\documentclass[a4paper, 11pt]{article}
\usepackage[left=2cm,right=2cm,
    top=2.5cm,bottom=2.5cm,bindingoffset=0cm]{geometry}
    \usepackage{amssymb, amsmath}
        \newcommand{\R}{\mathbb R}
                \newcommand{\Z}{\mathbb Z}
    \setlength{\unitlength}{1cm}
\thicklines

\usepackage{tikz}
\usetikzlibrary{positioning}
\usetikzlibrary{decorations.text}
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{decorations.markings}

\usetikzlibrary{arrows} 
\tikzset{
    %Define standard arrow tip
    >=stealth',
    % Define arrow style
    pil/.style={
           ->,
           thick,
           shorten <=2pt,
           shorten >=2pt,}
}



    \begin{document}
    \thispagestyle{empty}

    \begin{center}
    \textbf{ MATH534B, Exam 1}\\
     \textbf{ February 26, 2019}
    \end{center}
    \begin{enumerate}
    \item Let $X$ and $Y$ be homotopy equivalent topological spaces. Assume also that $X$ has the following property: any two continuous maps $S^2 \to X$ are homotopic to each other. Prove that $Y$ has the same property.
    \item Recall that the M\"obius band is the space obtained from a rectangle by identifying a pair of its opposite sides as shown in the figure:
    {
    \begin{center}
\begin{tikzpicture}[thick]
\draw [->] (0,0) -- (0,2);
\draw  (4,2) -- (0,2);
\draw (4,0) -- (0,0);
\draw [<-] (4,0) -- (4,2);
 \node  at (-0.2,1) () {$a$};
  \node  at (4.2,1) () {$a$};
%  \node  at (1.6,0.9) () {$a$};
%    \node  at (0.35,0.9) () {$a$};
\end{tikzpicture}
\end{center}
	}
Prove that there exists no retraction of the M\"obius band to its boundary.
\item Consider the space $X$ obtained from an octagon by identifying its sides as shown in the figure:
    {
    \begin{center}
\begin{tikzpicture}[thick]
\draw [->] (0,0) -- (0,1.5);
\draw [->] (0,1.5) -- (1,2.5);
\draw [<-] (1,2.5) -- (2.5,2.5);
\draw [dashed] (0,0) -- (2.5,2.5);
\draw [<-] (2.5,2.5) -- (3.5,1.5);
\draw [->] (3.5,1.5) -- (3.5,0);
\draw [->] (3.5,0) -- (2.5, -1);
\draw [<-] (2.5, -1) -- (1, -1);
\draw [<-]  (1, -1) -- (0,0);
 \node  at (-0.2,0.75) () {$a$};
  \node  at (0.3,2.1) () {$b$};
    \node  at (3.2,2.1) () {$b$};
      \node  at (0.2,-0.5) () {$d$};
    \node  at (3.2,-0.5) () {$d$};
     \node  at (3.65,0.75) () {$c$};
      \node  at (1.75,2.7) () {$a$};
            \node  at (1.75,-1.2) () {$c$};
             \node  at (-0.25,0) () {$x_0$};
             \draw[fill=black] (0,0) circle (.2ex);
%\draw  (4,2) -- (0,2);
%\draw (4,0) -- (0,0);
%\draw [<-] (4,0) -- (4,2);
% \node  at (-0.2,1) () {$a$};
%  \node  at (4.2,1) () {$a$};
%  \node  at (1.6,0.9) () {$a$};
%    \node  at (0.35,0.9) () {$a$};
\end{tikzpicture}
\end{center}
	}
	The dashed path in the figure determines a loop in $X$ based at $x_0$, and hence an element of $\pi_1(X, x_0)$. Is that element equal to the identity? 
    \end{enumerate}
    \end{document}