Clean up the lens space section

main.tex

@@ -24,12 +24,13 @@
 \DeclareMathOperator{\lensop}{L}
 \DeclareMathOperator{\rotmatop}{R}
 \DeclareMathOperator{\soop}{SO}
-\newcommand*{\lens}[2]{\textcolor{darkyellow}{\lensop\left(\textcolor{black}{#1},\textcolor{black}{#2}\right)}} %
+\newcommand*{\lens}[2]{\lensop\left(#1,#2\right)} %
 \newcommand*{\so}[1]{\soop\left(#1\right)}
 \newcommand*{\rotmat}[1]{\textcolor{red}{\rotmatop\left(\textcolor{black}{#1}\right)}}
 \renewcommand{\S}{\mathbb{S}}
 \newcommand{\R}{\mathbb{R}}
 \newcommand{\sparkles}{\includegraphics[height=0.9em]{sparkles.png}}
+\newcommand{\ghostzero}{\textcolor{lightgray}{0}}
 
 % cool color
 

prerequisites.tex

@@ -27,30 +27,35 @@
 	\end{itemize}
 \end{frame}
 
-\begin{frame}{Lens Spaces}
+\begin{frame}{Lens Spaces — the Explicit Construction}
 	\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}
+	\pause
 	\begin{definition}[Lens space]
-		Given $q \in \mathbb Z$ and $s \in \mathbb Z ^n$ elementwise coprime with $q$
+		Given coprime $p, z \in \mathbb Z$, we define
 		\begin{align*}
-			\lens q s \coloneq \S^{2n-1}/\langle M_{q,s} \rangle,
+			\lens p q \coloneq \S^3/\langle M \rangle,
 		\end{align*}
-		where $\langle M_{q,s} \rangle$ is the group generated by
+		where $\langle M \rangle$ 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}.
+			M \coloneq
+			\begin{bmatrix*}[r]
+				\cos{(2 \pi / p)} & \sin{(-2 \pi / p)} & \ghostzero          & \ghostzero           \\
+				\sin{(2 \pi / p)} & \cos{(2 \pi / p)}  & \ghostzero          & \ghostzero           \\
+				\ghostzero        & \ghostzero         & \cos{(2 \pi q / p)} & \sin{(-2 \pi q / p)} \\
+				\ghostzero        & \ghostzero         & \sin{(2 \pi q / p)} & \cos{(2 \pi q / p)}
+			\end{bmatrix*}.
+			% \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}
 
-	\pause
-	In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb Z$.
+	% \pause
+	% In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb Z$.
 \end{frame}