54 lines
2.1 KiB
TeX
54 lines
2.1 KiB
TeX
\section{Prerequisites}
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\begin{frame}{Manifolds \& Homotopy groups}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
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\caption{The prototypical example of a manifold a mug [13].}
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\label{fig:mug-neighbourhoods}
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\end{figure}
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\begin{figure}[H]
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\centering
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%$ \captionsetup{width=.75\linewidth}
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\includegraphics[width=0.2\linewidth]{Contractible loops.png}
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\caption{Two double tori with (non)-contractible paths [7].}
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\label{fig:CoLoop}
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\end{figure}
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\end{frame}
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\begin{frame}{Quotients of the $3$-sphere}
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\begin{itemize}
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\item $\so 4$ is isomorphic to the isometry group of $\S^3$.
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\item Subgroups $\Gamma \leqslant \so 4$ define equivalence classes of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^3$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
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\item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$.
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\item This can be easily generalised to the $n$-sphere.
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\end{itemize}
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\end{frame}
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\begin{frame}{Lens spaces}
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\begin{itemize}
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\item Lens spaces are obtained by taking the quotient of some $n$-sphere by a cyclic group.
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\item They cannot be distinguished by their homotopy group alone.
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\end{itemize}
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\end{frame}
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\begin{frame}{Lens spaces — the explicit construction}
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\begin{definition}[Lens space]
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Given $q \in \mathbb Z$ and $s \in \mathbb Z ^n$ elementwise coprime with $q$
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\begin{align*}
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\lens q s \coloneq \S^{2n-1}/\langle M_{q,s} \rangle,
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\end{align*}
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where $\langle M_{q,s} \rangle$ is the group generated by
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\begin{align*}
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M_{q,s} \coloneq
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\begin{pmatrix}
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\rotmat{2 \pi s_1 / q} & & & \\
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& \rotmat{2 \pi s_2 / q} & & \\
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& & \ddots & \\
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& & & \rotmat{2 \pi s_n / q}
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\end{pmatrix}
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\end{align*}
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\end{definition}
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In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb Z$.
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\end{frame}
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