Commit 3f5be997 authored by Jan-Bernhard Kordaß's avatar 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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