Commit 3f5be997 by Jan-Bernhard Kordaß

### Added draft of Romans fifth lecture (minus a figure).

parent ce5c6561
 ... ... @@ -865,6 +865,158 @@ A map $r \colon Y \to X$ such that $r \circ j = \id_X$ is called \CmMark{element \end{enumerate} \end{proof} The reverse statement $\tau(f) = 0 \implies f$ simple'' follows from a geometric description of the Whitehead group. To this end, we need some lemmas about elementary collapsed and expansions. \textbf{Notation.} Write \begin{itemize} \item $X \nearrow Y$, if there is a sequence of elementary expansions from $X$ to $Y$. \item $X \searrow Y$, if there is a sequence of elementary collapses from $X$ to $Y$. \item $X \doglegarrowright Y$, if there is a sequence of elementary expansions or collapses from $X$ to $Y$. \end{itemize} \begin{lemma*}[The relativity principle.] Given cellular pushouts \begin{equation*} \begin{tikzcd} X \ar[hook]{d} \ar{r}{f} & Y\ar[hook]{d} & X \ar{r}{f}\ar[hook]{d} & Y \ar[hook]{d}\\ Z \ar{r} & W & Z' \ar{r} & W' \end{tikzcd} \end{equation*} where we assume that $Z \doglegarrowright Z'$ rel X. Then $W \doglegarrowright W'$ rel $Y$ (by the same'' sequence of elementary expansions or collapses). \end{lemma*} \begin{lemma*}[The cylinder lemma] Let $f \colon X \to Y$ be cellular and let $A \subset X$ be a subcomplex. Then the inclusion $\cyl(f|_A) \hookrightarrow \cyl(f)$ is a composition of elementary expansions. (Special case: $A = \emptyset, Y \hookrightarrow \cyl(f)$.) \end{lemma*} \begin{proof} First consider the case $X = D^n \cup_q A$ for $q \colon S^{n-1} \to A$, i.e. $X$ is obtained by gluing an $n$-ball to $A$. This yields a pushout \begin{equation*} \begin{tikzcd} S^{n-1} \times [0,1] \cup_{S^{n-1} \times \{0\}} D^n \times \{0\} \ar{r} \ar[hook]{d}[swap]{\simeq} & \cyl(f|_A) \ar[hook]{d}[swap]{\simeq}\\ D^n \times [0,1] \ar{r} & \cyl(f). \end{tikzcd} \end{equation*} For the left hand side of the diagram, we have $(D^n \times [0,1], S^{n-1} \times [0,1] \cup_{S^{n-1} \times \{0\}} D^n \times \{0\}) \cong (D^{n+1}, S^n_+)$, where $S^n_+$ is a hemisphere. Thus the pushout describes one elementar expansion. \end{proof} \begin{lemma*}[Relative isomorphism lemma] If we have \begin{equation*} \begin{tikzcd} & Y_1 \ar{dd}{\cong}[swap]{h} \\ X \ar[hook]{ru} \ar[hook]{rd} \\ & Y_2 \end{tikzcd}, \end{equation*} where $h$ is a CW isomorphism, then $Y_1 \doglegarrowright Y_2$ rel $X$. \end{lemma*} \begin{proof} By the cylinder lemma we have $X \times [0,1] \cup Y_2 \times \{1\} = \cyl(h|_X) \nearrow \cyl(h)$. By the same proof, $X \times [0,1] \cup Y_1 \times \{0\} \nearrow \cyl(h)$. Applying the relativity principle to $\pr \colon X \times [0,1] \to X$ \begin{equation*} \begin{tikzcd} X \times [0,1] \ar{r}{\pr} \ar[hook]{d} & X \ar[hook]{d} & X \times [0,1] \ar{r}{\pr} \ar[hook]{d} & X \ar[hook]{d} \\ \cyl(h|_X) \ar{r} \ar[bend right]{rr}[swap]{\doglegarrowright} & Y \ar[bend right,dashed]{rr}[swap]{\doglegarrowright} & X \times [0,1] \cup Y_1 \times \{0\} \ar{r} & Y \end{tikzcd} \end{equation*} \end{proof} \begin{lemma*}[Homotopy lemma] Let cellular maps $f, g \colon K \to L$ be homotopic. Then $\cyl(f) \doglegarrowright \cyl(g)$ rel $K \cup L$ (top, bottom). \end{lemma*} \begin{proof} Let $H$ be a cellular homotopy with $H_0 = f, H_1 = g$. We have to show that \begin{align*} \cyl(H_0) \cup K \times [0,1] \nearrow \cyl(H) \nwarrow \cyl(H_1) \cup K \times [0,1]. \end{align*} This is implied by the general fact for $X = K \times [0,1]$ and $X_0 = K \times \{0\}$ and $f = H$. \begin{enumerate} \item[($*$)] If $f \colon X \to Y$ is cellular and $X \supset X_0 \nearrow X$, then $X \cup \cyl(f|_{X_0}) \nearrow \cyl(f)$. \end{enumerate} Now apply the relativity principle with respect to $\pr \colon K \times [0,1] \to K$ \begin{equation*} \begin{tikzcd} K \times [0,1] \ar{r}{\pr} \ar[hook]{d} & K \ar{d} & K \times [0,1] \ar{r}{\pr} \ar[hook]{d} & K \ar[hook]{d} \\ \cyl(H_0) \cup K \times [0,1] \ar{r} \ar[bend right]{rr}[swap]{\doglegarrowright} & \cyl(H_0) & \cyl(H_1) \cup K \times [0,1] \ar{r} & \cyl(H_1) \end{tikzcd} \end{equation*} Thus, $\cyl(f) \doglegarrowright \cyl(g)$. \end{proof} Let $(Y,X)$ and $(Z,X)$ be paris of finite CW complexes such that $X \overset{\simeq}{\hookrightarrow} Y$ and $X \overset{\simeq}{\hookrightarrow} Z$ are homotopy equivalences. We say that $(Y,X)$ and $(Z,X)$ are \CmMark{equivalent}, if $Y \doglegarrowright X$ rel $X$. \begin{dfn*} The \CmMark{geometric Whitehead group} $\Wh^{\text{geo}}(X)$ of $X$ is the set of equivalence classes of such pairs $(Y,X)$ of finite CW complexes with $X \overset{\simeq}{\hookrightarrow} Y$. $\Wh^{\text{geo}}(X)$ carries the structure of an abelian group. \begin{enumerate} \item Abelian addition: $[Y,X] + [Z,X] = [Y \cup_X Z, X]$ This is well-defined since for $(Y',X) \sim (Y,X)$ \begin{equation*} \begin{tikzcd} X \ar[hook]{r}{\simeq} \ar[hook]{d}{\simeq} & Z \ar[hook]{d}{\simeq} & X \ar{r}{\simeq} \ar[hook]{d}{\simeq} & Z \ar[hook]{d} \\ Y \ar[hook]{r} \ar[bend right]{rr}[swap]{\doglegarrowright} & Y \cup_X Z & Y' \ar{r} & Y' \cup_X Z \end{tikzcd} \end{equation*} Thus, $(Y \cup_X Z, X) \sim (Y' \cup_X Z, X)$. \item Zero element: $[X,X]$. \item Inverse: Take a pair $(Y,X)$ and let $D \colon Y \to X$ be a cellular strong deformation retract. \begin{figure}[h!] \centering \caption{TODO: Add figure for $2\cyl(D)$.} \end{figure} Claim: $[2\cyl(D),X] = -[Y,X]$. \begin{align*} [2 \cyl(D),X] + [Y,X] = [2 \cyl(D) \cup_X Y, X] = [\cyl(i \circ D) \cup \tilde{\cyl}(D), X] \end{align*} for $i \circ D \colon Y \to Y \simeq \id$, the homotopy lemma yields $\cyl(i \circ D) \doglegarrowright Y \times [0,1]$ rel $Y \times \{0\} \cup Y$ \begin{align*} = [Y \times [0,1], \tilde{\cyl}(D),X] \end{align*} $Y \times [0,1] \searrow X \times [0,1] \cup Y \times \{0\}$ which follows from $(*)$ for $\id_Y$. \begin{align*} = [Y \times [0,1] \cup \tilde{\cyl}(D), X]. \end{align*} $\tilde \cyl(D) \searrow X \times [-1,0]$ by the cylinder lemma for $D$. \begin{align*} = [X \times [-1,1], X] = [X,X] = 0 \end{align*} \end{enumerate} \end{dfn*} \begin{thm*} \begin{align*} \Wh^{\text{geo}}(X) \xrightarrow{\cong} \Wh(\pi(X)), \quad [Y,X] \mapsto i_{*}^{-1}(\tau(i \colon X \hookrightarrow Y)) \end{align*} is an isomorphism of abelian groups. \end{thm*} In some sense this is a topological version of the s-cobordism theorem. \chapter{Harmonic Maps [Andy Sanders]} ... ... @@ -1529,3 +1681,9 @@ By uniformization, we have a section \mathcal M/C^{\infty}(\Sigma) \to \mathcal \mathrm{QD}(\sigma) \simeq \mathcal T(\Sigma) \simeq \mathcal C(\Sigma)/\Diff_0 \simeq \mathcal M_{-1}/\Diff_0(\Sigma) \simeq \C^{3g-3}, \end{align*} where\mathcal T(\Sigma)\$ is Teichmüller space. %%%Local Variables: %%% mode: latex %%% TeX-master: "skript-rtg-lectures-ws1617" %%% End:
 ... ... @@ -182,6 +182,7 @@ \newcommand{\modulo}[1]{\ensuremath{/_{\displaystyle #1}}} \newcommand\doglegarrowright{\,\begin{tikzpicture}[scale=0.2]\draw[arrows=-,bend right=0,solid] (0.3,0) to (1,1);\draw[->,bend right=0,solid] (1,1) to (1.7,0);\end{tikzpicture}\,} %% Theorems \newcounter{thmglobal} ... ...
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