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\begin{document}
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\rhead{1300Y\ \ Geometry and Topology, Assignment 2}



\begin{exercise}Let $M,N$ be compact manifolds with boundary, and let $\psi:\del M\lra \del N$ be a
diffeomorphism.  Explain how to define a smooth structure on
$M\sqcup_{\psi} N = M\sqcup N / \sim$, where $x\sim y$ iff
$\psi(x)=y$ or $x = y$.  Is the manifold resulting from your
procedure uniquely specified (up to diffeomorphism) by the data
$(M,N,\psi)$ provided above? Give a simple example where the
diffeomorphism class of $M\sqcup_{\psi} N$ depends on $\psi$.
 \end{exercise}

\begin{exercise}
Let $f:M\lra M$ be a smooth map and suppose $p$ is a fixed point
under $f$, i.e. $f(p)=p$.  The point $p$ is called a \emph{Lefschetz
fixed point} when the derivative map $Df(p):T_pM\lra T_pM$ does not
have $+1$ as an eigenvalue.

Show that if $M$ is compact and all fixed points for $f$ are
Lefschetz, then there are only finitely many fixed points for $f$.
\end{exercise}

\begin{exercise}
Prove that there are no smooth functions on a compact manifold $M$
without critical points.
\end{exercise}

\begin{exercise}
A \emph{Morse function} on a manifold $M$ is a real-valued function
all of whose critical points are nondegenerate, in the sense that
the Hessian matrix at every critical point $p$ is nondegenerate
(this is independent of which chart is used to compute the Hessian).

{\it Note: Morse functions are important because, while they are not
regular everywhere, they do have a local classification near each
critical point - look up the ``Morse Lemma'' if you are interested,
it says that near each critical point there is a coordinate chart
for which $f = \sum_i \pm x_i^2$.}

Show that if $U\subset \RR^n$ is an open set and $f:U\lra \RR$ is a
smooth function, then for almost all $n$-tuples
$a=(a_1,\ldots,a_n)\in\RR^n$, the modified function
\[
f_a = f + \sum_{i=1}^n a_i x_i
\]
is a Morse function.  (Note: ``almost all'' means that the set of
$a$ for which $f_a$ fails to be Morse is of measure zero in
$\RR^n$.)
\end{exercise}

\begin{exercise}[Stability of Morse functions]
Let $f_0$ be a Morse function on a compact manifold $M$, and suppose
that $f:(-1,1)\times M\lra \RR$ is a smooth function with
$f|_{\{0\}\times M} = f_0$.  Show that for $\eps$ sufficiently
small, $f|_{\{\eps\}\times M}$ is also Morse.

\emph{(Intuitively, we think of $f$ as giving a smooth family of
maps $M\lra \RR$, parametrized by $t\in(-1,1)$.}
\end{exercise}
\begin{exercise}
Consider the function $\CC^2\backslash\{0\}\lra \CC$ given by
$f(z_1,z_2) = z_1^p + z_2^q$, for integers $p,q$ which are
relatively prime and $\geq 2$. Describe the regular and critical
points and values of this map.

Show that the intersection $K=f^{-1}(0)\cap S^3$ is transverse,
where we view $S^3\subset \RR^4\cong\CC^2$, and identify the
manifold $K$. By considering the 2-tori $\{(z_1,z_2)\ :\
|z_1|=c_1\text{ and } |z_2|=c_2\}$ for constants $c_1,c_2$, describe
the way in which $K$ is embedded in $S^3$, perhaps including a
diagram.\end{exercise}



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