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% Wednesday, November 7, 2012


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\date[November 7, 2012]
 {S\'{e}minaire g\'{e}n\'{e}ral\\Institut de Math\'{e}matiques de Bourgogne\\November 7, 2012}

  
\title[Classifying Foliations]{Classifying Foliations\\{\normalsize after Bott, Haefliger \& Thurston}}


\author{Steven Hurder}

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



\begin{document}

\frame{\titlepage} % # 1

% def and applications
% examples - list keeps growing
% classifying - topological, dynamical, ergodic, descriptive
% BGamma - C1 verses C2
% exponential dichotomy
% Godbillon Vey vanishing
% solenoids
% Bott Heitsch




\section{Introduction}
 
 \frame % # 2
{
  \frametitle{Foliations...}
 
Origins  of modern theory began around with   theses of   Reeb [1952] and   Haefliger [1956], and work of   Ehresmann in 1940's and 50's. Now:
 

  \begin{enumerate}
   \item  Generalized dynamical systems\\ 
   ( Reeb, Godbillon, Sacksteder, Anosov, Smale, Hector, Ghys \ldots )
   

   
  \item  Representation theory:  cocycles, co-orbit spaces, $W^*$ \& $C^*$-algebras\\ 
  ( Murray - von Neumann, Mackey, Kirillov, Kasparov, Renault, \ldots )
   
  
  
  \item  Topology of classifying spaces\\ 
  ( Bott, Haefliger,  Gelfand-Fuks, Mather,   Thurston, Tsuboi, \ldots )    
   
  
     
   
 \item  Physics \& Non-Commutative Geometry, quasicrystals \\ 
 ( Bellisard, Connes, Gambaudo, Kellendonk, Barge, \ldots )
 
  
 
    \item  Descriptive Set Theory \& Complexity\\ 
    ( Kechris, Foreman, Hjorth, Louveau, Simon, \ldots )
  \end{enumerate}
  
\vfill   
}
 
   
 
\frame % 4
{
  \frametitle{Classifying foliations}
 
  
 {\bf Problem:}      ``Classify'' the foliations on a given manifold $M$.
  
  \bigskip
   
Multiple classification schemes have been developed:
\begin{enumerate}
\item ``homotopy properties'' and classifying spaces;
\item  ``dynamical properties'' and invariants;
\item  ``classify''  von~Neumann and $C^*$-algebras of foliations;
\item ``complexity theory'' of Borel equivalence relations.
\end{enumerate}

      
 
All  are long-term research topics.  

  \bigskip
   


 {\bf Question:}   
How are these classifications schemes related to one another?

 
\medskip


See ``Classifying foliations'',  in  {\it Foliations, Topology and Geometry},  Contemp. Math. vol 498,  American Math. Soc.,  2009.  

\vfill

}


\frame % 1
{
  \frametitle{}
 
  A foliation $\mathcal F$ of dimension $p$ on a manifold $M^m$ is a 
 decomposition  into ``uniform layers'' -- the leaves -- which are immersed submanifolds of codimension $q$:  there is an open covering of $M$ by   coordinate charts 
so that the leaves are mapped into linear planes of dimension $p$, and the transition function preserves these planes.
\begin{center}
\includegraphics[width=0.3\textwidth]{dijon_pix/chart.pdf}
\end{center}
 
 A leaf of $\F$ is a connected component of the manifold $M$ in the ``fine'' topology induced by charts. 
 
 \vfill   

}


 

\frame % 2
{
  \frametitle{Foliations by surfaces}
 
 
\begin{center}
\includegraphics[width=0.6\textwidth]{dijon_pix/torus_reeb.png}\\
{\it Reeb Foliation of the solid torus}
\end{center}
 
 How do you go from this seminal example, to an understanding of foliations and their geometry? Even more, how to classify?
 \vfill   

}

\frame % 2
{
  \frametitle{Construct examples --  surgery}
 
Surgery on knots + Reeb foliation yields:
\medskip

 {\bf Theorem:} [Lickorish 1965, Novikov \& Zieschang 1965, Wood 1969]\\
Every compact $3$-manifold  $M$ admits a codimension-one foliation.
 
 \bigskip
 
  Spinnable structures (existence of fibered knots)  + Reeb foliation yields:
 \medskip
 
  {\bf Theorem:} [Durfee \& H.B. Lawson 1973, Tamura 1973]\\
  Every (m-1)-connected smooth closed (2m+1)-manifold, $m \geq 3$, admits a codimension-one foliation.
 
 \bigskip
 
 Other surgery methods   for codimension-one on $3$-manifolds [Gabai 1986].
 
  \vfill   

}

\frame % 2
{
  \frametitle{Construct examples -- Lie group actions}
 
  $\bullet$~ Non-singular smooth flow $\phi_t \colon M \to M$ defines  a foliation. \\Same as a \emph{locally-free} smooth action $\phi \colon \mR \times M \to M$.
 
 
 \bigskip
 
  $\bullet$~ \emph{Locally-free} smooth action $\phi \colon \mR^p \times M \to M$ defines foliation on $M$.
 
 
 \bigskip
 
  $\bullet$~ $G$ connected Lie group, \emph{locally-free} smooth action $\phi \colon G \times M \to M$ defines foliation on $M$.
 
 \bigskip
 
{\bf Example:}  
  $\G \subset \mS\mL(2, \mR)$ a cocompact lattice, $\mP \subset \mS\mL(2, \mR)$ the parabolic subgroup consisting of upper triangular matrices. Then $M = \mS\mL(2, \mR)/\G$ and $\mP$ acts on right, to define the \emph{Roussarie foliation} $\F$ of $M$. 
 
 
  \vfill   

}

\frame % 2
{
  \frametitle{Construct examples -- suspensions}
 
 Given a connected, closed manifold $B$ with $b_0 \in B$,   then $\G = \pi_1(B, b_0)$ is a finitely-presented group, acts freely on the universal covering $\widetilde{B} \to B$.

\medskip
 
Given closed manifold $N^q$, assume there is $C^r$-action $\phi \colon \G \times N \to N$.
 
 $$M = \left\{ \widetilde{B} \times N \right\} / (b \cdot  \gamma , x) \sim (b, \phi(\gamma)  \cdot x)$$
 
 Leaves of $\F_{\phi}$ defined by images of $\widetilde{B} \times \{x\}$ in $M$.
 
 \bigskip
 
 {\bf Example:} $\G$ finitely-presented, $\phi \colon \G \times \mS^1 \to \mS^1$ a $C^r$-action, suspension yields a codimension-one foliation $\F_{\phi}$
 \bigskip
 
 These are the traditional examples...
 
  \vfill   

}

\frame % 2
{
  \frametitle{Construct examples -- inverse limits}
 
Introduce a fourth class of examples:

\medskip

 Foliations obtained by \emph{inverse limit constructions}.

\bigskip

In some sense, these are discretized versions of the previous examples.

   \vfill   

}

 
 
\frame % 2
{
  \frametitle{Matchbox manifolds}
 
  
    
\bigskip

 {\bf Definition:}     $\fM$ is   an \emph{$n$-dimensional matchbox manifold} if:

$\bullet$  $\fM$  is a continuum: a compact, connected metrizable space; 

$\bullet$ $\fM$  admits a covering by  foliated coordinate charts $\ds \cU = \left\{ \varphi_{i} \colon U_i \to [-1,1]^p \times \fT_i \mid i \in \cI \right\}$; 

$\bullet$  each  $\fT_i$ is a     clopen subset of  a totally disconnected   space   $\fX$. 

\medskip

Then the  arc-components of $\fM$ are locally Euclidean: 
$$
\fT_i \quad \text{are totally disconnected} ~   \Longleftrightarrow  ~ \fM \quad \text{is a matchbox manifold}$$


  \bigskip
     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. 

  

  

 
 \vfill   

}

\frame % 2
{
  \frametitle{}
 
  
 
 
The ``matchbox manifold'' concept is  much more general than minimal sets for foliations:   they also appear in study of tiling spaces,  subshifts of finite type, graph constructions, generalized solenoids, pseudogroup actions on totally disconnected spaces, \ldots 

\bigskip

{\bf Examples:  }
\smallskip

Minimal $\mZ^p$-actions on Cantor set $K$.
 
 $\bullet$ Adding machines (minimal equicontinuous systems)
 
  $\bullet$ Toeplitz subshifts over $\mZ^p$
  
  $\bullet$ Minimal subshifts over $\mZ^p$
  
  $\bullet$ Tiling spaces for aperiodic, repetitive tilings of $\mR^p$ of finite local complexity.
   
 \bigskip

First, a detour through the title of the talk:

\medskip

``Interest level''  of examples $\Leftrightarrow$    their properties  +   ``classification theory''

 \vfill   

}
 

\frame % 4
{
  \frametitle{Classification in the 1970's}

A ``new era'' for  foliation theory: 

\medskip

$\bullet$~  Bott: \emph{On a topological obstruction to integrability}, 1970 ICM

\smallskip

$\bullet$~  Haefliger: \emph{Feuilletages sur les vari\'{e}t\'{e}s ouvertes,} Topology 1970.

\smallskip

$\bullet$~  Bott \& Haefliger: \emph{On characteristic classes of $\Gamma$-foliations}, \\
Bulletin AMS, 1972

\smallskip

$\bullet$~  Thurston: \emph{Theory of foliations of codimension greater than one}, Comment. Math. Helv. 1974.

\smallskip

$\bullet$~  Thurston: \emph{Existence of codimension-one foliations}, \\ Annals of Math. 1976.

\smallskip

$\bullet$~   Lawson: \emph{Quantitative theory of foliations},  CBMS Lectures 1977.

\vfill

}



   \frame % 5
{
  \frametitle{Classification -- first steps (Reeb \& Haefliger)}

 
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.  The   holonomy of $\F$ on $\cT$ yields a compactly generated pseudogroup $\cGF$:

\medskip

$g \in \cGF ~ \Longrightarrow ~ g \colon U  \to V$ $C^r$-diffeomorphism, for $U, V \subset \cT$, $r \geq 0$
 
 $g \in \cGF ~ \Longrightarrow ~ g^{-1} \in \cGF$  
 
  $g \in \cGF, ~ {\rm open} ~ W   \subset U  ~ \Longrightarrow ~ g|W \in \cGF$  

and elements satisfy gluing and composition rules.

\bigskip

 
{\bf Definition:}
  $\cGF$   is \emph{compactly generated}     if there is 
\begin{itemize}
\item  relatively compact open subset $\cT_0 \subset \cT$ meeting all orbits of $\cGF$;
\item  a finite set $\G = \{g_1, \ldots , g_k\} \subset    \cGF$  which generates $\cGF | \cT_0$; 
\item   $g_i \colon D({g_i}) \to R({g_i})$ is the restriction of  $\widetilde{g}_i \in   \cGF$ with   
$\overline{D(g)} \subset D({\widetilde{g}_i})$.
\end{itemize}

 
\vfill

}

 
 
 
 
 

 \frame % 5
{
  \frametitle{Derivative cocycle}


     
    Assume that the holonomy maps in $\cGF$ are uniformly $C^1$.
    
    \bigskip
    
  Assume $(\cGF, \cT)$ is a compactly generated pseudogroup, and $\cT$ has a uniform Riemannian metric. Choose a uniformly bounded, Borel trivialization, ~ $T\cT \cong \cT \times \mR^q$,  ~ $T_x\cT \cong_x  \mR^q$ ~ for all $x \in \cT$.
  
  \medskip
  
  
 {\bf Definition:}   \emph{Normal derivative cocycle} $D  \colon \GF \times \cT \to {\mG\mL(\mR^q)}$
 $$D ([g]_x) =  D_x g   \colon T_x \cT \cong_x \mR^q  \to T_{y}\cT \cong_y \mR^q$$
Satisfies the  \emph{cocycle law} (chain rule of derivatives)
 $$ D( [h]_y \circ [g]_x ) = D([h]_y) \cdot D ([g]_x) ~ , ~ y = g(x)$$
 
  
     
 \vfill

}

   

 
\frame % 
{
  \frametitle{Classifying spaces}


   ``Milnor join''    construction $\Rightarrow$  classifying space $BG$ of a Lie group $G$ 
 
 \medskip 
$B \mS\mO(1) \cong B\mS^1 \cong \mC\mP^{\infty}$, ~ 
 $B \mS\mO(q)$ $\cong$  infinite Grassmannian manifold of oriented $q$-planes in $\mR^{\infty}$.
 
 
     
\medskip

 
$q \geq 1$, $r \geq 1$, then  $\G^r_q$ denotes the topological \emph{groupoid}:  
 \smallskip
 
 $\bullet$~  Objects are points of $\mR^q$.
 \smallskip
  
 $\bullet$~ Morphisms are germs of local $C^r$-diffeomorphisms of $\mR^q$. 

\medskip
 
   [Haefliger 1970, Segal 1975] generalized Milnor join construction to obtain  classifying spaces for a \emph{topological groupoid} $\G$. 

\medskip

$B\G \equiv \| \G \|$ is   the ``semi-simplicial fat realization'' of   $\G$.
 

  
 
 
   

 \vfill
 
}

 


\frame % 5
{
  \frametitle{Homotopy classification}

 
 
   $B\Gamma^r_q$ is the   ``classifying space''  of  codimension-$q$ $C^r$-foliations.
  
  
 \medskip
Derivative at a point, $g \mapsto D_x g$,   defines functor, hence     $\nu \colon   B\G^r_q \to B\mO(q)$.
 
   
 \medskip
 
   {\bf Theorem:} [Haefliger 1970]  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 depends only on concordance class of  $\mathcal F$.

 \medskip
 
 {\bf Remark:} The homotopy fiber $F\Gamma^r_q \to B\Gamma^r_q \to BO(q)$ classifies foliations with \emph{framed} normal bundles. 

 \bigskip
   
  {\bf Theorem:} (Bott)
 For $r \geq 2$ and $q \geq 2$, the fiber $F\Gamma^r_q$ is not contractible. 
 
  \medskip

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


 
{\bf  Problem:} How to calculate the homotopy type of the fiber $F\Gamma^r_q$? 


  
 \vfill
 
}

   \begin{frame}{Godbillon-Vey classes}
  
   
  
 $\F$   a $C^2$-foliation and the normal bundle $Q = TM/T\F$ is oriented. 
  \medskip
  
    $\omega$   a $q$-form defining $T\F$,   then $d \omega = \eta \wedge \omega$ for a $1$-form $\eta$.
  
  \medskip
  
  The form $\eta \wedge (d \eta)^q$ is then a closed $2q+1$-form on $M$.   
  
  \medskip
  
     {\bf Theorem:} (Godbillon-Vey [1971])  The \emph{Godbillon-Vey class}  cohomology class 
 $GV(\F) = [\eta \wedge (d \eta)^q] \in H^{2q+1}(M; \mR)$ is independent of choices.

 \bigskip
 
     {\bf Theorem:} [Roussarie 1972] $GV(\F) \ne 0$ for the Roussarie foliation. 
   
   \medskip
   
   {\bf Corollary:} Homotopy type of $F\G_1^r$ is very non-trivial for $r \geq 2$.
   
\vfill
\end{frame}

  \begin{frame}{Non-vanishing of $GV(\F)$}
  
 
  {\bf Question:} [Moussu-Pelletier 1974, Sullivan 1975] If $GV(\F) \ne 0$,  what does this imply about the dynamical properties of $\F$?    
    
  \bigskip

{\bf Theorem:} [Duminy 1982]  $GV(\F) \ne 0$  for $C^2$-foliation $\F$ of codimension-one $\Rightarrow$ $\F$ has  resilient (homoclinic) leaves.

\bigskip
{\bf Theorem:} [Hurder 1986]   $GV(\F) \ne 0$  for $C^2$-foliation of codimension-$q$ $\Rightarrow$ $\F$ has  leaves of exponential growth.

\bigskip

{\bf Theorem:} [Hurder \& Langevin 2004]   $GV(\F) \ne 0$  for $C^1$-foliation of codimension-one $\Rightarrow$ $\F$ has   non-zero foliation entropy.

\vfill
\end{frame}


\frame % 6
{
  \frametitle{Higher secondary classes}

 
   {\bf Theorem:} (Bott-Haefliger, Gelfand-Fuks, Kamber-Tondeur [1972]) \\
For each codimension $q$, and $r \geq 2$, 
  there is a non-trivial space of secondary invariants $H^*(W_q)$ and functorial  characteristic map 
    whose image contains the Godbillon-Vey class
  
\begin{picture}(100,60)
\put(140,40){$H^*(F\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}$}
\put(124,8){$\Delta$}
\put(80,0){$H^*(W_q)   \longrightarrow  H^*(M; \mR)  $}
\end{picture}
  
     % \pause
\medskip

In this notation: $GV(\F) =   [\eta \wedge (d \eta)^q] = \Delta(h_1 \wedge c_1^q)$.

 
\medskip

\begin{center}
Classification of $\F$ on $M$ $\leftrightarrow$ calculate secondary classes for examples
\end{center}

 
 \vfill
 
}

  
 
 
\frame
{
  \frametitle{Non-vanishing secondary classes}

 Almost all examples are based on variations of a simple scheme:
 \medskip
 
 Let $G$ be a semi-simple connected Lie group.
 
 \smallskip
 
 $P \subset G$ a parabolic subgroup.
 
 \smallskip
 
 $\G \subset G$ a cocompact lattice.
 
 \smallskip
 
 Then right cosets of $P$ foliate the compact manifold $M = \G \slash G$.
 
 \smallskip
 
 Then calculate, and obtain that the secondary classes can be non-zero over ranges of classes in $H^*(W_q)$ of the form
 
 $$\Delta(h_1 \wedge h_{\cI} \wedge c_{\cJ}] ~ {\rm where}~ \cI = \{i_2, \ldots , i_k\}, i_2 \geq 3, i_{\ell} \leq 2q ~ {\rm odd}$$
 
 \medskip
 
 Last   new examples   by   Heitsch [1978],  Rasmussen [1980],   Hurder [1985]. 
 
  
  \vfill
 
}

\frame
{
  \frametitle{Vanishing secondary classes}

{\bf Theorem:} [Hurder \& Katok 1987]  $\Delta(h_1 \wedge h_{\cI} \wedge c_{\cJ}] \ne 0$ then the algebraic hull of the derivative cocycle   cannot be \emph{amenable} (i.e., parabolic).


\bigskip

There seems no hope of extending the classification scheme using Lie group examples.
Theme switched in 1980's from using secondary classes to study classification problems, to showing   the classification scheme fails!

  \vfill
 
}

  
\frame
{

  \frametitle{Foliation dynamics}

The 1980's --    study of foliations as dynamical systems:

\begin{itemize}

 
\item  A 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 trajectory of the point $x \in M$.  \\

\item  In foliation dynamics, we replace the concept of time-ordered trajectories with multi-dimensional futures for points. \\
\item Study the   minimal sets, attractors, rates of expansion, etc. for the leaves of $\F$.
\end{itemize}

 \bigskip

 \vfill

}

  

 
  \begin{frame}{Closed invariant sets}

  As an alternative to asking for the ``big picture'' of foliation dynamics,  study   the dynamics near closed invariant sets.   
   \bigskip
 
 $\cZ \subset M$   \emph{minimal} $\Longleftrightarrow$ closed and every leaf $L \subset \cZ$ is dense. 
 
 \medskip
 
$\cW \subset M$ is \emph{transitive} $\Longleftrightarrow$  closed and there exists a dense leaf $L \subset \cW$

% % \pause
\medskip

$M$ compact, then minimal sets for foliations always exist.
 
 \medskip
 
Transitive sets are most important for flows -- Axiom A attractors  are transitive sets, while the minimal sets include the  periodic orbits in the domain of attraction.
  
      
   \medskip
   
 $\cZ$ is  \emph{exceptional minimal set} if $\cZ \cap \cT$ is a Cantor set.
     
   \medskip
   
   $\cZ$ is a matchbox manifold, embedded in a foliated neighborhood. 
\vfill
\end{frame}


 
 \begin{frame}{Structure of exceptional minimal sets}

{\bf Conjecture:}   $\F$  a codimension-one, $C^2$-foliation,   $\fM \subset M$ is an exceptional minimal set, then  $\fM$ has the structure of a \emph{Markov Minimal Set}. 

\medskip
That is, there is a covering of $\fX = \fM \cap \cT$ by clopen sets, such that the restricted holonomy pseudogroup $\cGF^{\fX}$ is generated by a free semigroup.

    \bigskip
  % \pause
   
{\bf Theorem:} [Clark--H--Lukina, 2012]  Let  $\fM$ be an exceptional minimal set of dimension $p$ for a $C^r$-foliation, $r \geq 1$. Then  there exists a presentation  $\ds \cP = \{ p_{\ell} \colon M_{\ell} \to M_{\ell -1}  \mid \ell \geq \ell_0\}$ consisting of   branched $p$-manifolds $M_\ell$ and local    covering $C^{r-1}$-maps $p_\ell$, such that $\fM$   is   homeomorphic to the inverse system, 
$$
  \fM ~  \cong  ~  \cS_{\cP} = \varprojlim ~ \{ p_{\ell} \colon M_{\ell} \to M_{\ell -1} \mid \ell \geq \ell_0\}. 
$$
  
 \medskip
 
Generalizes  result for the case where $\fM$ is the tiling space associated to a  
   repetitive, aperiodic tiling of $\mR^q$ with      finite local complexity. 
  
\vfill
\end{frame}



 
\begin{frame}{Williams solenoids}

Williams (1970, 1974) introduced a class of inverse limit spaces, called generalized solenoids, which model Smale's Axiom A attractors.

\medskip
Let $K$ be  a branched manifold, i.e. each $x \in K$ has a neighborhood homeomorphic to the disjoint union of a finite number of Euclidean disks modulo some identifications.

\medskip
$f: K \to K$ is an expansive immersion of branched manifolds satisfying a flattening condition. Then  
 $$\fM = \lim_{\longleftarrow}\left\{ f: K \to K \right\}$$
each point $x$ has a neighborhood homeomorphic to $[-1,1]^n \times \textrm{Cantor set}$. Moreover, $f$ extends to a hyperbolic map defined on some open neighborhood of $\fM \subset V \subset \mR^m$ for $m$ sufficiently large. 

\smallskip

 $\fM$ is an exceptional minimal set for the expanding foliation $\F_u$ of $V$. 

\end{frame}



 

   
    
\frame % 
{
  \frametitle{Shape}


 $\cZ \subset M$ a minimal set of $\F$ always has a ``neighborhood system''
 $$\cZ \subset \cdots U_{i} \subset \cdots \subset U_1 \quad , \quad  K = \bigcap_{i =1}^{\infty} ~U_i$$
where the $U_i$ are open. The system    defines the \emph{shape} of $K$. 
 
   
 \bigskip
  
 
 {\bf Definition:} A minimal set $\cZ \subset M$ is \emph{moveable} (or \emph{stable}) if it has a neighborhood system such that for $i \gg 1$ the inclusions $U_{i+1} \subset U_i$ are homotopy equivalences. 
 
  \bigskip
  
 
{\bf Remark:} Stable is almost the same as the tower  above being sequence of homotopy equivalences.

\medskip

{\bf Example:}  The Denjoy minimal set in $\mT^2$ is stable.

\vfill
 
}

 
\frame % 
{
  \frametitle{Classification, revisted}

$\cZ \subset U \subset M$ a closed $\F$-saturated set 
$\Rightarrow$ classifying  map $B\G_{U | \F}  \to B\Gamma_q$
 
 \bigskip
 
 {\bf Conjecture:} Let $\cZ \subset M$ be an exceptional minimal set which is stable, and $\cZ \subset U$ a stable neighborhood.  Then the classifying  map 
 $B\G_{U | \F}  \to B\Gamma_q$ is homotopically trivial.
 
\bigskip
In contrast, we have:
 \smallskip
 
 {\bf Theorem:} [Hurder2012] For all $q \geq 2$, there exists open foliated spaces $U$ containing an \emph{unstable exceptional minimal set} $\cZ$ such that  $H^k(B\Gamma_q ; \mR) \to H^k(B\G_{U | \F} ; \mR)$ in non-zero in arbitrarily high degrees.
 
 \medskip
 
 These examples are a new type, and are motivated by constructions in dynamical systems, not in Lie theory.
Use sequence of ``period doubling maps'' to generate these minimal sets via limiting construction.

   \vfill
 
}


\frame % 
{
  \frametitle{Bott Vanishing Theorem}

We give the ideas behind this types of result.
\bigskip
 
 {\bf Theorem:}   [Bott 1970]  $\F$ be a $C^2$-foliation of codimension-$q$ on manifold $M$, with oriented normal bundle. Then 
 \begin{center}
 $\nu_{\mE}^* \colon H^{*}(BSO(2m) ; \mR) \to H^{*}(M ; \mR)$ is trivial for $* > 2q = 4m$.
 \end{center}

 \medskip
 
{\bf Corollary:}   Let $p \colon \mE \to M$ be an oriented smooth vector bundle with fibers of dimension $q$,    $\F$ a  $C^2$- foliation $\F$  transverse to the fibers, then 
 \begin{center}
 $\nu_{\mE}^* \colon H^{*}(BSO(2m) ; \mR) \to H^{*}(M ; \mR)$ is trivial for $* > 2q = 4m$.
 \end{center}

 
 
 \vfill
 
}
 
    
     
\frame % 
{
  \frametitle{Heitsch Thesis}




{\bf Theorem:} [Heitsch 1970]   Bott Vanishing Theorem is false for $\mZ$ coefficients! That is, there is an \emph{injective} map
  $$\nu_{\mE}^* \colon H^{*}(BSO(2m) ; \mZ) \to H^{*}(B\G_{2m}^r ; \mZ)$$ 
    
 

 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 $\mD^2$.
Form the suspension flat bundle 
$$\mE = \mS^{\infty} \times \mD^{2m}/ \varphi \to \mS^{\infty}/(\mZ/ p \mZ) \cong B(\mZ/ p \mZ)$$
 Then the composition 
  $$\nu_{\mE}^* \colon H^{*}(BSO(2m) ; \mZ/ p \mZ) \to  H^{*}(B\G_{2m}^+ ; \mZ/ p \mZ)  \to H^{*}(\mE ; \mZ/ p \mZ)$$ 
  is injective.  Then let $p \to \infty$ and use Universal Coefficient Formula.

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  \frametitle{A construction}
  

{\bf Theorem:} [Clark \& Hurder 2008] For $p \geq 1$ and $q \geq 2m-1$, 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 $\cZ$, such that:

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

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$\bullet$ ~ Action of $\mZ^p$ on   Cantor set $\mK = \cS \cap \mS^q$ has a unique invariant probability measure (action is equivalent to generalized odometer)

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

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$\bullet$ ~ The isotropy groups of periodic orbits form a profinite series
$$ \cdots \G_i \subset \cdots \G_1 \subset \G_0 = \mZ^n \quad , \quad  \G_i = \vec{n}_i \cdot \mZ^p$$

 
 

$\bullet$ ~ $K$ is an ``$\{\vec{n}_i\}$-adic'' completion of $\mZ^p$: $\ds K \cong \lim_{\leftarrow} ~ (\G_0/\G_i)$.


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  \frametitle{Application}

   
Take a shape neighborhood sequence of the minimal set $\cZ$ for the construction. 
 Each open neighborhood $U_i$    yields is a natural map $B\G_{U_{i} \mid \F}  \to B\Gamma_q$, hence an induced map of the limit space 
$$h_{Z} \colon \widehat{Z} \equiv \lim_{\leftarrow} ~ \{B\G_{U_{i+1} \mid \F} \to B\G_{U_{i} \mid \F}  \}  \longrightarrow B\Gamma_q$$

{\bf Theorem:}   The   induced map  $h_{\cZ} \colon \widehat{\cZ} \to B\Gamma_q$ is non-trivial:  
$$h_{\cS}^* \colon H^{4\ell -1}(B\G_q ; \mR) \to \cH^{4\ell -1}(\cS, \F ; \mR) ~ \mbox{is non-trivial for} ~    \ell > q/2$$
  
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{\bf Proof:}   The  Cheeger-Simons classes for $C^r$-foliations, $r \geq 2$, derived from  $H^*(BSO(q); \mR)$ are in the image of   $h_{\cS}^*$.  

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 These characteristic class of the shape of $\cZ$ cane thought of as ``non-commutative winding numbers''.   
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  \frametitle{Remarks}

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

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Relation between cohomology invariants and foliation dynamics has only been explored at the edges.

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 {\bf Problems:} 
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 $\bullet$~ When does a minimal matchbox manifold embed as a minimal set in a $C^r$-foliation with codimension $q \geq 2$, for some $r \geq 1$?
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 $\bullet$~ What are the cohomology invariants of these embeddings?
 
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{\it Thank you for your attention!}
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