memoire-m2

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -164,7 +164,7 @@ to consider how the geometric relationship between \(S\) and other surfaces
 affects \(\Mod(S)\). Indeed, different embeddings \(R \hookrightarrow S\)
 translate to homomorphisms at the level of mapping class groups.
 
-\begin{example}[Inclusion homomorphism]
+\begin{example}[Inclusion homomorphism]\label{ex:inclusion-morphism}
   Let \(R \subset S\) be a closed subsurface. Given some \(\phi \in \Homeo^+(R,
   \partial R)\), we may extend \(\phi\) to \(\tilde{\phi} \in \Homeo^+(S,
   \partial S)\) by setting \(\tilde{\phi}(x) = x\) for \(x \in S\) outside of

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -264,7 +264,7 @@ representations.
   We claim that it suffices to find a \(m\)-dimensional
   \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of
   $\Mod(S_g^b)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^b)$ from
-  Lemma~\ref{thm:inclusion-morphism}, which can be shown to be injective in
+  Example~\ref{ex:inclusion-morphism}, which can be shown to be injective in
   this particular case.} subspace \(W \subset \mathbb{C}^n\) with \(2 \le m \le
   n - 2\). Indeed, in this case \(m < 2(g - 1)\) and \(\dim
   \mfrac{\mathbb{C}^n}{W} = n - m < 2(g - 1)\). Thus both representations

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -245,7 +245,7 @@ the graph \(\Gamma\), we consider\dots
   \(\alpha\) and \(\beta\).
 \end{definition}
 
-It is clear from Example~\ref{ex:change-of-coordinates} that the actions of
+It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of
 \(\Mod(S_{g, r}^b)\) on \(V(\hat{\mathcal{N}}(S_{g, r}^b))\) and \(\{([\alpha],
 [\beta]) \in V(\hat{\mathcal{N}}(S_{g, r}^b))^2 : \#(\alpha \cap \beta) = 1
 \}\) are both transitive. But why should \(\hat{\mathcal{N}}(S_{g, r}^b)\) be