Commit fd650c4f authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß

Added pictures.

parent a07bad41
......@@ -733,10 +733,17 @@ where $P$ is a permutation matrix.
Hence we obtain a well-defined element $\tau(C_{*}^{\text{CW}}(\tilde f)) \in \Wh(\pi)$ independent of the choices of cellular bases of $\tilde X, \tilde Y$.
It may still depend on the choice of basepoints.
Let $y,y' \in Y$ be connected by two paths $\alpha$ and $\beta$.\marginpar{TODO}
\begin{equation*}
\text{figure: paths}
\end{equation*}
Let $y,y' \in Y$ be connected by two paths $\alpha$ and $\beta$ as in the figure below.
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\fill (0,0) circle (0.05);
\fill (5,1) circle (0.05);
\draw (0,0) node[left]{$y$} to[out=0,in=215] node[below] {$\beta$} (5,1) node[right]{$y'$};
\draw (0,0) to[out=100,in=174] node[above]{$\alpha$} (5,1);
\end{tikzpicture}
\end{figure}
Then $\pi_1(Y,y) \xrightarrow{\alpha_{*}, \beta_{*}} \pi_1(Y,y')$ differ by an inner automorphism of $\pi_1(Y,y')$, which induces the identity on $\Wh(\pi_1(Y,y'))$.
So there is a canonical isomorphism
\begin{align*}
......@@ -769,7 +776,22 @@ One verifies that $\tau(C_{*}^{\text{CW}}(\tilde f)) \in \Wh(\pi(Y))$ is indepen
\end{rem*}
In the following we will discuss the topological meaning of $\tau(f)$.
For this let $S^{n-2} \subset S_+^{n-1} \subset S^{n-1} \subset D^n$, where $S_+^{n-1}$ denotes the upper hemisphere.\marginpar{TODO: figure}
For this let $S^{n-2} \subset S_+^{n-1} \subset S^{n-1} \subset D^n$, where $S_+^{n-1}$ denotes the upper hemisphere.
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\draw [thick,domain=0:180] plot ({cos(\x)}, {sin(\x)});
\node[above] at (0,1) {$S^1_+$};
\draw[pattern=north west lines,pattern color=gray] (0,0) circle (1);
\fill (-1,0) circle (0.05) node[left]{$S^0$};
\fill (1,0) circle (0.05);
\draw[pattern=north west lines,pattern color=gray] (3,-1) to[out=60,in=202] (6,0) to[out=100,in=30] (3.5,1) to[out=210,in=85] (3,-1);
\draw[thick] (3,-1) to[out=60,in=202] node[below] {$X$} (6,0);
\end{tikzpicture}
\end{figure}
\begin{equation*}
\begin{tikzcd}
S_+^{n-1} \ar{r}{q} \ar[hook]{d}{\simeq} & X \ar[hook]{d}{\simeq} \\
......@@ -804,7 +826,7 @@ A map $r \colon Y \to X$ such that $r \circ j = \id_X$ is called \CmMark{element
\end{thm*}
\begin{proof}
\begin{enumerate}[label=ad (\arabic*),leftmargin=1.5cm]
\begin{enumerate}[label=ad (\arabic*).,leftmargin=1.5cm]
\item We will only prove the direction ``simple $\implies \tau(f) = 0$''.
We may assume that $f$ is an elementary expansion
\begin{equation*}
......
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