Commit c4118ec6 authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß
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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 $\Z/5 \cong \left<t\right>$ 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}[3][]{
\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}[3][]{
\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}[2][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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