Commit 72b86f98 authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß
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Added lecture of the 7th RTG day.

parent 5960319c
......@@ -1206,6 +1206,181 @@ Since $a \equiv \pm 2 \mod 7$, $a=3,4$ are the only possible choices, for both o
\end{enumerate}
\end{expl*}
% 2017-02-07
Recall that we considered $D_{*} \simeq 0$ a finite dimensional Hilbert space and observed
\begin{align*}
\ln |\det \rho(D_{*})| = -\frac{1}{2} \sum_{p \geq 0} (-1)^p \ln \det \Delta_p.
\end{align*}
We want to have this in the $\infty$-dimensional setting, specifically
\begin{align*}
D_{*} = l^2 \Gamma \otimes_{\Z\Gamma} C_{*}(\tilde M),
\quad
\Gamma = \pi_1M
\end{align*}
provided $b_p^{(2)}(M) = \dim_{\Gamma}(\ker(\Delta_p^{(n)})) = 0$ for all $p \geq 0$.
This actually happens in quite a few interesting cases, as highlighted by the following conjecture.
\begin{conj*}[Hopf-Singer]
Every closed odd-dimensional aspherical manifold is $l^2$-acyclic.
(It is verified, e.g., for locally symmetric spaces.)
\end{conj*}
We need to define a determinant in the $\infty$-dimensional setting.
\subsection{Digression on spectral calculus}
Let $A \colon l^2\Gamma^n \to l^2\Gamma^n$ be a bounded positive equvariant operator.
Then we can exhibit the following map
\begin{align*}
\{ \text{polyn. functions on } [0, \|A\|] \} & \to \{ \text{ bdd., $\Gamma$-equiv. operators } l^2\Gamma \to l^2\Gamma\} =: L(\Gamma)\\
P & \mapsto P(A),
\end{align*}
where $L(\Gamma)$ is called \CmMark{von Neumann algebra} of $P$.
This homeomorphism extends to bounded Borel functions on $[0,\|A\|]$ and we obtain a \emph{spectral measure}.
By the Riesz representation theorem, there exists a unique Borel probability measure $\mu_A$ supported on $[0,\|A\|]$ such that
\begin{align*}
\int_{\R} f \dop\mu_{A} = \tr_{\Gamma}(f(A)).
\end{align*}
\begin{expl*}
Let $\Gamma = \{1\}$ and $A \colon \C^n \to \C^n$ be a positive matrix we can express as $A = S^{-1} \diag(\lambda_1, \ldots, \lambda_n) S$.
For a bounded Borel function, we have $f(A) = S^{-1}\diag(f(\lambda_1), \ldots, f(\lambda_n))$.
In this case, $\tr_{\Gamma}$ is the ordinary trace and the spectral probability measure obtained is
\begin{align*}
\mu_A = \frac{1}{n} \left(\sum_{i = 1}^n \delta_{\lambda_i}\right).
\end{align*}
Here we have
\begin{align*}
\ln \det A = \sum_{i=1}^n \ln(\lambda_i) = n\int_{o^+}^{\infty}\ln(\lambda) \dop \mu_A(\lambda),
\end{align*}
which motivates the following general definition.
\end{expl*}
\begin{dfn*}
The \CmMark{Fuglede-Kadison determinant} $\det^{(2)}(A)$ is defined as
\begin{align*}
{\textstyle \det^{(2)}(A)} := \exp \left( \int_{0^+}^{\infty} \ln(\lambda) \dop \mu_A(\lambda) \right) \in \R,
\end{align*}
provided $\int_0^{\infty} \ln(\lambda) \dop \mu_A$ exists.
In this case we say that $A$ is \CmMark{determinant class}.
\end{dfn*}
\begin{rem*}
If $A$ is right multiplication by a matrix in $M_n(\Z[\Gamma])$ and $\Gamma$ is sofic, then $A$ is determinant class (Elek-Szabo).
\end{rem*}
\begin{dfn*}
Let $M$ be a finite CW-complex.
Assume that $\Gamma = \pi_1M$ is sofic and that $M$ is $l^2$-acyclic.
Then the \CmMark{$l^2$-torsion} $\rho^{(2)}(M)$ is defined as
\begin{align*}
\rho^{(2)}(M) = -\frac{1}{2} \sum_{p \geq 0} (-1)^p p \cdot \ln {\textstyle \det^{(2)}}(\Delta_p^{(2)} \colon l^2 \Gamma \otimes_{\Z\Gamma} C_p(\tilde M) \to l^2 \Gamma \otimes_{\Z\Gamma} C_p(\tilde M)).
\end{align*}
\end{dfn*}
\begin{rem*}
Note that the following remarks are actually propositions.
\begin{itemize}
\item $l^2$-torsion is a homotopy invariant
\item Let $M$ be a closed hyperbolic $3$-manifold.
Then $\rho^{(2)}(M) = \frac{1}{6\pi} \operatorname{vol} (M, g_{\text{hyp}})$.
This implies, in particular, the homotopy invariance of the hyperbolic volume, which also follows non-trivially by Mostow rigidity.
\end{itemize}
\end{rem*}
\subsection{Approximation}
\begin{thm*}[Lück]
Let $M$ be a finite CW-complex with $\Gamma = \pi_1M$ residually finite.
Let $\Gamma = \Gamma_0 > \Gamma_1 > \Gamma_2 > \cdots$ be a \CmMark{residual chain}, i.e. $[\Gamma: \Gamma] = 0$, $\Gamma_i \vartriangleleft \Gamma$, $\bigcup \Gamma_i \{1\}$.
Then
\begin{align*}
b_p^{(2)}(M) = \lim_{i \to \infty} \frac{b_p(M_i)}{[\Gamma : \Gamma_i]},
\end{align*}
where $\Gamma_i\backslash M = M_i$.
\end{thm*}
The following is a bold conjecture (that probably is only true up to a few more adjectives added, e.g. (arithmetic) hyperbolic manifold).
\begin{conj*}
Let $M^{2n+1}$ be a closed odd-dimeninsional aspherical manifold with residually finite $\Gamma = \pi_1M$ and let $(\Gamma_i)$ be a residual chain in $\Gamma$.
Then $\rho^{(2)}(M) = \lim_{i \to \infty} \frac{\ln |\operatorname{tors} H_n(M_i; \Z)|}{[\Gamma : \Gamma_i]}$.
\end{conj*}
In particular, this would tell us that, if $M$ is a closed hyperbolic 3-manifold, then
\begin{align*}
\frac{1}{6\pi} \operatorname{vol}(M) = \lim_{i \to \infty} \frac{\ln |\operatorname{tors}(\Gamma_i)_{\text{ab}}|}{[\Gamma : \Gamma_i]}.
\end{align*}
(There is a result of quite similar nature if one consideres suitably twisted coefficients by Bergeron-.)
\begin{proof}[Sketch of Lück's approximation theorem]
Consider the Laplace operators
\begin{align*}
& \Delta^{(2)}_p \colon l^2 \Gamma \otimes_{\Z\Gamma} C_p(\tilde M) \cong l^2\Gamma^n \to l^2\Gamma^n\\
& \Delta_p(i) \colon C_p(M_i) \cong \C[\Gamma/\Gamma_i]^n \to \C[\Gamma/\Gamma_i]^n.
\end{align*}
Both come from a multiplication with $A \in M_n(\Z[\Gamma])$ (resp. $A_i \in M_n(\Z[\Gamma/\Gamma_i])$), where $A_i$ is the ``reduction'' of $A$ mod $\Gamma_i$.
We want to show that
\begin{align*}
b_p^{(2)}(M) = \tr_{\Gamma}(\pr_{\Ker A}) = \lim_{i \to \infty} \tr_{\Gamma/\Gamma_i}(\pr_{\Ker A_i}) = \lim \frac{b_p(M_i)}{[\Gamma:\Gamma_i]}.
\end{align*}
This follows from spectral calculus, by using the equalities $\tr_{\Gamma}(\pr_{\Ker A}) = \mu_A(\{0\})$ and $\lim_{i \to \infty} \tr_{\Gamma/\Gamma_i}(\pr_{\Ker A_i}) = \lim_{i \to \infty} \mu_{A_i}(\{0\})$.
To prove the equality on this level it is important to consider the spectral measure in a neighbourhood of $\{0\}$.
Recall that measures $\nu_i$ on the real line \CmMark[weak convergence of measures]{converge weakly}, if $\int_{\R}f\dop \nu_i \to \int_{\R} f\dop \nu$ converges for all $f \in C_c(\R)$.
(In good situations, it is enough to check this for monomials.)
To check this for $(A_i)$, we regard
\begin{align*}
\int_{\R} x^m \dop \mu_A = \tr_{\Gamma}(A^m)
\text{ and }
\int_{\R} x^m \dop \mu_{A_i} = \tr_{\Gamma/\Gamma_i}(A_i^m).
\end{align*}
As the right-hand-sides are purely combinatorial, we get $\tr_{\Gamma/\Gamma_i}(A^m) \to \tr_{\Gamma}(A^m)$ and conclude $\mu_{A_i} \to \mu_A$ weakly.
Caveat: For the Borel probability measure on $\R$ we have $\nu_i = i \chi_{(0,\frac{1}{i}]}\dop \lambda \to \nu = \delta_{\{0\}}$ weakly, but $0 = \nu(\{0\}) \not\to \delta_{\{0\}}(\{0\}) = 1$.
Without loss of ideas (needed for the general case), we consider
\begin{align*}
A_i \colon \C[\Gamma/\Gamma_i] \to \C[\Gamma/\Gamma_i].
\end{align*}
coming from a matrix over $\Z[\Gamma/\Gamma_i]$.
Further, fix $i$, let $n = [\Gamma : \Gamma_i]$ and denote by $0 = \lambda_1 = \cdots \lambda_m < \lambda_{m+1} \leq \dots \leq \lambda_n$ the eigenvalues of $A_i$.
The characteristic polynomial of $A_i$ is given by $p(z) = z^mq(z)$ for $q \in \Z[t]$ and this yields
\begin{align*}
\lambda_{m+1} \cdots \lambda_n - q(0) \geq 1.
\end{align*}
Small eigenvaolues: $N(\varepsilon) = \#$ eigenvalues of $A_i$ in $(0,\varepsilon)$.
We get $1 \leq \lambda_{m+1} \cdots \lambda_n \leq \varepsilon^{N(\varepsilon)}\|A_i\|^n \leq \varepsilon^{N(\varepsilon)} \cdot \text{const}^n$.
Thus taking a logarithm, we obtain $\frac{N(\varepsilon)}{n} \leq \frac{\text{const}}{|\log \varepsilon|}$ (independent of $i$).
Since $\mu_{A_i}((0,\varepsilon)) = \frac{N(\varepsilon)}{n}$, we have found an upper bound for it.
Basic measure theory shows that weak convergence yields $\limsup \mu_{A_i}(W) \leq \mu_A(W)$ for a closed set $W$ and $\liminf \mu_{A_i}(U) \geq \mu_A(U)$ for $U$ open.
Now consider
\begin{align*}
\liminf \mu_i(\{0\}) & = \liminf(\mu_i([0,\varepsilon)) - \mu((0,\varepsilon)))\\
& \geq \liminf(\mu_i((-\varepsilon,\varepsilon)) - \frac{\text{const}}{|\log \varepsilon|}\\
& \geq \mu((\varepsilon, \varepsilon)) - \frac{\text{const}}{|\log \varepsilon|}\\
& \geq \mu(\{0\}) - \frac{\text{const}}{|\log \varepsilon|}.
\end{align*}
As this does not depend on $i$, it shows the claim.
\end{proof}
\chapter{Harmonic Maps [Andy Sanders]}
Also consider the notes \url{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}.
......@@ -1871,7 +2046,157 @@ By uniformization, we have a section $\mathcal M/C^{\infty}(\Sigma) \to \mathcal
where $\mathcal T(\Sigma)$ is Teichmüller space.
%%%Local Variables:
\todo[inline]{Add missing lecture?!}
% Lecture 2017-02-07
\section{Why Harmonic flat bundle}
Recall that lat time we considered a manifold $M$ with a complex vector bundle $E \to M$ of rank $n$.
By mapping a flat connection $\nabla$ to its holonomy $\hol(\nabla)$ we motivated a correspondence
\begin{align*}
\Hom(\pi_1 M, \Gl_n\C)/\Gl_n\C & \leftrightarrow \coprod_{\substack{\text{finite over}\\\text{dist top. types}}} \mathcal F(E)/\mathcal A(E)\\
\rho & \mapsto \tilde M \times_{\rho} \C^n,
\end{align*}
where $\mathcal F(E)$ are flat connections on $E$ and $\mathcal A(E)$ denote automorphisms.
Then we introduced a hermitian metric $h$ and a flat connection $\nabla$.
A map $\rho \colon \tilde M \to M$ yields a pullback $\tilde h \colon \tilde M \to \Gl_n\C/U(n)$, which is $\hol(\nabla)$-equivariant.
Fix a riemannian metric $g$ on $M$.
A bundle $(E,h,\nabla) \to (M,g)$ is called a \CmMark{Harmonic flat bundle}, if $\tilde M \colon (\tilde M, \rho^{*}g) \to \Gl_n\C/U(n)$ is harmonic.
\begin{thm*}[Donaldson ($n=2$), Corlette (general case)]
A flat bundle $(E,\nabla) \to (M,g)$ admits a harmonic metric if and only if $\hol(\nabla)$ is semi-simple.
\end{thm*}
\begin{rem*}
\begin{itemize}
\item $\Gl_n\C/U(n)$ is a non-compacat (continuous) non-positively curved homogeneous riemannian manifold.
\item If $\rho$ is discrete and the image is acting freely on $\Gl_n\C/U(n)$, then the equivariant map $\tilde h$ descends to a map
\begin{align*}
\tilde h \colon M \to \Im(\rho)\backslash(\Gl_n\C/U(n))
\end{align*}
\end{itemize}
\end{rem*}
Now let us fix a hermitian metric $h$ on $E$.
\begin{thm*}[Restatement]
Given a bundle $(E,h) \to (M,g)$ a flat connection $\nabla \in \mathcal F(E)$, there exists an automorphism $\phi \in \mathcal A(E)$ such that $(E,h,\phi^{*}\nabla)$ ($\phi^{*}\nabla = \phi \circ \nabla \circ \phi^{-1}$) is a harmonic flat bundle if and only if $\hol(\nabla)$ is semi-simple.
Moreover, $\phi$ is unique up to $\mathcal A_h(E) = \{ \phi \in \mathcal A(E) \mid \phi^{*}h = h\}$.
\end{thm*}
\begin{dfn*}
The space of Harmonic flat bundles is
\begin{align*}
\mathcal{HF}(E, h) := \{ \nabla \in \mathcal F(E) \mid (E, \nabla, h) \text{ is a Harmonic flat bundle} \}/\mathcal A_h(E).
\end{align*}
We also call $\mathcal A_h(E)$ the group of \CmMark{gauge transformations} of $(E,h)$.
\end{dfn*}
\begin{thm*}[Global version]
There is a homeomorphism
\begin{align*}
\mathcal{HF}(E,h) & \longleftrightarrow \mathcal F_{\text{ss}}(E)/\mathcal A(E) \subset \Hom_{\text{ss}}(\pi_1M, \Gl_n\C)/\Gl_n\C \\
[\nabla]_{\mathcal A_h(E)} & \mapsto [\nabla]_{\mathcal A(E)}\\
[\phi^{*}\nabla]_{\mathcal A_h(E)} & \mathrel{\reflectbox{\ensuremath{\mapsto}}} [\nabla]_{\mathcal A(E)}.
\end{align*}
\end{thm*}
The space $\mathcal{HF}(E,h)$, while homeomorphic to the space on the right-hand-side, has significantly more structure, i.e. we can make several techniques available to the other side.
Every element $\nabla \in \mathcal{HF}(E,h)$ corresponds to a $\hol(\nabla)$-equivariant map
\begin{align*}
h_{\nabla} \colon \tilde M \to \Gl_n\C/U(n),
\end{align*}
such that $E(h_{\nabla}) = \frac{1}{2} \int_F \|\dop h_{\nabla}\|^2\dop V_g \in \R_{\geq}$, where $F$ is a fundamental domain for the $\pi_1M$ action on the universal cover $\tilde M$.
This assignment defines a continuous function
\begin{align*}
E \colon \mathcal{HF}(E,h) \to \R_{\geq 0}
\end{align*}
called the enery function.
Now we will specialize to $M = \Sigma$ a closed, oriented surface of genus $> 1$ equipped with a conformal structure $\sigma$.
In this case, the space $\mathcal HF(E,h)$ is remarkably rich!
Given a connection $\nabla$ on $(E,h)$, there exists a unique decomposition $\nabla = \nabla^h + \psi \in \Omega^1(\End(E))$, where $\nabla^h$ is a unitary connection, i.e. $\dop h(s,s') = h(\nabla^hs,s') + h(s,\nabla^hs')$, and such that $\psi$ is hermitian, i.e. $h(\psi(s),s') = h(s,\psi(s'))$.
Thus, we have
\begin{align*}
\mathcal{HF}(E,h) = \{ (\nabla^H, \psi) \mid \substack{\nabla^H \text{ is unitary}, \psi \text{ is hermitian},\\ (E,h,\nabla^H + \psi) \text{ is a Harmonic flat bundle}}\}/\mathcal A_h(E).
\end{align*}
In these terms, the harmonic and flat conditions have a nice description.
\begin{align*}
0 = F(\nabla^H + \psi) = F(\nabla^H) + \dop^{\nabla^H}\psi + \frac{1}{2} [\psi,\psi],
\end{align*}
where if $\psi = \sum \gamma_i \otimes s_i$ for $\gamma_i \in \Omega^1(\Sigma), s_i \in \End(E)$ we put $[\psi,\psi] = \sum \gamma_i \wedge \gamma_i \otimes [s_i, s_i]$, wherein $[\blank,\blank]$ is the commutator.
We observe:
\begin{itemize}
\item If $\nabla^H$ is unitary, then $F(\nabla^H)$ is skew-hermitian,
\item $F(nabla^h) + \frac{1}{2}[\psi,\psi] = 0$ skew-hermitian,
\item $\dop^{\nabla^H}\psi = 0$ hermitian
\item $\delta^{\nabla^H}\psi = 0$
\end{itemize}
Recall for the last observation, that we introduced for $\alpha, \beta \in \Omega^1(X,\End(E))$ a pairing $\left<\alpha, \beta\right> = \int \left<\alpha, \beta\right> \dop V_g$.
(Note that, on a surface this expression in 1-forms does only depend on the conformal class of $g$)
Now $\delta^{\nabla^H}$ was the adjoint of $\dop^{\nabla^H}$ with respect to this pairing.
Thus we can write
\begin{align*}
\mathcal{HF}(E,h) = \{ (\nabla^H, \psi) \mid F(\nabla^H) + \frac{1}{2}[\psi,\psi] = 0,\ \dop^{\nabla^H}\psi = 0,\ \delta^{\nabla^H}\psi = 0\}/\mathcal A_h(E).
\end{align*}
Recall that on a Riemann surface $X$ with $z$ the local complex coordinate, a holomorphic $\kappa$ differential on $X$ is
\begin{itemize}
\item a holomorphic section $\alpha$ of $S^k\T_{\hol}^{*}X$ ($S^k$ denotes the $k$th symmetric power),
\item $\alpha = f(z) \dop z^k$, where $f(z)$ is a holomorphic function.
\end{itemize}
For $\psi \in \Omega^1(X,\End(E))$ we have $\psi^k \in \Gamma(\T^{*}X^{\otimes k} \otimes \End(E))$ and we can take its trace $\tr(\psi^k) \in \Gamma(\T^{*}X^{\otimes k})$, for which we have the following proposition.
\begin{prop*}
If $(\nabla^H, \psi) \in \mathcal{HF}(E,h)$, then
\begin{align*}
\alpha_k := \tr(\psi^k)^{k,0} = f(z) \dop z^k,
\end{align*}
(where $f(z)$ is only $C^{\infty}$ a priori) is actually holomorphic.
Moreover, $\alpha_2$ is the Hopf differential of the corresponding harmonic map.
\end{prop*}
\begin{rem*}
Earlier we saw that the map
\begin{equation*}
\begin{tikzcd}
\{ \substack{\text{discrete, faithful}\\\text{repr. } \pi_1\Sigma \to \SL_2\R} \}/\SL_2(\R) \ar{r}{\sim} \ar[hook]{d} & \{ \text{hyp. metrics on } \Sigma \}/\Diff_0(\Sigma) \ar{r}{\sim} & \mathrm{QD}(X)\\
\mathcal F(E)/\mathcal A(E) \ar{r}{\sim} & \mathcal{HF}(E,h)/\mathcal A_h(E) \ar{r}{\tr(\psi^2)} & \mathrm{QD}(X).
\end{tikzcd}
\end{equation*}
\end{rem*}
\subsection{Another unique feature of $\mathcal{HF}(E,h)$ for a Riemann surface}
$U(1)$-action of $\mathcal{HF}(E,h) \ni (\nabla^H, \psi)$ with $F(\nabla^H) + \frac{1}{2}[\psi,\psi] = 0$, $\dop^{\nabla^h}\psi = 0$ and $\delta^{\nabla^H}\psi = 0$.
Given $X, I$, we get a Hodge $*$-operator
\begin{align*}
* \colon \Omega^1(X, \R) \to \Omega^1(X, \R).
\end{align*}
We write $\psi = sum_{i = 1}^k \gamma_i \otimes s_i$ and define $I\psi = \sum_{i = 1}^k *\gamma_i \otimes s_i$ and $\theta \cdot \psi = e^{I\theta}\psi = \cos \theta \psi + \sin \theta I\psi$.
\begin{prop*}
We have $(\nabla^H, \psi) \in \mathcal{HF}(E,h) \iff (\nabla^H, \theta \cdot \psi) \in \mathcal{HF}(E,h)$, i.e. this defines a $U(1)$-action on $\mathcal{HF}(E,h)$.
\end{prop*}
The enery $E$ is invariant under this $U(1)$-action $U(1) \curvearrowright \mathcal{HF}(E,h) \xrightarrow{E} \R$ and in fact, we have $\dop E|_{(\nabla^H, \psi)} = 0$ if and only if $(\nabla^H, \psi)$ is fixed by the $U(1)$-action.
It was first realized by Atiyah-Bott and used in the exact context by Hitchin, that $E$ can be used as a Morse function to compute the Betti numbers of $\mathcal{HF}(E,h)$.
This has been completely carred out for $\SpecialLinear_2\C$ by Hitchin and still today, this is the only method to understand the Betti numbers of $\Hom_{\text{ss}}(\pi_1\Sigma, \Gl_n\C)/\Gl_n\C$.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "skript-rtg-lectures-ws1617"
%%% End:
......@@ -96,6 +96,7 @@
\DeclareMathOperator{\End}{End} % endomorphisms
\DeclareMathOperator{\Isom}{Isom} % isometriess
\DeclareMathOperator{\Imm}{Imm} % immersions
\DeclareMathOperator{\hol}{hol} % holonomy
\DeclareMathOperator{\Wh}{Wh} % Whitehead group
......@@ -198,6 +199,7 @@
\newtheorem{dfn}[thmglobal]{Definition}
\newtheorem{expl}[thmglobal]{Example}
\newtheorem{conj}[thmglobal]{Conjecture}
\theoremstyle{remark}
......@@ -218,6 +220,7 @@
\renewtheorem*{dfn*}{Definition}
\renewtheorem*{expl*}{Example}
\renewtheorem*{conj*}{Conjecture}
\theoremstyle{remark}
\renewtheorem*{rem*}{Remark}
......
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