% UIUC Colloquium
% October 17, 2013
% slides for talk "Foliations --   Thurston Zebras to Cantor Geometries"
% slides completed October 17, 2013


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\title[Thurston Zebras to Cantor Geometries]{Foliations \\   Thurston Zebras to Cantor Geometries}


\author{Steve Hurder}

\date{October 17th, 2013}

\institute[UIC] {University of Illinois at Chicago\\www.math.uic.edu/$\sim$hurder}



\begin{document}

\frame{\titlepage} % # 1



 
  \section{Zebras}
 
  
  \frame % 1
{
  \frametitle{Foliations}
 
 
In Spring 1974, William Thurston gave a colloquium   at Rice University,    and spent 20 minutes drawing a picture of a zebra on the blackboard.  

The message: this is what foliations are   about,    a very geometric subject!  

 
 \bigskip
  % \pause
 
Later, in  graduate courses  by \\
 \medskip
 
 \emph{Franz Kamber},   \emph{Philippe Tondeur}, and    also \emph{Kuo-Tsai Chen} in 1977-79;

\medskip

plus  Riemannian geometry courses by \\

\medskip
\emph{Richard Bishop} and \emph{Stephanie Alexander};

  \medskip
  more formal approaches emerged. 
   
   \medskip
   % \pause
   
    Still,  the definition by Thurston has its attractions, though later his Zebra was   replaced by a Chicago Institution:  
   \vfill   

}

 
  \frame % 1
{
  \frametitle{}
 
  
 

\begin{center}
\includegraphics[width=0.5\textwidth]{pix/cow_foliated.eps}
\end{center}
 
 \begin{center}
A Foly Cow!
 \end{center}
 
 % \pause
 
{\bf Wikipedia:} A foliation is a kind of clothing worn on a manifold, cut from a stripy fabric. On each sufficiently small piece of the manifold, these stripes give the manifold a local product structure. This product structure does not have to be consistent outside local patches: a stripe followed around long enough might return to a different, nearby stripe.

  
   \vfill   

}

 
  \frame % 1
{
  \frametitle{}
 
\medskip
Forty years since the subject was energized by Thurston's works,  1971-75.

\medskip

This was one theme at the   meeting on {\it Geometry and Foliations 2013}   last September in Tokyo: the contributions of Thurston and the subsequent developments of his ideas in the field. We ignored his famous comment, \emph{that after his work [on foliations], all that was left to do was trivial}.

\medskip
 


Historical paper by {Andr\'{e}~Haefliger}, available at {\it foliations.org}, 

 
  {\it Naissance des feuilletages, d'{Ehresmann}-Reeb \`a {Novikov}}

  
 \begin{center}
\includegraphics[width=0.4\textwidth]{pix/reebcomp_01.jpg}
\end{center}

    
   \vfill   

}

 
  \frame % 1
{
  \frametitle{Formal Definition}
 
 
  
{\bf Definition:}  A foliation $\F$ of   a   manifold $M$ is  a ``uniform partition'' of $M$ into submanifolds of constant dimension $p$ and codimension $q$.
More precisely, a    smooth manifold of dimension $n$  is \emph{foliated}  if there is a covering of $M$ by coordinate charts whose change of coordinate functions preserve the leaves:
 \begin{center}
\includegraphics[width=0.48\textwidth]{pix/chart.pdf}
\end{center}
 
 
   \vfill   

}

  \frame % 1
{
  \frametitle{}
 A foliation $\F$ of   a compact manifold $M$ is also \ldots
 
 \bigskip


$\bullet$ a local geometric structure on $M$, given by a $\G_{\mR^q}$-cocycle for a ``good covering''. (Ehresmann, Haefliger)

 
 \medskip

$\bullet$ a  dynamical system on $M$ with multi-dimensional time.

\medskip

$\bullet$ a groupoid   $ \cGF \to M$ with fibers complete manifolds, the holonomy covers of leaves.  
 
 \bigskip
    

 Each point of view has  advantages and disadvantages.
 
 \medskip
 
 {\bf Problem:}  How to distinguish and classify foliations?
 
 \medskip
 
 Convert to a discrete problem:
   
    \vfill   

}

 
  \frame % 1
{
  \frametitle{Pseudogroups}
 

A section $\cT \subset M$ for $\F$ is an embedded submanifold of dimension $q$ which intersects each leaf of $\F$ at least once, and always transversally. 

\medskip

Covering by coordinate charts provides local sections, whose union is a global transversal $\cT$.
 The transverse   holonomy along \emph{leafwise paths} for $\F$ between points in $\cT$     generates the pseudogroup $\cGF$.

 \begin{center}
\includegraphics[width=0.45\textwidth]{pix/chart.pdf}
\end{center}
 
    \vfill   

}

 
  \frame % 1
{
  \frametitle{Groupoids}

   
    
{\bf Definition:} The groupoid of $\cGF$ is the space of germs 
$$\G_{\F} = \{[g]_x \mid g \in \cGF ~ \& ~ x \in D(g)\} ~ , ~ \cGF$$
with source map $s[g]_x = x$ and range map $r[g]_x = g(x) = y$.

\bigskip

{\bf Definition:} 
The \emph{Haefliger   groupoid} $\Gamma^r_q $ is the collection of all germs of local $C^r$-diffeomorphisms between open subsets of $\mR^q$, endowed with the sheaf topology. 
The  objects of $\G^r_q$ are points of $\mR^q$, and morphisms are germs of local $C^r$-diffeomorphisms of $\mR^q$. 

\medskip

The derivative defines a groupoid transformation $\nu \colon \Gamma^r_q \to GL(\mR^q)$. 

\medskip

This space is universal, it is non-Hausdorff, and it is simply huge!


   \vfill   

}

   
 
\frame % 5
{
  \frametitle{Haefliger-Thurston classification}

 \bigskip
 \bigskip
 
 
  
      $B\Gamma^r_q$ =  ``classifying space''  of (smooth) codimension $q$-foliations with transverse differentiability $C^r$,       introduced by Andr\'{e}  Haefliger  in 1970. 

 \bigskip
      
 $B\Gamma^r_q \cong   \| \Gamma^r_q \|$ $\equiv$   the ``semi-simplicial fat realization'' of the groupoid $\Gamma^r_q$.
    
      \bigskip
       
      
      For $r \geq 1$,  there is a natural map  $B\nu \colon B\Gamma^r_q \to BGL(\mR^q) \cong BO(\mR^q)$   classifying the normal bundle to the universal foliation on $B\Gamma^r_q$.
   
  \bigskip
  
 
  {\bf Observation:} [Haefliger] The foliation on $B\Gamma^r_q$ has a single leaf. 
  
  \medskip
  
  $\Rightarrow$ this is a really strange  (non-Hausdorff) space. 
    
 

  \vfill
 
}

   
 
\frame % 5
{
  \frametitle{}

  
   \bigskip
 \bigskip
 
 

$\bullet$~ {\bf Theorem:} (Haefliger) Each $C^r$-foliation $\F$ on $M$ of codimension $q$ determines a well-defined map $h_{\F} \colon M \to B\Gamma^r_q$ whose homotopy class in uniquely defined by $\mathcal F$. 

\smallskip
 {\it Proof:} Phillips Transversality, Gromov ``h-Principle''

\medskip
 
  
 % \pause


$\bullet$~   {\bf Theorem:} (Thurston) Each ``natural'' map  $h_{\F} \colon M \to B\Gamma^r_q \times BO_p$ corresponds to a $C^r$-foliation $\F$ of codimension $q$ on $M$, whose  concordance class  is determined by $h_{\F}$. 

\smallskip
{\it Proof:} Draw lots of zebras; that is, very geometric at key steps...
 
 
 % \pause


\begin{center}
Classification of $\F$ on $M$ $\leftrightarrow$ calculate homotopy sets $[M, B\Gamma^r_q]$
\end{center}

 
  \vfill
 
}

   
 
\frame % 5
{
  \frametitle{The Quest}

\medskip

 {\bf Problem:} Understand the homotopy type of the mysterious space  $B\Gamma^r_q$!

\bigskip
% \pause

Introduce      $F\G_q^{r}$ , the homotopy fiber of the   map $B\nu \colon B\Gamma^r_q \to  BO(\mR^q)$. 

\medskip

$F\G_q^{r}$ is a topological measure of the difference between having a tangent subbundle   $F \subset TM$ of rank $p$, and a $C^r$-foliation $\F$ on $M$ with $F = T\F$.

\medskip
% \pause


{\bf Theorem:} [Bott Vanishing Theorem, 1968] 
 $$B\nu^* \colon H^{\ell}(BO(\mR^q); \mR)  \longrightarrow H^{\ell}(B\G_q^r , \mR)$$
 is the zero map for $r \geq 2$ and $\ell > 2q$.
\medskip

$F\G_q^{r}$ for $r \geq 2$ is definitely not trivial. % \pause
In contrast, there is the amazing
\medskip

{\bf Theorem:} [Tsuboi, 1989] $F\G_q^{1}$ has the weak homotopy type of a point.

 
 

 \vfill
 
}

  

  
 
  \section{Lie Groups}
 
  
  \frame % 1
{
  \frametitle{Godbillon-Vey class}
 
 
 {\bf Theorem:} (Godbillon-Vey [1971])  For a $C^2$-foliation of codimension $q$, there is a cohomology class  
 $GV(\F) = \Delta(h_1 \cdot c_1^q) \in H^{2q+1}(M; \mR)$, whose  value  depends only on the homotopy class of   $h_{\F} \colon M \to B\Gamma^2_q$.

\smallskip
 

{\it Proof:}

 \begin{itemize}
 \item Assume $\F$ has oriented normal bundle $\Rightarrow$  there is a $q$-form $\omega$ with $ker(\omega) = T\F$.
\item Froebenius Theorem implies the differential $d\omega = \eta \wedge \omega$ for   $1$-form $\eta$.
\item Calculation shows $\eta \wedge (d\eta)^{q}$ is a closed form.
\item  $GV(\F) = [\eta \wedge (d\eta)^{q}] \in H^{2q+1}_{deR}(M) \cong H^{2q+1}(M; \mR)$.
\end{itemize}
 
\medskip
 

$\bullet$  ~ $\eta$ is the \emph{Reeb modular form} for $\F$. 

$\bullet$ ~     $h_1 = [\eta] \in H_{deR}^1(\F)$ in leafwise cohomology is well-defined.

$\bullet$  ~  $c_1= [d\eta]$ represents the first \emph{Chern class} of the normal bundle to $\F$. 


   \vfill   

}

  \frame % 1
{
  \frametitle{}
 
 \medskip
 
 {\bf Theorem:} [Roussarie, 1971] The Godbillon-Vey class of the weak-stable foliation $\F$ on the unit sphere bundle $M^3$ for a Riemann surface $\Sigma$ with curvature $\kappa = -1$ is non-zero.
 
 \medskip
  

 {\it Proof:} $\F$ is also described as the foliation by cosets of the parabolic subgroup of upper triangular matrices in $SL(\mR^2)$, acting on the unit tangent bundle viewed as the coset space $M = T^1\Sigma_g = \Gamma\backslash SL(\mR^2)$. 
 
 $\bullet$~  $\eta$ is the contact form associated to the geodesic flow on $M$. 
   
  $\bullet$~  Calculate   $\eta \wedge d\eta$ is a multiple of the volume form. 
   
\bigskip
% \pause


{\bf Theorem:} [Thurston, 1971]  
For each $\alpha \in \mR$, 
there exists a   foliation $\F_{\alpha}$ of codimension-one on $\mS^3$ with $GV(\F_{\alpha})   = \alpha \in \mR \cong     H^{3}(M; \mR)$. 
 
 \medskip
 
 {\it Proof:} Glue foliations as above together, using clever a observation about  flows on unit tangent bundles to  Riemann surfaces defined by fundamental domains in $\mH^2$ (the slopes on boundary tori can be controlled...) 

 \vfill   

}

  \frame % 1
{
  \frametitle{Secondary Classes}
 \bigskip
 
   {\bf Theorem:} (Bott-Haefliger, Gelfand-Fuks, Kamber-Tondeur [1972]) 
   \smallskip
   
   For each codimension $q$, and $r \geq 2$, 
  there is a non-trivial space of secondary invariants $H^*(WO_q)$ and functorial  characteristic map 
    whose image contains the Godbillon-Vey class
  
\begin{picture}(100,60)
\put(140,40){$H^*(B\Gamma^r_q ; \mR) $}
\put(170,30){\vector(0,-1){17}}
\put(110,13){\vector(3,2){30}}
\put(180,20){$h_{\F}^*$}
\put(110,25){$\tilde{\Delta}$}
\put(134,8){$\Delta$}
\put(80,0){$H^*(WO_q)  ~ \longrightarrow  H^*(M; \mR)  $}
\end{picture}
  
  \medskip
  
The image of $\Delta$ depends only on  the homotopy class of  $h_{\F} \colon M \to B\Gamma^2_q$.

   \vfill   

}


  \frame % 1
{
  \frametitle{Cartan approach of Kamber \& Tondeur }
 
 $\bullet$  ~   $\F$ determines a ``partially flat connection'' $\nabla$ on the $GL(\mR^q)$-bundle of frames orthogonal to the foliation tangent bundle $T\F \subset TM$.  
 
 
 \medskip
 
   $\bullet$  ~ Construct a  graded DGA, the``truncated Weil algebra''
   
   $W(gl(\mR^q), O(q))_{2q}$ 
   with $H^*(W(gl(\mR^q), O(q))_{2q}) \cong  H^*(WO_q)$. 
   
 
\medskip
 
   $\bullet$  ~ Calculate     characteristic  map $H^*(W(gl(\mR^q), O(q))_{2q}) \to H^*(B\Gamma_q^2; \mR)$ using foliations defined by locally homogeneous spaces $M = \Gamma \backslash G$,  where $G$ is a semi-simple Lie group of higher rank, with foliation defined by right cosets of a Borel subgroup   $P \subset G$.
   
 
\medskip
 
   $\bullet$  ~ Key observation is that the non-triviality of the characteristic map follows from non-vanishing of the normal Euler class associated to the action of $\Gamma$ on the compact quotient space $X = G/P$. 
   
  
  \medskip
 
 \begin{center}
 {\it  This became the model for almost all examples constructed     in the 1970's of foliations which had non-trivial maps to $B\Gamma_q^2$. }
 \end{center}
 
   
   \vfill   

}


  \section{Ergodic Theory}
 
  
  \frame % 1
{
  \frametitle{Foliation dynamics}
 
 $\bullet$ ~ 
A \emph{continuous dynamical system} on a compact manifold $M$ is a flow $\varphi \colon M \times \mR \to M$, where the orbit $L_x = \{\varphi_t(x) = \varphi(x,t) \mid t \in \mR\}$ is thought of as the ``time evolution'' of the point $x \in M$.  The trajectories of the points of $M$ are necessarily \emph{points}, \emph{circles} or \emph{lines} immersed in $M$, and the study of their aggregate and statistical behavior is the subject of foliation dynamics and ergodic theory for flows.

 \bigskip
   


 $\bullet$ ~ 
\emph{Foliation dynamics,  ergodic theory} for foliations of leaf dimension $\geq 2$.    

Replace the concept of time-ordered trajectories with multi-dimensional futures for points. The study of the \emph{dynamics} of $\F$  asks for properties of the   limiting and statistical behavior of the collection of its leaves, and for ergodic theory look for measurable invariants. 
 \bigskip
  
  
   
   \vfill   

}

 
  \frame % 1
{
  \frametitle{Growth of leaves}
 
An ergodic property of foliations: Riemannian metric on $M$ yields complete Riemannian metrics on leaves.
 
 \medskip
 
 For $x \in M$ and leaf $L_x$, consider leafwise balls of radius $R$, 
 $$B_{\F}(x, R) = \{y \in L_x \mid d_{\F}(x, y) \leq R\}$$
$$ Gr(\F, x) = \limsup_{R \to \infty} ~ \frac{\ln \{{\rm Vol}_{L_x}(B_{\F}(x, R))\}}{R} < \infty$$
 
$ Gr(\F, x)$ is measurable orbit invariant.

\medskip
 

$L_x$ has \emph{exponential growth} if $Gr(\F, x) > 0$.

\medskip
  

{\bf Sullivan Conjecture:}[1975] If $GV(\F) \ne 0$ then the set of leaves with exponential growth is non-empty.

   \vfill   

}

 
  \frame % 1
{
  \frametitle{}
\bigskip

{\bf Theorem:} [Hurder, Jour. Diff. Geom. 1984]  

\medskip

If $GV(\F) \ne 0$ for $\F$ a codimension $q \geq 1$ foliation, then the set of leaves with exponential growth has positive Lebesgue measure.
 
 \medskip
  
 
{\it Idea of   Proof:} Uses joint work with James Heitsch, inspired by work of Gerard Duminy, and \emph{cocycle tempering} \`{a} la \emph{Pesin Theory}.

 \medskip
  

The \emph{Reeb modular class} 
 $[\eta] \in H_{deR}^1(\F)$ is an ergodic property of $\F$,  and the operator norm of $[\eta]$ has estimate, 
$$ \| [\eta] \| \sim \int_M  ~ Gr(\F, x) ~ dvol_M(x) $$ 
   
   \vfill   

}

 
  \frame % 1
{
  \frametitle{}
 
 \bigskip
 
  More refined ergodic invariants, use cocycles in spirit of Zimmer program.

\medskip
  
  Derivative map gives measurable cocycle $D\nu  \colon \G_{\F} \to GL(\mR^n)$
   
   \medskip
For any parabolic subgroup $P \subset  GL(\mR^n)$ get an action on homogeneous space $X_P = GL(\mR^n)/P$

   \medskip
% \pause


{\bf Theorem:}[Hurder \& Katok, Annals of Math 1987] 

\medskip

$\bullet$ ~ $[D\nu]_e \in H_{erg}^1(\G_{\F} ; X_P)$ is ergodic invariant

\medskip

$\bullet$ ~ Equivalence relation defined by $\F$   measurably amenable $\Rightarrow$  $[D\nu]_e = 0$.

\medskip

$\bullet$ ~ If $[D\nu]_e = 0$ then many of the secondary classes vanish.

\bigskip
% \pause

{\bf Conclusion:} 

\medskip $B\G_q^2$  Foliation Classification $\Leftrightarrow$    Foliation Dynamics \& Ergodic Properties. 
   \vfill   

}

   \frame % 1
{
  \frametitle{Foliation Dynamics \& Ergodic Theory}
 
 \medskip
 
 Major theme of work in the field since 1980:
 
 \medskip

$\bullet$ ~ Random walks on leaves, leafwise harmonic measures 

[Garnett, 1980] , [Ghys, 1995], [Candel, 2003], [Deroin \& Klepsyn, 2007], [Deroin, Klepsyn \& Navas, 2007]
 

\medskip

$\bullet$ ~ Foliation geometric entropy [Ghys, Langevin \& Walczak, 1988]

\medskip
 

$\bullet$ ~ Lyapunov spectrum and foliation dynamics [Hurder, Walczak,  1988]

\medskip
 

$\bullet$ ~ Rigidity of Anosov systems [Hurder \& Katok, 1990], [Hurder, 1992],  [Katok, Spatzier, et al, 1993 onwards]

\medskip
 

$\bullet$ ~ Regularity of weak stable foliations [Hasselblatt, 1992 onwards], [Pugh, Shub \&Wilkinson, 1997 onwards], etc
   
   \vfill   

}

     

  \section{Cantor Geometries}
 
  
  \frame % 1
{
  \frametitle{Mather-Thurston Theorem}

\medskip
 
 ${\rm Diff}_c(\mR^q)$ =  compactly supported $C^{\infty}$-diffeomorphisms of $\mR^q$.
 
\medskip
 
 ${\rm Diff}_c^d(\mR^q)$ = ${\rm Diff}_c(\mR^q)$  with \underline{discrete topology}. HUGE!
 
 \bigskip
 % \pause

\emph{Most amazing} ``unknown'' theorem:

\smallskip

 {\bf Theorem:} [Mather \& Thurston, 1974] ~ 
 There is a natural map $\sigma \colon {\rm Diff}_c^d(\mR^q) \to \Omega^q F\G_q^{\infty}$ which induces isomorphisms on homology
 
 $$  H^*(\Sigma^q {\rm Diff}_c^d(\mR^q); \mZ) \cong  H^*(F\G_q^{\infty})$$
 
   $F\G_q^{\infty}$ is the homotopy fiber of the universal map $B\nu \colon B\Gamma^r_q \to  BO(\mR^q)$. 
 
\bigskip
% \pause

{\bf Question:} What are the cycles in $\ds H^*(\Sigma^q {\rm Diff}_c^d(\mR^q); \mZ)$?

 
\medskip

Propose to construct cycles using dynamical systems!

   \vfill   

}
 
  \frame % 1
{
  \frametitle{Lipschitz Cantor   Geometries}
 
   
Only one ``Cantor set'', call it $\fX$, but has     many metrics:

\medskip

  $\dX$ and $\dXp$ are \emph{Lipschitz equivalent},  if they satisfy a Lipschitz condition:
$$
C^{-1} \cdot \dX(x,y) ~ \leq ~ \dXp(x,y) ~ \leq ~ C \cdot \dX(x,y) \quad {\rm for ~ all} ~ x,y \in \fX
$$
 
 {\bf Definition:}
A compactly generated pseudogroup  $\cGX = \langle g_1, \ldots, g_k\rangle$  acting on a Cantor set $\fX$ is \emph{Lipschitz}  with respect to a metric $d_{\fX}$ on $\fX$, if there exists $C \geq 1$ such that for each   $1 \leq i \leq k$ then  
$$
C^{-1} \cdot d_{\fX}(w,w') \leq d_{\fX}(h_{i}(w), h_{i}(w')) \leq C \cdot d_{\fX}(w,w') ~ ,~ \forall ~   w, w' \in   \Dom(g_{i}). 
$$
 
{\bf Definition:}  \emph{Lipschitz  Cantor Geometry}   investigates     properties of \emph{minimal Lipschitz pseudogroups actions}  on metric Cantor sets $(\fX, \dX)$. 

  \medskip
  
  Work with coauthors Alex Clark and Olga Lukina during past 5 years has studied properties of \emph{matchbox manifolds}, which are \emph{foliated spaces}  with  transversal Cantor model $\fX$, and holonomy is pseudogroup action on $\fX$.
  
   \vfill   

}


  \frame % 1
{
  \frametitle{}
 
 \bigskip
 
 {\bf Definition:}  \emph{$C^r$  Cantor Geometry}, $r \geq 1$,    investigates the   properties of   Cantor actions satisfying:
 
\begin{itemize}
\item $\fX \subset \mR^q$ is an embedding, 
\item  $\cGX = \langle g_1, \ldots, g_k\rangle$ is generated by restrictions of $C^r$-diffeomorphisms on some open neighborhood of $\fX \subset U \subset \mR^q$. 
\end{itemize}
 
 
 \medskip
 

{\bf Why study these objects?} 

\medskip


$\bullet$~ Interesting in their own right --  invariant sets for $C^2$-dynamical systems. 

\medskip
 

$\bullet$~  For $r \geq 2$, examples which yield non-trivial classes   in $H^*(F\G_q^{r}; \mR)$, not in the image of the usual secondary characteristic maps. 
   
   \vfill   

}

  \frame % 1
{
  \frametitle{Smoothly embedded solenoids}
 
    
\begin{center}
\includegraphics[width=0.25\textwidth]{pix/disksT.png}  \quad\quad\quad \includegraphics[width=0.4\textwidth]{pix/Torusintorus2.png}\\
  {\it Cantor Set $\fX$}  \quad \quad \quad \quad       $\to$  \quad \quad \quad \quad \quad    {\it Solenoid in $\mR^3$} \quad \quad
\end{center}
 
 

The holonomy along leaves of $\F$  ``twist'' to the points of $\fX$.

\medskip
 Pseudogroup   maps on $\fX$ extend   to   open neighborhoods of $\fX$.

 
 
\medskip
 

Solenoids naturally arise  as hyperbolic  attractors of smooth flows. 

[Smale, 1967], [Gambaudo, Sullivan,  Tresser, 1990 -- onwards]

 
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{\bf Theorem:} [Clark \& Hurder, 2011] For $p \geq 1$ and $q \geq 2n$, there exists commuting   diffeomorphisms  $\varphi_i \colon \mS^q \to \mS^q$, $1 \leq i \leq p$, so that the suspension of the induced action $\mZ^p$ on $\mS^q$ yields a smooth foliation $\F$ with solenoidal minimal set $\cS$, such that:

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$\bullet$ ~ The leaves  of $\F$ restricted to $\cS$ are all isometric to $\mR^p$

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$\bullet$ ~ The isotropy groups of periodic orbits form a profinite series
$$ \cdots \G_i \subset \cdots \G_1 \subset \G_0 = \mZ^n $$
$\bullet$ ~ $\ds \fX \cong \lim_{\leftarrow} ~ (\G_0/\G_i)$  is an ``adic-completion'' of $\mZ^p$.

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$\bullet$ ~ Every open neighborhood of $\fX$ contains periodic \underline{open domains} for the action of $\mZ^p$ on $\mS^q$

 
   
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{\bf Question:} What is relation to the quest?
  
 
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{\bf Theorem:} [Heitsch's Thesis, 1970] 

The    Bott Vanishing Theorem is false for $\mZ$ coefficients. 
 

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{\bf Example:} Let $(\mZ/ p \mZ)^m$ act on $\mD^{2m}$ via rotations $\{\varphi_1, \ldots , \varphi_m\}$ with period $p$ on each of  the $m$-factors of $\mD^2$.
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Form the suspension flat bundle 
$\ds \mE = \mS^{\infty} \times \mD^{2m}/ \varphi$.
 Then 
  $$\nu_{\mE}^* \colon H^{*}(BSO(2m) ; \mZ/ p \mZ) \to  H^{*}(B\G_{2m}^+ ; \mZ/ p \mZ)  \to H^{*}(\mE ; \mZ/ p \mZ)$$ 
  is injective. Let $p \to \infty$, then observe associated $\ds {\rm lim}_1$-terms not zero, so   Pontrjagin classes do not vanish in $H^*(B\G_q^2; \mZ)$.


 
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For any open neighborhood $Z \subset U$   of the solenoidal minimal set,  there are infinite sequences of ``Heitsch Examples'' embedded in   $\F | U$.

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 The restricted foliation $\F | U$ yields a map  $h_Z \colon B\G_{U \mid \F}  \to B\Gamma_q^{\infty}$.
 
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{\bf Theorem:}   Let $\cS$ be the solenoidal minimal set above. Then the homotopy class of $h_Z \colon B\G_{U \mid \F}  \to B\Gamma_q^{2}$ is non-trivial. 

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Moreover, for   $\ell > q/2$  there is a natural surjection 
$$h_{Z}^* \colon H^{4\ell -1}(B\G_q^2 ; \mR) \to \cH^{4\ell -1}(\cS, \F ; \mR) $$
  
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{\bf Proof:}   The   Cheeger-Simons classes   derived from  $H^*(BSO(q); \mR)$ are in the image of   $h_{Z}^*$ \ldots

   
    
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    These solenoidal cycles are just the simplest type of iterated braids. 
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    There are many more $C^2$ Cantor geometries...
  

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  After 40 years, the study of the classification problem for foliations has produced wide-ranging new techniques,  and new perspectives on   traditional and more recent topics in dynamics. 

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As for the quest to   ``find''  $B\G_q^r$   -- not even close to understanding this mysterious space!

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 {\large Thank you for your attention!}
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  \frametitle{References}
{\tiny   

{R.~Bott}, {\it On a topological obstruction to integrability},
  In {\bf Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968)},
  {Amer. Math. Soc.},  1970:127--131.

{R.~Bott and J.~Heitsch}, {\it A remark on the integral cohomology of $B\Gamma_q$},
 {\bf Topology}, 11:141-146, 1972.

 {K.-T.Chen}, {\it Iterated integrals of differential forms and loop space       homology},
{\bf Ann. of Math. (2)}, 97:217--246, 1973.

{A.~Clark and S.~Hurder},
 {\it Embedding solenoids in foliations},
 {\bf Topology Appl.}, 158:1249--1270, 2011.


{A.~Clark and S.~Hurder},
 {\it Homogeneous matchbox manifolds},
 {\bf Trans. Amer. Math. Soc.}, 365:3151--3191, 2013,  {arXiv:1006.5482v2}.

{C.~Godbillon and J.~Vey},  {\it Un invariant des feuilletages de codimension 1},
 {\bf  C.R. Acad. Sci. Paris}, 273:92--95, 1971.


{M.~Gromov}, {\it  Stable mappings of foliations into manifolds}
 {\bf Math. USSR Izv.},   33:671--694, 1969.


{A.~Haefliger}, {\it Feuilletages sur les vari\'et\'es ouvertes}, {\bf Topology}, 9:183--194, 1970.

{A.~Haefliger}, {\it Homotopy and integrability},
 In {\bf  Manifolds--Amsterdam 1970},
 {Lect. Notes in Math.} Vol.  197,    Springer--Verlag, 1971:133--163.

{A.~Haefliger},
 {\it Naissance des feuilletages, d'{Ehresmann}-Reeb \`a {Novikov}},
 In {\bf G\'eom\'etrie au vingti\`eme si\`ecle : 1930-2000}, {Hermann, \'Editeurs des Sciences et des Arts}, Paris.
Available at  http://www.unige.ch/math/folks/haefliger/Feuilletages.pdf


 {S.~Hurder}, {\it Dual homotopy invariants for {G}-foliations}, {\bf  Topology}, 20:365--387, 1981.

{S.~Hurder}, {\it The {Godbillon} measure of amenable foliations}, {\bf Jour. Differential Geom.}, 23:347--365, 1986.

 
{S.~Hurder and A.~Katok}, {\it Ergodic  Theory and {Weil} measures for foliations}, {\bf Ann. of Math. (2)}, 126:221--275, 1987.

{S.~Hurder},
 {\it Lectures on Foliation Dynamics: Barcelona 2010},
 {\bf Foliations: Dynamics, Geometry and Topology}, Advanced Courses in Mathematics CRM Barcelona,  to appear  2013.

{S.~Hurder},
 {\it Lipshitz matchbox manifolds}, {preprint}, {arXiv:1309.1512}.

{J.N.~Mather}, {\it Loops and foliations},
 In {\bf Manifolds---Tokyo 1973},
 {Univ. Tokyo Press, Tokyo}, 1975, 175--180.

{J.N.~Mather}, {\it On the homology of Haefliger's classifying space},
 In  {\bf Differential topology (Varenna, 1976)},  1979, 71--116.


 
{A.~Phillips}, {\it Foliations on open manifolds. {I,II}},
{\bf Comment. Math. Helv.}, 43:204--211, 1968; 44:367--370, 1969.

     
{F.W.~Kamber and Ph.~Tondeur},
 {\bf Foliated bundles and characteristic classes},
 {Lect. Notes in Math. Vol. 493},   Springer-Verlag, 1975.


{W.P.~Thurston},
 {\it Noncobordant foliations of ${S}\sp{3}$}
 {\bf Bull. Amer. Math. Soc.} 78:511--514, 1972.

{W.P.~Thurston},
 {\it Foliations and groups of diffeomorphisms}
 {\bf Bull. Amer. Math. Soc.} 80:304--307, 1974.

{W.P.~Thurston},
 {\it The theory of foliations of codimension greater than one}
 {\bf Comment. Math. Helv.} 49:214--231, 1974.

{W.P.~Thurston},
 {\it Existence of codimension-one foliations}
 {\bf Ann. of Math. (2)} 104: 249--268, 1976. 

 
 
{T.~Tsuboi},
 {\it On the foliated products of class {$C^1$}},
 {\bf Ann. of Math. (2)}, 130:227--271, 1989.
 
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