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\date[July 29, 2011]
   
\title{Introduction to Matchbox Manifolds}


\author{Steven Hurder}

\institute[UIC] {University of Illinois at Chicago\\www.math.uic.edu/$\sim$hurder \\ ...\\
{\small Report on works with Alex Clark and Olga Lukina}}



\begin{document}

\frame{\titlepage} % # 1




 \section{Matchbox Manifolds}
 
 
\frame % 2
{
  \frametitle{Matchbox manifolds}
 
Let $\fM$ be a continuum.
  
  \pause
Suppose that  each $x \in \fM$ has an open neighborhood homeomorphic to $(-1,1)^n \times \fT_x$ ~ , ~ where $\fT_x$ is a totally disconnected clopen subset of some Polish space $\fX$.  $\Longrightarrow$ 
  arc-components are locally Euclidean.
  
\medskip
\pause

 {\bf Definition:}     $\fM$ is   an \emph{$n$-dimensional matchbox manifold} $\Longleftrightarrow$  $\fM$  admits a covering by  foliated coordinate charts $\ds \cU = \left\{ \vp_{i} \colon U_i \to [-1,1]^n \times \fT_i \mid i \in \cI \right\}$ where $\fT_i$ are   is a totally disconnected clopen subsets of    $\fX$.

 The transition functions are assumed to be $C^r$, for $1 \leq r\leq \infty$,   along leaves, and the derivatives depend (uniformly) continuously on the transverse parameter.
 
 
  
 
    \vfill   

}
  
 
\frame % 1
{
  \frametitle{Why Matchbox Manifolds?}
 
$L_0$ is  a connected, complete Riemannian manifold, 
``marked''    with a metric,   a net, a tiling, or other local structure.
 

 
 

``Compactify'' this data  by looking for a   \emph{continuum} $\fM$ in which $L_0$ embeds as a leaf of a ``foliation'' and respecting   the local structure. 

   

A simple example which embeds in foliation on $\mT^2$ ~:
 \begin{center}
\includegraphics[width=0.4\textwidth]{pix/fig3.png}
\end{center}

``Slinky model'' gives even more compact model, and is  continuum.



  \vfill   

}
 
\frame % 1a
{
  \frametitle{Triply periodic manifold}
 

 \begin{center}
\includegraphics[width=0.4\textwidth]{pix/fig8.png}
\end{center}

 

``Compactifying'' gives a foliation of a compact 4-manifold

$M = (L \times \mS^1)/\mZ^3$ with leaf $L$

  \vfill   

}
 
 
 
\frame % 1a
{
  \frametitle{Penrose tiling stripped of decorations}
 
Closure of $\mR^2$-translates of the graph below yields a continuum $\fM$ foliated by action of $\mR^2$: 
 \begin{center}
\includegraphics[width=.4 \textwidth]{pix/penrose_tiling1.png}
\end{center}



  \vfill   

}
 
 
\frame % 1a
{
  \frametitle{Graph closures \& Ghys-Kenyon examples}
 
Closure of space of subtrees of given graph, yields a Cantor set with pseudogroup action, which generates a foliated continuum $\fM$.

 \begin{center}
\includegraphics[width=.5 \textwidth]{pix/KenyonMod2.png}
\end{center}



  \vfill   

}

\frame % 2
{
  \frametitle{}
 
   
 
 \medskip
 {\bf Definition:}  An $n$-dimensional  \emph{matchbox manifold} is a   continuum  $\fM$   which is a smooth foliated space with codimension zero and leaf   dimension $n$. Similar concept to laminations.
 \begin{eqnarray*}
\fT_i \quad \text{are totally disconnected} ~&  \Longleftrightarrow &~ \fM \quad \text{is a matchbox manifold}\\\pause
\text{leaves of}~ \F  ~ & \Longleftrightarrow & ~ \text{orbits of group action}
\end{eqnarray*}


\pause

$\bullet$  A ``smooth matchbox manifold'' $\fM$ is analogous to a compact manifold,  and the pseudogroup dynamics of the foliation $\F$ on the transverse   fibers $\fT_i$ represents \emph{intrinsic} fundamental groupoid. 

  
 
   
$\bullet$   They appear in study of tiling spaces, leaves of foliations, graph constructions, inverse limit spaces, pseudogroup actions on totally disconnected spaces, et cetera.





  \vfill   

}

 
  \section{Goals}


\frame % 4
{
  \frametitle{Questions that we would like to understand}

$\bullet$ How does a matchbox manifold $\fM$ differ from an invariant set of a smooth dynamical system? of a foliation dynamical system?

\pause

$\bullet$ What is the group of homeomorphisms of $\fM$? is it ``big'' or ``small''? is it ``algebraic''?

\pause
$\bullet$ Can you ``count''  the matchbox manifolds? How do you distinguish one from another? with K-Theory invariants? using cohomology invariants? systems of approximations?

\pause
$\bullet$  What classification scheme   works to understand these spaces?

\pause
$\bullet$  Restrict attention to one of two cases:\\
\hspace{1cm}  $\fM$  is \emph{transitive} if there exists a dense leaf. \\
\hspace{1cm}   $\fM$ is \emph{minimal} if every leaf in $\fM$  is dense.


 \vfill   

}

\frame % 4
{
  \frametitle{Embeddings}

 
 If $f \colon \bf K \to \bf K$ is minimal action on Cantor set,
  then classical problem asks, when does this action arise as the restriction of a $C^r$-diffeomorphism of $f \colon \mS^1 \to \mS^1$?, where $r < 2$? How about as invariant set for some diffeomorphism on $N^k$ where $k \geq 1$?


\medskip

 Solutions to this problem for solenoids modeled on $\mT^n$, $n=1$ by Gambaudo, Tressier, et al in 1990's. For $n \geq 1$ by  Clark \&   Hurder,   ``Embedding solenoids in foliations'',   {\bf Topology Appl.}, 2011. 

The criteria for embedding depend on the degree of smoothness required. 
These are very special cases, and problem is wide open.


 \vfill   

}


\frame % 4
{
  \frametitle{Homeomorphisms}


 Let $\fM$ be a matchbox manifold of dimension $n$.  


 
{\bf Lemma:} A homeomorphism $\phi \colon \fM \to \fM'$ of matchbox manifolds must map leaves to leaves $\Rightarrow$  is a foliated homeomorphism. 

{\it Proof:} $\text{Leaves of } ~ \F ~ \Longleftrightarrow  ~\text{path components of} ~ \fM$

\medskip


{\bf Corollary:}  ${\bf Homeo}(\fM) = {\bf  Homeo}(\fM, \F) $ -- all homeomorphisms are leaf preserving.



 \vfill   

}


\frame % 4
{
  \frametitle{Bing Bling}



{\bf Bing Question:} Let $X$ be a homogeneous continuum, and suppose   every proper subcontinuum of $X$ is an arc.  Must
$X$ then be a circle or a solenoid?
\pause
\emph{Yes!     Hagopian, 1977}. 

\pause  
Mislove \& Rogers, 1989.\\
\pause    
Aarts,  Hagopian \& Oversteegen, 1991. \\
\pause  
  Clark, 2002.\\
\pause
  Clark \& Hurder, 2010.\\
\pause

\medskip

Proofs vary in their degrees of ``abstractness'', suggesting:

\medskip

{\bf Bing Conjecture:} Suppose that $\fM$ is homogeneous continuum, and $\fM$ is a matchbox manifold of dimension $n \geq 1$. Then either $\fM$ is homeomorphic to a compact manifold, or to a McCord solenoid.


 \vfill   

}

 
\frame % 4
{
  \frametitle{The Theorem}


 

\medskip


 {\bf Theorem:} [C \& H, 2010] Bing Conjecture is true for all $n \geq 1$. 

\medskip

\pause

 
Sketch of proof of this,   introduces many   ideas from foliation and topological dynamical systems of matchbox manifolds -- \textit{a.k.a.}
\emph{dynamics of pseudogroups acting on totally disconnected spaces}.

\pause
\medskip

 
 \begin{enumerate}
\item detour through weak solenoids
\item dynamics of pseudogroups
\item shape and transverse foliations 
\item codings and solenoids
\item automorphisms...
\end{enumerate}

 

 \vfill   

}


\section{Solenoids}


\frame % 6
{
 \frametitle{Weak solenoids}
  
  Let $B_{\ell}$ be compact, orientable manifolds of dimension $n \geq 1$ for $\ell \geq 0$, with   orientation-preserving covering maps
  $$ \stackrel{p_{\ell +1}}{\longrightarrow} B_{\ell} \stackrel{p_{\ell}}{\longrightarrow} B_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  B_1 \stackrel{p_1}{\longrightarrow}  B_0 $$
   
The $p_{\ell}$ are the \emph{bonding maps} for the weak solenoid
$$\cS = \lim_{\leftarrow} ~ \{p_{\ell} \colon B_{\ell} \to B_{\ell -1}\} ~ \subset ~ \prod_{\ell =0}^{\infty} ~ B_{\ell} $$ 
 
 {\bf Proposition:} $\cS$ has natural structure of a matchbox manifold, with every leaf dense.
 
 
  \vfill   

}


\frame % 7
{
  \frametitle{From Vietoris solenoids to McCord solenoids}
  
  Basepoints $x_{\ell} \in B_{\ell}$ with $p_{\ell}(x_{\ell}) = x_{\ell -1}$, set $G_{\ell} = \pi_1(B_{\ell} , x_{\ell})$. 


There is  a descending chain of groups and injective maps
  $$ \stackrel{p_{\ell +1}}{\longrightarrow} G_{\ell} \stackrel{p_{\ell}}{\longrightarrow} G_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  G_1 \stackrel{p_1}{\longrightarrow}  G_0 $$

Set $q_{\ell} = p_{\ell} \circ \cdots \circ p_1 \colon B_{\ell} \longrightarrow  B_0$.

{\bf Definition:} $\cS$ is a \emph{McCord solenoid}  for some fixed $\ell_0 \geq 0$,    for all $\ell \geq \ell_0$ the image 
$G_{\ell} \to H_{\ell} \subset G_{\ell_0}$ is a normal subgroup of $G_{\ell_0}$. 

 \medskip
 
 {\bf Theorem} [McCord 1965] Let $B_0$ be an oriented  smooth closed manifold. Then a  McCord solenoid $\cS$ is an orientable,  homogeneous, equicontinuous   smooth matchbox manifold.
  
 
   \vfill   

}
 

\frame % 8
{
  \frametitle{Classifying weak solenoids}
  
A weak solenoid is determined by the base manifold $B_0$ and the tower equivalence of the descending chain

 $$\cP \equiv \left\{  \stackrel{p_{\ell +1}}{\longrightarrow} G_{\ell} \stackrel{p_{\ell}}{\longrightarrow} G_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  G_1 \stackrel{p_1}{\longrightarrow}  G_0 \right\}$$
  

{\bf Theorem:}[Pontragin 1934; Baer 1937] For $G_0 \cong \mZ$, the homeomorphism types of McCord solenoids is uncountable. 
  

{\bf Theorem:}[Kechris 2000; Thomas2001] For $G_0 \cong \mZ^k$ with $k \geq 2$, the homeomorphism types of McCord solenoids is not classifiable, \emph{in the sense of Descriptive Set Theory}.

The number of such is not just huge, but indescribably large.


  \vfill   


}

 
 
  
  

 \section{Pseudogroups}


 \frame % 12
{
  \frametitle{Pseudogroups}
  
  


Covering of $\fM$ by foliation charts $\Longrightarrow$ transversal  $\cT \subset \fM$ for $\F$ 


 Holonomy of $\F$ on $\cT$ $\Longrightarrow$  compactly generated pseudogroup $\cG_{\F}$:

 

\begin{itemize}
\item  relatively compact open subset $\cT_0 \subset \cT$ meeting all leaves of $\F$
\item  a finite set $\G = \{g_1, \ldots , g_k\} \subset    \cG_{\F}$  such that $\langle \G \rangle =  \cG_{\F} | \cT_0$; 
\item   $g_i \colon D({g_i}) \to R({g_i})$ is the restriction of  $\widetilde{g}_i \in   \cG_{\F}$,    
$\overline{D(g)} \subset D({\widetilde{g}_i})$.
\end{itemize}

 

  Dynamical properties of $\F$ formulated in terms of $\cG_{\F}$;  e.g.,
 

 
 $\F$ has no leafwise holonomy if for $g \in \cG_{\F}$, $x \in Dom(g)$, $g(x) = x$ implies $g | V = Id$ for some open neighborhood $x \in V \subset  \cT$.


  \vfill

}

    
\frame % 13
{

  \frametitle{Topological dynamics}


{\bf Definition:} $\fM$ is an \emph{equicontinuous matchbox manifold}  if it admits some covering by foliation charts as above, such that for all $\e > 0$, there exists $\delta > 0$ so that for all $h_{\cI} \in \cGF$ we have
$$x,x' \in D(h_{\cI}) ~ {\rm with} ~ d_{\cT}(x,x') < \delta \quad \Longrightarrow  \quad d_{\cT}(h_{\cI}(x), h_{\cI}(c')) < \e$$

  
  \bigskip
 

 {\bf Theorem:}  Let $\fM$ be an equicontinuous matchbox manifold. Then $\fM$ is   minimal.  

  
\pause
\bigskip


 {\bf Theorem:}   If $\fM$ is a homogeneous  matchbox manifold, then the pseudogroup  $\cGF$ is   equicontinuous.  





  \vfill

}

\frame % 13
{

  \frametitle{Topological dynamics of pseudogroups}

 
Can also define and study pseudogroup dynamics which are distal, expansive, proximal, etc. 


\emph{ADVERT}:    
See 

$\bullet$ {\bf Lectures on Foliation Dynamics: Barcelona 2010}  \\
S.~H., [2011 arXiv]

$\bullet$    {\bf Dynamics of foliations, groups and pseudogroups}, \\
{P.~Walczak}, [2004, Birkh\"auser, 2004]


  \vfill

}


\section{Shape}

 \frame % 14
{
  \frametitle{Shape theory}
 


The \emph{shape} of a set $\fM \subset \cB$ is defined by a co-final descending chain $\{U_{\ell} \mid \ell \geq 1\}$ of open neighborhoods in Banach space $\cB$, 
$$ U_1 \supset U_2 \supset \cdots \supset U_{\ell} \supset \cdots \supset \fM \quad ; \quad \bigcap_{\ell = 1}^{\infty} ~ U_{\ell} = \fM$$
Such a tower is called a shape approximation to $\fM$.

 
\medskip
 

Homeomorphism $h \colon \fM \to \fM'$ induces maps $h_{\ell, \ell'} \colon U_{\ell} \to U_{\ell'}'$ of shape approximations.



  \vfill

}




 \frame % 15
{
  \frametitle{Main technical result}


{\bf Theorem:} Let $\fM$ be a transitive matchbox manifold. Then $\fM$ has a shape approximation such that each $U_{\ell}$ admits a quotient map   $\pi_{\ell} \colon U_{\ell} \to B_{\ell}$ for $\ell \geq 0$ where $B_{\ell}$ is a ``branched $n$-manifold'', covered by a leaf of $\F$.

The system of induced maps $p_{\ell} \colon B_{\ell} \to B_{\ell -1}$ yields an inverse limit space homeomorphic to $\fM$.



 \pause
  \medskip
   
  
This study is part of sequence of   papers by Clark, H. \& Lukina:

$\bullet$  {\it Voronoi tessellations for matchbox manifolds}, July 2011 (arXiv).
  
$\bullet$  {\it  Shape of   matchbox manifolds}, September 2011, to appear. 
 
$\bullet$  {\it  Classification of   matchbox manifolds},  2011, to appear. 
 
    
     


  \vfill

}




 \frame % 16
{
  \frametitle{Remarks}

     
     
     $\bullet$ For $\fM$ a tiling space on $\mR^n$, this is just the presentation of $\fM$ as inverse limit in usual methods.

\pause

$\bullet$ For  $\fM$ with foliation defined by free $G$-action and tiling on orbits, as in Benedetti \& Gambaudo, same as their result.

\pause

 For general $\fM$, the problem is to find good local product structures, which are stable under transverse perturbation. The leaves are not assumed to have flat structures, so this adds an extra level of difficulty, as compared to the methods  in paper of  Giordano, Matui, Hiroki, Putnam, \& Skau: ``Orbit equivalence for Cantor minimal $\mZ^d$-systems'',  Invent. Math. 179 (2010)
 
   \vfill

}




 \frame % 16
{
  \frametitle{}


 
 The difficulties   depends on the dimension:

$\bullet$ For $n=1$, it is trivial.
 
\pause

$\bullet$ For $n=2$,  given a uniformly spaced net in $L_0$, the volumes of triangles in the associated  Delaunay triangulation in the plane are \emph{a priori} bounded by the net spacing estimates.

 \pause
 
 $\bullet$  For $n \geq 3$, there are no \emph{a priori} estimates on simplicial volumes, and the method becomes much more involved. 
 
\pause

In terms of \emph{leaf dimensions}, we have the fundamental observation:
$$1 ~  \ll ~ 2 ~ \ll 3 < n $$


  \vfill

}




 

  
\section{Coding}
 
 
 \frame % 18
{
  \frametitle{Coding orbits}
 
 Coding the orbits of a dynamical system is about as old an idea as exists in dynamics.
 \pause 
Ahlfors,  Gottschalk,  ...
 
 \pause
E. Thomas in 1970 paper for $1$-dimensional matchbox manifolds applied these ideas to matchbox manifolds.


  \vfill

}



 


\frame % 19
{
  \frametitle{}


{\bf Theorem:} [Clark \& Hurder] Suppose that $\F$ is equicontinuous.  Then for all $\epsilon > 0$, there is a decomposition into $k_{\epsilon}$ disjoint clopen sets,  for $k_{\epsilon} \gg 0$, 
$$\cT = \cT_{1} \cup \cdots \cup \cT_{k_{\epsilon}}$$
such that $diam(\cT_i) < \epsilon$ for all $i$, and the sets $\cT_i$ are permuted by the action of $\cG_{\F}$. 

\pause 

We obtain a ``good coding'' of the orbits of the pseudogroup $\cG_{\F}$.


\pause
Moreover, the coding respects the inverse limit structure defined by shape approximations.

  \vfill

}



 


\frame % 19
{
  \frametitle{}
  
  
    
   
  
 {\bf Theorem:} [C \& H, 2010] Let $\fM$ be a equicontinuous matchbox manifold. Then $\fM$ is minimal,  and  homeomorphic to a weak solenoid. 
 
    \medskip
   
  
 {\bf Corollary:}  Let $\fM$ be a equicontinuous matchbox manifold. Then $\fM$ is homeomorphic to the suspension of an minimal action of a countable group  on a Cantor space $\mK$.


   \vfill   

}
 

\frame % 20
{
  \frametitle{Homogeneous matchbox manifolds}
  
 {\bf Definition:} A matchbox manifold $\fM$ is \emph{homogeneous} if the group of homeomorphisms of $\fM$ acts transitively.
 
 
  \medskip
   
    
  
 {\bf Theorem:} [C \& H, 2010] Let $\fM$ be a homogeneous matchbox manifold. Then $\fM$ is equicontinuous, minimal,  without holonomy; and $\fM$ is homeomorphic to a McCord solenoid. 
 
    \medskip

   
   
  
 {\bf Corollary:}  Let $\fM$ be a homogeneous matchbox manifold. Then $\fM$ is homeomorphic to the suspension of an minimal action of a countable group  on a Cantor \emph{group} $\mK$.



    \vfill   

}


   
\section{Morphisms}


\frame % 21
{
  \frametitle{Leeuwenbrug Program}
  
 {\bf Question:} To what extent is an element of  ${\bf Homeo}(\fM)$ determined by its restriction to a complete transversal $\cT$ to $\F$?
 
  
 \medskip
 \pause
 
 {\bf Question':} Let $\fM, \fM'$ be matchbox manifolds of leaf dimension $n$, with transversals $\cT, \cT'$ and associated pseudogroups $\cG_{\F}$ and $\cG'_{\F'}$.
 Given a homeomorphism $h \colon \cT \to \cT'$ which intertwines actions of $\cG_{\F}$ and $\cG'_{\F'}$, when does there exists a homeomorphism $H \colon \fM \to \fM'$ which induces $h$?
 
 \medskip
  \pause
  
 
 {\bf Theorem:} True for $n=1$, i.e., for oriented  flows.
 
 J.M. Aarts and M. Martens, ``Flows on one-dimensional spaces'', Fund. Math., 131:39�58, 1988.




    \vfill   

}


 

\frame % 21
{
  \frametitle{co-Hopfian}




Example of Alex Clark shows this is false for $n=2$! 

False even for solenoids built over a surface $B_0$ of higher genus.

The problem comes up from the fact that   covers of the base $B_0$ need not be homeomorphic to the base.

     
    
    \bigskip
    
    {\bf Problem:}  Understand equivalence between matchbox manifolds in terms of their holonomy pseudogroups, and other invariants of their dynamics and geometry.
    
     
    
    \emph{Long way to go}...


    \vfill   

}





     
 
\end{document}


 