Commit fd650c4f by Jan-Bernhard Kordaß

 ... ... @@ -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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