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% talk for S�minaire GT3 in Strasbourg
% October 22, 2012
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\title[Measured LS category]{LS category of foliations and F\"{o}lner properties\footnote{ S\'{e}minaire GT3,  IRMA, Universit\'{e} de Strasbourg }\\
{\normalsize  Joint work with Carlos Meni\~no-Cot\'on}
}
\author{Steven Hurder}
\institute{University of Illinois at Chicago}
\date{October 22, 2012}
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\begin{document}

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\section{Complexity of foliations}


\begin{frame}{Foliation charts}


Let $M$ be a smooth manifold of dimension $n$.

\medskip
{\bf Definition:}  $M$   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 leaves
 \begin{center}
\includegraphics[width=0.35\textwidth]{strasbourg_pix/chart.pdf}
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$p$ is leaf dimension, $q = n-p$ is codimension.
 
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\begin{frame}{Average Euler characteristic}

A.~Phillips and D.~Sullivan

{\it Geometry of leaves}, [Topology, 1981]

\bigskip

Suppose $L$ is leaf of dimension $p=2$   and there is a sequence of connected submanifolds $K_{\ell} \subset L$ with  $length(\partial K_{\ell})/area(K_{\ell}) \longrightarrow 0$. 

 
 
 \bigskip
 
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\includegraphics[width=0.6\textwidth]{strasbourg_pix/JacobsLadder.png}
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\begin{frame}{}

The sets $\{K_{\ell}\}$ define an averaging sequence for $\F$. 

\medskip

Pass to a subsequence to obtain convergence, then define:
 

\begin{eqnarray*}
E_{\mu} & = &  {\rm  Average ~Euler}~  (\{K_{\ell} \mid \ell = 1,2, \ldots \})  \\
& = & \lim_{\ell \to \infty} ~ \frac{\chi(K_{\ell}) }{Area(K_{\ell})}\\
  & = & \lim_{\ell \to \infty} ~ \frac{1}{Area(K_{\ell})} \cdot \int_{K_{\ell}} ~ e(T\F) \\
  & = &  \langle C_\mu ,  e(T\F) \rangle 
\end{eqnarray*}
where $ e(T\F) $ is the closed Euler   2-form for $T\F \to M$, and
\medskip

$\bullet$~ $\mu$ is the invariant measure defined by the averaging sequence $\{K_{\ell}\}$

\medskip

$\bullet$~  $C_{\mu}$ is the Ruelle-Sullivan current on $2$-forms associated to it. 
 
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 It is easy to get zero for an answer:
 \bigskip
 
  \begin{center}
\includegraphics[width=0.7\textwidth]{strasbourg_pix/lochness.png}
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\bigskip
\pause

 T.~Januszkiewicz
 
 {\it Characteristic invariants of noncompact Riemannian manifolds}, [Topology, 1984].

\bigskip

The Phillips-Sullivan  ideas extend to   \emph{average Pontrjagin invariants} of a foliation.
  

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\end{frame}
 

\begin{frame}{Average leaf topology}

 The Phillips-Sullivan and Januszkiewicz results suggest: 
 
 \medskip
 
 {\bf Question:} How do you measure the ``average topology'' of the leaves of   foliations, not just their average characteristic invariants?
 
  \bigskip
  
  First, you need an invariant measure to average with.
  
  \medskip
  
  Second, you need a way to ``count'' the topology of a space, like a leaf. 
 \vfill
 
\end{frame}


\begin{frame}{Covering number}

 Here is a question that Reeb must have wondered at some point:
 
 \medskip
 
 {\bf Question:} Let $\F$ be a foliation of closed manifold $M$ with     $p,q > 0$. What is the least number of foliation charts required to cover $M$?
 \medskip
 
 This can be asked for topological and smooth foliations. Let $Cov(M,\F)$ be the minimum number of foliation charts required.
 
 \bigskip
 \pause

  {\bf Theorem:} [Foulon, 1994] $Cov(M,\F) > 2$.
 
 \bigskip
 
 There does not seem to be much more known about  $Cov(\F)$, and its relation to the topology of $M$ or dynamics of $\F$. One obvious relation:
\bigskip

$\bullet$~  $Cov(M,\F) \geq cat(M) + 1$, the Lusternik-Schnirelmann category of $M$. 
 
 
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\end{frame}

 
\begin{frame}{Lusternik-Schnirelmann category}


L. Lusternik and L. Schnirelmann, 

{\it M\'{e}thodes topologiques dans les probl\`{e}mes variationnels},  [Hermann, Paris, 1934]. 

\bigskip

$X$ a connected topological space, $x_0 \in X$ a basepoint. 

\medskip

 $U \subset X$ is \emph{categorical} (in $X$) if there exists a homotopy $H_t \colon U \to X$ with $H_0 = Id$ and $H_1(U) = x_0$.
 
 \bigskip
 
 {\bf Definition:} $cat(X) \leq k$ if there is an open covering $\{U_0, U_1, \ldots , U_k\}$ of $X$ where each $U_i$ is an open set which is categorical in $X$. 

\bigskip
\pause


$\bullet$ ~ $cat(\mS^n) = 1$. 

\smallskip 

$\bullet$ ~ $cat(\mT^n) = n$.

\smallskip 

$\bullet$ ~ $cat(M^n) \leq n$.


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\end{frame}


\begin{frame}{Tangential Lusternik-Schnirelmann category}

 
H. Colman and E. Macias-Virg\'{o}s

{\it Tangential Lusternik-Schnirelmann category of foliations}, [Journal L.M.S.,  2002]

\medskip
 $H_t$ is \emph{foliated homotopy} map if the curves $t \mapsto H_t(x)$ remains in the leaf $L_x$ containing $x$, for all $x \in M$. In particular, $H_t(L_x) \subset L_x$ for all $x \in M$, $0 \leq t \leq 1$. 

\medskip
 


 $U \subset M$ is \emph{$\F$-categorical}   if there exists a  foliated  homotopy $H_t \colon U \to M$ with $H_0 = Id$ and $H_1(L'_y| U) = x_y$ where $L'_y$ is the \underline{connected} leaf   of the restricted foliation $\F| U$ which contains $y \in U$.
 
   
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\includegraphics[width=0.7\textwidth]{strasbourg_pix/categorical_set.png}
\end{center}

  
 
  
 
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\end{frame}


\begin{frame}{$cat_{\F}(M)$ calculations}

{\bf Definition:} $cat_{\F}(M) \leq k$ if there is an open covering $\{U_0, U_1, \ldots , U_k\}$ of $M$ where each $U_i$ is an open set which is $\F$-categorical. 

\bigskip

  Calculating  $cat_{\F}(M)$ is surprisingly subtle.  A basic tool remains the observation:
  
  
  \medskip
  
{\bf Remark:}~  $cat_{\F}(M) \geq cat(L_x)$ for all leaves $L_x$  of $\F$. 

\bigskip

 $\bullet$ ~  $cat_{\F}(\mT^2) = 1$  where  $\F$  is the Reeb foliation of $\mT^2$. 

\smallskip

 $\bullet$ ~  $cat_{\F}(\mS^3) = 2$  where  $\F$  is the Reeb foliation of $\mS^3$. 

\smallskip

 $\bullet$ ~  $cat_{\F}(M) = p$ where  $\F$  is a linear foliation of $\mT^n$ with   dense leaves. 
 
 \bigskip
 
 
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\end{frame}


\begin{frame}{Relation to topology}


W.~Singhof and E.~Vogt

{\it Tangential category of foliations}, [Topology, 2003]


\medskip

{\bf Theorem:}  ~  $\F$ is $C^2$-foliation implies that $cat_{\F}(M) \leq p$. 
 
 
\medskip

{\bf Corollary:}  ~  $\F$ is $C^2$,       $L \subset M$ with $cat(L) =p$, then  $cat_{\F}(M) = p$. 

\medskip

{\bf Corollary:}   $\F$ is $C^2$, $q=1$, $p \geq 2$,   $cat_{\F}(M) = 1$ ~ $\Rightarrow$ ~ $\F$ is foliation by spheres.
 
 \bigskip
 \pause

There has been recent work relating it to more standard notions of topology. 
  
  \medskip
  
 J.-P.~Doeraene,  E.~Macias-Virg\'{o}s, D.~Tanr\'{e}, 
{\it Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations}, [Top. Apps., 2010].


 \bigskip
 
{\bf Remark:}  $cat_{\F}(M)$ does not depend (much) on the dynamical properties of $\F$.   
 
 

 \bigskip
 
  
 
 
 
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\end{frame}

 
 
\begin{frame}{Foliated cohomology}

 H.~Colman and S.~Hurder, {\it  Tangential LS category and cohomology for foliations}, [Contemp. Math. Vol. 316,   2002]. 
 
 \medskip
 
  $\Omega^r(\F)$   the space of smooth $r$-forms along the leaves. 
  \medskip
 
$\ds d_{\F} \colon \Omega^r(\F) \to \Omega^{r+1}(\F)$ leafwise differential.

\medskip

The  {\it foliated cohomology}  $H_{\F}^r(M)$ is the cohomology of the complex $\ds \left( \Omega^r(\F), d_{\F} \right)$. 

\medskip

Product of forms yields product of foliated cohomology 
$$\wedge \colon H_{\F}^r(M) \otimes H_{\F}^s(M)  \longrightarrow H_{\F}^{r+s}(M)$$
 
 \pause


 {\bf Theorem:} $cat_{\F}(M) \geq 1 + {\rm nil} \;    H_{\F}^+(M) .$
 
 \bigskip
 
  {\bf Theorem:}   $\F$   a $C^2$-foliation:  $0 \ne GV(\F) \in H^{2q+1}(M; \mR)  ~ \Rightarrow~ cat_{\F}(M) \geq q+2$.
 
 \bigskip
 
\emph{Leafwise cohomology   yields     tangential categorical invariants}. 
 
 
 \vfill
 
\end{frame}

 
 
  
\begin{frame}{Holonomy pseudogroup of $\F$}

   
 
{\bf Idea:}     Use the tangential LS category   to define an ``average  topology of leaves''.  
Need some preliminary notions.
\pause

\bigskip
\bigskip

 Let $\cT = \cT_1 \cup \cdots \cup \cT_k$ be a complete transversal for $\F$ associated to an open covering of $M$ by foliation charts 
 $\{ \vp_i \colon U_i \to (-1,1)^p \times (-1,1)^q \mid 1 \leq i \leq k \}$.
 
 \medskip
 
Let $\gamma \colon [0,1] \to M$ be a leafwise path with $\gamma(0) = x \in \cT$ and $\gamma(1) = y \in \cT$. 

\medskip

Let $h_{\gamma} \colon U_x \to V_y$ denote the holonomy map defined by $\gamma$, where $U_x , V_y \subset \cT$.

\medskip

The   holonomy maps   between points in $\cT$ defines pseudogroup $\cGF$ acting on $\cT$.  

 

 \vfill
 
\end{frame}


\begin{frame}{Transverse invariant measures}

  
A Borel   measure $\mu$ on $\cT$ is \emph{holonomy invariant} if $\mu(E) = \mu (h_{\gamma}(E))$ for every leafwise path $\gamma$ and Borel set $E \subset {\rm domain} (h_{\gamma})$.

\bigskip


{\bf Remark:} A holonomy invariant Borel   measure $\mu$ extends to a measure on Borel transversals $f \colon X \to M$.  
   $\mu$ is \emph{probability} measure means that $\mu(\cT) = 1$.

\bigskip

{\bf Theorem:} [Plante 1975] An averaging sequence  $\{K_{\ell} \}$ determines a Borel probability measure $\mu$ on $\cT$ which is holonomy invariant.  

\bigskip

{\bf Definition:} For $X$ compact topological space,   a Borel map $f \colon X \to M$  is a transversal for $\F$ if the intersection $f(X) \cap L$ is discrete for all leaves $L$.

  \vfill
 
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\begin{frame}{Measured LS category}

Carlos Meni\~no-Cot\'on:

{\it LS category, foliated spaces and transverse invariant measure}, [Thesis,  2012].

\bigskip

Suppose the foliation $\F$ admits a transverse invariant measure $\mu$. 
\medskip

{\bf Question:} Is there a way to average the leafwise     LS category?

\medskip

Can the idea of LS category be used to define numerical invariants of $(\F, \mu)$? 
What would they measure?

\pause

 
\bigskip

 Let $U \subset M$ be categorical set and $H_t \colon U \to M$ be foliated homotopy. 
 
 \medskip
 
 $\Rightarrow$ ~ $T_U = H_1(U)$ intersects each leaf in a discrete set of points, 
 
 \medskip
 
  $\Rightarrow$ ~  $\mu(T_U) < \infty$.

 

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\begin{frame}{}

Let $\mu$ be a holonomy invariant,  Borel probability measure $\mu$ on $\cT$.

\medskip
{\bf Definition:} ($\mu$-category of $\F$)
$$cat_{\F, \mu}(M) = \inf \left\{\sum_{i=0}^k ~ \mu(T_{U_i}) \mid  \{ U_0 , U_1 , \ldots , U_k \}~ {\rm is ~ an} ~ \F ~ {\rm categorical ~ cover}   \right\}$$

 

$\bullet$ ~ $cat_{\F, \mu}(M)$ measures how ``efficiently'' the space $M$ can be decomposed and squeezed into transversals.

\medskip
\pause

$\bullet$ ~ If all leaves of $\F$ are compact and bounded, then $cat_{\F, \mu}(M) > 0$ depends on the orbifold quotient $M/\F$ and LS-category of fibers.

\medskip
 

$\bullet$ ~ If all leaves of $\F$ are dense, then $cat_{\F, \mu}(M) = 0$. 

\medskip
\pause

This last result seems surprising, but follows because there is no ``price to pay'' for moving parts of leaves long distances.
The ``price'' should be a measure of how far a point in $U_i$ has to travel to the basepoint $x_0$. 

\medskip  

$\bullet$ ~The highest ``prices'' should be along the boundary $\partial U_i$  of each $U_i$. 

 \vfill
 
 
 
 
\end{frame}

 
 
  
  
\begin{frame}{Isoperimetric measured LS category}

  
 {\bf Definition:} (Iso-$\mu$-category of $\F$)
$$cat^{\partial}_{\F, \mu}(M) = \inf \left\{\sum_{i=0}^k ~ \mu(\partial U_i) \mid  \{ U_0 , U_1 , \ldots , U_k \}~ {\rm is ~ an} ~ \F ~ {\rm categorical ~ cover}   \right\}$$

 

The notion of $\mu(\partial U_i)$ requires some explanation. Cover $U_i$ by flow charts, then take the measure of the transversals corresponding to the boundary plaques of $U_i$. 

\medskip

Each $U_i$ defines a Borel subset $E_i \subset \cT$ for the plaques in $U_i$. 
Let $E_i^{\partial}$ denote the transversal points for the boundary plaques.  Then $\mu(\partial U_i) = \mu(E_i^{\partial})$.


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\includegraphics[width=0.6\textwidth]{strasbourg_pix/boundary.png}
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\begin{frame}{}

\medskip

  $cat^{\partial}_{\F, \mu}(M)$ is well-defined, for:
  \medskip
  
  $\bullet$~ fixed covering of $M$ by foliation charts
  
   \smallskip
  
  $\bullet$~ fixed transverse invariant measure $\mu$. 
\bigskip


Finally, take the infimum over all open coverings of $M$ by foliation charts.
 
 \bigskip
 \pause


 {\bf Remarks:}
\medskip

$\bullet$~  $cat^{\partial}_{\F, \mu}(M)$ is always finite: take finite cover of $M$ by   foliation flow boxes.

\medskip

$\bullet$~ Suppose that $L = \mS^p$ is compact leaf supporting $\mu$. Then $cat^{\partial}_{\F, \mu}(M) = 0$.

\medskip  


$\bullet$~ Suppose that $L = \mT^p$ is compact leaf supporting $\mu$. Then $cat^{\partial}_{\F, \mu}(M) > 0$.

\medskip  


$\bullet$~  $cat^{\partial}_{\F, \mu}(\fM)$ defined for foliated spaces $\fM$ and \emph{matchbox manifolds} (transversally Cantor foliated spaces).
  

   \vfill
 
\end{frame}


 
 
  
 
\begin{frame}{Suspensions}

 Let $\G = \pi_1(B, b_0)$ be fundamental group of closed manifold $B$, and $\phi \colon \G \times X \to X$ an action on compact space $X$ preserving a Borel probability measure $\mu$ on $X$.
 The \emph{suspension} of $\phi$ is the foliated space 
 $$\fM_{\phi} = \widetilde{B} \times X / (b, x) \sim (b \cdot \gamma^{-1} , \phi(\gamma)(x)$$ 
 
 \smallskip
 \pause

 {\bf Problem:} How is $cat^{\partial}_{\F, \mu}(\fM)$ related to:
 
 \smallskip
 
 $\bullet$~  $cat(B)$?
 
 \smallskip
 
 $\bullet$~  properties of $\G$? 
 
 \smallskip
 
 $\bullet$~  dynamics of $\phi$?
 
  \bigskip
 Suppose that $\fM_{\phi}$ is foliated space defined by suspension of minimal action   $\phi \colon \G \times X \to X$ on Cantor set $X$, preserving measure $\mu$.

   
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\end{frame}


\begin{frame}{ }

  
 
For free, minimal Cantor actions by $\G = \mZ^p$, all such actions are \emph{topologically hyperfinite}, 
or \emph{affable}, a topological form of the F\"{o}lner condition. 

\medskip

{\bf Theorem:} [Forrest, 1999]  Let  $\phi \colon  \mZ^p \times X \to X$ be a free, minimal action, then $\phi$  is \emph{topologically F\"{o}lner} on a subset $X_0 \subset X$, whose complement $Z = X - X_0$ has measure zero for any transverse invariant measure $\mu$ on $X$. 

\medskip

{\bf Corollary:} Let $\fM_{\phi}$ be the suspension for $B = \mT^p$ of a  minimal action of $\mZ^p$ on a Cantor set $X$, preserving a measure $\mu$. Then $cat^{\partial}_{\F, \mu}(\fM) = 0$.
 
  \pause

  \bigskip
 In the case where the leaves of $\F$ are contractible, $cat^{\partial}_{\F, \mu}(\fM)$ is a form of average \emph{Cheeger isoperimetric constant} $\lambda(\G) $  for $\G$. 
% Suppose that $\G$ is a finitely generated group with isoperimetric constant $\lambda(\G) > 0$. 
 
 \bigskip
 \pause

{\bf Theorem:}  Let $\fM_{\phi}$  be   the suspension of a  free, minimal action $\phi \colon \G \times X \to X$ which preserves a probability measure $\mu$. 
Then $cat^{\partial}_{\F, \mu}(M) \geq \lambda(\G)$.
 

 \vfill
 
\end{frame}


\begin{frame}{Average Euler class, revisited}

Consider again the example of ``Jacob's Ladder''

 \begin{center}
\includegraphics[width=.6 \textwidth]{strasbourg_pix/JacobsLadder.png}
\end{center}

\medskip

A decomposition of this leaf into categorical sets for  a foliation $\F$ must have ``very long edges'', which have uniform fraction of total mass.

\medskip

Let $U \subset M$ be $\F$-categorical set, and $\cP \subset U$ a leaf for $\F | U$. Then the Euler form $e(T\F) | \cP = d_{\F}(T(e|\cP))$ for a \emph{uniformly bounded} $1$-form on $U$.

$$\int_{\cP} ~ e(T\F) = \int_{\cP} ~ d_{\F}(T(e|\cP)) = \int_{\partial \cP}~ T(e|\cP) \leq C_e \cdot length (\partial \cP)$$
 \vfill
 
 
 
 
 
\end{frame}

 
\begin{frame}{Semi-locality}

DeRham cohomology invariants of bundles $\cE \to M$ have a semi-local property:

\medskip

{\bf Proposition:} [Folklore] Let $L$ have bounded geometry and let $U \subset L$ be a ``nice'' categorical set. Then for any geometric class $P(\cE)$ for $\cE$ of degree $p$, formed from products of  the Euler, Chern and Pontrjagin classes,    there exists a bounded $(p-1)$-form $TP(\cE)$ on $U$ such that $d_{\F}TP(\cE) = P(\cE)$, where the bound depends on the geometry of $M$ and $\cE$ but not on $U$. 


\bigskip
\pause

{\bf Theorem:} Let $(M, \F)$ have invariant measure $\mu$, and let $P(T\F)$ be a geometric form of degree $p$.
If $\langle C_{\mu} , P(\cE) \rangle \ne 0$ then  $cat^{\partial}_{\F, \mu}(M) > 0$.

 \bigskip
 
 {\bf Corollary:}  If $\F$ contains a leaf $L$ which is quasi-isometric to the Jacobs Ladder, and $\mu$ is a transverse invariant measured associated to an averaging sequence defined by $L$, then  $cat^{\partial}_{\F, \mu}(M) > 0$.
 
  
 \vfill
 
\end{frame}

 
\begin{frame}{Average topology of leaves}

 The invariant $cat^{\partial}_{\F, \mu}(M) > 0$ depends on:
 
 \medskip
 
 $\bullet$ ~ The isoperimetric constant of  leaves of $\F$.
 
 \medskip
 
 $\bullet$ ~ The topology of     leaves in the support of $\mu$.
 
  
  \bigskip
 \pause


 {\bf Remark:} $cat^{\partial}_{\F, \mu}(M)$ is analogous to the invariants that   Robert Brooks studied:
 
 \medskip
 
 
 
 {\it The spectral geometry of foliations}, [Amer. Journal Math., 1984]

\bigskip
 \pause


 {\bf Problem:} Relate $cat^{\partial}_{\F, \mu}(M)$   to the spectrum of leafwise elliptic operators for $\F$.
 


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Thank you for your attention!
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