memoire-m2

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -8,9 +8,7 @@ intuition behind them, culminating in the statement of a presentation for
 geometry of curves in \(\Sigma_g\) -- see
 Theorem~\ref{thm:wajnryb-presentation}.
 
-We start by the so called \emph{lantern relation}.
-
-\begin{fundamental-observation}
+\begin{fundamental-observation}[Lantern relation]
   Let \(\Sigma_0^4\) be the surface the of genus \(0\) with \(4\) boundary
   components and \(\alpha, \beta, \gamma, \delta_1, \ldots, \delta_4 \subset
   \Sigma_0^4\) be as in Figure~\ref{fig:latern-relation}. Consider the
@@ -320,7 +318,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
   \cite[Section~9.4]{farb-margalit}.
 \end{observation}
 
-\begin{fundamental-observation}
+\begin{fundamental-observation}[$k$-chain relations]
   The Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^1) \isoto \Mod(\Sigma_{0,
   2\ell+1}^1)\) takes the twists \(\tau_\delta \in \SMod(\Sigma_\ell^1)\) about
   the boundary \(\delta = \partial \Sigma_\ell^1\) to \(\tau_{\bar\delta}^2 \in
@@ -368,7 +366,7 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
   2})\).
 \end{theorem}
 
-\begin{fundamental-observation}
+\begin{fundamental-observation}[Hyperelliptic relations]
   Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset \Sigma_g\) be as in
   Figure~\ref{fig:hyperellipitic-relations}. Then
   \begin{equation}\label{eq:hyperelliptic-eq}

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -161,7 +161,7 @@ too.
   f^{-1}\).
 \end{observation}
 
-\begin{observation}
+\begin{observation}[Disjointness relations]
   Given \(f \in \Mod(\Sigma)\), \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] =
   [\alpha]\). In particular, \([\tau_\alpha, \tau_\beta] = 1\) for \(\alpha\)
   and \(\beta\) disjoint.
@@ -174,7 +174,7 @@ too.
   [\alpha] = [\beta]\) and then apply Observation~\ref{ex:conjugate-twists}.
 \end{observation}
 
-\begin{fundamental-observation}\label{ex:braid-relation}
+\begin{fundamental-observation}[Braid relations]\label{ex:braid-relation}
   Given \(\alpha, \beta \subset \Sigma\) with \(\#(\alpha \cap \beta) = 1\), it
   is not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] =
   [\alpha]\). From Observation~\ref{ex:conjugate-twists} we then get