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\theoremstyle{definition}
\newtheorem{problem}{}
\newcommand{\bp}{\begin{problem}}
\newcommand{\ep}{\end{problem}\bigskip}

\newcommand{\Aut}{\operatorname{\mathrm{Aut}}}
\newcommand{\R}{\mathbb{R}}

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\begin{document}
\pagestyle{fancy}
\fancyfoot[R,C,L]{}
\fancyhead[R]{\large \textbf{Fall 2022}}
\fancyhead[L]{\large \textbf{Topology Homework 7}}


\bp Let $B=S^1 \vee S^1$ and consider the cover
$p:E \to B$ pictured below 
\begin{center}
\includegraphics[scale=.7]{cover4bw.pdf}
\end{center}
I've tried to sketch the covering space in $\R^3$ and the base space in the $xy$-plane with the covering map being project $(x,y,z)\mapsto (x,y)$.  The seeming self intersections in $E$ are an illusion of the way I'm picturing this, like the way a drawing of a Klein bottle appears to intersect itself.  You might find it more helpful to have this other picture of $E$ below.  Here, the space $E$ is pictured as a graph and the map $E\to B$ is described by the labelling and arrows.

\begin{center}
\begin{tikzpicture}
\tikzstyle{main node}=[draw,shape=circle,minimum size=0pt,inner
  sep=2]

\tikzset{->-/.style={draw, thick, decoration=
{markings, mark=at position .625 with {\arrow[scale=1.5,black]{latex}}},postaction={decorate}}}

\def \radius {2cm}
\def \n {4}
\foreach \s in {1,...,\n}
{
  \node[main node] (\s) at ({360/\n * (\s - 1)}:\radius) {$x_\s$};
% \draw[->,>= latex] ({360/\n * (\s - 1)+\margin}:\radius) 
%arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}
\path[-] 
(1) edge[->-] node[above right] {$a$} (2)
(2) edge[->-] node[above left ] {$a$} (3)
(3) edge[->-] node[below left] {$a$} (4)
(4) edge[->-] node[below right] {$a$} (1)
(1) edge[->-,bend left=60] node[above right] {$b$} (2)
(2) edge[->-,bend left=60] node[above right] {$b$} (1)
(3) edge[->-,bend left=60] node[below left] {$b$} (4)
(4) edge[->-,bend left=60] node[below left] {$b$} (3)
;
\end{tikzpicture}
\end{center}

\begin{enumerate}[label=(\alph*)] 
\item Compute $\pi_1(E,x_1)$ and $\pi_1(E,x_2)$ as subgroups of $\pi_1(B, x).$  

\emph{Hint:}  Use the picture.  Let $a$ be the class of the loop in $B$ starting at $x$ and proceeding counterclockwise once around the circle on the left, and let $b$ be the class of the loop in $B$ starting at $x$ and proceeding counterclockwise once around the circle on the right.  So, $\pi_1(B,x)=F(a,b)$.  Now, both $\pi_1(E,x_1)$ and $\pi_1(E,x_2)$ are isomorphic to a free group on $5$ generators, since $E$ is homotopic to $S^1\vee S^1\vee S^1\vee S^1\vee S^1$, which you can see by contracting the ``$a$'' edges.
  \item Describe the action of $\pi_1(X,x)$ on the fiber $p^{-1}(x)$.
\item Compute $\Aut(E)$, the group of automorphisms of the cover $E\overset{p}{\to}B$.
\end{enumerate}
\ep

\bp The free group on two generators $F(a,b)$ is a subgroup of the
free group on three generators $F(c,d,e)$ and vice versa.   Find
covering maps $p:E\to X$ with
\begin{enumerate}[label=(\alph*)] 
\item $F(c,d,e)=\pi_1(B,b)$ and $F(a,b)=\pi_1(E,e)$
\item $F(a,b)=\pi_1(B,b)$ and $F(c,d,e)=\pi_1(B,e)$
\end{enumerate}
\ep



\bp Consider the topological space $B=\{a,b,c,d\}$ with topology
$$\tau=\{\emptyset, \{a\},\{c\}, \{a,c\}, \{a,b, c\}, \{c, d,a\},\{a,b,c,d\}\}.$$
The universal cover of $B$ is the space $\tilde{E}=\{a_i, b_i, c_i,
d_i:i\in \mathbb{Z}\}$ with topology generated by the basic open sets 
$\{a_i\}$, $\{c_i\}$, $\{a_i, b_i, c_i\}$ and $\{c_i,
d_{i+1}, a_{i+1}\}$.  
The covering map is given by \begin{align*}p:&\tilde{E} \to
  B\\ x_i&\mapsto x\end{align*}
Use a covering space argument to compute the fundamental group of $B$.
\ep


\bp Suppose $(E,e)\overset{p}{\to}(B,b)$ is any based cover.  Prove that the induced map in homotopy groups $p_*: \pi_n(E,e) \to \pi_n(B,b)$ is an isomorphism for $n\geq 2$.
\ep

\bp \textbf{[Problem 4 Spring 2014]}
\bigskip

\noindent
Find all connected three-fold covers of the wedge sum of two
projective planes, carefully justifying your answer.
\ep

\bp \textbf{Problem 3 Spring 2011]}
\bigskip

\noindent
Use a covering space argument to prove that any map $\R P^2 \times \R
P^2 \to S^1$ is null-homotopic.
\ep


\end{document}
