Commit f7e4dc1a authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß

Aded draft of the first rtg lecture on the third rtg day.

parent 422d5e6f
......@@ -442,6 +442,222 @@ 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*}
\section{Whitehead torsion for chain complexes}
In the following let us repeat some preliminaries on chain complexes.
Let $R$ be a (not necessarily commutative) ring.
Let $f_{*} \colon C_{*} \to D_{*}$ be an $R$-chain map.
The \CmMark{mapping cylinder} $\cyl(f_{*})$ is an $R$-chain complex with p-th differential
\begin{align*}
C_p \oplus C_{p-1} \oplus D_p
\xrightarrow{%
\begin{pmatrix}
c_p & -\id & 0\\
0 & -c_{p-1} & 0\\
0 & f_{p-1} & f
\end{pmatrix}}
C_{p-1} \oplus C_{p-2} \oplus D_{p-1},
\quad
\end{align*}
\begin{rem*}
For a continous map $f \colon X \to Y$ we have the topological mapping cylinder.
\begin{equation*}
\begin{tikzcd}
X \ar{r}{f} \ar[hook]{d}{i_0} & Y \ar[hook]{d}\\
X \times [0,1] \ar{r} & \cyl(f)
\end{tikzcd}
\end{equation*}
If $X,Y$ are CW-complexes and $f$ is cellular, then
\begin{align*}
\cyl(C_{*}(f) \colon C_{*}(X) \to C_{*}(Y)) = C_{*}(\cyl(f)).
\end{align*}
\end{rem*}
The \CmMark{mapping cone} $\cone(f_{*} \colon C_{*} \to D_{*})$ is a quotient of $\cyl(f_{*})$ by the obvious copy of $C_{*}$, so its differential is
\begin{align*}
C_{p-1} \oplus D_+
\xrightarrow{%
\begin{pmatrix}
-c_{p-1} & 0 \\
f_{p-1} & d_p
\end{pmatrix}}
C_{p-2} \oplus D_{p-1}
\end{align*}
\begin{rem*}
Again there is a topological analogue, the topological mapping cone $\cone(f) = \cyl(f)/X \times \{1\}$.
These are related via
\begin{align*}
\cone_i(C_{*}(f)) = C_i(\cone(f)) \text{ for } i > 0.
\end{align*}
\end{rem*}
The \CmMark{suspension} $\Sigma C_{*}$ of an $R$-chain complex $C_{*}$ is a chain complex with $p$-th differential
\begin{align*}
C_{p-1} \xrightarrow{-c_{p-1}} C_{p-2},
\end{align*}
which is isomorphic to a quotient of $\cone(\id_{C_{*}})$ by $C_{*}$.
We have two exact sequences
\begin{align*}
& 0 \to C_{*} \to \cyl(f_{*}) \to \cone(f_{*}) \to 0
& 0 \to D_{*} \to \cone(f_{*}) \to \Sigma C_{*} \to 0
\end{align*}
\begin{dfn*}
An $R$-chain complex $C_{*}$ is \CmMark{finite}, if $|C_p| = 0$ for $p >> 0$ and each $C_p$ is finitely generated.
It is called \CmMark{projective}, if each $C_p$ is projective; \CmMark{free}, if each $C_p$ is free, and \CmMark{based free}, if each $C_p$ is based free with a preferred basis.
\end{dfn*}
\begin{rem*}
Let $f_{*} \colon C_{*} \to D_{*}$ be a chain map between projective chain complexes.
Then the following statements are equivalent
\begin{enumerate}
\item $f_{*}$ is a homology isomorphism (i.e. $H_i(f_{*})$ is an isomorphism for all $i$)
\item $f_{*}$ is a chain homotopy equivalence
\item $\cone(f_{*})$ is contractible (i.e. $\cone(f_{*}) \simeq 0$).
\end{enumerate}
This can be seen from the following sequence (together with the fundamental lemma in homological algebra to show the equivalence of the first two statements).
\begin{equation*}
\begin{tikzcd}
0 \ar{r} & C_{*} \ar{r} & \cyl(f) \ar[<->]{d}{\simeq} \ar{r} & \cone(f_{*}) \ar{r} & 0\\
& & D_{*} & &
\end{tikzcd}
\end{equation*}
A short exact sequence of chain complexes induces a long exact sequence in homology associated to $C_{*}$, which implies $\cone(f_{*}) = 0 \rightsquigarrow H_{*}(\cone(f_{*})) = 0\rightsquigarrow f_{*}$ is homology isomorphism.
\end{rem*}
\begin{lemma*}
Let $C_+$ be a based free, finite $R$-chain complex that is contractible.
Let $\gamma_p \colon C_p \to C_{p+1}$ for $p \in \Z$ be a chain contraction, i.e.
\begin{align*}
c_{p+1} \circ \gamma_p + \gamma_{p-1} \circ c_p = \id - 0.
\end{align*}
Then the $R$-homomorphism $(c_{*} + \gamma_{*}) \colon C_{\text{odd}} \to C_{\text{ev}}$ (where $C_{\text{odd}} = \bigoplus_p C_{2p+1}$ and $C_{\text{ev}} = \bigoplus_p C_{2p}$) is an isomorphism.
Let $A$ be its representing matrix.
Its class $[A] \in K_1(R)$ is independent of the choice of $\gamma_{*}$.
\end{lemma*}
\begin{expl*}
Let $C_p = 0$ unless $i \in \{0,1,2\}$.
Then
\begin{equation*}
\begin{tikzcd}
0 \ar{r} & C_2 \ar{r}[swap]{c_2} & C_1 \ar{r}[swap]{c_1} \ar[bend right]{l}[swap]{\gamma_1} & C_0 \ar{r} \ar[bend right]{l}[swap]{\gamma_0} & 0
\end{tikzcd}
\end{equation*}
is the full complex and thus ``contractibe'' means ``short exact''.
$C_1 \xrightarrow{\cong} C_0 \oplus C_2$ via $x \mapsto c_1 (x) + \gamma_1(x)$ with inverse $C_0 \oplus C_2 \xrightarrow{\cong} C_1, (x,y) \mapsto \gamma_0(x) + c_2(y)$.
Let $\tilde\gamma$ be another chain contraction
\begin{equation*}
\begin{tikzcd}
C_0 \oplus C_2 \ar{r}{\cong}[swap]{(\tilde \gamma_1, c_2)} &
C_1 \ar{r}{\cong}[swap]{(c_1,\gamma_1)} & C_0 \oplus C_2
\end{tikzcd}
\end{equation*}
Let $x + y \in C_0 \oplus C_2$.
Then
\begin{align*}
\tilde \gamma_0(x) + c_2(y) \mapsto \ & c_1\tilde \gamma_0(x) + \gamma_1 \tilde\gamma_0(x) + \gamma_1 c_2(y)\\
& = (x - \underbrace{\tilde\gamma_2c_0(x))}_{=0} + \gamma_1\tilde\gamma_0(x) + (y - \underbrace{c_3\gamma_2(y)}_{=0})\\
& = x + y + \gamma_1 \tilde\gamma_0(x),
\end{align*}
which is represented by a matrix $\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}$.
\end{expl*}
\begin{proof}
Let $\gamma_{*}, \delta_{*}$ be two chain contractions of $C_{*}$.
Then we consider
\begin{align*}
(c_{*} + \delta_{*})_{\text{odd}} \colon C_{\text{odd}} \xrightarrow{A} C_{\text{ev}}
\text{ and }
(c_{*} + \delta_{*})_{\text{ev}} \colon C_{\text{ev}} \xrightarrow{B} C_{\text{odd}}
\end{align*}
represented by matrices $A$ and $B$.
Define $\mu_n = (\gamma_{n+1} - \delta_{n+1}) \circ \delta_n$ and $\nu_n = (\delta_{n+1} - \gamma_{n+1}) \circ \gamma_n$.
Then $(\id + \mu_{*})_{\text{odd}}$, $(\id + \nu_{*})_{\text{ev}}$ and $(c_{*} + \gamma_{*})_{\text{odd}} \circ (\id + \mu_{*})_{\text{odd}} \circ (c_{*} + \delta_{*})_{\text{ev}}$ are represented by upper triangular matrices with ones on the diagonal.
Thus $[A] = -[B]$ in $K_1(R)$ and $B$ is independent of the choice of $\gamma_{*}$, hence $A$ is independed of the choice of $\gamma_{*}$.
\end{proof}
\begin{dfn*}
\begin{enumerate}[label=(\roman*)]
\item For a contractible, based free, finite $R$-chain complex $C_{*}$ define
\begin{align*}
\tau(C_{*}) := [(c_{*} + \gamma_{*})_{\text{odd}}] \in K_1(R)
\end{align*}
(for some or every) choice of $\gamma_{*} \colon C_{*} \simeq 0$).
\item Let $f_{*} \colon C_{*} \to D_{*}$ be a chain homotopy equivalence of based free, finite $R$-chain complexes.
The \CmMark{Whitehead torsion} of $f_{*}$ is
\begin{align*}
\tau(f_{*}) := \tau(\cone(f_{*})) \in K_1(R).
\end{align*}
\end{enumerate}
\end{dfn*}
We say that a short exact sequence of based free modules
\begin{align*}
0 \to A \xrightarrow{j} B \xrightarrow{p} C \to 0
\end{align*}
is \CmMark{based exact}, if $\text{basis}_B = B_1 \coprod B_2$, such that $B_1 = j(\text{basis}_{A)}$ and $p(B_2) = \text{basis}_C$.
\begin{lemma*}
Consider the following diagram with based exact rows.
\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 $\tau(g_{*}) = \tau(f_{*}) + \tau(h_{*})$.
\end{lemma*}
\begin{proof}
We have the following diagrams with exact rows and columns.
\begin{equation*}
\begin{tikzcd}
& 0 \ar{d} & 0 \ar{d} & 0 \ar{d} & \\
0 \ar{r} & C_{*}' \ar{r} \ar{d}{\simeq} & D_{*}' \ar{r} \ar{d}{\simeq} & E_{*}' \ar{r} \ar{d}{\simeq} & 0\\
0 \ar{r} & \cyl(f_{*}) \ar{r} \ar{d} & \cyl(g_{*}) \ar{r} \ar{d} & \cyl(h_{*}) \ar{r} \ar{d} & 0\\
0 \ar{r} & \cone(f_{*}) \ar{r} \ar{d} & \cone(g_{*}) \ar{r} \ar{d} & \cone(h_{*}) \ar{r} \ar{d} & 0\\
& 0 & 0 & 0
\end{tikzcd}
\end{equation*}
We may assume that the short exact sequence given by the columns
\begin{align}\label{eq:ses-1}
0 \to C_{*} \xrightarrow{j_{*}} D_{*} \xrightarrow{p_{*}} E_{*} \to 0
\end{align}
is a based exact sequence of contractible based exact sequence of contractible based free, finite chain complexes and then have to prove that $\tau(D_{*}) = \tau(C_{*}) + \tau(E_{*})$.
The sequence \eqref{eq:ses-1} splits as chain complexes.\footnote{This heavily depends on contractibility and is not true for arbitrary chain complexes.}
Let $e_*$ be a contraction of $E_{*}$ and let $\sigma_i \colon E_i \to D_i$ be a split of $p_i$ for all $i$.
Set $s_i \colon E_i \to D_i, s_i := d_{i+1} \circ \sigma_{i+1} \circ \varepsilon_i + \sigma_i \circ \varepsilon_{i-1} \circ e_i$.
Claim: $s_{*}$ is a chain map and $p_{*} \circ s_{*} = \id_{E_{*}}$.
Hence we obtain an isomorphism of chain complexes
\begin{align*}
(j_{*},s_{*}) \colon C_{*} \oplus E_{*} \xrightarrow{\cong} D_{*},
\end{align*}
which has a corresponding matrix of the form $\begin{pmatrix}\Id & * \\ 0 & \Id \end{pmatrix}$.
We can finish the proof with the following remark.
If $u_{*} \colon C_{*} \to D_{*}$ is a chain isomorphism of then
\begin{align*}
\tau(C_{*}) - \tau(D_{*}) = \sum_p (-1)^p [u_p] \in K_1(R).
\end{align*}
This can be shown by transporting a chain contraction $\gamma_{*}$ for $C_{*}$ to one for $D_{*}$ via $u_{*}$, which yields a diagram:
\begin{equation*}
\begin{tikzcd}
C_{\text{odd}} \ar{r} \ar{d}{u_{\text{odd}}}[swap]{\simeq} & C_{\text{ev}} \ar{d}{u_{\text{ev}}}[swap]{\simeq}\\
D_{\text{odd}} \ar{r} & D_{\text{ev}}.
\end{tikzcd}
\end{equation*}
\end{proof}
\chapter{Harmonic Maps [Andy Sanders]}
......
......@@ -55,6 +55,9 @@
\DeclareMathOperator{\Characteristic}{char}
\DeclareMathOperator{\aff}{aff} % affine Hülle
\DeclareMathOperator{\cone}{cone} %
\DeclareMathOperator{\cyl}{cyl} %
% Lie Theory
\DeclareMathOperator{\Ad}{Ad} % Ad
\DeclareMathOperator{\ad}{ad} % ad
......@@ -68,6 +71,7 @@
\DeclareMathOperator{\lk}{lk} % link
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\rg}{rank}
\DeclareMathOperator{\rank}{rank}
......
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