% 36th Summer Topology Conference
% Foliation theory special session (Olga Lukina, Hiraku Nozawa)
% July 22, 2022
 
 
% slides for talk "Type invariants for solenoidal manifolds"
 
 
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\date[July 22, 2022]


  
\title{Type invariants for solenoidal manifolds}


\author{Steve Hurder\\ University of Illinois at Chicago\\   {\footnotesize  joint work with}   \\ Olga Lukina\\ Leiden University, The Netherlands} 
 
 
\begin{document}
 
   \frame{\titlepage} % # 1

        \frame % 1
{
  \frametitle{Motivation}

 
 {\bf Problem 1:} Study   properties of foliated spaces which are transversally totally disconnected. These are called matchbox manifolds, or solenoidal manifolds in works of Sullivan.
 
 \medskip
 
{\bf Problem 2:} Determine when a foliated space is homeomorphic to a minimal set of a $C^r$-foliation.
 
 \medskip
 
   
 {\bf Problem 3:} Classify the  foliated spaces  $(\fM, \F)$    for which  the transverse holonomy maps are equicontinuous, so preserve an ultra-metric on the transverse Cantor space. 
 
  \bigskip
  
$\star$~ We focus on some recent contributions to Problem 3.
 \vfill   

}
 
   

       \frame % 1
{
  \frametitle{}
  
 Problem 3 asks to study the properties of     Riemannian foliated spaces, analogous to Riemannian foliations on smooth   manifolds.
 
 \smallskip 
 
 Series of works by Jes\'{u}s  \'{A}lvarez L\'{o}pez,  Alberto Candel, Ram\'{o}n  Barral Lij\'{o},  and Manuel Moreira Galicia, including:  

\bigskip

 {\small
 
 $\bullet$~ {\it Equicontinuous foliated spaces}, by  \'{A}lvarez L\'{o}pez \& Candel, {\bf Mathematische Zeitschrift}, vol. 263, 2009.

  \smallskip
   

$\bullet$~ {\it Topological {M}olino's theory}, by  \'{A}lvarez L\'{o}pez   \& Moreira Galicia, {\bf Pacific Journal of Math.}, vol. 280, 2016.

  \smallskip
 

$\bullet$~ {\it Molino's description and foliated homogeneity}, by  \'{A}lvarez L\'{o}pez \& Barral Lij\'{o}, {\bf Topology and its Applications}, vol. 260, 2019.

  }  

\medskip

Here we focus on the structure of these spaces.
 \vfill   

}
 
   

       \frame % 1
{
  \frametitle{Weak solenoids}

{\bf Theorem:} [Clark \& Hurder 2013] Let  $(\fM, \F)$ be an equicontinuous foliated space. Then $\fM$ is homeomorphic to a \emph{weak solenoid}, in the sense of 

 
$\bullet$~ {\it Inverse limit sequences with covering maps}, by M.C. McCord,  {\bf Trans. Amer. Math. Soc.}, vol. 114, 1965


\medskip
 That is, there is given a collection  $\cP = \{\, q_{\ell} \colon  M_{\ell} \to M_{\ell -1} \mid  \ell \geq 1\}$, 
 where each $M_{\ell}$ is a compact connected manifold without boundary of dimension $n \geq 1$, and $q_{\ell}$ is a proper covering map.
The inverse limit     
$$
\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ q_{\ell } \colon M_{\ell } \to M_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ M_{\ell} ~  
$$
is   the  \emph{solenoidal manifold}   associated to $\cP$. 

\medskip
The theorem asserts that $\fM$ is foliated homeomorphic to $\cS_{\cP}$ for some choice of $M_0$ and tower of coverings $\cP$.


 \vfill   

}
 
   

       \frame % 1
{
  \frametitle{}
{\bf Problem:} Let $\cP$ be a presentation,  
$$M_0 \leftarrow M_1 \leftarrow M_2 \leftarrow \cdots $$
What are the invariants of homeomorphism between two spaces which are the inverse limits of such  towers of coverings?

\bigskip

Facts about the inverse limit $\cS_{\cP}$
\medskip

 
$\star$~ $\cS_{\cP}$ is compact and connected but not locally connected, 

$\star$~ There is a fibration map $p_{\ell} \colon \cS_{\cP} \to M_{\ell}$ for each $\ell \geq 0$ 

$\star$~ The fiber $\fX = p_0^{-1}(x_0)$ is a Cantor space

$\star$~  $\cS_{\cP}$ is foliated by leaves which cover $M_0$

$\star$~  The holonomy pseudogroup of the foliation $\F_{\cP}$  preserves an ultra-metric on the Cantor transversals.


\medskip

 
 \vfill   

}
 
  
       \frame % 1
{
  \frametitle{}

{\bf Example:} ~  The {Vietoris - van Dantzig Solenoid:}
$$\cS(\vec{m}) =  \underleftarrow{\lim} ~ \{\, q_{\ell}:\mS^1 \to \mS^1 \mid  \ell \geq 1\} 
$$
where $q_{\ell}$ is  a covering map  of the circle $\mS^1$   of degree  $m_{\ell} > 1$, and   $\vec{m} = (m_1, m_2,\dots)$  denotes the collection of covering degrees. 

\medskip
When $m_i =2$ for all $i \geq 1$ we get the \emph{Smale attractor}, which is a minimal set for a $C^{\infty}$-foliation:
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.4\textwidth]{Smale-Williams_Solenoid_Large}
\label{figure-originalhirsch}
\end{center}
\end{figure}

 
 \vfill   

}
 
     

       \frame % 1
{
  \frametitle{Supernatural numbers}


A \emph{Steinitz}   (or \emph{supernatural}) number  is a formal infinite   product
$$\Pi[\vec{m}] = \prod_{0 \leq i < \infty} m_i   = \prod_{p \in \pi} \ p^{n(p)} \quad , \quad 0 \leq n(p) \leq \infty$$
where $\pi$ denotes the collection of all primes.

\medskip

  Steinitz numbers $\Pi$ and $\Pi'$ are \emph{asymptotically equivalent}, written $\Pi \mor \Pi'$, if there exists integers $m, m' >0$ such that $m \cdot \Pi  = m' \cdot \Pi'$. 

The asymptotic equivalence class of $\Pi$ is denote by $\Pi_a$

\medskip

{\bf Definition:}  A \underline{type} is a class $\Pi_a$ of a Steinitz number $\Pi$.
 \vfill   

}
 
  
  

       \frame % 1
{
  \frametitle{}
 

{\bf Definition:}  Given Steinitz number $\Pi = \prod_{p \in \pi} \ p^{n(p)}$, define:
 
$\star$~ $\pi(\Pi)    =     \{ p \in \pi \mid 0 < n(p)   \}$    ,   \emph{prime spectrum of} \ $\Pi$ 

$\star$~ $\pi_f(\Pi)    =     \{ p \in \pi \mid 0 < n(p) < \infty \}$  ,  \emph{finite prime spectrum of} \ $\Pi$  

$\star$~ $\pi_{\infty}(\Pi)    =     \{ p \in \pi \mid  n(p) = \infty \}$  ,    \emph{infinite prime spectrum of} \ $\Pi$ 
 
\bigskip


  Note that if $\Pi \mor \Pi'$, then $\pi_{\infty}(\Pi) = \pi_{\infty}(\Pi')$. The property that $\pi_f(\Pi)$ is an \emph{infinite} set is also preserved by asymptotic equivalence.
 
 \vfill   

}
 
   
   

       \frame % 1
{
  \frametitle{}


{\bf Definition:} Let $\cS(\vec{m})$ be a $1$-dimensional solenoid defined by a sequence of covering degrees $\vec{m}$. Then the type of  $\cS(\vec{m})$ is the asymptotic equivalence class    $\Pi_a[\vec{m}]$.


\bigskip

[Bing, 1960] and [McCord, 1965]   showed:

{\bf Theorem:} Solenoids $\cS(\vec{m}) $ and $\cS(\vec{m'}) $ are homeomorphic  if and only if they have the same type. 

\medskip

 [Aarts \& Fokkink, 1991] gave a new proof of this result.

Their novel method of proof suggests the following:


 \medskip

{\bf Question:}   Does ``type'' make sense for all weak solenoids, and if so, what does it tell you about these foliated spaces?

 
\vfill   

}
 
     

       \frame % 1
{
  \frametitle{Types of weak solenoids}

Here is our first result.

Let $\cS_{\cP}$ be a weak solenoid defined by a presentation $\cP = \{\, q_{\ell} \colon  M_{\ell} \to M_{\ell -1} \mid  \ell \geq 1\}$.

Let $m_{\ell}$ denote the degree of the covering map $q_{\ell}$.

Set $\vec{m} = \{m_1, m_2, \ldots\}$.
\medskip

{\bf Definition:} The \emph{type} of $\cS_{\cP}$ is the type of  $\Pi[\vec{m}]$.

\bigskip

{\bf Theorem 1 [Hurder \& Lukina 2021]:}   The type of $\cS_{\cP}$   is an invariant of the homeomorphism class of the continuum  $\cS_{\cP}$. 
 
  \bigskip
  
  {\bf Corollary:} The infinite prime spectrum $\pi_{\infty}(\Pi[\vec{m}])$  is an invariant of the homeomorphism class of  $\cS_{\cP}$. Also, the property that $\pi_f(\Pi[\vec{m}])$ is an \emph{infinite} set is   preserved by homeomorphism. 
 
 
}
 
     

       \frame % 1
{
  \frametitle{}

For  $\cP = \{\, q_{\ell} \colon  M_{\ell} \to M_{\ell -1} \mid  \ell \geq 1\}$, set $\G = \pi_1(M_0,x_0)$.

For   non-trivial $\gamma \in \pi_1(M_0,x_0)$ there is a well defined type $\Pi_a[\gamma]$, defined as the type of the $1$-dimensional solenoid in $\cS_{\cP}$ that covers a closed curve representing $\gamma$.

\medskip

{\bf Definition:} The collection 
$${\bf TS}[\cS_{\cP}] = \{ \Pi_a[\gamma] \mid \gamma \in \pi_1(M_0, x_0)\}$$
is called the \emph{type set} of $\cS_{\cP}$.

\bigskip

{\bf Theorem 2 [Hurder \& Lukina 2022]:}   The type set ${\bf TS}[\cS_{\cP}]$    is an invariant of the homeomorphism class of $\cS_{\cP}$. 


 \vfill   

}
 
     

       \frame % 1
{
  \frametitle{}

The theorems follow from the results in the works
\medskip


$\bullet$~ {\it Classifying matchbox manifolds}, by Clark, Hurder \& Lukina, {\bf Geometry \& Topology}, vol. 23, 2019.

\bigskip

$\bullet$~ {\it Pro-groups and generalizations of a theorem of Bing}, by Clark, Hurder \& Lukina, {\bf Topology and its Applications}, vol. 271, 2020.

 \bigskip
 We explain how this works.
 
\vfill   

}
 
     

       \frame % 1
{
  \frametitle{Cantor actions}

Given the weak solenoid
$$
\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ q_{\ell } \colon M_{\ell } \to M_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ M_{\ell} ~  
$$
 for $x_0 \in M_0$, the fiber $\fX = p_0^{-1}(x_0)$ is a Cantor set.
 
 \smallskip
 
 The monodromy along leaves of the foliation of $\cS_{\cP}$  gives an action of $\G = \pi_1(M_0, x_0)$ on $\fX$.
 The action $\Phi \colon \G \times \fX \to \fX$ is minimal and equicontinuous.
 
 \medskip
 
 {\bf Theorem:} $\cS_{\cP}$ is homeomorphic to the suspension foliation for the action $(\fX,\G,\Phi)$.
 
 \medskip
 
 $\star$~  Suffices to study the properties of   minimal equicontinuous Cantor actions of finitely generated groups.
  \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}
Assume that   $(\fX,\G,\Phi)$  is a minimal equicontinuous Cantor action.

\medskip
 
{\bf Definition:} $U \subset \fX$  is \emph{adapted} to the action $(\fX,\G,\Phi)$ if $U$ is a   \emph{non-empty clopen} subset, and for any $g \in \G$, 
if $\Phi(g)(U) \cap U \ne \emptyset$ implies that  $\Phi(g)(U) = U$.   

\smallskip
$\bullet$ ~ Given  $x \in \fX$ and clopen set $x \in W$, there is an adapted clopen set $U$ with $x \in U \subset W$. 

\smallskip
$\bullet$ ~ For   $U$ adapted,   the set of ``return times'' to $U$, 
$$
\G_U = \left\{g \in \G \mid g \cdot U  \cap U \ne \emptyset  \right\}  
$$
is a subgroup of   $\G$, called the \emph{stabilizer} of $U$.      

\medskip

$\bullet$ ~ $\cH_U = \Phi(\G_U) \subset \Homeo(U)$ is the monodromy of $\G_U$

 \vfill   

}


       \frame % 1
{
  \frametitle{}
  
  
{\bf Definition:} 
Minimal equicontinuous Cantor  actions $(\fX_1, \G_1, \Phi_1)$ and $(\fX_2, \G_2, \Phi_2)$ are  \underline{return equivalent} if there exists 
  an adapted set $U_1 \subset \fX_1$ for the action $\Phi_1$   and  
  an adapted set $U_2 \subset \fX_2$ for the action $\Phi_2$,
and a homeomorphism $h \colon U_1 \to U_2$ which induces an isomorphism of the monodromy groups $\cH_{U_1}$ with $\cH_{U_2}$.


  \medskip
This is the version of Morita equivalence appropriate for equicontinuous actions on Cantor sets.

  
  \vfill   

}
 
   
  
    
 

       \frame % 1
{
  \frametitle{}

{\bf Theorem [Clark, Hurder \& Lukina, 2019]:}  If solenoidal manifolds $\cS_{\cP}$ and $\cS_{\cP'}$ are homeomorphic, then their monodromy Cantor actions   are \emph{return equivalent}.

 \bigskip
 
  If the actions $(\fX_1, \G_1, \Phi_1)$ and $(\fX_2, \G_2, \Phi_2)$ are \emph{topologically free}, then this result implies that the   groups $\G_1$ and $\G_2$ are commensurable. 
  
  \medskip
  
  That is, there are subgroups $H_1 \subset \G_1$ and $H_2 \subset \G_2$ such that $H_1 \cong H_2$.  For example, for the 1-dimensional solenoid, two subgroups of $\mZ$ are commensurable.

  
  \vfill   

}
 
   
  
    
 

       \frame % 1
{
  \frametitle{Maps between inverse limits}
 The second ingredient we need are some properties of homeomorphisms between inverse limits, derived from:
 
 
$\bullet$~ {\it Mappings of inverse limits},   Mioduszewski, {\bf Coll. Math.},  vol. 10, 1963.
\smallskip

 $\bullet$~  {\it Approximation of maps of inverse limit spaces by induced maps},  Fort \& McCord, {\bf Fund. Math.}, vol. 59, 1966.
   

 
 $\bullet$~  {\it Maps between weak solenoidal spaces},   Rogers \& Tollefson,
  {\bf Coll. Math.}, vol. 23, 1971.

  $\bullet$~  {\it Homogeneous inverse limit spaces with non-regular covering maps as bonding maps},   Rogers \& Tollefson, {\bf Proc. A.M.S}, vol. 29, 1971.


 
  \vfill   

}
 
   
  
    
 

       \frame % 1
{
  \frametitle{}
 
{\bf Theorem:}  Suppose there are  two presentations as weak solenoids
 \begin{eqnarray}
\cP & = &  \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell}  \mid  \ell \geq 0 \} \\
\cQ & = & \{ q_{\ell +1} \colon N_{\ell +1} \to N_{\ell}  \mid  \ell \geq 0 \} ~ , 
\end{eqnarray}
 a homeomorphism $\Phi  \colon    \cS_{\cP}\to     \cS_{\cQ}$ and $\e > 0$.
Then there exists a decreasing sequence $\widehat{\epsilon} = \{\e_0>  \e_1 > \e_2 > \cdots\}$ with $\e_0 \leq \e$,  
\begin{enumerate}
\item an intertwined increasing sequence $0 \leq  j_0 < i_0 <j_1 <i_1 < j_2 < i_2 < \cdots$, 
\item covering maps $\lambda_{\ell} \colon M_{i_{\ell +1}} \to N_{j_{\ell}}$ for $\ell \geq 0$,
\item covering maps $\mu_{\ell} \colon N_{j_{\ell}} \to M_{i_{\ell}}$ for $\ell \geq 0$ ~ , 
\end{enumerate}
such that  the following diagram $\widehat{\epsilon}$-commutes:
    \begin{align} \label{eq-commutativediagram}
 \xymatrix{
 M_{i_{0}}    &   \ar[l]  M_{i_1} \ar[ld]_{\lambda_0}  &  \ar[l]  M_{j_1}     &  \ar[l]  M_{i_2} \ar[ld]_{\lambda_1}     &  \ar[l]  M_{j_2}  \cdots       \\
N_{j_{0}}    &  \ar[l]     N_{i_1}   &  \ar[l]   N_{j_1} \ar[lu]_{\mu_1}      &  \ar[l]  N_{i_2}   &  \ar[l]  N_{j_2}  \ar[lu]_{\mu_2}  \cdots       
} 
\end{align}
 

  
  \vfill   

}
 
   
  
    
 

       \frame % 1
{
  \frametitle{}
 {\bf Corollary:} There is a commutative diagram of   groups:
 
{\tiny 
    \begin{align} \label{eq-commutativediagrampi1}
 \xymatrix{
   &      \pi_1(M_{i_1}  , x_{i_1})  \ar[ld]_{\lambda_0}  &  \ar[l]  \pi_1(M_{j_1}  , x_{j_1})      &  \ar[l]  \pi_1(M_{i_2} , x_{i_2})   \ar[ld]_{\lambda_1}     &  \ar[l]  \pi_1(M_{j_2}  , x_{j_2})  \cdots       \\
\pi_1(N_{j_0}   , y_{j_0})   &  \ar[l]     \pi_1(N_{i_1}   , y_{i_1})  &  \ar[l]   \pi_1(N_{j_1}  , y_{j_1}) \ar[lu]_{\mu_1}      &  \ar[l]  \pi_1(N_{i_2}   , y_{i_2})  &  \ar[l]  \pi_1(N_{j_2}   , y_{j_2}) \ar[lu]_{\mu_2}  \cdots       
} 
\end{align}
}

 That is, for $\G_{\ell} = \pi_1(M_{i_{\ell}} , x_{i_{\ell}})$ and $\G_{\ell}' = \pi_1(N_{j_{\ell}}   , y_{j_{\ell}})$, the descending group   chains $\{\G_{\ell} \mid \ell > 0\}$ and $\{\G_{\ell}' \mid \ell > 0\}$ are intertwined.
  
  The proof of Theorem~1 then follows almost immediately.
  
  
\medskip

$\bullet$~  {\it The prime spectrum of solenoidal manifolds}, Hurder \& Lukina, {arXiv:2103.06825}.


  \vfill   

}
 
   
  
  


   

       \frame % 1
{
  \frametitle{}
 
 Theorem~2 is motivated by a result from the theory of finite rank subgroups of $\mQ^k$, which has a massive theory:


$\bullet$~ {\bf Abelian groups}, by L\'{a}szl\'{o} Fuchs, {Springer Monographs in Mathematics},  2015.

\medskip
A finite rank subgroup of $\mQ^k$ corresponds to a descending chain in $\mZ^k$ by Pontrjagin duality, as described in 

\smallskip

$\bullet$~ {\it $\Bbb{Z}^d$-odometers and cohomology},  {Giordano,  Putnam,\& Skau}, {\bf  Groups, Geometry, and Dynamics},
vol. 13, 2019.
   \medskip
     
     So one asks, whether any of this classification theory for subgroups of $\mQ^k$ is applicable to the study of weak solenoids?
\medskip

Some, and maybe more than some.
 
 \vfill   

}
 
 
 
 

       \frame % 1
{
  \frametitle{}
 
 Denote by $\cG =  \{\G = \G_0 \supset \G_1 \supset \G_2 \supset \cdots\}$   a properly descending chain of finite index subgroups
 
 Choose $\gamma \in \G$ 
 such that  $H_0 = \langle \gamma \rangle \subset \G$ is free abelian. Then for $H_{\ell} = \G_{\ell} \cap H_0$, we get a descending chain
 $$\mZ \cong   H_0 \supset H_1   \supset H_2 \supset H_3 \supset \cdots$$
 
 {\bf Definition:} Set  $\Pi[\gamma]= LCM \ \{ \#(H_0/H_{\ell}) \mid \ell > 0\}$.   
 \medskip
 
{\bf Definition:} The collection of types 
 $${\bf TS}[\cG] = \{ \Pi_a[\gamma] \mid \gamma \in \G\}$$
is called the \emph{type set} of $\cG$.

 \vfill   

}
 
 
 
 

       \frame % 1
{
  \frametitle{}
 


For $\cG$ the group chain associated to a tower of coverings $\cP$, it follows in a straightforward manner that the type set
 ${\bf TS}[\cS_{\cP}] = \{ \Pi_a[\gamma] \mid \gamma \in \pi_1(M_0, x_0)\}$ is an invariant for homeomorphisms between weak solenoids.

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The story does not stop here, as the set of types have a partial order, yielding what is called the \emph{type graph}:

      
$\bullet$~  {\it Type graph},  Mutzbauer,  {\bf Abelian group theory ({H}onolulu, {H}awaii, 1983)},  
  {Lecture Notes in Math.}, Vol. 1006,   1983.
     
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{\bf Question:} Do the type invariants impose any restrictions on whether a weak solenoid can be the minimal set of a foliation? 

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  \frametitle{Realization}
 
 {\bf Theorem:} Let $M_0 = \mT^k$ and $\ds  \cP = \{\, q_{\ell} \colon  M_{\ell} \to M_{\ell -1} \mid  \ell \geq 1\}$ be a tower of coverings of $M_0$. Then there exists a $C^0$ foliation $\F_{\cP}$ of $M_0 \times \mD^{q}$, for $q \geq 2k$,   with minimal set homeomorphic to $\cS_{\cP}$. 
 
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 Moreover, for $r \geq 1$, if $\cP$ satisfies certain technical growth conditions, then $\F_{\cP}$ can be constructed as a $C^r$   foliation. 

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$\bullet$~ {\it Embedding solenoids in foliations}, by Clark \& Hurder, {\bf Topology and its Applications}, vol. 158, 2011.

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$\star$~ This suggests that any type set can be realized as a foliated minimal set, for abelian groups. For non-abelian groups, such as torsion-free nilpotent groups, the story seems more complicated.
     
 
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