\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}
                           \newcommand{\C}{\mathbb C}
    \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 2}\\
     \textbf{ April 9, 2019}
    \end{center}
    \begin{enumerate}
    \item Describe deck transformations of the covering $\C \setminus \{-1\} \to \C \setminus \{-1\} $ given by $z \mapsto z^2 + 2z$. 
    
    \item Consider the subspace $X \subset \R^2$ given by $X = \{y = 0\} \cup \{ x \in \Z\}$. Define a $\Z$-action on $X$ by $(x,y) \mapsto (x + m, y)$, where $m \in \Z$. \begin{enumerate} \item Describe the quotient space $X / \Z$. \item Prove that the quotient map $X \to X / \Z$ is a universal covering. \item Compute the fundamental group of $X / \Z$.
    \end{enumerate}

\item Consider the space $X$ obtained from a hexagon by identifying its sides as shown in the figure:
    {
    \begin{center}
\begin{tikzpicture}[thick, xscale = 1.6]
\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 [->] (1,2.5) -- (2,1.5);
\draw [<-] (2,1.5) -- (2,0);
\draw [<-] (2,0) -- (1, -1);
%\draw [<-] (2.5, -1) -- (1, -1);
\draw [<-]  (1, -1) -- (0,0);
 \node  at (-0.2,0.75) () {$a$};
  \node  at (0.4,2.2) () {$b$};
    \node  at (1.6,2.2) () {$c$};
    \node  at (1.6,-0.7) () {$b$};
        \node  at (0.4,-0.7) () {$c$};
%    \node  at (3.2,2.1) () {$b$};
%      \node  at (0.2,-0.5) () {$d$};
%    \node  at (3.2,-0.5) () {$d$};
     \node  at (2.15,0.75) () {$a$};
%      \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}
	}
	Compute the simplicial homology of $X$ with coefficients in $\Z$.
    \end{enumerate}
    \end{document}