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\begin{document}
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\lhead{1300F Geometry and Topology, Assignment 3}
\rhead{Due date: November 16, 2012}
\cfoot{}

\begin{exercise}\mbox{}
  Let $K,L$ be submanifolds of a manifold $M$, and suppose that their
  intersection $K\cap L$ is also a submanifold.  Then $K,L$ are said
  to have \emph{clean} intersection when, for each $p\in K\cap L$, we
  have $T_p(K\cap L) = T_pK\cap T_pL$.  Show that there are
  coordinates near $p\in K\cap L$ such that $K$, $L$, and $K\cap L$
  are given by linear subspaces of $\RR^n$ of the form
  $V(x^{i_1},\ldots, x^{i_k})$ for some subset of the coordinates.  It
  is useful to use the algebraic geometry notation $V(x^1,\ldots,
  x^k)$ to mean the ``vanishing'' subspace $x^1 = \cdots = x^k = 0$.

  Also, can the intersection of submanifolds be transverse but not
  clean? Can it be clean but not transverse? Give examples or proofs
  as necessary.
\end{exercise}

\begin{exercise}\mbox{}
  Compute the mod 2 self-intersection number of the zero section $X\to
  TX$ for the manifolds $X\in\{ S^1,S^2,\RR P^2\}$, showing your
  reasoning.  Deduce that every smooth vector field on $\RR P^2$ must
  have a zero. Produce an explicit example of a vector field on $\RR
  P^2$ with a single transverse zero.
\end{exercise}
\begin{exercise}
  Let $X$ be compact and $f:X\to Y$ smooth with $\dim X = \dim Y$ and
  $Y$ connected. Recall that the mod 2 degree of $f$ is defined in
  terms of the mod 2 intersection number as follows: $\deg_2(f) =
  I_2(f,\iota)$, where $\iota:y\mapsto Y$ is the inclusion map of a
  point $y\in Y$.
  \begin{enumerate}\setlength{\itemsep}{-0.5ex}
  \item Prove that $\deg_2(f)$ is independent of the point $y\in Y$.
  \item If $Y$ is non-compact, prove that $\deg_2(f)=0$.
  \item A map $f:X\to Y$ is called \emph{essential} when it is not
    homotopic to a constant map.  Prove that if $\deg_2(f)=1$, then
    $f$ is essential.
  \item Give example of a smooth surjective map $f:S^2\to S^2$ with
    $\deg_2(f)=0$.
  \item Can there exist a smooth map $f:S^2\to T^2$ with
    $\deg_2(f)=1$?  [Hint: consider two embedded circles $C_1,C_2$ in
    $T^2$ intersecting transversally at a single point.]  Can there
    exist a smooth map of $\deg_2(f)=1$ in the opposite direction? In
    each case, give proofs.
  \end{enumerate}
\end{exercise}

\begin{exercise}[Jordan curve theorem]
  Let $f:S^1\to \RR^2$ be an embedding and choose $p\in
  \RR^2\backslash f(S^1)$.  Define $f_p: S^1\to S^1$ by $f_p(z) =
  \frac{f(z)-p}{|f(z)-p|}.$ Then we define the mod 2 winding number of
  $f$ about $p$ to be the degree of $f_p$, i.e. $w_2(f,p) =
  \deg_2(f_p)$.  Warm up by computing $w_2(f,p)$ for the standard
  embedding of $S^1$ in $\RR^2$, and for any $p$, with justifications.
  \begin{enumerate}\setlength{\itemsep}{-.5ex}
  \item Let $R_p(v)$ be the ray starting at $p$ with direction $v\in
    S^1$.  Prove that $v\in S^1$ is a critical value of $f_p$ if and
    only if $R_p(v)$ is somewhere tangent to $f(S^1)$.
  \item Show that $w_2(f,p)$ coincides with the number of points mod 2
    in $R_p(v)\cap f(S^1)$, whenever $v$ is a regular value of $f_p$.
  \item Show that there are points $p,q\in\RR^2\backslash f(S^1)$ such
    that $w_2(f,p) = 0$ and $w_2(f,q)=1$.  Show that this implies that
    $\RR^2\backslash f(S^1)$ has at least two components.
  \item Fix $a\in f(S^1)$.  Show that it is possible to choose a
    coordinate chart $(U,\varphi)$ containing $a$ such that
    $\varphi(U)$ contains $(-2,2)\times (-2,2)$, $\varphi(a) = (0,0)$,
    and $\varphi(U\cap f(S^1)) = \{(x,y)\ :\ y=0\}$. \item Prove that
    each point $p\in \RR^2\backslash f(S^1)$ may be connected by a
    continuous path to either $\varphi^{-1}(0,1)$ or
    $\varphi^{-1}(0,-1)$. [Hint: recall the tubular neighbourhood
    theorem].  Conclude that $\RR^2\backslash f(S^1)$ has two
    connected components. 
  \end{enumerate}
\end{exercise}

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