### Added draft of the lectures from the second RTG day.

parent 1627ea30
 ... ... @@ -157,10 +157,295 @@ Thus, this number only depends on the homotopy class of $f$. \end{proof} \section{Whitehead torsion} \subsection{Introduction/Motivation} Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-complexes, Whitehead torsion is an assignment $\tau(f) \in \Wh(\pi_1(Y))$ living in the so-called Whitehead group. \begin{thm*}[Properties of Whitehead torsion] \begin{enumerate}[label=(\arabic*)] \item homotopy invariance \item\footnote{This is a deep theorem of Chapman.} If $f \colon X \to Y$ is a homeomorphism, then $\tau(f) = 0$. \item additivity: A cellular pushout is a diagram \begin{equation*} \begin{tikzcd} X_0 \ar{r}{f} \ar[hook]{d}{i} & X_2 \ar{d}\\ X_1 \ar{r} & X \end{tikzcd} \end{equation*} with $X_i$ be CW-complexes, where $f$ is cellular and $i$ is an inclusion of a subcomplex. If the diagram \begin{equation*} \begin{tikzcd} X_0 \ar{rr} \ar[hook]{dd} \ar{rd}{f_0}[swap]{\simeq} &[0.8cm] &[-0.2cm] X_2 \ar[bend left=10]{rd}{f_2}[swap]{\simeq} &[0.8cm] \\ & Y_0 \ar[near start]{rr}{\phi} \ar[near end,hook]{dd}{j} & & Y_2 \ar{dd}{i}\\ X_1 \ar[crossing over]{rr} \ar[bend right=10]{rd}{f_1}[swap]{\simeq} & & X \ar[dashed]{rd}{f} \ar[leftarrow,crossing over]{uu} & \\ & Y_1 \ar{rr}{\psi} & & Y \end{tikzcd} \end{equation*} is a map of cellular pshouts such that $f_i$ are homotopy equivalences. Then $f$ is a homotopy equivalence and \begin{align*} \tau(f) & = "\tau(f_1) + \tau(f_2) - \tau(f_0)"\\ & = \psi_{*}(\tau(f_1)) + j_{*}(\tau(f_2)) - (\psi \circ i)_{*}(\tau(f_0)) \in \Wh(\pi_1(Y)). \end{align*} A similar additivity holds for the Lefschetz number.\footnote{Idea. $0 \to C_1(X_0) \to C_{*}(X_1) \oplus C_{*}(X_2) \to C_{*}(X) \to 0$ exact.} \item composition formula. If we have $\begin{tikzcd} X \ar{r}{f}[swap]{\simeq} & Y \ar{r}{g}[swap]{\simeq} & Z \end{tikzcd}$, then \begin{align*} \tau(g \circ f) & = "\tau(f) + \tau(g)"\\ & = g_{*}(\tau(f) + \tau(g)) \in \Wh(\pi_1Z). \end{align*} \end{enumerate} \end{thm*} \begin{thm*}[s-cobordism theorem (Mazur, Barden, Stallings, Smale)] Let $M$ be a closed smooth manifold of dimension $\geq 5$. Let $(W, i, j)$ be an s-cobordism \begin{figure}[h!] \centering \begin{tikzpicture}[scale=0.8] \coordinate (L1) at (2,0); \coordinate (L2) at (2,3); \coordinate (LL1) at ($(L1)+(-4,0)$); \coordinate (LL2) at ($(L2)+(-4,0)$); \coordinate (R1) at (8,0.5); \coordinate (R2) at (8,2); \coordinate (RR1) at ($(R1)+(3,0)$); \coordinate (RR2) at ($(R2)+(3,0)$); % left inclusion \ellipsebetweenvert{LL1}{LL2} \node[below] at (LL1) {$M$}; \draw[right hook->] ($(LL1) + (1.2,1.5)$) to node[above] {$i$} node[below]{$\simeq$} ($(L1) + (-1.2,1.5)$); % cobordism \ellipsebetweenvert{L1}{L2} \node[below] at (L1) {$M_0$}; \draw[out=0,in=180] (L1) to (5,-1) to (R1); \draw[out=0,in=180] (L2) to (R2); \topgenus[0.35]{5,0} \topgenus[0.22]{6,1.2} \node at (6,3) {$W$}; \ellipsebetweenvert[left]{R1}{R2} \node[below] at (R1) {$M_1$}; % right inclusion \draw[left hook->] ($(RR1) + (-1,0.75)$) to node[above] {$j$} node[below]{$\simeq$} ($(R1) + (1,0.75)$); \ellipsebetweenvert{RR1}{RR2} \node[below] at (RR1) {$N$}; \end{tikzpicture} % sketch: \includegraphics[width=0.8\textwidth]{img/1.png} \end{figure} i.e. $\partial W = M_0 \coprod M_1$ and $i \colon M \hookrightarrow W$, $j \colon N \hookrightarrow W$ are homotopy equivalences. Then $\tau(M \xrightarrow{i} W) = 0$ if and only if $(W,i_0,i_1)$ is trivial, i.e. \begin{figure}[h!] \centering \begin{tikzpicture}[every node/.style={scale=0.8}] \coordinate (LU1) at (2.4,1.5); \coordinate (LU2) at (2.4,2.5); \coordinate (RU1) at (5,1.3); \coordinate (RU2) at (5,2.3); \coordinate (LD1) at (2.4,0.5); \coordinate (LD2) at (2.4,-0.5); \coordinate (RD1) at (5,0.5); \coordinate (RD2) at (5,-0.5); % leftmost part \node at (0,3) {$\exists$}; \node[scale=1.5] at (-1,1) {$M$}; \draw[right hook->] (-0.4,1.2) to node[above] {$i$} (2,2); \draw[right hook->] (-0.4,0.8) to node[below] {$\operatorname{incl}$} (2,0); % upper cobordism \ellipsebetweenvert{LU1}{LU2} \draw[out=0,in=180] (LU1) to (3,1.6) to (RU1); \draw[out=0,in=180] (LU2) to (4,2.2) to (RU2); \ellipsebetweenvert[left]{RU1}{RU2} % arrow from the cobordism to the cylinder \draw[->] ($(LU1)!0.5!(RU1) + (0,-0.1)$) to node[left]{$\cong$} node[right]{diffeo} ($(LU1)!0.5!(RU1) + (0,-0.7)$); % cylinder \ellipsebetweenvert{LD1}{LD2} \draw (LD1) to (RD1); \draw (LD2) to (RD2); \ellipsebetweenvert[left]{RD1}{RD2} \end{tikzpicture} % sketch: \includegraphics[width=0.5\textwidth]{img/2.png} \end{figure} \end{thm*} This theorem implies the Poincaré conjecture in dimensions at least 6, which says that if $M$ is a closed (smooth) manifold that is homotopy equivalent to $S^n$, then $M$ is homeomorphic to $S^n$. The proof of this implication is basically along these lines: Pick disjoint embedded $n$-disks in $M$ and remove them. The result is a manifold $W$ with two boundary components. Consider the s-cobordism theorem for this manifold as depicted in the figure below, where $f$ is a diffeomorphism of $(n-1)$-spheres. \begin{figure}[h!] \centering \begin{tikzpicture} \coordinate (U1) at (0,3); \coordinate (U2) at (2.2,3); \coordinate (D1) at (-0.2,0); \coordinate (D2) at (1.8,0); % bordism on the left \ellipsebetweenhor{U1}{U2} \draw[out=270,in=90] (U1) to (0.3,1.8) to (D1); \draw[out=270,in=90] (U2) to (2.3,1.7) to (D2); \ellipsebetweenhor[upper]{D1}{D2} \node[left] at (0,1.5) {$W$}; % arrows and lower disk \draw[bend left=15,->] (2.8,3.3) to node[above] {$f$} (4.2,3.3); \draw[->] (3,1.5) to node[above] {$\cong$} (4,1.5); \draw[left hook->] (2.5,-1) to (1.3,-0.5); \draw[right hook->] (4.2,-1) to (5.5,-0.5); \draw[pattern=north west lines,pattern color=gray] (3.35,-1) ellipse (0.6 and 0.3); \node at (3.8,-0.4) {$D^n$}; % cylinder on the right \ellipsebetweenhor{5,3}{7,3} \draw (5,3) to (5,0); \draw (7,3) to (7,0); \ellipsebetweenhor[upper]{5,0}{7,0} \node[right] at (7,1.5) {$S^{n-1} \times [0,1]$}; \end{tikzpicture} % sketch: \includegraphics[width=0.7\textwidth]{img/3.png} \end{figure} By filling top and bottom, the Poincaré conjecture is implied, if we can extend a diffeomorphism $f \colon S^{n-1} \xrightarrow{\cong} S^{n-1}$ to a homeomorphism $F \colon D^n \xrightarrow{\cong} D^n$. This can be done by the so-called Alexander trick ($F(t x) = tf(x)$ for $t \in [0,1], x\in S^{n-1}$). \subsection{Whitehead group and lower K-theory} Let $R$ be a unital ring. Then \begin{align*} K_0(R) := \left< G \mid R \right>_{\text{ab}} \end{align*} with generators $G = \{$ isomorphism classes $[P]$ of fin. gen. projective $R$-modules $\}$ and relations $R = \{\ [P_1] = [P_0] + [P_2]$ whenever $0 \to P_0 \to P_1 \to P_2 \to 0$ is exact $\}$. (Recall that a direct summand of in a free $R$-module is called a \CmMark{projective module}.) This can be understood as a kind of universal dimension for projective $R$-modules. \begin{align*} K_1(R) := \left< G \mid R \right>_{\text{ab}} \end{align*} with generators $G = \{$ conjugacy classes $[f]$ of automorphisms $f \colon P \to P$ of fin. gen. projective $R$-modules $\}$ and relations $R$ given as follows. \begin{enumerate}[label=(\roman*)] \item Every commuting diagram \begin{equation*} \begin{tikzcd} 0 \ar{r} & P_0 \ar{r} \ar{d}{f_0}[swap]{\cong} & P_1 \ar{r} \ar{d}{f_1}[swap]{\cong} & P_2 \ar{r} \ar{d}{f_2}[swap]{\cong} & 0\\ 0 \ar{r} & P_0 \ar{r} & P_1 \ar{r} & P_2 \ar{r} & 0. \end{tikzcd} \end{equation*} gives rise to a relation $[f_1] = [f_0] + [f_2]$. \item $f,g \colon P \xrightarrow{\cong} P$ yield a relation $[f \circ g] = [f] + [g]$. \end{enumerate} This can be understood as an attempt to define a universal determinant of an automorphism. There is a more common definition of $K_1(R)$ in terms of the general linear groups with coefficients in $R$. Recall that $\GL(R) = \colim_{n \to \infty} \GL_n(R)$ with respect to the inclusion $\GL_n(R) \hookrightarrow \GL_{n+1}(R)$ to the upper left block. Then \begin{align*} K_1(R) = \GL(R)_{\text{ab}} = \GL(R)/[\GL(R),\GL(R)]. \end{align*} The so-called \CmMark{Whitehead lemma} states that $[\GL(R),\GL(R)] = E(R)$, where $E(R)$ is the subgroup of $\GL(R)$ generated by all elementary upper triangular matrices with ones on the diagonal. As a consequence, if $R$ is a field then the determinant defines an isomorphism $\det \colon K_1(R) \xrightarrow{\cong} \R\setminus\{0\}$. To see the equivalence of these two definitions, we can use the map \begin{align*} \GL(R) & \to K_1(R)\\ A & \mapsto [R^n \to R^n, \ x \mapsto Ax] \end{align*} and the fact that is descents to $\GL(R)_{\text{ab}} \to K_1(R)$. The inverse homomorphism is given by $R^n \cong \{ P \oplus Q \xrightarrow{f \otimes \id} P \oplus Q \mid f \text{ iso } \}$. \begin{dfn*} Let $\Gamma$ be a group. Then the \CmMark{Whitehead group} is defined as \begin{align*} \Wh(\Gamma) := \coker(\Gamma \times \{\pm 1\} \to K_1(\Z[\Gamma]), \ (\gamma, \pm 1) \mapsto \pm[\gamma]) \end{align*} \end{dfn*} \begin{expl*} The Whitehead group $\Wh(\{1\})$ is trivial, since the determinant yields an isomorphism $K_1(\Z) \xrightarrow{\det} \{\pm 1\}$. It is a conjecture that torsion-free groups $\Gamma$ have vanishing Whitehead group. Assertion: $\Wh(\Z/5) \cong \Z$. Here only prove that $\Wh(\Z/5)$ is infinite. We have a map $\phi_{*} \colon K_1(\Z[\Z/5]) \to K_1(\C)$ induced by $\Z[\Z/5] \xrightarrow{\phi} \C, \ t \mapsto \xi$, where $\Z/5 \cong \left$ and $\xi = \exp(2\pi i/5) \in \C$. Thus \begin{equation*} \begin{tikzcd} K_1(\Z[\Z/5]) \ar{r}{\phi_{*}} \ar{d} & K_1(\C) \ar{r}{\det} & \C^{\times} \ar{r}{|\blank|} & \R_{> 0}\\ \Wh(\Z/5) \ar[bend right=10]{rrru}{\tau} & & & \end{tikzcd} \end{equation*} One can see that $1 - t - t^{-1}$ is a unit in $\Z[\Z/5]$, since $(1 - t - t^{-1})( - t^2 - t^3) = 1$ and thus $\tau([1 - t - t^{-1}]) \neq 1$ \end{expl*} \chapter{Harmonic Maps [Andy Sanders]} Also consider the notes at \href{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}. Also consider the notes \url{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}. \section{Basics of harmonic maps} ... ...
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 ... ... @@ -81,6 +81,8 @@ \DeclareMathOperator{\Bild}{Im} \DeclareMathOperator{\Rang}{Rank} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\colim}{colim} % colimit \DeclareMathOperator{\Hom}{Hom} % homomorphisms \DeclareMathOperator{\Diff}{Diff} % diffeomorphisms ... ... @@ -90,6 +92,8 @@ \DeclareMathOperator{\Isom}{Isom} % isometriess \DeclareMathOperator{\Imm}{Imm} % immersions \DeclareMathOperator{\Wh}{Wh} % Whitehead group \DeclareMathOperator{\Vol}{Vol} % volume \DeclareMathOperator{\Ext}{Ext} % Ext term ... ...
 ... ... @@ -19,6 +19,47 @@ \usepackage{tikz} \usepackage{tikz-cd} \usetikzlibrary{matrix,arrows,calc,intersections, through, positioning, patterns, decorations.text, decorations.pathmorphing, decorations.markings, decorations.pathreplacing} % (optional) [left|right], one point, the other point \newcommand{\ellipsebetweenvert}[]{ \ifthenelse{\equal{#1}{right}}{ \draw[looseness=0.5,out=0,in=0,dashed] (#2) to (#3); }{ \draw[looseness=0.5,out=0,in=0] (#2) to (#3); } \ifthenelse{\equal{#1}{left}}{ \draw[looseness=0.5,out=180,in=180,dashed] (#2) to (#3); }{ \draw[looseness=0.5,out=180,in=180] (#2) to (#3); } } % (optional) [upper|lower], one point, the other point \newcommand{\ellipsebetweenhor}[]{ \ifthenelse{\equal{#1}{upper}}{ \draw[looseness=0.5,out=90,in=90,dashed] (#2) to (#3); }{ \draw[looseness=0.5,out=90,in=90] (#2) to (#3); } \ifthenelse{\equal{#1}{lower}}{ \draw[looseness=0.5,out=270,in=270,dashed] (#2) to (#3); }{ \draw[looseness=0.5,out=270,in=270] (#2) to (#3); } } % scale, point \newcommand{\topgenus}[0.5]{ \node at (#2) { \begin{tikzpicture}[scale=#1, every node/.style={scale=#1}] \draw (-2,.2) .. controls (-1.5,-0.3) and (-1,-0.5) .. (0,-.5) .. controls (1,-0.5) and (1.5,-0.3) .. (2,0.2); \draw (-1.75,0) .. controls (-1.5,0.3) and (-1,0.5) .. (0,.5) .. controls (1,0.5) and (1.5,0.3) .. (1.75,0); \end{tikzpicture} }; } \usepackage{setspace} \renewcommand{\baselinestretch}{1.1} ... ...
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