memoire-m2

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -26,35 +26,39 @@ notes, all surfaces considered will be of the form \(\Sigma = \Sigma_{g,
 r}^b\). Any such \(\Sigma\) admits a natural compactification
 \(\widebar\Sigma\) obtained by filling its punctures. We denote \(\Sigma_{g, r}
 = \Sigma_{g, r}^0\). All closed curves \(\alpha, \beta \subset \Sigma\) we
-consider lie in the interior of \(\Sigma\) and intersect transversely.
+consider lie in the interior of \(\Sigma\) and intersect transversely. Unless
+explicitly stated otherwise, the curves \(\alpha, \beta\) are assumed to
+be \emph{unoriented} -- i.e. we regard them as subsets of \(\Sigma\).
 
 Despite the apparent clarity of the picture painted by
 Theorem~\ref{thm:classification-of-surfaces}, there are still plenty of
 interesting, sometimes unanswered, questions about surfaces and their
-homeomorphisms. For instance, it is interesting to consider how the
-classification of surfaces informs the geometry of the curves in \(\Sigma\).
+homeomorphisms. For instance, we can use the classification of surfaces to
+deduce information about how different curves in \(\Sigma\) are related by its
+homeomorphisms.
 
 \begin{observation}[Change of coordinates principle]
-  Given oriented nonseparating simple closed curves \(\alpha, \beta \subset
-  \Sigma = \Sigma_{g, r}^b\), we can find an orientation-preserving
-  homeomorphism \(\phi : \Sigma \isoto \Sigma\) fixing \(\partial \Sigma\)
-  pointwise such that \(\phi(\alpha) = \beta\) with orientation. To see this,
-  we consider the surface \(\Sigma_\alpha\) obtained by cutting \(\Sigma\)
-  across \(\alpha\): we subtract the curve \(\alpha\) from \(\Sigma\) and then
-  add one additional boundary component \(\delta_i\) in each side of
-  \(\alpha\), as shown in Figure~\ref{fig:change-of-coordinates}. By
-  identifying \(\delta_1\) with \(\delta_2\) we can see \(\Sigma\) as a
-  quotient of \(\Sigma_\alpha\). Since \(\alpha\) is nonseparating,
-  \(\Sigma_\alpha\) is a connected surface of genus \(g - 1\). In other words,
-  \(\Sigma_\alpha \cong \Sigma_{g-1,r}^{b+2}\). Similarly, \(\Sigma_\beta \cong
-  \Sigma_{g-1, r}^{b+2}\) also has two additional boundary components
-  \(\delta_1', \delta_2' \subset \partial \Sigma_\beta\). Now by the
-  classification of surfaces we can find an orientation-preserving
-  homeomorphism \(\tilde\phi : \Sigma_\alpha \isoto \Sigma_\beta\). Even more
-  so, we can choose \(\tilde\phi\) taking \(\delta_i\) to \(\delta_i'\). The
-  homeomorphism \(\tilde\phi\) then descends to a self-homeomorphism \(\phi\)
-  the quotient surface \(\Sigma \cong \mfrac{\Sigma_\alpha}{\sim} \cong
-  \mfrac{\Sigma_\beta}{\sim}\) with \(\phi(\alpha) = \beta\), as desired.
+  Given oriented nonseparating simple closed curves \(\alpha, \beta :
+  \mathbb{S}^1 \to \Sigma = \Sigma_{g, r}^b\), we can find an
+  orientation-preserving homeomorphism \(\phi : \Sigma \isoto \Sigma\) fixing
+  \(\partial \Sigma\) pointwise such that \(\phi(\alpha) = \beta\) with
+  orientation. To see this, we consider the surface \(\Sigma_\alpha\) obtained
+  by cutting \(\Sigma\) across \(\alpha\): we subtract the curve \(\alpha\)
+  from \(\Sigma\) and then add one additional boundary component \(\delta_i\)
+  in each side of \(\alpha\), as shown in
+  Figure~\ref{fig:change-of-coordinates}. By identifying \(\delta_1\) with
+  \(\delta_2\) we can see \(\Sigma\) as a quotient of \(\Sigma_\alpha\). Since
+  \(\alpha\) is nonseparating, \(\Sigma_\alpha\) is a connected surface of
+  genus \(g - 1\). In other words, \(\Sigma_\alpha \cong
+  \Sigma_{g-1,r}^{b+2}\). Similarly, \(\Sigma_\beta \cong \Sigma_{g-1,
+  r}^{b+2}\) also has two additional boundary components \(\delta_1', \delta_2'
+  \subset \partial \Sigma_\beta\). Now by the classification of surfaces we can
+  find an orientation-preserving homeomorphism \(\tilde\phi : \Sigma_\alpha
+  \isoto \Sigma_\beta\). Even more so, we can choose \(\tilde\phi\) taking
+  \(\delta_i\) to \(\delta_i'\). The homeomorphism \(\tilde\phi\) then descends
+  to a self-homeomorphism \(\phi\) the quotient surface \(\Sigma \cong
+  \mfrac{\Sigma_\alpha}{\sim} \cong \mfrac{\Sigma_\beta}{\sim}\) with
+  \(\phi(\alpha) = \beta\), as desired.
 \end{observation}
 
 \begin{figure}[ht]
@@ -65,34 +69,41 @@ classification of surfaces informs the geometry of the curves in \(\Sigma\).
   \label{fig:change-of-coordinates}
 \end{figure}
 
-A very similar argument goes to show\dots
+By splitting \(\Sigma\) across curves \(\alpha, \alpha' \subset \Sigma\)
+crossing once, we can also show\dots
 
 \begin{observation}\label{ex:change-of-coordinates-crossing}
-  Let \(\alpha, \beta, \alpha', \beta' \subset \Sigma\) be nonseparating curve
+  Let \(\alpha, \beta, \alpha', \beta' \subset \Sigma\) be nonseparating curves
   such that each pair \((\alpha, \alpha'), (\beta, \beta')\) crosses exactly
   once. Then we can find an orientation-preserving \(\phi : \Sigma \isoto
-  \Sigma\) fixing \(\partial \Sigma\) poitwise such that \(\phi(\alpha) =
-  \beta\) and \(\phi(\alpha') = \beta'\).
+  \Sigma\) fixing \(\partial \Sigma\) pointwise such that \(\phi(\alpha) =
+  \beta\) and \(\phi(\alpha') = \beta'\) -- without orientation.
 \end{observation}
 
 Given a surface \(\Sigma\), the group \(\Homeo^+(\Sigma, \partial \Sigma)\) of
-orientation-preserving homeomorphism of \(\Sigma\) fixing each point in
+orientation-preserving homeomorphisms of \(\Sigma\) fixing each point in
 \(\partial \Sigma\) is a topological group\footnote{Here we endow
 \(\Homeo^+(\Sigma, \partial \Sigma)\) with the compact-open topology.} with a
 rich geometry, but its algebraic structure is often regarded as too complex to
 tackle. More importantly, all of this complexity is arguably unnecessary for
 most topological applications, in the sense that usually we are only really
-interested in considering \emph{homeomorphisms up to isotopy}.
-
-For example, given \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\), it is well
-known that the diffeomorphism class of the mapping torus \(M_\phi =
-\mfrac{\Sigma \times [0, 1]}{(x, 0) \sim (\phi(x), 1)}\) -- a fundamental
-construction in low-dimensional topology -- is invariant under isotopy. This
-fact underspins some of the steps in Thurston's geometrization of
-\(3\)-manifolds. It is thus more natural to consider the group of connected
-components of \(\Homeo^+(\Sigma, \partial \Sigma)\), a countable discrete group
-known as \emph{the mapping class group}. This will be the focus of the
-dissertation at hand.
+interested in considering \emph{homeomorphisms up to isotopy}. For example\dots
+\begin{enumerate}
+  \item Isotopic \(\phi \simeq \psi \in \Homeo^+(\Sigma, \partial \Sigma)\)
+    determine the same application \(\phi_* = \psi_* : \pi_1(\Sigma, x) \to
+    \pi_1(\Sigma, x)\) and \(\phi_* = \psi_* : H_1(\Sigma, \mathbb{Z}) \to
+    H_1(\Sigma, \mathbb{Z})\)
+    at the levels of homotopy and homology.
+
+  \item The diffeomorphism class of the mapping torus \(M_\phi = \mfrac{\Sigma
+    \times [0, 1]}{(x, 0) \sim (\phi(x), 1)}\) -- a fundamental construction in
+    low-dimensional topology -- is invariant under isotopy.
+\end{enumerate}
+
+It is thus more natural to consider the group of connected components of
+\(\Homeo^+(\Sigma, \partial \Sigma)\), a countable discrete group known as
+\emph{the mapping class group}. This will be the focus of the dissertation at
+hand.
 
 \begin{definition}\label{def:mcg}
   The \emph{mapping class group \(\Mod(\Sigma)\) of an orientable surface
@@ -106,7 +117,7 @@ dissertation at hand.
 
 There are many variations of Definition~\ref{def:mcg}. For example\dots
 
-\begin{example}\label{ex:action-on-punctures}
+\begin{observation}\label{ex:action-on-punctures}
   Any \(\phi \in \Homeo^+(\Sigma, \partial \Sigma)\) extends uniquely to a
   homeomorphism \(\tilde\phi\) of \(\widebar\Sigma\) that permutes the set
   \(\{x_1, \ldots, x_r\} = \widebar\Sigma \setminus \Sigma\) of punctures of
@@ -114,7 +125,7 @@ There are many variations of Definition~\ref{def:mcg}. For example\dots
   \ldots, x_r\}\) via \(f \cdot x_i = \tilde\phi(x_i)\) for \(f = [\phi] \in
   \Mod(\Sigma)\) -- which is independent of the choice of representative
   \(\phi\) of \(f\).
-\end{example}
+\end{observation}
 
 % TODO: Change this notation?
 \begin{definition}
@@ -125,27 +136,28 @@ There are many variations of Definition~\ref{def:mcg}. For example\dots
   the subgroup of mapping classes that fix every puncture of \(\Sigma\).
 \end{definition}
 
-\begin{example}\label{ex:action-on-curves}
-  Given a simple closed curve \(\alpha \subset \Sigma\), denote by
-  \(\vec{[\alpha]}\) and \([\alpha]\) the isotopy classes of \(\alpha\) with
-  and without orientation, respectively -- i.e \(\vec{[\alpha]} =
-  \vec{[\beta]}\) if \(\alpha \simeq \beta\) as functions and \([\alpha] =
+\begin{observation}\label{ex:action-on-curves}
+  Given an oriented simple closed curve \(\alpha : \mathbb{S}^1 \to \Sigma\),
+  denote by \(\vec{[\alpha]}\) and \([\alpha]\) the isotopy classes of
+  \(\alpha\) with and without orientation, respectively -- i.e \(\vec{[\alpha]}
+  = \vec{[\beta]}\) if \(\alpha \simeq \beta\) as functions and \([\alpha] =
   [\beta]\) if \(\vec{[\alpha]} = \vec{[\beta]}\) or \(\vec{[\alpha]} =
   \vec{[\beta^{-1}]}\). There are natural actions \(\Mod(\Sigma) \leftaction \{
-  \vec{[\alpha]} : \alpha \subset \Sigma \}\) and \(\Mod(\Sigma) \leftaction \{
-  [\alpha] : \alpha \subset \Sigma \}\) given by
+    \vec{[\alpha]} \, | \, \alpha : \mathbb{S}^1 \to \Sigma \}\) and
+  \(\Mod(\Sigma) \leftaction \{ [\alpha] \, | \, \alpha \subset \Sigma \}\)
+  given by
   \begin{align*}
     f \cdot \vec{[\alpha]} & = \vec{[\phi(\alpha)]} &
     f \cdot [\alpha]       & = [\phi(\alpha)]
   \end{align*}
   for \(f = [\phi] \in \Mod(\Sigma)\).
-\end{example}
+\end{observation}
 
 \begin{definition}
   Given a simple closed curve \(\alpha \subset \Sigma\), we denote by
   \(\Mod(\Sigma)_{\vec{[\alpha]}}\) and \(\Mod(\Sigma)_{[\alpha]}\) the
-  subgroups of mapping classes that fix \(\vec{[\alpha]}\) and \([\alpha]\),
-  respectively.
+  subgroups of mapping classes that fix \(\vec{[\alpha]}\) -- for any given
+  choice of orientation of \(\alpha\) -- and \([\alpha]\), respectively.
 \end{definition}
 
 While trying to understand the mapping class group of some surface \(\Sigma\),
@@ -169,10 +181,10 @@ mapping class groups.
 \end{example}
 
 \begin{example}[Capping homomorphism]\label{ex:capping-morphism}
-  Let \(\delta \subset \partial \Sigma\) be an oriented boundary component of
-  \(\Sigma\). We refer to the inclusion homomorphism \(\operatorname{cap} :
-  \Mod(\Sigma) \to \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}))\) as
-  \emph{the capping homomorphism}.
+  Let \(\delta \subset \partial \Sigma\) be a boundary component of \(\Sigma\).
+  We refer to the inclusion homomorphism \(\operatorname{cap} : \Mod(\Sigma)
+  \to \Mod(\Sigma \cup_\delta (\mathbb{D}^2 \setminus \{0\}))\) as \emph{the
+  capping homomorphism}.
 \end{example}
 
 \begin{example}[Cutting homomorphism]\label{ex:cutting-morphism}
@@ -191,8 +203,9 @@ mapping class groups.
 
 As goes for most groups, another approach to understanding the mapping class
 group of a given surface \(\Sigma\) is to study its actions. We have already
-seen simple example of such actions in Example~\ref{ex:action-on-punctures} and
-Example~\ref{ex:action-on-curves}. A particularly important class of actions of
+seen simple examples of such actions in
+Observation~\ref{ex:action-on-punctures} and
+Observation~\ref{ex:action-on-curves}. An important class of actions of
 \(\Mod(\Sigma)\) are its \emph{linear representations} -- i.e. the group
 homomorphisms \(\Mod(\Sigma) \to \GL_n(\mathbb{C})\). These may be seen as
 actions \(\Mod(\Sigma) \leftaction \mathbb{C}^n\) where each \(f \in
@@ -234,7 +247,7 @@ Here we collect a few fundamental examples of linear representations of
   \(\Mod(\Sigma_g) \leftaction H_1(\Sigma_g, \mathbb{Z}) \cong
   \mathbb{Z}^{2g}\) given by \(f \cdot [\alpha] = \phi_*([\alpha]) =
   [\phi(\alpha)]\). Since pushforwards by orientation-preserving homeomorphisms
-  preserve the index of intersection points, \((f \cdot [\alpha]) \cdot (f
+  preserve the indices of intersection points, \((f \cdot [\alpha]) \cdot (f
   \cdot [\beta]) = [\alpha] \cdot [\beta]\) for all \(\alpha, \beta \subset
   \Sigma_g\) and \(f \in \Mod(\Sigma_g)\). In light of
   (\ref{eq:symplectic-form}), this implies \(\Mod(\Sigma_g)\) acts on
@@ -266,7 +279,7 @@ The symplectic representation already allows us to compute some important
 examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
 \Sigma_1\) and the once-punctured torus \(\Sigma_{1, 1}\).
 
-\begin{example}[Alexander trick]\label{ex:alexander-trick}
+\begin{observation}[Alexander trick]\label{ex:alexander-trick}
   The group \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) of homeomorphisms of the
   unit disk \(\mathbb{D}^2 \subset \mathbb{C}\) is contractible. In particular,
   \(\Mod(\mathbb{D}^2) = 1\). Indeed, for any \(\phi \in
@@ -282,13 +295,13 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   that ``fixes the band \(\{ z \in \mathbb{D}^2 : |z| \ge 1 - t \}\) and does
   \(\phi\) inside the sub-disk \(\{ z \in \mathbb{D}^2 : |z| \le 1 - t\}\)''
   joins \(\phi = \phi_0\) and \(1 = \phi_1\).
-\end{example}
+\end{observation}
 
-\begin{example}\label{ex:mdg-once-punctured-disk}
+\begin{observation}\label{ex:mdg-once-punctured-disk}
   By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\).
-\end{example}
+\end{observation}
 
-\begin{example}[$\Mod(\mathbb{T}^2)$]\label{ex:torus-mcg}
+\begin{observation}[Linearity of $\Mod(\mathbb{T}^2)$]\label{ex:torus-mcg}
   The symplectic representation \(\psi : \Mod(\mathbb{T}^2) \to
   \operatorname{Sp}_2(\mathbb{Z}) = \operatorname{SL}_2(\mathbb{Z})\) is a
   group isomorphism. In particular, \(\Mod(\mathbb{T}^2) \cong
@@ -307,11 +320,11 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   of the surface \(\mathbb{T}_{\alpha_1 \beta_1}^2 \cong \mathbb{D}^2\)
   obtained by cutting \(\mathbb{T}^2\) across \(\alpha_1\) and \(\beta_1\), as
   in Figure~\ref{fig:cut-torus-across}. Now by the Alexander trick from
-  Example~\ref{ex:alexander-trick}, \(\tilde\phi\) must be isotopic to the
+  Observation~\ref{ex:alexander-trick}, \(\tilde\phi\) must be isotopic to the
   identity. The isotopy \(\tilde\phi \simeq 1 \in \Homeo^+(\mathbb{D}^2,
   \mathbb{S}^1)\) then descends to an isotopy \(\phi \simeq 1 \in
   \Homeo^+(\mathbb{T}^2)\), so \(f = 1 \in \Mod(\mathbb{T}^2)\) as desired.
-\end{example}
+\end{observation}
 
 \begin{figure}[ht]
   \centering
@@ -322,10 +335,10 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   \label{fig:cut-torus-across}
 \end{figure}
 
-\begin{example}\label{ex:punctured-torus-mcg}
+\begin{observation}\label{ex:punctured-torus-mcg}
   By the same token, \(\Mod(\Sigma_{1, 1}) \cong
   \operatorname{SL}_2(\mathbb{Z})\).
-\end{example}
+\end{observation}
 
 \begin{remark}
   Despite the fact \(\psi : \Mod(\mathbb{T}^2) \to
@@ -333,7 +346,7 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   representation is \emph{not} injective for surfaces of genus \(g \ge 2\) --
   see \cite[Section~6.5]{farb-margalit} for a description of its kernel.
   Korkmaz and Bigelow-Budney \cite{korkmaz-linearity, bigelow-budney} showed
-  there exists injective linear representations of \(\Mod(\Sigma_2)\), but the
+  there exist injective linear representations of \(\Mod(\Sigma_2)\), but the
   question of linearity of \(\Mod(\Sigma_g)\) remains wide-open for \(g \ge
   3\). Recently, Korkmaz \cite[Theorem~3]{korkmaz} established the lower bound
   of \(3 g - 3\) for the dimension of an injective representation of

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -174,7 +174,7 @@ sequence
   \end{tikzcd}
 \end{center}
 given that \(\Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) is contractible by
-Example~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}^2) = 1\). Hence we
+Observation~\ref{ex:alexander-trick}. But \(\Mod(\mathbb{D}^2) = 1\). Hence we
 get\dots
 
 \begin{proposition}

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -12,9 +12,9 @@ a convenient generating set for \(\Mod(\Sigma)\), known as the set of
 The idea here is to reproduce the proof of injectivity in
 Example~\ref{ex:torus-mcg}: by cutting across curves and arcs, we can always
 decompose a surface into copies of \(\mathbb{D}^2\) and \(\mathbb{D}^2
-\setminus \{0\}\). Example~\ref{ex:alexander-trick} and
-Example~\ref{ex:mdg-once-punctured-disk} then imply the triviality of mapping
-classes fixing such arcs and curves. Formally, this translates to\dots
+\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\dots
 
 \begin{proposition}[Alexander method]\label{thm:alexander-method}
   Let \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) be essential simple closed
@@ -425,12 +425,12 @@ Theorem~\ref{thm:mcg-is-fg}.
   \setminus \{0\}))\) to Dehn twists about the corresponding curves in
   \(\Sigma_{g, r}^b\) and add \(\tau_{\delta_1}\) to the generating set.
 
-  It thus suffices to consider the boundaryless case \(\Sigma_{g, r}\). As promised,
-  we proceed by double induction on \(r\) and \(g\). For the base case, it is
-  clear from Example~\ref{ex:torus-mcg} and Example~\ref{ex: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
+  It thus suffices to consider the boundaryless case \(\Sigma_{g, r}\). As
+  promised, we proceed by double induction on \(r\) and \(g\). 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*}