% slides completed July 2, 2012
%
% for Matchbox Dynamics - Mini-symposium
% http://www.auburn.edu/~kuperkm/dynamicsKrakow2012/
% 6th European Congress of Mathematics 
% Krak—w, July 2-7, 2012
%

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\title[Cohomology and smooth embeddings]{Cohomology and smooth embeddings\\ for matchbox manifolds\footnote{\scriptsize   Matchbox Dynamics - Mini-Symposium}
}
\author{Steven Hurder}
\institute{University of Illinois at Chicago}
\date{July 2, 2012}
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\begin{document}

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\frame{\titlepage} % # 1

% \addtocounter{framenumber}{-1}

 
 
\frame % 2
{
  \frametitle{Topics}
 
\medskip

 $\bullet$ ~   Matchbox manifolds --    smooth foliated spaces $\fM$ with codimension zero. 
 
 \bigskip
   
  $\bullet$ ~ Smooth foliated embeddings of $\fM$ into smooth foliated manifolds $M$.
  
 
 \bigskip
   
  $\bullet$ ~ Shape and the  \emph{fixed-point property}.
  
 
 \bigskip
   
  $\bullet$ ~    \emph{Non-commutative}  self-linking invariants.
  
    
  
   
    \vfill   

}
  


 \section{Matchbox Manifolds}
 
 
\frame % 2
{
  \frametitle{Matchbox manifolds}
 
Let $\fM$ be a continuum --  a nonempty compact connected metric space.
  
 
\medskip
 
 {\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 $\fT_i$     is a totally disconnected \emph{clopen} subset  of  a   Polish space  $\fX$.

\medskip
 
 Generalizes concept of continuum with      every proper subcontinuum    an arc, and 
 
 arc-components are leaves of a foliation $\F_{\fM}$, where local charts are ``matchboxes''. 

\medskip
 

 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.
 
  
 \medskip
 
 $\fM$     is a smooth foliated space with codimension zero and leaf   dimension $n$. 
 
   \medskip
  
  Similar concept to laminations, but not necessarily $1$ or $2$ dimensional, and not given as embedded subspace.
    \vfill   

}
  
 
\frame % 2
{
  \frametitle{}
 
   
 $\fM$ denotes a smooth matchbox manifold. 
 
  \bigskip
 
   
$\bullet$   Arise in study of tiling spaces, dynamics of foliations, graph constructions, inverse limit spaces, pseudogroup actions on Cantor  spaces, et cetera.

  \bigskip
 
   $\bullet$   Analogous to a compact manifold, with  unique quasi-isometry class of   Riemannian metrics on the leaves of $\F_{\fM}$.
  
    \bigskip
 
   $\bullet$ 
The holonomy pseudogroup $\cGF$ for $\fM$, acting on a complete (Cantor) transversal $\fX$, yields a well-defined dynamical system associated to $\fM$.


\bigskip
 

Study of matchbox manifolds has been active for fifty years at least in various contexts. The current research  program looks at the overall picture for this class of spaces.  In some ways, it  is analogous to study  of properties of embedding Riemannian manifolds $L \subset M$, though with much greater complexity. 




  \vfill   

}

 
\frame % 4
{
  \frametitle{Intrinsic properties}


 Study  $\fM$ in terms of its \emph{intrinsic}  and \emph{extrinsic} properties: 
 
 \bigskip
 
$\bullet$ Quasi-isometry class of leafwise metrics, yields ``large scale'' geometric invariant, such as growth properties of the leaves of $\F_{\fM}$ and foliation   entropy $0 \leq h(\cGF) \leq \infty$ as defined by Ghys-Langevin-Walczak.

\bigskip 
 

$\bullet$   [Clark \& H,  2011]  Action of $\cGF$ on $\fX$ equicontinuous implies  $\F_{\fM}$ is minimal, and $\fM$ is (foliated) homeomorphic to a generalized solenoid (described below).


\bigskip 
 

$\bullet$ If $\fM$ is the space of tilings associated to an \emph{aperiodic} tiling of $\mR^n$ with \emph{finite local complexity},
then the action of $\cGF$ on $\fX$ is expansive. 


\bigskip
 

 

$\bullet$ Growth rates of \v{C}ech cohomology groups for $\fM$, as studied by Gambaudo \& Martens for tiling spaces.

  
 \vfill   

}

 
\frame % 4
{
  \frametitle{Invariant measures}

  $\fX \subset \fM$ is  a compact transversal to $\F_{\fM}$ intersecting every leaf.
  
  \medskip
  
{\bf Definition:} Probability measure $\mu$   on $\fX$ is invariant if for each $\phi \in \cGF$ and compact Borel set $E \subset \text{Domain} (\phi)$, 
$\mu(E) = \mu(\phi(E))$. 

\bigskip

$\bullet$ Exists if $\cGF$ is equivalent to  suspension of amenable group action on Cantor set. 

\bigskip 

$\bullet$  Exists if there is a leaf with   subexponential volume growth,  such as defined by a locally free   $\mR^n$-action, or $G$-action of a simply connected nilpotent Lie group $G$.

 \bigskip 
 

{\bf Question:}  when does invariant measure $\mu$ for $\fM$ exist?  and if so, can you describe the space of all such, in terms of topological of dynamical properties of $\fM$?

 \vfill   

}



\frame % 4
{
  \frametitle{Extrinsic properties}


Consider $\fM$    embedded as a closed invariant set  of a smooth foliation  for a (possibly open) foliated manifold $M$.
The \emph{intrinsic pseudogroup dynamics}     reflected in   \emph{extrinsic properties of the embedding}.

\bigskip

$\bullet$    Invariant measure $\mu$ $\Rightarrow$   codimension-one embedding must be   Denjoy type.

\bigskip
$\bullet$  $f \colon \bf K \to \bf K$ a minimal action on Cantor set ${\bf K} \subset \mS^1$,  
 when does this action arise as the restriction of a $C^r$-diffeomorphism of $f \colon \mS^1 \to \mS^1$?

\medskip
 Denjoy Theorem implies $r < 2$, with restrictions on gap lengths  for $1 \leq r < 2$. 
   
\bigskip

{\bf Problem:}  Suppose given embedding of Cantor set ${\bf K} \subset \mR^k$  for $k \geq 2$, when does a minimal action of finitely-generated group on $\bf K$  extend to a $C^r$-map of an open neighborhood of $\bf K$, for $r \geq 1$?

 

 

 \vfill   

}

  
\section{Embeddings}

 \frame % 14
{
  \frametitle{Fixed-Point Property (FPP)}
 

 {\bf Definition:} 
The \emph{shape} of  $\fM \subset M$ is defined by a co-final descending chain $\{U_{\ell} \mid \ell \geq 1\}$ of open neighborhoods  (called a shape approximation to $\fM$)
$$M \supset  U_1 \supset U_2 \supset \cdots \supset U_{\ell} \supset \cdots \supset \fM \quad ; \quad \bigcap_{\ell = 1}^{\infty} ~ U_{\ell} = \fM$$
 
  
 \medskip
 
  {\bf Definition:} A \emph{foliated embedding} $\fM \subset M$  has the \emph{fixed-point property} if there is a shape approximation $\{U_{\ell} \mid \ell \geq 1\}$ to $\fM$
so that for each $\ell \geq 1$ there is a \emph{compact leaf} $L_{\ell} \subset U_{\ell}$ which is a homotopy retract. 

 
  \vfill

}
   
   \frame % 14
{
  \frametitle{}
 
 
\medskip
 The \emph{fixed-point property} is extrinsic to $\fM$, though may depend on the embedding. 
 
 \bigskip 
 
{\bf Remark:} If one shape approximation to $\fM \subset M$ has the FPP, then it holds for all.
 
 \bigskip
 
 {\bf Problem:} If $\fM$ has a $C^r$-foliated embedding $\fM \subset M$ with the FPP, must every $C^r$-foliated embedding have the FPP,  for $r \geq 0$?



 

\bigskip

Here is one application of the FPP:

\medskip

{\bf Lemma:}  If $\fM \subset M$  has the \emph{fixed-point property}, then there is invariant measure $\mu_{\cP}$ on $\fM$ defined by compact approximations  $\cP$.

  \vfill

}
   
  

 

\frame % 6
{
 \frametitle{McCord solenoids}
  
  Let $B_{\ell}$ be compact, orientable manifolds of dimension $n \geq 1$ for $\ell \geq 0$, with   orientation-preserving covering maps
  $$\cP ~ \equiv ~  \stackrel{p_{\ell +1}}{\longrightarrow} B_{\ell} \stackrel{p_{\ell}}{\longrightarrow} B_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  B_1 \stackrel{p_1}{\longrightarrow}  B_0 $$
 
  
   
The $p_{\ell}$ are the \emph{bonding maps} for the weak solenoid
$$\cS = \lim_{\leftarrow} ~ \{p_{\ell} \colon B_{\ell} \to B_{\ell -1}\} ~ \subset ~ \prod_{\ell =0}^{\infty} ~ B_{\ell} $$ 
 
  $\cS$ has natural structure of a matchbox manifold, with every leaf dense.
 
 \medskip
 
  
 
 $\cS$ is a \emph{McCord solenoid} if every finite composition of coverings is normal. 
 
 \medskip
 
 If the base $B_0 = \mT^n$ then every tower is a McCord solenoid.
 
  \vfill   

}

  
      
\frame % 17
{
  \frametitle{Smooth embeddings}
  
     
{\bf Theorem:}  
Let $\cS$ be a generalized solenoid   over the base space $\mT^n$, with presentation tower $\cP$. Let $r \geq 2$ and let $q \geq 2n$. Subject to technical restrictions on the growth rates of the degree so the coverings in the tower,     there exists a   $C^r$-foliation $\whF$ of $\mT^n \times \mD^q$ transverse to the disk fibers  such that:

\begin{enumerate}
\item  $\whF$  is a distal foliation, with    smooth transverse invariant volume form;\\
\item $L_0 = \mT^n \times \{ \vec0\}$ is a leaf of $\whF$, and  $\whF = \F_0$ near the boundary of $M$;\\
\item the solenoid $\cS$ embeds as  a minimal set $\whF$; \\
\item the embedding $\cS$ has the FPP, realizing the compact manifolds $\{B_{\ell} \mid \ell \geq 1\}$ as leaves $L_{\ell}$ which are the homotopy retracts of the approximations to $\cS$.
\end{enumerate}
 
 [Clark \& H, 2010] \emph{Embedding solenoids in foliations}, Topology \& its Apps, 2011.
\medskip

 
  {\bf Problem:} For  McCord solenoids, when  is smooth  embedding  with FPP possible?
  
    \vfill   

}

    
\section{Cohomology}


\frame % 17
{
  \frametitle{Self-linking invariants}
  
  Let $\vec{X}$ be non-vanishing vector field on Riemannian   $3$-manifold $M$.
  
  \medskip
  
  Defines foliation $\F$ of codimension $2$.
  
  \medskip
 
 Assume there is closed invariant set $\fM \subset M$. 
 \medskip
 
 Flow $\vp_t$ of $\vec{X}$ defines transverse invariant measure $\mu$ on $\fM$.
 
  
 
 \medskip
 
 Assume the normal bundle $Q = T\F^{\perp}$ restricted to open neighborhood $\fM \subset U \subset M$ is trivial: $TU \cong \langle \vec{X} \rangle \oplus Q$. 
 $\Pi \colon TU \to Q$ orthogonal projection.
 
 \medskip
 
 Define connection $\ds \nabla_{\vec{X}} (\vec{Y}) = \Pi \left( L_{\vec{X}} \vec{Y}\right)$ for section $\vec{Y} \colon U \to Q$. 
 
 \medskip
 
 Extend to  matrix-valued connection $1$-form $\omega \colon TU \to \frak{gl}(2, \mR)$.
 
   
 
    \vfill   

}

\frame % 17
{
  \frametitle{Ruelle invariant}
  
  
 Choose ``generic'' point $x \in \fM$, so that $\mu$ is defined by averaging over orbit of $x$.
 
 \medskip
 
 $\bullet$  Define  an extrinsic invariant of $\fM \subset M$ (and the choice of $\mu$ and framing of $Q$)
 $$ R_{\mu} = \lim_{n \to \infty} ~ \frac{1}{2n} \cdot \int_{-n}^n ~  Tr \{\omega\}(\dot{\vp}_t)  \,  dt ~ \in \mR$$
 
   $R_{\mu}$ measures the average infinitesimal rotation number of a unit section of the normal bundle to the flow, in the support of $\mu$. 
 
 \medskip
 

 If the support of $\mu$ is all of $M$, then $R_{\mu}$ equals the \emph{Ruelle} invariant. 
 
 \medskip
 
 More generally, $R_{\mu}$ is the \emph{first measured leaf class} for $\F$, as defined in  
 \medskip
 
[Hurder]  \emph{Global invariants for measured foliations}, Trans. AMS 1983.
 
     \vfill   

}


      
\frame % 17
{
  \frametitle{Non-commutative  self-linking invariants}
  

 {\bf Definition:} \emph{Chern polynomials} $\cI_q \equiv \mR[c_1, \ldots , c_q]$ where $\text{deg}(c_i) = 2i$. 
 
 \medskip
 
 
 {\bf Theorem:}  Let $\fM$  be  embedded as a minimal   set  of a $C^2$-foliation $\F$ of  a  foliated manifold $M$, with leaves of dimension $n$ and codimension $q \geq 2$. 
 
 \medskip
 Assume the  the  normal bundle  $Q = T\F^{\perp}$  to the leaves of $\F$ in $M$ is  trivial, and the embedding has the FPP with tower $\cP = \{L_{\ell} \mid \ell \geq 1\}$.
% Let    $\mu_{\cP}$ be the invariant measure on $\fM$ defined by the tower $\cP$. 
 \medskip
 
  Then for each homogeneous polynomial $C \in \cI_q$ with $2\ell= \text{deg}(C) \geq 2$,  
 there exists a well-defined  \emph{non-commutative  self-linking invariant},
 $\ds  \widehat{C}_{\cP} \in \mR$. 
 
  
 \medskip
 
$\bullet$   For the case of a flow with $n=1$, take $C = c_1$ then ${d(\cP)} = 1$ and $\widehat{C}_{\cP} = R_{\mu_{\cP}}$.

  
 \medskip
 
$\bullet$   $\widehat{C}_{\cP}$ measures the ``non-commutative'' rotation of the normal bundle along $\F_{\fM}$. 
 
    \vfill   

}

      
\frame % 17
{
  \frametitle{Non-triviality}


 {\bf Theorem:}
For each $n\geq 1$ and $q=2n$, and homogeneous polynomial $C \in \cI_q$ with $2\ell= \text{deg}(C) \geq 2$,  there exists a 
 a generalized solenoid  $\cS$ over  $B_0 = \mT^n$  with   tower $\cP$ which satisfies 
   the conditions in the embedding theorem above are satisfied for $r=2$, so that $\cS \subset \mT^n \times \mD^q$ is a $C^2$-embedding with the FPP with respect to $\cP$, and the   non-commutative  self-linking invariant 
 $ \widehat{C}_{\cP} \neq 0$. 

\bigskip

That is, we can find smooth embeddings of generalized solenoids over $\mT^n$ so that all of the higher-order linking numbers are non-trivial. 

 \bigskip
 \bigskip
  
  
    \vfill   

}
      

      
\frame % 17
{
  \frametitle{Application to classifying spaces}

Let $B\Gamma_q^r$ denote the \emph{Haefliger classifying space} of $C^r$ codimension-$q$ foliations.
\medskip

 Let $\fM$  be  embedded as a minimal   set  of a $C^r$-foliation $\F$ of  a  foliated manifold $M$, with leaves of dimension $n$ and codimension $q \geq 2$. 
 
 \medskip
 
 Let $B\F$ denote the classifying space of $\F$ on $M$, and $B\F_{\fM}$ its restriction to morphisms with source and target in $\fM$. Alternately, $B\F_{\fM}$ is the classifying space of the germ of $\F$ to an open neighborhood of $\fM$ in $M$.
 
 \medskip
 
 
  Then   there exists a well-defined ``classifying map'' $f_{\fM} \colon B\F_{\fM} \to B\Gamma_q^r$ defined by restricting the classifying map for $B\F$. 
  
  \bigskip
  
  {\bf Theorem:} For each $n \geq 1$, $q=2n$, let $\cS$ be a generalized solenoid which admits a $C^2$ embedding    $\fM \subset \mT^n \times \mD^q$  as above. 
  Then   $f_{\fM} \colon B\F_{\fM} \to B\Gamma_q^r$ is non-trivial in cohomology for all degrees $2\ell -1 > 2q$ for which the associated invariant $ \widehat{C}_{\cP} \neq 0$. 
  

  
  
    \vfill   

}
      
       
\frame % 17
{
  \frametitle{Ideas of proof}

 
   Bott and Heitsch observed that the cohomology of torsion flat bundles injects into the integral cohomology of the classifying space $B\Gamma_q$:
  \medskip
  
$\bullet$  [Bott \& Heitsch] \emph{A remark on the integral cohomology of $B\Gamma_q$}, Topology, 1972.
  
  \bigskip
  
 Cheeger and Simons observed that the Chern-Weil homomorphism for foliations and flat bundles can be used to calculate the torsion classes of the bundle:
   
  \medskip
  $\bullet$ [Cheeger \& Simons1985] \emph{Differential characters and geometric invariants}, {preprint}, Stony Brook University, 1972.  
     
  
 
  $\bullet$ [Simons] \emph{Characteristic forms and transgression: characters associated to a connection}, 
  {preprint}, Stony Brook University, 1972.
  
  \medskip
  
  In the last paragraph of this second preprint, Simons conjectures that the $\mR$-valued differential characters associated to the normal bundle of foliations is non-trivial  in all degrees above $2q$. We show this using embedded solenoids with the FPP.

    \vfill   

}
      


      
\frame % 17
{
  \frametitle{Key steps}

    
     
      $\bullet$   Normal bundle to each approximation $L_{\ell}$ is flat with finite order  $p$, so the Cheeger-Simons classes mod (p) inject into its classifying space.
 
    \medskip
    
      $\bullet$ Classifying spaces for the leaves $L_{\ell}$ are embedded continuously into that for $\F$.
 
    \medskip
    
      $\bullet$ Chern-Weil forms are defined on all of $M$, so vary continuously with the embedded cycles from the compact leaves.
 
  \medskip
  
      $\bullet$ Limit of the compact cycles is a higher dimensional closed current on the semi-simplicial model for the classifying space   $B\F_{\fM}$. Thus, the evaluation of the Chern-Weil forms on these asymptotic cycles is non-trivial.
 
 \medskip
 
      $\bullet$   Finally, one notes that  the classifying space of the germ of the embedding  $\fM \subset M$ carries the inverse limit of the approximating tower $\cP$ if the FPP condition holds, and that the Chern-Weil construction depends smoothly on data. 

  
    \vfill   

}
      
            
\frame % 17
{
  \frametitle{Conclusion}

    $\bullet$ Construction of smooth $C^r$-foliated embeddings of measured matchbox manifolds generates a large variety of normal invariants, for $r \geq 2$.    
    \medskip 
     
    
    $\bullet$
     Normal invariants can be interpreted universally as arising from classifying spaces,  revealing more about homotopy properties of the spaces $B\Gamma_q^r$.
    
            \medskip 
     
     
    
    $\bullet$ Normal invariants generalize invariants for classical Vietoris solenoids in $\mR^3$,  so may have physical interpretations.
    
     
    
    \bigskip
    \bigskip
    
    \begin{center}
    \emph{Thank you for your attention.}
   
   \medskip
   slides available at\\
    http://www.math.uic.edu/$\sim$hurder
    \end{center}
    
    
    \vfill   

}




     
 
\end{document}


  
\section{Proof}


      
\frame % 17
{
  \frametitle{Characteristic classes}
  
  Assume the  normal bundle to the leaves of $\F$ in $M$ is $Q = T\F^{\perp}$   is trivial. 
 
$\Pi \colon Q \cong M \times \mR^q \to \mR^q$ , smooth sections $Y \colon M \to \mR^q$. 
Define covariant derivative along leaves for $\vec{X}$ tangent field to leaves,
$$\nabla^b_{\vec{X}} (Y) = \Pi \left( L_{\vec{X}} Y \right)$$
 Superscript  ``b''  stands for either ``basic'' or ``Bott''. 

\medskip
Represented by $\frak{gl}(q)$ Maurer-Cartan $1$-form $\omega^b$ on $M$.

\medskip

$\nabla^B_{\vec{X}} (Y) = 0$ means that $Y$ is flow-invariant. 
For $q=2$, this means that $Y$ rotates in synchronization with $\vec{X}$. (very extrinsic property of $\fM$)

\medskip

$\nabla^t$ denotes product connection, represented by $\frak{gl}(q)$ Maurer-Cartan $1$-form $\omega^t$ on $M$.
$$\nabla^t_{\vec{X}} (Y) = \Pi \left( L_{\vec{X}} Y \right)$$


    \vfill   

}

  
      
\frame % 17
{
  \frametitle{}
  Form connection $1$-form on $M \times \mR$
   $$\omega_t = t \cdot \omega^b + (1-t) \cdot \omega^t$$
 $\Omega_t $ is curvature $2$-form of $\omega_t$ on $M \times \mR$
 
 \medskip
 $\sigma_i (\Omega_t) = Tr \left( \Omega_t^{\ell} \right) $ is $2i$-closed form on $M \times \mR$
 
 \medskip
 
 $\cI_q = \mR[\sigma_1(\Omega_t) , \ldots , \sigma_q(\Omega_t)]$ ring of ``Chern classes'' on $M \times \mR$
 
 \medskip
 For each $\sigma(\Omega_t) \in \cI_q$ of degree $2\ell$, define the transgression $(2q-1)$-form
 $$\widehat{\sigma} = \int_0^1 ~ \iota(\partial/\partial t) \cdot \sigma(\Omega_t) ~ dt$$
 
 For example, $\ell =1$ we have $ \sigma_1(\Omega_t) = Tr \left( \Omega_t \right) $ then $\widehat{\sigma_1}$ is $1$-form on $M$ that measures ``infinitesimal rotation'' of normal bundle.
  
    \vfill   

}
  
 
 
\frame
{
  \frametitle{Bott-Heitsch revisited}

 
  For the construction of $\cS$, we go back to the beginning:
  \medskip
  
    {\bf Theorem:} (Bott-Heitsch [1972])  
   \begin{center}
$h_Q^* \colon H^{*}(BO_q ; \mZ) \to H^{*}(M ; \mZ)$ is injective for all $*$.
 \end{center}

 \bigskip
 
We recall the proof for the case of oriented normal bundles and $q =2$. 
\medskip


 $H^*(BSO_2 ; \mZ) \cong \mZ[e]$
 
 \medskip
 
 Let $n > 2$, and set $\mZ_n = \mZ/n \mZ$. Embed $\mZ_n \subset SO_2$, acts isometrically on $\mR^2$
 
 \medskip
 
 $\mZ_n$ acts freely on $\mS^{2k+1}$ for $k > 0$.
 
  
 
 $$\mE_{n,k} = \mS^{2k} \times \mR^2 / \mZ_n ~, ~ \F_{n,k} = ~{\rm flat ~ bundle ~ foliation}$$
   \vfill
  
}
  
 \frame
{
  
 
For $* \leq 2k$ have injection:
 $$\mZ_n[e] \to H^*(BSO_2 ; \mZ_n) \to H^*(B\G_2 ; \mZ_n) \to H^*(\mE_{n,\ell} ; \mZ_n)$$
 So $H^*(BSO_2 ; \mZ_n) \to H^*(B\G_2 ; \mZ_n)$ is injective for all $*$ and all $n \to \infty$.
 \medskip
 
 $H^*(BSO_2 ; \mZ) \to H^*(B\G_2 ; \mZ)$   injective follows  from this. 
 
 \medskip
 
 
 General case for $q > 2$ uses splitting principle, for torsion subgroups of maximal torus, $\mZ_n^{k} \subset \mT^{k} \subset SO_{2k}$
 
 \bigskip
 
 
 {\bf Question:} Can we realize this limit process $(n, k) \to \infty$ with foliation?
  \vfill
  
}


 