memoire-m2

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -9,12 +9,12 @@ geometry of curves in \(\Sigma_g\) -- see
 Theorem~\ref{thm:wajnryb-presentation}.
 
 \begin{fundamental-observation}[Lantern relation]
-  Let \(\Sigma_0^4\) be the surface the of genus \(0\) with \(4\) boundary
+  Let \(\Sigma_0^4\) be the surface 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
-  surfaces \(\Sigma_0^3 = \Sigma_0^4 \cup_{\delta_1} \mathbb{D}^2\) and
-  \(\Sigma_{0,1}^3 = \Sigma_0^4 \cup_{\delta_1} (\mathbb{D}^2 \setminus \{ 0
-  \})\), as well as the map \(\operatorname{push} : \pi_1(\Sigma_0^3, 0) \to
+  \Sigma_0^4\) be as in Figure~\ref{fig:latern-relation}. Consider the surfaces
+  \(\Sigma_0^3 = \Sigma_0^4 \cup_{\delta_1} \mathbb{D}^2\) and \(\Sigma_{0,1}^3
+  = \Sigma_0^4 \cup_{\delta_1} (\mathbb{D}^2 \setminus \{ 0 \})\), as well as
+  the map \(\operatorname{push} : \pi_1(\Sigma_0^3, 0) \to
   \Mod(\Sigma_{0,1}^3)\). Let \(\eta_1, \eta_2, \eta_3 \subset \Sigma_0^3\) be
   the loops from Figure~\ref{fig:lantern-relation-capped}, so that \([\eta_1]
   \cdot [\eta_2] = [\eta_3]\). From Observation~\ref{ex:push-simple-loop} we