memoire-m2

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -192,7 +192,7 @@ A perhaps less obvious fact about Dehn twists is\dots
 The proof of Theorem~\ref{thm:mcg-is-fg} is simple in nature: we proceed by
 indution in \(g\) and \(r\). On the other hand, the indutction steps are
 somewhat involved and require two ingrediantes we have not encountered so far,
-namely the \emph{Birman exact sequence} and the \emph{modified complex of
+namely the \emph{Birman exact sequence} and the \emph{modified graph of
 curves}.
 
 \section{The Birman Exact Sequence}
@@ -286,7 +286,7 @@ show\dots
   \label{fig:push-map}
 \end{figure}
 
-\section{The Modified Complex of Curves}
+\section{The Modified Graph of Curves}
 
 Having established Theorem~\ref{thm:birman-exact-seq}, we now need to adress
 the induction step in the genus \(g\) of \(S_{g, r}^b\). Our strategy is to
@@ -305,7 +305,7 @@ We are interested, of course, in the group \(G = \PMod(S_{g, r}^b)\). As for
 the graph \(\Gamma\), we consider\dots
 
 \begin{definition}
-  The \emph{modified complex of nonseparating curves \(\hat{\mathcal{N}}(S)\)
+  The \emph{modified graph of nonseparating curves \(\hat{\mathcal{N}}(S)\)
   of a surface \(S\)} is the graph whose vertices are (un-oriented) isotopy
   classes of nonseparating simple closed curves in \(S\) and
   \[
@@ -322,18 +322,18 @@ It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of
 \}\) are both transitive. But why should \(\hat{\mathcal{N}}(S_{g, r}^b)\) be
 connected?
 
-Historically, the modified complex of nonseparating curves first arised as a
+Historically, the modified graph of nonseparating curves first arised as a
 \emph{modified} version of another graph of curves, known as\dots
 
 \begin{definition}
-  Given a surface \(S\), the \emph{complex of curves \(\mathcal{C}(S)\) of
+  Given a surface \(S\), the \emph{graph of curves \(\mathcal{C}(S)\) of
   \(S\)} is the graph whose vertices are (un-oriented) isotopy classes of
   essential simple closed curves in \(S\) and
   \[
     \text{\([\alpha]\) --- \([\beta]\) in \(\mathcal{C}(S)\)}
     \iff \#(\alpha \cap \beta) = 0.
   \]
-  The \emph{complex of nonseparating curves \(\mathcal{N}(S)\)} is the subgraph
+  The \emph{graph of nonseparating curves \(\mathcal{N}(S)\)} is the subgraph
   of \(\mathcal{C}(S)\) whose vertices consist of nonseparating curves.
 \end{definition}
 
@@ -351,13 +351,13 @@ find a path \([\alpha] = [\alpha_1] \text{---} \cdots \text{---} [\alpha_n] =
 [\beta]\) in \(\mathcal{C}(S_{g, r})\). Now if \(\alpha\) and \(\beta\) are
 nonseparating, by inductively adjusting this path we then get\dots
 
-\begin{corollary}\label{thm:mofied-complex-is-connected}
+\begin{corollary}\label{thm:mofied-graph-is-connected}
   If \(g \ge 2\) then both \(\mathcal{N}(S_{g, r})\) and
   \(\hat{\mathcal{N}}(S_{g, r})\) are connected.
 \end{corollary}
 
 See \cite[Section~4.1]{farb-margalit} for a proof of
-Corollary~\ref{thm:mofied-complex-is-connected}. We are now ready to show
+Corollary~\ref{thm:mofied-graph-is-connected}. We are now ready to show
 Theorem~\ref{thm:mcg-is-fg}.
 
 \begin{proof}[Proof of Theorem~\ref{thm:mcg-is-fg}]