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% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
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\begin{document}
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\lhead{1300F Geometry and Topology, Assignment 1}
\rhead{Due date: October 6, 2017}
\cfoot{}
\begin{exercise}
Prove that $\CC P^1$ is homeomorphic to $S^2$. Try proving this using the given coordinate charts.
\end{exercise}
\begin{exercise}
Consider the 3-sphere $S^3\subset \RR^4$. Using the isomorphism
$\RR^4 \cong \CC^2$, we obtain the inclusion $\iota: S^3\to
\CC^2\setminus\{0\}$. Composing with the projection map $\pi:\CC^2\setminus\{0\}\to
\CC P^1$, we obtain
\begin{equation*}
p = \pi\circ\iota: S^3\to \CC P^1,
\end{equation*}
known as the Hopf fibration. Using the coordinate charts given in the notes for $S^3$ and $\CC P^1$,
compute $p$ in coordinates (one chart on each of the domain and codomain will suffice).
\end{exercise}
\begin{exercise}\label{firstone}
Let $\Gamma$ be a group, and give it the discrete topology. Suppose
$\Gamma$ acts continuously on the topological $n$-manifold $M$, meaning
that the action map
\begin{equation*}
\xymatrix@R=1ex{\Gamma\times M\ar[r]^-\rho & M\\
(h,x)\ar@{|->}[r]& h\cdot x}
\end{equation*}
is continuous. Suppose also that the action is \emph{free},
i.e. the stabilizer of each point is trivial. Finally, suppose the
action is \emph{properly discontinuous}, meaning that each $x\in M$
has a neighbourhood $U$ such that $h\cdot U$ is disjoint from
$U$ for all nontrivial $h\in \Gamma$, that is, for all $h\neq 1$.
\begin{enumerate}[i)]
\item Show that the quotient map $\pi:M\to M/\Gamma$ is a local
homeomorphism, where $M/\Gamma$ is given the quotient
topology. Conclude that $M/\Gamma$ is locally homeomorphic to
$\RR^n$.
%\item Show that the quotient topology on $M/\Gamma$ is uniquely
% determined by the requirement that $\pi$ is a local homeomorphism.
\item Show that $\pi$ is an open map.
\item Give an example where $M/\Gamma$ is not Hausdorff.
\end{enumerate}
\end{exercise}
\begin{exercise}
Let $(\Gamma, M, \rho)$ be as in Exercise~\ref{firstone}, and let
$f:M\to N$ be a continuous map such that
\begin{equation*}
f(h\cdot x) = f(x)
\end{equation*}
for all $x\in M$ and $h\in\Gamma$. Show that there is a unique map
$\bar f:M/\Gamma\to N$ such that $\bar f(\pi(x)) = f(x)$ for all
$x\in M$, and show that it is continuous.
\end{exercise}
\begin{exercise}
Let $(\Gamma, M, \rho)$ be as in Exercise~\ref{firstone}. Prove
that $M/\Gamma$ is Hausdorff if and only if the image of the map
\begin{equation*}
\xymatrix@R=1ex{\Gamma\times M\ar[r]& M\times M\\
(g, x)\ar@{|->}[r]&(gx, x)}
\end{equation*}
is closed in $M\times M$.
\end{exercise}
\begin{exercise}\mbox{}
Let the group of order two, $C_2=\{1,-1\}$, act on $S^n$ via $x\mapsto
-x$. Show that $S^n/C_2$ is homeomorphic to the projective space
$\RR P^n$, as it was defined in class.
\end{exercise}
\begin{exercise}
Recall that in the description of $\RR P^3$, the space of 1-dimensional subspaces of $\RR^4$, we represented each point of $\RR P^3$ as the equivalence class
\[
[x_0:x_1:x_2:x_3] = [(x_0,x_1, x_2,x_3)]
\]
for the relation on 4-vectors defined by the action of the group $\RR^*$: that is, $x\sim y \Leftrightarrow y = \lambda x$ for $\lambda\in\RR^*$.
Each coordinate defines a hyperplane $H_i = \{x\in\RR^4\ :\ x_i=0\}$ and therefore an open set $U_i = \RR^4\backslash H_i$. We made these into coordinate charts by sending $x\in U_i$ to the 3-vector obtained by rescaling $x$ by $x_i^{-1}$ and deleting the $i^{\text{th}}$ coordinate (which has value 1 due to the rescaling).
We now apply the same strategy to study $Gr(2,4)$, the Grassmannian of 2-dimensional linear subspaces of $\RR^4$. Every point $P$ in the Grassmannian is a 2-dimensional subspace of $\RR^4$ and so we can choose a basis for it: write this basis as a $2\times 4$ matrix where the rows are the basis vectors:
\[
\begin{bmatrix}
a_1 & a_2 & a_3 & a_4\\
b_1 & b_2 & b_3 & b_4
\end{bmatrix}
\]
Notice that we are going back to the traditional way of numbering coordinates $(x_1, x_2, x_3, x_4)$ starting from $1$ rather than $0$.
\begin{enumerate}
\item Describe precisely what condition on the above $2\times 4$ matrix guarantees that its rows span a 2-dimensional subspace. Prove that such matrices form an open subset of all $2\times 4$ matrices.
\item What is the appropriate equivalence relation for such $2\times 4$ matrices? That is, when do two matrices represent the same point $P\in Gr(2,4)$? Express this equivalence relation as the action of a group.
\item Suppose we focus on the first coordinate $x_1$: it defines a hyperplane $H_1 = \{x\in\RR^4\ : x_1=0\}$. Note that the intersection of $P$ with $H_1$ must have dimension either 1 or 2. Suppose that $\dim P\cap H_1 = 1$. Show that this condition defines an open set in $Gr(2,4)$, and prove that any element of this open set can be described by a matrix of the form
\[
\begin{bmatrix}
1 & a_2 & a_3 & a_4\\
0 & b_2 & b_3 & b_4
\end{bmatrix}
\]
\item Suppose that $\dim P\cap H_1 = 1$. Now consider the other coordinate $x_2$ and think about the hyperplane $H_{12}$ it defines \emph{inside} $H_1$ -- this has dimension 2. Notice that $P\cap H_1$ has dimension 1 and $H_{12}$ has dimension 2 in the 3-dimensional space $H_1$. As a result their intersection must have dimension 0 or 1. Show that the simultaneous requirements
\[
\dim (P\cap H_1) = 1 \text{ and } \dim ( P\cap H_{12}) = 0
\]
define an open set $U_{12}\subset Gr(2,4)$, and show that any element of this open set may be described uniquely by a matrix of the form
\[
\begin{bmatrix}
1 & 0 & a_3 & a_4\\
0 & 1 & b_3 & b_4
\end{bmatrix}
\]
\item Generalize the above by considering other pairs of coordinates besides $(x_1,x_2)$, i.e. consider also $(13), (14), (23), (24),$ and $(34)$. In this way construct an atlas of six coordinate charts for $Gr(2,4)$, and prove that it is a smooth 4-dimensional manifold.
\end{enumerate}
\end{exercise}
\end{document}