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}]