Commit a07bad41 authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß

Added Romans fourth lecture.

parent 15e2fdd0
......@@ -442,6 +442,7 @@ The inverse homomorphism is given by $R^n \cong \{ P \oplus Q \xrightarrow{f \ot
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*}
\subsection{Whitehead torsion for chain complexes}
In the following let us repeat some preliminaries on chain complexes.
......@@ -659,14 +660,194 @@ Then $\tau(g_{*}) = \tau(f_{*}) + \tau(h_{*})$.
\end{proof}
Recall that last time we proved the following Lemma.
\begin{lemma*}
\begin{enumerate}[label=(\arabic*)]
\item Additivity
\begin{equation*}
\begin{tikzcd}
0 \ar{r} & C_{*}' \ar{r} \ar{d}{f_{*}}[swap]{\simeq} & D_{*}' \ar{r} \ar{d}{g_{*}}[swap]{\simeq} & E_{*}' \ar{r} \ar{d}{h_{*}}[swap]{\simeq} & 0\\
0 \ar{r} & C_{*} \ar{r} & D_{*} \ar{r} & E_{*} \ar{r} & 0
\end{tikzcd}
\end{equation*}
Then $\tau(g_{*}) = \tau(f_{*}) + \tau(h_{*})$.
\item Homotopy invariance.
If $f_{*} \simeq g_{*} \colon C_{*} \xrightarrow{\simeq} D_{*}$, then $\tau(f_{*}) = \tau(g_{*})$.
\item Composition formula.
$\tau(g_{*} \circ f_{*}) = \tau(g_{*}) + \tau(f_{*})$.
\end{enumerate}
\end{lemma*}
\begin{proof}
\begin{enumerate}[label=ad (\arabic*),leftmargin=1.5cm]
\setcounter{enumi}{2}
\item If $h_{*} \colon f_{*} \simeq g_{*}$, then we have an isomorphism
\begin{align*}
\cone(f_{*}) \colon C_{*-1} \oplus D_{*} \xrightarrow{\begin{pmatrix}\id & 0 \\ h_{*-1} & \id\end{pmatrix}} C_{*-1} \oplus D_{*} = \cone(g_{*}).
\end{align*}
Thus
\begin{align*}
& \tau(f_{*}) - \tau(g_{*}) = \tau(\cone(f_{*})) - \tau(\cone(g_{*}))\\
& \qquad = \sum_{p \geq 0} (-1)^p \begin{pmatrix}\id & 0 \\ h_{*-1} & \id\end{pmatrix} = 0 \in K_1(R).
\end{align*}
\end{enumerate}
\end{proof}
\subsection{Whitehead torsion of maps between CW-complexes}
Let $X,Y$ be connected finite CW-complexes and let $f \colon X \xrightarrow{\simeq} Y$ be a homotopy equivalence.
Further pick base points $x \in X, y = f(x) \in Y$ and set $\pi := \pi_1(Y,y)$.
We identify $\pi_1(X,x)$ with $\pi$ via $\pi_1(f)$ and considering the universal covers we obtain the diagram
\begin{equation*}
\begin{tikzcd}
\tilde X \ar{r}{\tilde f}[swap]{\simeq} \ar{d}{\pr_X} & \tilde Y \ar{d}{\pr_Y}\\
X \ar{r}{f}[swap]{\simeq} & Y
\end{tikzcd}
\end{equation*}
There is a unique lift $\tilde f$ with $\tilde f(\tilde x) = \tilde y$.
Further, $\tilde f$ is $\pi$-equivariant.
$\tilde X$ caries a CW structure $\tilde X^n = \pr_X^{-1}(X^n)$.
Thus $C^{\text{CW}}_{*}(\tilde f) \colon C_{*}^{\text{CW}}(\tilde X) \xrightarrow{\simeq} C_{*}^{\text{CW}}(\tilde Y)$, where $C_{n}^{CW}(\tilde X) = H_n(\tilde X^n, \tilde X^{n-1})$, are chain homotopy equivalences of $\Z\pi$-chain complexes.
We obtain a $\Z\pi$-basis of $C_n^{\text{CW}}(\tilde X)$ by choosing ($\pi$-)push-outs:
\begin{equation*}
\begin{tikzcd}
\coprod_{I_n} \pi \times S^{n-1} \ar{r} \ar[hook]{d} & \tilde X^{n-1} \ar[hook]{d} \\
\coprod_{I_n} \pi \times D^n \ar{r} & \tilde X^n
\end{tikzcd}.
\end{equation*}
This yields a cellular basis
\begin{align*}
C_n^{\text{CW}}(\tilde X) = H_n(\tilde X^n, \tilde X^{n-1}) \xleftarrow{\cong} \bigoplus_{I_n} H_n(\pi \times (D^n, S^{n-1})) = \bigoplus_{I_n}\Z\pi
\end{align*}
The matrix of base change when we take another push-out looks like
\begin{align*}
\bigoplus_{I_n} \Z\pi
\xrightarrow{%
\begin{pmatrix}\pm g_1 & & \\ & \ddots &\\ & & \pm g_n\end{pmatrix}P}
\bigoplus_{I_n} \Z\pi,
\end{align*}
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*}
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*}
\phi_{y,y'} \colon \Wh(\pi_1(Y,y)) \xrightarrow{\cong} \Wh(\pi_1(Y,y'))
\end{align*}
induced by any choice of path from $y$ to $y'$.
This yields a basepoint free definition of the Whitehead group.
\begin{dfn*}
$\Wh(\pi_1(Y)) := \colim_{y \in Y} \Wh(\pi_1(Y,y)) = \coprod_{y \in Y}\Wh(\pi_1(Y,y))/\sim$
for $z \sim \phi_{y,y'}$.
\end{dfn*}
One verifies that $\tau(C_{*}^{\text{CW}}(\tilde f)) \in \Wh(\pi(Y))$ is independent of all choices of basepoints.
\begin{dfn*}
Let $X \xrightarrow{f} Y$ be a homotopy equivalence of finite CW-complexes.
Define
\begin{align*}
\Wh(\pi(Y)) := \bigoplus_{C \in \pi_0(Y)} \Wh(\pi(C))
\end{align*}
and
\begin{align*}
\tau(f) := \left(\tau(C_{*}^{\text{CW}}(\tilde f \colon \tilde{f^{-1}(C)} \to \tilde C))\right)_{C \in \pi_0(Y)},
\end{align*}
which is called the \CmMark{Whitehead torsion} of $f$.
\end{dfn*}
\begin{rem*}
The elementary properties of $\tau(f)$ stated in the beginning of chapter 2 are now direct consequences of the lemma in section 2.3.
\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}
\begin{equation*}
\begin{tikzcd}
S_+^{n-1} \ar{r}{q} \ar[hook]{d}{\simeq} & X \ar[hook]{d}{\simeq} \\
D^n \ar{r}{\bar q} & Y
\end{tikzcd}
\end{equation*}
Such a homotopy equivalence $X \to Y$ as in the diagram is called \CmMark{elementary expansion}.
A map $r \colon Y \to X$ such that $r \circ j = \id_X$ is called \CmMark{elementary collapse} ($r$ will be a homotopy inverse of $j$).
\begin{rem*}
If $X$ is a CW-complex and $q(S^{n-2}) \subset X^{n-2}$ with $q(S^{n-1}_+) \subset X^{n-1}$, then $Y$ inherits a natural CW-structure.
Then $Y$ is the result of attaching an $(n-1)$-cell and then an $n$-cell to $X$.
\end{rem*}
\begin{dfn*}
Let $f \colon X \to Y$ be a map of finite CW-complexes.
We call $f$ a \CmMark{simple homotopy equivalence}, if it is homotopic to a zig-zag of elementary expansions and collapses, i.e. there exists maps $f(i)$ such that
\begin{equation*}
\begin{tikzcd}
X = X(0) \ar{r}{f(0)} \ar[bend right]{rrr}{f} & X(1) \ar{r} & \cdots \ar{r}{f(n-1)} & X(n) = Y
\end{tikzcd},
\end{equation*}
commutes up to homotopy, where each $f(i)$ is an elementary expansion or collapse.
\end{dfn*}
\begin{thm*}
\begin{enumerate}[label=(\arabic*)]
\item A homotopy equivalence $f \colon X \to Y$ is simple if and only if $\tau(f) = 0$.
\item For every element $a \in \Wh(\pi(X))$ there is a homotopy equivalence $X \xrightarrow{f} Y$ such that $f_{*}^{-1}(\tau(f)) \in \Wh(\pi(X))$.
\end{enumerate}
\end{thm*}
\begin{proof}
\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*}
\begin{tikzcd}
0 \ar{r} & C_{*}(\tilde X) \ar{r}{f_{*}} & C_{*}(\tilde Y) \ar{r} & C_{*}(\tilde Y, \tilde X) \ar{r} & 0\\
0 \ar{r} & C_{*}(\tilde X) \ar{r}{\id} \ar{u}{\id} & C_{*}(\tilde X) \ar{r} \ar{u}{f_{*}} & 0 \ar{r} \ar{u}{0} & 0
\end{tikzcd}
\end{equation*}
Thus, by additivity, we have $\tau(f) = \tau(C_{*}(\tilde Y, \tilde X))$.
But this is a very simple CW-complex with differential $\id \colon \Z\pi \to \Z\pi$ (for $\pi = \pi_1(Y)$) from degree $n$ to degree $n-1$ and thus $\tau(C_{*}(\tilde Y,\tilde X)) = 0$.
The converse is more complicated.
\item Let $A \in \GL_n(\Z\pi)$ ($n \geq 3$) and let $X' = X \vee \bigvee_{j = 1}^nS^{n-1}$, where we denote the $j$-th inclusion of $S^{n-1}$ into the wedge by $b_j$.
We attach $n$ $n$-cells to $X'$ via attaching maps $f_j \colon S^{n-1} \to X'$ (relevant: $[f_j] \in \i_{n-1}(X')$ which yields $Y$.
\begin{align*}
\begin{pmatrix}
f_1 \\ \vdots \\ f_n
\end{pmatrix}
= A \cdot
\begin{pmatrix}
b_1 \\ \vdots \\ b_n
\end{pmatrix}
\ni \bigoplus_{j = 1}^n \pi_{n-1}(X')
\text{ for } b_j \in \pi_{n-1}(X') \curvearrowleft \pi_1(X') = \pi_1(X) = \pi.
\end{align*}
One sees by closer inspection that
\begin{align*}
\tau(\tilde X \hookrightarrow \tilde Y) = \tau(C_{*}^{\text{CW}}(\tilde Y, \tilde X))
= \tau\left(\bigoplus_{j=1}^n \Z\pi \xrightarrow{A} \bigoplus_{j=1}^n \Z\pi\right)
\end{align*}
for $[A] \in \Wh(\pi)$.
\end{enumerate}
\end{proof}
\chapter{Harmonic Maps [Andy Sanders]}
Also consider the notes \url{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}.
\section{Basics of harmonic maps}
In the following let every manifold be oriented (for integration safety reasons).
\subsection{Background differential geometry}
Let $E \to M$ be an $\R$-vector bundle over $M$ (second countable, hausdorff manifold) of rank $r$.
......
......@@ -87,6 +87,7 @@
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\colim}{colim} % colimit
\DeclareMathOperator{\pr}{pr} % projection
\DeclareMathOperator{\Hom}{Hom} % homomorphisms
\DeclareMathOperator{\Diff}{Diff} % diffeomorphisms
......
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