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\begin{document}
\pagestyle{fancy} \rhead{1300Y\ \ Geometry and Topology, Assignment
3}

Convince yourself that you understand the classification of
1-dimensional manifolds: namely, if the manifold $M$ is connected
and 1-dimensional, then it is diffeomorphic either to $[0,1]$,
$(0,1]$, $(0,1)$, or $S^1$.  

\begin{exercise}
Let $X_1,X_2$ be sets of $n_1, n_2$ points respectively.  What
conditions on $n_1,n_2$ are there which guarantee that there exists
a (compact!) cobordism between $X_1,X_2$?  How many cobordisms are
there between $X_1,X_2$ up to diffeomorphism?
\end{exercise}

\begin{exercise}
A smooth map $f:X\lra Y$ is said to be transverse to a regular
submanifold $Z\subset Y$ when $f$ is transverse to the inclusion map
$\iota:Z\subset Y$.  Assuming $X$ is compact, show that the
transversality of $f$ to $Z$ is stable under perturbations of $f$.
\end{exercise}


\begin{exercise}
Let $f:M\lra N$ be a smooth map of manifolds with the same
dimension, and suppose $M$ is compact.  Let $\iota:[0,1]\lra N$ be
an embedding such that both $\iota$ and $\del \iota$ are transverse
to $f$. Show first that $f^{-1}(\iota(0))$ and $f^{-1}(\iota(1))$
are finite sets and second that $\sharp (f^{-1}(\iota(0)))\equiv
\sharp (f^{-1}(\iota(1))) \pmod 2$.

For any $m\equiv n\pmod 2$, give an example of a map $f:S^1\lra S^1$
such that $f^{-1}(1)$ has $n$ elements and $f^{-1}(-1)$ has $m$
elements, and give an example of an embedding $\iota$ as above with
$\iota(0)=1$ and $\iota(1)=-1$.
\end{exercise}


\begin{exercise}
Show that having an isolated zero of a smooth function $f:T^2\lra
\RR$ is not stable under perturbations of $f$.  Show also that if a
function $g:T^2\lra \RR^2$ has only regular zeros, then there must
be a finite even number of them. Give an example of such a function
with 4 zeros.
\end{exercise}

\begin{exercise}
Show that a compact $n$-manifold $M$ cannot be embedded in $\RR^n$.
Can it be immersed? Can it be submerged?
\end{exercise}
 \begin{exercise}
Show using the Brouwer fixed point theorem that any $n\times n$
matrix $A$ with non-negative real entries has a non-negative
eigenvalue.
 \end{exercise}
 \begin{exercise}
Show that the fixed point in the Brouwer fixed point theorem need
not be an interior point.
 \end{exercise}



\begin{exercise}Let $X$ be a manifold with boundary and $x\in\del X$ be a boundary point.
Show there exists a smooth non-negative function on some
neighbourhood $U$ of $x$ with $0$ as a regular value and
$f^{-1}(0)=\del U$.   Then show that there exists a smooth
non-negative function $F$ on all of $X$ with $0$ a regular value and
such that $F^{-1}(0)=\del X$. Hint: in going from local to global, a
partition of unity is often useful.
 \end{exercise}

\end{document}
