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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}
\end{document}