memoire-m2

My M2 Memoire on mapping class groups & their representations

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sections/introduction.tex	23839B	-rw-r--r--

\chapter{Introduction}\label{ch:introduction}

Ever since ancestral humans first stepped foot on the surface of Earth, Mankind
has pondered the shape of the planet we inhabit. More recently, mathematicians
have spent the past centuries trying to understand the topology of manifolds
and, in particular, surfaces. Orientable compact surfaces were perhaps first
classified by Gauss in the early 19th century. The proof of the following
formulation of the classification, often attributed to Möbius, was completed in
the 1920s with the work of Radò and others.

\begin{theorem}[Classification of surfaces]\label{thm:classification-of-surfaces}
  Any closed connected orientable surface is homeomorphic to the connected sum
  \(\Sigma_g\) of the sphere \(\mathbb{S}^2\) with \(g \ge 0\) copies of the
  torus \(\mathbb{T}^2 = \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\). Any compact
  connected orientable surface \(\Sigma\) is homeomorphic to the surface
  \(\Sigma_g^b\) obtained from \(\Sigma_g\) by removing \(b \ge 0\) open disks
  with disjoint closures.
\end{theorem}

The integer  \(g \ge 0\) in Theorem~\ref{thm:classification-of-surfaces} is
called \emph{the genus of \(\Sigma\)}. We also have the noncompact surface
\(\Sigma_{g, r}^b = \Sigma_g^b \setminus \{x_1, \ldots, x_r\}\), where \(x_1,
\ldots, x_r\) lie in the interior of \(\Sigma_g^b\). The points \(x_1, \ldots,
x_r\) are called the \emph{punctures} of \(\Sigma_{g, r}^b\). Throughout these
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. 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, 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 \(\alpha, \beta :
  \mathbb{S}^1 \to \Sigma = \Sigma_{g, r}^b\), 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 \(\Sigma\) along \(\alpha\): 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 \(\phi\) of 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]
  \centering
  \includegraphics[width=.5\linewidth]{images/change-of-coords-cut.eps}
  \caption{The surface $\Sigma_\alpha \cong \Sigma_{g-1, r}^{b+2}$ for a
  certain $\alpha \subset \Sigma$.}
  \label{fig:change-of-coordinates}
\end{figure}

By cutting \(\Sigma\) along curves \(\alpha, \alpha' \subset \Sigma\) crossing
once, we can also show the following result.

\begin{observation}\label{ex:change-of-coordinates-crossing}
  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 \(\partial \Sigma\) 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 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,
\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
  \(\Sigma\)} is the group of isotopy classes of orientation-preserving
  homeomorphisms \(\Sigma \isoto \Sigma\), where both the homeomorphisms and
  the isotopies are assumed to fix \(\partial \Sigma\) pointwise.
  \[
    \Mod(\Sigma) = \mfrac{\Homeo^+(\Sigma, \partial \Sigma)}{\simeq}
  \]
\end{definition}

There are many variations of Definition~\ref{def:mcg}.

\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
  \(\Sigma\). We may thus define an action \(\Mod(\Sigma) \leftaction \{x_1,
  \ldots, x_r\}\) 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{observation}

\begin{definition}
  Given an orientable surface \(\Sigma\) and a puncture \(x \in
  \widebar\Sigma\) of \(\Sigma\), denote by \(\Mod(\Sigma, x) \subset
  \Mod(\Sigma)\) the subgroup of mapping classes that fix \(x\). The \emph{pure
  mapping class group \(\PMod(\Sigma) \subset \Mod(\Sigma)\) of \(\Sigma\)} is
  the subgroup of mapping classes that fix every puncture of \(\Sigma\).
\end{definition}

\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 \(\Mod(\Sigma) \leftaction \{
    \vec{[\alpha]} \, | \, \alpha : \mathbb{S}^1 \to \Sigma \}\) 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{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 \(\vec{[\alpha]}\) -- for any given
  choice of orientation of \(\alpha\) -- and \([\alpha]\), respectively.
\end{definition}

While trying to understand the mapping class group of some surface \(\Sigma\),
it is interesting to consider how the geometric relationship between \(\Sigma\)
and other surfaces affects \(\Mod(\Sigma)\). Indeed, different embeddings
\(\Sigma' \hookrightarrow \Sigma\) translate to homomorphisms at the level of
mapping class groups.

\begin{example}[Inclusion homomorphism]\label{ex:inclusion-morphism}
  Let \(\Sigma' \subset \Sigma\) be a closed subsurface. Given \(\phi \in
  \Homeo^+(\Sigma', \partial \Sigma')\), we may extend \(\phi\) to
  \(\tilde{\phi} \in \Homeo^+(\Sigma, \partial \Sigma)\) by setting
  \(\tilde{\phi}(x) = x\) for \(x \in \Sigma\) outside of \(\Sigma'\) -- which
  is well defined since \(\phi\) fixes every point in \(\partial \Sigma'\).
  This construction yields a group homomorphism
  \begin{align*}
    \Mod(\Sigma') & \to \Mod(\Sigma) \\
     [\phi] & \mapsto [\tilde\phi],
  \end{align*}
  known as \emph{the inclusion homomorphism}.
\end{example}

\begin{example}[Capping homomorphism]\label{ex:capping-morphism}
  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}
  Given a simple closed curve \(\alpha \subset \Sigma\), any \(f \in
  \Mod(\Sigma_{g+1})_{\vec{[\alpha]}}\) has a representative \(\phi \in
  \Homeo^+(\Sigma, \partial \Sigma)\) fixing \(\alpha\) point-wise -- so that
  \(\phi\) restricts to a homeomorphism of \(\Sigma \setminus \alpha\).
  Furthermore, if \(\phi\!\restriction_{\Sigma \setminus \alpha} \simeq 1\) in
  \(\Sigma \setminus \alpha\) then \(\phi \simeq 1 \in \Homeo^+(\Sigma,
  \partial \Sigma)\) -- see \cite[Proposition~3.20]{farb-margalit}. There is
  thus a group homomorphism
  \begin{align*}
    \operatorname{cut} : \Mod(\Sigma)_{\vec{[\alpha]}}
    & \to \Mod(\Sigma\setminus\alpha) \\
    [\phi] & \mapsto [\phi\!\restriction_{\Sigma \setminus \alpha}],
  \end{align*}
  known as \emph{the cutting homomorphism}.
\end{example}

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 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 \(f \in
\Mod(\Sigma)\) acts via some linear isomorphism \(\mathbb{C}^n \isoto
\mathbb{C}^n\).

\section{Representations}

Here we collect a few fundamental examples of linear representations of
\(\Mod(\Sigma)\).

\begin{observation}
  Recall \(H_1(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}^{2g}\), with standard
  basis given by \([\alpha_1], [\beta_1], \ldots, [\alpha_g], [\beta_g] \in
  H_1(\Sigma_g, \mathbb{Z})\) for \(\alpha_1, \ldots, \alpha_g, \beta_1,
  \ldots, \beta_g\) as in Figure~\ref{fig:homology-basis}. The Abelian group
  \(H_1(\Sigma_g, \mathbb{Z})\) is endowed with a natural
  \(\mathbb{Z}\)-bilinear alternating form given by the \emph{algebraic
  intersection number} \([\alpha] \cdot [\beta] = \sum_{x \in \alpha \cap
  \beta} \operatorname{ind}\,x\) -- where the index \(\operatorname{ind}\,x =
  \pm 1\) of an intersection point is given by
  Figure~\ref{fig:intersection-index}. In terms of the standard basis of
  \(H_1(\Sigma_g, \mathbb{Z})\), this form is given by
  \begin{align}\label{eq:symplectic-form}
    [\alpha_i] \cdot [\beta_j]  & = \delta_{i j} &
    [\alpha_i] \cdot [\alpha_j] & = 0            &
    [\beta_i]  \cdot [\beta_j]  & = 0
  \end{align}
  and thus coincides with the pullback of the standard \(\mathbb{Z}\)-bilinear
  symplectic form in \(\mathbb{Z}^{2g}\).
\end{observation}

\begin{example}[Symplectic representation]\label{ex:symplectic-rep}
  Given \(f = [\phi] \in \Mod(\Sigma_g)\), we may consider the map \(\phi_* :
  H_1(\Sigma_g, \mathbb{Z}) \to H_1(\Sigma_g, \mathbb{Z})\) induced at the
  level of singular homology. By homotopy invariance, the map \(\phi_*\) is
  independent of the choice of representative \(\phi\) of \(f\). By the
  functoriality of homology groups we then get a \(\mathbb{Z}\)-linear action
  \(\Mod(\Sigma_g) \leftaction H_1(\Sigma_g, \mathbb{Z}) \cong
  \mathbb{Z}^{2g}\) given by \(f \cdot [\alpha] = \phi_*([\alpha]) =
  [\phi(\alpha)]\). Since pushforwards by orientation-preserving homeomorphisms
  preserve the 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
  \(\mathbb{Z}^{2g}\) via symplectomorphisms. We thus obtain a group
  homomorphism \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})
  \subset \GL_{2g}(\mathbb{C})\), known as \emph{the symplectic representation
  of \(\Mod(\Sigma_g)\)}.
\end{example}

\noindent
\begin{minipage}[b]{.47\linewidth}
  \centering
  \includegraphics[width=.9\linewidth]{images/homology-generators.eps}
  \captionof{figure}{The curves $\alpha_1, \beta_1, \ldots, \alpha_g, \beta_g
  \subset \Sigma_g$ that generate its first homology group.}
  \label{fig:homology-basis}
\end{minipage}
\hspace{.6cm} %
\begin{minipage}[b]{.47\linewidth}
  \centering
  \includegraphics[width=.9\linewidth]{images/intersection-index.eps}
  \vspace*{.75cm}
  \captionof{figure}{The index of an intersection point $x \in \alpha \cap
  \beta$.}
  \label{fig:intersection-index}
\end{minipage}

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{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 \(\phi \in
  \Homeo^+(\mathbb{D}^2, \mathbb{S}^1)\) the isotopy
  \begin{align*}
    \phi_t : \mathbb{D}^2 & \to     \mathbb{D}^2 \\
                        z & \mapsto
    \begin{cases}
      (1 - t) \phi(\sfrac{z}{1 - t}) & \text{if } 0 \le |z| \le 1 - t \\
      z                              & \text{otherwise}
    \end{cases}
  \end{align*}
  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{observation}

\begin{observation}\label{ex:mdg-once-punctured-disk}
  By the same token, \(\Mod(\mathbb{D}^2 \setminus \{0\}) = 1\).
\end{observation}

\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, \(\Mod(\mathbb{T}^2) \cong
  \operatorname{SL}_2(\mathbb{Z})\). To see \(\psi\) is surjective, first
  observe \(\mathbb{Z}^2 \subset \mathbb{R}^2\) is
  \(\operatorname{SL}_2(\mathbb{Z})\)-invariant. Hence any matrix \(g \in
  \operatorname{SL}_2(\mathbb{Z})\) descends to an orientation-preserving
  homeomorphism \(\phi_g\) of the quotient \(\mathbb{T}^2 =
  \mfrac{\mathbb{R}^2}{\mathbb{Z}^2}\), which satisfies \(\psi([\phi_g]) = g\).
  To see \(\psi\) is injective we consider the curves \(\alpha_1\) and
  \(\beta_1\) from Figure~\ref{fig:homology-basis}. Given \(f = [\phi] \in
  \Mod(\mathbb{T}^2)\) with \(\psi(f) = 1\), \(f \cdot \vec{[\alpha_1]} =
  \vec{[\alpha_1]}\) and \(f \cdot \vec{[\beta_1]} = \vec{[\beta_1]}\), so we
  may choose a representative \(\phi\) of \(f\) fixing \(\alpha_1 \cup
  \beta_1\) pointwise. Such \(\phi\) determines a homeomorphism \(\tilde \phi\)
  of the surface \(\mathbb{T}_{\alpha_1 \beta_1}^2 \cong \mathbb{D}^2\)
  obtained by cutting \(\mathbb{T}^2\) along \(\alpha_1\) and \(\beta_1\), as
  in Figure~\ref{fig:cut-torus-along}. Now by the Alexander trick from
  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{observation}

\begin{figure}[ht]
  \centering
  \includegraphics[width=.55\linewidth]{images/torus-cut.eps}
  \caption{By cutting $\mathbb{T}^2$ along $\alpha_1$ we obtain a cylinder,
  where $\beta_1$ determines a yellow arc joining the two boundary components.
  Now by cutting along this yellow arc we obtain a disk.}
  \label{fig:cut-torus-along}
\end{figure}

\begin{observation}\label{ex:punctured-torus-mcg}
  By the same token, \(\Mod(\Sigma_{1, 1}) \cong
  \operatorname{SL}_2(\mathbb{Z})\).
\end{observation}

\begin{remark}
  Despite the fact \(\psi : \Mod(\mathbb{T}^2) \to
  \operatorname{SL}_2(\mathbb{Z})\) is an isomorphism, the symplectic
  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 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
  \(\Mod(\Sigma_g)\) in the \(g \ge 3\) case -- if one such representation
  exists.
\end{remark}

Another fundamental class of examples of representations are the so-called
\emph{TQFT representations}.

\begin{definition}
  A \emph{cobordism} between closed oriented surfaces \(\Sigma\) and
  \(\Sigma'\) is a triple \((W, \phi_+, \phi_-)\) where \(W\) is a smooth
  oriented compact \(3\)-manifold with \(\partial W = \partial_+ W \amalg
  \partial_- W\), \(\phi_+ : \Sigma \isoto \partial_+ W\) is an orientation
  preserving diffeomorphism and \(\phi_- : \Sigma' \isoto \partial_- W\) is an
  orientation-reversing diffeomorphism. We may abuse the notation and denote
  \(W = (W, \phi_+, \phi_-)\).
\end{definition}

\begin{definition}
  We denote by \(\Cob\) the category whose objects are (possibly disconnected)
  closed oriented surfaces and whose morphisms \(\Sigma \to \Sigma'\) are
  diffeomorphism classes\footnote{Here we only consider orientation-preserving
  diffeomorphisms $\varphi : W \isoto W'$ that are compatible with the boundary
  identifications in the sense that $\varphi(\partial_\pm W) = \partial_\pm W'$
  and $\psi_\pm = \varphi \circ \phi_\pm$.} of cobordisms between \(\Sigma\)
  and \(\Sigma'\), with composition given by
  \[
    [W, \phi_-, \phi_+] \circ [W', \psi_-, \psi_+]
    = [W \cup_{\psi_- \circ \phi_+^{-1}} W', \phi_-, \psi_+]
  \]
  for \([W, \phi_-, \phi_+] : \Sigma \to \Sigma'\) and \([W', \psi_-, \phi_+] :
  \Sigma' \to \Sigma''\). We endow \(\Cob\) with the monoidal structure given
  by
  \begin{align*}
    \Sigma \otimes \Sigma'
    & = \Sigma \amalg \Sigma' &
    [W,\phi_+,\phi_-] \otimes [W',\psi_+,\psi_-]
    & = [W \amalg W', \phi_+ \amalg \psi_+, \phi_- \amalg \psi_-].
  \end{align*}
\end{definition}

\begin{definition}[TQFT]\label{def:tqft}
  A \emph{topological quantum field theory} (abbreviated by \emph{TQFT})
  is a functor \(\mathcal{F} : \Cob \to \Vect\) satisfying
  \begin{align*}
    \mathcal{F}(\emptyset) & = \mathbb{C} &
    \mathcal{F}(\Sigma \otimes \Sigma')
    & = \mathcal{F}(\Sigma) \otimes \mathcal{F}(\Sigma') &
    \mathcal{F}([W] \otimes [W'])
    & = \mathcal{F}([W]) \otimes \mathcal{F}([W']),
  \end{align*}
  where \(\Vect\) denotes the category of finite-dimensional complex vector
  spaces.
\end{definition}

\begin{observation}
  Given \(\phi \in \Homeo^+(\Sigma_g)\), we may consider the so-called
  \emph{mapping cylinder} \(C_\phi = (\Sigma_g \times [0, 1], \phi, 1)\), a
  cobordism between \(\Sigma_g\) and itself -- where \(\partial_+ (\Sigma_g
  \times [0, 1]) = \Sigma_g \times 0\) and \(\partial_- (\Sigma_g \times [0,
  1]) = \Sigma_g \times 1\). The diffeomorphism class of \(C_\phi\) is
  independent of the choice of representative of \(f = [\phi] \in
  \Mod(\Sigma_g)\), so \(C_f = [C_\phi] : \Sigma_g \to \Sigma_g\) is a well
  defined morphism in \(\Cob\).
\end{observation}

\begin{example}[TQFT representations]\label{ex:tqft-reps}
  It is clear that \(C_1\) is the identity morphism \(\Sigma_g \to \Sigma_g\)
  in \(\Cob\). In addition, \(C_{f \cdot g} = C_f \circ C_g\) for all \(f, g
  \in \Mod(\Sigma_g)\) -- see \cite[Lemma~2.5]{costantino}. Now given a TQFT
  \(\mathcal{F} : \Cob \to \Vect\), by functoriality we obtain a linear
  representation
  \begin{align*}
    \rho_{\mathcal{F}} : \Mod(\Sigma_g) & \to \GL(\mathcal{F}(\Sigma_g)) \\
                                      f & \mapsto \mathcal{F}(C_f).
  \end{align*}
\end{example}

As simple as the construction in Example~\ref{ex:tqft-reps} is, in practice it
is not that easy to come across functors as the ones in
Definition~\ref{def:tqft}. This is because, in most interesting examples, we
are required to attach some extra data to our surfaces to get a well defined
association \(\Sigma_g \mapsto \mathcal{F}(\Sigma_g)\). Moreover, the condition
\(\mathcal{F}([W] \circ [W']) = \mathcal{F}([W]) \circ \mathcal{F}([W'])\) may
only hold up to multiplication by scalars.

Hence constructing an actual functor typically requires \emph{extending}
\(\Cob\) and \emph{tweaking} \(\Vect\). Such functors give rise to linear and
projective representations of the \emph{extended mapping class groups}
\(\Mod(\Sigma_g) \times \mathbb{Z}\). We refer the reader to \cite{costantino,
julien} for constructions of one such TQFT and its corresponding
representations: the so-called \emph{\(\operatorname{SU}_2\) TQFT of level
\(r\)}, first introduced by Witten and Reshetikhin-Tuarev \cite{witten,
reshetikhin-turaev} in their foundational papers on quantum topology.

Besides Example~\ref{ex:symplectic-rep} and Example~\ref{ex:tqft-reps}, not a
lot of other linear representations of \(\Mod(\Sigma_g)\) are known. Indeed,
the representation theory of mapping class groups remains a mystery at large.
In Chapter~\ref{ch:representations} we provide a brief overview of the field,
as well as some recent developments. More specifically, we highlight Korkmaz'
\cite{korkmaz} proof of the triviality of low-dimensional representations and
comment on his classification of \(2g\)-dimensional representations. To that
end, in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations} we survey
the group structure of mapping class groups: its relations and known
presentations.