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\title{Differential forms and the Hölder homeomorphism problem, after Gromov and Rumin}
\author{P. Pansu}
\date{February 1st, 2012}

\newtheorem{thm}{Theorem}
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\newtheorem{prop}[lem]{Proposition}
\newtheorem{cor}[lem]{Corollary}
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\newtheorem{nota}[lem]{Notation}
\newtheorem{ques}[lem]{Question}
\newtheorem{prob}[lem]{Problem}
\newtheorem{sol}[lem]{Solution}
\newtheorem{conc}[lem]{Conclusion}
\newtheorem{ex}[lem]{Example}
\newtheorem{exs}[lem]{Examples}
\newtheorem{res}[lem]{Results}
\newtheorem{plan}[lem]{Plan of lecture}
\newtheorem{scheme}[lem]{Scheme of proof}

\def\proof{\par\medskip\noindent {\bf Proof}}
\def\qed{~q.e.d.}

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\begin{document}


\begin{frame}
\titlepage
\end{frame}

\section{Gromov's H\"older homeomorphism problem}

\begin{frame}

\begin{ques}
\textcolor{red}{\em (Gromov 1993)}. Let $M$ be an $n$-dimensional sub-Riemannian manifold. For which $\alpha\in(0,1)$ does there exist locally a homeomorphism $\R^n \to M$ which is $C^{\alpha}$-H\"older continuous ?
\end{ques}\pause

\begin{defi}
Let $\alpha(M)=\sup\{\alpha\in(0,1)\,|\,\exists$ locally a homeomorphism $\R^n \to M\}$.
\end{defi}\pause

\begin{ex}
If $G$ is a $r$-step Carnot group, the exponential map $\mathfrak{g}=Lie(G)\to G$ is locally $C^{1/r}$-H\"older continuous. Thus $\alpha(M)\geq 1/r$.
\end{ex}\pause

\begin{thm}[Gromov 1993]
Let $M$ be sub-Riemannian, with Hausdorff dimension $Q$. Then $\alpha(M)\leq\frac{n-1}{Q-1}$.

Let $M$ be a $2m+1$-dimensional contact manifold. Then $\alpha(M)\leq\frac{m+1}{m+2}$ ($\leq\frac{2m}{2m+1}$).
\end{thm}

\end{frame}

\begin{frame}

Gromov's proof uses Hausdorff dimension of subset of given topological dimension: if all subsets of topological dimension $k$ have Hausdorff dimension $\geq k'$, then $\alpha(M)\leq\frac{k}{k'}$.\pause

\medskip

To get lower bounds on Hausdorff dimension of subsets, Gromov constructs local foliations by horizontal submanifolds. If there are enough such dimension $k$ foliations, all subsets of topological dimension $n-k$ have Hausdorff dimension $\geq Q-k$, therefore $\alpha(M)\leq\frac{n-k}{Q-k}$.\pause

\medskip

Constructing horizontal curves amounts to solving a system of PDE's. If $k=1$, it is an ODE, the method applies to all (equiregular) sub-Riemannian manifolds. Gromov solves the relevant PDE for contact $2m+1$-manifolds and $k=m$, and, more generally, for generic $h$-dimensional distributions, and $k$ such that $h-k\geq(n-h)k$. \pause

\medskip

Today, I describe an alternative method, due again to Gromov, but based on Rumin's theory of differential forms on sub-Riemannian manifolds. A motivation to further study this theory in this seminar.

\end{frame}

\section{Cochains}
\subsection{Definition}

\begin{frame}

\begin{defi}
On a metric space $X$, a (straight) {\em $q$-cochain of size} $\eps$ is a function $c$ on $q+1$-uples of diameter $\leq \eps$. Its $\eps$-absolute value is
\begin{eqnarray*}
|c|_{\eps}=\sup\{c(\Delta)\,;\,diam(\Delta)\leq \eps\}.
\end{eqnarray*} 
\end{defi}
In other words, straight cochains of size $\eps$ coincide with simplicial cochains on the simplicial complex whose vertices are points of $X$ and a $q$-face joins $q+1$ vertices as soon as all pairwise distances are $\leq \eps$. Therefore, they form a complex $\mathcal{C}_{\eps}^{.}$. There is a dual complex of chains $\mathcal{C}_{.,\eps}$. 

\begin{lemma}
Assume $X$ is a manifold with boundary, or bi-H\"older homeomorphic to such, then the inductive limit complex $\underrightarrow{\lim}$ $\mathcal{C}_{\eps}^{\cdot}$ computes cohomology. 
\end{lemma}

\begin{defi}
Given a cohomology class $\kappa$ and a number $\nu>0$, one can define the \emph{$\nu$-norm}
\begin{eqnarray*}
\n{\kappa}_{\nu}=\liminf_{\epsilon\to 0}\epsilon^{-\nu}\inf\{|c|_{\epsilon}\,|\,\textrm{ cochains } c \textrm{ of size } \epsilon \textrm{ representing } \kappa\}.
\end{eqnarray*}
\end{defi}

\end{frame}

\subsection{Metric weights}

\begin{frame}

\begin{defi}
Let $X$ be a metric space, let $q\in\mathbb{N}$. Define the {\em metric weight} $MW_q (X)$ as the supremum of numbers $\nu$ such that there exist arbitrarily small open sets $U\subset M$ and nonzero straight cohomology classes $\kappa\in H^q (U,\R)$ with finite $\nu$-norm $\n{\kappa}_{\nu}<+\infty$.
\end{defi}

\begin{prop}
In a Riemannian manifold with boundary, all straight cocycles $c$ representing a nonzero class $\kappa$ of degree $q$ satisfy
$|c|_{\epsilon}\geq \textrm{const.}(\kappa)\,\epsilon^q$.
In other words, $\n{\kappa}_{q}>0$.
\end{prop}

\proof.
Fix a cycle $c'$ such that $\kappa(c')>0$. Subdivide it as follows : fill simplices with geodesic singular simplices, subdivide them and keep only their vertices. This does not change the homology class. The number of simplices of size $\epsilon$ thus generated is $\leq \textrm{const.}(c')\,\epsilon^{-q}$. For any representative $c$ of size $\epsilon$ of $\kappa$,
\begin{eqnarray*}
\kappa(c')=c(c')\leq \textrm{const.}\,\epsilon^{-q} |c|_{\epsilon}.\qed
\end{eqnarray*}

\begin{cor}
Euclidean $n$-space has $MW_q \leq q$ for all $q=1,\ldots,n-1$.
\end{cor}

\end{frame}

\subsection{H\"older covariance}

\begin{frame}

\begin{prop}
\label{holderweight}
Let $f:X\to Y$ be a $C^{\alpha}$-H\"older continuous homeomorphism. Let $\kappa\in H^q (Y,\R)$. Then
\begin{eqnarray*}
\n{\kappa}_{\nu}<+\infty\Rightarrow\n{f^{*}\kappa}_{\nu\alpha}<+\infty.
\end{eqnarray*}
In particular, $MW_q (X)\geq \alpha MW_q (Y)$.
\end{prop}

\proof.
If $\sigma$ is a straight simplex of size $\epsilon$ in $X$, $f(\sigma)$ has size $\epsilon' \leq \n{f}_{C^{\alpha}}\,\epsilon^{\alpha}$ in $Y$. If $c$ is a representative of $\kappa$, $f^* c$ is a representative of  $f^{*}\kappa$, and
\begin{eqnarray*}
\epsilon'^{-\nu}|c|_{\epsilon'}
&\geq&\epsilon'^{-\nu}|c(f(\sigma))|\\
&=&\epsilon'^{-\nu}|f^* c(\sigma)|\\
&\geq&\n{f}_{C^{\alpha}}^{-\nu}\,\epsilon^{-\nu\alpha}|f^* c(\sigma)|.
\end{eqnarray*}
Therefore
\begin{eqnarray*}
\epsilon^{-\nu\alpha}|f^* c|_{\epsilon}
\leq \n{f}_{C^{\alpha}}^{\nu}\,\epsilon'^{-\nu}|c|_{\epsilon'}.
\end{eqnarray*}
This leads to
\begin{eqnarray*}
\n{f^* \kappa}_{\nu\alpha}\leq \n{f}_{C^{\alpha}}^{\nu}\,\n{\kappa}_{\nu}.\qed
\end{eqnarray*}

\end{frame}

\section{Differential forms}
\subsection{Weights of differential forms}

\begin{frame}

Let $G$ be a Carnot group with Lie algebra $\mathfrak{g}$. Left-invariant differential forms on $G$ split into homogeneous components under the dilations $\delta_\epsilon$,
\begin{eqnarray*}
\Lambda^* \mathfrak{g}^* =\bigoplus_{w}\Lambda^{*,w} \quad \textrm{where}\quad \Lambda^{*,w} =\{\alpha\,|\,\delta_{\epsilon}^{*}\alpha=\epsilon^{w}\alpha\}.
\end{eqnarray*}
Therefore Lie algebra cohomology splits $H^{q}(\mathfrak{g})=\bigoplus_{w}H^{q,w}(\mathfrak{g})$.

\begin{ex}
If $G=Heis^{2m+1}$ is the Heisenberg group, for each degree $q\not=0$, $2m+1$,
\begin{eqnarray*}
\Lambda^q \mathcal{G}^* =\Lambda^{q,q}\oplus\Lambda^{q,q+1},
\end{eqnarray*}
where $\Lambda^{q,q}=\Lambda^{q}(V^1 )^*$ and $\Lambda^{q,q+1}=\Lambda^{q-1}(V^1 )^* \otimes (V^2 )^*$.
\end{ex}

This gradation by weight depends on the group structure. What remains for general sub-Riemannian manifolds is a filtration.

\begin{defi}
Let $(M,\Delta)$ be a sub-Riemannian manifold, $m\in M$. Say a $q$-form $\alpha$ on $T_m M$ has \emph{weight} $\geq w$ if it vanishes on $q$-vectors of $\Delta^{i_1}\otimes\cdots \otimes \Delta^{i_q}$ whenever $i_1 + \cdots +i_q <w$. If $(M,\Delta)$ is equiregular, such forms constitute a subbundle $\Lambda^{q,\geq w}T^* M$. The space of its smooth sections is \emph{denoted by} $\Omega^{*,\geq w}$.
\end{defi}
Note that each $\Omega^{*,\geq w}$ is a differential ideal in $\Omega^*$. 

\end{frame}

\subsection{Algebraic versus metric weights}

\begin{frame}

\begin{prop}
\label{uppernorm}
Let $M$ be an equiregular sub-Riemannian manifold. Let $U\subset M$ be a bounded open set with smooth boundary. Let $\omega$ be a closed differential form on $U$ of weight $\geq w$. Then, for every $\epsilon$ small enough, the cohomology class $\kappa\in H^q (U,\R)$ of $\omega$ can be represented by a straight cocycle $c_{\epsilon}$ (maybe defined on a slightly smaller homotopy equivalent open set) such that $|c_{\epsilon}|_{\epsilon}\leq \textrm{const.}\,\epsilon^w $. In other words, $\n{\kappa}_{w}<+\infty$.
\end{prop}

\proof.
In the case of a Carnot group $G$. Use the exponential map to push affine simplices in the Lie algebra to the group. Fill in all straight simplices in $G$ of unit Carnot-Carath\'eodory size with such affine singular simplices. Apply $\delta_{\epsilon}$ and obtain a filling $\sigma_{\epsilon}$ for each straight simplex $\sigma$ in $G$ of Carnot-Carath\'eodory size $\epsilon$. Define a straight cochain $c_{\epsilon}$ of size $\epsilon$ on $U$ by
\begin{eqnarray*}
c_{\epsilon}(\sigma)=\int_{\sigma_{\epsilon}}\omega.
\end{eqnarray*}
Since $\omega$ is closed, Stokes theorem shows that $c_{\epsilon}$ is a cocycle. Its cohomology class in $H^q (U',\R)\simeq H^q (U,\R)$ is the same as $\omega$'s. Furthermore,
\begin{eqnarray*}
|c_{\epsilon}(\sigma)|
=\int_{\sigma_1}\delta_{\epsilon}^{*}\omega
\leq V\,\n{\delta_{\epsilon}^{*}\omega}_{\infty}
\leq\textrm{const.}(\omega)\,\epsilon^{w}.\qed
\end{eqnarray*}

\end{frame}

\subsection{Algebraic weights}

\begin{frame}

\begin{defi}
Let $M$ be a sub-Riemannian manifold. Define the \emph{algebraic weight} $AW_q (M)$ as the largest $w$ such that there exists arbitrarily small open sets with smooth boundary $U\subset M$ and nonzero classes in $H^q (U,\R)$ which can be represented by closed differential forms of weight $\geq w$.
\end{defi}

\begin{rem}
Equiregular sub-Riemannian manifolds satisfy $MW_q \geq AW_q$. 
\end{rem}

\begin{cor}
Let $M$ be a sub-Riemannian manifold. Then for all $q=1,\ldots,n-1$, $\alpha(M)\leq\frac{q}{AW_q}$.
\end{cor}

So our goal now is to show that for certain sub-Riemannian manifolds, for certain degrees $q$, in every open set, every closed differential $q$-form is cohomologous to a form of high weight.

\end{frame}

\section{Estimates on algebraic weights}
\subsection{Codimension 1}

\begin{frame}

As a warm up, let us treat codimension 1 forms.

\begin{prop}[Gromov]
Let $(M,\Delta)$ be a sub-Riemannian manifold of Hausdorff dimension $Q$. Every closed $n-1$-form is cohomologous to a form of weight $\geq Q-1$. Thus $AW_{n-1}(M)=Q-1$.
\end{prop}

\proof.
The filtration of $\Delta\subset \Delta^{2}\subset \cdots\subset TM$ induces a filtration $A^{n-1}=\mathcal{F}^{1}\subset \mathcal{F}^{2}\subset \cdots\subset \Omega^{n-1}$ as follows : $\alpha\in\mathcal{F}^j$ if and only if there exist an $n$-form $\omega$ and a vectorfield $Z\subset \Delta^j$ such that $\alpha=\iota_Z \omega$. We show that for all $j\geq 1$, $\mathcal{F}^{j+1} \subset \mathcal{F}^{j}+im(d)$. 

Let $\omega$ be an $n$-form and $X$, $Y$ be vectorfields such that $X\in \Delta$, $Y\in \Delta^{j}$. Then, using Lie derivatives,
$\mathcal{L}_{X}(\iota_{Y}\omega)
=\iota_{\mathcal{L}_{X}(Y)} \omega+\iota_{Y}(\mathcal{L}_{X}\omega)\iota_{[X,Y]}\omega \textrm{ mod }\mathcal{F}^j $.
Thanks to Cartan's formula,
$\mathcal{L}_{X}(\iota_{Y}\omega)
= d(\iota_{X}\iota_{Y}\omega)+\iota_{X}d(\iota_{Y}\omega)\in im(d)+\mathcal{F}^j $.
Therefore $\iota_{[X,Y]}\omega\in im(d)+\mathcal{F}^j$.

Let $\alpha\in \mathcal{F}^{j+1}$, $\alpha=\iota_Z \omega$ with $Z\in \Delta^{j+1}$. Write $Z=\sum_\ell a_\ell [X_\ell ,Y_\ell ]$ where $a_\ell$ are functions, $X_\ell$, $Y_\ell$ are vectorfields, $X_\ell \in \Delta$, $Y_\ell \in \Delta^{j}$. Then $\alpha=\sum_\ell \iota_{[X_\ell ,Y_\ell ]}\omega_\ell$ (where $\omega_\ell =a_\ell \omega$), therefore $\alpha\in im(d)+\mathcal{F}^j$. This shows that $\mathcal{F}^{j+1} \subset \mathcal{F}^{j}+im(d)$. 

The bracket generating assumption, $\Delta^r =TM$, implies that $\Omega^{n-1}=\mathcal{F}^{r}\subset im(d)+\mathcal{F}^{1}=im(d)+A^{n-1}$.
Given a closed $n-1$-form $\phi$, the equation 
\begin{eqnarray*}
d\psi=-\phi \quad\textrm{mod}\quad A^{n-1}
\end{eqnarray*}
admits a smooth solution $\beta\in \Omega^{n-2}$. Then $\phi+d\psi\in A^{n-1}$ is a horizontal form.\qed

\end{frame}

\subsection{Rumin's complex}

\begin{frame}

Rumin's complex is a subcomplex of the de Rham complex, homotopic to it, consisting of differential forms of preferably high weights. The construction requires to invert the weight $0$ component $d_0$ of $d$. $d_0$ identifies with the exterior differential on left-invariant forms on tangent Lie algebras $\mathfrak{g}_m$. So one needs that the cohomology $m\mapsto H^{q,w}(\mathfrak{g}_m)$ be constant, whence the word {\em equihomological}. It turns out that the obstruction for cohomologing $q$-forms towards weight $>w$ is $H^{q,w}(\mathfrak{g}_m)$.

\begin{thm}[Rumin 2005]
Let $M$ be a equihomological sub-Riemannian manifold. Assume that there exists a point $m\in M$ such that, in the cohomology of the tangent Lie algebra $\mathfrak{g}_m$, $H^{q,w'}(\mathfrak{g}_m)=0$ for all $w'<w$. Then $AW_q (M)\geq w$.
\end{thm}

On Carnot groups, the grading of cohomology is compatible with Poincar\'e duality, $H^{q,w}(\mathfrak{g})=H^{n-q,Q-w}(\mathfrak{g})$. So 
\begin{eqnarray*}
\exists m~H^{n-q}(\mathfrak{g}_m)=H^{n-q,\leq Q-w}(\mathfrak{g}_m)\quad\Rightarrow\quad AW_q (M)\geq w.
\end{eqnarray*}

\begin{ex}
Degree $n-1$. On any Carnot Lie algebra $\mathfrak{g}$, closed $1$-forms belong to $(V^1)^* =\Lambda^{1,1}$, so $H^1 (\mathfrak{g})=H^{1,1}(\mathfrak{g})$, and $AW_{n-1}(M)\geq Q-1$.
\end{ex}

\end{frame}

\subsection{More examples}

\begin{frame}

\begin{ex}
Contact case. Closed $m$-forms belong to $\Lambda^{m,m}$. Therefore $H^m (\mathfrak{g})=H^{m,m}(\mathfrak{g})$, and $AW_{m+1}(M)\geq m+2$.
\end{ex}
Indeed, if $\omega\in\Lambda^{m,m+1}$, $\omega=\theta\wedge\phi$ where $\theta\in (V^2)^*$, $\phi\in\Lambda^{m-1,m-1}$, $(d\omega)^{m+1,m+1}=(d\theta)\wedge\phi\not=0$ since $d\theta$ is symplectic on $\Delta$.\pause
\begin{ex}
Generic sub-Riemannian case. Let $h=dim(\Delta)$ and $k$ be such that $h-k\geq(n-h)k$. Then $H^{k}(\mathfrak{g}_m)=H^{k,k}(\mathfrak{g}_m)$, thus $AW(M)\geq Q-k$.
\end{ex}
Let $\theta$ be a $\R^{n-h}$-valued $1$-form defining $\Delta$. Say a $k$-plane $S\subset \Delta_m$ is {\em isotropic} if $d\theta_{|S}=0$. Say $S$ is
{\em regular} if the map $\Delta_m \to Hom(S,\R^{n-h})$, $u\mapsto (\iota_u d\theta)_{|S}$ is onto. $h-k\geq(n-h)k$ is a necessary condition for existence of regular isotropic horizontal $k$-planes. It is generically sufficient. When it holds, closed left-invariant $k$-forms have to be of weight $k$, so $H^{k}(\mathfrak{g}_m)=H^{k,k}(\mathfrak{g}_m)$ for $m\in M$.

\begin{rem}
The method just exposed seems to cover all presently known results on the H\"older homeomorphism problem.
\end{rem}

\end{frame}


\end{document}