\section{Prerequisites}
\begin{frame}{Manifolds \& Homotopy Groups}
	\begin{figure}[H]
		\centering
		\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
		\caption{The prototypical example of a manifold a mug [13].}
		\label{fig:mug-neighbourhoods}
	\end{figure}
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	\begin{figure}[H]
		\centering
		%$ \captionsetup{width=.75\linewidth}
		\includegraphics[width=0.2\linewidth]{Contractible loops.png}
		\caption{Two double tori with (non)-contractible paths [7].}
		\label{fig:CoLoop}
	\end{figure}
\end{frame}

\begin{frame}{Quotients of the $3$-sphere}
	\begin{itemize}
		\item $\so 4$ is isomorphic to the isometry group of $\S^3$.
		\item Subgroups $\Gamma \leqslant \so 4$ define \textcolor{darkyellow}{equivalence classes} of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^4$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
		      \pause
		\item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$.
		\item This can be easily generalized to the $n$-sphere.
	\end{itemize}
\end{frame}

\begin{frame}{Lens Spaces}
	\begin{itemize}
		\item Lens spaces are obtained by taking the quotient of some $n$-sphere by a cyclic group.
		\item They cannot be distinguished by their homotopy group alone.
	\end{itemize}
\end{frame}
\begin{frame}{Lens Spaces — the Explicit Construction}
	\begin{definition}[Lens space]
		Given $q \in \mathbb Z$ and $s \in \mathbb Z ^n$ elementwise coprime with $q$
		\begin{align*}
			\lens q s \coloneq \S^{2n-1}/\langle M_{q,s} \rangle,
		\end{align*}
		where $\langle M_{q,s} \rangle$ is the group generated by
		\begin{align*}
			M_{q,s} \coloneq
			\begin{pmatrix}
				\rotmat{2 \pi s_1 / q} &                        &        &                        \\
				                       & \rotmat{2 \pi s_2 / q} &        &                        \\
				                       &                        & \ddots &                        \\
				                       &                        &        & \rotmat{2 \pi s_n / q}
			\end{pmatrix}.
		\end{align*}
	\end{definition}

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	In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb Z$.
\end{frame}