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\begin{document}
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\begin{center}
Homework 7.
\end{center}

\begin{enumerate}

\item  a.) Consider  $f:X\to \R$  continuous and assume $X$ is connected.
Let $a,b\in f(X)$ and assume $a<b$. Show that for all $c\in (a,b)$ $\exists z\in X$ such that $f(z)=c$. {\it (This is the so called Generalized Intermediate Value Theorem.)}


b.) Show that if $f:S^1\to\R$ is continuous, then there is an $\underline{x}=(x_1,x_2)\in S^1$ for which $f(\underline{x})=f(-\underline{x})$.



\item a.) Find all compact connected subsets of $\R$. Justify your answer.


\noindent
b.) Find all subsets of $\R$ that $S^1$ is homeomorphic to. Justify your answer.

\item Identify which surfaces in the classification theorem the following ones are homeomorphic to 

a.) The connected sum $K\# T$, where $K$ is the Klein bottle and $T$ is the torus.

b.) A disk sewn onto the edge of a Mobius strip. \newline
{\it (We did this problem already, using a cut-and-paste argument. But now use diagrams, as well as the Euler characteristic and orientability.)}

c.) The surface determined by the labelling scheme of a polygon 

as $\ \ abcdec^{-1}d^{-1}a^{-1}b^{-1}e^{-1}$

d.) The surface which is called the "Purse of Fortunatus"
in Lewis Caroll's novel {\it Sylvie and Bruno Concluded}, where Mein Herr teaches Lady Muriel 
how to construct such a purse by sewing together three handkerchiefs along their edges according to  
$cb^{-1}e^{-1}a^{-1}$, $\ bda^{-1}f^{-1}$ and $d^{-1}c^{-1}f^{-1}e^{-1}$.

He then justifies the name of the purse as follows: 
{\it "Whatever is inside that purse is outside it; and whatever is outside it, is inside it. So you have all the wealth of the world in that little purse!"}



\item This exercise is related to platonic solids. A platonic solid is a convex polyhedron all of whose
faces are regular $n$-gons, i.e. all the faces have the same number of sides, which in platonic solids are also of the same length. Note that $n\geq 3$.
The word "convex" means that the inner angles between adjacent faces  are less then 180 degrees.

It is well-known that there are five platonic solids: the cube, the tetrahedron,  the octahedron, the
icosahedron and the dodecahedron.

In this problem, we will prove, using the Euler characteristic, that these 5 are the only ones possible.

\smallskip
Consider a platonic solid M. If it is placed inside a sphere and then expanded from within (ie it is mapped onto the sphere along rays from the center of the sphere), then it determines a triangulation   of the sphere, where

$V=$number of vertices of $M$, as well as of the triangulation;

$E=$number of edges of $M$, as well as of the triangulation;

$F=$number of faces of $M$, as well as of the triangulation.

 Let $p$ denote the number of edges meeting at a vertex.
 Note that since $M$ is a platonic solid $p$ is the same for every vertex and  $p\geq 3$.

a.)  We have $pV = 2E$
and that $nF = 2E$ - why?

b.) Conclude that
$${1\over n}-{1\over 2}+{1\over p}={1\over E} $$

and because $E > 0$ this implies 

$${1\over n}+{1\over p} > {1\over 2}$$

c.) Use the last inequality to find upper bounds on $n$ and $p$
and thus, since these numbers are natural numbers, there is a finite list of possible values for them. What is this list? Use it to find all possible platonic solids. Identify them with the five known ones.




\item[Extra Credit] 

Wolfram Mathematica says that the doughnut surface (a subset of $\R^3$) can be parametrized by

\smallskip
\begin{align*}
    x &= (a+b \cos (v))\cos u \\
y &= (a+b \cos(v))\sin u \\
z &= b\sin v
\end{align*}



\bigskip\noindent
for $(u,v)\in [0,2\pi)$, which is a continuous bijection, if $a>b$.
Assume this.

\smallskip\noindent
Use the above info to prove that the torus, as we defined it, is homeomorphic to the doughnut surface.

{(Hint: use an appropriate commutative triangle.)}




\end{enumerate}


\end{document}
\bye

