\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 {Seifert-van Kampen}}

\begin{enumerate}[1.]

\item Let $X,Y$ be spaces and $f:X\to Y$ a map. The
  {\it mapping cone} of $f$ is the space $C_f$ obtained from the
  mapping cylinder $X\times I \sqcup Y/(x,1)\sim f(x)$ by collapsing
  $X\times \{0\}$ to a point.

  \begin{enumerate}[(a)]
    \item
  Now assume $X,Y$ are path-connected. Express $\pi_1(C_f)$ as a
  pushout involving $\pi_1(X), \pi_1(Y)$ and
  $f_*:\pi_1(X)\to\pi_1(Y)$.
\item
  Let $X=Y=S^1\vee S^1$ with the 1-cells labeled $a,b$ and let $f:X\to
  Y$ be given by $a\mapsto aba^{-1}b^{-1}$ and $b\mapsto
  a^2b^3$. Compute $\pi_1(C_f)$ explicitly.
  \end{enumerate}

  \item Let $X$ be a space and $f:X\to X$ a map. The {\it mapping
    torus} of $f$ is the space $$T_f=X\times I/(x,1)\sim (f(x),0)$$

    Now suppose that $X$ is a connected cell complex with $x_0\in X$
    the only
    0-cell and $f:X\to X$ is a cellular map with $f(x_0)=x_0$. Then
    $T_f$ is also a cell complex with cells $e\times \{0\}$ and
    $e\times (0,1)$ for cells $e$ of $X$ (you don't have to prove
    this). Assume that $\pi_1(X,x_0)=\< A\mid R\>$. Prove that
    $$\pi_1(T_f)=\< A,t\mid R, tat^{-1}=f_*(a), a\in A\>$$
    Hint: $t$ corresponds to the loop $\{x_0\}\times I$. N.B.: The
    assumption that there is only one 0-cell is for convenience, the
    general case can be reduced to this.

    \item Let $A,B$ be nontrivial groups. Prove that $A*B$ has trivial
      center.

    \item Let $X$ be the space obtained from $S^2$ by identifying 3
      points. Find an explicit cell structure and from it compute
      $\pi_1(X)$.
\end{enumerate}

\noindent
The following questions are harder (and more rewarding).

\noindent
\begin{enumerate}[1.]
  \setcounter{enumi}{4}

\item Problem 9 from Hatcher, p.53.

  \item Problem 10 from Hatcher, p.53. Hint: Look at Problem 22 for
    inspiration. 

\item Problem 14 from Hatcher, p.54.

\item Problem 15 from Hatcher, p.54.

  \item Problem 16 from Hatcher, p.54. Hint: Show that the surface
    deformation retracts to a graph by representing it as an
    increasing union of compact subsurfaces $X_1\subset
    X_2\subset\cdots$ and showing that $X_{n+1}$ deformation retracts
    to $X_n$ union a graph.
    


 \end{enumerate}

\end{document}
