\documentclass[11pt]{article}

  \usepackage{latexsym}
  \usepackage{comment}
  \usepackage{showlabels}
  \usepackage{amsmath,amsthm,amsfonts,amssymb,graphicx,epsfig,latexsym,color,amssymb}
  %\usepackage{showkeys}
  \usepackage{epsf,enumerate}
  \usepackage{tikz}
  \def\rank{\operatorname{rank}}

\newcommand{\bbfamily}{\fontencoding{U}\fontfamily{bbold}\selectfont}
\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n}

  \newtheorem{thm}{Theorem}[section]
  \newtheorem{lemma}[thm]{Lemma}
  \newtheorem{claim}[thm]{Claim}
  \newtheorem{cor}[thm]{Corollary}
  \newtheorem{prop}[thm]{Proposition}
  \newtheorem*{main}{Main Theorem}{}
  \newtheorem*{rrt}{Rank Rigidity Theorem}{}

  \theoremstyle{remark}
  \newtheorem{example}[thm]{Example}
  \newtheorem{exercise}[thm]{Exercise}
  \newtheorem{definition}[thm]{Definition}
  \newtheorem{remark}[thm]{Remark}
  \newtheorem{examples}[thm]{Examples}
  \newtheorem*{definition*}{Definition}
  \newtheorem*{remark*}{Remark}
  \newtheorem{question}[thm]{Question}

  \newcommand{\mb}[1]{\marginpar{\tiny #1 --mb}}

  \bibliographystyle{plain}   %plain

  \def\R{{\mathbb R}}
  \def\Q{{\mathbb Q}}

  \def\Z{{\mathbb Z}}
  \def\N{{\mathbb N}}
  \def\H{{\mathbb H}}  
  \def\G{{\Gamma}}
  \def\S{{\mathcal S}}
  \def\s{{\sigma}}
  \def\t{{\tau}}
  \def\e{{\varepsilon}}
  \def\F{{\mathcal F}}
  \def\X{{\mathcal X}}
  \def\P{{\mathcal P}}
  \def\pa{pseudo-Anosov\ }
  \def\diam{\operatorname{diam}}
  \def\spec{\operatorname{spec}}
  \def\stab{\operatorname{Stab}}
  \def\diag{\operatorname{diag}}
  \def\SL{\operatorname{SL}}
  \def\GL{\operatorname{GL}}
  \def\Sym{\operatorname{Sym}}
  \def\FF{{\mathbb F}}
  \def\A{{\mathcal A}}
  \def\l{{\lambda}}
  \def\C{{\cal C}}

  \newcommand{\<}{\langle}
  \renewcommand{\>}{\rangle}

  \begin{document}

  \centerline{\Large {Covering spaces}}

\begin{enumerate}[1.]

\item Let $(X,Y)$ be a CW pair with both $X,Y$ connected and $x_0\in
  X$ a basepoint. Assume that inclusion induced homomorphism
  $\pi_1(Y,x_0)\to\pi_1(X,x_0)=G$ is injective and denote its image by $H<G$.
  Let $p:(\tilde X,\tilde x_0)\to (X,x_0)$ be the
  universal cover and let $p_H:(\tilde X_H,\tilde x_H)\to (X,x_0)$ be
  the covering corresponding to $H$.
  \begin{enumerate}[(a)]
    \item
  Prove that each component of $p^{-1}(Y)$ is
  simply connected (and hence is a universal cover of $Y$).
\item Show that there is a natural bijection between the cosets $gH$
  of $H$ in $G$ and the components of $p^{-1}(Y)$ given by
  path lifting (which you should describe).
  \item Assume that $H$ is {\it malnormal} in $G$, i.e. $g\not\in H$
    implies $gHg^{-1}\cap H=\{1\}$. For example, $\<a\><\<a,b\>$
    is malnormal, and so is any subgroup of the fundamental group of
    an orientable surface defined by an embedded circle that doesn't
    bound a disk (this is called an {\it essential simple closed
      curve}). Show that one of the components of $p_H^{-1}(Y)$ is
    homeomorphic to $Y$ and the others are simply-connected, and that the
    set of all components is in natural bijection with the set of
    double cosets $HgH$.
  \end{enumerate}

  
\item Do the following problems from Hatcher, Section 1.3:
  4,8,9,12,14,18,20,21,30. On \# 8 the hint should refer to Exercise
  11 in Chapter 0. I am not going to assign problems regarding point
  set topology, but you may want to go over problems 5-7 for
  counterexamples to standard theorems when assumptions like local
  path connectedness are dropped.

  \item This is optional, but if you are interested in
    geometry/topology you should read sections 1A and 1B in
    Hatcher. In particular, read the construction of the
    contractible $\Delta$-complex $EG$ on which a group $G$ acts
    freely (in algebra, $EG$ -- or more precisely its chain complex --
    is called the {\it bar resolution} of $G$). 


 \end{enumerate}

\end{document}
