((#1)+0.5*(#2))*\s},{(#2)*\s*sqrt(3)/2})}
+\def\A{\punkt{0}{0}}
+\def\B{\punkt{1}{0}}
+\def\C{\punkt{2}{0}}
+\def\D{\punkt{3}{0}}
+\def\E{\punkt{4}{0}}
+\def\F{\punkt{0}{1}}
+\def\G{\punkt{1}{1}}
+\def\H{\punkt{2}{1}}
+\def\I{\punkt{3}{1}}
+\def\J{\punkt{0}{2}}
+\def\K{\punkt{1}{2}}
+\def\L{\punkt{2}{2}}
+\def\M{\punkt{0}{3}}
+\def\N{\punkt{1}{3}}
+\def\O{\punkt{0}{4}}
+
+%\def\knoten#1#2#3{
+%	\fill[color=white] \punkt{#1}{#2} circle[radius=0.3];
+%	\node at \punkt{#1}{#2} {$#3$\strut};
+%	\draw \punkt{#1}{#2} circle[radius=0.3];
+%}
+\def\dreieck#1#2#3{
+	\fill[color=gray] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+		-- \punkt{#1}{(#2)+1} -- cycle;
+%	\node at \punkt{#1+0.3333}{#2+0.3333} {$#3$\strut};
+%	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+%		\punkt{#1}{#2} -- \punkt{#1+1}{#2};
+%	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+%		\punkt{#1+1}{#2} -- \punkt{#1}{#2+1};
+%	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+%		\punkt{#1}{#2+1} -- \punkt{#1}{#2};
+}
+
+%\def\Dreieck#1#2#3{
+%	\fill[color=gray!50] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
+%		-- \punkt{#1+1}{(#2)-1} -- cycle;
+%	\node at \punkt{#1+0.3333}{#2+0.3333} {$#3$\strut};
+%}
+
+%\def\kante#1#2#3{
+%	\fill[color=white,opacity=0.8] \punkt{#1}{#2} circle[radius=0.15];
+%	\node at \punkt{#1}{#2} {$\scriptstyle #3$};
+%}
+
+\def\gebiet{
+	\dreieck{0}{0}{1}
+	\dreieck{1}{0}{2}
+	\dreieck{2}{0}{3}
+	\dreieck{3}{0}{4}
+	\dreieck{0}{1}{5}
+	\dreieck{2}{1}{6}
+	\dreieck{0}{2}{7}
+	\dreieck{1}{2}{8}
+	\dreieck{0}{3}{9}
+}
+
+\begin{scope}
+\gebiet
+\draw[color=red] \A -- \B -- \F -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_1$};
+\end{scope}
+
+\begin{scope}[xshift=2cm]
+\gebiet
+\draw[color=red] \B -- \C -- \G -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_2$};
+\end{scope}
+
+\begin{scope}[xshift=4cm]
+\gebiet
+\draw[color=red] \C -- \D -- \H -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_3$};
+\end{scope}
+
+\begin{scope}[xshift=6cm]
+\gebiet
+\draw[color=red] \D -- \E -- \I -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_4$};
+\end{scope}
+
+\begin{scope}[xshift=8cm]
+\gebiet
+\draw[color=red] \A -- \B -- \G -- \F -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_5$};
+\end{scope}
+
+\begin{scope}[xshift=10cm]
+\gebiet
+\draw[color=red] \C -- \D -- \I -- \H -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_6$};
+\end{scope}
+
+\begin{scope}[xshift=12cm]
+\gebiet
+\draw[color=red] \A -- \B -- \G -- \J -- \F -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_7$};
+\end{scope}
+
+\begin{scope}[xshift=0cm,yshift=-3cm]
+\gebiet
+\draw[color=red] \C -- \D -- \I -- \L -- \H -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_8$};
+\end{scope}
+
+\begin{scope}[xshift=2cm,yshift=-3cm]
+\gebiet
+\draw[color=red] \A -- \B -- \C -- \H -- \L -- \K -- \J -- \F -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_9$};
+\end{scope}
+
+\begin{scope}[xshift=4cm,yshift=-3cm]
+\gebiet
+\draw[color=red] \J -- \K -- \M -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_{10}$};
+\end{scope}
+
+\begin{scope}[xshift=6cm,yshift=-3cm]
+\gebiet
+\draw[color=red] \A -- \B -- \C -- \H -- \L -- \N -- \K -- \J -- \F -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_{11}$};
+\end{scope}
+
+\begin{scope}[xshift=8cm,yshift=-3cm]
+\gebiet
+\draw[color=red] \J -- \K -- \N -- \M -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_{12}$};
+\end{scope}
+
+\begin{scope}[xshift=10cm,yshift=-3cm]
+\gebiet
+\draw[color=red] \J -- \K -- \N -- \O -- \M -- cycle;
+\node[color=red] at ({2*\s},-0.5) {$z_{13}$};
+\end{scope}
+
+
+\end{tikzpicture}
+\end{document}
+