\chapter{Dehn Twists}\label{ch:dehn-twists}

With the goal of studying the linear representations of mapping class groups in
mind, we now start investigating the group structure of \(\Mod(\Sigma)\). We
begin by computing some fundamental examples and then explore how we can use
these examples to understand the structure of the mapping class groups of other
surfaces. Namely, we compute \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong
\mathbb{Z}\), and discuss how its generator gives rise to a convenient
generating set for \(\Mod(\Sigma)\), known as the set of \emph{Dehn twists}.

The idea here is to reproduce the proof of injectivity in
Observation~\ref{ex:torus-mcg}: by cutting along curves and arcs, we can always
decompose a surface into copies of \(\mathbb{D}^2\) and \(\mathbb{D}^2
\setminus \{0\}\). Observation~\ref{ex:alexander-trick} and
Observation~\ref{ex:mdg-once-punctured-disk} then imply the triviality of
mapping classes fixing such arcs and curves. Formally, this translates to the
following result.

\begin{proposition}[Alexander method]\label{thm:alexander-method}
  Let \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) be essential simple closed
  curves or proper arcs satisfying the following conditions.
  \begin{enumerate}
    \item \([\alpha_i] \ne [\alpha_j]\) for \(i \ne j\).
    \item Each pair \((\alpha_i, \alpha_j)\) crosses at most once.
    \item Given distinct \(i, j, k\), at least one of \(\alpha_i \cap \alpha_j,
      \alpha_i \cap \alpha_k, \alpha_j \cap \alpha_k\) is empty.
    \item The surface obtained by cutting \(\Sigma\) along the \(\alpha_i\) is a
      disjoint union of disks and once-punctured disks.
  \end{enumerate}
  Suppose \(f \in \Mod(\Sigma)\) is such that \(f \cdot \vec{[\alpha_i]} =
  \vec{[\alpha_i]}\) for all \(i\). Then \(f = 1 \in \Mod(\Sigma)\).
\end{proposition}

See \cite[Proposition~2.8]{farb-margalit} for a proof of
Proposition~\ref{thm:alexander-method}. We now state some \emph{fundamental}
applications of the Alexander method.

\begin{example}\label{ex:mcg-annulus}
  The mapping class group \(\Mod(\mathbb{S}^1 \times [0, 1])\) is freely
  generated by \(f = [\phi]\), where
  \begin{align*}
    \phi : \mathbb{S}^1 \times [0, 1] & \isoto  \mathbb{S}^1 \times [0, 1] \\
                   (e^{2 \pi i t}, s) & \mapsto (e^{2 \pi i (t - s)}, s)
  \end{align*}
  is the map illustrated in Figure~\ref{fig:dehn-twist-cylinder}. In
  particular, \(\Mod(\mathbb{S}^1 \times [0, 1]) \cong \mathbb{Z}\).
\end{example}

\begin{example}\label{ex:mcg-twice-punctured-disk}
  The mapping class group \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2},
  \sfrac{1}{2}\})\) of the twice punctured unit disk in \(\mathbb{C}\) is
  freely generated by \(f = [\phi]\), where
  \begin{align*}
    \phi : \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}
    & \isoto \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\} \\
    z & \mapsto -z
  \end{align*}
  is the map from Figure~\ref{fig:hald-twist-disk}. In particular,
  \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}) \cong
  \mathbb{Z}\).
\end{example}

\noindent
\begin{minipage}[b]{.47\linewidth}
  \centering
  \includegraphics[width=.7\linewidth]{images/dehn-twist-cylinder.eps}
  \captionof{figure}{The generator $f$ of $\Mod(\mathbb{S}^1 \times [0, 1])
  \cong \mathbb{Z}$ takes the yellow arc on the left-hand side to the arc on
  the right-hand side that winds about the curve $\alpha$.}
  \label{fig:dehn-twist-cylinder}
\end{minipage}
\hspace{.6cm} %
\begin{minipage}[b]{.47\linewidth}
  \centering
  \includegraphics[width=.4\linewidth]{images/half-twist-disk.eps}
  \captionof{figure}{The generator $f$ of $\Mod(\mathbb{D}^2 \setminus
  \{-\sfrac{1}{2}, \sfrac{1}{2}) \cong \mathbb{Z}$ corresponds to the
  clockwise rotation by $\pi$ about the origin.}
  \label{fig:hald-twist-disk}
\end{minipage}

Let \(\Sigma\) be an orientable surface, possibly with punctures and non-empty
boundary. Given some closed \(\alpha \subset \Sigma\), we may envision doing
something similar to Example~\ref{ex:mcg-annulus} in \(\Sigma\) by looking at
annular neighborhoods of \(\alpha\). These are precisely the \emph{Dehn twists},
illustrated in Figure~\ref{fig:dehn-twist-bitorus} in the case of the bitorus
\(\Sigma_2\).

\begin{definition}
  Given a simple closed curve \(\alpha \subset \Sigma\), fix a closed annular
  neighborhood \(A \subset \Sigma\) of \(\alpha\) -- i.e. \(A \cong
  \mathbb{S}^1 \times [0, 1]\). Let \(f \in \Mod(A) \cong \Mod(\mathbb{S}^1
  \times [0, 1]) \cong \mathbb{Z}\) be the generator from
  Example~\ref{ex:mcg-annulus}. The \emph{Dehn twist \(\tau_\alpha \in
  \Mod(\Sigma)\) about \(\alpha\)} is defined as the image of \(f\) under the
  inclusion homomorphism \(\Mod(A) \to \Mod(\Sigma)\).
\end{definition}

\begin{figure}[ht]
  \centering
  \includegraphics[width=.6\linewidth]{images/dehn-twist-bitorus.eps}
  \caption{The Dehn twist about the curve $\alpha$ takes the peanut-shaped curve
  on the left-hand side to the yellow curve on the right-hand side.}
  \label{fig:dehn-twist-bitorus}
\end{figure}

Similarly, using the description of the mapping class group of the
twice-puncture disk derived in Example~\ref{ex:mcg-twice-punctured-disk}, the
generator of \(\Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\})\)
gives rise the so-called \emph{half-twists}. These are examples of mapping
classes that permute the punctures of \(\Sigma\).

\begin{definition}
  Given an arc \(\alpha \subset \Sigma\) joining two punctures in the interior
  of \(\Sigma\), fix a closed neighborhood \(D \subset \Sigma\) of \(\alpha\)
  with \(D \cong \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). Let
  \(f \in \Mod(D) \cong \Mod(\mathbb{D}^2 \setminus \{-\sfrac{1}{2},
  \sfrac{1}{2}\}) \cong \mathbb{Z}\) be the generator from
  Example~\ref{ex:mcg-twice-punctured-disk}. The \emph{half-twist \(h_\alpha
  \in \Mod(\Sigma)\) about \(\alpha\)} is defined as the image of \(f\) under
  the inclusion homomorphism \(\Mod(D) \to \Mod(\Sigma)\).
\end{definition}

We can use the Alexander method to describe the kernel of capping and cutting
morphisms in terms of Dehn twists.

\begin{observation}[Capping exact sequence]\label{ex:capping-seq}
  Let \(\delta \subset \partial \Sigma\) be a boundary component of \(\Sigma\)
  and \(\operatorname{cap} : \Mod(\Sigma) \to \Mod(\Sigma \cup_\delta
  (\mathbb{D}^2 \setminus \{0\}))\) be the corresponding the capping
  homomorphism from Example~\ref{ex:capping-morphism}. There is an exact
  sequence
  \begin{center}
    \begin{tikzcd}
      1 \rar &
      \langle \tau_\delta \rangle \rar &
      \Mod(\Sigma) \rar{\operatorname{cap}} &
      \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}), 0) \rar &
      1,
    \end{tikzcd}
  \end{center}
  known as \emph{the capping exact sequence} -- see
  \cite[Proposition~3.19]{farb-margalit} for a proof.
\end{observation}

\begin{observation}\label{ex:cutting-morphism-kernel}
  Let \(\alpha \subset \Sigma\) be a simple closed curve and
  \(\operatorname{cut} : \Mod(\Sigma)_{\vec{[\alpha]}} \to \Mod(\Sigma
  \setminus \alpha)\) be the cutting homomorphism from
  Example~\ref{ex:cutting-morphism}. Then \(\ker \operatorname{cut} = \langle
  \tau_\alpha \rangle \cong \mathbb{Z}\).
\end{observation}

It is also interesting to study how the geometry of two curves affects the
relationship between their corresponding Dehn twists. For instance,
by investigating the geometric intersection number
\[
  \#(\alpha \cap \beta) = \min
  \left\{
    |\alpha' \cap \beta'| : [\alpha'] = [\alpha]
    \text{ and }
    [\beta'] = [\beta]
  \right\}
\]
we can distinguish between powers of Dehn twists
\cite[Proposition~3.2]{farb-margalit}.

\begin{proposition}\label{thm:twist-intersection-number}
  Let \(\alpha \subset \Sigma\) be a simple closed curve and \(T_\alpha\) be a
  representative of \(\tau_\alpha \in \Mod(\Sigma)\). Then \(\#
  (T_\alpha^k(\beta) \cap \beta) = |k| \cdot \#(\alpha \cap \beta)^2\) for any
  \(k \in \mathbb{Z}\). In particular, if \(\alpha\) is nontrivial then
  \(\tau_\alpha\) has infinite order.
\end{proposition}

\begin{observation}
  Given \(\alpha, \beta \subset \Sigma\), \(\tau_\alpha = \tau_\beta \iff
  [\alpha] = [\beta]\). Indeed, if \(\alpha\) and \(\beta\) are non-isotopic,
  we can find \(\gamma\) with \(\#(\gamma \cap \alpha) > 0\) and \(\#(\gamma
  \cap \beta) = 0\). It thus follows from
  Proposition~\ref{thm:twist-intersection-number} that \(\#(T_\alpha(\gamma)
  \cap \gamma) > \#(T_\beta(\gamma) \cap \gamma)\), so \(\tau_\alpha \ne
  \tau_\beta\).
\end{observation}

Many other relations between Dehn twists can derived be in a geometric fashion
too.

\begin{observation}\label{ex:conjugate-twists}
  Given \(f = [\phi] \in \Mod(\Sigma)\), \(\tau_{\phi(\alpha)} = f \tau_\alpha
  f^{-1}\).
\end{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, for we can choose a representative of \(\tau_\beta\)
  whose support is disjoint from \(\alpha\).
\end{observation}

\begin{observation}
  If \(\alpha, \beta \subset \Sigma\) are both nonseparating then \(\tau_\alpha,
  \tau_\beta \in \Mod(\Sigma)\) are conjugate. Indeed, by the change of
  coordinates principle we can find \(f \in \Mod(\Sigma)\) with \(f \cdot
  [\alpha] = [\beta]\) and then apply Observation~\ref{ex:conjugate-twists}.
\end{observation}

\begin{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
  \((\tau_\alpha \tau_\beta) \tau_\alpha (\tau_\alpha \tau_\beta)^{-1} =
  \tau_\beta\), from which follows the \emph{braid relation}
  \[
    \tau_\alpha \tau_\beta \tau_\alpha = \tau_\beta \tau_\alpha \tau_\beta.
  \]
\end{observation}

A perhaps less obvious fact about Dehn twists is the following.

\begin{theorem}\label{thm:mcg-is-fg}
  Let \(\Sigma_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with
  \(r\) punctures and \(b\) boundary components. Then the pure mapping class
  group \(\PMod(\Sigma_{g, r}^b)\) is generated by finitely many Dehn twists
  about nonseparating curves or boundary components.
\end{theorem}

The proof of Theorem~\ref{thm:mcg-is-fg} is simple in nature: we proceed by
induction in \(g\), \(b\) and \(r\). On the other hand, the induction steps
require two ingredients we have not encountered so far, namely the \emph{Birman
exact sequence} and the \emph{modified graph of curves}. We now provide a
concise account of these ingredients.

\section{The Birman Exact Sequence}

Having the proof of Theorem~\ref{thm:mcg-is-fg} in mind, it is interesting to
consider the relationship between the mapping class group of \(\Sigma_{g,
r}^b\) and that of \(\Sigma_{g, r+1}^b = \Sigma_{g, r}^b \setminus \{ x \}\)
for some \(x\) in the interior \((\Sigma_{g, r}^b)\degree\) of \(\Sigma_{g,
r}^b\). Indeed, this will later allow us to establish the induction step on the
number of punctures \(r\).

Given an orientable surface \(\Sigma\) and \(x_1, \ldots, x_n \in
\Sigma\degree\), denote by \(\Mod(\Sigma \setminus \{x_1, \ldots,
x_n\})_{\{x_1, \ldots, x_n\}} \subset \Mod(\Sigma \setminus \{x_1, \ldots,
x_n\})\) the subgroup of mapping classes \(f\) that permute \(x_1, \ldots,
x_n\) -- i.e. \(f \cdot x_i = x_{\sigma(i)}\) for some permutation \(\sigma \in
S_n\). We certainly have a surjective homomorphism
\begin{align*}
  \operatorname{forget} :
  \Mod(\Sigma \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots, x_n\}}
  & \to \Mod(\Sigma) \\
  [\phi] & \mapsto [\tilde\phi]
\end{align*}
which ``\emph{forgets} the additional punctures \(x_1, \ldots,
x_n\) of \(\Sigma \setminus \{x_1, \ldots, x_n\}\),'' but what is its kernel?

To answer this question, we consider the configuration space \(C(\Sigma, n) =
\mfrac{C^{\operatorname{ord}}(\Sigma, n)}{S_n}\) of \(n\) (unordered) points in
the interior of \(\Sigma\) -- where \(C^{\operatorname{ord}}(\Sigma, n) = \{
  (x_1, \ldots, x_n) \in (\Sigma\degree)^n : x_i \ne x_j \ \text{for}\ i \ne j
\}\). Denote \(\Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n} = \{\phi
\in \Homeo^+(\Sigma, \partial \Sigma) : \phi(x_i) = x_i \}\). From the
fibration\footnote{See \cite[Chapter~4]{hatcher} for a reference.}
\[
  \arraycolsep=1.4pt
  \begin{array}{ccrcl}
    \Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n}
    & \hookrightarrow & \Homeo^+(\Sigma, \partial \Sigma)
    & \relbar\joinrel\twoheadrightarrow & C(\Sigma, n) \\
    & & \phi & \mapsto & [\phi(x_1), \ldots, \phi(x_n)]
  \end{array}
\]
and its long exact sequence in homotopy we then obtain the following
fundamental result.

\begin{theorem}[Birman exact sequence]\label{thm:birman-exact-seq}
  Suppose \(\pi_1(\Homeo^+(\Sigma, \partial \Sigma), 1) = 1\). Then there is an
  exact sequence
  \begin{center}
    \begin{tikzcd}[cramped]
      1 \rar
      & \pi_1(C(\Sigma, n), [x_1, \ldots, x_n]) \rar{\operatorname{push}}
      & \Mod(\Sigma \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots, x_n\}}
        \rar{\operatorname{forget}}
      & \Mod(\Sigma) \rar
      & 1.
    \end{tikzcd}
  \end{center}
\end{theorem}

\begin{remark}
  Notice that \(C(\Sigma, 1) = \Sigma\degree \simeq \Sigma\). Hence for \(n =
  1\) Theorem~\ref{thm:birman-exact-seq} gives us a sequence
  \begin{center}
    \begin{tikzcd}
      1 \rar
      & \pi_1(\Sigma, x) \rar{\operatorname{push}}
      & \Mod(\Sigma \setminus \{x\}, x) \rar{\operatorname{forget}}
      & \Mod(\Sigma) \rar
      & 1.
    \end{tikzcd}
  \end{center}
\end{remark}

We may regard a simple loop \(\alpha : \mathbb{S}^1 \to C(\Sigma, n)\) based at
\([x_1, \ldots, x_n]\) as \(n\) disjoint curves \(\alpha_1, \ldots, \alpha_n :
[0, 1] \to \Sigma\) with \(\alpha_i(0) = x_i\) and \(\alpha_i(1) =
x_{\sigma(i)}\) for some \(\sigma \in S_n\). The element
\(\operatorname{push}([\alpha]) \in \Mod(\Sigma)\) can then be seen as the
mapping class that ``\emph{pushes} a neighborhood of \(x_{\sigma(i)}\) towards
\(x_i\) along the curve \(\alpha_i^{-1}\),'' as shown in
Figure~\ref{fig:push-map} for the case \(n = 1\). Indeed, this goes to
show \(\operatorname{push}([\alpha])\) can be descrived as a product of Dehn
twists.

\begin{fundamental-observation}\label{ex:push-simple-loop}
  Using the notation of Figure~\ref{fig:push-map},
  \(\operatorname{push}([\alpha]) = \tau_{\delta_1} \tau_{\delta_2}^{-1} \in
  \Mod(\Sigma)\).
\end{fundamental-observation}

\begin{figure}[ht]
  \centering
  \includegraphics[width=.35\linewidth]{images/push-map.eps}
  \caption{The inclusion $\operatorname{push} : \pi_1(\Sigma, x) \to
  \Mod(\Sigma)$ maps a simple loop $\alpha : \mathbb{S}^1 \to \Sigma$ to the
  mapping class supported at an annular neighborhood $A$ of $\alpha$. Inside
  this neighborhood, $\operatorname{push}([\alpha])$ takes the arc joining the
  boundary components $\delta_i \subset \partial A$ on the left-hand side to
  the yellow arc on the right-hand side.}
  \label{fig:push-map}
\end{figure}

\section{The Modified Graph of Curves}

Having established Theorem~\ref{thm:birman-exact-seq}, we now need to address
the induction step in the genus \(g\) of \(\Sigma_{g, r}^b\). Our strategy is
to apply the following lemma from geometric group theory.

\begin{lemma}\label{thm:ggt-lemma}
  Let \(G\) be a group and \(\Gamma\) be a \emph{connected} graph with \(G
  \leftaction \Gamma\) via graph automorphisms. Suppose that \(G\) acts
  transitively on both \(V(\Gamma)\) and \(\{(v, w) \in V(\Gamma)^2 :
  v \text{ --- } w \text{ in } \Gamma \}\). If \(v, w \in V(\Gamma)\) are
  connected by an edge and \(g \in G\) is such that \(g \cdot w = v\) then
  \(G\) is generated by \(g\) and the stabilizer \(G_v\).
\end{lemma}

We are interested, of course, in the group \(G = \PMod(\Sigma_{g, r}^b)\). As
for the the role of \(\Gamma\), we consider the following graph.

\begin{definition}
  The \emph{modified graph of nonseparating curves
  \(\hat{\mathcal{N}}(\Sigma)\) of a surface \(\Sigma\)} is the graph whose
  vertices are (unoriented) isotopy classes of nonseparating simple closed
  curves in \(\Sigma\) and
  \[
    \text{\([\alpha]\) --- \([\beta]\) in \(\hat{\mathcal{N}}(\Sigma)\)}
    \iff \#(\alpha \cap \beta) = 1,
  \]
  where \(\#(\alpha \cap \beta)\) is the geometric intersection number of
  \(\alpha\) and \(\beta\).
\end{definition}

It is clear from the change of coordinates principle and
Observation~\ref{ex:change-of-coordinates-crossing} that the actions of
\(\Mod(\Sigma_{g, r}^b)\) on \(V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))\) and
\(\{([\alpha], [\beta]) \in V(\hat{\mathcal{N}}(\Sigma_{g, r}^b))^2 : \#(\alpha
\cap \beta) = 1 \}\) are both transitive. But why should
\(\hat{\mathcal{N}}(\Sigma_{g, r}^b)\) be connected?
Historically, the modified graph of nonseparating curves first arose as a
\emph{modified} version of another graph, known as \emph{the graph of of
curves}.

\begin{definition}
  Given a surface \(\Sigma\), the \emph{graph of curves \(\mathcal{C}(\Sigma)\)
  of \(\Sigma\)} is the graph whose vertices are (unoriented) isotopy classes
  of essential simple closed curves in \(\Sigma\) and
  \[
    \text{\([\alpha]\) --- \([\beta]\) in \(\mathcal{C}(\Sigma)\)}
    \iff \#(\alpha \cap \beta) = 0.
  \]
  The \emph{graph of nonseparating curves \(\mathcal{N}(\Sigma)\)} is the
  subgraph of \(\mathcal{C}(\Sigma)\) whose vertices consist of nonseparating
  curves.
\end{definition}

Lickorish \cite{lickorish} essentially showed that, apart from a small number
of sporadic cases, \(\mathcal{C}(\Sigma_{g, r})\) is connected.

\begin{theorem}
  If \(\Sigma_{g, r}\) is not one \(\Sigma_0 = \mathbb{S}^2, \Sigma_{0, 1},
  \ldots, \Sigma_{0, 4}, \Sigma_1 = \mathbb{T}^2\) and \(\Sigma_{1, 1}\) then
  \(\mathcal{C}(\Sigma_{g, r})\) is connected.
\end{theorem}

In other words, given simple closed curves \(\alpha, \beta \subset \Sigma_{g,
r}\), we can find closed \(\alpha = \alpha_1, \alpha_2, \ldots, \alpha_n =
\beta\) in \(\Sigma_{g, r}\) with \(\alpha_i\) disjoint from \(\alpha_{i+1}\).
Now if \(\alpha\) and \(\beta\) are nonseparating, by inductively adjusting
this sequence of curves we obtain the following corollary.

\begin{corollary}\label{thm:mofied-graph-is-connected}
  If \(g \ge 2\) then both \(\mathcal{N}(\Sigma_{g, r})\) and
  \(\hat{\mathcal{N}}(\Sigma_{g, r})\) are connected.
\end{corollary}

See \cite[Section~4.1]{farb-margalit} for a proof of
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}]
  Let \(\Sigma_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with
  \(r\) punctures and \(b\) boundary components. We want to establish that
  \(\PMod(\Sigma_{g, r}^b)\) is generated by a finite number of Dehn twists
  about nonseparating simple closed curves or boundary components. As promised,
  we proceed by triple induction on \(r\), \(g\) and \(b\).

  For the base case, it is clear from Observation~\ref{ex:torus-mcg} and
  Observation~\ref{ex:punctured-torus-mcg} that \(\Mod(\mathbb{T}^2) \cong
  \Mod(\Sigma_{1, 1}) \cong \operatorname{SL}_2(\mathbb{Z})\) are generated by
  the Dehn twists about the curves \(\alpha\) and \(\beta\) from
  Figure~\ref{fig:torus-mcg-generators}, each corresponding to one of the
  standard generators
  \begin{align*}
    \begin{pmatrix}
      1 & 1 \\
      0 & 1
    \end{pmatrix}
    &&
    \begin{pmatrix}
       1 & 0 \\
      -1 & 1
    \end{pmatrix}
  \end{align*}
  of \(\operatorname{SL}_2(\mathbb{Z})\).

  \begin{figure}[ht]
    \centering
    \includegraphics[width=.5\linewidth]{images/torus-mcg-generators.eps}
    \caption{The curves $\alpha$ and $\beta$ whose Dehn twists generate
    $\Mod(\mathbb{T}^2)$ and $\Mod(\Sigma_{1, 1})$.}
    \label{fig:torus-mcg-generators}
  \end{figure}

  Now suppose \(\PMod(\Sigma_{g, r})\) is finitely-generated by twists about
  nonseparating curves for \(g \ge 2\) or \(g = 1\) and \(r > 1\). In both
  case, \(\chi(\Sigma_{g, r}) = 2 - 2g - r < 0\) and thus
  \(\pi_1(\Homeo^+(\Sigma_{g, r})) = 1\) -- see
  \cite[Theorem~1.14]{farb-margalit}. The Birman exact sequence from
  Theorem~\ref{thm:birman-exact-seq} then gives us
  \begin{center}
    \begin{tikzcd}
      1 \rar
      & \pi_1(\Sigma_{g, r}, x) \rar{\operatorname{push}}
      & \PMod(\Sigma_{g, r + 1}) \rar{\operatorname{forget}}
      & \PMod(\Sigma_{g, r}) \rar
      & 1,
    \end{tikzcd}
  \end{center}
  where \(\Sigma_{g, r + 1} = \Sigma_{g, r} \setminus \{x\}\). Since \(g \ge
  1\), \(\pi_1(\Sigma_{g, r}, x)\) is generated by finitely many nonseparating
  loops. We have seen in Observation~\ref{ex:push-simple-loop} that
  \(\operatorname{push} : \pi_1(\Sigma_{g, r}, x) \to \Mod(\Sigma_{g, r+1},
  x)\) takes nonseparating simple loops to products of twists about
  nonseparating simple curves. Furthermore, we may lift the
  generators of \(\PMod(\Sigma_{g, r})\) to Dehn twists about the corresponding
  curves in \(\Sigma_{g, r + 1}\). This goes to show that
  \(\PMod(\Sigma_{g, r + 1})\) is also generated by finitely many twists about
  simple curves, concluding the induction step on \(r\).

  As for the induction step on \(g\), fix \(g \ge 1\) and suppose that, for
  each \(r \ge 0\), \(\PMod(\Sigma_{g, r})\) is finitely generated by twists
  about nonseparating curves or boundary components. Let us show that the same
  holds for \(\Mod(\Sigma_{g + 1})\). To that end, we consider the action
  \(\Mod(\Sigma_{g + 1}) \leftaction \hat{\mathcal{N}}(\Sigma_{g + 1})\). Since
  \(g + 1 \ge 2\), \(\hat{\mathcal{N}}(\Sigma_{g + 1})\) is connected and the
  conditions of Lemma~\ref{thm:ggt-lemma} are met. Now recall from
  Observation~\ref{ex:braid-relation} that, given nonseparating \(\alpha, \beta
  \subset \Sigma_{g + 1}\) crossing once, \(\tau_\beta \tau_\alpha \cdot
  [\beta] = [\alpha]\). It thus follows from Lemma~\ref{thm:ggt-lemma} that
  \(\Mod(\Sigma_{g + 1})\) is generated by \(\tau_\beta \tau_\alpha\) and
  \(\Mod(\Sigma_{g + 1})_{[\alpha]} = \{ f \in \Mod(\Sigma_{g + 1}) : f \cdot
  [\alpha] = [\alpha]\}\).

  In turn, \(\Mod(\Sigma_{g + 1})_{[\alpha]}\) has its index \(2\) subgroup
  \[
    \Mod(\Sigma_{g + 1})_{\vec{[\alpha]}}
    = \{ f \in \Mod(\Sigma_{g + 1}) : f \cdot \vec{[\alpha]} = \vec{[\alpha]}\}
  \]
  of mapping classes fixing any given choice of orientation of \(\alpha\). One
  can check that \(\tau_\beta \tau_\alpha^2 \tau_\beta \in \Mod(\Sigma_{g +
  1})_{[\alpha]}\) inverts the orientation of \(\alpha\) and is thus a
  representative of the nontrivial
  \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\)-coset in
  \(\Mod(\Sigma_{g+1})_{[\alpha]}\). In particular, \(\Mod(\Sigma_{g+1})\) is
  generated by \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\), \(\tau_\beta
  \tau_\alpha\) and \(\tau_\beta \tau_\alpha^2 \tau_\beta\).

  We now claim \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is generated by
  finitely many twists about nonseparating curves. First observe that
  \(\Sigma_{g+1} \setminus \alpha \cong \Sigma_{g,2}\), as shown in
  Figure~\ref{fig:cut-along-nonseparating-adds-two-punctures}.
  Observation~\ref{ex:cutting-morphism-kernel} then gives us an exact sequence
  \begin{equation}\label{eq:cutting-seq}
    \begin{tikzcd}
      1 \rar &
      \langle \tau_\alpha \rangle \rar &
      \Mod(\Sigma_{g+1})_{\vec{[\alpha]}} \rar{\operatorname{cut}} &
      \PMod(\Sigma_{g,2}) \rar &
      1.
    \end{tikzcd}
  \end{equation}
  But by the induction hypothesis, \(\PMod(\Sigma_{g, 2})\) is
  finitely-generated by twists about nonseparating simple closed curves. As
  before, these generators may be lifted to appropriate twists in
  \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\). Now by (\ref{eq:cutting-seq}) we get
  that \(\Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) is finitely generated by twists
  about nonseparating curves, as desired. This concludes the induction step in
  \(g\).

  \begin{figure}[ht]
    \centering
    \includegraphics[width=.75\linewidth]{images/cutting-homeo.eps}
    \caption{The homeomorphism $\Sigma_{g + 1} \setminus \alpha \cong
    \Sigma_{g, 2}$: removing the curve $\alpha$ has the same effect as cutting
    along $\alpha$ and then capping the two resulting boundary components with
    once-punctured disks, which gives us $\Sigma_{g, 2}$.}
    \label{fig:cut-along-nonseparating-adds-two-punctures}
  \end{figure}

  Finally, we handle the induction in \(b\). The boundaryless case \(b = 0\)
  was already dealt with before. Now suppose \(\PMod(\Sigma_{g, s}^b)\) is
  finitely generated by twists about simple closed curves or boundary
  components for all \(g\) and \(s\). Fix some boundary component \(\delta
  \subset \partial \Sigma_{g, r}^{b+1}\). From the homeomorphism \(\Sigma_{g,
  r+1}^b \cong \Sigma_{g, r}^{b+1} \cup_\delta (\mathbb{D}^2 \setminus \{ 0
  \})\) and the capping exact sequence from Observation~\ref{ex:capping-seq}
  we obtain a sequence
  \begin{center}
    \begin{tikzcd}
      1 \rar                                              &
      \langle \tau_\delta \rangle \rar                    &
      \PMod(\Sigma_{g, r}^{b+1}) \rar{\operatorname{cap}} &
      \PMod(\Sigma_{g, r+1}^b) \rar                       &
      1.
    \end{tikzcd}
  \end{center}
  Now by induction hypothesis we may once again lift the generators of
  \(\PMod(\Sigma_{g, r+1}^b)\) to Dehn twists about the corresponding curves in
  \(\Sigma_{g, r}^{b+1}\) and add \(\tau_\delta\) to the generating set,
  concluding the induction in \(b \ge 0\). We are done.
\end{proof}

There are many possible improvements to this last result. For instance, in
\cite[Section~4.4]{farb-margalit} Farb-Margalit exhibit an explicit set of
generators of \(\Mod(\Sigma_g^b)\) by adapting the induction steps in the
proof of Theorem~\ref{thm:mcg-is-fg}. These are known as the \emph{Lickorish
generators}.

\begin{theorem}[Lickorish generators]\label{thm:lickorish-gens}
  If \(g \ge 1\) then \(\Mod(\Sigma_g^b)\) is generated by the Dehn twists
  about the curves \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g,
  \gamma_1, \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{b-1}\) as in
  Figure~\ref{fig:lickorish-gens}
\end{theorem}

In the boundaryless case \(b = 0\), we can write \(\tau_{\alpha_3}, \ldots,
\tau_{\alpha_g} \in \Mod(\Sigma_g)\) as products of the twists about the
remaining curves, from which we get the so-called \emph{Humphreys generators}.

\begin{corollary}[Humphreys generators]\label{thm:humphreys-gens}
  If \(g \ge 2\) then \(\Mod(\Sigma_g)\) is generated by the Dehn twists about the
  curves \(\alpha_0, \ldots, \alpha_{2g}\) as in
  Figure~\ref{fig:humphreys-gens}.
\end{corollary}

\noindent
\begin{minipage}[b]{.47\linewidth}
  \centering
  \includegraphics[width=\linewidth]{images/lickorish-gens.eps}
  \captionof{figure}{The curves from Lickorish generators of
  $\Mod(\Sigma_g^b)$.}
  \label{fig:lickorish-gens}
\end{minipage}
\hspace{.6cm} %
\begin{minipage}[b]{.47\textwidth}
  \centering
  \includegraphics[width=\linewidth]{images/humphreys-gens.eps}
  \captionof{figure}{The curves from Humphreys generators of $\Mod(\Sigma_g)$.}
  \label{fig:humphreys-gens}
\end{minipage}
