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\begin{document}

\lhead{Mathematics 468}
\rhead{J.~Palmieri}

\begin{center}
{\large \textbf{Final Exam}}
\end{center}

This exam is due on Thursday, May 6, at 5:30 pm, in my office.

This exam is being conducted under the honor code: you may not discuss
it with anyone except me.  You may use books and (your own) notes.  If
you get stuck on some problem, feel free to ask me for hints; the
easiest way to do this is to come by my office.  I should be in during
most of the day Monday through Thursday, and available by e-mail other
times.  (Note that I'm giving another exam Thursday, 1:45-3:45, so I'm
busy then)

The questions with only one part should be straight-forward, as should
part~(a) of each of the others.  The later parts of the other problems
are, for the most part, harder; some of them are quite hard.  Please
do \underline{four} of the one-part problems and/or the (a) parts, and
try the rest of \underline{two} of the multi-part problems.  (If you
can't completely solve them, make as much progress as you can.)

Contact me if you have any questions.

\begin{questions}

\question
Answer the first sentence of Exercise 6 from Section 10 (p.~67).  Go
into as much detail as you can.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\question
Do Exercise 6 in Section 13 (p.~83).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\question
Prove that the connected sum $P \# T$ is topologically equivalent to $P
\# P \# P$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\question

\begin{parts}

\part
Let $S$ be a subset of $\mathbf{R}^{n}$ for some $n$ (if you need to,
assume that $n$ is 1 or 2, but try to work more generally if you can).
The \emph{closure} of $S$, written $\overline{S}$ or $\closure{S}$, is
the union of $S$ with all of the near points of $S$.  Show that
$\overline{S}$ is closed for any $S$.

\uplevel{Write $\mycomp{S}$ for the complement of $S$.  Together with
closure, this gives two operations $S \mapsto \mycomp{S}$ and $S
\mapsto \closure{S}$ we can perform on a set, and we can iterate them
and combine them in different ways.  For example, starting with the
open interval $S=(0,1)$ viewed as a subset of the line, the closure is
$\closure{S}=[0,1]$, the complement is $\mycomp{S}=(-\infty,0] \cup
[1,\infty)$, the complement of the closure is
$\mycomp{\closure{S}}=(-\infty,0) \cup (1,\infty)$, and the closure of
the complement is the same as the complement, and also the same as the
closure of the complement of the closure:
$\closure{\mycomp{S}}=\mycomp{S}= \closure{\mycomp{\closure{S}}} =
(-\infty,0] \cup [1,\infty)$.  Note also that for any set $S$,
$\closure{\closure{S}} = \closure{S}$ (since $\closure{S}$ is closed),
and $\mycomp{\mycomp{S}} = S$, so you don't get anything very
interesting by doing either operation twice in a row---you have to
alternate them.}

\uplevel{Kuratowski showed that starting with any set $S$, you can get
at most 14 different sets by combining these two operations.}

\part 
The set $S=(0,1)$ gave 4 different sets under these operations.  Find
a subset of $\mathbf{R}$ which gives 14 different sets, or if you
can't manage that, as close to 14 as you can.

\part
Prove Kuratowski's result.  [Hint: first show that if $A$ is the
closure of an open set, then $\closure{\mycomp{\closure{\mycomp{A}}}}
= A$.]

\end{parts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\question 
\begin{parts}

\part
Let $D$ be a cell.  Show that the Brouwer fixed point theorem (p.~37
in the book) is equivalent to this: every continuous vector field $v$
on $D$, for which the vector $v(P)$ lies entirely within $D$ for each
point $P \in D$, has a critical point.

% \part
% ham sandwich in two dimensions.  [hint]  

\part
Does the sphere have the fixed point property?  What about the torus?
The projective plane?  If no, give an example.  if yes, give a proof,
or as much evidence as you can.
\end{parts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\question 
Let $S$ be any compact connected surface, and suppose it has been
triangulated.  Let $v$ be the number of vertices, $e$ the number of
edges, $f$ the number of faces, and $\chi = v-e+f$ the Euler
characteristic.

\begin{parts}
\part
Show that $3f=2e$, $e=3(v-\chi)$, and $\displaystyle v \geq
\frac{7+\sqrt{49-24\chi}}{2}$.

\part 
Find lower bounds for the number of vertices for triangulations of the
sphere, the torus, and the projective plane.  What are the
corresponding numbers of edges and faces?

\part
Find examples of triangulations of these surfaces that have exactly
these numbers of vertices, edges, and faces.

\end{parts}


\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\question

\begin{parts}

\part
Let $\theta$ denote the subset of the plane consisting of two vertices
and three edges (no faces), arranged as in the first figure below.
Compute the homology groups of $\theta$.

\part
Here is another subset of the plane: take a hexagon and glue it to
another hexagon along an edge: see the second figure below.  Compute
its homology groups.  (Just as in part~(a), there are no faces
here---just vertices and edges.)

\part
According to what I said in class on Wednesday, $H_{0}(X) \cong
\mathbf{Z}/2$ for any connected complex $X$, and $H_{2}(S) \cong
\mathbf{Z}/2$ for any compact connected surface $S$.  So if $S$ is a
compact connected surface, then $H_{1}(S)$ is the only ``interesting''
homology group.  Describe as best you can these groups:
\[
H_{1}(\underbrace{P \# \dots \# P}_{m}), \ \ 
H_{1}(\underbrace{T \# \dots \# T}_{n})
\]
for all numbers $m, n \geq 1$.

\end{parts}

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\end{questions}

\end{document}
