Commit ce7a32e5 authored by Jannes Bantje's avatar Jannes Bantje

add content for coarse index theory

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%!TEX root = ../index_theory.tex
\chapter{Coarse Index Theory} % (fold)
\label{cha:coarse_index_theory}
In this chapter we cover some of the basics of coarse index theory.
The material in here is for most parts a recollection of my discussions with Johannes.
They might be enriched by
\begin{itemize}
\item the book \citetitle{WilletYu} by \textcite{WilletYu}
\item the notes from Rudi's lecture on that topic
\item references to \citetitle{higson_roe} by \textcite{higson_roe}
\item references to \citetitle{Higson_guentner} by \textcite{Higson_guentner}
\end{itemize}
\section{Motivation} % (fold)
\label{sec:motivation}
Clearly we motivate this geometrically via manifolds and their Dirac operators.
So let $M^d$ be a complete Riemannian manifold with a $\Spin$-structure and $D$ the Dirac operator, which we view as a self-adjoint operator acting on $L^2(M;S)$, where $S$ denotes the Spinor bundle.\footnote{a more general setting using some operator of Dirac type is also possible}
Now let $u$ be a smooth section with compact support and consider the \Index{wave equation}\footnote{the classical one from physics is recovered from this by taking the second order derivatives and plugging in the definition of $D$}
\[
\diff{u}{t} u = i D u
\]
We write $u_t = e^{i D t} u$ for the solution of the wave equation.
The main feature of a solution -- which is well known to physicists -- is the property of having \Index{finite propagation}, i.e. the support of $u_t$ is contained in the $t$-neighbourhood of the support of $u$,
\[
\supp (u_t) \subseteq N_t \enbrace*{\supp u}
\]
See \cite[Prop.~10.3.1]{higson_roe}.
In this construction we used the functional calculus with $e^{ixt}$ and we want to generalise the finite propagation property to other functions, namely to functions $\varphi \in \mathcal{S} = C_0(\mathbb{R})$.
These are the functions that come up in the spectral picture of \K-theory.
Since $\varphi$ is a $C_0$ function we know, that $\varphi(D) \cdot f$ and $ f \cdot \varphi(D)$ are compact for $f \in C_c(\mathbb{R})$.
For the time being we assume, that $\varphi$ is of Schwartz class such that its Fourier transform has compact support, $\supp(\widehat{\varphi}) \subseteq \benbrace*{-r,r}$ for some $r>0$.
Hence we may write
\[
\varphi(t) = \frac{1}{\sqrt{2 \pi}} \int\limits_{-\infty}^\infty \widehat{\varphi}(\xi) e^{i \xi t} \, \mathd \xi = \frac{1}{\sqrt{2 \pi}} \int\limits_{-r}^r \widehat{\varphi}(\xi) e^{i \xi t} \, \mathd \xi
\]
This formula also applies to the functional calculus
\[
\varphi(D) = \frac{1}{\sqrt{2 \pi}} \int\limits_{-r}^r \widehat{\varphi}(\xi) e^{i D \xi} \, \mathd \xi
\]
Hence we get
\[
\supp \enbrace*{\varphi(D) u} \subseteq N_r \enbrace*{\supp (u)}
\]
For the general case we use, that $\varphi \in C_0(\mathbb{R})$ can be approximated arbitrarily well in the $C_0$-norm by a Schwartz function $\psi$, such that $\widehat{\psi}$ has compact support.
We summarise.
\begin{lemma}
$\forall \varepsilon > 0$ and $\varphi \in C_0(\mathbb{R})$ there is an operator $T$ of $L^2(M,S)$ such that $\norm*{\varphi(D) - T} < \varepsilon$ and there exists $r >0$ with
\[
\supp(T u) \subseteq N_R \enbrace*{\supp u}
\]
\end{lemma}
% section motivation (end)
\section{The General Theory} % (fold)
\label{sec:the_general_theory}
% section the_general_theory (end)
\section{Applications} % (fold)
\label{sec:applications_coarse}
% section applications (end)
% chapter coarse_index_theory (end)
\ No newline at end of file
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