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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Intersection and Degree Theory}}
\vskip 1cm
  


\subsection*{Degree Theory}

In this section all manifolds are compact, connected, oriented and
have the same dimension $n$.

\begin{enumerate}[1.]

\item If $f:X\to Y$ and $g:Y\to Z$ are smooth, show that
  $\deg(gf)=\deg(g)\deg(f)$. Hint: First find some $z\in Z$ which is a
  regular value for $g$ and so that every point in $g^{-1}(z)$ is a
  regular value for $f$.

\item Let $f,g:S^n\to S^n$ be two smooth maps such that $f(x)\neq
  -g(x)$ for every $x\in S^n$. Prove that $\deg f=\deg g$.

\item Show that for every smooth map $f:S^{2k}\to S^{2k}$ there exists
  a point $x\in S^{2k}$ such that either $f(x)=x$ or $f(x)=-x$.

\end{enumerate}


\subsection*{Degree Theory for continuous maps}


\begin{enumerate}[resume]

  \item Prove the Boundary Theorem for continuous maps: if $W$ is
    compact oriented with boundary, $Y$ is compact oriented connected, $f:W\to
    Y$ is continuous, then $\deg \partial f:\partial W\to Y=0$. 

  \item Prove that there is no continuous retraction of any compact
    manifold to its boundary. You can assume that both the manifold
    and its boundary are connected.
  

\end{enumerate}


\subsection*{Sections of bundles and self-intersection of the
  0-section}


\begin{enumerate}[resume]

\item \label{ample}
  Let $\pi:E\to B$ be a vector bundle 
    and for convenience assume that $B$ is compact. Show
    that there are finitely many smooth sections
    $\sigma_1,\cdots,\sigma_N:B\to E$ such that for every $b\in B$ the
    vectors $\sigma_1(b),\cdots,\sigma_N(b)$ span the vector space
    $\pi^{-1}(b)$. Hint: Working in a chart, show that this is
    possible in a neighborhood of every point.

  \item Let $\pi:E\to B$ be a vector bundle 
    and for convenience assume that $B$ is compact. Show that there is
    a section $\sigma:B\to E$ which is transverse to the submanifold
    of $E$ consisting of 0's in each fiber (i.e. the
    0-section). Moreover, every section can be approximated by a
    section transverse to the 0-section.
    Hint:
    Use Problem \ref{ample} and the transversality theorem.

  \item Let $\pi:E\to B$ be a vector bundle 
    and for convenience assume that $B$ is compact. Also asume that
    it's an $n$-dimensional bundle and that the base $B$ is an
    $n$-manifold (with the same $n$). Let $\sigma:B\to E$ be a section
    transverse to the 0-section $Z$, and assume $B$ and the bundle are
    oriented (so in particular $Z$ is transversally oriented).
    Show that $I(\sigma, Z)$ is independent of the choice of
    $\sigma$. This is called the {\it Euler number} of the bundle. It
    also equals $I(Z,Z)$ (so it's 0 when the dimension is odd).

  \item Show that every odd dimensional oriented vector bundle admits
    an orientation reversing automorphism. Hint: $v\mapsto -v$.

\end{enumerate}


\section*{Miscallenious}

\begin{enumerate}[resume]
  \item Let $X$ be a manifold, and identify it with the diagonal
    $\Delta\subset X\times X$ via $x\mapsto (x,x)$. Prove that the
    normal bundle of $\Delta$ in $X\times X$ is isomorphic to the
    tangent bundle $TX$. Hint: $(v,-v)\mapsfrom v$.

 \item Compute the self-intersection mod 2 of $\R P^n\subset
   \R P^{2n}$. Here, a point in $\R P^{2n}$ is given in homogeneous
   coordinates as $[x_0:x_1:\cdots:x_{2n}]$ and the submanifold
   $\R P^n$ is defined as the set of such points where
   $x_{n+1}=\cdots=x_{2n}=0$. Hint: Explicitly isotope $\R P^n$ so it
   is transverse to the initial copy. E.g. when $n=1$ you can move the
   point $[a:b:0]$ to $[a:b:b]$ via $[a:b:tb]$, then to $[a:0:b]$. Now
   the intersection is only $[1:0:0]$. Comment: this also works in $\C
   P^{2n}$ and there you can do the oriented intersection number.


\end{enumerate}

\end{document}
