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\title{Metric Geometry}
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\section{Second approach to the Homeomorphism Theorem}
Let $(\mathbf{e},Z)$ be a normalized Brownian excursion and the Head of the Brownian snake driven by $\mathbf{e}$. We associate a pseudo-distance on $[0,1]$ to $\mathbf{e}$ by setting for $a,b \in [0,1]$
$$ \op{d}_{\mathbf{e}}(a,b) = \mathbf{e}(a)+\mathbf{e}(b) - 2 \inf \big\{\mathbf{e}(u) : u \in [a \wedge b, a \vee b]\big\}.$$ Let $L_{\mathbf{e}}$ be the union of segments $[xy]$ where $x = \exp(2i\pi a)$ and $y=\exp(2i\pi b)$ with $\op{d}_{\mathbf{e}}(a,b)=0$. Finally, we define an equivalence relation on $\overline{\mathbb{D}}$ using $\mathbf{e}$: if $x,y \in \overline{\mathbb{D}}$, we put $x \sim_{\mathbf{e}}y$  if $x$ and $y$ belongs to an chord $[e^{2i\pi a}e^{2i\pi b}]$ with $\op{d}_{\mathbf{e}}(a,b)=0$ or if $x$ and $y$ belong to the closure of some open triangle of $\overline{\mathbb{D}} \backslash L_{\mathbf{e}}$. We proved that the quotient set $\overline{\mathbb{D}} / \sim_{\mathbf{e}}$ is homeomorphic to the $\R$-tree $T_{\mathbf{e}}$ coded by the function $\mathbf{e}$.\\
We can do exactly the same procedure for $Z$ (in particular we admit that the local minima of $Z$ are distinct). Thus $Z$ furnishes an equivalence relation $\sim_{Z}$ on $\overline{\mathbb{D}}$ is a similar manner as to $\mathbf{e}$. We consider $\mathbb{S}_{2}$ the standard Euclidean sphere of radius $1$ in $\mathbb{R}^3$ and put $$H_{+} = \big\{ (x_{1},x_{2},x_{3}) \in \mathbb{R}^3,x_{1}^2+x_{2}^2+x_{3}^2=1 \mbox{ and } x_{3} \geq 0 \big\},$$ for the closed North hemisphere of $\mathbb{S}_{2}$ and similarly $H_{-}$ denotes the closed South hemisphere. The stereographic projections from the North and South poles enable us to identify $H_{+}$ and $H_{-}$ with $\overline{\mathbb{D}}$. We will associate the function $\mathbf{e}$ (resp.\,$Z$) to the North (resp.\,South) part of the ball, hence we can define $\sim_{\mathbf{e}}$ on $H_{+}$ and $\sim_{Z}$ on $H_{-}$.
\begin{enumerate}
\item Check that $H_{+}/\sim_{\mathbf{e}}$ is still homeomorphic to $T_{\mathbf{e}}$.
\end{enumerate}
We put a relation on $x,y\in \mathbb{S}_{2}$ by $x \sim y$ if and only if $x,y \in H_{+}$ and $ x\sim_{\mathbf{e}} y$ or $x,y \in H_{-}$ and $x\sim_{Z}y$. We admit the following  fact about the process $(\mathbf{e}_{t},Z_{t})_{t \in[0,1]}$. Almost surely, for every $s \in\,]0,1[$ such that for some $\varepsilon>0$ if we have $$\mathbf{e}_{s}= \min_{r \in [s-\varepsilon,s]}\mathbf{e}_{r}\mbox{\ \ \  or \ \ \ } \mathbf{e}_{s}= \min_{r \in [s,s+\varepsilon]}\mathbf{e}_{r}$$  then $$Z_{s}> \min_{r \in [s-\delta,s]}Z_{r}\mbox{, for every }0<\delta <s \mbox{\ \ \ and \ \ \ } Z_{s}> \min_{r \in [s,s+\delta]}Z_{r}\mbox{, for every }0<\delta <1-s.$$


 \begin{enumerate}
\setcounter{enumi}{1}
\item Prove that a.s.\,$\sim$ is a closed equivalence relation.
\end{enumerate}
\begin{theorem}[Moore (1925)] Let $\sim$ be a closed equivalence relation on the two dimensional sphere $\mathbb{S}_{2}$. Assume that every equivalence class of $\sim$ is a compact path-connected subset of the sphere whose complement is connected. The quotient space $\mathbb{S}_{2}/\sim$ is homeomorphic to $\mathbb{S}_{2}$.
\end{theorem}
\begin{enumerate}
\setcounter{enumi}{2}
\item Give an example of a closed equivalence relation $\simeq$ such that the quotient $\mathbb{S}_{2}/ \simeq$ is not homeomorphic to $\mathbb{S}_{2}$.
\item Prove that in our setting $\sim$ a.s.\,verifies all hypotheses of Moore's Theorem and deduce that almost surely $\mathbb{S}_2 / \sim$ is homeomorphic to $\mathbb{S}_2$.
\item Does anybody see a link with scaling limits of random planar quadrangulations ?
\end{enumerate}

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\begin{figure}
\includegraphics[height=10cm]{thurstonbis}
(Taken from Thurston)
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