memoire-m2

 \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}