\documentclass[11pt]{article}

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  \newtheorem*{main}{Main Theorem}{}
  \newtheorem*{rrt}{Rank Rigidity Theorem}{}

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  \def\N{{\mathbb N}}
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  \def\G{{\Gamma}}
  \def\S{{\mathcal S}}
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  \def\t{{\tau}}
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  \def\X{{\mathcal X}}
  \def\P{{\mathcal P}}
  \def\pa{pseudo-Anosov\ }
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  \begin{document}

  \centerline{\Large {Take Home Final}}

\begin{enumerate}[1.]

\item Let $X$ be the space obtained from the 3-sphere $S^3$ by
  identifying antipodal points on the equator $S^2$. Compute
  $\pi_1(X)$.

\item Let $p:X\to Y$ be a covering map between path-connected and
  locally path-connected spaces. Suppose $p$ is homotopic to a
  constant map. Prove that $X$ is contractible.


\item Let $p:\tilde X\to X$ be a $d$-sheeted covering map for some
  $1\leq d<\infty$.
  \begin{enumerate}[(a)]
    \item Show that for every singular simplex $\sigma:\Delta^n\to X$
      there are exactly $d$ lifts $\tilde\sigma_i:\Delta^n\to\tilde
      X$.
      \item Show that the homomorphism $\tau_n:C_n(X)\to C_n(\tilde
        X)$ defined by
        $$\tau_n(\sigma)=\tilde\sigma_1+\cdots+\tilde\sigma_d$$
        is a chain morphism $C_n(X)\to C_n(\tilde X)$.
        \item Show that the induced homomorphism $$\tau_*:H_n(X)\to
          H_n(\tilde X)$$ satisfies $$p_* \tau_* (x)=dx$$ for every
          $x\in H_n(X)$. ($\tau_*$ is called {\it transfer}, a generic
          name for homomorphisms that go {\it the wrong way}).
          \item Assuming $H_n(X), H_n(\tilde X)$ are finitely
            generated, prove that $$rank(H_n(\tilde X))\geq
            rank(H_n(X))$$. Thus passing to a finite-sheeted covering
            space can only increase homology.
  \end{enumerate}

  \item Let $X$ be the space obtained from two copies of $\R P^2$ by
    gluing them along standard copies of $\R P^1$.
    \begin{enumerate}[(a)]
      \item State Seifert-van Kampen's theorem and use it to compute
        $\pi_1(X)$.
        \item Using any method you like ($\Delta$-homology, cellular
          homology, Mayer-Vietoris,...) compute homology and
          cohomology groups of $X$ with both $\Z$ and $\Z/2\Z$
          coefficients.
    \end{enumerate}

    \item Prove that every map $S^4\to S^2\times S^2$ has degree 0,
      i.e. the induced homomorphism in $H_4$ is 0.

    \item Prove that no closed orientable 3-manifold is homotopy
      equivalent to $S_g\vee S^3$, where $S_g$ is the orientable
      surface of genus $g\geq 0$.

 \end{enumerate}

\end{document}
